ENCYCLOPEDIA OF TRIANGLE CENTERS Part7

+ ENCYCLOPEDIA OF TRIANGLE CENTERS Part7

Encyclopedia of Triangle Centers - ETC

This is PART 7: Centers X(12001) - X(14000)

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(12001) = HOMOTHETIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS AND OUTER-YFF TANGENTS

Trilinears (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+(b^2-4*b*c+c^2)^2*a-(b^2-c^2)*(b-c)*(b^2-8*b*c+c^2))*a : :
X(12001) = 4*R*X(1)-(R+r)*X(3)

X(12001) lies on these lines: {1,3}, {5,10529}, {11,11929}, {30,10806}, {104,5734}, {140,10587}, {145,6911}, {381,10532}, {405,10283}, {474,5844}, {956,5901}, {1056,6842}, {1058,7491}, {1478,10949}, {1483,3149}, {1537,10941}, {1598,11401}, {1616,5398}, {1656,10527}, {3244,11499}, {3560,10595}, {3621,6946}, {3622,6883}, {3623,6905}, {3843,10742}, {4317,5840}, {5070,9711}, {5093,9026}, {5288,9624}, {5434,10525}, {5790,10916}, {6959,10530}, {6985,7967}, {7517,10835}, {9301,10879}, {9654,10957}, {9655,10738}, {9669,10959}, {10804,11842}, {10931,11916}, {10932,11917}, {11911,11915}, {11949,11957}, {11950,11958}

X(12001) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,10680,3), (1,10966,3295), (3,10247,12000), (56,10679,3), (999,1482,3), (3304,3338,999), (10529,10597,5), (10532,10943,381)

X(12002) = X(4)X(52)∩X(51)X(1657)

Barycentrics (SB+SC)*(5*SA^2-(14*R^2+3*SW) *SA+8*S^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25355.

X(12002) lies on these lines: {4,52}, {51,1657}, {140,6688}, {511,3850}, {550,5462}, {1216,3851}, {1656,5447}, {3522,5892}, {3523,11465}, {3854,5891}, {3858,10263}, {5056,10625}, {5059,9730}, {5068,10170}, {10219,11592}, {10575,11002}

X(12002) =

X(12003) = POINT BEID 1

Trilinears ((4*cos(2*A)+5)*cos(B-C)+cos( A)*cos(2*(B-C))-9*cos(A)-cos( 3*A))*((cos(2*A)+5/2)*cos(B-C) -2*cos(A)+cos(3*A))*sec(A) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25355.

X(12003) lies on this line: {6000, 10295}

X(12004) = POINT BEID 2

Barycentrics a^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^2-2 a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) (a^10 b^2-5 a^8 b^4+10 a^6 b^6-10 a^4 b^8+5 a^2 b^10-b^12+a^10 c^2+4 a^8 b^2 c^2-8 a^6 b^4 c^2+6 a^4 b^6 c^2-9 a^2 b^8 c^2+6 b^10 c^2-5 a^8 c^4-8 a^6 b^2 c^4+8 a^4 b^4 c^4+4 a^2 b^6 c^4-15 b^8 c^4+10 a^6 c^6+6 a^4 b^2 c^6+4 a^2 b^4 c^6+20 b^6 c^6-10 a^4 c^8-9 a^2 b^2 c^8-15 b^4 c^8+5 a^2 c^10+6 b^2 c^10-c^12) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25358.

X(12004) lies on these lines: {3,49} et al

X(12005) = MIDPOINT OF X(1) AND X(5884)

Trilinears (b+c)*a^5-(b-c)^2*a^4-(b+c)*( 2*b^2-b*c+2*c^2)*a^3+(2*b^2+b* c+2*c^2)*(b-c)^2*a^2+(b^3-c^3) *(b^2-c^2)*a-(b^2-c^2)*(b-c)*( b^3+c^3) : :
X(12005) = (3*R+2*r)*X(1)+(R-2*r)*X(104)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25365.

X(12005) lies on these lines: {1,104}, {3,3874}, {4,5557}, {5,2801}, {10,10202}, {12,10265}, {30,6583}, {40,3873}, {48,1729}, {57,6796}, {65,4311}, {72,10165}, {84,11020}, {140,3678}, {354,946}, {355,5883}, {515,942}, {517,548}, {518,5771}, {551,5887}, {581,982}, {631,5904}, {758,1385}, {912,1125}, {938,6256}, {944,4317}, {950,5570}, {952,3754}, {1006,6763}, {1064,3953}, {1210,10958}, {1482,3892}, {1483,2802}, {1490,10980}, {2771,5901}, {3149,4860}, {3218,10902}, {3333,6261}, {3336,11491}, {3337,6905}, {3555,11362}, {3576,3868}, {3577,9845}, {3616,5693}, {3651,5536}, {3742,5777}, {3833,9956}, {3878,10246}, {3889,7982}, {3894,7987}, {4015,11231}, {5045,6001}, {5253,6326}, {5439,10175}, {5542,6245}, {5708,11500}, {5728,6260}, {5770,10198}, {6705,11018}, {6952,11219}, {9948,10569}, {10573,10805}, {11025,11372}

X(12005) = midpoint of X(i) and X(j) for these {i,j}: {1,5884}, {3,3874}, {65,5882}, {3555,11362}, {11570,11715}
X(12005) = reflection of X(i) in X(j) for these (i,j): (3678,140), (3754,5885), (6684,9940)

X(12006) = MIDPOINT OF X(3) AND X(143)

Trilinears a*((b^2+c^2)*a^6-(3*b^4-2*b^2* c^2+3*c^4)*a^4+3*(b^4-3*b^2*c^ 2+c^4)*(b^2+c^2)*a^2-(b^4-b^2* c^2+c^4)*(b^2-c^2)^2) : :
X(12006) = (4*R^2+OH^2)*X(5)-OH^2*X(113)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25365.

X(12006) lies on these lines: {2,6102}, {3,143}, {5,113}, {30,5462}, {51,550}, {52,549}, {54,1511}, {140,389}, {156,6642}, {182,1658}, {186,6152}, {381,10574}, {382,5640}, {511,3530}, {546,5943}, {547,5907}, {548,5446}, {568,631}, {632,5562}, {1112,3520}, {1199,1493}, {1539,3521}, {1656,5876}, {1657,9781}, {1986,6143}, {3523,6243}, {3526,5889}, {3528,11002}, {3628,10219}, {3845,10575}, {3850,6000}, {3851,6241}, {3858,11381}, {5012,5944}, {5054,11412}, {5055,11465}, {5070,11459}, {6146,9827}, {7514,9786}, {7526,10601}, {9703,11423}, {10272,11806}, {11245,11264}

X(12006) = midpoint of X(i) and X(j) for these {i,j}: {3,143}, {52,10627}, {125,11561}, {140,389}, {548,5446}, {5462,9729}, {6102,11591}, {8254,11802}, {10272,11806}
X(12006) = reflection of X(i) in X(j) for these (i,j): (3628,11695), (10095,5462), (10627,11592)
X(12006) = complement of X(11591)

X(12007) = REFLECTION OF X(5) IN X(6329)

Trilinears (6*a^6-9*(b^2+c^2)*a^4+4*(b^2- c^2)^2*a^2-(-c^4+b^4)*(b^2-c^ 2))/a : :
X(12007) = X(4) - 5*X(6)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25365.

X(12007) lies on these lines: {3,3629}, {4,6}, {5,6329}, {20,5102}, {30,5097}, {69,10303}, {98,9300}, {125,11245}, {140,3631}, {141,3526}, {182,524}, {193,5085}, {511,548}, {542,5066}, {575,3564}, {578,6696}, {597,1352}, {1350,1992}, {1351,3534}, {2854,9826}, {3398,7789}, {3523,11008}, {3567,9973}, {3618,7486}, {3815,9755}, {3818,3857}, {5306,9744}, {6144,10519}, {6247,11426}, {6279,11314}, {6280,11313}, {6676,11225}, {10168,11540}, {10192,11433}, {11064,11422}

X(12007) = midpoint of X(i) and X(j) for these {i,j}: {3,3629}, {6,8550}, {182,1353}, {5480,6776}, {8584,11179}
X(12007) = reflection of X(i) in X(j) for these (i,j): (5,6329), (3589,575), (3631,140)

X(12008) = POINT BEID 3

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25373.

X(12008) lies on the cubic K040 and this line: {1,1030}

X(12009) = X(1)X(3)∩X(3988)X(10124)

Trilinears 3*(b+c)*a^5-(3*b^2-4*b*c+3* c^2)*a^4-(b+c)*(6*b^2-5*b*c+6* c^2)*a^3+2*(3*b^4+3*c^4-b*c*( 5*b^2-b*c+5*c^2))*a^2+(b^2-c^ 2)*(b-c)*(3*b^2+b*c+3*c^2)*a- 3*(b^2-c^2)^2*(b-c)^2 : :
Trilinears (2*sin(A/2)-3*sin(3*A/2))*cos( (B-C)/2)+3*(cos(A)-1)*cos(B-C) -2*cos(A)+1/2 : :
X(12009) = (13*R+6*r)*X(1) + 3*(R-2*r)*X(3)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25379.

X(12009) lies on these lines: {1,3}, {3988,10124}, {5550,5694}

X(12010) = EULER LINE INTERCEPT OF X(11557)X(11591)

Barycentrics 2*a^10-7*(b^2+c^2)*a^8+2*(3*b^ 4+b^2*c^2+3*c^4)*a^6+(b^2+c^2) *(4*b^4-5*b^2*c^2+4*c^4)*a^4-( b^2-c^2)^2*(8*b^4+b^2*c^2+8*c^ 4)*a^2+3*(b^4-c^4)*(b^2-c^2)^3 : :
Trilinears (cos(2*A)+5/2)*cos(B-C)-3*cos( A)*cos(2*(B-C))-3*cos(A)+cos( 3*A) : :
X(12010) = (37*R^2-10*SW)*X(3) + 3*(9*R^2- 2*SW)*X(4)

As a point on the Euler line, X(12010) has Shinagawa coefficients [3*E+40*F, -9*E+8*F].

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25379.

X(12010) lies on these lines: {2,3}, {11557,11591}

X(12011) = POINT BEID 4

Barycentrics a^2 (a^6-a^4 (3 b^2+c^2)+a^2 (3 b^4+b^2 c^2-c^4)-(b^2-c^2)^3)(2 a^14-a^12 (9 b^2+13c^2)+2 a^10 (7 b^4+19 b^2 c^2+18 c^4)-a^8 (5 b^6+42 b^4 c^2+56 b^2 c^4+55 c^6)+a^6 (-10 b^8+31 b^6 c^2+28 b^4 c^4+27 b^2 c^6+50 c^8)+a^4 (13 b^10-26 b^8 c^2-b^6 c^4-5 b^4 c^6+10 b^2 c^8-27 c^10)-a^2 (b^2-c^2)^2 (6 b^8-3 b^6 c^2-4 b^4 c^4-3 b^2 c^6-8 c^8)+(b^2-c^2)^5 (b^2+c^2)^2 ) : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25380.

X(12011) lies on these lines: {186,1291}, {550,1263}

X(12012) = REFLECTION OF X(10184) IN X(2)

Barycentrics (S^2+SB*SC)*(4*SA^2+(8*R^2-6* SW)*SA-S^2+SW^2-4*R^2*SW) : :
Trilinears cos(B-C)*(4*(cos(A)-cos(3*A))* cos(B-C)+cos(2*(B-C))-cos(2*A) -3*cos(4*A)+3)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25381.

X(12012) lies on these lines: {2,10184}, {3,275}, {418,10003}, {549,1154}

X(12012) = reflection of X(10184) in X(2)

X(12013) = POINT BEID 5

Barycentrics (10*SA^2-8*(R^2+SW)*SA+21*S^2- SW^2+4*R^2*SW)*(3*S^2-SB*SC) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25381.

X(12013) lies on these lines: {547,11197}, {1656,3462}

X(12014) = POINT BEID 6

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25389.

X(12014) lies on these lines:

X(12015) = POINT BEID 7

Trilinears ((b+c)*a^10-2*(b^2+4*b*c+c^2)* a^9-(b+c)*(3*b^2-14*b*c+3*c^2) *a^8+(8*b^4+8*c^4+b*c*(7*b^2+ 6*b*c+7*c^2))*a^7+(b+c)*(2*b^ 4+2*c^4-b*c*(23*b^2+3*b*c+23* c^2))*a^6-(12*b^6+12*c^6-b*c*( 7*b^2+13*b*c+7*c^2)*(b-c)^2)* a^5+(b+c)*(2*b^6+2*c^6+b*c*( 13*b^2-10*b*c+13*c^2)*(b+c)^2) *a^4+(8*b^6+8*c^6-(3*b^4+3*c^ 4+2*b*c*(17*b^2+25*b*c+17*c^2) )*b*c)*(b-c)^2*a^3-(b+c)*(3*b^ 8+3*c^8-(3*b^6+3*c^6-(15*b^4+ 15*c^4+b*c*(3*b^2-4*b*c+3*c^2) )*b*c)*b*c)*a^2-(b^2-c^2)^2*( 2*b^6+2*c^6-(13*b^4+13*c^4+b* c*(13*b^2-8*b*c+13*c^2))*b*c)* a+(b^3-c^3)*(b^2-c^2)^3*(b^2- 6*b*c+c^2))/(-a+b+c) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25394.

X(12015) lies on this line: {7,2475}

X(12016) = INCIRCLE-INVERSE OF X(104)

Trilinears q*p^3*(4*p^2-4*p*q+4*q^2-3)+( 4*p^4-4*p^2+p*q+1)*(1-q^2) : : , where p = sin(A/2) and q = cos(B/2 - C/2)
Trilinears (b+c)*a^8-2*(b^2+c^2)*a^7-(2* b-c)*(b-2*c)*(b+c)*a^6+2*(3*b^ 2+4*b*c+3*c^2)*(b-c)^2*a^5-7*( b^2-c^2)*(b-c)*b*c*a^4-2*(3*b^ 4+3*c^4+2*b*c*(b^2+c^2))*(b-c) ^2*a^3+(b^2-c^2)*(b-c)*(2*b^4+ 2*c^4+3*b*c*(b-c)^2)*a^2+2*(b^ 2-c^2)^2*(b^4+c^4-2*b*c*(b-c)^ 2)*a-(b^2-c^2)^2*(b-c)^2*(b^3+ c^3) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25395.

X(12016) lies on these lines:
{1,104}, {7,151}, {56,11713}, {57,102}, {65,1359}, {117,226}, {124,1210}, {354,1361}, {518,3040}, {928,11028}, {942,2818}, {974,2779}, {1845,5902}, {2807,3664}, {3042,3812}, {3340,10696}, {3586,10732}, {3738,10015}, {3911,6711}, {4654,10709}, {5722,10747}, {9579,10726}

X(12016) = midpoint of X(65) and X(1364)
X(12016) = reflection of X(3042) in X(3812)
X(12016) = incircle-inverse-of-X(104)
X(12016) = X(131)-of-intouch-triangle

X(12017) = INVERSE-IN-SCHOUTE-CIRCLE OF X(5013)

Trilinears    a + 3R cot ω cos A : :
Trilinears    5 cos(A - ω) + cos(A + ω) : :
Barycentrics    (3 cot A + 2 tan ω)sin²A : :
Barycentrics    a^2 (5 a^4-4 a^2 b^2-b^4-4 a^2 c^2-10 b^2 c^2-c^4) : :
X(12017) = 3 X[3] + 2 X[6], X[6] - 6 X[182], X[3] + 4 X[182], X[69] - 6 X[549], 7 X[6] - 12 X[575], 7 X[182] - 2 X[575], 7 X[3] + 8 X[575], 11 X[6] - 6 X[576], 11 X[182] - X[576], 11 X[3] + 4 X[576], 7 X[3] - 2 X[1350], 14 X[182] + X[1350], 4 X[575] + X[1350], 7 X[6] + 3 X[1350], 14 X[576] + 11 X[1350], 16 X[576] - 11 X[1351], 8 X[6] - 3 X[1351], 16 X[182] - X[1351], 4 X[3] + X[1351], 8 X[1350] + 7 X[1351]

The Schoute circle is here defined as the radical circle of the Schoute coaxal system; that is, the circle with diameter X(15)X(16) and center X(187).

X(12017) lies on these lines:

X(12017) = reflection of X(1351) in X(11482)
X(12017) = Brocard-circle-inverse of X(33878)
X(12017) = Schoute-circle-inverse of X(5013)
X(12017) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,182,5050), (3,5050,1351), (3,5093,1350), (6,5085,5092), (6,5092,3), (15,16,5013), (182,5085,3), (182,5092,6), (575,1350,5093), (1353,3530,10519), (5012,7484,3167), (5085,10541,182), (6200,8375,6221), (6221,6398,5024), (6396,8376,6398), (11485,11486,9605).

X(12018) = POINT BEID 8

Trilinears (b+c)*a^8-2*(3*b^2+4*b*c+3*c^ 2)*a^7-(b+c)*(2*b^2-35*b*c+2* c^2)*a^6+2*(9*b^4+9*c^4-2*b*c* (4*b^2+15*b*c+4*c^2))*a^5-b*c* (b+c)*(77*b^2-137*b*c+77*c^2)* a^4-2*(9*b^6+9*c^6-2*(12*b^4+ 12*c^4+b*c*(9*b^2-14*b*c+9*c^ 2))*b*c)*a^3+(b+c)*(2*b^6+2*c^ 6+(41*b^4+41*c^4-b*c*(165*b^2- 229*b*c+165*c^2))*b*c)*a^2+6*( b^2-c^2)^2*(b^4+c^4-b*c*(4*b^ 2-3*b*c+4*c^2))*a+(b^2-c^2)^2* (b+c)^3*(3*b*c-b^2-c^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25401.

X(12018) lies on this line: {2475,2802}

X(12019) = MIDPOINT OF X(11) AND X(80)

Barycentrics 2*a^4-2*(b+c)*a^3+(b^2+c^2)*a^ 2+2*(b^2-c^2)*(b-c)*a-3*(b^2- c^2)^2 : :
X(12019) = (R+r)*X(1) - 3*r*X(5)

See Tran Quang Hung and César Lozada, Hyacinthos 25418.

X(12019) lies on these lines: {1,5}, {2,10609}, {4,653}, {8,4767}, {10,528}, {30,1155}, {44,5179}, {46,3627}, {65,546}, {100,405}, {104,3149}, {140,10572}, {149,1145}, {153,6835}, {214,6667}, {381,1159}, {382,1788}, {429,1862}, {497,5790}, {515,5126}, {517,11545}, {519,5087}, {632,3612}, {938,9654}, {942,2801}, {950,9956}, {960,2802}, {1086,6788}, {1320,3621}, {1478,4860}, {1479,5690}, {1482,10591}, {1656,3486}, {1698,6174}, {1728,5128}, {1770,3853}, {1836,3845}, {1985,3240}, {2646,3628}, {2771,7687}, {2800,6797}, {2829,6245}, {3035,3634}, {3245,3583}, {3295,5818}, {3419,3820}, {3474,3830}, {3485,3851}, {3526,4305}, {3579,5840}, {3586,10993}, {3622,10031}, {3625,5854}, {3654,9580}, {3679,4679}, {3843,4295}, {4187,5086}, {4304,11231}, {4663,5848}, {4870,11737}, {4997,6790}, {5204,10090}, {5217,10058}, {5220,5856}, {5225,6928}, {5229,5708}, {5550,6224}, {5560,10483}, {5657,9668}, {5691,11219}, {5714,9803}, {5855,11813}, {6147,10895}, {6914,11502}, {9779,11041}, {10246,10589}, {10573,10896}, {11604,11684}

X(12019) = midpoint of X(i) and X(j) for these {i,j}: {11,80}, {149,1145}, {1317,9897}
X(12019) = reflection of X(i) in X(j) for these (i,j): (214,6667), (1387,11), (3035,6702), (9945,3035)
X(12019) = complement of X(10609)
X(12019) = Fuhrmann circle-inverse-of-X(5722)
X(12019) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (355,9581,496), (1837,10826,5), (5587,5722,495), (7741,10950,5901)

X(12020) = X(3)X(6)∩X(76,2546)

Barycentrics a^2 (S + SA Tan[w/2]^2 Cot[w]) : :
X(12020) = tan[w/2]^2 X[3] + X[6]

See Angel Montesdeoca, Hechos Geométricos en el Triángulo (2015) and Circunferencias de Apolonio. See also X(8160) and X(12021).

X(12020) lies on these lines: {3,6}, {76,2546}, {1428,3238}, {1503,5403}, {1676,3934}, {2330,3237}, {3589,5404}

X(12020) = reflection of X(12021) in X(182)
X(12020) = {X(2030),X(3094)}-harmonic conjugate of X(12021)
X(12020) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,1343,8160), (1343,1671,39)

X(12021) = X(3)X(6)∩X(76,2547)

Barycentrics a^2 (S + SA Cot[w/2]^2 Cot[w]) : :
X(12020) = X[1671] - 3 X[5085]

See Angel Montesdeoca, Hechos Geométricos en el Triángulo (2015) and Circunferencias de Apolonio. See also X(8160) and X(12020).

X(12021) lies on these lines: {3,6}, {76,2547}, {1428,3237}, {1503,5404}, {1677,3934}, {2330,3238}, {3589,5403}

X(12021) = reflection of X(12020) in X(182)
X(12021) = {X(2030),X(3094)}-harmonic conjugate of X(12020)
X(12021) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,1342,8161), (12020,12021,2030), (1342,1670,39)

X(12022) = X(4)X(6)∩X(5)X(49)

Trilinears (cos(2*A)+2)*cos(B-C)-cos(A)*( cos(2*(B-C))+1)-cos(3*A) : :
Barycentrics 2*a^10-5*(b^2+c^2)*a^8+2*(2*b^ 4+3*b^2*c^2+2*c^4)*a^6-2*(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-(b^4- c^4)*(b^2-c^2)^3 : :
X(12022) = (3*R^2-SW)*X(4)+SW*X(6) = 2*(5*R^2-SW)*X(5)-(7*R^2-2*SW) *X(49)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25420 and Hyacinthos 25493 .

X(12022) lies on these lines:
{3,3580}, {4,6}, {5,49}, {30,568}, {51,7576}, {68,7503}, {125,11430}, {141,7550}, {184,403}, {185,1986}, {235,1614}, {378,1899}, {381,11402}, {389,6240}, {436,6761}, {468,11464}, {511,11660}, {539,5891}, {546,11423}, {550,3581}, {569,9927}, {578,1594}, {1593,11457}, {1885,6241}, {1994,3153}, {3448,7527}, {3542,9707}, {3564,11459}, {3567,3575}, {3628,11704}, {5446,11750}, {5562,5965}, {5876,11264}, {6193,6816}, {6756,9781}, {7507,11426}, {9545,9820}, {9818,11442}, {9833,10594}, {10018,10182}, {10127,11451}, {10282,10619}, {10295,11438}, {10297,11422}

X(12022) = reflection of X(i) in X(j) for these (i,j): (5890,11245), (7576,51)
X(12022) = X(5692)-of-orthic-triangle if ABC is acute
X(12022) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,6776,11456), (265,567,5)

X(12023) = ISOGONAL CONJUGATE OF X(13620)

Trilinears 1/((10*cos(2*A)+5)*cos(B-C)-4* cos(A)-5*cos(3*A)) : :
Barycentrics 1/(5*a^8-10*(b^2+c^2)*a^6+9*b^ 2*c^2*a^4+2*(b^2+c^2)*(5*b^4- 7*b^2*c^2+5*c^4)*a^2-5*(b^6-c^ 6)*(b^2-c^2)) : :
X(12023) = |OH|²*X(3) - R²*X(4)

See X(7688) and Antreas Hatzipolakis and César Lozada, Hyacinthos 25420 and Hyacinthos 25493 . See also Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25511 .

X(12023) lies on the Jerabek hyperbola.

X(12023) = isogonal conjugate of X(13620)

X(12024) = POINT BEID 9

Trilinears (cos(2*A)+2)*cos(B-C)-cos(A)*( cos(2*(B-C))+1)-cos(3*A) : :
Barycentrics 10*a^10-25*(b^2+c^2)*a^8+4*(5* b^4+6*b^2*c^2+5*c^4)*a^6-10*( b^4-c^4)*(b^2-c^2)*a^4+2*(b^2- c^2)^2*(5*b^4-2*b^2*c^2+5*c^4) *a^2-5*(b^4-c^4)*(b^2-c^2)^3 : :
X(12024) = (12*R^2-5*SW)*X(4)+5*SW*X(6)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25420 and Hyacinthos 25493 .

X(12024) lies on these lines: {4,6}, {30,11225}, {1899,11410}, {3628,5972}

X(12025) = POINT BEID 10

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25422.

X(12024) lies on this line: {1,5}

X(12026) = MIDPOINT OF X(3) AND X(1263)

Trilinears (3*cos(2*A)+cos(4*A)+1/2)*cos( B-C)+(-2*cos(A)-2*cos(3*A))*co s(2*(B-C))+(-cos(2*A)+1)*cos(3 *(B-C))-cos(5*A)-3*cos(A)-cos( 3*A) : :
Barycentrics 8*S^4-2*(3*R^4+(SB+SC)*(2*SA+2 *SW-5*R^2))*S^2-SA*(27*R^4-22* SW*R^2+4*SW^2)*(SB+SC) : :

See Tran Quang Hung and César Lozada, Hyacinthos 25424.

X(12026) lies on these lines: {3,1263}, {5,49}, {30,137}, {128,3628}, {140,6592}, {549,930}

X(12026) = midpoint of X(i) and X(j) for these {i,j}: {3,1263}, {5,1141}
X(12026) = reflection of X(i) in X(j) for these (i,j): (128,3628), (6592,140)

X(12027) = PERSPECTOR OF THESE TRIANGLES: 1ST EHRMANN AND INTRIANGLE OF X(6)

Trilinears a*(a^4-b^4+4*b^2*c^2-c^4)*(a^10- 7*(b^2+c^2)*a^8+6*(b^4+9*b^2*c^2+c^4)*a^6+2*(b^2+c^2)*(3*b^4-34*b^2*c^2+3*c^4)*a^4-(7*b^8+7*c^8+2*b^2*c^2*(3*b^4-29*b^2*c^2+3*c^4))*a^2+(-c^4+b^4)*(b^2-c^2)*(b^4+6*b^2*c^2+c^4)) : :
Barycentrics (SB+SC)*(3*SA^2-2*SW*SA+3*S^2)*((9*R^2-SW)*S^2+3*(3*SA+SW)*(3*SA-2*SW)*R^2-(3*SA^2-SW^2)*SW) : :

The intriangle of a point given by trilinears x : y : z is the central triangle having A-vertex 0 : y + z cos A : z + y cos A. (See TCCT, p. 196). Thus, the A-vertex of the intriangle of X(6) is 0 : b + c cos A : c + b cos A. Contributed by César Lozada, February 11, 2017.

X(12027) lies on these lines: {3,5913}, {1296,9465}, {1995,5512}, {6776,7464}

X(12028) = X(30)X(50)∩X(186)X(476)

Trilinears (sin 2A csc 3A)/(1 + cos 2B + cos 2C) : :
Barycentrics (SA/(((b^2+c^2) a^4-2 (b^4-b^2 c^2+c^4)a^2+ b^6-b^4 c^2-b^2 c^4+c^6) (4 SA^2-b^2 c^2)) : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25429.

X(12028) lies on these lines: {2, 5627}, {30, 50}, {94, 2071}, {186, 476}, {265, 2072}, {1141, 3153}

X(12028) = isogonal conjugate of X(1986)

X(12029) = X(1)X(6079)∩X(100)X(1149)

Trilinears a/((b+c)*(a^3+5*b*c*a-b*c*(b+ c))+(b^2-10*b*c+c^2)*a^2) : :

See Tran Quang Hung and César Lozada, Hyacinthos 25436.

X(12029) lies on the circumncircle and these lines: {1,6079}, {100,1149}, {901,3915}, {995,2748} , {7292,9059}

X(12030) = TRILINEAR POLE OF X(6)X(2610)

Trilinears 1/((a^5-(b^2+c^2)*a^3+(b^2-c^ 2)*(b-c)*a^2+b^2*c^2*a-(b^4-c^ 4)*(b-c))*(b+c)) : :

See Tran Quang Hung and César Lozada, Hyacinthos 25436.

X(12030) lies on the circumncircle and these lines: {12,2222}, {21,1290}, {23,9070}, {28,2766}, {30,6011}, {74,6003}, {100,1325}, {101,4053}, {108,2074}, {109,5127}, {110,758}, {476,6757}, {523,759}, {842,7427}, {2651,4588}, {2691,4221}, {2701,4653}, {4227,10100}, {6012,7481}, {7469,9058}

X(12030) = trilinear pole of X(6)X(2610)
X(12030) = Λ(X(1), X(110))

X(12031) = X(98)X(6002)∩X(99)X(740)

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

See Tran Quang Hung and César Lozada, Hyacinthos 25436.

X(12031) lies on the circumncircle and these lines: {58,2702}, {98,6002}, {99,740}, {100,1931}, {101,1326}, {110,3747}, {511,6010}, {512,741}, {789,5209}, {813,1500}, {825,5006}, {2703,3736}

X(12032) = REFLECTION OF X(813) IN X(3)

Trilinears a/((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) : :

See Tran Quang Hung and César Lozada, Hyacinthos 25436.

X(12032) lies on the circumncircle and these lines: {1,927}, {3,813}, {41,919}, {100,2340}, {101,7193}, {103,9320}, {105,663}, {108,1429}, {109,2223}, {112,5009}, {741,7254}, {929,990}, {934,1458}, {991,1308}, {1305,3100}, {2222,5091}, {2704,11012}, {2737,5732}

X(12032) = reflection of X(813) in X(3)
X(12032) = circumcircle-antipode of X(813)

X(12033) = X(55)X(2316)∩X(3196)X(6600)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25437.

X(12033) lies on these lines: {55,2316}, {3196,6600}

X(12034) = MIDPOINT OF X(165) AND X(9355)

Trilinears 3*a^4-4*(b+c)*a^3-(b^2-9*b*c+c ^2)*a^2+(b+c)*(4*b^2-9*b*c+4*c ^2)*a-2*(b^2-c^2)^2 : :
X(12034) = 2(4R^2 - 15Rr - 4r^2)*X(9) - (8R^2 + 6Rr + r^2 - 3s^2)*X(48)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25437.

X(12034) lies on these lines:
{3,3196}, {6,10247}, {9,48}, {44,517}, {45,10246}, {165,2246}, {952,4370}, {1635,2827}, {1743,2170}, {1766,3973}, {2291,2348}, {2792,10175}

X(12034) = midpoint of X(165) and X(9355)
X(12034) = X(3163)-of-excentral-triangle

X(12035) = REFLECTION OF X(3756) IN X(2)

Barycentrics (2*a-b-c)*(2*a^2-3*(b+c)*a -b^2+6*b*c-c^2) : :
X(12035) = 2 X(3699) + X(3756)

For P on the circumcircle of a triangle ABC, let G(P) denote then centroid of the pedal triangle of P. The locus of G(P) is an ellipse, E, with center G = X(2), and the following pass-through points as shown here:

P	G(P)
X(74)	X(125)
X(106)	X(3756)
X(110)	X(5642)
X(98)	X(6784)
X(99)	X(6786)
X(111)	X(6791)
X(112)	X(6793)

The ellipse E, described at X(6784), also passes through the vertices of the (pedal triangle of X(376)) = X(2)-of-antipedal-triangle-of-X(2), as well as the the following reflections:

X(5642) = reflection of X(125) in X(2)
X(6786) = reflection of X(6784) in X(2)
X(12035) = reflection of X(3756) in X(2)
X(12036) = reflection of X(6791) in X(2)
X(12037) = reflection of X(6793) in X(2)

See César Lozada, Hyacinthos 25463.

X(12035) lies on these lines:
{2,1280}, {121,519}, {524,5205}, {900,1635}, {952,10713}, {1086,9458}, {1213,6791}, {1647,4152}, {3679,5854}

X(12035) = midpoint of X(2) and X(3699)
X(12035) = reflection of X(3756) in X(2)
X(12035) = tripolar centroid of X(2415)
X(12035) = centroid of (degenerate) pedal triangle of X(1293)

X(12036) = REFLECTION OF X(6791) IN X(2)

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

See X(12035) and César Lozada, Hyacinthos 25463.

X(12036) lies on these lines:
{2,5503}, {125,599}, {126,524}, {351,690}, {538,9127}, {542,10717}, {543,5108}, {1992,4563}, {5477,8030}, {5650,6784}, {5969,9172}, {6786,9023}

X(12036) = midpoint of X(2) and X(9146)
X(12036) = reflection of X(6791) in X(2)
X(12036) = tripolar centroid of X(2418)
X(12036) = centroid of (degenerate) pedal triangle of X(1296)

X(12037) = REFLECTION OF X(6793) IN X(2)

Barycentrics (b^2-c^2)^2*(-a^2+b^2+c^2)*(a^ 4+(2*(b^2+c^2))*a^2-2*b^2*c^2- 3*c^4-3*b^4) : :
X(12037) = 4 X(127) - X(1562)

See X(12035) and César Lozada, Hyacinthos 25463.

X(12037) lies on these lines:
{2,6793}, {122,125}, {127,525}, {599,5642}, {2777,10718}, {2871,3917}, {6054,10519}

X(12037) = reflection of X(6793) in X(2)
X(12037) = tripolar centroid of X(2419)
X(12037) = centroid of (degenerate) pedal triangle of X(1297)

X(12038) = MIDPOINT OF X(3) AND X(1147)

Trilinears    (2*a^6-3*(b^2+c^2)*a^4+2*b^2* c^2*a^2+(b^4-c^4)*(b^2-c^2))*( -a^2+b^2+c^2) *a : :
Trilinears    cos(A)*(1+2*cos(2*A)+cos(2*B)+ cos(2*C)) : :
Barycentrics    a^2 (a^2-b^2-c^2) (2 a^6+2 a^2 b^2 c^2-3 a^4 (b^2+c^2)+(b^2-c^2)^2 (b^2+c^2)) : :
X(12038) = 3*X(2)-X(9927) = 3*X(3)+X(155) = X(3)+X(1147) = 5*X(3)+3*X(3167) = 3*X(3)-X(7689) = X(20)+3*X(5654) = X(155)-3*X(1147) = 5*X(155)-9*X(3167) = X(155)+X(7689) = 5*X(1147)-3*X(3167)

See Antreas Hatzipolakis and Angel Montesdeoca, and César Lozada, Hyacinthos 25470 and Hyacinthos 25471 .

Let N_AN_BN_C be the reflection triangle of X(5). Let O_A be the circumcenter of AN_BN_C, and define O_B and O_C cyclically. Triangle O_AO_BO_C is orthologic to the orthic triangle at X(12038). (Randy Hutson, June 7, 2019)

X(12038) lies on these lines:
{2, 9927}, {3, 49}, {4, 11449}, {5, 1511}, {20, 5654}, {24, 5446}, {26, 11202}, {30, 5448}, {52, 186}, {54, 5504}, {68, 631}, {74, 9705}, {110, 3520}, {140, 5449}, {156, 6000}, {182, 8548}, {378, 10539}, {382, 1495}, {511, 1658}, {539, 549}, {541, 5894}, {550, 5944}, {567, 2931}, {569, 5892}, {578, 5462}, {1069, 5217}, {1152, 8909}, {1614, 2071}, {3043, 11562}, {3157, 5204}, {3523, 6193}, {3524, 11411}, {3530, 3564}, {3576, 9928}, {3855, 10546}, {5010, 6238}, {5646, 7393}, {5657, 9933}, {5663, 10226}, {5890, 9545}, {6146, 10257}, {6200, 10666}, {6241, 9544}, {6396, 10665}, {6418,8912}, {6642, 11425}, {6689, 7399}, {6699, 10116}, {7280, 7352}, {7488, 10625}, {7503, 10170}, {7506, 11424}, {7514, 9938}, {7526, 9306}, {7575, 10263}, {8546, 8681}, {9707, 11413}, {10020, 10182}, {10298, 11412}, {10540, 11381}, {10645, 10662}, {10646, 10661}

X(12038) = midpoint of X(i) and X(j) for these {i,j}: {3,1147}, {155,7689}, {156,11250}
X(12038) = reflection of X(i) in X(j) for these {i,j}: {5448,9820}, {5449,140}
X(12038) = complement of X(9927)
X(12038) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,49,185), (3,155,7689), (3,1092,1216), (578,6644,5462), (1147,7689,155), (1614,2071,10575)

X(12039) = MIDPOINT OF X(6) AND X(8542)

Trilinears (2*a^6-3*(b^2+c^2)*a^4-2*(b^4+ b^2*c^2+c^4)*a^2+(b^2-3*c^2)*( 3*b^2-c^2)*(b^2+c^2))*a : :
X(12039) = X(6)+X(8542) = 3*X(182)-X(8547) = X(8547)+9*X(9813)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25471 .

X(12039) lies on these lines:
{6,373}, {39,9145}, {182,2393}, {193,7605}, {523,7804}, {524,547}, {575,2854}, {576,10170}, {597,5972}, {1843,2916}, {3618,5486}, {5092,8705}, {5650,10510}, {9730,11579}, {11003,11188}

X(12039) = midpoint of X(6) and X(8542)

X(12040) = MIDPOINT OF X(2) AND X(11165)

Trilinears (8*a^4-17*(b^2+c^2)*a^2-2*b^2* c^2+5*c^4+5*b^4)/a : :
X(12040) = 5*X(2)-X(5485) = 7*X(2)+X(11148) = X(2)+X(11165) = X(5)-2*X(9771) = 2*X(140)-X(7610) = 3*X(549)-2*X(5569) = X(549)-2*X(7622) = 7*X(5485)+5*X(11148) = X(5485)+5*X(11165) = X(5569)-3*X(7622) = X(7618)+X(11184) = X(11148)-7*X(11165)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25471 .

X(12040) lies on these lines:
{2,2418}, {3,9770}, {5,543}, {30,7618}, {39,9167}, {83,5503}, {99,3363}, {140,7610}, {182,524}, {538,7619}, {547,7615}, {550,7775}, {597,620}, {631,9740}, {1007,5077}, {2482,3815}, {2549,8355}, {3845,8176}, {3849,8703}, {5013,8360}, {5055,7620}, {5215,5306}, {7763,8359}, {7769,9166}, {7777,8598}, {7870,8362}, {8182,9766}, {8667,11812}

X(12040) = midpoint of X(i) and X(j) for these {i,j}: {2,11165}, {3,9770}, {7615,8716}, {7618,11184}, {8182,9766}
X(12040) = reflection of X(i) in X(j) for these (i,j): (5,9771), (549,7622), (3845,8176), (7610,140), (7615,547)

X(12041) = MIDPOINT OF X(3) AND X(74)

Trilinears    (2*a^8-3*(b^2+c^2)*a^6-3*(b^4- 4*b^2*c^2+c^4)*a^4+(b^2+c^2)*( 7*b^4-15*b^2*c^2+7*c^4)*a^2-( 3*b^4+7*b^2*c^2+3*c^4)*(b^2-c^ 2)^2)*a : :
Trilinears    (3*cos(2*A)+7/2)*cos(B-C)-6* cos(A)-cos(3*A) : :
Barycentrics    (a^2 (2 a^8-3 a^6 (b^2+c^2)-3 a^4 (b^4-4 b^2 c^2+c^4)-(b^2-c^2)^2 (3 b^4+7 b^2 c^2+3 c^4)+a^2 (7 b^6-8 b^4 c^2-8 b^2 c^4+7 c^6)) : :
X(12041) = 3*X(2)-X(7728) = X(3)+X(74) = 3*X(3)-X(110) = 5*X(3)-X(399) = 2*X(3)-X(1511) = 4*X(3)-X(5609) = 3*X(3)+X(10620) = 2*X(5)-X(1539) = X(5)-2*X(6699) = 3*X(5)-4*X(6723) = 3*X(74)+X(110) = 5*X(74)+X(399) = X(1539)-4*X(6699) = 3*X(1539)-8*X(6723) = 3*X(6699)-2*X(6723)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25471 . Also see Tran Quang Hung and Angel Montesdeoca, Hyacinthos 25680.

X(12041) lies on these lines:
{2,7728}, {3,74}, {5,1539}, {20,265}, {30,125}, {35,3028}, {55,10081}, {56,10065}, {64,9934}, {113,140}, {146,631}, {182,2781}, {185,10226}, {376,3448}, {378,1112}, {381,10721}, {511,11806}, {517,11709}, {541,549}, {542,8703}, {550,10264}, {567,1986}, {974,1204}, {1154,2071}, {1350,5621}, {1351,5622}, {1657,10733}, {2420,3269}, {2771,9943}, {2780,9208}, {2854,3098}, {2935,7526}, {3521,6143}, {3524,5655}, {3530,10272}, {3532,5504}, {3534,9140}, {3576,9904}, {3581,7464}, {3627,7687}, {3818,6698}, {5050,10752}, {5054,10706}, {5085,9970}, {5092,6593}, {5204,10091}, {5217,10088}, {5462,11807}, {5544,9818}, {6101,7689}, {6409,10819}, {6410,10820}, {6642,9919}, {6644,10117}, {6689,11805}, {7280,7727}, {7502,8717}, {7583,8994}, {7722,11003}, {7731,10574}, {7978,10246}, {8718,11559}, {9729,11557}, {10610,10628}, {11438,11746}

X(12041) = complement of X(7728)
X(12041) = circumcircle-inverse of X(10620)
X(12041) = X(11)-of-Trinh-triangle if ABC is acute

X(12042) = MIDPOINT OF X(3) AND X(98)

Trilinears 2*a^8-3*(b^2+c^2)*a^6+3*(b^4+ c^4)*a^4-(b^2+c^2)*(2*b^4-3*b^ 2*c^2+2*c^4)*a^2-(b^2-c^2)^2* b^2*c^2)/a : :
X(12042) = 3*X(2)-X(6033) = X(3)+X(98) = 3*X(3)-X(99) = X(5)-2*X(6036) = 3*X(5)-4*X(6722) = X(20)+X(6321) = 3*X(98)+X(99) = X(114)-2*X(140) = X(114)+X(10991) = X(115)-3*X(6055) = 2*X(140)+X(10991) = 3*X(6036)-2*X(6722)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25471 .

X(12042) lies on these lines:
{2,5191}, {3,76}, {5,2794}, {20,6321}, {30,115}, {32,2023}, {35,3027}, {36,3023}, {55,10069}, {56,10053}, {114,140}, {141,542}, {147,631}, {148,376}, {157,1605}, {182,10007}, {262,11842}, {378,5186}, {381,3972}, {404,5985}, {517,11710}, {543,8703}, {550,11623}, {632,6721}, {671,3534}, {1657,10723}, {1916,7793}, {2080,5999}, {2784,6684}, {3095,7766}, {3098,5969}, {3111,5663}, {3329,3398}, {3523,5984}, {3524,8289}, {3576,9860}, {3830,9166}, {3845,5461}, {4027,7824}, {5027,11176}, {5050,10753}, {5054,6054}, {5149,7815}, {5182,12017}, {5204,10089}, {5217,10086}, {5569,9830}, {5961,7502}, {5986,7485}, {5987,7496}, {6642,9861}, {6671,6771}, {6672,6774}, {7583,8980}, {7776,8781}, {7798,9737}, {7857,9873}, {7970,10246}, {8667,9888}, {8725,11606}, {9167,11812}, {10352,11285}

X(12042) = midpoint of X(i) and X(j) for these {i,j}: {3,98}, {20,6321}, {114,10991}, {376,11632}, {671,3534}, {1657,10723}, {1916,9821}, {2080,5999}, {6033,9862}, {6295,6582}, {8667,9888}, {8724,11177}, {8725,11606}
X(12042) = reflection of X(i) in X(j) for these (i,j): (5,6036), (114,140), (3845,5461), (5026,5092)
X(12042) = complement of X(6033)
X(12042) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,9862,6033), (1078,5152,5976), (3524,11177,8724)

X(12043) = 8th HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics 6 a^10-17 a^8 b^2+10 a^6 b^4+12 a^4 b^6-16 a^2 b^8+5 b^10-17 a^8 c^2+26 a^6 b^2 c^2-13 a^4 b^4 c^2+19 a^2 b^6 c^2-15 b^8 c^2+10 a^6 c^4-13 a^4 b^2 c^4-6 a^2 b^4 c^4+10 b^6 c^4+12 a^4 c^6+19 a^2 b^2 c^6+10 b^4 c^6-16 a^2 c^8-15 b^2 c^8+5 c^10 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25476 .

X(12043) lies on these lines:
{2,3} et al

X(12043) = {X(140),X(2072)}-harmonic conjugate of X(3530)

X(12044) = POINT BEID 11

Trilinears cos(B-C)*sec(2*(B-C))/(1-2* cos(2*A)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25472 .

X(12044) lies on this line: {252,5449}

X(12045) = MIDPOINT OF X(3819) AND X(5640)

Trilinears a*((b^2+c^2)*a^2-b^4+24*b^2*c^ 2-c^4) : :
Trilinears (cos(2*A)-12)*cos(B-C)-11*cos( A) : :
X(12045) = 11*X(2)+X(51) = 3*X(2)+X(373) = 7*X(2)-X(3819) = 13*X(2)-X(3917) = 7*X(2)+X(5640) = 5*X(2)+X(5943) = 9*X(2)-X(7998) = 15*X(2)+X(11002) = 8*X(3628)+X(9729) = X(6102)+5*X(10170) = X(6102)-10*X(11695)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25490 .

X(12045) lies on these lines:
{2,51}, {575,6090}, {576,5544}, {3589,9027}, {3848,9026}, {5663,6723}, {6102,10170}, {8705,9822}

X(12045) = midpoint of X(i) and X(j) for these {i,j}: {3819,5640}, {5650,5943}
X(12045) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,10219,6688), (373,5650,11002), (373,11002,5943)

X(12046) = MIDPOINT OF X(11017) AND X(12006)

Trilinears a*((b^2+c^2)*a^6-(3*b^4-2*b^2* c^2+3*c^4)*a^4+3*(b^2+c^2)*(b^ 2+3*b*c+c^2)*(b^2-3*b*c+c^2)*a ^2-(b^4-13*b^2*c^2+c^4)*(b^2-c ^2)^2) : :
Trilinears (2*cos(2*A)-13)*cos(B-C)-4*cos (A) : :
X(12046) = 3*X(2)-X(11592) = 15*X(5)+X(185) = 7*X(5)+9*X(373) = 13*X(5)+3*X(9730) = 3*X(5)-X(11017) = 3*X(5)+X(12006) = X(143)+7*X(3090) = X(185)+5*X(11017) = X(185)-5*X(12006) = 27*X(373)-7*X(12006)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25490 .

X(12046) lies on these lines:
{2,11592}, {5,113}, {143,3090}, {156,11484}, {1216,10095}, {2979,9781}, {3567,11591}, {5447,10110}, {5876,11451}

X(12046) = midpoint of X(11017) and X(12006)
X(12046) = complement of X(11592)
X(12046) = {X(5),X(12006)}-harmonic conjugate of X(11017)

X(12047) = MIDPOINT OF X(1) AND X(3585)

Barycentrics (b+c)*a^3+(b^2+c^2)*a^2-(b^2- c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(12047) = 2*X(1125)-X(5267) = X(2646)-3*X(4870) = 3*X(3584)-X(11010) = X(3916)-2*X(4999)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25491 .

X(12047) lies on these lines:
{1,4}, {2,46}, {3,1770}, {5,65}, {7,90}, {8,6871}, {10,908}, {11,113}, {12,517}, {19,5747}, {20,3612}, {21,36}, {30,2646}, {35,411}, {40,498}, {55,6985}, {56,3560}, {57,499}, {72,2886}, {80,7548}, {115,2653}, {124,1845}, {140,1155}, {142,3624}, {165,6988}, {191,5745}, {235,1905}, {238,1780}, {284,1839}, {354,496}, {355,2099}, {376,4333}, {377,997}, {381,1837}, {386,3914}, {431,1829}, {442,960}, {474,5880}, {484,6684}, {486,2362}, {495,3057}, {519,5086}, {527,6763}, {551,4311}, {553,1776}, {595,3011}, {631,3474}, {758,6734}, {938,6870}, {952,11011}, {962,3085}, {999,10404}, {1001,7742}, {1111,3674}, {1156,5557}, {1158,6833}, {1159,3851}, {1193,3120}, {1210,3671}, {1319,5901}, {1329,3753}, {1385,7354}, {1388,9657}, {1420,4317}, {1452,3542}, {1454,6862}, {1470,7702}, {1482,5252}, {1532,7686}, {1538,5806}, {1565,4059}, {1697,10056}, {1698,2093}, {1708,6832}, {1709,6847}, {1717,3100}, {1723,5746}, {1727,6888}, {1728,6846}, {1738,3216}, {1756,4357}, {1768,6705}, {1788,3090}, {1892,11399}, {1940,7551}, {2051,4424}, {2098,3656}, {2475,4511}, {2800,8068}, {3091,10826}, {3136,10974}, {3146,4305}, {3149,11507}, {3179,5243}, {3304,11373}, {3306,10200}, {3333,4654}, {3336,3911}, {3339,6855}, {3340,5587}, {3428,5812}, {3434,3811}, {3555,3813}, {3576,4299}, {3579,5432}, {3584,11010}, {3601,4302}, {3614,9956}, {3616,4293}, {3634,5445}, {3635,7972}, {3670,8229}, {3683,6675}, {3687,4647}, {3697,9710}, {3698,3820}, {3702,3936}, {3720,4303}, {3746,10624}, {3754,3814}, {3755,5312}, {3812,4187}, {3816,5439}, {3822,3878}, {3841,10176}, {3850,12019}, {3868,10916}, {3899,5837}, {3916,4999}, {3925,5044}, {3931,5718}, {3947,4301}, {4002,9711}, {4047,5742}, {4294,5703}, {4297,10483}, {4298,5563}, {4309,9580}, {4640,7483}, {4679,11108}, {4847,5904}, {4848,6874}, {4867,6737}, {5010,6876}, {5045,7743}, {5083,5533}, {5123,10107}, {5173,5777}, {5218,6361}, {5250,10198}, {5274,11036}, {5328,11024}, {5398,7299}, {5425,6738}, {5433,11230}, {5506,6666}, {5542,10394}, {5657,10588}, {5690,10592}, {5722,10896}, {5726,11531}, {5730,5794}, {5763,7957}, {5905,10527}, {6001,6831}, {6866,9581}, {6875,7280}, {6911,11509}, {6982,7982}, {6990,10395}, {7284,10586}, {7680,10523}, {7965,11018}, {8069,11496}, {9596,9620}, {9597,9619}, {9655,10246}, {10042,11372}, {10057,10698}, {10264,11670}, {10265,11571}, {10679,11501}, {10883,11019}

X(12047) = midpoint of X(1) and X(3585)
X(12047) = reflection of X(i) in X(j) for these (i,j): (3916,4999), (5267,1125), (10039,12)
X(12047) = X(49)-of-intouch-triangle
X(12047) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4,10572), (1,1699,1479), (1,3583,950), (1,5270,10106), (1,9612,1478), (4,3485,1), (4,3487,10393), (226,946,1), (497,3487,1), (1058,3475,1), (5290,11522,1)

X(12048) = X(3)X(6)∩X(237)X(8881)

Barycentrics a^2 (SA+S Cot[w]^2 (Cot[w]-Csc[w])) : :

See Angel Montedeoca, Hechos Geométricos en el Triángulo (2015) and Circunferencias de Apolonio .

X(12048) lies on these lines: {3,6}, {237,8881}

X(12048) = {X(3),X(32)}-harmonic conjugate of X(12049)
X(12048) = {X(32),X(39)}-harmonic conjugate of X(1343)

X(12049) = X(3)X(6)∩X(237)X(8880)

Barycentrics a^2 (SA+S Cot[w]^2 (Cot[w]+Csc[w])) : :

See Angel Montedeoca, Hechos Geométricos en el Triángulo (2015) and Circunferencias de Apolonio .

X(12049) lies on these lines: {3,6}, {237,8880}

X(12049) = {X(3),X(32)}-harmonic conjugate of X(12048)
X(12049) = {X(32),X(39)}-harmonic conjugate of X(1342)

X(12050) = X(3)X(6)∩X(1501)X(8880)

Barycentrics = a^2 (SA+S (Csc[w]+Tan[w])) : :

See Angel Montedeoca, Hechos Geométricos en el Triángulo (2015) and Circunferencias de Apolonio .

X(12050) lies on these lines:
{3,6}, {1501,8880}, {1676,3767}, {1677,2548}, {1701,9593}, {2546,5286}

X(12050) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,182,12051), (6,1691,1343), (32,2035,1342), (182,12020,1343)

X(12051) = X(3)X(6)∩X(1501)X(8881)

Barycentrics = a^2*(-a^2 + b^2 + c^2 + 2*S*(-Csc[ω] + Tan[ω])) : :

See Angel Montedeoca, Hechos Geométricos en el Triángulo (2015) and Circunferencias de Apolonio .

X(12051) lies on these lines: {3, 6}, {1501, 8881}, {1673, 16502}, {1676, 2548}, {1677, 3767}, {1700, 9593}, {2547, 5286}, {8880, 20965}

X(12051) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): ): (6,182,12050), (6,1691,1342), (32,2036,1343), (182,12021,1342)

X(12052) = MIDPOINT OF X(1112) AND X(3154)

Trilinears (cos(2*A)+7*cos(4*A)+cos(6*A)- 15/2)*cos(B-C)+(10*cos(A)-2* cos(3*A)-2*cos(5*A))*cos(2*(B- C))+(-cos(2*A)+cos(4*A)-3/2)* cos(3*(B-C))-2*cos(5*A)+2*cos( A)-6*cos(3*A) : :
X(12052) = 3*X(51)+X(3258) = X(476)-9*X(5640) = X(477)+7*X(9781)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25498 .

X(12052) lies on these lines:
{30,9826}, {51,3258}, {476,1316}, {477,9781}, {523,11746}, {1112,3154}

X(12052) = midpoint of X(1112) and X(3154)

X(12053) = X(1)X(4)∩X(10)X(11)

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

See Antreas Hatzipolakis and Angel Montedeoca, Hyacinthos 25502 .

The line AX(8) meets the incircle in two points, A' and A'', where A' is the point closer to A. Let σ be the affine transformation that carries A'B'C' onto A''B''C''. The finite fixed point of σ is X(12053). (Angel Montesdeoca, July 5, 2021)

X(12053) lies on these lines: {1, 4}, {2, 1697}, {3, 10624}, {5, 7743}, {7, 738}, {8, 3452}, {10, 11}, {12, 3817}, {20, 1420}, {21, 3254}, {30, 4311}, {35, 6940}, {40, 3086}, {46, 10072}, {55, 474}, {56, 516}, {57, 962}, {63, 10529}, {65, 4301}, {78, 5853}, {102, 1067}, {142, 390}, {145, 908}, {165, 7288}, {329, 6762}, {354, 3671}, {355, 9669}, {411, 2078}, {495, 9955}, {496, 517}, {498, 6983}, {499, 5119}, {518, 10392}, {519, 1837}, {527, 11240}, {550, 5126}, {551, 2646}, {553, 3333}, {595, 1936}, {758, 10959}, {936, 5082}, {938, 3340}, {960, 3813}, {993, 10966}, {999, 4292}, {1000, 5818}, {1071, 1537}, {1108, 8804}, {1155, 5493}, {1193, 3755}, {1201, 3914}, {1319, 4297}, {1329, 3880}, {1385, 1387}, {1388, 9670}, {1482, 5722}, {1616, 3772}, {1698, 9819}, {1737, 5697}, {1770, 5563}, {1776, 6763}, {1788, 7991}, {1836, 3304}, {1858, 3874}, {1864, 3555}, {1898, 2801}, {2066, 8983}, {2099, 6738}, {2136, 7080}, {2269, 5257}, {2321, 3702}, {2478, 3872}, {2550, 8583}, {2551, 4853}, {3023, 11599}, {3085, 6964}, {3091, 9578}, {3146, 4308}, {3243, 5809}, {3244, 5048}, {3295, 5886}, {3303, 11375}, {3306, 10586}, {3338, 4031}, {3361, 3474}, {3478, 10570}, {3501, 8568}, {3576, 4294}, {3577, 5804}, {3582, 11010}, {3600, 9579}, {3612, 4309}, {3622, 4313}, {3624, 5218}, {3649, 4890}, {3660, 9943}, {3663, 3665}, {3687, 4673}, {3741, 10480}, {3746, 6946}, {3753, 9843}, {3814, 10915}, {3816, 5836}, {3847, 5123}, {3877, 5837}, {3878, 10916}, {3885, 4193}, {3889, 10394}, {3895, 5552}, {3913, 6745}, {3953, 7004}, {4035, 4742}, {4310, 4907}, {4315, 7354}, {4425, 8240}, {4654, 11037}, {4668, 8275}, {4863, 6743}, {5045, 10391}, {5049, 6147}, {5068, 7320}, {5084, 9623}, {5086, 10707}, {5128, 5435}, {5250, 5745}, {5252, 10863}, {5261, 9779}, {5265, 9778}, {5281, 5550}, {5289, 6737}, {5433, 10164}, {5533, 10265}, {5536, 7098}, {5570, 5884}, {5587, 10591}, {5687, 6700}, {5703, 10389}, {5758, 10396}, {5768, 7971}, {5794, 11235}, {6705, 10785}, {6767, 11374}, {6796, 11508}, {6975, 7741}, {7988, 10588}, {8715, 11502}, {8808, 10373}, {9956, 10593}, {10043, 10051}, {10543, 11263}, {10580, 11518}

X(12053) = midpoint of X(i) and X(j) for these {i,j}: {1,1479}, {1837,2098}
X(12053) = reflection of X(i) in X(j) for these {i,j}: {10,3825}, {1210,496}, {4848,1210}, {5687,6700}, {6736,1329}
X(12053) = inner-Johnson-to-ABC similarity image of X(10)
X(12053) = Ursa-minor-to-Ursa-major similarity image of X(10)

X(12054) = X(3)X(6)∩X(30)X(83)

Trilinears 2 cos A + cos(A - 2ω) : :
Barycentrics a^2 (a^6+a^4 b^2-2 a^2 b^4+a^4 c^2-5 a^2 b^2 c^2-3 b^4 c^2-2 a^2 c^4-3 b^2 c^4) : :
X(12054) = 3 X[3]+2 X[5041].

See Angel Montedeoca, Hechos Geométricos en el Triángulo (2015) , where circles Oa, Ob, Oc are defined. Let A' be the point, other than A, in which the circles Ob and Oc intersect. Define B' and C' cyclically. Then X(12054) = X(3)-of-A'B'C'. (Peter Moses, February 25, 2017)

X(12054) lies on these lines:
{2,10131}, {3,6}, {5,7859}, {20,10359}, {30,83}, {36,10799}, {98,140}, {376,7787}, {378,11380}, {381,7808}, {382,10358}, {524,6308}, {538,8150}, {542,6292}, {549,1078}, {631,7836}, {1176,9407}, {1503,6287}, {2782,8290}, {3329,7470}, {3406,7709}, {3522,10788}, {3524,7793}, {3526,7915}, {4027,7824}, {4299,10797}, {4302,10798}, {5054,7815}, {5182,8359}, {5217,10801}, {5999,11272}, {6033,6656}, {6054,7944}, {6309,8177}, {7779,10357}, {7789,8724}, {7791,10349}, {7800,11179}, {7876,9996}, {7889,10168}, {8356,10350}, {9862,10333}

X(12054) = inverse-in-Brocard-circle of X(9821)
X(12054) = inverse-in-circle-{{X(1687),X(1688),PU(1),PU(2)}} of X(5007)
X(12054) = center of inverse-in-circle-{{X(1687),X(1688),PU(1),PU(2)}}-of-Moses-circle
X(12054) = harmonic center of Gallatly circle and circle {{X(1687),X(1688),PU(1),PU(2)}}
X(12054) = midpoint of centers of circles {{X(1379),PU(1)}} and {{X(1380),PU(1)}}
X(12054) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,9821), (3,182,3398), (3,3398,2080), (3,11842,5171), (20,10359,10796), (39,5092,3), (182,5092,1691), (1342,1343,5092), (1687,1688,5007), (5085,5116,5092)

X(12055) = X(3)X(6)∩X(99)X(3589)

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

See Angel Montedeoca, Hechos Geométricos en el Triángulo (2015) , where circles Oa, Ob, Oc are defined. Let A' be the point, other than A, in which the circles Ob and Oc intersect. Define B' and C' cyclically. Then X(12055) = X(6)-of-A'B'C'. (Peter Moses, February 25, 2017)

X(12055) lies on these lines:
{3,6}, {99,3589}, {141,7799}, {323,8041}, {732,7824}, {1495,10329}, {2023,8290}, {2502,7711}, {3231,5888}, {3619,7836}, {3763,7880}, {4048,7786}, {5103,7847}, {5254,7859}, {7757,8177}

X(12055) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,5092,1691), (6,5116,5092), (39,5092,6), (39,5116,1691), (3094,5038,5111).

X(12056) = POINT BEID 12

Trilinears 9*(cos(2*A)+cos(4*A)-11/6)* cos(B-C)-10*(cos(A)-cos(3*A))* cos(2*(B-C))+(cos(2*A)-6)*cos( 3*(B-C))-cos(5*A)-11*cos(A)+ 11*cos(3*A) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25504 .

See another construction: Antreas Hatzipolakis and Peter Moses, Euclid 102 .

X(12056) lies on this line: (2,3}

X(12057) = MIDPOINT OF X(140) AND X(10289)

Trilinears 11*(cos(2*A)+cos(4*A)-3/2)* cos(B-C)-2*(5*cos(A)-6*cos(3* A))*cos(2*(B-C))-(cos(2*A)+5)* cos(3*(B-C))+cos(5*A)-9*cos(A) +11*cos(3*A) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25504 .

X(12057) lies on this line: (2,3}

X(12057) = midpoint of X(140) and X(10289)

X(12058) = POINT BEID 13

Trilinears a*((b^2+c^2)*a^8-2*(b^4+c^4)* a^6+4*b^2*c^2*(b^2+c^2)*a^4+2* (b^8+c^8-2*(b^4+b^2*c^2+c^4)* b^2*c^2)*a^2-(b^4-c^4)^2*(b^2+ c^2)) : :
Trilinears (4*cos(2*A)+cos(4*A)-1)*cos(B- C)-(3*cos(A)+cos(3*A))*cos(2*( B-C))+9*cos(A)-cos(3*A) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25506 .

X(12058) lies on these lines: {20,2979}, {22,1495}, {51,858}, {161,1350}, {185,1993}, {394,1619}, {511,1370}, {1216,11414}, {1843,7391}, {2071,5012}, {3060,7396}, {3819,7493}, {5447,9715}, {7667,9967}, {7998,10565}, {10625,11750}

X(12059) = MIDPOINT OF X(1479) AND X(5904)

Trilinears (b+c)*a^5-(b^2+c^2)*a^4-(b+c)* (2*b^2-3*b*c+2*c^2)*a^3+(2*b^ 4+2*c^4+b*c*(b+c)^2)*a^2+(b+c) *(b^4+c^4-3*b*c*(b^2+c^2))*a-( b+c)*(b^2-c^2)*(b^3-c^3) : :
Trilinears csc(A/2)^2*((3*sin(A/2)-4*sin( 3*A/2)+sin(5*A/2))*cos((B-C)/ 2)+(cos(A)-cos(2*A))*cos(B-C)- cos(A)+1) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25506 .

X(12059) lies on these lines: {63,3678}, {72,515}, {78,2801}, {144,3648}, {200,7992}, {329,1479}, {518,10392}, {758,3436}, {908,3825}, {1898,5853}, {2802,3632}, {2975,10176}, {3421,5693}, {3585,5176}, {3680,9951}, {3681,4882}, {3868,11678}, {3927,11499}, {4847,5777}, {5442,5744}, {5883,11681}, {6001,6736}, {6763,10090}

X(12059) = midpoint of X(1479) and X(5904)
X(12059) = reflection of X(3874) in X(3825)

X(12060) = POINT BEID 14

Trilinears ((cos(2*A)+3/2)*cos(2*(B-C))+ 3*cos(2*A)-cos(4*A))*sec(B-C) : :
X(12060) = X(54) + 3 X(1157)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25510 .

X(12060) lies on this line: {3,54}

X(12061) = POINT BEID 15

Trilinears (5*cos(2*A)-2*cos(4*A)-3)*cos( B-C)+(-cos(A)+2*cos(3*A))*cos( 2*(B-C))+cos(A)-2*cos(3*A) : :
X(12061) = 4*X(389)-3*X(8550) = X(6241)-9*X(6403) = X(6241)+9*X(9973) = 9*X(11188)-5*X(11444)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25510 .

X(12061) lies on this lines:
{3,8705}, {5,11649}, {6,3518}, {52,2854}, {143,8584}, {156,576}, {235,1843}, {389,2393}, {511,3627}, {524,6243}, {575,5944}, {1192,8549}, {1503,6240}, {2781,11381}, {3517,11216}, {5449,8262}, {9019,10625}, {9781,9971}, {11188,11444}, {11441,11477}

X(12061) = midpoint of X(i) and X(j) for these {i,j}: {3,11663}, {6403,9973}
X(12061) = reflection of X(5480) in X(1843)

X(12062) = POINT BEID 16

Trilinears (3*cos(2*A)-2*cos(4*A)+7/4)* cos(B-C)+(-cos(A)+2*cos(3*A))* cos(2*(B-C))-4*cos(A)-cos(3*A) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25510 .

X(12062) lies on these lines:
{3518,11935}, {3627,6243}

X(12063) = POINT BEID 17

Trilinears (9*cos(2*A)-2*cos(4*A)+13/2)* cos(B-C)+(-3*cos(A)+2*cos(3*A) )*cos(2*(B-C))-19/2*cos(A)-3* cos(3*A) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25510 .

X(12063) lies on these lines: {24,3431}, {7530,9716}

X-parabola and related centers: X(12064)-X(12079)

This preamble and centers X(12064)-X(12079) were contributed by César Eliud Lozada, February 27, 2017.

Let A*B*C* be the side triangle of the medial and orthic triangles of ABC, and let A'B'C' be the medial triangle of A*B*C*. Then A, B, C, A', B', C' lie on a parabola here named the X-parabola of ABC. Some properties of this parabola are:

It has barycentric equation: (b^2-c^2)^2*y*z+(c^2-a^2)^2*z*x+(a^2-b^2)^2*x*y=0
It passes through the vertices of the antipedal triangle of X(477) and centers X(476), X(523), X(685), X(850), X(892), X(2395), X(2501), X(4024), X(4036), X(4581), X(4608), X(5466), X(8599), X(10412) and X(12079).
Its directrix and the Euler line of ABC are parallel, therefore its axis and the Euler line of ABC are perpendicular.
Its 4th intersection with the circumcircles of ABC and A'B'C' is X(476)
The focus and vertex are X(12064) and X(12065), respectively.
Its axis is the line {523, 5972}, trilinear polar of X(12066).
Its directrix is the line {30, 10279}, trilinear polar of X(12067).
The perspector is X(115) and the center is X(523).
The dual conic of the X-parabola has center X(620), perspector X(4590) and passes through the vertices of the cevian triangle of X(4590) and centers X(2), X(32), X(439), X(593), X(1509), X(2482), X(3926), X(4027), X(7058), X(7794), X(11128), X(11129).

Let t_a, t_b, t_c be the tangents to the X-parabola at A, B, C, respectively; the triangle A_tB_tC_t bounded by these tangents is here named the X-parabola-tangential triangle of ABC. Barycentric coordinates of A-vertex are:
A_t = -(b^2-c^2)^2 : (a^2-c^2)^2 : (a^2-b^2)^2

The appearance of (T,i) in the following list means that triangles T and X-parabola-tangential are perspective with perspector X(i): (ABC, 115), (extouch, 12069), (2nd Hatzipolakis, 12070), (incentral, 12071), (intouch, 12072), (Lemoine, 12073), (Macbeath, 12075), (medial, 523), (orthic, 512), (Steiner, 12076), (symmedial, 12077), (Yff contact, 12078).

The X-parabola is the isogonal conjugate of line X(110)X(351) (the tangent to circumcircle at X(110)), and the isotomic conjugate of line X(99)X(110) (the tangent to Steiner circumellipse at X(99)). (Randy Hutson, March 9, 2017)

The X-parabola-tangential triangle is the anticevian triangle of X(115). (Randy Hutson, June 27, 2018)

X(12064) = FOCUS OF THE X-PARABOLA

Barycentrics (a^10-2*(b^2+c^2)*a^8+6*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-b^2*c^2*(5*b^4-9*b^2*c^2+5*c^4))*a^2-(b^4-c^4)*(b^2-c^2)^3)*(b^2-c^2) : :
Trilinears ((cos(A)+cos(3*A))*cos(B-C)-(cos(2*A)+cos(4*A)-5/2)*cos(2*(B-C))-(cos(A)-cos(3*A))*cos(3*(B-C))+1/2*cos(6*A)+cos(2*A)+cos(4*A)-1)*sin(B-C) : :
X(12064) = X(110)+3*X(8029) = X(125)-3*X(10278) = X(3448)-9*X(5466) = 2*X(6723)-3*X(10189)

X(12064) lies on the curve Q077 and these lines: {110,8029}, {125,10278}, {523,5972}, {1112,2501}, {3448,5466}, {5663,10279}, {6723,10189}

X(12065) = VERTEX OF THE X-PARABOLA

Barycentrics (SB-SC)/(7*S^4+(8*SA^2-5*SB*SC-SW^2)*S^2+3*(SA^2-7*SB*SC-SW^2)*SA^2) : :
Trilinears sin(B-C)/((2*cos(A)-cos(3*A))*cos(B-C)-(2*cos(2*A)+3/2)*cos(2*(B-C))+3*cos(2*A)+cos(4*A)+1/2) : :
X(12065) = X(3233)+3*X(8029)

X(12065) lies on these lines: {523,5972}, {3233,8029}

X(12066) = TRILINEAR POLE OF THE AXIS OF THE X-PARABOLA

Barycentrics 1/(7*S^4+(8*SA^2-5*SB*SC-SW^2)*S^2+3*(SA^2-7*SB*SC-SW^2)*SA^2) : :

X(12066) is the trilinear pole of X(523)X(5972) which is the locus of radical centers of the circles centered at the vertices of ABC and tangent to lines through X(30) (i.e., parallel to Euler line). (Randy Hutson, March 9, 2017)

X(12066) lies on the Kiepert hyperbola and these lines: {98,10733}, {5466,12065}

X(12066) = Trilinear pole of the line {523,5972}

X(12067) = TRILINEAR POLE OF THE DIRECTRIX OF THE X-PARABOLA

Barycentrics 1/((2*cos(A)+3*cos(3*A))*cos(B-C)-2*(cos(2*A)+cos(4*A)+3/4)*cos(2*(B-C))-(2*cos(A)+cos(3*A))*cos(3*(B-C))+1/2*cos(6*A)+2*cos(2*A)+2*cos(4*A)+3) : :

X(12067) = isogonal conjugate of {6,647}∩{3292,11063}
X(12067) = trilinear pole of the line {30,10279}

X(12068) = EULER LINE ∩ AXIS OF THE X-PARABOLA

Trilinears (2*cos(2*A)+4*cos(4*A)-7/2)*cos(B-C)+(12*cos(A)+cos(3*A))*cos(2*(B-C))-(2*cos(2*A)+5/2)*cos(3*(B-C))-2*cos(5*A)-6*cos(A)-7*cos(3*A) : :
X(12068) = 3*X(2)+X(7471)

X(12068) lies on these lines: {2,3}, {125,3233}, {523,5972}, {5642,6070}, {11064,11657}

X(12068) = midpoint of X(i) and X(j) for these {i,j}: {125,3233}, {3154,7471}, {11064,11657}
X(12068) = complement of X(3154)
X(12068) = orthogonal projection of X(5972) on the Euler line
X(12068) = {X(2), X(7471)}-harmonic conjugate of X(3154)

X(12069) = PERSPECTOR OF THESE TRIANGLES: X-PARABOLA-TANGENTIAL AND EXTOUCH

Barycentrics (b^2-c^2)*(b+c)*(a^4-(b^2+b*c+c^2)*a^2+(b+c)*(b^2-3*b*c+c^2)*a+(b^2-c^2)^2) : :
X(12069) = X(4041)-3*X(8029)

X(12069) lies on these lines: {523,8045}, {4041,8029}, {4770,6367}

X(12070) = PERSPECTOR OF THESE TRIANGLES: X-PARABOLA-TANGENTIAL AND 2nd HATZIPOLAKIS

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

X(12070) lies on no lines {X(i), X(j)} for i, j ≤ 12069

X(12071) = PERSPECTOR OF THESE TRIANGLES: X-PARABOLA-TANGENTIAL AND INCENTRAL

Barycentrics (b^2-c^2)*(b+c)^2*(a^3+(b+c)*a^2-(b^2-3*b*c+c^2)*a-(b^2-c^2)*(b-c)) : :
X(12071) = X(4705)-3*X(8029)

X(12071) lies on these lines: {512,12069}, {523,8043}, {4041,4838}, {4705,8029}

X(12072) = PERSPECTOR OF THESE TRIANGLES: X-PARABOLA-TANGENTIAL AND INTOUCH

Barycentrics (b^2-c^2)*(b+c)*(a^3+(b+c)*a^2+b*c*a-(b^2-c^2)*(b-c)) : :
X(12072) = X(661)-3*X(8029)

X(12072) lies on these lines: {512,12069}, {523,2487}, {661,8029}

X(12072) = reflection of X(12069) in X(12071)

X(12073) = PERSPECTOR OF THESE TRIANGLES: X-PARABOLA-TANGENTIAL AND LEMOINE

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

X(12073) lies on these lines: {30,511}, {83,5466}, {1637,3288}, {1649,3005}, {4108,9189}, {4808,4822}, {5027,9185}, {8371,11183}, {8723,9751}, {9123,9208}, {9180,11606}, {9485,9889}, {10183,10278}

X(12073) = crossdifference of every pair of points on line X(6)X(5888)
X(12073) = isogonal conjugate of X(12074)

X(12074) = ISOGONAL CONJUGATE OF X(12073)

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

X(12074) lies on the circumcircle and these lines: {39,111}, {98,549}, {662,2748}, {691,1634}, {827,5467}, {843,2076}, {2396,9069}, {9145,11636}

X(12074) = reflection of X(11638) in X(7711)
X(12074) = isogonal conjugate of X(12073)
X(12074) = trilinear pole of the line {6,5888}

X(12075) = PERSPECTOR OF THESE TRIANGLES: X-PARABOLA-TANGENTIAL AND MACBEATH

Barycentrics ((b^2+c^2)*a^2+(b^2-c^2)^2)*(b^2-c^2) : :
X(12075) = X(669)-3*X(1637) = X(850)-3*X(9134) = X(2525)-3*X(9148) = X(3005)+3*X(8029)

X(12075) lies on these lines: {83,5466}, {460,512}, {523,4885}, {669,1637}, {826,850}, {2525,9148}, {3005,8029}, {6562,9209}

X(12075) = radical center of {nine-point circle, nine-point circle of medial triangle, orthosymmedial circle}

X(12076) = PERSPECTOR OF THESE TRIANGLES: X-PARABOLA-TANGENTIAL AND STEINER

Barycentrics (b^2-c^2)^3*(3*a^4-3*(b^2+c^2)*a^2-b^2*c^2+2*c^4+2*b^4) : :
X(12076) = X(115)-3*X(8029) = 2*X(6722)-3*X(10278)

X(12076) lies on these lines: {115,8029}, {148,690}, {523,620}, {2079,7669}, {6036,10279}, {6721,8151}, {6722,10278}

X(12076) = reflection of X(i) in X(j) for these (i,j): (6036,10279), (8151,6721)

X(12077) = PERSPECTOR OF THESE TRIANGLES: X-PARABOLA-TANGENTIAL AND SYMMEDIAL

Trilinears sin A sin(2B - 2C) : :
Barycentrics ((b^2+c^2)*a^2-(b^2-c^2)^2)*(b^2-c^2) : :
X(12077) = 2*X(647)-3*X(1637) = 3*X(647)-4*X(6587) = 3*X(1637)-4*X(2501) = 3*X(2501)-2*X(6587) = X(3005)-3*X(8029) = 3*X(8029)-2*X(12075)

Let A'B'C' be the anticevian triangle of X(4). Let A"B"C" be the tangential triangle, wrt A'B'C', of the bianticevian conic of X(4) and X(6). The lines A'A", B'B", C'C" concur in X(12077). (Randy Hutson, March 9, 2017)

X(12077) lies on these lines: {6,2623}, {230,231}, {251,2395}, {648,9514}, {661,2171}, {826,3569}, {850,2525}, {1640,12073}, {2081,2600}, {3005,8029}, {3288,7927}, {5466,7608}

X(12077) = reflection of X(i) in X(j) for these (i,j): (647,2501), (2525,850), (3005,12075)
X(12077) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (647,2501,1637), (3005,8029,12075)
X(12077) = intersection of trilinear polars of X(4) and X(5)
X(12077) = perspector of hyperbola {A,B,C,X(4),X(5)} (circumconic centered at X(137))
X(12077) = crossdifference of every pair of points on line X(3)X(54)
X(12077) = center of circumconic that is locus of trilinear poles of lines passing through X(137)
X(12077) = X(2)-Ceva conjugate of X(137)
X(12077) = polar conjugate of isotomic conjugate of X(6368)
X(12077) = X(63)-isoconjugate of X(933)
X(12077) = X(95)-isoconjugate of X(163)
X(12077) = perspector of ABC and orthocevian triangle of X(930)
X(12077) = barycentric product X(5)*X(523)
X(12077) = intersection of orthic axes of ABC and reflection triangle

X(12078) = PERSPECTOR OF THESE TRIANGLES: X-PARABOLA-TANGENTIAL AND YFF CONTACT

Barycentrics (b^2-c^2)^2*(b-c)*(a^4+2*(b+c)*a^3-2*(b^2+b*c+c^2)*a^2-(b+c)*(b^2+c^2)*a+2*b^4+2*c^4+b*c*(b^2-b*c+c^2)) : :
X(12078) = X(3120)-3*X(8029)

X(12078) lies on these lines: {148,690}, {3120,8029}

X(12079) = X(30)X(74)∩X(125)X(523)

Barycentrics (b^2-c^2)^2/(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(12079) = X(74)+3*X(5627) = 3*X(125)-X(3258) = X(476)+3*X(9140) = 3*X(3154)-2*X(3258) = X(3258)+3*X(6070)

Let M_aM_bM_c = medial triangle. Let (𝒫_a) be the parabola, tangent to Euler line, to NM_a, and to the line BC at its vertex, so that its directrix, d_a, is parallel to BC. Define the lines d_b and d_c cyclically. Let T be the triangle bounded by the lines d_a, d_b, d_c. Then T is homothetic to ABC, and the center of homothety is X(12079). For a construction, see Paris Pamfilos, A Gallery of Conics by Five Elements, Forum Geometricorum 14 (2014) 295-348, paragraph 13.3, page 346: construct a conic tangent to the line at infinity, i.e. a parabola, tangent to three lines a, b, c and passing through [D], i.e. with given axis-direction. (Angel Montesdeoca, March 1, 2022)

X(12079) lies on the X-parabola, Gibert's cubics K217, K741, Gibert's curve Q078 and these lines: {2,9717}, {30,74}, {98,468}, {110,12068}, {115,2501}, {125,523}, {325,892}, {339,850}, {542,3233}, {868,2394}, {1503,11657}, {1552,10151}, {1648,2395}, {2452,5094}, {3448,7471}, {3470,3628}, {7473,9862}, {8749,8791}, {10257,10419}

X(12079) = midpoint of X(i) and X(j) for these {i,j}: {125,6070}, {3448,7471}
X(12079) = reflection of X(i) in X(j) for these (i,j): (110,12068), (3154,125)
X(12079) = reflection of X(476) in the axis of the X-parabola
X(12079) = vertex of inscribed parabola with focus X(74) (and perspector X(1494), axis X(30)X(74) and directrix X(4)X(523))

X(12080) = POINT BEID 18

Barycentrics (b+c) (-2 a^5+a^4 b+3 a^3 b^2-a^2 b^3-a b^4+a^4 c-4 a^3 b c+b^4 c+3 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(12080) = X[1109]-3 X[1962]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25539.

X(12080) lies on these lines: {1109,1962}, {2650,3635}, {3957 ,6758}

X(12081) = POINT BEID 19

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25539.

X(12081) lies on these lines: {1,21}, {517,3724}, {523,663}, {740,4511}, {5844,10459}

X(12082) = REFLECTION OF X(378) IN X(22)

Trilinears (2*cos(2*A)-4)*cos(B-C)+11* cos(A)-cos(3*A) : :
X(12082) = 2*X(3)-3*X(22) = 4*X(3)-3*X(378) = 5*X(3)-6*X(7502) = 3*X(3)-4*X(7555) = 5*X(22)-4*X(7502) = 9*X(22)-8*X(7555)

As a point on the Euler line, X(12082) has Shinagawa coefficients (2*E+2*F, -5*E-2*F).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25532.

X(12082) lies on these lines: {2,3}, {159,5656}, {316,9723}, {511,11456}, {575,10984}, {576,7592}, {944,9911}, {1181,8718}, {1199,11482}, {1350,11459}, {1498,2781}, {1633,6361}, {3068,9695}, {3284,8743}, {3292,6759}, {4293,10833}, {4296,9645}, {8717,9730}, {10625,11441}

X(12082) = reflection of X(378) in X(22)
X(12082) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,23,24), (3,1598,11284), (3,5198,3090), (3,7387,23), (3,7530,1995), (3,11284,631), (4,10323,7509), (4,11414,10323), (20,23,3), (20,7387,24), (26,1657,11413), (1995,7530,10594), (3146,7492,7527), (3529,7556,7464), (7464,7556,3), (7492,7527,3)

X(12083) = REFLECTION OF X(3) IN X(22)

Trilinears (2*cos(2*A)-2)*cos(B-C)+7*cos( A)-cos(3*A) : :
X(12083) = X(3) - 2 X(22) = 3*X(3)-2*X(378) = 3*X(3)-4*X(7502) = 5*X(3)-8*X(7555) = 3*X(22)-X(378) = 3*X(22)-2*X(7502) = 5*X(22)-4*X(7555)

As a point on the Euler line, X(12083) has Shinagawa coefficients (3*E+4*F, -7*E-4*F).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25532.

X(12083) lies on these lines: {2,3}, {35,9658}, {36,9673}, {115,8553}, {159,399}, {161,6000}, {195,11577}, {265,5621}, {394,10540}, {567,3796}, {999,4351}, {1154,11456}, {1181,6243}, {1351,8547}, {1482,9911}, {2917,5895}, {3070,9683}, {3098,5891}, {3295,4354}, {3579,8185}, {3581,10605}, {5446,10984}, {5889,8718}, {6101,11441}, {6449,8276}, {6450,8277}, {6759,10625}, {7592,10263}, {7737,9609}, {8148,8192}, {9655,10831}, {9659,10483}, {9668,10832}, {9914,9920}, {10564,11202}, {10620,11820}

X(12083) = reflection of X(i) in X(j) for these (i,j): (3,22), (7391,5)
X(12083) = Stammler-circle-inverse-of-X(7574)
X(12083) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1598,1656), (3,3843,7395), (3,5073,1593), (3,5899,25), (3,7387,7517), (3,7517,7506), (3,9909,2070), (4,6636,7514), (20,26,3), (22,378,7502), (23,376,6644), (25,5899,7517), (378,7502,3), (1657,2937,3), (3146,7512,7526), (3627,7525,7503), (5198,7393,3851), (7387,11414,3), (7503,7525,3), (7512,7526,3), (7556,11001,2071)

X(12084) = MIDPOINT OF X(64) AND X(155)

Trilinears (2*cos(2*A)+4)*cos(B-C)-7*cos( A)-cos(3*A) : :
Barycentrics a^2[S^2 + SA(SA + 2 SW - 14 R^2)] : :
X(12084) = 3*X(3)-2*X(1658) = 3*X(3)-X(7387) = 15*X(3)-7*X(10244) = 17*X(3)-9*X(10245) = 3*X(5654)-X(5878)

As a point on the Euler line, X(12084) has Shinagawa coefficients (E-4*F, -3*E+4*F).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25532.

Let La be the polar of X(3) wrt the A-power circle, and define Lb, Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. Triangle A'B'C' is homothetic to ABC at X(5094) and to the anticomplementary triangle at X(22). X(4)-of-A'B'C' = X(1657), and X(5)-of-A'B'C' = X(12084). (Randy Hutson, March 9, 2017)

X(12084) lies on these lines: {2,3}, {49,11456}, {52,1204}, {56,8144}, {64,155}, {74,5889}, {143,9786}, {156,1498}, {184,10575}, {394,5876}, {511,7689}, {542,9925}, {1069,10060}, {1092,10564}, {1147,6000}, {1151,11265}, {1152,11266}, {1154,10606}, {1236,1975}, {1288,1294}, {2883,9820}, {3157,10076}, {3357,9938}, {3796,10610}, {4299,9672}, {4302,9659}, {4550,11793}, {5204,9645}, {5446,11438}, {5584,8141}, {5621,11255}, {5654,5878}, {5946,10982}, {6102,10605}, {6759,12038}, {7747,9608}, {7756,9609}, {9730,11424}, {10263,12041}, {10539,11381}, {11267,11480}, {11268,11481}, {11412,11440}

X(12084) = midpoint of X(64) and X(155)
X(12084) = reflection of X(i) in X(j) for these (i,j): (3,11250), (26,3), (1498,156), (1658,10226), (2883,9820), (6759,12038), (7387,1658), (11477,11255)
X(12084) = 1st-Droz-Farny-circle-inverse-of-X(403)
X(12084) = midpoint of X(3) and X(12085)
X(12084) = harmonic center of circumcircle and first Droz-Farny circle
X(12084) = harmonic center of tangential circle and Trinh circle
X(12084) = center of inverse-in-first-Droz-Farny-circle-of-nine-point-circle
X(12084) = reflection in X(5) of [center of inverse-in-second-Droz-Farny-circle-of-nine-point-circle]
X(12084) = center of circle that is the circumperp conjugate of the nine-point circle
X(12084) = circumperp conjugate of X(2072)
X(12084) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,4,6644), (3,382,24), (3,1593,5), (3,1597,6642), (3,5073,2070), (3,7387,1658), (3,7395,549), (3,7503,7516), (3,7517,186), (3,7526,7514), (4,2071,3), (4,3548,5), (24,382,7530), (186,3146,7517), (1597,6642,546), (1658,7387,26), (1658,10226,3), (2041,2042,11799), (7503,7516,7514), (7516,7526,7503), (7529,11403,3845)

X(12085) = EULER LINE INTERCEPT OF X(36)X(9645)

Trilinears (2*cos(2*A)+6)*cos(B-C)-11* cos(A)-cos(3*A) : :
X(12085) = 3*X(3)-2*X(26) = 5*X(3)-4*X(1658) = 11*X(3)-7*X(10244) = 13*X(3)-9*X(10245) = 3*X(3)-4*X(11250)

As a point on the Euler line, X(12085) has Shinagawa coefficients (E-2*F, -3*E+2*F).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25532.

X(12085) lies on these lines: {2,3}, {36,9645}, {52,10605}, {56,9629}, {68,6247}, {154,12038}, {155,6000}, {511,3357}, {999,8144}, {1069,6285}, {1092,11381}, {1147,1498}, {1181,10575}, {1350,9973}, {1351,6102}, {1619,5878}, {1853,9927}, {1993,6241}, {2777,9914}, {2883,5654}, {2935,9937}, {3157,7355}, {3260,3964}, {3527,5946}, {4299,10832}, {4302,10831}, {4550,5447}, {5446,9786}, {5907,11472}, {6001,9928}, {6221,11265}, {6238,10060}, {6398,11266}, {6800,8718}, {7352,10076}, {7689,10606}, {8778,10317}, {9730,10982}, {9908,9938}, {10539,10564}

X(12085) = reflection of X(i) in X(j) for these (i,j): (3,12084), (26,11250), (68,6247), (1498,1147), (7387,3), (9908,9938)
X(12085) = exsimilicenter of tangential circle and Trinh circle; the insimilicenter is X(3)
X(12085) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,4,6642), (3,382,25), (3,1597,5), (3,1598,6644), (3,3830,7506), (3,5073,7517), (3,7517,3515), (3,9714,186), (3,9909,1658), (4,3546,5), (4,7464,11413), (4,11413,3), (22,3520,3), (26,11250,3), (376,7503,3), (550,7526,3), (2071,3146,24), (3522,7527,7509), (3627,6644,1598), (3830,7506,5198), (9715,11410,3)

X(12086) = EULER LINE INTERCEPT OF X(52)X(74)

Trilinears (2*cos(2*A)+5)*cos(B-C)-8*cos( A)-cos(3*A) : :
X(12086) = 5*X(3)-3*X(2937) = 2*X(3)-3*X(3520) = 4*X(3)-3*X(7488)

As a point on the Euler line, X(12086) has Shinagawa coefficients (E-4*F, -4*E+4*F).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25532.

X(12086) lies on these lines: {2,3}, {52,74}, {54,10575}, {56,9539}, {64,1993}, {110,11381}, {185,1994}, {324,1105}, {511,11440}, {1204,3060}, {1498,9544}, {2935,3448}, {3357,5889}, {3580,6696}, {4550,7999}, {5584,9536}, {5866,7773}, {7355,9637}, {9306,11439}, {9545,11456}, {9786,11002}, {10539,11455}, {10574,11424}, {11003,11425}

X(12086) = reflection of X(7488) in X(3520)
X(12086) = 1st-Droz-Farny-circle-inverse-of-X(11563)
X(12086) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,3146,23), (3,3627,3518), (3,11403,1995), (382,11250,186), (1593,11413,2)

X(12087) = REFLECTION OF X(3520) IN X(2937)

Trilinears (2*cos(2*A)-3)*cos(B-C)+8*cos( A)-cos(3*A) : :
X(12087) = 3*X(3)-5*X(2937) = 6*X(3)-5*X(3520) = 4*X(3)-5*X(7488)

As a point on the Euler line, X(12087) has Shinagawa coefficients (3*E+4*F, -8*E-4*F).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25532.

X(12087) lies on these lines: {2,3}, {52,8718}, {145,9911}, {161,6225}, {323,6759}, {3600,10833}, {7691,11381}, {8185,9778}

X(12087) = reflection of X(3520) in X(2937)
X(12087) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (20,7387,23), (26,3529,2071), (382,7512,7527), (2937,3520,7488), (5198,7485,5068)

X(12088) = REFLECTION OF X(7488) IN X(2937)

Trilinears (2*cos(2*A)-1)*cos(B-C)+4*cos( A)-cos(3*A) : :
X(12088) = 3*X(3)-5*X(2937) = 6*X(3)-5*X(3520) = 4*X(3)-5*X(7488)

As a point on the Euler line, X(12088) has Shinagawa coefficients (2*E+4*F, -5*E-4*F).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25532.

X(12088) lies on these lines: {2,3}, {110,10625}, {156,323}, {182,9781}, {185,8718}, {511,1614}, {515,9591}, {516,9626}, {575,1173}, {576,11423}, {1058,10833}, {1199,3060}, {1994,10263}, {2883,2917}, {2916,5480}, {2979,10539}, {3068,9683}, {3085,9658}, {3086,9673}, {3098,7999}, {3567,10984}, {3746,4354}, {4297,9625}, {4351,5563}, {5012,5446}, {5657,8185}, {6101,10540}, {6759,11412}, {7712,9545}, {7737,9700}, {8744,10316}, {9934,10628}

X(12088) = reflection of X(7488) in X(2937)
X(12088) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,23,3518), (3,1995,3525), (3,3091,7550), (3,3627,7527), (3,3628,7496), (3,7530,3091), (3,7545,3628), (3,10594,3090), (4,22,7512), (20,26,186), (22,7387,4), (24,11414,376), (25,10323,631), (1598,7509,3545), (3091,7492,3), (3529,7556,3), (3547,7500,4), (3627,7555,3), (7485,7529,5067), (7492,7530,7550), (9909,11414,24)

X(12089) = CENTER OF ASHRAFOV-MONTESDEOCA CONIC

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25540.

Let ABC be a triangle with incenter X(1)=I and let a' be perpendicular line to AI through I. Denote as A'_b the intersection of a' and the perpendicular line to AB through B and denote as A'_c the intersection of a' and the perpendicular line through C to AC. Perpendicular lines to a' through A'_b and A'_c cut BC at A_b and A_c, respectively. Points B_c, B_a, C_a, C_b are built cyclically. Then these six points lie on a conic here named the Ashrafov-Montesdeoca conic. (See: Angel Montesdeoca, HGT-Feb 17, 2017).

An alternative construction of A_b and A_c: Let ABC be a triangle with incenter I=X(1) and let A'B'C' be the antipedal triangle of I (excentral triangle). The parallel lines to AI through C', B' cut BC at A_b and A_c, respectively. (Antreas Hatzipolakis, Hyacinthos 25529).

X(12089) lies on these lines: {65, 603}, {73, 2292}, {1071, 3931}, {1254, 1400}

X(12090) = CENTER OF HATZIPOLAKIS-MONTESDEOCA-DE LONGCHAMPS CONIC

Barycentrics 3 (b^2-c^2)^8 (25 b^8+188 b^6 c^2+342 b^4 c^4+188 b^2 c^6+25 c^8)
-2 (b^2-c^2)^6 (167 b^10+123 b^8 c^2-1826 b^6 c^4-1826 b^4 c^6+123 b^2 c^8+167 c^10) a^2
+4 (b^2-c^2)^4 (69 b^12-1094 b^10 c^2-277 b^8 c^4+3628 b^6 c^6-277 b^4 c^8-1094 b^2 c^10+69 c^12) a^4
+2 (b^2-c^2)^4 (467 b^10+2599 b^8 c^2-7930 b^6 c^4-7930 b^4 c^6+2599 b^2 c^8+467 c^10) a^6
-(b^2-c^2)^2 (2073 b^12-9110 b^10 c^2-12937 b^8 c^4+44044 b^6 c^6-12937 b^4 c^8-9110 b^2 c^10+2073 c^12) a^8
+4 (b^2-c^2)^2 (165 b^10-3599 b^8 c^2+5098 b^6 c^4+5098 b^4 c^6-3599 b^2 c^8+165 c^10) a^10
+32 (b^2-c^2)^2 (63 b^8+68 b^6 c^2-782 b^4 c^4+68 b^2 c^6+63 c^8) a^12
-4 (b^2-c^2)^2 (549 b^6-1877 b^4 c^2-1877 b^2 c^4+549 c^6) a^14
+(201 b^8-5252 b^6 c^2+9846 b^4 c^4-5252 b^2 c^6+201 c^8) a^16
+22 (39 b^6-23 b^4 c^2-23 b^2 c^4+39 c^6) a^18
-4 (125 b^4-98 b^2 c^2+125 c^4) a^20
+78 (b^2+c^2) a^22
+5 a^24 : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25540.

Let ABC be a triangle, P = X(20) = De Longchamps point of ABC and let A'B'C' be the antipedal triangle of P. The parallel lines to AP through C', B' cut BC at A_b and A_c, respectively. Build B_c, B_a, C_a, C_b cyclically. Then these six points lie on a conic where named the Hatzipolakis-Montesdeoca-De Longchamps conic. (Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25540).

X(12091) = X(3)X(476)∩X(30)X(1986)

Trilinears cos(A)*((80*cos(A)+30*cos(3*A) +2*cos(5*A))*cos(B-C)+(-24* cos(2*A)-2*cos(4*A)-23)*cos(2* (B-C))+10*cos(A)*cos(3*(B-C))- cos(4*(B-C))-8*cos(4*A)-36* cos(2*A)-27) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25543.

X(12091) lies on these lines: {3,476}, {30,1986}, {131,2072}, {523,7723}, {1368,11749}

X(12092) = POINT BEID 20

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25543.

X(12092) lies on the circumcircle and this line: {74,11250}

X(12092) = Ψ(X(4), X(49))

X(12093) = POINT BEID 21

Trilinears a*((b^2+c^2)*a^2-b^4-c^4)*((a^ 2+b^2+c^2)^2-9*b^2*c^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25550.

X(12093) lies on these lines: {2,2854}, {114,325}, {183,9775}, {526,9185}, {1995,9145}, {2871,7998} , {5640,11163}, {5663,6054}, {5968,9155}, {9770,11002}, {9872,11580}, {10748,11185}

X(12094) = POINT BEID 22

Trilinears ((15*cos(A)-16*cos(3*A)+7*cos( 5*A))*cos(B-C)+(-3*cos(2*A)-5* cos(4*A)-4)*cos(2*(B-C))-6* cos(A)*cos(3*(B-C))+6*cos(4*A) -cos(6*A)+18*cos(2*A)-11)*csc( A) : :
Barycentrics 2*a^10+3*(b^2+c^2)*a^8-(7*b^4+ 4*b^2*c^2+7*c^4)*a^6-(b^2+c^2) *(3*b^4-11*b^2*c^2+3*c^4)*a^4+ (b^2-c^2)^2*(5*b^4+13*b^2*c^2+ 5*c^4)*a^2-6*(b^4-c^4)*(b^2-c^ 2)*b^2*c^2 : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25550.

X(12094) lies on this line: {543,3629}

X(12095) = MIDPOINT OF X(186) AND X(10420)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25553.

X(12095) lies on the cubic K038 and these lines: {2,5962}, {3,49}, {30,136}, {186,1299}

X(12095) = midpoint of X(186)) and X(10420)
X(12095) = complement of X(5962)
X(12095) = circumcircle-inverse-of-X(155)
X(12095) = inverse-in-complement-of-polar-circle of X(1216)

X(12096) = CIRCUMCIRCLE-INVERSE OF X(1498)

Trilinears (-a^2+b^2+c^2)^2*(2*a^10-(b^2+ c^2)*a^8-8*(b^2-c^2)^2*a^6+10* (b^4-c^4)*(b^2-c^2)*a^4-2*(b^ 2-c^2)^2*(b^4+6*b^2*c^2+c^4)* a^2-(b^4-c^4)*(b^2-c^2)^3)*a : :
Trilinears cos(A)^2*((5*cos(2*A)+7)*cos( B-C)-cos(A)*cos(2*(B-C))-10* cos(A)-cos(3*A)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25553.

X(12096) lies on the cubic K038 and these lines: {3,64}, {30,122}, {131,10257}, {520,4091}, {631,6761}, {1304,2071}, {2060,3346}

X(12096) = midpoint of X(i) and X(j) for these {i,j}: {3,6760}, {1304,2071}
X(12096) = reflection of X(11589) in X(3)
X(12096) = complement of X(34170)
X(12096) = circumcircle-inverse-of-X(1498)

X(12097) = X(2)X(17)∩X(6671)X(8014)

Barycentrics 3*sqrt(3)*(2*a^6-4*(b^2+c^2)* a^4+(b^4-12*b^2*c^2+c^4)*a^2+( b^4-c^4)*(b^2-c^2)) -2*S*(2*a^4+20*(b^2+c^2)*a^2+2 0*b^2*c^2-7*b^4-7*c^4) : :

See Tran Quang Hung and César Lozada, Hyacinthos 25555.

X(12097) lies on these lines: {2, 17}, {6671, 8014}

X(12098) = X(2)X(18)∩X(6672)X(8015)

Barycentrics 3*sqrt(3)*(2*a^6-4*(b^2+c^2)* a^4+(b^4-12*b^2*c^2+c^4)*a^2+( b^4-c^4)*(b^2-c^2)) +2*S*(2*a^4+20*(b^2+c^2)*a^2+2 0*b^2*c^2-7*b^4-7*c^4) : :

See Tran Quang Hung and César Lozada, Hyacinthos 25555.

X(12098) lies on these lines: {2, 18}, {6672, 8015}

X(12099) = MIDPOINT OF X(51) AND X(125)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25559.

X(12099) lies on the Hutson centroidal ellipse and these lines: {4,10293}, {6,5505}, {25,5622}, {51,125}, {54,5643}, {74,1597}, {373,597}, {381,5640}, {526,1637}, {542,5943} et al.

X(12099) = midpoint of X(51) and X(125)
X(12099) = centroid of pedal triangle of X(125)
X(12099) = intersection of tangents to Walsmith rectangular hyperbola at X(6) and X(125)

X(12100) = MIDPOINT OF X(5) AND X(376)

Barycentrics 10*a^4-11*(b^2+c^2)*a^2+(b^2- c^2)^2 : :
X(12100) = X(2)+3*X(3) = 2*X(3)+X(140) = 11*X(2)-3*X(4) = 5*X(2)-3*X(5) = 13*X(2)+3*X(20) = 2*X(2)-3*X(140) = 5*X(2)+3*X(376) = 7*X(2)-3*X(381) = 23*X(2)-3*X(382) = X(40)+3*X(3653) = 3*X(165)+X(3656)

As a point on the Euler line, X(12100) has Shinagawa coefficients: (11, -9).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25561.

Let A' be the circumcenter of BCX(2), and define B' and C' cyclically. The centroid of A'B'C'X(2) is X(12100). (Randy Hutson, March 9, 2017)

Let Oa be the circle centered at A with radius equal to the distance between X(2) and the midpoint of BC, and define Ob and Oc cyclically. X(12100) is the radical center of Oa, Ob, Oc. (Randy Hutson, March 9, 2017)

X(12100) lies on these lines:
{2,3}, {35,5298}, {36,4995}, {40,3653}, {165,3656}, {182,8584}, {187,9300}, {230,8589}, {395,10645}, {396,10646}, {524,5092}, {539,10213}, {541,10272}, {551,3579}, {553,5122}, {574,5306}, {597,3098}, {952,4669}, {1216,11592}, {1327,8253}, {1328,8252}, {1503,10193}, {1587,6497}, {1588,6496}, {1992,12017}, {2482,12042}, {3055,6781}, {3058,5010}, {3068,6452}, {3069,6451}, {3576,3654}, {3655,4677}, {3793,7837}, {3815,8588}, {3819,5663}, {4316,5326}, {4324,7294}, {4745,6684}, {5204,10056}, {5217,10072}, {5434,7280}, {5442,10543}, {5585,7737}, {5609,11693}, {5642,12041}, {6390,7771}, {6410,8981}, {6445,7586}, {6446,7585}, {6456,9540}, {7288,10386}, {7618,8667}, {7767,7799}, {7811,7871}, {8182,9766}, {9729,10627}, {9774,11149}, {10192,11204}

X(12100) =midpoint of X(i) and X(j) for these {i,j}: {2,8703}, {3,549}, {5,376}, {381,550}, {547,548}, {551,3579}, {597,3098}, {2482,12042}, {3534,3845}, {3655,5690}, {5642,12041}, {8182,12040}, {10192,11204}, {10304,11539}
X(12100) = reflection of X(i) in X(j) for these (i,j): (2,11812), (4,11737), (5,10124), (140,549), (381,3628), (546,547), (547,140), (549,3530), (3543,3861), (3830,3860), (3845,10109), (3853,381), (5066,2), (10109,11540)
X(12100) = complement of X(3845)
X(12100) = anticomplement of X(10109)
X(12100) = X(140)-Gibert-Moses centroid; see the preamble just before X(21153)
X(12100) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,8703), (2,3830,5), (2,3845,10109), (2,5066,547), (3,381,10304), (5,3830,3860), (5,3860,5066), (140,3853,3628), (140,5066,2), (376,3839,1657), (381,631,11539), (381,5055,3544), (381,10304,550), (381,11539,3628), (3146,3523,631), (3146,10304,376), (3830,5054,2), (3845,8703,3534), (3845,10109,5066), (3853,11539,547), (3860,11812,10124), (5067,5073,3857)

X(12101) = MIDPOINT OF X(5) AND X(3543)

Barycentrics 14*a^4-(b^2+c^2)*a^2-13*(b^2- c^2)^2 : :
X(12101) = 13*X(2)-9*X(3) = X(2)-9*X(4) = 7*X(2)-9*X(5) = 25*X(2)-9*X(20) = 10*X(2)-9*X(140) = 17*X(2)-9*X(376) = 5*X(2)-9*X(381) = 11*X(2)+9*X(382)

As a point on the Euler line, X(12101) has Shinagawa coefficients: (1, -27).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25561.

X(12101) lies on these lines: {2,3}, {1327,6441}, {1328,6442}

X(12101) =midpoint of X(i) and X(j) for these {i,j}: {5,3543}, {381,3627}, {382,549}, {3830,3845}
X(12101) = reflection of X(i) in X(j) for these (i,j): (3,11737), (140,381), (376,3628), (381,3861), (547,546), (549,3850), (550,10124), (3534,11812), (5066,3845), (8703,10109), (10124,3856)
X(12101) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3845,3860), (4,3627,3861), (4,3830,3845), (4,3853,546), (4,5076,5), (5,3627,5073), (140,3853,3627), (140,3861,546), (381,5073,3524), (382,3839,549), (3146,3858,3530), (3524,3543,5073), (3534,3830,3543), (3534,5076,3830), (3543,3839,3522), (3543,3845,11812), (3627,3845,8703), (3627,3861,140), (3845,5066,546), (3856,11541,140), (5070,11541,550)

X(12102) = MIDPOINT OF X(140) AND X(382)

Barycentrics 10*a^4-(b^2+c^2)*a^2-9*(b^2-c^ 2)^2 : :
X(12102) = 27*X(2)-19*X(3) = 3*X(2)-19*X(4) = 15*X(2)-19*X(5) = 21*X(2)-19*X(140) = 35*X(2)-19*X(376) = 11*X(2)-19*X(381) = 9*X(2)-19*X(546) = 17*X(2)-19*X(547)

As a point on the Euler line, X(12102) has Shinagawa coefficients: (1, -19).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25561.

X(12102) lies on these lines: {2,3}, {517,4536}, {5447,11017}, {11565,11645}

X(12102) = midpoint of X(i) and X(j) for these {i,j}: {4,3853}, {140,382}, {546,3627}, {3543,5066}
X(12102) = reflection of X(i) in X(j) for these (i,j): (140,3856), (3530,3850), (3628,546), (3850,3861), (3861,4), (5447,11017), (11737,3845), (11812,381)
X(12102) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,382,11541), (4,382,3845), (4,3543,3843), (4,3627,546), (4,3830,5), (4,5076,3627), (5,3627,3146), (382,3845,140), (382,5055,5059), (546,3853,3627), (3091,3146,376), (3091,11541,3), (3146,3523,3529), (3146,3525,1657), (3146,3830,3627), (3146,3839,3525), (3543,3843,550), (3627,5076,3853), (3628,3861,546), (3830,5054,3543), (3832,5073,549)

X(12103) = MIDPOINT OF X(20) AND X(550)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25562.

X(12103) lies on these lines: {2,3} et al

X(12103) =midpoint of X(20) and X(550)

X(12104) = MIDPOINT OF X(21) AND X(5428)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25562.

X(12104) lies on these lines: {2,3} et al

X(12104) =midpoint of X(21) and X(5428)

X(12105) = MIDPOINT OF X(23) AND X(7575)

Barycentrics a^2 (4 a^8+2 a^4 b^2 c^2-8 a^6 (b^2+c^2)-(b^2-c^2)^2 (4 b^4-b^2 c^2+4 c^4)+a^2 (8 b^6-3 b^4 c^2-3 b^2 c^4+8 c^6)) : :
X(12105) = X(3) + 3 X(23)

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25562.

X(12105) lies on these lines: {2,3} et al

X(12105) =midpoint of X(23) and X(7575)
X(12105) = {X(3),X(23)}-harmonic conjugate of X(37967)

X(12106) = MIDPOINT OF X(25) AND X(6644)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25562.

X(12106) lies on these lines: {2,3} et al

X(12106) =midpoint of X(25) and X(6644)

X(12107) = MIDPOINT OF X(26) AND X(1658)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25562.

X(12107) lies on these lines: {2,3} et al

X(12107) =midpoint of X(26) and X(1658)

X(12108) = MIDPOINT OF X(140) AND X(3530)

Barycentrics 10 a^4+3 (b^2-c^2)^2-13 a^2 (b^2+c^2) : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25562.

X(12108) lies on these lines: {2,3} et al

X(12108) =midpoint of X(140) and X(3530)

X(12109) = X(1)X(181)∩X(4)X(150)

Barycentrics a^2 (a^3 b^2+a^2 b^3-a b^4-b^5+3 a^2 b^2 c-3 b^4 c+a^3 c^2+3 a^2 b c^2+4 a b^2 c^2+2 b^3 c^2+a^2 c^3+2 b^2 c^3-a c^4-3 b c^4-c^5) : :
X(12109) = (a^3+b^3+c^3-a b c) X[1]-(a^3+b^3+c^3+3 a b c) X[181]
X(12109) = 3 X[51] + X[3868], 9 X[373] - 5 X[3876], 3 X[3819] - 5 X[5439], X[72] - 3 X[5943], 2 X[5044] - 3 X[6688]

Let A'B'C' be the orthic triangle of a triangle ABC. Let Ia be the incircle of B'C'A, and define Ib and Ic cyclically. Let U be the smallest circle tangent to each of the three circles Ia, Ib, Ic. Then X(12109) = center of U. Let A'' be the touch point of U and Ia. Barycentrics are given by

A'' = -a (a+b-c) (a-b+c) (a^3 b^2-a b^4-2 a b^3 c-2 b^4 c+a^3 c^2-2 a b c^3-a c^4-2 b c^4) : b^2 (a+b-c) (a+c)^2 (-a+b+c) (-a^2+b^2+c^2) : (a+b)^2 (a-b-c) c^2 (a-b+c) (a^2-b^2-c^2) .

Contributed by Thanh Oai Dao and Peter Moses, March 4, 2017.

X(12109) lies on these lines: {1,181}, {4,150}, {10,9052}, {51,3868}, {72,5943}, {373,3876}, {511,942}, {517,6738}, {518,9822}, {576,3157}, {674,3812}, {912,10110}, {916,5806}, {938,5933}, {1046,3271}, {1722,3779}, {2810,3874}, {2841,4757}, {3690,5047}, {3819,5439}, {4662,9049}, {5044,6688}

Orthologic centers: X(12110)-X(12269)

Centers X(12210)-X(12269) were contributed by César Eliud Lozada, March 10, 2017.

X(12110) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO ABC

Barycentrics a^8-3*(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(12110) = (S^2+SW^2)*X(4)-(2*(S^2-SW^2))*X(32)

The reciprocal orthologic center of these triangles is X(4).

X(12110) lies on these lines: {2,5171}, {3,83}, {4,32}, {5,316}, {6,11257}, {20,182}, {30,3398}, {39,11676}, {40,10791}, {51,401}, {55,10797}, {56,10798}, {99,3095}, {114,7785}, {194,576}, {211,11674}, {263,287}, {376,10359}, {381,10104}, {382,11842}, {384,511}, {385,6248}, {550,12054}, {631,7808}, {944,10800}, {946,11364}, {1003,10349}, {1351,1975}, {1478,10801}, {1479,10802}, {1513,7745}, {1614,3203}, {1632,9971}, {1656,7934}, {1691,5480}, {2782,7760}, {3090,7815}, {3091,7793}, {3098,10345}, {3146,9748}, {3552,9737}, {3575,6530}, {3818,9863}, {4027,6658}, {5034,7738}, {5039,6776}, {5097,7839}, {5188,7804}, {5476,7833}, {5691,10789}, {5870,10793}, {5871,10792}, {6054,7812}, {6284,10799}, {6785,10684}, {7608,11170}, {7709,7772}, {8541,11596}, {9821,10347}, {9838,11840}, {9939,11178}, {10795,11490}

X(12110) = X(4)-of-5th-anti-Brocard-triangle
X(12110) = 5th-anti-Brocard-to-ABC similarity image of X(4)
X(12110) = radical center of polar circles of ABC, 5th Brocard triangle, and 5th anti-Brocard triangle
X(12110) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,10796,83), (4,32,98), (4,10788,32), (5,2080,1078), (20,7787,182), (5171,10358,2), (7737,9993,10722), (7808,8722,631)

X(12111) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO ABC

Trilinears a*((b^2+c^2)*a^6-(3*b^4-b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^2-(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(12111) = 3*X(2)-4*X(5907) = 9*X(2)-8*X(9729) = 8*X(3)-9*X(7998) = 6*X(3)-7*X(7999) = 4*X(3)-5*X(11444) = 2*X(3)-3*X(11459) = 3*X(3)-4*X(11591) = 15*X(3)-16*X(11592)

The reciprocal orthologic center of these triangles is X(3).

X(12111) lies on these lines: {1,11446}, {2,185}, {3,74}, {4,52}, {5,5890}, {8,2807}, {15,11452}, {16,11453}, {20,2979}, {22,1498}, {30,11412}, {40,11445}, {51,3832}, {54,7526}, {64,394}, {69,6225}, {113,5449}, {143,3843}, {146,2888}, {155,378}, {186,7689}, {193,11469}, {235,3580}, {323,12086}, {343,2883}, {371,11447}, {372,11448}, {376,1216}, {381,3567}, {382,1154}, {389,3091}, {511,3146}, {546,568}, {567,11423}, {569,4550}, {576,11443}, {578,7527}, {631,5891}, {850,9242}, {858,6247}, {916,3868}, {930,6069}, {1092,2071}, {1147,3520}, {1181,5012}, {1204,9306}, {1351,11403}, {1593,1993}, {1594,7703}, {1657,6101}, {1658,10540}, {1870,6238}, {1885,3564}, {1994,11424}, {1995,9786}, {2060,5910}, {2779,5693}, {2781,5895}, {3090,9730}, {3100,6285}, {3101,6254}, {3167,3516}, {3193,4219}, {3522,3917}, {3523,11793}, {3525,10170}, {3528,5447}, {3529,10625}, {3534,10627}, {3545,5462}, {3574,5169}, {3627,6243}, {3830,10263}, {3839,10110}, {3851,5946}, {4296,7355}, {5055,11465}, {5067,5892}, {5068,5943}, {5422,11479}, {5448,7577}, {6198,7352}, {6696,7729}, {6759,7488}, {6895,10441}, {7486,11695}, {7592,9818}, {7728,7731}, {8549,11416}, {8718,8907}, {9545,11430}, {10282,10298}, {10546,11438}, {10675,11420}, {10676,11421}, {11220,11573}

X(12111) = reflection of X(i) in X(j) for these (i,j): (3,5876), (20,5562), (74,7723), (185,5907), (1657,6101), (3146,11381), (3529,10625), (5889,4), (6241,3), (6243,3627), (6293,2883), (7722,113), (7731,7728), (10575,1216)
X(12111) = anticomplement of X(185)
X(12111) = X(8)-of-1st-anti-circumperp-triangle if ABC is acute
X(12111) = X(4)-of-X(3)-Fuhrmann-triangle
X(12111) = pedal-isogonal conjugate of X(20)
X(12111) = X(20)-of-X(4)-anti-altimedial-triangle
X(12111) = X(20)-of-X(20)-anti-altimedial-triangle
X(12111) = X(20)-of-X(2)-adjunct-anti-altimedial-triangle
X(12111) = X(3)-of-X(4)-adjunct-anti-altimedial-triangle
X(12111) = homothetic center of Ehrmann side-triangle and 4th anti-Euler triangle
X(12111) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,185,10574), (3,110,11449), (3,156,11464), (3,399,156), (3,5876,11459), (3,11440,11454), (3,11441,110), (3,11444,7998), (3,11459,11444), (3,11591,7999), (4,5889,3060), (110,11440,3), (185,5907,2), (3060,11439,4), (5876,6241,11444), (6241,11459,3), (7999,11459,11591), (7999,11591,11444), (11440,11441,11449), (11449,11454,3), (12276,12277,12272)

X(12112) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL TO ABC

Trilinears a*(a^8-4*(b^2+c^2)*a^6+(6*b^4-5*b^2*c^2+6*c^4)*a^4-4*(b^4-c^4)*(b^2-c^2)*a^2+(b^4+7*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(12112) = 3*X(4)-4*X(1514) = 3*X(23)-2*X(3581) = 2*X(74)-3*X(186) = 3*X(186)-4*X(1495) = 4*X(1511)-3*X(2071) = 2*X(1531)-3*X(10706)

The reciprocal orthologic center of these triangles is X(74).

X(12112) lies on these lines: {3,7712}, {4,6}, {23,3581}, {30,146}, {74,186}, {110,841}, {156,12086}, {184,11455}, {352,1499}, {378,3426}, {394,11001}, {542,1533}, {1511,2071}, {1513,11580}, {1531,10706}, {1545,10658}, {1546,10657}, {1614,11381}, {1994,3830}, {2393,10752}, {3098,11459}, {3448,11799}, {3518,6241}, {3520,6759}, {3529,11441}, {3543,11004}, {5092,7550}, {5655,10989}, {5888,10170}, {5907,8718}, {6090,11820}, {6800,11472}, {7575,10620}, {7687,10821}, {7725,10814}, {7726,10815}, {7728,10296}, {7998,8717}, {8614,10308}, {9730,10545}, {12088,12111}

X(12112) = reflection of X(i) in X(j) for these (i,j): (74,1495), (323,399), (2071,10540), (3448,11799), (7464,110), (10296,7728), (10620,7575), (10989,5655)
X(12112) = {X(74), X(1495)}-harmonic conjugate of X(186)
X(12112) = X(74)-of-anti-orthocentroidal-triangle
X(12112) = 4th-Brocard-to-circumsymmedial similarity image of X(74)

X(12113) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO ABC

Barycentrics (-a^2+b^2+c^2)*(2*a^10-2*(b^2+c^2)*a^8-(5*b^4-12*b^2*c^2+5*c^4)*a^6+7*(b^4-c^4)*(b^2-c^2)*a^4-(b^2-c^2)^2*(b^4+8*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2)^3)*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2) : :
Trilinears cos(A)*(cos(B-C)-2*cos(A))*((4*cos(2*A)+5)*cos(B-C)-cos(A)*cos(2*(B-C))-7*cos(A)-cos(3*A)) : :
X(12113) = X(4)-3*X(11845) = 2*X(4)-3*X(11897) = X(382)-3*X(11911) = 2*X(402)-3*X(11845) = 4*X(402)-3*X(11897) = 2*X(946)-3*X(11831) = X(5691)-3*X(11852)

The reciprocal orthologic center of these triangles is X(4).

X(12113) lies on these lines: {2,3}, {40,11900}, {55,11905}, {56,11906}, {944,11910}, {946,11831}, {1478,11912}, {1479,11913}, {2777,7740}, {3184,9033}, {5691,11852}, {5870,11902}, {5871,11901}, {6284,11909}, {9838,11907}, {9839,11908}, {9873,11885}, {11500,11848}, {11839,12110}

X(12113) = midpoint of X(20) and X(4240)
X(12113) = reflection of X(i) in X(j) for these (i,j): (4,402), (1650,3), (11897,11845)
X(12113) = X(4)-of-Gossard-triangle

X(12114) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO ABC

Trilinears a^6-(b+c)*a^5-2*(b^2-3*b*c+c^2)*a^4+2*(b^2-c^2)*(b-c)*a^3+(b-c)^4*a^2-(b^2-c^2)*(b-c)^3*a-2*(b^2-c^2)^2*b*c : :
X(12114) = 3*X(1)-X(7971) = 3*X(1)+X(7992) = 3*X(84)+X(7971) = 3*X(84)-X(7992) = X(1490)-3*X(3576) = 3*X(3576)+X(10864) = 4*X(5450)-X(11500) = 3*X(5886)-X(6259)

The reciprocal orthologic center of these triangles is X(4).

X(12114) lies on these lines: {1,84}, {3,10}, {4,11}, {5,6256}, {8,6909}, {12,6833}, {20,2894}, {21,3427}, {28,963}, {30,10525}, {36,3149}, {40,956}, {48,5776}, {55,944}, {57,7686}, {119,6958}, {153,6972}, {165,5258}, {219,1765}, {280,1295}, {281,1436}, {376,5584}, {381,10199}, {382,11928}, {388,6847}, {405,1490}, {474,5587}, {499,1532}, {513,945}, {516,8666}, {517,1158}, {518,3358}, {519,10306}, {550,11495}, {601,5710}, {946,999}, {952,3913}, {960,7330}, {971,1001}, {997,5777}, {1006,8273}, {1125,6260}, {1191,3073}, {1317,10965}, {1329,6891}, {1468,5706}, {1470,1837}, {1476,10309}, {1478,6831}, {1479,10948}, {1482,2800}, {1519,11376}, {1537,10052}, {1593,5101}, {1617,4311}, {1699,5563}, {1706,10270}, {2077,5687}, {2096,4295}, {2551,6926}, {2716,2765}, {2886,6850}, {2950,6264}, {3035,6961}, {3058,10806}, {3072,4252}, {3085,6935}, {3091,5253}, {3295,5882}, {3303,7967}, {3304,3649}, {3339,3577}, {3359,5836}, {3436,6890}, {3486,5768}, {3523,5260}, {3575,11390}, {3614,6879}, {3632,5537}, {3655,4428}, {3656,12001}, {3816,6893}, {3897,11220}, {3925,6897}, {4018,7982}, {4321,11372}, {4413,5818}, {4423,5658}, {4999,6825}, {5080,6943}, {5120,10445}, {5204,6905}, {5217,6950}, {5229,6844}, {5231,7580}, {5251,7987}, {5288,7991}, {5289,5887}, {5432,6977}, {5433,6834}, {5434,10532}, {5538,5904}, {5542,7373}, {5552,6966}, {5693,5730}, {5870,10920}, {5871,10919}, {5886,6259}, {6244,11362}, {6253,6934}, {6257,11371}, {6258,11370}, {6284,6938}, {6667,6981}, {6690,6892}, {6691,6944}, {6713,6959}, {6762,6769}, {6830,10895}, {6836,10522}, {6845,9657}, {6848,7288}, {6914,10267}, {6925,10527}, {6956,10590}, {6968,7173}, {6971,10742}, {7171,9943}, {7966,7990}, {8071,10572}, {9838,10945}, {9839,10946}, {9873,10871}, {9910,11365}, {10043,10058}, {10165,11108}, {10609,11517}, {10794,12110}, {10950,11509}, {11903,12113}

X(12114) = midpoint of X(i) and X(j) for these {i,j}: {1,84}, {1490,10864}, {2950,6264}, {5882,9948}, {6762,6769}, {7971,7992}
X(12114) = reflection of X(i) in X(j) for these (i,j): (3,5450), (10,6705), (3913,11248), (6256,5), (6260,1125), (6261,1385), (10525,10943), (11500,3)
X(12114) = X(4)-of-inner-Johnson-triangle
X(12114) = inverse-in-Feuerbach-hyperbola of X(56)
X(12114) = Ursa-minor-to-Ursa-major similarity image of X(4)
X(12114) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1012,11496), (1,7992,7971), (3,355,1376), (3,9708,6684), (4,11,10893), (4,104,56), (4,3086,7681), (4,10785,11), (8,6909,10310), (20,2975,3428), (20,3434,11826), (36,5691,3149), (84,7971,7992), (944,6906,55), (993,4297,3), (1385,3560,1001), (1478,6831,10894), (2077,5881,5687), (3576,10864,1490), (10525,10943,11235)

X(12115) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO ABC

Barycentrics a^7-(b+c)*a^6-(b^2-8*b*c+c^2)*a^5+(b+c)*(b^2-6*b*c+c^2)*a^4-(b^2+6*b*c+c^2)*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)*a^2+(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :
Trilinears (12*sin(A/2)-2*sin(3*A/2))*cos((B-C)/2)+(2*cos(A)-4)*cos(B-C)+2*sin(A/2)*cos(3*(B-C)/2)+8*cos(A)-cos(2*A)-5 : :
X(12115) = 4*X(226)-3*X(5603) = 5*X(631)-4*X(993) = 7*X(3090)-8*X(3822) = 2*X(7680)-3*X(11237) = 2*X(10679)-3*X(11239)

The reciprocal orthologic center of these triangles is X(4).

X(12115) lies on these lines: {1,4}, {2,104}, {3,3436}, {5,10584}, {8,912}, {10,6897}, {11,6968}, {12,6833}, {30,10679}, {36,6880}, {40,10915}, {55,2829}, {56,6834}, {57,1512}, {63,2096}, {84,9578}, {100,6948}, {355,377}, {376,535}, {382,12000}, {390,10728}, {443,5818}, {495,1012}, {496,10598}, {498,5450}, {517,5905}, {529,3428}, {631,993}, {952,3434}, {956,6907}, {958,6889}, {999,1532}, {1001,6976}, {1125,6898}, {1158,10039}, {1181,9370}, {1317,10947}, {1329,6967}, {1376,6955}, {1385,2478}, {1389,5555}, {1470,4293}, {1621,6930}, {1837,11047}, {2550,2801}, {2975,6825}, {3085,6906}, {3086,6941}, {3090,3822}, {3091,10586}, {3304,7681}, {3575,11400}, {3576,6947}, {3577,4654}, {3600,6848}, {3616,6893}, {3913,11826}, {4190,11499}, {4297,6899}, {4299,6796}, {5080,5731}, {5193,6969}, {5218,6950}, {5251,6878}, {5252,6001}, {5253,6944}, {5260,6989}, {5261,6847}, {5437,5587}, {5768,6826}, {5804,11037}, {5870,10930}, {5871,10929}, {5884,10573}, {5886,6957}, {6259,10935}, {6284,10965}, {6830,10590}, {6831,9654}, {6836,10526}, {6838,10530}, {6842,10527}, {6862,10585}, {6872,10267}, {6879,7951}, {6891,11681}, {6929,10246}, {6934,7354}, {6935,8164}, {6949,7288}, {6952,10588}, {6982,11680}, {7680,11237}, {7686,10404}, {7966,9580}, {7992,10970}, {9838,11955}, {9839,11956}, {9873,10878}, {10085,10827}, {10803,12110}, {11914,12113}

X(12115) = reflection of X(i) in X(j) for these (i,j): (4,1478), (956,6907), (1012,495), (3434,6923), (6938,55)
X(12115) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4,10531), (1,1519,5603), (1,6256,4), (3,10942,5552), (4,388,10532), (4,1056,5603), (4,7967,497), (4,10597,946), (4,10805,1), (4,10806,1479), (119,10269,2), (498,5450,6977), (1479,5882,10806), (3359,6735,5657), (3421,6916,5657), (5080,5731,6827), (6831,9654,10599), (7354,10955,11509), (7354,11500,6934), (10246,10742,6929)

X(12116) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO ABC

Barycentrics a^7-(b+c)*a^6-(b^2+4*b*c+c^2)*a^5+(b+c)^3*a^4-(b-c)^4*a^3+(b^2-c^2)*(b-c)^3*a^2+(-c^4+b^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : :
Trilinears (4*sin(A/2)+2*sin(3*A/2))*cos((B-C)/2)-2*cos(A)*cos(B-C)-2*sin(A/2)*cos(3*(B-C)/2)+4*cos(A)+cos(2*A)-3 : :
X(12116) = 7*X(3090)-8*X(3825) = 3*X(5603)-4*X(12053) = 2*X(7681)-3*X(11238) = 2*X(10680)-3*X(11240)

The reciprocal orthologic center of these triangles is X(4).

X(12116) lies on these lines: {1,4}, {2,10267}, {3,3434}, {5,10585}, {8,6827}, {10,6947}, {11,6834}, {20,104}, {30,10680}, {35,6977}, {40,6899}, {55,6833}, {56,5842}, {84,9580}, {100,6891}, {119,5187}, {145,6840}, {355,392}, {376,11012}, {377,1385}, {382,12001}, {390,6847}, {411,6585}, {495,10599}, {496,3149}, {498,6879}, {499,6796}, {517,6836}, {528,10310}, {631,2550}, {908,5534}, {912,11415}, {952,3436}, {958,6936}, {962,5768}, {1001,6832}, {1125,6854}, {1191,5721}, {1376,6967}, {1512,9581}, {1532,9669}, {1621,6824}, {1836,11048}, {2078,6927}, {2551,6902}, {2802,6903}, {2886,6889}, {2975,6868}, {3058,11496}, {3072,11269}, {3085,6830}, {3086,6905}, {3090,3825}, {3091,10587}, {3254,10305}, {3295,6831}, {3303,7680}, {3421,3984}, {3428,3813}, {3555,5812}, {3575,11401}, {3576,6897}, {3616,6826}, {3622,6839}, {3816,6983}, {3871,6943}, {4190,10269}, {4294,6906}, {4302,5450}, {5082,5657}, {5084,5818}, {5218,6952}, {5231,10268}, {5253,6885}, {5274,6848}, {5284,6887}, {5552,6882}, {5587,6898}, {5687,6922}, {5709,6361}, {5731,6850}, {5759,6601}, {5787,10936}, {5870,10932}, {5871,10931}, {5886,6835}, {6245,10624}, {6284,6938}, {6825,11680}, {6890,10530}, {6896,8227}, {6917,10246}, {6925,10525}, {6941,10591}, {6942,7288}, {6949,10589}, {6959,10584}, {6968,10896}, {7681,11238}, {7966,9578}, {7992,10971}, {9838,11957}, {9839,11958}, {9873,10879}, {10804,12110}, {10944,10953}, {11915,12113}

X(12116) = reflection of X(i) in X(j) for these (i,j): (4,1479), (3149,496), (3436,6928), (5687,6922), (6934,56)
X(12116) = anticomplement of X(11499)
X(12116) = X(4)-of-outer-Yff-tangents-triangle
X(12116) = inner-Yff-to-outer-Yff similarity image of X(4)
X(12116) = 1st-Johnson-Yff-to-2nd-Johnson-Yff similarity image of X(4)
X(12116) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4,10532), (3,10943,10527), (4,497,10531), (4,944,12115), (4,1058,5603), (4,7967,388), (4,10596,946), (4,10805,1478), (4,10806,1), (20,10529,11249), (499,6796,6880), (1478,5882,10805), (1532,9669,10598), (3583,6256,4), (5082,6865,5657), (6284,10949,10966), (6284,12114,6938)

X(12117) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO ANTI-MCCAY

Barycentrics 7*a^8-15*(b^2+c^2)*a^6+13*(b^4+b^2*c^2+c^4)*a^4-(b^2+c^2)*(3*b^4+2*b^2*c^2+3*c^4)*a^2-(b^2-c^2)^2*(2*b^2-c^2)*(b^2-2*c^2) : :
X(12117) = X(20)+2*X(10992) = 5*X(99)-2*X(6033) = 3*X(99)-2*X(8724) = 4*X(99)-X(10722) = 2*X(115)-3*X(3524) = 3*X(165)-X(9875) = 4*X(6033)-5*X(6054) = 3*X(6033)-5*X(8724)

The reciprocal orthologic center of these triangles is X(9855).

X(12117) lies on these lines: {2,9734}, {3,671}, {4,2482}, {20,542}, {30,99}, {35,10054}, {36,10070}, {98,376}, {114,3543}, {115,3524}, {148,6055}, {165,9875}, {262,11159}, {381,10723}, {511,8593}, {515,9881}, {517,9884}, {530,5474}, {531,5473}, {549,6321}, {620,3545}, {631,5461}, {1350,9830}, {1632,5648}, {2782,3534}, {2794,11001}, {2796,4297}, {2936,12082}, {3098,9878}, {3522,8596}, {3528,11623}, {3655,7983}, {4558,9214}, {5071,9167}, {5182,12110}, {5503,9744}, {5969,11257}, {7417,10717}, {8703,11632}, {8787,11477}, {9876,11414}, {9882,11824}, {9883,11825}, {10754,11179}

X(12117) = midpoint of X(20) and X(8591)
X(12117) = reflection of X(i) in X(j) for these (i,j): (4,2482), (98,376), (148,6055), (671,3), (3543,114), (6054,99), (6321,549), (7983,3655), (8591,10992), (10722,6054), (10723,381), (10754,11179), (11477,8787), (11632,8703)
X(12117) = anticomplement of X(9880)
X(12117) = orthologic center of these triangles: ABC-X3 reflections to McCay.
X(12117) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (148,10304,6055), (549,6321,9166)

X(12118) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO ARIES

Barycentrics (-a^2+b^2+c^2)*(3*a^8-4*(b^2+c^2)*a^6+4*b^2*c^2*a^4+(b^2-c^2)^4) : :
X(12118) = 3*X(2)-4*X(12038) = 3*X(4)-4*X(5448) = 2*X(4)-3*X(5654) = 2*X(20)+X(9936) = 3*X(165)-X(9896) = 3*X(376)-2*X(7689) = 3*X(376)-X(11411) = 3*X(1147)-2*X(5448) = 4*X(1147)-3*X(5654) = 8*X(5448)-9*X(5654)

The reciprocal orthologic center of these triangles is X(9833).

X(12118) lies on these lines: {2,9927}, {3,68}, {4,110}, {5,11425}, {20,6193}, {30,155}, {35,10055}, {36,10071}, {165,9896}, {265,6640}, {376,539}, {381,9820}, {382,3167}, {515,9928}, {517,9933}, {542,3357}, {550,1350}, {569,6815}, {631,5449}, {912,3962}, {1060,9931}, {1069,6284}, {1181,4846}, {1216,11821}, {1352,7526}, {1370,11750}, {1503,12085}, {1657,11820}, {1993,6240}, {2071,11457}, {2929,2931}, {3070,8909}, {3098,9923}, {3157,7354}, {3520,11442}, {4299,7352}, {4302,6238}, {4549,5562}, {5446,7487}, {6560,10665}, {6561,10666}, {7505,11449}, {7528,11424}, {8548,11179}, {9908,11414}, {9929,11824}, {9930,11825}, {10112,11438}, {10619,10984}

X(12118) = midpoint of X(20) and X(6193)
X(12118) = reflection of X(i) in X(j) for these (i,j): (4,1147), (68,3), (9927,12038), (9936,6193), (11411,7689)
X(12118) = anticomplement of X(9927)
X(12118) = orthologic center of these triangles: ABC-X3 reflections to 2nd Hyacinth.
X(12118) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1147,5654), (376,11411,7689), (9927,12038,2)

X(12119) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO FUHRMANN

Trilinears (8*sin(A/2)-4*sin(3*A/2))*cos((B-C)/2)+(2*cos(A)-2)*cos(B-C)+sin(A/2)*cos(3*(B-C)/2)+11/2*cos(A)-3/2*cos(2*A)-3 : :
X(12119) = 2*X(11)-3*X(3576) = X(149)-3*X(5731) = 5*X(631)-4*X(6702) = 3*X(1699)-4*X(11729) = 4*X(3035)-3*X(5587) = 3*X(5660)-2*X(10742)

The reciprocal orthologic center of these triangles is X(3).

X(12119) lies on these lines: {1,5840}, {3,80}, {4,214}, {11,3576}, {20,2800}, {30,6265}, {35,10057}, {36,10073}, {40,550}, {100,515}, {104,3651}, {119,5691}, {149,5731}, {165,9897}, {516,10698}, {517,4316}, {528,5732}, {631,6702}, {944,2802}, {946,10724}, {1145,5881}, {1317,7982}, {1320,5882}, {1385,4857}, {1490,2829}, {1699,11729}, {2801,5759}, {2932,11500}, {3035,5587}, {3612,8068}, {4293,5083}, {4299,11570}, {4996,5450}, {5444,6980}, {5531,6282}, {5660,10742}, {6262,11825}, {6263,11824}, {6713,7987}, {6869,9946}, {8988,9540}, {9613,10956}, {9912,11414}, {10058,10902}, {10090,10572}, {10768,11711}, {10769,11710}, {10770,11714}, {10771,11700}, {10772,11712}, {10777,11713}, {10778,11709}, {11014,11826}

X(12119) = midpoint of X(20) and X(6224)
X(12119) = reflection of X(i) in X(j) for these (i,j): (4,214), (80,3), (104,4297), (149,11715), (1320,5882), (5541,10993), (5691,119), (5881,1145), (6326,10609), (7982,1317), (10724,946), (10738,1385), (10768,11711), (10769,11710), (10770,11714), (10771,11700), (10772,11712), (10777,11713), (10778,11709)
X(12119) = anticomplement of X(6246)
X(12119) = {X(149), X(5731)}-harmonic conjugate of X(11715)

X(12120) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO HUTSON EXTOUCH

Trilinears (41*sin(A/2)-10*sin(3*A/2)-3*sin(5*A/2))*cos((B-C)/2)+(-4*cos(A)+3*cos(2*A)+1)*cos(B-C)+(sin(A/2)+sin(3*A/2))*cos(3*(B-C)/2)-cos(3*A)/2-7*cos(A)/2-6*cos(2*A)-22 : :

The reciprocal orthologic center of these triangles is X(40).

X(12120) lies on these lines: {1,5759}, {3,7091}, {35,10059}, {36,10075}, {40,3555}, {165,9898}, {517,8000}, {944,11519}, {956,10864}, {1490,3428}, {4326,6766}, {5584,9850}, {5732,6762}, {8726,11037}

X(12120) = midpoint of X(20) and X(9874)
X(12120) = reflection of X(7160) in X(3)

X(12121) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO 1st HYACINTH

Barycentrics (-a^2+b^2+c^2)*(3*a^8-3*(b^2+c^2)*a^6-(2*b^4-7*b^2*c^2+2*c^4)*a^4+(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4) : :
X(12121) = 3*X(3)-2*X(125) = 5*X(3)-4*X(6699) = 4*X(110)-3*X(5655) = 5*X(110)-3*X(10706) = 3*X(110)-X(10721) = 4*X(125)-3*X(265) = 5*X(125)-6*X(6699) = 5*X(265)-8*X(6699) = 3*X(5655)-2*X(7728) = 5*X(5655)-4*X(10706) = 9*X(5655)-4*X(10721) = 5*X(7728)-6*X(10706)

The reciprocal orthologic center of these triangles is X(6102).

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Euler line. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the Euler line. The triangle A"B"C" is homothetic to ABC, with center of homothety X(125) and circumcenter X(12121). (See Hyacinthos #16741/16782, Sep 2008.) (Randy Hutson, July 21, 2017)

X(12121) lies on these lines: {74,550}, {113,382}, {146,3529}, {376,3448}, {381,5972}, {399,1498}, {511,11562}, {541,11820}, {542,1350}, {548,10264}, {549,11801}, {1154,7722}, {1539,3146}, {1656,7687}, {1986,6243}, {2771,12119}, {3028,4299}, {3070,10819}, {3071,10820}, {3520,6288}, {3581,10295}, {3627,10272}, {3830,5642}, {4324,7727}, {4549,7723}, {5054,6723}, {5648,11645}, {6146,10816}, {6240,11597}, {6284,10091}, {6449,8994}, {6759,11744}, {7354,10088}, {7574,10564}, {8703,9140}, {9143,11001}, {9730,11800}, {10117,12083}, {10263,11561}, {10625,10628}

X(12121) = midpoint of X(i) and X(j) for these {i,j}: {146,3529}, {399,1657}, {9143,11001}
X(12121) = reflection of X(i) in X(j) for these (i,j): (4,1511), (67,3098), (74,550), (146,5609), (265,3), (382,113), (3146,1539), (3448,12041), (3581,10295), (3627,10272), (3830,5642), (6243,1986), (7574,10564), (7728,110), (9140,8703), (10263,11561), (10264,548), (10733,5), (11744,6759)
X(12121) = anticomplement of X(10113)
X(12121) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (110,7728,5655), (376,3448,12041)

X(12122) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO 2nd NEUBERG

Barycentrics a^8+5*(b^2+c^2)*a^6-(3*b^4+b^2*c^2+3*c^4)*a^4-(b^2+c^2)*(3*b^4+2*b^2*c^2+3*c^4)*a^2-(b^2-c^2)^2*b^2*c^2 : :
X(12122) = 5*X(631)-4*X(6704) = 5*X(6308)-4*X(7780)

The reciprocal orthologic center of these triangles is X(3).

X(12122) lies on these lines: {2,6249}, {3,83}, {4,6292}, {20,1352}, {30,6287}, {35,10064}, {36,10080}, {98,5188}, {99,550}, {165,9903}, {376,754}, {382,7910}, {511,7839}, {517,7977}, {574,3528}, {631,6704}, {732,1350}, {3522,9737}, {3529,3734}, {6274,11825}, {6275,11824}, {8150,8722}, {8993,9540}, {9918,11414}

X(12122) = midpoint of X(20) and X(2896)
X(12122) = reflection of X(i) in X(j) for these (i,j): (4,6292), (83,3), (8725,550)
X(12122) = anticomplement of X(6249)
X(12122) = X(83)-of-ABC-X3-reflections-triangle
X(12122) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,83,9751), (5188,7470,98)

X(12123) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO INNER-VECTEN

Barycentrics a^2*(a^4+2*(b^2+c^2)*a^2-2*b^2*c^2-3*c^4-3*b^4)-2*(4*a^4-3*(b^2+c^2)*a^2-(b^2-c^2)^2)*S : :
X(12123) = 2*X(20)+X(6281) = 6*X(376)-X(6280)

The reciprocal orthologic center of these triangles is X(3).

X(12123) lies on these lines: {2,6251}, {3,486}, {4,642}, {20,487}, {30,6290}, {35,10067}, {36,10083}, {99,489}, {165,9906}, {371,7738}, {376,5860}, {485,8997}, {517,7980}, {550,1350}, {631,6119}, {1151,2549}, {2043,5473}, {2044,5474}, {3098,9986}, {6399,8182}, {6560,9732}, {9921,11414}

X(12123) = midpoint of X(20) and X(487)
X(12123) = reflection of X(i) in X(j) for these (i,j): (4,642), (486,3), (6281,487)
X(12123) = anticomplement of X(6251)

X(12124) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO OUTER-VECTEN

Barycentrics a^2*(a^4+2*(b^2+c^2)*a^2-2*b^2*c^2-3*c^4-3*b^4)+2*(4*a^4-3*(b^2+c^2)*a^2-(b^2-c^2)^2)*S : :
X(12124) = 2*X(20)+X(6278) = 3*X(165)-X(9907) = 6*X(376)-X(6279)

The reciprocal orthologic center of these triangles is X(3).

X(12124) lies on these lines: {2,6250}, {3,485}, {4,641}, {20,488}, {30,6289}, {35,10068}, {36,10084}, {99,490}, {165,9907}, {372,7738}, {376,5861}, {486,9739}, {517,7981}, {550,1350}, {631,6118}, {1152,2549}, {2043,5474}, {2044,5473}, {3098,9987}, {6222,8182}, {6561,9733}, {9922,11414}

X(12124) = midpoint of X(20) and X(488)
X(12124) = reflection of X(i) in X(j) for these (i,j): (4,641), (485,3), (6278,488)
X(12124) = anticomplement of X(6250)

X(12125) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-CONWAY TO ANDROMEDA

Trilinears (b+c)*a^5-(b^2-5*b*c+c^2)*a^4-2*(b^2+c^2)*(b+c)*a^3+2*(b^4+c^4-b*c*(b^2-6*b*c+c^2))*a^2+(b+c)*(b^4-10*b^2*c^2+c^4)*a-(b^2+3*b*c+c^2)*(b^2-c^2)^2 : :
X(12125) = 4*X(938)-3*X(3873)

The reciprocal orthologic center of these triangles is X(1).

X(12125) lies on these lines: {1,11678}, {2,9850}, {8,971}, {9,9846}, {63,4882}, {78,9845}, {100,9841}, {145,9848}, {200,9851}, {329,9797}, {377,5176}, {452,3890}, {519,3869}, {936,2975}, {938,3436}, {3877,12059}, {5744,9858}, {5815,6865}, {5828,9940}, {6736,11220}, {9842,11680}, {9843,11681}, {9849,11686}, {9852,11688}, {9853,11690}

X(12125) = reflection of X(i) in X(j) for these (i,j): (145,9848), (9797,9844), (9846,9), (9859,4882)
X(12125) = anticomplement of X(9850)
X(12125) = excentral-to-inner-Conway similarity image of X(4882)

X(12126) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO ANDROMEDA

Trilinears (8*q^2-1)*p^6-(8*q^2-13)*q*p^5-(9*q^2+5)*p^4+(11*q^2-9)*q*p^3+(q^2-2)*(2*q^2-1)*p^2-(3*q^2-4)*q*p-2*(q^2-1)*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(1).

X(12126) lies on these lines: {1,971}, {936,10882}, {938,11021}, {1764,4882}, {9797,10446}, {9841,10434}, {9842,10886}, {9843,10887}, {9844,10888}, {9846,10889}, {9849,11893}, {9852,10892}, {9853,11894}, {9858,10856}, {9859,10444}, {11679,12125}

X(12126) = excentral-to-3rd-Conway similarity image of X(4882)

X(12127) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS TO ANDROMEDA

Trilinears a^3-(b+c)*a^2-(b^2-26*b*c+c^2)*a+(b+c)*(b^2-10*b*c+c^2) : :
X(12127) = 3*X(1)-2*X(936) = 4*X(936)-3*X(4882)

The reciprocal orthologic center of these triangles is X(1).

X(12127) lies on these lines: {1,2}, {269,4460}, {517,9845}, {944,2951}, {971,7982}, {1317,1467}, {2136,3361}, {3339,3880}, {3340,9850}, {3555,9851}, {3813,5726}, {3875,7271}, {3879,7274}, {4900,5836}, {5045,11525}, {5223,9957}, {6762,9819}, {7962,9848}, {7991,9841}, {9842,11522}, {9844,11523}, {9846,11526}, {9849,11528}, {9852,11533}, {9858,11518}, {9859,11520}, {10914,10980}, {11521,12126}, {11682,12125}

X(12127) = midpoint of X(i) and X(j) for these {i,j}: {145,9797}, {9851,11531}
X(12127) = reflection of X(i) in X(j) for these (i,j): (4882,1), (7991,9841)
X(12127) = excentral-to-excenters-reflections similarity image of X(4882)
X(12127) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3632,8580), (1,11519,4915), (3241,4853,1), (3635,9623,1)

X(12128) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO ANDROMEDA

Trilinears (3*sin(A/2)+sin(3*A/2))*cos((B-C)/2)+(-cos(A)-1)*cos(B-C)+4*cos(A)-10 : :
X(12128) = 5*X(3616)-X(12125)

The reciprocal orthologic center of these triangles is X(1).

X(12128) lies on these lines: {1,971}, {145,10569}, {355,938}, {495,9843}, {496,3817}, {517,3600}, {519,942}, {936,999}, {3295,9841}, {3333,4882}, {3487,9844}, {3616,12125}, {3655,4313}, {4297,9957}, {4321,8158}, {4853,10855}, {5082,9797}, {6244,7091}, {9581,11237}, {9846,11038}, {9849,11040}, {9852,11043}, {11529,12127}

X(12128) = midpoint of X(1) and X(9850)
X(12128) = reflection of X(938) in X(5045)
X(12128) = excentral-to-incircle-circles similarity image of X(4882)

X(12129) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO ANDROMEDA

Trilinears -2*b*c*((b+c)*a^3-(b^2-14*b*c+c^2)*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2)*S+(b+c)*a^7-(b^2-6*b*c+c^2)*a^6-(b+c)*(b^2+b*c+c^2)*a^5+(b^4+c^4+b*c*(b^2+20*b*c+c^2))*a^4-(b+c)*(b^4+c^4-2*b*c*(b^2-7*b*c+c^2))*a^3+(b^4+c^4-2*b*c*(b^2+b*c+c^2))*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(b^4+c^4+b*c*(b^2-8*b*c+c^2))*a-(b^2-c^2)^2*(b^4+c^4+3*b*c*(b^2+c^2)) : :

The reciprocal orthologic center of these triangles is X(1).

X(12129) lies on these lines: {1,10867}, {519,9808}, {936,8225}, {938,11030}, {971,7596}, {4882,8231}, {8224,9841}, {8228,9842}, {8230,9843}, {8233,9844}, {8234,9845}, {8237,9846}, {8239,9848}, {8243,9850}, {8244,9851}, {8246,9852}, {8247,9853}, {8248,9854}, {9789,9797}, {9849,11925}, {9858,10858}, {9859,10885}, {10891,12126}, {11042,12128}, {11532,12127}, {11687,12125}

X(12129) = excentral-to-2nd-Pamfilos-Zhou similarity image of X(4882)

X(12130) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO ANDROMEDA

Trilinears 2*(a+b+c)*(a^3-(b+c)*a^2-(b^2-10*b*c+c^2)*a+(b^2-c^2)*(b-c))*b*c*sin(A/2)+(a+b-c)*(a-b+c)*((b+c)*a^3-(b^2-6*b*c+c^2)*a^2-(b+c)^3*a+(b+c)^4) : :

The reciprocal orthologic center of these triangles is X(1).

X(12130) lies on these lines: {1,11860}, {174,9850}, {936,7587}, {938,8083}, {971,8351}, {8126,12125}, {8382,9843}, {8389,9846}, {8423,9851}, {8425,9852}, {8729,9858}, {9797,11891}, {9848,11924}, {9859,11890}, {11535,12127}, {11896,12126}, {11996,12129}

X(12130) = excentral-to-Yff-central similarity image of X(4882)

X(12131) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO 1st ANTI-BROCARD

Barycentrics ((b^2+c^2)*a^8-2*(b^4+c^4)*a^6+(b^6+c^6)*a^4+2*(b^2-c^2)^2*b^2*c^2*a^2+(b^4-c^4)*(b^2-c^2)*b^2*c^2)/(-a^2+b^2+c^2) : :
X(12131) = (6*R^2-SW)*X(25)-(4*R^2-SW)*X(98)

The reciprocal orthologic center of these triangles is X(5999).

X(12131) lies on these lines: {4,147}, {24,12042}, {25,98}, {33,3027}, {34,3023}, {99,1593}, {114,136}, {115,235}, {132,8754}, {232,2023}, {264,5976}, {428,542}, {458,10352}, {468,6036}, {1785,5988}, {1862,2783}, {2784,5185}, {2794,3575}, {5064,6054}, {5090,9864}, {5984,6995}, {6226,11389}, {6227,11388}, {7487,9862}, {7713,9860}, {7714,11177}, {7970,11396}, {10053,11398}, {10069,11399}, {11363,11710}

X(12131) = reflection of X(5186) in X(4)
X(12131) = polar circle-inverse-of-X(147)
X(12131) = orthologic center of these triangles: anti-Ara to6th anti-Brocard, anti-Ara and 1st Brocard, anti-Ara to 6th Brocard

X(12132) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO ANTI-MCCAY

Barycentrics (4*a^6-5*(b^2+c^2)*a^4+2*(b^2+c^2)^2*a^2+(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2))/(-a^2+b^2+c^2) : :
X(12132) = (6*R^2-SW)*X(25)-(4*R^2-SW)*X(671)

The reciprocal orthologic center of these triangles is X(9855).

X(12132) lies on these lines: {4,8591}, {25,671}, {30,12131}, {99,5064}, {148,7714}, {235,9880}, {427,2482}, {428,543}, {468,5461}, {542,3575}, {1593,12117}, {1843,9830}, {1907,10992}, {2782,7576}, {5090,9881}, {6995,8596}, {7713,9875}, {8541,8787}, {9878,11386}, {9882,11388}, {9883,11389}, {9884,11396}, {10054,11398}, {10070,11399}

X(12132) = reflection of X(5186) in X(428)
X(12132) = orthologic center of these triangles: anti-Ara to 2nd Hyacinth

X(12133) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO ANTI-ORTHOCENTROIDAL

Trilinears ((8*cos(A)+cos(3*A))*cos(B-C)-4*cos(2*A)-5)*sec(A) : :
X(12133) = 3*X(4)-X(1986) = 5*X(4)-X(7722) = X(74)+3*X(11455) = X(110)-5*X(11439) = 2*X(974)-3*X(12099) = 3*X(1112)-2*X(1986) = 5*X(1112)-2*X(7722) = 5*X(1986)-3*X(7722) = 4*X(7687)-3*X(12099)

The reciprocal orthologic center of these triangles is X(12112).

Let A'B'C' be the orthocentroidal triangle. Let A" be the orthogonal projection of A on line B'C', and define B" and C" cyclically. Triangle A"B"C" is perspective to the orthic triangle at X(12133). (Randy Hutson, July 21, 2017)

X(12133) lies on these lines: {4,94}, {24,12041}, {25,74}, {33,3028}, {66,11744}, {110,1593}, {113,427}, {125,235}, {185,11746}, {378,1511}, {381,9826}, {382,7723}, {399,1597}, {428,541}, {468,6699}, {542,5186}, {690,12131}, {974,1514}, {1596,10264}, {1598,10620}, {1843,2781}, {1862,2771}, {1900,2779}, {2772,5185}, {2777,3575}, {2931,11472}, {5064,10706}, {6143,11017}, {7713,9904}, {7725,11388}, {7726,11389}, {7978,11396}, {9984,11386}, {10065,11398}, {10081,11399}, {10628,11576}, {10721,11387}, {11363,11709}

X(12133) = midpoint of X(i) and X(j) for these {i,j}: {125,11381}, {382,7723}
X(12133) = reflection of X(i) in X(j) for these (i,j): (185,11746), (974,7687), (1112,4)
X(12133) = polar circle-inverse-of-X(146)
X(12133) = intersection of tangents to Walsmith rectangular hyperbola at X(74) and X(113)
X(12133) = orthologic center of these triangles: anti-Ara to orthocentroidal
X(12133) = {X(974), X(7687)}-harmonic conjugate of X(12099)

X(12134) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO ARIES

Barycentrics 2*a^10-5*(b^2+c^2)*a^8+4*(b^4+b^2*c^2+c^4)*a^6-2*(b^2+c^2)*(b^4+c^4)*a^4+2*(b^4-c^4)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(12134) = 3*X(568)-4*X(11745) = 5*X(3091)-3*X(12022) = 11*X(5072)-6*X(12024) = 4*X(5462)-3*X(11245) = X(5889)-3*X(7576) = 3*X(5891)-X(11750) = X(6243)-3*X(7540)

The reciprocal orthologic center of these triangles is X(9833).

X(12134) lies on these lines: {2,9707}, {3,66}, {4,155}, {5,156}, {6,7528}, {23,2888}, {24,11442}, {25,68}, {26,343}, {30,5562}, {49,5576}, {52,1843}, {54,5133}, {110,1594}, {113,137}, {140,5944}, {154,3549}, {182,7405}, {235,9927}, {381,11426}, {389,542}, {427,1147}, {428,539}, {468,5449}, {511,7553}, {524,6243}, {568,11745}, {578,3818}, {1069,11393}, {1092,11550}, {1154,11819}, {1209,6676}, {1568,11572}, {1593,12118}, {1625,7745}, {1656,8780}, {1853,3548}, {1885,12133}, {1899,6642}, {3091,12022}, {3157,11392}, {3410,7488}, {3518,3580}, {3547,11206}, {5072,12024}, {5090,9928}, {5169,9545}, {5447,7667}, {5462,10116}, {5654,7507}, {5889,7576}, {5891,11750}, {5921,7487}, {6240,12111}, {6288,10024}, {6639,10192}, {6776,7401}, {6800,7558}, {7491,10454}, {7542,10282}, {7544,7592}, {7565,9143}, {7713,9896}, {9306,11585}, {9730,9825}, {9923,11386}, {9929,11388}, {9930,11389}, {9933,11396}, {10055,11398}, {10071,11399}, {10095,11264}, {10110,10112}, {10111,11746}, {10295,11440}

X(12134) = midpoint of X(6240) and X(12111)
X(12134) = reflection of X(i) in X(j) for these (i,j): (52,6756), (6146,5), (10111,11746), (10112,10110), (10116,5462), (11264,10095)
X(12134) = complement of X(34224)
X(12134) = crosspoint, wrt excentral or tangential triangle, of X(155) and X(2918)
X(12134) = orthologic center of these triangles: anti-Ara to 2nd Hyacinth
X(12134) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (110,1594,9820), (578,3818,7403), (1352,9833,3), (5462,10116,11245), (5921,7487,11411), (6288,10540,10024)

X(12135) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO EXCENTERS-MIDPOINTS

Barycentrics (2*a^3-(b^2+c^2)*a+(b+c)*(b^2+c^2))/(-a^2+b^2+c^2) : :
X(12135) = 3*X(428)-2*X(1829)

The reciprocal orthologic center of these triangles is X(10).

X(12135) lies on these lines: {1,427}, {4,145}, {8,25}, {10,468}, {24,5690}, {27,6542}, {28,4720}, {29,7140}, {33,1904}, {34,5101}, {235,355}, {407,5174}, {428,519}, {429,6198}, {431,5086}, {469,4393}, {515,1885}, {517,3575}, {594,1474}, {944,1593}, {1398,3476}, {1483,1595}, {1594,5901}, {1824,1891}, {1826,1990}, {1843,5846}, {1870,1883}, {1876,10106}, {1892,3340}, {1906,5881}, {1973,4390}, {2098,11393}, {2099,11392}, {2204,5291}, {2356,10459}, {3088,7967}, {3189,11406}, {3241,5064}, {3486,7071}, {3515,5657}, {3516,5731}, {3541,10246}, {3542,5790}, {3616,5094}, {3617,6353}, {3621,6995}, {3622,8889}, {3623,7378}, {3632,7713}, {3913,11383}, {4232,4678}, {5603,7507}, {5844,6756}, {10573,11399}, {10912,11390}

X(12135) = reflection of X(1885) in X(1902)
X(12135) = orthologic center of these triangles: anti-Ara to 2nd Schiffler
X(12135) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5090,427), (4,145,11396), (8,7718,25), (10,11363,468), (33,5130,1904), (5174,7009,407)

X(12136) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO EXTOUCH

Trilinears ((2*sin(A/2)+sin(3*A/2)-sin(5*A/2))*cos((B-C)/2)+(2*cos(A)+cos(2*A)+1)*cos(B-C)+2*cos(A)-2*cos(2*A)-4)*sec(A) : :
X(12136) = (6*R^2-SW)*X(25)-(4*R^2-SW)*X(84)

The reciprocal orthologic center of these triangles is X(40).

X(12136) lies on these lines: {4,7}, {25,84}, {34,1854}, {185,1829}, {235,6245}, {406,10167}, {427,6260}, {429,9942}, {451,11227}, {468,6705}, {475,5927}, {515,1885}, {1490,1593}, {1709,11398}, {1870,9856}, {3088,5658}, {4194,11220}, {6257,11389}, {6258,11388}, {7713,7992}, {7971,11396}, {10085,11399}, {11363,12114}

X(12137) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO FUHRMANN

Barycentrics (2*a^6-(b+c)*a^5-(3*b^2-2*b*c+3*c^2)*a^4+(b+c)*(2*b^2-b*c+2*c^2)*a^3-b*c*(b^2+c^2)*a^2-(b^4-c^4)*(b-c)*a+(b^4-c^4)*(b^2-c^2))/(-a^2+b^2+c^2) : :
X(12137) = (6*R^2-SW)*X(25)-(4*R^2-SW)*X(80)

The reciprocal orthologic center of these triangles is X(3).

X(12137) lies on these lines: {4,6224}, {11,11363}, {25,80}, {100,5090}, {149,7718}, {214,427}, {468,6702}, {515,1878}, {952,1829}, {1593,12119}, {1862,1900}, {1902,5840}, {2800,3575}, {2802,12135}, {2829,12136}, {6262,11389}, {6263,11388}, {7713,9897}, {7972,11396}, {10057,11398}, {10073,11399}

X(12138) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO INNER-GARCIA

Barycentrics (2*a^6-2*(b+c)*a^5-(3*b^2-8*b*c+3*c^2)*a^4+4*(b^2-c^2)*(b-c)*a^3-8*b*c*(b-c)^2*a^2-2*(b^2-c^2)*(b-c)^3*a+(b^4-c^4)*(b^2-c^2))/(-a^2+b^2+c^2) : :
Trilinears ((6*sin(A/2)-2*sin(3*A/2))*cos((B-C)/2)+cos(A)*cos(B-C)+4*cos(A)-cos(2*A)-4)*sec(A) : :
X(12138) = (6*R^2-SW)*X(25)-(4*R^2-SW)*X(104)

The reciprocal orthologic center of these triangles is X(40).

X(12138) lies on these lines: {4,145}, {11,34}, {25,104}, {33,1317}, {80,1041}, {100,1593}, {119,427}, {468,6713}, {515,1878}, {1112,2771}, {1387,1870}, {1484,1596}, {1595,11698}, {1768,7713}, {1828,2829}, {1829,2800}, {1861,3036}, {1885,5840}, {1890,5851}, {1902,2802}, {1904,6265}, {1905,11570}, {1907,5130}, {2783,5186}, {2787,12131}, {2801,5185}, {4219,9945}, {5064,10711}, {5155,6326}, {6154,11471}, {10058,11398}, {10074,11399}, {11363,11715}

X(12138) = reflection of X(1862) in X(4)
X(12138) = polar circle-inverse-of-X(153)

X(12139) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO HUTSON EXTOUCH

Barycentrics p*(2*q*p^6-(4*q^2-3)*p^5+(2*q^2-5)*q*p^4+(5*q^2-3)*p^3-(3*q^2+1)*q*p^2-(q^2-1)*p+(q^2+3)*q)*tan(A) : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(40).

X(12139) lies on these lines: {4,9874}, {25,7160}, {1593,12120}, {1824,12136}, {7713,9898}, {8000,11396}, {10059,11398}, {10075,11399}

X(12140) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO 1st HYACINTH

Trilinears (2*(cos(A)+cos(3*A))*cos(B-C)-(cos(2*A)+1)*cos(2*(B-C))-2*cos(2*A)-cos(4*A))*sec(A) : :
X(12140) = 3*X(1112)-4*X(11566) = X(1986)-3*X(7576)

The reciprocal orthologic center of these triangles is X(6102).

X(12140) lies on these lines: {4,110}, {24,125}, {25,265}, {30,12133}, {66,74}, {186,6699}, {235,10113}, {378,3818}, {403,1495}, {427,1511}, {542,1843}, {974,1503}, {1112,6756}, {1593,12121}, {1594,5972}, {2771,12137}, {2777,6240}, {3448,7487}, {3575,5663}, {6146,11746}, {6403,7731}, {6723,10018}, {7505,11750}, {7577,10546}, {7722,11387}, {10088,11392}, {10091,11393}

X(12140) = reflection of X(i) in X(j) for these (i,j): (1112,6756), (6146,11746)
X(12140) = polar-circle inverse of X(39118)

X(12141) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO INNER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(3).

X(12141) lies on these lines: {4,617}, {14,25}, {24,6774}, {115,10641}, {235,5479}, {427,619}, {428,531}, {462,6110}, {468,6670}, {530,12132}, {542,1843}, {1593,5474}, {3439,3456}, {5064,5464}, {5471,10642}, {6269,11389}, {6271,11388}, {6773,7487}, {7713,9900}, {7974,11396}, {9113,11409}, {9981,11386}, {10061,11398}, {10077,11399}, {11363,11706}

X(12141) = {X(1843),X(7576)}-harmonic conjugate of X(12142)

X(12142) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO OUTER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(3).

X(12142) lies on these lines: {4,616}, {13,25}, {24,6771}, {115,10642}, {235,5478}, {427,618}, {428,530}, {463,6111}, {468,6669}, {531,12132}, {542,1843}, {1593,5473}, {3438,3456}, {5064,5463}, {5472,10641}, {6268,11389}, {6270,11388}, {6770,7487}, {7713,9901}, {7975,11396}, {9112,11408}, {9982,11386}, {10062,11398}, {10078,11399}, {11363,11705}

X(12142) = {X(1843),X(7576)}-harmonic conjugate of X(12141)

X(12143) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO 1st NEUBERG

Barycentrics (b^2+c^2)*(a^4+b^2*c^2)/(-a^2+b^2+c^2) : :
X(12143) = (6*R^2-SW)*X(25)-(4*R^2-SW)*X(76)

The reciprocal orthologic center of these triangles is X(3).

X(12143) lies on these lines: {4,147}, {25,76}, {39,427}, {235,6248}, {262,7507}, {264,11325}, {384,11380}, {428,538}, {468,3934}, {511,3575}, {730,1829}, {732,1843}, {1593,11257}, {1594,11272}, {2790,9873}, {3088,7709}, {3186,9983}, {3541,11171}, {3542,7697}, {5064,7757}, {5094,7786}, {5969,12132}, {6272,11389}, {6273,11388}, {7713,9902}, {7976,11396}, {10063,11398}, {10079,11399}

X(12143) = polar-circle-inverse of X(32528)
X(12143) = X(76)-of-anti-Ara-triangle

X(12144) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO 2nd NEUBERG

Barycentrics (b^2+c^2)*(3*a^4+2*(b^2+c^2)*a^2+b^2*c^2)/(-a^2+b^2+c^2) : :
X(12144) = (6*R^2-SW)*X(25)-(4*R^2-SW)*X(83)

The reciprocal orthologic center of these triangles is X(3).

X(12144) lies on these lines: {4,2896}, {25,83}, {235,6249}, {427,6292}, {428,754}, {468,6704}, {732,1843}, {1593,12122}, {3199,10301}, {3515,9751}, {6274,11389}, {6275,11388}, {6756,12131}, {7713,9903}, {7977,11396}, {10064,11398}, {10080,11399}

X(12144) = X(83)-of-anti-Ara-triangle

X(12145) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO 1st ORTHOSYMMEDIAL

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

The reciprocal orthologic center of these triangles is X(4).

X(12145) lies on these lines: {4,339}, {25,1073}, {33,3320}, {34,6020}, {51,125}, {112,1593}, {127,235}, {428,9530}, {511,1529}, {1862,2831}, {1885,2794}, {2799,12131}, {2806,12138}, {2825,5185}, {9517,12133}, {10734,10735}

X(12146) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO 1st SCHIFFLER

Trilinears (64*p^7*(p-q)-16*(2*q^2+5)*p^6+96*q*p^5+4*(8*q^4+4*q^2+7)*p^4-4*(4*q^2+3)*q*p^3-(16*q^4+6*q^2+3)*p^2+2*(4*q^2-7)*q*p-q^2)*sec(A) : :
where p=sin(A/2), q=cos((B-C)/2)
X(12146) = (6*R^2-SW)*X(25)-(4*R^2-SW)*X(76)

The reciprocal orthologic center of these triangles is X(79).

X(12146) lies on the line {25,10266}

X(12147) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO INNER-VECTEN

Barycentrics (4*(b^2+c^2)*a^2*S+2*a^6-3*(b^2+c^2)*a^4-4*b^2*c^2*a^2+(b^4-c^4)*(b^2-c^2))/(-a^2+b^2+c^2) : :
X(12147) = (6*R^2-SW)*X(25)-(4*R^2-SW)*X(486)

The reciprocal orthologic center of these triangles is X(3).

X(12147) lies on these lines: {4,487}, {25,486}, {30,6406}, {52,1843}, {235,6251}, {371,8967}, {427,642}, {468,6119}, {1593,12123}, {6280,11389}, {6281,11388}, {6995,8948}, {7713,9906}, {7980,11396}, {9986,11386}, {10067,11398}, {10083,11399}

X(12147) = {X(1843),X(6756)}-harmonic conjugate of X(12148)

X(12148) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO OUTER-VECTEN

Barycentrics (-4*(b^2+c^2)*a^2*S+2*a^6-3*(b^2+c^2)*a^4-4*b^2*c^2*a^2+(b^4-c^4)*(b^2-c^2))/(-a^2+b^2+c^2) : :
X(12148) = (6*R^2-SW)*X(25)-(4*R^2-SW)*X(485)

The reciprocal orthologic center of these triangles is X(3).

X(12148) lies on these lines: {4,488}, {25,485}, {30,6291}, {52,1843}, {235,6250}, {427,641}, {468,6118}, {1593,12124}, {6278,11389}, {6279,11388}, {6995,8946}, {7713,9907}, {7981,11396}, {9987,11386}, {10068,11398}, {10084,11399}

X(12148) = {X(1843),X(6756)}-harmonic conjugate of X(12147)

X(12149) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARTZT TO 4th ANTI-BROCARD

Trilinears a*((2*b^4-b^2*c^2+2*c^4)*a^2+2*(b^2+c^2)*(b^4-5*b^2*c^2+c^4))*(c^2-a^2)*(a^2-b^2) : :
X(12149) = (9*R^2-2*SW)*X(110)+2*SW*X(1296)

The reciprocal orthologic center of these triangles is X(9870).

X(12149) lies on these lines: {2,9869}, {110,1296}, {512,9146}, {2854,2979}, {5077,7998}

X(12149) = 1st-tri-squares-to-anti-Artzt similarity image of X(13641)

X(12150) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO ANTI-ARTZT

Barycentrics 3*a^4+(b^2+c^2)*a^2+b^2*c^2 : :
X(12150) = X(384)+2*X(5007) = 2*X(384)+X(7760) = 2*X(3398)+X(12110) = 4*X(5007)-X(7760) = X(6655)-4*X(7829) = X(6658)+2*X(7765) = X(7768)-4*X(7819) = 2*X(7873)-5*X(7948)

The reciprocal orthologic center of these triangles is X(2).

X(12150) lies on these lines: {2,32}, {3,7878}, {4,7856}, {6,99}, {30,3398}, {76,11286}, {98,381}, {148,5355}, {182,376}, {187,3329}, {316,7792}, {325,8368}, {384,538}, {385,5008}, {428,11380}, {524,6661}, {530,11299}, {531,11300}, {543,4027}, {549,2080}, {551,11364}, {597,1691}, {671,3407}, {1186,3224}, {1384,7771}, {1651,11839}, {1975,7894}, {1992,5039}, {3053,7786}, {3058,10799}, {3241,10800}, {3524,5171}, {3545,10358}, {3552,7772}, {3589,7831}, {3679,10789}, {3734,7766}, {3788,7921}, {3849,7924}, {4234,4279}, {4421,11490}, {5038,8598}, {5041,7783}, {5055,10104}, {5304,11185}, {5306,8370}, {5475,7806}, {5860,10793}, {5861,10792}, {6179,7770}, {6573,6579}, {6655,7829}, {6658,7765}, {7669,11327}, {7737,7790}, {7745,7828}, {7747,7797}, {7748,7920}, {7750,7859}, {7759,7892}, {7761,7875}, {7762,7832}, {7768,7819}, {7773,7942}, {7774,7835}, {7776,7930}, {7778,7926}, {7779,7820}, {7782,9605}, {7784,7943}, {7789,7905}, {7795,7877}, {7799,8369}, {7801,7837}, {7802,7803}, {7807,7858}, {7816,7839}, {7822,7893}, {7823,7834}, {7825,7932}, {7836,7838}, {7840,7880}, {7841,7884}, {7842,7923}, {7843,7901}, {7845,7931}, {7847,8353}, {7850,7868}, {7852,7885}, {7854,10159}, {7860,7866}, {7869,7946}, {7870,9766}, {7873,7948}, {7874,7941}, {7881,7949}, {7898,7913}, {7903,7945}, {7915,7939}, {8703,12054}, {9909,10790}, {10056,10801}, {10072,10802}, {10794,11235}, {10795,11236}, {10797,11237}, {10798,11238}, {10803,11239}, {10804,11240}, {11057,11287}, {11163,11288}, {11207,11837}, {11208,11838}

X(12150) = reflection of X(7883) in X(2)
X(12150) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,7812,7809), (2,9939,7865), (6,1003,7757), (6,3972,99), (32,83,1078), (32,7787,83), (32,7808,7793), (32,10348,10347), (315,7846,7944), (316,7792,7919), (384,5007,7760), (1003,7757,99), (1384,11174,7771), (3972,7757,1003), (5008,7804,385), (5309,11361,671), (6680,7785,7899), (7759,7892,7909), (7762,7832,7917), (10796,11842,98)

X(12150) = X(2)-of-5th-anti-Brocard-triangle

X(12151) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-BROCARD TO ANTI-ARTZT

Barycentrics 3*a^6-4*(b^2+c^2)*a^4+(2*b^4-3*b^2*c^2+2*c^4)*a^2+2*b^2*c^2*(b^2+c^2) : :
X(12151) = 2*X(99)+X(8586) = 4*X(5026)-X(5104) = X(7840)+2*X(8787)

The reciprocal orthologic center of these triangles is X(8593).

X(12151) lies on these lines: {2,2056}, {6,538}, {83,11054}, {99,8586}, {182,599}, {183,10485}, {249,524}, {542,2456}, {575,9466}, {2080,2482}, {3398,7801}, {4027,7840}, {5026,5104}, {5111,5969}, {7809,8593}, {7810,12054}, {7839,9887}, {8584,12150}, {9939,10131}

X(12151) = midpoint of X(7809) and X(8593)
X(12151) = reflection of X(1691) in X(5182)

X(12152) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO ANTI-ARTZT

Barycentrics 4*(10*a^6-27*(b^2+c^2)*a^4-48*b^2*c^2*a^2+(b^2+c^2)^3)*S+9*a^8+24*(b^2+c^2)*a^6-26*(3*b^4+8*b^2*c^2+3*c^4)*a^4+8*(b^2+c^2)*(6*b^4-11*b^2*c^2+6*c^4)*a^2-3*(b^4-c^4)^2 : :

The reciprocal orthologic center of these triangles is X(2).

X(12152) lies on these lines: {2,493}, {30,9838}, {376,11828}, {381,8212}, {428,11394}, {551,11377}, {1651,11907}, {3058,11947}, {3241,8210}, {3679,8188}, {4421,11503}, {5860,8218}, {5861,8216}, {6461,12153}, {7811,10875}, {8194,9909}, {8201,11207}, {8208,11208}, {10056,11951}, {10072,11953}, {10945,11235}, {10951,11236}, {11237,11930}, {11238,11932}, {11239,11955}, {11240,11957}, {11840,12150}

X(12153) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO ANTI-ARTZT

Barycentrics -4*(10*a^6-27*(b^2+c^2)*a^4-48*b^2*c^2*a^2+(b^2+c^2)^3)*S+9*a^8+24*(b^2+c^2)*a^6-26*(3*b^4+8*b^2*c^2+3*c^4)*a^4+8*(b^2+c^2)*(6*b^4-11*b^2*c^2+6*c^4)*a^2-3*(b^4-c^4)^2 : :

The reciprocal orthologic center of these triangles is X(2).

X(12153) lies on these lines: {2,494}, {30,9839}, {376,11829}, {381,8213}, {428,11395}, {551,11378}, {1505,8222}, {1651,11908}, {3058,11948}, {3241,8211}, {3679,8189}, {4421,11504}, {5860,8219}, {5861,8217}, {6461,12152}, {7811,10876}, {8195,9909}, {8202,11207}, {8209,11208}, {10056,11952}, {10072,11954}, {10946,11235}, {10952,11236}, {11237,11931}, {11238,11933}, {11239,11956}, {11240,11958}, {11841,12150}

X(12154) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARTZT TO INNER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(9761).

X(12154) lies on these lines: {2,14}, {6,543}, {13,11317}, {16,8598}, {17,10809}, {61,8370}, {99,9113}, {395,9886}, {396,3363}, {398,8369}, {530,8593}, {542,11295}, {597,6775}, {1003,9114}, {2482,5471}, {5339,11318}, {5475,9117}, {5476,11296}, {6772,9830}, {8595,9116}

X(12154) = reflection of X(6775) in X(597)
X(12154) = Napoleon-outer circle-inverse-of-X(9760)
X(12154) = {X(6), X(11159)}-harmonic conjugate of X(12155)

X(12155) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARTZT TO OUTER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(9763).

X(12155) lies on these lines: {2,13}, {6,543}, {14,11317}, {15,8598}, {18,10808}, {62,8370}, {99,9112}, {395,3363}, {396,9885}, {397,8369}, {531,8593}, {542,11296}, {597,6772}, {1003,9116}, {2482,5472}, {5340,11318}, {5475,9115}, {5476,11295}, {6775,9830}, {8594,9114}

X(12155) = reflection of X(6772) in X(597)
X(12155) = Napoleon-inner circle-inverse-of-X(9762)
X(12155) = {X(6), X(11159)}-harmonic conjugate of X(12154)

X(12156) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARTZT TO 2nd NEUBERG

Barycentrics 7*a^4+3*(b^2+c^2)*a^2+3*b^2*c^2-2*c^4-2*b^4 : :
X(12156) = 2*X(2)-3*X(83) = 5*X(2)-3*X(2896) = 5*X(83)-2*X(2896) = 7*X(83)-4*X(6292) = 7*X(2896)-10*X(6292) = 11*X(2896)-20*X(6704) = 4*X(5066)-3*X(6287) = 4*X(8703)-3*X(12122)

The reciprocal orthologic center of these triangles is X(9766).

X(12156) lies on these lines: {2,32}, {3845,11632}, {3972,9766}, {4677,9903}, {5066,6287}, {5097,10723}, {5306,9166}, {7760,11361}, {7878,11287}, {8584,8593}, {8703,12122}, {9300,11155}, {9751,12100}, {11055,11159}, {11149,11163}

X(12156) = {X(7812), X(12150)}-harmonic conjugate of X(7809)

X(12157) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARTZT TO 3rd PARRY

Trilinears ((2*b^4+b^2*c^2+2*c^4)*a^10-(b^2+c^2)*(5*b^4+b^2*c^2+5*c^4)*a^8+b^2*c^2*(7*b^4+24*b^2*c^2+7*c^4)*a^6+(b^2+c^2)*(5*b^8+5*c^8-2*b^2*c^2*(8*b^4-3*b^2*c^2+8*c^4))*a^4-(2*b^8+2*c^8-b^2*c^2*(7*b^4+3*b^2*c^2+7*c^4))*(b^4-b^2*c^2+c^4)*a^2-(b^2+c^2)*(2*b^8+2*c^8-b^2*c^2*(2*b^4-b^2*c^2+2*c^4))*b^2*c^2)*a : :

The reciprocal orthologic center of these triangles is X(2).

X(12157) lies on the anti-Artzt circle and these lines: {99,511}, {110,5104}, {512,11161}, {805,2770}, {6787,11178}, {10717,12149}

X(12157) = circumsymmedial-to-anti-Artzt similarity image of X(2698)

X(12158) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARTZT TO INNER-VECTEN

Barycentrics 7*a^4-(b^2+c^2)*a^2+8*b^2*c^2-2*c^4-2*b^4-2*(8*a^2-b^2-c^2)*S : :

The reciprocal orthologic center of these triangles is X(591).

X(12158) lies on these lines: {2,371}, {8584,11159}

X(12158) = X(1328)-of-anti-Artzt-triangle
X(12158) = {X(8584),X(11159)}-harmonic conjugate of X(12159)

X(12159) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARTZT TO OUTER-VECTEN

Barycentrics 7*a^4-(b^2+c^2)*a^2+8*b^2*c^2-2*c^4-2*b^4+2*(8*a^2-b^2-c^2)*S : :

The reciprocal orthologic center of these triangles is X(1991).

X(12159) lies on these lines: {2,372}, {1991,6390}, {8584,11159}

X(12159) = X(1327)-of-anti-Artzt-triangle
X(12159) = {X(8584),X(11159)}-harmonic conjugate of X(12158)

X(12160) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO 1st ANTI-CIRCUMPERP

Trilinears (a^8-6*(b^2+c^2)*a^6+4*(3*b^4+2*b^2*c^2+3*c^4)*a^4-2*(b^2+c^2)*(5*b^4-6*b^2*c^2+5*c^4)*a^2+(3*b^4+2*b^2*c^2+3*c^4)*(b^2-c^2)^2)*a : :
X(12160) = 2*X(3)-3*X(11402)

The reciprocal orthologic center of these triangles is X(11412).

X(12160) lies on these lines: {2,11432}, {3,54}, {4,193}, {5,6515}, {6,5562}, {24,3167}, {25,52}, {68,7507}, {69,7399}, {110,3517}, {143,7529}, {156,9714}, {184,9715}, {389,394}, {427,11411}, {511,1181}, {568,6090}, {576,5907}, {912,11396}, {1092,9786}, {1147,3515}, {1199,5050}, {1216,7484}, {1350,10984}, {1398,7352}, {1498,2393}, {1597,12111}, {1598,3060}, {1614,9909}, {1656,3580}, {1994,7503}, {3091,3527}, {3410,7566}, {3518,8780}, {3567,5020}, {3575,6193}, {5059,11820}, {5093,11459}, {5198,5446}, {5410,10665}, {5411,10666}, {5422,11444}, {5462,11284}, {5640,11484}, {6146,10602}, {6237,11406}, {6238,7071}, {6243,7387}, {6643,11245}, {7689,11410}, {8548,11405}, {9936,12134}, {10601,11793}, {10661,11408}, {10662,11409}

X(12160) = reflection of X(11414) in X(1181)
X(12160) = orthologic center of anti-Ascella triangle to these triangles: anti-Conway, 2nd anti-Conway, 3rd anti-Euler, 3rd anti-Euler, anti-excenters-reflections, anti-Hutson intouch, anti-incircle-circles, anti-inverse-in-incircle, 6th anti-mixtilinear, circumorthic, 2nd Ehrmann, 2nd Euler, extangents, intangents, 1st Kenmotu diagonals, 2nd Kenmotu diagonals, Kosnita, orthic, submedial, tangential, inner tri-equilateral, outer tri-equilateral, Trinh.
X(12160) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,5562,7395), (52,155,25), (576,5907,10982), (1199,7509,5050), (1993,5889,3), (1994,7503,11426), (3060,11441,1598), (7592,11412,3)

X(12161) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-CONWAY TO ANTI-ASCELLA

Trilinears (a^8-4*(b^2+c^2)*a^6+2*(3*b^4+2*b^2*c^2+3*c^4)*a^4-2*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^2+(b^4+c^4)*(b^2-c^2)^2)*a : :
X(12161) = X(3)-3*X(11402) = 3*X(3796)-2*X(7525) = 3*X(11402)+X(12160)

The reciprocal orthologic center of these triangles is X(12160).

X(12161) lies on these lines: {2,1199}, {3,54}, {4,1994}, {5,6}, {20,11004}, {22,6243}, {24,49}, {25,143}, {26,52}, {30,1181}, {51,10539}, {81,6862}, {110,3567}, {140,394}, {182,1216}, {185,12084}, {186,9545}, {193,3547}, {265,7547}, {323,631}, {381,11441}, {382,11456}, {389,1147}, {399,3843}, {546,10982}, {567,7503}, {569,5562}, {575,11793}, {576,2393}, {578,7526}, {895,3527}, {1092,9730}, {1184,10011}, {1351,7387}, {1498,3627}, {1593,5663}, {1598,5093}, {1614,3060}, {1656,5422}, {2070,9704}, {2914,3448}, {2937,6800}, {3167,5946}, {3193,6928}, {3518,9544}, {3549,6515}, {3574,7564}, {3580,6639}, {3618,11487}, {3628,10601}, {3796,7525}, {5050,7393}, {5097,10110}, {5462,9306}, {5576,11442}, {5876,9818}, {6237,11428}, {6238,11429}, {7395,11591}, {7507,11264}, {7512,11003}, {7529,9777}, {7689,11430}, {9587,9625}, {9590,9622}, {9706,11464}, {9833,11819}, {9927,10112}, {10115,10274}, {10540,10594}, {10602,11255}, {10605,11250}, {10625,10984}, {11245,11585}, {11411,11427}, {11438,12038}, {11818,12134}

X(12161) = midpoint of X(3) and X(12160)
X(12161) = reflection of X(7526) in X(578)
X(12161) = X(3)-of-2nd-anti-extouch-triangle
X(12161) = X(4)-of-anti-Conway-triangle
X(12161) = X(5)-of-anti-Ascella-triangle
X(12161) = anti-Conway-isogonal conjugate of X(32046)
X(12161) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,195,1993), (6,155,5), (49,568,24), (52,184,26), (54,5889,3), (110,3567,7506), (143,156,25), (182,1216,7516), (389,1147,6644), (569,5562,7514), (576,6759,5446), (1351,7387,10263), (1614,3060,7517), (1993,7592,3), (2070,9704,9707), (5012,11412,3), (5446,6759,7530), (5889,11422,54), (11402,12160,3), (11412,11423,5012)

X(12162) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO ANTI-ASCELLA

Trilinears ((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^4-4*b^2*c^2+3*c^4)*a^2-(b^4+4*b^2*c^2+c^4)*(b^2-c^2)^2)*a : :
X(12162) = 5*X(3)-6*X(3819) = 2*X(3)-3*X(5891) = 3*X(3)-4*X(11793) = 5*X(4)-3*X(3060) = 3*X(4)-X(5889) = 3*X(4)-5*X(11439) = 11*X(4)-8*X(12002) = 5*X(52)-6*X(3060) = 3*X(52)-2*X(5889) = 4*X(3819)-5*X(5891) = 3*X(3819)-5*X(5907) = 12*X(3819)-5*X(10575)

The reciprocal orthologic center of these triangles is X(12160).

Let T be the triangle whose vertices are the orthocenters of the altimedial triangles; then X(12162) = X(20)-of-T. (Randy Hutson, July 21, 2017)

X(12162) lies on these lines: {2,6241}, {3,64}, {4,52}, {5,113}, {20,1216}, {24,7689}, {30,5562}, {33,7352}, {34,6238}, {39,1625}, {49,399}, {51,546}, {54,7527}, {67,3521}, {110,3520}, {143,3845}, {155,1593}, {184,7526}, {186,11440}, {355,2807}, {376,5447}, {378,1147}, {381,389}, {382,511}, {394,12085}, {403,5449}, {517,6253}, {550,3917}, {568,3843}, {569,1181}, {631,10170}, {912,1902}, {1060,7355}, {1062,6285}, {1092,10564}, {1154,3627}, {1204,6644}, {1209,2883}, {1352,5878}, {1495,1658}, {1503,9967}, {1511,10226}, {1531,11572}, {1594,5448}, {1656,9729}, {2772,5884}, {2777,7723}, {2979,3529}, {3090,5892}, {3091,5462}, {3146,11412}, {3522,7999}, {3528,7998}, {3530,5650}, {3541,5654}, {3544,11451}, {3547,5656}, {3567,3832}, {3830,6243}, {3839,9781}, {3850,5946}, {3851,5943}, {3853,10263}, {3855,5640}, {3858,10095}, {4550,7503}, {4846,6815}, {5055,11695}, {5079,6688}, {5498,10272}, {6193,11469}, {6237,11471}, {6247,11585}, {6254,8251}, {6288,7728}, {6636,8718}, {6642,10605}, {6696,10257}, {7506,11438}, {7512,12112}, {7514,10984}, {7529,9786}, {7544,7706}, {7691,12088}, {7722,11557}, {8538,8549}, {8548,11470}, {10116,12022}, {10634,10675}, {10635,10676}, {10661,11475}, {10662,11476}, {10665,11473}, {10666,11474}, {10996,11487}, {11403,12160}, {11424,12161}

X(12162) = midpoint of X(i) and X(j) for these {i,j}: {4,12111}, {3146,11412}, {5562,11381}
X(12162) = reflection of X(i) in X(j) for these (i,j): (3,5907), (20,1216), (52,4), (185,5), (550,11591), (5562,5876), (5889,5446), (6102,546), (7722,11557), (10263,3853), (10575,3), (10625,5562), (11562,113)
X(12162) = complement of X(6241)
X(12162) = X(4)-of-X(4)-Brocard-triangle
X(12162) = X(10)-of-Ehrmann-side-triangle if ABC is acute" to X(12162)
X(12162) = X(10)-of-Ehrmann-side-triangle if ABC is acute
X(12162) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,5907,5891), (3,10540,10282), (4,5889,5446), (4,11442,9927), (5,185,9730), (5,12006,373), (20,11459,1216), (110,3520,12038), (155,11472,1593), (376,11444,5447), (378,11441,1147), (546,6102,51), (550,11591,3917), (568,3843,10110), (3357,9306,3), (5446,5889,52), (5876,11381,10625), (5889,11439,4), (5891,10575,3), (11439,12111,5889)

X(12163) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH TO ANTI-ASCELLA

Trilinears (-a^2+b^2+c^2)*(a^6+(b^2+c^2)*a^4-(5*b^4-6*b^2*c^2+5*c^4)*a^2+3*(b^4-c^4)*(b^2-c^2))*a : :
X(12163) = 3*X(3)-2*X(1147) = 5*X(3)-3*X(3167) = 5*X(3)-4*X(12038) = 4*X(140)-3*X(5654) = 3*X(154)-4*X(1658) = 3*X(155)-4*X(1147) = 5*X(155)-6*X(3167) = X(155)-4*X(7689) = 3*X(376)-X(6193) = 3*X(381)-4*X(5449)

The reciprocal orthologic center of these triangles is X(12160).

X(12163) lies on these lines: {3,49}, {4,3580}, {5,9786}, {6,6102}, {20,11411}, {22,6241}, {24,12111}, {25,12162}, {26,1498}, {30,64}, {35,3157}, {36,1069}, {40,912}, {52,1593}, {55,7352}, {56,6238}, {74,9938}, {140,5646}, {154,1658}, {186,11441}, {376,6193}, {378,5889}, {381,5449}, {382,9927}, {389,9818}, {511,3357}, {539,3534}, {548,9936}, {550,1350}, {568,10982}, {631,9820}, {1151,10665}, {1152,10666}, {1154,10606}, {1192,5876}, {1597,5446}, {1656,5448}, {1657,10620}, {1993,3520}, {3066,3851}, {3515,10539}, {3516,12160}, {3532,5504}, {3579,9928}, {3581,7517}, {4550,5462}, {4846,6823}, {5584,6237}, {5890,7503}, {5907,6642}, {6000,7387}, {6200,8909}, {6240,11442}, {6284,10071}, {6285,9645}, {6445,8912}, {7354,10055}, {7393,9729}, {7488,7712}, {7509,10574}, {7691,10323}, {8548,11477}, {8567,11250}, {9707,10298}, {9937,10575}, {10661,11480}, {10662,11481}, {11425,12161}

X(12163) = midpoint of X(20) and X(11411)
X(12163) = reflection of X(i) in X(j) for these (i,j): (3,7689), (155,3), (382,9927), (1498,26), (5504,12041), (9928,3579), (11477,8548), (12085,3357), (12118,550)
X(12163) = ABC-X3-reflections-isogonal conjugate of X(33495)
X(12163) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,3167,12038), (74,11412,11413), (1204,5562,3), (4550,5462,11479), (5889,11440,378), (5907,11438,6642), (6102,7526,6)

X(12164) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO ANTI-ASCELLA

Trilinears (-a^2+b^2+c^2)*(a^6-5*(b^2+c^2)*a^4+(7*b^4-6*b^2*c^2+7*c^4)*a^2-3*(b^4-c^4)*(b^2-c^2))*a : :
Trilinears (cos A)(cos^2 B + cos^2 C - cos^2 A - sec A sec B sec C) : :
X(12164) = 3*X(3)-4*X(1147) = 2*X(3)-3*X(3167) = 5*X(3)-4*X(7689) = 7*X(3)-8*X(12038) = 3*X(3)-2*X(12163) = 2*X(68)-3*X(381) = 3*X(155)-2*X(1147) = 4*X(155)-3*X(3167) = X(382)+2*X(9936) = 5*X(1656)-6*X(5654)

Let A'B'C' be the orthic triangle. Let Oa be the A-power circle of triangle AB'C', and define Ob and Oc cyclically. X(12164) is the radical center of circles Oa, Ob, Oc. (Randy Hutson, July 31 2018)

The reciprocal orthologic center of these triangles is X(12160).

X(12164) lies on these lines: {3,49}, {4,193}, {5,11411}, {6,5907}, {24,8780}, {25,5889}, {30,6193}, {52,1598}, {68,381}, {69,6823}, {110,3515}, {235,6515}, {323,11413}, {382,9936}, {389,5020}, {399,7517}, {511,1498}, {524,2883}, {539,3830}, {568,7529}, {912,1482}, {916,2293}, {999,1069}, {1154,7387}, {1593,1993}, {1597,12162}, {1614,9715}, {1619,6293}, {1656,5544}, {1657,11820}, {2070,9932}, {2781,9914}, {3060,5198}, {3091,9777}, {3311,10665}, {3312,10666}, {3517,10539}, {3526,9820}, {3843,9927}, {3851,5448}, {5050,7395}, {5055,5449}, {5093,10982}, {5462,11484}, {5504,10620}, {5663,12085}, {5876,9818}, {6102,6642}, {6221,8909}, {6237,10306}, {6759,9909}, {6800,7691}, {6816,11245}, {7393,11591}, {7484,11444}, {7503,11402}, {7507,11442}, {7509,12017}, {8548,11482}, {8681,11477}, {8718,11412}, {9306,9786}, {9654,10055}, {9669,10071}, {9714,10540}, {9925,12082}, {10661,11485}, {10662,11486}, {11410,11440}

X(12164) = reflection of X(i) in X(j) for these (i,j): (3,155), (1657,12118), (6391,1351), (10620,5504), (11411,5), (12163,1147)
X(12164) = X(64)-Ceva conjugate of X(3)
X(12164) = X(10864)-of-orthic-triangle if ABC is acute
X(12164) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,155,3167), (4,12160,1351), (6,5907,11479), (155,12163,1147), (185,394,3), (1069,7352,999), (1092,10605,3), (1147,12163,3), (1181,5562,3), (1993,12111,1593), (3157,6238,3295), (5889,11441,25), (7395,7592,5050), (7592,11459,7395), (11412,11456,11414)

X(12165) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO ANTI-ORTHOCENTROIDAL

Trilinears (6*(3*cos(A)+cos(3*A))*cos(B-C)-7*cos(2*A)+cos(4*A)-6)*sec(A) : :
X(12165) = (6*R^2-SW)*X(25)-2*(4*R^2-SW)*X(399)

The reciprocal orthologic center of these triangles is X(3581).

X(12165) lies on these lines: {3,3043}, {4,11703}, {25,399}, {74,11410}, {110,3515}, {155,11562}, {378,2914}, {1112,5198}, {1181,10628}, {1351,10733}, {1593,5663}, {2771,11396}, {3448,7507}, {3516,10620}, {5094,10264}, {7071,7727}, {7395,7723}, {7687,9777}, {7724,11406}, {7731,9919}, {9826,11284}, {9976,11405}, {10657,11408}, {10658,11409}

X(12165) = orthologic center of these triangles: anti-Ascella to orthic
X(12165) = {X(399), X(1986)}-harmonic conjugate of X(25)

X(12166) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO ARIES

Trilinears cos(A)*(4*(3*cos(A)+cos(3*A))*cos(B-C)-2*(cos(2*A)-1)*cos(2*(B-C))+6*cos(2*A)-cos(4*A)-5) : :
X(12166) = 3*X(3167)-2*X(12161)

The reciprocal orthologic center of these triangles is X(7387).

X(12166) lies on these lines: {3,69}, {25,52}, {68,7395}, {578,8681}, {912,8192}, {1147,5892}, {1154,7387}, {3167,5946}, {3515,9932}, {6391,11426}, {7071,9931}, {9715,9908}, {9820,11284}, {9926,11405}, {9938,11410}, {10659,11408}, {10660,11409}

X(12166) = reflection of X(12160) in X(155)
X(12166) = orthologic center of these triangles: anti-Ascella to 2nd Hyacinth
X(12166) = {X(155), X(9937)}-harmonic conjugate of X(25)

X(12167) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO 1st EHRMANN

Trilinears    (a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^2-3*b^2-3*c^2)*a : :
Trilinears    (sin A)(tan A + 2 tan ω) : :
Trilinears    (sin A)(2 + tan A cot ω) : :
X(12167) = 4*X(6)-3*X(11402) = 4*X(3867)-3*X(5064) = 3*X(5093)-2*X(12161)

The reciprocal orthologic center of these triangles is X(576).

X(12167) lies on these lines: {3,6403}, {4,193}, {6,25}, {24,5050}, {69,427}, {141,5094}, {182,3515}, {186,12017}, {399,2971}, {428,1992}, {458,3186}, {460,3087}, {468,3618}, {511,1593}, {518,11396}, {524,3867}, {542,12165}, {576,5198}, {895,1112}, {1154,1597}, {1350,3516}, {1352,7507}, {1353,6756}, {1398,1469}, {1598,5093}, {1829,3751}, {1862,10755}, {2207,5052}, {3056,7071}, {3089,3527}, {3098,11410}, {3575,6776}, {3620,8889}, {3779,11406}, {5017,8778}, {5020,11416}, {5032,7714}, {5039,11380}, {5090,5847}, {5102,11470}, {5107,5140}, {5185,10756}, {5186,10754}, {6090,11188}, {7395,9967}, {7484,9813}, {7487,11432}, {7529,11255}, {8593,12132}, {9307,12110}, {9732,11395}, {9733,11394}, {9737,10607}, {9822,11284}, {10594,11482}, {10752,12133}, {10753,12131}, {10759,12138}, {11382,11433}, {11403,11477}

X(12167) = reflection of X(12160) in X(1351)
X(12167) = homothetic center of orthic triangle and reflection of tangential triangle in X(6)
X(12167) = {X(12171),X(12172)}-harmonic conjugate of X(1593)
X(12167) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,1843,25), (6,7716,1974), (6,9924,184), (6,9973,159), (25,8541,11405), (1351,6391,193), (1843,1974,7716), (1843,8541,6), (1974,7716,25), (5410,11389,25), (5411,11388,25)

X(12168) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO 1st HYACINTH

Trilinears 2*(9*cos(2*A)+2*cos(4*A)+9)*cos(B-C)-2*(3*cos(A)+cos(3*A))*cos(2*(B-C))-4*cos(3*A)-cos(5*A)-19*cos(A) : :
X(12168) = (6*R^2-SW)*X(25)-4*R^2*X(113)

The reciprocal orthologic center of these triangles is X(10112).

X(12168) lies on these lines: {3,74}, {22,146}, {25,113}, {125,7395}, {159,2935}, {265,9818}, {1597,10733}, {1657,8907}, {2777,11414}, {3028,10832}, {3043,3167}, {3448,7503}, {6644,10272}, {6699,7484}, {7387,7728}, {7514,10264}, {9715,10117}, {9909,10706}, {10663,11408}, {10664,11409}, {10982,11800}, {11562,12163}, {12085,12121}

X(12168) = {X(113), X(2931)}-harmonic conjugate of X(25)

X(12169) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO LUCAS ANTIPODAL

Trilinears a*((16*a^8-16*(b^2+c^2)*a^6-48*b^2*c^2*a^4+16*(b^2+c^2)*(b^4+4*b^2*c^2+c^4)*a^2-16*b^8+16*b^6*c^2-128*b^4*c^4+16*b^2*c^6-16*c^8)*S+a^10-(b^2+c^2)*a^8-2*(b^4+30*b^2*c^2+c^4)*a^6+2*(b^2+c^2)*(b^4+34*b^2*c^2+c^4)*a^4+(b^8+c^8-6*b^2*c^2*(6*b^4-b^2*c^2+6*c^4))*a^2+(b^4-c^4)*(b^2-c^2)*(-b^4+26*b^2*c^2-c^4)) : :

The reciprocal orthologic center of these triangles is X(3).

X(12169) lies on these lines: {25,487}, {486,7484}, {642,11284}, {3564,11414}, {5198,6290}

X(12169) = orthic-to-anti-Ascella similarity image of X(487)

X(12170) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO LUCAS(-1) ANTIPODAL

Trilinears a*(-(16*a^8-16*(b^2+c^2)*a^6-48*b^2*c^2*a^4+16*(b^2+c^2)*(b^4+4*b^2*c^2+c^4)*a^2-16*b^8+16*b^6*c^2-128*b^4*c^4+16*b^2*c^6-16*c^8)*S+a^10-(b^2+c^2)*a^8-2*(b^4+30*b^2*c^2+c^4)*a^6+2*(b^2+c^2)*(b^4+34*b^2*c^2+c^4)*a^4+(b^8+c^8-6*b^2*c^2*(6*b^4-b^2*c^2+6*c^4))*a^2+(b^4-c^4)*(b^2-c^2)*(-b^4+26*b^2*c^2-c^4)) : :

The reciprocal orthologic center of these triangles is X(3).

X(12170) lies on these lines: {25,488}, {485,7484}, {641,11284}, {3564,11414}, {5198,6289}

X(12170) = orthic-to-anti-Ascella similarity image of X(488)

X(12171) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO LUCAS CENTRAL

Trilinears (a^2+b^2-c^2)*(a^2-b^2+c^2)*((-a^2+3*b^2+3*c^2)*S+a^4-2*(b^2+c^2)*a^2+6*b^2*c^2+c^4+b^4)*a : :
X(12171) = (6*R^2-SW)*X(25)-2*(4*R^2-SW)*X(1151)

The reciprocal orthologic center of these triangles is X(3).

X(12171) lies on these lines: {3,6239}, {25,1151}, {511,1593}, {1398,7362}, {5023,5413}, {5411,8778}, {6200,8948}, {6252,11406}, {6283,7071}, {7690,11410}, {9732,11394}, {9823,11284}, {9974,11405}, {10667,11408}, {10668,11409}

X(12171) = {X(1593),X(12167)}-harmonic conjugate of X(12172)
X(12171) = X(176)-of-anti-Ascella-triangle if ABC is acute
X(12171) = orthic-to-anti-Ascella similarity image of X(6291)

X(12172) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO LUCAS(-1) CENTRAL

Trilinears (a^2+b^2-c^2)*(a^2-b^2+c^2)*(-(-a^2+3*b^2+3*c^2)*S+a^4-2*(b^2+c^2)*a^2+6*b^2*c^2+c^4+b^4)*a : :
X(12172) = (6*R^2-SW)*X(25)-2*(4*R^2-SW)*X(1152)

The reciprocal orthologic center of these triangles is X(3).

X(12172) lies on these lines: {3,6400}, {25,1152}, {511,1593}, {1398,7353}, {5023,5412}, {5410,8778}, {6396,8946}, {6404,11406}, {6405,7071}, {7692,11410}, {9733,11395}, {9824,11284}, {9975,11405}, {10671,11408}, {10672,11409}

X(12172) = {X(1593),X(12167)}-harmonic conjugate of X(12171)
X(12172) = X(175)-of-anti-Ascella-triangle if ABC is acute
X(12172) = orthic-to-anti-Ascella similarity image of X(6406)

X(12173) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO MACBEATH

Barycentrics (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))*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
X(12173) = 3*X(4)-2*X(1595) = 4*X(4)-3*X(5064) = 3*X(381)-2*X(7526) = 3*X(1593)-4*X(1595) = 2*X(1593)-3*X(5064) = 8*X(1595)-9*X(5064)

As a point on the Euler line, X(12173) has Shinagawa coefficients: (-F, E+5*F).

X(12173) lies on these lines: {2,3}, {33,4348}, {34,7221}, {64,6145}, {70,3426}, {125,1192}, {515,11396}, {516,5090}, {950,1892}, {962,12135}, {1112,10733}, {1204,1853}, {1398,7354}, {1503,12167}, {1699,11363}, {1829,5691}, {1843,5895}, {1862,10724}, {1870,9655}, {1876,9579}, {2207,7747}, {3070,5410}, {3071,5411}, {3172,7737}, {3574,11425}, {3583,11399}, {3585,11398}, {5185,10725}, {5186,10723}, {5318,11408}, {5321,11409}, {5339,8739}, {5340,8740}, {5890,6746}, {6198,9668}, {6241,7730}, {6253,11391}, {6256,11400}, {6284,7071}, {6403,12111}, {7718,9812}, {7728,12140}, {7823,9308}, {8550,11405}, {10721,11387}, {10722,12131}, {10728,12138}, {11432,12022}

X(12173) = reflection of X(i) in X(j) for these (i,j): (20,6823), (1593,4)
X(12173) = homothetic center of orthic triangle and reflection of tangential triangle in X(4)
X(12173) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,4,7507), (4,20,427), (4,24,381), (4,186,7547), (4,403,3843), (4,1593,5064), (4,1885,11403), (4,3089,10151), (4,3146,1885), (4,3542,546), (4,3575,25), (4,6240,3), (4,6353,3832), (4,6622,3839), (4,6756,5198), (4,6995,1906), (4,7487,235), (4,7576,1598), (1596,3853,4), (1598,3830,4), (3627,6756,4)

X(12174) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO MIDHEIGHT

Trilinears (a^8-6*(b^2+c^2)*a^6+4*(3*b^4-2*b^2*c^2+3*c^4)*a^4-10*(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^2*(3*b^2+c^2)*(b^2+3*c^2))*a : :
X(12174) = 2*X(578)-3*X(1181) = 4*X(578)-3*X(1593) = 8*X(578)-9*X(11402) = 4*X(1181)-3*X(11402) = 2*X(1593)-3*X(11402)

The reciprocal orthologic center of these triangles is X(389).

X(12174) lies on these lines: {3,74}, {4,3527}, {6,9968}, {20,12164}, {25,185}, {30,12160}, {64,184}, {154,1204}, {155,10575}, {221,3270}, {235,5656}, {381,11457}, {389,5198}, {569,11472}, {578,1181}, {1192,1495}, {1351,3146}, {1398,7355}, {1425,2192}, {1503,12167}, {1597,7592}, {1598,5890}, {1885,6225}, {1899,2883}, {1906,11433}, {2777,12165}, {2807,8192}, {3167,11413}, {3357,11410}, {3426,11426}, {3515,6759}, {3529,11820}, {4846,12134}, {5020,10574}, {5093,11458}, {5094,6247}, {5095,5895}, {5422,11439}, {5878,6146}, {5907,7484}, {6001,11396}, {6199,11462}, {6254,11406}, {6285,7071}, {6293,9914}, {6395,11463}, {6767,11461}, {7395,12162}, {7722,9919}, {8549,11405}, {9715,12163}, {9729,11284}, {10594,12112}, {10675,11408}, {10676,11409}, {11414,12166}

X(12174) = reflection of X(1593) in X(1181)
X(12174) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,11441,6090), (6,11381,11403), (74,9707,3), (185,1498,25), (1181,1593,11402), (6241,11456,3), (6759,10605,3515), (6800,11440,3)

X(12175) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO REFLECTION

Trilinears ((2*cos(A)+6*cos(3*A))*cos(B-C)+5*cos(2*A)+cos(4*A)-2)*sec(A) : :
X(12175) = (6*R^2-SW)*X(25)-2*(4*R^2-SW)*X(195)

The reciprocal orthologic center of these triangles is X(6243).

X(12175) lies on these lines: {3,6242}, {25,195}, {54,3515}, {539,12160}, {1154,1593}, {1351,5198}, {1398,7356}, {1614,9920}, {2888,7507}, {2914,10594}, {5965,12167}, {6255,11406}, {6286,7071}, {7691,11410}, {9827,11284}, {9977,11405}, {10677,11408}, {10678,11409}, {12165,12173}

X(12175) = {X(195), X(6152)}-harmonic conjugate of X(25)

X(12176) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO 1st ANTI-BROCARD

Barycentrics (a^6-(b^2+c^2)*a^4-c^2*(b^2-2*c^2)*a^2-(b^2-c^2)*b^2*c^2)*(a^6-(b^2+c^2)*a^4+b^2*(2*b^2-c^2)*a^2+(b^2-c^2)*b^2*c^2) : :
X(12176) = (SW^2+S^2)*X(99)-4*SW^2*X(182)

The reciprocal orthologic center of these triangles is X(5999).

X(12176) lies on these lines: {3,1916}, {4,32}, {83,114}, {99,182}, {147,7787}, {384,2782}, {542,12150}, {1078,6036}, {1691,11676}, {2080,5999}, {2966,6784}, {3027,10799}, {3407,9755}, {5025,10104}, {5039,10753}, {6033,10796}, {6226,10793}, {6227,10792}, {7970,10800}, {9860,10789}, {9861,10790}, {9864,10791}, {10053,10801}, {10069,10802}, {10352,10359}, {11361,11632}, {11364,11710}, {11380,12131}

X(12176) = midpoint of X(98) and X(12110)
X(12176) = reflection of X(4027) in X(3398)

X(12177) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-BROCARD TO 1st ANTI-BROCARD

Barycentrics a^10-4*(b^2+c^2)*a^8+(4*b^4+b^2*c^2+4*c^4)*a^6-(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^4+(b^8+c^8-3*b^2*c^2*(b^4+c^4))*a^2+(b^4-c^4)*(b^2-c^2)*b^2*c^2 : :
X(12177) = X(98)-3*X(5182) = 2*X(182)-3*X(5182) = 3*X(5085)-2*X(12042)

The reciprocal orthologic center of these triangles is X(147).

X(12177) lies on these lines: {2,98}, {3,5026}, {5,5038}, {6,2782}, {30,12151}, {32,5477}, {83,575}, {99,511}, {115,5034}, {194,576}, {381,9830}, {385,9772}, {524,2080}, {597,11632}, {611,3023}, {613,3027}, {671,5476}, {690,9970}, {1351,5969}, {1428,10069}, {1469,10089}, {1503,2456}, {1569,5028}, {1691,3564}, {1992,10788}, {2330,10053}, {2482,8722}, {2770,6233}, {2793,5652}, {3056,10086}, {3398,8550}, {3926,5171}, {5085,12042}, {5286,10358}, {5480,6321}, {5655,10748}, {5999,8289}, {7808,11623}, {8787,11842}, {9863,10131}, {10350,11257}

X(12177) = midpoint of X(i) and X(j) for these {i,j}: {99,10753}, {147,6776}, {6054,8593}
X(12177) = reflection of X(i) in X(j) for these (i,j): (3,5026), (98,182), (671,5476), (1352,114), (6321,5480), (10754,576), (11161,11178), (11632,597), (11646,5)
X(12177) = X(4)-of-6th-anti-Brocard triangle
X(12177) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (98,5182,182), (147,4027,98)
X(12177) = perspector of 6th anti-Brocard triangle and 1st Brocard-reflected triangle

X(12178) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 1st ANTI-BROCARD

Barycentrics a^2*(-a+b+c)-4*(SA*SC-SB^2)*(SA*SB-SC^2)*R/(S*(3*S^2-SW^2)) : :
X(12178) = (R+r)*X(55)-R*X(98)

The reciprocal orthologic center of these triangles is X(5999).

X(12178) lies on these lines: {3,11711}, {35,9860}, {55,98}, {56,7970}, {99,10310}, {100,147}, {114,1376}, {115,11496}, {197,9861}, {542,4421}, {1001,6036}, {2782,11248}, {2784,8715}, {2794,11500}, {3023,11509}, {3295,11710}, {4428,6055}, {5687,9864}, {6033,11499}, {6226,11498}, {6227,11497}, {9862,11491}, {10053,11507}, {10069,11508}, {10267,12042}, {11383,12131}, {11490,12176}

X(12179) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO 1st ANTI-BROCARD

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

The reciprocal orthologic center of these triangles is X(5999).

X(12179) lies on these lines: {55,12180}, {98,5597}, {99,11822}, {114,5599}, {115,8196}, {147,5601}, {542,11207}, {2782,11252}, {3027,11873}, {5598,7970}, {6033,8200}, {6226,8199}, {6227,8198}, {8190,9861}, {8197,9864}, {9862,11843}, {10053,11877}, {10069,11879}, {11366,11710}, {11492,12178}, {11837,12176}

X(12179) = reflection of X(12180) in X(55)

X(12180) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO 1st ANTI-BROCARD

The reciprocal orthologic center of these triangles is X(5999).

X(12180) lies on these lines: {55,12179}, {98,5598}, {99,11823}, {114,5600}, {115,8203}, {147,5602}, {542,11208}, {2782,11253}, {3027,11874}, {5597,7970}, {6033,8207}, {6226,8206}, {6227,8205}, {8187,9860}, {8191,9861}, {8204,9864}, {9862,11844}, {10053,11878}, {10069,11880}, {11367,11710}, {11493,12178}, {11838,12176}

X(12180) = reflection of X(12179) in X(55)

X(12181) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO 1st ANTI-BROCARD

Trilinears (2*cos(A)-cos(B-C))*((11*cos(A)-8*cos(3*A)+cos(5*A)+cos(7*A))*cos(B-C)+(4*cos(2*A)-2*cos(4*A)-cos(6*A)+3)*cos(2*(B-C))-5*cos(A)*cos(3*(B-C))+(-2*cos(2*A)-1/2*cos(4*A)-1)*cos(4*(B-C))-cos(6*A)+2*cos(2*A)+13/2*cos(4*A)-8) : :
X(12181) = 2*X(115)-3*X(11897) = X(9862)-3*X(11845) = 2*X(11710)-3*X(11831)

The reciprocal orthologic center of these triangles is X(5999).

X(12181) lies on these lines: {30,99}, {98,402}, {114,1650}, {115,11897}, {147,4240}, {542,1651}, {2782,11251}, {2794,12113}, {3027,11909}, {6226,11902}, {6227,11901}, {7970,11910}, {9860,11852}, {9861,11853}, {9862,11845}, {9864,11900}, {10053,11912}, {10069,11913}, {11710,11831}, {11832,12131}, {11839,12176}, {11848,12178}

X(12181) = midpoint of X(147) and X(4240)
X(12181) = reflection of X(i) in X(j) for these (i,j): (98,402), (1650,114)

X(12182) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO 1st ANTI-BROCARD

Barycentrics S*(3*S^2-SW^2)*(-a+b+c)*(b-c)^2-4*(R-r)*(SB^2-SA*SC)*(SC^2-SA*SB) : :
X(12182) = (R-2*r)*X(11)-(R-r)*X(98)

The reciprocal orthologic center of these triangles is X(5999).

X(12182) lies on these lines: {11,98}, {99,11826}, {114,1376}, {115,10893}, {147,3434}, {355,6033}, {542,11235}, {2782,10525}, {2794,12114}, {3027,10947}, {6226,10920}, {6227,10919}, {7970,10944}, {9860,10826}, {9861,10829}, {9862,10785}, {9864,10914}, {10053,10523}, {10069,10948}, {10794,12176}, {11373,11710}, {11390,12131}, {11865,12179}, {11866,12180}, {11903,12181}

X(12182) = reflection of X(12178) in X(114)
X(12182) = X(98)-of-inner-Johnson-triangle
X(12182) = X(12189)-of-outer-Johnson-triangle

X(12183) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO 1st ANTI-BROCARD

Barycentrics S*(3*S^2-SW^2)*(b+c)^2*(a-b+c)*(a+b-c)-4*(R+r)*(SB^2-SA*SC)*(SC^2-SA*SB)*(a+b+c) : :
X(12183) = (R+2*r)*X(12)-(R+r)*X(98)

The reciprocal orthologic center of these triangles is X(5999).

X(12183) lies on these lines: {10,2792}, {12,98}, {72,9864}, {99,11827}, {114,958}, {115,10894}, {147,3436}, {355,6033}, {542,11236}, {2782,10526}, {2794,11500}, {3027,10953}, {6226,10922}, {6227,10921}, {6253,10722}, {7970,10950}, {9860,10827}, {9861,10830}, {9862,10786}, {10053,10954}, {10069,10523}, {10795,12176}, {11374,11710}, {11391,12131}, {11867,12179}, {11868,12180}, {11904,12181}

X(12183) = reflection of X(12182) in X(6033)
X(12183) = X(98)-of-outer-Johnson-triangle
X(12183) = X(12190)-of-inner-Johnson-triangle

X(12184) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO 1st ANTI-BROCARD

Barycentrics a^8+2*b*c*a^6-(b^4+c^4+b*c*(2*b^2+b*c+2*c^2))*a^4+(b^4-b^2*c^2+c^4)*(b+c)^2*a^2-(b^4+c^4)*(b^2-c^2)^2 : :
X(12184) = (R+2*r)*X(12)-r*X(98) = (R-r)*X(56)+2*r*X(114)

The reciprocal orthologic center of these triangles is X(5999).

X(12184) lies on the inner-Johnson-Yff circle and these lines: {1,6033}, {4,3027}, {5,10069}, {12,98}, {55,2794}, {56,114}, {65,9864}, {99,7354}, {115,9650}, {147,388}, {148,5229}, {226,2784}, {495,10053}, {498,12042}, {542,611}, {620,5204}, {1317,10768}, {1388,11724}, {1478,2782}, {1569,9651}, {2023,9596}, {3028,11005}, {3029,9553}, {3044,9653}, {3085,9862}, {3585,6321}, {5261,5984}, {5434,6054}, {6226,10924}, {6227,10923}, {6284,10722}, {7970,10944}, {9578,9860}, {9861,10831}, {10797,12176}, {11375,11710}, {11392,12131}, {11501,12178}, {11869,12179}, {11870,12180}, {11905,12181}

X(12184) = reflection of X(10053) in X(495)
X(12184) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,6033,12185), (147,388,3023)

X(12185) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO 1st ANTI-BROCARD

Barycentrics a^8-2*b*c*a^6-(b^4+c^4-b*c*(2*b^2-b*c+2*c^2))*a^4+(b^4-b^2*c^2+c^4)*(b-c)^2*a^2-(b^4+c^4)*(b^2-c^2)^2 : :
X(12185) = (R-2*r)*X(11)+r*X(98) = (R+r)*X(55)-2*r*X(114)

The reciprocal orthologic center of these triangles is X(5999).

X(12185) lies on the outer-Johnson-Yff circle and these lines: {1,6033}, {4,3023}, {5,10053}, {11,98}, {30,10089}, {55,114}, {56,2794}, {99,6284}, {115,9665}, {147,497}, {148,5225}, {496,10069}, {499,12042}, {542,613}, {620,5217}, {1479,2782}, {1569,9664}, {2023,9599}, {2784,12053}, {3029,9554}, {3044,9666}, {3057,9864}, {3058,6054}, {3086,9862}, {3583,6321}, {3845,10054}, {5274,5984}, {5985,11680}, {6226,10926}, {6227,10925}, {7354,10722}, {7970,10950}, {9581,9860}, {9861,10832}, {10798,12176}, {11376,11710}, {11393,12131}, {11502,12178}, {11871,12179}, {11872,12180}, {11906,12181}

X(12185) = reflection of X(10069) in X(496)
X(12185) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,6033,12184), (147,497,3027)

X(12186) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO 1st ANTI-BROCARD

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

The reciprocal orthologic center of these triangles is X(5999).

X(12186) lies on these lines: {98,493}, {99,11828}, {114,8222}, {115,8212}, {147,6462}, {542,12152}, {2782,10669}, {2794,9838}, {3027,11947}, {6033,8220}, {6226,8218}, {6227,8216}, {6461,12187}, {7970,8210}, {8188,9860}, {8194,9861}, {8201,12179}, {8208,12180}, {8214,9864}, {9862,10875}, {10053,11951}, {10069,11953}, {10945,12182}, {10951,12183}, {11377,11710}, {11394,12131}, {11503,12178}, {11840,12176}, {11907,12181}, {11930,12184}, {11932,12185}

X(12187) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO 1st ANTI-BROCARD

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

The reciprocal orthologic center of these triangles is X(5999).

X(12187) lies on these lines: {98,494}, {99,11829}, {114,8223}, {115,8213}, {147,6463}, {542,12153}, {2782,10673}, {2794,9839}, {3027,11948}, {6033,8221}, {6226,8219}, {6227,8217}, {6461,12186}, {7970,8211}, {8189,9860}, {8195,9861}, {8202,12179}, {8209,12180}, {8215,9864}, {9862,10876}, {10053,11952}, {10069,11954}, {10946,12182}, {10952,12183}, {11378,11710}, {11395,12131}, {11504,12178}, {11841,12176}, {11908,12181}, {11931,12184}, {11933,12185}

X(12188) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO 1st ANTI-BROCARD

Barycentrics a^8-3*b^2*c^2*a^4-(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-2*b^2*c^2*(b^2-c^2)^2 : :
X(12188) = 3*X(3)-2*X(99) = 3*X(3)-4*X(12042) = 3*X(98)-X(99) = 3*X(98)-2*X(12042) = 4*X(114)-5*X(1656) = 3*X(114)-4*X(6722) = 4*X(115)-3*X(381) = 2*X(115)-3*X(11632) = X(148)+3*X(11177) = 3*X(381)-2*X(6033) = 15*X(1656)-16*X(6722) = 5*X(1656)-8*X(11623) = X(6033)-3*X(11632) = 2*X(6722)-3*X(11623)

The reciprocal orthologic center of these triangles is X(5999).

X(12188) lies on the 2nd Neuberg circle, Stammler circle and these lines: {2,7711}, {3,76}, {4,5984}, {5,147}, {6,13}, {25,5986}, {30,148}, {114,1656}, {182,7697}, {355,2784}, {382,2794}, {405,5985}, {517,9860}, {538,8178}, {543,3534}, {620,5054}, {621,6770}, {622,6773}, {671,3830}, {690,10620}, {868,3448}, {999,3023}, {1281,4385}, {1569,5013}, {1597,5186}, {1598,12131}, {1657,10991}, {1916,7754}, {1995,5987}, {2023,9605}, {2070,5938}, {2407,9512}, {2482,8556}, {2793,11258}, {2925,2926}, {3027,3295}, {3029,9566}, {3044,9703}, {3095,7798}, {3398,6248}, {3407,9755}, {3526,6036}, {3564,5207}, {3673,7061}, {3934,12054}, {4027,7770}, {5026,12017}, {5050,12177}, {5055,6054}, {5070,7943}, {5073,10723}, {5092,9466}, {5093,10753}, {5790,9864}, {6226,11917}, {6227,11916}, {7470,8782}, {7517,9861}, {7751,9821}, {7790,9996}, {7803,9478}, {7902,10356}, {7913,11178}, {7970,10247}, {7983,8148}, {8591,8703}, {8596,11001}, {9418,10540}, {9654,12184}, {9669,12185}, {10246,11710}, {11849,12178}, {11875,12179}, {11876,12180}, {11911,12181}, {11928,12182}, {11929,12183}, {11949,12186}, {11950,12187}

X(12188) = midpoint of X(i) and X(j) for these {i,j}: {4,5984}, {148,9862}, {8596,11001}
X(12188) = reflection of X(i) in X(j) for these (i,j): (3,98), (99,12042), (114,11623), (147,5), (381,11632), (382,6321), (3830,671), (5073,10723), (5655,11656), (6033,115), (8148,7983), (8591,8703), (8724,6055), (9301,385)
X(12188) = circumcircle-inverse-of-X(12042)
X(12188) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (98,99,12042), (99,12042,3), (115,6033,381), (148,11177,9862), (3023,10069,999), (3027,10053,3295), (6033,11632,115), (6055,8724,5054), (10104,11257,3)

X(12189) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO 1st ANTI-BROCARD

Trilinears (SB^2-SA*SC)*(SC^2-SA*SB)*b*c-4*R^2*S^3*(3*S^2-SW^2)/(2*R*s-S) : :
X(12189) = 2*R*X(1)-(R-r)*X(98)

The reciprocal orthologic center of these triangles is X(5999).

X(12189) lies on these lines: {1,98}, {12,12182}, {99,11248}, {114,5552}, {115,10531}, {119,10768}, {147,10528}, {542,11239}, {2782,10679}, {2794,12115}, {3023,11509}, {3027,10965}, {6033,10942}, {6226,10930}, {6227,10929}, {6256,10722}, {9861,10834}, {9862,10805}, {9864,10915}, {10803,12176}, {10955,12183}, {10956,12184}, {10958,12185}, {11400,12131}, {11881,12179}, {11882,12180}, {11914,12181}, {11955,12186}, {11956,12187}, {12000,12188}

X(12189) = reflection of X(98) in X(10053)
X(12189) = {X(98),X(7970)}-harmonic conjugate of X(12190)

X(12190) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO 1st ANTI-BROCARD

Trilinears (SB^2-SA*SC)*(SC^2-SA*SB)*b*c-4*S^3*R^2*(3*S^2-SW^2)/(2*R*s+S) : :
X(12190) = 2*R*X(1)-(R+r)*X(98)

The reciprocal orthologic center of these triangles is X(5999).

X(12190) lies on these lines: {1,98}, {11,12183}, {99,11249}, {114,10527}, {115,10532}, {147,10529}, {542,11240}, {2782,10680}, {2792,12053}, {2794,12116}, {3027,10966}, {6033,10943}, {6226,10932}, {6227,10931}, {9861,10835}, {9862,10806}, {9864,10916}, {10804,12176}, {10949,12182}, {10957,12184}, {10959,12185}, {11401,12131}, {11510,12178}, {11883,12179}, {11884,12180}, {11915,12181}, {11957,12186}, {11958,12187}, {12001,12188}

X(12190) = reflection of X(98) in X(10069)
X(12190) = {X(98),X(7970)}-harmonic conjugate of X(12189)

X(12191) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO ANTI-MCCAY

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

The reciprocal orthologic center of these triangles is X(9855).

X(12191) lies on these lines: {6,11152}, {30,12176}, {32,671}, {83,2482}, {98,3543}, {148,5304}, {182,12117}, {384,5969}, {542,12110}, {543,4027}, {1003,1916}, {1078,5461}, {1691,9855}, {2080,8859}, {3407,11159}, {3552,9888}, {5032,12177}, {5039,8593}, {5182,7787}, {6034,7833}, {8724,10796}, {9875,10789}, {9876,10790}, {9881,10791}, {9882,10792}, {9883,10793}, {9884,10800}, {10054,10801}, {10070,10802}, {11380,12132}

X(12191) = reflection of X(4027) in X(12150)
X(12191) = orthologic center of these triangles: 5th anti-Brocard to McCay
X(12191) = X(671)-of-5th-anti-Brocard-triangle
X(12191) = {X(7787), X(8591)}-harmonic conjugate of X(5182)

X(12192) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO ANTI-ORTHOCENTROIDAL

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

The reciprocal orthologic center of these triangles is X(12112).

X(12192) lies on these lines: {2,98}, {32,74}, {83,113}, {146,7787}, {541,12150}, {690,12176}, {1078,6699}, {1511,12054}, {2080,12041}, {2777,12110}, {3028,10799}, {3043,3203}, {3398,5663}, {5039,10752}, {7725,10792}, {7728,10796}, {7978,10800}, {9904,10789}, {9919,10790}, {10065,10801}, {10081,10802}, {10620,11842}, {11364,11709}, {11380,12133}

X(12192) = orthologic center of these triangles: 5th anti-Brocard to orthocentroidal
X(12192) = X(74)-of-5th-anti-Brocard-triangle

X(12193) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO ARIES

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

The reciprocal orthologic center of these triangles is X(9833).

X(12193) lies on these lines: {32,68}, {83,1147}, {98,9927}, {155,10796}, {182,12118}, {539,12150}, {1069,10798}, {1078,5449}, {3157,10797}, {5654,10358}, {6193,7787}, {8548,12177}, {9896,10789}, {9908,10790}, {9928,10791}, {9929,10792}, {9930,10793}, {9933,10800}, {10055,10801}, {10071,10802}, {10788,11411}, {11380,12134}

X(12193) = orthologic center of these triangles: 5th anti-Brocard to 2nd Hyacinth
X(12193) = X(68)-of-5th-anti-Brocard-triangle

X(12194) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO ASCELLA

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

The reciprocal orthologic center of these triangles is X(3).

X(12194) lies on these lines: {1,32}, {3,11490}, {8,7787}, {10,82}, {31,239}, {40,182}, {55,11837}, {58,99}, {98,946}, {213,8300}, {291,5299}, {355,10794}, {384,730}, {515,12110}, {517,3398}, {519,12150}, {726,7760}, {731,904}, {944,10788}, {1078,1125}, {1385,2080}, {1386,1691}, {1428,3503}, {1482,11842}, {1582,2300}, {1698,7808}, {1829,11380}, {1837,10798}, {3057,10799}, {3097,7772}, {3576,5171}, {3579,12054}, {3616,7793}, {3624,7815}, {3640,10793}, {3641,10792}, {3734,9902}, {3751,5039}, {3795,8715}, {3972,7976}, {5034,9593}, {5182,9881}, {5252,10797}, {5315,8297}, {5587,10358}, {5657,10359}, {5886,10104}, {7987,8722}, {9798,10790}, {9857,10345}

X(12194) = orthologic center of triangle 5th anti-Brocard to these triangles: Atik, 1st circumperp, 2nd circumperp, inner-Conway, Conway, 2nd Conway, 3rd Conway, 1st Ehrmann, 3rd Euler, 4th Euler, excenters-reflections, excentral, 2nd extouch, hexyl, Honsberger, inner-Hutson, Hutson intouch, outer-Hutson, 2nd Hyacinth, intouch, inverse-in-incircle, 2nd Pamfilos-Zhou, 1st Sharygin, tangential-midarc, 2nd tangential-midarc, Yff central
X(12194) = X(1)-of-5th-anti-Brocard-triangle
X(12194) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,32,11364), (1,10789,32), (8,7787,10791), (32,10800,1), (32,10803,10801), (32,10804,10802), (10789,10800,11364), (10794,10795,10796)

X(12195) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO EXCENTERS-MIDPOINTS

Barycentrics a^5+(b+c)*a^4-(b^2+c^2)*a^3+(b^2+c^2)*(b+c)*a^2-b^2*c^2*a+b^2*c^2*(b+c) : :
X(12195) = (SW^2+S^2)*X(8)-2*(S^2-SW^2)*X(32)

The reciprocal orthologic center of these triangles is X(10).

X(12195) lies on these lines: {1,83}, {6,7976}, {8,32}, {10,1078}, {98,355}, {145,7787}, {182,944}, {517,12110}, {519,12150}, {730,7760}, {760,10350}, {952,3398}, {1482,10796}, {2080,5690}, {2098,10798}, {2099,10797}, {3616,7808}, {3617,7793}, {3632,10789}, {3913,11490}, {5171,5657}, {5603,10358}, {5790,10104}, {7815,9780}, {7967,10359}, {9941,10347}, {9997,10345}, {10573,10802}, {10794,10912}, {10799,10950}, {11380,12135}

X(12195) = orthologic center of these triangles: 5th anti-Brocard to 2nd Schiffler
X(12195) = X(8)-of-5th-anti-Brocard-triangle
X(12195) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,10791,83), (10,11364,1078), (145,7787,10800)

X(12196) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO EXTOUCH

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

The reciprocal orthologic center of these triangles is X(40).

X(12196) lies on these lines: {32,84}, {83,6260}, {98,6245}, {182,1490}, {515,12195}, {971,3398}, {1078,6705}, {1709,10801}, {5658,10359}, {6001,12194}, {6257,10793}, {6258,10792}, {6259,10796}, {7971,10800}, {7992,10789}, {9910,10790}, {10085,10802}, {11364,12114}, {11380,12136}

X(12196) = X(84)-of-5th-anti-Brocard-triangle

X(12197) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO 3rd EXTOUCH

Trilinears a^7-(b^2+c^2)*a^5+(b+c)*(b^2+c^2)*a^4-b*c*(2*b^2+3*b*c+2*c^2)*a^3-(b+c)*(b^4+c^4-b*c*(2*b^2-b*c+2*c^2))*a^2-b^2*c^2*(b+c)^2*a-(b^2-c^2)*(b-c)*b^2*c^2 : :
X(12197) = 2*(SW^2+S^2)*X(10)-(3*S^2-SW^2)*X(98) = 2*(S^2-SW^2)*X(32)-(SW^2+S^2)*X(40) = (S^2+5*SW^2)*X(83)-2*(SW^2+S^2)*X(946)

The reciprocal orthologic center of these triangles is X(4).

X(12197) lies on these lines: {1,182}, {3,11364}, {4,10791}, {10,98}, {32,40}, {46,10802}, {65,10799}, {83,946}, {165,5171}, {172,8924}, {515,12195}, {516,12110}, {517,3398}, {962,7787}, {1078,6684}, {1385,12054}, {1699,10358}, {1836,10797}, {1902,11380}, {2080,3579}, {3097,9737}, {5034,9575}, {5119,10801}, {5603,10359}, {5812,10795}, {6361,10788}, {7808,8227}, {7982,10800}, {7991,10789}, {8669,9751}, {9911,10790}, {10306,11490}

X(12197) = reflection of X(12194) in X(3398)
X(12197) = X(40)-of-5th-anti-Brocard-triangle

X(12198) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO FUHRMANN

Barycentrics 4*(R-2*r)*S*s*a^4-(SW^2+S^2)*(2*SB-a*c)*(2*SC-a*b) : :
X(12198) = 2*(S^2-SW^2)*X(32)-(SW^2+S^2)*X(80)

X(12198) lies on these lines: {11,11364}, {32,80}, {83,214}, {100,10791}, {182,12119}, {952,12194}, {1078,6702}, {2800,12110}, {2802,12195}, {2829,12196}, {5840,12197}, {6224,7787}, {6262,10793}, {6263,10792}, {6265,10796}, {7972,10800}, {9897,10789}, {9912,10790}, {10057,10801}, {10073,10802}, {11380,12137}

X(12198) = X(80)-of-5th-anti-Brocard-triangle

X(12199) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO INNER-GARCIA

Trilinears (a^5-b*a^4-(b^2-b*c+c^2)*a^3+b^3*a^2+b*c*(b-c)^2*a+(b^2-c^2)*b*c^2)*(a^5-c*a^4-(b^2-b*c+c^2)*a^3+c^3*a^2+b*c*(b-c)^2*a-(b^2-c^2)*c*b^2) : :
X(12199) = 2*(SW^2+S^2)*X(11)-(3*S^2-SW^2)*X(98)

The reciprocal orthologic center of these triangles is X(40).

X(12199) lies on these lines: {11,98}, {32,104}, {83,119}, {100,182}, {153,7787}, {515,12198}, {952,3398}, {1078,6713}, {1317,10799}, {1768,10789}, {2783,4027}, {2787,12176}, {2800,12194}, {2802,12197}, {2829,12110}, {5039,10759}, {9913,10790}, {10058,10801}, {10074,10802}, {10698,10800}, {10742,10796}, {11364,11715}, {11380,12138}

X(12199) = X(104)-of-5th-anti-Brocard-triangle

X(12200) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO HUTSON EXTOUCH

Trilinears S^2*(8*R*r+8*R^2+r^2)*(a+b+c)*a^3+(SW^2+S^2)*(S*(4*R+r)-SB*b)*(S*(4*R+r)-SC*c) : :
X(12200) = 2*(S^2-SW^2)*X(32)-(SW^2+S^2)*X(7160)

The reciprocal orthologic center of these triangles is X(40).

X(12200) lies on these lines: {32,7160}, {182,12120}, {7787,9874}, {8000,10800}, {9898,10789}, {10059,10801}, {10075,10802}, {11380,12139}

X(12200) = X(7160)-of-5th-anti-Brocard-triangle

X(12201) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO 1st HYACINTH

Barycentrics 2*a^4*S^2*(SW*S^2-9*SA*SB*SC)-(SW^2+S^2)*(4*SB^2-a^2*c^2)*(4*SC^2-a^2*b^2)*SA : :
X(12201) = 2*(S^2-SW^2)*X(32)-(SW^2+S^2)*X(265)

The reciprocal orthologic center of these triangles is X(6102).

X(12201) lies on these lines: {30,12192}, {32,265}, {83,1511}, {98,10113}, {110,10796}, {125,2080}, {182,12121}, {2771,12198}, {3448,10788}, {5663,12110}, {10088,10797}, {10091,10798}, {11380,12140}

X(12201) = X(265)-of-5th-anti-Brocard-triangle

X(12202) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO MIDHEIGHT

Trilinears 2*S^2*SA*SB*SC*a^3+(SW^2+S^2)*(S^2-2*SA*SC)*(S^2-2*SA*SB)*a : :
X(12202) = 2*(S^2-SW^2)*X(32)-(SW^2+S^2)*X(64)

The reciprocal orthologic center of these triangles is X(4).

X(12202) lies on these lines: {30,12193}, {32,64}, {83,2883}, {98,6247}, {182,1498}, {1078,6696}, {1503,6656}, {2080,3357}, {2777,12201}, {3398,6000}, {5171,10606}, {5656,10359}, {5878,10796}, {6001,12197}, {6225,7787}, {6266,10793}, {6267,10792}, {6759,12054}, {7355,10799}, {7973,10800}, {8567,8722}, {9899,10789}, {9914,10790}, {10060,10801}, {10076,10802}, {11380,11381}

X(12202) = X(64)-of-5th-anti-Brocard-triangle

X(12203) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO ANTICEVIAN-OF-X(4)

Barycentrics a^8+(b^2+c^2)*a^6-(b^4+3*b^2*c^2+c^4)*a^4-(b^2+c^2)*(b^4+c^4)*a^2-(b^2-c^2)^2*b^2*c^2 : :
X(12203) = X(20)+2*X(7765) = 4*X(3398)-3*X(12150) = 2*X(7470)+X(7760) = 2*X(12110)-3*X(12150)

The reciprocal orthologic center of these triangles is X(4).

X(12203) lies on these lines: {2,8721}, {3,76}, {4,83}, {5,7859}, {20,32}, {30,3398}, {39,5999}, {114,7899}, {147,626}, {315,6776}, {316,2456}, {376,5171}, {382,10796}, {385,5188}, {458,1629}, {511,7470}, {515,12195}, {516,12194}, {542,7883}, {550,2080}, {631,7835}, {962,10800}, {1342,10999}, {1343,11000}, {1350,7754}, {1351,7894}, {1352,3096}, {1503,6656}, {1513,7828}, {1657,11842}, {1691,5254}, {1885,11380}, {2794,4027}, {2896,5984}, {3091,7808}, {3098,12251}, {3146,7787}, {3407,7864}, {3522,6392}, {3523,7815}, {3529,10788}, {3564,7768}, {3978,7467}, {4297,11364}, {4299,10802}, {4302,10801}, {5025,10131}, {5038,7745}, {5050,7878}, {5085,7770}, {5092,6248}, {5182,7841}, {5691,10791}, {5840,12199}, {6179,9755}, {6194,6308}, {7354,10799}, {7697,9751}, {7709,9737}, {7748,10723}, {7752,9744}, {7761,9863}, {7762,8550}, {7810,11177}, {7812,11179}, {7830,10991}, {7856,9753}, {7911,12177}, {7924,10333}, {7933,10334}, {8703,11054}, {9166,9774}, {9756,11285}, {9821,12122}, {9873,10347}, {12192,12193}

X(12203) = reflection of X(i) in X(j) for these (i,j): (12110,3398), (12195,12197)
X(12203) = X(20)-of-5th-anti-Brocard-triangle
X(12203) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,98,1078), (3,11257,99), (4,182,83), (4,10359,10358), (182,10358,10359), (3398,12110,12150), (3522,7793,8722), (4027,6655,10350), (5025,10131,10352), (10358,10359,83)

X(12204) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO INNER-NAPOLEON

Barycentrics 2*sqrt(3)*a^2*(a^4+2*(b^2+c^2)*a^2+b^2*c^2)*S+3*a^8-7*(b^2+c^2)*a^6+(2*b^4-9*b^2*c^2+2*c^4)*a^4+(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^2+2*(b^2-c^2)^2*b^2*c^2 : :
X(12204) = (SW^2+S^2)*X(14)-2*(S^2-SW^2)*X(32)

The reciprocal orthologic center of these triangles is X(3).

X(12204) lies on these lines: {14,32}, {61,384}, {83,619}, {98,5469}, {182,5474}, {530,12191}, {531,11300}, {542,12201}, {617,7787}, {1078,6670}, {2080,6774}, {5182,9114}, {5613,10796}, {6269,10793}, {6271,10792}, {6773,10788}, {7974,10800}, {9900,10789}, {9915,10790}, {10061,10801}, {10077,10802}, {11364,11706}, {11380,12141}

X(12204) = X(14)-of-5th-anti-Brocard-triangle

X(12205) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO OUTER-NAPOLEON

Barycentrics -2*sqrt(3)*a^2*(a^4+2*(b^2+c^2)*a^2+b^2*c^2)*S+3*a^8-7*(b^2+c^2)*a^6+(2*b^4-9*b^2*c^2+2*c^4)*a^4+(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^2+2*(b^2-c^2)^2*b^2*c^2 : :
X(12205) = (SW^2+S^2)*X(13)-2*(S^2-SW^2)*X(32)

The reciprocal orthologic center of these triangles is X(3).

X(12205) lies on these lines: {13,32}, {62,384}, {83,618}, {98,5470}, {182,5473}, {530,11299}, {531,12191}, {542,12201}, {616,7787}, {1078,6669}, {2080,6771}, {5182,9116}, {5617,10796}, {6268,10793}, {6270,10792}, {6770,10788}, {7975,10800}, {9901,10789}, {9916,10790}, {10062,10801}, {11364,11705}, {11380,12142}

X(12205) = X(13)-of-5th-anti-Brocard-triangle

X(12206) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(12206) lies on these lines: {2,32}, {6,10131}, {98,6249}, {182,12122}, {194,5039}, {384,732}, {3398,7470}, {3972,6309}, {4027,5007}, {5171,9751}, {5969,7839}, {6274,10793}, {6275,10792}, {6287,9863}, {7745,9478}, {7977,10800}, {9903,10789}, {9918,10790}, {10064,10801}, {10080,10802}, {11380,12144}

X(12206) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (83,1078,6704), (83,6308,2), (2896,7787,83), (10350,12150,7787)
X(12206) = X(83)-of-5th-anti-Brocard-triangle

X(12207) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO 1st ORTHOSYMMEDIAL

Barycentrics a^2*(a^8-(b^2+c^2)*a^6-(b^4-2*c^4)*a^4+(b^2-c^2)*(b^4+c^4)*a^2+(b^2-c^2)*c^2*(2*b^4+b^2*c^2+c^4))*(a^8-(b^2+c^2)*a^6+(2*b^4-c^4)*a^4-(b^2-c^2)*(b^4+c^4)*a^2-(b^2-c^2)*b^2*(b^4+b^2*c^2+2*c^4)) : :
X(12207) = 2*(S^2-SW^2)*X(32)-(SW^2+S^2)*X(1297)

The reciprocal orthologic center of these triangles is X(4).

X(12207) lies on these lines: {32,1297}, {83,132}, {98,127}, {112,182}, {2794,4027}, {2799,12176}, {2806,12199}, {3320,10799}, {9517,12192}, {9530,12150}, {11380,12145}

X(12207) = X(1297)-of-5th-anti-Brocard-triangle

X(12208) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO REFLECTION

Trilinears (36*cos(2*A)-2*cos(6*A)-33)*cos(B-C)-2*(5*cos(A)-5*cos(3*A)+cos(5*A))*cos(2*(B-C))-cos(3*(B-C))+cos(7*A)-16*cos(A)+26*cos(3*A)-9*cos(5*A) : :
X(12208) = 2*(S^2-SW^2)*X(32)-(SW^2+S^2)*X(54)

The reciprocal orthologic center of these triangles is X(4).

X(12208) lies on these lines: {32,54}, {83,1209}, {98,3574}, {182,7691}, {195,11842}, {539,12150}, {1078,6689}, {1154,3398}, {2080,10610}, {2888,7787}, {6276,10793}, {6277,10792}, {6288,10796}, {7979,10800}, {9905,10789}, {9920,10790}, {10066,10801}, {10082,10802}, {10628,12192}, {11380,11576}

X(12208) = X(54)-of-5th-anti-Brocard-triangle

X(12209) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO 1st SCHIFFLER

Barycentrics 2*s*S^2*(3*R+2*r)^2*a^4+2*(SW^2+S^2)*(4*(s-b)*SB+(R+2*r)*S)*(4*(s-c)*SC+(R+2*r)*S)*(s-a) : :
X(12209) = 2*(S^2-SW^2)*X(32)-(SW^2+S^2)*X(10266)

The reciprocal orthologic center of these triangles is X(79).

X(12209) lies on these lines: {32,10266}, {11380,12146}

X(12209) = X(10266)-of-5th-anti-Brocard-triangle

X(12210) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO INNER-VECTEN

Barycentrics 2*a^2*(a^4+2*(b^2+c^2)*a^2+b^2*c^2)*S+2*a^8-4*(b^2+c^2)*a^6+((b^2-c^2)^2-4*b^2*c^2)*a^4+(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2+(b^2-c^2)^2*b^2*c^2 : :
X(12210) = 2*(S^2-SW^2)*X(32)-(SW^2+S^2)*X(486)

The reciprocal orthologic center of these triangles is X(3).

X(12210) lies on these lines: {32,486}, {83,642}, {98,6251}, {182,12123}, {487,7787}, {1078,6119}, {3564,12193}, {6280,10793}, {6281,10792}, {6290,10796}, {7980,10800}, {9906,10789}, {9921,10790}, {10067,10801}, {10083,10802}, {11380,12147}

X(12210) = X(486)-of-5th-anti-Brocard-triangle

X(12211) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO OUTER-VECTEN

Barycentrics -2*a^2*(a^4+2*(b^2+c^2)*a^2+b^2*c^2)*S+2*a^8-4*(b^2+c^2)*a^6+((b^2-c^2)^2-4*b^2*c^2)*a^4+(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2+(b^2-c^2)^2*b^2*c^2 : :
X(12211) = 2*(S^2-SW^2)*X(32)-(SW^2+S^2)*X(485)

The reciprocal orthologic center of these triangles is X(3).

X(12211) lies on these lines: {32,485}, {83,641}, {98,6250}, {182,12124}, {488,7787}, {1078,6118}, {3564,12193}, {6278,10793}, {6279,10792}, {6289,10796}, {7981,10800}, {9907,10789}, {9922,10790}, {10068,10801}, {10084,10802}, {11380,12148}

X(12211) = X(485)-of-5th-anti-Brocard-triangle

X(12212) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO 1st EHRMANN

Trilinears    (a^4+2*(b^2+c^2)*a^2+b^2*c^2)*a : :
Trilinears    2 sin A + sin(A - 2ω) : :
Trilinears    a + R sin(A - 2ω) : :
X(12212) = 2*X(182)-3*X(3398)

The reciprocal orthologic center of these triangles is X(3).

X(12212) lies on these lines: {3,6}, {31,7077}, {69,7787}, {83,141}, {98,5306}, {110,251}, {159,10790}, {384,732}, {518,12194}, {524,6661}, {542,12201}, {611,10801}, {613,10802}, {698,7760}, {729,12074}, {755,11636}, {1078,3589}, {1184,3066}, {1352,10796}, {1353,12177}, {1386,11364}, {1469,5332}, {1501,11003}, {1503,12110}, {1613,5651}, {1843,11380}, {1992,12151}, {2211,10312}, {2781,12192}, {3056,7296}, {3124,5354}, {3242,10800}, {3329,10007}, {3407,7766}, {3416,10791}, {3564,12193}, {3618,7793}, {3751,10789}, {3763,7776}, {3972,4048}, {3981,5359}, {4027,5969}, {5031,7785}, {5103,7828}, {5182,8584}, {5846,12195}, {6179,8177}, {6308,8362}, {6636,11205}, {6776,10788}, {7837,10334}, {7893,10345}, {9225,9463}, {9830,12191}, {10358,10516}, {10359,10519}

X(12212) = reflection of X(i) in X(j) for these (i,j): (6,5007), (7768,141)
X(12212) = X(6)-of-5th-anti-Brocard-triangle
X(12212) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(9821)
X(12212) = inverse-in-circle-{{X(1687),X(1688),PU(1),PU(2)}} of X(5092)
X(12212) = X(23)-of-X(6)PU(1)
X(12212) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,32,1691), (6,1691,5038), (6,2076,39), (6,5017,3094), (32,5007,3398), (32,5039,6), (251,3051,1915), (371,372,9821), (1351,11842,182), (1687,1688,5092), (1915,3051,2056), (3094,5017,5104), (10792,10793,32)

X(12213) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-BROCARD TO INNER-NAPOLEON

Barycentrics 2*sqrt(3)*(a^8-(b^2+c^2)*a^6-3*b^2*c^2*a^4+(c^6+b^6)*a^2+b^2*c^2*(b^4+c^4))*S+3*a^10-8*(b^2+c^2)*a^8+(11*b^4+5*b^2*c^2+11*c^4)*a^6-7*(b^4-c^4)*(b^2-c^2)*a^4+(b^8+c^8-2*b^2*c^2*(5*b^4-2*b^2*c^2+5*c^4))*a^2+(b^4-c^4)*(b^2-c^2)*b^2*c^2 : :
X(12213) = 4*S^2*X(182)-(SW^2+S^2)*X(3642)

The reciprocal orthologic center of these triangles is X(5617).

X(12213) lies on these lines: {30,12214}, {182,3642}, {298,619}, {530,12151}, {531,5182}, {533,1691}, {623,6777}, {4027,5978}, {6109,10352}, {9988,10131}

X(12213) = X(13)-of-6th-anti-Brocard-triangle

X(12214) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-BROCARD TO OUTER-NAPOLEON

Barycentrics -2*sqrt(3)*(a^8-(b^2+c^2)*a^6-3*b^2*c^2*a^4+(c^6+b^6)*a^2+b^2*c^2*(b^4+c^4))*S+3*a^10-8*(b^2+c^2)*a^8+(11*b^4+5*b^2*c^2+11*c^4)*a^6-7*(b^4-c^4)*(b^2-c^2)*a^4+(b^8+c^8-2*b^2*c^2*(5*b^4-2*b^2*c^2+5*c^4))*a^2+(b^4-c^4)*(b^2-c^2)*b^2*c^2 : :
X(12214) = 4*S^2*X(182)-(SW^2+S^2)*X(3643)

The reciprocal orthologic center of these triangles is X(5613).

X(12214) lies on these lines: {30,12213}, {182,3643}, {299,618}, {530,5182}, {531,12151}, {532,1691}, {624,6778}, {4027,5979}, {6108,10352}, {9989,10131}

X(12214) = X(14)-of-6th-anti-Brocard-triangle

X(12215) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-BROCARD TO 1st NEUBERG

Barycentrics (-a^2+b^2+c^2)*(a^4-b^2*c^2) : :
X(12215) = X(385)-4*X(5026) = 2*X(1692)-3*X(5182) = 4*X(5031)-5*X(7925) = X(5111)-3*X(12151)

The reciprocal orthologic center of these triangles is X(98).

X(12215) lies on these lines: {3,69}, {6,194}, {23,10330}, {32,6309}, {63,7019}, {76,182}, {99,511}, {110,2868}, {141,5116}, {147,325}, {183,5085}, {184,305}, {193,3552}, {315,7470}, {323,4576}, {350,1428}, {385,732}, {419,3978}, {450,6331}, {524,2076}, {525,3049}, {538,1692}, {542,5152}, {736,2458}, {1003,1992}, {1078,5092}, {1352,7763}, {1570,10754}, {1909,2330}, {2024,10352}, {2396,5967}, {2456,2782}, {3094,7783}, {3098,7782}, {3292,4563}, {3329,10334}, {3589,7797}, {3618,5286}, {3619,11285}, {3734,5034}, {3763,7945}, {3818,7752}, {3972,5039}, {5012,8024}, {5028,7781}, {5031,7925}, {5033,7751}, {5052,7816}, {5058,6318}, {5062,6314}, {5111,5969}, {5162,7813}, {5651,11059}, {6230,8294}, {6231,8293}, {7757,10000}, {7779,10997}, {7809,11645}, {9464,11003}, {9983,10131}, {10007,12055}

X(12215) = reflection of X(i) in X(j) for these (i,j): (69,6393), (385,1691), (1691,5026), (5207,325), (6393,6390), (10754,1570), (11646,5031)
X(12215) = X(1916)-of-6th-anti-Brocard-triangle
X(12215) = crosspoint of X(147) and X(194) wrt both the excentral and anticomplementary triangles
X(12215) = crossdifference of every pair of points of line X(882)X(1843)
X(12215) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,4048,384), (141,5116,7824), (193,3552,5017), (325,5989,5999), (3926,6776,69), (4027,9865,385)

X(12216) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-BROCARD TO 2nd NEUBERG

Barycentrics a^10+(b^2+c^2)*a^8-2*(b^4+3*b^2*c^2+c^4)*a^6+((b^2-c^2)^2-4*b^2*c^2)*(b^2+c^2)*a^4+(b^8+c^8+b^2*c^2*(3*b^4+b^2*c^2+3*c^4))*a^2+(b^4+b^2*c^2+c^4)*(b^2+c^2)*b^2*c^2 : :
X(12216) = 4*S^2*X(182)-(SW^2+S^2)*X(2896)

The reciprocal orthologic center of these triangles is X(147).

X(12216) lies on these lines: {6,76}, {69,8150}, {182,2896}, {511,8290}, {754,2458}, {4027,9866}, {5039,10334}, {7779,10352}, {7905,12212}, {9990,10131}, {10722,12177}

X(12216) = X(11606)-of-6th-anti-Brocard-triangle

X(12217) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-BROCARD TO INNER-VECTEN

Barycentrics (2*a^8-2*(b^2+c^2)*a^6-6*b^2*c^2*a^4+2*(c^6+b^6)*a^2+2*b^2*c^2*(b^4+c^4))*S+2*a^10-6*(b^2+c^2)*a^8+4*(2*b^4+b^2*c^2+2*c^4)*a^6-5*(b^4-c^4)*(b^2-c^2)*a^4+(b^4+c^4)*(b^2+3*b*c+c^2)*(b^2-3*b*c+c^2)*a^2+(b^4-c^4)*(b^2-c^2)*b^2*c^2 : :
X(12217) = 4*S^2*X(182)-(SW^2+S^2)*X(6229)

The reciprocal orthologic center of these triangles is X(6231).

X(12217) lies on these lines: {182,6229}, {642,7769}, {2462,5182}, {4027,9867}, {9991,10131}

X(12218) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-BROCARD TO OUTER-VECTEN

Barycentrics -(2*a^8-2*(b^2+c^2)*a^6-6*b^2*c^2*a^4+2*(c^6+b^6)*a^2+2*b^2*c^2*(b^4+c^4))*S+2*a^10-6*(b^2+c^2)*a^8+4*(2*b^4+b^2*c^2+2*c^4)*a^6-5*(b^4-c^4)*(b^2-c^2)*a^4+(b^4+c^4)*(b^2+3*b*c+c^2)*(b^2-3*b*c+c^2)*a^2+(b^4-c^4)*(b^2-c^2)*b^2*c^2 : :
X(12218) = 4*S^2*X(182)-(SW^2+S^2)*X(6228)

The reciprocal orthologic center of these triangles is X(6230).

X(12218) lies on these lines: {182,6228}, {641,7769}, {2461,5182}, {4027,9868}, {9992,10131}

X(12219) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP TO ANTI-ORTHOCENTROIDAL

Trilinears (5*cos(2*A)+cos(4*A)+7/2)*cos(B-C)-(3*cos(A)+cos(3*A))*cos(2*(B-C))-3*cos(A)-3/2*cos(3*A) : :
X(12219) = 2*X(113)-3*X(11459) = 4*X(1112)-5*X(3091) = 3*X(3060)-4*X(7687) = 3*X(3543)-4*X(12133) = 3*X(5891)-2*X(11557) = X(7731)-3*X(11459)

The reciprocal orthologic center of these triangles is X(3581).

X(12219) lies on these lines: {2,1986}, {3,3043}, {4,7723}, {20,5663}, {22,399}, {69,146}, {74,323}, {110,5562}, {113,7731}, {125,5889}, {155,3047}, {185,9706}, {265,1154}, {511,10296}, {858,10264}, {1112,3091}, {1216,11562}, {1511,10298}, {2777,12111}, {3060,7687}, {3100,7727}, {3101,7724}, {3448,11411}, {3543,12133}, {5876,7728}, {5890,6699}, {5891,11557}, {5972,11444}, {6243,10113}, {9976,11416}, {10117,11441}, {10620,11413}, {10657,11420}, {10658,11421}, {10721,12162}

X(12219) = anticomplement of X(1986)
X(12219) = orthologic center of these triangles: 1st anti-circumperp to orthocentroidal
X(12219) = X(80)-of-1st-anti-circumperp-triangle if ABC is acute
X(12219) = reflection of X(i) in X(j) for these (i,j): (4,7723), (110,5562), (5889,125), (6243,10113), (7722,3), (7728,5876), (7731,113), (10721,12162), (11562,1216), (12121,6101)
X(12219) = {X(7731), X(11459)}-harmonic conjugate of X(113)

X(12220) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP TO 1st EHRMANN

Trilinears a*((b^2+c^2)*a^4-b^2*c^2*a^2-b^6-c^6) : :
X(12220) = 3*X(2)-4*X(11574) = 4*X(6)-3*X(3060) = 2*X(69)-3*X(2979) = 8*X(141)-9*X(7998) = 4*X(141)-3*X(11188) = 4*X(1352)-5*X(11444) = 3*X(2979)-4*X(3313) = 9*X(7998)-4*X(9973) = 3*X(7998)-2*X(11188) = 2*X(9973)-3*X(11188)

The reciprocal orthologic center of these triangles is X(576).

X(12220) lies on these lines: {2,1843}, {3,6403}, {4,9967}, {6,22}, {20,185}, {23,1974}, {51,10565}, {66,69}, {74,3565}, {110,159}, {141,858}, {160,3001}, {182,7488}, {394,9924}, {542,12219}, {805,2697}, {1205,3448}, {1350,7691}, {1351,7592}, {1352,11444}, {1353,6243}, {1469,4296}, {1503,12111}, {1995,7716}, {2071,3098}, {2876,4329}, {3056,3100}, {3101,3779}, {3153,3818}, {3564,11412}, {3567,5050}, {3589,9971}, {3618,5640}, {3620,3917}, {3867,5133}, {4260,7520}, {5092,10298}, {5093,10263}, {5562,5921}, {6101,11898}, {6563,9009}, {6636,8541}, {7401,11387}, {8538,12088}, {8681,12058}, {10625,11411}, {11470,12087}

X(12220) = reflection of X(i) in X(j) for these (i,j): (4,9967), (69,3313), (193,6467), (1843,11574), (3448,1205), (5889,6776), (5921,5562), (6243,1353), (6403,3), (9973,141), (11898,6101)
X(12220) = anticomplement of X(1843)
X(12220) = X(7)-of-1st-anti-circumperp-triangle if ABC is acute
X(12220) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (69,3313,2979), (141,9973,11188), (1843,11574,2), (3618,9969,5640), (12223,12224,20)

X(12221) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP TO LUCAS ANTIPODAL

Barycentrics 8*((b^2+c^2)*a^2+b^4-4*b^2*c^2+c^4)*S-13*a^6+17*(b^2+c^2)*a^4-(11*b^4+2*b^2*c^2+11*c^4)*a^2+7*(b^4-c^4)*(b^2-c^2) : :
X(12221) = 3*X(2)-4*X(486) = 9*X(2)-8*X(642) = 3*X(486)-2*X(642) = X(3146)+4*X(6280) = 5*X(3522)-4*X(12123) = 7*X(3832)-8*X(6251) = 7*X(3832)-4*X(6281)

The reciprocal orthologic center of these triangles is X(3).

X(12221) lies on these lines: {2,371}, {3,12169}, {4,193}, {8,9906}, {20,6463}, {23,9921}, {52,6239}, {69,3071}, {385,7000}, {488,6561}, {489,3053}, {490,5860}, {492,6337}, {1132,1271}, {1270,11294}, {1992,3070}, {1993,3092}, {3091,6290}, {3146,5870}, {3522,12123}, {3620,7388}, {3623,7980}, {3832,6202}, {5032,7581}, {6289,6462}, {6406,8681}, {6423,7586}, {6995,8948}, {7374,7774}, {7389,7582}, {7584,11291}

X(12221) = reflection of X(i) in X(j) for these (i,j): (8,9906), (487,486), (6281,6251), (12222,2996)
X(12221) = anticomplement of X(487)
X(12221) = {X(4),X(193)}-harmonic conjugate of X(12222)
X(12221) = orthic-to-1st-anti-circumperp similarity image of X(487)

X(12222) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP TO LUCAS(-1) ANTIPODAL

Barycentrics -8*((b^2+c^2)*a^2+b^4-4*b^2*c^2+c^4)*S-13*a^6+17*(b^2+c^2)*a^4-(11*b^4+2*b^2*c^2+11*c^4)*a^2+7*(b^4-c^4)*(b^2-c^2) : :
X(12222) = 3*X(2)-4*X(485) = 9*X(2)-8*X(641) = 5*X(3091)-4*X(6289) = X(3146)+4*X(6279) = 5*X(3522)-4*X(12124) = 5*X(3623)-4*X(7981) = 7*X(3832)-4*X(6278)

The reciprocal orthologic center of these triangles is X(3).

X(12222) lies on these lines: {2,372}, {3,12170}, {4,193}, {8,9907}, {20,6462}, {23,9922}, {52,6400}, {69,3070}, {385,7374}, {487,6560}, {489,5861}, {490,3053}, {491,6337}, {1131,1270}, {1271,11293}, {1992,3071}, {1993,3093}, {3091,6289}, {3146,5871}, {3522,12124}, {3620,7389}, {3623,7981}, {3832,6201}, {5032,7582}, {6290,6463}, {6291,8681}, {6424,7585}, {6995,8946}, {7000,7774}, {7388,7581}, {7583,11292}

X(12222) = reflection of X(i) in X(j) for these (i,j): (8,9907), (488,485), (6278,6250), (12221,2996)
X(12222) = anticomplement of X(488)
X(12222) = {X(4),X(193)}-harmonic conjugate of X(12221)
X(12222) = orthic-to-1st-anti-circumperp similarity image of X(488)

X(12223) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP TO LUCAS CENTRAL

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

The reciprocal orthologic center of these triangles is X(3).

X(12223) lies on these lines: {2,6291}, {3,6239}, {20,185}, {22,1151}, {489,2979}, {2071,7690}, {3060,6459}, {3100,6283}, {3101,6252}, {3565,9733}, {4296,7362}, {9974,11416}, {10667,11420}, {10668,11421}

X(12223) = reflection of X(6239) in X(3)
X(12223) = anticomplement of X(6291)
X(12223) = {X(20),X(12220)}-harmonic conjugate of X(12224)
X(12223) = X(176)-of-1st-anti-circumperp-triangle if ABC is acute
X(12223) = orthic-to-1st-anti-circumperp similarity image of X(6291)

X(12224) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP TO LUCAS(-1) CENTRAL

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

The reciprocal orthologic center of these triangles is X(3).

X(12224) lies on these lines: {2,6406}, {3,6400}, {20,185}, {22,1152}, {490,2979}, {2071,7692}, {3060,6460}, {3100,6405}, {3101,6404}, {3565,9732}, {4296,7353}, {9975,11416}, {10671,11420}, {10672,11421}

X(12224) = reflection of X(6400) in X(3)
X(12224) = anticomplement of X(6406)
X(12224) = {X(20),X(12220)}-harmonic conjugate of X(12223)
X(12224) = X(175)-of-1st-anti-circumperp-triangle if ABC is acute
X(12224) = orthic-to-1st-anti-circumperp similarity image of X(6406)

X(12225) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP TO MACBEATH

Barycentrics 2*a^10-3*(b^2+c^2)*a^8-2*(b^4-b^2*c^2+c^4)*a^6+4*(b^6+c^6)*a^4-2*(b^2-c^2)^2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(12225) = 3*X(4)-2*X(7553) = 2*X(52)-3*X(12022) = 3*X(381)-2*X(11819) = 6*X(428)-7*X(3832) = 4*X(546)-3*X(7540) = 5*X(3522)-6*X(7667) = 9*X(5640)-8*X(11745) = 3*X(11459)-2*X(12134)

The reciprocal orthologic center of these triangles is X(4). As a point of the Euler line, X(12225) has Shinagawa coefficients: (E+2*F, -2*E-6*F).

X(12225) lies on these lines: {2,3}, {52,12022}, {343,6145}, {1141,8800}, {1503,12111}, {1568,10282}, {2697,11635}, {3070,11417}, {3071,11418}, {3100,6284}, {3101,6253}, {3164,7823}, {4296,7354}, {5254,10313}, {5318,11420}, {5321,11421}, {5523,10316}, {5596,6225}, {5640,11745}, {5654,9707}, {5889,6146}, {6247,11440}, {6696,11454}, {8550,11416}, {9820,11464}, {9833,11441}, {11064,11449}, {11457,12163}, {11459,12134}

X(12225) = reflection of X(i) in X(j) for these (i,j): (3146,1885), (5889,6146), (6240,3)
X(12225) = anticomplement of X(3575)
X(12225) = X(65)-of-1st-anti-circumperp-triangle if ABC is acute
X(12225) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,7507,2), (4,20,22), (4,376,3547), (4,7404,7566), (4,7503,5133), (20,1370,11413), (20,2071,550), (20,3153,7488), (20,7396,3522), (22,858,7495), (2071,7574,858), (3153,7488,5), (3627,7403,4), (5094,7396,858), (6816,7487,1995), (7404,7566,5133), (7503,7566,7404)

X(12226) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP TO REFLECTION

Trilinears (cos(2*A)-cos(4*A)+1/2)*cos(B-C)+(cos(A)+cos(3*A))*cos(2*(B-C))+cos(A)-cos(3*A)/2 : :
X(12226) = 2*X(52)-3*X(54) = 5*X(52)-6*X(10115) = 5*X(54)-4*X(10115) = 6*X(1209)-7*X(7999) = 5*X(3091)-4*X(11576) = 3*X(6288)-4*X(11591) = 4*X(6689)-3*X(7730)

The reciprocal orthologic center of these triangles is X(6243).

X(12226) lies on these lines: {2,6152}, {3,6242}, {20,1154}, {22,195}, {52,54}, {69,1225}, {74,10625}, {539,11412}, {1209,7999}, {1493,6243}, {2071,7691}, {2914,12088}, {3091,11576}, {3100,6286}, {3101,6255}, {3153,6288}, {3519,6101}, {4296,7356}, {5889,10619}, {5965,12220}, {6689,7730}, {9977,11416}, {10298,10610}, {10677,11420}, {10678,11421}, {12219,12225}

X(12226) = reflection of X(i) in X(j) for these (i,j): (3519,6101), (5889,10619), (6242,3), (6243,1493)
X(12226) = anticomplement of X(6152)
X(12226) = X(79)-of-1st-anti-circumperp-triangle if ABC is acute

X(12227) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-CONWAY TO ANTI-ORTHOCENTROIDAL

Trilinears (10*cos(2*A)+8)*cos(B-C)+(-4*cos(A)-2*cos(3*A))*cos(2*(B-C))-2*cos(3*A)+cos(5*A)-11*cos(A) : :
X(12227) = 3*X(11402)+X(12165)

The reciprocal orthologic center of these triangles is X(3581).

X(12227) lies on these lines: {6,13}, {54,74}, {110,389}, {125,7592}, {155,5972}, {184,1986}, {195,12121}, {569,7723}, {578,5663}, {1112,6759}, {1147,1511}, {1181,2777}, {1994,10733}, {2904,11456}, {3043,5890}, {5012,12219}, {5609,11746}, {6467,9934}, {7724,11428}, {7727,11429}, {9306,9826}, {10620,11425}, {11402,12165}

X(12227) = orthologic center of these triangles: anti-Conway to orthocentroidal
X(12227) = {X(6), X(399)}-harmonic conjugate of X(7687)
X(12227) = X(80)-of-anti-Conway-triangle if ABC is acute

X(12228) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-CONWAY TO 1st HYACINTH

Trilinears (4*cos(2*A)+2*cos(4*A)+4)*cos(B-C)-2*cos(A)*cos(2*(B-C))-3*cos(A)-2*cos(3*A)-cos(5*A) : :
X(12228) = 3*X(11402)+X(12168)

The reciprocal orthologic center of these triangles is X(10112).

X(12228) lies on these lines: {2,3043}, {3,1986}, {5,49}, {6,1511}, {26,1112}, {74,5012}, {113,184}, {125,569}, {146,11003}, {182,6699}, {389,11536}, {399,9818}, {1147,5972}, {1176,10752}, {1181,5663}, {1539,9934}, {2914,12219}, {5622,10264}, {7503,7723}, {11402,12168}, {11818,12140}

X(12228) = X(104)-of-anti-Conway-triangle if ABC is acute
X(12228) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (49,567,12022), (265,11597,110), (567,11597,265)

X(12229) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-CONWAY TO LUCAS ANTIPODAL

Trilinears a*(-a^2+b^2+c^2)*(8*(2*a^6-2*(b^2+c^2)*a^4+2*(b^2-c^2)^2*a^2+b^2*c^2*(b^2+c^2))*S+a^8-(b^2+c^2)*a^6-(b^4+20*b^2*c^2+c^4)*a^4+(b^2+c^2)^3*a^2-6*(b^2-c^2)^2*b^2*c^2) : :
X(12229) = 3*X(11402)+X(12169)

The reciprocal orthologic center of these triangles is X(3).

X(12229) lies on these lines: {3,8908}, {182,486}, {184,487}, {642,9306}, {5012,12221}, {6290,6759}, {11402,12169}

X(12229) = orthic-to-anti-Conway similarity image of X(487)

X(12230) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-CONWAY TO LUCAS(-1) ANTIPODAL

Trilinears a*(-a^2+b^2+c^2)*(-8*(2*a^6-2*(b^2+c^2)*a^4+2*(b^2-c^2)^2*a^2+b^2*c^2*(b^2+c^2))*S+a^8-(b^2+c^2)*a^6-(b^4+20*b^2*c^2+c^4)*a^4+(b^2+c^2)^3*a^2-6*(b^2-c^2)^2*b^2*c^2) : :
X(12230) = 3*X(11402)+X(12170)

The reciprocal orthologic center of these triangles is X(3).

X(12230) lies on these lines: {182,485}, {184,488}, {641,9306}, {3564,12229}, {5012,12222}, {6289,6759}, {11402,12170}

X(12230) = orthic-to-anti-Conway similarity image of X(488)

X(12231) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-CONWAY TO LUCAS CENTRAL

Trilinears (a^8-3*(b^2+c^2)*a^6+(3*b^4+4*b^2*c^2+3*c^4)*a^4-(b^4-c^4)*(b^2-c^2)*a^2-2*(b^2-c^2)^2*b^2*c^2-S*(a^6-(b^2+c^2)*a^4-(b^2+c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)))*a : :
X(12231) = 3*X(11402)+X(12171)

The reciprocal orthologic center of these triangles is X(3).

X(12231) = {3,6}, {54,6239}, {184,6291}, {485,8909}, {5012,12223}, {6252,11428}, {6283,11429}, {9306,9823}, {11402,12171}

X(12231) = {X(6),X(578)}-harmonic conjugate of X(12232)
X(12231) = X(176)-of-anti-Conway-triangle if ABC is acute
X(12231) = orthic-to-anti-Conway similarity image of X(6291)

X(12232) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-CONWAY TO LUCAS(-1) CENTRAL

Trilinears (a^8-3*(b^2+c^2)*a^6+(3*b^4+4*b^2*c^2+3*c^4)*a^4-(b^4-c^4)*(b^2-c^2)*a^2-2*(b^2-c^2)^2*b^2*c^2+S*(a^6-(b^2+c^2)*a^4-(b^2+c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)))*a : :
X(12232) = 3*X(11402)+X(12172)

The reciprocal orthologic center of these triangles is X(3).

X(12232) lies on these lines: {3,6}, {54,6400}, {184,6406}, {5012,12224}, {6404,11428}, {6405,11429}, {9306,9824}, {11402,12172}

X(12232) = {X(6),X(578)}-harmonic conjugate of X(12231)
X(12232) = X(175)-of-anti-Conway-triangle if ABC is acute
X(12232) = orthic-to-anti-Conway similarity image of X(6406)

X(12233) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-CONWAY TO MACBEATH

Barycentrics 3*(b^2+c^2)*a^8-8*(b^4+c^4)*a^6+6*(b^4-c^4)*(b^2-c^2)*a^4-(b^4-c^4)*(b^2-c^2)^3 : :
X(12233) = X(20)-3*X(3796) = 3*X(5064)+X(12174) = 3*X(11402)+X(12173)

The reciprocal orthologic center of these triangles is X(4).

X(12233) lies on these lines: {2,9786}, {4,6}, {5,389}, {12,11436}, {20,3796}, {24,10192}, {25,11745}, {30,578}, {51,235}, {54,6240}, {64,3088}, {113,11746}, {115,8799}, {140,11438}, {141,5562}, {154,7487}, {184,3575}, {185,427}, {343,5889}, {378,5894}, {381,11432}, {382,11426}, {394,6815}, {403,3567}, {511,6823}, {524,12160}, {550,10610}, {568,10024}, {590,6810}, {615,6809}, {631,1192}, {858,10574}, {946,5173}, {1147,7706}, {1350,7400}, {1352,12164}, {1353,10112}, {1368,9729}, {1594,5890}, {1595,6000}, {1596,10110}, {1597,5878}, {1614,7576}, {1620,3524}, {1885,11424}, {1899,7507}, {1907,11381}, {3091,11433}, {3541,6696}, {3589,7395}, {3855,11431}, {3858,7687}, {4846,12085}, {5012,12225}, {5020,9815}, {5064,12174}, {5133,12111}, {5654,6642}, {5891,7405}, {6253,11428}, {6284,11429}, {6644,9820}, {6756,6759}, {6816,10601}, {6831,10478}, {7403,12162}, {7495,7691}, {7544,11441}, {9306,9825}, {9730,11585}, {11402,12173}, {11412,11660}

X(12233) = midpoint of X(4) and X(1181)
X(12233) = reflection of X(3867) in X(5480)
X(12233) = X(958)-of-orthic-triangle if ABC is acute
X(12233) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1199,12022), (4,7592,6146), (20,11427,11425), (185,427,6247), (185,3574,427), (2883,5480,4), (3541,10605,6696), (5448,5462,5), (5562,7399,141), (6146,7592,8550)

X(12234) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-CONWAY TO REFLECTION

Trilinears (2*cos(2*A)-4)*cos(B-C)+2*cos(3*A)*cos(2*(B-C))-cos(A)+2*cos(3*A)-cos(5*A) : :
X(12234) = 3*X(11402)+X(12175)

The reciprocal orthologic center of these triangles is X(6243).

X(12234) lies on these lines: {5,11536}, {6,17}, {54,186}, {184,6152}, {539,12161}, {578,1154}, {973,10274}, {1147,1493}, {1181,12173}, {1843,11808}, {1994,5562}, {2904,3574}, {5012,12226}, {6255,11428}, {6286,11429}, {6759,11576}, {7592,10619}, {7691,11430}, {8681,9972}, {9306,9827}, {10610,11438}, {11402,12175}, {11702,11746}, {12227,12233}

X(12234) = X(79)-of-anti-Conway-triangle if ABC is acute

X(12235) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CONWAY TO ARIES

Trilinears cos(A)*((cos(A)+cos(3*A))*cos(B-C)+(-cos(2*A)+1)*cos(2*(B-C))) : :
X(12235) = 3*X(51)-X(155) = 3*X(3060)+X(11411) = 5*X(3567)-X(6193) = 3*X(5892)-2*X(12038) = 3*X(9730)-X(12118)

The reciprocal orthologic center of these triangles is X(7387).

X(12235) lies on these lines: {4,52}, {6,1147}, {26,2393}, {51,155}, {143,3564}, {343,1216}, {389,10112}, {539,973}, {569,5892}, {578,9932}, {974,6146}, {1209,10170}, {1843,9908}, {3003,3133}, {3546,5447}, {3567,6193}, {5907,7687}, {5943,9820}, {6217,9930}, {6218,9929}, {7689,10606}, {9730,12118}, {9777,12166}, {9931,11436}, {9938,11438}, {10297,11692}

X(12235) = midpoint of X(52) and X(68)
X(12235) = reflection of X(i) in X(j) for these (i,j): (1147,5462), (1216,5449)
X(12235) = orthologic center of these triangles: 2nd anti-Conway to 2nd Hyacinth
X(12235) = {X(6), X(9937)}-harmonic conjugate of X(1147)
X(12235) = X(84)-of-2nd-anti-Conway-triangle if ABC is acute

X(12236) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CONWAY TO 1st HYACINTH

Trilinears (3*cos(2*A)+cos(4*A)+1)*cos(B-C)-cos(3*A)*cos(2*(B-C))-2*cos(A)-cos(3*A) : :
X(12236) = 3*X(51)-X(113) = X(74)+3*X(3060) = X(110)-5*X(3567) = X(146)-9*X(11002) = X(265)+3*X(568) = 3*X(568)-X(1986) = X(1511)-3*X(5946) = 3*X(5890)+X(10733) = X(7731)+3*X(9140)

The reciprocal orthologic center of these triangles is X(10112).

X(12236) lies on these lines: {4,94}, {5,11746}, {6,1511}, {30,974}, {51,113}, {52,125}, {74,3060}, {110,3567}, {389,11800}, {511,6699}, {541,11807}, {542,9969}, {567,12006}, {1154,2072}, {1216,6723}, {1353,2854}, {1493,11597}, {1994,3043}, {2071,3581}, {2777,5446}, {2781,10264}, {3047,3518}, {3548,6101}, {5462,5972}, {5889,7723}, {5890,10733}, {7530,9934}, {7731,9140}, {9777,12168}, {10111,12140}, {10114,11225}, {10263,12041}, {11262,11804}

X(12236) = midpoint of X(i) and X(j) for these {i,j}: {52,125}, {265,1986}, {389,11800}, {5446,11806}, {5889,7723}, {6102,10113}, {10111,12140}, {10263,12041}
X(12236) = reflection of X(i) in X(j) for these (i,j): (5,11746), (1112,143), (1216,6723), (1511,9826), (5972,5462)
X(12236) = 1st Droz-Farny circle-inverse-of-X(3448)
X(12236) = X(119)-of-orthic-triangle if ABC is acute
X(12236) = X(104)-of-2nd-anti-Conway-triangle if ABC is acute
X(12236) = anticenter of the cyclic quadrilateral consisting of the vertices of the orthic triangle and X(125)
X(12236) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (265,568,1986), (5504,6644,1511)

X(12237) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CONWAY TO LUCAS ANTIPODAL

Trilinears a*(2*SA*(SB + SC - S)*b^2*c^2 - 2*S*(S^2 + SB*SC)*(2*S - SW)) : :
X(12237) = 3*X(51)-X(487) = 2*X(642)-3*X(5943) = 3*X(3060)+X(12221)

The reciprocal orthologic center of these triangles is X(3).

X(12237) lies on these lines: {6,12229}, {51,487}, {486,511}, {642,5943}, {3060,12221}, {3564,5446}, {3819,6119}, {5907,6251}, {6290,10110}, {9729,12123}, {9777,12169}

X(12237) = reflection of X(i) in X(j) for these (i,j): (5907,6251), (6290,10110), (12123,9729)
X(12237) = orthic-to-2nd-anti-Conway similarity image of X(487)

X(12238) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CONWAY TO LUCAS(-1) ANTIPODAL

Trilinears a*(2*SA*(SB+SC+S)*b^2*c^2-2*S*(S^2+SB*SC)*(2*S+SW)) : :
X(12238) = 3*X(51)-X(488) = 2*X(641)-3*X(5943) = 3*X(3060)+X(12222)

The reciprocal orthologic center of these triangles is X(3).

X(12238) lies on these lines: {6,12230}, {51,488}, {485,511}, {641,5943}, {3060,12222}, {3564,5446}, {3819,6118}, {5907,6250}, {6289,10110}, {9729,12124}, {9777,12170}

X(12238) = reflection of X(i) in X(j) for these (i,j): (5907,6250), (6289,10110), (12124,9729)
X(12238) = orthic-to-2nd-anti-Conway similarity image of X(488)

X(12239) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CONWAY TO LUCAS CENTRAL

Trilinears a*((b^2+c^2)*a^6-3*(b^4+c^4)*a^4+3*(b^4-c^4)*(b^2-c^2)*a^2-(b^4+c^4)*(b^2-c^2)^2-4*S*a^2*b^2*c^2) : :
X(12239) = 3*X(51)-X(6291) = 3*X(3060)+X(12223) = 5*X(3567)-X(6239)

The reciprocal orthologic center of these triangles is X(3).

X(12239) lies on these lines: {3,6}, {51,3071}, {155,8276}, {185,3070}, {486,5462}, {590,5562}, {1147,9682}, {1154,8981}, {1216,5418}, {1587,5890}, {1588,3567}, {2781,8991}, {3060,6459}, {3068,5889}, {5420,5892}, {5446,6561}, {5891,10576}, {5943,9823}, {5946,7584}, {6102,7583}, {6252,11435}, {6283,11436}, {6457,8577}, {6460,10574}, {6564,12162}, {8252,11695}, {8253,11793}, {9540,11412}, {9683,9687}, {9777,12171}

X(12239) = {X(6),X(389)}-harmonic conjugate of X(12240)
X(12239) = X(176)-of-2nd-anti-Conway-triangle if ABC is acute
X(12239) = orthic-to-2nd-anti-Conway similarity image of X(6291)

X(12240) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CONWAY TO LUCAS(-1) CENTRAL

Trilinears a*((b^2+c^2)*a^6-3*(b^4+c^4)*a^4+3*(b^4-c^4)*(b^2-c^2)*a^2-(b^4+c^4)*(b^2-c^2)^2+4*S*a^2*b^2*c^2) : :
X(12240) = 3*X(51)-X(6406) = 3*X(3060)+X(12224) = 5*X(3567)-X(6400)

The reciprocal orthologic center of these triangles is X(3).

X(12240) lies on these lines: {3,6}, {51,3070}, {155,8277}, {185,3071}, {485,5462}, {615,5562}, {1216,5420}, {1587,3567}, {1588,5890}, {3060,6460}, {3069,5889}, {5418,5892}, {5446,6560}, {5891,10577}, {5943,9824}, {5946,7583}, {6102,7584}, {6404,11435}, {6405,11436}, {6458,8576}, {6459,10574}, {6565,12162}, {8252,11793}, {8253,11695}, {8981,12006}, {8998,9826}, {9777,12172}

X(12240) = {X(6),X(389)}-harmonic conjugate of X(12239)
X(12240) = X(175)-of-2nd-anti-Conway-triangle if ABC is acute
X(12240) = orthic-to-2nd-anti-Conway similarity image of X(6406)

X(12241) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CONWAY TO MACBEATH

Barycentrics 2*a^10-5*(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+2*(b^2-c^2)^4*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(12241) = X(4)+3*X(12022) = 2*X(4)+3*X(12024) = 3*X(51)-X(3575) = 3*X(51)-2*X(11745) = X(185)-3*X(11245) = X(1885)+3*X(11245) = 3*X(3060)+X(12225) = 5*X(3567)-X(6240) = X(6146)-3*X(12022)

The reciprocal orthologic center of these triangles is X(4).

Let A'B'C' be the midheight triangle. Let A_B, A_C be the orthogonal projections of A' on CA, AB, resp. Define B_C, B_A, C_A, C_B cyclically. Let A" = C_AA_C∩A_BB_A, and define B" nad C" cyclically. Triangle A"B"C" is homothetic to ABC at X(6). X(12241) = X(4)-of-A"B"C". (Randy Hutson, March 29, 2020)

X(12241) lies on these lines: {2,11425}, {4,6}, {5,578}, {12,11429}, {20,9786}, {30,143}, {51,3575}, {54,403}, {68,9818}, {140,11430}, {141,7395}, {154,3089}, {182,6823}, {184,235}, {185,1885}, {230,1970}, {265,5576}, {343,7503}, {376,1192}, {378,6696}, {381,11426}, {382,11432}, {394,6816}, {427,11424}, {524,5562}, {550,11438}, {567,10024}, {590,6809}, {615,6810}, {1211,7549}, {1352,11479}, {1495,10619}, {1593,1899}, {1596,6759}, {1598,9833}, {1620,3528}, {1746,6831}, {1853,3088}, {1907,11550}, {3060,12225}, {3091,11427}, {3542,10192}, {3564,5907}, {3567,6240}, {3589,7399}, {3629,12160}, {3850,7687}, {5085,7400}, {5462,9826}, {5894,10605}, {5943,9825}, {6253,11435}, {6284,11436}, {6523,6618}, {6642,12118}, {6756,10110}, {6815,10601}, {7553,11750}, {7576,9781}, {9777,12173}

X(12241) = midpoint of X(i) and X(j) for these {i,j}: {4,6146}, {185,1885}, {5907,10112}, {7553,11750}
X(12241) = reflection of X(i) in X(j) for these (i,j): (3575,11745), (6756,10110), (12024,12022)
X(12241) = X(960)-of-orthic-triangle if ABC is acute
X(12241) = X(65)-of-2nd-anti-Conway-triangle if ABC is acute
X(12241) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,6,12233), (4,1181,2883), (4,6776,1498), (4,10982,5480), (4,12022,6146), (20,11433,9786), (51,3575,11745), (397,398,1990), (1587,1588,1249), (1593,1899,6247), (1885,11245,185), (2883,8550,1181), (3070,3071,53)

X(12242) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CONWAY TO REFLECTION

Trilinears    (2*cos(2*A)-1)*cos(B-C)-cos(A)*cos(2*(B-C))-cos(A)+cos(3*A) : :
Barycentrics    (cos B) cos(C - A)[c cos(A - B) + a cos(B - C)] + (cos C) cos(A - B)[a cos(B - C) + b cos(C - A)] : :
Barycentrics    2a^10 - 8a^8(b^2 + c^2) + a^6(11b^4 + 11c^4 + 8b^2c^2) - 5a^4(b^2 - c^2)^2(b^2 + c^2) - a^2(b^2 - c^2)^2(b^4 + c^4 + 4b^2c^2) + (b^2 - c^2)^4(b^2 + c^2) : :
X(12242) = X(4)+3*X(54) = X(4)-3*X(3574) = 3*X(51)-X(6152) = 2*X(140)-3*X(6689) = X(140)-3*X(8254) = 3*X(195)+5*X(1656) = 3*X(195)+X(3519) = 3*X(1209)-5*X(1656) = 3*X(1209)-X(3519) = 5*X(1656)-X(3519) = 3*X(6689)+2*X(11803) = 3*X(8254)+X(11803)

The reciprocal orthologic center of these triangles is X(6243).

Let A'B'C' be the reflection triangle. Let Oa be the circle centered at A' and tangent to BC, and define Ob, Oc cyclically. X(12242) is the radical center of circles Oa, Ob, Oc. (Randy Hutson, July 21, 2017)

Let A'B'C' be the medial triangle. Let Ba and Ca be the orthogonal projections of B' and C' on line BC, resp. Let (Oa) be the circle with segment BaCa as diameter. Define (Ob) and (Oc) cyclically. X(12242) is the radical center of circles (Oa), (Ob), (Oc). (Randy Hutson, November 2, 2017)

X(12242) lies on these lines: {2,11431}, {4,54}, {5,539}, {6,17}, {51,6152}, {125,1199}, {140,389}, {397,6116}, {398,6117}, {468,973}, {542,5576}, {550,10610}, {575,11585}, {576,3549}, {974,10628}, {1487,7604}, {1657,11425}, {2888,5056}, {2917,3517}, {3060,12226}, {3090,11271}, {3523,7691}, {3567,6242}, {3628,10275}, {3850,7687}, {3851,6288}, {4857,11429}, {5449,11225}, {5462,5972}, {5476,7529}, {5943,9820}, {6217,6276}, {6218,6277}, {6255,11435}, {6286,11436}, {9777,12175}, {9813,9972}, {9905,11522}, {9969,11808}, {10110,11576}, {10114,11702}, {11064,11695}

X(12242) = midpoint of X(i) and X(j) for these {i,j}: {4,10619}, {5,1493}, {54,3574}, {125,2914}, {140,11803}, {195,1209}, {11576,11577}, {11702,11804}
X(12242) = reflection of X(i) in X(j) for these (i,j): (6689,8254), (11576,10110)
X(12242) = trilinear pole, wrt half-altitude triangle, of orthic axis
X(12242) = X(3647)-of-orthic-triangle if ABC is acute
X(12242) = X(79)-of-2nd-anti-Conway-triangle if ABC is acute
X(12242) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,54,10619), (6,195,12234), (17,18,233), (195,1656,3519), (1656,3519,1209), (3574,10619,4), (8254,11803,140)

X(12243) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO ANTI-MCCAY

Barycentrics a^8+3*(b^2+c^2)*a^6-(2*b^4+11*b^2*c^2+2*c^4)*a^4-(b^2-3*c^2)*(3*b^2-c^2)*(b^2+c^2)*a^2+(b^2-c^2)^2*((b^2+c^2)^2-9*b^2*c^2) : :
X(12243) = 3*X(98)-X(12117) = 2*X(99)-3*X(3524) = 4*X(114)-5*X(5071) = 2*X(114)-3*X(9166) = 4*X(115)-3*X(3545) = X(148)+2*X(12188) = 3*X(3524)-4*X(6055) = 3*X(3545)-2*X(6054) = 3*X(7709)-2*X(11152) = X(8724)-3*X(11632)

The reciprocal orthologic center of these triangles is X(9855).

X(12243) lies on these lines: {2,2782}, {3,7616}, {4,542}, {20,8596}, {24,9876}, {30,148}, {40,2796}, {76,9302}, {98,376}, {99,3524}, {110,11656}, {114,5071}, {115,3545}, {147,381}, {338,5648}, {388,10054}, {497,10070}, {511,11054}, {515,9875}, {530,6773}, {531,6770}, {631,2482}, {3090,5461}, {3455,7556}, {3528,10992}, {3529,10991}, {3543,5984}, {3564,8352}, {3839,6033}, {5182,10359}, {5286,6034}, {5523,6761}, {5657,9881}, {6248,7827}, {6776,7620}, {7487,12132}, {7615,9744}, {7790,11178}, {7967,9884}, {8550,10488}, {9755,11159}, {9882,10783}, {9883,10784}, {9890,11257}, {10053,10385}, {10304,12042}, {10788,12191}, {11179,11185}, {11180,11646}

X(12243) = midpoint of X(i) and X(j) for these {i,j}: {20,8596}, {148,11177}, {3543,5984}
X(12243) = reflection of X(i) in X(j) for these (i,j): (2,11632), (4,671), (99,6055), (110,11656), (147,381), (376,98), (2482,11623), (3543,6321), (6054,115), (8591,3), (9862,11177), (10488,8550), (11177,12188), (11180,11646)
X(12243) = anticomplement of X(8724)
X(12243) = orthologic center of these triangles: anti-Euler to McCay
X(12243) = X(671)-of-anti-Euler-triangle
X(12243) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (99,6055,3524), (114,9166,5071), (115,6054,3545), (148,12188,9862)

X(12244) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO ANTI-ORTHOCENTROIDAL

Trilinears (12*cos(2*A)+15)*cos(B-C)-2*cos(A)*cos(2*(B-C))-22*cos(A)-3*cos(3*A) : :
X(12244) = 3*X(2)-4*X(12041) = 3*X(4)-4*X(125) = 7*X(4)-8*X(7687) = 3*X(4)-2*X(10721) = X(4)-4*X(10990) = 3*X(20)-2*X(12121) = 3*X(74)-2*X(125) = 7*X(74)-4*X(7687) = 3*X(74)-X(10721) = 5*X(146)-8*X(10272)

The reciprocal orthologic center of these triangles is X(12112).

X(12244) lies on these lines: {2,7728}, {3,146}, {4,74}, {20,5663}, {24,9919}, {30,3448}, {67,11738}, {110,376}, {113,631}, {185,7731}, {186,10117}, {265,3146}, {382,10264}, {388,10065}, {399,550}, {477,1138}, {497,10081}, {515,9904}, {542,11001}, {690,9862}, {974,11431}, {1181,2914}, {1511,3522}, {1539,3091}, {2771,9961}, {2781,6776}, {2931,12088}, {2935,3520}, {3028,4294}, {3060,11806}, {3090,6699}, {3431,10293}, {3524,5972}, {3529,11411}, {3534,9143}, {3543,10113}, {3567,11807}, {3830,11801}, {4299,7727}, {5071,6723}, {5480,5621}, {5603,11709}, {5655,10304}, {6225,9934}, {6241,10628}, {7487,12133}, {7505,11270}, {7552,11454}, {7577,10606}, {7725,10783}, {7726,10784}, {7967,7978}, {10295,12112}, {10323,12168}, {10574,11557}, {10788,12192}

X(12244) = reflection of X(i) in X(j) for these (i,j): (4,74), (74,10990), (146,3), (382,10264), (399,550), (2935,5894), (3146,265), (3448,10620), (6225,9934), (7728,12041), (7731,185), (9143,3534), (10721,125), (12112,10295)
X(12244) = anticomplement of X(7728)
X(12244) = X(74)-of-anti-Euler-triangle
X(12244) = orthologic center of these triangles: anti-Euler to orthocentroidal
X(12244) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (74,10721,125), (125,10721,4), (7728,12041,2)

X(12245) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO EXCENTERS-MIDPOINTS

Barycentrics a^4-4*(b+c)*a^3+8*b*c*a^2+4*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(12245) = X = 4*X(1)-5*X(631) = 2*X(1)-3*X(5657) = 3*X(1)-4*X(6684) = 5*X(1)-7*X(9588) = 5*X(1)-6*X(10165) = 3*X(2)-4*X(5690) = 9*X(2)-8*X(5901) = 6*X(2)-5*X(10595) = 5*X(631)-6*X(5657) = 5*X(631)-8*X(11362) = 3*X(1482)-4*X(5901) = 4*X(1482)-5*X(10595) = 3*X(5690)-2*X(5901)

The reciprocal orthologic center of these triangles is X(10).

X(12245) lies on these lines: {1,631}, {2,1482}, {3,145}, {4,8}, {5,3617}, {7,7317}, {10,3090}, {20,952}, {21,10679}, {40,376}, {46,3476}, {55,6875}, {65,1056}, {78,6927}, {80,5225}, {100,6942}, {104,5854}, {140,3622}, {149,6928}, {165,3633}, {239,7397}, {390,5729}, {404,10680}, {443,10597}, {495,6937}, {496,6963}, {497,5697}, {498,11009}, {515,3529}, {516,3625}, {528,11827}, {529,11826}, {758,12115}, {920,3486}, {938,9957}, {946,3545}, {953,6079}, {956,6906}, {960,6898}, {999,6940}, {1006,3295}, {1058,3057}, {1075,3176}, {1125,3533}, {1145,5730}, {1159,11036}, {1210,7962}, {1317,5204}, {1320,6891}, {1350,9053}, {1385,3241}, {1389,6852}, {1512,6736}, {1697,3488}, {1698,11224}, {1699,4668}, {1766,5839}, {2077,8666}, {2093,10106}, {2095,6904}, {2098,3086}, {2099,3085}, {2550,6901}, {2551,3878}, {2800,5904}, {2802,6903}, {2886,6874}, {2975,6950}, {3088,11396}, {3091,4678}, {3149,8158}, {3242,10519}, {3244,3576}, {3245,4299}, {3296,5559}, {3340,3487}, {3428,3913}, {3485,8164}, {3523,3623}, {3525,3616}, {3526,10283}, {3528,3579}, {3544,9955}, {3600,6955}, {3626,3855}, {3634,9624}, {3635,10164}, {3656,5071}, {3661,7402}, {3817,4691}, {3820,6975}, {3868,6916}, {3876,6939}, {3877,5084}, {3880,6899}, {3885,6865}, {3889,10202}, {3893,7957}, {3940,6848}, {4004,9776}, {4007,10445}, {4189,11849}, {4293,10944}, {4294,10950}, {4295,5252}, {4311,5128}, {4323,11374}, {4345,5704}, {4511,6880}, {4677,5691}, {4816,9589}, {4853,6769}, {4861,6977}, {5044,5804}, {5067,5734}, {5087,7704}, {5126,6049}, {5288,5450}, {5289,8256}, {5550,11231}, {5552,6949}, {5601,11253}, {5602,11252}, {5604,10518}, {5605,10517}, {5656,7973}, {5658,7971}, {5687,6905}, {5714,9578}, {5722,9785}, {5727,10624}, {5759,5853}, {5761,6856}, {5763,6844}, {5768,6764}, {5789,6847}, {5836,6854}, {5837,9623}, {5846,6776}, {6734,6956}, {6735,6969}, {6743,6766}, {6825,10528}, {6873,7680}, {6883,12000}, {6896,7686}, {6920,9708}, {6932,10942}, {6943,10943}, {6946,9709}, {6952,10527}, {6989,10587}, {7487,12135}, {7512,8193}, {7709,7976}, {8128,11924}, {8192,10323}, {8715,11012}, {9669,11545}, {9798,12088}, {9997,10357}, {10175,11522}, {10359,10800}, {10588,11280}, {10785,10912}, {10788,12195}, {11822,11844}, {11823,11843}

X(12245) = midpoint of X(i) and X(j) for these {i,j}: {20,3621}, {3632,7991}, {3893,7957}
X(12245) = reflection of X(i) in X(j) for these (i,j): (1,11362), (4,8), (145,3), (944,40), (962,355), (1482,5690), (3241,3654), (3529,6361), (3633,5882), (4301,3626), (5881,3625), (6361,7991), (7982,10), (8148,5), (10698,1145), (11531,946)
X(12245) = anticomplement of X(1482)
X(12245) = orthologic center of these triangles: anti-Euler to 2nd Schiffler
X(12245) = X(8)-of-anti-Euler-triangle
X(12245) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5657,631), (1,9588,10165), (1,11362,5657), (2,1482,10595), (3,145,7967), (8,962,355), (8,3869,3421), (8,11415,5176), (10,5603,3090), (10,7982,5603), (40,944,376), (100,11249,6942), (140,10247,3622), (165,3633,5882), (355,962,4), (1482,5690,2), (3419,5758,4), (5080,10525,4), (5175,5812,4), (5697,10573,497)

X(12246) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO EXTOUCH

Trilinears (4*sin(A/2)-2*sin(3*A/2))*cos((B-C)/2)+(-2*cos(A)-6)*cos(B-C)+2*sin(A/2)*cos(3*(B-C)/2)+14*cos(A)-3*cos(2*A)-3 : :
X(12246) = 4*X(3)-3*X(5658) = 3*X(4)-4*X(6245) = 3*X(84)-2*X(6245) = 3*X(376)-2*X(1490) = 5*X(631)-4*X(6260) = 4*X(1158)-3*X(5657) = 7*X(3090)-8*X(6705) = 4*X(3358)-3*X(5817) = 3*X(5603)-4*X(12114) = 5*X(5818)-4*X(6256)

The reciprocal orthologic center of these triangles is X(40).

Let A'B'C' be the Hutson-extouch triangle. Let La be the tangent to the A-excircle at A', and define B' and C' cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. Triangle A"B"C" is homothetic to ABC at X(57), and X(12246) = X(4)-of-A"B"C". (Randy Hutson, July 21, 2017)

X(12246) lies on these lines: {1,7955}, {2,6259}, {3,5658}, {4,57}, {20,72}, {24,9910}, {30,9799}, {104,10309}, {376,1490}, {388,1709}, {443,3358}, {452,10167}, {497,10085}, {515,3529}, {516,6762}, {631,5316}, {944,3057}, {946,4355}, {1012,3487}, {1158,5657}, {1768,1788}, {2801,3189}, {2829,6253}, {3090,6705}, {3146,5787}, {3304,3649}, {3427,10308}, {3474,4848}, {3600,9856}, {3982,11522}, {4297,5698}, {4298,11372}, {5129,11227}, {5259,5450}, {5714,6847}, {5815,6244}, {5818,6256}, {5927,6904}, {6257,10784}, {6258,10783}, {6865,7171}, {6868,9960}, {6872,11220}, {6916,7330}, {6936,9942}, {7487,12136}, {7704,10785}, {7967,7971}, {10788,12196}, {10884,11111}

X(12246) = reflection of X(i) in X(j) for these (i,j): (4,84), (3146,5787), (5691,9948)
X(12246) = anticomplement of X(6259)
X(12246) = X(84)-of-anti-Euler-triangle

X(12247) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO FUHRMANN

Trilinears (20*sin(A/2)-6*sin(3*A/2))*cos((B-C)/2)+(6*cos(A)-5)*cos(B-C)+2*sin(A/2)*cos(3*(B-C)/2)+7*cos(A)-cos(2*A)-7 : :
X(12247) = 4*X(11)-3*X(5603) = 2*X(100)-3*X(5657) = 4*X(119)-5*X(5818) = 4*X(214)-5*X(631) = 3*X(376)-2*X(12119) = 7*X(3090)-8*X(6702) = 3*X(3679)-X(5531) = 3*X(5603)-2*X(10698) = 3*X(7967)-2*X(7972) = 3*X(7967)-4*X(11715) = X(7972)-3*X(11219) = 3*X(11219)-2*X(11715)

The reciprocal orthologic center of these triangles is X(3).

X(12247) lies on these lines: {1,6952}, {2,6265}, {3,8}, {4,80}, {10,6326}, {11,2099}, {24,9912}, {119,2476}, {145,6972}, {149,517}, {153,355}, {214,631}, {376,12119}, {377,9964}, {388,10044}, {443,9946}, {484,515}, {497,10051}, {519,6264}, {528,5759}, {912,5176}, {938,1387}, {962,10738}, {1056,5083}, {1320,6943}, {1389,6831}, {1478,11571}, {1482,1484}, {1532,11545}, {1537,12019}, {1788,10090}, {2550,2801}, {2802,6903}, {2829,6253}, {2949,5541}, {3036,6937}, {3090,6702}, {3476,10074}, {3485,8068}, {3486,10058}, {3617,10786}, {3632,7993}, {3679,5531}, {3754,6901}, {3878,6902}, {4214,12138}, {5218,7967}, {5289,6963}, {5790,11698}, {5805,6797}, {5840,6361}, {6262,10784}, {6263,10783}, {6906,10950}, {7487,12137}, {9809,10742}, {9963,10993}, {10788,12198}

X(12247) = midpoint of X(i) and X(j) for these {i,j}: {8,9803}, {1768,9897}, {3632,7993}
X(12247) = reflection of X(i) in X(j) for these (i,j): (1,10265), (4,80), (153,355), (944,104), (962,10738), (1482,1484), (1532,11545), (1537,12019), (5541,11362), (6224,3), (6326,10), (7967,11219), (7972,11715), (9809,10742), (9963,10993), (10698,11)
X(12247) = anticomplement of X(6265)
X(12247) = X(80)-of-anti-Euler-triangle
X(12247) = X(6326)-of-outer-Garcia-triangle
X(12247) = inner-Garcia-to-outer-Garcia similarity image of X(4)
X(12247) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11,10698,5603), (7972,11219,11715), (7972,11715,7967)

X(12248) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO INNER-GARCIA

Trilinears (16*sin(A/2)-6*sin(3*A/2))*cos((B-C)/2)+(2*cos(A)-5)*cos(B-C)+2*sin(A/2)*cos(3*(B-C)/2)+13*cos(A)-3*cos(2*A)-7 : :
X(12248) = 3*X(3)-2*X(11698) = 3*X(4)-4*X(11) = 3*X(4)-2*X(10728) = 2*X(11)-3*X(104) = 2*X(100)-3*X(376) = 3*X(104)-X(10728) = 4*X(119)-5*X(631) = 3*X(153)-4*X(11698) = 3*X(944)-2*X(7972) = 3*X(7967)-2*X(10698)

The reciprocal orthologic center of these triangles is X(40).

X(12248) lies on the cubic K542 and these lines: {2,10742}, {3,153}, {4,11}, {20,952}, {24,9913}, {30,149}, {80,1788}, {100,376}, {119,631}, {382,1484}, {388,10058}, {390,6938}, {484,515}, {495,6906}, {497,10074}, {516,6264}, {528,11001}, {944,2800}, {1317,4294}, {1387,3600}, {1537,6147}, {2096,11041}, {2771,3648}, {2787,9862}, {2801,5759}, {2802,6361}, {2828,5667}, {3035,3524}, {3090,6713}, {3146,10738}, {3486,11570}, {3488,5083}, {3529,5840}, {4297,6326}, {4996,6876}, {5071,6667}, {5218,6950}, {5225,5533}, {5229,8068}, {5450,6952}, {5603,11715}, {5691,10265}, {5731,6265}, {6256,6949}, {6845,9655}, {6930,11729}, {6965,10269}, {7487,12138}, {10788,12199}

X(12248) = reflection of X(i) in X(j) for these (i,j): (4,104), (153,3), (382,1484), (3146,10738), (5691,10265), (6326,4297), (9809,6265), (10728,11), (12247,1768)
X(12248) = anticomplement of X(10742)
X(12248) = X(104)-of-anti-Euler-triangle
X(12248) = Cundy-Parry Phi transform of X(3563)
X(12248) = Cundy-Parry Psi transform of X(3564)
X(12248) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11,10728,4), (104,10728,11), (5731,9809,6265)

X(12249) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO HUTSON EXTOUCH

Trilinears p^3*(3*p^3+12*q-11*p-4*p^2*q)+(q^4-3*q^2+12)*p^2+2*(q^2-5)*q*p-2+2*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(12249) = 3*X(376)-2*X(12120) = 3*X(7967)-2*X(8000)

The reciprocal orthologic center of these triangles is X(40).

X(12249) lies on these lines: {3,9874}, {4,1697}, {376,12120}, {388,10059}, {497,10075}, {515,9898}, {944,7957}, {2951,6361}, {5759,7674}, {7487,12139}, {7967,8000}, {10788,12200}

X(12249) = reflection of X(i) in X(j) for these (i,j): (4,7160), (9874,3)
X(12249) = X(7160)-of-anti-Euler-triangle

X(12250) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO MIDHEIGHT

Trilinears 8*(2*cos(2*A)+3)*cos(B-C)-2*cos(A)*cos(2*(B-C))-35*cos(A)-3*cos(3*A) : :
X(12250) = 3*X(2)-4*X(3357) = 4*X(3)-3*X(5656) = 5*X(4)-6*X(1853) = 3*X(4)-2*X(5895) = 3*X(4)-4*X(6247) = 5*X(64)-3*X(1853) = 3*X(64)-X(5895) = 3*X(64)-2*X(6247) = 9*X(1853)-5*X(5895) = 3*X(5656)-2*X(6225)

The reciprocal orthologic center of these triangles is X(4).

X(12250) lies on these lines: {2,3357}, {3,5656}, {4,64}, {20,2979}, {24,9914}, {30,11411}, {74,3542}, {154,3528}, {376,1498}, {388,10060}, {497,10076}, {515,9899}, {550,11206}, {631,2883}, {1204,3089}, {1294,3346}, {1503,3529}, {1515,6616}, {2777,3146}, {3088,3574}, {3090,6696}, {3091,7703}, {3426,6756}, {3522,6759}, {3524,8567}, {3545,5893}, {3566,5489}, {3962,6001}, {4293,6285}, {4294,7355}, {4846,7404}, {5663,6193}, {5890,11431}, {6145,11738}, {6241,6776}, {6266,10784}, {6267,10783}, {7401,11472}, {7487,11381}, {7967,7973}, {10192,10299}, {10282,10304}, {10788,12202}

X(12250) = reflection of X(i) in X(j) for these (i,j): (4,64), (1498,5894), (3529,5925), (5878,3357), (5895,6247), (6225,3)
X(12250) = anticomplement of X(5878)
X(12250) = X(64)-of-anti-Euler-triangle
X(12250) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6225,5656), (1498,5894,376), (2883,10606,631), (3357,5878,2), (5895,6247,4)

X(12251) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO 1st NEUBERG

Barycentrics (b^2+c^2)*a^6+3*b^2*c^2*a^4-(b^2+c^2)^3*a^2+(b^2-c^2)^2*b^2*c^2 : :
X(12251) = 9*X(2)-8*X(11272) = 2*X(3)-3*X(6194) = 4*X(3)-3*X(7709) = 3*X(4)-4*X(6248) = 4*X(39)-5*X(631) = 3*X(76)-2*X(6248) = X(194)-3*X(6194) = 2*X(194)-3*X(7709) = 6*X(262)-7*X(3090) = 3*X(262)-4*X(3934) = 7*X(3090)-8*X(3934) = 3*X(3095)-4*X(11272)

The reciprocal orthologic center of these triangles is X(3).

X(12251) lies on these lines: {2,3095}, {3,194}, {4,69}, {5,3314}, {6,10359}, {20,2782}, {24,9917}, {30,9863}, {39,631}, {40,726}, {83,576}, {98,7751}, {99,5171}, {114,7796}, {140,7806}, {182,7760}, {262,3090}, {343,5117}, {376,538}, {384,10788}, {388,10063}, {394,419}, {497,10079}, {515,9902}, {575,7894}, {698,1350}, {730,944}, {732,6776}, {1078,9737}, {1351,7770}, {1513,3933}, {1569,5206}, {1656,7931}, {1975,11676}, {2080,3552}, {2794,7826}, {3068,3103}, {3069,3102}, {3091,7697}, {3094,5286}, {3097,6684}, {3398,7766}, {3523,11171}, {3524,7757}, {3525,7786}, {3533,6683}, {3545,9466}, {3734,12110}, {3926,5976}, {5097,7878}, {5969,12243}, {6272,10784}, {6273,10783}, {7487,12143}, {7758,8149}, {7781,8722}, {7795,9753}, {7802,9991}, {7967,7976}, {10333,10796}, {10983,11285}

X(12251) = reflection of X(i) in X(j) for these (i,j): (4,76), (20,9821), (194,3), (7709,6194), (7758,8149), (11257,5188)
X(12251) = anticomplement of X(3095)
X(12251) = X(76)-of-anti-Euler-triangle
X(12251) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,194,7709), (194,6194,3), (262,3934,3090), (5188,11257,376), (9821,9983,9862)

X(12252) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO 2nd NEUBERG

Barycentrics 3*a^8+(b^2+c^2)*a^6-(2*b^2+c^2)*(b^2+2*c^2)*a^4-(b^2+c^2)^3*a^2-(b^6-c^6)*(b^2-c^2) : :
X(12252) = 3*X(4)-4*X(6249) = 3*X(83)-2*X(6249) = 3*X(376)-2*X(12122) = 5*X(631)-4*X(6292) = 3*X(7967)-2*X(7977) = X(11001)+2*X(12156)

The reciprocal orthologic center of these triangles is X(3).

X(12252) lies on these lines: {2,6287}, {3,147}, {4,83}, {20,3095}, {24,9918}, {98,8150}, {376,754}, {382,7864}, {388,10064}, {497,10080}, {515,9903}, {546,7923}, {550,7762}, {631,6292}, {732,6776}, {3090,6704}, {3528,6337}, {3529,7737}, {3796,5117}, {5569,9774}, {6274,10784}, {6275,10783}, {6655,10131}, {7487,12144}, {7791,10334}, {7869,10299}, {7967,7977}, {10788,12206}, {11001,12156}

X(12252) = reflection of X(i) in X(j) for these (i,j): (4,83), (20,8725), (2896,3)
X(12252) = anticomplement of X(6287)
X(12252) = X(83)-of-anti-Euler-triangle
X(12252) = X(6292), X(9751)}-harmonic conjugate of X(631)

X(12253) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO 1st ORTHOSYMMEDIAL

Barycentrics (R^2-SW)*S^4+((3*SA^2+SA*SW-8*SW^2)*R^2-(2*SA^2-SA*SW-2*SW^2)*SW)*S^2+3*(4*R^2-SW)*(SB+SC)*SA*SW^2 : :
X(12253) = 3*X(4)-4*X(127) = 2*X(112)-3*X(376) = 2*X(127)-3*X(1297) = 4*X(132)-5*X(631) = 9*X(3524)-8*X(6720)

The reciprocal orthologic center of these triangles is X(4).

X(12253) lies on these lines: {4,127}, {112,376}, {132,631}, {2781,5596}, {2794,3529}, {2799,9862}, {2806,12248}, {3146,10749}, {3320,4294}, {3524,6720}, {4293,6020}, {7487,12145}, {9517,12244}, {10788,12207}, {11641,12082}

X(12253) = reflection of X(i) in X(j) for these (i,j): (4,1297), (3146,10749)
X(12253) = X(1297)-of-anti-Euler-triangle

X(12254) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO REFLECTION

Trilinears (4*cos(2*A)+3)*cos(B-C)-2*cos(A)*cos(2*(B-C))-2*cos(A)-3*cos(3*A) : :
X(12254) = 3*X(2)-4*X(10610) = 3*X(4)-4*X(3574) = 2*X(20)+X(11271) = 3*X(54)-2*X(3574) = 3*X(376)-2*X(7691) = 3*X(381)-4*X(8254) = 4*X(389)-3*X(7730) = 5*X(631)-4*X(1209) = 4*X(1493)-X(3146) = 3*X(3060)-4*X(10115)

The reciprocal orthologic center of these triangles is X(4).

X(12254) lies on these lines: {2,6288}, {3,2888}, {4,54}, {20,1154}, {24,9920}, {30,195}, {49,3153}, {156,11597}, {186,2917}, {265,5944}, {376,539}, {381,8254}, {388,10066}, {389,7730}, {497,10082}, {515,9905}, {631,1209}, {973,11431}, {1141,3459}, {1199,3575}, {1493,3146}, {1511,11565}, {1568,9705}, {1885,12112}, {2914,5895}, {3060,10115}, {3090,6689}, {3431,6145}, {3518,12022}, {3519,3522}, {3520,6247}, {3567,11808}, {3581,11264}, {4299,7356}, {4302,6286}, {5073,11803}, {6153,9730}, {6241,10628}, {6242,6776}, {6276,10784}, {6277,10783}, {7487,11576}, {7552,9927}, {7728,11702}, {7967,7979}, {9862,9985}, {9977,11179}, {10574,11802}, {10788,12208}, {11464,11704}, {11577,12250}

X(12254) = reflection of X(i) in X(j) for these (i,j): (4,54), (54,10619), (2888,3), (6288,10610), (7728,11702)
X(12254) = anticomplement of X(6288)
X(12254) = X(54)-of-anti-Euler-triangle
X(12254) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (54,1614,10274), (6288,10610,2)

X(12255) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO 1st SCHIFFLER

Trilinears 16*p^5*(p+2*q)-8*(12*q^2-7)*p^4+8*(4*q^2-9)*q*p^3+(16*q^4+56*q^2-63)*p^2-4*(10*q^2-13)*q*p+9-7*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(79).

X(12255) lies on these lines: {4,5885}, {5330,12248}, {7487,12146}, {10788,12209}

X(12255) = reflection of X(4) in X(10266)
X(12255) = X(10266)-of-anti-Euler-triangle

X(12256) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO INNER-VECTEN

Barycentrics (-a^2+b^2+c^2)*(3*a^4+(b^2-c^2)^2-4*a^2*S) : :
X(12256) = 3*X(4)-4*X(6251) = 3*X(376)+2*X(6280) = 3*X(376)-2*X(12123) = 2*X(485)-3*X(7612) = 3*X(486)-2*X(6251) = 5*X(631)-4*X(642) = 5*X(631)-2*X(6281) = 3*X(7967)-2*X(7980)

The reciprocal orthologic center of these triangles is X(3).

X(12256) lies on these lines: {2,6290}, {3,69}, {4,372}, {20,6463}, {24,9921}, {98,638}, {147,8317}, {182,11292}, {184,1589}, {193,9732}, {376,5860}, {388,10067}, {485,7612}, {497,10083}, {515,9906}, {615,8406}, {631,642}, {637,9991}, {1151,8550}, {1152,1503}, {1181,1578}, {1352,11291}, {1587,6423}, {1588,6421}, {1590,1899}, {3090,6119}, {3102,6459}, {3155,11433}, {3156,11206}, {3592,12007}, {3594,5480}, {5408,7386}, {5871,6813}, {5965,7692}, {6215,11316}, {7374,9748}, {7487,12147}, {7494,11090}, {7967,7980}, {9862,9986}, {9863,11293}, {10788,12210}, {10984,12229}

X(12256) = midpoint of X(i) and X(j) for these {i,j}: {20,12221}, {6280,12123}
X(12256) = reflection of X(i) in X(j) for these (i,j): (4,486), (487,3), (6281,642)
X(12256) = anticomplement of X(6290)
X(12256) = X(486)-of-anti-Euler-triangle
X(12256) = {X(3),X(6776)}-harmonic conjugate of X(12257)

X(12257) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO OUTER-VECTEN

Barycentrics (-a^2+b^2+c^2)*(3*a^4+(b^2-c^2)^2+4*a^2*S) : :
X(12257) = 3*X(4)-4*X(6250) = 3*X(376)+2*X(6279) = 3*X(376)-2*X(12124) = 3*X(485)-2*X(6250) = 2*X(486)-3*X(7612) = 5*X(631)-4*X(641) = 5*X(631)-2*X(6278) = 3*X(7967)-2*X(7981)

The reciprocal orthologic center of these triangles is X(3).

X(12257) lies on these lines: {2,6222}, {3,69}, {4,371}, {20,6462}, {24,9922}, {98,637}, {147,8316}, {182,11291}, {184,1590}, {193,9733}, {376,5861}, {388,10068}, {486,7612}, {497,10084}, {515,9907}, {590,8414}, {631,641}, {638,9992}, {1151,1503}, {1152,8550}, {1181,1579}, {1352,11292}, {1587,6422}, {1588,6424}, {1589,1899}, {3069,8911}, {3090,6118}, {3103,6460}, {3155,11206}, {3156,11433}, {3592,5480}, {3594,12007}, {5409,7386}, {5870,6811}, {5871,9541}, {5965,7690}, {6214,11315}, {7000,9748}, {7487,12148}, {7494,11091}, {7967,7981}, {9862,9987}, {9863,11294}, {10788,12211}, {10984,12230}

X(12257) = midpoint of X(i) and X(j) for these {i,j}: {20,12222}, {6279,12124}
X(12257) = reflection of X(i) in X(j) for these (i,j): (4,485), (488,3), (6278,641)
X(12257) = anticomplement of X(6289)
X(12257) = X(485)-of-anti-Euler-triangle
X(12257) = {X(3),X(6776)}-harmonic conjugate of X(12256)

X(12258) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO ANTI-MCCAY

Barycentrics 2*a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+(b^2+c^2)*(b+c)*a^2+(5*b^4-8*b^2*c^2+5*c^4)*a+(b+c)*(2*b^2-c^2)*(b^2-2*c^2) : :
X(12258) = 3*X(3576)-X(12117) = 5*X(3616)-X(8591) = 7*X(3622)+X(8596) = X(4301)+2*X(11623) = X(5184)-3*X(8859) = 3*X(5603)+X(12243) = 3*X(5886)-X(8724) = 2*X(11599)+X(11711) = X(11599)+2*X(11725) = X(11711)-4*X(11725)

The reciprocal orthologic center of these triangles is X(9855).

X(12258) lies on these lines: {1,671}, {2,9881}, {10,5461}, {30,11710}, {115,519}, {350,1111}, {515,9880}, {530,11706}, {531,11705}, {542,946}, {543,551}, {1086,1125}, {1386,9830}, {3027,4870}, {3545,9864}, {3576,12117}, {3616,8591}, {3622,8596}, {3655,6321}, {3656,11632}, {3679,7983}, {4301,11623}, {5184,8859}, {5603,12243}, {5886,8724}, {9876,11365}, {9878,11368}, {9882,11370}, {9883,11371}, {11363,12132}, {11364,12191}

X(12258) = midpoint of X(i) and X(j) for these {i,j}: {1,671}, {551,11599}, {3655,6321}, {3656,11632}, {3679,7983}, {9875,9884}
X(12258) = reflection of X(i) in X(j) for these (i,j): (10,5461), (551,11725), (2482,1125), (11711,551)
X(12258) = complement of X(9881)
X(12258) = X(671)-of-anti-Aquila-triangle
X(12258) = orthologic center of these triangles: anti-Aquila to McCay
X(12258) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,9875,9884), (671,9884,9875), (7983,9166,3679), (11599,11725,11711)

X(12259) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO ARIES

Trilinears (-2*p^2+1)*(4*p^2*q*(p-q)+4*(q^2-1)*q*p+(2*q^2-1)^2) : :
where p=sin(A/2), q=cos((B-C)/2)
X(12259) = X(155)-3*X(5886) = 3*X(3576)-X(12118) = 5*X(3616)-X(6193) = 3*X(3817)-2*X(5448) = 3*X(5603)+X(11411) = 3*X(10165)-2*X(12038)

The reciprocal orthologic center of these triangles is X(9833).

X(12259) lies on these lines: {1,68}, {2,9928}, {5,226}, {10,5449}, {155,5886}, {515,9927}, {516,7689}, {539,551}, {1069,11376}, {1125,1147}, {1386,3564}, {3157,11375}, {3576,12118}, {3616,6193}, {3817,5448}, {4297,11709}, {5603,11411}, {5654,8227}, {7352,12047}, {9624,9936}, {9820,11230}, {9908,11365}, {9923,11368}, {9929,11370}, {9930,11371}, {10165,12038}, {11363,12134}, {11364,12193}

X(12259) = midpoint of X(i) and X(j) for these {i,j}: {1,68}, {9896,9933}
X(12259) = reflection of X(i) in X(j) for these (i,j): (10,5449), (1147,1125)
X(12259) = complement of X(9928)
X(12259) = X(68)-of-anti-Aquila-triangle
X(12259) = orthologic center of these triangles: anti-Aquila to 2nd Hyacinth

X(12260) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO HUTSON EXTOUCH

Trilinears a^6-(b+c)*a^5-2*(b^2+5*b*c+c^2)*a^4+2*(b+c)^3*a^3+(b^2+10*b*c+c^2)*(b+c)^2*a^2-(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)*a-2*(b^2-c^2)^2*b*c : :
Trilinears (10*sin(A/2)+2*sin(3*A/2))*cos((B-C)/2)+2*cos(B-C)+10*cos(A)+cos(2*A)+11 : :
X(12260) = 3*X(1)-X(8000) = 3*X(3576)-X(12120) = 3*X(5603)+X(12249) = 3*X(7160)+X(8000)

The reciprocal orthologic center of these triangles is X(40).

X(12260) lies on these lines: {1,5920}, {3,5542}, {10,6767}, {11,1058}, {55,3487}, {200,3646}, {405,4533}, {946,3295}, {954,1490}, {1001,3811}, {1125,6600}, {3576,12120}, {3616,9874}, {3913,10198}, {5603,12249}, {5763,10267}, {6147,11495}, {6743,11108}, {11363,12139}, {11364,12200}

X(12260) = midpoint of X(i) and X(j) for these {i,j}: {1,7160}, {8000,9898}
X(12260) = X(7160)-of-anti-Aquila-triangle
X(12260) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,9898,8000), (7160,8000,9898)

X(12261) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO 1st HYACINTH

Trilinears 16*p^4*(2*q*p-1)+16*(2*q^2-3)*q*p^3+8*(4*q^4-6*q^2+3)*p^2-2*(8*q^2-9)*q*p-(4*q^2-3)^2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(12261) = 3*X(1699)-X(7728) = X(3448)+3*X(5603) = 3*X(3576)-X(12121) = 3*X(3656)-X(7978) = 2*X(5972)-3*X(11230) = X(7978)+3*X(9140)

The reciprocal orthologic center of these triangles is X(6102).

X(12261) lies on these lines: {1,265}, {11,113}, {30,11709}, {110,5886}, {125,517}, {355,7984}, {515,10113}, {516,12041}, {542,1386}, {946,5663}, {952,11801}, {1125,1511}, {1385,11735}, {1699,7728}, {1836,10081}, {2807,11806}, {2948,8227}, {3448,5603}, {3576,12121}, {3579,6699}, {3656,7978}, {5901,11720}, {5972,11230}, {6265,10778}, {6723,11231}, {9812,12244}, {10088,11375}, {10091,11376}, {11363,12140}, {11364,12201}

X(12261) = midpoint of X(i) and X(j) for these {i,j}: {1,265}, {355,7984}, {3656,9140}, {6265,10778}
X(12261) = reflection of X(i) in X(j) for these (i,j): (113,9955), (1385,11735), (1511,1125), (3579,6699), (11699,11723), (11720,5901)
X(12261) = X(265)-of-anti-Aquila-triangle

X(12262) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO MIDHEIGHT

Trilinears 2*p^5*(2*p-q)+(8*q^2-9)*p^4-(2*q^2-3)*q*p^3+(-q^2+1)*(-q*p+7*p^2-2) : :
where p=sin(A/2), q=cos((B-C)/2)
X(12262) = 3*X(1)-X(7973) = 3*X(1)+X(9899) = 3*X(64)+X(7973) = 3*X(64)-X(9899) = 3*X(154)-5*X(7987) = 3*X(165)-5*X(8567) = X(1498)-3*X(3576) = 3*X(1699)-X(5895) = 3*X(1853)-X(5691) = 5*X(3616)-X(6225)

The reciprocal orthologic center of these triangles is X(4).

X(12262) lies on these lines: {1,64}, {3,960}, {10,6696}, {30,12259}, {40,10606}, {57,1854}, {65,4219}, {154,7987}, {165,8567}, {221,3601}, {515,6247}, {516,5894}, {517,3357}, {912,12084}, {1125,2883}, {1192,7713}, {1204,1829}, {1319,6285}, {1385,6000}, {1420,2192}, {1498,3576}, {1503,4297}, {1699,5895}, {1853,5691}, {2646,7355}, {2777,12261}, {3616,6225}, {3817,5893}, {5603,12250}, {5878,5886}, {6266,11371}, {6267,11370}, {7520,9961}, {9914,11365}, {11363,11381}, {11364,12202}

X(12262) = midpoint of X(i) and X(j) for these {i,j}: {1,64}, {7973,9899}
X(12262) = reflection of X(i) in X(j) for these (i,j): (10,6696), (2883,1125
X(12262) = X(64)-of-anti-Aquila-triangle

X(12263) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO 1st NEUBERG

Barycentrics (b^2+c^2)*a^3+2*b^2*c^2*a+b^2*c^2*(b+c) : :
X(12263) = 3*X(1)-X(7976) = 3*X(76)+X(7976) = X(194)-5*X(3616) = 3*X(262)-5*X(8227) = X(355)-3*X(7697) = X(962)+3*X(6194) = X(3095)-3*X(5886) = 3*X(3097)-7*X(3624) = 3*X(3576)-X(11257) = 3*X(5603)+X(12251)

The reciprocal orthologic center of these triangles is X(3).

X(12263) lies on these lines: {1,76}, {10,3934}, {37,39}, {194,3616}, {262,8227}, {355,7697}, {384,11364}, {385,12194}, {511,946}, {515,6248}, {516,5188}, {519,9466}, {538,551}, {731,9063}, {732,1386}, {962,6194}, {1269,1964}, {1385,2782}, {2140,3836}, {3095,5886}, {3097,3624}, {3576,11257}, {4093,4647}, {5603,12251}, {5969,12258}, {6179,10789}, {6272,11371}, {6273,11370}, {7751,10800}, {7770,10791}, {9917,11365}, {9983,11368}, {11230,11272}, {11363,12143}

X(12263) = midpoint of X(i) and X(j) for these {i,j}: {1,76}, {7976,9902}
X(12263) = reflection of X(i) in X(j) for these (i,j): (10,3934), (39,1125)
X(12263) = X(76)-of-anti-Aquila-triangle
X(12263) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,9902,7976), (76,7976,9902), (3097,3624,7786)

X(12264) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO 2nd NEUBERG

Barycentrics 2*a^5+(b+c)*a^4+4*(b^2+c^2)*a^3+(b+c)*(b^2+c^2)*a^2+(b^4+4*b^2*c^2+c^4)*a+b^2*c^2*(b+c) : :
X(12264) = X(2896)-5*X(3616) = 3*X(3576)-X(12122) = 3*X(5603)+X(12252) = 3*X(5886)-X(6287)

The reciprocal orthologic center of these triangles is X(3).

X(12264) lies on these lines: {1,83}, {10,6704}, {40,9751}, {515,6249}, {551,754}, {732,1386}, {1125,1279}, {2896,3616}, {3576,12122}, {5603,12252}, {5886,6287}, {5901,11710}, {6274,11371}, {6275,11370}, {8150,10800}, {9918,11365}, {11363,12144}, {11364,12206}

X(12264) = midpoint of X(i) and X(j) for these {i,j}: {1,83}, {7977,9903}
X(12264) = reflection of X(i) in X(j) for these (i,j): (10,6704), (6292,1125)
X(12264) = X(83)-of-anti-Aquila-triangle
X(12264) = X(3)-of-1st-Hyacinth-triangle
X(12264) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,9903,7977), (83,7977,9903)

X(12265) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO 1st ORTHOSYMMEDIAL

Trilinears SB*SC*(SA-SC)*(SA-SB)*a+8*(s^3-s*(a*s+b*c)+(R+r)*S)*((3*S^2-4*SW^2)*R^2-SW*(S^2-SW^2)) : :
X(12265) = X(112)-3*X(3576) = 3*X(5603)+X(12253) = 2*X(6720)-3*X(10165)

The reciprocal orthologic center of these triangles is X(4).

X(12265) lies on these lines: {1,1297}, {40,10705}, {112,3576}, {127,515}, {132,1125}, {214,2831}, {551,9530}, {1319,6020}, {1385,11722}, {2646,3320}, {2781,11720}, {2794,4297}, {2799,11710}, {2806,11715}, {2825,11712}, {2853,11713}, {5603,12253}, {6720,10165}, {9517,11709}, {9518,11714}, {9523,11716}, {9527,11717}, {9532,11700}, {10780,12119}, {11363,12145}, {11364,12207}

X(12265) = midpoint of X(i) and X(j) for these {i,j}: {1,1297}, {40,10705}, {10780,12119}
X(12265) = reflection of X(i) in X(j) for these (i,j): (132,1125), (11722,1385)
X(12265) = X(1297)-of-anti-Aquila-triangle

X(12266) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO REFLECTION

Trilinears S^2*(3*SW*S^2+5*SA*SB*SC)+(S^2+SA*SC)*(S^2+SA*SB)*a*(a+b+c) : :
X(12266) = 3*X(1)-X(7979) = 3*X(54)+X(7979) = X(195)+3*X(10246) = X(2888)-5*X(3616) = 3*X(3576)-X(7691) = 3*X(5603)+X(12254) = X(5882)+2*X(12242) = 3*X(5886)-X(6288)

The reciprocal orthologic center of these triangles is X(4).

X(12266) lies on these lines: {1,54}, {10,6689}, {195,10246}, {515,3574}, {517,10610}, {539,551}, {952,8254}, {960,1493}, {1125,1209}, {1154,1385}, {2888,3616}, {3576,7691}, {5603,12254}, {5882,12242}, {5886,6288}, {5901,11720}, {6276,11371}, {6277,11370}, {9920,11365}, {9985,11368}, {10628,11709}, {11363,11576}, {11364,12208}

X(12266) = midpoint of X(i) and X(j) for these {i,j}: {1,54}, {7979,9905}
X(12266) = reflection of X(i) in X(j) for these (i,j): (10,6689), (1209,1125)
X(12266) = X(54)-of-anti-Aquila-triangle

X(12267) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO 1st SCHIFFLER

Barycentrics S^2*(3*R+2*r)^2*a+(S*(R+2*r)+2*SB*(a-b+c))*(S*(R+2*r)+2*SC*(a+b-c))*(-a+b+c) : :
X(12267) = 3*X(5603)+X(12255)

The reciprocal orthologic center of these triangles is X(79).

X(12267) lies on these lines: {1,5180}, {11,11263}, {5603,12255}, {11363,12146}, {11364,12209}

X(12267) = midpoint of X(1) and X(10266)
X(12267) = X(10266)-of-anti-Aquila-triangle

X(12268) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO INNER-VECTEN

Barycentrics 2*a*(2*a^2+(b+c)*a+b^2+c^2)*S+(a+b+c)*(2*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(12268) = X(487)-5*X(3616) = 3*X(3576)-X(12123) = 7*X(3622)+X(12221) = 3*X(5603)+X(12256) = 3*X(5886)-X(6290) = X(6281)-7*X(9624)

The reciprocal orthologic center of these triangles is X(3).

X(12268) lies on these lines: {1,486}, {10,6119}, {56,481}, {487,3616}, {515,6251}, {642,1125}, {1386,3564}, {3576,12123}, {3622,12221}, {5603,12256}, {5886,6290}, {6280,11371}, {6281,9624}, {9921,11365}, {9986,11368}, {11363,12147}, {11364,12210}

X(12268) = midpoint of X(1) and X(486)
X(12268) = reflection of X(642) in X(1125)
X(12268) = X(486)-of-anti-Aquila-triangle
X(12268) = {X(1386),X(5901)}-harmonic conjugate of X(12269)

X(12269) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO OUTER-VECTEN

Barycentrics -2*a*(2*a^2+(b+c)*a+b^2+c^2)*S+(a+b+c)*(2*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(12269) = 3*X(1)-X(7981) = 3*X(1)+X(9907) = 3*X(485)+X(7981) = 3*X(485)-X(9907) = X(488)-5*X(3616) = 3*X(3576)-X(12124) = 7*X(3622)+X(12222) = 3*X(5603)+X(12257) = 3*X(5886)-X(6289)

The reciprocal orthologic center of these triangles is X(3).

X(12269) lies on these lines: {1,485}, {10,6118}, {56,482}, {488,3616}, {515,6250}, {641,1125}, {1386,3564}, {3576,12124}, {3622,12222}, {5603,12257}, {5886,6289}, {6278,9624}, {6279,11370}, {9922,11365}, {9987,11368}, {11363,12148}, {11364,12211}

X(12269) = midpoint of X(1) and X(485)
X(12269) = reflection of X(641) in X(1125)
X(12269) = X(485)-of-anti-Aquila-triangle
X(12269) = {X(1386),X(5901)}-harmonic conjugate of X(12268)

Orthologic centers: X(12270)-X(12431)

Centers X(12270)-X(12431) were contributed by César Eliud Lozada, March 16, 2017.

X(12270) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO ANTI-ORTHOCENTROIDAL

Trilinears (18*cos(2*A)+2*cos(4*A)+15)*cos(B-C)-2*(4*cos(A)+cos(3*A))*cos(2*(B-C))-20*cos(A)-7*cos(3*A) : :
X(12270) = 3*X(4)-4*X(11557) = 2*X(265)-3*X(5890) = 3*X(381)-4*X(11561) = 4*X(1511)-3*X(11459) = 4*X(1539)-3*X(11455) = 4*X(1986)-3*X(3060) = 3*X(2979)-2*X(12219) = 3*X(3060)-2*X(10733) = 4*X(7723)-5*X(11444) = 2*X(11557)-3*X(11562)

The reciprocal orthologic center of these triangles is X(3581).

X(12270) lies on these lines: {3,74}, {4,11557}, {20,10628}, {30,7731}, {113,7577}, {125,10574}, {146,1531}, {185,3448}, {265,5890}, {381,11561}, {974,9140}, {1176,5621}, {1539,11455}, {1986,3060}, {1993,12165}, {2781,12220}, {2979,12219}, {3543,11807}, {3567,10113}, {4846,11442}, {5640,7687}, {5889,7722}, {6143,12162}, {7547,11439}, {7724,11445}, {7727,11446}, {9826,11451}, {9976,11443}, {10575,12244}, {10657,11452}, {10658,11453}, {11412,12121}, {11422,12227}

X(12270) = reflection of X(i) in X(j) for these (i,j): (4,11562), (3448,185), (5889,7722), (10733,1986), (11412,12121), (12111,110), (12244,10575)
X(12270) = orthologic center of these triangles: 3rd anti-Euler to orthocentroidal
X(12270) = X(80)-of-3rd-anti-Euler-triangle if ABC is acute
X(12270) = {X(1986), X(10733)}-harmonic conjugate of X(3060)

X(12271) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO ARIES

Trilinears cos(A)*((6*cos(A)+2*cos(3*A))*cos(B-C)+(-2*cos(2*A)+1)*cos(2*(B-C))+3*cos(2*A)-2) : :
X(12271) = 4*X(68)-5*X(11444) = 4*X(155)-3*X(3060) = 6*X(3167)-5*X(3567)

The reciprocal orthologic center of these triangles is X(7387).

X(12271) lies on these lines: {68,11444}, {110,9937}, {155,3060}, {1147,1199}, {1993,12166}, {2979,11411}, {3167,3567}, {3564,11412}, {5562,8681}, {5640,12235}, {5889,6193}, {6391,7395}, {6403,12160}, {9820,11451}, {9926,11443}, {9931,11446}, {9932,11449}, {9938,11454}, {10659,11452}, {10660,11453}

X(12271) = reflection of X(5889) in X(6193)
X(12271) = X(84)-of-3rd-anti-Euler-triangle if ABC is acute
X(12271) = orthologic center of these triangles: 3rd anti-Euler to 2nd Hyacinth

X(12272) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO 1st EHRMANN

Trilinears ((b^2+c^2)*a^4-3*b^2*c^2*a^2-(b^4-3*b^2*c^2+c^4)*(b^2+c^2))*a : :
X(12272) = 4*X(69)-3*X(2979) = 3*X(69)-2*X(3313) = 2*X(193)-3*X(3060) = 4*X(1353)-5*X(3567) = 4*X(1843)-3*X(3060) = 9*X(2979)-8*X(3313) = 3*X(2979)-2*X(12220) = 4*X(3313)-3*X(12220)

The reciprocal orthologic center of these triangles is X(576).

X(12272) lies on these lines: {2,6467}, {4,12271}, {6,110}, {22,9924}, {25,6391}, {52,11387}, {66,69}, {157,4558}, {182,11449}, {193,1843}, {489,12224}, {490,12223}, {511,3146}, {524,9973}, {542,12270}, {1350,11440}, {1351,5198}, {1353,3567}, {1992,9969}, {1993,12167}, {3056,11446}, {3098,11454}, {3564,3575}, {3620,7998}, {3629,9971}, {3630,8705}, {3779,11445}, {5093,9781}, {5157,8542}, {5181,6697}, {6515,11382}, {6776,10574}, {9027,11008}, {9822,11451}, {9967,11444}, {10733,12133}, {11412,11898}

X(12272) = reflection of X(i) in X(j) for these (i,j): (193,1843), (5889,6403), (11412,11898), (12111,5921), (12220,69)
X(12272) = anticomplement of X(6467)
X(12272) = X(7)-of-3rd-anti-Euler-triangle if ABC is acute
X(12272) = {X(12276),X(12277)}-harmonic conjugate of X(12111)
X(12272) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (69,12220,2979), (193,1843,3060), (3620,11574,7998)

X(12273) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO 1st HYACINTH

Trilinears (10*cos(2*A)+2*cos(4*A)+9)*cos(B-C)+(-4*cos(A)-2*cos(3*A))*cos(2*(B-C))-8*cos(A)-cos(3*A) : :
X(12273) = 2*X(74)-3*X(2979) = 3*X(110)-2*X(1986) = 4*X(125)-5*X(11444) = 2*X(265)-3*X(11459) = 5*X(631)-4*X(11806) = 4*X(1511)-3*X(5890) = 4*X(1986)-3*X(5889) = 9*X(5640)-8*X(12236) = 8*X(6699)-9*X(7998) = 3*X(10733)-4*X(12133)

The reciprocal orthologic center of these triangles is X(10112).

X(12273) lies on these lines: {24,110}, {74,2979}, {113,3060}, {125,11444}, {146,511}, {265,11459}, {399,1154}, {542,12219}, {568,10272}, {631,11806}, {1511,5890}, {1657,5663}, {1993,12168}, {2781,9924}, {3091,11800}, {3448,5562}, {5640,12236}, {6101,10620}, {6241,12121}, {6699,7998}, {9833,10628}, {10625,12244}, {10663,11452}, {10664,11453}, {10733,12133}, {11422,12228}

X(12273) = reflection of X(i) in X(j) for these (i,j): (3448,5562), (5889,110), (6241,12121), (7731,399), (10620,6101), (12244,10625)
X(12273) = X(104)-of-3rd-anti-Euler-triangle if ABC is acute

X(12274) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO LUCAS ANTIPODAL

Trilinears a*(4*SA*(S-SB-SC)*b^2*c^2+S*(3*S^2+(SA-2*SW)*SA)*(2*S-SW)) : :
X(12274) = 8*X{486} - 9*X{7998} = 4*X{487} - 3*X{3060} = 16*X{642} - 15*X{11451} = 3*X{2979} - 2*X{12221} = 9*X{5640} - 8*X{12237}

The reciprocal orthologic center of these triangles is X(3).

X(12274) lies on these lines: {486,7998}, {487,3060}, {642,11451}, {2979,12221}, {3564,12275}, {5640,12237}, {11422,12229}

X(12274) = orthic-to-3rd-anti-Euler similarity image of X(487)

X(12275) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO LUCAS(-1) ANTIPODAL

Trilinears a*(-4*SA*(S+SB+SC)*b^2*c^2+(3*S^2+S*(SA-2*SW)*SA)*(2*S+SW)) : :
X(12275) = 8*X(485)-9*X(7998) = 4*X(488)-3*X(3060) = 16*X(641)-15*X(11451) = 3*X(2979)-2*X(12222) = 9*X(5640)-8*X(12238)

The reciprocal orthologic center of these triangles is X(3).

X(12275) lies on these lines: {485,7998}, {488,3060}, {641,11451}, {1993,12170}, {2979,12222}, {3564,12274}, {5640,12238}, {11422,12230}

X(12275) = orthic-to-3rd-anti-Euler similarity image of X(488)

X(12276) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO LUCAS CENTRAL

Trilinears a*((3*S^2+SA^2+2*SB*SC)*(SW+2*S)+8*S*(SA^2-SB*SC-(4*SA+2*S)*R^2)) : :
X(12276) = 3*X(2979)-2*X(12223) = 3*X(3060)-4*X(6291) = 9*X(5640)-8*X(12239)

The reciprocal orthologic center of these triangles is X(3).

X(12276) lies on these lines: {110,1151}, {489,2979}, {511,3146}, {1993,12171}, {3060,6291}, {5640,12239}, {5889,6239}, {6252,11445}, {6283,11446}, {7690,11454}, {9823,11451}, {9974,11443}, {10667,11452}, {10668,11453}, {11422,12231}

X(12276) = reflection of X(5889) in X(6239)
X(12276) = {X(12111),X(12272)}-harmonic conjugate of X(12277)
X(12276) = X(176)-of-3rd-anti-Euler-triangle if ABC is acute
X(12276) = orthic-to-3rd-anti-Euler similarity image of X(6291)

X(12277) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO LUCAS(-1) CENTRAL

Trilinears a*((3*S^2+SA^2+2*SB*SC)*(SW-2*S)-8*S*(SA^2-SB*SC-(4*SA-2*S)*R^2)) : :
X(12277) = 3*X(2979)-2*X(12224) = 3*X(3060)-4*X(6406) = 9*X(5640)-8*X(12240)

The reciprocal orthologic center of these triangles is X(3).

X(12277) lies on these lines: {110,1152}, {490,2979}, {511,3146}, {1993,12172}, {3060,6406}, {5640,12240}, {5889,6400}, {6404,11445}, {6405,11446}, {7692,11454}, {9824,11451}, {9975,11443}, {10671,11452}, {10672,11453}, {11422,12232}

X(12277) = reflection of X(5889) in X(6400)
X(12277) = {X(12111),X(12272)}-harmonic conjugate of X(12276)
X(12277) = X(175)-of-3rd-anti-Euler-triangle if ABC is acute
X(12277) = orthic-to-3rd-anti-Euler similarity image of X(6406)

X(12278) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO MACBEATH

Barycentrics 2*a^10-4*(b^2+c^2)*a^8+(b^4+7*b^2*c^2+c^4)*a^6+(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^4+(b^2-c^2)^2*(b^4-b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(12278) = X = 3*X(376)-2*X(11750) = 4*X(1885)-5*X(11439) = 3*X(2979)-2*X(12225) = 3*X(3060)-4*X(3575) = 9*X(5640)-8*X(12241)

The reciprocal orthologic center of these triangles is X(4).

X(12278) lies on these lines: {4,110}, {5,11449}, {20,2888}, {24,9938}, {30,11412}, {186,9927}, {376,11750}, {382,11441}, {550,11454}, {1092,3153}, {1199,7706}, {1204,3448}, {1503,12272}, {1511,10255}, {1885,11439}, {1993,12173}, {2979,12225}, {3060,3575}, {3070,11447}, {3071,11448}, {5059,5921}, {5318,11452}, {5321,11453}, {5640,12241}, {5889,6240}, {6146,10574}, {6253,11445}, {6284,11446}, {7526,8907}, {7577,12038}, {8550,11443}, {9825,11451}, {10024,11464}, {10619,11003}, {11250,12121}, {11422,12233}, {11550,12086}

X(12278) = reflection of X(5889) in X(6240)
X(12278) = X(65)-of-3rd-anti-Euler-triangle if ABC is acute
X(12278) = {X(20), X(11442)}-harmonic conjugate of X(11440)

X(12279) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO MIDHEIGHT

Barycentrics a^2*((b^2+c^2)*a^6-(3*b^4-7*b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(3*b^4-8*b^2*c^2+3*c^4)*a^2-(b^2-c^2)^2*(b^4+5*b^2*c^2+c^4)) : :
X(12279) = 6*X(2)-5*X(11439) = 7*X(4)-8*X(5462) = 8*X(4)-9*X(5640) = 5*X(4)-6*X(9730) = 4*X(5)-3*X(11455) = 4*X(5462)-7*X(10575) = 15*X(5640)-16*X(9730) = 9*X(5640)-16*X(10575) = 3*X(9730)-5*X(10575) = 4*X(11381)-5*X(11439)

The reciprocal orthologic center of these triangles is X(389).

X(12279) lies on these lines: {30,5889}, {110,1498}, {143,382}, {184,12086}, {185,3060}, {373,3854}, {376,5447}, {389,3543}, {511,5059}, {548,7999}, {550,11459}, {858,2883}, {1147,7464}, {1181,11422}, {1370,6225}, {1425,9539}, {1499,11450}, {1503,12272}, {1593,5012}, {1614,12084}, {1657,5663}, {1658,11468}, {1993,12174}, {2071,6759}, {2777,12270}, {2918,10323}, {3091,11695}, {3100,7355}, {3357,7488}, {3426,7395}, {3516,6800}, {3522,5907}, {3528,5891}, {3529,12271}, {3534,5876}, {3567,3627}, {3830,9781}, {3832,9729}, {3850,11465}, {3855,5892}, {4296,6285}, {5073,6102}, {5076,5946}, {5422,11403}, {6254,11445}, {7509,11472}, {7527,10984}, {7689,12088}, {7691,9920}, {8549,11443}, {10170,10299}, {10304,11793}, {10539,12112}, {10625,11001}, {10675,11452}, {10676,11453}, {11250,11464}, {11456,12085}, {12082,12163}

X(12279) = reflection of X(i) in X(j) for these (i,j): (4,10575), (3146,185), (5073,6102), (5889,6241), (11412,1657), (12111,20)
X(12279) = anticomplement of X(11381)
X(12279) = X(8)-of-3rd-anti-Euler-triangle if ABC is acute
X(12279) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,11381,11439), (4,10574,5640), (20,12111,2979), (185,3146,3060), (376,12162,11444), (1498,11413,110), (2071,6759,11449), (3357,7488,11454), (3522,5907,7998), (3832,9729,11451)

X(12280) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO REFLECTION

Trilinears (2*cos(2*A)+2*cos(4*A)+3)*cos(B-C)-2*cos(3*A)*cos(2*(B-C))+cos(3*A) : :
X(12280) = 4*X(143)-3*X(195) = 8*X(143)-9*X(7730) = 2*X(195)-3*X(7730) = 4*X(1493)-5*X(3567) = 3*X(2888)-2*X(5562) = 3*X(2979)-2*X(12226) = 3*X(3060)-4*X(6152) = 9*X(5640)-8*X(12242)

The reciprocal orthologic center of these triangles is X(6243).

X(12280) lies on these lines: {4,12273}, {52,11271}, {54,6644}, {110,143}, {155,3060}, {382,1154}, {539,5889}, {1350,7691}, {1493,3567}, {1595,11664}, {1993,12175}, {2888,3153}, {2914,10539}, {2979,12226}, {3519,11412}, {5640,12242}, {5965,12272}, {6255,11445}, {6286,11446}, {9827,11451}, {9977,11443}, {10574,10619}, {10677,11452}, {10678,11453}, {11422,12234}, {12270,12278}

X(12280) = reflection of X(i) in X(j) for these (i,j): (5889,6242), (11271,52), (11412,3519)
X(12280) = X(79)-of-3rd-anti-Euler-triangle if ABC is acute

X(12281) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO ANTI-ORTHOCENTROIDAL

Trilinears (14*cos(2*A)+2*cos(4*A)+11)*cos(B-C)+(-8*cos(A)-2*cos(3*A))*cos(2*(B-C))-12*cos(A)-5*cos(3*A) : :
X(12281) = 2*X(110)-3*X(11459) = 4*X(125)-3*X(5890) = 3*X(568)-4*X(11801) = 4*X(1511)-5*X(11444) = 4*X(1539)-5*X(11439) = 5*X(1656)-4*X(11561) = 4*X(1986)-5*X(3567) = 3*X(1986)-4*X(11746) = 3*X(5890)-2*X(7722) = 4*X(7723)-3*X(11459) = 5*X(7731)-8*X(11807)

X(12281) lies on these lines: {2,11562}, {3,74}, {4,7730}, {125,5890}, {146,12162}, {185,6143}, {265,5889}, {568,11801}, {578,2914}, {1539,11439}, {1656,11561}, {1986,3567}, {2781,6403}, {2918,8718}, {2979,12121}, {3060,10113}, {3091,11557}, {3153,3448}, {6000,12244}, {7592,12165}, {7687,9781}, {7724,11460}, {7727,11461}, {9826,11465}, {9976,11458}, {10224,10264}, {10657,11466}, {10658,11467}, {11412,12219}, {11423,12227}

X(12281) = reflection of X(i) in X(j) for these (i,j): (110,7723), (146,12162), (399,5876), (5889,265), (6241,74), (7722,125), (7731,4), (11412,12219), (12270,3)
X(12281) = anticomplement of X(11562)
X(12281) = X(80)-of-4th-anti-Euler-triangle if ABC is acute
X(12281) = orthologic center of these triangles: 4th anti-Euler to orthocentroidal
X(12281) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (110,7723,11459), (125,7722,5890), (6241,11459,11464)

X(12282) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO ARIES

Trilinears cos(A)*((10*cos(A)+2*cos(3*A))*cos(B-C)+(-2*cos(2*A)+1)*cos(2*(B-C))+cos(2*A)-4) : :
X(12282) = 4*X(68)-3*X(11459) = 4*X(155)-5*X(3567) = 3*X(3060)-2*X(12164)

The reciprocal orthologic center of these triangles is X(7387).

X(12282) lies on these lines: {3,12271}, {52,6995}, {68,11459}, {155,1995}, {185,8681}, {1147,11423}, {1370,11411}, {1593,6391}, {1614,9937}, {3060,5198}, {3564,3575}, {5890,6193}, {7592,12166}, {9781,12235}, {9820,11465}, {9926,11458}, {9931,11461}, {9932,11464}, {9938,11468}, {10659,11466}, {10660,11467}

X(12282) = reflection of X(i) in X(j) for these (i,j): (11412,11411), (12271,3)
X(12282) = orthologic center of these triangles: 4th anti-Euler to 2nd Hyacinth
X(12282) = X(84)-of-4th-anti-Euler-triangle if ABC is acute
X(12282) = {X(5889), X(12272)}-harmonic conjugate of X(3575)

X(12283) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO 1st EHRMANN

Trilinears 2*(2*cos(2*A)-cos(4*A)-1)*cos(B-C)-2*(2*cos(A)-cos(3*A))*cos(2*(B-C))+9*cos(A)-7*cos(3*A) : :
X(12283) = 4*X(182)-3*X(11188) = 4*X(1353)-3*X(3060) = 4*X(1843)-5*X(3567) = 3*X(2979)-2*X(11898) = 3*X(5890)-2*X(6403) = 2*X(5921)-3*X(11459) = 3*X(9971)-4*X(12007)

The reciprocal orthologic center of these triangles is X(576).

X(12283) lies on these lines: {3,12272}, {4,6467}, {6,1173}, {20,2013}, {24,9924}, {69,11457}, {74,1296}, {154,11746}, {182,11188}, {511,3529}, {542,12281}, {1351,11456}, {1353,3060}, {1843,3567}, {2393,5890}, {2979,11898}, {3056,11461}, {3098,11468}, {3564,11412}, {3779,11460}, {5050,9707}, {5921,9967}, {6391,11414}, {7592,12167}, {7998,10300}, {7999,11574}, {8550,9973}, {9822,11465}, {9971,12007}, {11387,11432}

X(12283) = reflection of X(i) in X(j) for these (i,j): (4,6467), (5921,9967), (6403,6776), (9973,8550), (11412,12220), (12272,3)
X(12283) = X(7)-of-4th-anti-Euler-triangle if ABC is acute
X(12283) = {X(12287),X(12288)}-harmonic conjugate of X(6241)
X(12283) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5921,9967,11459), (6403,6776,5890)

X(12284) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO 1st HYACINTH

Trilinears (14*cos(2*A)+2*cos(4*A)+13)*cos(B-C)-2*(2*cos(A)+cos(3*A))*cos(2*(B-C))-16*cos(A)-3*cos(3*A) : :
X(12284) = 3*X(2)-4*X(11806) = 3*X(4)-4*X(11800) = 2*X(110)-3*X(5890) = 4*X(113)-5*X(3567) = 4*X(125)-3*X(11459) = 4*X(1112)-3*X(10706) = 3*X(2979)-4*X(12041) = 3*X(3060)-2*X(7728) = 8*X(6699)-7*X(7999) = 2*X(7723)-3*X(9140)

The reciprocal orthologic center of these triangles is X(10112).

X(12284) lies on these lines: {2,11806}, {3,12273}, {4,11800}, {52,146}, {74,9938}, {110,5890}, {113,3567}, {125,11459}, {265,12111}, {382,5663}, {399,6102}, {511,12244}, {542,6403}, {1112,10706}, {1154,10620}, {1511,9704}, {1614,2931}, {1986,10594}, {2979,12041}, {3047,11464}, {3060,7728}, {3153,3448}, {6699,7999}, {7592,12168}, {7723,9140}, {9781,12236}, {10663,11466}, {10664,11467}, {11423,12228}, {12270,12278}

X(12284) = reflection of X(i) in X(j) for these (i,j): (146,52), (399,6102), (7731,5889), (11412,74), (12111,265), (12273,3), (12281,3448)
X(12284) = X(104)-of-4th-anti-Euler-triangle if ABC is acute

X(12285) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO LUCAS ANTIPODAL

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

The reciprocal orthologic center of these triangles is X(3).

X(12285) lies on these lines: {3,12274}, {486,7999}, {487,3567}, {642,11465}, {3564,12286}, {7592,12169}, {9781,12237}, {11412,12221}, {11423,12229}

X(12285) = orthic-to-4th-anti-Euler similarity image of X(487)

X(12286) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO LUCAS(-1) ANTIPODAL

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

The reciprocal orthologic center of these triangles is X(3).

X(12286) lies on these lines: {3,12275}, {485,7999}, {488,3567}, {641,11465}, {3564,12285}, {7592,12170}, {9781,12238}, {11412,12222}, {11423,12230}

X(12286) = orthic-to-4th-anti-Euler similarity image of X(488)

X(12287) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO LUCAS CENTRAL

Trilinears ((16*R^2-5*SW)*S^2-2*S*(S^2-(20*R^2-3*SA-2*SW)*SA)+(4*R^2-3*SA+2*SW)*SA*SW)*a : :
X(12287) = 5*X(3567)-4*X(6291) = 3*X(5890)-2*X(6239)

The reciprocal orthologic center of these triangles is X(3).

X(12287) lies on these lines: {3,12276}, {511,3529}, {1151,1614}, {3567,6291}, {5890,6239}, {6252,11460}, {6283,11461}, {7592,12171}, {7690,11468}, {9781,12239}, {9823,11465}, {9974,11458}, {10667,11466}, {10668,11467}, {11412,12223}, {11423,12231}

X(12287) = {X(6241),X(12283)}-harmonic conjugate of X(12288)
X(12287) = X(176)-of-4th-anti-Euler-triangle if ABC is acute
X(12287) = orthic-to-4th-anti-Euler similarity image of X(6291)

X(12288) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO LUCAS(-1) CENTRAL

Trilinears ((16*R^2-5*SW)*S^2+2*S*(S^2-(20*R^2-3*SA-2*SW)*SA)+(4*R^2-3*SA+2*SW)*SA*SW)*a : :
X(12288) = 5*X(3567)-4*X(6406) = 3*X(5890)-2*X(6400) = 7*X(9781)-8*X(12240)

The reciprocal orthologic center of these triangles is X(3).

X(12288) lies on these lines: {3,12277}, {511,3529}, {1152,1614}, {3567,6406}, {5890,6400}, {6404,11460}, {6405,11461}, {7592,12172}, {7692,11468}, {9781,12240}, {9824,11465}, {9975,11458}, {10671,11466}, {10672,11467}, {11412,12224}, {11423,12232}

X(12288) = {X(6241),X(12283)}-harmonic conjugate of X(12287)
X(12288) = X(175)-of-4th-anti-Euler-triangle if ABC is acute
X(12288) = orthic-to-4th-anti-Euler similarity image of X(6406)

X(12289) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO MACBEATH

Trilinears (4*cos(2*A)+3)*cos(B-C)-2*cos(A)*cos(2*(B-C))-3*cos(A)-2*cos(3*A) : :
X(12289) = 4*X(1885)-3*X(11455) = 5*X(3567)-4*X(3575) = 5*X(3567)-6*X(12022) = 2*X(3575)-3*X(12022) = 3*X(5890)-4*X(6146) = 3*X(5890)-2*X(6240) = 3*X(7576)-4*X(12241)

The reciprocal orthologic center of these triangles is X(4).

X(12289) lies on these lines: {3,12278}, {4,54}, {5,10546}, {20,68}, {30,5889}, {185,12063}, {265,1658}, {381,9707}, {382,11456}, {550,11468}, {1147,3153}, {1503,12283}, {1885,11455}, {2072,11449}, {3070,11462}, {3071,11463}, {3567,3575}, {3583,9638}, {3627,11422}, {5073,12174}, {5318,11466}, {5321,11467}, {5448,9544}, {5449,10298}, {5654,9705}, {5878,10721}, {5890,6146}, {5944,10254}, {6253,11460}, {6284,11461}, {6293,7731}, {6776,8537}, {7488,9927}, {7576,9781}, {7592,12173}, {8550,11458}, {9825,11465}, {9932,11413}, {10018,11704}, {11270,11564}, {11412,12225}, {11423,12233}, {11430,11572}, {12273,12281}

X(12289) = reflection of X(i) in X(j) for these (i,j): (20,11750), (6240,6146), (11412,12225), (12278,3)
X(12289) = X(65)-of-4th-anti-Euler-triangle if ABC is acute

X(12289) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,12254,184), (20,3448,7689), (20,11457,74), (3575,12022,3567), (6146,6240,5890), (7576,12241,9781)

X(12290) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO MIDHEIGHT

Trilinears (2*cos(2*A)+5)*cos(B-C)-7*cos(A) : :
Barycentrics a^2[b^14 + c^14 + a^12(b^2 + c^2) - a^10(6b^4 + b^2c^2 + 6c^4) + a^8(15b^6 - 4b^4c^2 - 4b^2c^4 + 15c^6) - a^6(20b^8 - 6b^6c^2 - 12b^4c^4 - 6b^2c^6 + 20c^8) + a^4(15b^10 + b^8c^2 - 12b^6c^4 - 12b^4c^6 + b^2c^8 + 15c^10) - a^2(6b^12 + 5b^10c^2 - 18b^8c^4 + 14b^6c^6 - 18b^4c^8 + 5b^2c^10 + 6c^12) + 2b^12c^2 - 8b^10c^4 + 5b^8c^6 + 5b^6c^8 - 8b^4c^10 + 2b^2c^12]/(b^2 + c^2 - a^2) : :
X(12290) = 7*X(4)-6*X(51) = 3*X(4)-2*X(185) = 5*X(4)-4*X(389) = 6*X(4)-5*X(3567) = 4*X(4)-3*X(5890) = 8*X(4)-7*X(9781) = 9*X(4)-8*X(10110) = 4*X(5)-5*X(11439) = 3*X(20)-4*X(1216) = 2*X(20)-3*X(11459) = 8*X(1216)-9*X(11459) = 2*X(1216)-3*X(12162) = 3*X(11459)-4*X(12162)

The reciprocal orthologic center of these triangles is X(389).

X(12290) lies on these lines: {30,11412}, {52,3543}, {54,1593}, {110,12084}, {143,5076}, {186,3357}, {376,5907}, {378,1498}, {381,10574}, {382,5663}, {403,6247}, {477,6080}, {548,7998}, {550,11444}, {568,3853}, {1092,7464}, {1147,12086}, {1154,5073}, {1181,11423}, {1204,3518}, {1503,12283}, {1514,7729}, {1594,2883}, {1597,7592}, {1656,11017}, {1657,2979}, {1658,11454}, {1870,6285}, {1907,10938}, {2013,3146}, {2071,10539}, {2777,12281}, {3060,3627}, {3090,10219}, {3516,9707}, {3520,6759}, {3522,5891}, {3528,11793}, {3529,5562}, {3534,11591}, {3541,5656}, {3544,11695}, {3545,9729}, {3830,6102}, {3832,9730}, {3839,5462}, {3843,5640}, {3850,11451}, {4846,7544}, {5059,10625}, {5068,5892}, {5072,12046}, {5870,12288}, {5871,12287}, {5894,10295}, {5895,6152}, {6198,7355}, {6254,11460}, {6696,10018}, {7503,11472}, {7691,12083}, {7728,12270}, {8549,11458}, {10540,11250}, {10594,10605}, {10675,11466}, {10676,11467}, {11270,11738}, {11441,12085}

X(12290) = reflection of X(i) in X(j) for these (i,j): (4,11381), (20,12162), (1657,5876), (3529,5562), (5059,10625), (5889,382), (5890,11455), (6241,4), (7731,10721), (11412,12111), (12270,7728), (12279,3), (12284,10733)
X(12290) = anticomplement of X(10575)
X(12290) = X(8)-of-4th-anti-Euler-triangle if ABC is acute
X(12290) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,185,3567), (4,5890,9781), (4,6241,5890), (4,11381,11455), (20,12162,11459), (185,3567,5890), (186,3357,11468), (376,5907,7999), (378,1498,1614), (1593,11456,54), (1597,12174,7592), (1657,5876,2979), (1870,6285,11461), (3520,6759,11464), (3520,12112,6759), (3545,9729,11465), (3567,6241,185), (6241,11455,4), (10540,11250,11449)

X(12291) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO REFLECTION

Trilinears (2*cos(2*A)-2*cos(4*A)+1)*cos(B-C)+2*cos(3*A)*cos(2*(B-C))-3*cos(3*A) : :
X(12291) = 5*X(54)-4*X(973) = 6*X(54)-5*X(3567) = 3*X(54)-2*X(6152) = 4*X(54)-3*X(7730) = 3*X(54)-4*X(11577) = 2*X(185)-3*X(12254) = 3*X(195)-2*X(10263) = 6*X(973)-5*X(6152) = 16*X(973)-15*X(7730) = 3*X(973)-5*X(11577)

The reciprocal orthologic center of these triangles is X(6243).

X(12291) lies on these lines: {3,12280}, {6,24}, {20,12284}, {185,12254}, {195,1614}, {511,11271}, {539,11412}, {1154,1657}, {1205,11457}, {1216,2888}, {1493,3060}, {2013,12163}, {2914,6759}, {2979,3519}, {5890,6242}, {5965,12283}, {6255,11460}, {6286,11461}, {7592,12175}, {7691,11468}, {9781,12242}, {9827,11465}, {9977,11458}, {10677,11466}, {10678,11467}, {11423,12234}, {12273,12281}

X(12291) = reflection of X(i) in X(j) for these (i,j): (6152,11577), (6242,10619), (11412,12226), (12280,3)
X(12291) = X(79)-of-4th-anti-Euler-triangle if ABC is acute
X(12291) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (54,6152,3567), (3567,6152,7730), (6152,11577,54)

X(12292) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO ANTI-ORTHOCENTROIDAL

Trilinears ((6*cos(A)+cos(3*A))*cos(B-C)-3*cos(2*A)-7/2)*sec(A) : :
X(12292) = 3*X(4)-2*X(1112) = 3*X(4)-X(7722) = 4*X(1112)-3*X(1986) = X(1112)-3*X(12133) = 3*X(1986)-2*X(7722) = 3*X(5890)-4*X(11746) = X(10721)-3*X(11455) = 5*X(11439)-X(12270) = 3*X(11455)+X(12281)

X(12292) lies on these lines: {4,94}, {24,64}, {25,10620}, {30,7723}, {34,7727}, {70,11744}, {110,378}, {113,1594}, {125,403}, {185,7687}, {186,12041}, {235,10264}, {399,1593}, {541,7576}, {974,6241}, {1511,3520}, {1902,2771}, {1905,11670}, {2777,6240}, {2781,6403}, {3028,6198}, {3043,5609}, {3091,9826}, {3146,12219}, {5504,11441}, {5890,11746}, {6152,10628}, {6699,10018}, {7547,11439}, {7724,11471}, {9976,11470}, {10151,11801}, {10657,11475}, {10658,11476}, {10733,12111}, {11403,12165}, {11424,12227}

X(12292) = midpoint of X(i) and X(j) for these {i,j}: {74,12290}, {3146,12219}, {10721,12281}, {10733,12111}
X(12292) = reflection of X(i) in X(j) for these (i,j): (4,12133), (185,7687), (1986,4), (6240,12140), (6241,974), (7722,1112), (10575,6699)
X(12292) = polar circle-inverse-of-X(7728)
X(12292) = orthologic center of these triangles: anti-excenters-reflections to orthocentroidal
X(12292) = X(80)-of-anti-excenters-reflections-triangle if ABC is acute
X(12292) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,7722,1112), (1112,7722,1986), (11455,12281,10721)

X(12293) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO ARIES

Trilinears (6*cos(A)*cos(B-C)-2*cos(2*(B-C))-3*cos(2*A)-3)*cos(A) : :
X(12293) = 3*X(3)-4*X(5449) = 3*X(4)-X(6193) = 3*X(155)-2*X(6193) = 3*X(381)-2*X(1147) = 4*X(546)-3*X(5654) = 5*X(1656)-4*X(12038) = 3*X(1853)-2*X(12084) = X(2013)+3*X(11455) = 3*X(3830)-X(12164) = 2*X(5449)-3*X(9927)

The reciprocal orthologic center of these triangles is X(7387).

X(12293) lies on these lines: {3,125}, {4,155}, {5,11425}, {22,12289}, {24,9938}, {30,64}, {34,9931}, {52,12173}, {70,12225}, {185,12235}, {378,9932}, {381,1147}, {382,6243}, {539,3830}, {546,5654}, {912,3901}, {1069,3583}, {1593,9937}, {1656,12038}, {1657,7689}, {1853,12084}, {1885,11472}, {2013,11455}, {3091,9820}, {3146,11411}, {3157,3585}, {3167,3843}, {3564,3627}, {3853,9936}, {5504,10113}, {5663,5895}, {6284,10055}, {6564,8909}, {6800,12254}, {7354,10071}, {7706,11432}, {9926,11470}, {10659,11475}, {10660,11476}, {10733,12111}, {11403,12166}, {11414,11750}, {11439,12271}

X(12293) = midpoint of X(3146) and X(11411)
X(12293) = reflection of X(i) in X(j) for these (i,j): (3,9927), (155,4), (185,12235), (1657,7689), (5504,10113), (12118,5), (12163,68)
X(12293) = orthologic center of these triangles: anti-excenters-reflections to 2nd Hyacinth
X(12293) = X(84)-of-anti-excenters-reflections-triangle if ABC is acute
X(12293) = {X(3167), X(3843)}-harmonic conjugate of X(5448)

X(12294) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO 1st EHRMANN

Trilinears ((3*cos(A)+cos(3*A))*cos(B-C)+4)*sec(A) : :
X(12294) = 3*X(4)-X(6403) = 3*X(51)-4*X(5480) = 2*X(1350)-3*X(3917) = 3*X(1843)-2*X(6403) = 5*X(3618)-4*X(9729) = 5*X(11439)-X(12272) = 3*X(11455)+X(12283)

The reciprocal orthologic center of these triangles is X(576).

X(12294) lies on these lines: {2,12058}, {3,1974}, {4,69}, {6,64}, {20,11574}, {24,3098}, {25,1350}, {30,9967}, {33,1469}, {34,3056}, {39,2211}, {51,125}, {52,1595}, {141,235}, {182,378}, {184,1619}, {193,11469}, {232,3094}, {373,5094}, {389,3088}, {468,5650}, {518,1902}, {542,12292}, {1205,2777}, {1216,1598}, {1351,1597}, {1353,5095}, {1503,1885}, {1596,5891}, {1907,3867}, {2063,9306}, {2807,3751}, {2854,12133}, {2979,6995}, {3060,7378}, {3089,10519}, {3091,9822}, {3146,12220}, {3313,3575}, {3516,5085}, {3517,5447}, {3520,5092}, {3564,12162}, {3618,9729}, {3619,6622}, {3779,11471}, {3819,6353}, {4219,4260}, {4232,7998}, {5017,10311}, {5097,7722}, {5104,10985}, {5198,7716}, {5921,8681}, {5943,8889}, {5969,12131}, {6000,6776}, {6756,10625}, {7507,9969}, {7715,10627}, {9024,12138}, {10628,10752}, {11403,11477}, {11439,12272}, {11455,12283}

X(12294) = midpoint of X(i) and X(j) for these {i,j}: {193,12111}, {3146,12220}, {6467,11381}
X(12294) = reflection of X(i) in X(j) for these (i,j): (20,11574), (69,5907), (185,6), (1843,4)
X(12294) = X(7)-of-anti-excenters-reflections-triangle if ABC is acute
X(12294) = X(20)-of-1st-orthosymmedial-triangle
X(12294) = {X(12298),X(12299)}-harmonic conjugate of X(4)

X(12295) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO 1st HYACINTH

Trilinears (6*cos(2*A)+7)*cos(B-C)-3*cos(A)*cos(2*(B-C))-9*cos(A)-2*cos(3*A) : :
X(12295) = 3*X(3)-4*X(6723) = 3*X(4)-X(110) = 2*X(110)-3*X(113) = X(110)+3*X(10733) = X(113)+2*X(10733) = 3*X(125)-4*X(11801) = 3*X(125)-2*X(12041) = 3*X(265)-X(10620) = 3*X(382)+X(10620) = 2*X(6723)-3*X(7687) = 3*X(10113)-X(12041)

The reciprocal orthologic center of these triangles is X(10112).

X(12295) lies on these lines: {3,6723}, {4,110}, {20,6699}, {30,125}, {52,3627}, {64,265}, {74,3146}, {115,2420}, {185,12236}, {381,5972}, {399,5076}, {511,7723}, {541,3448}, {542,1351}, {546,1511}, {974,10575}, {1112,11562}, {1539,3853}, {1593,2931}, {1699,11723}, {1986,5446}, {3060,7722}, {3818,5181}, {3839,11693}, {3845,5642}, {3861,10272}, {5449,11454}, {5609,12102}, {6000,11800}, {6564,8998}, {7978,9812}, {9140,12244}, {9730,11746}, {9880,11656}, {10264,10990}, {10297,10564}, {10663,11475}, {10664,11476}, {10723,11005}, {10728,10778}, {11403,12168}, {11424,12228}, {11439,12273}, {11455,12284}, {12133,12162}

X(12295) = midpoint of X(i) and X(j) for these {i,j}: {4,10733}, {74,3146}, {265,382}, {3448,10721}, {10723,11005}, {10728,10778}
X(12295) = reflection of X(i) in X(j) for these (i,j): (3,7687), (20,6699), (113,4), (125,10113), (185,12236), (1511,546), (1539,3853), (1986,5446), (5181,3818), (5642,3845), (10272,3861), (10564,10297), (10575,974), (10990,10264), (11562,1112), (11656,9880), (11693,3839), (12041,11801), (12121,5972), (12162,12133)
X(12295) = anticomplement of X(38726)
X(12295) = X(10698)-of-orthic-triangle if ABC is acute
X(12295) = X(104)-of-anti-excenters-reflections-triangle if ABC is acute
X(12295) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381,12121,5972), (3448,3543,10721), (10113,12041,11801), (11801,12041,125)

X(12296) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO LUCAS ANTIPODAL

Barycentrics (SA-2*SW)*S^2+S*(4*S^2+5*SA^2-5*SA*SW)-2*(SA-SW)*SA*SW : :
X(12296) = 3*X(2)-4*X(6251) = 3*X(4)-2*X(6290) = 4*X(642)-5*X(3091) = 3*X(3146)+2*X(6280) = 3*X(5731)-4*X(12268)

The reciprocal orthologic center of these triangles is X(3).

X(12296) lies on these lines: {2,6251}, {4,487}, {20,486}, {30,12256}, {148,5871}, {185,12237}, {382,3564}, {488,6231}, {516,9906}, {642,3091}, {3071,8406}, {3146,5870}, {3523,6119}, {4293,10083}, {4294,10067}, {5731,12268}, {6459,8375}, {11403,12169}, {11424,12229}, {11439,12274}, {11455,12285}

X(12296) = midpoint of X(3146) and X(12221)
X(12296) = reflection of X(i) in X(j) for these (i,j): (20,486), (185,12237), (487,4), (12123,6251)
X(12296) = anticomplement of X(12123)
X(12296) = orthic-to-anti-excenters-reflections similarity image of X(487)

X(12297) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO LUCAS(-1) ANTIPODAL

Barycentrics (SA-2*SW)*S^2-S*(4*S^2+5*SA^2-5*SA*SW)-2*(SA-SW)*SA*SW : :
X(12297) = 3*X(2)-4*X(6250) = 3*X(4)-2*X(6289) = 4*X(641)-5*X(3091) = 3*X(3146)+2*X(6279) = 3*X(5731)-4*X(12269)

The reciprocal orthologic center of these triangles is X(3).

X(12297) lies on these lines: {2,6250}, {4,488}, {20,485}, {30,12257}, {148,5870}, {185,12238}, {382,3564}, {487,6230}, {516,9907}, {641,3091}, {671,8982}, {3070,8414}, {3146,5871}, {3523,6118}, {4293,10084}, {4294,10068}, {5731,12269}, {6460,8376}, {11403,12170}, {11424,12230}, {11439,12275}, {11455,12286}

X(12297) = midpoint of X(3146) and X(12222)
X(12297) = reflection of X(i) in X(j) for these (i,j): (20,485), (185,12238), (488,4), (12124,6250)
X(12297) = anticomplement of X(12124)
X(12297) = orthic-to-anti-excenters-reflections similarity image of X(488)

X(12298) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO LUCAS CENTRAL

Trilinears a*SB*SC*((SB+SC)*S^2+2*S*(S^2+SA^2)+2*SW*SA^2) : :
X(12298) = 3*X(4)-X(6239) = 2*X(6239)-3*X(6291) = 3*X(11455)+X(12287)

The reciprocal orthologic center of these triangles is X(3).

X(12298) lies on these lines: {4,69}, {24,7690}, {33,7362}, {34,6283}, {185,3070}, {1151,1593}, {1160,8948}, {3091,9823}, {3092,9974}, {3146,12223}, {6252,11471}, {10311,11474}, {10667,11475}, {10668,11476}, {11403,12171}, {11424,12231}, {11439,12276}, {11455,12287}

X(12298) = midpoint of X(3146) and X(12223)
X(12298) = reflection of X(i) in X(j) for these (i,j): (185,12239), (6291,4)
X(12298) = {X(4),X(12294)}-harmonic conjugate of X(12299)
X(12298) = X(176)-of-anti-excenters-reflections-triangle if ABC is acute
X(12298) = orthic-to-anti-excenters-reflections similarity image of X(6291)

X(12299) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO LUCAS(-1) CENTRAL

Trilinears a*SB*SC*((SB+SC)*S^2-2*S*(S^2+SA^2)+2*SW*SA^2) : :
X(12299) = 3*X(4)-X(6400) = 2*X(6400)-3*X(6406) = 5*X(11439)-X(12277)

The reciprocal orthologic center of these triangles is X(3).

X(12299) lies on these lines: {4,69}, {24,7692}, {33,7353}, {34,6405}, {185,3071}, {1152,1593}, {1161,8946}, {3091,9824}, {3093,9975}, {3146,12224}, {6404,11471}, {10311,11473}, {10671,11475}, {10672,11476}, {11403,12172}, {11424,12232}, {11439,12277}, {11455,12288}

X(12299) = midpoint of X(3146) and X(12224)
X(12299) = reflection of X(i) in X(j) for these (i,j): (185,12240), (6406,4)
X(12299) = {X(4),X(12294)}-harmonic conjugate of X(12298)
X(12299) = X(175)-of-anti-excenters-reflections-triangle if ABC is acute
X(12299) = orthic-to-anti-excenters-reflections similarity image of X(6406)

X(12300) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO REFLECTION

Trilinears ((4*cos(A)+cos(3*A))*cos(B-C)-cos(2*A)+1/2)*sec(A) : :
X(12300) = 3*X(4)-X(6242) = 3*X(4)-2*X(11576) = 3*X(54)-X(6241) = 2*X(389)-3*X(3574) = 6*X(973)-7*X(9781) = 3*X(6152)-2*X(6242) = 3*X(6152)-4*X(11576) = 3*X(7691)-5*X(11444) = 3*X(7730)-4*X(11743) = 5*X(11439)-X(12280)

The reciprocal orthologic center of these triangles is X(6243).

X(12300) lies on these lines: {4,93}, {24,7691}, {33,7356}, {34,6286}, {54,64}, {125,389}, {185,12242}, {195,1593}, {403,1209}, {539,12162}, {546,7723}, {973,7547}, {1493,2914}, {1885,12292}, {2904,11426}, {3091,9827}, {3146,12226}, {3518,11591}, {3520,10610}, {3541,10937}, {5562,7576}, {5965,12294}, {6000,10619}, {6240,10625}, {6255,11471}, {7730,11743}, {9977,11470}, {10594,11459}, {10677,11475}, {10678,11476}, {11271,11469}, {11403,12175}, {11424,12234}, {11439,12280}, {11455,12291}, {11472,12111}, {11577,12290}

X(12300) = midpoint of X(3146) and X(12226)
X(12300) = reflection of X(i) in X(j) for these (i,j): (185,12242), (6152,4), (6242,11576)
X(12300) = X(79)-of-anti-excenters-reflections-triangle if ABC is acute
X(12300) = {X(4), X(6242)}-harmonic conjugate of X(11576)

X(12301) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH TO ARIES

Trilinears (4*(5*cos(A)+cos(3*A))*cos(B-C)-2*(cos(2*A)+3)*cos(2*(B-C))-14*cos(2*A)-cos(4*A)-9)*cos(A) : :
X(12301) = 3*X(3)-2*X(9932)

The reciprocal orthologic center of these triangles is X(7387).

X(12301) lies on these lines: {3,68}, {25,12293}, {30,9908}, {56,9931}, {64,12085}, {74,2013}, {155,1593}, {378,6193}, {1147,9818}, {1350,7689}, {3516,12166}, {3564,12084}, {5646,7393}, {6642,9927}, {7387,10117}, {7503,11487}, {9786,12235}, {9820,11479}, {9926,11477}, {10625,12163}, {10659,11480}, {10660,11481}, {11411,11413}, {11440,12271}

X(12301) = reflection of X(i) in X(j) for these (i,j): (3,9938), (9937,3), (11477,9926)
X(12301) = X(84)-of-anti-Hutson-intouch-triangle if ABC is acute
X(12301) = orthologic center of these triangles: anti-Hutson intouch to 2nd Hyacinth

X(12302) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH TO 1st HYACINTH

Trilinears ((22*cos(A)+4*cos(3*A))*cos(B-C)-2*(cos(2*A)+2)*cos(2*(B-C))-11*cos(2*A)-cos(4*A)-10)*cos(A) : :

The reciprocal orthologic center of these triangles is X(10112).

X(12302) lies on these lines: {3,125}, {24,10733}, {25,12295}, {30,10117}, {64,155}, {68,10264}, {74,9938}, {110,378}, {113,1593}, {146,12086}, {394,7723}, {399,1147}, {1069,7727}, {1350,5621}, {1511,7526}, {1993,7722}, {2071,3448}, {2771,9928}, {2777,9914}, {3047,11456}, {3516,12168}, {5646,7514}, {5972,9818}, {6642,7687}, {6644,10113}, {7464,12244}, {9786,12236}, {9908,10990}, {10663,11480}, {10664,11481}, {11250,12118}, {11425,12228}, {11438,11800}, {11440,12273}

X(12302) = reflection of X(i) in X(j) for these (i,j): (68,10264), (155,5504), (399,1147), (2931,3), (2935,12084), (12163,74), (12293,265)
X(12302) = X(104)-of-anti-Hutson-intouch-triangle if ABC is acute

X(12303) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH TO LUCAS ANTIPODAL

Trilinears (4*R^2*S^2-2*S*(S^2-SA*(10*R^2-SA-SW))-(8*R^2-SW)*SA*SW)*a : :

The reciprocal orthologic center of these triangles is X(3).

X(12303) lies on these lines: {3,486}, {25,12296}, {74,12285}, {487,1593}, {642,11479}, {1597,6290}, {3516,12169}, {3564,12085}, {5020,6251}, {9786,12237}, {11413,12221}, {11425,12229}, {11440,12274}

X(12303) = orthic-to-anti-Hutson-intouch similarity image of X(487)

X(12304) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH TO LUCAS(-1) ANTIPODAL

Trilinears (4*R^2*S^2+2*S*(S^2-SA*(10*R^2-SA-SW))-(8*R^2-SW)*SA*SW)*a : :

The reciprocal orthologic center of these triangles is X(3).

X(12304) lies on these lines: {3,485}, {25,12297}, {74,12286}, {488,1593}, {641,11479}, {1597,6289}, {3516,12170}, {3564,12085}, {5020,6250}, {9786,12238}, {11413,12222}, {11425,12230}, {11440,12275}

X(12304) = orthic-to-anti-Hutson-intouch similarity image of X(488)

X(12305) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH TO LUCAS CENTRAL

Trilinears (-4*(-a^2+b^2+c^2)*S+a^4+2*(b^2+c^2)*a^2-2*b^2*c^2-3*c^4-3*b^4)*a : :
Trilinears (S^2-2*SA*S-2*SA*SW)*a : :
X(12305) = X(1151)-4*X(7690)

The reciprocal orthologic center of these triangles is X(3).

X(12305) lies on these lines: {3,6}, {20,492}, {22,5406}, {25,12298}, {30,6289}, {55,7362}, {56,6283}, {74,12287}, {154,5408}, {325,490}, {378,6239}, {488,1503}, {524,12257}, {548,12123}, {1593,6291}, {1853,11090}, {2979,5407}, {3516,12171}, {5480,11292}, {5584,6252}, {6312,6399}, {6813,7778}, {8982,9766}, {9823,11479}, {11413,12223}, {11440,12276}

X(12305) = {X(3),X(1350)}-harmonic conjugate of X(12306)
X(12305) = X(176)-of-anti-Hutson-intouch-triangle if ABC is acute
X(12305) = orthic-to-anti-Hutson-intouch similarity image of X(6291)
X(12305) = reflection of X(i) in X(j) for these (i,j): (3,7690), (1151,3), (11477,9974)

X(12306) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH TO LUCAS(-1) CENTRAL

Trilinears (4*(-a^2+b^2+c^2)*S+a^4+2*(b^2+c^2)*a^2-2*b^2*c^2-3*c^4-3*b^4)*a : :
Trilinears (S^2+2*SA*S-2*SA*SW)*a : :
X(12306) = X(1152)-4*X(7692)

The reciprocal orthologic center of these triangles is X(3).

X(12306) lies on these lines: {3,6}, {20,491}, {22,5407}, {25,12299}, {30,6290}, {55,7353}, {56,6405}, {74,12288}, {154,5409}, {325,489}, {376,1991}, {378,6400}, {487,1503}, {524,12256}, {548,12124}, {1593,6406}, {1853,11091}, {2979,5406}, {3516,12172}, {5480,11291}, {5584,6404}, {6222,6316}, {6811,7778}, {9824,11479}, {11413,12224}, {11440,12277}

X(12306) = reflection of X(i) in X(j) for these (i,j): (3,7692), (1152,3)
X(12306) = {X(3),X(1350)}-harmonic conjugate of X(12305)
X(12306) = X(175)-of-anti-Hutson-intouch-triangle if ABC is acute
X(12306) = orthic-to-anti-Hutson-intouch similarity image of X(6406)

X(12307) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH TO REFLECTION

Trilinears (6*cos(2*A)+4)*cos(B-C)-cos(3*A) : :
X(12307) = 3*X(3)-2*X(54) = 7*X(3)-4*X(1493) = 5*X(3)-4*X(10610) = 4*X(54)-3*X(195) = 7*X(54)-6*X(1493) = X(54)-3*X(7691) = 3*X(381)-4*X(1209) = 4*X(548)-X(11271) = 3*X(568)-4*X(11802) = X(1657)+2*X(3519)

The reciprocal orthologic center of these triangles is X(6243).

X(12307) lies on these lines: {3,54}, {5,7693}, {20,10620}, {25,12300}, {30,2888}, {55,7356}, {56,6286}, {64,1657}, {74,12291}, {378,6242}, {381,1209}, {382,6288}, {399,2917}, {539,3534}, {548,11271}, {550,12254}, {568,11802}, {631,8254}, {973,3527}, {1092,11597}, {1216,3581}, {1350,5965}, {1593,6152}, {1597,11576}, {1656,3574}, {2070,5562}, {3516,12175}, {3523,11803}, {3526,5646}, {3579,9905}, {5054,6689}, {5584,6255}, {5663,5898}, {5876,5899}, {6243,11424}, {7666,10274}, {7689,12302}, {7730,10263}, {7979,8148}, {9786,12242}, {9827,11479}, {9914,9920}, {9977,11477}, {10605,10619}, {10677,11480}, {10678,11481}, {11413,12226}, {11425,12234}, {11440,12280}

X(12307) = reflection of X(i) in X(j) for these (i,j): (3,7691), (195,3), (382,6288), (8148,7979), (9905,3579), (11477,9977), (12254,550)
X(12307) = X(79)-of-anti-Hutson-intouch-triangle if ABC is acute

X(12308) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO ANTI-ORTHOCENTROIDAL

Trilinears (6*cos(2*A)+10)*cos(B-C)-12*cos(A)+cos(3*A) : :
X(12308) = 5*X(3)-4*X(74) = 3*X(3)-4*X(110) = 7*X(3)-8*X(1511) = 5*X(3)-8*X(5609) = 3*X(3)-2*X(10620) = 9*X(3)-8*X(12041) = 3*X(74)-5*X(110) = 8*X(113)-7*X(3851) = 8*X(125)-9*X(5055) = 2*X(125)-3*X(5655) = 3*X(5055)-4*X(5655)

The reciprocal orthologic center of these triangles is X(3581).

X(12308) lies on these lines: {3,74}, {4,11703}, {25,7722}, {113,3851}, {125,5055}, {146,382}, {265,3527}, {378,11935}, {381,3448}, {541,11820}, {542,1351}, {567,12162}, {1482,2771}, {1498,5898}, {1597,12292}, {1598,1986}, {1656,10264}, {3028,7373}, {3043,3516}, {3066,11806}, {3167,12302}, {3295,7727}, {3303,6126}, {3304,7343}, {3526,10272}, {3534,9143}, {5070,6053}, {5073,12164}, {5093,9970}, {5169,11804}, {6407,10819}, {6408,10820}, {7687,11432}, {7724,10306}, {9704,11559}, {9826,11484}, {9976,11482}, {10113,10706}, {10145,10817}, {10146,10818}, {10246,11699}, {10657,11485}, {10658,11486}, {10733,12160}, {11414,12219}, {11426,12227}

X(12308) = reflection of X(i) in X(j) for these (i,j): (3,399), (74,5609), (382,146), (3534,9143), (9919,1498), (10620,110)
X(12308) = Stammler circle-inverse-of-X(110)
X(12308) = orthologic center of these triangles: anti-incircle-circles to orthocentroidal
X(12308) = X(80)-of-anti-incircle-circles-triangle if ABC is acute
X(12308) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (110,10620,3), (399,10620,110)

X(12309) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO ARIES

Trilinears (4*(cos(A)-cos(3*A))*cos(B-C)+2*(cos(2*A)-3)*cos(2*(B-C))-10*cos(2*A)+cos(4*A)-3)*cos(A) : :
X(12309) = 3*X(3)-4*X(9932) = 3*X(3)-2*X(12301) = 2*X(9932)-3*X(9937) = 3*X(9937)-X(12301)

The reciprocal orthologic center of these triangles is X(7387).

X(12309) lies on these lines: {3,68}, {4,12166}, {25,6193}, {155,1351}, {159,10243}, {539,9908}, {567,5544}, {1147,5020}, {1597,12293}, {2013,11456}, {3167,3527}, {3295,9931}, {3564,5596}, {6243,12164}, {6759,8681}, {8193,9896}, {9820,11484}, {9926,11482}, {9927,11479}, {10659,11485}, {10660,11486}, {11411,11414}, {11432,12235}, {11441,12271}

X(12309) = reflection of X(i) in X(j) for these (i,j): (3,9937), (12301,9932)
X(12309) = orthologic center of these triangles: anti-incircle-circles to 2nd Hyacinth
X(12309) = X(84)-of-anti-incircle-circles-triangle if ABC is acute

X(12310) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO 1st HYACINTH

Trilinears (2*(cos(A)-2*cos(3*A))*cos(B-C)+2*(cos(2*A)-1)*cos(2*(B-C))-cos(2*A)+cos(4*A)-2)*cos(A) : :
X(12310) = 3*X(3)-2*X(12302) = 4*X(110)-3*X(3167) = 3*X(2931)-X(12302)

The reciprocal orthologic center of these triangles is X(10112).

X(12310) lies on these lines: {3,125}, {4,12168}, {6,11800}, {22,3448}, {23,3564}, {25,110}, {26,9920}, {68,2937}, {74,11414}, {113,1598}, {155,5898}, {159,1177}, {161,542}, {373,12038}, {382,9932}, {399,7517}, {1511,6642}, {1593,10733}, {1597,12295}, {1995,7693}, {2079,6388}, {2771,9913}, {2930,6144}, {2948,8185}, {3527,5504}, {5020,5972}, {5594,7733}, {5595,7732}, {5609,12166}, {5654,7545}, {5663,7387}, {5889,12165}, {5899,12308}, {6800,8548}, {7514,11801}, {7687,11479}, {7984,8192}, {8276,8912}, {8277,10820}, {9517,11641}, {9714,12309}, {9818,10113}, {10037,10088}, {10046,10091}, {10620,11820}, {10663,11485}, {10664,11486}, {11365,11720}, {11426,12228}, {11432,12236}, {11441,12273}, {11456,12284}, {12082,12244}

X(12310) = reflection of X(i) in X(j) for these (i,j): (3,2931), (9919,7387), (12164,399)
X(12310) = X(104)-of-anti-incircle-circles-triangle if ABC is acute

X(12311) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO LUCAS ANTIPODAL

Trilinears (8*R^2*S^2-2*S*(2*S^2-SA*(-2*SA+SW+8*R^2))-SW*SA*(SW+4*R^2))*a : :

The reciprocal orthologic center of these triangles is X(3).

X(12311) lies on these lines: {3,486}, {4,12169}, {487,1598}, {642,11484}, {1597,12296}, {3564,12312}, {11414,12221}, {11426,12229}, {11432,12237}, {11441,12274}, {11456,12285}

X(12311) = orthic-to-anti-incircle-circles similarity image of X(487)

X(12312) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO LUCAS(-1) ANTIPODAL

Trilinears (8*R^2*S^2+2*S*(2*S^2-SA*(-2*SA+SW+8*R^2))-SW*SA*(SW+4*R^2))*a : :

The reciprocal orthologic center of these triangles is X(3).

X(12312) lies on these lines: {3,485}, {4,12170}, {488,1598}, {641,11484}, {1597,12297}, {3564,12311}, {11414,12222}, {11426,12230}, {11432,12238}, {11441,12275}, {11456,12286}

X(12312) = orthic-to-anti-incircle-circles similarity image of X(488)

X(12313) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO LUCAS CENTRAL

Trilinears (-4*(-a^2+b^2+c^2)*S+a^4-4*(b^2+c^2)*a^2-2*b^2*c^2+3*c^4+3*b^4)*a : :
Trilinears 2 sin A + (2 - cot ω) cos A : :
X(12313) = 5*X(3)-4*X(7690) = 3*X(3)-2*X(12305) = 5*X(1151)-2*X(7690) = 3*X(1151)-X(12305) = 6*X(7690)-5*X(12305)

The reciprocal orthologic center of these triangles is X(3).

X(12313) lies on these lines: {3,6}, {4,12171}, {5,487}, {25,6239}, {30,12257}, {51,5407}, {999,7362}, {1353,12256}, {1597,12298}, {1598,6291}, {1600,9777}, {3155,3167}, {3295,6283}, {3564,6462}, {5020,5409}, {6252,10306}, {8964,11427}, {9823,11484}, {9909,10132}, {11414,12223}, {11441,12276}, {11456,12287}, {11949,12311}

X(12313) = reflection of X(3) in X(1151)
X(12313) = {X(3),X(1351)}-harmonic conjugate of X(12314)
X(12313) = X(176)-of-anti-incircle-circles-triangle if ABC is acute
X(12313) = orthic-to-anti-incircle-circles similarity image of X(6291)
X(12313) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,3311,5050), (3,5093,372), (6,9738,3)

X(12314) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO LUCAS(-1) CENTRAL

Trilinears (4*(-a^2+b^2+c^2)*S+a^4-4*(b^2+c^2)*a^2-2*b^2*c^2+3*c^4+3*b^4)*a : :
Trilinears 2 sin A + (2 + cot ω) cos A : :
X(12314) = 5*X(3)-4*X(7692) = 3*X(3)-2*X(12306) = 5*X(1152)-2*X(7692) = 3*X(1152)-X(12306) = 6*X(7692)-5*X(12306)

The reciprocal orthologic center of these triangles is X(3).

X(12314) lies on these lines: {3,6}, {4,12172}, {5,488}, {25,6400}, {30,12256}, {51,5406}, {999,7353}, {1353,12257}, {1597,12299}, {1598,6406}, {1599,9777}, {3156,3167}, {3295,6405}, {3564,6463}, {5020,5408}, {6404,10306}, {9824,11484}, {9909,10133}, {11414,12224}, {11441,12277}, {11456,12288}, {11950,12312}

X(12314) = reflection of X(3) in X(1152)
X(12314) = {X(3),X(1351)}-harmonic conjugate of X(12313)
X(12314) = X(175)-of-anti-incircle-circles-triangle if ABC is acute
X(12314) = orthic-to-anti-incircle-circles similarity image of X(6406)
X(12314) = {X(6), X(9739)}-harmonic conjugate of X(3)

X(12315) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO MIDHEIGHT

Trilinears (6*cos(2*A)+14)*cos(B-C)-21*cos(A)+cos(3*A) : :
X(12315) = 3*X(3)-2*X(64) = 5*X(3)-6*X(154) = 5*X(3)-4*X(3357) = 11*X(3)-10*X(8567) = 7*X(3)-8*X(10282) = 11*X(3)-12*X(11202) = 13*X(3)-12*X(11204) = 2*X(5)-3*X(5656) = 3*X(381)-4*X(2883) = 2*X(550)-3*X(11206) = 3*X(11206)-X(12250)

The reciprocal orthologic center of these triangles is X(389).

X(12315) lies on these lines: {3,64}, {4,3527}, {5,5544}, {20,11820}, {24,12112}, {25,6241}, {30,6193}, {54,1593}, {185,1598}, {221,6767}, {381,2883}, {382,1351}, {550,11206}, {999,7355}, {1181,1597}, {1482,6001}, {1614,3516}, {1656,6247}, {1657,9833}, {1853,3851}, {2192,7373}, {2777,12308}, {3146,12160}, {3167,12085}, {3295,6285}, {3517,10605}, {3579,9899}, {5054,6696}, {5073,5895}, {5198,5890}, {5663,7387}, {6254,10306}, {6449,10533}, {6450,10534}, {7592,11403}, {7973,8148}, {8549,9968}, {9707,11410}, {9729,11484}, {9909,12163}, {9914,9920}, {9934,10620}, {10076,10535}, {10675,11485}, {10676,11486}, {10721,12165}, {11414,12111}, {11441,12279}

X(12315) = reflection of X(i) in X(j) for these (i,j): (3,1498), (64,6759), (382,5878), (1657,9833), (5073,5895), (8148,7973), (8549,9968), (9899,3579), (10620,9934), (12250,550)
X(12315) = Stammler circle-inverse-of-X(6760)
X(12315) = X(8)-of-anti-incircle-circles-triangle if ABC is acute
X(12315) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (154,3357,3), (1181,1597,11426), (1181,11381,1597), (1593,12290,3426), (7592,11455,11403), (8567,11202,3), (10282,10606,3), (11206,12250,550), (11456,12290,1593)

X(12316) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO REFLECTION

Trilinears (6*cos(2*A)+2)*cos(B-C)+cos(3*A) : :
X(12316) = 3*X(3)-4*X(54) = 5*X(3)-8*X(1493) = 5*X(3)-4*X(7691) = 7*X(3)-8*X(10610) = 3*X(3)-2*X(12307) = 2*X(54)-3*X(195) = 5*X(54)-6*X(1493) = 3*X(381)-2*X(2888) = X(382)+2*X(11271) = 8*X(1209)-9*X(5055)

The reciprocal orthologic center of these triangles is X(6243).

X(12316) lies on these lines: {3,54}, {4,12175}, {24,2914}, {25,6242}, {64,10628}, {146,382}, {155,5898}, {381,2888}, {394,10115}, {399,10263}, {539,3830}, {999,7356}, {1209,5055}, {1351,3818}, {1482,5693}, {1597,12300}, {1598,6152}, {1656,11803}, {1657,12254}, {2937,7712}, {3295,6286}, {3519,3527}, {3526,8254}, {4550,11424}, {5070,5544}, {5073,5895}, {5899,6243}, {6255,10306}, {6515,10255}, {9703,10274}, {9827,11484}, {9977,11482}, {10677,11485}, {10678,11486}, {11414,12226}, {11426,12234}, {11432,12242}, {11441,12280}, {11456,12291}

X(12316) = reflection of X(i) in X(j) for these (i,j): (3,195), (1657,12254), (3519,3574), (7691,1493), (12307,54)
X(12316) = Stammler circle-inverse-of-X(1157)
X(12316) = X(79)-of-anti-incircle-circles-triangle if ABC is acute
X(12316) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (54,12307,3), (195,12307,54)

X(12317) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO ANTI-ORTHOCENTROIDAL

Trilinears (2*cos(2*A)+7/2)*cos(B-C)+cos(A)*cos(2*(B-C))-5*cos(A)+cos(3*A)/2 : :
X(12317) = 3*X(4)-2*X(146) = 3*X(4)-4*X(265) = 9*X(4)-8*X(1539) = 5*X(4)-4*X(7728) = 7*X(4)-8*X(10113) = 4*X(74)-3*X(376) = 4*X(110)-5*X(631) = 3*X(110)-4*X(6699) = 3*X(146)-4*X(1539) = 15*X(631)-16*X(6699)

The reciprocal orthologic center of these triangles is X(3581).

X(12317) lies on these lines: {2,399}, {3,5900}, {4,94}, {5,12308}, {8,2771}, {20,10620}, {69,74}, {110,631}, {113,3545}, {125,3090}, {427,12165}, {497,7727}, {541,6515}, {1056,3028}, {1370,12219}, {1511,3524}, {1553,5627}, {1992,9976}, {2550,7724}, {2930,10519}, {2931,7556}, {2948,5657}, {3525,5609}, {3528,12041}, {3529,11411}, {3533,5972}, {3564,7464}, {3580,12112}, {3616,11699}, {3818,5890}, {3832,11801}, {4295,11670}, {4846,11442}, {5071,5655}, {5422,10821}, {5946,7693}, {5984,7422}, {6126,10056}, {6193,12302}, {6361,9904}, {6643,7723}, {6776,8546}, {7343,10072}, {7392,9826}, {7408,11566}, {7552,11456}, {7687,10706}, {10628,12284}, {10657,11488}, {10658,11489}, {11003,11597}, {11061,11579}, {11382,12140}, {11427,12227}, {11440,12254}, {11457,12281}, {12088,12310}

X(12317) = reflection of X(i) in X(j) for these (i,j): (4,3448), (20,10620), (146,265), (399,10264), (3529,12244), (6193,12302), (6361,9904), (11061,11579), (12112,3580), (12308,5)
X(12317) = anticomplement of X(399)
X(12317) = X(80)-of-anti-inverse-in-incircle-triangle if ABC is acute
X(12317) = antipode of X(4) in rectangular hyperbola passing through X(4), X(8), and the extraversions of X(8)
X(12317) = anticomplementary circle-inverse-of-X(265)
X(12317) = orthologic center of these triangles: anti-inverse-in-incircle to orthocentroidal
X(12317) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (146,265,4), (146,3448,265), (399,10264,2)

X(12318) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO ARIES

Trilinears (2*(4*cos(A)+3*cos(3*A))*cos(B-C)-2*(3*cos(2*A)-1)*cos(2*(B-C))+2*cos(A)*cos(3*(B-C))+6*cos(2*A)-cos(4*A)-1)*cos(A) : :

The reciprocal orthologic center of these triangles is X(7387).

X(12318) lies on these lines: {2,9937}, {4,155}, {5,12309}, {20,12301}, {54,6815}, {68,69}, {376,9938}, {427,12166}, {497,9931}, {631,9932}, {1147,7401}, {1370,2013}, {1992,9926}, {3147,8907}, {3167,7528}, {6403,11382}, {6816,11487}, {7392,9820}, {10659,11488}, {10660,11489}, {11433,12235}, {11442,12271}

X(12318) = reflection of X(i) in X(j) for these (i,j): (20,12301), (12309,5)
X(12318) = anticomplement of X(9937)
X(12318) = X(84)-of-anti-inverse-in-incircle-triangle if ABC is acute
X(12318) = orthologic center of these triangles: anti-inverse-in-incircle to 2nd Hyacinth

X(12319) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO 1st HYACINTH

Trilinears (6*(2*cos(A)+cos(3*A))*cos(B-C)-3*(2*cos(2*A)+1)*cos(2*(B-C))+2*cos(A)*cos(3*(B-C))-3*cos(2*A)-cos(4*A)-3)*cos(A) : :
X(12319) = (2*R^2-SW)*X(4)+2*R^2*X(110)

The reciprocal orthologic center of these triangles is X(10112).

X(12319) lies on these lines: {2,2931}, {4,110}, {5,12310}, {20,12302}, {30,9919}, {69,265}, {74,1370}, {125,6643}, {146,7391}, {323,3153}, {427,12168}, {3448,11411}, {3564,7574}, {5972,7401}, {6699,7386}, {9927,11444}, {10272,11818}, {10663,11488}, {10664,11489}, {11427,12228}, {11433,12236}, {11442,12273}, {11457,12284}

X(12319) = reflection of X(i) in X(j) for these (i,j): (20,12302), (11411,3448), (12310,5)
X(12319) = anticomplement of X(2931)
X(12319) = X(104)-of-anti-inverse-in-incircle-triangle if ABC is acute
X(12319) = anticomplementary-circle-inverse-of-X(1300)
X(12319) = {X(110), X(10733)}-harmonic conjugate of X(12140)

X(12320) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO LUCAS ANTIPODAL

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

The reciprocal orthologic center of these triangles is X(3).

X(12320) lies on these lines: {4,487}, {5,12311}, {20,12303}, {427,12169}, {486,7386}, {642,7392}, {1370,12221}, {3564,12321}, {10996,12123}, {11427,12229}, {11433,12237}, {11442,12274}, {11457,12285}

X(12320) = orthic-to-anti-inverse-in-incircle similarity image of X(487)

X(12321) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO LUCAS(-1) ANTIPODAL

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

The reciprocal orthologic center of these triangles is X(3).

X(12321) lies on these lines: {4,488}, {5,12312}, {20,12304}, {427,12170}, {485,7386}, {641,7392}, {1370,12222}, {3564,12320}, {10996,12124}, {11427,12230}, {11433,12238}, {11442,12275}, {11457,12286}

X(12321) = orthic-to-anti-inverse-in-incircle similarity image of X(488)

X(12322) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO LUCAS CENTRAL

Barycentrics    (a^2-b^2+c^2)*(a^2+b^2-c^2)+(-a^2+b^2+c^2)*S : :
Barycentrics    2*SB*SC+SA*S : :
Barycentrics    sin 2B + sin 2C - sin 2A + sin^2 B + sin^2 C - sin^2 A : :
X(12322) = 3*X(376)-4*X(7690)

The reciprocal orthologic center of these triangles is X(3).

X(12322) lies on these lines: {2,489}, {4,69}, {5,487}, {6,12221}, {20,492}, {30,488}, {183,7000}, {193,3070}, {325,7374}, {376,7690}, {388,7362}, {427,12171}, {486,11291}, {490,1270}, {491,3091}, {497,6283}, {524,12222}, {615,5023}, {639,6561}, {641,11147}, {1007,6811}, {1271,3832}, {1370,12223}, {1587,1992}, {1588,3618}, {2550,6252}, {3069,11293}, {3522,3593}, {3595,5068}, {3619,7388}, {5491,6251}, {5590,11294}, {6214,12296}, {6289,6337}, {6460,7823}, {7392,9823}, {8979,9306}, {10667,11488}, {10668,11489}, {11427,12231}, {11433,12239}, {11442,12276}, {11457,12287}

X(12322) = reflection of X(i) in X(j) for these (i,j): (20,12305), (12313,5)
X(12322) = anticomplement of X(1151)
X(12322) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,69,12323), (4,637,69)
X(12322) = X(176)-of-anti-inverse-in-incircle-triangle if ABC is acute
X(12322) = orthic-to-anti-inverse-in-incircle similarity image of X(6291)

X(12323) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO LUCAS(-1) CENTRAL

Barycentrics    (a^2-b^2+c^2)*(a^2+b^2-c^2)-(-a^2+b^2+c^2)*S : :
Barycentrics    2*SB*SC-SA*S : :
Barycentrics    sin 2B + sin 2C - sin 2A - sin^2 B - sin^2 C + sin^2 A : :
X(12323) = 3*X(376)-4*X(7692)

The reciprocal orthologic center of these triangles is X(3).

X(12323) lies on these lines: {2,490}, {4,69}, {5,488}, {6,12222}, {20,491}, {30,487}, {183,7374}, {193,3071}, {325,7000}, {376,7692}, {388,7353}, {427,12172}, {485,11292}, {489,1271}, {492,3091}, {497,6405}, {524,12221}, {590,5023}, {640,6560}, {642,11147}, {1007,6813}, {1270,3832}, {1370,12224}, {1587,3618}, {1588,1992}, {2550,6404}, {3068,11294}, {3522,3595}, {3593,5068}, {3619,7389}, {5490,6250}, {5591,11293}, {6215,12297}, {6290,6337}, {6459,7823}, {7392,9824}, {10671,11488}, {10672,11489}, {11427,12232}, {11433,12240}, {11442,12277}, {11457,12288}

X(12323) = reflection of X(i) in X(j) for these (i,j): (20,12306), (12314,5)
X(12323) = anticomplement of X(1152)
X(12323) = X(175)-of-anti-inverse-in-incircle-triangle if ABC is acute
X(12323) = orthic-to-anti-inverse-in-incircle similarity image of X(6406)
X(12323) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,69,12322), (4,638,69)

X(12324) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO MIDHEIGHT

Trilinears 4*(cos(2*A)+3)*cos(B-C)+2*cos(A)*cos(2*(B-C))-19*cos(A)+cos(3*A) : :
X(12324) = 3*X(2)-4*X(6247) = 4*X(3)-3*X(11206) = 3*X(4)-2*X(5878) = 4*X(5)-3*X(5656) = 3*X(20)-4*X(5894) = 3*X(64)-2*X(5894) = 6*X(154)-7*X(3523) = 3*X(154)-4*X(6696) = 7*X(3523)-8*X(6696) = 3*X(5656)-2*X(12315) = 4*X(5878)-3*X(6225)

The reciprocal orthologic center of these triangles is X(389).

X(12324) lies on these lines: {2,1498}, {3,11206}, {4,51}, {5,5544}, {8,6001}, {20,64}, {30,11411}, {66,6815}, {125,6622}, {154,3523}, {376,3357}, {388,7355}, {427,12174}, {497,6285}, {511,2013}, {516,9899}, {631,5651}, {1158,6350}, {1181,3088}, {1352,10996}, {1370,12111}, {1559,6526}, {1593,6776}, {1853,2883}, {1895,10365}, {1992,8549}, {2550,6254}, {2777,12317}, {2917,7492}, {3146,6515}, {3332,7513}, {3522,10606}, {3524,10282}, {3527,11431}, {3538,11793}, {3541,11456}, {3543,5895}, {3575,11382}, {3839,5893}, {4293,10076}, {4294,10060}, {4295,7282}, {5059,5925}, {5596,7503}, {5663,12319}, {5731,12262}, {5907,7386}, {6193,12085}, {6643,12162}, {6995,9786}, {6997,10574}, {7288,10535}, {7378,12233}, {7392,9729}, {7408,11745}, {7487,10605}, {7505,12112}, {7544,7729}, {7667,11821}, {8567,10304}, {10192,10303}, {10299,11202}, {10675,11488}, {10676,11489}, {11245,11403}, {11442,12279}

X(12324) = reflection of X(i) in X(j) for these (i,j): (20,64), (1498,6247), (5059,5925), (6193,12085), (6225,4), (9833,3357), (12315,5)
X(12324) = anticomplement of X(1498)
X(12324) = anticomplementary-circle-inverse of X(34170)
X(12324) = X(8)-of-anti-inverse-in-incircle-triangle if ABC is acute
X(12324) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,12315,5656), (154,6696,3523), (1181,3088,11427), (1498,6247,2), (1853,2883,3091), (1899,11381,4), (3357,9833,376), (11457,12290,4)

X(12325) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO REFLECTION

Trilinears (4*cos(2*A)+3)*cos(B-C)+2*cos(A)*cos(2*(B-C))+2*cos(A)+cos(3*A) : :
X(12325) = 9*X(2)-8*X(8254) = X(4)-4*X(3519) = 3*X(4)-4*X(6288) = 4*X(54)-5*X(631) = 3*X(195)-4*X(8254) = 3*X(376)-4*X(7691) = 3*X(376)-2*X(12254) = 5*X(631)-2*X(11271) = 3*X(2888)-2*X(6288) = 3*X(3519)-X(6288)

The reciprocal orthologic center of these triangles is X(6243).

X(12325) lies on these lines: {2,195}, {4,93}, {5,12316}, {8,6951}, {20,10620}, {24,11898}, {54,69}, {68,12319}, {155,7552}, {184,10203}, {323,6143}, {376,539}, {388,7356}, {427,12175}, {497,6286}, {1205,11457}, {1209,3090}, {1352,7730}, {1370,12226}, {1493,3525}, {1992,9977}, {2550,6255}, {2895,6853}, {2914,7505}, {2917,2930}, {2937,5898}, {3060,6153}, {3448,6101}, {3524,10610}, {3529,12324}, {3533,6689}, {3545,3574}, {3564,7512}, {5056,11803}, {5067,5645}, {5657,9905}, {5878,10628}, {5889,7706}, {7392,9827}, {9920,12088}, {10677,11488}, {10678,11489}, {11427,12234}, {11433,12242}, {11442,12280}

X(12325) = reflection of X(i) in X(j) for these (i,j): (4,2888), (20,12307), (2888,3519), (11271,54), (12254,7691), (12316,5)
X(12325) = anticomplement of X(195)
X(12325) = X(79)-of-anti-inverse-in-incircle-triangle if ABC is acute
X(12325) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3410,6243,4), (7691,12254,376)

X(12326) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO ANTI-MCCAY

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

The reciprocal orthologic center of these triangles is X(9855).

X(12326) lies on these lines: {30,12178}, {35,9875}, {55,671}, {56,9884}, {100,8591}, {115,4428}, {197,9876}, {542,11500}, {543,4421}, {1001,5461}, {1376,2482}, {2796,8715}, {3295,12258}, {5687,9881}, {8724,11499}, {9878,11494}, {9880,11496}, {9882,11497}, {9883,11498}, {10054,11507}, {10070,11508}, {10310,12117}, {11383,12132}, {11490,12191}, {11491,12243}

X(12326) = orthologic center of these triangles: anti-Mandart-incircle to McCay
X(12326) = X(671)-of-anti-Mandart-incircle-triangle

X(12327) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO ANTI-ORTHOCENTROIDAL

Trilinears a*((-a+b+c)*S*(SW*S^2-9*SA*SB*SC)+4*R*(S^2-3*SA*SC)*(S^2-3*SA*SB)) : :
X(12327) = (R+r)*X(55)-R*X(74)

The reciprocal orthologic center of these triangles is X(12112).

X(12327) lies on these lines: {3,11720}, {35,9904}, {40,2778}, {55,74}, {56,7978}, {100,146}, {110,10310}, {113,1376}, {125,11496}, {197,9919}, {541,4421}, {690,12178}, {1001,6699}, {2771,3811}, {2777,11500}, {2779,10620}, {2948,5537}, {3295,11709}, {5663,11248}, {7725,11497}, {7726,11498}, {7728,11499}, {9984,11494}, {10065,11507}, {10081,11508}, {10267,12041}, {11383,12133}, {11490,12192}, {11491,12244}

X(12327) = orthologic center of these triangles: anti-Mandart-incircle to orthocentroidal
X(12327) = X(74)-of-anti-Mandart-incircle-triangle

X(12328) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO ARIES

Barycentrics a^2*(-a+b+c)*S*SA*SB*SC-R*(S^2-SC^2)*(S^2-SB^2)*SA : :
X(12328) = (R+r)*X(55)-R*X(68)

The reciprocal orthologic center of these triangles is X(9833).

X(12328) lies on these lines: {3,914}, {35,9896}, {40,912}, {55,68}, {56,9933}, {100,6193}, {155,11499}, {197,9908}, {539,4421}, {1001,5449}, {1069,11502}, {1147,1376}, {3157,11501}, {3295,12259}, {5687,9928}, {9923,11494}, {9927,11496}, {9929,11497}, {9930,11498}, {10055,11507}, {10071,11508}, {10310,12118}, {11383,12134}, {11411,11491}, {11490,12193}

X(12328) = orthologic center of these triangles: anti-Mandart-incircle to 2nd Hyacinth
X(12328) = X(68)-of-anti-Mandart-incircle-triangle

X(12329) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 1st EHRMANN

Trilinears (a^3-(b+c)*(b^2+c^2-a*(-a+b+c)))*a : :
X(12329) = 3*X(165)-X(7289)

The reciprocal orthologic center of these triangles is X(3).

X(12329) lies on these lines: {3,518}, {6,31}, {9,1486}, {10,7535}, {22,3681}, {25,210}, {35,3751}, {40,3827}, {41,4878}, {44,7083}, {48,2340}, {56,976}, {69,100}, {72,3556}, {141,1376}, {144,1633}, {159,197}, {165,7289}, {182,9052}, {198,480}, {206,219}, {220,1973}, {241,1037}, {354,7484}, {511,11248}, {517,9818}, {524,4421}, {573,2876}, {597,4428}, {611,4259}, {613,11508}, {1001,3589}, {1260,3185}, {1350,8679}, {1351,9047}, {1352,11499}, {1386,3295}, {1428,11510}, {1469,11509}, {1503,11500}, {1593,7957}, {1621,3618}, {1757,7295}, {1804,2283}, {1843,11383}, {1974,3690}, {2164,7077}, {2175,2911}, {2182,3059}, {2187,2318}, {2781,12327}, {2810,3098}, {3085,5800}, {3094,11494}, {3220,5223}, {3416,5687}, {3564,12328}, {3740,5020}, {3763,4413}, {3844,9709}, {3870,5314}, {3873,7485}, {3913,5846}, {3941,5120}, {3961,5329}, {4097,5847}, {4265,5217}, {4420,11337}, {4661,6636}, {5044,11365}, {5085,9049}, {5480,11496}, {5777,9911}, {5845,11495}, {5849,6776}, {6601,7397}, {8177,9055}, {9041,11194}, {9830,12326}, {10477,11517}, {11490,12212}

X(12329) = X(6)-of-anti-Mandart-incircle-triangle
X(12329) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (72,8193,3556), (198,480,4557), (200,5285,197), (1631,4557,198), (2330,3779,6), (3242,5096,56), (5227,5285,159)

X(12330) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO EXTOUCH

Trilinears S^3*a*(-a+b+c)-4*R*(-S*r+SB*b)*(-S*r+SC*c)*(a+b+c) : :
X(12330) = (R+r)*X(55)-R*X(84)

The reciprocal orthologic center of these triangles is X(40).

X(12330) lies on these lines: {3,960}, {35,7992}, {55,84}, {56,7971}, {109,1498}, {197,9910}, {268,3197}, {515,3913}, {516,8730}, {971,6600}, {999,5884}, {1001,6705}, {1012,3486}, {1035,2956}, {1260,1490}, {1376,6260}, {1657,2829}, {1709,11507}, {1768,7742}, {3149,3474}, {3295,5882}, {5658,10309}, {5880,6918}, {6244,11500}, {6245,11496}, {6257,11498}, {6258,11497}, {6259,11499}, {6796,11495}, {10085,11508}, {11383,12136}, {11490,12196}, {11491,12246}

X(12330) = X(84)-of-anti-Mandart-incircle-triangle

X(12331) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO FUHRMANN

Barycentrics a^2*(-a+b+c)*(a+b+c)*(R-2*r)+2*R*(2*SB-a*c)*(2*SC-a*b) : :
X(12331) = 3*X(3)-2*X(104) = 4*X(11)-5*X(1656) = 2*X(80)-3*X(5790) = 3*X(100)-X(104) = 4*X(214)-3*X(10246) = X(382)+4*X(6154) = 2*X(1320)-3*X(10247) = X(1657)-4*X(10993) = 8*X(3035)-7*X(3526) = 2*X(6154)+X(10742)

The reciprocal orthologic center of these triangles is X(3).

X(12331) lies on the Stammler circle and these lines: {1,6797}, {2,1484}, {3,8}, {4,11698}, {5,149}, {11,498}, {30,153}, {35,9897}, {40,2771}, {55,80}, {56,7972}, {119,381}, {145,6924}, {197,9912}, {214,1376}, {355,8715}, {382,5840}, {404,1483}, {517,3689}, {550,12248}, {971,2950}, {999,1317}, {1001,6702}, {1012,9963}, {1320,6911}, {1351,9024}, {1385,6264}, {1387,6767}, {1482,2802}, {1597,12138}, {1598,1862}, {1657,2829}, {1768,3579}, {2095,8730}, {2346,6881}, {2783,12188}, {2800,11500}, {2801,11495}, {2805,11258}, {3032,9567}, {3035,3526}, {3036,9708}, {3045,9704}, {3149,8148}, {3158,3577}, {3434,6980}, {3534,6244}, {3576,7993}, {3621,6942}, {3746,9956}, {3830,10711}, {4678,6875}, {5054,6174}, {5055,10707}, {5073,10728}, {5082,6863}, {5083,5708}, {5093,10755}, {5552,6971}, {5603,9802}, {5694,11010}, {5844,6905}, {5848,11898}, {5854,10680}, {6262,11498}, {6263,11497}, {6361,9809}, {6917,10528}, {6918,11729}, {6928,7080}, {6946,10283}, {9913,12083}, {10057,11507}, {10073,11508}, {10310,12119}, {11383,12137}, {11490,12198}

X(12331) = midpoint of X(i) and X(j) for these {i,j}: {40,5531}, {5541,6326}, {6361,9809}
X(12331) = reflection of X(i) in X(j) for these (i,j): (3,100), (4,11698), (149,5), (382,10742), (1482,6265), (1768,3579), (3830,10711), (5073,10728), (6264,1385), (8148,10698), (10738,119), (12247,5690), (12248,550)
X(12331) = X(80)-of-anti-Mandart-incircle-triangle
X(12331) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11,10087,3295), (55,5790,7489), (100,6224,2932), (119,10738,381), (355,8715,11849), (1317,10090,999), (3913,11499,1482), (5690,11491,3)
X(12331) = anticomplement of X(1484)

X(12332) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO INNER-GARCIA

Trilinears a*(-a+b+c)*S^2*(R-2*r)+2*R*(-2*S*r+SB*b)*(-2*S*r+SC*c) : :
X(12332) = (R+r)*X(55)-R*X(104)

The reciprocal orthologic center of these triangles is X(40).

X(12332) lies on these lines: {3,214}, {11,6833}, {20,100}, {55,104}, {56,10698}, {80,1012}, {84,5531}, {119,1376}, {197,9913}, {515,12331}, {528,8730}, {952,3913}, {1001,6713}, {1158,2771}, {1537,10090}, {2077,2932}, {2787,12178}, {2801,6600}, {2802,10306}, {3035,6825}, {3295,11715}, {3428,4996}, {5450,11849}, {5537,5541}, {5722,10265}, {6224,6909}, {6256,11698}, {6259,6796}, {6702,6913}, {6906,10950}, {8069,11570}, {10058,11507}, {10074,11508}, {10742,11499}, {11383,12138}, {11490,12199}, {11491,12248}

X(12332) = midpoint of X(i) and X(j) for these {i,j}: {84,5531}, {2950,6326}
X(12332) = reflection of X(i) in X(j) for these (i,j): (6256,11698), (11500,100)
X(12332) = X(104)-of-anti-Mandart-incircle-triangle

X(12333) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO HUTSON EXTOUCH

Trilinears a*(-a+b+c)*S*(a+b+c)*(8*R*r+8*R^2+r^2)-4*R*((4*R+r)*S-SB*b)*((4*R+r)*S-SC*c) : :
X(12333) = (R+r)*X(55)-R*X(7160)

The reciprocal orthologic center of these triangles is X(40).

X(12333) lies on these lines: {35,9898}, {55,84}, {56,8000}, {100,9874}, {946,3295}, {1750,3746}, {3035,3526}, {3913,6684}, {6600,10267}, {8715,8730}, {10059,11507}, {10075,11508}, {10306,11495}, {10310,12120}, {11383,12139}, {11490,12200}, {11491,12249}

X(12333) = X(7160)-of-anti-Mandart-incircle-triangle

X(12334) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 1st HYACINTH

Barycentrics 2*a^2*(-a+b+c)*S^3*(9*R^2-2*SW)+R*(-a^2*c^2+4*SB^2)*(-a^2*b^2+4*SC^2)*SA : :
X(12334) = (R+r)*X(55)-R*X(265)

The reciprocal orthologic center of these triangles is X(6102).

X(12334) lies on these lines: {30,12327}, {40,2771}, {55,265}, {110,11499}, {125,10267}, {542,12329}, {1376,1511}, {3295,12261}, {3448,11491}, {5663,11500}, {6911,11720}, {10088,11501}, {10091,11502}, {10113,11496}, {10310,12121}, {11383,12140}, {11490,12201}

X(12334) = X(265)-of-anti-Mandart-incircle-triangle

X(12335) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO MIDHEIGHT

Trilinears a*((-a+b+c)*S*SC*SA*SB-2*R*(S^2-2*SA*SC)*(S^2-2*SA*SB)) : :
X(12335) = (R+r)*X(55)-R*X(64)

The reciprocal orthologic center of these triangles is X(4).

X(12335) lies on these lines: {30,12328}, {35,9899}, {40,197}, {55,64}, {56,7973}, {100,6225}, {199,5584}, {204,11471}, {1001,6696}, {1376,2883}, {1466,2192}, {1498,3682}, {1802,3197}, {2777,12334}, {3295,12262}, {3357,10267}, {3811,6001}, {3827,6769}, {5878,11499}, {6000,11248}, {6247,11496}, {6266,11498}, {6267,11497}, {6285,11509}, {8273,8567}, {10060,11507}, {10076,11508}, {11381,11383}, {11490,12202}, {11491,12250}

X(12335) = X(64)-of-anti-Mandart-incircle-triangle

X(12336) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO INNER-NAPOLEON

Barycentrics a^2*(-a+b+c)*(sqrt(3)*SW-3*S)+2*R*(sqrt(3)*SB-S)*(sqrt(3)*SC-S) : :
X(12336) = (R+r)*X(55)-R*X(14)

The reciprocal orthologic center of these triangles is X(3).

X(12336) lies on these lines: {14,55}, {35,9900}, {56,7974}, {100,617}, {197,9915}, {530,12326}, {531,4421}, {542,12329}, {619,1376}, {1001,6670}, {3295,11706}, {4428,5460}, {5474,10310}, {5479,11496}, {5613,11499}, {6269,11498}, {6271,11497}, {6773,11491}, {6774,10267}, {9981,11494}, {10061,11507}, {10077,11508}, {11383,12141}, {11490,12204}

X(12336) = X(14)-of-anti-Mandart-incircle-triangle

X(12337) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO OUTER-NAPOLEON

Barycentrics a^2*(-a+b+c)*(sqrt(3)*SW+3*S)-2*R*(sqrt(3)*SB+S)*(sqrt(3)*SC+S) : :
X(12337) = (R+r)*X(55)-R*X(13)

The reciprocal orthologic center of these triangles is X(3).

X(12337) lies on these lines: {13,55}, {35,9901}, {56,7975}, {100,616}, {197,9916}, {530,4421}, {531,12326}, {542,12329}, {618,1376}, {1001,6669}, {3295,11705}, {4428,5459}, {5473,10310}, {5478,11496}, {5617,11499}, {6268,11498}, {6270,11497}, {6770,11491}, {6771,10267}, {9982,11494}, {10062,11507}, {10078,11508}, {11383,12142}, {11490,12205}

X(12337) = X(13)-of-anti-Mandart-incircle-triangle

X(12338) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 1st NEUBERG

Trilinears (b^2+c^2)*a^4-(b+c)*(b^2+c^2)*a^3+b^2*c^2*a^2-b^2*c^2*(b+c)*a+2*b^3*c^3 : :
X(12338) = (R+r)*X(55)-R*X(76)

The reciprocal orthologic center of these triangles is X(3).

X(12338) lies on these lines: {3,730}, {35,9902}, {39,1376}, {55,76}, {56,7976}, {100,194}, {197,9917}, {384,11490}, {511,11500}, {538,4421}, {726,8715}, {732,12329}, {1001,3934}, {2782,11248}, {3095,11499}, {3295,12263}, {4413,7786}, {4428,9466}, {5969,12326}, {6248,11496}, {6272,11498}, {6273,11497}, {9983,11494}, {10063,11507}, {10079,11508}, {10310,11257}, {11383,12143}, {11491,12251}

X(12338) = X(76)-of-anti-Mandart-incircle-triangle

X(12339) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 2nd NEUBERG

Trilinears a^6-(b+c)*a^5+(3*b^2+2*b*c+3*c^2)*a^4-3*(b+c)*(b^2+c^2)*a^3+(b^2+b*c+c^2)^2*a^2-(b+c)*(b^4+3*b^2*c^2+c^4)*a+2*b^3*c^3 : :
X(12339) = (R+r)*X(55)-R*X(83)

The reciprocal orthologic center of these triangles is X(3).

X(12339) lies on these lines: {35,9903}, {55,83}, {56,7977}, {100,2896}, {197,9918}, {732,12329}, {754,4421}, {1001,6704}, {1376,6292}, {3295,12264}, {6249,11496}, {6274,11498}, {6275,11497}, {6287,11499}, {10064,11507}, {10080,11508}, {10310,12122}, {11383,12144}, {11490,12206}, {11491,12252}

X(12339) = X(83)-of-anti-Mandart-incircle-triangle

X(12340) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 1st ORTHOSYMMEDIAL

Trilinears a*((-a+b+c)*S^3*((3*S^2-4*SW^2)*R^2-(S^2-SW^2)*SW)-R*((SC+SA)*S^2-2*SA*SC*SW)*((SA+SB)*S^2-2*SA*SB*SW)) : :
X(12340) = (R+r)*X(55)-R*X(1297)

The reciprocal orthologic center of these triangles is X(4).

X(12340) lies on these lines: {3,11722}, {55,1297}, {112,10310}, {127,11496}, {132,1376}, {2799,12178}, {2806,12332}, {3295,12265}, {4421,9530}, {6020,11509}, {9517,12327}, {11383,12145}, {11490,12207}, {11491,12253}

X(12340) = X(1297)-of-anti-Mandart-incircle-triangle

X(12341) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO REFLECTION

Trilinears a*((-a+b+c)*S^3*(-2*SW+5*R^2)+R*(S^2+SA*SC)*(S^2+SA*SB)) : :
X(12341) = (R+r)*X(55)-R*X(54)

The reciprocal orthologic center of these triangles is X(4).

X(12341) lies on these lines: {35,9905}, {54,55}, {56,7979}, {100,2888}, {195,11849}, {197,9920}, {539,4421}, {692,10274}, {1001,6689}, {1154,11248}, {1209,1376}, {3295,12266}, {3574,11496}, {6276,11498}, {6277,11497}, {6288,11499}, {7691,10310}, {9985,11494}, {10066,11507}, {10082,11508}, {10267,10610}, {10628,12327}, {11383,11576}, {11490,12208}, {11491,12254}

X(12341) = X(54)-of-anti-Mandart-incircle-triangle

X(12342) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 1st SCHIFFLER

Barycentrics (-a+b+c)*((a+b+c)*S*(3*R+2*r)^2*a^2-4*R*(S*(R+2*r)+2*SB*(a-b+c))*(S*(R+2*r)+2*SC*(a+b-c))) : :
X(12342) = (R+r)*X(55)-R*X(10266)

The reciprocal orthologic center of these triangles is X(79).

X(12342) lies on these lines: {55,10266}, {149,2975}, {3295,12267}, {3913,5904}, {11383,12146}, {11490,12209}, {11491,12255}

X(12342) = X(10266)-of-anti-Mandart-incircle-triangle

X(12343) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO INNER-VECTEN

Barycentrics a^2*(-a+b+c)*(2*S-SW)+2*R*(-2*S^2+(SB+SC)*(S+SA)) : :
X(12343) = (R+r)*X(55)-R*X(486)

The reciprocal orthologic center of these triangles is X(3).

X(12343) lies on these lines: {35,9906}, {55,486}, {56,7980}, {100,487}, {197,9921}, {642,1376}, {1001,6119}, {3295,12268}, {3564,12328}, {6251,11496}, {6280,11498}, {6281,11497}, {6290,11499}, {9986,11494}, {10067,11507}, {10083,11508}, {10310,12123}, {11383,12147}, {11490,12210}, {11491,12256}

X(12343) = X(486)-of-anti-Mandart-incircle-triangle

X(12344) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO OUTER-VECTEN

Barycentrics a^2*(-a+b+c)*(2*S+SW)-2*R*(2*S^2+(SB+SC)*(S-SA)) : :
X(12344) = (R+r)*X(55)-R*X(485)

The reciprocal orthologic center of these triangles is X(3).

X(12344) lies on these lines: {35,9907}, {55,485}, {56,7981}, {100,488}, {197,9922}, {641,1376}, {1001,6118}, {3295,12269}, {3564,12328}, {6250,11496}, {6278,11498}, {6279,11497}, {6289,11499}, {9987,11494}, {10068,11507}, {10084,11508}, {10310,12124}, {11383,12148}, {11490,12211}, {11491,12257}

X(12344) = X(485)-of-anti-Mandart-incircle-triangle

X(12345) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO ANTI-MCCAY

Barycentrics (4*a^5+(b+c)*a^4-4*(b^2+c^2)*a^3-(b+c)*(b^2+c^2)*a^2+(b^2+c^2)^2*a-(b+c)*(2*b^2-c^2)*(b^2-2*c^2))*D-3*a^2*(-a+b+c)*(a+b+c)*(a^4-(b^2+c^2)*a^2-b^2*c^2+c^4+b^4) : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(9855).

X(12345) lies on these lines: {30,12179}, {543,11207}, {671,5597}, {2482,5599}, {5598,9884}, {5601,8591}, {8190,9876}, {8196,9880}, {8197,9881}, {8198,9882}, {8199,9883}, {8200,8724}, {9878,11861}, {10054,11877}, {10070,11879}, {11366,12258}, {11384,12132}, {11492,12326}, {11822,12117}, {11837,12191}, {11843,12243}

X(12345) = X(671)-of-1st-Auriga-triangle
X(12345) = X(9884)-of-2nd-Auriga-triangle

X(12346) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO ANTI-MCCAY

Barycentrics -(4*a^5+(b+c)*a^4-4*(b^2+c^2)*a^3-(b+c)*(b^2+c^2)*a^2+(b^2+c^2)^2*a-(b+c)*(2*b^2-c^2)*(b^2-2*c^2))*D-3*a^2*(-a+b+c)*(a+b+c)*(a^4-(b^2+c^2)*a^2-b^2*c^2+c^4+b^4) : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(9855).

X(12346) lies on these lines: {30,12180}, {55,12345}, {543,11208}, {671,5598}, {2482,5600}, {5597,9884}, {5602,8591}, {8187,9875}, {8191,9876}, {8203,9880}, {8204,9881}, {8205,9882}, {8206,9883}, {8207,8724}, {9878,11862}, {10054,11878}, {10070,11880}, {11367,12258}, {11385,12132}, {11493,12326}, {11823,12117}, {11838,12191}, {11844,12243}

X(12346) = X(671)-of-2nd-Auriga-triangle
X(12346) = X(9884)-of-1st-Auriga-triangle

X(12347) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO ANTI-MCCAY

Barycentrics (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(4*a^12-8*(b^2+c^2)*a^10-4*(b^4-8*b^2*c^2+c^4)*a^8+2*(b^2+c^2)*(11*b^4-26*b^2*c^2+11*c^4)*a^6-(19*b^8+19*c^8+6*b^2*c^2*(2*b^4-11*b^2*c^2+2*c^4))*a^4+2*(b^4-c^4)*(b^2-c^2)*(2*b^4+15*b^2*c^2+2*c^4)*a^2+(b^4-8*b^2*c^2+c^4)*(b^4-c^4)^2) : :
X(12347) = 3*X(11845)-X(12243)

The reciprocal orthologic center of these triangles is X(9855).

X(12347) lies on these lines: {30,99}, {402,671}, {542,12113}, {543,1651}, {1650,2482}, {4240,8591}, {9875,11852}, {9876,11853}, {9878,11885}, {9880,11897}, {9881,11900}, {9882,11901}, {9883,11902}, {9884,11910}, {10054,11912}, {10070,11913}, {11831,12258}, {11832,12132}, {11839,12191}, {11845,12243}, {11848,12326}, {11863,12345}, {11864,12346}

X(12347) = midpoint of X(4240) and X(8591)
X(12347) = reflection of X(i) in X(j) for these (i,j): (671,402), (1650,2482)
X(12347) = X(671)-of-Gossard-triangle

X(12348) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO ANTI-MCCAY

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

The reciprocal orthologic center of these triangles is X(9855).

X(12348) lies on these lines: {11,671}, {30,12182}, {355,8724}, {542,12114}, {543,11235}, {1376,2482}, {3434,8591}, {9875,10826}, {9876,10829}, {9878,10871}, {9880,10893}, {9881,10914}, {9882,10919}, {9883,10920}, {9884,10944}, {10054,10523}, {10070,10948}, {10785,12243}, {10794,12191}, {11373,12258}, {11390,12132}, {11826,12117}, {11865,12345}, {11866,12346}, {11903,12347}

X(12348) = reflection of X(12326) in X(2482)
X(12348) = reflection of X(12349) in X(8724)
X(12348) = X(671)-of-inner-Johnson-triangle

X(12349) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO ANTI-MCCAY

Barycentrics 3*(a-b+c)*(a+b-c)*(b+c)^2*(-3*S^2+SW^2)+4*(R+r)*(3*SB-SW)*(3*SC-SW)*S*(a+b+c) : :
X(12349) = (R+2*r)*X(12)-(R+r)*X(671)

The reciprocal orthologic center of these triangles is X(9855).

X(12349) lies on these lines: {12,671}, {30,12183}, {72,9881}, {355,8724}, {542,11500}, {543,11236}, {958,2482}, {3436,8591}, {9875,10827}, {9876,10830}, {9878,10872}, {9880,10894}, {9882,10921}, {9883,10922}, {9884,10950}, {10054,10954}, {10070,10523}, {10786,12243}, {10795,12191}, {11374,12258}, {11391,12132}, {11827,12117}, {11867,12345}, {11868,12346}, {11904,12347}

X(12349) = reflection of X(12348) in X(8724)
X(12349) = X(671)-of-outer-Johnson-triangle

X(12350) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO ANTI-MCCAY

Barycentrics 3*(a-b+c)*(a+b-c)*(b+c)^2*(-3*S^2+SW^2)+4*r*(3*SB-SW)*(3*SC-SW)*S*(a+b+c) : :
X(12350) = (R-2*r)*X(11)-(R-r)*X(671)

The reciprocal orthologic center of these triangles is X(9855).

X(12350) lies on these lines: {1,8724}, {2,3027}, {5,10070}, {12,671}, {55,542}, {56,2482}, {65,9881}, {98,4995}, {99,5434}, {114,11238}, {147,10385}, {226,2796}, {388,8591}, {495,10054}, {543,11237}, {2276,6034}, {2782,10056}, {3028,11006}, {3058,6054}, {3085,12243}, {3584,11632}, {5261,8596}, {7354,12117}, {9578,9875}, {9657,10992}, {9876,10831}, {9878,10873}, {9880,10895}, {9882,10923}, {9883,10924}, {9884,10944}, {10797,12191}, {11375,12258}, {11392,12132}, {11501,12326}, {11869,12345}, {11870,12346}, {11905,12347}

X(12350) = reflection of X(10054) in X(495)
X(12350) = X(671)-of-1st-Johnson-Yff-triangle
X(12350) = {X(1), X(8724)}-harmonic conjugate of X(12351)
X(12350) = {X(3058), X(6054)}-harmonic conjugate of X(12185)

X(12351) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO ANTI-MCCAY

Barycentrics 3*(-a+b+c)*(b-c)^2*(-3*S^2+SW^2)-4*r*(3*SB-SW)*(3*SC-SW)*S : :
X(12351) = (R-2*r)*X(11)+r*X(671)

The reciprocal orthologic center of these triangles is X(9855).

X(12351) lies on these lines: {1,8724}, {2,3023}, {5,10054}, {11,671}, {30,10089}, {55,2482}, {56,542}, {98,5298}, {99,3058}, {114,11237}, {496,10070}, {497,8591}, {543,11238}, {549,10053}, {2275,6034}, {2782,10072}, {2796,12053}, {3057,9881}, {3086,12243}, {3582,11632}, {5182,10799}, {5274,8596}, {5434,6054}, {6284,12117}, {9581,9875}, {9670,10992}, {9876,10832}, {9878,10874}, {9880,10896}, {9882,10925}, {9883,10926}, {9884,10950}, {10798,12191}, {11376,12258}, {11393,12132}, {11502,12326}, {11871,12345}, {11872,12346}, {11906,12347}

X(12351) = reflection of X(10070) in X(496)
X(12351) = X(671)-of-2nd-Johnson-Yff-triangle
X(12351) = {X(1), X(8724)}-harmonic conjugate of X(12350)

X(12352) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO ANTI-MCCAY

Barycentrics 9*SA^2*S^2+S*(4*R^2*(9*S^2+18*SA^2-12*SA*SW-SW^2)-3*(3*S^2-SW^2)*(SB+SC))+(6*SA^2-6*SA*SW+SW^2)*SW^2 : :

The reciprocal orthologic center of these triangles is X(9855).

X(12352) lies on these lines: {30,12186}, {493,671}, {542,9838}, {543,12152}, {2482,8222}, {6461,12353}, {6462,8591}, {8188,9875}, {8194,9876}, {8201,12345}, {8208,12346}, {8210,9884}, {8212,9880}, {8214,9881}, {8216,9882}, {8218,9883}, {8220,8724}, {9878,10875}, {10054,11951}, {10070,11953}, {10945,12348}, {10951,12349}, {11377,12258}, {11394,12132}, {11503,12326}, {11828,12117}, {11840,12191}, {11846,12243}, {11907,12347}, {11930,12350}, {11932,12351}

X(12352) = X(671)-of-Lucas-homothetic-triangle

X(12353) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO ANTI-MCCAY

Barycentrics 9*SA^2*S^2-S*(4*R^2*(9*S^2+18*SA^2-12*SA*SW-SW^2)-3*(3*S^2-SW^2)*(SB+SC))+(6*SA^2-6*SA*SW+SW^2)*SW^2 : :

The reciprocal orthologic center of these triangles is X(9855).

X(12353) lies on these lines: {30,12187}, {494,671}, {542,9839}, {543,12153}, {2482,8223}, {6461,12352}, {6463,8591}, {8189,9875}, {8195,9876}, {8202,12345}, {8209,12346}, {8211,9884}, {8213,9880}, {8215,9881}, {8217,9882}, {8219,9883}, {8221,8724}, {9878,10876}, {10054,11952}, {10070,11954}, {10946,12348}, {10952,12349}, {11378,12258}, {11395,12132}, {11504,12326}, {11829,12117}, {11841,12191}, {11847,12243}, {11908,12347}, {11931,12350}, {11933,12351}

X(12353) = X(671)-of-Lucas(-1)-homothetic-triangle

X(12354) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO ANTI-MCCAY

Barycentrics (-a+b+c)*(4*a^6-(5*b^2-2*b*c+5*c^2)*a^4+2*(b^2+c^2)*(b^2-b*c+c^2)*a^2+(2*b^2-c^2)*(b^2-2*c^2)*(b-c)^2) : :
Barycentrics 3*a^2*(-a+b+c)*(-3*S^2+SW^2)+4*r*(3*SB-SW)*(3*SC-SW)*S : :
X(12354) = (R+r)*X(55)-r*X(671)

The reciprocal orthologic center of these triangles is X(9855).

X(12354) lies on these lines: {3,10070}, {4,12350}, {11,2482}, {12,9880}, {33,12132}, {55,671}, {56,12117}, {99,11238}, {115,4995}, {148,10385}, {390,8596}, {497,8591}, {542,6284}, {543,3023}, {950,2796}, {1479,8724}, {1697,9875}, {1837,9881}, {2098,9884}, {2646,12258}, {3056,9830}, {3295,10054}, {4294,12243}, {5182,10798}, {5432,5461}, {6034,9598}, {6321,10056}, {9876,10833}, {9878,10877}, {9882,10927}, {9883,10928}, {10799,12191}, {10947,12348}, {10953,12349}, {11873,12345}, {11874,12346}, {11909,12347}, {11947,12352}, {11948,12353}

X(12354) = reflection of X(3023) in X(3058)
X(12354) = X(671)-of-Mandart-incircle-triangle
X(12354) = {X(497), X(8591)}-harmonic conjugate of X(12351)

X(12355) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO ANTI-MCCAY

Barycentrics 5*a^8-12*(b^2+c^2)*a^6+(8*b^4+17*b^2*c^2+8*c^4)*a^4+(b^2+c^2)*(3*b^4-13*b^2*c^2+3*c^4)*a^2-2*(b^2-c^2)^2*(2*b^2-c^2)*(b^2-2*c^2) : :
Barycentrics 3*SA*a^2*(3*S^2-SW^2)-4*(3*SB-SW)*(3*SC-SW)*S^2 : :
X(12355) = 3*X(3)-2*X(12117) = 2*X(99)-3*X(5055) = 4*X(114)-5*X(381) = 2*X(114)-5*X(6321) = 5*X(148)-X(11177) = 4*X(148)-X(12188) = 3*X(148)-X(12243) = 3*X(671)-X(12117) = 4*X(11177)-5*X(12188) = 3*X(11177)-5*X(12243) = 3*X(12188)-4*X(12243)

The reciprocal orthologic center of these triangles is X(9855).

X(12355) lies on these lines: {3,671}, {4,8596}, {5,8591}, {30,148}, {99,5055}, {114,381}, {115,5054}, {355,2796}, {382,542}, {517,9875}, {576,10488}, {999,10070}, {1351,9830}, {1598,12132}, {1656,2482}, {2782,3830}, {2936,7545}, {3295,10054}, {3526,5461}, {3534,11632}, {3655,11599}, {5093,8593}, {5790,9881}, {7517,9876}, {8787,11482}, {9654,12350}, {9669,12351}, {9882,11916}, {9883,11917}, {9884,10247}, {10246,12258}, {11152,11317}, {11656,12121}, {11842,12191}, {11849,12326}, {11875,12345}, {11876,12346}, {11911,12347}, {11928,12348}, {11929,12349}, {11949,12352}, {11950,12353}

X(12355) = midpoint of X(4) and X(8596)
X(12355) = reflection of X(i) in X(j) for these (i,j): (3,671), (381,6321), (3534,11632), (3655,11599), (8591,5), (8724,9880), (10488,576), (10992,5461), (12121,11656)
X(12355) = X(671)-of-X3-ABC-reflections-triangle
X(12355) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6321,8724,9880), (8724,9880,381), (10054,12354,3295)

X(12356) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO ANTI-MCCAY

Barycentrics 6*R*(3*S^2-SW^2)*a-(R-r)*(3*SB-SW)*(3*SC-SW)*(a+b+c) : :
X(12356) = 2*R*X(1)-(R-r)*X(671)

The reciprocal orthologic center of these triangles is X(9855).

X(12356) lies on these lines: {1,671}, {12,12348}, {30,12189}, {542,12115}, {543,11239}, {2482,5552}, {8591,10528}, {8724,10942}, {9876,10834}, {9878,10878}, {9880,10531}, {9881,10915}, {9882,10929}, {9883,10930}, {10803,12191}, {10805,12243}, {10955,12349}, {10956,12350}, {10958,12351}, {10965,12354}, {11248,12117}, {11400,12132}, {11509,12326}, {11881,12345}, {11882,12346}, {11914,12347}, {11955,12352}, {11956,12353}, {12000,12355}

X(12356) = reflection of X(671) in X(10054)
X(12356) = X(671)-of-inner-Yff-tangents-triangle

X(12357) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO ANTI-MCCAY

Barycentrics 6*R*(3*S^2-SW^2)*a-(R+r)*(3*SB-SW)*(3*SC-SW)*(a+b+c) : :
X(12357) = 2*R*X(1)-(R+r)*X(671)

The reciprocal orthologic center of these triangles is X(9855).

X(12357) lies on these lines: {1,671}, {11,12349}, {30,12190}, {542,12116}, {543,11240}, {2482,10527}, {8591,10529}, {8724,10943}, {9876,10835}, {9878,10879}, {9880,10532}, {9881,10916}, {9882,10931}, {9883,10932}, {10804,12191}, {10806,12243}, {10949,12348}, {10957,12350}, {10959,12351}, {10966,12354}, {11249,12117}, {11401,12132}, {11510,12326}, {11883,12345}, {11884,12346}, {11915,12347}, {11957,12352}, {11958,12353}, {12001,12355}

X(12357) = reflection of X(671) in X(10070)
X(12357) = X(671)-of-outer-Yff-tangents-triangle

X(12358) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR TO ANTI-ORTHOCENTROIDAL

Trilinears (2*(4*cos(A)+cos(3*A))*cos(B-C)-(2*cos(2*A)+3)*cos(2*(B-C))-2*cos(2*A)-1)*cos(A) : :
X(12358) = X(74)+3*X(11459) = X(110)-5*X(11444) = X(113)-3*X(5891) = 5*X(631)-X(7722) = 3*X(2979)+X(10733) = 9*X(7998)-X(12270) = 7*X(7999)+X(12281)

The reciprocal orthologic center of these triangles is X(3581).

X(12358) lies on these lines: {2,1986}, {3,74}, {5,1112}, {20,12292}, {30,12133}, {52,11746}, {69,265}, {113,127}, {125,5562}, {143,10255}, {182,12227}, {389,6723}, {394,5504}, {511,7687}, {526,6334}, {542,11574}, {631,7722}, {974,6699}, {1040,7727}, {1060,3028}, {1154,2072}, {1368,10264}, {2777,5907}, {2854,9967}, {2914,7550}, {2979,10733}, {3448,6643}, {3564,10111}, {5076,11387}, {5894,12162}, {5972,7542}, {6101,10113}, {6102,6640}, {6676,10272}, {6746,10224}, {7386,12317}, {7484,12165}, {7514,12228}, {7724,10319}, {7728,11487}, {9140,12273}, {9976,11511}, {10170,11557}, {10625,12295}, {10657,11515}, {10658,11516}, {11821,12121}

X(12358) = midpoint of X(i) and X(j) for these {i,j}: {3,7723}, {20,12292}, {125,5562}, {1986,12219}, {5876,12041}, {6101,10113}, {10625,12295}
X(12358) = reflection of X(i) in X(j) for these (i,j): (52,11746), (389,6723), (974,6699), (1112,5), (1986,9826), (5972,11793)
X(12358) = anticomplement of X(9826)
X(12358) = complement of X(1986)
X(12358) = orthologic center of these triangles: 6th anti-mixtilinear to orthocentroidal
X(12358) = X(80)-of-6th-anti-mixtilinear-triangle if ABC is acute
X(12358) = {X(2), X(12219)}-harmonic conjugate of X(1986)

X(12359) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR TO ARIES

Trilinears (2*cos(A)*cos(B-C)+cos(2*(B-C))-1)*cos(A) : :
Barycentrics ((b^2+c^2)*a^6-(3*b^4-2*b^2*c^2+3*c^4)*a^4+3*(b^4-c^4)*(b^2-c^2)*a^2-(b^2-c^2)^4)*(-a^2+b^2+c^2) : :
X(12359) = 3*X(2)+X(11411) = 3*X(3)-X(12118) = 3*X(5)-2*X(5448) = 3*X(68)+X(12118) = 2*X(156)-3*X(10192) = 5*X(631)-X(6193) = 5*X(1656)-3*X(5654) = 5*X(1656)-X(12164) = X(5448)-3*X(5449) = 3*X(5654)-X(12164)

The reciprocal orthologic center of these triangles is X(7387).

Let A'B'C' be as described at X(11585). Then X(12359) = X(4)-of-A'B'C'. (Randy Hutson, July 21, 2017)

X(12359) lies on these lines: {2,155}, {3,68}, {4,3580}, {5,389}, {10,912}, {11,6238}, {12,7352}, {20,12293}, {22,11457}, {24,11442}, {26,1503}, {30,3357}, {51,7403}, {52,427}, {55,10071}, {56,10055}, {69,3546}, {110,10018}, {125,5562}, {135,432}, {140,141}, {143,5480}, {156,10020}, {184,7542}, {235,12162}, {394,3548}, {403,12111}, {468,10539}, {498,3157}, {499,1069}, {511,12235}, {517,12259}, {524,8548}, {525,10279}, {539,549}, {542,10282}, {550,10264}, {568,5576}, {569,11245}, {590,10665}, {615,10666}, {631,6193}, {858,11412}, {1040,9931}, {1092,10257}, {1181,3549}, {1209,7399}, {1216,1368}, {1352,6642}, {1594,5889}, {1595,5446}, {1656,5544}, {2013,7999}, {2080,12193}, {2883,5663}, {2918,2931}, {3167,3526}, {3519,5504}, {3541,6515}, {3567,5133}, {3576,9896}, {3925,6237}, {5094,12160}, {5392,8800}, {5418,8909}, {5447,11574}, {6640,11064}, {6643,11821}, {6696,12084}, {7386,12318}, {7404,11433}, {7484,12166}, {7505,11441}, {7526,12241}, {7553,11550}, {7691,9140}, {7998,12271}, {8546,9925}, {9926,11511}, {9933,10246}, {10112,11430}, {10267,12328}, {10295,12278}, {10659,11515}, {10660,11516}, {11745,11818}

X(12359) = midpoint of X(i) and X(j) for these {i,j}: {3,68}, {4,12163}, {20,12293}, {155,11411}, {2931,3448}, {7689,9927}
X(12359) = reflection of X(i) in X(j) for these (i,j): (5,5449), (155,9820), (156,10020), (1147,140), (12084,6696)
X(12359) = anticomplement of X(9820)
X(12359) = complement of X(155)
X(12359) = X(68)-of-6th-anti-mixtilinear-triangle if ABC is acute
X(12359) = orthologic center of these triangles: 6th anti-mixtilinear to 2nd Hyacinth
X(12359) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,155,9820), (2,11411,155), (5,6102,12233), (24,11442,12134), (125,5562,11585), (156,10020,10192), (1209,9730,7399), (1656,12164,5654)

X(12360) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR TO LUCAS CENTRAL

Trilinears (4*a^2*b^2*c^2+((b^2+c^2)*a^2+b^4+c^4)*S)*(-a^2+b^2+c^2)*a : :
X(12360) = 3*X(2)+X(12223) = 5*X(631)-X(6239) = 9*X(7998)-X(12276) = 7*X(7999)+X(12287)

The reciprocal orthologic center of these triangles is X(3).

X(12360) lies on these lines: {2,6291}, {3,6}, {20,12298}, {488,8681}, {631,6239}, {1038,7362}, {1040,6283}, {6252,10319}, {7386,12322}, {7484,12171}, {7998,12276}, {7999,12287}, {8909,12230}, {9822,11292}

X(12360) = midpoint of X(i) and X(j) for these {i,j}: {20,12298}, {6291,12223}
X(12360) = reflection of X(6291) in X(9823)
X(12360) = anticomplement of X(9823)
X(12360) = complement of X(6291)
X(12360) = {X(3),X(11574)}-harmonic conjugate of X(12361)
X(12360) = X(176)-of-6th-anti-mixtilinear-triangle if ABC is acute
X(12360) = orthic-to-6th-anti-mixtilinear similarity image of X(6291)

X(12361) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR TO LUCAS(-1) CENTRAL

Trilinears (4*a^2*b^2*c^2-((b^2+c^2)*a^2+b^4+c^4)*S)*(-a^2+b^2+c^2)*a : :
X(12361) = 3*X(2)+X(12224) = 5*X(631)-X(6400) = 9*X(7998)-X(12277) = 7*X(7999)+X(12288)

The reciprocal orthologic center of these triangles is X(3).

X(12361) lies on these lines: {2,6406}, {3,6}, {20,12299}, {487,8681}, {631,6400}, {1038,7353}, {1040,6405}, {5943,8964}, {6404,10319}, {7386,12323}, {7484,12172}, {7998,12277}, {7999,12288}, {9822,11291}

X(12361) = midpoint of X(i) and X(j) for these {i,j}: {20,12299}, {6406,12224}
X(12361) = reflection of X(6406) in X(9824)
X(12361) = anticomplement of X(9824)
X(12361) = complement of X(6406)
X(12361) = {X(3),X(11574)}-harmonic conjugate of X(12360)
X(12361) = X(175)-of-6th-anti-mixtilinear-triangle if ABC is acute
X(12361) = orthic-to-6th-anti-mixtilinear similarity image of X(6406)

X(12362) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR TO MACBEATH

Trilinears (6*cos(A)*cos(B-C)-cos(2*(B-C))-2*cos(2*A)+1)*cos(A) : :
X(12362) = 3*X(2)+X(12225) = 2*X(3)-3*X(10691) = 3*X(5)-X(11819) = X(20)-3*X(7667) = 3*X(381)-X(7553) = 3*X(428)-5*X(3091) = X(5889)-3*X(11245) = 3*X(5891)+X(11750) = 3*X(5891)-X(12134) = 3*X(5943)-2*X(11745)

The reciprocal orthologic center of these triangles is X(4).
As a point on the Euler line, X(12362) has Shinagawa coefficients: (E+F, -E-3*F).

X(12362) lies on these lines: {2,3}, {69,11821}, {182,12233}, {206,2883}, {216,7745}, {511,12241}, {524,10112}, {577,5254}, {1038,7354}, {1040,6284}, {1060,4320}, {1062,4319}, {1352,9924}, {1353,12160}, {1503,5907}, {1578,6560}, {1579,6561}, {2968,5015}, {3070,11513}, {3071,11514}, {3292,10619}, {3564,4173}, {3580,7691}, {4549,12163}, {4911,6356}, {5305,10316}, {5318,11515}, {5321,11516}, {5889,11245}, {5891,11750}, {5943,11745}, {5965,12024}, {6253,10319}, {6389,7784}, {6776,12164}, {7583,10897}, {7584,10898}, {7998,12278}, {7999,12289}, {8550,11511}, {10634,11542}, {10635,11543}, {11412,12022}

X(12362) = midpoint of X(i) and X(j) for these {i,j}: {20,1885}, {3575,12225}, {5562,6146}, {11750,12134}
X(12362) = reflection of X(i) in X(j) for these (i,j): (3575,9825), (6756,5), (7576,10128)
X(12362) = anticomplement of X(9825)
X(12362) = complement of X(3575)
X(12362) = X(65)-of-6th-anti-mixtilinear-triangle if ABC is acute
X(12362) = X(4)-of-3rd-pedal-triangle-of-X(3)
X(12362) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,12225,3575), (3,4,6823), (3,5,6676), (3,381,3547), (3,2072,7542), (4,7395,5), (4,7509,7399), (5,550,26), (5,3627,11818), (5,7715,7529), (5,10154,3542), (20,6816,25), (25,6816,5), (376,3542,9715), (1885,7667,20), (2043,2044,9909), (2072,7542,3628), (3091,7539,5), (3542,9715,10154), (6804,7487,5020), (10024,10297,3850)

X(12363) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR TO REFLECTION

Trilinears ((4*cos(A)-2*cos(3*A))*cos(B-C)+(2*cos(2*A)+1)*cos(2*(B-C))-2*cos(2*A)+3)*cos(A) : :
X(12363) = 3*X(2)+X(12226) = 3*X(54)+X(11412) = 5*X(631)-X(6242) = 3*X(973)-4*X(5462) = 2*X(1216)+X(11577) = 2*X(5462)-3*X(6689) = X(6102)-3*X(10610) = 9*X(7730)-17*X(11465) = 9*X(7998)-X(12280) = 7*X(7999)+X(12291)

The reciprocal orthologic center of these triangles is X(6243).

X(12363) lies on these lines: {2,6152}, {3,54}, {5,11576}, {20,12300}, {69,3519}, {140,6746}, {141,1209}, {182,12234}, {511,12242}, {539,1216}, {631,6242}, {973,5462}, {1038,7356}, {1040,6286}, {1656,6403}, {2888,6643}, {3917,12359}, {5447,6699}, {5562,10619}, {5894,10575}, {5907,12134}, {5965,11574}, {6193,11821}, {6243,11427}, {6255,10319}, {6288,11487}, {6676,8254}, {7386,12325}, {7484,12175}, {7730,11465}, {7998,12280}, {7999,12291}, {9977,11511}, {10625,12233}, {10677,11515}, {10678,11516}, {12358,12362}

X(12363) = midpoint of X(i) and X(j) for these {i,j}: {20,12300}, {1493,6101}, {5562,10619}, {6152,12226}
X(12363) = reflection of X(i) in X(j) for these (i,j): (973,6689), (6152,9827), (11576,5)
X(12363) = anticomplement of X(9827)
X(12363) = complement of X(6152)
X(12363) = X(79)-of-6th-anti-mixtilinear-triangle if ABC is acute
X(12363) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,12226,6152), (54,1993,1493)

X(12364) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL TO ARIES

Trilinears (12*cos(A)*cos(B-C)-(2*cos(2*A)+1)*cos(2*(B-C))+cos(2*A)+cos(4*A)-5)*cos(A) : :
X(12364) = 6*R^2*(6*R^2-SW)*X(5)-SW*(11*R^2-2*SW)*X(6) = R^2*X(74)-(5*R^2-SW)*X(323)

The reciprocal orthologic center of these triangles is X(9934).

X(12364) lies on these lines: {5,6}, {74,323}, {113,539}, {186,12273}, {399,1514}, {974,10816}, {1147,10574}, {9938,12164}, {11456,12118}

X(12364) = orthologic center of these triangles: anti-orthocentroidal to 2nd Hyacinth
X(12364) = X(5504)-of-anti-orthocentroidal-triangle

X(12365) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO ANTI-ORTHOCENTROIDAL

Trilinears ((b+c)*a^8+(b-c)^2*a^7-2*(b^3+c^3)*a^6-(3*b^2+4*b*c+3*c^2)*(b-c)^2*a^5-b*c*(2*b-c)*(b-2*c)*(b+c)*a^4+(b^3-c^3)*(b-c)*(3*b^2+5*b*c+3*c^2)*a^3+(b^2-c^2)*(b-c)*(2*b^4+2*c^4+b*c*(2*b^2-b*c+2*c^2))*a^2-(b^2-c^2)^2*(b^2+b*c+c^2)^2*a-(b^4-c^4)*(b^2-c^2)^2*(b-c))*D-4*a*S^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))*(-a+b+c) : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(12112).

X(12365) lies on these lines: {55,12366}, {74,5597}, {110,11822}, {113,5599}, {125,8196}, {146,5601}, {541,11207}, {690,12179}, {3028,11873}, {5598,7978}, {5663,11252}, {7725,8198}, {7726,8199}, {7728,8200}, {10065,11877}, {10081,11879}, {10620,11875}, {11366,11709}, {11492,12327}, {11837,12192}, {11843,12244}

X(12365) = reflection of X(12366) in X(55)
X(12365) = X(74)-of-1st-Auriga-triangle
X(12365) = X(7978)-of-2nd-Auriga-triangle

X(12366) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO ANTI-ORTHOCENTROIDAL

Trilinears -((b+c)*a^8+(b-c)^2*a^7-2*(b^3+c^3)*a^6-(3*b^2+4*b*c+3*c^2)*(b-c)^2*a^5-b*c*(2*b-c)*(b-2*c)*(b+c)*a^4+(b^3-c^3)*(b-c)*(3*b^2+5*b*c+3*c^2)*a^3+(b^2-c^2)*(b-c)*(2*b^4+2*c^4+b*c*(2*b^2-b*c+2*c^2))*a^2-(b^2-c^2)^2*(b^2+b*c+c^2)^2*a-(b^4-c^4)*(b^2-c^2)^2*(b-c))*D-4*a*S^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))*(-a+b+c) : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(12112).

X(12366) lies on these lines: {55,12365}, {74,5598}, {110,11823}, {113,5600}, {125,8203}, {146,5602}, {541,11208}, {690,12180}, {3028,11874}, {5597,7978}, {5663,11253}, {7725,8205}, {7726,8206}, {7728,8207}, {8187,9904}, {10065,11878}, {10081,11880}, {10620,11876}, {11367,11709}, {11493,12327}, {11838,12192}, {11844,12244}

X(12366) = reflection of X(12365) in X(55)
X(12366) = X(74)-of-2nd-Auriga-triangle
X(12366) = X(7978)-of-1st-Auriga-triangle

X(12367) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL TO 1st EHRMANN

Trilinears (a^6+2*(b^2+c^2)*a^4-(b^4+3*b^2*c^2+c^4)*a^2-2*(b^4-c^4)*(b^2-c^2))*a : :
X(12367) = 9*R^2*X(67)-2*SW*X(74)

The reciprocal orthologic center of these triangles is X(9970).

X(12367) lies on these lines: {6,25}, {23,2854}, {30,5648}, {50,5191}, {67,74}, {110,8705}, {156,11663}, {187,5938}, {237,9142}, {323,9019}, {399,511}, {512,5104}, {542,3581}, {597,10545}, {599,3098}, {1614,12061}, {1995,8547}, {2781,12112}, {3448,8262}, {5640,8546}, {7575,11579}, {10540,11649}

X(12367) = reflection of X(i) in X(j) for these (i,j): (6,1495), (3448,8262), (10510,110), (11579,7575)
X(12367) = X(67)-of-anti-orthocentroidal-triangle

X(12368) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO ANTI-ORTHOCENTROIDAL

Barycentrics 4*(b+c)*S^4*(9*R^2-2*SW)+(S^2-3*SA*SC)*(S^2-3*SA*SB)*a^2*(b+c+a) : :
X(12368) = 3*X(1)-4*X(11723) = 3*X(113)-2*X(11723) = 2*X(125)-3*X(5587) = 3*X(381)-2*X(12261) = 5*X(1698)-4*X(6699) = 3*X(3576)-4*X(5972) = 3*X(5655)-2*X(11699) = 3*X(5657)-X(12244) = 3*X(5790)-X(10620) = X(7978)-3*X(10706)

The reciprocal orthologic center of these triangles is X(12112).

X(12368) lies on these lines: {1,113}, {2,11709}, {8,146}, {10,74}, {40,2777}, {65,79}, {72,2778}, {110,515}, {125,5587}, {355,5663}, {381,12261}, {516,10721}, {517,7728}, {519,7978}, {541,3679}, {542,3751}, {690,9864}, {944,11720}, {946,7984}, {1698,6699}, {1737,10081}, {1837,3028}, {2779,5086}, {2781,3416}, {2931,8185}, {2948,5691}, {3465,4551}, {3576,5972}, {3822,5494}, {5090,12133}, {5655,11699}, {5657,12244}, {5687,12327}, {5688,7726}, {5689,7725}, {5777,10693}, {5790,10620}, {5847,10752}, {8193,9919}, {8197,12365}, {8204,12366}, {8227,11735}, {8998,9583}, {9798,12168}, {9857,9984}, {10039,10065}, {10088,10572}, {10791,12192}

X(12368) = midpoint of X(i) and X(j) for these {i,j}: {8,146}, {2948,5691}
X(12368) = reflection of X(i) in X(j) for these (i,j): (1,113), (74,10), (944,11720), (7984,946), (10693,5777)
X(12368) = anticomplement of X(11709)
X(12368) = X(74)-of-outer-Garcia-triangle
X(12368) = X(1)-of-X(30)-Fuhrmann-triangle

X(12369) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO ANTI-ORTHOCENTROIDAL

Trilinears (cos(B-C)-2*cos(A))*((2*cos(2*A)+3)*cos(B-C)-4*cos(A))*((4*cos(2*A)+5)*cos(B-C)-cos(A)*cos(2*(B-C))-7*cos(A)-cos(3*A)) : :
X(12369) = 2*X(125)-3*X(11897) = X(10620)-3*X(11911) = 2*X(11709)-3*X(11831) = 3*X(11845)-X(12244)

The reciprocal orthologic center of these triangles is X(12112).

X(12369) lies on these lines: {30,110}, {74,402}, {113,1650}, {125,11897}, {146,4240}, {541,1651}, {690,12181}, {2777,7740}, {3028,11909}, {5663,11251}, {7725,11901}, {7726,11902}, {7978,11910}, {9904,11852}, {9919,11853}, {9984,11885}, {10065,11912}, {10081,11913}, {10620,11911}, {11709,11831}, {11832,12133}, {11839,12192}, {11845,12244}, {11848,12327}, {11900,12368}

X(12369) = midpoint of X(146) and X(4240)
X(12369) = reflection of X(i) in X(j) for these (i,j): (74,402), (1650,113)
X(12369) = X(74)-of-Gossard-triangle

X(12370) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st HYACINTH TO ANTI-ORTHOCENTROIDAL

Trilinears (cos(2*A)+2)*cos(B-C)-cos(A)*cos(2*(B-C))-2*cos(A)-cos(3*A) : :
X(12370) = X(3)-3*X(12022) = 3*X(568)-X(6240) = X(1885)+2*X(11264) = 3*X(3060)+X(12289) = 5*X(3567)-X(12278)

The reciprocal orthologic center of these triangles is X(399).

X(12370) lies on these lines: {3,3580}, {4,1994}, {5,578}, {6,12293}, {23,12254}, {30,52}, {49,403}, {54,10024}, {68,7526}, {113,137}, {143,3575}, {156,235}, {265,1594}, {382,12174}, {389,11800}, {539,5907}, {568,6240}, {576,1353}, {1352,9925}, {1614,11799}, {1885,5663}, {1899,12084}, {2777,11232}, {3060,12289}, {3564,5876}, {3567,12278}, {5133,6288}, {5446,10115}, {5449,11430}, {6000,10116}, {6101,12362}, {6243,12225}, {6644,12118}, {6676,10610}, {6696,10264}, {7530,9833}, {9818,12166}, {10274,11563}, {10982,11818}, {11536,12227}

X(12370) = midpoint of X(6243) and X(12225)
X(12370) = reflection of X(i) in X(j) for these (i,j): (5,12241), (3575,143), (6101,12362), (11819,5446), (12134,546)
X(12370) = X(1)-of-1st-Hyacinth-triangle if ABC is acute
X(12370) = {X(578), X(9927)}-harmonic conjugate of X(5)

X(12371) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO ANTI-ORTHOCENTROIDAL

Barycentrics (-a+b+c)*(b-c)^2*S^3*(9*R^2-2*SW)+(R-r)*(S^2-3*SA*SC)*(S^2-3*SA*SB)*a^2 : :
X(12371) = (R-2*r)*X(11)-(R-r)*X(74)

The reciprocal orthologic center of these triangles is X(12112).

X(12371) lies on these lines: {11,74}, {110,11826}, {113,1376}, {125,10893}, {146,3434}, {355,7728}, {541,11235}, {690,12182}, {2777,12114}, {3028,10947}, {5663,10525}, {7725,10919}, {7726,10920}, {7978,10944}, {9904,10826}, {9919,10829}, {9984,10871}, {10065,10523}, {10081,10948}, {10620,11928}, {10785,12244}, {10794,12192}, {10914,12368}, {11373,11709}, {11390,12133}, {11865,12365}, {11866,12366}, {11903,12369}

X(12371) = reflection of X(12327) in X(113)
X(12371) = reflection of X(12372) in X(7728)
X(12371) = X(74)-of-inner-Johnson-triangle
X(12371) = X(12381)-of-outer-Johnson-triangle

X(12372) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO ANTI-ORTHOCENTROIDAL

Barycentrics (a-b+c)*(a+b-c)*(b+c)^2*S^3*(9*R^2-2*SW)+(R+r)*(S^2-3*SA*SC)*(S^2-3*SA*SB)*a^2*(a+b+c) : :
X(12372) = (R+2*r)*X(12)-(R+r)*X(74)

The reciprocal orthologic center of these triangles is X(12112).

X(12372) lies on these lines: {12,74}, {72,2778}, {110,11827}, {113,958}, {125,10894}, {146,3436}, {265,2779}, {355,7728}, {541,11236}, {690,12183}, {2777,11500}, {3028,10953}, {5663,10526}, {6253,10721}, {7725,10921}, {7726,10922}, {7978,10950}, {9904,10827}, {9919,10830}, {9984,10872}, {10065,10954}, {10081,10523}, {10620,11929}, {10786,12244}, {10795,12192}, {11374,11709}, {11391,12133}, {11867,12365}, {11868,12366}, {11904,12369}

X(12372) = reflection of X(12371) in X(7728)
X(12372) = X(74)-of-outer-Johnson-triangle
X(12372) = X(12382)-of-inner-Johnson-triangle

X(12373) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO ANTI-ORTHOCENTROIDAL

Trilinears (4*cos(A)-6*cos(2*A)-8)*cos(B-C)+2*cos(A)*cos(2*(B-C))+10*cos(A)-2*cos(2*A)+cos(3*A)-3 : :
X(12373) = (R+2*r)*X(12)-r*X(74)

The reciprocal orthologic center of these triangles is X(12112).

X(12373) lies on the inner-Johnson-Yff-circle and these lines: {1,7728}, {4,3028}, {5,10081}, {12,74}, {30,10088}, {55,2777}, {56,113}, {65,79}, {73,9627}, {110,7354}, {125,10895}, {146,388}, {495,10065}, {498,12041}, {541,11237}, {690,12184}, {1388,11723}, {1478,5663}, {1479,1539}, {1511,4299}, {2931,9658}, {2948,9579}, {3031,9553}, {3043,9652}, {3047,9653}, {3085,12244}, {3448,5229}, {5204,5972}, {5270,7727}, {5434,10706}, {6284,10721}, {7725,10923}, {7726,10924}, {7978,10944}, {9578,9904}, {9647,10819}, {9648,10817}, {9654,10620}, {9659,10117}, {9919,10831}, {9984,10873}, {10082,11805}, {10483,12121}, {10797,12192}, {11375,11709}, {11392,12133}, {11501,12327}, {11905,12369}

X(12373) = reflection of X(10065) in X(495)
X(12373) = X(74)-of-1st-Johnson-Yff-triangle
X(12373) = {X(1),X(7728)}-harmonic conjugate of X(12374)

X(12374) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO ANTI-ORTHOCENTROIDAL

Trilinears (4*cos(A)+6*cos(2*A)+8)*cos(B-C)-2*cos(A)*cos(2*(B-C))-10*cos(A)-2*cos(2*A)-cos(3*A)-3 : :
X(12374) = (R-2*r)*X(11)+r*X(74)

The reciprocal orthologic center of these triangles is X(12112).

X(12374) lies on the outer-Johnson-Yff-circle and these lines: {1,7728}, {5,10065}, {11,74}, {30,10091}, {55,113}, {56,2777}, {110,6284}, {125,10896}, {146,497}, {265,3583}, {399,9668}, {496,10081}, {499,12041}, {541,11238}, {690,12185}, {1478,1539}, {1479,5663}, {1511,4302}, {2931,9673}, {2948,9580}, {3031,9554}, {3043,9667}, {3047,9666}, {3057,12368}, {3058,10706}, {3086,12244}, {3448,5225}, {5217,5972}, {7354,10721}, {7725,10925}, {7726,10926}, {7978,10950}, {9581,9904}, {9630,10118}, {9660,10819}, {9663,10817}, {9669,10620}, {9672,10117}, {9919,10832}, {9984,10874}, {10066,11805}, {10798,12192}, {10833,12168}, {11376,11709}, {11393,12133}, {11502,12327}, {11871,12365}, {11872,12366}, {11906,12369}

X(12374) = reflection of X(10081) in X(496)
X(12374) = X(74)-of-2nd-Johnson-Yff-triangle
X(12374) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7728,12373), (3583,7727,265)

X(12375) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS TO ANTI-ORTHOCENTROIDAL

Trilinears a*((3*SA+S)*S^2*(9*R^2-2*SW)+(S^2-3*SA*SC)*(S^2-3*SA*SB)) : :
X(12375) = (3*S+SW)*X(6200)-2*S*X(74)

Let Ka, Kb, Kc be the free vertices of the Kenmotu squares. Triangle KaKbKc is here named the 1st Kenmotu free vertices triangle. KaKbKc is the anti-Kosnita triangle of the 1st Kenmotu diagonals triangle. KaKbKc is homothetic to ABC at X(372). X(12375) is the perspector of the 1st Kenmotu diagonals triangle and the reflection of KaKbKc in X(371). (Randy Hutson, July 21, 2017)

The reciprocal orthologic center of these triangles is X(3581).

X(12375) lies on these lines: {6,13}, {74,6200}, {110,372}, {125,10576}, {146,6561}, {371,5663}, {485,3448}, {590,10264}, {615,10272}, {1151,10620}, {1511,6396}, {1986,5412}, {2066,7727}, {2771,7969}, {3068,12317}, {3311,12308}, {5410,12165}, {5415,7724}, {5609,6420}, {6453,10817}, {7722,10880}, {7723,10897}, {7968,11699}, {8909,12302}, {9826,10961}, {11417,12219}, {11447,12270}, {11462,12281}, {11473,12292}, {11513,12358}

X(12375) = {X(6),X(399)}-harmonic conjugate of X(12376)
X(12375) = X(80)-of-1st-Kenmotu-diagonals-triangle if ABC is acute
X(12375) = X(110)-of-1st-Kenmotu-free-vertices-triangle

X(12376) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS TO ANTI-ORTHOCENTROIDAL

Trilinears a*((3*SA-S)*S^2*(9*R^2-2*SW)+(S^2-3*SA*SC)*(S^2-3*SA*SB)) : :
X(12376) = (3*S-SW)*X(6396)-2*S*X(74)

The reciprocal orthologic center of these triangles is X(3581).

Let Ka', Kb', Kc' be the free vertices of the 2nd Kenmotu squares. Triangle Ka'Kb'Kc' is here named the 2nd Kenmotu free vertices triangle. Ka'Kb'Kc' is the anti-Kosnita triangle of the 2nd Kenmotu diagonals triangle. Ka'Kb'Kc' is homothetic to ABC at X(371). X(12376) is the perspector of the 2nd Kenmotu diagonals triangle and the reflection of Ka'Kb'Kc' in X(372). (Randy Hutson, July 21, 2017)

X(12376) lies on these lines: {6,13}, {74,6396}, {110,371}, {125,10577}, {146,6560}, {372,5663}, {486,3448}, {590,10272}, {615,10264}, {1152,10620}, {1511,6200}, {1986,5413}, {2771,7968}, {3069,12317}, {3312,12308}, {5411,12165}, {5414,7727}, {5416,7724}, {5609,6419}, {5642,8994}, {6126,8973}, {6454,10818}, {7722,10881}, {7723,10898}, {7969,11699}, {9826,10963}, {11418,12219}, {11448,12270}, {11463,12281}, {11474,12292}, {11514,12358}

X(12376) = {X(6),X(399)}-harmonic conjugate of X(12375)
X(12376) = X(80)-of-2nd-Kenmotu-diagonals-triangle if ABC is acute
X(12376) = X(110)-of-2nd-Kenmotu-free-vertices-triangle

X(12377) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO ANTI-ORTHOCENTROIDAL

Trilinears a*(3*SA*SW^2*(12*R^2-SA-2*SW)+4*S*(72*SA*R^4-(20*S^2+15*SA^2+12*SA*SW)*R^2+2*(2*S^2+SA^2)*SW)-(36*(SA+SW)*R^2-(8*SA+7*SW)*SW)*S^2) : :

The reciprocal orthologic center of these triangles is X(12112).

X(12377) lies on these lines: {74,493}, {110,11828}, {113,8222}, {125,8212}, {146,6462}, {541,12152}, {690,12186}, {2777,9838}, {3028,11947}, {5663,10669}, {6461,12378}, {7725,8216}, {7726,8218}, {7728,8220}, {7978,8210}, {8188,9904}, {8194,9919}, {8214,12368}, {9984,10875}, {10065,11951}, {10081,11953}, {10620,11949}, {10945,12371}, {10951,12372}, {11377,11709}, {11394,12133}, {11503,12327}, {11840,12192}, {11846,12244}, {11907,12369}, {11930,12373}, {11932,12374}

X(12377) = X(74)-of-Lucas-homothetic-triangle

X(12378) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO ANTI-ORTHOCENTROIDAL

Trilinears a*(3*SA*SW^2*(12*R^2-SA-2*SW)-4*S*(72*SA*R^4-(20*S^2+15*SA^2+12*SA*SW)*R^2+2*(2*S^2+SA^2)*SW)-(36*(SA+SW)*R^2-(8*SA+7*SW)*SW)*S^2) : :

The reciprocal orthologic center of these triangles is X(12112).

X(12378) lies on these lines: {74,494}, {110,11829}, {113,8223}, {125,8213}, {146,6463}, {541,12153}, {690,12187}, {2777,9839}, {3028,11948}, {5663,10673}, {6461,12377}, {7725,8217}, {7726,8219}, {7728,8221}, {7978,8211}, {8189,9904}, {8195,9919}, {8215,12368}, {9984,10876}, {10065,11952}, {10081,11954}, {10620,11950}, {10946,12371}, {10952,12372}, {11378,11709}, {11395,12133}, {11504,12327}, {11841,12192}, {11847,12244}, {11908,12369}, {11931,12373}, {11933,12374}

X(12378) = X(74)-of-Lucas(-1)-homothetic-triangle

X(12379) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL TO MIDHEIGHT

Trilinears 6*(11*cos(2*A)+cos(4*A)+12)*cos(B-C)-4*(5*cos(A)+cos(3*A))*cos(2*(B-C))-20*cos(3*A)-cos(5*A)-99*cos(A) : :
X(12379) = (7*R^2-SW)*X(74)-3*R^2*X(403) = 4*X(1514)-3*X(11744)

The reciprocal orthologic center of these triangles is X(974).

X(12379) lies on these lines: {6,64}, {74,403}, {399,2935}, {974,10821}, {1656,3357}, {2777,3581}, {4550,10606}, {5663,12364}, {5895,11438}, {7687,10816}

X(12380) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL TO REFLECTION

Trilinears 3*(2*cos(4*A)+1)*cos(B-C)-2*(cos(A)+2*cos(3*A))*cos(2*(B-C))+cos(3*A)-cos(5*A)-3*cos(A) : :

The reciprocal orthologic center of these triangles is X(7731).

X(12380) lies on these lines: {6,24}, {23,12364}, {26,12280}, {74,10421}, {186,10821}, {399,1154}, {1495,2914}, {1614,6242}, {1657,7691}, {9707,12175}, {9920,11456}, {10628,12112}, {11438,12254}, {12290,12307}

X(12380) = reflection of X(2914) in X(1495)
X(12380) = {X(24), X(12291)}-harmonic conjugate of X(54)

X(12381) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO ANTI-ORTHOCENTROIDAL

Trilinears 8*R*S^4*(9*R^2-2*SW)+(R-r)*(S^2-3*SA*SC)*(S^2-3*SA*SB)*a*(a+b+c) : :
X(12381) = 2*R*X(1)-(R-r)*X(74)

The reciprocal orthologic center of these triangles is X(12112).

X(12381) lies on these lines: {1,74}, {12,12371}, {110,11248}, {113,5552}, {125,10531}, {146,10528}, {541,11239}, {690,12189}, {2777,12115}, {3028,10965}, {5663,10679}, {6256,10721}, {7725,10929}, {7726,10930}, {7728,10942}, {9919,10834}, {9984,10878}, {10620,12000}, {10803,12192}, {10805,12244}, {10915,12368}, {10955,12372}, {10956,12373}, {10958,12374}, {11400,12133}, {11509,12327}, {11881,12365}, {11882,12366}, {11914,12369}, {11955,12377}, {11956,12378}

X(12381) = reflection of X(74) in X(10065)
X(12381) = {X(74),X(7978)}-harmonic conjugate of X(12382)
X(12381) = X(74)-of-inner-Yff-tangents-triangle

X(12382) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO ANTI-ORTHOCENTROIDAL

Trilinears 8*R*S^4*(9*R^2-2*SW)+(R+r)*(S^2-3*SA*SC)*(S^2-3*SA*SB)*a*(a+b+c) : :
X(12382) = 2*R*X(1)-(R+r)*X(74)

The reciprocal orthologic center of these triangles is X(12112).

X(12382) lies on these lines: {1,74}, {11,12372}, {110,11249}, {113,10527}, {125,10532}, {146,10529}, {541,11240}, {690,12190}, {2777,12116}, {2779,10091}, {3028,10966}, {5663,10680}, {7725,10931}, {7726,10932}, {7728,10943}, {9919,10835}, {9984,10879}, {10620,12001}, {10804,12192}, {10806,12244}, {10916,12368}, {10949,12371}, {10957,12373}, {10959,12374}, {11401,12133}, {11510,12327}, {11883,12365}, {11884,12366}, {11915,12369}, {11957,12377}, {11958,12378}

X(12382) = reflection of X(74) in X(10081)
X(12382) = {X(74),X(7978)}-harmonic conjugate of X(12381)
X(12382) = X(74)-of-outer-Yff-tangents-triangle

X(12383) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO 1st HYACINTH

Trilinears (4*cos(2*A)+3)*cos(B-C)-2*cos(A)*cos(2*(B-C))-6*cos(A)-3*cos(3*A) : :
X(12383) = 3*X(2)-4*X(1511) = 3*X(4)-4*X(113) = 3*X(4)-2*X(10733) = 5*X(4)-4*X(12295) = 3*X(110)-2*X(113) = 3*X(110)-X(10733) = 5*X(110)-2*X(12295) = X(146)-3*X(9143) = 2*X(399)-3*X(9143) = 4*X(12121)-X(12244)

Let A'B'C' be the dual of orthic triangle (a.k.a 1st anti-circumperp triangle). Let L, M, N be lines through A', B', C', respectively, parallel to the Euler line. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L', M', N' concur in X(12383). (cf. X(74), X(113), X(399), X(1511), X(5504), X(10692), X(14094), X(30714)) (Randy Hutson, March 21, 2019)

The reciprocal orthologic center of these triangles is X(6102).

X(12383) lies on the cubics K544, K611, K753 and these lines: {2,265}, {3,2888}, {4,110}, {20,5663}, {24,12310}, {30,146}, {67,10519}, {68,5963}, {69,74}, {100,12334}, {125,631}, {147,7422}, {185,12284}, {186,2931}, {193,1986}, {378,12168}, {381,10272}, {388,10088}, {497,10091}, {511,7731}, {515,2948}, {541,11001}, {550,2889}, {568,11561}, {974,2854}, {1092,12289}, {1112,7487}, {1503,2892}, {1539,3543}, {1656,11801}, {1657,12308}, {1993,2914}, {2771,3648}, {2777,3529}, {2781,5596}, {3028,4293}, {3060,11557}, {3068,10819}, {3069,10820}, {3090,5972}, {3091,10113}, {3146,5609}, {3520,12302}, {3522,12041}, {3524,6699}, {3533,6723}, {3545,5642}, {3564,10295}, {3567,11800}, {3616,12261}, {4302,7727}, {5055,11694}, {5157,5622}, {5562,12281}, {5603,11720}, {5648,11180}, {5656,11744}, {5667,9033}, {5889,11271}, {6053,10706}, {6143,12038}, {6193,7722}, {6560,12375}, {6561,12376}, {7552,11464}, {7706,11422}, {7732,10783}, {7733,10784}, {7787,12201}, {7967,7984}, {8907,9938}, {9919,12082}, {9927,11449}, {9934,11206}, {9976,11179}, {10114,11438}, {10117,12088}, {10574,11806}, {10628,11412}, {11469,12292}, {12270,12273}

X(12383) = midpoint of X(i) and X(j) for these {i,j}: {1657,12308}, {12270,12273}
X(12383) = reflection of X(i) in X(j) for these (i,j): (4,110), (20,12121), (146,399), (265,1511), (3146,7728), (3448,3), (3543,5655), (5889,11562), (7728,5609), (10620,550), (10733,113), (11180,5648), (12244,20), (12281,5562), (12284,185), (12317,74), (12319,5504)
X(12383) = isogonal conjugate of X(35372)
X(12383) = anticomplementary-circle-inverse of X(39118)
X(12383) = cevapoint of X(399) and X(2931)
X(12383) = crossdifference of every pair of points on line X(686)X(14398)
X(12383) = anticomplement of X(265)
X(12383) = X(265)-of-anticomplementary-triangle
X(12383) = X(110)-of-anti-Euler-triangle
X(12383) = crosspoint, wrt excentral or tangential triangle, of X(399) and X(2931)
X(12383) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (110,5504,3043), (110,10733,113), (113,10733,4), (146,9143,399), (265,1511,2), (376,12317,74), (1147,12278,4)

X(12384) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO 1st ORTHOSYMMEDIAL

Barycentrics (R^2-SW)*S^4+((4*SA-8*SW)*R^2-SA^2+2*SW^2)*SW*S^2+2*(4*R^2-SW)*(SB+SC)*SA*SW^2 : :
X(12384) = 3*X(2)-4*X(132) = 3*X(4)-2*X(10749) = 4*X(127)-5*X(3091) = 7*X(3523)-8*X(6720) = 3*X(3543)-2*X(10735) = 5*X(3616)-4*X(12265) = 3*X(3839)-2*X(10718) = 3*X(5731)-4*X(11722)

The reciprocal orthologic center of these triangles is X(4).

X(12384) lies on the anticomplementary circle and these lines: {2,107}, {3,12253}, {4,339}, {20,112}, {100,12340}, {127,3091}, {146,9517}, {147,2799}, {148,2794}, {149,2831}, {150,2825}, {151,2853}, {152,9518}, {153,2806}, {388,6020}, {497,3320}, {2781,3448}, {3523,6720}, {3543,10735}, {3616,12265}, {3839,10718}, {5731,11722}, {7787,12207}

X(12384) = reflection of X(i) in X(j) for these (i,j): (20,112), (1297,132), (12253,3)
X(12384) = anticomplement of X(1297)
X(12384) = orthoptic circle of Steiner inellipse-inverse-of-X(6716)
X(12384) = polar circle-inverse-of-X(12145)
X(12384) = X(1297)-of-anticomplementary-triangle
X(12384) = X(12918)-of-anti-Euler-triangle
X(12384) = de-Longchamps-circle-inverse of X(34168)
X(12384) = {X(132), X(1297)}-harmonic conjugate of X(2)

X(12385) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ASCELLA TO ANTLIA

Trilinears 4*p^8*(2*q*p-1)+4*(4*q^2-7)*q*p^7+4*(2*q^2+7)*p^6+4*(2*q^4-8*q^2+11)*q*p^5+(12*q^4+4*q^2-53)*p^4-4*(q^4-5*q^2+5)*q*p^3-2*(13*q^2-22)*p^2+4*(q^2-2)*q*p-(q^2-4)^2+4*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(12385) = 6*R*X(10855)-(2*R+r)*X(12386)

The reciprocal orthologic center of these triangles is X(1).

X(12385) lies on these lines: {3,1279}, {10855,12386}

X(12386) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ATIK TO ANTLIA

Barycentrics 4*a^10-13*(b+c)*a^9+(19*b^2+62*b*c+19*c^2)*a^8-2*(b+c)*(13*b^2+31*b*c+13*c^2)*a^7+2*(b^2+3*b*c+c^2)*(13*b^2+6*b*c+13*c^2)*a^6+2*(b+c)*(2*b^4+2*c^4-b*c*(73*b^2-78*b*c+73*c^2))*a^5-2*(16*b^6+16*c^6-(61*b^4+61*c^4+2*b*c*(2*b^2-25*b*c+2*c^2))*b*c)*a^4+2*(b^2-c^2)*(b-c)^3*(13*b^2+19*b*c+13*c^2)*a^3-2*(7*b^6+7*c^6-(5*b^4+5*c^4-b*c*(5*b^2+18*b*c+5*c^2))*b*c)*(b-c)^2*a^2+(b^2-c^2)*(b-c)^5*(9*b^2+8*b*c+9*c^2)*a-(-c^4+b^4)*(b^2-c^2)*(b-c)^2*(3*b^2-2*b*c+3*c^2) : :
X(12386) = 6*R*X(10855)-(4*R-r)*X(12385)

The reciprocal orthologic center of these triangles is X(1).

X(12386) lies on these lines: {10855,12385}, {10860,12387}

X(12387) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st CIRCUMPERP TO ANTLIA

Trilinears a^9-(b+c)*a^8-4*b*c*a^7+18*b*c*(b+c)*a^6-2*(b^4+c^4+8*b*c*(2*b^2+b*c+2*c^2))*a^5+2*(b+c)*(b^4+c^4+b*c*(17*b^2-12*b*c+17*c^2))*a^4-4*(5*b^4+5*c^4+2*b*c*(b^2-b*c+c^2))*b*c*a^3+2*(b^2-c^2)*(b-c)*b*c*(3*b^2+2*b*c+3*c^2)*a^2+(b^2-c^2)^2*(b^4+c^4-2*b*c*(4*b^2-5*b*c+4*c^2))*a-(-c^4+b^4)*(b-c)^5 : :
X(12387) = (2*R-r)*X(10860)+2*R*X(12386)

The reciprocal orthologic center of these triangles is X(1).

X(12387) lies on these lines: {3,1279}, {10860,12386}

X(12388) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TO ANTLIA

Trilinears a^9-3*(b+c)*a^8+4*(b^2+4*b*c+c^2)*a^7-2*(b+c)*(2*b^2+9*b*c+2*c^2)*a^6+2*(b^4+c^4+2*b*c*(3*b^2+8*b*c+3*c^2))*a^5+2*(b+c)*(b^4+c^4-b*c*(9*b^2-8*b*c+9*c^2))*a^4-4*(b^4+c^4-4*b*c*(b^2+b*c+c^2))*(b-c)^2*a^3+2*(b^2-c^2)*(b-c)*(2*b^4+2*c^4-b*c*(7*b^2-2*b*c+7*c^2))*a^2-(3*b^4-2*b^2*c^2+3*c^4)*(b-c)^4*a+(-c^4+b^4)*(b-c)^5 : :
X(12388) = (6*R-r)*X(8583)-2*R*X(12386)

The reciprocal orthologic center of these triangles is X(1).

X(12388) lies on these lines: {1,7056}, {3,1279}, {2961,7084}, {8583,12386}

X(12388) = reflection of X(12387) in X(3)

X(12389) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-CONWAY TO ANTLIA

Trilinears (b+c)*a^8-(2*b^2+9*b*c+2*c^2)*a^7+(b+c)*(2*b^2+15*b*c+2*c^2)*a^6+(b+c)*(19*b^2-26*b*c+19*c^2)*a^4*b*c-(2*b^4+2*c^4+(17*b^2+14*b*c+17*c^2)*b*c)*a^5+(2*b^6+2*c^6-(15*b^4+15*c^4-2*(5*b^2+13*b*c+5*c^2)*b*c)*b*c)*a^3-(b+c)*(2*b^6+2*c^6-(9*b^4+9*c^4-2*(5*b^2+3*b*c+5*c^2)*b*c)*b*c)*a^2+(2*b^6+2*c^6-(3*b^2-4*b*c+3*c^2)*(b+c)^2*b*c)*(b-c)^2*a-(b^4-c^4)*(b-c)*(b^4+c^4-3*(b-c)^2*b*c) : :
X(12389) = (R-2*r)*X(100)+2*r*X(12387)

The reciprocal orthologic center of these triangles is X(1).

X(12389) lies on these lines: {100,12387}, {2975,12388}, {5744,12385}, {11678,12386}

X(12390) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CONWAY TO ANTLIA

Trilinears (b+c)*a^11-3*(b^2+3*b*c+c^2)*a^10+(b+c)*(3*b^2+20*b*c+3*c^2)*a^9-(b^2+22*b*c+c^2)*(b^2+b*c+c^2)*a^8-2*(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a^7+2*(3*b^6+3*c^6-(b^4+c^4-(9*b^2+40*b*c+9*c^2)*b*c)*b*c)*a^6-2*(b+c)*(3*b^6+3*c^6+(b^2+4*b*c+c^2)*(2*b^2+b*c+2*c^2)*b*c)*a^5+2*(b^8+c^8+(9*b^6+9*c^6+(10*b^4+10*c^4-(15*b^2-14*b*c+15*c^2)*b*c)*b*c)*b*c)*a^4+(b^2-c^2)*(b-c)*(b^6+c^6-(10*b^4+10*c^4+(17*b^2-4*b*c+17*c^2)*b*c)*b*c)*a^3-(3*b^8+3*c^8-(5*b^6+5*c^6+(14*b^4+14*c^4-(17*b^2+30*b*c+17*c^2)*b*c)*b*c)*b*c)*(b-c)^2*a^2+(b^2-c^2)*(b-c)^3*(3*b^6+3*c^6-(4*b^4+4*c^4+3*(b^2+c^2)*b*c)*b*c)*a-(b^4-c^4)*(b^2-c^2)*(b-c)^2*(b^4+c^4-3*(b-c)^2*b*c) : :
X(12390) = 3*X(2)-4*X(12385)

The reciprocal orthologic center of these triangles is X(1).

X(12390) lies on these lines: {2,12385}, {21,12388}, {63,12389}, {7411,12387}, {10861,12386}

X(12391) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CONWAY TO ANTLIA

Barycentrics a^10-4*(b+c)*a^9+(7*b^2+34*b*c+7*c^2)*a^8-8*(b+c)*(b^2+6*b*c+c^2)*a^7-8*(b+c)*(11*b^2-13*b*c+11*c^2)*a^5*b*c+2*(3*b^4+3*c^4+4*(7*b^2+8*b*c+7*c^2)*b*c)*a^6-2*(3*b^6+3*c^6-(38*b^4+38*c^4-(3*b^2+40*b*c+3*c^2)*b*c)*b*c)*a^4+8*(b^2-c^2)*(b-c)*(b^4+c^4-2*(b+c)^2*b*c)*a^3-(7*b^6+7*c^6-(10*b^4+10*c^4-(5*b^2+28*b*c+5*c^2)*b*c)*b*c)*(b-c)^2*a^2+4*(b^2-c^2)*(b-c)^3*(b^4+c^4-2*(b^2+c^2)*b*c)*a-(b^4-c^4)^2*(b-c)^2 : :
X(12391) = (4*R+r)*X(8)-8*R*X(12386) = 5*X(3616)-4*X(12388) = 3*X(9778)-4*X(12387)

The reciprocal orthologic center of these triangles is X(1).

X(12391) lies on these lines: {7,12390}, {8,12386}, {329,12389}, {962,4645}, {3616,12388}, {9776,12385}, {9778,12387}

X(12392) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO ANTLIA

Trilinears (2*b^2+3*b*c+2*c^2)*a^10+2*(9*b^2+22*b*c+9*c^2)*a^8*b*c-(b+c)*(2*b^2+15*b*c+2*c^2)*a^9-4*(b+c)*(b^2+3*b*c+c^2)*a^7*b*c-2*(2*b^6+2*c^6-(3*b^4+3*c^4+4*(b^2-8*b*c+c^2)*b*c)*b*c)*a^6-4*(4*b^6+4*c^6-(7*b^4+7*c^4+2*(5*b^2+b*c+5*c^2)*b*c)*b*c)*a^4*b*c+2*(b+c)*(2*b^6+2*c^6+(b^4+c^4-12*(b-c)^2*b*c)*b*c)*a^5+4*(b+c)*(b^6+c^6-(5*b^4+5*c^4+(3*b^2-2*b*c+3*c^2)*b*c)*b*c)*a^3*b*c+(b^2-c^2)^2*(2*b^6+2*c^6-(9*b^4+9*c^4-2*(5*b^2+7*b*c+5*c^2)*b*c)*b*c)*a^2-2*(b^4-c^4)*(b^2-c^2)*(b-c)^4*b*c-(b^2-c^2)*(b-c)*(2*b^8+2*c^8-3*(3*b^6+3*c^6-(2*b^2-3*b*c+2*c^2)*(3*b^2+4*b*c+3*c^2)*b*c)*b*c)*a : :
X(12392) = (-3*s^2+SW)*X(10862)+(2*(r^2+s^2))*X(12386)

The reciprocal orthologic center of these triangles is X(1).

X(12392) lies on these lines: {1,5575}, {10434,12387}, {10444,12390}, {10446,12391}, {10856,12385}, {10862,12386}, {10882,12388}, {11679,12389}

X(12393) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd EULER TO ANTLIA

Barycentrics (b+c)*a^9+18*(b+c)*a^7*b*c-(b^2+10*b*c+c^2)*a^8-16*(b^2+b*c+c^2)*a^6*b*c-2*(b+c)*(b^4+c^4-(13*b^2-16*b*c+13*c^2)*b*c)*a^5+2*(b^2-c^2)*(b-c)*(11*b^2+2*b*c+11*c^2)*a^3*b*c+2*(b^4+c^4-2*(7*b^2+8*b*c+7*c^2)*b*c)*(b-c)^2*a^4-8*(b^3-c^3)*(b-c)*(b^2+c^2)*a^2*b*c+(b^2-c^2)*(b-c)^5*(b^2+4*b*c+c^2)*a-(b^4-c^4)^2*(b-c)^2 : :
X(12393) = (5*R-r)*X(10863)-R*X(12386)

The reciprocal orthologic center of these triangles is X(1).

X(12393) lies on these lines: {2,12387}, {4,12388}, {5,3823}, {8727,12385}, {9779,12391}, {10863,12386}, {10883,12390}, {10886,12392}, {11680,12389}

X(12393) = midpoint of X(4) and X(12388)
X(12393) = complement of X(12387)

X(12394) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th EULER TO ANTLIA

Barycentrics (b+c)*a^9-(b+3*c)*(3*b+c)*a^8+2*(b+c)*(2*b^2+9*b*c+2*c^2)*a^7-4*(b^2+b*c+c^2)*(b^2+6*b*c+c^2)*a^6+2*(b+c)*(b^4+c^4+(13*b^2-4*b*c+13*c^2)*b*c)*a^5+2*(b^4+c^4-8*(b^2-b*c+c^2)*b*c)*(b+c)^2*a^4-2*(b^2-c^2)*(b-c)*(2*b^4+2*c^4+(b+c)^2*b*c)*a^3+4*(b^2-c^2)^2*(b^4+c^4-3*(b-c)^2*b*c)*a^2-(b^2-c^2)*(b-c)^5*(3*b^2+4*b*c+3*c^2)*a+(b^4-c^4)^2*(b-c)^2 : :
X(12394) = (3*R-r)*X(8582)+R*X(12386)

The reciprocal orthologic center of these triangles is X(1).

X(12394) lies on these lines: {2,12388}, {4,12387}, {5,3823}, {10,1541}, {4197,12390}, {8582,12386}, {8728,12385}, {9780,12391}, {10887,12392}, {11681,12389}

X(12394) = midpoint of X(4) and X(12387)
X(12394) = reflection of X(12393) in X(5)
X(12394) = complement of X(12388)

X(12395) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS TO ANTLIA

Trilinears a^9-3*(b+c)*a^8+4*(b^2+b*c+c^2)*a^7-4*(b+c)*(b^2+c^2)*a^6+2*(b^4+c^4-2*(3*b^2-8*b*c+3*c^2)*b*c)*a^5+2*(b+c)*(b^4+c^4+4*(3*b-2*c)*(2*b-3*c)*b*c)*a^4-4*(b^2+3*b*c+c^2)*(b^2+8*b*c+c^2)*(b-c)^2*a^3+4*(b^2-c^2)*(b-c)*(b^4+c^4-2*(b^2-8*b*c+c^2)*b*c)*a^2-(3*b^4-2*b^2*c^2+3*c^4)*(b-c)^4*a+(b^4-c^4)*(b-c)^3*(b^2+4*b*c+c^2) : :
X(12395) = 3*X(1)-2*X(12388) = 3*X(3679)-4*X(12394)

The reciprocal orthologic center of these triangles is X(1).

X(12395) lies on these lines: {1,7056}, {145,12391}, {1721,7982}, {3679,12394}, {7991,12387}, {11518,12385}, {11519,12386}, {11520,12390}, {11521,12392}, {11522,12393}, {11682,12389}

X(12395) = midpoint of X(145) and X(12391)
X(12395) = reflection of X(7991) in X(12387)

X(12396) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTRAL TO ANTLIA

Trilinears a^9-3*(b+c)*a^8+4*(b^2+5*b*c+c^2)*a^7-4*(b+c)*(b^2+6*b*c+c^2)*a^6+2*(b^4+c^4+2*(5*b^2+8*b*c+5*c^2)*b*c)*a^5+2*(b+c)*(b^4+c^4-4*(5*b^2-7*b*c+5*c^2)*b*c)*a^4-4*(b^4+c^4-(9*b^2+14*b*c+9*c^2)*b*c)*(b-c)^2*a^3+4*(b^2-c^2)*(b-c)*(b^4+c^4-4*(b^2+b*c+c^2)*b*c)*a^2-(3*b^4-2*b^2*c^2+3*c^4)*(b-c)^4*a+(b^4-c^4)*(b-c)^3*(b^2-4*b*c+c^2) : :
X(12396) = 3*X(165)-2*X(12387) = 5*X(1698)-4*X(12394) = 3*X(1699)-4*X(12393) = 4*X(12388)-X(12395)

The reciprocal orthologic center of these triangles is X(1).

X(12396) lies on these lines: {1,7056}, {2,12391}, {40,238}, {57,12385}, {63,12389}, {165,12387}, {1698,12394}, {1699,12393}, {1764,12392}, {8580,12386}

X(12396) = midpoint of X(12389) and X(12390)
X(12396) = reflection of X(i) in X(j) for these (i,j): (1,12388), (12395,1)
X(12396) = complement of X(12391)
X(12396) = Ursa-minor-to-excentral similarity image of X(17633)

X(12397) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH TO ANTLIA

Trilinears (b+c)*a^11-(b+3*c)*(3*b+c)*a^10+(b+c)*(3*b^2+22*b*c+3*c^2)*a^9-(b^4+c^4+6*(4*b^2+5*b*c+4*c^2)*b*c)*a^8-2*(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a^7+2*(3*b^4+3*c^4-2*(3*b^2-13*b*c+3*c^2)*b*c)*(b+c)^2*a^6-2*(b+c)*(3*b^6+3*c^6+(4*b^4+4*c^4+(13*b^2+12*b*c+13*c^2)*b*c)*b*c)*a^5+2*(b^6+c^6+(8*b^4+8*c^4-(13*b^2-20*b*c+13*c^2)*b*c)*b*c)*(b+c)^2*a^4+(b^2-c^2)*(b-c)*(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*(3*b^6+3*c^6-5*(2*b^4+2*c^4-(b^2+4*b*c+c^2)*b*c)*b*c)*a^2+(b^4-c^4)*(b-c)^3*(3*b^4+3*c^4-2*(b^2+3*b*c+c^2)*b*c)*a-(b^4-c^4)*(b^2-c^2)*(b-c)^2*(b^4+c^4-2*(b^2-3*b*c+c^2)*b*c) : :
X(12397) = 3*r*X(5927)-2*(2*R+r)*X(12386)

The reciprocal orthologic center of these triangles is X(1).

X(12397) lies on these lines: {2,12385}, {4,341}, {9,12396}, {329,12389}, {405,12388}, {442,12394}, {5927,12386}, {7580,12387}, {8226,12393}, {10888,12392}, {11523,12395}

X(12397) = midpoint of X(12389) and X(12391)
X(12397) = reflection of X(12390) in X(12385)
X(12397) = anticomplement of X(12385)
X(12397) = complement of X(12390)

X(12398) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HEXYL TO ANTLIA

Trilinears a^9-8*a^7*b*c-(b+c)*a^8+24*(b+c)*a^6*b*c-2*(b^4+c^4+4*(5*b^2+2*b*c+5*c^2)*b*c)*a^5-8*(5*b^4+5*c^4+(b^2-6*b*c+c^2)*b*c)*a^3*b*c+2*(b+c)*(b^4+c^4+4*(7*b^2-8*b*c+7*c^2)*b*c)*a^4+8*(b^2-c^2)*(b-c)*(b^2+3*b*c+c^2)*a^2*b*c+(b^2-c^2)^2*(b^4+c^4-2*(4*b^2-5*b*c+4*c^2)*b*c)*a-(b^4-c^4)*(b-c)^3*(b^2-4*b*c+c^2) : :
X(12398) = 3*X(3576)-2*X(12388) = 3*X(5587)-4*X(12394) = 5*X(8227)-4*X(12393)

The reciprocal orthologic center of these triangles is X(1).

X(12398) lies on these lines: {1,5575}, {3,12396}, {20,12391}, {40,12387}, {78,12389}, {517,12395}, {1490,12397}, {3576,12388}, {5587,12394}, {8227,12393}, {8726,12385}, {10864,12386}, {10884,12390}

X(12398) = midpoint of X(20) and X(12391)
X(12398) = reflection of X(i) in X(j) for these (i,j): (40,12387), (12396,3)

X(12399) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO ANTLIA

Trilinears (b+c)*a^10-(4*b^2+11*b*c+4*c^2)*a^9+(b+c)*(7*b^2+27*b*c+7*c^2)*a^8-4*(2*b^4+2*c^4+(14*b^2+15*b*c+14*c^2)*b*c)*a^7-2*(33*b^4+33*c^4+2*(7*b^2-10*b*c+7*c^2)*b*c)*a^5*b*c+2*(b+c)*(3*b^4+3*c^4+2*(15*b^2-4*b*c+15*c^2)*b*c)*a^6-2*(b+c)*(3*b^6+3*c^6-(31*b^4+31*c^4-(31*b^2-12*b*c+31*c^2)*b*c)*b*c)*a^4+4*(2*b^6+2*c^6-(6*b^4+6*c^4+(7*b^2+4*b*c+7*c^2)*b*c)*b*c)*(b-c)^2*a^3-(b^2-c^2)*(b-c)*(7*b^6+7*c^6-(22*b^4+22*c^4-(13*b^2+12*b*c+13*c^2)*b*c)*b*c)*a^2+(4*b^6+4*c^6-(3*b^4+3*c^4+2*(2*b^2-b*c+2*c^2)*b*c)*b*c)*(b-c)^4*a-(b^4-c^4)*(b-c)^3*(b^4+c^4-3*(b-c)^2*b*c) : :

The reciprocal orthologic center of these triangles is X(1).

X(12399) lies on these lines: {7,12390}, {9,12389}, {1445,12396}, {7675,12398}, {7676,12387}, {7677,12388}, {7678,12393}, {7679,12394}, {8232,12397}, {8732,12385}, {10865,12386}, {10889,12392}, {11526,12395}

X(12399) = reflection of X(12389) in X(9)

X(12400) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON INTOUCH TO ANTLIA

Trilinears (b+c)*a^8-2*(b^2+6*b*c+c^2)*a^7+2*(b+c)*(b^2+11*b*c+c^2)*a^6+2*(b+c)*(17*b^2-10*b*c+17*c^2)*a^4*b*c-2*(b^4+c^4+2*(7*b^2+9*b*c+7*c^2)*b*c)*a^5+2*(b^6+c^6-(14*b^4+14*c^4-(7*b^2-12*b*c+7*c^2)*b*c)*b*c)*a^3-2*(b^2-c^2)*(b-c)*(b^4+c^4-(7*b^2-4*b*c+7*c^2)*b*c)*a^2+2*(b^6+c^6-(4*b^4+4*c^4+(b^2+c^2)*b*c)*b*c)*(b-c)^2*a-(b^4-c^4)*(b-c)^5 : :

The reciprocal orthologic center of these triangles is X(1).

X(12400) lies on these lines: {1,5575}, {11,12394}, {12,12393}, {55,12388}, {56,12387}, {145,12389}, {950,12397}, {1697,12396}, {3601,12385}, {4313,12390}, {7962,12395}, {8236,12399}, {9785,12391}, {10866,12386}

X(12400) = midpoint of X(145) and X(12389)

X(12401) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO ANTLIA

Trilinears (b+c)*a^8-2*(b^2+3*b*c+c^2)*a^7+2*(b+c)*(b^2+5*b*c+c^2)*a^6+10*(b^2-c^2)*(b-c)*a^4*b*c-2*(b^4+c^4+5*(b^2+c^2)*b*c)*a^5+2*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2-5*b*c+c^2)*a^3-2*(b^2-c^2)*(b-c)*(b^4+c^4-(b^2+8*b*c+c^2)*b*c)*a^2+2*(b^6+c^6-(b^4+c^4+(b^2-6*b*c+c^2)*b*c)*b*c)*(b-c)^2*a-(b^4-c^4)*(b-c)^5 : :
X(12401) = 5*X(3616)-X(12389)

The reciprocal orthologic center of these triangles is X(1).

X(12401) lies on these lines: {1,5575}, {495,12394}, {496,12393}, {942,12385}, {999,12388}, {3295,12387}, {3333,12396}, {3487,12397}, {3616,12389}, {11035,12386}, {11036,12390}, {11037,12391}, {11038,12399}, {11529,12395}

X(12402) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INTOUCH TO ANTLIA

Trilinears ((b+c)*a^6-2*(b^2+4*b*c+c^2)*a^5-12*(b+c)^2*a^3*b*c+(b+c)*(b^2+16*b*c+c^2)*a^4-(b^2-6*b*c+c^2)*(b+c)^3*a^2+2*(b^2-c^2)^2*(b-c)^2*a-(b^4-c^4)*(b-c)^3)*(a^2+(b-c)^2) : :
X(12402) = X(12400)-4*X(12401)

The reciprocal orthologic center of these triangles is X(1).

X(12402) lies on these lines: {1,5575}, {2,12389}, {7,12390}, {11,12393}, {12,12394}, {55,12387}, {56,12388}, {57,12385}, {226,12397}, {3340,12395}, {8581,12386}

X(12402) = midpoint of X(i) and X(j) for these {i,j}: {7,12399}, {12390,12391}
X(12402) = reflection of X(i) in X(j) for these (i,j): (1,12401), (12396,12385), (12400,1)
X(12402) = complement of X(12389)

X(12403) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE TO ANTLIA

Barycentrics (4*a^7-7*(b+c)*a^6+2*(5*b^2+6*b*c+5*c^2)*a^5-(b+c)*(19*b^2-20*b*c+19*c^2)*a^4+8*(b^2+b*c+c^2)*(2*b^2-3*b*c+2*c^2)*a^3-(b^2-c^2)*(b-c)*(5*b^2+2*b*c+5*c^2)*a^2+2*(b-c)^2*(b^2+c^2)^2*a-(b^4-c^4)*(b-c)^3 : :
X(12403) = 3*X(3873)+X(12389)

The reciprocal orthologic center of these triangles is X(1).

X(12403) lies on these lines: {1,7056}, {57,12387}, {65,12400}, {226,12393}, {354,12402}, {1210,12394}, {3333,12398}, {3873,12389}, {5045,12401}, {5728,12397}, {10580,12391}, {11018,12385}, {11019,12386}, {11020,12390}, {11021,12392}, {11025,12399}

X(12403) = midpoint of X(65) and X(12400)
X(12403) = reflection of X(12401) in X(5045)

X(12404) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR TO ANTLIA

Trilinears a^9+(b+c)*a^8-4*(b^2+7*b*c+c^2)*a^7+4*(b+c)*(b^2+15*b*c+c^2)*a^6-2*(3*b^4+3*c^4+2*(21*b^2+16*b*c+21*c^2)*b*c)*a^5+2*(b+c)*(b^4+c^4+2*(27*b^2-26*b*c+27*c^2)*b*c)*a^4+4*(b^6+c^6-(21*b^4+21*c^4-(b^2+14*b*c+c^2)*b*c)*b*c)*a^3-4*(b^3-c^3)*(b^2-c^2)*(b^2-8*b*c+c^2)*a^2+(5*b^6+5*c^6-3*(6*b^4+6*c^4+(3*b^2-4*b*c+3*c^2)*b*c)*b*c)*(b-c)^2*a-(b^4-c^4)*(b-c)^3*(3*b^2-8*b*c+3*c^2) : :
X(12404) = 3*X(165)-4*X(12387) = 3*X(165)-2*X(12396) = 9*X(7988)-8*X(12393) = 7*X(7989)-8*X(12394)

The reciprocal orthologic center of these triangles is X(1).

X(12404) lies on these lines: {1,5575}, {165,12387}, {200,12389}, {516,12391}, {1750,12397}, {3062,12386}, {4326,12399}, {5732,12390}, {7987,12388}, {7988,12393}, {7989,12394}, {10857,12385}, {10980,12403}, {11531,12395}

X(12404) = reflection of X(i) in X(j) for these (i,j): (1,12398), (11531,12395), (12396,12387)

X(12405) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO ANTLIA

Trilinears (b^2+6*b*c+c^2)*a^10-2*(b+c)*(b^2+6*b*c+c^2)*a^9-2*(b+c)*(13*b^2-3*b*c+13*c^2)*a^7*b*c+(b^4+c^4+2*(11*b^2+15*b*c+11*c^2)*b*c)*a^8-2*(b^6+c^6-(b^4+c^4+4*(b^2-b*c+c^2)*b*c)*b*c)*a^6+2*(b+c)*(2*b^6+2*c^6+(9*b^4+9*c^4-(13*b^2-30*b*c+13*c^2)*b*c)*b*c)*a^5+2*(b^2-c^2)*(b-c)*(5*b^4+5*c^4-(b^2+4*b*c+c^2)*b*c)*a^3*b*c-2*(b^6+c^6+(7*b^4+7*c^4-2*(10*b^2-19*b*c+10*c^2)*b*c)*b*c)*(b+c)^2*a^4+(b^2-c^2)^2*(b^6+c^6-(8*b^4+8*c^4-3*(3*b^2-4*b*c+3*c^2)*b*c)*b*c)*a^2-2*(b^4-c^4)*(b-c)^3*(b^4+c^4-(b^2+b*c+c^2)*b*c)*a+(b^4-c^4)*(b^2-c^2)*(b-c)^2*(b^4+c^4-2*(b-c)^2*b*c) : :

The reciprocal orthologic center of these triangles is X(1).

X(12405) lies on these lines: {21,12388}, {846,12396}, {1284,12402}, {4199,12397}, {4220,12387}, {5051,12394}, {8229,12393}, {8235,12398}, {8238,12399}, {8240,12400}, {8245,12404}, {8246,12405}, {8731,12385}, {9791,12391}, {10868,12386}, {10892,12392}, {11031,12403}, {11043,12401}, {11533,12395}, {11688,12389}

X(12405) = 2nd-Sharygin-to-1st-Sharygin similarity image of X(13220)

X(12406) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO ANTLIA

Trilinears -2*(a+b+c)*(a^6-2*(b+c)*a^5+3*(b+c)^2*a^4-4*(b+c)*(b^2+c^2)*a^3+(3*b^4+3*c^4-2*b*c*(2*b^2-7*b*c+2*c^2))*a^2-2*(b^2-c^2)*(b-c)^3*a+(b-c)^2*(b^2+c^2)^2)*b*c*sin(A/2)-(a^2+(b-c)^2)*((b+c)*a^6-2*(b^2+4*b*c+c^2)*a^5+(b+c)*(b^2+16*b*c+c^2)*a^4-12*b*c*(b+c)^2*a^3-(b^2-6*b*c+c^2)*(b+c)^3*a^2+2*(b^2-c^2)^2*(b-c)^2*a-(b^4-c^4)*(b-c)^3) : :

The reciprocal orthologic center of these triangles is X(1).

X(12406) lies on these lines: {174,12402}, {7587,12388}, {8126,12389}, {8382,12394}, {8389,12399}, {8423,12404}, {8425,12405}, {8729,12385}, {11535,12395}, {11860,12386}, {11890,12390}, {11891,12391}, {11924,12400}

X(12407) = ORTHOLOGIC CENTER OF THESE TRIANGLES: AQUILA TO 1st HYACINTH

Barycentrics 2*a^2*(a^2-c^2)*(a^2-b^2)-(2*S^2+2*b*c*(SB+SC)-2*a*(2*S*(R+r)-a*SA)+6*SB*SC)*(9*R^2-2*SW) : :
X(12407) = 2*X(110)-3*X(5587) = 4*X(125)-3*X(3576) = 3*X(165)-2*X(12121) = 3*X(381)-2*X(11699) = 4*X(1511)-5*X(1698) = 5*X(8227)-4*X(11720)

The reciprocal orthologic center of these triangles is X(6102).

X(12407) lies on these lines: {1,265}, {10,12383}, {30,9904}, {35,12334}, {110,5587}, {125,3576}, {165,12121}, {355,2948}, {381,11699}, {515,3448}, {542,3751}, {1511,1698}, {1699,10113}, {2777,9899}, {3028,9613}, {5663,5691}, {5886,11801}, {6264,10778}, {7713,12140}, {7724,8274}, {8227,11720}, {9140,11709}, {9578,10088}, {9581,10091}, {10789,12201}

X(12407) = reflection of X(i) in X(j) for these (i,j): (1,265), (2948,355), (6264,10778), (12383,10)
X(12407) = X(265)-of-Aquila-triangle

X(12408) = ORTHOLOGIC CENTER OF THESE TRIANGLES: AQUILA TO 1st ORTHOSYMMEDIAL

Trilinears -SB*SC*(a^2-c^2)*(a^2-b^2)*a+(-a*b-a*c+b*c-2*SA+SW)*(a+b+c)*(-SW*(S^2-SW^2)+(-4*SW^2+3*S^2)*R^2) : :
X(12408) = 3*X(1)-4*X(12265) = 2*X(112)-3*X(165) = 4*X(127)-3*X(1699) = 4*X(132)-5*X(1698) = 3*X(1297)-2*X(12265) = 5*X(7987)-4*X(11722)

The reciprocal orthologic center of these triangles is X(4).

X(12408) lies on the Bevan circle and these lines: {1,1297}, {10,12384}, {35,12340}, {57,6020}, {112,165}, {127,1699}, {132,1698}, {515,12253}, {1054,9527}, {1282,2825}, {1697,3320}, {1768,2806}, {2781,2948}, {2799,9860}, {2831,5541}, {3679,9530}, {5540,9523}, {7713,12145}, {7987,11722}, {9517,9904}, {10705,11531}, {10789,12207}

X(12408) = reflection of X(i) in X(j) for these (i,j): (1,1297), (11531,10705), (12384,10)
X(12408) = X(1297)-of-Aquila-triangle

X(12409) = ORTHOLOGIC CENTER OF THESE TRIANGLES: AQUILA TO 1st SCHIFFLER

Barycentrics 3*a^7-(7*b^2-8*b*c+7*c^2)*a^5-2*(b^3+c^3)*a^4+(5*b^4+5*c^4-3*(2*b^2+b*c+2*c^2)*b*c)*a^3+2*(b+c)*(2*b^4+2*c^4-(3*b^2-b*c+3*c^2)*b*c)*a^2-(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(79).

X(12409) lies on these lines: {1,5180}, {5,1768}, {35,12342}, {515,12255}, {7713,12146}, {10789,12209}

X(12409) = reflection of X(1) in X(10266)
X(12409) = X(10266)-of-Aquila-triangle

X(12410) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARA TO EXCENTERS-MIDPOINTS

Trilinears (a^5+(b+c)*a^4-(-4*b^2*c^2+(b^2-c^2)^2)*a-(b+c)*(b^2+c^2)^2)*a : :
X(12410) = 4*R^2*X(1)-SW*X(3) = 2*R^2*X(8)-(6*R^2-SW)*X(25)

The reciprocal orthologic center of these triangles is X(10).

X(12410) lies on these lines: {1,3}, {8,25}, {10,5020}, {22,145}, {23,3621}, {24,12245}, {26,5844}, {28,5082}, {42,1036}, {159,5846}, {197,3913}, {219,1973}, {355,1598}, {515,9910}, {518,3556}, {519,9798}, {859,1792}, {944,11414}, {946,11479}, {952,7387}, {958,1486}, {960,12329}, {961,4339}, {962,1593}, {970,7074}, {1037,1042}, {1398,4318}, {1610,3189}, {1616,5096}, {1995,3617}, {2802,9912}, {3220,6762}, {3421,4222}, {3434,4185}, {3435,8668}, {3436,4186}, {3616,7484}, {3622,7485}, {3623,6636}, {3632,8185}, {3633,9591}, {3871,11337}, {5247,7083}, {5250,7085}, {5603,7395}, {5690,6642}, {5790,7529}, {5901,7393}, {7465,10587}, {7509,10595}, {7516,10283}, {7967,10323}, {7978,12168}, {8132,11924}, {9780,11284}, {9812,11403}, {9956,11484}, {10046,10573}, {10790,12195}, {10829,10912}, {10833,10950}

X(12410) = X(8)-of-Ara-triangle
X(12410) = X(1)-of-3rd-antipedal-triangle-of-X(3)
X(12410) = orthologic center of these triangles: Ara to 2nd Schiffler
X(12410) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,8193,3), (10,11365,5020), (22,145,8192)

X(12411) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARA TO HUTSON EXTOUCH

Trilinears (a^11-(b+c)*a^10-(3*b^2+14*b*c+3*c^2)*a^9+(b+c)*(3*b^2-2*b*c+3*c^2)*a^8+2*(b^4+c^4+2*(7*b^2+8*b*c+7*c^2)*b*c)*a^7-2*(b+c)*(b^4+c^4-2*(b^2+7*b*c+c^2)*b*c)*a^6+2*(b^6+c^6+5*(b^2+4*b*c+c^2)*b^2*c^2)*a^5-2*(b^4-c^4)*(b^2-c^2)*(b+c)*a^4-(b^2+c^2)*(3*b^6+3*c^6+(28*b^4+28*c^4+(37*b^2-8*b*c+37*c^2)*b*c)*b*c)*a^3+(b^2-c^2)^2*(b+c)*(3*b^4+3*c^4-2*(2*b^2+15*b*c+2*c^2)*b*c)*a^2+(b^2-c^2)^2*(b+c)^2*(b^4+c^4+2*(6*b^2-11*b*c+6*c^2)*b*c)*a-(b^2-c^2)^5*(b-c))*a : :
X(12411) = (6*R^2-SW)*X(25)-2*R^2*X(7160) = (3*R^2-SW)*X(22)+R^2*X(9874)

The reciprocal orthologic center of these triangles is X(40).

X(12411) lies on these lines: {22,9874}, {24,12249}, {25,7160}, {197,12333}, {8000,8192}, {8185,9898}, {10037,10059}, {10046,10075}, {10790,12200}, {11365,12260}, {11414,12120}

X(12411) = X(7160)-of-Ara-triangle

X(12412) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARA TO 1st HYACINTH

Trilinears (SA*(18*R^4+(3*SA-15*SW)*R^2+2*SW^2)+S^2*R^2)*a : :
X(12412) = (6*R^2-SW)*X(3)-2*R^2*X(74)

The reciprocal orthologic center of these triangles is X(6102).

X(12412) lies on these lines: {3,74}, {6,11557}, {22,12383}, {24,3448}, {25,265}, {26,9920}, {30,9919}, {69,7502}, {113,9818}, {125,6642}, {146,378}, {155,10628}, {159,542}, {186,12317}, {197,12334}, {541,2935}, {1181,11562}, {1539,1597}, {1593,7728}, {1598,10113}, {1619,9934}, {1993,7731}, {2070,3580}, {2771,3556}, {2777,9914}, {2781,5504}, {3763,5621}, {5622,9826}, {5961,7669}, {5972,7393}, {6644,10264}, {7387,10117}, {7514,10272}, {8185,12407}, {9786,11806}, {10088,10831}, {10091,10832}, {10790,12201}, {11365,12261}, {11413,12244}, {11414,12121}, {12167,12236}

X(12412) = reflection of X(i) in X(j) for these (i,j): (7387,10117), (12085,12302), (12310,26)
X(12412) = circumcircle-inverse-of-X(12358)
X(12412) = X(265)-of-Ara-triangle
X(12412) = {X(74), X(110)}-harmonic conjugate of X(12358)

X(12413) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARA TO 1st ORTHOSYMMEDIAL

Trilinears (-SA*SW^2*(4*R^2-SW)*(4*R^2-SA+SW)+(SW-2*R^2)*S^4+(8*SW*R^4+(-2*SA^2+SA*SW+2*SW^2)*R^2+(SA^2-SA*SW-SW^2)*SW)*S^2)*a : :
X(12413) = SW*X(3)-4*R^2*X(132)

The reciprocal orthologic center of these triangles is X(4).

X(12413) lies on these lines: {3,132}, {22,12384}, {24,12253}, {25,1073}, {112,11414}, {127,1598}, {197,12340}, {1661,9530}, {2781,12310}, {2799,9861}, {2806,9913}, {3320,10833}, {7387,11641}, {8185,12408}, {9517,9919}, {10790,12207}, {11365,12265}

X(12413) = reflection of X(11641) in X(7387)
X(12413) = X(1297)-of-Ara-triangle

X(12414) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARA TO 1st SCHIFFLER

Barycentrics SB*SC*a^2*(a+b+c)*(3*R+2*r)^2-4*R^2*(4*a*s*c-(7*R+2*r)*S)*(4*a*b*s-(7*R+2*r)*S)*(-a+b+c) : :
X(12414) = (6*R^2-SW)*X(25)-2*R^2*X(10266)

The reciprocal orthologic center of these triangles is X(79).

X(12414) lies on these lines: {3,7701}, {24,12255}, {25,10266}, {197,12342}, {8185,12409}, {10790,12209}, {11365,12267}

X(12414) = X(10266)-of-Ara-triangle

X(12415) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO ARIES

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

The reciprocal orthologic center of these triangles is X(9833).

X(12415) lies on these lines: {55,12416}, {68,5597}, {155,8200}, {539,11207}, {1147,5599}, {3157,11869}, {5598,9933}, {5601,6193}, {8190,9908}, {8196,9927}, {8197,9928}, {10055,11877}, {10071,11879}, {11366,12259}, {11411,11843}, {11822,12118}, {11837,12193}

X(12415) = reflection of X(12416) in X(55)
X(12415) = X(68)-of-1st-Auriga-triangle
X(12415) = X(9933)-of-2nd-Auriga-triangle

X(12416) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO ARIES

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

The reciprocal orthologic center of these triangles is X(9833).

X(12416) lies on these lines: {55,12415}, {68,5598}, {155,8207}, {539,11208}, {1147,5600}, {3157,11870}, {5597,9933}, {5602,6193}, {8187,9896}, {8191,9908}, {8203,9927}, {8204,9928}, {10055,11878}, {11367,12259}, {11411,11844}, {11823,12118}, {11838,12193}

X(12416) = reflection of X(12415) in X(55)
X(12416) = X(68)-of-2nd-Auriga-triangle
X(12416) = X(9933)-of-1st-Auriga-triangle

X(12417) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXTANGENTS TO ARIES

Trilinears (8*p^8+8*q*p^7+4*(4*q^2-5)*p^6+4*(4*q^2-7)*q*p^5+2*(4*q^4-18*q^2+9)*p^4+2*(4*q^4-14*q^2+13)*q*p^3-(8*q^4-22*q^2+7)*p^2-(8*q^4-14*q^2+7)*q*p+1-2*q^2)*cos(A) : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(7387).

X(12417) lies on these lines: {19,155}, {40,9896}, {55,9931}, {65,921}, {68,71}, {1147,11428}, {2013,11460}, {2550,12318}, {3101,11411}, {3564,8141}, {5584,12301}, {6193,6197}, {7688,9938}, {8539,9926}, {9816,9820}, {9932,10902}, {10306,12309}, {10319,12359}, {10636,10659}, {10637,10660}, {11406,12166}, {11435,12235}, {11445,12271}, {11471,12293}

X(12417) = reflection of X(9931) in X(9937)
X(12417) = X(84)-of-extangents-triangle if ABC is acute

X(12418) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO ARIES

Barycentrics (2*S^2+3*(SA-SW)*SA)*SA*((12*R^4+(14*SA+2*SW)*R^2-(SA+SW)^2)*S^2-4*(9*R^4-7*R^2*SW+SW^2)*(SA-SW)*SA) : :
X(12418) = X(11411)-3*X(11845)

The reciprocal orthologic center of these triangles is X(9833).

X(12418) lies on the Jerabek hyperbola and these lines: {30,155}, {68,402}, {539,1651}, {1069,11906}, {1147,1650}, {3157,11905}, {4240,6193}, {9896,11852}, {9908,11853}, {9923,11885}, {9927,11897}, {9928,11900}, {9929,11901}, {9930,11902}, {9933,11910}, {10055,11912}, {10071,11913}, {11411,11845}, {11831,12259}, {11832,12134}, {11839,12193}, {11848,12328}, {11863,12415}, {11864,12416}

X(12418) = midpoint of X(4240) and X(6193)
X(12418) = reflection of X(i) in X(j) for these (i,j): (68,402), (1650,1147)
X(12418) = isogonal conjugate of X(13621)
X(12418) = X(68)-of-Gossard-triangle

X(12419) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARIES TO 1st HYACINTH

Barycentrics (12*R^4+(14*SA+2*SW)*R^2-(3*SA+SW)*SW)*S^2-(18*R^4-27*SW*R^2+5*SW^2)*(SA-SW)*SA : :
X(12419) = (14*R^2-3*SW)*X(265)-2*(9*R^2-2*SW)*X(403)

The reciprocal orthologic center of these triangles is X(1147).

X(12419) lies on these lines: {20,5663}, {25,10111}, {110,11585}, {159,542}, {265,403}, {1353,11566}, {1498,11744}, {1503,5504}, {3147,3448}, {6776,9826}

X(12420) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARIES TO 2nd HYACINTH

Barycentrics ((SB+SC)*(2*R^4+(SA-5*SW)*R^2+SW^2)-S^2*(SW-4*R^2))*SA : :
X(12420) = (4*R^4-6*SW*R^2+SW^2)*X(26)+SW*(5*R^2-SW)*X(159)

The reciprocal orthologic center of these triangles is X(12421).

X(12420) lies on these lines: {20,6193}, {26,159}, {68,3542}, {155,6146}, {186,11411}, {1147,3546}, {6623,9927}

X(12420) = X(4)-of-Aries-triangle
X(12420) = Aries-isogonal conjugate of X(32048)

X(12421) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd HYACINTH TO ARIES

Barycentrics ((SB+SC)*(4*R^4+(-2*SA-6*SW)*R^2+SW^2)-(2*SW-12*R^2)*S^2)*SA : :

The reciprocal orthologic center of these triangles is X(12420).

X(12421) lies on these lines: {5,6}, {378,11411}, {539,11802}, {1092,10257}, {1147,11245}, {5878,12293}, {6515,9908}, {9927,10151}, {12134,12235}

X(12421) = reflection of X(12134) in X(12235)
X(12421) = X(4)-of-2nd-Hyacinth-triangle

X(12422) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO ARIES

Trilinears (-8*p^6+8*q*p^5+4*(2*q^2-1)*p^4-8*q^3*p^3-2*(4*q^2-5)*p^2+2*(4*q^2-3)*q*p-(2*q^2-1)^2)*cos(A) : :
where p=sin(A/2), q=cos((B-C)/2)
X(12422) = (R-2*r)*X(11)-(R-r)*X(68)

The reciprocal orthologic center of these triangles is X(9833).

X(12422) lies on these lines: {11,68}, {155,355}, {539,11235}, {912,1482}, {1147,1376}, {3157,9933}, {3434,6193}, {3564,10943}, {9896,10826}, {9908,10829}, {9923,10871}, {9927,10893}, {9928,10914}, {9929,10919}, {9930,10920}, {10055,10523}, {10071,10948}, {10785,11411}, {10794,12193}, {11373,12259}, {11390,12134}, {11826,12118}, {11865,12415}, {11866,12416}, {11903,12418}

X(12422) = reflection of X(12328) in X(1147)
X(12422) = reflection of X(12423) in X(155)
X(12422) = X(68)-of-inner-Johnson-triangle
X(12422) = X(12430)-of-outer-Johnson-triangle

X(12423) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO ARIES

Trilinears (8*p^6-8*q*p^5-4*(2*q^2+3)*p^4+8*(q^2+1)*q*p^3+6*p^2-2*q*p-(2*q^2-1)^2)*cos(A) : :
where p=sin(A/2), q=cos((B-C)/2)
X(12423) = (R+2*r)*X(12)-(R+r)*X(68)

The reciprocal orthologic center of these triangles is X(9833).

X(12423) lies on these lines: {3,63}, {12,68}, {155,355}, {539,11236}, {958,1147}, {1069,9933}, {3436,6193}, {9896,10827}, {9908,10830}, {9923,10872}, {9927,10894}, {9929,10921}, {9930,10922}, {10055,10954}, {10071,10523}, {10786,11411}, {10795,12193}, {11374,12259}, {11391,12134}, {11500,12328}, {11827,12118}, {11867,12415}, {11868,12416}, {11904,12418}

X(12423) = reflection of X(12422) in X(155)
X(12423) = X(68)-of-outer-Johnson-triangle
X(12423) = X(12431)-of-inner-Johnson-triangle

X(12424) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS TO ARIES

Trilinears a*SA*(-S*(4*R^4+(2*SA-4*SW)*R^2-S^2-SA^2+SW^2)+(S^2+SA^2-SA*SW)*(4*R^2-SW)) : :
X(12424) = (4*R^2+S-SW)*SW*X(6)+2*S*(2*R^2-SW)*X(1147)

The reciprocal orthologic center of these triangles is X(7387).

X(12424) lies on these lines: {6,1147}, {68,6413}, {155,5412}, {372,9932}, {1151,12301}, {2013,11462}, {2066,9931}, {3068,12318}, {3311,12309}, {3564,11265}, {5410,12166}, {5415,12417}, {6193,10880}, {6200,9938}, {9820,10961}, {11411,11417}, {11447,12271}, {11473,12293}, {11513,12359}

X(12424) = X(84)-of-1st-Kenmotu-diagonals-triangle if ABC is acute
X(12424) = {X(6),X(9937)}-harmonic conjugate of X(12425)

X(12425) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS TO ARIES

Trilinears a*SA*(S*(4*R^4+(2*SA-4*SW)*R^2-S^2-SA^2+SW^2)+(S^2+SA^2-SA*SW)*(4*R^2-SW)) : :
X(12425) = (4*R^2-S-SW)*SW*X(6)-2*S*(2*R^2-SW)*X(1147)

The reciprocal orthologic center of these triangles is X(7387).

X(12425) lies on these lines: {6,1147}, {68,6414}, {155,5413}, {371,9932}, {1152,12301}, {2013,11463}, {3069,12318}, {3312,12309}, {3564,11266}, {5411,12166}, {5414,9931}, {5416,12417}, {6193,10881}, {6396,9938}, {9820,10963}, {11411,11418}, {11448,12271}, {11474,12293}, {11514,12359}

X(12425) = X(84)-of-2nd-Kenmotu-diagonals-triangle if ABC is acute
X(12425) = {X(6),X(9937)}-harmonic conjugate of X(12424)

X(12426) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO ARIES

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

The reciprocal orthologic center of these triangles is X(9833).

X(12426) lies on these lines: {68,493}, {155,8220}, {539,12152}, {1147,8222}, {3157,11930}, {6193,6462}, {6461,12427}, {8188,9896}, {8194,9908}, {8210,9933}, {8212,9927}, {8214,9928}, {8216,9929}, {8218,9930}, {8408,9936}, {9923,10875}, {10055,11951}, {10071,11953}, {10945,12422}, {10951,12423}, {11377,12259}, {11394,12134}, {11411,11846}, {11503,12328}, {11828,12118}, {11840,12193}, {11907,12418}

X(12426) = X(68)-of-Lucas-homothetic-triangle

X(12427) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO ARIES

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

The reciprocal orthologic center of these triangles is X(9833).

X(12427) lies on these lines: {68,494}, {155,8221}, {539,12153}, {1147,8223}, {3157,11931}, {6193,6463}, {6461,12426}, {8189,9896}, {8195,9908}, {8211,9933}, {8213,9927}, {8215,9928}, {8217,9929}, {8219,9930}, {8420,9936}, {9923,10876}, {10055,11952}, {10071,11954}, {10946,12422}, {10952,12423}, {11378,12259}, {11395,12134}, {11411,11847}, {11504,12328}, {11829,12118}, {11841,12193}, {11908,12418}

X(12427) = X(68)-of-Lucas(-1)-homothetic-triangle

X(12428) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO ARIES

Barycentrics (-a^2+b^2+c^2)*(2*a^8-(3*b^2+2*b*c+3*c^2)*a^6+(b^2+c^2)^2*a^4-(b^2-c^2)^2*(b-c)^2*a^2+(b^2-c^2)^4) : :
X(12428) = (R+r)*X(55)-r*X(68)

The reciprocal orthologic center of these triangles is X(9833).

X(12428) lies on these lines: {1,9931}, {3,10071}, {4,651}, {5,11429}, {11,1147}, {12,9927}, {30,7352}, {33,12134}, {35,12359}, {55,68}, {56,12118}, {155,1479}, {497,1069}, {539,3058}, {912,10572}, {1062,6146}, {1478,12293}, {1594,9637}, {1697,9896}, {1837,9928}, {2098,9933}, {2646,12259}, {3028,7354}, {3056,3564}, {3167,9669}, {3295,10055}, {4294,11411}, {4302,12163}, {5432,5449}, {5433,12038}, {5654,10896}, {7741,9820}, {9645,9833}, {9668,12164}, {9670,9936}, {9908,10833}, {9923,10877}, {9929,10927}, {9930,10928}, {10799,12193}, {10947,12422}, {10953,12423}, {11909,12418}, {11947,12426}, {11948,12427}

X(12428) = X(68)-of-Mandart-incircle-triangle

X(12429) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO ARIES

Barycentrics (-a^2+b^2+c^2)*(3*a^8-5*(b^2+c^2)*a^6+(3*b^4+2*b^2*c^2+3*c^4)*a^4-3*(b^4-c^4)*(b^2-c^2)*a^2+2*(b^2-c^2)^4) : :
X(12429) = 3*X(3)-2*X(12118) = 3*X(3)-4*X(12359) = 4*X(5)-3*X(3167) = 3*X(68)-X(12118) = 3*X(68)-2*X(12359) = 2*X(155)-3*X(381) = 3*X(155)-4*X(5448) = 3*X(568)-4*X(12235) = 4*X(1147)-5*X(1656) = 3*X(3167)-2*X(6193)

The reciprocal orthologic center of these triangles is X(9833).

X(12429) lies on these lines: {3,68}, {4,193}, {5,3167}, {6,10112}, {26,9920}, {30,11411}, {69,11821}, {155,195}, {382,6243}, {517,9896}, {542,1498}, {567,1147}, {568,12235}, {912,4018}, {999,10071}, {1069,9669}, {1216,11850}, {1352,11479}, {1503,9914}, {1593,11442}, {1598,12134}, {1657,10620}, {1993,7507}, {2013,12111}, {2888,7503}, {3060,11576}, {3157,9654}, {3295,10055}, {3448,11413}, {3515,3580}, {3526,5449}, {3527,7528}, {3534,7689}, {3542,8780}, {3575,6515}, {3843,9936}, {3851,5654}, {5050,7399}, {5054,12038}, {5055,9820}, {5489,8057}, {5562,11898}, {5790,9928}, {5889,12173}, {5907,8681}, {6238,9668}, {6776,6823}, {6815,11245}, {7352,9655}, {7383,12017}, {7395,12022}, {7517,9908}, {7544,9777}, {7592,8548}, {8909,8976}, {8912,8981}, {9301,9923}, {9818,12166}, {9825,11433}, {9833,9909}, {9929,11916}, {9930,11917}, {9933,10247}, {10246,12259}, {11459,12271}, {11842,12193}, {11849,12328}, {11875,12415}, {11876,12416}, {11911,12418}, {11928,12422}, {11929,12423}, {11949,12426}, {11950,12427}

X(12429) = midpoint of X(2013) and X(12111)
X(12429) = reflection of X(i) in X(j) for these (i,j): (3,68), (155,9927), (382,12293), (1657,12163), (6193,5), (12118,12359), (12164,4)
X(12429) = homothetic center of Ehrmann side-triangle and X3-ABC reflections triangle
X(12429) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (68,12118,12359), (155,9927,381), (1352,12241,11479), (10055,12428,3295), (12118,12359,3)

X(12430) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO ARIES

Barycentrics 8*R*a*S^4*(4*R^2-SW)+(R-r)*(S^2-SB^2)*(S^2-SC^2)*SA*(a+b+c) : :
X(12430) = 2*R*X(1)-(R-r)*X(68)

The reciprocal orthologic center of these triangles is X(9833).

X(12430) lies on these lines: {1,68}, {12,12422}, {119,5654}, {155,10942}, {539,11239}, {952,1854}, {1069,10958}, {1147,5552}, {3157,10956}, {6193,10528}, {9908,10834}, {9923,10878}, {9927,10531}, {9928,10915}, {9929,10929}, {9930,10930}, {10803,12193}, {10805,11411}, {10955,12423}, {10965,12428}, {11248,12118}, {11400,12134}, {11509,12328}, {11881,12415}, {11882,12416}, {11914,12418}, {11955,12426}, {11956,12427}, {12000,12429}

X(12430) = reflection of X(68) in X(10055)
X(12430) = X(68)-of-inner-Yff-tangents-triangle
X(12430) = {X(68),X(9933)}-harmonic conjugate of X(12431)

X(12431) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO ARIES

Barycentrics 8*R*a*S^4*(4*R^2-SW)+(R+-r)*(S^2-SB^2)*(S^2-SC^2)*SA*(a+b+c) : :
X(12431) = 2*R*X(1)-(R+r)*X(68)

The reciprocal orthologic center of these triangles is X(9833).

X(12431) lies on these lines: {1,68}, {11,12423}, {155,10943}, {539,11240}, {912,1479}, {1069,10959}, {1147,10527}, {3157,10957}, {6193,10529}, {9908,10835}, {9923,10879}, {9927,10532}, {9928,10916}, {9929,10931}, {9930,10932}, {10804,12193}, {10806,11411}, {10949,12422}, {10966,12428}, {11249,12118}, {11401,12134}, {11510,12328}, {11883,12415}, {11884,12416}, {11915,12418}, {11957,12426}, {11958,12427}, {12001,12429}

X(12431) = reflection of X(68) in X(10071)
X(12431) = X(68)-of-outer-Yff-tangents-triangle
X(12431) = {X(68),X(9933)}-harmonic conjugate of X(12430)

X(12432) = X(1)X(1170)∩X(10)X(12)

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

Let A'B'C' be the orthic triangle of a triangle ABC. Let (Oa) be the incircle of AB'C', and define (Ob) and (Oc) cyclically. Then X(12432) = radical center of (Oa), (Ob), (Oc); see figure 1 and figure 2 . (Contributed by Thanh Oai Dao, March 4, 2017)

X(12432) lies on these lines: {1,1170}, {6,4347}, {7,5904}, {10,12}, {35,10122}, {46,10884}, {56,3881}, {57,3811}, {200,3339}, {201,3743}, {517,6738}, {518,4298}, {653,1844}, {942,6684}, {960,6666}, {962,1479}, {1125,5173}, {1203,4318}, {1254,3293}, {1400,3970}, {1420,3892}, {1448,3751}, {1708,5248}, {1724,4332}, {1788,5883}, {1825,1873}, {1902,5185}, {2099,3884}, {2171,3294}, {2800,6797}, {2801,4292}, {3085,5902}, {3256,7098}, {3305,3869}, {3340,3878}, {3361,3873}, {3485,10176}, {3555,4315}, {3681,5290}, {4294,10399}, {4314,5728}, {5435,5442}, {5884,11500}, {5905,12059}

X(12432) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (65,72,3671), (65,4848,3754), (3678,3754,3841), (5728,7957,4314).

X(12433) = MIDPOINT OF X(942) AND X(950)

Barycentrics 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(12433) = 3 X[553] - 5 X[942], 3 X[553] + 5 X[950], 9 X[553] - 5 X[4292], 3 X[942] - X[4292], 3 X[950] + X[4292], 3 X[3058] + X[5903], 3 X[5902] + X[6284], 3 X[5049] - X[10106], 3 X[354] + X[10572], 5 X[1] - X[10944], 3 X[1] + X[10950], 3 X[10944] + 5 X[10950], X[3868] + 3 X[11113], 9 X[5728] - X[11662].

Let A'B'C' be the orthic triangle of a triangle ABC. Let (Oa) be the incircle of AB'C', and define (Ob) and (Oc) cyclically. Then X(12433) = center of the circle that is externally tangent to (Oa), (Ob), (Oc); i.e., the outer Apollonian circle of (Oa), (Ob), (Oc), which passes through X(12019). See figure 1 and figure 2 , (Contributed by Thanh Oai Dao, March 4, 2017)

Let A'B'C' be the orthic triangle. Let Oa be the circle centered at A' and tangent to the internal angle bisector of angle A, and define Ob and Oc cyclically. Then X(12433) is the radical center of circles Oa, Ob, Oc. (Angel Montesdeoca, August 31, 2019)

X(12433) lies on these lines: {1,5}, {3,938}, {4,6147}, {7,382}, {8,5284}, {20,5708}, {30,553}, {36,10543}, {40,10386}, {57,550}, {140,1210}, {145,3940}, {226,546}, {354,10572}, {381,3487}, {404,9945}, {452,3927}, {515,5045}, {517,6738}, {519,4015}, {528,3754}, {529,3881}, {548,4304}, {549,3601}, {944,5804}, {962,1159}, {999,3486}, {1056,6849}, {1058,1482}, {1385,11019}, {1656,5703}, {1844,1852}, {1895,7510}, {2095,6868}, {2310,5492}, {2829,12005}, {3058,5903}, {3189,9709}, {3244,3452}, {3295,5690}, {3303,10573}, {3337,5441}, {3419,8728}, {3475,9654}, {3485,9669}, {3526,5704}, {3530,3911}, {3579,4314}, {3583,3649}, {3586,3627}, {3622,6856}, {3623,6919}, {3626,6666}, {3632,7308}, {3748,10039}, {3811,3820}, {3843,5714}, {3845,9612}, {3851,5226}, {3868,11113}, {3884,5855}, {4295,9668}, {4299,4860}, {4302,5221}, {4342,11278}, {4857,5425}, {4995,5445}, {5049,10106}, {5253,10609}, {5274,6866}, {5436,5791}, {5572,7686}, {5728,5762}, {5761,10247}, {5790,6887}, {5818,10578}, {5840,5885}, {5841,6583}, {5844,9957}, {5882,7682}, {5902,6284}, {6261,7956}, {6675,6734}, {6825,10246}, {6844,10595}, {6848,7967}, {8148,9785}, {10051,11510}, {11544,11551}

X(12433) = midpoint of X(942) and X(950)
X(12433) = reflection of X(5045) in X(6744)
X(12433) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5,5719), (1,11,37737), (1,496,5901), (1,1837,495), (1,5722,5), (1,9581,11374), (1,11373,10283), (145,5084,3940), (938,3488,3), (944,10580,7373), (1482,6827,5763), (5722,11374,9581), (9581,11374,5), (9785,11041,8148)

Orthologic centers: X(12434)-X(12624)

Centers X(12434)-X(12624) were contributed by César Eliud Lozada, March, 22, 2017.

X(12434) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARTZT TO 3rd PARRY

Trilinears a*((S^2+SB*SW)*(S^2+SC*SW)*(3*S^2-SW^2)^2-3*(3*S^2+SW^2)*(b^2-c^2)^2*(S^2-SA*SW)*(a^4-b^2*c^2)) : :
X(12434) = SW^2*X(263)-(3*S^2+SW^2)*X(2679)

The reciprocal orthologic center of these triangles is X(2).

X(12434) lies on the Artzt circle and these lines: {2,12157}, {98,512}, {111,9831}, {263,2679}, {511,9877}

X(12434) = circumsymmedial-to-Artzt similarity image of X(2698)

X(12435) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO ASCELLA

Trilinears (2*b^2+3*b*c+2*c^2)*a^4+(b+c)*(2*b^2-b*c+2*c^2)*a^3-(2*b^2-3*b*c+2*c^2)*(b+c)^2*a^2-2*(b^2-c^2)^2*b*c-(b+c)*(2*b^4+2*c^4-(b-c)^2*b*c)*a : :
X(12435) = 4*X(970)-5*X(1698) = 3*X(5587)-2*X(5752)

The reciprocal orthologic center of these triangles is X(942).

X(12435) lies on these lines: {1,3}, {8,10435}, {10,10478}, {63,10451}, {72,10888}, {145,10465}, {511,5691}, {516,10454}, {518,10442}, {519,12126}, {946,10479}, {962,10449}, {970,1698}, {975,994}, {2292,10892}, {3216,9549}, {3632,10825}, {3741,4301}, {3868,10444}, {3869,11679}, {5587,5752}, {5836,10456}, {7672,10889}, {8093,11894}, {9780,10440}, {9808,10891}

X(12435) = reflection of X(1) in X(10441)
X(12435) = Conway circle-inverse-of-X(1319)
X(12435) = X(4)-of-3rd-Conway-triangle
X(12435) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,40,10434), (1,165,10470), (1,1764,10882), (1,10441,10439), (10,10478,10887), (55,10474,1), (65,10480,1), (946,10479,10886), (1764,11521,1), (2098,10475,1), (3057,10473,1), (7982,10476,1), (10446,10447,10435)

X(12436) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ASCELLA TO 5th CONWAY

Barycentrics 2*a^4+(b+c)*a^3-(b^2-6*b*c+c^2)*a^2-(b+c)*(b^2-6*b*c+c^2)*a-(b^2-c^2)^2 : :
X(12436) = 3*X(2)+X(4292) = X(72)+3*X(553) = X(950)+3*X(11112) = X(1770)+7*X(3624) = 5*X(3616)-X(10624) = 5*X(3698)+3*X(5434) = 7*X(3922)+X(10944) = 3*X(5902)+X(6737)

The reciprocal orthologic center of these triangles is X(1).

X(12436) lies on these lines: {1,6904}, {2,4292}, {3,142}, {4,5437}, {5,6692}, {7,936}, {10,57}, {20,10857}, {58,3008}, {72,553}, {84,6864}, {140,3824}, {226,474}, {376,5436}, {377,1210}, {386,3664}, {404,5249}, {442,3911}, {515,3812}, {519,942}, {527,5044}, {535,11575}, {551,3601}, {758,10855}, {950,5439}, {975,3663}, {997,3671}, {1054,5530}, {1056,1706}, {1329,3634}, {1467,4315}, {1478,8582}, {1698,5744}, {1770,3624}, {2095,11362}, {2550,3333}, {2999,4340}, {3243,3296}, {3244,11518}, {3338,4847}, {3361,8732}, {3487,5438}, {3600,9623}, {3616,10624}, {3626,5708}, {3646,5698}, {3678,5850}, {3698,5434}, {3752,5717}, {3753,10106}, {3811,5542}, {3817,6847}, {3825,8727}, {3828,5791}, {3833,11227}, {3838,6691}, {3874,6743}, {3922,10944}, {4190,4304}, {4208,5435}, {4255,4675}, {4294,10582}, {4295,8583}, {4297,8726}, {4301,6282}, {4355,8580}, {4413,10404}, {4511,9782}, {5045,5853}, {5084,9579}, {5087,5122}, {5691,11407}, {5715,6926}, {5883,6738}, {5902,6737}, {6245,6256}, {6259,9842}, {6260,6918}, {6678,6693}, {6744,11018}, {6765,11037}, {6824,10171}, {6849,7171}, {6850,7682}, {6935,8227}, {7330,8257}, {10164,10198}

X(12436) = midpoint of X(i) and X(j) for these {i,j}: {10,4298}, {3874,6743}
X(12436) = X(389)-of-Ascella-triangle
X(12436) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,142,1125), (4,5437,9843), (57,443,10), (226,474,6700), (377,3306,1210), (3600,11024,9623), (4208,5435,5705), (5438,6173,3487), (5439,11112,950), (5745,8728,3634), (6904,9776,1)

X(12437) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ASCELLA TO EXCENTERS-MIDPOINTS

Trilinears (-a+b+c)*(4*a^3+(b+c)*a^2-2*(b^2+c^2)*a+(b^2-c^2)*(b-c)) : :
X(12437) = X(8)-3*X(3158) = 3*X(551)-2*X(3813) = 3*X(3241)-X(3680) = 5*X(3522)-3*X(3928) = 3*X(10165)-2*X(10916)

The reciprocal orthologic center of these triangles is X(1).

X(12437) lies on these lines: {1,142}, {3,519}, {8,3158}, {9,4313}, {10,6675}, {20,527}, {21,5325}, {55,5837}, {57,145}, {72,4304}, {78,950}, {100,4848}, {200,3486}, {210,10543}, {226,2475}, {284,1043}, {515,3811}, {517,9942}, {518,4297}, {522,5592}, {528,4301}, {551,3813}, {553,4190}, {579,3169}, {936,3488}, {938,5438}, {942,3244}, {944,6282}, {952,6245}, {958,6600}, {960,4314}, {1210,5440}, {1265,2325}, {1376,6738}, {1483,9940}, {1837,6745}, {2646,4847}, {2802,9946}, {3241,3680}, {3243,3600}, {3522,3928}, {3555,4311}, {3621,5744}, {3626,5791}, {3679,6857}, {3689,6736}, {3870,10106}, {3879,7176}, {3911,4855}, {3939,5247}, {3984,6872}, {4035,7270}, {4320,8271}, {4511,12053}, {4685,8731}, {5175,5219}, {5436,6666}, {5720,9842}, {5722,6700}, {5727,7080}, {5730,10624}, {5731,6762}, {5836,11018}, {5854,9945}, {5881,6847}, {6049,8732}, {7967,8726}, {9843,12433}, {10165,10916}, {10857,11519}

X(12437) = midpoint of X(i) and X(j) for these {i,j}: {1,3189}, {20,11523}, {145,2136}, {944,6765}, {3243,7674}
X(12437) = reflection of X(i) in X(j) for these (i,j): (10912,3635), (11362,8715)
X(12437) = orthologic center of these triangles: Ascella to 2nd Schiffler
X(12437) = X(64)-of-Ascella-triangle
X(12437) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,3601,5745), (55,6737,5837), (78,950,3452), (200,3486,5795), (938,5438,6692), (3241,6904,11518), (3555,10609,4311), (3689,10950,6736), (4190,11520,553)

X(12438) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO ASCELLA

Barycentrics (2*S^2-3*(SB+SC)*SA)^2*(S^2-3*SB*SC)*(a+b+c)+(5*b+5*c-4*a)*(S^2-3*SB*SC)*(S^2-3*SA*SC)*(S^2-3*SA*SB) : :
X(12438) = X(1)-3*X(11852) = X(944)-3*X(11845) = 2*X(946)-3*X(11897) = X(1482)-3*X(11911) = X(3081)+2*X(4669)

The reciprocal orthologic center of these triangles is X(3).

X(12438) lies on these lines: {1,402}, {3,11848}, {8,4240}, {10,1650}, {30,40}, {55,11863}, {515,12113}, {517,11251}, {519,1651}, {944,11845}, {946,11897}, {1482,11911}, {1829,11832}, {1837,11906}, {3057,11909}, {3081,4669}, {3640,11902}, {3641,11901}, {5252,11905}, {9798,11853}, {9941,11885}, {11839,12194}

X(12438) = midpoint of X(i) and X(j) for these {i,j}: {8,4240}, {11903,11904}
X(12438) = reflection of X(i) in X(j) for these (i,j): (1,402), (1650,10), (11831,11852)
X(12438) = X(1)-of-Gossard-triangle

X(12439) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ASCELLA TO HUTSON EXTOUCH

Trilinears 2*q*p^5-(4*q^2-3)*p^4+(2*q^2-13)*q*p^3+11*(q^2-1)*p^2-(q^2-16)*q*p-3*q^2+7-2*q/p : :
where p=sin(A/2), q=cos((B-C)/2)
X(12439) = 5*X(3889)-X(9874)

The reciprocal orthologic center of these triangles is X(3555).

X(12439) lies on these lines: {3,12333}, {142,5045}, {518,12260}, {3555,7160}, {3601,5920}, {3889,9874}, {8001,10857}, {9776,9804}, {9953,10855}

X(12439) = midpoint of X(3555) and X(7160)

X(12440) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO ASCELLA

Trilinears (4*R^2-a*(a+2*b+2*c))*S^2+(SA^2-(b+c)*a*(SW+SA))*S-(b+c)*a*SA^2 : :

The reciprocal orthologic center of these triangles is X(3).

X(12440) lies on these lines: {1,493}, {3,11503}, {8,6462}, {40,11828}, {55,8201}, {355,8220}, {515,9838}, {517,10669}, {519,12152}, {944,11846}, {946,8212}, {1482,11949}, {1829,11394}, {1837,11932}, {2292,8393}, {3057,11947}, {3640,8218}, {3641,8216}, {5252,11930}, {6339,8215}, {6461,12441}, {8194,9798}, {9941,10875}, {11840,12194}, {11907,12438}

X(12440) = X(1)-of-Lucas-homothetic-triangle
X(12440) = {X(8201),X(8208)}-harmonic conjugate of X(55)

X(12441) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO ASCELLA

Trilinears (4*R^2-a*(a+2*b+2*c))*S^2-(SA^2-(b+c)*a*(SW+SA))*S-(b+c)*a*SA^2 : :

The reciprocal orthologic center of these triangles is X(3).

X(12441) lies on these lines: {1,494}, {3,11504}, {8,6463}, {10,8223}, {40,11829}, {55,8202}, {355,8221}, {515,9839}, {517,10673}, {519,12153}, {944,11847}, {946,8213}, {1482,11950}, {1829,11395}, {1837,11933}, {2292,8394}, {3057,11948}, {3640,8219}, {3641,8217}, {5252,11931}, {6339,8214}, {6461,12440}, {8195,9798}, {9941,10876}, {11841,12194}, {11908,12438}

X(12441) = X(1)-of-Lucas(-1)-homothetic-triangle
X(12441) = {X(8202),X(8209)}-harmonic conjugate of X(55)

X(12442) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ASCELLA TO MANDART-EXCIRCLES

Trilinears 8*p^8-4*q*p^7+2*(2*q^2-13)*p^6-4*(q^2-5)*q*p^5-2*(2*q^4+3*q^2-13)*p^4+(4*q^2-21)*q*p^3+10*(q^2-1)*p^2-(3*q^2-7)*q*p-q^2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(12442) = 3*X(2)+X(12538)

The reciprocal orthologic center of these triangles is X(3555).

X(12442) lies on these lines: {2,12538}, {3,12517}, {5744,12534}, {8727,12613}, {8728,12621}, {9776,12542}, {10855,12449}, {10856,12553}

X(12443) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ASCELLA TO MIDARC

Trilinears 2*a*b*c*sin(A/2)-(a^2+2*a*b-b^2+2*b*c-c^2)*c*sin(B/2)-(a^2+2*a*c-b^2+2*b*c-c^2)*b*sin(C/2) : :
X(12443) = (2*R-r)*X(57)+(2*R+r)*X(164)

The reciprocal orthologic center of these triangles is X(1).

X(12443) lies on these lines: {1,8733}, {57,164}, {167,10857}, {3601,8422}, {5571,11018}, {5744,11691}, {7670,8732}, {9776,9807}

X(12443) = orthologic center of these triangles: Ascella to 2nd midarc
X(12443) = X(1)-of-Ascella-triangle
X(12443) = {X(8733),X(8734)}-harmonic conjugate of X(1)

X(12444) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ASCELLA TO 1st SCHIFFLER

Barycentrics (R+2*r)*(a-b+c)*(a+b-c)*(b+c)*(a+b+c)*(3*R+2*r)-16*R^2*(R*S*s+SB*SC) : :
X(12444) = (R+2*r)*(R+r)*X(226)-R^2*X(2475)

The reciprocal orthologic center of these triangles is X(21).

X(12444) lies on these lines: {3,12342}, {226,2475}, {942,3838}, {6841,9946}

X(12445) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO ASCELLA

Trilinears 2*(a+b+c)*b*c*sin(A/2)+(b+c)*(a+b-c)*(a-b+c) : :

The reciprocal orthologic center of these triangles is X(942).

X(12445) lies on these lines: {10,8382}, {57,7588}, {65,174}, {258,3339}, {517,8130}, {519,12130}, {2292,8425}, {3057,10502}, {3868,11890}, {3869,8126}, {5902,11217}, {7672,8389}, {7991,8423}, {9808,11996}, {11896,12435}

X(12445) = {X(3057), X(10502)}-harmonic conjugate of X(11924)

X(12446) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ATIK TO 4th CONWAY

Trilinears (b+c)*a^5-(b-c)^2*a^4-(b+c)*(2*b^2-3*b*c+2*c^2)*a^3+(2*b^2-3*b*c+2*c^2)*(b+c)^2*a^2+(b+c)*(b^4+c^4-3*(b^2-4*b*c+c^2)*b*c)*a-(b^2+3*b*c+c^2)*(b^2-c^2)^2 : :
X(12446) = X(3555)-3*X(3671) = 3*X(4295)+X(5904)

The reciprocal orthologic center of these triangles is X(1).

X(12446) lies on these lines: {1,9859}, {8,79}, {10,5927}, {516,960}, {1125,10855}, {3062,5234}, {3555,3671}, {3841,6702}, {3878,9589}, {3884,10624}, {4301,5784}, {4314,10609}, {5248,8583}, {6001,9947}

X(12446) = X(578)-of-Atik-triangle

X(12447) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ATIK TO 5th CONWAY

Barycentrics 2*a^4-3*(b+c)*a^3-(b^2-6*b*c+c^2)*a^2+(b+c)*(3*b^2+2*b*c+3*c^2)*a-(b^2-c^2)^2 : :
X(12447) = 3*X(2)+X(6737) = 3*X(553)+X(3962) = 3*X(3740)-X(5795) = 7*X(3983)+X(10944) = 5*X(4005)+3*X(5434) = X(4292)+3*X(5692)

The reciprocal orthologic center of these triangles is X(1).

X(12447) lies on these lines: {1,2}, {3,9948}, {9,4297}, {20,3062}, {72,4298}, {210,10106}, {220,5783}, {392,10866}, {405,10392}, {443,3671}, {515,5044}, {516,960}, {518,11035}, {553,3962}, {758,10855}, {993,8273}, {1001,12437}, {1376,5837}, {1706,6766}, {1837,5316}, {2550,4301}, {3035,9952}, {3160,5232}, {3452,5794}, {3488,3646}, {3600,5223}, {3678,9954}, {3740,5795}, {3874,10569}, {3876,11678}, {3878,7957}, {3923,9950}, {3983,10944}, {4005,5434}, {4292,5692}, {4308,5686}, {4314,10384}, {4342,5082}, {4413,4848}, {5234,5731}, {5273,7987}, {5328,7989}, {5438,10164}, {5542,11523}, {5791,10165}, {5833,11036}, {5882,9708}, {5927,10176}, {8158,9709}, {9858,9943}, {9949,10860}

X(12447) = midpoint of X(i) and X(j) for these {i,j}: {1,6743}, {72,4298}, {6737,6738}
X(12447) = reflection of X(6744) in X(1125)
X(12447) = complement of X(6738)
X(12447) = X(389)-of-Atik-triangle
X(12447) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3632,9797), (2,6737,6738), (8,8580,10), (8,8583,11019), (10,997,1125), (10,3244,9623), (10,6700,3634), (1125,3626,10916), (8583,11019,1125)

X(12448) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ATIK TO EXCENTERS-MIDPOINTS

Trilinears (-a+b+c)*((b+c)*a^4-2*b*c*a^3-2*(b+c)*(b^2-5*b*c+c^2)*a^2+2*(5*b^2-18*b*c+5*c^2)*b*c*a+(b^4-c^4)*(b-c)) : :
X(12448) = 2*X(3913)-3*X(10179) = 3*X(10178)-4*X(11260)

The reciprocal orthologic center of these triangles is X(1).

X(12448) lies on these lines: {8,210}, {145,8581}, {517,9948}, {518,3062}, {519,9856}, {2136,8580}, {2802,9952}, {3244,11035}, {3340,10912}, {3621,11678}, {3813,8582}, {3878,9953}, {3913,8583}, {4853,10384}, {5836,11019}, {5854,9951}, {9957,12447}, {10178,11260}, {10855,12437}

X(12448) = orthologic center of these triangles: Atik to 2nd Schiffler
X(12448) = X(64)-of-Atik-triangle

X(12449) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ATIK TO MANDART-EXCIRCLES

Trilinears 8*q*p^9-4*(4*q^2-1)*p^8+4*(4*q^2-5)*q*p^7-2*(8*q^4-24*q^2+5)*p^6+2*(4*q^4-20*q^2+3)*q*p^5+2*(10*q^4-16*q^2+11)*p^4-(12*q^4-22*q^2-5)*q*p^3-2*(3*q^4-2*q^2+10)*p^2+(4*q^4-7*q^2+14)*q*p-2*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(12449) = 6*R*X(10855)-(4*R-r)*X(12442)

The reciprocal orthologic center of these triangles is X(3555).

X(12449) lies on these lines: {8,12542}, {8582,12621}, {8583,12522}, {10855,12442}, {10860,12517}, {10861,12538}, {10862,12553}, {10863,12613}, {11678,12534}

X(12450) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ATIK TO MIDARC

Trilinears ((b+c)*a^2-2*(b^2+c^2)*a+(b+c)^3)*sin(A/2)-(a^2-(b-c)*(2*a-b-3*c))*b*sin(B/2)-(a^2+(b-c)*(2*a-3*b-c))*c*sin(C/2) : :
X(12450) = (4*R-r)*X(164)-(8*R-r)*X(8580)

The reciprocal orthologic center of these triangles is X(1).

X(12450) lies on these lines: {1,9853}, {8,9807}, {164,8580}, {167,3062}, {177,8581}, {5571,11019}, {7670,10865}, {8422,10866}, {10855,12443}, {11678,11691}

X(12450) = orthologic center of these triangles: Atik to 2nd midarc
X(12450) = X(1)-of-Atik-triangle
X(12450) = {X(11858),X(11859)}-harmonic conjugate of X(1)

X(12451) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ATIK TO 1st SCHIFFLER

Barycentrics (4*R^2+7*R*r+2*r^2)*(-a+b+c)*S^2*(3*R+2*r)^2-(4*R^2-9*R*r+2*r^2)*(4*a*c*s-7*R*S-2*S*r)*(4*a*b*s-7*R*S-2*S*r)*(-a+b+c) : :
X(12451) = (4*R^2+7*R*r+2*r^2)*X(8)-(4*R^2-9*R*r+2*r^2)*X(10266)

The reciprocal orthologic center of these triangles is X(21).

X(12451) lies on these lines: {8,10266}, {3062,6597}, {10855,12444}

X(12452) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO 1st EHRMANN

Trilinears (-a+b+c)*(a+b+c)*(a^2+b^2+c^2)*a+((b+c)*a-b^2-c^2)*D : : , where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(12452) lies on these lines: {6,5597}, {55,63}, {69,5601}, {141,5599}, {159,8190}, {511,11252}, {524,11207}, {611,11877}, {613,11879}, {1350,11822}, {1351,11875}, {1352,8200}, {1386,11366}, {1843,11384}, {2781,12365}, {3056,11873}, {3094,11861}, {3242,5598}, {3416,8197}, {3564,12415}, {5480,8196}, {6776,11843}, {9041,11208}, {9830,12345}, {11492,12329}, {11837,12212}

X(12452) = reflection of X(12453) in X(55)
X(12452) = {X(8198),X(8199)}-harmonic conjugate of X(5597)
X(12452) = X(6)-of-1st-Auriga-triangle
X(12452) = X(3242)-of-2nd-Auriga-triangle

X(12453) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO 1st EHRMANN

Trilinears (-a+b+c)*(a+b+c)*(a^2+b^2+c^2)*a+(b^2+c^2-(b+c)*a)*D : : , where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(12453) lies on these lines: {6,5598}, {55,63}, {69,5602}, {141,5600}, {159,8191}, {511,11253}, {524,11208}, {611,11878}, {613,11880}, {1350,11823}, {1351,11876}, {1352,8207}, {1386,11367}, {1843,11385}, {3056,11874}, {3094,11862}, {3242,5597}, {3416,8204}, {3564,12416}, {3751,8187}, {5480,8203}, {6776,11844}, {9041,11207}, {9830,12346}, {11493,12329}, {11838,12212}

X(12453) = reflection of X(12452) in X(55)
X(12453) = {X(8205),X(8206)}-harmonic conjugate of X(5598)
X(12453) = X(6)-of-2nd-Auriga-triangle
X(12453) = X(3242)-of-1st-Auriga-triangle

X(12454) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO EXCENTERS-MIDPOINTS

Barycentrics (a+b+c)*(-a+b+c)*a^2-(2*a-b-c)*D : :
where D=4*S*sqrt(R*(4*R+r))
X(12454) = 2*X(55)-3*X(11207) = 4*X(55)-3*X(11208)

The reciprocal orthologic center of these triangles is X(10).

X(12454) lies on these lines: {1,5599}, {8,5597}, {10,11366}, {55,519}, {145,5598}, {355,8196}, {944,11822}, {1482,8200}, {2098,11871}, {2099,11869}, {3244,11367}, {3621,5602}, {3633,8187}, {3913,11492}, {5844,11253}, {5846,12452}, {8190,12410}, {8207,11875}, {9053,12453}, {10573,11879}, {10912,11865}, {10950,11873}, {11384,12135}, {11823,11843}, {11837,12195}

X(12454) = X(12454) = reflection of X(i) in X(j) for these (i,j): (11208,11207), (12455,55)
X(12454) = X(8)-of-1st-Auriga-triangle
X(12454) = X(145)-of-2nd-Auriga-triangle

X(12455) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO EXCENTERS-MIDPOINTS

Barycentrics (a+b+c)*(-a+b+c)*a^2+(2*a-b-c)*D : : , where D=4*S*sqrt(R*(4*R+r))
X(12455) = 4*X(55)-3*X(11207) = 2*X(55)-3*X(11208)

The reciprocal orthologic center of these triangles is X(10).

X(12455) lies on these lines: {1,5600}, {8,5598}, {10,11367}, {55,519}, {145,5597}, {355,8203}, {944,11823}, {1482,8196}, {2098,11872}, {2099,11870}, {3244,11366}, {3621,5601}, {3632,8187}, {3913,11493}, {5844,11252}, {5846,12453}, {8191,12410}, {8200,11876}, {9053,12452}, {10573,11880}, {10912,11866}, {10950,11874}, {11385,12135}, {11822,11844}, {11838,12195}

X(12455) = reflection of X(i) in X(j) for these (i,j): (11207,11208), (12454,55)
X(12455) = X(8)-of-2nd-Auriga-triangle
X(12455) = X(145)-of-1st-Auriga-triangle

X(12456) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO EXTOUCH

Trilinears 4*S^2*(a+b-c)*(a-b+c)*(-a+b+c)^2*a+(2*(b+c)*a^5-2*(b^2+c^2)*a^4-4*(b^2-c^2)*(b-c)*a^3+4*(b^2-c^2)^2*a^2+2*(b^2-c^2)*(b-c)^3*a-2*(b^4-c^4)*(b^2-c^2))*D : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(40).

X(12456) lies on these lines: {55,6001}, {84,5597}, {515,12454}, {971,11252}, {1490,11822}, {1709,11877}, {5598,7971}, {5599,6260}, {6245,8196}, {6257,8199}, {6258,8198}, {6259,8200}, {8190,9910}, {10085,11879}, {11366,12114}, {11492,12330}, {11837,12196}, {11843,12246}

X(12456) = reflection of X(12457) in X(55)
X(12456) = X(84)-of-1st-Auriga-triangle
X(12456) = X(7971)-of-2nd-Auriga-triangle

X(12457) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO EXTOUCH

Trilinears 4*S^2*(a+b-c)*(a-b+c)*(-a+b+c)^2*a-(2*(b+c)*a^5-2*(b^2+c^2)*a^4-4*(b^2-c^2)*(b-c)*a^3+4*(b^2-c^2)^2*a^2+2*(b^2-c^2)*(b-c)^3*a-2*(b^4-c^4)*(b^2-c^2))*D : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(40).

X(12457) lies on these lines: {55,6001}, {84,5598}, {515,12455}, {971,11253}, {1490,11823}, {1709,11878}, {5597,7971}, {5600,6260}, {6245,8203}, {6257,8206}, {6258,8205}, {6259,8207}, {7992,8187}, {8191,9910}, {10085,11880}, {11367,12114}, {11493,12330}, {11838,12196}, {11844,12246}

X(12457) = reflection of X(12456) in X(55)
X(12457) = X(84)-of-2nd-Auriga-triangle
X(12457) = X(7971)-of-1st-Auriga-triangle

X(12458) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO 3rd EXTOUCH

Trilinears 2*S^2*(-a+b+c)*a-((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c))*D : : , where D=4*S*sqrt(R*(4*R+r))
X(12458) = (2*R*S-D)*X(1)+(2*S*r+D)*X(3)

The reciprocal orthologic center of these triangles is X(4).

X(12458) lies on these lines: {1,3}, {4,8197}, {10,8196}, {515,12454}, {946,5599}, {962,5601}, {1836,11869}, {2800,12457}, {4301,8203}, {5600,11362}, {5812,11867}, {6361,11843}, {8190,9911}, {8204,12245}, {11837,12197}

X(12458) = reflection of X(i) in X(j) for these (i,j): (55,11252), (12459,55)
X(12458) = X(40)-of-1st-Auriga-triangle
X(12458) = X(7982)-of-2nd-Auriga-triangle

X(12459) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO 3rd EXTOUCH

2*S^2*(-a+b+c)*a+((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c))*D, where D=4*S*sqrt(R*(4*R+r))
X(12459) = (2*R*S+D)*X(1)+(2*S*r-D)*X(3)

The reciprocal orthologic center of these triangles is X(4).

X(12459) lies on these lines: {1,3}, {4,8204}, {10,8203}, {515,12455}, {946,5600}, {962,5602}, {1836,11870}, {2800,12456}, {4301,8196}, {5599,11362}, {5812,11868}, {6361,11844}, {8191,9911}, {8197,12245}, {11838,12197}

X(12459) = reflection of X(i) in X(j) for these (i,j): (55,11253), (12458,55)
X(12459) = X(40)-of-2nd-Auriga-triangle
X(12459) = X(7982)-of-1st-Auriga-triangle

X(12460) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO FUHRMANN

Barycentrics (a+b+c)*(-a+b+c)*(a^3-(b+c)*a^2-(b^2-3*b*c+c^2)*a+(b^2-c^2)*(b-c))*a^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)*D : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(12460) lies on these lines: {11,11366}, {55,952}, {80,5597}, {100,8197}, {214,5599}, {1317,11367}, {2802,12454}, {2829,12456}, {5598,7972}, {5601,6224}, {5840,12458}, {6262,8199}, {6263,8198}, {6265,8200}, {8190,9912}, {10057,11877}, {10073,11879}, {11384,12137}, {11492,12331}, {11822,12119}, {11837,12198}, {11843,12247}

X(12460) = reflection of X(12461) in X(55)
X(12460) = X(80)-of-1st-Auriga-triangle
X(12460) = X(7972)-of-2nd-Auriga-triangle

X(12461) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO FUHRMANN

Barycentrics (a+b+c)*(-a+b+c)*(a^3-(b+c)*a^2-(b^2-3*b*c+c^2)*a+(b^2-c^2)*(b-c))*a^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)*D : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(12461) lies on these lines: {11,11367}, {55,952}, {80,5598}, {100,8204}, {214,5600}, {1317,11366}, {2802,12455}, {2829,12457}, {5597,7972}, {5602,6224}, {5840,12459}, {6262,8206}, {6263,8205}, {6265,8207}, {8187,9897}, {8191,9912}, {10057,11878}, {10073,11880}, {11385,12137}, {11493,12331}, {11823,12119}, {11838,12198}, {11844,12247}

X(12461) = reflection of X(12460) in X(55)
X(12461) = X(80)-of-2nd-Auriga-triangle
X(12461) = X(7972)-of-1st-Auriga-triangle

X(12462) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO INNER-GARCIA

Trilinears 4*S^2*(a^3-(b+c)*a^2-(b^2-3*b*c+c^2)*a+(b^2-c^2)*(b-c))*(-a+b+c)*a+(-(b+c)*a^5+(b+c)^2*a^4+(2*b-c)*(b-2*c)*(b+c)*a^3-(b+2*c)*(2*b+c)*(b-c)^2*a^2-(b^2-c^2)*(b-c)*(b^2-3*b*c+c^2)*a+(b^2-c^2)*(b-c)*(b^3+c^3))*D : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(40).

X(12462) lies on these lines: {11,8196}, {55,2800}, {100,11822}, {104,5597}, {119,5599}, {153,5601}, {515,12460}, {1317,11873}, {1537,8203}, {2787,12179}, {2802,12458}, {5598,10698}, {8190,9913}, {10058,11877}, {10074,11879}, {11366,11715}, {11492,12332}, {11837,12199}, {11843,12248}

X(12462) = reflection of X(12463) in X(55)
X(12462) = X(104)-of-1st-Auriga-triangle
X(12462) = X(10698)-of-2nd-Auriga-triangle

X(12463) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO INNER-GARCIA

Trilinears 4*S^2*(a^3-(b+c)*a^2-(b^2-3*b*c+c^2)*a+(b^2-c^2)*(b-c))*(-a+b+c)*a-(-(b+c)*a^5+(b+c)^2*a^4+(2*b-c)*(b-2*c)*(b+c)*a^3-(b+2*c)*(2*b+c)*(b-c)^2*a^2-(b^2-c^2)*(b-c)*(b^2-3*b*c+c^2)*a+(b^2-c^2)*(b-c)*(b^3+c^3))*D : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(40).

X(12463) lies on these lines: {11,8203}, {55,2800}, {100,11823}, {104,5598}, {119,5600}, {153,5602}, {515,12461}, {1317,11874}, {1537,8196}, {1768,8187}, {2787,12180}, {2802,12459}, {5597,10698}, {8191,9913}, {10058,11878}, {10074,11880}, {11367,11715}, {11493,12332}, {11838,12199}, {11844,12248}

X(12463) = reflection of X(12462) in X(55)
X(12463) = X(104)-of-2nd-Auriga-triangle
X(12463) = X(10698)-of-1st-Auriga-triangle

X(12464) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO HUTSON EXTOUCH

Trilinears (2*(b+c)*a^5-2*(b^2+c^2)*a^4-4*(b+c)^3*a^3+4*(b^4+6*b^2*c^2+c^4)*a^2+2*(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)*a-2*(b^4-c^4)*(b^2-c^2))*D+(-a+b+c)*(a+b+c)*(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)*a : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(40).

X(12464) lies on these lines: {55,12465}, {5597,7160}, {5598,8000}, {5601,9874}, {8190,12411}, {10059,11877}, {10075,11879}, {11366,12260}, {11492,12333}, {11822,12120}, {11837,12200}, {11843,12249}

X(12464) = reflection of X(12465) in X(55)
X(12464) = X(7160)-of-1st-Auriga-triangle
X(12464) = X(8000)-of-2nd-Auriga-triangle

X(12465) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO HUTSON EXTOUCH

Trilinears (2*(b+c)*a^5-2*(b^2+c^2)*a^4-4*(b+c)^3*a^3+4*(b^4+6*b^2*c^2+c^4)*a^2+2*(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)*a-2*(b^4-c^4)*(b^2-c^2))*D-(-a+b+c)*(a+b+c)*(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)*a : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(40).

X(12465) lies on these lines: {55,12464}, {5597,8000}, {5598,7160}, {5602,9874}, {8187,9898}, {8191,12411}, {10059,11878}, {10075,11880}, {11367,12260}, {11493,12333}, {11823,12120}, {11838,12200}, {11844,12249}

X(12465) = reflection of X(12464) in X(55)
X(12465) = X(7160)-of-2nd-Auriga-triangle
X(12465) = X(8000)-of-1st-Auriga-triangle

X(12466) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO 1st HYACINTH

Barycentrics ((48*a^2+16*(b-c)*a+16*b^2+16*c^2)*S^4+4*a*(5*a^5-2*c*a^4-(23*b^2-2*b*c+5*c^2)*a^3-(11*b^3-2*c^3-b*c*(9*b-4*c))*a^2+2*b*c*(b^2-8*b*c+c^2)*a+2*(b^2-c^2)*(b-c)*b^2)*S^2-9*a^3*b^2*(2*a^5+(b-c)*a^4-2*(2*b^2+c^2)*a^3-(2*b^3-c^3-b*c*(2*b-3*c))*a^2+2*b*(b^3-3*b*c^2+c^3)*a+(b^2-c^2)*(b-c)*b^2))*D-16*a^2*(a-b-c)*(a^2+b^2+c^2)*S^4+36*a^4*b^2*c^2*(a-b-c)*S^2 : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(6102).

X(12466) lies on these lines: {30,12365}, {55,12467}, {110,8200}, {542,12452}, {1511,5599}, {3448,11843}, {5601,12383}, {11822,12121}

X(12466) = reflection of X(12467) in X(55)
X(12466) = X(265)-of-1st-Auriga-triangle
X(12466) = X(12898)-of-2nd-Auriga-triangle

X(12467) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO 1st HYACINTH

Barycentrics (-2*a^10+(b+c)*a^9+2*(2*b^2-b*c+2*c^2)*a^8-2*(b^3+c^3)*a^7-(b^4+c^4-2*b*c*(b^2-4*b*c+c^2))*a^6-b*c*(2*b-c)*(b-2*c)*(b+c)*a^5-(b^6+c^6-(2*b^4+2*c^4+3*b*c*(b-c)^2)*b*c)*a^4+(b^2-c^2)*(b-c)*(2*b^4+2*c^4+b*c*(2*b^2-b*c+2*c^2))*a^3-(b^2-c^2)^2*(b^4+c^4+2*b*c*(b^2-b*c+c^2))*a^2-(b^4-c^4)*(b^2-c^2)^2*(b-c)*a+(b^4-c^4)*(b^2-c^2)^3)*D+4*S^2*a^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))*(-a+b+c) : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(6102).

X(12467) lies on these lines: {30,12366}, {55,12466}, {110,8207}, {542,12453}, {1511,5600}, {2771,12461}, {3448,11844}, {5602,12383}, {8187,12407}, {8191,12412}, {10091,11872}, {11367,12261}, {11493,12334}, {11823,12121}, {11838,12201}

X(12467) = reflection of X(12466) in X(55)
X(12467) = X(265)-of-2nd-Auriga-triangle
X(12467) = X(12898)-of-1st-Auriga-triangle

X(12468) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO MIDHEIGHT

Trilinears ((b+c)*a^7+(b^2-4*b*c+c^2)*a^6-3*(b^2-c^2)*(b-c)*a^5-(3*b^2+4*b*c+3*c^2)*(b-c)^2*a^4+(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^3+3*(b^4-c^4)*(b^2-c^2)*a^2-(b^2-c^2)^2*(b+c)^3*a-(b^4-c^4)*(b^2-c^2)*(b-c)^2)*D+a*(a+b-c)*(a-b+c)*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(-a^2+b^2+c^2)*(-a+b+c)^2 : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(4).

X(12468) lies on these lines: {30,12415}, {55,12469}, {64,5597}, {1498,11822}, {2777,12466}, {2883,5599}, {5598,7973}, {5601,6225}, {5878,8200}, {6000,11252}, {6001,12458}, {6247,8196}, {6266,8199}, {6267,8198}, {7355,11873}, {8190,9914}, {10060,11877}, {10076,11879}, {11366,12262}, {11381,11384}, {11492,12335}, {11837,12202}, {11843,12250}

X(12468) = reflection of X(12469) in X(55)
X(12468) = X(64)-of-1st-Auriga-triangle
X(12468) = X(7973)-of-2nd-Auriga-triangle

X(12469) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO MIDHEIGHT

Trilinears -((b+c)*a^7+(b^2-4*b*c+c^2)*a^6-3*(b^2-c^2)*(b-c)*a^5-(3*b^2+4*b*c+3*c^2)*(b-c)^2*a^4+(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^3+3*(b^4-c^4)*(b^2-c^2)*a^2-(b^2-c^2)^2*(b+c)^3*a-(b^4-c^4)*(b^2-c^2)*(b-c)^2)*D+a*(a+b-c)*(a-b+c)*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(-a^2+b^2+c^2)*(-a+b+c)^2 : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(4).

X(12469) lies on these lines: {30,12416}, {55,12468}, {64,5598}, {1498,11823}, {2777,12467}, {2883,5600}, {5597,7973}, {5602,6225}, {5878,8207}, {6000,11253}, {6001,12459}, {6247,8203}, {6266,8206}, {6267,8205}, {7355,11874}, {8187,9899}, {8191,9914}, {10060,11878}, {10076,11880}, {11367,12262}, {11381,11385}, {11493,12335}, {11838,12202}, {11844,12250}

X(12469) = reflection of X(12468) in X(55)
X(12469) = X(64)-of-2nd-Auriga-triangle
X(12469) = X(7973)-of-1st-Auriga-triangle

X(12470) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO INNER-NAPOLEON

Barycentrics 3 Sqrt[3] a^2 (a-b-c)^2 (a+b-c) (a-b+c) (a+b+c)^2-Sqrt[3] (a+b+c) (4 a^4-3 a^3 b-2 a^2 b^2+3 a b^3-2 b^4-3 a^3 c+6 a^2 b c-3 a b^2 c-2 a^2 c^2-3 a b c^2+4 b^2 c^2+3 a c^3-2 c^4) D+6 a^2 (a-b-c) (a+b+c) (a^2+b^2+c^2) S+6 a (a b-b^2+a c-c^2) D S : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(12470) lies on these lines: {14,5597}, {55,12471}, {530,12345}, {531,11207}, {542,12452}, {617,5601}, {619,5599}, {5474,11822}, {5479,8196}, {5598,7974}, {5613,8200}, {6269,8199}, {6271,8198}, {6773,11843}, {9981,11861}, {10061,11877}, {10077,11879}, {11366,11706}, {11492,12336}, {11837,12204}

X(12470) = reflection of X(12471) in X(55)
X(12470) = X(14)-of-1st-Auriga-triangle
X(12470) = X(7974)-of-2nd-Auriga-triangle

X(12471) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO INNER-NAPOLEON

Barycentrics 3 Sqrt[3] a^2 (a-b-c)^2 (a+b-c) (a-b+c) (a+b+c)^2+Sqrt[3] (a+b+c) (4 a^4-3 a^3 b-2 a^2 b^2+3 a b^3-2 b^4-3 a^3 c+6 a^2 b c-3 a b^2 c-2 a^2 c^2-3 a b c^2+4 b^2 c^2+3 a c^3-2 c^4) D+6 a^2 (a-b-c) (a+b+c) (a^2+b^2+c^2) S-6 a (a b-b^2+a c-c^2) D S : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(12471) lies on these lines: {14,5598}, {55,12470}, {530,12346}, {531,11208}, {542,12453}, {617,5602}, {619,5600}, {5474,11823}, {5479,8203}, {5597,7974}, {5613,8207}, {6269,8206}, {6271,8205}, {6773,11844}, {8187,9900}, {9981,11862}, {10061,11878}, {10077,11880}, {11367,11706}, {11493,12336}, {11838,12204}

X(12471) = reflection of X(12470) in X(55)
X(12471) = X(14)-of-2nd-Auriga-triangle
X(12471) = X(7974)-of-1st-Auriga-triangle

X(12472) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO OUTER-NAPOLEON

Barycentrics 3 Sqrt[3] a^2 (a-b-c)^2 (a+b-c) (a-b+c) (a+b+c)^2+Sqrt[3] (a+b+c) (4 a^4-3 a^3 b-2 a^2 b^2+3 a b^3-2 b^4-3 a^3 c+6 a^2 b c-3 a b^2 c-2 a^2 c^2-3 a b c^2+4 b^2 c^2+3 a c^3-2 c^4) D-6 a^2 (a-b-c) (a+b+c) (a^2+b^2+c^2) S+6 a (a b-b^2+a c-c^2) D S : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(12472) lies on these lines: {13,5597}, {55,12473}, {530,11207}, {531,12345}, {542,12452}, {616,5601}, {618,5599}, {5473,11822}, {5478,8196}, {5598,7975}, {5617,8200}, {6268,8199}, {6270,8198}, {6770,11843}, {9982,11861}, {10062,11877}, {10078,11879}, {11366,11705}, {11492,12337}, {11837,12205}

X(12472) = reflection of X(12473) in X(55)
X(12472) = X(13)-of-1st-Auriga-triangle
X(12472) = X(7975)-of-2nd-Auriga-triangle

X(12473) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO OUTER-NAPOLEON

Barycentrics 3 Sqrt[3] a^2 (a-b-c)^2 (a+b-c) (a-b+c) (a+b+c)^2-Sqrt[3] (a+b+c) (4 a^4-3 a^3 b-2 a^2 b^2+3 a b^3-2 b^4-3 a^3 c+6 a^2 b c-3 a b^2 c-2 a^2 c^2-3 a b c^2+4 b^2 c^2+3 a c^3-2 c^4) D-6 a^2 (a-b-c) (a+b+c) (a^2+b^2+c^2) S-6 a (a b-b^2+a c-c^2) D S : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(12473) lies on these lines: {13,5598}, {55,12472}, {530,11208}, {531,12346}, {542,12453}, {616,5602}, {618,5600}, {5473,11823}, {5478,8203}, {5597,7975}, {5617,8207}, {6268,8206}, {6270,8205}, {6770,11844}, {8187,9901}, {9982,11862}, {10062,11878}, {10078,11880}, {11367,11705}, {11493,12337}, {11838,12205}

X(12473) = reflection of X(12472) in X(55)
X(12473) = X(13)-of-2nd-Auriga-triangle
X(12473) = X(7975)-of-1st-Auriga-triangle

X(12474) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO 1st NEUBERG

Barycentrics ((b^2+c^2)*a^3-b^2*c^2*(b+c))*D-a^2*(-a+b+c)*(a+b+c)*((b^2+c^2)*a^2+b^2*c^2) : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(12474) lies on these lines: {39,5599}, {55,730}, {76,5597}, {194,5601}, {384,11837}, {538,11207}, {732,12452}, {2782,11252}, {3095,8200}, {5598,7976}, {5969,12345}, {6248,8196}, {6272,8199}, {6273,8198}, {8190,9917}, {9983,11861}, {10063,11877}, {10079,11879}, {11257,11822}, {11366,12263}, {11384,12143}, {11492,12338}, {11843,12251}

X(12474) = reflection of X(12475) in X(55)
X(12474) = X(76)-of-1st-Auriga-triangle
X(12474) = X(7976)-of-2nd-Auriga-triangle

X(12475) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO 1st NEUBERG

Barycentrics ((b^2+c^2)*a^3-b^2*c^2*(b+c))*D-a^2*(-a+b+c)*(a+b+c)*((b^2+c^2)*a^2+b^2*c^2) : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(12475) lies on these lines: {39,5600}, {55,730}, {76,5598}, {194,5602}, {384,11838}, {538,11208}, {732,12453}, {2782,11253}, {3095,8207}, {5597,7976}, {5969,12346}, {6248,8203}, {6272,8206}, {6273,8205}, {8187,9902}, {8191,9917}, {9983,11862}, {10063,11878}, {10079,11880}, {11257,11823}, {11367,12263}, {11385,12143}, {11493,12338}, {11844,12251}

X(12475) = reflection of X(12474) in X(55)
X(12475) = X(76)-of-2nd-Auriga-triangle
X(12475) = X(7976)-of-1st-Auriga-triangle

X(12476) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO 2nd NEUBERG

Barycentrics ((b+c)*(a^2+c^2)*(a^2+b^2)-a*(b^2+c^2)*(b^2+c^2+2*a^2))*D+a^2*(a+b+c)*(-a+b+c)*(a^4+3*(b^2+c^2)*a^2+3*b^2*c^2+c^4+b^4) : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(12476) lies on these lines: {55,12477}, {83,5597}, {732,12452}, {754,11207}, {2896,5601}, {5598,7977}, {5599,6292}, {6249,8196}, {6274,8199}, {6275,8198}, {6287,8200}, {8190,9918}, {10064,11877}, {10080,11879}, {11366,12264}, {11384,12144}, {11492,12339}, {11822,12122}, {11837,12206}, {11843,12252}

X(12476) = reflection of X(12477) in X(55)
X(12476) = X(83)-of-1st-Auriga-triangle
X(12476) = X(7977)-of-2nd-Auriga-triangle

X(12477) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(12477) lies on these lines: {55,12476}, {83,5598}, {732,12453}, {754,11208}, {2896,5602}, {5597,7977}, {5600,6292}, {6249,8203}, {6274,8206}, {6275,8205}, {6287,8207}, {8187,9903}, {8191,9918}, {10064,11878}, {10080,11880}, {11367,12264}, {11385,12144}, {11493,12339}, {11823,12122}, {11838,12206}, {11844,12252}

X(12477) = reflection of X(12476) in X(55)
X(12477) = X(83)-of-2nd-Auriga-triangle
X(12477) = X(7977)-of-1st-Auriga-triangle

X(12478) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO 1st ORTHOSYMMEDIAL

Trilinears (-(b+c)*a^12-(b-c)^2*a^11+2*(b^3+c^3)*a^10+(b^2+c^2)*(b-c)^2*a^9-(b+c)*(b^2-b*c+c^2)^2*a^8+(2*b^4+2*c^4+b*c*(4*b^2+3*b*c+4*c^2))*(b-c)^2*a^7+(b^2-c^2)*(b-c)*b^2*c^2*a^6-(b^2-c^2)^2*(2*b^4+b^2*c^2+2*c^4)*a^5+(b^6-c^6)*(b^2-c^2)*(b+c)*a^4-(b^6-c^6)*(b^2-c^2)*(b+c)^2*a^3-(b^2-c^2)^2*(b+c)*(2*b^6+2*c^6-b*c*(2*b^2+b*c+2*c^2)*(b-c)^2)*a^2+(b^2-c^2)^2*(b+c)^4*(b^2-b*c+c^2)^2*a+(b^8-c^8)*(b^2-c^2)^2*(b-c))*D+4*S^2*a*(a^10-(b^2+c^2)*a^8+b^2*c^2*a^6-(b^6-c^6)*(b^2-c^2)*a^2+(b^8-c^8)*(b^2-c^2))*(-a+b+c) : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(4).

X(12478) lies on these lines: {55,12479}, {112,11822}, {127,8196}, {2799,12179}, {2806,12462}, {3320,11873}, {5601,12384}, {9517,12365}, {9530,11207}, {11366,12265}, {11492,12340}, {11837,12207}

X(12478) = reflection of X(12479) in X(55)
X(12478) = X(1297)-of-1st-Auriga-triangle

X(12479) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO 1st ORTHOSYMMEDIAL

The reciprocal orthologic center of these triangles is X(4).

X(12479) lies on these lines: {55,12478}, {112,11823}, {127,8203}, {2799,12180}, {2806,12463}, {3320,11874}, {5602,12384}, {8187,12408}, {9517,12366}, {9530,11208}, {11367,12265}, {11493,12340}, {11838,12207}

X(12479) = reflection of X(12478) in X(55)
X(12479) = X(1297)-of-2nd-Auriga-triangle

X(12480) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO REFLECTION

Trilinears ((b+c)*a^8-(b+c)^2*a^7-2*(b^3+c^3)*a^6+(3*b^4+3*c^4+2*(b^2+b*c+c^2)*b*c)*a^5-b*c*(b+c)*(2*b^2-b*c+2*c^2)*a^4-(b-c)^2*(3*b^4+3*c^4+(4*b^2+5*b*c+4*c^2)*b*c)*a^3+(b^2-c^2)*(b-c)*(2*b^4+2*c^4+(2*b^2+3*b*c+2*c^2)*b*c)*a^2+(b^2-c^2)^2*(b^4+c^4-b*c*(2*b^2-b*c+2*c^2))*a-(b^4-c^4)*(b^2-c^2)^2*(b-c))*D-4*S^2*a*(a^6-(b^2+c^2)*a^4-(b^4+b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(-a+b+c) : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(4).

X(12480) lies on these lines: {54,5597}, {55,12481}, {195,11875}, {539,11207}, {1154,11252}, {1209,5599}, {2888,5601}, {3574,8196}, {5598,7979}, {7691,11822}, {10066,11877}, {10082,11879}, {10628,12365}, {11366,12266}, {11492,12341}, {11837,12208}, {11843,12254}

X(12480) = reflection of X(12481) in X(55)
X(12480) = X(54)-of-1st-Auriga-triangle
X(12480) = X(7979)-of-2nd-Auriga-triangle

X(12481) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO REFLECTION

Trilinears -((b+c)*a^8-(b+c)^2*a^7-2*(b^3+c^3)*a^6+(3*b^4+3*c^4+2*(b^2+b*c+c^2)*b*c)*a^5-b*c*(b+c)*(2*b^2-b*c+2*c^2)*a^4-(b-c)^2*(3*b^4+3*c^4+(4*b^2+5*b*c+4*c^2)*b*c)*a^3+(b^2-c^2)*(b-c)*(2*b^4+2*c^4+(2*b^2+3*b*c+2*c^2)*b*c)*a^2+(b^2-c^2)^2*(b^4+c^4-b*c*(2*b^2-b*c+2*c^2))*a-(b^4-c^4)*(b^2-c^2)^2*(b-c))*D-4*S^2*a*(a^6-(b^2+c^2)*a^4-(b^4+b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(-a+b+c) : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(4).

X(12481) lies on these lines: {54,5598}, {55,12480}, {195,11876}, {539,11208}, {1154,11253}, {1209,5600}, {2888,5602}, {3574,8203}, {5597,7979}, {7691,11823}, {8187,9905}, {10066,11878}, {10082,11880}, {10628,12366}, {11367,12266}, {11493,12341}, {11838,12208}, {11844,12254}

X(12481) = reflection of X(12480) in X(55)
X(12481) = X(54)-of-2nd-Auriga-triangle
X(12481) = X(7979)-of-1st-Auriga-triangle

X(12482) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO 1st SCHIFFLER

Barycentrics (a^3-(b+c)*a^2-(b^2-3*b*c+c^2)*a+(b^2-c^2)*(b-c))*(2*a^4+(b+c)*a^3-(b^2+c^2)*a^2-(b^3+c^3)*a-(b^2-c^2)^2)*D-a^2*(-a+b+c)*(a+b+c)*(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+(b^2-c^2)*(b-c))^2 : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(79).

X(12482) lies on these lines: {55,12483}, {5597,10266}, {8190,12414}, {11366,12267}, {11492,12342}, {11837,12209}, {11843,12255}

X(12482) = reflection of X(12483) in X(55)
X(12482) = X(10266)-of-1st-Auriga-triangle

X(12483) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO 1st SCHIFFLER

Barycentrics -(a^3-(b+c)*a^2-(b^2-3*b*c+c^2)*a+(b^2-c^2)*(b-c))*(2*a^4+(b+c)*a^3-(b^2+c^2)*a^2-(b^3+c^3)*a-(b^2-c^2)^2)*D-a^2*(-a+b+c)*(a+b+c)*(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+(b^2-c^2)*(b-c))^2 : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(79).

X(12483) lies on these lines: {55,12482}, {5598,10266}, {8187,12409}, {8191,12414}, {11367,12267}, {11493,12342}, {11838,12209}, {11844,12255}

X(12483) = reflection of X(12482) in X(55)
X(12483) = X(10266)-of-2nd-Auriga-triangle

X(12484) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO INNER-VECTEN

Barycentrics (-2*a*((b+c)*a-b^2-c^2)*S+(a+b+c)*(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))*D-2*S*a^2*(a+b+c)*(-a+b+c)*(a^2+b^2+c^2-4*S) : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(12484) lies on these lines: {55,12485}, {486,5597}, {487,5601}, {642,5599}, {3564,12415}, {5598,7980}, {6251,8196}, {6280,8199}, {6281,8198}, {6290,8200}, {9986,11861}, {10067,11877}, {10083,11879}, {11366,12268}, {11492,12343}, {11822,12123}, {11837,12210}, {11843,12256}

X(12484) = reflection of X(12485) in X(55)
X(12484) = X(486)-of-inner-Vecten-triangle
X(12484) = X(7980)-of-outer-Vecten-triangle

X(12485) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO INNER-VECTEN

Barycentrics -(-2*a*((b+c)*a-b^2-c^2)*S+(a+b+c)*(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))*D-2*S*a^2*(a+b+c)*(-a+b+c)*(a^2+b^2+c^2-4*S) : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(12485) lies on these lines: {55,12484}, {486,5598}, {487,5602}, {642,5600}, {3564,12416}, {5597,7980}, {6251,8203}, {6280,8206}, {6281,8205}, {6290,8207}, {8187,9906}, {9986,11862}, {10067,11878}, {10083,11880}, {11367,12268}, {11493,12343}, {11823,12123}, {11838,12210}, {11844,12256}

X(12485) = reflection of X(12484) in X(55)
X(12485) = X(486)-of-outer-Vecten-triangle
X(12485) = X(7980)-of-inner-Vecten-triangle

X(12486) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO OUTER-VECTEN

Barycentrics (2*a*((b+c)*a-b^2-c^2)*S+(a+b+c)*(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))*D+2*S*a^2*(a+b+c)*(-a+b+c)*(a^2+b^2+c^2+4*S) : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(12486) lies on these lines: {55,12487}, {485,5597}, {488,5601}, {641,5599}, {3564,12415}, {5598,7981}, {6250,8196}, {6278,8199}, {6279,8198}, {6289,8200}, {9987,11861}, {10068,11877}, {10084,11879}, {11366,12269}, {11492,12344}, {11822,12124}, {11837,12211}, {11843,12257}

X(12486) = reflection of X(12487) in X(55)
X(12486) = X(485)-of-inner-Vecten-triangle
X(12486) = X(7981)-of-outer-Vecten-triangle

X(12487) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO OUTER-VECTEN

Barycentrics -(2*a*((b+c)*a-b^2-c^2)*S+(a+b+c)*(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))*D+2*S*a^2*(a+b+c)*(-a+b+c)*(a^2+b^2+c^2+4*S) : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(12487) lies on these lines: {55,12486}, {485,5598}, {488,5602}, {641,5600}, {3564,12416}, {5597,7981}, {6250,8203}, {6278,8206}, {6279,8205}, {6289,8207}, {8187,9907}, {9987,11862}, {10068,11878}, {10084,11880}, {11367,12269}, {11493,12344}, {11823,12124}, {11838,12211}, {11844,12257}

X(12487) = reflection of X(12486) in X(55)
X(12487) = X(485)-of-outer-Vecten-triangle
X(12487) = X(7981)-of-inner-Vecten-triangle

X(12488) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-HUTSON TO AYME

Trilinears (-a+b+c)*((b-c)^2*a^4+2*(b+c)*(b^2-4*b*c+c^2)*a^3-4*b*c*(2*b^2-b*c+2*c^2)*a^2-2*((b^2-c^2)^2-4*b^2*c^2)*(b+c)*a-(b^2-c^2)^2*(b-c)^2)*sin(A/2)+(a-b+c)*((b-c)*a^5+(2*b^2+b*c-5*c^2)*a^4+2*c*(2*b-c)*(2*b+c)*a^3-2*(b^4-3*c^4-b*c*(3*b^2+c^2))*a^2-(b^3-3*c^3-b*c*(3*b-7*c))*(b+c)^2*a+(b^2-c^2)*(b+c)^3*c)*sin(B/2)-(a+b-c)*((b-c)*a^5+(5*b^2-b*c-2*c^2)*a^4+2*b*(b-2*c)*(b+2*c)*a^3-2*(3*b^4-c^4+b*c*(b^2+3*c^2))*a^2-(3*b^3-c^3-b*c*(7*b-3*c))*(b+c)^2*a+(b^2-c^2)*(b+c)^3*b)*sin(C/2) : :

The reciprocal orthologic center of these triangles is X(10).

X(12488) lies on these lines: {3,363}, {4,9783}, {40,8140}, {72,11685}, {517,9805}, {942,8113}, {1071,11886}, {1385,8109}, {1482,11527}, {3579,8107}, {5045,11026}, {5728,8385}, {5777,5934}, {6732,8100}, {8099,8133}, {8377,9955}, {8380,9956}, {8390,9957}, {8391,9959}, {9940,11854}, {9947,11856}, {10441,11892}

X(12488) = midpoint of X(9805) and X(9836)
X(12488) = reflection of X(12489) in X(40)
X(12488) = X(5)-of-inner-Hutson-triangle

X(12489) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-HUTSON TO AYME

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

The reciprocal orthologic center of these triangles is X(10).

X(12489) lies on these lines: {3,168}, {4,9787}, {40,8140}, {72,11686}, {178,946}, {517,9806}, {942,8114}, {1071,11887}, {1385,8110}, {1482,11528}, {3579,8108}, {5045,11027}, {5728,8386}, {5777,5935}, {8099,8135}, {8100,8138}, {8378,9955}, {8381,9956}, {8392,9957}, {9940,11855}, {9947,11857}, {9959,11926}, {10441,11893}

X(12489) = midpoint of X(9806) and X(9837)
X(12489) = reflection of X(12488) in X(40)
X(12489) = X(5)-of-outer-Hutson-triangle

X(12490) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO AYME

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

The reciprocal orthologic center of these triangles is X(10).

X(12490) lies on these lines: {3,8231}, {4,9789}, {5,3739}, {40,8244}, {72,11687}, {517,7596}, {942,8243}, {1335,7133}, {1385,8225}, {1482,11532}, {3579,8224}, {5045,11030}, {5728,8237}, {5777,8233}, {8099,8247}, {8100,8248}, {8228,9955}, {8230,9956}, {8239,9957}, {8246,9959}, {9940,10858}, {9947,10867}, {10441,10891}

X(12490) = midpoint of X(7596) and X(9808)
X(12490) = X(5)-of-2nd-Pamfilos-Zhou-triangle

X(12491) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO AYME

Trilinears 4*(a+b+c)*b*c*sin(A/2)+(b+c)*a^2+2*b*c*a-(b^2-c^2)*(b-c) : :
X(12491) = X(8351)-3*X(11195)

The reciprocal orthologic center of these triangles is X(10).

X(12491) lies on these lines: {1,10502}, {4,11891}, {40,8423}, {72,8126}, {174,942}, {258,5708}, {517,8130}, {1159,11899}, {1385,7587}, {1482,11535}, {5045,8083}, {5439,8125}, {5728,8389}, {8129,8729}, {8382,9956}, {8425,9959}, {9947,11860}, {9957,11924}, {10441,11896}, {11996,12490}

X(12491) = midpoint of X(8351) and X(12445)
X(12491) = X(5)-of-Yff-central-triangle
X(12491) = {X(11195), X(12445)}-harmonic conjugate of X(8351)

X(12492) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MIDARC TO BCI

Barycentrics F(a, b, c)*sin(A/2)+G(a, b, c)*sin(B/2)-G(a, c, b)*sin(C/2)+H(a, b, c) : :
where
F(a,b,c)=-8*(b-c)*(-a+b+c)*((b+c)*a-(b-c)^2)*a*S+2*(b-c)*(a+b-c)*(-a+b+c)*(a-b+c)*(a^3+2*(b+c)*b*c-(b-c)^2*a)
G(a,b,c)=8*(a-b+c)*(a^2*b-(b^2-b*c+c^2)*a+(2*b-c)*(b-c)*c)*a*S-2*(a+b-c)*(-a+b+c)*(a-b+c)*((b+c)*a^2+2*(b-2*c)*a*c-(b^2-c^2)*(b-c))*a
H(a,b,c)=(b-c)*(a+b-c)*(-a+b+c)*(a-b+c)*(-4*a*S+5*a^3-3*(b+c)*a^2-(b^2-6*b*c+c^2)*a-(b^2-c^2)*(b-c))

The reciprocal orthologic center of these triangles is X(1).

X(12492) lies on these lines: {1,483}, {177,481}, {8083,8093}

X(12492) = reflection of X(12493) in X(1)
X(12492) = X(485)-of-mid-arc-triangle
X(12492) = X(12124)-of-2nd-mid-arc-triangle

X(12493) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd MIDARC TO BCI

Barycentrics F(a, b, c)*sin(A/2)+G(a, b, c)*sin(B/2)-G(a, c, b)*sin(C/2)+H(a, b, c) : :
where
F(a,b,c)=2*(b-c)*(-a+b+c)*(4*a*((b+c)*a^2-10*b*c*a-(b+c)*(b^2-4*b*c+c^2))*S-(a+b+c)*(a+b-c)*(a-b+c)*(a^3-(b^2+6*b*c+c^2)*a+2*(b+c)*b*c))
G(a,b,c)=-2*(a-b+c)*((4*(b+2*c)*a^3-4*c*(12*b+c)*a^2-4*(b^3+2*c^3-b*c*(2*b+11*c))*a+4*(b^2-c^2)*c*(2*b-c))*S-(a+b+c)*(a+b-c)*(-a+b+c)*((b-7*c)*a^2+2*c*(b+2*c)*a-(b^2-c^2)*(b-c)))*a
H(a,b,c)=4*S^2*(b-c)*(4*S*a+11*a^3-13*(b+c)*a^2+(b^2+10*b*c+c^2)*a+(b^2-c^2)*(b-c))

The reciprocal orthologic center of these triangles is X(1).

X(12493) lies on these lines: {1,483}, {8094,10968}

X(12493) = reflection of X(12492) in X(1)
X(12493) = X(485)-of-2nd-mid-arc-triangle
X(12493) = X(12124)-of-mid-arc-triangle

X(12494) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th EULER TO 2nd BROCARD

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

The reciprocal orthologic center of these triangles is X(6232).

X(12494) lies on the nine-points circle and these lines: {2,6233}, {4,6323}, {114,9771}, {543,11569}

X(12494) = midpoint of X(4) and X(6323)
X(12494) = complement of X(6233)
X(12494) = reflection of X(13234) in X(5)
X(12494) = 2nd-Brocard-to-5th-Euler similarity image of X(6232)

X(12495) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO EXCENTERS-MIDPOINTS

Barycentrics 2*a^5+(b^2+c^2)*a^3-(b+c)*(b^2+c^2)*a^2+(b^4+b^2*c^2+c^4)*a-(b^3+c^3)*(b^2+b*c+c^2) : :
X(12495) = (S^2-3*SW^2)*X(8)-2*(S^2-SW^2)*X(32)

The reciprocal orthologic center of these triangles is X(10).

X(12495) lies on these lines: {1,3096}, {8,32}, {10,7846}, {145,2896}, {355,9993}, {517,9873}, {519,7811}, {944,3098}, {952,9821}, {1482,9996}, {2098,10874}, {2099,10873}, {3094,5846}, {3099,3632}, {3241,7865}, {3616,7914}, {3617,10583}, {3913,11494}, {5603,10356}, {7967,10357}, {9862,12245}, {10047,10573}, {10345,10800}, {10348,12194}, {10828,12410}, {10871,10912}, {10877,10950}, {11386,12135}, {11861,12454}, {11862,12455}

X(12495) = orthologic center of these triangles: 5th Brocard to 2nd Schiffler
a X(12495) = X(8)-of-5th-Brocard-triangle
X(12495) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,9857,3096), (10,11368,7846), (145,2896,9997)

X(12496) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO EXTOUCH

Trilinears S^4*a^3+(S^2-3*SW^2)*(SB*b-S*r)*(SC*c-S*r)*(a+b+c) : :
X(12496) = 2*(S^2-SW^2)*X(32)-(S^2-3*SW^2)*X(84)

The reciprocal orthologic center of these triangles is X(40).

X(12496) lies on these lines: {32,84}, {515,12495}, {971,9821}, {1490,3098}, {1709,10038}, {3096,6260}, {3099,7992}, {5658,10357}, {6001,9941}, {6245,9993}, {6257,9995}, {6258,9994}, {6259,9996}, {6705,7846}, {7971,9997}, {9862,12246}, {9910,10828}, {10047,10085}, {11368,12114}, {11386,12136}, {11494,12330}, {11861,12456}, {11862,12457}

X(12496) = X(84)-of-5th-Brocard-triangle

X(12497) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO 3rd EXTOUCH

Trilinears S^2*a^3+(S^2-3*SW^2)*(SA*a-S*r) : :
X(12497) = 2*(S^2-SW^2)*X(32)-(S^2-3*SW^2)*X(40)

The reciprocal orthologic center of these triangles is X(4).

X(12497) lies on these lines: {1,3098}, {3,11368}, {4,9857}, {10,9993}, {32,40}, {46,10047}, {65,10877}, {484,7132}, {515,12495}, {516,9873}, {517,9821}, {946,3096}, {962,2896}, {1699,10356}, {1836,10873}, {1902,11386}, {3099,7991}, {5119,10038}, {5184,9301}, {5603,10357}, {5812,10872}, {6361,9862}, {6684,7846}, {7914,8227}, {7982,9997}, {9911,10828}, {10306,11494}, {11861,12458}, {11862,12459}

X(12497) = reflection of X(9941) in X(9821)
X(12497) = X(40)-of-5th-Brocard-triangle

X(12498) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO FUHRMANN

Barycentrics 2*a^4*(a+b+c)*S*(R-2*r)-(S^2-3*SW^2)*(2*SB-a*c)*(2*SC-a*b) : :
X(12498) = 2*(S^2-SW^2)*X(32)-(S^2-3*SW^2)*X(80)

The reciprocal orthologic center of these triangles is X(3).

X(12498) lies on these lines: {11,11368}, {32,80}, {100,9857}, {214,3096}, {952,9941}, {2800,9873}, {2802,12495}, {2829,12496}, {2896,6224}, {3098,12119}, {3099,9897}, {5840,12497}, {6262,9995}, {6263,9994}, {6265,9996}, {6702,7846}, {7972,9997}, {9862,12247}, {9912,10828}, {10038,10057}, {10047,10073}, {11386,12137}, {11494,12331}, {11861,12460}, {11862,12461}

X(12498) = X(80)-of-5th-Brocard-triangle

X(12499) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO INNER-GARCIA

Trilinears 2*a^3*S^3*(R-2*r)-(S^2-3*SW^2)*(b*SB-2*S*r)*(c*SC-2*S*r) : :
X(12499) = 2*(S^2-SW^2)*X(32)-(S^2-3*SW^2)*X(104)

The reciprocal orthologic center of these triangles is X(40).

X(12499) lies on these lines: {11,9993}, {32,104}, {100,3098}, {119,3096}, {153,2896}, {214,3061}, {515,12498}, {952,9821}, {1317,10877}, {1768,3099}, {2783,8782}, {2787,9862}, {2800,9941}, {2802,12497}, {2829,9873}, {6713,7846}, {7865,10711}, {9913,10828}, {9978,9999}, {9980,9998}, {9996,10742}, {9997,10698}, {10038,10058}, {10047,10074}, {11368,11715}, {11386,12138}, {11494,12332}, {11861,12462}, {11862,12463}

X(12499) = X(104)-of-5th-Brocard-triangle

X(12500) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO HUTSON EXTOUCH

Trilinears a^3*(a+b+c)*S^2*(8*R*r+8*R^2+r^2)+(S^2-3*SW^2)*(b*SB-(4*R+r)*S)*(SC*c-(4*R+r)*S) : :
X(12500) = 2*(S^2-SW^2)*X(32)-(S^2-3*SW^2)*X(7160)

The reciprocal orthologic center of these triangles is X(40).

X(12500) lies on these lines: {32,7160}, {2896,9874}, {3098,12120}, {3099,9898}, {8000,9997}, {9862,12249}, {10038,10059}, {10047,10075}, {10828,12411}, {11368,12260}, {11386,12139}, {11494,12333}, {11861,12464}, {11862,12465}

X(12500) = X(7160)-of-5th-Brocard-triangle

X(12501) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO 1st HYACINTH

Barycentrics 8*a^4*S^4*(9*R^2-2*SW)-(S^2-3*SW^2)*(4*SB^2-a^2*c^2)*(4*SC^2-a^2*b^2)*SA : :
X(12501) = 2*(S^2-SW^2)*X(32)-(S^2-3*SW^2)*X(265)

The reciprocal orthologic center of these triangles is X(6102).

X(12501) lies on these lines: {32,265}, {67,3098}, {110,9996}, {542,1569}, {1511,3096}, {2771,12498}, {2896,12383}, {3099,12407}, {3448,9862}, {5663,9873}, {9993,10113}, {10088,10873}, {10091,10874}, {10828,12412}, {11368,12261}, {11386,12140}, {11494,12334}, {11861,12466}, {11862,12467}

X(12501) = X(265)-of-5th-Brocard-triangle

X(12502) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO MIDHEIGHT

Trilinears a*(2*a^2*S^4*(4*R^2-SW)-(S^2-3*SW^2)*(S^2-2*SA*SC)*(S^2-2*SA*SB)) : :
X(12502) = 2*(S^2-SW^2)*X(32)-(S^2-3*SW^2)*X(64)

The reciprocal orthologic center of these triangles is X(4).

X(12502) lies on these lines: {30,9923}, {32,64}, {1498,3098}, {2777,12501}, {2883,3096}, {2896,6225}, {3099,9899}, {5656,10357}, {5878,9996}, {6000,9821}, {6001,12497}, {6247,9993}, {6266,9995}, {6267,9994}, {6696,7846}, {7355,10877}, {7973,9997}, {9862,12250}, {9914,10828}, {10038,10060}, {10047,10076}, {11368,12262}, {11381,11386}, {11494,12335}, {11861,12468}, {11862,12469}

X(12502) = X(64)-of-5th-Brocard-triangle

X(12503) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO 1st ORTHOSYMMEDIAL

Trilinears a*(4*a^2*S^4*(-SW*(S^2-SW^2)+(-4*SW^2+3*S^2)*R^2)+(S^2-3*SW^2)*((SA-SC)*S^2-2*(SA*SB-SC^2)*SA)*((SA-SB)*S^2-2*(SA*SC-SB^2)*SA)) : :
X(12503) = 2*(S^2-SW^2)*X(32)-(S^2-3*SW^2)*X(1297)

The reciprocal orthologic center of these triangles is X(4).

X(12503) lies on these lines: {32,1297}, {112,3098}, {127,9993}, {132,3096}, {2794,8782}, {2799,9862}, {2806,12499}, {2896,12384}, {3099,12408}, {3320,10877}, {7811,9530}, {9157,9999}, {9517,9984}, {10828,12413}, {11368,12265}, {11386,12145}, {11494,12340}, {11861,12478}, {11862,12479}

X(12503) = X(1297)-of-5th-Brocard-triangle

X(12504) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO 1st SCHIFFLER

Barycentrics a^4*(a+b+c)*S^2*(3*R+2*r)^2+(S^2-3*SW^2)*(2*SB*(a-b+c)+(R+2*r)*S)*(2*SC*(a+b-c)+(R+2*r)*S)*(-a+b+c) : :
X(12504) = 2*(S^2-SW^2)*X(32)-(S^2-3*SW^2)*X(10266)

The reciprocal orthologic center of these triangles is X(79).

X(12504) lies on these lines: {32,10266}, {3099,12409}, {9862,12255}, {10828,12414}, {11368,12267}, {11386,12146}, {11494,12342}, {11861,12482}, {11862,12483}

X(12504) = X(10266)-of-5th-Brocard-triangle

X(12505) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CIRCUMMEDIAL TO 5th EULER

Barycentrics 6*(9*R^2-2*SW)*S^4+(9*(6*R^2-SW)*SA^2-6*(9*R^2-SW)*SW*SA-SW^3)*S^2+2*(SB+SC)*SA*SW^3 : :
X(12505) = 4*X(5)-3*X(6032) = X(20)-3*X(6031) = 7*X(3090)-6*X(10162)

The reciprocal orthologic center of these triangles is X(12506).

X(12505) lies on these lines: {3,9829}, {4,3849}, {5,5913}, {20,6031}, {631,10163}, {3090,10162}, {5067,10173}, {6232,7770}

X(12505) = X(4)-of-circummedial-triangle

X(12506) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th EULER TO CIRCUMMEDIAL

Barycentrics 6*SW*S^4+(9*(6*R^2-SW)*SA^2-6*(9*R^2-SW)*SW*SA+SW^3)*S^2+2*(SB+SC)*SA*SW^3 : :
X(12506) = X(4)-3*X(6032) = 2*X(5)-3*X(10162) = 7*X(3523)-3*X(6031)

The reciprocal orthologic center of these triangles is X(12505).

X(12506) lies on these lines: {2,12505}, {3,3849}, {4,6032}, {5,9172}, {140,10163}, {631,9829}, {1656,10173}, {3523,6031}

X(12506) = complement of X(12505)
X(12506) = orthoptic-circle-of-Steiner-inellipse-inverse of X(39157)

X(12507) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CIRCUMMEDIAL TO 2nd ORTHOSYMMEDIAL

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

The reciprocal orthologic center of these triangles is X(12508).

X(12507) lies on the circumcircle and these lines: {2697,8705}, {2781,6325}, {6236,9517}

X(12508) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ORTHOSYMMEDIAL TO CIRCUMMEDIAL

Barycentrics 2*(b^4+c^4)*a^14-2*(b^2+c^2)*(2*b^4+3*b^2*c^2+2*c^4)*a^12+3*b^2*c^2*(5*b^4+b^2*c^2+5*c^4)*a^10+(b^2+c^2)*(4*b^8+4*c^8+b^2*c^2*(b^4+18*b^2*c^2+c^4))*a^8-(2*b^12+2*c^12+(13*b^8+13*c^8-2*b^2*c^2*(7*b^4+9*b^2*c^2+7*c^4))*b^2*c^2)*a^6+(b^4-c^4)*(b^2-c^2)*b^2*c^2*(b^4-8*b^2*c^2+c^4)*a^4+(b^2-c^2)^2*(2*b^8+2*c^8+b^2*c^2*(3*b^4-2*b^2*c^2+3*c^4))*b^2*c^2*a^2-2*(b^4-c^4)*(b^2-c^2)^3*b^4*c^4 : :

The reciprocal orthologic center of these triangles is X(12507).

X(12508) lies on the line {1316,6232}

X(12508) = X(12507)-of-1st-orthosymmedial-triangle

X(12509) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC TO LUCAS ANTIPODAL

Barycentrics (2*SA-SW)*S^2-2*S*(S^2-2*SB*SC)+(SB+SC)*SA*SW : :
X(12509) = 3*X(4)-4*X(6290) = 3*X(4)-2*X(12296) = 9*X(376)-4*X(6280) = 3*X(376)-4*X(12123) = 3*X(376)-2*X(12256) = 4*X(486)-5*X(631) = 8*X(642)-7*X(3090) = 9*X(3545)-8*X(6251) = X(6280)-3*X(12123) = 2*X(6280)-3*X(12256)

The reciprocal orthologic center of these triangles is X(3).

X(12509) lies on these lines: {3,12169}, {4,487}, {20,3564}, {25,12311}, {54,12229}, {69,9991}, {376,5860}, {378,12303}, {486,631}, {637,6337}, {642,3090}, {3533,6119}, {3545,6251}, {3567,12237}, {5657,9906}, {5889,12274}, {5890,12285}, {7582,12210}, {7612,10851}, {9738,12322}, {9921,12088}, {10625,12223}

X(12509) = midpoint of X(5889) and X(12274)
X(12509) = reflection of X(i) in X(j) for these (i,j): (4,487), (12221,3), (12256,12123), (12296,6290)
X(12509) = orthic-to-circumorthic similarity image of X(487)

X(12510) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC TO LUCAS(-1) ANTIPODAL

Barycentrics (2*SA-SW)*S^2+2*S*(S^2-2*SB*SC)+(SB+SC)*SA*SW : :
X(12510) = 3*X(4)-4*X(6289) = 3*X(4)-2*X(12297) = 9*X(376)-4*X(6279) = 3*X(376)-4*X(12124) = 3*X(376)-2*X(12257) = 4*X(485)-5*X(631) = 8*X(641)-7*X(3090) = 9*X(3545)-8*X(6250) = X(6279)-3*X(12124) = 2*X(6279)-3*X(12257)

The reciprocal orthologic center of these triangles is X(3).

X(12510) lies on these lines: {3,12170}, {4,488}, {20,3564}, {25,12312}, {54,12230}, {69,9992}, {376,5861}, {378,12304}, {485,631}, {638,6337}, {641,3090}, {3533,6118}, {3545,6250}, {3567,12238}, {5210,9540}, {5657,9907}, {5889,12275}, {5890,12286}, {7581,12211}, {7612,10852}, {9739,12323}, {9922,12088}, {10625,12224}

X(12510) = midpoint of X(5889) and X(12275)
X(12510) = reflection of X(i) in X(j) for these (i,j): (4,488), (12222,3), (12257,12124), (12297,6289)
X(12510) = orthic-to-circumorthic similarity image of X(488)

X(12511) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st CIRCUMPERP TO 4th CONWAY

Trilinears a^6-(b+c)*a^5-2*(b+c)^2*a^4+(b+c)*(2*b^2-3*b*c+2*c^2)*a^3+(b^2+b*c+c^2)*(b+c)^2*a^2-(b^3+c^3)*(b-c)^2*a+(b^2-c^2)^2*b*c : :
X(12511) = 3*X(3)-X(11496) = 5*X(3522)-X(4294) = X(4295)+3*X(9778) = 3*X(5248)-2*X(11496)

The reciprocal orthologic center of these triangles is X(1).

X(12511) lies on these lines: {1,7411}, {3,142}, {4,3841}, {10,5584}, {20,993}, {35,4295}, {36,3522}, {40,758}, {55,3671}, {56,4314}, {58,1742}, {72,7964}, {100,3984}, {165,411}, {191,9961}, {376,11012}, {386,9441}, {404,4512}, {550,5450}, {551,8273}, {1490,3678}, {1621,9589}, {1699,6986}, {1709,3647}, {1754,4300}, {2077,6876}, {3146,5251}, {3149,10164}, {3357,3579}, {3361,4326}, {3428,4297}, {3528,10596}, {3587,6261}, {3635,8158}, {3814,6838}, {3822,6908}, {3825,6865}, {3874,10884}, {3916,5918}, {4229,4278}, {5259,9812}, {5715,6701}, {6361,10902}, {6681,6926}, {6684,6985}, {6763,11220}, {7742,10624}, {10393,12432}, {10860,12446}

X(12511) = midpoint of X(3671) and X(5493)
X(12511) = reflection of X(i) in X(j) for these (i,j): (4,3841), (5248,3)
X(12511) = X(578)-of-1st-circumperp-triangle
X(12511) = complement, wrt 1st circumperp triangle, of X(12514)
X(12511) = complement, wrt excentral triangle, of X(12514)
X(12511) = excentral-to-1st-circumperp similarity image of X(12514)
X(12511) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3428,4297,8666), (5584,7580,10)

X(12512) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st CIRCUMPERP TO 5th CONWAY

Barycentrics 6*a^4+(b+c)*a^3-(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(12512) = X(1)-5*X(3522) = X(1)+3*X(9778) = 15*X(2)-7*X(10248) = 3*X(3)-X(946) = 7*X(3)-3*X(5886) = 5*X(3)-3*X(10165) = 2*X(946)-3*X(1125) = 7*X(946)-9*X(5886) = 5*X(946)-9*X(10165) = 7*X(1125)-6*X(5886) = 5*X(3522)+X(5493) = 5*X(3522)+3*X(9778) = X(5493)-3*X(9778)

The reciprocal orthologic center of these triangles is X(1).

As a point P moves on the circumcircle, the centroid of the 12 excenters of triangles ABC, BCP, CAP, ABP traces a curvilinear triangle, T. Let A', B', C' be the vertices of T, and (Oa), (Ob), (Oc) the circles whose arcs form the sides of T; the triangle A'B'C' is also the orthic triangle of the anticomplementary triangle of OaObOc, and OaObOc the medial triangle of the excentral triangle of A'B'C'. Then A'B'C' is homothetic to the medial triangle at X(12512). Let A" be the intersection, other than A', of circles (Ob) and (Oc), and define B" and C" cyclically. Then A"B"C" is the excentral triangle of A'B'C', and the anticomplementary triangle of OaObOc. Also, A"B"C" is homothetic to the extraversion triangle of X(10) (i.e. the complement of the excentral triangle) at X(12512). (Randy Hutson, July 21, 2017)

X(12512) lies on these lines: {1,3522}, {2,10248}, {3,142}, {4,3634}, {10,20}, {30,3828}, {35,4292}, {36,10624}, {40,376}, {46,4304}, {55,4298}, {57,4314}, {58,4229}, {63,6743}, {72,5918}, {140,10171}, {226,5217}, {355,3534}, {382,10175}, {386,1742}, {390,3361}, {411,6700}, {498,4333}, {515,550}, {517,548}, {546,10172}, {551,962}, {631,3817}, {726,5188}, {758,9943}, {936,2951}, {942,10178}, {950,1155}, {960,9858}, {971,3678}, {975,1721}, {993,5584}, {1040,4347}, {1158,3587}, {1210,4302}, {1385,8703}, {1420,4342}, {1587,9582}, {1697,4315}, {1698,3146}, {1699,3523}, {1703,9541}, {1737,4324}, {1770,5010}, {2077,3651}, {2093,4305}, {3244,5731}, {3339,4313}, {3474,3601}, {3486,5128}, {3524,8227}, {3528,3576}, {3529,5587}, {3530,9955}, {3543,7989}, {3616,9589}, {3624,9812}, {3627,11231}, {3755,4252}, {3811,5732}, {3833,5806}, {3841,8727}, {3874,7957}, {3911,6284}, {3916,7964}, {3947,5218}, {3956,9947}, {4192,6686}, {4294,11019}, {4308,9819}, {4311,5119}, {4312,5703}, {4316,10039}, {4330,5131}, {4353,5266}, {4355,10578}, {4512,6904}, {4652,4847}, {4691,5657}, {4701,11362}, {4746,5881}, {5059,9780}, {5204,12053}, {5267,6909}, {5281,5290}, {5438,5698}, {5692,9961}, {5818,11001}, {5842,6705}, {5904,11220}, {6244,8715}, {6409,8983}, {6460,9616}, {6872,8582}, {6906,7688}, {6987,10270}, {7288,9580}, {7988,10303}, {9899,11206}, {9949,10860}, {10391,12432}

X(12512) = midpoint of X(i) and X(j) for these {i,j}: {1,5493}, {10,20}, {40,4297}, {550,3579}, {3244,7991}, {3874,7957}, {4301,6361}
X(12512) = reflection of X(i) in X(j) for these (i,j): (4,3634), (1125,3), (4301,3636), (4701,11362), (5881,4746), (9955,3530)
X(12512) = X(389)-of-1st-circumperp-triangle
X(12512) = X(10110)-of-hexyl-triangle
X(12512) = X(11793)-of-excentral-triangle
X(12512) = excentral-to-1st-circumperp similarity image of X(10)
X(12512) = excentral-to-2nd-Conway similarity image of X(12571)
X(12512) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,9778,5493), (3,11495,12511), (4,10164,3634), (20,165,10), (40,376,4297), (46,4304,6738), (57,4314,6744), (962,7987,551), (962,10304,7987), (3474,3601,3671), (3522,9778,1), (3528,6361,3576), (3576,4301,3636), (3576,6361,4301), (5218,9579,3947), (5248,12436,1125), (5731,7991,3244), (7957,10167,3874)

X(12513) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st CIRCUMPERP TO EXCENTERS-MIDPOINTS

Trilinears a^3-(b^2-4*b*c+c^2)*a-2*b*c*(b+c) : :
X(12513) = 3*X(1)-X(11523) = 4*X(3)-3*X(4421) = 3*X(3)-2*X(8715) = 2*X(3)-3*X(11194) = 2*X(4)-3*X(11235) = 4*X(3813)-3*X(11235) = 2*X(3913)-3*X(4421) = X(3913)-4*X(8666) = 3*X(3913)-4*X(8715) = 6*X(11260)-X(11523)

The reciprocal orthologic center of these triangles is X(1).

X(12513) lies on these lines: {1,6}, {2,3304}, {3,519}, {4,529}, {5,11236}, {8,56}, {10,999}, {11,3436}, {12,6933}, {20,528}, {21,3241}, {35,3633}, {36,3632}, {40,3880}, {46,10914}, {55,145}, {57,4853}, {63,3057}, {75,7176}, {78,1319}, {100,3621}, {104,5854}, {105,6553}, {106,8688}, {144,8163}, {165,2136}, {198,5839}, {200,1420}, {241,6167}, {312,9369}, {355,10680}, {377,5434}, {382,535}, {388,2886}, {391,1696}, {443,9710}, {474,3679}, {480,6049}, {516,8158}, {517,1158}, {524,9840}, {527,4301}, {604,3713}, {672,4513}, {758,1482}, {908,11376}, {936,4662}, {940,10459}, {944,3428}, {952,11249}, {961,1219}, {988,4646}, {993,3244}, {1005,3486}, {1012,7982}, {1043,3286}, {1125,7373}, {1145,10074}, {1155,3893}, {1201,4383}, {1259,1317}, {1329,3086}, {1385,3811}, {1388,4511}, {1398,1861}, {1407,9363}, {1457,9370}, {1468,5710}, {1475,4390}, {1483,5428}, {1610,8301}, {1617,6737}, {1621,3623}, {1697,4640}, {1706,3361}, {1727,5697}, {1776,2098}, {1818,4322}, {2099,3868}, {2319,11051}, {2321,5120}, {2475,9657}, {2476,11237}, {2478,11240}, {2550,3600}, {2551,3816}, {2646,3870}, {2802,11256}, {3035,7080}, {3058,6872}, {3085,4999}, {3091,3829}, {3149,5881}, {3158,7987}, {3189,5584}, {3207,3684}, {3219,3890}, {3306,3698}, {3333,3812}, {3338,3753}, {3339,10107}, {3434,7354}, {3475,11281}, {3501,5022}, {3509,4051}, {3576,6765}, {3616,8167}, {3617,4413}, {3622,4423}, {3626,9709}, {3635,5248}, {3680,3928}, {3689,4855}, {3740,8583}, {3741,5793}, {3754,5708}, {3820,10200}, {3838,5290}, {3871,5217}, {3878,3927}, {3895,4652}, {3901,11009}, {3911,6736}, {3916,5119}, {3962,5048}, {4187,10072}, {4252,5255}, {4293,5082}, {4297,5853}, {4298,5880}, {4313,9797}, {4317,11112}, {4361,6647}, {4673,5695}, {4847,5794}, {4882,5438}, {4921,7419}, {4930,7489}, {5046,11238}, {5080,10896}, {5130,11401}, {5231,9578}, {5250,5919}, {5252,6734}, {5298,6921}, {5432,10528}, {5433,5552}, {5450,10306}, {5690,10269}, {5698,9785}, {5732,9845}, {5734,6912}, {5784,9850}, {5795,11019}, {5844,11248}, {5886,12001}, {6668,8164}, {6910,11239}, {7483,10056}, {7966,10268}, {8240,8424}, {9053,12329}, {9670,11114}, {9671,10707}, {10475,11679}, {10522,10949}, {10526,10943}, {10530,10955}, {10860,12448}, {10895,11680}, {10950,10966}, {10953,10959}, {11492,12455}, {11493,12454}, {11827,12116}

X(12513) = midpoint of X(i) and X(j) for these {i,j}: {1,6762}, {2136,11519}, {3189,6764}, {3680,7991}
X(12513) = reflection of X(i) in X(j) for these (i,j): (1,11260), (3,8666), (4,3813), (355,10916), (3811,1385), (3913,3), (4421,11194), (10306,5450), (10526,10943), (11500,11249)
X(12513) = orthologic center of these triangles: 1st circumperp to 2nd Schiffler
X(12513) = X(64)-of-1st-circumperp-triangle
X(12513) = X(1498)-of-2nd-circumperp-triangle
X(12513) = X(2883)-of-excentral-triangle
X(12513) = X(6247)-of-hexyl-triangle
X(12513) = excentral-to-1st-circumperp similarity image of X(2136)
X(12513) = excentral-to-2nd-circumperp similarity image of X(6762)
X(12513) = excentral-to-hexyl similarity image of X(3913)
X(12513) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,72,5289), (1,238,1616), (1,956,958), (1,958,1001), (1,5247,1191), (1,5258,405), (1,5288,956), (1,5904,5730), (3,3913,4421), (3,8666,11194), (4,3813,11235), (8,56,1376), (8,1788,8256), (21,3241,3303), (21,3303,4428), (405,956,5258), (405,5258,958), (1476,5435,56), (3436,10529,11), (3913,11194,3)

X(12514) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st CIRCUMPERP TO 4th EXTOUCH

Trilinears a^3+(b+c)*a^2-(b+c)^2*a-(b+c)*(b^2+c^2) : :
Barycentrics a + (a + b + c) cos(A) : :
X(12514) = X(1)-3*X(4512) = 3*X(165)-2*X(12511) = X(1697)+3*X(3929) = 5*X(1698)-4*X(3841) = X(3486)-3*X(11111) = 3*X(4512)-2*X(5248) = X(9949)+2*X(12512)

The reciprocal orthologic center of these triangles is X(65).

X(12514) lies on these lines: {1,21}, {2,46}, {3,960}, {4,9}, {6,3931}, {8,90}, {20,1709}, {29,1748}, {30,5794}, {35,78}, {36,4652}, {37,5711}, {44,4646}, {55,72}, {56,392}, {57,1125}, {65,405}, {84,4297}, {92,3559}, {100,3876}, {109,1038}, {165,411}, {171,975}, {190,4385}, {200,1005}, {201,1395}, {210,1898}, {214,1768}, {221,1214}, {226,10198}, {238,986}, {329,3085}, {355,5842}, {377,1770}, {386,1245}, {387,1723}, {406,1452}, {442,1836}, {443,3474}, {452,1728}, {474,1155}, {484,1698}, {495,5857}, {497,10916}, {498,908}, {515,5837}, {517,958}, {518,3295}, {519,1697}, {535,9613}, {551,3333}, {560,6042}, {612,5264}, {614,3670}, {902,976}, {912,10267}, {940,6051}, {942,1001}, {946,5709}, {956,3057}, {962,5273}, {984,5255}, {988,995}, {1107,1572}, {1150,3702}, {1156,4606}, {1193,4414}, {1203,5256}, {1329,6842}, {1334,5282}, {1376,3579}, {1385,5289}, {1445,3339}, {1454,7483}, {1479,6734}, {1490,10268}, {1571,1575}, {1610,4221}, {1656,5087}, {1699,5705}, {1714,3914}, {1724,4424}, {1727,3612}, {1737,2478}, {1741,5706}, {1743,4868}, {1759,2198}, {1760,5263}, {1761,5327}, {1782,10319}, {1788,5084}, {1837,7082}, {2093,3754}, {2136,3625}, {2245,4205}, {2257,4356}, {2646,5730}, {2802,4853}, {2886,5791}, {2950,10270}, {2951,5785}, {3052,5266}, {3086,5744}, {3158,4134}, {3218,3338}, {3244,6762}, {3303,3555}, {3306,3336}, {3358,9948}, {3359,3452}, {3361,4973}, {3416,3695}, {3419,6284}, {3421,10915}, {3436,10039}, {3550,5293}, {3576,5267}, {3587,6869}, {3632,3895}, {3634,5128}, {3646,5437}, {3650,10404}, {3654,8256}, {3679,5086}, {3681,3871}, {3682,4300}, {3685,10449}, {3689,4005}, {3697,3715}, {3698,5183}, {3704,5814}, {3711,4533}, {3714,5774}, {3740,9709}, {3742,5708}, {3746,3870}, {3812,8257}, {3817,6855}, {3822,9612}, {3831,4011}, {3911,10200}, {3913,5220}, {4008,11683}, {4067,11523}, {4084,5436}, {4187,4679}, {4197,4338}, {4199,10974}, {4304,6737}, {4307,5279}, {4326,5223}, {4423,5221}, {4426,9620}, {4450,5300}, {4647,5271}, {4666,4880}, {4668,5541}, {4847,10624}, {4855,5010}, {4999,5886}, {5046,10826}, {5080,10827}, {5217,5440}, {5227,5847}, {5231,9614}, {5234,6912}, {5247,7262}, {5259,5902}, {5290,8545}, {5295,5695}, {5302,5836}, {5438,6876}, {5506,10129}, {5535,6852}, {5536,11522}, {5693,10902}, {5720,6796}, {5731,10085}, {5732,7992}, {5743,5955}, {5755,5799}, {5777,11500}, {5795,6930}, {5812,7680}, {5815,6172}, {5880,8728}, {6690,11374}, {6700,6988}, {6867,10175}, {6870,9812}, {6871,9780}, {6913,7686}, {6932,9588}, {7085,8193}, {7162,10528}, {7373,10179}, {7411,9961}, {7548,7989}, {7969,9678}, {8273,10167}, {8424,9959}, {8580,12446}, {9589,10883}, {9778,9800}, {9949,10860}, {9957,12513}, {11344,11507}

X(12514) = midpoint of X(i) and X(j) for these {i,j}: {8,4294}, {3295,3927}, {4326,5223}
X(12514) = reflection of X(i) in X(j) for these (i,j): (1,5248), (3671,1125)
X(12514) = complement of X(4295)
X(12514) = X(578)-of-excentral-triangle
X(12514) = complement, wrt excentral triangle, of X(12565)
X(12514) = anticomplement, wrt 1st circumperp triangle, of X(12511)
X(12514) = anticomplement, wrt excentral triangle, of X(12511)
X(12514) = 1st-circumperp-to-excentral similarity image of X(12511)
X(12514) = 2nd-circumperp-to-excentral similarity image of X(5248)
X(12514) = intouch-to-excentral similarity image of X(3671)
X(12514) = inner-Conway-to-excentral similarity image of X(12526)
X(12514) = orthologic center of these triangles: excentral to 4th Conway
X(12514) = Ursa-major-to-excentral similarity image of X(17646)
X(12514) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,191,63), (1,1707,58), (1,3899,11682), (1,3901,11520), (1,4512,5248), (2,11415,12047), (3,960,997), (3,5887,6261), (9,3496,169), (21,3869,1), (31,2292,1), (38,3915,1), (63,5250,1), (71,2354,573), (960,4640,3), (993,3878,1), (1621,3868,1), (1621,11684,3868), (2975,3877,1), (3647,3878,993), (3884,8666,1), (6212,6213,573)

X(12515) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st CIRCUMPERP TO INNER-GARCIA

Trilinears a^6-(3*b^2-b*c+3*c^2)*a^4+3*b*c*(b+c)*a^3+(3*b^4+3*c^4-2*b*c*(b+c)^2)*a^2-3*(b^2-c^2)*(b-c)*b*c*a-(b^2-c^2)*(b-c)*(b^3+c^3) : :
X(12515) = 3*X(3)-2*X(214) = 3*X(40)-X(5541) = 3*X(104)-X(1320) = X(153)-3*X(5657) = 3*X(165)-X(6326) = 4*X(214)-3*X(6265) = 3*X(355)-4*X(3036) = 2*X(1145)-3*X(3654) = 4*X(1387)-3*X(3656) = 3*X(1768)+X(5541)

The reciprocal orthologic center of these triangles is X(3869).

X(12515) lies on these lines: {3,214}, {8,12248}, {9,119}, {10,3652}, {11,46}, {20,12247}, {30,80}, {35,11571}, {40,550}, {55,11570}, {57,1387}, {63,1145}, {65,10058}, {72,74}, {104,517}, {149,6361}, {153,5657}, {165,6326}, {191,11698}, {355,1158}, {376,6224}, {381,6702}, {516,10265}, {912,3689}, {1155,10090}, {1317,3655}, {1385,10698}, {1482,4757}, {1484,5535}, {1537,3306}, {1728,5128}, {1782,2828}, {1836,8068}, {2077,4867}, {2320,6950}, {2801,11495}, {2802,11256}, {3035,12514}, {3057,10074}, {3219,10711}, {3295,5083}, {3587,9945}, {5884,11849}, {6001,6100}, {6264,7991}, {6284,10073}, {6797,7098}, {6905,10225}, {7354,10057}, {7411,9964}, {7972,11010}, {9778,9803}, {9952,10860}

X(12515) = midpoint of X(i) and X(j) for these {i,j}: {8,12248}, {20,12247}, {40,1768}, {149,6361}, {6264,7991}
X(12515) = reflection of X(i) in X(j) for these (i,j): (100,3579), (1482,11715), (1537,6713), (6265,3), (6905,10225), (10698,1385), (10738,10265), (10742,10), (12119,550)
X(12515) = X(265)-of-1st-circumperp-triangle
X(12515) = X(12121)-of-2nd-circumperp-triangle
X(12515) = X(1511)-of-excentral-triangle
X(12515) = X(10113)-of-hexyl-triangle
X(12515) = X(1387)-of-tangential-of-excentral-triangle
X(12515) = excentral-to-1st-circumperp similarity image of X(6326)

X(12516) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st CIRCUMPERP TO HUTSON EXTOUCH

Trilinears a^9-(b+c)*a^8-4*(b^2+3*b*c+c^2)*a^7+2*(b+c)*(2*b^2-b*c+2*c^2)*a^6+6*(b^4+c^4+6*b*c*(b^2+b*c+c^2))*a^5-2*(b+c)*(3*b^4+3*c^4-b*c*(3*b^2+8*b*c+3*c^2))*a^4-4*(b^2+c^2)*(b^4+c^4+b*c*(9*b^2+4*b*c+9*c^2))*a^3+2*(b^2-c^2)*(b-c)*(2*b^4+2*c^4+b*c*(b^2-10*b*c+c^2))*a^2+(b^2-c^2)^2*(b^4+c^4+2*b*c*(6*b^2-5*b*c+6*c^2))*a-(b^4-c^4)*(b^2-c^2)^2*(b-c) : :
X(12516) = 3*X(165)+X(8001)

The reciprocal orthologic center of these triangles is X(3555).

X(12516) lies on these lines: {3,12333}, {9,946}, {40,6764}, {56,5920}, {165,8001}, {1158,5493}, {3333,3523}, {3361,9898}, {3651,12120}, {5045,12260}, {9778,9804}, {9953,10860}

X(12516) = reflection of X(12521) in X(3)

X(12517) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st CIRCUMPERP TO MANDART-EXCIRCLES

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

The reciprocal orthologic center of these triangles is X(3555).

X(12517) lies on these lines: {3,12442}, {19,1598}, {522,8668}, {946,6911}, {10860,12449}

X(12517) = reflection of X(12522) in X(3)

X(12518) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st CIRCUMPERP TO MIDARC

Trilinears -(-a+b+c)^2*a*sin(A/2)+(a-c)*(a-b+c)^2*sin(B/2)+(a-b)*(a+b-c)^2*sin(C/2) : :
X(12518) = X(164)-3*X(165) = 3*X(165)+X(167)

The reciprocal orthologic center of these triangles is X(1).

X(12518) lies on these lines: {3,12443}, {55,177}, {56,8422}, {57,5571}, {100,11691}, {164,165}, {7670,7676}, {9778,9807}, {10860,12450}

X(12518) = midpoint of X(164) and X(167)
X(12518) = orthologic center of these triangles: 1st circumperp to 2nd midarc
X(12518) = reflection of X(12523) in X(3)
X(12518) = X(1)-of-1st-circumperp-triangle
X(12518) = X(40)-of-2nd-circumperp-triangle
X(12518) = X(10)-of-excentral-triangle
X(12518) = X(946)-of-hexyl-triangle
X(12518) = {X(165), X(167)}-harmonic conjugate of X(164)

X(12519) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st CIRCUMPERP TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(21).

X(12519) lies on these lines: {3,12342}, {2475,3925}, {10860,12451}

X(12519) = reflection of X(12524) in X(3)

X(12520) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TO 4th EXTOUCH

Trilinears a^6-2*(b+c)*a^5-(b+c)^2*a^4+2*(b+c)*(2*b^2-3*b*c+2*c^2)*a^3-(b^2-4*b*c+c^2)*(b+c)^2*a^2-2*(b^3+c^3)*(b-c)^2*a+(b^4-c^4)*(b^2-c^2) : :
X(12520) = 3*X(3576)-2*X(5248) = 4*X(3841)-3*X(5587) = X(4294)-3*X(5731) = 3*X(4512)-5*X(7987)

The reciprocal orthologic center of these triangles is X(65).

X(12520) lies on these lines: {1,7}, {3,960}, {10,1490}, {21,1709}, {40,758}, {46,411}, {56,10167}, {65,7580}, {72,480}, {78,165}, {84,993}, {103,1310}, {224,3869}, {355,9710}, {392,8273}, {515,6850}, {572,1973}, {936,10164}, {946,6851}, {958,971}, {1001,9856}, {1040,10571}, {1125,6847}, {1214,1854}, {1319,10866}, {1385,11496}, {1467,11019}, {1699,6895}, {1708,1858}, {1737,6838}, {1750,5177}, {1768,4652}, {2475,5691}, {2551,5658}, {2646,5918}, {2886,5787}, {2975,10085}, {3243,6766}, {3359,6796}, {3522,4511}, {3576,5248}, {3601,10860}, {3612,6909}, {3616,9800}, {3624,6888}, {3841,5587}, {3870,7991}, {3878,7971}, {3962,7964}, {4189,4512}, {4666,11522}, {5302,5779}, {5436,11372}, {5450,7171}, {5493,6769}, {5534,11362}, {5693,7688}, {5709,5884}, {5715,11263}, {5720,6684}, {5745,9948}, {5768,10916}, {6282,12512}, {6836,12047}, {6845,8227}, {6892,10165}, {6925,10572}, {6932,10826}, {8583,9949}, {10864,12446}

X(12520) = midpoint of X(20) and X(4295)
X(12520) = reflection of X(i) in X(j) for these (i,j): (40,12511), (11496,1385), (12514,3)
X(12520) = complement, wrt hexyl triangle, of X(12705)
X(12520) = anticomplement, wrt 2nd circumperp triangle, of X(5248)
X(12520) = excentral-to-2nd-circumperp similarity image of X(12565)
X(12520) = excentral-to-1st-circumperp similarity image of X(12526)
X(12520) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1044,1448), (1,5732,4297), (3,6261,997), (21,9961,1709), (2975,11220,10085)

X(12521) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TO HUTSON EXTOUCH

Trilinears a^6-2*(b+c)*a^5-(b^2+14*b*c+c^2)*a^4+2*(b+2*c)*(2*b+c)*(b+c)*a^3-(b^4+c^4-2*b*c*(7*b^2+19*b*c+7*c^2))*a^2-2*(b^3+c^3)*(b^2+6*b*c+c^2)*a+(b^4-c^4)*(b^2-c^2) : :
X(12521) = 5*X(7987)-X(8001)

The reciprocal orthologic center of these triangles is X(3555).

X(12521) lies on these lines: {3,12333}, {21,3870}, {55,5920}, {100,3333}, {224,11036}, {1001,3811}, {3528,12120}, {3616,9804}, {3913,5045}, {3957,9874}, {4301,6261}, {5732,6769}, {7987,8001}, {8583,9953}, {10385,10393}

X(12521) = reflection of X(12516) in X(3)

X(12522) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TO MANDART-EXCIRCLES

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

The reciprocal orthologic center of these triangles is X(3555).

X(12522) lies on these lines: {3,12442}, {8583,12449}

X(12522) = reflection of X(12517) in X(3)

X(12523) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TO MIDARC

Barycentrics Sin[A]*(Sin[A/2]*Sin[A] - Sin[B/2]*(Sin[C] + Sin[A]) - Sin[C/2]*(Sin[A] + Sin[B])) : :
Trilinears -a*sin(A/2)+(a+c)*sin(B/2)+(a+b)*sin(C/2) : :
X(12523) = 3 X[1] - X[12656], X[1] + 3 X[55168], 3 X[1] + X[55169], 2 X[1] + X[55170], X[1] + 2 X[55171], X[1] - 3 X[55175], 2 X[1] - 3 X[55176], 3 X[164] + X[12656], X[164] - 3 X[55168], 3 X[164] - X[55169], X[164] + 2 X[55172], 2 X[164] + X[55173], X[164] + 3 X[55175], 2 X[164] + 3 X[55176], X[12656] + 9 X[55168], and many others

The reciprocal orthologic center of the 2nd circumperp and midarc triangles is X(1).

X(12523) lies on the cubics K838 and K1271 and these lines: {1, 164}, {2, 12622}, {3, 12443}, {4, 12614}, {21, 12539}, {55, 8422}, {56, 177}, {57, 31768}, {167, 7987}, {188, 3659}, {260, 8241}, {363, 10233}, {388, 31734}, {405, 12694}, {497, 31769}, {503, 21214}, {504, 7707}, {958, 18258}, {999, 12908}, {1125, 21633}, {1385, 53810}, {1697, 31767}, {2646, 17641}, {2975, 11691}, {3295, 32183}, {3303, 11234}, {3304, 11191}, {3576, 12844}, {3616, 9807}, {4293, 31735}, {4294, 31770}, {5666, 52999}, {6244, 31800}, {7587, 13092}, {7670, 7677}, {7991, 8108}, {8091, 10496}, {8109, 12879}, {8110, 12884}, {8225, 13090}, {8583, 12450}, {10215, 42614}, {10882, 12554}, {12513, 47303}, {17614, 17657}

X(12523) = midpoint of X(i) and X(j) for these {i,j}: {1, 164}, {7991, 11528}, {12656, 55169}, {55168, 55175}, {55170, 55173}, {55171, 55172}
X(12523) = reflection of X(i) in X(j) for these {i,j}: {1, 55172}, {4, 12614}, {164, 55171}, {12518, 3}, {21633, 1125}, {55170, 164}, {55173, 1}, {55176, 55175}
X(12523) = anticomplement of X(12622)
X(12523) = orthologic center of these triangles: 2nd circumperp to 2nd midarc
X(12523) = X(1)-of-2nd-circumperp-triangle
X(12523) = X(40)-of-1st-circumperp-triangle
X(12523) = X(10)-of-hexyl-triangle
X(12523) = X(946)-of-excentral-triangle
X(12523) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 55168, 164}, {1, 55169, 12656}, {1, 55171, 55170}, {1, 55172, 55176}, {1, 55175, 55172}, {164, 12656, 55169}, {164, 55168, 55171}, {164, 55172, 55173}, {164, 55175, 1}, {7588, 8077, 1}, {55168, 55172, 55170}, {55170, 55176, 55173}, {55171, 55175, 55173}, {55173, 55176, 1}

X(12524) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(21).

X(12524) lies on these lines: {1,6597}, {3,12342}, {12,100}, {21,10266}, {1001,12267}, {5443,6599}, {8583,12451}

X(12524) = reflection of X(12519) in X(3)

X(12525) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MCCAY TO CIRCUMSYMMEDIAL

Trilinears (-a^2*(2*b^4+b^2*c^2+2*c^4)*(b^2+c^2-a^2)+4*b^2*c^2*(b^4-3*b^2*c^2+c^4))*a : :
X(12525) = 5*X(1656)+4*X(6310)

The reciprocal orthologic center of these triangles is X(99).

X(12525) lies on the McCay circumcircle and these lines: {2,9879}, {3,5106}, {183,6787}, {263,3363}, {381,511}, {512,7610}, {1656,6310}, {5650,11287}, {7841,7998}, {11317,11673}

X(12525) = X(6323)-of-McCay-triangle
X(12525) = circumsymmedial-to-McCay similarity image of X(99)
X(12525) = anti-McCay-to-McCay similarity image of X(9879)

X(12526) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-CONWAY TO 4th CONWAY

Trilinears a^3+3*(b+c)*a^2-(b+c)^2*a-(b+c)*(3*b^2-2*b*c+3*c^2) : :
X(12526) = 2*X(1)-3*X(4512) = 3*X(1)-4*X(5248) = 3*X(165)-2*X(12520) = 2*X(958)-3*X(3929) = X(3340)-3*X(3929) = 4*X(3927)-X(4853) = 9*X(4512)-8*X(5248) = 3*X(4512)-4*X(12514) = 2*X(5248)-3*X(12514)

The reciprocal orthologic center of these triangles is X(1).

X(12526) lies on these lines: {1,21}, {2,3339}, {8,144}, {9,65}, {10,329}, {19,3958}, {20,6737}, {40,64}, {46,936}, {55,3962}, {56,3928}, {57,960}, {78,165}, {92,4647}, {100,3984}, {145,4314}, {201,2324}, {210,1706}, {219,221}, {377,4312}, {388,527}, {392,3333}, {405,4018}, {452,6738}, {517,3927}, {518,1697}, {519,4294}, {610,1761}, {899,8951}, {908,1698}, {942,10582}, {946,5231}, {950,5698}, {956,7982}, {958,3340}, {962,4847}, {986,2999}, {1001,11518}, {1125,5744}, {1155,5438}, {1158,6282}, {1191,3677}, {1260,5584}, {1376,5128}, {1420,5289}, {1695,3687}, {1699,6734}, {1788,3452}, {1854,7070}, {2263,5279}, {2551,4848}, {2951,9961}, {3057,6762}, {3091,5775}, {3190,4300}, {3218,3361}, {3219,5234}, {3243,3303}, {3338,4880}, {3421,6256}, {3428,7971}, {3434,9589}, {3436,3585}, {3485,5745}, {3556,5285}, {3576,3916}, {3579,3940}, {3601,4640}, {3612,4867}, {3616,10980}, {3617,4866}, {3634,5748}, {3646,5439}, {3680,7285}, {3681,4882}, {3683,5436}, {3698,3715}, {3811,4067}, {3812,7308}, {3827,5227}, {3841,11681}, {3876,8580}, {3885,11519}, {4005,5183}, {4127,8715}, {4298,9965}, {4511,4652}, {4643,5835}, {4668,5176}, {4861,11224}, {5119,5904}, {5219,6668}, {5220,5836}, {5221,5437}, {5252,5857}, {5290,5905}, {5493,6743}, {5552,9588}, {5694,5720}, {5697,10050}, {5705,12047}, {5709,5887}, {5710,7174}, {5794,9579}, {5815,6736}, {5842,5881}, {5884,8726}, {6180,7273}, {6904,12447}, {7688,11517}, {7962,12513}, {9614,10916}, {10527,11522}, {11678,12446}

X(12526) = reflection of X(i) in X(j) for these (i,j): (1,12514), (145,4314), (388,5837), (3340,958), (4295,10), (7982,11496), (9579,5794), (9800,9949)
X(12526) = anticomplement of X(3671)
X(12526) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,12514,4512), (8,3951,5223), (40,72,200), (40,5693,1490), (46,5692,936), (55,3962,11523), (57,960,8583), (63,3869,1), (63,11682,2975), (405,4018,11529), (968,2650,1), (1621,11520,1), (2975,3869,11682), (2975,11682,1), (3340,3929,958), (3868,5250,1), (3869,11684,63), (3899,6763,1), (5223,7991,8), (6734,11415,1699)

X(12526) = X(578)-of-inner-Conway-triangle
X(12546) = Conway-circle-inverse of X(37743)
X(12526) = Conway-to-inner-Conway similarity image of X(1)
X(12526) = excentral-to-inner-Conway similarity image of X(12514)
X(12526) = 1st-circumperp-to-excentral similarity image of X(12520)
X(12526) = complement, wrt inner-Conway triangle, of X(12529)

X(12527) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-CONWAY TO 5th CONWAY

Barycentrics 2*a^4+(b+c)*a^3-(b-c)^2*a^2-(b+c)^3*a-(b^2-c^2)^2 : :
X(12527) = X = 3*X(210)-X(7354) = 3*X(553)-4*X(3812) = 5*X(1698)-4*X(12436) = X(1770)-3*X(3679) = X(3555)-3*X(11113) = 5*X(3697)-3*X(11112) = 5*X(3698)-3*X(11246) = 3*X(3873)-4*X(6744) = 5*X(3876)-4*X(12447)

The reciprocal orthologic center of these triangles is X(1).

X(12527) lies on these lines: {1,329}, {2,3361}, {3,6745}, {4,4847}, {8,144}, {9,388}, {10,46}, {12,5745}, {20,200}, {36,6700}, {40,2123}, {56,3452}, {57,2551}, {65,527}, {72,515}, {78,4297}, {100,12512}, {142,10404}, {165,7080}, {191,10039}, {210,7354}, {219,5930}, {226,958}, {355,3927}, {497,6762}, {518,950}, {519,3869}, {529,960}, {535,3678}, {553,3812}, {908,1125}, {936,4293}, {946,956}, {962,4853}, {997,4311}, {1145,3650}, {1210,10629}, {1220,4357}, {1329,3911}, {1697,5698}, {1698,5744}, {1706,3474}, {1737,6763}, {1759,8074}, {1770,3679}, {1788,3928}, {2321,10371}, {2478,11019}, {2550,9579}, {3091,5231}, {3244,11682}, {3245,3626}, {3304,4679}, {3333,5084}, {3338,9843}, {3339,9965}, {3428,6260}, {3475,5436}, {3486,11523}, {3555,11113}, {3600,8583}, {3624,5748}, {3634,11681}, {3671,5905}, {3681,6743}, {3687,6999}, {3697,11112}, {3698,11246}, {3715,9657}, {3717,7270}, {3811,4304}, {3817,10527}, {3868,5850}, {3870,4314}, {3873,6744}, {3876,11678}, {3916,6684}, {3929,9578}, {3962,10950}, {4294,6765}, {4295,9623}, {4353,5262}, {4355,9776}, {4388,9369}, {4643,5793}, {4652,5552}, {4915,9589}, {5022,8568}, {5080,5536}, {5129,10582}, {5220,5794}, {5227,8804}, {5249,5260}, {5252,5837}, {5258,12047}, {5261,5273}, {5265,5328}, {5325,11237}, {5435,8165}, {5534,6868}, {5705,10590}, {5716,7174}, {5730,5882}, {5791,9654}, {5853,6284}, {6904,8580}, {7406,11679}, {10578,11106}, {10860,12246}, {12053,12513}

X(12527) = midpoint of X(i) and X(j) for these {i,j}: {3962,10950}, {5904,10572}
X(12527) = reflection of X(i) in X(j) for these (i,j): (65,5795), (3868,6738), (4292,10), (6737,72), (10106,960)
X(12527) = anticomplement of X(4298)
X(12527) = X(329)-of-inner-Conway-triangle
X(12527) = excentral-to-inner-Conway similarity image of X(10)
X(12527) = Conway-to-inner-Conway similarity image of X(4292)
X(12527) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,144,12526), (20,5815,200), (40,3421,6736), (57,2551,8582), (63,3436,10), (908,2975,1125), (3870,6872,4314), (4652,5552,10164), (5129,11037,10582), (5223,5691,8), (5234,5290,2)

X(12528) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-CONWAY TO EXTOUCH

Trilinears (b+c)*a^5-(b^2-b*c+c^2)*a^4-2*(b^3+c^3)*a^3+2*(b^4+c^4)*a^2+(b^4-c^4)*(b-c)*a-(b+c)*(b^2-c^2)*(b^3-c^3) : :
X(12528) = 4*X(3)-5*X(3876) = 4*X(3)-3*X(11220) = 2*X(40)-3*X(3681) = 3*X(165)-4*X(3678) = 4*X(942)-5*X(3091) = 2*X(942)-3*X(5927) = 2*X(944)-3*X(3877) = 5*X(3091)-6*X(5927) = 5*X(3876)-3*X(11220) = 3*X(3877)-4*X(5887)

The reciprocal orthologic center of these triangles is X(72).

X(12528) lies on these lines: {1,651}, {3,3219}, {4,912}, {5,9964}, {7,6835}, {8,6001}, {9,6986}, {20,72}, {21,7330}, {33,3562}, {40,3681}, {57,6915}, {63,411}, {65,5229}, {78,84}, {100,1158}, {110,11107}, {119,7705}, {153,355}, {165,3678}, {185,2808}, {200,7992}, {210,9943}, {226,6828}, {255,3465}, {329,6836}, {388,1858}, {392,11106}, {404,5720}, {405,5779}, {443,10861}, {497,1898}, {515,3869}, {516,5904}, {517,3146}, {518,962}, {758,5691}, {908,6245}, {916,5889}, {938,1864}, {942,3091}, {944,3877}, {946,3873}, {952,3885}, {960,5731}, {984,4300}, {997,10085}, {1210,6945}, {1699,3874}, {1709,3811}, {1736,4306}, {1837,9803}, {1854,9370}, {1870,8757}, {1871,6994}, {1902,5921}, {2096,4190}, {2478,5768}, {2800,5881}, {2975,6261}, {3090,10202}, {3100,7078}, {3149,3218}, {3157,6198}, {3305,8726}, {3419,6259}, {3487,6837}, {3523,5044}, {3555,9856}, {3753,9947}, {3839,5806}, {3871,5534}, {3881,11522}, {3889,5603}, {3890,5882}, {3927,7580}, {3935,10306}, {3984,6282}, {4005,5918}, {4015,9588}, {4134,12512}, {4295,7672}, {4297,5692}, {4312,12432}, {4420,10310}, {4511,12114}, {5056,5439}, {5086,6256}, {5174,5906}, {5220,5584}, {5226,6860}, {5249,6991}, {5279,5776}, {5450,6326}, {5531,8715}, {5570,10591}, {5587,5884}, {5658,6838}, {5696,6743}, {5703,6974}, {5728,11036}, {5744,6962}, {5758,10431}, {5770,6834}, {5787,6840}, {5812,6895}, {5817,6886}, {5883,7989}, {6147,8226}, {6260,6734}, {6888,11374}, {6938,11015}, {7282,7331}, {7414,9928}, {7548,9612}, {7987,10176}, {8095,11690}, {8227,12005}, {8581,11037}, {9948,11678}, {10303,11227}, {10826,11570}, {11444,11573}

X(12528) = reflection of X(i) in X(j) for these (i,j): (20,72), (944,5887), (3555,9856), (3868,4), (3869,5693), (9960,1490), (9961,40)
X(12528) = X(68)-of-inner-Conway-triangle
X(12528) = excentral-to-inner-Conway similarity image of X(1490)
X(12528) = Conway-to-inner-Conway similarity image of X(9960)
X(12528) = inner-Conway-isotomic conjugate of X(12530)
X(12528) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9,10884,6986), (63,1490,411), (78,84,6909), (329,9799,6836), (908,6245,6943), (942,5927,3091), (944,5887,3877), (3681,9961,40), (3876,11220,3), (5044,10167,3523), (5439,10157,5056)

X(12529) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-CONWAY TO 4th EXTOUCH

Trilinears (b+c)*a^5-(b^2-b*c+c^2)*a^4-2*(b^3+c^3)*a^3+2*(b^2-c^2)^2*a^2+(b+c)*(b^4+c^4-2*b*c*(b^2-3*b*c+c^2))*a-(b+c)*(b^2-c^2)*(b^3-c^3) : :
X(12529) = 4*X(3671)-3*X(3873) = 3*X(3681)-2*X(12526) = 3*X(3877)-2*X(4294) = 5*X(3890)-4*X(4314) = 3*X(11220)-4*X(12520)

The reciprocal orthologic center of these triangles is X(65).

X(12529) lies on these lines: {8,6001}, {63,7992}, {65,5175}, {72,6361}, {100,3876}, {224,1621}, {329,9800}, {516,3869}, {758,3632}, {912,5082}, {1858,2550}, {1898,2551}, {2801,4853}, {2975,10085}, {3434,3868}, {3671,3873}, {3681,4882}, {3877,4294}, {3890,4314}, {4511,11496}, {4512,4855}, {5086,10573}, {5174,6327}, {5744,9943}, {5777,7080}, {9949,11678}

X(12529) = reflection of X(3868) in X(4295)
X(12529) = anticomplement, wrt inner-Conway triangle, of X(12526)
X(12529) = {X(4882), X(12059)}-harmonic conjugate of X(3681)

X(12530) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-CONWAY TO 5th EXTOUCH

Trilinears (b+c)*a^3-(b^2-b*c+c^2)*a^2+(b^2-c^2)*(b-c)*a-b^4-c^4-b*c*(b-c)^2 : :
X(12530) = 4*X(3663)-3*X(3873) = 3*X(3681)-2*X(3729) = 5*X(3876)-4*X(3923)

The reciprocal orthologic center of these triangles is X(65).

X(12530) lies on these lines: {63,1721}, {100,1766}, {200,7996}, {329,9801}, {516,3869}, {990,2975}, {1633,1760}, {1742,1959}, {3663,3873}, {3681,3729}, {3876,3923}, {5744,9944}, {9950,11678}

X(12530) = reflection of X(9962) in X(1721)
X(12530) = X(317)-of-inner-Conway-triangle
X(12530) = excentral-to-inner-Conway similarity image of X(1721)
X(12530) = inner-Conway-isotomic conjugate of X(12528)
X(12530) = anticomplement, wrt inner-Conway triangle, of X(3729)

X(12531) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-CONWAY TO FUHRMANN

Barycentrics 3*a^4-4*(b+c)*a^3-(b^2-9*b*c+c^2)*a^2+(b+c)*(4*b^2-9*b*c+4*c^2)*a-2*(b^2-c^2)^2 : :
X(12531) = 3*X(1)-4*X(6702) = 3*X(2)-4*X(3036) = 3*X(8)-2*X(1145) = 3*X(8)-X(6224) = 5*X(8)-2*X(10609) = 3*X(100)-4*X(1145) = 3*X(100)-2*X(6224) = 5*X(100)-4*X(10609) = 3*X(4677)-X(5541) = 6*X(4677)-X(9963)

The reciprocal orthologic center of these triangles is X(8).

X(12531) lies on these lines: {1,6702}, {2,1317}, {3,8}, {10,7972}, {11,145}, {21,10087}, {63,4677}, {78,6264}, {80,519}, {119,11680}, {144,528}, {149,3436}, {153,3434}, {200,7993}, {214,3679}, {329,9802}, {355,10698}, {404,10074}, {517,10724}, {1156,5853}, {1387,3241}, {2771,10914}, {2800,5881}, {2802,3632}, {3035,3617}, {3555,6797}, {3622,6667}, {3625,11684}, {3871,10058}, {4193,5533}, {4853,5531}, {4861,6265}, {5080,5844}, {5253,10944}, {5818,11729}, {5840,12245}, {5846,10755}, {6713,7967}, {6735,10265}, {8097,11690}, {8197,12461}, {8204,12460}, {9951,11678}, {11362,12119}

X(12531) = midpoint of X(i) and X(j) for these {i,j}: {149,3621}, {3632,9897}
X(12531) = reflection of X(i) in X(j) for these (i,j): (100,8), (145,11), (1317,3036), (1320,80), (3555,6797), (6224,1145), (7972,10), (9963,5541), (10031,3679), (10698,355), (12119,11362)
X(12531) = anticomplement of X(1317)
X(12531) = X(74)-of-inner-Conway-triangle
X(12531) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,6224,1145), (80,1320,10707), (956,12331,4996), (1145,6224,100), (1317,3036,2), (4996,12331,100)

X(12532) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-CONWAY TO INNER-GARCIA

Trilinears (b+c)*a^5-(b^2+b*c+c^2)*a^4-(b+c)*(2*b^2-3*b*c+2*c^2)*a^3+(2*b^4+2*c^4+b*c*(b^2-b*c+c^2))*a^2+(b+c)*(b^4+c^4-3*b*c*(b^2-b*c+c^2))*a-(b^4-c^4)*(b^2-c^2) : :
X(12532) = 2*X(214)-3*X(5692) = 2*X(1145)-3*X(3681) = 2*X(1317)-3*X(3877) = 4*X(1387)-3*X(3873) = 4*X(3035)-5*X(3876) = 5*X(3616)-4*X(5083) = 3*X(5902)-4*X(6702)

The reciprocal orthologic center of these triangles is X(3869).

X(12532) lies on these lines: {2,11570}, {8,153}, {10,11571}, {11,3868}, {63,4996}, {72,74}, {78,1768}, {80,758}, {104,912}, {144,2801}, {149,11415}, {214,5692}, {329,9803}, {517,10724}, {518,1156}, {908,10265}, {952,3869}, {1145,3681}, {1317,3877}, {1387,3873}, {2802,3621}, {2829,12528}, {2932,3940}, {2975,5694}, {3035,3876}, {3218,10090}, {3436,12247}, {3616,5083}, {3648,4127}, {3878,7972}, {4018,6797}, {4861,5887}, {5046,10073}, {5057,10738}, {5086,10742}, {5531,12526}, {5744,9946}, {5902,6702}, {6264,11682}, {9952,11678}

X(12532) = reflection of X(i) in X(j) for these (i,j): (100,72), (3868,11), (4018,6797), (6265,5694), (7972,3878), (9964,6326), (10698,5887), (11571,10)
X(12532) = anticomplement of X(11570)
X(12532) = {X(63), X(6326)}-harmonic conjugate of X(4996)
X(12532) = X(265)-of-inner-Conway-triangle
X(12532) = excentral-to-inner-Conway similarity image of X(6326)

X(12533) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-CONWAY TO HUTSON EXTOUCH

Trilinears (b+c)*a^8-(2*b^2+b*c+2*c^2)*a^7-(b+c)*(2*b^2+13*b*c+2*c^2)*a^6+3*(2*b^2+b*c+2*c^2)*(b+c)^2*a^5+b*c*(b+c)*(27*b^2+22*b*c+27*c^2)*a^4-(6*b^6+6*c^6+(27*b^4+27*c^4+2*b*c*(7*b^2+33*b*c+7*c^2))*b*c)*a^3+(b+c)*(2*b^6+2*c^6-(15*b^4+15*c^4+2*b*c*(11*b^2-19*b*c+11*c^2))*b*c)*a^2+(b^2-c^2)^2*(2*b^4+2*c^4+b*c*(13*b^2+2*b*c+13*c^2))*a-(b^3-c^3)*(b^2-c^2)^3 : :

The reciprocal orthologic center of these triangles is X(3555).

X(12533) lies on these lines: {8,6835}, {100,12516}, {145,5920}, {200,8001}, {329,9804}, {2975,12521}, {5744,12439}, {9953,11678}

X(12533) = reflection of X(145) in X(5920)
X(12533) = excentral-to-inner-Conway similarity image of X(12658)

X(12534) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-CONWAY TO MANDART-EXCIRCLES

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

The reciprocal orthologic center of these triangles is X(3555).

X(12534) lies on these lines: {4,6735}, {100,12517}, {2975,12522}, {3729,5082}, {5744,12442}, {11678,12449}

X(12535) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-CONWAY TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(21).

X(12535) lies on these lines: {2,10044}, {100,12519}, {191,7161}, {2975,12524}, {3648,4127}, {5744,12444}, {11678,12451}

X(12535) = reflection of X(10266) in X(191)

X(12536) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CONWAY TO EXCENTERS-MIDPOINTS

Barycentrics (-a+b+c)*(7*a^3+(b+c)*a^2-(3*b^2+2*b*c+3*c^2)*a+3*(b^2-c^2)*(b-c)) : :
X(12536) = 3*X(2)-4*X(12437) = 3*X(8)-4*X(3913) = 3*X(145)-2*X(3680) = 6*X(3158)-5*X(3617) = 3*X(3189)-2*X(3913) = 9*X(5731)-8*X(8666)

The reciprocal orthologic center of these triangles is X(1).

X(12536) lies on these lines: {1,4208}, {2,12437}, {7,145}, {8,21}, {20,519}, {35,5775}, {63,2136}, {78,5328}, {80,5828}, {377,3241}, {390,6737}, {474,938}, {517,9960}, {527,5059}, {944,6764}, {952,9799}, {1004,4308}, {2802,9964}, {2900,5175}, {3146,11523}, {3158,3617}, {3244,11036}, {3419,5703}, {3434,4323}, {3476,9797}, {3488,11108}, {3623,5249}, {3632,4304}, {3633,4292}, {3813,4197}, {3868,3880}, {3893,10391}, {4188,5435}, {4853,7675}, {4866,6743}, {5260,6600}, {5440,5704}, {5731,8666}, {5732,11519}, {5734,6839}, {5794,10578}, {5815,10572}, {5836,11020}, {5854,9963}, {6172,6872}, {7411,12513}, {10861,12448}

X(12536) = reflection of X(i) in X(j) for these (i,j): (8,3189), (3146,11523), (3621,2136), (6764,944)
X(12536) = X(64)-of-Conway-triangle
X(12536) = X(6293)-of-2nd-Conway-triangle
X(12536) = excentral-to-Conway similarity image of X(2136)
X(12536) = excentral-to-2nd-Conway similarity image of X(12625)
X(12536) = orthologic center of these triangles: Conway to 2nd Schiffler
X(12536) = {X(8), X(4313)}-harmonic conjugate of X(5273)

X(12537) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CONWAY TO HUTSON EXTOUCH

Trilinears (b+c)*a^8-(2*b^2-b*c+2*c^2)*a^7-(b+c)*(2*b^2+17*b*c+2*c^2)*a^6+(6*b^4+6*c^4+b*c*(13*b^2-18*b*c+13*c^2))*a^5+b*c*(b+c)*(35*b^2+46*b*c+35*c^2)*a^4-(6*b^6+6*c^6+(29*b^4+29*c^4-6*b*c*(3*b^2+11*b*c+3*c^2))*b*c)*a^3+(b+c)*(2*b^6+2*c^6-(19*b^4+19*c^4+2*b*c*(23*b^2-15*b*c+23*c^2))*b*c)*a^2+(b^2-c^2)^2*(2*b^4+2*c^4+3*b*c*(5*b^2+2*b*c+5*c^2))*a-(b^3-c^3)*(b^2-c^2)^3 : :
X(12537) = 3*X(2)-4*X(12439) = 3*X(3681)-4*X(12260)

The reciprocal orthologic center of these triangles is X(3555).

X(12537) lies on these lines: {2,12439}, {7,3555}, {21,3870}, {63,12533}, {3681,12260}, {4313,5920}, {5732,8001}, {7411,12516}, {9953,10861}

X(12537) = reflection of X(9874) in X(3555)

X(12538) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CONWAY TO MANDART-EXCIRCLES

Barycentrics 3*a^10+2*(b+c)*a^9-(5*b^2+18*b*c+5*c^2)*a^8-2*(b+c)*(2*b^2-b*c+2*c^2)*a^7-2*(b^4+c^4-b*c*(13*b^2+24*b*c+13*c^2))*a^6-2*b*c*(b+c)*(5*b^2-2*b*c+5*c^2)*a^5+2*(3*b^6+3*c^6-b*c*(b^2+b*c+c^2)*(7*b^2+4*b*c+7*c^2))*a^4+2*(b+c)*(2*b^6+2*c^6-(b^4+c^4-2*b*c*(3*b^2+b*c+3*c^2))*b*c)*a^3-(b^2-c^2)^2*(b^4+c^4-2*b*c*(3*b^2-13*b*c+3*c^2))*a^2-2*(b^2-c^2)^3*(b-c)*(b^2-3*b*c+c^2)*a-(b^4-c^4)*(b^2-c^2)^3 : :
X(12538) = 3*X(2)-4*X(12442)

The reciprocal orthologic center of these triangles is X(3555).

X(12538) lies on these lines: {2,12442}, {21,12522}, {63,12534}, {1266,6361}, {7411,12517}, {10861,12449}

X(12539) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CONWAY TO MIDARC

Trilinears -(a+b+c)*b*c*sin(A/2)+(a+c)*(a+b-c)*c*sin(B/2)+(a+b)*(a-b+c)*b*sin(C/2) : :
X(12539) = (4*R+r)*X(7)-2*(r+2*R)*X(177)

The reciprocal orthologic center of these triangles is X(1).

X(12539) lies on these lines: {1,11888}, {2,12443}, {7,177}, {21,12523}, {63,164}, {167,5732}, {4313,8422}, {5571,11020}, {7411,12518}, {8080,8733}, {10861,12450}

X(12539) = reflection of X(i) in X(j) for these (i,j): (9807,177), (11691,164)
X(12539) = orthologic center of these triangles: Conway to 2nd midarc
X(12539) = X(1)-of-Conway-triangle
X(12539) = {X(11888), X(11889)}-harmonic conjugate of X(1)

X(12540) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CONWAY TO 1st SCHIFFLER

Barycentrics (-a+b+c)*(a^9+(b+c)*a^8-2*(2*b^2+b*c+2*c^2)*a^7-4*(b^2+b*c+c^2)*(b+c)*a^6+(6*b^4+6*c^4-b*c*(2*b^2+15*b*c+2*c^2))*a^5+(b+c)*(6*b^4+6*c^4+b*c*(4*b^2-13*b*c+4*c^2))*a^4-(4*b^6+4*c^6-(10*b^4+10*c^4+b*c*(7*b^2-18*b*c+7*c^2))*b*c)*a^3-(b^2-c^2)*(b-c)*(4*b^4+4*c^4+b*c*(4*b^2-5*b*c+4*c^2))*a^2+(b^2-c^2)^2*(b-c)^2*(b^2-4*b*c+c^2)*a+(b^2-c^2)^3*(b-c)^3) : :
X(12540) = 3*X(2)-4*X(12444)

The reciprocal orthologic center of these triangles is X(21).

X(12540) lies on these lines: {2,12444}, {7,6597}, {20,5538}, {21,10266}, {63,12535}, {1836,3868}, {5905,12536}, {7411,12519}, {10861,12451}

X(12541) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CONWAY TO EXCENTERS-MIDPOINTS

Barycentrics (-a+b+c)*(a^3+3*(b+c)*a^2+(3*b^2-14*b*c+3*c^2)*a+(b^2-c^2)*(b-c)) : :
X(12541) = 6*X(3158)-7*X(3622) = 2*X(3189)-3*X(3241) = 3*X(3241)-4*X(10912) = 5*X(3616)-4*X(3913) = 4*X(3680)-X(12536) = 4*X(3811)-5*X(5734) = 2*X(7674)-3*X(8236) = 3*X(9778)-4*X(12513)

The reciprocal orthologic center of these triangles is X(1).

X(12541) lies on these lines: {1,11024}, {2,2136}, {7,145}, {8,210}, {65,9797}, {72,9804}, {78,4345}, {329,3621}, {390,4853}, {516,11519}, {517,6764}, {519,962}, {938,10914}, {1697,5273}, {2802,9803}, {3158,3622}, {3169,5296}, {3189,3241}, {3244,11037}, {3616,3913}, {3632,5815}, {3633,4295}, {3811,5734}, {3813,9780}, {3870,4323}, {4298,12127}, {4342,4882}, {4513,5838}, {5176,7319}, {5274,6736}, {5328,12053}, {5758,5844}, {5828,10591}, {5836,10580}, {5854,9802}, {6601,7320}, {7674,8236}, {9778,12513}, {10578,11281}

X(12541) = reflection of X(i) in X(j) for these (i,j): (145,3680), (3057,12448), (3189,10912), (12536,145)
X(12541) = anticomplement of X(2136)
X(12541) = orthologic center of these triangles: 2nd Conway to 2nd Schiffler
X(12541) = X(64)-of-2nd-Conway-triangle
X(12541) = excentral-to-2nd-Conway similarity image of X(2136)
X(12541) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (497,3893,8), (3189,10912,3241)

X(12542) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CONWAY TO MANDART-EXCIRCLES

Barycentrics a^10+2*(b+c)*a^9-(b^2+6*b*c+c^2)*a^8-2*(b+c)*(2*b^2+3*b*c+2*c^2)*a^7-2*(b^4+c^4-b*c*(b^2+24*b*c+c^2))*a^6+2*b*c*(b+c)*(b^2+10*b*c+c^2)*a^5+2*(b^6+c^6-b*c*(b^2+4*b*c+c^2)*(b^2+9*b*c+c^2))*a^4+2*(b+c)*(2*b^6+2*c^6-(b^4+c^4+2*b*c*(3*b^2-13*b*c+3*c^2))*b*c)*a^3+(b^2-c^2)^2*(b^4+c^4+2*b*c*(3*b^2-11*b*c+3*c^2))*a^2-2*(b^2-c^2)^2*(b-c)^2*(b^3+c^3)*a-(b^4-c^4)*(b^2-c^2)^3 : :
X(12542) = 5*X(3616)-4*X(12522)

The reciprocal orthologic center of these triangles is X(3555).

X(12542) lies on these lines: {7,12538}, {8,12449}, {329,12534}, {3616,12522}, {3935,5758}, {9776,12442}, {9778,12517}

X(12542) = anticomplement of X(12659)

X(12543) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CONWAY TO 1st SCHIFFLER

Barycentrics (-a+b+c)*(a^9+(b+c)*a^8-2*(2*b^2-3*b*c+2*c^2)*a^7-4*(b^2-c^2)*(b-c)*a^6+(6*b^4+6*c^4-b*c*(14*b^2-17*b*c+14*c^2))*a^5+(b+c)*(6*b^4+6*c^4-b*c*(20*b^2-23*b*c+20*c^2))*a^4-(4*b^6+4*c^6-b*c*(10*b^2+11*b*c+10*c^2)*(b-c)^2)*a^3-(b^2-c^2)*(b-c)*(4*b^4+4*c^4-b*c*(8*b^2+b*c+8*c^2))*a^2+(b^4-c^4)*(b^2-c^2)*(b-c)^2*a+(b^2-c^2)^3*(b-c)^3) : :
X(12543) = 5*X(3616)-4*X(12524)

The reciprocal orthologic center of these triangles is X(21).

X(12543) lies on these lines: {7,6597}, {8,10266}, {329,12535}, {2476,9782}, {3616,12524}, {5046,6599}, {9776,12444}, {9778,12519}, {9799,10525}, {9802,10912}

X(12543) = anticomplement of X(12660)

X(12544) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO 4th CONWAY

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

The reciprocal orthologic center of these triangles is X(1).

X(12544) lies on these lines: {1,7}, {10,10888}, {40,7413}, {758,12435}, {1125,10856}, {1330,5691}, {1695,4384}, {1709,10461}, {1722,9535}, {1764,12514}, {3841,10887}, {5208,9961}, {5248,10882}, {6001,10441}, {9800,10453}, {10434,12511}, {10862,12446}, {11679,12526}

X(12544) = X(578)-of-3rd-Conway-triangle
X(12544) = excentral-to-3rd-Conway similarity image of X(12514)

X(12545) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO 5th CONWAY

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

The reciprocal orthologic center of these triangles is X(1).

X(12545) lies on these lines: {1,7}, {4,3741}, {10,1764}, {40,3980}, {515,10441}, {519,12126}, {894,2944}, {946,4425}, {950,10473}, {978,9535}, {1125,10478}, {3146,10453}, {3244,11521}, {3634,10887}, {5247,6996}, {5691,10449}, {6744,11021}, {7406,11679}, {10106,10480}, {10434,12512}, {10439,10454}, {10452,10464}, {10475,12053}, {10856,12436}, {10862,12447}

X(12545) = Conway circle-inverse-of-X(5018)
X(12545) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,10442,12544), (4,10476,3741), (10446,10465,1), (10478,10882,1125)

X(12545) = X(389)-of-3rd-Conway-triangle

X(12546) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO EXCENTERS-MIDPOINTS

Barycentrics a^4-9*(b+c)*a^3-(5*b^2-2*b*c+5*c^2)*a^2+(b+c)*(5*b^2-6*b*c+5*c^2)*a+4*b*c*(b+c)^2 : :
X(12546) = (4*r^2+3*s^2)*X(1)-3*(r^2+s^2)*X(2)

The reciprocal orthologic center of these triangles is X(1).

X(12546) lies on these lines: {1,2}, {740,11531}, {1764,2136}, {3680,10435}, {3813,10887}, {3880,12435}, {3893,10473}, {3913,10882}, {5836,11021}, {5853,10442}, {10434,12513}, {10444,12536}, {10446,12541}, {10856,12437}, {10862,12448}, {10912,11369}

X(12546) = orthologic center of these triangles: 3rd Conway to 2nd Schiffler
X(12546) = X(64)-of-3rd-Conway-triangle
X(12546) = excentral-to-3rd-Conway similarity image of X(2136)

X(12547) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO EXTOUCH

Trilinears (2*b^2+3*b*c+2*c^2)*a^7+4*(b+c)*b*c*a^6-(6*b^4+6*c^4-b*c*(b+c)^2)*a^5-2*b*c*(b+c)*(b^2+c^2)*a^4+(6*b^4+6*c^4+b*c*(9*b^2+10*b*c+9*c^2))*(b-c)^2*a^3-(b^2-c^2)^2*(2*b^4+2*c^4+b*c*(b^2+6*b*c+c^2))*a-2*(b^4-c^4)*(b^2-c^2)*(b+c)*b*c : :
X(12547) = (2*R*r+r^2+s^2)*X(1)-(r^2+s^2)*X(84)

The reciprocal orthologic center of these triangles is X(72).

X(12547) lies on these lines: {1,84}, {4,10435}, {515,12435}, {517,12546}, {944,10890}, {946,11021}, {971,10441}, {1158,10434}, {1490,1764}, {5691,10825}, {6245,10478}, {6260,10479}, {6261,10882}, {9799,10446}, {9942,10856}, {9948,10862}, {9960,10444}, {11679,12528}

X(12547) = X(68)-of-3rd-Conway-triangle
X(12547) = excentral-to-3rd-Conway similarity image of X(1490)
X(12547) = 3rd-Conway-isotomic conjugate of X(12549)

X(12548) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO 4th EXTOUCH

Trilinears (2*b^2+3*b*c+2*c^2)*a^7+4*(b+c)*b*c*a^6-(6*b^4+6*c^4-b*c*(b^2+6*b*c+c^2))*a^5-2*b*c*(b+c)^3*a^4+3*(b^2-c^2)^2*(2*b^2-b*c+2*c^2)*a^3+4*(b^2-c^2)*(b-c)*b^2*c^2*a^2-(b^2-c^2)^2*(2*b^4+2*c^4+b*c*(b^2+6*b*c+c^2))*a-2*(b^4-c^4)*(b^2-c^2)*(b+c)*b*c : :
X(12548) = (4*R^2+2*R*r+r^2+s^2)*X(1)-(r^2+s^2)*X(84)

The reciprocal orthologic center of these triangles is X(65).

X(12548) lies on these lines: {1,84}, {516,10454}, {3671,11021}, {4512,10470}, {9800,10446}, {9943,10856}, {9949,10862}, {9961,10444}, {10434,12514}, {10439,12544}, {10882,12520}, {11679,12529}

X(12548) = excentral-to-3rd-Conway similarity image of X(12565)

X(12549) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO 5th EXTOUCH

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

The reciprocal orthologic center of these triangles is X(65).

X(12549) lies on these lines: {1,7175}, {516,10454}, {968,1766}, {990,10882}, {1721,1764}, {3663,11021}, {3729,3869}, {4061,10445}, {5208,9962}, {9801,10446}, {9944,10856}, {9950,10862}, {11679,12530}

X(12549) = X(317)-of-3rd-Conway-triangle
X(12549) = excentral-to-3rd-Conway similarity image of X(1721)
X(12549) = 3rd-Conway-isotomic conjugate of X(12547)
X(12549) = anticomplement, wrt 3rd Conway triangle, of X(10444)

X(12550) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO FUHRMANN

Barycentrics a^7-6*(b+c)*a^6-(b^2-11*b*c+c^2)*a^5+(b+c)*(10*b^2-17*b*c+10*c^2)*a^4-(3*b^4+3*c^4+b*c*(7*b^2-24*b*c+7*c^2))*a^3-(b+c)*(4*b^2-3*b*c+c^2)*(b^2-3*b*c+4*c^2)*a^2+(b^2-c^2)^2*(3*b^2-4*b*c+3*c^2)*a+2*(b^2-c^2)^2*(b+c)*b*c : :
X(12550) = (-R*r+4*r^2+2*s^2)*X(1)-2*(r^2+s^2)*X(5)

The reciprocal orthologic center of these triangles is X(8).

X(12550) lies on these lines: {1,5}, {100,10882}, {104,10434}, {528,10442}, {1320,10435}, {1764,5541}, {2800,12547}, {2802,12435}, {5854,12546}, {8097,11894}, {9802,10446}, {9945,10856}, {9951,10862}, {9963,10444}, {10825,11521}, {11679,12531}

X(12550) = Conway circle-inverse-of-X(1317)
X(12550) = X(74)-of-3rd-Conway-triangle
X(12550) = excentral-to-3rd-Conway similarity image of X(5541)

X(12551) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO INNER-GARCIA

Trilinears (2*b^2+3*b*c+2*c^2)*a^7-(6*b^4+6*c^4+(b^2-b*c+c^2)*b*c)*a^5+b*c*(2*b-c)*(b-2*c)*(b+c)*a^4+(6*b^6+6*c^6-b*c*(3*b^2-5*b*c+3*c^2)*(b+c)^2)*a^3+3*(b^2-c^2)*(b-c)*b^2*c^2*a^2-(b^2-c^2)^2*(2*b^4+2*c^4-b*c*(b^2-6*b*c+c^2))*a-2*(b^3+c^3)*(b^2-c^2)^2*b*c : :
X(12551) = (R*r+r^2+s^2)*X(1)+(r^2+s^2)*X(104)

The reciprocal orthologic center of these triangles is X(3869).

X(12551) lies on these lines: {1,104}, {11,11369}, {80,10435}, {517,12550}, {952,12435}, {1387,11021}, {1764,6326}, {2771,10441}, {2801,10442}, {2802,12546}, {2829,12547}, {6264,11521}, {6265,10882}, {7972,10890}, {9803,10446}, {9809,10449}, {9897,10825}, {9946,10856}, {9952,10862}, {9964,10444}, {10265,10478}, {10434,12515}, {11679,12532}

X(12551) = Conway circle-inverse-of-X(11700)
X(12551) = X(265)-of-3rd-Conway-triangle
X(12551) = excentral-to-3rd-Conway similarity image of X(6326)

X(12552) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO HUTSON EXTOUCH

Trilinears 2*(2*r^2+s^2)*b*c*a*(8*R*r+8*R^2+r^2)+(r^2+s^2)*(-a+b+c)*(2*b*c*(20*R^2+10*R*r+r^2)-(b+c)*(24*R^2*s+8*R*S+S*r)+2*(4*R*s+S)*R*s) : :
X(12552) = (2*r^2+s^2)*X(1)-(r^2+s^2)*X(5920)

The reciprocal orthologic center of these triangles is X(3555).

X(12552) lies on these lines: {1,5920}, {9804,10446}, {9953,10862}, {10434,12516}, {10444,12537}, {10856,12439}, {10882,12521}, {11679,12533}

X(12553) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO MANDART-EXCIRCLES

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

The reciprocal orthologic center of these triangles is X(3555).

X(12553) lies on these lines: {1266,6361}, {10434,12517}, {10446,12542}, {10856,12442}, {10862,12449}, {10882,12522}, {11679,12534}

X(12554) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO MIDARC

Trilinears -(-a+b+c)*(a+b+c)^2*a*b*c*sin(A/2)+(a-b+c)*((b+2*c)*a^3+2*b^2*a^2+(b-c)*(b^2+b*c+2*c^2)*a+2*(b^2-c^2)*b*c)*c*sin(B/2)+(a+b-c)*((2*b+c)*a^3+2*c^2*a^2-(b-c)*(2*b^2+b*c+c^2)*a-2*(b^2-c^2)*b*c)*b*sin(C/2) : :
X(12554) = (-2*s^2+SW)*X(1)+(r^2+s^2)*X(167)

The reciprocal orthologic center of these triangles is X(1).

X(12554) lies on these lines: {1,167}, {164,1764}, {5571,11021}, {7670,10889}, {9807,10446}, {10434,12518}, {10444,12539}, {10856,12443}, {10862,12450}, {10882,12523}, {11679,11691}

X(12554) = orthologic center of these triangles: 3rd Conway to 2nd midarc
X(12554) = X(1)-of-3rd-Conway-triangle
X(12554) = {X(11894),X(11895)}-harmonic conjugate of X(1)

X(12555) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO MIXTILINEAR

Trilinears a^5-3*(b+c)*a^4-6*(b^2+c^2)*a^3+2*(b^2-c^2)*(b-c)*a^2+(5*b^4+6*b^2*c^2+5*c^4)*a-(b^2-c^2)*(b-c)*(-c^2-6*b*c-b^2) : :
X(12555) = (2*R*r+r^2+2*s^2)*X(1)-2*(r^2+s^2)*X(3)

The reciprocal orthologic center of these triangles is X(1).

X(12555) lies on these lines: {1,3}, {329,4416}, {511,1750}, {527,10442}, {966,3452}, {1396,1753}, {1999,9965}, {3781,8580}, {3820,10887}, {7682,10479}, {7956,10886}, {8101,11894}, {9954,10862}

X(12555) = Conway circle-inverse-of-X(3660)
X(12555) = X(25)-of-3rd-Conway-triangle
X(12555) = excentral-to-3rd-Conway similarity image of X(57)
X(12555) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1764,10856), (10446,11679,10888)

X(12556) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO 1st SCHIFFLER

Trilinears 8*p^3*(6*p^3-8*p^2*q-7*p+9*q)+(16*q^4-8*q^2+27)*p^2-2*(4*q^2+5)*q*p+q^2-9 : :
where p=sin(A/2), q=cos((B-C)/2)
X(12556) = 3*X(3)-X(13126) = 3*X(165)-X(12409) = 3*X(376)-X(12255) = 3*X(3576)-2*X(12267) = 3*X(10266)-2*X(13126)

The reciprocal orthologic center of these triangles is X(79)

X(12556) lies on these lines: {2,12600}, {3,10266}, {4,13089}, {20,5694}, {30,12798}, {35,13128}, {36,13129}, {40,12660}, {56,13080}, {100,3648}, {165,12409}, {182,12209}, {376,12255}, {515,12786}, {517,13100}, {1593,12146}, {2771,12535}, {3098,12504}, {3576,12267}, {3651,12519}, {5732,12845}, {5840,6595}, {6284,12957}, {7354,12947}, {10310,12342}, {11248,13130}, {11249,13131}, {11414,12414}, {11822,12482}, {11823,12483}, {11824,12807}, {11825,12808}, {11826,12927}, {11827,12937}, {11829,13001}

X(12556) = reflection of X(i) in X(j) for these (i,j): (4,13089), (10266,3)
X(12556) = anticomplement of X(12600)
X(12556) = X(10266)-of-ABC-X3-reflections-triangle

X(12557) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO 1st SCHIFFLER

Barycentrics a^10+5*(b+c)*a^9+2*(2*b^2+3*b*c+2*c^2)*a^8-4*(b+c)*(3*b^2+b*c+3*c^2)*a^7-(18*b^4+18*c^4+13*b*c*(2*b^2+3*b*c+2*c^2))*a^6+(b+c)*(6*b^4+6*c^4-b*c*(8*b^2+13*b*c+8*c^2))*a^5+(20*b^6+20*c^6+(14*b^4+14*c^4+b*c*(7*b^2+6*b*c+7*c^2))*b*c)*a^4+(b+c)*(4*b^6+4*c^6+(12*b^4+12*c^4+b*c*(9*b^2-14*b*c+9*c^2))*b*c)*a^3-(7*b^6+7*c^6-(24*b^4+24*c^4-b*c*(17*b^2-4*b*c+17*c^2))*b*c)*(b+c)^2*a^2-3*(b^2-c^2)^4*(b+c)*a-4*(b^2-c^2)^4*b*c : :

The reciprocal orthologic center of these triangles is X(21).

X(12557) lies on these lines: {1,5180}, {6597,10435}, {10434,12519}, {10444,12540}, {10446,12543}, {10856,12444}, {10862,12451}, {10882,12524}, {11679,12535}

X(12558) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd EULER TO 4th CONWAY

Barycentrics (b-c)^2*a^5-(b+c)*(b^2+b*c+c^2)*a^4-(2*b^2-3*b*c+2*c^2)*(b+c)^2*a^3+(b^2-c^2)*(b-c)*(2*b^2+3*b*c+2*c^2)*a^2+(b^2-c^2)^2*(b^2+3*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :
X(12558) = 3*X(1699)+X(12514) = 7*X(3832)+X(4294) = 5*X(8227)-X(12520)

The reciprocal orthologic center of these triangles is X(1).

X(12558) lies on these lines: {1,10883}, {2,12511}, {4,3822}, {5,516}, {10,7957}, {11,3671}, {12,4314}, {35,6894}, {40,6990}, {165,6991}, {226,1898}, {758,946}, {1699,5705}, {3814,5537}, {3817,3825}, {3925,5493}, {4294,7951}, {4295,5704}, {4421,11496}, {5885,6001}, {8227,12520}, {10395,12432}, {11680,12526}

X(12558) = midpoint of X(4) and X(5248)
X(12558) = reflection of X(3841) in X(5)
X(12558) = complement of X(12511)
X(12558) = X(578)-of-3rd-Euler-triangle
X(12558) = excentral-to-3rd-Euler similarity image of X(12514)
X(12558) = {X(3817), X(6831)}-harmonic conjugate of X(3825)

X(12559) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS TO 4th CONWAY

Trilinears a^3-3*(b+c)*a^2-(b+c)^2*a+(b+c)*(3*b^2-4*b*c+3*c^2) : :
X(12559) = 5*X(1)-3*X(4512) = 3*X(1)-2*X(5248) = 3*X(1)-X(12526) = 3*X(3241)-X(4294) = 3*X(3679)-4*X(3841) = 9*X(4512)-10*X(5248) = 6*X(4512)-5*X(12514) = 9*X(4512)-5*X(12526) = 4*X(5248)-3*X(12514)

The reciprocal orthologic center of these triangles is X(1).

X(12559) lies on these lines: {1,21}, {9,4067}, {10,3487}, {40,4084}, {55,4018}, {65,3689}, {72,3715}, {78,5902}, {145,4295}, {200,3754}, {214,3361}, {354,5730}, {377,11551}, {388,519}, {405,3962}, {516,944}, {517,12520}, {936,5883}, {942,997}, {1125,11518}, {1159,5836}, {1482,6001}, {1698,3984}, {1706,3919}, {2093,4757}, {2099,3555}, {3158,4744}, {3218,3612}, {3241,4294}, {3336,4855}, {3338,4511}, {3419,3649}, {3485,10916}, {3635,4314}, {3679,3841}, {3711,4002}, {3812,3940}, {3928,5267}, {3951,5251}, {4301,7971}, {4305,9965}, {4333,11015}, {4430,4861}, {4652,4880}, {4917,5541}, {4930,7373}, {4973,7987}, {5045,5289}, {5221,5440}, {5425,5904}, {5791,11281}, {5794,6147}, {5905,10572}, {6668,11374}, {7991,12511}, {9851,11224}, {11519,12446}, {11521,12544}, {11522,12558}

X(12559) = X(578)-of-excenters-reflections-triangle
X(12559) = excentral-to-excenters-reflections similarity image of X(12514)
X(12559) = midpoint of X(145) and X(4295)
X(12559) = reflection of X(i) in X(j) for these (i,j): (4314,3635), (5794,6147), (7991,12511), (12514,1), (12526,5248)
X(12559) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3901,63), (1,12526,5248), (3243,7982,3244), (4757,8715,2093), (5248,12526,12514), (11523,11529,10)

X(12560) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO 4th CONWAY

Trilinears (a-b+c)*(a+b-c)*(a^3-3*(b+c)*a^2+3*(b+c)^2*a-(b+c)*(b^2-6*b*c+c^2)) : :
X(12560) = 4*X(1001)-3*X(4512)

The reciprocal orthologic center of these triangles is X(1).

X(12560) lies on these lines: {1,7}, {9,65}, {10,8232}, {40,954}, {57,1001}, {85,3886}, {142,3485}, {200,226}, {388,5853}, {480,1706}, {518,3340}, {528,4654}, {673,2258}, {758,5223}, {942,3358}, {948,3755}, {1125,8732}, {1159,5779}, {1445,3339}, {1449,1456}, {1768,10980}, {1788,6666}, {2099,3243}, {3059,11523}, {3062,10394}, {3333,11496}, {3361,5248}, {3475,10388}, {3487,6769}, {3601,11495}, {3826,5219}, {3841,7679}, {3883,6604}, {4882,5261}, {5045,7171}, {5226,8580}, {5228,7290}, {5290,6765}, {5572,10384}, {5728,6001}, {5809,6738}, {7091,10390}, {7673,9819}, {7676,12511}, {7678,12558}, {10860,11018}, {10865,12446}, {11520,12529}, {11526,12559}

X(12560) = reflection of X(i) in X(j) for these (i,j): (7,3671), (2951,12520), (4326,1), (12526,9)
X(12560) = X(578)-of-Honsberger-triangle
X(12560) = excentral-to-Honsberger similarity image of X(12514)
X(12560) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7,4321), (1,2951,7675), (1,4312,5732), (1,7271,1458), (1,7274,4327), (7,4323,11038), (7,8236,3600), (2099,8581,3243), (4318,7190,1), (7672,8545,5223), (10384,11518,5572), (11372,11529,5728)

X(12561) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-HUTSON TO 4th CONWAY

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=2*(b+c)*(a+b-c)*(3*a+b+c)*(a-b+c)*b*c
G(a,b,c)=-2*(a+b-c)*(-a+b+c)*(a^3+(b+c)*a^2+(b+c)*(2*b-c)*a+(b^2-c^2)*c)*c
H(a,b,c)=a^6-2*(b+c)*a^5-(b^2+6*b*c+c^2)*a^4+4*(b^2-c^2)*(b-c)*a^3-(b^2-6*b*c+c^2)*(b+c)^2*a^2-2*(b^2-c^2)*(b-c)^3*a+(b^2-c^2)^2*(b+c)^2

The reciprocal orthologic center of these triangles is X(1).

X(12561) lies on these lines: {1,11886}, {10,5934}, {363,12514}, {516,9836}, {758,9805}, {1125,11854}, {3671,8113}, {3841,8380}, {4295,9783}, {4314,8390}, {5248,8109}, {8107,12511}, {8111,12520}, {8385,12560}, {11527,12559}, {11685,12526}, {11856,12446}, {11892,12544}

X(12561) = X(578)-of-inner-Hutson-triangle
X(12561) = excentral-to-inner-Hutson similarity image of X(12514)

X(12562) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-HUTSON TO 4th CONWAY

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)-H(a,b,c) : :
where
F(a,b,c)=2*(b+c)*(a+b-c)*(3*a+b+c)*(a-b+c)*b*c
G(a,b,c)=-2*(a+b-c)*(-a+b+c)*(a^3+(b+c)*a^2+(b+c)*(2*b-c)*a+(b^2-c^2)*c)*c
H(a,b,c)=a^6-2*(b+c)*a^5-(b^2+6*b*c+c^2)*a^4+4*(b^2-c^2)*(b-c)*a^3-(b^2-6*b*c+c^2)*(b+c)^2*a^2-2*(b^2-c^2)*(b-c)^3*a+(b^2-c^2)^2*(b+c)^2

The reciprocal orthologic center of these triangles is X(1).

X(12562) lies on these lines: {1,11887}, {10,5935}, {516,9837}, {758,9806}, {1125,11855}, {3671,8114}, {3841,8381}, {4295,9787}, {4314,8392}, {5248,8110}, {8108,12511}, {8112,12520}, {8140,12561}, {8378,12558}, {8386,12560}, {11528,12559}, {11686,12526}, {11857,12446}, {11893,12544}

X(12562) = X(578)-of-outer-Hutson-triangle
X(12562) = excentral-to-outer-Hutson similarity image of X(12514)

X(12563) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO 4th CONWAY

Barycentrics 2*a^4-5*(b+c)*a^3-3*(b+c)^2*a^2+5*(b^2-c^2)*(b-c)*a+(b^2-c^2)^2 : :
X(12563) = 5*X(1)-X(4294) = 3*X(1)+X(4295) = 3*X(1)-X(4314) = 3*X(551)-X(12514) = X(3340)+3*X(3475) = X(3486)+3*X(4654) = 5*X(3671)+X(4294) = 3*X(3671)-X(4295) = 3*X(3671)+X(4314) = 3*X(4294)+5*X(4295)

The reciprocal orthologic center of these triangles is X(1).

X(12563) lies on these lines: {1,7}, {10,3487}, {142,12447}, {226,1837}, {495,3626}, {496,12558}, {515,6147}, {519,5794}, {551,3333}, {553,2646}, {758,942}, {938,3817}, {946,5787}, {950,3649}, {958,5850}, {999,3636}, {1056,3244}, {1159,11362}, {1210,10171}, {3295,12511}, {3339,5703}, {3340,3475}, {3485,11019}, {3486,4654}, {3616,10980}, {3622,4512}, {3625,11041}, {3634,11374}, {3982,7354}, {4031,5204}, {4847,11520}, {5045,6001}, {5249,6737}, {5572,9856}, {5708,10165}, {5719,6684}, {5789,5886}, {5880,12437}, {5883,6700}, {6598,11263}, {7373,11496}, {7991,10578}, {10569,10866}, {10580,11522}, {11035,12446}, {11039,12561}, {11040,12562}

X(12563) = midpoint of X(i) and X(j) for these {i,j}: {1,3671}, {10,12559}, {4295,4314}, {4301,12520}
X(12563) = reflection of X(i) in X(j) for these (i,j): (3626,3841), (5248,3636)
X(12563) = X(578)-of-incircle-circles-triangle
X(12563) = excentral-to-incircle-circles similarity image of X(12514)
X(12563) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7,4297), (1,4295,4314), (1,4312,4313), (1,4355,5731), (1,11036,5542), (1,11551,4292), (3339,5703,10164), (3485,11518,11019), (3487,11529,10), (3671,4314,4295), (4323,11038,1), (5745,11281,1125)

X(12564) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE TO 4th CONWAY

Trilinears (b+c)*a^5-(b-c)^2*a^4-(b+c)*(2*b^2+b*c+2*c^2)*a^3+(2*b^2-7*b*c+2*c^2)*(b+c)^2*a^2+(b^2-c^2)*(b-c)*(b^2+3*b*c+c^2)*a+(b^2-c^2)*(b-c)*(-b^3-c^3) : :
X(12564) = 3*X(354)-X(3671) = 3*X(3833)-2*X(3841) = X(3868)+3*X(4512) = 3*X(3873)+X(12526) = 3*X(3892)-X(12559) = X(4294)+3*X(5902)

The reciprocal orthologic center of these triangles is X(1).

X(12564) lies on these lines: {1,21}, {10,3059}, {55,12432}, {57,12511}, {65,4314}, {226,1898}, {354,3671}, {516,942}, {938,5883}, {1125,11018}, {1210,3833}, {1864,3947}, {3085,4015}, {3333,12520}, {3339,4326}, {3754,6738}, {4208,5696}, {4294,5902}, {4295,10580}, {4298,10391}, {4355,11220}, {5045,6001}, {5290,10394}, {5703,10176}, {5842,12433}, {5884,11496}, {5904,10578}, {8255,8728}, {9949,10569}, {11019,12446}, {11021,12544}, {11025,12560}, {11026,12561}, {11027,12562}

X(12564) = midpoint of X(i) and X(j) for these {i,j}: {65,4314}, {3874,12514}, {5884,11496}
X(12564) = reflection of X(12563) in X(5045)
X(12564) = X(578)-of-inverse-in-incircle-triangle
X(12564) = excentral-to-inverse-in-incircle similarity image of X(12514)
X(12564) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,774,3743), (942,5572,6744)

X(12565) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR TO 4th CONWAY

Trilinears a^6-2*(b+c)*a^5-(b^2+6*b*c+c^2)*a^4+4*(b^2-c^2)*(b-c)*a^3-(b^2-6*b*c+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(12565) = 4*X(3)-3*X(4512) = 3*X(165)-4*X(12511) = 3*X(165)-2*X(12514) = 3*X(3576)-2*X(11496) = 2*X(4314)-3*X(5731) = 4*X(5248)-5*X(7987)

The reciprocal orthologic center of these triangles is X(1).

Let A' be the trilinear product of the circumcircle intercepts of the A-excircle. Define B' and C' cyclically. Triangle A'B'C' is perspective to the excentral triangle at X(12565). (Randy Hutson, July 31 2018)

X(12565) lies on these lines: {1,7}, {2,9800}, {3,4512}, {9,5584}, {10,1750}, {40,64}, {56,5918}, {57,9943}, {63,7992}, {78,9778}, {84,3428}, {165,411}, {221,7070}, {255,2956}, {497,1467}, {515,4853}, {610,3556}, {758,6765}, {946,8726}, {956,10864}, {960,11495}, {997,12512}, {1103,1745}, {1125,10857}, {1245,2999}, {1764,12548}, {2093,12432}, {3062,5234}, {3174,7957}, {3333,10167}, {3555,6766}, {3576,11496}, {3579,5720}, {3587,5887}, {3811,5493}, {4847,9799}, {5223,12528}, {5231,6245}, {5248,7987}, {5691,9623}, {6261,6282}, {6361,6769}, {7171,11249}, {8580,9949}, {10980,12564}, {11531,12559}

X(12565) = midpoint of X(i) and X(j) for these {i,j}: {9961,12529}, {12561,12562}
X(12565) = reflection of X(i) in X(j) for these (i,j): (1,12520), (962,3671), (4294,4297), (4326,5732), (11531,12559), (12514,12511), (12526,40)
X(12565) = complement of X(9800)
X(12565) = X(578)-of-6th-mixtilinear-triangle
X(12565) = excentral-to-6th-mixtilinear similarity image of X(12514)
X(12565) = 2nd-extouch-to-hexyl similarity image of X(40)
X(12565) = 2nd-circumperp-to-excentral similarity image of X(12520)
X(12565) = anticomplement, wrt excentral triangle, of X(12514)
X(12565) = orthologic center of these triangles: excentral to 4th extouch
X(12565) = Ursa-minor-to-excentral similarity image of X(17634)
X(12565) = Ursa-major-to-excentral similarity image of X(17650)
X(12565) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1044,269), (1,2951,20), (1,4295,12560), (40,1490,200), (56,5918,9841), (63,9961,7992), (946,8726,10582), (962,10884,1), (1042,4319,1), (3811,5493,7994), (12511,12514,165)

X(12566) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO 4th CONWAY

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

The reciprocal orthologic center of these triangles is X(1).

X(12566) lies on these lines: {1,10885}, {3,142}, {10,8233}, {758,9808}, {3671,8243}, {3841,8230}, {4295,9789}, {4314,8239}, {6001,12490}, {8228,12558}, {8231,12514}, {8234,12520}, {8237,12560}, {10867,12446}, {10891,12544}, {11030,12564}, {11042,12563}, {11532,12559}, {11687,12526}

X(12566) = X(578)-of-2nd-Pamfilos-Zhou-triangle
X(12566) = excentral-to-2nd-Pamfilos-Zhou similarity image of X(12514)

X(12567) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO 4th CONWAY

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

The reciprocal orthologic center of these triangles is X(1).

X(12567) lies on these lines: {1,21}, {10,4199}, {56,10180}, {740,958}, {1284,3671}, {4068,12513}, {4295,9791}, {4314,8240}, {4647,5251}, {6001,9959}, {8235,12520}, {8238,12560}, {11043,12563}, {11926,12562}

X(12567) = X(578)-of-1st-Sharygin-triangle
X(12567) = excentral-to-1st-Sharygin similarity image of X(12514)
X(12567) = 2nd-Sharygin-to-1st-Sharygin similarity image of X(13245)

X(12568) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL-MIDARC TO 4th CONWAY

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

The reciprocal orthologic center of these triangles is X(1).

X(12568) lies on these lines: {1,11888}, {10,8079}, {516,8091}, {758,8093}, {1125,8733}, {2089,3671}, {3841,8087}, {4295,9793}, {4314,8241}, {5248,8077}, {6001,8099}, {8075,12511}, {8078,12514}, {8081,12520}, {8085,12558}, {8089,12565}, {8133,12561}, {8135,12562}, {8249,12567}, {8387,12560}, {11032,12564}, {11690,12526}, {11894,12544}

X(12568) = X(578)-of-tangential-midarc-triangle
X(12568) = excentral-to-tangential-midarc similarity image of X(12514)
X(12568) = reflection of X(12569) in X(1)

X(12569) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO 4th CONWAY

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

The reciprocal orthologic center of these triangles is X(1).

X(12569) lies on these lines: {1,11888}, {10,8080}, {516,8092}, {758,8094}, {3841,8088}, {4295,9795}, {4314,8242}, {6001,8100}, {8076,12511}, {8082,12520}, {8086,12558}, {8090,12565}, {8138,12562}, {8248,12566}, {8250,12567}, {8388,12560}, {11033,12564}, {11895,12544}

X(12569) = reflection of X(12568) in X(1)
X(12569) = X(578)-of-2nd-tangential-midarc-triangle
X(12569) = excentral-to-2nd-tangential-midarc similarity image of X(12514)

X(12570) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO 4th CONWAY

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

The reciprocal orthologic center of these triangles is X(1).

X(12570) lies on these lines: {1,11890}, {174,3671}, {516,8351}, {758,12445}, {1125,8729}, {3841,8382}, {4295,11891}, {4314,11924}, {5248,7587}, {6001,12491}, {8083,12564}, {8126,12526}, {8423,12565}, {8425,12567}, {11535,12559}, {11860,12446}, {11896,12544}, {11996,12566}

X(12570) = X(578)-of-Yff-central-triangle
X(12570) = excentral-to-Yff-central similarity image of X(12514)

X(12571) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd EULER TO 5th CONWAY

Barycentrics 2*a^4+(b+c)*a^3+(3*b^2-2*b*c+3*c^2)*a^2-(b^2-c^2)*(b-c)*a-5*(b^2-c^2)^2 : :
X(12571) = X(1)+7*X(3832) = 9*X(2)+7*X(10248) = X(3)-3*X(10171) = 5*X(4)+3*X(3576) = X(4)+3*X(3817) = 3*X(4)+X(4297) = 3*X(4)+5*X(8227) = 5*X(1125)-3*X(3576) = X(1125)-3*X(3817) = 3*X(1125)-X(4297) = 7*X(10248)+3*X(12512)

The reciprocal orthologic center of these triangles is X(1).

X(12571) lies on these lines: {1,3832}, {2,10248}, {3,10171}, {4,1125}, {5,516}, {8,3854}, {10,962}, {11,4298}, {20,7988}, {40,3545}, {165,5056}, {226,6744}, {355,519}, {497,3947}, {515,3636}, {517,4015}, {551,5691}, {758,5806}, {908,5178}, {1698,5493}, {3244,11522}, {3626,4301}, {3635,5603}, {3671,9581}, {3678,10157}, {3825,12436}, {3833,9943}, {3874,5927}, {3911,7173}, {4292,7741}, {4312,5704}, {4314,5219}, {4315,5229}, {4342,9578}, {4347,9817}, {4669,11531}, {4701,7982}, {4745,5818}, {5274,5290}, {5425,6738}, {5542,5714}, {5715,5811}, {5722,12563}, {5726,9785}, {5789,5805}, {7951,10624}, {9579,10589}, {9580,10588}, {9589,9780}, {9612,10591}, {9614,10590}, {10895,12053}, {11680,12527}

X(12571) = midpoint of X(i) and X(j) for these {i,j}: {4,1125}, {546,9955}, {3626,4301}, {3754,9856}, {4701,7982}
X(12571) = reflection of X(3634) in X(5)
X(12571) = complement of X(12512)
X(12571) = X(389)-of-3rd-Euler-triangle
X(12571) = 2nd-Conway-to-excentral similarity image of X(12512)
X(12571) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,3817,1125), (4,8227,4297), (5,3579,10172), (962,3091,7989), (962,7989,10), (1698,9812,5493), (1699,3091,10), (1699,7989,962), (3817,4297,8227), (3832,9779,1), (4297,8227,1125), (4301,5587,3626), (5068,9812,1698), (5219,5225,4314), (9612,10591,11019)

X(12572) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH TO 5th CONWAY

Barycentrics 2*a^4+(b+c)*a^3-(b+c)^2*a^2-(b+c)^3*a-(b^2-c^2)^2 : :
X(12572) = X(72)+3*X(11113) = 3*X(442)-X(10123) = 3*X(553)-5*X(5439) = X(950)-3*X(11113) = 5*X(1698)-X(1770) = 5*X(3876)+3*X(11114)

The reciprocal orthologic center of these triangles is X(1).

X(12572) lies on these lines: {1,329}, {2,4292}, {3,3452}, {4,9}, {5,5745}, {7,5129}, {8,3586}, {12,3683}, {20,936}, {21,908}, {30,5044}, {35,1005}, {37,5717}, {44,1834}, {46,8582}, {56,226}, {57,5084}, {63,1210}, {72,519}, {78,4304}, {84,6865}, {142,11108}, {144,938}, {191,1737}, {200,4294}, {201,1877}, {204,7952}, {210,6284}, {355,5837}, {376,5438}, {377,3305}, {381,5325}, {387,1743}, {390,5815}, {392,10106}, {440,3454}, {442,1155}, {443,7308}, {474,5316}, {515,960}, {517,5795}, {522,11247}, {527,942}, {528,4662}, {551,3487}, {553,5439}, {758,6738}, {846,5530}, {946,958}, {956,12053}, {962,9623}, {997,1490}, {1006,5267}, {1058,6762}, {1104,4415}, {1167,1785}, {1260,8715}, {1329,4640}, {1330,3912}, {1479,4847}, {1697,3421}, {1698,1770}, {1699,5234}, {1901,4205}, {2049,5257}, {2321,5814}, {2325,3695}, {2816,3042}, {2886,5302}, {3085,4512}, {3091,5273}, {3219,5046}, {3244,3488}, {3419,3626}, {3436,5250}, {3523,5328}, {3579,3820}, {3601,11111}, {3678,6743}, {3679,5175}, {3686,5295}, {3687,7283}, {3710,5016}, {3717,5015}, {3811,4314}, {3817,5715}, {3868,10399}, {3874,5728}, {3876,11114}, {3883,4385}, {3911,3916}, {3927,5722}, {3929,9581}, {3940,12437}, {3947,10198}, {4186,7085}, {4199,6685}, {4222,5285}, {4293,8583}, {4301,5758}, {4357,4911}, {4387,10371}, {4416,10449}, {4703,5928}, {4863,9670}, {4999,5087}, {5047,5249}, {5051,5294}, {5057,5260}, {5082,9580}, {5119,6736}, {5219,6857}, {5223,5809}, {5231,10591}, {5251,12047}, {5289,5882}, {5290,8232}, {5692,6737}, {5703,11106}, {5709,6893}, {5720,6868}, {5744,6919}, {5762,5806}, {5779,5787}, {5927,10176}, {6245,6827}, {6666,8728}, {6705,6922}, {6832,10171}, {6908,10164}, {6920,11813}, {6992,10884}, {7007,8806}, {7082,10953}, {7580,12512}, {8226,12571}, {8983,9678}, {9841,12246}, {10888,12545}

X(12572) = midpoint of X(i) and X(j) for these {i,j}: {1,12527}, {8,10624}, {72,950}, {6737,10572}
X(12572) = reflection of X(i) in X(j) for these (i,j): (3874,6744), (4292,12436), (4298,1125), (6743,3678)
X(12572) = anticomplement of X(12436)
X(12572) = complement of X(4292)
X(12572) = X(389)-of-2nd-extouch-triangle
X(12572) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4292,12436), (3,3452,6700), (4,9,10), (40,2551,10), (57,5084,9843), (63,2478,1210), (72,11113,950), (78,6872,4304), (226,405,1125), (329,452,1), (390,5815,6765), (1329,4640,6684), (1490,6987,4297), (2551,5698,40), (3091,5273,5705), (3219,5046,6734), (3487,5436,551), (3488,11523,3244), (5812,6913,946), (7308,9579,443)

X(12573) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO 5th CONWAY

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

The reciprocal orthologic center of these triangles is X(1).

X(12573) lies on these lines: {1,7}, {9,388}, {10,1445}, {12,6666}, {56,142}, {57,2550}, {65,5853}, {85,3883}, {226,1001}, {278,1890}, {515,5728}, {518,4032}, {519,7672}, {527,5434}, {528,553}, {673,1416}, {948,7290}, {950,5572}, {999,5805}, {1056,5759}, {1125,7677}, {1471,3008}, {2257,5819}, {3243,3476}, {3244,11526}, {3361,8732}, {3634,7679}, {3755,5228}, {3826,3911}, {3886,6604}, {4067,5850}, {4989,5723}, {5263,9436}, {5269,7365}, {5290,8232}, {5691,5809}, {5716,7273}, {6594,10956}, {6601,7091}, {6744,11025}, {7676,12512}, {7678,12571}, {9579,10384}, {9613,10398}, {10865,12447}

X(12573) = reflection of X(i) in X(j) for these (i,j): (7,4298), (950,5572), (12527,9)
X(12573) = X(389)-of-Honsberger-triangle
X(12573) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7,390,12560), (7,3600,4321), (7,4308,11038), (4327,4331,3663)

X(12574) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-HUTSON TO 5th CONWAY

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

The reciprocal orthologic center of these triangles is X(1).

X(12574) lies on these lines: {1,9783}, {10,363}, {20,8140}, {3244,11527}, {4292,11886}, {4297,8111}, {4298,8113}, {5934,12572}, {6744,11026}, {8107,12512}, {8385,12573}, {11685,12527}, {11856,12447}, {11892,12545}

X(12574) = X(389)-of-inner-Hutson-triangle

X(12575) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON INTOUCH TO 5th CONWAY

Barycentrics (-a+b+c)*(2*a^3+3*(b+c)*a^2+2*(b-c)^2*a+(b^2-c^2)*(b-c)) : :
X(12575) = 5*X(1)-X(1770) = 3*X(1)-X(4292) = X(950)-3*X(3058) = 3*X(1770)-5*X(4292) = 2*X(1770)-5*X(4298) = X(1770)+5*X(10624) = X(3057)+3*X(3058) = 3*X(3057)+X(10950) = 2*X(4292)-3*X(4298) = 3*X(6738)-4*X(12433)

The reciprocal orthologic center of these triangles is X(1).

X(12575) lies on these lines: {1,7}, {8,4082}, {10,497}, {11,3634}, {12,12571}, {30,10105}, {40,1058}, {55,474}, {56,12512}, {57,5493}, {65,6744}, {72,519}, {144,9797}, {145,12527}, {226,3303}, {388,9580}, {392,10866}, {452,4853}, {496,6684}, {498,10171}, {515,9856}, {517,6738}, {551,3601}, {726,11997}, {938,7991}, {946,3295}, {960,5853}, {1000,5881}, {1191,3755}, {1210,5119}, {1385,10386}, {1479,6957}, {1617,12511}, {1698,5274}, {1699,3947}, {1837,3626}, {2098,3635}, {2136,2551}, {2269,3294}, {2478,3895}, {2646,3636}, {3085,3817}, {3086,10164}, {3244,3486}, {3333,6361}, {3339,10580}, {3361,9778}, {3452,3913}, {3485,10389}, {3488,7982}, {3555,5850}, {3621,8275}, {3624,5281}, {3625,5727}, {3718,3883}, {3746,5443}, {3811,10388}, {3813,5745}, {3828,11238}, {3832,5726}, {3871,6745}, {3877,6737}, {3880,5795}, {3881,10391}, {4652,11240}, {4656,5813}, {4847,5250}, {4857,10039}, {4915,12541}, {5048,10543}, {5173,12564}, {5223,6764}, {5225,9578}, {5252,9670}, {5289,12437}, {5290,9812}, {5698,6762}, {5703,11522}, {5704,9588}, {5722,11362}, {5759,6766}, {5919,6284}, {6666,9710}, {6700,8715}, {8162,10404}, {8390,12574}, {9669,10175}, {9799,9949}, {9804,9898}, {9845,12246}, {10165,11373}, {10172,10593}

X(12575) = midpoint of X(i) and X(j) for these {i,j}: {1,10624}, {145,12527}, {950,3057}, {6284,10106}
X(12575) = reflection of X(i) in X(j) for these (i,j): (65,6744), (4298,1), (6743,960)
X(12575) = X(389)-of-Hutson-intouch-triangle

X(12575) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,20,4315), (1,390,4314), (1,962,3671), (1,4294,4297), (1,4295,5542), (1,4302,4311), (1,4309,4304), (1,4312,11037), (1,9589,7), (1,9785,4342), (40,1058,11019), (55,12053,1125), (390,9785,1), (497,1697,10), (2478,3895,6736), (3057,3058,950), (3057,9848,72), (3085,9614,3817), (3486,7962,3244), (4314,4342,1)

X(12576) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-HUTSON TO 5th CONWAY

The reciprocal orthologic center of these triangles is X(1).

X(12576) lies on these lines: {20,8140}, {1125,8110}, {3244,11528}, {4292,11887}, {4297,8112}, {4298,8114}, {5935,12572}, {6744,11027}, {8108,12512}, {8386,12573}, {8392,12575}, {9837,12562}, {11686,12527}, {11855,12436}, {11857,12447}, {11893,12545}

X(12576) = reflection of X(12574) in X(20)
X(12576) = X(389)-of-outer-Hutson-triangle

X(12577) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO 5th CONWAY

Barycentrics 2*a^4+(b+c)*a^3-(b^2-14*b*c+c^2)*a^2-(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(12577) = 7*X(1)+X(1770) = 3*X(1)+X(4292) = 5*X(1)-X(10624) = 3*X(1)-X(12575) = 9*X(354)-X(10950) = 3*X(551)-X(12572) = 3*X(553)+X(3057) = 3*X(1770)-7*X(4292) = X(1770)-7*X(4298) = 5*X(1770)+7*X(10624)

The reciprocal orthologic center of these triangles is X(1).

X(12577) lies on these lines: {1,7}, {4,9845}, {8,10980}, {10,1056}, {65,10569}, {85,10520}, {142,12513}, {226,3304}, {354,6738}, {388,9581}, {495,3634}, {496,12571}, {515,5045}, {518,11035}, {519,942}, {551,3487}, {553,3057}, {946,6259}, {950,5434}, {958,999}, {960,5850}, {1210,10827}, {1385,5763}, {1420,3475}, {3086,3947}, {3189,3244}, {3295,12512}, {3306,6736}, {3361,10164}, {3476,11518}, {3555,6743}, {3616,12527}, {3742,5795}, {3817,5290}, {3873,6737}, {4848,4860}, {4853,9776}, {4915,11024}, {5253,6745}, {5444,5563}, {5691,10580}, {5704,5726}, {5708,11362}, {5728,9850}, {7987,10578}, {10404,12053}, {11039,12574}, {11040,12576}

X(12577) = midpoint of X(i) and X(j) for these {i,j}: {1,4298}, {3555,6743}, {4292,12575}, {6738,10106}
X(12577) = reflection of X(6744) in X(5045)
X(12577) = X(389)-of-incircle-circles-triangle
X(12577) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7,4301), (1,3600,4297), (1,4292,12575), (1,4293,4314), (1,4295,4342), (1,4312,9785), (1,4317,4304), (1,4321,12520), (1,4355,962), (1,5542,12563), (1,11037,5542), (354,10106,6738), (1056,3333,10), (4298,12575,4292), (4308,11038,1)

X(12578) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO 5th CONWAY

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

The reciprocal orthologic center of these triangles is X(1).

X(12578) lies on these lines: {1,9789}, {3,142}, {10,8231}, {20,8244}, {515,12490}, {519,9808}, {3244,11532}, {3634,8230}, {4292,10885}, {4297,8234}, {4298,8243}, {6744,11030}, {8228,12571}, {8233,12572}, {8237,12573}, {8239,12575}, {10867,12447}, {10891,12545}, {11042,12577}, {11687,12527}, {11922,12574}

X(12578) = X(389)-of-2nd-Pamfilos-Zhou-triangle

X(12579) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO 5th CONWAY

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

The reciprocal orthologic center of these triangles is X(1).

X(12579) lies on these lines: {1,6646}, {10,846}, {20,8245}, {21,36}, {405,3821}, {515,9959}, {516,9840}, {519,2292}, {1284,4298}, {2392,3884}, {3244,11533}, {3634,5051}, {3647,8258}, {4085,5302}, {4297,8235}, {4656,8669}, {6685,12572}, {6744,11031}, {8238,12573}, {8240,12575}, {11043,12577}, {11688,12527}

X(12579) = X(389)-of-1st-Sharygin-triangle
X(12579) = 2nd-Sharygin-to-1st-Sharygin similarity image of X(13246)
X(12579) = {X(21), X(4425)}-harmonic conjugate of X(1125)

X(12580) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL-MIDARC TO 5th CONWAY

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

The reciprocal orthologic center of these triangles is X(1).

X(12580) lies on these lines: {1,9793}, {10,8078}, {20,8089}, {515,8099}, {516,8091}, {519,8093}, {950,10503}, {1125,8077}, {2089,4298}, {3244,11534}, {3634,8087}, {4292,11888}, {4297,8081}, {6744,11032}, {10106,10506}

X(12580) = reflection of X(12581) in X(1)
X(12580) = X(389)-of-tangential-midarc-triangle

X(12581) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO 5th CONWAY

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

The reciprocal orthologic center of these triangles is X(1).

X(12581) lies on these lines: {1,9793}, {10,258}, {20,8090}, {174,4298}, {515,8100}, {516,8092}, {519,8094}, {942,5571}, {950,10501}, {1125,7588}, {3244,11899}, {3634,8088}, {4292,11889}, {4297,8082}, {4355,11891}, {5542,7590}, {6744,11033}, {8423,11037}

X(12581) = reflection of X(12580) in X(1)
X(12581) = X(389)-of-2nd-tangential-midarc-triangle

X(12582) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO 5th CONWAY

Barycentrics 8*(a+b+c)*a*b*c*sin(A/2)+(a+b-c)*(a-b+c)*(2*a^2+(b+c)*a+(b+c)^2) : :

The reciprocal orthologic center of these triangles is X(1).

X(12582) lies on these lines: {1,11891}, {20,8423}, {174,4298}, {515,12491}, {519,12130}, {950,10502}, {1125,7587}, {3244,11535}, {3634,8382}, {6744,8083}, {8126,12527}, {8425,12579}, {8729,12436}, {11860,12447}, {11996,12578}

X(12583) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO 1st EHRMANN

Barycentrics (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*((b^2+c^2)*a^8-4*b^2*c^2*a^6-(b^2+c^2)*(2*b^2+b*c-2*c^2)*(2*b^2-b*c-2*c^2)*a^4+4*(b^4-c^4)^2*a^2+(b^4-c^4)*(b^2-c^2)*(-b^4-3*b^2*c^2-c^4)) : :
X(12583) = X(1351)-3*X(11911) = X(3751)-3*X(11852) = 2*X(5480)-3*X(11897)

The reciprocal orthologic center of these triangles is X(3).

X(12583) lies on these lines: {6,402}, {30,599}, {69,4240}, {141,1650}, {159,11853}, {511,11251}, {518,12438}, {524,1651}, {611,11912}, {613,11913}, {1351,11911}, {1386,11831}, {1503,12113}, {1843,11832}, {2781,12369}, {3056,11909}, {3094,11885}, {3242,11910}, {3416,11900}, {3564,12418}, {3751,11852}, {5181,9033}, {5480,11897}, {6776,11845}, {9830,12347}, {11839,12212}, {11848,12329}, {11863,12452}

X(12583) = midpoint of X(69) and X(4240)
X(12583) = reflection of X(i) in X(j) for these (i,j): (6,402), (1650,141)
X(12583) = X(6)-of-Gossard-triangle
X(12583) = {X(11901),X(11902)}-harmonic conjugate of X(402)

X(12584) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st EHRMANN TO 1st HYACINTH

Trilinears ((9*R^2-3*SW)*S^2-(9*R^2+3*SA-4*SW)*SA*SW)*a : :
X(12584) = 5*X(3)-3*X(5621) = 5*X(110)-X(10752) = 5*X(2930)+3*X(5621)

The reciprocal orthologic center of these triangles is X(12585).

X(12584) lies on these lines: {3,67}, {6,11935}, {23,110}, {24,5095}, {54,575}, {74,12074}, {143,576}, {159,2777}, {182,1511}, {399,1350}, {524,7575}, {526,8723}, {597,11694}, {690,11616}, {1177,10282}, {1352,12383}, {1385,2836}, {1995,5476}, {2781,5609}, {2892,9833}, {3043,6403}, {3098,5663}, {5092,11579}, {5480,10272}, {5562,8718}, {5972,11284}, {7464,11645}, {7492,9143}, {7496,9140}, {7556,11061}, {9925,9932}, {10510,11649}

X(12584) = midpoint of X(i) and X(j) for these {i,j}: {3,2930}, {399,1350}, {1352,12383}, {2892,9833}
X(12584) = reflection of X(i) in X(j) for these (i,j): (182,1511), (576,6593), (597,11694), (895,575), (1177,10282), (5476,5642), (5480,10272), (9976,182), (11579,5092)
X(12584) = circumcircle-inverse-of-X(8724)
X(12584) = circummedial-to-1st-Ehrmann similarity image of X(14682)

X(12585) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st HYACINTH TO 1st EHRMANN

Barycentrics (12*R^2*SA-5*SA*SW+SW^2)*S^2+(2*R^2-SW)*(SA-SW)*SA*SW : :

The reciprocal orthologic center of these triangles is X(12584).

X(12585) lies on these lines: {6,5449}, {69,569}, {141,575}, {193,8538}, {389,3564}, {511,12370}, {524,1216}, {542,6102}, {1147,5181}, {2393,10116}

X(12585) = {X(141), X(575)}-harmonic conjugate of X(6689)
X(12585) = X(1156)-of-1st-Hyacinth-triangle if ABC is acute
X(12585) = orthic-to-1st-Hyacinth similarity image of X(5095)

X(12586) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO 1st EHRMANN

Barycentrics a^5-(b+c)*a^4+2*b*c*a^3+(b^2+c^2)*(b-c)^2*a-(-c^4+b^4)*(b-c) : :
X(12586) = (R-r)*X(6)-(R-2*r)*X(11)

The reciprocal orthologic center of these triangles is X(3).

X(12586) lies on these lines: {1,5820}, {4,8679}, {6,11}, {12,12594}, {66,1439}, {69,674}, {141,1376}, {159,10829}, {354,1899}, {355,518}, {375,7392}, {511,10525}, {524,11235}, {611,10523}, {613,10948}, {1350,11826}, {1351,11928}, {1386,11373}, {1503,12114}, {1709,7289}, {1843,11390}, {2781,12371}, {2810,3818}, {3056,10947}, {3094,10871}, {3242,10944}, {3410,4430}, {3416,10914}, {3564,10943}, {3618,10584}, {3751,10826}, {3873,11442}, {5480,10893}, {5810,10916}, {5846,10912}, {5927,9004}, {6776,10785}, {7595,9043}, {9018,10446}, {9830,12348}, {10794,12212}, {10945,12590}, {10946,12591}, {10949,12595}, {11865,12452}, {11866,12453}, {11903,12583}

X(12586) = reflection of X(i) in X(j) for these (i,j): (12329,141), (12587,1352)
X(12586) = X(6)-of-inner-Johnson-triangle
X(12586) = Ursa-minor-to-Ursa-major similarity image of X(6)
X(12586) = {X(10919),X(10920)}-harmonic conjugate of X(11)
X(12586) = {X(12928),X(12929)}-harmonic conjugate of X(10943)

X(12587) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO 1st EHRMANN

Barycentrics a^6-(b^2+c^2)*a^4-2*b*c*(b+c)*a^3+(b^2+c^2)*(b+c)^2*a^2-(-c^4+b^4)*(b^2-c^2) : :
X(12587) = (R+r)*X(6)-(R+2*r)*X(12)

The reciprocal orthologic center of these triangles is X(3).

X(12587) lies on these lines: {4,674}, {6,12}, {10,9028}, {11,12595}, {66,72}, {69,313}, {141,958}, {159,10830}, {210,1899}, {355,518}, {375,11433}, {498,5135}, {511,10526}, {524,11236}, {611,10954}, {613,10523}, {1350,11827}, {1351,11929}, {1386,11374}, {1478,4259}, {1503,11500}, {1843,11391}, {2321,2385}, {2781,12372}, {3056,10953}, {3094,10872}, {3242,10950}, {3410,4661}, {3618,10585}, {3681,11442}, {3751,5820}, {3818,9052}, {3844,5791}, {5220,5845}, {5480,10894}, {5810,5847}, {6776,10786}, {9830,12349}, {10795,12212}, {10951,12590}, {10952,12591}, {10955,12594}, {11867,12452}, {11868,12453}, {11904,12583}

X(12587) = reflection of X(12586) in X(1352)
X(12587) = X(6)-of-outer-Johnson-triangle
X(12587) = {X(10921),X(10922)}-harmonic conjugate of X(12)
X(12587) = {X(12938),X(12939)}-harmonic conjugate of X(10942)

X(12588) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO 1st EHRMANN

Barycentrics (a^4+(b^2+c^2)*(b+c)^2)*(a-b+c)*(a+b-c) : :
X(12588) = r*X(6)+R*X(12)

The reciprocal orthologic center of these triangles is X(3).

X(12588) lies on these lines: {1,1352}, {2,1428}, {4,3056}, {5,613}, {6,12}, {7,8}, {11,10516}, {55,1503}, {56,141}, {66,73}, {67,3028}, {159,10831}, {182,498}, {193,5261}, {226,4362}, {495,611}, {511,1478}, {524,11237}, {542,10053}, {599,5434}, {612,1899}, {1330,1431}, {1350,7354}, {1351,9654}, {1386,11375}, {1460,11358}, {1479,3818}, {1843,11392}, {2099,5846}, {2330,3085}, {2781,12373}, {3027,11646}, {3094,9597}, {3098,4299}, {3242,10944}, {3600,3620}, {3618,10588}, {3619,7288}, {3745,5712}, {3751,9578}, {3763,5433}, {3961,5018}, {4260,9552}, {4293,10519}, {5052,9650}, {5085,5432}, {5480,10895}, {5848,10956}, {6284,10387}, {8540,10590}, {9830,12350}, {10072,11178}, {10797,12212}, {10957,12595}, {11501,12329}, {11869,12452}, {11870,12453}, {11905,12583}, {11930,12590}, {11931,12591}

X(12588) = reflection of X(611) in X(495)
X(12588) = X(6)-of-1st-Johnson-Yff-triangle
X(12588) = outer-Johnson-to-ABC similarity image of X(6)
X(12588) = {X(10923),X(10924)}-harmonic conjugate of X(12)
X(12588) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1352,12589), (69,388,1469), (3085,6776,2330), (12941,12942,10056)

X(12589) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO 1st EHRMANN

Barycentrics (-a+b+c)*(a^4+(b^2+c^2)*(b-c)^2) : :
X(12589) = r*X(6)+(R-2*r)*X(11)

The reciprocal orthologic center of these triangles is X(3).

X(12589) lies on these lines: {1,1352}, {2,2330}, {4,1469}, {5,611}, {6,11}, {7,4459}, {12,10516}, {55,141}, {56,1503}, {69,350}, {159,10832}, {182,499}, {193,5274}, {354,5738}, {390,3620}, {496,613}, {511,1479}, {518,1837}, {524,11238}, {542,10069}, {599,3058}, {614,1899}, {1350,6284}, {1351,9669}, {1386,5820}, {1428,3086}, {1478,3818}, {1843,11393}, {2098,5846}, {2781,12374}, {2892,10118}, {3023,11646}, {3057,3416}, {3094,9598}, {3098,4302}, {3242,10950}, {3486,5484}, {3582,11179}, {3618,10589}, {3619,5218}, {3751,9581}, {3763,5432}, {4260,9555}, {4294,10519}, {5052,9665}, {5085,5433}, {5480,10896}, {5596,10535}, {5716,10372}, {5847,12053}, {5849,10959}, {7191,11442}, {7194,7281}, {9830,12351}, {10056,11178}, {10798,12212}, {10958,12594}, {11502,12329}, {11871,12452}, {11872,12453}, {11906,12583}, {11932,12590}, {11933,12591}

X(12589) = reflection of X(613) in X(496)
X(12589) = X(6)-of-2nd-Johnson-Yff-triangle
X(12589) = inner-Johnson-to-ABC similarity image of X(6)
X(12589) = {X(10925),X(10926)}-harmonic conjugate of X(11)
X(12589) = Ursa-major-to-Ursa-minor similarity image of X(6)
X(12589) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1352,12588), (69,497,3056), (3086,6776,1428), (12951,12952,10072)

X(12590) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO 1st EHRMANN

Trilinears (2*SW*SA^2+2*S*SW*SA-(8*R^2*S-4*S*SW-SW^2)*S)*a : :

The reciprocal orthologic center of these triangles is X(3).

X(12590) lies on these lines: {6,493}, {69,6462}, {141,8222}, {159,8194}, {511,10669}, {518,12440}, {524,12152}, {611,11951}, {613,11953}, {1350,11828}, {1351,11949}, {1352,8220}, {1386,11377}, {1503,9838}, {1843,11394}, {2781,12377}, {3056,11947}, {3094,10875}, {3242,8210}, {3416,8214}, {3564,12426}, {3751,8188}, {5013,6461}, {5480,8212}, {6776,11846}, {8201,12452}, {8208,12453}, {9830,12352}, {10945,12586}, {10951,12587}, {11503,12329}, {11840,12212}, {11907,12583}, {11930,12588}, {11932,12589}, {11955,12594}, {11957,12595}

X(12590) = X(6)-of-Lucas-homothetic-triangle

X(12591) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO 1st EHRMANN

Trilinears (2*SW*SA^2-2*S*SW*SA+(-8*R^2*S+4*S*SW-SW^2)*S)*a : :

The reciprocal orthologic center of these triangles is X(3).

X(12591) lies on these lines: {6,494}, {69,6463}, {141,8223}, {159,8195}, {511,10673}, {518,12441}, {524,12153}, {611,11952}, {613,11954}, {1350,11829}, {1351,11950}, {1352,8221}, {1386,11378}, {1503,9839}, {1843,11395}, {2781,12378}, {3056,11948}, {3094,10876}, {3242,8211}, {3416,8215}, {3564,12427}, {3751,8189}, {5013,6461}, {5480,8213}, {6776,11847}, {8202,12452}, {8209,12453}, {9830,12353}, {10946,12586}, {10952,12587}, {11504,12329}, {11841,12212}, {11908,12583}, {11931,12588}, {11933,12589}, {11956,12594}, {11958,12595}

X(12591) = X(6)-of-Lucas(-1)-homothetic-triangle

X(12592) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st EHRMANN TO MANDART-EXCIRCLES

Trilinears 16*p^10+32*(q^2-2)*p^8+8*q*p^7+8*(q^2-1)*(2*q^2-9)*p^6+2*(4*q^2-5)*q*p^5-2*(12*q^4-33*q^2+5)*p^4-2*(7*q^2+3)*q*p^3+(6*q^4-4*q^2-15)*p^2+2*(q^2+6)*q*p-(2*q^2+1)*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(3779).

X(12592) lies on these lines: {}

X(12593) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st EHRMANN TO 2nd ORTHOSYMMEDIAL

Trilinears (a^16-(8*b^4+9*b^2*c^2+8*c^4)*a^12+(b^2+c^2)*(6*b^4+7*b^2*c^2+6*c^4)*a^10+(12*b^8+12*c^8-b^2*c^2*(11*b^4+8*b^2*c^2+11*c^4))*a^8-4*(b^2+c^2)*(3*b^8+3*c^8-b^2*c^2*(4*b^4-3*b^2*c^2+4*c^4))*a^6-(b^2-c^2)^2*(4*b^4+5*b^2*c^2+4*c^4)*(b^4-4*b^2*c^2+c^4)*a^4+(b^4-c^4)*(b^2-c^2)*(b^2-2*c^2)*(2*b^2-c^2)*(3*b^4+2*b^2*c^2+3*c^4)*a^2+(b^4-c^4)^2*(b^2+c^2)^2*(-b^4+3*b^2*c^2-c^4))*a : :

The reciprocal orthologic center of these triangles is X(6).

X(12593) lies on the line {576,2781}

X(12594) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO 1st EHRMANN

Trilinears (a^4-2*(b^2+b*c+c^2)*a^2+2*b*c*(b+c)*a+b^4+c^4-4*b^3*c-2*b^2*c^2-4*b*c^3)*a : :
X(12594) = 2*R*X(1)-(R-r)*X(6)

The reciprocal orthologic center of these triangles is X(3).

X(12594) lies on these lines: {1,6}, {12,12586}, {55,8679}, {69,10528}, {119,10516}, {141,5552}, {159,10834}, {221,5252}, {495,5820}, {511,10679}, {524,11239}, {1350,11248}, {1351,12000}, {1352,10942}, {1469,11509}, {1470,5096}, {1503,12115}, {1843,11400}, {2097,3359}, {2781,12381}, {3056,10965}, {3094,10878}, {3416,10915}, {3564,12430}, {3618,10586}, {5085,10269}, {5480,10531}, {5848,10956}, {6776,10805}, {9830,12356}, {10803,12212}, {10955,12587}, {10958,12589}, {11881,12452}, {11882,12453}, {11914,12583}, {11955,12590}, {11956,12591}

X(12594) = reflection of X(i) in X(j) for these (i,j): (6,611), (5820,495)
X(12594) = X(6)-of-inner-Yff-tangents-triangle
X(12594) = outer-Yff-to-inner-Yff similarity image of X(6)
X(12594) = {X(10929),X(10930)}-harmonic conjugate of X(1)
X(12594) = {X(6), X(3242)}-harmonic conjugate of X(12595)

X(12595) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO 1st EHRMANN

Trilinears (a^4-2*(b^2-b*c+c^2)*a^2-2*b*c*(b+c)*a+b^4+c^4+4*b^3*c-2*b^2*c^2+4*b*c^3)*a : :
X(12595) = 2*R*X(1)-(R+r)*X(6)

The reciprocal orthologic center of these triangles is X(3).

X(12595) lies on these lines: {1,6}, {11,12587}, {56,674}, {69,10529}, {141,10527}, {159,10835}, {511,10680}, {524,11240}, {999,4259}, {1350,11249}, {1351,12001}, {1352,10943}, {1428,11510}, {1503,12116}, {1843,11401}, {3056,10966}, {3094,10879}, {3295,5135}, {3416,10916}, {3564,12431}, {3618,10587}, {4265,10387}, {5085,10267}, {5480,10532}, {5849,10959}, {6776,10806}, {9028,12053}, {9830,12357}, {10804,12212}, {10949,12586}, {10957,12588}, {11883,12452}, {11915,12583}, {11957,12590}, {11958,12591}

X(12595) = reflection of X(6) in X(613)
X(12595) = X(6)-of-outer-Yff-tangents-triangle
X(12595) = inner-Yff-to-outer-Yff similarity image of X(6)
X(12595) = {X(10931),X(10932)}-harmonic conjugate of X(1)
X(12595) = {X(6), X(3242)}-harmonic conjugate of X(12594)

X(12596) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EHRMANN TO 1st HYACINTH

Trilinears (-SW*(-2*SW^2-6*SW*SA+7*SA^2)+27*S^2*R^2+18*R^4*SW+(-12*SW^2-24*SW*SA+27*SA^2)*R^2-7*S^2*SW)*SA*a : :
X(12596) = 5*X(11482)-X(12310)

The reciprocal orthologic center of these triangles is X(10112).

X(12596) lies on these lines: {6,1511}, {74,11416}, {110,8537}, {113,8541}, {125,8538}, {265,895}, {1351,1986}, {1539,9970}, {1992,12319}, {5663,8549}, {6699,11511}, {11405,12168}, {11443,12273}, {11458,12284}, {11470,12295}, {11477,12302}, {11482,12310}

X(12596) = midpoint of X(11477) and X(12302)

X(12597) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EHRMANN TO LUCAS ANTIPODAL

Trilinears ((8*R^2-3*SW)*SW*S^2-2*S*(SA*SW*(2*R^2-SA+SW)+(12*R^2-4*SW)*S^2)+SA*SW^3)*a : :

The reciprocal orthologic center of these triangles is X(3).

X(12597) lies on these lines: {6,12229}, {486,11511}, {487,8541}, {642,9813}, {1992,12320}, {3564,12598}, {8537,12509}, {8538,12601}, {11405,12169}, {11416,12221}, {11443,12274}, {11458,12285}, {11470,12296}, {11477,12303}, {11482,12311}

X(12597) = orthic-to-2nd-Ehrmann similarity image of X(487)

X(12598) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EHRMANN TO LUCAS(-1) ANTIPODAL

Trilinears ((8*R^2-3*SW)*SW*S^2+2*S*(SA*SW*(2*R^2-SA+SW)+(12*R^2-4*SW)*S^2)+SA*SW^3)*a : :

The reciprocal orthologic center of these triangles is X(3).

X(12598) lies on these lines: {6,12230}, {485,11511}, {488,8541}, {641,9813}, {1992,12321}, {3564,12597}, {8537,12510}, {8538,12602}, {11405,12170}, {11416,12222}, {11443,12275}, {11458,12286}, {11470,12297}, {11477,12304}, {11482,12312}

X(12598) = orthic-to-2nd-Ehrmann similarity image of X(488)

X(12599) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EULER TO HUTSON EXTOUCH

Trilinears q*p^5-(3*q^2-2)*p^4+3*(q^2-1)*q*p^3-(q^4-6*q^2+3)*p^2-(5*q^2-1)*q*p+2-2*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(12599) = 3*X(4)+X(12249) = 5*X(3091)-X(9874) = 3*X(7160)-X(12249)

The reciprocal orthologic center of these triangles is X(40).

X(12599) lies on these lines: {2,12120}, {4,1697}, {10,5805}, {98,12200}, {235,12139}, {515,12260}, {946,10157}, {1478,10075}, {1479,10059}, {1598,12411}, {1699,9898}, {3091,9874}, {3851,12620}, {4866,7682}, {5290,7992}, {5534,12521}, {5603,8000}, {6245,7680}, {6841,12612}, {8196,12464}, {8203,12465}, {9993,12500}, {11496,12333}

X(12599) = midpoint of X(4) and X(7160)
X(12599) = complement of X(12120)
X(12599) = X(7160)-of-Euler-triangle

X(12600) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EULER TO 1st SCHIFFLER

Trilinears (4*sin(3*A/2)-2*sin(5*A/2))*cos((B-C)/2)+5*cos(B-C)+(-2*sin(A/2)+2*sin(3*A/2))*cos(3*(B-C)/2)+(-2*cos(A)+2)*cos(2*(B-C))-6*cos(A)+2*cos(2*A)-cos(3*A)+1 : :
X(12600) = 3*X(4)+X(12255) = 3*X(1699)+X(12409) = 3*X(10266)-X(12255)

The reciprocal orthologic center of these triangles is X(79).

X(12600) lies on these lines: {4,5885}, {11,79}, {98,12209}, {235,12146}, {515,12267}, {1598,12414}, {1699,12409}, {6265,6599}, {6841,12615}, {8196,12482}, {8203,12483}, {9993,12504}, {11496,12342}

X(12600) = midpoint of X(4) and X(10266)
X(12600) = X(10266)-of-Euler-triangle

X(12601) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EULER TO LUCAS ANTIPODAL

Barycentrics (SB+SC)*(S^2-SA^2+SB*SC)-4*S*SB*SC : :
X(12601) = 3*X(3)-2*X(12123) = 3*X(381)-4*X(6251) = 3*X(381)-2*X(6290) = 3*X(486)-X(12123) = 4*X(642)-5*X(1656) = 3*X(3830)+2*X(6280) = 2*X(7980)-3*X(10247) = 3*X(10246)-4*X(12268)

The reciprocal orthologic center of these triangles is X(3).

X(12601) lies on these lines: {2,12509}, {3,486}, {4,193}, {5,487}, {30,12256}, {52,12237}, {355,7596}, {381,1991}, {494,8036}, {517,9906}, {569,12229}, {642,1656}, {999,10083}, {1587,11482}, {1588,5050}, {1598,12147}, {3070,5093}, {3295,10067}, {3526,6119}, {3830,6280}, {3843,6281}, {5139,8946}, {5446,6291}, {6565,9732}, {6643,12320}, {7395,12169}, {7517,9921}, {7980,10247}, {8538,12597}, {9301,9986}, {10246,12268}, {11444,12274}, {11459,12285}, {11842,12210}, {11849,12343}, {11875,12484}, {11876,12485}

X(12601) = midpoint of X(i) and X(j) for these {i,j}: {4,12221}, {12256,12296}
X(12601) = reflection of X(i) in X(j) for these (i,j): (3,486), (52,12237), (487,5), (6290,6251)
X(12601) = complement of X(12509)
X(12601) = orthic-to-2nd-Euler similarity image of X(487)
X(12601) = {X(4),X(1351)}-harmonic conjugate of X(12602)

X(12602) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EULER TO LUCAS(-1) ANTIPODAL

Barycentrics (SB+SC)*(S^2-SA^2+SB*SC)+4*S*SB*SC : :
X(12602) = 3*X(3)-2*X(12124) = 3*X(485)-X(12124) = 4*X(641)-5*X(1656) = 3*X(3830)+2*X(6279) = 5*X(3843)-2*X(6278) = 2*X(7981)-3*X(10247) = 3*X(10246)-4*X(12269)

The reciprocal orthologic center of these triangles is X(3).

X(12602) lies on these lines: {2,12510}, {3,485}, {4,193}, {5,488}, {30,12257}, {52,12238}, {493,8035}, {517,9907}, {569,12230}, {641,1656}, {999,10084}, {1587,5050}, {1588,11482}, {1598,12148}, {3071,5093}, {3295,10068}, {3526,6118}, {3830,6279}, {3843,6278}, {5139,8948}, {5200,8780}, {5446,6406}, {6564,9733}, {6643,12321}, {7395,12170}, {7517,9922}, {7981,10247}, {8538,12598}, {8982,10846}, {9301,9987}, {10246,12269}, {11444,12275}, {11459,12286}, {11842,12211}, {11849,12344}, {11875,12486}, {11876,12487}

X(12602) = midpoint of X(i) and X(j) for these {i,j}: {4,12222}, {12257,12297}
X(12602) = reflection of X(i) in X(j) for these (i,j): (3,485), (52,12238), (488,5), (6289,6250)
X(12602) = complement of X(12510)
X(12602) = orthic-to-2nd-Euler similarity image of X(488)
X(12602) = {X(4),X(1351)}-harmonic conjugate of X(12601)

X(12603) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EULER TO LUCAS CENTRAL

Trilinears SA*(2*R^2*(SW-2*S)+SB*SC-SW^2)*a : :
X(12603) = 5*X(11444)-X(12276) = 3*X(11459)+X(12287)

The reciprocal orthologic center of these triangles is X(3).

X(12603) lies on these lines: {2,6239}, {3,6}, {4,12223}, {5,6291}, {30,12298}, {487,1216}, {1060,7362}, {1062,6283}, {1656,9823}, {6252,8251}, {6413,10670}, {6643,12322}, {7395,12171}, {11444,12276}, {11459,12287}

X(12603) = midpoint of X(4) and X(12223)
X(12603) = reflection of X(i) in X(j) for these (i,j): (3,12360), (52,12239), (6291,5)
X(12603) = complement of X(6239)
X(12603) = X(176)-of-2nd-Euler-triangle if ABC is acute
X(12603) = orthic-to-2nd-Euler similarity image of X(6291)
X(12603) = {X(3),X(9967)}-harmonic conjugate of X(12604)

X(12604) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EULER TO LUCAS(-1) CENTRAL

Trilinears SA*(2*R^2*(SW+2*S)+SB*SC-SW^2)*a : :
X(12604) = 5*X(11444)-X(12277) = 3*X(11459)+X(12288)

The reciprocal orthologic center of these triangles is X(3).

X(12604) lies on these lines: {2,6400}, {3,6}, {4,12224}, {5,6406}, {30,12299}, {51,8964}, {488,1216}, {1060,7353}, {1062,6405}, {1656,9824}, {6404,8251}, {6414,10674}, {6643,12323}, {7395,12172}, {11444,12277}, {11459,12288}

X(12604) = midpoint of X(4) and X(12224)
X(12604) = reflection of X(i) in X(j) for these (i,j): (3,12361), (52,12240), (6406,5)
X(12604) = complement of X(6400)
X(12604) = X(175)-of-2nd-Euler-triangle if ABC is acute
X(12604) = orthic-to-2nd-Euler similarity image of X(6406)
X(12604) = {X(3),X(9967)}-harmonic conjugate of X(12603)

X(12605) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EULER TO MACBEATH

Barycentrics (-a^2+b^2+c^2)*(2*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^2-c^2)^4) : :
X(12605) = 5*X(3)-6*X(10691) = 3*X(428)-4*X(546) = 2*X(550)-3*X(7667) = X(5889)-3*X(12022) = 2*X(6102)-3*X(11245) = 5*X(11444)-X(12278) = 3*X(11459)+X(12289)

The reciprocal orthologic center of these triangles is X(4).

X(12605) lies on these lines: {2,3}, {52,12241}, {68,4549}, {131,10600}, {216,7747}, {339,7767}, {343,9927}, {394,12118}, {569,12233}, {577,7748}, {973,5446}, {1038,10483}, {1060,7354}, {1062,6284}, {1154,12370}, {1176,3521}, {1216,12358}, {1503,9967}, {1568,9820}, {1899,12163}, {3070,10897}, {3071,10898}, {3284,7765}, {5254,10316}, {5305,10317}, {5318,10634}, {5321,10635}, {5596,12315}, {5889,12022}, {5907,12134}, {6102,11245}, {6146,10116}, {6253,8251}, {7723,12606}, {8538,8550}, {11064,12038}, {11444,12278}, {11459,12289}

X(12605) = midpoint of X(i) and X(j) for these {i,j}: {4,12225}, {11750,12162}
X(12605) = reflection of X(i) in X(j) for these (i,j): (3,12362), (52,12241), (3575,5), (7553,4), (11819,546), (12134,5907)
X(12605) = complement of X(6240)
X(12605) = anticomplement of X(31833)
X(12605) = X(65)-of-2nd-Euler-triangle if ABC is acute
X(12605) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,7547,5), (3,5,7542), (3,381,3549), (3,2072,140), (3,10024,6676), (3,11585,10257), (4,20,7387), (4,3529,7500), (4,5133,546), (4,7404,381), (4,7503,5), (4,7566,3845), (5,550,1658), (5,1658,468), (381,3534,10245), (381,9714,3089), (546,6676,10024), (546,11819,428), (1556,6656,546), (3091,7569,5), (7542,10297,5)

X(12606) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EULER TO REFLECTION

Trilinears a*SA*(S^2+2*R^2*(7*R^2-2*SA-5*SW)+SA^2-2*SB*SC+2*SW^2) : :
X(12606) = 3*X(54)-X(5889) = 3*X(381)-2*X(11576) = 3*X(1209)-4*X(11793) = 5*X(11444)-X(12280) = 3*X(11459)+X(12291)

The reciprocal orthologic center of these triangles is X(6243).

X(12606) lies on these lines: {2,6242}, {3,54}, {4,12226}, {5,6152}, {30,12300}, {52,12242}, {68,3519}, {125,1216}, {381,11576}, {539,5562}, {569,12234}, {973,6639}, {1060,7356}, {1062,6286}, {1209,2072}, {1352,6288}, {1656,9827}, {2914,7512}, {3574,5446}, {4549,9936}, {5876,12289}, {5965,9967}, {6255,8251}, {6643,12325}, {7395,12175}, {7542,8254}, {7723,12605}, {8538,9977}, {10634,10677}, {10635,10678}, {11444,12280}, {11459,12291}

X(12606) = midpoint of X(4) and X(12226)
X(12606) = reflection of X(i) in X(j) for these (i,j): (3,12363), (52,12242), (6152,5)
X(12606) = X(79)-of-2nd-Euler-triangle if ABC is acute
X(12606) = complement of X(6242)

X(12607) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd EULER TO EXCENTERS-MIDPOINTS

Barycentrics (b^2+4*b*c+c^2)*a^2-2*b*c*(b+c)*a-(b^2-c^2)^2 : :
X(12607) = 4*X(5)-3*X(3829) = X(20)-3*X(4421) = 5*X(631)-3*X(11194) = 3*X(1699)+X(2136) = 5*X(3091)-3*X(11235) = 3*X(3158)+X(5691) = 3*X(3679)+X(11523) = X(3680)-5*X(11522) = 2*X(3813)-3*X(3829) = 3*X(3928)-7*X(9588)

The reciprocal orthologic center of these triangles is X(1).

X(12607) lies on these lines: {1,1329}, {2,3304}, {3,529}, {4,528}, {5,519}, {8,12}, {10,141}, {11,145}, {20,4421}, {30,8715}, {55,3436}, {56,3035}, {65,6735}, {72,10039}, {78,5252}, {100,7354}, {119,1482}, {120,6552}, {140,8666}, {200,5794}, {226,5836}, {341,3932}, {355,3811}, {377,11237}, {388,1376}, {404,5434}, {405,10056}, {442,3679}, {452,4428}, {496,3244}, {498,956}, {517,10915}, {535,550}, {631,11194}, {758,5499}, {908,3057}, {938,5828}, {946,3880}, {958,3085}, {976,5724}, {999,6691}, {1001,2551}, {1125,3820}, {1210,5123}, {1215,5835}, {1259,11501}, {1478,5687}, {1532,7982}, {1698,6762}, {1699,2136}, {1706,5290}, {1737,3555}, {1837,3870}, {1904,3175}, {2098,10958}, {2478,3303}, {2550,5261}, {2802,11698}, {2829,11248}, {2975,5432}, {3036,10573}, {3058,5046}, {3086,6667}, {3091,11235}, {3158,5691}, {3241,4193}, {3419,10827}, {3428,10786}, {3434,10895}, {3584,5258}, {3614,3621}, {3617,3925}, {3625,10592}, {3626,3822}, {3632,7951}, {3633,7741}, {3635,3825}, {3671,10107}, {3680,11522}, {3698,5249}, {3703,4696}, {3704,4385}, {3742,8582}, {3746,11113}, {3754,6147}, {3782,4642}, {3838,3947}, {3841,4691}, {3871,5080}, {3928,9588}, {3935,5086}, {3991,5179}, {4004,11551}, {4030,5016}, {4188,6174}, {4189,4995}, {4190,9657}, {4423,10587}, {4511,10944}, {4640,12527}, {4853,5219}, {4882,5726}, {4930,6980}, {5082,8168}, {5087,12053}, {5176,10950}, {5187,11238}, {5220,5815}, {5270,11112}, {5587,6765}, {5603,10912}, {5657,5852}, {5718,10459}, {5734,6945}, {5842,10526}, {5881,6831}, {5882,6922}, {6067,7679}, {6256,10306}, {6675,10197}, {6692,12577}, {6745,10106}, {6764,7958}, {6869,11500}, {6907,11362}, {6931,11240}, {7373,10200}, {7988,11519}, {8668,11496}, {8727,12437}, {9565,10408}, {9708,10198}, {9712,10037}, {9713,10831}, {9779,12541}, {9947,12617}, {9956,10916}, {10310,12115}, {10863,12448}, {10883,12536}, {10886,12546}, {10914,12047}, {11491,11827}

X(12607) = midpoint of X(i) and X(j) for these {i,j}: {4,3913}, {355,3811}, {6256,10306}
X(12607) = reflection of X(i) in X(j) for these (i,j): (3813,5), (8666,140), (10916,9956), (11260,1125)
X(12607) = complement of X(12513)
X(12607) = X(64)-of-3rd-Euler-triangle
X(12607) = excentral-to-3rd-Euler similarity image of X(2136)
X(12607) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1329,3816), (5,3813,3829), (8,12,2886), (56,5552,3035), (65,6735,8256), (119,1482,7681), (145,11681,11), (200,9578,5794), (226,6736,5836), (388,7080,1376), (442,3679,9710), (498,956,4999), (958,3085,6690), (1706,5290,5880), (2478,11239,3303), (3085,3421,958), (3244,3814,496), (3436,10528,55), (3584,5258,7483), (3913,11236,4)

X(12608) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd EULER TO EXTOUCH

Barycentrics (b+c)*a^6-(b+c)*(3*b^2-4*b*c+3*c^2)*a^4+2*b*c*(b-c)^2*a^3+3*(b^4-c^4)*(b-c)*a^2-2*(b^2-c^2)^2*b*c*a-(b^2-c^2)^3*(b-c) : :
X(12608) = X(84)-5*X(8227) = X(1490)+3*X(1699) = 3*X(3817)-X(6245) = 3*X(5587)+X(7971) = 3*X(5886)+X(6259) = 3*X(5886)-X(12114) = 9*X(7988)-X(7992)

The reciprocal orthologic center of these triangles is X(72).

Let Na be the nine-point center of BCI, and define Nb and Nc cyclically. Triangle NaNbNc is perspective to the 3rd Euler triangle at X(12608). (Randy Hutson, July 21, 2017)

X(12608) lies on these lines: {1,4}, {2,1158}, {5,3812}, {10,119}, {21,10165}, {40,908}, {46,6834}, {65,1532}, {84,5249}, {90,499}, {142,3358}, {153,4861}, {411,2077}, {516,6796}, {517,10915}, {912,10916}, {920,3911}, {942,1538}, {960,6907}, {962,10528}, {971,9955}, {997,6850}, {1012,11375}, {1125,3560}, {1210,1858}, {1385,2829}, {1470,4292}, {1537,3057}, {1709,6833}, {1737,6941}, {1770,6905}, {1788,6969}, {1836,3149}, {2096,7288}, {2360,3559}, {2476,7705}, {2886,5777}, {2950,5316}, {3359,3452}, {3474,6927}, {3576,6872}, {3612,6938}, {3657,6003}, {3671,7682}, {3816,9940}, {3817,6245}, {3869,6735}, {4295,6848}, {4297,7491}, {5086,12531}, {5087,6922}, {5119,10786}, {5261,10935}, {5440,11826}, {5554,5587}, {5693,6734}, {5698,6988}, {5722,10893}, {5768,10591}, {5880,6918}, {5886,6259}, {5905,10530}, {6147,7956}, {6247,6708}, {6827,12520}, {6828,9948}, {6856,10172}, {6867,9842}, {6943,9961}, {6968,10826}, {7680,9856}, {7988,7992}, {8085,8095}, {8086,8096}, {8727,9942}, {9779,9799}, {9960,10883}, {10085,10785}, {10679,11500}, {10724,11015}, {10886,12547}, {11019,12005}, {11372,11919}, {11374,11496}, {11680,12528}

X(12608) = midpoint of X(i) and X(j) for these {i,j}: {1,6256}, {4,6261}, {946,6260}, {6259,12114}
X(12608) = reflection of X(i) in X(j) for these (i,j): (5450,1125), (10915,10942), (12616,5)
X(12608) = complement of X(1158)
X(12608) = X(68)-of-3rd-Euler-triangle
X(12608) = excentral-to-3rd-Euler similarity image of X(1490)
X(12608) = 3rd-Euler-isotomic conjugate of X(12610)
X(12608) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1519,946), (1,1699,10531), (4,12047,946), (942,1538,7681), (946,5882,12053), (5087,9943,6922), (5603,10805,1), (5886,6259,12114), (5887,6842,10), (6825,12514,6684), (6838,11415,40)

X(12609) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd EULER TO 4th EXTOUCH

Barycentrics (a^3+(b+c)*a^2-(b^2-4*b*c+c^2)*a-(b^2-c^2)*(b-c))*(b+c) : :
X(12609) = 3*X(2)+X(4295) = 5*X(1698)-X(12526) = 3*X(1699)+X(12565) = 7*X(3624)-3*X(4512) = X(3671)+2*X(3841)

The reciprocal orthologic center of these triangles is X(65).

Let (Oa), (Ob), (Oc) be the Odehnal tritangent circles. Let La be the polar of A wrt (Oa), and define Lb, Lc cyclically. La is also the line through the touchpoints of (Oa) and CA and AB, and cyclically for Lb and Lc. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. Triangle A'B'C' is homothetic to the extraversion triangle of X(10) at X(12609). (Randy Hutson, July 21, 2017)

X(12609) lies on these lines: {1,224}, {2,46}, {3,142}, {4,12520}, {5,3812}, {8,12559}, {10,12}, {11,5439}, {21,1770}, {40,6889}, {79,5251}, {191,11552}, {306,4647}, {386,1738}, {405,1836}, {443,997}, {474,11375}, {495,5836}, {496,3742}, {499,3306}, {515,6917}, {517,3824}, {518,6147}, {519,5794}, {551,2646}, {908,1698}, {936,12560}, {942,2886}, {956,10404}, {960,8728}, {993,4292}, {1004,10624}, {1089,4054}, {1155,6681}, {1158,6824}, {1159,3626}, {1210,5883}, {1213,4047}, {1376,11374}, {1385,5842}, {1454,3911}, {1519,5437}, {1699,6836}, {1709,6837}, {1737,2476}, {1788,6856}, {1858,10395}, {2245,5257}, {2475,10572}, {2550,3487}, {2551,5714}, {3011,5264}, {3086,9776}, {3159,4078}, {3333,6173}, {3338,10044}, {3339,5705}, {3452,3634}, {3474,6857}, {3475,5082}, {3556,7535}, {3576,6934}, {3579,6690}, {3612,3616}, {3624,4512}, {3772,5711}, {3813,5045}, {3814,8582}, {3816,9955}, {3817,3825}, {3826,5044}, {3827,9895}, {3868,11551}, {3869,4197}, {3874,4847}, {3881,5542}, {3884,4301}, {3916,11246}, {4298,8666}, {4324,5426}, {4425,12567}, {4640,6675}, {5047,5057}, {5086,6175}, {5123,10592}, {5226,11024}, {5290,9623}, {5302,11544}, {5554,10827}, {5587,6984}, {5603,6897}, {5690,10107}, {5887,6881}, {5902,6734}, {6261,6826}, {6667,12611}, {6668,11231}, {6691,11230}, {6692,6862}, {6860,7988}, {6871,10826}, {6887,8257}, {6907,7686}, {6955,9624}, {8727,9943}, {9614,10582}, {9779,9800}, {9949,10863}, {9961,10883}, {10478,12544}, {10886,12548}, {11019,12446}, {11680,12529}

X(12609) = midpoint of X(i) and X(j) for these {i,j}: {4,12520}, {8,12559}, {10,3671}, {4295,12514}, {12446,12564}
X(12609) = reflection of X(i) in X(j) for these (i,j): (10,3841), (5248,1125), (12617,5)
X(12609) = complement of X(12514)
X(12609) = excentral-to-3rd-Euler similarity image of X(12565)
X(12609) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4295,12514), (10,3919,4848), (10,4138,3454), (10,11263,226), (12,3753,10), (65,442,10), (72,3925,10), (142,946,1125), (443,3485,997), (495,5836,10915), (942,2886,10916), (2550,3487,3811), (3616,4190,3612), (3649,3925,72), (3754,3822,10), (3754,6701,3822), (3812,3838,5), (3817,9843,3825), (3825,3833,9843), (5437,8227,10200)

X(12610) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd EULER TO 5th EXTOUCH

Barycentrics (b+c)*a^4-2*b*c*(b-c)^2*a-(b^4-c^4)*(b-c) : :
X(12610) = 9*X(7988)-X(7996) = 9*X(9779)-X(9801)

The reciprocal orthologic center of these triangles is X(65).

X(12610) lies on these lines: {2,1766}, {3,142}, {4,990}, {5,3739}, {8,11532}, {10,8230}, {57,1848}, {75,7377}, {116,2823}, {141,517}, {226,1465}, {355,4361}, {497,10383}, {515,3946}, {573,4357}, {908,3729}, {942,5799}, {952,4852}, {971,5480}, {1418,1565}, {1482,4851}, {1699,1721}, {1826,4858}, {1890,3220}, {2050,3772}, {2345,7402}, {3662,10446}, {3817,8228}, {4104,10440}, {4353,11042}, {4384,5816}, {4425,8246}, {4648,5603}, {5249,10444}, {5393,7133}, {5405,7595}, {6003,10099}, {6245,7683}, {6707,11230}, {7988,7996}, {8239,12053}, {8727,9944}, {9779,9801}, {9950,10863}, {9962,10883}, {10867,11019}, {10886,12549}, {11680,12530}

X(12610) = midpoint of X(i) and X(j) for these {i,j}: {4,990}, {3663,10445}
X(12610) = reflection of X(12618) in X(5)
X(12610) = complement of X(1766)
X(12610) = X(317)-of-3rd-Euler-triangle
X(12610) = excentral-to-3rd-Euler similarity image of X(1721)
X(12610) = 3rd-Euler-isotomic conjugate of X(12608)

X(12611) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd EULER TO INNER-GARCIA

Barycentrics ((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c))*(a^4-(2*b^2-3*b*c+2*c^2)*a^2+(b^2-c^2)^2) : :
X(12611) = 3*X(4)+X(6224) = 3*X(5)-2*X(6702) = X(80)-3*X(381) = X(104)-3*X(5886) = 3*X(119)-X(1145) = X(908)+3*X(1519) = X(1145)+3*X(1537) = X(6224)-3*X(6265) = 4*X(6702)-3*X(12619) = 2*X(6713)-3*X(11230)

The reciprocal orthologic center of these triangles is X(3869).

X(12611) lies on these lines: {1,10742}, {2,12515}, {4,6224}, {5,2800}, {11,113}, {30,214}, {80,381}, {104,5886}, {119,517}, {142,6713}, {153,5603}, {226,1387}, {355,10698}, {382,12119}, {496,5083}, {546,946}, {1320,3656}, {1385,2829}, {1484,2801}, {1699,6326}, {1768,8227}, {1836,10090}, {2802,11698}, {3035,3579}, {3091,12247}, {3616,12248}, {3817,10265}, {4996,5057}, {5316,11231}, {5660,12331}, {5840,9945}, {5854,11278}, {5901,11715}, {6264,11522}, {6667,12609}, {6911,12332}, {7704,12528}, {8727,9946}, {9779,9803}, {9818,9912}, {9952,10863}, {9957,10956}, {9964,10883}, {10057,10895}, {10058,11375}, {10073,10896}, {10074,11376}, {10284,10942}, {10886,12551}, {11680,12532}

X(12611) = midpoint of X(i) and X(j) for these {i,j}: {1,10742}, {4,6265}, {119,1537}, {355,10698}, {382,12119}, {3656,10711}, {6326,10738}
X(12611) = reflection of X(i) in X(j) for these (i,j): (11,9955), (1385,11729), (3579,3035), (11715,5901), (12619,5)
X(12611) = complement of X(12515)
X(12611) = X(265)-of-3rd-Euler-triangle
X(12611) = X(12121)-of-4th-Euler-triangle
X(12611) = excentral-to-3rd-Euler similarity image of X(6326)
X(12611) = {X(1699), X(6326)}-harmonic conjugate of X(10738)

X(12612) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd EULER TO HUTSON EXTOUCH

Trilinears 2*p^4*q*(p-q)-(2*q^2+7)*q*p^3+(2*q^4+3)*p^2+(7*q^2+3)*q*p-4+2*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(12612) = 9*X(7988)-X(8001)

The reciprocal orthologic center of these triangles is X(3555).

X(12612) lies on these lines: {2,12516}, {4,12521}, {5,4662}, {12,5920}, {142,5709}, {226,9589}, {946,6765}, {6838,7160}, {6841,12599}, {7988,8001}, {8727,12439}, {9779,9804}, {9953,10863}, {10883,12537}, {10886,12552}, {11680,12533}

X(12612) = midpoint of X(4) and X(12521)
X(12612) = reflection of X(12620) in X(5)
X(12612) = complement of X(12516)

X(12613) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd EULER TO MANDART-EXCIRCLES

Trilinears 4*(2*q^2-1)*p^6-8*(q^2-1)*q*p^5+2*(4*q^4-10*q^2+1)*p^4-4*(2*q^4-7*q^2+4)*q*p^3-2*(2*q^2+1)*(2*q^2-3)*p^2+(4*q^4-6*q^2-3)*q*p+q^2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(12613) = 3*X(2)-X(12517)

The reciprocal orthologic center of these triangles is X(3555).

X(12613) lies on these lines: {2,12517}, {4,12522}, {5,12621}, {3825,6684}, {8727,12442}, {9779,12542}, {10863,12449}, {10883,12538}, {10886,12553}, {11680,12534}

X(12613) = midpoint of X(4) and X(12522)
X(12613) = reflection of X(12621) in X(5)
X(12613) = complement of X(12517)

X(12614) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd EULER TO MIDARC

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

The reciprocal orthologic center of these triangles is X(1).

X(12614) lies on these lines: {1,8085}, {5,12622}, {11,177}, {12,8422}, {164,1699}, {167,7988}, {226,5571}, {3679,8381}, {7670,7678}, {9779,9807}, {11680,11691}

X(12614) = midpoint of X(4) and X(12523)
X(12614) = reflection of X(12622) in X(5)
X(12614) = complement of X(12518)
X(12614) = X(1)-of-3rd-Euler-triangle

X(12615) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd EULER TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(21).

X(12615) lies on these lines: {2,12519}, {4,12524}, {5,12623}, {6841,12600}, {6949,12342}, {8727,12444}, {9779,12543}, {10863,12451}, {10883,12540}, {10886,12557}, {11680,12535}

X(12615) = midpoint of X(4) and X(12524)
X(12615) = reflection of X(12623) in X(5)
X(12615) = complement of X(12519)

X(12616) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th EULER TO EXTOUCH

Barycentrics (b+c)*a^6-2*(b^2+c^2)*a^5-(b+c)*(b^2-4*b*c+c^2)*a^4+2*(2*b^2+3*b*c+2*c^2)*(b-c)^2*a^3-(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)*a^2-2*(b^2-c^2)*(b-c)*(b^3+c^3)*a+(b^2-c^2)^3*(b-c) : :
X(12616) = X(84)+3*X(5587) = X(1490)-5*X(1698) = 3*X(5587)-X(6256) = X(6260)-3*X(10175) = X(7971)-5*X(8227) = 7*X(7989)+X(7992)

The reciprocal orthologic center of these triangles is X(72).

Let (Oa), (Ob), (Oc) be the Odehnal tritangent circles. Let La be the polar of A wrt (Oa), and define Lb and Lc cyclically. La is also the line through the touchpoints of (Oa) and CA and AB, and cyclically for Lb, Lc. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. Let Ma be the polar of I wrt (Oa), and define Mb, Mc cyclically. Let A" = Mb∩Mc, B" = Mc∩Ma, C" = Ma∩Mb. Triangles A'B'C' and A"B"C" are homothetic at X(12616). (Randy Hutson, July 21, 2017)

Let A'B'C' be the excentral triangle. X(12616) is the radical center of the 1st Droz-Farny circles of triangles A'BC, B'CA, C'AB. (Randy Hutson, June 27, 2018)

X(12616) lies on these lines: {1,6833}, {2,6261}, {3,10}, {4,46}, {5,3812}, {8,6890}, {11,65}, {40,3434}, {63,10522}, {84,377}, {104,10057}, {224,5552}, {225,1735}, {226,5884}, {442,5927}, {516,10525}, {517,3813}, {581,5530}, {908,5693}, {942,7680}, {944,3612}, {950,11507}, {952,10915}, {960,6922}, {971,3826}, {997,6891}, {1012,1837}, {1072,3670}, {1125,6862}, {1329,5777}, {1385,6690}, {1454,4292}, {1490,1698}, {1512,4316}, {1519,7741}, {1699,10598}, {1715,1869}, {1765,1826}, {1768,3585}, {1771,3215}, {1777,1877}, {1898,10958}, {2096,5229}, {2245,10445}, {2646,5882}, {2829,12619}, {3057,10949}, {3085,5768}, {3338,10532}, {3339,5715}, {3419,10310}, {3485,6956}, {3486,6935}, {3576,6910}, {3579,5842}, {3869,6943}, {3916,11827}, {4197,9960}, {4295,6844}, {4511,6972}, {4847,10914}, {5086,6909}, {5119,12116}, {5563,11219}, {5657,6899}, {5722,11496}, {5761,12559}, {5818,6897}, {5881,6735}, {5887,6882}, {5905,10524}, {6825,12520}, {6827,12514}, {6830,12047}, {6860,7971}, {6906,10572}, {6907,9943}, {6913,12330}, {6932,9961}, {6984,7989}, {7483,10165}, {7681,9856}, {7682,10893}, {7686,8727}, {8087,8095}, {8088,8096}, {9780,9799}, {10044,10599}, {10085,10827}, {10624,10947}, {10887,12547}, {10948,12053}, {11019,11373}, {11681,12528}

X(12616) = midpoint of X(i) and X(j) for these {i,j}: {4,1158}, {10,6245}, {84,6256}, {355,12114}, {5787,11500}, {6260,9948}
X(12616) = reflection of X(i) in X(j) for these (i,j): (5450,6705), (6796,6684), (12608,5)
X(12616) = complement of X(6261)
X(12616) = X(68)-of-4th-Euler-triangle
X(12616) = excentral-to-4th-Euler similarity image of X(1490)
X(12616) = 4th-Euler-isotomic conjugate of X(12618)
X(12616) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (65,6831,946), (84,5587,6256), (944,6977,3612), (946,10265,1210), (1709,10826,4), (9948,10175,6260), (10085,10827,12115)

X(12617) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th EULER TO 4th EXTOUCH

Barycentrics (b+c)*a^6-2*(b^2+c^2)*a^5-(b+c)*(b^2-4*b*c+c^2)*a^4+2*(2*b^2-3*b*c+2*c^2)*(b+c)^2*a^3-(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)*a^2-2*(b+c)*(b^2-c^2)*(b^3-c^3)*a+(b^2-c^2)^3*(b-c) : :
X(12617) = 5*X(1698)-X(12565) = 3*X(1699)+X(12526) = X(3671)-3*X(3817) = 2*X(3841)-3*X(10175) = 3*X(4512)+X(5691)

The reciprocal orthologic center of these triangles is X(65).

X(12617) lies on these lines: {1,6837}, {2,12520}, {4,9}, {5,3812}, {12,1898}, {21,4297}, {46,6835}, {65,8226}, {90,1478}, {118,5517}, {142,9948}, {226,1858}, {355,3913}, {377,1709}, {411,10164}, {515,3560}, {758,946}, {920,4292}, {960,8727}, {997,6847}, {1001,5787}, {1125,6245}, {1158,6826}, {1210,3671}, {1329,10157}, {1490,10198}, {1698,6838}, {1699,6734}, {1737,3091}, {1770,6839}, {2476,8582}, {2801,12564}, {2886,9856}, {3485,11019}, {3486,10389}, {3612,6974}, {3634,6825}, {3746,4314}, {3822,6260}, {3841,6842}, {3869,4301}, {4197,9961}, {4294,10039}, {4512,5691}, {5086,6736}, {5439,7958}, {5603,12559}, {5777,7680}, {6678,12262}, {6684,6985}, {6855,9843}, {6866,7682}, {6869,12512}, {6871,7989}, {6957,10826}, {8728,9943}, {9780,9800}, {9947,12607}, {10394,10865}, {10479,12544}, {10887,12548}, {11681,12529}

X(12617) = midpoint of X(i) and X(j) for these {i,j}: {4,12514}, {355,11496}
X(12617) = reflection of X(i) in X(j) for these (i,j): (946,12558), (12511,6684), (12609,5)
X(12617) = complement of X(12520)
X(12617) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5887,6841,946), (6261,6824,1125), (6828,12047,3817), (6870,11415,1699)

X(12618) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th EULER TO 5th EXTOUCH

Barycentrics (b+c)*a^5-(b-c)^2*a^4-2*b*c*(b+c)^2*a^2-(b^2-c^2)^2*(b+c)*a+(b^4-c^4)*(b^2-c^2) : :
X(12618) = 5*X(1698)-X(1721) = 7*X(7989)+X(7996)

ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th EULER TO 5th EXTOUCH

X(12618) lies on these lines: {1,5807}, {2,990}, {4,9}, {5,3739}, {118,123}, {141,971}, {307,1210}, {321,4712}, {517,5480}, {726,10916}, {962,5772}, {991,3912}, {1041,4347}, {1211,5927}, {1698,1721}, {1754,5294}, {2298,4349}, {3332,5749}, {3454,6260}, {3677,4353}, {3729,6734}, {4197,9962}, {4220,10164}, {4363,5805}, {4643,5779}, {5016,6736}, {5051,8582}, {5101,7085}, {5743,10157}, {7989,7996}, {8728,9944}, {9780,9801}, {10444,10479}, {10887,12549}, {11681,12530}

X(12618) = midpoint of X(4) and X(1766)
X(12618) = reflection of X(12610) in X(5)
X(12618) = complement of X(990)
X(12618) = X(317)-of-4th-Euler-triangle
X(12618) = excentral-to-4th-Euler similarity image of X(1721)
X(12618) = 4th-Euler-isotomic conjugate of X(12616)
X(12618) = {X(9), X(1861)}-harmonic conjugate of X(10)

X(12619) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th EULER TO INNER-GARCIA

Barycentrics (b+c)*a^6-2*(b^2+b*c+c^2)*a^5-(b+c)*(b^2-5*b*c+c^2)*a^4+2*(2*b^4+2*c^4-b*c*(b^2+3*b*c+c^2))*a^3-(b^2-c^2)*(b-c)*(b^2+5*b*c+c^2)*a^2-2*(b^2-c^2)^2*(b-c)^2*a+(b^2-c^2)^3*(b-c) : :
X(12619) = 3*X(2)+X(12247) = 3*X(3)-X(12119) = 3*X(80)+X(12119) = X(149)+3*X(5657) = X(153)-5*X(5818) = 5*X(631)-X(6224) = 5*X(1698)-X(6326) = X(1768)+3*X(5587) = 2*X(3035)-3*X(11231) = 3*X(5587)-X(10742) = 4*X(6702)-X(12611)

The reciprocal orthologic center of these triangles is X(3869).

X(12619) lies on these lines: {2,6265}, {3,80}, {4,12515}, {5,2800}, {10,140}, {11,517}, {12,5885}, {24,12137}, {40,10738}, {55,10073}, {56,10057}, {65,8068}, {100,1006}, {104,355}, {119,125}, {149,5657}, {153,5818}, {495,5083}, {496,10284}, {631,6224}, {912,5123}, {1145,6734}, {1210,1387}, {1317,10039}, {1329,5694}, {1484,2802}, {1537,9955}, {1698,6326}, {1768,5587}, {1788,10526}, {1837,10058}, {2080,12198}, {2801,3826}, {2829,12616}, {3057,5533}, {3560,12332}, {3576,9897}, {3579,5840}, {3653,10031}, {3654,10707}, {3679,6264}, {4197,9964}, {4413,5790}, {5221,11929}, {5252,10074}, {5428,6684}, {5444,7972}, {5499,12623}, {5854,10916}, {5886,10698}, {6642,9912}, {6667,11230}, {6958,10573}, {7583,8988}, {7951,11571}, {8256,10943}, {8582,9952}, {8728,9946}, {9780,9803}, {10267,12331}, {10887,12551}, {11681,12532}

X(12619) = midpoint of X(i) and X(j) for these {i,j}: {3,80}, {4,12515}, {10,10265}, {40,10738}, {104,355}, {1484,5690}, {1768,10742}, {3654,10707}, {5790,11219}, {6265,12247}
X(12619) = reflection of X(i) in X(j) for these (i,j): (5,6702), (119,9956), (214,140), (1385,6713), (1537,9955), (11570,5885), (11729,6667), (12611,5)
X(12619) = complement of X(6265)
X(12619) = K798i-isogonal conjugate of X(3)
X(12619) = X(265)-of-4th-Euler-triangle
X(12619) = X(12121)-of-3rd-Euler-triangle
X(12619) = excentral-to-4th-Euler similarity image of X(6326)
X(12619) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,12247,6265), (1768,5587,10742), (6667,11729,11230)

X(12620) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th EULER TO HUTSON EXTOUCH

Barycentrics (b+c)*a^6-2*(b^2+c^2)*a^5-(b+c)*(b^2+12*b*c+c^2)*a^4+2*(2*b^4+2*c^4+b*c*(7*b^2-2*b*c+7*c^2))*a^3-(b+c)*(b^4+c^4-2*b*c*(7*b^2+11*b*c+7*c^2))*a^2-2*(b^2-c^2)^2*(b^2+7*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :
X(12620) = 7*X(7989)+X(8001)

The reciprocal orthologic center of these triangles is X(3555).

X(12620) lies on these lines: {2,12521}, {4,12516}, {5,4662}, {10,6767}, {11,3983}, {442,3555}, {497,10395}, {3826,10916}, {3851,12599}, {4197,12537}, {5187,9874}, {5220,5812}, {7989,8001}, {8582,9953}, {8728,12439}, {9780,9804}, {10887,12552}, {11681,12533}

X(12620) = midpoint of X(4) and X(12516)
X(12620) = reflection of X(12612) in X(5)
X(12620) = complement of X(12521)

X(12621) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th EULER TO MANDART-EXCIRCLES

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

The reciprocal orthologic center of these triangles is X(3555).

X(12621) lies on these lines: {2,12522}, {4,12517}, {5,12613}, {4197,12538}, {5521,5687}, {8582,12449}, {8728,12442}, {9780,12542}, {10887,12553}, {11681,12534}

X(12621) = midpoint of X(4) and X(12517)
X(12621) = reflection of X(12613) in X(5)
X(12621) = complement of X(12522)

X(12622) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th EULER TO MIDARC

Barycentrics -(b+c)^2*sin(A/2)+(-a*c+b^2-c^2)*sin(B/2)+(-a*b-b^2+c^2)*sin(C/2) : :
X(12622) = X(164)-5*X(1698) = X(167)+7*X(7989)

The reciprocal orthologic center of these triangles is X(1).

X(12622) lies on these lines: {1,8087}, {2,12523}, {4,12518}, {5,12614}, {11,8422}, {12,177}, {164,1698}, {167,7989}, {1210,5571}, {7670,7679}, {9780,9807}, {11681,11691}

X(12622) = midpoint of X(4) and X(12518)
X(12622) = orthologic center of these triangles: 4th Euler to 2nd midarc
X(12622) = reflection of X(12614) in X(5)
X(12622) = X(1)-of-4th-Euler-triangle
X(12622) = {X(8087), X(8088)}-harmonic conjugate of X(1)

X(12623) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th EULER TO 1st SCHIFFLER

Barycentrics (-a+b+c)*((b^2+4*b*c+c^2)*a^7-(b^2-c^2)*(b-c)*a^6-(3*b^4+3*c^4+2*b*c*(4*b^2-b*c+4*c^2))*a^5+(b+c)*(3*b^4+3*c^4-2*b*c*(4*b^2-b*c+4*c^2))*a^4+(3*b^6+3*c^6+2*(2*b^4+2*c^4-b*c*(3*b^2+2*b*c+3*c^2))*b*c)*a^3-(b^2-c^2)*(b-c)*(3*b^4+3*c^4-b*c*(4*b^2+7*b*c+4*c^2))*a^2-(b^2-c^2)^4*a+(b^2-c^2)^3*(b-c)^3) : :
X(12623) = (R^2-4*r^2)*X(11)-(6*R*r+R^2)*X(21)

The reciprocal orthologic center of these triangles is X(21).

X(12623) lies on these lines: {2,12524}, {4,12519}, {5,12615}, {10,12267}, {11,21}, {442,1749}, {4197,12540}, {5046,12342}, {5499,12619}, {6599,7161}, {8582,12451}, {8728,12444}, {9780,12543}, {10887,12557}, {11681,12535}

X(12623) = midpoint of X(4) and X(12519)
X(12623) = reflection of X(12615) in X(5)
X(12623) = complement of X(12524)

X(12624) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th EULER TO 2nd ORTHOSYMMEDIAL

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

The reciprocal orthologic center of these triangles is X(12508).

X(12624) lies on the nine-points circle and the line {2,12507}

X(12624) = complement of X(12507)

Orthologic centers: X(12625)-X(12808)

Centers X(12625)-X(12808) were contributed by César Eliud Lozada, March, 26, 2017.

X(12625) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH TO EXCENTERS-MIDPOINTS

Barycentrics (-a+b+c)*(3*a^3-(b+c)^2*a+2*(b^2-c^2)*(b-c)) : :
X(12625) = 3*X(1)-4*X(3813) = 3*X(8)-X(12632) = 3*X(8)-2*X(12640) = 4*X(10)-3*X(3158) = 2*X(20)-3*X(3928) = 2*X(442)-3*X(6598) = 3*X(2136)-2*X(12632) = 3*X(2136)-4*X(12640) = 3*X(3158)-2*X(3189) = 3*X(3679)-2*X(3913)

The reciprocal orthologic center of these triangles is X(1).

X(12625) lies on these lines: {1,442}, {2,12437}, {4,519}, {8,9}, {10,3158}, {20,3928}, {57,4190}, {72,3586}, {78,4193}, {145,226}, {149,11682}, {200,1837}, {329,3621}, {355,6765}, {377,6173}, {388,3243}, {405,3679}, {497,6737}, {515,6762}, {517,5924}, {518,5691}, {527,3146}, {528,7991}, {674,12435}, {936,5722}, {938,5437}, {952,1490}, {1006,8715}, {1210,5438}, {1266,11851}, {1449,5716}, {1482,5715}, {1699,12635}, {1750,11519}, {1864,3893}, {2475,4654}, {2550,6738}, {2551,6743}, {2646,5231}, {2654,3190}, {2802,12691}, {2893,3875}, {3244,3487}, {3340,3434}, {3486,4847}, {3555,9613}, {3576,10916}, {3601,6734}, {3633,9612}, {3651,8666}, {3811,5587}, {3868,9579}, {3869,9580}, {3870,5086}, {3929,6872}, {3951,11114}, {3984,5046}, {4199,12642}, {4313,5745}, {4333,4880}, {4421,9588}, {4652,11015}, {4677,11113}, {4853,4863}, {5141,5219}, {5325,11106}, {5728,5836}, {5730,9614}, {5768,9841}, {5777,12645}, {5812,5844}, {5839,8804}, {5854,9897}, {5855,11531}, {5882,6908}, {5927,12448}, {5934,12633}, {5935,12634}, {6284,12526}, {6735,10395}, {6829,9624}, {6987,11362}, {7580,12513}, {8226,12607}, {8232,12630}, {8233,12638}, {8668,11517}, {10888,12546}, {11235,11522}

X(12625) = midpoint of X(3621) and X(12541)
X(12625) = reflection of X(i) in X(j) for these (i,j): (2136,8), (2900,3419), (3189,10), (3243,6601), (3633,10912), (6765,355), (11523,4), (12536,12437), (12632,12640)
X(12625) = anticomplement of X(12437)
X(12625) = complement of X(12536)
X(12625) = X(64)-of-2nd-extouch-triangle
X(12625) = X(6293)-of-excentral-triangle
X(12625) = excentral-to-2nd-extouch similarity image of X(2136)
X(12625) = 2nd-Conway-to-excentral similarity image of X(12536)
X(12625) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,12536,12437), (8,390,5837), (8,950,9), (8,12632,12640), (10,3189,3158), (10,3488,5436), (145,5175,226), (377,11518,6173), (2321,5802,9), (2475,11520,4654), (3586,3632,72), (3870,5086,9578), (4863,10950,4853), (12632,12640,2136)

X(12626) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO EXCENTERS-MIDPOINTS

Barycentrics (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(2*a^9-2*(b^2+c^2)*a^7-(5*b^4-12*b^2*c^2+5*c^4)*a^5+(b^2-c^2)^2*(b+c)*a^4+8*(b^4-c^4)*(b^2-c^2)*a^3-2*(b^4-c^4)*(b^2-c^2)*(b+c)*a^2-(b^2-c^2)^2*(3*b^4+8*b^2*c^2+3*c^4)*a+(b^4-c^4)^2*(b+c)) : :
X(12626) = 2*X(10)-3*X(11831) = X(3632)-3*X(11852) = 3*X(11845)-X(12245)

The reciprocal orthologic center of these triangles is X(10).

X(12626) lies on these lines: {1,1650}, {8,402}, {10,11831}, {30,944}, {145,4240}, {355,11897}, {515,12668}, {517,12113}, {519,1651}, {952,11251}, {2098,11906}, {2099,11905}, {2802,12729}, {3632,11852}, {3913,11848}, {5846,12583}, {10573,11913}, {10912,11903}, {10950,11909}, {11832,12135}, {11839,12195}, {11845,12245}, {11853,12410}, {11863,12454}, {11864,12455}, {11885,12495}, {11901,12627}, {11902,12628}, {11904,12635}, {11907,12636}, {11908,12637}, {11911,12645}, {11912,12647}, {11914,12648}, {11915,12649}

X(12626) = midpoint of X(145) and X(4240)
X(12626) = reflection of X(i) in X(j) for these (i,j): (8,402), (1650,1)
X(12626) = X(8)-of-Gossard-triangle

X(12627) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO EXCENTERS-MIDPOINTS

Barycentrics 2*a^3-(-a+b+c)*(b^2+c^2-S) : :
X(12627) = 2*X(3641)-3*X(5861)

The reciprocal orthologic center of these triangles is X(10).

X(12627) lies on these lines: {1,5591}, {6,8}, {10,11370}, {145,1271}, {355,6202}, {515,6258}, {517,5871}, {519,3641}, {944,11824}, {952,1161}, {1482,6215}, {2098,10925}, {2099,10923}, {2802,6263}, {3632,5589}, {3913,11497}, {5595,12410}, {5603,10514}, {5604,10513}, {5844,5875}, {7967,10517}, {8198,12454}, {8205,12455}, {8216,12636}, {8217,12637}, {9994,12495}, {10040,12647}, {10048,10573}, {10783,12245}, {10792,12195}, {10912,10919}, {10921,12635}, {10927,10950}, {10929,12648}, {10931,12649}, {11388,12135}, {11901,12626}, {11916,12645}

X(12627) = reflection of X(12628) in X(8)
X(12627) = X(8)-of-inner-Grebe-triangle
X(12627) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5689,5591), (145,1271,5605)

X(12628) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO EXCENTERS-MIDPOINTS

Barycentrics 2*a^3-(-a+b+c)*(b^2+c^2+S) : :
X(12628) = 2*X(3640)-3*X(5860)

The reciprocal orthologic center of these triangles is X(10).

X(12628) lies on these lines: {1,5590}, {6,8}, {10,11371}, {145,1270}, {355,6201}, {515,6257}, {517,5870}, {519,3640}, {944,11825}, {952,1160}, {1482,6214}, {2098,10926}, {2099,10924}, {2802,6262}, {3632,5588}, {3913,11498}, {5594,12410}, {5603,10515}, {5605,10513}, {5844,5874}, {7967,10518}, {8199,12454}, {8206,12455}, {8218,12636}, {8219,12637}, {9995,12495}, {10041,12647}, {10049,10573}, {10784,12245}, {10793,12195}, {10912,10920}, {10922,12635}, {10928,10950}, {10930,12648}, {10932,12649}, {11389,12135}, {11902,12626}, {11917,12645}

X(12628) = reflection of X(12627) in X(8)
X(12628) = X(8)-of-outer-Grebe-triangle
X(12628) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5688,5590), (145,1270,5604)

X(12629) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HEXYL TO EXCENTERS-MIDPOINTS

Trilinears a^3-(b+c)*a^2-(b^2-10*b*c+c^2)*a+(b^2-6*b*c+c^2)*(b+c) : :
X(12629) = 3*X(1)-2*X(3811) = 3*X(165)-4*X(8666) = 4*X(1385)-3*X(3158) = 5*X(1698)-4*X(10915) = 3*X(3576)-2*X(3913) = 3*X(3576)-4*X(11260) = 3*X(3679)-4*X(10916) = 2*X(3811)+3*X(11519) = 4*X(3813)-3*X(5587) = 3*X(3928)-2*X(12702)

The reciprocal orthologic center of these triangles is X(1).

X(12629) lies on these lines: {1,2}, {3,2136}, {9,9957}, {20,12541}, {40,3880}, {56,3893}, {57,10914}, {63,3885}, {72,7962}, {84,517}, {165,8666}, {355,7956}, {518,5693}, {726,12652}, {937,1222}, {944,5732}, {952,1490}, {956,1697}, {999,1706}, {1000,5837}, {1058,5795}, {1320,11682}, {1385,3158}, {1388,3689}, {1420,5687}, {1449,5782}, {1476,3361}, {1482,5777}, {1768,2802}, {2077,8668}, {2324,5839}, {2975,3895}, {3189,5882}, {3243,5784}, {3333,5836}, {3340,3555}, {3421,12053}, {3434,9613}, {3436,9614}, {3576,3913}, {3646,10179}, {3754,10980}, {3813,5587}, {3878,5223}, {3928,12702}, {3984,5330}, {4298,9874}, {4512,5258}, {4863,10944}, {4866,10176}, {5082,10106}, {5119,5288}, {5436,6767}, {5437,7373}, {5657,12640}, {5697,10050}, {5720,12645}, {5731,12632}, {5780,10247}, {5854,6264}, {6282,12245}, {7675,12630}, {7967,8726}, {7987,8715}, {7997,11224}, {8111,12633}, {8112,12634}, {8227,12607}, {8234,12638}, {8235,12642}, {8951,10700}, {9785,12572}, {9819,12514}, {9845,9943}, {10864,12448}, {10884,12536}, {11526,12559}

X(12629) = midpoint of X(i) and X(j) for these {i,j}: {1,11519}, {20,12541}, {145,6764}, {3680,6762}
X(12629) = reflection of X(i) in X(j) for these (i,j): (40,12513), (2136,3), (3189,5882), (3913,11260), (6264,11256), (6765,1), (7982,10912), (11523,1482)
X(12629) = X(64)-of-hexyl-triangle
X(12629) = excentral-to-hexyl similarity image of X(2136)
X(12629) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,8,936), (1,3632,200), (1,3679,8583), (1,4853,9623), (1,4882,997), (1,4915,10), (1,12127,3244), (997,3625,4882), (3333,11525,5836), (3870,4861,1), (3913,11260,3576)

X(12630) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO EXCENTERS-MIDPOINTS

Barycentrics 7*a^3-13*(b+c)*a^2+(9*b^2-2*b*c+9*c^2)*a-3*(b^2-c^2)*(b-c) : :
X(12630) = 3*X(7)-4*X(3243) = 3*X(8)-4*X(1001) = 2*X(8)-3*X(8236) = 4*X(142)-5*X(3623) = 3*X(145)-2*X(3243) = 3*X(390)-2*X(5223) = 4*X(390)-3*X(6172) = 2*X(2550)-3*X(3241) = 4*X(3244)-3*X(11038) = 8*X(5223)-9*X(6172)

The reciprocal orthologic center of these triangles is X(1).

X(12630) lies on these lines: {7,145}, {8,344}, {9,3621}, {100,1617}, {142,3623}, {390,519}, {516,3633}, {517,12669}, {518,3644}, {956,4313}, {1445,2136}, {2550,3241}, {2802,12755}, {3189,4308}, {3244,11038}, {3632,5686}, {3813,7679}, {3870,5226}, {3880,7672}, {3893,5572}, {3913,7677}, {3935,5328}, {4321,12127}, {4326,11519}, {4344,4649}, {4413,10580}, {4678,6666}, {4779,4899}, {4863,10578}, {5759,5844}, {5817,12645}, {5836,11025}, {5854,12730}, {6049,8732}, {6737,7320}, {7675,12629}, {7676,12513}, {7678,12607}, {8232,12625}, {8237,12638}, {8238,12642}, {8385,12633}, {8386,12634}, {8389,12646}, {10865,12448}, {10889,12546}

X(12630) = reflection of X(i) in X(j) for these (i,j): (7,145), (3621,9), (3893,5572)
X(12630) = X(64)-of-Honsberger-triangle
X(12630) = excentral-to-Honsberger similarity image of X(2136)
X(12630) = {X(3189), X(9797)}-harmonic conjugate of X(4308)

X(12631) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-MIDPOINTS TO HUTSON EXTOUCH

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

The reciprocal orthologic center of these triangles is X(12632).

X(12631) lies on these lines: {3,12333}, {9,3295}, {10,6767}, {55,9898}, {100,5558}, {119,3851}, {142,3913}, {214,7373}, {442,5082}, {938,1145}, {999,8000}, {3303,3983}, {3870,5920}, {5687,9874}, {6184,9605}, {6244,12120}, {6260,12699}, {6744,12640}, {8001,8273}, {10679,12684}, {11530,12654}

X(12631) = midpoint of X(7160) and X(12658)
X(12631) = reflection of X(3) in X(12333)

X(12632) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON EXTOUCH TO EXCENTERS-MIDPOINTS

Barycentrics (-a+b+c)*(3*a^3+3*(b+c)*a^2+(b^2-10*b*c+c^2)*a+(b^2-c^2)*(b-c)) : :
X(12632) = 9*X(2)-8*X(3813) = 3*X(2)-4*X(3913) = 3*X(8)-2*X(12625) = 3*X(8)-4*X(12640) = 3*X(2136)-X(12625) = 3*X(2136)-2*X(12640) = 6*X(3158)-5*X(3616) = 4*X(3174)-3*X(11038) = 3*X(3241)-2*X(3680) = 3*X(3241)-4*X(12437) = 2*X(3813)-3*X(3913)

The reciprocal orthologic center of these triangles is X(12631).

X(12632) lies on these lines: {1,11024}, {2,3303}, {8,9}, {20,519}, {40,6764}, {57,9797}, {65,145}, {100,5265}, {144,12125}, {200,9785}, {442,5082}, {497,8165}, {518,9961}, {528,3146}, {529,5059}, {952,12684}, {962,1750}, {1706,10580}, {2551,8168}, {2899,6552}, {3158,3616}, {3174,11038}, {3241,3680}, {3244,11034}, {3434,5261}, {3486,3893}, {3522,12513}, {3523,8715}, {3621,11684}, {3623,10912}, {3632,4294}, {3633,4293}, {3832,12607}, {3871,5281}, {4193,5274}, {4297,11519}, {4309,4677}, {4313,4853}, {4314,4915}, {4315,12127}, {4452,7195}, {4673,7172}, {4882,12575}, {5068,11235}, {5141,10528}, {5177,11239}, {5731,12629}, {5815,10624}, {5919,12448}, {6743,9819}, {6762,9778}, {8666,10304}, {10385,11106}, {10465,12546}

X(12632) = reflection of X(i) in X(j) for these (i,j): (8,2136), (145,3189), (390,7674), (962,6765), (3680,12437), (6764,40), (11519,4297), (12541,1), (12625,12640)
X(12632) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,5250,5686), (2136,12625,12640), (3680,12437,3241), (12625,12640,8)

X(12633) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-HUTSON TO EXCENTERS-MIDPOINTS

Trilinears -4*(3*a-b-c)*b*c*sin(A/2)-2*(-a+b+c)*(a-3*b+c)*c*sin(B/2)-2*(-a+b+c)*(a+b-3*c)*b*sin(C/2)+a^3-(b+c)*a^2-(b^2-18*b*c+c^2)*a+(b+c)*(b^2-10*b*c+c^2) : :

The reciprocal orthologic center of these triangles is X(1).

X(12633) lies on these lines: {8,8390}, {145,8113}, {363,2136}, {519,9836}, {2802,12759}, {3244,11039}, {3621,11685}, {3680,11527}, {3813,8380}, {3913,8109}, {5836,11026}, {5854,12733}, {5934,12625}, {8107,12513}, {8111,12629}, {8140,11519}, {8377,12607}, {8385,12630}, {8391,12642}, {9783,12541}, {11854,12437}, {11856,12448}, {11886,12536}, {11892,12546}, {11922,12638}

X(12633) = reflection of X(12634) in X(11519)
X(12633) = X(64)-of-inner-Hutson-triangle
X(12633) = excentral-to-inner-Hutson similarity image of X(2136)

X(12634) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-HUTSON TO EXCENTERS-MIDPOINTS

Trilinears 4*(3*a-b-c)*b*c*sin(A/2)+2*(-a+b+c)*(a-3*b+c)*c*sin(B/2)+2*(-a+b+c)*(a+b-3*c)*b*sin(C/2)+a^3-(b+c)*a^2-(b^2-18*b*c+c^2)*a+(b+c)*(b^2-10*b*c+c^2) : :

The reciprocal orthologic center of these triangles is X(1).

X(12634) lies on these lines: {145,8114}, {519,9837}, {2802,12760}, {3244,11040}, {3621,11686}, {3813,8381}, {3913,8110}, {5836,11027}, {5854,12734}, {5935,12625}, {8108,12513}, {8112,12629}, {8140,11519}, {8378,12607}, {8386,12630}, {11855,12437}, {11857,12448}, {11887,12536}, {11893,12546}, {11925,12638}, {11926,12642}

X(12634) = reflection of X(12633) in X(11519)
X(12634) = X(64)-of-outer-Hutson-triangle
X(12634) = excentral-to-outer-Hutson similarity image of X(2136)

X(12635) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO EXCENTERS-MIDPOINTS

Trilinears a^3-2*(b+c)*a^2-(b^2+c^2)*a+2*c^3+2*b^3 : :
X(12635) = 3*X(1)-2*X(11260) = 2*X(40)-3*X(4421) = 2*X(355)-3*X(11236) = 4*X(946)-3*X(11235) = 4*X(1385)-3*X(11194) = X(1482)-3*X(4930) = 6*X(4930)-X(10912) = 2*X(11260)+3*X(11523) = 4*X(11260)-3*X(12513) = 2*X(11523)+X(12513)

The reciprocal orthologic center of these triangles is X(10).

X(12635) lies on these lines: {1,6}, {2,11281}, {3,758}, {8,12}, {10,3940}, {11,12649}, {36,3901}, {40,4421}, {46,4018}, {55,3869}, {56,1259}, {63,2646}, {65,78}, {142,12447}, {145,497}, {200,3340}, {210,3984}, {226,5794}, {320,7185}, {329,3486}, {354,11520}, {355,381}, {377,3649}, {404,5221}, {474,5902}, {480,7672}, {515,5812}, {516,12437}, {517,3811}, {527,4297}, {528,962}, {529,944}, {908,1837}, {912,12114}, {936,3812}, {938,3816}, {940,2650}, {942,997}, {952,10526}, {959,1257}, {965,2294}, {976,5710}, {986,4255}, {993,3927}, {999,3874}, {1012,5693}, {1042,1818}, {1043,5327}, {1046,4252}, {1125,5791}, {1155,4855}, {1159,3754}, {1215,5793}, {1265,3932}, {1320,7319}, {1385,11194}, {1389,10599}, {1698,5425}, {1699,12625}, {1706,10107}, {1788,3035}, {1848,5130}, {2136,11531}, {2171,3713}, {2271,3735}, {2800,10306}, {2802,8148}, {2932,11571}, {3057,3870}, {3149,6326}, {3158,7991}, {3190,10571}, {3207,3509}, {3218,5204}, {3241,5330}, {3295,3878}, {3303,3877}, {3304,3873}, {3339,5438}, {3419,12047}, {3452,6738}, {3496,4258}, {3560,5694}, {3601,4640}, {3612,3916}, {3617,3711}, {3632,10827}, {3633,9614}, {3671,5880}, {3678,9708}, {3680,11224}, {3715,5260}, {3742,8583}, {3746,3899}, {3813,5603}, {3880,6765}, {3881,7373}, {3884,6767}, {3890,3957}, {3894,5563}, {3924,4383}, {3928,7987}, {3930,4513}, {4101,10371}, {4189,11684}, {4190,11246}, {4299,10609}, {4301,5853}, {4313,5698}, {4345,9797}, {4428,5250}, {4662,9623}, {4848,6745}, {4860,5253}, {4880,7280}, {5086,10895}, {5087,9581}, {5703,6690}, {5704,6667}, {5719,10198}, {5720,7686}, {5731,5852}, {5734,6764}, {5761,7680}, {5780,10175}, {5844,10942}, {5846,12587}, {5851,12246}, {5854,10698}, {5886,10916}, {5887,11496}, {5905,7354}, {6049,6068}, {6265,10680}, {6282,9943}, {6284,11415}, {6734,11375}, {6769,7971}, {6872,10543}, {6943,9803}, {7080,8256}, {8168,10914}, {8666,10246}, {8715,12702}, {8834,10699}, {9669,11813}, {9812,12536}, {10176,11108}, {10474,11679}, {10522,10944}, {10523,10573}, {10786,12245}, {10795,12195}, {10830,12410}, {10872,12495}, {10921,12627}, {10922,12628}, {10951,12636}, {10952,12637}, {10954,12647}, {10955,12648}, {11391,12135}, {11495,12520}, {11868,12455}, {11904,12626}

X(12635) = midpoint of X(i) and X(j) for these {i,j}: {1,11523}, {962,3189}, {2136,11531}, {6765,7982}, {6769,7971}
X(12635) = reflection of X(i) in X(j) for these (i,j): (8,12607), (3913,3811), (6762,11260), (10912,1482), (12513,1), (12702,8715)
X(12635) = X(8)-of-outer-Johnson-triangle
X(12635) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,72,958), (1,960,1001), (1,4867,5730), (1,5692,405), (1,5730,5289), (1,5904,956), (1,6762,11260), (8,3485,2886), (65,78,1376), (72,958,5220), (145,3436,10950), (200,3340,5836), (226,6737,5794), (936,11529,3812), (2646,3962,63), (3868,4511,56), (4018,5440,46), (6762,11260,12513), (11929,12645,355), (12447,12563,142)

X(12636) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO EXCENTERS-MIDPOINTS

Barycentrics ((-12*R^2+SA-SW)*S+SA^2-2*SW^2)*a+(b+c)*((4*R^2+SA-SW)*S+SA^2) : :

The reciprocal orthologic center of these triangles is X(10).

X(12636) lies on these lines: {1,8214}, {8,493}, {10,11377}, {145,6462}, {355,8212}, {517,9838}, {519,12152}, {944,11828}, {952,10669}, {1482,8220}, {2098,11932}, {2099,11930}, {2802,12741}, {3632,8188}, {3913,11503}, {5846,12590}, {6339,8211}, {6461,12637}, {8194,12410}, {8201,12454}, {8208,12455}, {8216,12627}, {8218,12628}, {10573,11953}, {10875,12495}, {10912,10945}, {10950,11947}, {10951,12635}, {11394,12135}, {11840,12195}, {11846,12245}, {11907,12626}, {11949,12645}, {11951,12647}, {11955,12648}, {11957,12649}

X(12636) = X(8)-of-Lucas-homothetic-triangle

X(12637) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO EXCENTERS-MIDPOINTS

Barycentrics (-(-12*R^2+SA-SW)*S+SA^2-2*SW^2)*a+(b+c)*(-(4*R^2+SA-SW)*S+SA^2) : :

The reciprocal orthologic center of these triangles is X(10).

X(12637) lies on these lines: {1,8215}, {8,494}, {10,11378}, {145,6463}, {355,8213}, {517,9839}, {519,12153}, {944,11829}, {952,10673}, {1482,8221}, {2098,11933}, {2099,11931}, {2802,12742}, {3632,8189}, {3913,11504}, {5846,12591}, {6339,8210}, {6461,12636}, {8195,12410}, {8202,12454}, {8209,12455}, {8217,12627}, {8219,12628}, {10573,11954}, {10876,12495}, {10912,10946}, {10950,11948}, {10952,12635}, {11395,12135}, {11841,12195}, {11847,12245}, {11908,12626}, {11950,12645}, {11952,12647}, {11956,12648}, {11958,12649}

X(12637) = X(8)-of-Lucas(-1)-homothetic-triangle

X(12638) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO EXCENTERS-MIDPOINTS

Barycentrics (-a+b+c)*((2*(b+c)*a^2+4*(b^2-4*b*c+c^2)*a+2*(b^2-c^2)*(b-c))*S+(a+b-c)*(a-b+c)*(6*a^3-3*(b+c)*a^2+4*(b^2+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(12638) lies on these lines: {8,7090}, {145,8243}, {517,12681}, {519,7596}, {2136,8231}, {2802,12768}, {3244,11042}, {3621,11687}, {3680,7595}, {3813,8230}, {3880,9808}, {3913,8225}, {5836,11030}, {5854,12744}, {8224,12513}, {8228,12607}, {8233,12625}, {8234,12629}, {8237,12630}, {8244,11519}, {8246,12642}, {9789,12541}, {10858,12437}, {10867,12448}, {10885,12536}, {10891,12546}, {11922,12633}, {11925,12634}, {11996,12646}

X(12638) = X(64)-of-2nd-Pamfilos-Zhou-triangle
X(12638) = excentral-to-2nd-Pamfilos-Zhou similarity image of X(2136)

X(12639) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-MIDPOINTS TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(6598).

X(12639) lies on these lines: {2,6597}, {3,12342}, {9,10266}, {10,12267}, {100,6599}, {214,11263}, {11530,12657}

X(12639) = midpoint of X(i) and X(j) for these {i,j}: {100,6599}, {10266,12660}
X(12639) = complement of X(6597)

X(12640) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-MIDPOINTS TO 2nd SCHIFFLER

Barycentrics (-a+b+c)*(3*a-b-c)*((b+c)*a+(b-c)^2) : :
X(12640) = 3*X(8)-X(12625) = 3*X(8)+X(12632) = 3*X(10)-2*X(3813) = X(145)-3*X(3158) = 3*X(2136)+X(12625) = 3*X(2136)-X(12632) = 5*X(3617)-X(12541) = 3*X(3740)-X(12448) = 3*X(5657)-X(12629) = 3*X(10164)-2*X(11260)

The reciprocal orthologic center of these triangles is X(12641).

X(12640) lies on these lines: {1,6692}, {2,3680}, {3,519}, {8,9}, {10,496}, {65,10427}, {100,1476}, {119,946}, {142,5836}, {145,1420}, {214,3244}, {442,10914}, {517,6260}, {527,7991}, {529,5493}, {936,1000}, {993,8668}, {1125,10912}, {1145,1210}, {1329,4342}, {2098,6745}, {2551,9819}, {3057,3452}, {3189,3632}, {3617,12541}, {3625,3647}, {3679,5084}, {3740,12448}, {3885,4193}, {3890,5316}, {3893,4847}, {4190,10106}, {4301,12607}, {4677,11111}, {4853,5745}, {5542,10107}, {5657,12629}, {5919,8582}, {6556,8055}, {6600,6738}, {6743,8168}, {6744,12631}, {6765,12245}, {6848,7982}, {7080,7962}, {8256,11019}, {10164,11260}

X(12640) = midpoint of X(i) and X(j) for these {i,j}: {8,2136}, {100,12641}, {3189,3632}, {6765,12245}, {12625,12632}
X(12640) = reflection of X(i) in X(j) for these (i,j): (946,10915), (4301,12607), (5882,8715), (10912,1125), (12437,3913)
X(12640) = complement of X(3680)
X(12640) = X(4)-of-excenters-midpoints-triangle
X(12640) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,1697,5795), (8,3895,950), (8,12632,12625), (2136,12625,12632), (3057,6736,3452), (3885,6735,12053)

X(12641) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd SCHIFFLER TO EXCENTERS-MIDPOINTS

Barycentrics (-a+b+c)/(a^3-(b+c)*a^2-(b^2-5*b*c+c^2)*a+(b^2-3*b*c+c^2)*(b+c)) : :
X(12641) = 2*X(1317)-3*X(3158) = 3*X(11219)-2*X(11256)

The reciprocal orthologic center of these triangles is X(12640).

X(12641) lies on the Feuerbach hyperbola and these lines: {1,1145}, {4,2802}, {7,12648}, {8,4939}, {9,4534}, {11,3680}, {79,12749}, {80,3880}, {84,952}, {90,3632}, {100,1476}, {104,519}, {119,3577}, {149,7319}, {392,5559}, {528,3062}, {1000,3898}, {1156,5853}, {1317,3158}, {1320,6735}, {1389,10915}, {1392,5552}, {2320,5281}, {2800,10309}, {2801,10307}, {2932,3913}, {3036,4900}, {3893,6598}, {5541,7284}, {5554,7320}, {5665,10956}, {10305,12245}, {11219,11256}

X(12641) = reflection of X(i) in X(j) for these (i,j): (100,12640), (3680,11), (7972,3913)
X(12641) = isogonal conjugate of X(5193)
X(12641) = antigonal conjugate of X(3680)
X(12641) = X(4)-of-2nd-Schiffler-triangle
X(12641) = antipode of X(3680) in Feuerbach hyperbola

X(12642) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO EXCENTERS-MIDPOINTS

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

The reciprocal orthologic center of these triangles is X(1).

X(12642) lies on these lines: {8,21}, {145,1284}, {256,3680}, {517,12683}, {519,9840}, {846,2136}, {1469,11520}, {2292,3880}, {2802,12770}, {3244,11043}, {3621,11688}, {3813,5051}, {4199,12625}, {4220,12513}, {4685,8731}, {5836,11031}, {5854,12746}, {8229,12607}, {8235,12629}, {8238,12630}, {8245,11519}, {8246,12638}, {8391,12633}, {8425,12646}, {9791,12541}, {10868,12448}, {10892,12546}, {11926,12634}

X(12642) = X(64)-of-1st-Sharygin-triangle
X(12642) = 2nd-Sharygin-to-1st-Sharygin similarity image of X(13252)
X(12642) = excentral-to-1st-Sharygin similarity image of X(2136)

X(12643) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL-MIDARC TO EXCENTERS-MIDPOINTS

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

The reciprocal orthologic center of these triangles is X(1).

X(12643) lies on the cubic K201 and these lines: {1,236}, {8,188}, {145,2089}, {177,3680}, {517,8095}, {519,8091}, {2136,8078}, {3244,11044}, {3621,11690}, {3813,8087}, {3880,8093}, {3893,10503}, {3913,8077}, {5836,11032}, {5854,8097}, {5881,9836}, {6553,10490}, {7028,8422}, {8075,12513}, {8085,12607}, {8089,11519}, {8733,12437}, {9793,12541}, {11858,12448}, {11888,12536}, {11894,12546}

X(12643) = reflection of X(12644) in X(1)
X(12643) = X(64)-of-tangential-midarc-triangle
X(12643) = excentral-to-tangential-midarc similarity image of X(2136)
X(12643) = {X(8), X(8241)}-harmonic conjugate of X(188)

X(12644) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO EXCENTERS-MIDPOINTS

Barycentrics (a+b+c)*sin(A/2)-3*a+b+c : :

The reciprocal orthologic center of these triangles is X(1).

X(12644) lies on the cubic K201 and these lines: {1,236}, {145,174}, {1483,8130}, {2802,12772}, {3241,11924}, {3243,11535}, {3244,8351}, {3621,8125}, {3623,8126}, {3680,11899}, {3913,7588}, {5836,11033}, {5844,8129}, {8734,12437}, {11859,12448}, {11895,12546}

X(12644) = reflection of X(12643) in X(1)
X(12644) = X(64)-of-2nd-tangential-midarc-triangle
X(12644) = excentral-to-2nd-tangential-midarc similarity image of X(2136)
X(12644) = {X(145), X(174)}-harmonic conjugate of X(12646)

X(12645) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO EXCENTERS-MIDPOINTS

Barycentrics 3*a^4-4*(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(12645) = 4*X(1)-5*X(1656) = 3*X(3)-2*X(944) = 5*X(3)-6*X(5657) = 3*X(3)-4*X(5690) = 4*X(5)-3*X(10247) = 3*X(8)-X(944) = 5*X(8)-3*X(5657) = 3*X(8)-2*X(5690) = 7*X(8)-3*X(5731) = 2*X(145)-3*X(10247) = 3*X(145)-5*X(10595) = 2*X(3621)+X(8148) = 9*X(10247)-10*X(10595)

The reciprocal orthologic center of these triangles is X(10).

X(12645) lies on the cubic K201 and these lines: {1,1656}, {2,1483}, {3,8}, {4,3621}, {5,145}, {10,3526}, {30,12245}, {40,3534}, {80,2098}, {119,3813}, {140,3617}, {355,381}, {382,517}, {388,1159}, {499,1317}, {515,1657}, {518,11898}, {631,4678}, {912,10914}, {962,3830}, {999,10573}, {1351,5846}, {1352,9053}, {1385,3679}, {1388,7972}, {1484,4193}, {1598,12135}, {1699,11278}, {2099,9654}, {2136,7330}, {2802,12747}, {2937,9798}, {3086,11545}, {3090,3623}, {3167,9933}, {3241,5055}, {3244,5079}, {3295,7489}, {3421,6928}, {3445,6788}, {3576,4668}, {3579,4816}, {3616,5070}, {3622,3628}, {3626,5882}, {3633,5072}, {3635,10175}, {3653,4745}, {3654,4297}, {3655,4669}, {3851,5603}, {3871,6914}, {3880,5887}, {3913,11849}, {4691,10165}, {4701,11362}, {4853,5534}, {5048,10826}, {5076,12699}, {5082,6923}, {5176,5730}, {5531,11014}, {5694,5697}, {5708,10106}, {5720,12629}, {5722,5780}, {5727,9957}, {5777,12625}, {5779,5853}, {5811,12541}, {5817,12630}, {5854,10738}, {6147,11041}, {6265,11256}, {6862,10528}, {6913,12000}, {6918,12001}, {6941,11698}, {6958,7080}, {6959,10529}, {6971,10943}, {6980,10942}, {7517,12410}, {8168,12114}, {8200,11876}, {8207,11875}, {9301,12495}, {9858,10202}, {10525,10742}, {10827,11011}, {10895,11009}, {11499,12513}, {11842,12195}, {11911,12626}, {11916,12627}, {11917,12628}, {11949,12636}, {11950,12637}

X(12645) = midpoint of X(i) and X(j) for these {i,j}: {4,3621}, {3632,5881}
X(12645) = reflection of X(i) in X(j) for these (i,j): (3,8), (145,5), (944,5690), (1482,355), (1657,12702), (3655,4669), (5697,5694), (5882,3626), (8148,4), (11362,4701)
X(12645) = anticomplement of X(1483)
X(12645) = X(8)-of-X3-ABC-reflections-triangle
X(12645) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5790,1656), (5,145,10247), (8,944,5690), (10,10246,3526), (80,2098,9669), (355,1482,381), (355,10912,11928), (355,12635,11929), (944,5690,3), (3090,3623,10283), (3241,5818,5901), (3617,7967,140), (5818,5901,5055), (10573,10944,999), (10950,12647,3295)

X(12646) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO EXCENTERS-MIDPOINTS

Barycentrics (a+b+c)*sin(A/2)+3*a-b-c : :

The reciprocal orthologic center of these triangles is X(1).

X(12646) lies on the cubic K201 and these lines: {1,188}, {8,178}, {145,174}, {173,2136}, {177,3680}, {517,12685}, {519,8351}, {1483,8129}, {2802,12774}, {3621,8126}, {3623,8125}, {3813,8382}, {3880,12445}, {3893,10502}, {3913,7587}, {5836,8083}, {5844,8130}, {5854,12748}, {7593,12625}, {8389,12630}, {8423,11519}, {8425,12642}, {8729,12437}, {11860,12448}, {11890,12536}, {11891,12541}, {11896,12546}, {11996,12638}

X(12646) = X(64)-of-Yff-central-triangle
X(12646) = excentral-to-Yff-central similarity image of X(2136)
X(12646) = {X(483),X(3082)}-harmonic conjugate of X(236)
X(12646) = {X(145), X(174)}-harmonic conjugate of X(12644)

X(12647) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO EXCENTERS-MIDPOINTS

Barycentrics a^4-2*(b+c)*a^3+6*b*c*a^2+2*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(12647) = X(145)-3*X(11239) = 3*X(1478)-2*X(1836) = X(1836)-3*X(5252) = 3*X(3679)-2*X(4847)

The reciprocal orthologic center of these triangles is X(10).

X(12647) lies on these lines: {1,2}, {3,10944}, {4,5559}, {5,2098}, {11,5790}, {12,1482}, {20,11010}, {35,944}, {36,3476}, {40,4299}, {46,4317}, {47,5255}, {55,952}, {56,5690}, {65,10044}, {79,7317}, {80,497}, {140,1388}, {329,3899}, {355,1479}, {390,9897}, {474,8256}, {484,4293}, {495,2099}, {515,1709}, {517,1478}, {611,5846}, {912,12430}, {942,11045}, {946,6968}, {950,6976}, {956,8069}, {958,11508}, {962,3585}, {982,1772}, {1056,5902}, {1145,1376}, {1155,3654}, {1317,5432}, {1320,11680}, {1621,12531}, {1697,4309}, {1699,8275}, {1734,2401}, {1770,7991}, {1788,5563}, {1837,9957}, {2478,3884}, {2800,12115}, {2802,3434}, {2886,5854}, {3036,3816}, {3245,3474}, {3295,7489}, {3336,3600}, {3338,4848}, {3419,3880}, {3421,5692}, {3436,3878}, {3475,5425}, {3485,11009}, {3486,3746}, {3586,9819}, {3612,5882}, {3753,5570}, {3877,5176}, {3885,5086}, {3898,10073}, {4295,5270}, {4316,9778}, {4333,5493}, {4351,8270}, {4421,10609}, {4857,9785}, {5010,5731}, {5048,5886}, {5082,10629}, {5218,7967}, {5261,11280}, {5281,9803}, {5330,8070}, {5443,10588}, {5445,7288}, {5587,6973}, {5599,11880}, {5600,11879}, {5603,7951}, {5687,8071}, {5691,9898}, {5722,5919}, {5726,11224}, {5730,12607}, {5794,10914}, {5818,7741}, {5884,10805}, {6361,10483}, {6702,10584}, {6825,11014}, {6982,7982}, {7354,12702}, {8148,9654}, {8200,11874}, {8207,11873}, {9612,11531}, {9956,11376}, {10037,12410}, {10038,12495}, {10040,12627}, {10041,12628}, {10074,10269}, {10801,12195}, {10826,12053}, {10954,12635}, {10966,11499}, {11011,11374}, {11238,12019}, {11249,11501}, {11252,11870}, {11253,11869}, {11398,12135}, {11877,12454}, {11878,12455}, {11912,12626}, {11951,12636}, {11952,12637}, {12751,12758}

X(12647) = midpoint of X(8) and X(12648)
X(12647) = reflection of X(i) in X(j) for these (i,j): (1478,5252), (2099,495), (4302,5119)
X(12647) = X(8)-of-inner-Yff-triangle
X(12647) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,8,10573), (1,10,499), (1,1737,10072), (1,3679,1737), (1,10039,498), (46,10106,4317), (145,3085,1), (355,3057,1479), (1317,5432,10246), (1697,5881,10572), (1697,10572,4309), (3085,10527,10320), (3295,12645,10950), (3475,11041,5425), (3476,5657,36), (3632,3679,4915), (6929,10947,1479), (7991,9613,1770), (10106,11362,46), (10320,10527,499)

X(12648) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO EXCENTERS-MIDPOINTS

Barycentrics a^4-2*(b+c)*a^3+10*b*c*a^2+2*(b+c)*(b^2-3*b*c+c^2)*a-(b^2-c^2)^2 : :
X(12648) = 2*X(1)-3*X(11239) = 5*X(3617)-4*X(4847)

The reciprocal orthologic center of these triangles is X(10).

X(12648) lies on these lines: {1,2}, {4,3885}, {7,12641}, {12,10912}, {40,11919}, {65,10940}, {100,1470}, {119,1320}, {355,6957}, {377,10914}, {388,7702}, {497,5176}, {515,3895}, {517,5905}, {908,7962}, {942,11047}, {944,3871}, {952,1012}, {962,6256}, {999,1145}, {1000,3421}, {1478,2802}, {1482,1532}, {1697,6872}, {2077,5731}, {2098,10958}, {2099,5854}, {2478,9957}, {2551,3890}, {3057,3436}, {3218,3359}, {3304,8256}, {3434,3880}, {3680,6871}, {3868,6916}, {3893,5794}, {3913,10944}, {4188,4308}, {4190,10106}, {4345,5748}, {4917,12437}, {5046,9785}, {5123,10584}, {5175,12541}, {5187,12053}, {5193,5435}, {5559,5904}, {5657,10269}, {5697,11415}, {5844,6907}, {5846,12594}, {5853,8545}, {5902,11046}, {6913,12000}, {6931,11373}, {6939,10596}, {7982,12608}, {7991,10970}, {10247,11729}, {10524,10827}, {10803,12195}, {10834,12410}, {10878,12495}, {10929,12627}, {10930,12628}, {10950,10965}, {10955,12635}, {11400,12135}, {11881,12454}, {11882,12455}, {11914,12626}, {11955,12636}, {11956,12637}

X(12648) = reflection of X(i) in X(j) for these (i,j): (8,12647), (145,3870), (3434,5252)
X(12648) = X(8)-of-inner-Yff-tangents-triangle
X(12648) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,8,5554), (1,6735,2), (1,10915,5552), (8,145,12649), (145,10528,1), (1000,3421,3877), (10528,10530,5552)

X(12649) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO EXCENTERS-MIDPOINTS

Barycentrics a^4-2*(b+c)*a^3-2*b*c*a^2+2*(b^3+c^3)*a-(b^2-c^2)^2 : :
Barycentrics a/(1 - sec A) - b/(1 - sec B) - c/(1 - sec C) : :
X(12649) = 2*X(1)-3*X(11240) = 3*X(2)-4*X(1210) = 9*X(2)-8*X(6700) = 3*X(78)-4*X(6700) = 3*X(1210)-2*X(6700) = 5*X(3617)-4*X(6736)

The reciprocal orthologic center of these triangles is X(10).

X(12649) lies on these lines: {1,2}, {4,912}, {7,2475}, {11,12635}, {20,3218}, {21,3488}, {29,1069}, {40,11920}, {57,4190}, {63,950}, {65,3434}, {69,5016}, {72,2478}, {75,5738}, {81,5716}, {100,1788}, {144,5809}, {149,151}, {224,2900}, {225,11851}, {226,6871}, {273,5174}, {307,3875}, {329,5046}, {346,8557}, {354,5794}, {355,3555}, {376,11015}, {377,942}, {382,12690}, {388,3873}, {405,12433}, {411,944}, {452,3219}, {496,5730}, {497,3869}, {515,12687}, {517,6836}, {518,1837}, {758,1479}, {894,5807}, {908,5187}, {946,6870}, {952,3149}, {956,11344}, {1056,3889}, {1058,3877}, {1068,1897}, {1229,4696}, {1265,4358}, {1320,6943}, {1331,1724}, {1445,4848}, {1446,6604}, {1478,3874}, {1482,6831}, {1512,5534}, {1895,5081}, {1936,7538}, {1993,3562}, {2098,5855}, {2099,3813}, {2287,5839}, {2476,3487}, {2550,5178}, {2551,3681}, {2802,12750}, {2899,3952}, {2975,3486}, {3057,10936}, {3091,5804}, {3146,9799}, {3152,3210}, {3243,9578}, {3254,7319}, {3452,3984}, {3485,11680}, {3583,3901}, {3585,3894}, {3711,9711}, {3832,5715}, {3871,5657}, {3876,5084}, {3885,6865}, {3895,11362}, {3911,4855}, {3913,11510}, {3927,11113}, {3940,4187}, {3951,12572}, {4018,12699}, {4188,5435}, {4189,4313}, {4304,4652}, {4430,6894}, {4452,5932}, {4661,5815}, {4863,5836}, {4881,5265}, {5057,5225}, {5059,10430}, {5141,5226}, {5154,5748}, {5177,11036}, {5249,11518}, {5279,5802}, {5440,6921}, {5603,6828}, {5698,11684}, {5708,11112}, {5720,6953}, {5727,6762}, {5731,11012}, {5758,6840}, {5761,6830}, {5770,6906}, {5777,6957}, {5787,10431}, {5818,6991}, {5844,6922}, {5846,12595}, {5887,10531}, {6224,10074}, {6585,11491}, {6601,7672}, {6855,10595}, {6864,10597}, {6897,10202}, {6918,12001}, {6933,11374}, {6988,7967}, {7466,7718}, {7991,10971}, {10524,10826}, {10804,12195}, {10835,12410}, {10879,12495}, {10912,10949}, {10931,12627}, {10932,12628}, {10950,10966}, {11401,12135}, {11682,12053}, {11883,12454}, {11884,12455}, {11915,12626}, {11957,12636}, {11958,12637}, {12047,12559}

X(12649) = reflection of X(i) in X(j) for these (i,j): (8,10573), (78,1210), (3436,1837), (5730,496), (6224,10074), (11415,1479), (11682,12053)
X(12649) = isogonal conjugate of X(34430)
X(12649) = complement of X(20013)
X(12649) = anticomplement of X(78)
X(12649) = X(8)-of-outer-Yff-tangents-triangle
X(12649) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,6734,2), (1,10916,10527), (4,3868,5905), (7,5175,2475), (8,145,12648), (8,938,2), (8,6764,3621), (10,3870,10528), (63,950,6872), (72,5722,2478), (78,1210,2), (145,10528,3870), (145,10529,1), (908,9581,5187), (942,3419,377), (1737,3811,5552), (1788,3189,100), (3873,5086,388), (9581,11523,908), (10529,10530,10527)

X(12650) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS TO EXTOUCH

Trilinears a^6-2*(b+c)*a^5-(b^2-14*b*c+c^2)*a^4+4*(b+c)*(b^2-3*b*c+c^2)*a^3-(b^2+10*b*c+c^2)*(b-c)^2*a^2-2*(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a+(b^2-6*b*c+c^2)*(b^2-c^2)^2 : :
X(12650) = 3*X(1)-2*X(6261) = 3*X(165)-4*X(5450) = 3*X(1490)-4*X(6261) = 3*X(1699)-2*X(6256) = 3*X(3576)-2*X(11500) = 3*X(3679)-4*X(12616) = 3*X(5603)-2*X(6260) = 3*X(5657)-4*X(6705) = 3*X(5658)-5*X(10595) = 5*X(11522)-4*X(12608)

The reciprocal orthologic center of these triangles is X(72).

X(12650) lies on these lines: {1,4}, {3,1706}, {8,6245}, {10,6926}, {30,12700}, {40,956}, {84,517}, {145,9799}, {165,5450}, {200,5881}, {355,936}, {474,3576}, {519,6769}, {942,3577}, {952,5787}, {971,1482}, {993,10268}, {1012,1697}, {1125,6964}, {1158,6763}, {1385,6918}, {1420,3149}, {1467,4311}, {1512,10785}, {1698,6967}, {1709,5697}, {2057,5176}, {2098,12688}, {2099,12680}, {2136,10306}, {2800,3901}, {2802,2950}, {2829,6264}, {3057,12705}, {3062,12666}, {3295,7966}, {3333,7686}, {3427,6737}, {3555,6001}, {3624,6983}, {3679,12616}, {4187,5587}, {4915,11362}, {5657,6705}, {5731,6904}, {5732,5832}, {5758,5924}, {5795,6865}, {5806,7373}, {5842,12565}, {6735,6890}, {6796,6940}, {6831,9578}, {6975,7989}, {7962,12672}, {7994,12245}, {8148,12684}, {9845,11529}, {9942,11518}, {9948,11519}, {9960,11520}, {11521,12547}, {11523,12664}, {11526,12669}, {11532,12681}, {11533,12683}, {11535,12685}, {11682,12528}

X(12650) = midpoint of X(i) and X(j) for these {i,j}: {145,9799}, {7982,10864}, {7992,11531}, {8148,12684}
X(12650) = reflection of X(i) in X(j) for these (i,j): (8,6245), (40,12114), (1490,1), (2136,10306), (7971,1482), (7991,1158), (12667,946)
X(12650) = X(68)-of-excenters-reflections-triangle
X(12650) = excentral-to-excenters-reflections similarity image of X(1490)
X(12650) = excenters-reflections-isotomic conjugate of X(12652)
X(12650) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,944,10106), (5691,9614,4), (9845,11529,12675)

X(12651) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS TO 4th EXTOUCH

Trilinears a^6-2*(b+c)*a^5-(b^2-10*b*c+c^2)*a^4+4*(b+c)*(b^2+c^2)*a^3-(b+c)^4*a^2-2*(b^2-c^2)^2*(b+c)*a+(b^2-6*b*c+c^2)*(b^2-c^2)^2 : :
X(12651) = 3*X(1)-2*X(12520) = 2*X(40)-3*X(4512) = 3*X(165)-4*X(5248) = 3*X(3679)-4*X(12617) = 8*X(3841)-9*X(7988) = 3*X(4512)-4*X(11496) = 5*X(5734)-4*X(12563) = 5*X(7987)-4*X(12511) = 7*X(7989)-8*X(12558) = 4*X(12520)-3*X(12565)

The reciprocal orthologic center of these triangles is X(65).

X(12651) lies on these lines: {1,7}, {3,10582}, {4,200}, {9,7957}, {10,7994}, {40,405}, {72,11372}, {78,9812}, {145,9800}, {165,3833}, {354,9841}, {382,5534}, {388,10388}, {443,946}, {517,3927}, {936,1699}, {942,10860}, {956,6766}, {1467,3474}, {1490,5842}, {1698,6886}, {1750,3811}, {1998,6895}, {2098,9850}, {2999,6996}, {3062,12528}, {3091,8580}, {3146,3870}, {3174,6253}, {3243,12680}, {3340,12711}, {3361,6909}, {3522,4666}, {3555,6001}, {3679,12617}, {3841,7988}, {3868,7992}, {3957,5059}, {4420,10248}, {5231,6847}, {5234,6912}, {5268,7385}, {5290,6925}, {5436,5584}, {5531,10724}, {5691,6765}, {5806,6244}, {7962,12709}, {7987,12511}, {7989,12558}, {9851,11224}, {9943,11518}, {9949,11519}, {9961,11520}, {10398,12432}, {10857,12512}, {11521,12548}, {11522,12609}, {11523,12688}, {11526,12706}, {11527,12707}, {11528,12708}, {11529,12710}, {11532,12712}, {11533,12713}, {11535,12716}, {11682,12529}, {11899,12715}

X(12651) = midpoint of X(145) and X(9800)
X(12651) = reflection of X(i) in X(j) for these (i,j): (20,4314), (40,11496), (4295,4301), (7991,12514), (12526,12705), (12565,1)
X(12651) = excentral-to-excenters-reflections similarity image of X(12565)
X(12651) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,2951,10884), (1,4292,4321), (1,4294,4326), (4,6769,200), (40,11496,4512), (946,6282,8583), (4319,4332,1), (4336,4348,1), (7982,10864,3555)

X(12652) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS TO 5th EXTOUCH

Trilinears a^5-(b+c)*a^4+10*b*c*a^3-6*b*c*(b+c)*a^2-(b^2+c^2)*(b-c)^2*a+(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2) : :
X(12652) = 3*X(1)-2*X(990) = 4*X(990)-3*X(1721) = 3*X(3679)-4*X(12618) = 5*X(11522)-4*X(12610)

The reciprocal orthologic center of these triangles is X(65).

X(12652) lies on these lines: {1,7}, {40,238}, {43,7994}, {105,165}, {145,9801}, {200,4388}, {517,1351}, {612,9812}, {614,9778}, {651,7673}, {726,12629}, {936,4660}, {982,10860}, {984,11372}, {1038,12701}, {1279,11495}, {1697,9440}, {1699,5268}, {1743,1766}, {1750,3961}, {3057,6180}, {3177,3729}, {3339,8915}, {3340,12723}, {3679,12618}, {3749,7580}, {3923,9623}, {3976,9841}, {5223,9355}, {7290,9441}, {7962,12721}, {7996,11531}, {8270,9580}, {9944,11518}, {9950,11519}, {9962,11520}, {11521,12549}, {11522,12610}, {11523,12689}, {11526,12718}, {11527,12719}, {11528,12720}, {11529,12722}, {11532,12724}, {11533,12725}, {11535,12728}, {11682,12530}, {11899,12727}

X(12652) = midpoint of X(i) and X(j) for these {i,j}: {145,9801}, {7996,11531}
X(12652) = reflection of X(i) in X(j) for these (i,j): (1721,1), (7991,1766)
X(12652) = X(317)-of-excenters-reflections-triangle
X(12652) = excentral-to-excenters-reflections similarity image of X(1721)
X(12652) = excenters-reflections-isotomic conjugate of X(12650)
X(12652) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (390,2263,1), (1448,12575,1), (4318,4319,1), (4320,9785,1)

X(12653) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS TO FUHRMANN

Trilinears a^3-3*(b+c)*a^2-(b^2-11*b*c+c^2)*a+(b+c)*(3*b^2-8*b*c+3*c^2) : :
X(12653) = 3*X(1)-2*X(100) = 5*X(1)-4*X(214) = 4*X(11)-3*X(3679) = 5*X(100)-6*X(214) = X(100)-3*X(1320) = 4*X(100)-3*X(5541) = 3*X(165)-4*X(11715) = 2*X(214)-5*X(1320) = 3*X(1768)-4*X(12773) = 3*X(6264)-2*X(12773)

The reciprocal orthologic center of these triangles is X(8).

X(12653) lies on these lines: {1,88}, {8,11524}, {11,3679}, {40,12737}, {80,3632}, {104,7991}, {119,11522}, {145,9802}, {149,519}, {153,4301}, {165,11715}, {191,956}, {517,1768}, {528,3243}, {952,3627}, {1023,4919}, {1145,1698}, {1317,3340}, {1387,3624}, {1482,6326}, {1699,12751}, {2093,10074}, {2170,4752}, {2771,8148}, {2800,3901}, {2829,9589}, {2932,5563}, {3057,5251}, {3244,6224}, {3577,5660}, {3656,11698}, {3884,5506}, {3894,11571}, {3899,5223}, {4413,6797}, {4677,10707}, {4816,12019}, {5531,10698}, {5881,10738}, {6154,11034}, {6713,9588}, {6762,11256}, {9612,12749}, {9898,12654}, {9945,11518}, {9951,11519}, {9963,11520}, {10265,12245}, {10825,11521}, {11523,12690}, {11526,12730}, {11527,12733}, {11528,12734}, {11532,12744}, {11533,12746}, {11535,12748}, {11682,12531}, {12409,12657}

X(12653) = midpoint of X(i) and X(j) for these {i,j}: {145,9802}, {7993,11531}
X(12653) = reflection of X(i) in X(j) for these (i,j): (1,1320), (40,12737), (153,4301), (1768,6264), (3632,80), (4677,10707), (5531,10698), (5541,1), (5881,10738), (6154,12735), (6224,3244), (6326,1482), (6762,11256), (7991,104), (9897,149), (12245,10265)
X(12653) = X(74)-of-excenters-reflections-triangle
X(12653) = excentral-to-excenters-reflections similarity image of X(5541)
X(12653) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (244,10700,1), (4792,10700,244), (5531,11224,10698)

X(12654) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS TO HUTSON EXTOUCH

Trilinears a^6-2*(b+c)*a^5-(b+c)^2*a^4+4*(b+c)*(b^2+b*c+c^2)*a^3-(b^4+c^4-2*b*c*(4*b^2-23*b*c+4*c^2))*a^2-2*(b^2-4*b*c+c^2)*(b+c)*(b^2+6*b*c+c^2)*a+(b^2-6*b*c+c^2)*(b^2-c^2)^2 : :
X(12654) = 3*X(1)-2*X(12521) = 4*X(12521)-3*X(12658)

The reciprocal orthologic center of these triangles is X(3555).

X(12654) lies on these lines: {1,12521}, {72,4853}, {145,9804}, {200,9624}, {936,1387}, {1482,12670}, {3090,4882}, {3243,5784}, {3679,12620}, {3680,7160}, {4002,10582}, {5920,7962}, {6264,10609}, {6765,11374}, {7991,12516}, {8001,11531}, {9898,12653}, {9953,11519}, {11224,12756}, {11518,12439}, {11520,12537}, {11521,12552}, {11522,12612}, {11523,12692}, {11525,12260}, {11530,12631}, {11682,12533}

X(12654) = midpoint of X(i) and X(j) for these {i,j}: {145,9804}, {8001,11531}
X(12654) = reflection of X(i) in X(j) for these (i,j): (7991,12516), (12658,1)

X(12655) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS TO MANDART-EXCIRCLES

Trilinears 4*p^7*(p-4*q)+4*(4*q^2-1)*p^6-2*(8*q^2-17)*q*p^5+6*(2*q^4-5*q^2-2)*p^4+2*(7*q^2-2)*q*p^3-2*(3*q^4-q^2-5)*p^2+(2*q^2-7)*q*p+q^2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(12655) = 3*X(1)-2*X(12522) = 5*X(11522)-4*X(12613) = 4*X(12522)-3*X(12659)

The reciprocal orthologic center of these triangles is X(3555).

X(12655) lies on these lines: {1,12522}, {145,12542}, {3679,12621}, {7991,12517}, {11518,12442}, {11519,12449}, {11520,12538}, {11521,12553}, {11522,12613}, {11523,12693}, {11682,12534}

X(12655) = midpoint of X(145) and X(12542)
X(12655) = reflection of X(i) in X(j) for these (i,j): (7991,12517), (12659,1)

X(12656) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS TO MIDARC

Trilinears (-a+3*b+3*c)*sin(A/2)+(a-3*b+c)*sin(B/2)+(a+b-3*c)*sin(C/2) : :
X(12656) = 3*X(1)-2*X(12523) = 3*X(164)-4*X(12523) = 3*X(3679)-4*X(12622)

The reciprocal orthologic center of these triangles is X(1).

X(12656) lies on these lines: {1,164}, {145,9807}, {167,11531}, {177,3340}, {3679,12622}, {7670,11526}, {7962,8422}, {7991,12518}, {11519,12450}, {11520,12539}, {11521,12554}, {11682,11691}

X(12656) = midpoint of X(i) and X(j) for these {i,j}: {145,9807}, {167,11531}
X(12656) = reflection of X(i) in X(j) for these (i,j): (164,1), (7991,12518)
X(12656) = X(1)-of-excenters-reflections-triangle
X(12656) = excentral-to-excenters-reflections similarity image of X(164)
X(12656) = orthologic center of these triangles: excenters-reflections to 2nd midarc

X(12657) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS TO 1st SCHIFFLER

Trilinears 16*p^5*(p-5*q)+8*(14*q^2-5)*p^4-16*(3*q^2-5)*q*p^3-(56*q^2-17)*p^2+(16*q^2-19)*q*p+3-q/p : :
where p=sin(A/2), q=cos((B-C)/2)
X(12657) = 3*X(1)-2*X(12524) = 4*X(12524)-3*X(12660)

The reciprocal orthologic center of these triangles is X(21).

X(12657) lies on these lines: {1,6597}, {145,12543}, {3679,12623}, {3680,10266}, {6599,10950}, {7991,12519}, {11518,12444}, {11519,12451}, {11520,12540}, {11521,12557}, {11522,12615}, {11523,12695}, {11525,12267}, {11530,12639}, {11682,12535}, {12409,12653}

X(12657) = midpoint of X(145) and X(12543)
X(12657) = reflection of X(i) in X(j) for these (i,j): (7991,12519), (12660,1)

X(12658) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTRAL TO HUTSON EXTOUCH

Trilinears a^6-2*(b+c)*a^5-(b^2+18*b*c+c^2)*a^4+4*(b+c)*(b^2+3*b*c+c^2)*a^3-(b^4+c^4-2*b*c*(8*b^2+33*b*c+8*c^2))*a^2-2*(b+c)*(b^2+c^2)*(b^2+6*b*c+c^2)*a+(b^2-c^2)^2*(b+c)^2 : :
X(12658) = 3*X(165)-X(8001) = 3*X(165)-2*X(12516) = 3*X(1699)-4*X(12612) = 4*X(12521)-X(12654)

The reciprocal orthologic center of these triangles is X(3555).

X(12658) lies on these lines: {1,12521}, {2,9804}, {9,3295}, {40,3555}, {57,12439}, {63,12533}, {145,8726}, {165,8001}, {191,9898}, {200,3646}, {942,2136}, {962,1490}, {1697,5920}, {1698,12620}, {1699,12612}, {1764,12552}, {2951,6361}, {3174,5542}, {3339,5083}, {5531,11379}, {8580,9953}, {9776,9874}

X(12658) = midpoint of X(12533) and X(12537)
X(12658) = reflection of X(i) in X(j) for these (i,j): (1,12521), (7160,12631), (8001,12516), (12654,1)
X(12658) = complement of X(9804)
X(12658) = Ursa-minor-to-excentral similarity image of X(17639)

X(12659) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTRAL TO MANDART-EXCIRCLES

Trilinears 2*p^5*(2*p^3-6*p+q)-2*(2*q^4-q^2-8)*p^4-2*(q^2+2)*q*p^3+2*(q^4+q^2-5)*p^2-(2*q^2-7)*q*p-q^2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(12659) = 3*X(165)-2*X(12517) = 5*X(1698)-4*X(12621) = 3*X(1699)-4*X(12613) = 4*X(12522)-X(12655)

The reciprocal orthologic center of these triangles is X(3555).

X(12659) lies on these lines: {1,12522}, {2,12542}, {9,12693}, {40,1739}, {57,12442}, {63,12534}, {165,12517}, {1698,12621}, {1699,12613}, {1731,1766}, {1764,12553}, {5709,6361}, {8580,12449}

X(12659) = midpoint of X(12534) and X(12538)
X(12659) = reflection of X(i) in X(j) for these (i,j): (1,12522), (12655,1)
X(12659) = complement of X(12542)

X(12660) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTRAL TO 1st SCHIFFLER

Trilinears (-a+b+c)*(a^8-2*(2*b^2-b*c+2*c^2)*a^6+2*b*c*(b+c)*a^5+(6*b^4+6*c^4-b*c*(4*b^2-13*b*c+4*c^2))*a^4-2*b*c*(2*b-c)*(b-2*c)*(b+c)*a^3-(4*b^6+4*c^6-b*c*(2*b^2-b*c+c^2)*(b^2-b*c+2*c^2))*a^2+2*(b^2-c^2)*(b-c)^3*b*c*a+(b^2-c^2)^4) : :
X(12660) = 3*X(165)-2*X(12519) = 3*X(1699)-4*X(12615) = 4*X(12524)-X(12657)

The reciprocal orthologic center of these triangles is X(21).

X(12660) lies on these lines: {1,6597}, {2,12543}, {5,6599}, {9,10266}, {57,12444}, {63,12535}, {165,12519}, {191,12409}, {1698,12623}, {1699,12615}, {1764,12557}, {2949,6907}, {2950,10942}, {3646,12267}, {3871,6595}, {5506,7483}, {6326,6906}, {8580,12451}

X(12660) = midpoint of X(12535) and X(12540)
X(12660) = reflection of X(i) in X(j) for these (i,j): (1,12524), (10266,12639), (12657,1)
X(12660) = complement of X(12543)

X(12661) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXTANGENTS TO 1st HYACINTH

Trilinears (-a^2+b^2+c^2)*(a^13+(b+c)*a^12-2*(b^2+c^2)*a^11-2*(b+c)*(b^2+c^2)*a^10-(b^4+c^4+b*c*(2*b^2-3*b*c+2*c^2))*a^9-(b+c)*(b^4+c^4-b*c*(2*b^2+3*b*c+2*c^2))*a^8+(4*b^6+4*c^6+(5*b^4+5*c^4-b*c*(b+c)^2)*b*c)*a^7+(b+c)*(4*b^6+4*c^6-(5*b^4+5*c^4+b*c*(b-c)^2)*b*c)*a^6-(b^6+c^6+(b^2-c^2)^2*b*c)*(b+c)^2*a^5-(b^2-c^2)*(b-c)*(b^6+c^6-(b^2-c^2)^2*b*c)*a^4-(b^2-c^2)^2*(b+c)^2*(2*b^4+2*c^4-b*c*(3*b^2-b*c+3*c^2))*a^3-(b^2-c^2)^3*(b-c)*(2*b^4+2*c^4+b*c*(3*b^2+b*c+3*c^2))*a^2+(b^3+c^3)*(b+c)*(b^2-c^2)^4*a+(b^3-c^3)*(b^2-c^2)^5) : :

The reciprocal orthologic center of these triangles is X(10112).

X(12661) lies on the extangents circle and these lines: {19,113}, {40,12407}, {55,2931}, {65,5504}, {71,265}, {74,3101}, {110,6197}, {125,8251}, {146,9536}, {2550,12319}, {2948,9572}, {3448,9537}, {5584,12302}, {5663,6254}, {6699,10319}, {8539,12596}, {9573,9904}, {10306,12310}, {10636,10663}, {10637,10664}, {11406,12168}, {11428,12228}, {11445,12273}, {11460,12284}, {11471,12295}

X(12661) = reflection of X(10119) in X(8141)
X(12661) = antipode of X(10119) in extangents circle

X(12661) = X(104)-of-extangents-triangle if ABC is acute
X(12661) = orthic-to-extangents similarity image of X(113)

X(12662) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXTANGENTS TO LUCAS ANTIPODAL

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

The reciprocal orthologic center of these triangles is X(3).

X(12662) lies on these lines: {19,487}, {40,9906}, {486,10319}, {642,9816}, {2550,12320}, {3101,12221}, {3564,12663}, {5584,12303}, {8251,12601}, {8539,12597}, {10306,12311}, {11406,12169}, {11428,12229}, {11435,12237}, {11445,12274}, {11460,12285}, {11471,12296}

X(12662) = reflection of X(12910) in X(12978)
X(12662) = orthic-to-extangents similarity image of X(487)

X(12663) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXTANGENTS TO LUCAS(-1) ANTIPODAL

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

The reciprocal orthologic center of these triangles is X(3).

X(12663) lies on these lines: {19,488}, {40,9907}, {485,10319}, {641,9816}, {2550,12321}, {3101,12222}, {3564,12662}, {5584,12304}, {8251,12602}, {8539,12598}, {10306,12312}, {11406,12170}, {11428,12230}, {11435,12238}, {11445,12275}, {11460,12286}, {11471,12297}

X(12663) = reflection of X(12911) in X(12979)
X(12663) = orthic-to-extangents similarity image of X(488)

X(12664) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH TO EXTOUCH

Trilinears (b+c)*(a^2-b^2+c^2)*(a^2+b^2-c^2)*(a^4+(b^2-c^2)^2)-2*a^6*(b^2+c^2)*(a+b+c)+2*(3*b^4-2*b^2*c^2+3*c^4)*a^5-2*a*(b^2+c^2)*(b^2-c^2)^2*(3*a^2-(b+c)*a-b^2-c^2) : :
X(12664) = 3*X(210)-2*X(11500) = 3*X(5927)-2*X(6260) = 4*X(6705)-3*X(10167)

Let A'B'C' be the orthic triangle. X(12664) is the radical center of the Bevan circles of triangles AB'C', BC'A', CA'B'. (Randy Hutson, July 31 2018)

The reciprocal orthologic center of these triangles is X(72).

X(12664) lies on these lines: {2,9942}, {3,9}, {4,65}, {19,9786}, {33,1498}, {46,1750}, {64,1753}, {72,515}, {185,1824}, {210,11500}, {329,6836}, {388,3427}, {389,1871}, {405,6261}, {442,5927}, {517,5924}, {518,5758}, {912,5787}, {942,5715}, {946,5728}, {950,12672}, {960,6987}, {999,12687}, {1012,10393}, {1064,10396}, {1158,7580}, {1478,12677}, {1532,10395}, {1699,10399}, {1708,3149}, {1709,11507}, {1848,12233}, {1861,6247}, {1872,6000}, {1890,11745}, {2261,11425}, {2646,12114}, {2800,12690}, {2829,12691}, {2900,10306}, {3059,5759}, {3487,8581}, {3488,9848}, {3651,5918}, {3812,6843}, {4185,12136}, {4199,12683}, {5658,6889}, {5794,12667}, {5811,12666}, {5842,7957}, {6259,6917}, {6705,7483}, {6847,10391}, {6908,9943}, {6910,11220}, {6934,12246}, {7971,9856}, {8079,8095}, {8080,8096}, {8226,12608}, {8232,12669}, {8233,12681}, {10445,10974}, {10888,12547}, {11523,12650}

X(12664) = midpoint of X(9799) and X(12528)
X(12664) = reflection of X(i) in X(j) for these (i,j): (1490,5777), (7971,9856), (9960,9942), (12671,3), (12680,12114)
X(12664) = anticomplement of X(9942)
X(12664) = complement of X(9960)
X(12664) = X(68)-of-2nd-extouch-triangle
X(12664) = excentral-to-2nd-extouch similarity image of X(1490)
X(12664) = 2nd-extouch-isotomic conjugate of X(12689)
X(12664) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (389,1871,2262), (1864,12688,4), (6260,12616,442)

X(12665) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXTOUCH TO INNER-GARCIA

Trilinears (a^5-(b+c)*a^4-(2*b^2-3*b*c+2*c^2)*a^3+(b+c)*(2*b^2-b*c+2*c^2)*a^2+(b^4+c^4-3*b*c*(b^2+c^2))*a-(b^2-c^2)*(b^3-c^3))*((b+c)*a^3-(b^2+c^2)*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2) : :
X(12665) = 3*X(5587)-2*X(12736)

The reciprocal orthologic center of these triangles is X(12666).

X(12665) lies on the Mandart hyperbola and these lines: {8,153}, {9,48}, {11,5777}, {40,12059}, {72,2829}, {100,1158}, {119,912}, {200,2950}, {518,1537}, {952,1898}, {971,6068}, {1145,6001}, {1388,6265}, {1768,10270}, {1858,10956}, {2802,5881}, {3086,5083}, {3419,12761}, {3711,12515}, {3811,12775}, {5217,12738}, {5534,10087}, {5587,12736}, {5660,5770}, {5720,10090}, {5854,12672}, {6797,9947}, {7330,10058}

X(12665) = midpoint of X(i) and X(j) for these {i,j}: {100,12528}, {153,12532}, {5693,12751}
X(12665) = reflection of X(i) in X(j) for these (i,j): (40,14740), (11,5777), (6797,9947), (11570,119), (12757,6326), (12758,5887)
X(12665) = antipode of X(40) in the Mandart hyperbola

X(12666) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GARCIA TO EXTOUCH

Trilinears (b+c)*a^8-(2*b^2+b*c+2*c^2)*a^7-(2*b-c)*(b-2*c)*(b+c)*a^6+(6*b^4+6*c^4-b*c*(3*b^2-2*b*c+3*c^2))*a^5-b*c*(b+c)*(9*b^2-14*b*c+9*c^2)*a^4-(6*b^4+6*c^4+b*c*(3*b^2-2*b*c+3*c^2))*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(2*b^4+2*c^4+b*c*(7*b^2-2*b*c+7*c^2))*a^2+(b^2-c^2)^2*(2*b^4+2*c^4-b*c*(5*b^2-2*b*c+5*c^2))*a-(b^3-c^3)*(b^2-c^2)^3 : :

The reciprocal orthologic center of these triangles is X(12665).

X(12666) lies on the Feuerbach hyperbola of the inner Garcia triangle and on these lines: {8,6001}, {40,12059}, {84,997}, {90,104}, {165,191}, {515,5697}, {971,5698}, {1737,6260}, {1898,5768}, {2771,6259}, {2800,3632}, {2801,7971}, {2829,3869}, {3062,12650}, {3419,12676}, {3811,12686}, {5693,6737}, {5811,12664}, {6256,10573}, {9961,11500}, {12688,12701}

X(12666) = reflection of X(9961) in X(11500)
X(12665) = antipode of X(9) in the Mandart hyperbola
X(12665) = extouch-isogonal conjugate of X(13528)

X(12667) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO EXTOUCH

Barycentrics a^7-(b+c)*a^6-(b^2-10*b*c+c^2)*a^5+(b+c)*(b^2-6*b*c+c^2)*a^4-(b^2+6*b*c+c^2)*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)*a^2+(b^2-c^2)^2*(b^2-6*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :
X(12667) = 3*X(153)-2*X(12762) = 5*X(631)-4*X(5450) = X(944)-3*X(5658) = 2*X(1158)-3*X(5657) = 5*X(1698)-4*X(6705) = 3*X(5587)-2*X(6245) = 3*X(5587)-X(10864) = 3*X(5603)-4*X(12608) = 3*X(5657)-X(12246) = 3*X(5658)-2*X(6261)

The reciprocal orthologic center of these triangles is X(40).

X(12667) lies on these lines: {1,4}, {2,12114}, {3,1603}, {7,7686}, {8,6001}, {10,84}, {12,6847}, {20,100}, {30,10306}, {36,6927}, {40,2123}, {46,2096}, {56,6848}, {65,12678}, {72,12677}, {80,10305}, {104,6834}, {119,6891}, {149,12761}, {354,5804}, {355,971}, {376,6796}, {377,9799}, {443,5587}, {498,6935}, {499,6969}, {517,6259}, {519,7971}, {631,5251}, {938,12675}, {958,6908}, {960,5811}, {962,3885}, {993,6988}, {1012,3085}, {1125,6939}, {1158,5657}, {1329,6926}, {1385,6893}, {1466,7354}, {1498,9370}, {1512,1788}, {1532,3086}, {1538,11373}, {1698,6705}, {1709,10039}, {1737,10085}, {1837,5768}, {2478,5731}, {2800,5904}, {2975,6838}, {3057,12679}, {3146,5842}, {3149,4293}, {3189,5534}, {3333,7682}, {3419,12777}, {3529,5537}, {3576,5084}, {3577,3671}, {3616,6957}, {3679,7992}, {3822,6855}, {4297,6700}, {5080,6836}, {5082,5881}, {5086,9960}, {5090,12136}, {5176,9961}, {5218,6906}, {5223,11362}, {5234,6684}, {5252,12688}, {5253,6953}, {5261,7680}, {5274,10893}, {5552,6909}, {5660,6903}, {5687,12330}, {5688,6257}, {5689,6258}, {5787,6826}, {5790,12684}, {5794,12664}, {5818,6897}, {6831,10590}, {6833,10588}, {6844,10895}, {6845,10599}, {6851,10526}, {6864,9843}, {6888,10585}, {6890,11681}, {6928,10742}, {6930,10267}, {6932,10527}, {6938,11491}, {6941,10589}, {6942,12248}, {6944,10269}, {6948,11499}, {6956,7951}, {7373,7956}, {7501,8185}, {7966,12575}, {8193,9910}, {8197,12456}, {8204,12457}, {8727,9654}, {9857,12496}, {10431,11015}, {10791,12196}, {10902,11111}, {10914,12676}, {10915,12686}, {10916,12687}, {11900,12668}

X(12667) = reflection of X(i) in X(j) for these (i,j): (1,6260), (4,6256), (20,11500), (84,10), (149,12761), (944,6261), (3189,5534), (6851,10526), (10864,6245), (12246,1158), (12650,946), (12680,9942)
X(12667) = anticomplement of X(12114)
X(12667) = X(84)-of-outer-Garcia-triangle
X(12667) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,944,497), (4,1056,946), (4,7967,10531), (4,10805,5603), (4,12115,388), (4,12116,5225), (20,153,3436), (20,7080,10310), (104,6834,7288), (355,6850,2550), (944,5658,6261), (1478,5691,4), (1478,10572,10629), (1837,12680,5768), (5587,10864,6245), (5657,12246,1158), (6906,10786,5218), (6941,10785,10589), (10572,10629,497)

X(12668) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO EXTOUCH

Trilinears (2*cos(A)-cos(B-C))*(16*q*p^9+16*p^8+8*(2*q^2-5)*q*p^7-8*(2*q^4-8*q^2+7)*p^6-(16*q^4+8*q^2-33)*q*p^5+(48*q^4-120*q^2+73)*p^4+(16*q^4-8*q^2-9)*q*p^3-6*(q^2-1)*(6*q^2-7)*p^2-5*(q^2-1)*q^3*p+9*(q-1)^2*(q+1)^2) : :
where p=sin(A/2), q=cos((B-C)/2)
X(12668) = 2*X(6245)-3*X(11897) = X(7992)-3*X(11852) = 3*X(11831)-2*X(12114) = 3*X(11845)-X(12246) = 3*X(11911)-X(12684)

The reciprocal orthologic center of these triangles is X(40).

X(12668) lies on these lines: {30,1490}, {84,402}, {515,12626}, {971,11251}, {1650,6260}, {1709,11912}, {2829,12729}, {6001,12438}, {6245,11897}, {6257,11902}, {6258,11901}, {7971,11910}, {7992,11852}, {9910,11853}, {10085,11913}, {11831,12114}, {11832,12136}, {11839,12196}, {11845,12246}, {11848,12330}, {11885,12496}, {11900,12667}, {11903,12676}, {11904,12677}, {11905,12678}, {11906,12679}, {11909,12680}, {11911,12684}, {11914,12686}, {11915,12687}

X(12668) = reflection of X(i) in X(j) for these (i,j): (84,402), (1650,6260)
X(12668) = X(84)-of-Gossard-triangle

X(12669) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO EXTOUCH

Trilinears (-10*sin(A/2)+10*sin(3*A/2)+2*sin(5*A/2))*cos((B-C)/2)+(-6*cos(A)-4*cos(2*A)-6)*cos(B-C)-2*sin(3*A/2)*cos(3*(B-C)/2)+14*cos(A)+cos(2*A)+1 : :
X(12669) = 4*X(946)-5*X(11025) = 2*X(5732)-3*X(11220) = 3*X(7671)-2*X(11372) = 3*X(8236)-2*X(12672) = 3*X(11038)-4*X(12675)

The reciprocal orthologic center of these triangles is X(72).

X(12669) lies on these lines: {4,7}, {9,6986}, {20,518}, {21,10085}, {63,100}, {84,1803}, {142,6991}, {390,6001}, {515,7672}, {516,3868}, {517,12630}, {912,5759}, {946,11025}, {1158,7676}, {1445,1490}, {2800,7673}, {2829,12755}, {3059,9943}, {3475,8581}, {3873,10431}, {4197,7705}, {4326,7992}, {5273,10167}, {5542,11020}, {5572,11036}, {5779,6883}, {5817,6887}, {5851,9964}, {6261,7677}, {6839,12678}, {7671,11372}, {7678,12608}, {7679,12616}, {8095,8387}, {8096,8388}, {8232,12664}, {8236,12672}, {8237,12681}, {8238,12683}, {8389,12685}, {8732,9942}, {9948,10865}, {10429,11037}, {10889,12547}, {11038,12675}, {11526,12650}

X(12669) = reflection of X(i) in X(j) for these (i,j): (3059,9943), (12528,9), (12688,5572)
X(12669) = X(68)-of-Honsberger-triangle
X(12669) = excentral-to-Honsberger similarity image of X(1490)
X(12669) = Honsberger-isotomic conjugate of X(12718)

X(12670) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXTOUCH TO HUTSON EXTOUCH

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

he reciprocal orthologic center of these triangles is X(12671).

X(12670) lies on these lines: {8,6835}, {9,3295}, {40,12671}, {1482,12654}, {3057,5920}, {3870,12260}, {8000,9623}

X(12670) = midpoint of X(i) and X(j) for these {i,j}: {9874,12533}, {12756,12777}

X(12671) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON EXTOUCH TO EXTOUCH

Trilinears (b+c)*a^8-2*(b-c)^2*a^7-2*(b+c)^3*a^6+2*(3*b^2+2*b*c+3*c^2)*(b-c)^2*a^5+8*(b^3+c^3)*b*c*a^4-2*(b^2+c^2)*(3*b^2+4*b*c+3*c^2)*(b-c)^2*a^3+2*(b^4-c^4)*(b^2+c^2)*(b-c)*a^2+2*(b^4-c^4)^2*a-(b^2-c^2)^4*(b+c) : :
X(12671) = 2*X(1158)-3*X(5918) = 2*X(6245)-3*X(10167)

The reciprocal orthologic center of these triangles is X(12670).

X(12671) lies on these lines: {1,10045}, {3,9}, {4,9942}, {7,12675}, {20,3869}, {40,12670}, {63,11500}, {65,515}, {210,6796}, {377,9799}, {442,6245}, {1012,2646}, {1158,5918}, {1864,3149}, {2096,6934}, {4304,12672}, {5658,6833}, {5715,11018}, {5787,6917}, {5794,6916}, {5882,9850}, {5927,7483}, {6260,6831}, {7675,11496}, {7682,9844}, {10884,12114}

X(12671) = midpoint of X(20) and X(9960)
X(12671) = reflection of X(i) in X(j) for these (i,j): (4,9942), (12664,3), (12688,6261)

X(12672) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON INTOUCH TO EXTOUCH

Trilinears (b+c)*a^5-(b+c)^2*a^4-2*(b+c)*(b^2-3*b*c+c^2)*a^3+2*(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^4-c^4)*(b^2-c^2) : :
X(12672) = 3*X(1)-2*X(12675) = 2*X(3)-3*X(392) = 4*X(5)-3*X(3753) = X(20)-3*X(3877) = 3*X(72)-4*X(5694) = 3*X(210)-2*X(11362) = 3*X(354)-2*X(5884) = 2*X(355)-3*X(5927) = 2*X(5694)-3*X(5887) = 3*X(5927)-X(10914)

The reciprocal orthologic center of these triangles is X(72).

X(12672) lies on these lines: {1,84}, {3,392}, {4,8}, {5,1519}, {10,1532}, {11,65}, {12,12608}, {20,3877}, {40,936}, {55,6261}, {56,1158}, {78,10306}, {104,1476}, {145,12528}, {210,11362}, {354,5884}, {388,12676}, {390,944}, {474,3359}, {515,3057}, {516,3878}, {518,5693}, {758,4301}, {912,1482}, {942,5603}, {950,12664}, {956,7330}, {997,10310}, {1058,5768}, {1064,3931}, {1108,1765}, {1156,1389}, {1319,5450}, {1385,1621}, {1478,10043}, {1479,10051}, {1490,1697}, {1512,3697}, {1538,4002}, {1737,7681}, {1766,5782}, {1768,5563}, {1858,2099}, {1898,10950}, {2057,5687}, {2096,3600}, {2771,7984}, {2778,12371}, {2801,3244}, {2829,12758}, {2943,5293}, {3337,12767}, {3428,12514}, {3556,10829}, {3576,9943}, {3579,6905}, {3585,10057}, {3601,9942}, {3616,6935}, {3656,11240}, {3698,10175}, {3742,9624}, {3754,3817}, {3812,8227}, {3827,12586}, {3873,5734}, {3880,5881}, {3884,4297}, {3890,5731}, {3899,9589}, {3913,12703}, {3916,11249}, {3927,8158}, {3987,5400}, {4004,6830}, {4018,8727}, {4304,12671}, {4313,9960}, {4342,9949}, {4640,11012}, {4848,7682}, {5044,5657}, {5045,10595}, {5119,11500}, {5252,6256}, {5439,5886}, {5440,11248}, {5554,6957}, {5587,5836}, {5691,5697}, {5692,7991}, {5722,10531}, {5780,12702}, {5787,10936}, {5806,6844}, {5818,10157}, {5854,12665}, {5882,5919}, {5901,10202}, {5902,11522}, {5904,11531}, {6245,12053}, {6259,10935}, {6265,12775}, {6949,11231}, {6952,11230}, {6956,10584}, {6969,9780}, {7373,10569}, {7680,10523}, {7701,11260}, {7962,12650}, {8095,8241}, {8096,8242}, {8236,12669}, {8239,12681}, {8240,12683}, {9785,9799}, {9948,10866}, {10366,11254}, {10947,12701}, {11924,12685}

X(12672) = midpoint of X(i) and X(j) for these {i,j}: {145,12528}, {962,3869}, {3057,12688}, {5691,5697}, {5693,7982}, {5904,11531}
X(12672) = reflection of X(i) in X(j) for these (i,j): (4,9856), (8,5777), (40,960), (65,946), (72,5887), (944,9957), (3555,1482), (4297,3884), (10914,355), (12680,5882), (12711,11496)
X(12672) = anticomplement of X(31788)
X(12672) = X(68)-of-Hutson-intouch-triangle
X(12672) = X(12118)-of-intouch-triangle
X(12672) = excentral-to-Hutson-intouch similarity image of X(1490)
X(12672) = Hutson-intouch-isotomic conjugate of X(12721)
X(12672) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1709,12114), (1,1777,1455), (1,12705,1012), (355,12699,10525), (355,12700,3434), (946,12616,11), (962,3434,12700), (1538,9956,6941), (3890,9961,5731), (5603,10785,11373), (5919,12680,5882), (5927,10914,355)

X(12673) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-HUTSON TO EXTOUCH

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=(-a+b+c)*(a^7+(b+c)*a^6-(5*b^2-6*b*c+5*c^2)*a^5-(b^2-c^2)*(b-c)*a^4+(7*b^2+6*b*c+7*c^2)*(b-c)^2*a^3-(b^2-c^2)^2*(b+c)*a^2-(b^2-c^2)^2*(3*b^2-2*b*c+3*c^2)*a+(b^2-c^2)^3*(b-c))*b*c
G(a,b,c)=2*(a-b+c)*(a^2-b^2+c^2)*(a^3-(b-c)*a^2-(b-c)^2*a+(b+c)*(b^2-c^2))*a*b^2*c
H(a,b,c)=-2*S^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))

The reciprocal orthologic center of these triangles is X(72).

X(12673) lies on these lines: {84,266}, {363,1490}, {515,9805}, {946,11026}, {971,12488}, {1071,8113}, {1158,8107}, {6001,9836}, {6261,8109}, {6732,8096}, {7992,8140}, {8377,12608}, {8380,12616}, {9783,9799}, {9942,11854}, {9948,11856}, {9960,11886}, {11685,12528}, {11892,12547}

X(12673) = reflection of X(12674) in X(7992)
X(12673) = X(68)-of-inner-Hutson-triangle
X(12673) = excentral-to-inner-Hutson similarity image of X(1490)
X(12673) = inner-Hutson-isotomic conjugate of X(12719)

X(12674) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-HUTSON TO EXTOUCH

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=(-a+b+c)*(a^7+3*(b+c)*a^6-(5*b^2-6*b*c+5*c^2)*a^5-7*(b^2-c^2)*(b-c)*a^4+7*(b^2-c^2)^2*a^3+(b^2-c^2)*(b-c)*(5*b^2-6*b*c+5*c^2)*a^2-3*(b^2-c^2)^2*(b+c)^2*a-(b^2-c^2)^3*(b-c))*b*c
G(a,b,c)=2*(a-b+c)*(a^6+(b+2*c)*a^5-(4*b^2-5*b*c+5*c^2)*a^4-2*(b-c)*(b^2-2*c^2)*a^3+(b-c)*(5*b^3-7*c^3+b*c*(b-3*c))*a^2+(b^2-c^2)*(b-c)^2*(b-2*c)*a-(2*b^2+b*c+3*c^2)*(b^2-c^2)^2)*a*b*c
H(a,b,c)=2*S^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))

The reciprocal orthologic center of these triangles is X(72).

X(12674) lies on these lines: {4,8372}, {84,7590}, {168,1490}, {515,9806}, {946,11027}, {971,12489}, {1071,8114}, {1158,8108}, {6001,9837}, {6261,8110}, {7992,8140}, {8378,12608}, {8381,12616}, {9787,9799}, {9942,11855}, {9948,11857}, {9960,11887}, {11686,12528}, {11893,12547}

X(12674) = reflection of X(12673) in X(7992)
X(12674) = X(68)-of-outer-Hutson-triangle
X(12674) = excentral-to-outer-Hutson similarity image of X(1490)
X(12674) = outer-Hutson-isotomic conjugate of X(12720)

X(12675) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO EXTOUCH

Trilinears (b+c)*a^5-(b^2-4*b*c+c^2)*a^4-2*(b+c)*(b^2+c^2)*a^3+2*(b^2+c^2)*(b-c)^2*a^2+(b^2-c^2)^2*(b+c)*a-(b^4-c^4)*(b^2-c^2) : :
X(12675) = 3*X(1)-X(12672) = X(4)-3*X(354) = 2*X(5)-3*X(3742) = X(20)+3*X(3873) = X(40)-3*X(10167) = X(72)-3*X(3576) = 4*X(140)-3*X(3740) = 3*X(210)-5*X(631) = 3*X(354)+X(12680) = X(3555)+3*X(10167)

The reciprocal orthologic center of these triangles is X(72).

X(12675) lies on these lines: {1,84}, {3,518}, {4,354}, {5,3742}, {7,12671}, {10,9940}, {20,3873}, {40,3555}, {48,9119}, {52,9037}, {57,11500}, {65,944}, {72,3576}, {104,943}, {140,3740}, {210,631}, {355,3812}, {375,11695}, {376,7957}, {388,5768}, {389,8679}, {392,5693}, {442,12757}, {495,12616}, {496,12608}, {515,942}, {516,3881}, {517,550}, {601,3744}, {602,4641}, {774,4322}, {912,960}, {938,12667}, {946,971}, {950,2829}, {952,5836}, {962,3889}, {999,6261}, {1001,7330}, {1056,9850}, {1125,2801}, {1155,11491}, {1158,3295}, {1279,3073}, {1319,1858}, {1376,5534}, {1458,7138}, {1478,11045}, {1479,11046}, {1490,3333}, {1656,3848}, {1768,3746}, {1836,11048}, {1837,11047}, {1864,3086}, {1898,11376}, {2096,4294}, {2771,5609}, {2800,9957}, {2810,9729}, {3057,4305}, {3149,3338}, {3158,10270}, {3243,6769}, {3359,3913}, {3428,10884}, {3475,6847}, {3487,8581}, {3522,4430}, {3523,3681}, {3579,10178}, {3616,12528}, {3753,5881}, {3868,5731}, {3870,10310}, {3892,4301}, {3916,10902}, {3928,10268}, {4292,5173}, {4640,10267}, {4719,5396}, {5044,10165}, {5049,9856}, {5123,10942}, {5252,10805}, {5290,10894}, {5302,6883}, {5439,5587}, {5570,10572}, {5603,12688}, {5691,11034}, {5722,6256}, {5887,10246}, {5904,7987}, {5918,6361}, {5927,8227}, {6245,7680}, {6260,7681}, {6684,11227}, {6907,10916}, {7580,12704}, {7966,7991}, {8550,9004}, {9047,10625}, {9799,11020}, {9844,10893}, {9845,11529}, {9947,10175}, {9948,11035}, {9956,10265}, {9960,11036}, {10531,12679}, {10569,10864}, {10785,11375}, {10806,12701}, {11038,12669}, {11042,12681}, {11043,12683}, {12564,12577}

X(12675) = midpoint of X(i) and X(j) for these {i,j}: {4,12680}, {40,3555}, {65,944}, {3874,4297}, {5882,5884}
X(12675) = reflection of X(i) in X(j) for these (i,j): (10,9940), (355,3812), (942,12005), (946,5045), (960,1385), (5777,1125), (7680,11018), (7686,942)
X(12675) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,84,11496), (1,10085,1012), (1,10391,12710), (354,12680,4), (355,10202,3812), (3243,9841,6769), (3555,10167,40), (3889,11220,962), (6260,11019,7681), (9845,11529,12650)
X(12675) = X(68)-of-incircle-circles-triangle
X(12675) = excentral-to-incircle-circles similarity image of X(1490)
X(12675) = incircle-circles-isotomic conjugate of X(12722)

X(12676) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO EXTOUCH

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

The reciprocal orthologic center of these triangles is X(40).

X(12676) lies on these lines: {4,10309}, {11,84}, {12,12686}, {355,5836}, {388,12672}, {515,10912}, {946,999}, {971,10525}, {1158,1329}, {1376,6260}, {1490,11826}, {1709,9612}, {2829,12699}, {3419,12666}, {5880,12616}, {6245,10893}, {6257,10920}, {6258,10919}, {7704,10785}, {7971,10944}, {7992,10826}, {9910,10829}, {10085,10948}, {10794,12196}, {10871,12496}, {10914,12667}, {10947,12680}, {10949,12687}, {11390,12136}, {11865,12456}, {11866,12457}, {11903,12668}, {11928,12684}

X(12676) = midpoint of X(4) and X(10309)
X(12676) = X(84)-of-inner-Johnson-triangle
X(12676) = reflection of X(i) in X(j) for these (i,j): (12330,6260), (12677,6259)

X(12677) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO EXTOUCH

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

The reciprocal orthologic center of these triangles is X(40).

X(12677) lies on these lines: {4,5173}, {11,12687}, {12,84}, {72,12667}, {355,5836}, {515,5812}, {958,6260}, {971,10526}, {1478,12664}, {1490,11827}, {1709,10954}, {2829,12738}, {5080,9960}, {6244,11500}, {6245,10894}, {6257,10922}, {6258,10921}, {7971,10950}, {7992,10827}, {9910,10830}, {10085,10523}, {10786,12246}, {10795,12196}, {10872,12496}, {10953,12680}, {10955,12686}, {11374,12114}, {11391,12136}, {11867,12456}, {11868,12457}, {11904,12668}, {11929,12684}

X(12677) = reflection of X(12676) in X(6259)
X(12677) = X(84)-of-outer-Johnson-triangle

X(12678) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO EXTOUCH

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

The reciprocal orthologic center of these triangles is X(40).

X(12678) lies on these lines: {1,6259}, {4,354}, {5,10085}, {12,84}, {56,6260}, {65,12667}, {210,6916}, {495,1709}, {515,1836}, {518,6925}, {944,5048}, {971,1478}, {1155,2096}, {1483,12699}, {1490,7354}, {1538,10072}, {1768,11698}, {2801,3419}, {2829,12739}, {3085,12246}, {3436,9943}, {3576,4679}, {3585,5787}, {3742,6957}, {3877,9809}, {4293,5658}, {4860,7682}, {5080,11220}, {5229,9799}, {5252,6001}, {5534,11826}, {5584,12527}, {5691,11529}, {5794,12528}, {6245,10895}, {6253,9579}, {6257,10924}, {6258,10923}, {6839,12669}, {7971,10944}, {7992,9578}, {8273,12572}, {9612,10864}, {9614,9845}, {9654,12684}, {9910,10831}, {10797,12196}, {10873,12496}, {10956,12686}, {10957,12687}, {11375,12114}, {11376,12608}, {11392,12136}, {11501,12330}, {11869,12456}, {11870,12457}, {11905,12668}

X(12678) = reflection of X(i) in X(j) for these (i,j): (1709,495), (5252,12115)
X(12678) = {X(1), X(6259)}-harmonic conjugate of X(12679)
X(12678) = X(84)-of-1st-Johnson-Yff-triangle

X(12679) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO EXTOUCH

Barycentrics a^7-2*(b-c)^2*a^5-(b^2-c^2)*(b-c)*a^4+(b^2-c^2)^2*a^3+2*(b^4-c^4)*(b-c)*a^2-4*(b^2-c^2)^2*b*c*a-(b^2-c^2)^3*(b-c) : :
X(12679) = 2*X(496)-3*X(1699) = 3*X(1699)-X(10085)

The reciprocal orthologic center of these triangles is X(40).

X(12679) lies on these lines: {1,6259}, {3,4679}, {4,65}, {5,1709}, {11,84}, {12,12705}, {55,6260}, {64,1842}, {79,3062}, {210,5811}, {329,7957}, {480,516}, {496,1699}, {499,1538}, {515,2098}, {946,3304}, {952,3627}, {960,6925}, {962,3880}, {971,1479}, {1012,11375}, {1155,6848}, {1158,1532}, {1456,7952}, {1478,9856}, {1490,6284}, {1519,11376}, {1547,1892}, {1750,5812}, {1854,1877}, {2478,9943}, {2829,12740}, {3057,12667}, {3086,12246}, {3091,5880}, {3146,5057}, {3338,7956}, {3583,5787}, {3683,6908}, {3812,6957}, {3838,6837}, {3868,9809}, {4294,5658}, {4640,6838}, {5046,9961}, {5087,6890}, {5221,7682}, {5225,9799}, {5252,6256}, {5556,10429}, {5584,12572}, {5715,7965}, {5720,11826}, {5881,12700}, {5918,6865}, {6245,7702}, {6257,10926}, {6258,10925}, {7971,10950}, {7992,9581}, {9612,11372}, {9614,10864}, {9669,12684}, {9797,9812}, {9910,10832}, {10531,12675}, {10798,12196}, {10863,12436}, {10874,12496}, {10958,12686}, {10959,12687}, {11113,12520}, {11393,12136}, {11502,12330}, {11871,12456}, {11872,12457}, {11906,12668}

X(12679) = reflection of X(i) in X(j) for these (i,j): (1837,4), (10085,496)
X(12679) = X(84)-of-2nd-Johnson-Yff-triangle
X(12679) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,6259,12678), (1699,10085,496)

X(12680) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO EXTOUCH

Trilinears (b+c)*a^5-(b^2-6*b*c+c^2)*a^4-2*(b+c)*(b^2+c^2)*a^3+2*(b^2+c^2)*(b-c)^2*a^2+(b^2-c^2)^2*(b+c)*a-(b^2-c^2)^2*(b+c)^2 : :
X(12680) = 4*X(3)-3*X(210) = 2*X(4)-3*X(354) = X(8)-3*X(11220) = 2*X(10)-3*X(10167) = 2*X(40)-3*X(5918) = 3*X(65)-4*X(5884) = 3*X(354)-4*X(12675) = 4*X(355)-5*X(3698) = 2*X(960)-3*X(5731) = 3*X(5731)-X(12528)

The reciprocal orthologic center of these triangles is X(40).

X(12680) lies on these lines: {1,971}, {3,210}, {4,354}, {8,9859}, {9,8273}, {10,10167}, {11,6260}, {12,6245}, {20,518}, {33,12136}, {40,5918}, {48,1903}, {55,84}, {56,1490}, {65,515}, {72,2801}, {145,9961}, {185,8679}, {198,963}, {200,9841}, {227,7004}, {355,3698}, {388,9799}, {516,3555}, {517,1657}, {912,3962}, {942,4355}, {944,3057}, {952,3893}, {956,12520}, {958,5784}, {960,5731}, {1040,9370}, {1056,12710}, {1125,5927}, {1155,11500}, {1208,2183}, {1319,1898}, {1385,5259}, {1478,5787}, {1479,6259}, {1697,7992}, {1698,5789}, {1699,5045}, {1709,3295}, {1750,3333}, {1768,3579}, {1837,5768}, {2098,7971}, {2099,12650}, {2310,4322}, {2646,12114}, {2829,12743}, {2951,8001}, {3059,5584}, {3086,5658}, {3091,3742}, {3146,3873}, {3243,12651}, {3303,12705}, {3486,9960}, {3522,3681}, {3523,3740}, {3576,5777}, {3600,10394}, {3624,10157}, {3683,7330}, {3689,5534}, {3697,10164}, {3748,11496}, {3848,5056}, {3889,9812}, {3983,6684}, {4298,5728}, {4430,5059}, {4662,10178}, {4679,5811}, {5044,7987}, {5049,11522}, {5173,9579}, {5290,11018}, {5302,6986}, {5432,6705}, {5572,11037}, {5587,9940}, {5882,5919}, {5889,9037}, {6257,10928}, {6258,10927}, {6744,10569}, {6762,12565}, {6765,10860}, {8580,9858}, {9844,11019}, {9910,10833}, {10106,12711}, {10443,10823}, {10480,12547}, {10544,12721}, {10799,12196}, {10877,12496}, {10947,12676}, {10953,12677}, {10965,12686}, {10966,12687}, {11873,12456}, {11874,12457}, {11909,12668}

X(12680) = midpoint of X(145) and X(9961)
X(12680) = reflection of X(i) in X(j) for these (i,j): (4,12675), (8,9943), (72,4297), (3057,944), (3059,5732), (5691,942), (6253,4292), (7957,20), (9848,9845), (12528,960), (12664,12114), (12667,9942), (12672,5882), (12688,1)
X(12680) = X(84)-of-Mandart-incircle-triangle
X(12680) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,12675,354), (8,11220,9943), (944,12246,4294), (3295,12684,1709), (5534,7171,10310), (5534,10310,3689), (5731,12528,960), (5882,12672,5919), (9947,11227,1698)

X(12681) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO EXTOUCH

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

The reciprocal orthologic center of these triangles is X(72).

X(12681) lies on these lines: {4,7595}, {84,2067}, {515,9808}, {517,12638}, {946,11030}, {971,12490}, {1158,8224}, {1490,8231}, {2800,12744}, {2829,12768}, {6001,7596}, {6245,7683}, {6261,8225}, {7992,8244}, {8095,8247}, {8096,8248}, {8228,12608}, {8230,12616}, {8233,12664}, {8237,12669}, {8239,12672}, {8246,12683}, {9789,9799}, {9942,10858}, {9948,10867}, {9960,10885}, {10891,12547}, {11042,12675}, {11532,12650}, {11687,12528}, {11996,12685}

X(12681) = X(68)-of-2nd-Pamfilos-Zhou-triangle
X(12681) = excentral-to-2nd-Pamfilos-Zhou similarity image of X(1490)
X(12681) = 2nd-Pamfilos-Zhou-isotomic conjugate of X(12724)

X(12682) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXTOUCH TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(10308).

X(12682) lies on these lines: {8,12535}, {9,10266}, {72,3648}, {1145,11684}, {3337,11263}

X(12682) = midpoint of X(12769) and X(12786)

X(12683) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO EXTOUCH

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

The reciprocal orthologic center of these triangles is X(72).

X(12683) lies on these lines: {4,240}, {21,84}, {40,12530}, {515,2292}, {517,12642}, {846,1490}, {946,11031}, {971,9959}, {1158,4220}, {2800,12746}, {2829,12770}, {4199,12664}, {4425,6245}, {5051,12616}, {6001,9840}, {7992,8245}, {8095,8249}, {8096,8250}, {8229,12608}, {8238,12669}, {8240,12672}, {8246,12681}, {8425,12685}, {8731,9942}, {9791,9799}, {9948,10868}, {10892,12547}, {11043,12675}, {11533,12650},

X(12683) = X(68)-of-1st-Sharygin-triangle
X(12683) = excentral-to-1st-Sharygin similarity image of X(1490)
X(12683) = 2nd-Sharygin-to-1st-Sharygin similarity image of X(13256)
X(12683) = 1st-Sharygin-isotomic conjugate of X(12725)
{11688,12528}

X(12684) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO EXTOUCH

Trilinears 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+(b^4-c^4)*(b-c)*a-2*(b^2-c^2)^2*(b+c)^2 : :
X(12684) = 3*X(3)-2*X(1490) = 3*X(84)-X(1490) = 4*X(140)-3*X(5658) = 3*X(381)-4*X(6245) = 3*X(381)-2*X(6259) = 5*X(1656)-4*X(6260) = 7*X(3526)-8*X(6705) = 3*X(5790)-2*X(12667) = 2*X(7971)-3*X(10247) = 3*X(10246)-4*X(12114)

The reciprocal orthologic center of these triangles is X(40).

X(12684) lies on these lines: {3,9}, {4,5708}, {20,3927}, {30,9799}, {140,5658}, {355,9948}, {381,6245}, {382,2095}, {405,11220}, {515,1657}, {517,7992}, {944,10386}, {952,12632}, {956,9961}, {999,10085}, {1156,5265}, {1482,6001}, {1598,12136}, {1656,6260}, {1709,3295}, {2829,12747}, {3062,3333}, {3146,12690}, {3526,6705}, {3560,9960}, {3940,12528}, {5045,11372}, {5220,12512}, {5558,5603}, {5758,5843}, {5789,6907}, {5790,12667}, {6257,11917}, {6258,11916}, {6767,12705}, {7373,9856}, {7517,9910}, {7971,10247}, {8148,12650}, {9301,12496}, {9654,12678}, {9669,12679}, {9708,9943}, {10167,11108}, {10246,12114}, {10679,12631}, {11842,12196}, {11849,12330}, {11875,12456}, {11876,12457}, {11911,12668}, {11928,12676}, {11929,12677}, {12000,12686}, {12001,12687}

X(12684) = midpoint of X(i) and X(j) for these {i,j}: {7992,10864}, {9799,12246}
X(12684) = reflection of X(i) in X(j) for these (i,j): (3,84), (355,9948), (382,5787), (6259,6245), (8148,12650)
X(12684) = X(84)-of-X3-ABC-reflections-triangle
X(12684) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1709,12680,3295), (5044,9841,3), (5777,7171,3), (6245,6259,381), (10085,12688,999)

X(12685) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO EXTOUCH

Trilinears -8*S^2*b*c*sin(A/2)-(-a^2+b^2+c^2)*((b+c)*a^3-(b-c)^2*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(72).

X(12685) lies on these lines: {515,12445}, {517,12646}, {912,8130}, {946,8083}, {971,12491}, {2800,12748}, {2829,12774}, {6001,8351}, {6261,7587}, {7992,8423}, {8126,12528}, {8382,12616}, {8389,12669}, {8425,12683}, {8729,9942}, {9799,11891}, {9948,11860}, {9960,11890}, {10502,12688}, {11033,12005}, {11535,12650}, {11896,12547}, {11924,12672}, {11996,12681}

X(12685) = X(68)-of-Yff-central-triangle
X(12685) = excentral-to-Yff-central similarity image of X(1490)
X(12685) = Yff-central-isotomic conjugate of X(12728)

X(12686) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO EXTOUCH

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

The reciprocal orthologic center of these triangles is X(40).

X(12686) lies on these lines: {1,84}, {2,1158}, {9,119}, {12,12676}, {40,3436}, {46,1532}, {57,1519}, {515,3895}, {971,10679}, {1490,11248}, {1697,6938}, {1706,6923}, {2077,10860}, {2829,5119}, {3085,10309}, {3358,10202}, {3811,12666}, {4640,10270}, {5552,6260}, {6245,10531}, {6257,10930}, {6258,10929}, {6259,10942}, {6261,6909}, {6916,12514}, {6957,12616}, {7330,9623}, {9910,10834}, {10803,12196}, {10805,12246}, {10878,12496}, {10915,12667}, {10955,12677}, {10956,12678}, {10958,12679}, {10965,12680}, {11400,12136}, {11509,12330}, {11881,12456}, {11882,12457}, {11914,12668}, {12000,12684}

X(12686) = reflection of X(84) in X(1709)
X(12686) = X(84)-of-inner-Yff-tangents-triangle
X(12686) = {X(84),X(7971)}-harmonic conjugate of X(12687)

X(12687) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO EXTOUCH

Trilinears a^9-(b+c)*a^8-2*(2*b-c)*(b-2*c)*a^7+4*(b^2-4*b*c+c^2)*(b+c)*a^6+2*(3*b^4+3*c^4-b*c*(7*b^2-16*b*c+7*c^2))*a^5-2*(b^2-c^2)*(b-c)*(b-3*c)*(3*b-c)*a^4-2*(2*b^4+2*c^4+b*c*(5*b^2+18*b*c+5*c^2))*(b-c)^2*a^3+4*(b^2-c^2)*(b-c)*(b^4+c^4-2*b*c*(b^2-3*b*c+c^2))*a^2+(b^2-c^2)^2*(b^4+c^4+6*b*c*(b^2-b*c+c^2))*a-(b^2-c^2)^4*(b+c) : :

The reciprocal orthologic center of these triangles is X(40).

X(12687) lies on these lines: {1,84}, {11,12677}, {496,5715}, {515,12649}, {952,5709}, {956,9942}, {971,10680}, {999,12664}, {1158,5731}, {1490,11249}, {2829,12750}, {2975,6261}, {6245,10532}, {6257,10932}, {6258,10931}, {6259,10943}, {6260,10527}, {9910,10835}, {10804,12196}, {10806,12246}, {10879,12496}, {10916,12667}, {10949,12676}, {10957,12678}, {10959,12679}, {10966,12680}, {11401,12136}, {11510,12330}, {11883,12456}, {11884,12457}, {11915,12668}, {12001,12684}

X(12687) = reflection of X(84) in X(10085)
X(12687) = X(84)-of-outer-Yff-tangents-triangle
X(12687) = {X(84),X(7971)}-harmonic conjugate of X(12686)

X(12688) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH TO 4th EXTOUCH

Trilinears (b+c)*a^5-(b-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^2-c^2)^2*(b+c)^2 : :
X(12688) = 4*X(3)-3*X(5918) = 3*X(4)-2*X(7686) = 2*X(10)-3*X(5927) = 2*X(40)-3*X(210) = 3*X(65)-4*X(7686) = 3*X(165)-4*X(5044) = 3*X(210)-4*X(5777) = 3*X(354)-4*X(946) = 9*X(354)-8*X(12005) = 3*X(946)-2*X(12005)

The reciprocal orthologic center of these triangles is X(65).

X(12688) lies on these lines: {1,971}, {2,9943}, {3,1709}, {4,65}, {7,10429}, {9,5584}, {10,5927}, {11,6245}, {12,6260}, {19,64}, {20,960}, {28,12262}, {30,5887}, {33,221}, {34,1854}, {37,4300}, {40,210}, {55,1490}, {56,84}, {57,7992}, {72,516}, {104,10308}, {142,7958}, {165,5044}, {185,1839}, {207,7008}, {226,12711}, {227,2635}, {241,1044}, {329,9800}, {354,946}, {382,517}, {392,4297}, {405,12520}, {411,4640}, {442,12617}, {497,9799}, {515,3057}, {518,962}, {774,1427}, {912,12699}, {936,10860}, {942,1699}, {944,5919}, {950,12709}, {991,6051}, {999,10085}, {1001,10884}, {1012,2646}, {1042,2310}, {1125,10167}, {1155,1158}, {1192,5338}, {1204,2355}, {1210,9948}, {1319,12114}, {1385,5426}, {1425,1547}, {1478,6259}, {1479,5787}, {1532,12616}, {1538,7741}, {1593,2182}, {1698,10157}, {1824,11381}, {1829,5895}, {1848,2883}, {1871,6000}, {1872,2818}, {1902,3827}, {2098,12650}, {2099,7971}, {2264,5776}, {2771,7728}, {2777,10693}, {2778,10721}, {2801,3555}, {3085,5658}, {3091,3812}, {3146,3869}, {3427,10309}, {3428,7330}, {3485,9960}, {3487,12710}, {3523,10178}, {3616,11220}, {3624,11227}, {3646,10857}, {3671,5728}, {3678,5493}, {3679,9947}, {3689,10306}, {3698,5587}, {3817,5439}, {3838,6828}, {3868,9812}, {3876,9778}, {3983,5657}, {4005,6361}, {4199,12713}, {4293,12246}, {4423,8726}, {4679,6865}, {4731,5818}, {4882,9954}, {5045,11522}, {5057,6895}, {5087,6943}, {5247,9355}, {5252,12667}, {5433,6705}, {5572,11036}, {5603,12675}, {5720,10310}, {5732,8273}, {5794,6925}, {5806,5902}, {5880,6835}, {5883,12571}, {5934,12707}, {5935,12708}, {6738,9844}, {6831,12608}, {6847,9942}, {7580,12514}, {7701,11012}, {8079,12714}, {8226,12609}, {8227,9940}, {8232,12706}, {8233,12712}, {8582,9842}, {8583,9841}, {8727,12047}, {9843,10863}, {9955,10202}, {10176,12512}, {10431,11415}, {10445,10822}, {10473,12547}, {10477,12544}, {10502,12685}, {10888,12548}, {11263,12558}, {11406,12335}, {11509,12330}, {11523,12651}, {12666,12701}

X(12688) = midpoint of X(i) and X(j) for these {i,j}: {962,12528}, {3146,3869}, {5904,9589}, {9800,12529}
X(12688) = reflection of X(i) in X(j) for these (i,j): (1,9856), (20,960), (40,5777), (65,4), (3057,12672), (3555,4301), (3893,5881), (3962,5693), (5493,3678), (7957,72), (9961,9943), (12669,5572), (12671,6261), (12680,1)
X(12688) = anticomplement of X(9943)
X(12688) = complement of X(9961)
X(12688) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,9961,9943), (9,12565,5584), (40,5777,210), (65,1898,1864), (999,12684,10085), (1012,6261,2646), (1158,3149,1155), (1490,12705,55), (1836,1858,65), (3649,7965,946), (8581,9848,1), (9850,10866,1)

X(12688) = excentral-to-2nd-extouch similarity image of X(12565)

X(12689) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH TO 5th EXTOUCH

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

The reciprocal orthologic center of these triangles is X(65).

X(12689) lies on these lines: {2,9944}, {4,75}, {9,1721}, {72,516}, {226,12723}, {307,1827}, {329,9801}, {405,990}, {442,12618}, {950,12721}, {971,10444}, {1490,12717}, {1750,7996}, {1766,3693}, {3487,12722}, {3663,5728}, {4199,12725}, {5927,9950}, {5934,12719}, {5935,12720}, {8079,12726}, {8226,12610}, {8232,12718}, {8233,12724}, {10888,12549}, {11523,12652}

X(12689) = midpoint of X(9801) and X(12530)
X(12689) = reflection of X(9962) in X(9944)
X(12689) = anticomplement of X(9944)
X(12689) = complement of X(9962)
X(12689) = X(317)-of-2nd-extouch-triangle
X(12689) = excentral-to-2nd-extouch similarity image of X(1721)
X(12689) = 2nd-extouch-isogonal conjugate of X(5928)
X(12689) = 2nd-extouch-isotomic conjugate of X(12664)
X(12689) = anticomplement, wrt 2nd extouch triangle, of X(10445)
X(12689) = {X(2), X(9962)}-harmonic conjugate of X(9944)

X(12690) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH TO FUHRMANN

Barycentrics 4*a^4-2*(b+c)*a^3-(b+c)^2*a^2+2*(b^3+c^3)*a-3*(b^2-c^2)^2 : :
X(12690) = 3*X(11)-2*X(214) = 5*X(11)-4*X(1125) = X(145)-5*X(149) = 3*X(145)-5*X(1320) = 3*X(149)-X(1320) = 5*X(214)-6*X(1125) = 3*X(3583)-X(4867) = X(6224)-3*X(10707)

The reciprocal orthologic center of these triangles is X(8).

X(12690) lies on these lines: {2,9945}, {4,145}, {8,4756}, {9,80}, {10,6154}, {11,214}, {30,3218}, {72,2802}, {100,405}, {104,7580}, {119,8226}, {140,11015}, {226,1317}, {329,9802}, {355,3895}, {382,12649}, {517,12691}, {900,4707}, {1387,3488}, {1479,5289}, {1484,6907}, {1490,6264}, {1750,7993}, {2320,11680}, {2475,12433}, {2800,12664}, {2829,10864}, {3065,6598}, {3146,12684}, {3306,5722}, {3487,12735}, {3583,4867}, {3627,3868}, {3830,5905}, {4199,12746}, {4746,12572}, {4999,5441}, {5225,5730}, {5436,6667}, {5728,12736}, {5790,6976}, {5840,12515}, {5854,9897}, {5927,9951}, {5934,12733}, {5935,12734}, {6174,6702}, {6913,12331}, {7972,9612}, {8079,8097}, {8080,8098}, {8232,12730}, {8233,12744}, {9024,10477}, {9803,10724}, {10888,12550}, {10993,12619}, {11523,12653}

X(12690) = midpoint of X(i) and X(j) for these {i,j}: {9802,12531}, {9803,10724}
X(12690) = reflection of X(i) in X(j) for these (i,j): (100,12019), (1145,80), (1537,10738), (5541,3036), (6154,10), (6224,1387), (9963,9945), (10609,11), (10993,12619), (12732,1145)
X(12690) = anticomplement of X(9945)
X(12690) = complement of X(9963)
X(12690) = X(74)-of-2nd-extouch-triangle
X(12690) = excentral-to-2nd-extouch similarity image of X(5541)
X(12690) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,9963,9945), (80,5541,3036), (3036,5541,1145), (3419,3586,11113), (6224,10707,1387)

X(12691) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH TO INNER-GARCIA

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

The reciprocal orthologic center of these triangles is X(3869).

X(12691) lies on these lines: {2,9946}, {4,80}, {9,48}, {72,952}, {119,125}, {149,5758}, {226,8068}, {329,9803}, {404,5720}, {405,6265}, {517,12690}, {908,912}, {944,5692}, {950,12758}, {1387,5728}, {1490,1768}, {1512,6001}, {1708,10090}, {1750,12767}, {2802,12625}, {2829,12664}, {3487,5083}, {3651,12695}, {3678,11491}, {3754,5587}, {4199,12770}, {5884,7951}, {5904,12116}, {5927,9952}, {5934,12759}, {5935,12760}, {6224,6987}, {6264,11523}, {6702,6829}, {7580,12515}, {8000,10698}, {8079,12771}, {8226,12611}, {8232,12755}, {8233,12768}, {9612,11571}, {10058,10393}, {10888,12551}

X(12691) = midpoint of X(9803) and X(12532)
X(12691) = reflection of X(i) in X(j) for these (i,j): (9964,9946), (11570,10265), (12757,214)
X(12691) = anticomplement of X(9946)
X(12691) = complement of X(9964)

X(12691) = X(265)-of-2nd-Extouch-triangle
X(12691) = excentral-to-2nd-Extouch similarity image of X(6326)
X(12691) = X(4)-of-A"B"C", as defined at X(8068)

X(12692) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH TO HUTSON EXTOUCH

Trilinears (b+c)*a^8-2*(b^2+c^2)*a^7-2*(b+c)*(b^2+8*b*c+c^2)*a^6+2*(3*b^4+3*c^4+2*(4*b^2-b*c+4*c^2)*b*c)*a^5+16*b*c*(b+c)*(2*b^2+3*b*c+2*c^2)*a^4-2*(3*b^4+3*c^4+2*(5*b^2-9*b*c+5*c^2)*b*c)*(b+c)^2*a^3+2*(b+c)*(b^6+c^6-(8*b^4+8*c^4+25*b*c*(b^2+c^2))*b*c)*a^2+2*(b^2-c^2)^2*(b^4+c^4+2*(4*b^2+5*b*c+4*c^2)*b*c)*a-(b^2-c^2)^4*(b+c) : :
X(12692) = 3*X(210)-2*X(12260)

The reciprocal orthologic center of these triangles is X(3555).

X(12692) lies on these lines: {2,12439}, {4,4863}, {9,3295}, {210,12260}, {329,9804}, {405,12521}, {442,3555}, {518,12777}, {950,5920}, {1750,8001}, {3085,3983}, {5927,9953}, {7580,12516}, {8226,12612}, {10888,12552}, {11523,12654}

X(12692) = midpoint of X(9804) and X(12533)
X(12692) = reflection of X(12537) in X(12439)
X(12692) = anticomplement of X(12439)
X(12692) = complement of X(12537)

X(12693) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH TO MANDART-EXCIRCLES

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

The reciprocal orthologic center of these triangles is X(3555).

X(12693) lies on these lines: {2,12442}, {4,4723}, {9,12659}, {329,12534}, {405,12522}, {442,12621}, {2325,10445}, {5927,12449}, {7580,12517}, {8226,12613}, {10888,12553}, {11523,12655}

X(12693) = midpoint of X(12534) and X(12542)
X(12693) = reflection of X(12538) in X(12442)
X(12693) = anticomplement of X(12442)
X(12693) = complement of X(12538)

X(12694) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH TO MIDARC

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

The reciprocal orthologic center of these triangles is X(1).

X(12694) lies on these lines: {1,8079}, {2,12443}, {9,164}, {167,1750}, {177,226}, {329,9807}, {405,12523}, {442,12622}, {950,8422}, {5571,5728}, {5927,12450}, {7670,8232}, {10888,12554}

X(12694) = midpoint of X(9807) and X(11691)
X(12694) = orthologic center of these triangles: 2nd extouch to 2nd midarc
X(12694) = X(1)-of-2nd-extouch-triangle
X(12694) = excentral-to-2nd-extouch similarity image of X(164)
X(12694) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,12539,12443), (8079,8080,1)

X(12695) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(21).

X(12695) lies on these lines: {2,12444}, {4,5535}, {9,10266}, {35,72}, {329,12535}, {405,12524}, {442,1749}, {3065,6598}, {3651,12691}, {3652,12600}, {5927,12451}, {7580,12519}, {8226,12615}, {10888,12557}, {11523,12657}

X(12695) = midpoint of X(12535) and X(12543)
X(12695) = reflection of X(12540) in X(12444)
X(12695) = anticomplement of X(12444)
X(12695) = complement of X(12540)

X(12696) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO 3rd EXTOUCH

Barycentrics (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*((b+c)*a^11+(b-c)^2*a^10-2*(b^3+c^3)*a^9-(3*b^2+4*b*c+3*c^2)*(b-c)^2*a^8-(b+c)*(b^2+2*b*c-2*c^2)*(2*b^2-2*b*c-c^2)*a^7+(2*b^4+2*c^4+b*c*(10*b^2+17*b*c+10*c^2))*(b-c)^2*a^6+(b^2-c^2)*(b-c)*(8*b^4+8*c^4+b*c*(10*b^2+3*b*c+10*c^2))*a^5+(b^2-c^2)^2*(2*b^4+2*c^4-5*b*c*(2*b^2+b*c+2*c^2))*a^4-(b^2-c^2)^2*(b+c)*(7*b^4+7*c^4-5*b*c*(2*b^2-3*b*c+2*c^2))*a^3-(b^2-c^2)^2*(3*b^6+3*c^6-2*(2*b^4+2*c^4+b*c*(b^2+5*b*c+c^2))*b*c)*a^2+(b^2-c^2)^3*(b-c)*(b^2+2*c^2)*(2*b^2+c^2)*a+(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^4) : :
X(12696) = 2*X(3)-3*X(11831) = 2*X(10)-3*X(11897) = X(6361)-3*X(11845) = X(7991)-3*X(11852) = 3*X(11911)-X(12702)

The reciprocal orthologic center of these triangles is X(4).

X(12696) lies on these lines: {1,30}, {3,11831}, {4,11900}, {10,11897}, {40,402}, {46,11913}, {65,11909}, {515,12626}, {516,12113}, {517,11251}, {946,1650}, {962,4240}, {1902,11832}, {2802,12752}, {5119,11912}, {5812,11904}, {5840,12729}, {6001,12791}, {6361,11845}, {7982,11910}, {7991,11852}, {9911,11853}, {10306,11848}, {11839,12197}, {11863,12458}, {11864,12459}, {11885,12497}, {11901,12697}, {11902,12698}, {11903,12700}, {11911,12702}, {11914,12703}, {11915,12704}

X(12696) = midpoint of X(962) and X(4240)
X(12696) = X(40)-of-Gossard-triangle
X(12696) = reflection of X(i) in X(j) for these (i,j): (40,402), (1650,946), (12438,11251)

X(12697) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO 3rd EXTOUCH

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

The reciprocal orthologic center of these triangles is X(4).

X(12697) lies on these lines: {1,11824}, {3,11370}, {4,5689}, {6,40}, {10,6202}, {46,10048}, {65,10927}, {515,6258}, {516,5871}, {517,1161}, {946,5591}, {962,1271}, {1699,10514}, {1836,10923}, {1902,11388}, {2802,12753}, {5119,10040}, {5589,7991}, {5595,9911}, {5603,10517}, {5605,7982}, {5812,10921}, {5840,6263}, {6001,6267}, {6215,12699}, {6281,9589}, {6361,10783}, {8198,12458}, {8205,12459}, {9994,12497}, {10306,11497}, {10792,12197}, {10919,12700}, {10925,12701}, {10929,12703}, {10931,12704}, {11901,12696}, {11916,12702}

X(12697) = reflection of X(i) in X(j) for these (i,j): (3641,1161), (12698,40)
X(12697) = X(40)-of-inner-Grebe-triangle

X(12698) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO 3rd EXTOUCH

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

The reciprocal orthologic center of these triangles is X(4).

X(12698) lies on these lines: {1,11825}, {3,11371}, {4,5688}, {6,40}, {10,6201}, {46,10049}, {65,10928}, {515,6257}, {516,5870}, {517,1160}, {946,5590}, {962,1270}, {1699,10515}, {1836,10924}, {1902,11389}, {2802,12754}, {5119,10041}, {5588,7991}, {5594,9911}, {5603,10518}, {5604,7982}, {5812,10922}, {5840,6262}, {6001,6266}, {6214,12699}, {6278,9589}, {6361,10784}, {8199,12458}, {8206,12459}, {9995,12497}, {10306,11498}, {10793,12197}, {10920,12700}, {10926,12701}, {10930,12703}, {10932,12704}, {11902,12696}, {11917,12702}

X(12698) = reflection of X(i) in X(j) for these (i,j): (3640,1160), (12697,40)
X(12698) = X(40)-of-outer-Grebe-triangle

X(12699) = ORTHOLOGIC CENTER OF THESE TRIANGLES: JOHNSON TO 3rd EXTOUCH

Barycentrics a^4+(b+c)*a^3-2*b*c*a^2-(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(12699) = 4*X(1)-3*X(3655) = 2*X(1)-3*X(3656) = 3*X(3)-4*X(1125) = 2*X(3)-3*X(5886) = 5*X(3)-6*X(10165) = 5*X(3)-4*X(12512) = 3*X(4) - X(8) = 3*X(946)-2*X(1125) = 4*X(946)-3*X(5886) = 5*X(946)-3*X(10165) = X(6361)-4*X(9955)

The reciprocal orthologic center of these triangles is X(4).

X(12699) lies on these lines: {1,30}, {2,3579}, {3,142}, {4,8}, {5,40}, {7,1058}, {10,381}, {11,46}, {12,5119}, {19,7359}, {20,1385}, {35,11375}, {36,11376}, {52,2807}, {55,6985}, {56,1770}, {57,496}, {63,3650}, {65,1479}, {74,12261}, {80,5560}, {84,3254}, {85,5195}, {100,12611}, {113,12778}, {140,165}, {145,3543}, {146,149}, {219,1839}, {224,1537}, {226,3295}, {238,582}, {320,10446}, {347,10400}, {376,3616}, {377,392}, {382,515}, {388,9957}, {390,3487}, {442,5250}, {484,7741}, {495,1697}, {497,942}, {499,1155}, {519,3830}, {528,3811}, {546,5587}, {548,7987}, {549,3624}, {550,3576}, {551,3534}, {595,3772}, {631,9778}, {908,5687}, {912,12688}, {943,8543}, {944,3146}, {950,9668}, {952,3627}, {999,4292}, {1012,11249}, {1056,9785}, {1159,6738}, {1210,9669}, {1319,4299}, {1320,10728}, {1330,4673}, {1387,1420}, {1478,3057}, {1480,5710}, {1483,12678}, {1484,1768}, {1519,3149}, {1538,6848}, {1539,12368}, {1571,3815}, {1572,5254}, {1596,7713}, {1597,12410}, {1621,3651}, {1656,3817}, {1657,4297}, {1658,9625}, {1702,7583}, {1703,7584}, {1706,3820}, {1709,10943}, {1737,10896}, {1745,5399}, {1750,5534}, {1788,10591}, {1837,3583}, {1892,4318}, {2077,6924}, {2093,9581}, {2095,6245}, {2099,10572}, {2102,10737}, {2103,10736}, {2325,10445}, {2475,3877}, {2478,3753}, {2550,5044}, {2646,4302}, {2775,4010}, {2800,10738}, {2802,10742}, {2809,10741}, {2817,10747}, {2829,12676}, {2886,5791}, {3062,5843}, {3073,5398}, {3086,3474}, {3090,9779}, {3091,5657}, {3120,3915}, {3333,4312}, {3338,4338}, {3340,3586}, {3359,6922}, {3416,3818}, {3428,3560}, {3452,9709}, {3485,4294}, {3524,5550}, {3526,10164}, {3529,5731}, {3545,9780}, {3555,5905}, {3585,5252}, {3587,8728}, {3617,3839}, {3628,7988}, {3634,5055}, {3679,3845}, {3702,6327}, {3832,5818}, {3838,10198}, {3843,4691}, {3847,5955}, {3850,7989}, {3851,10175}, {3853,5844}, {3878,5794}, {3897,12600}, {3916,10527}, {3927,4847}, {3944,5255}, {3966,4647}, {4018,12649}, {4298,7373}, {4512,6675}, {4677,12101}, {4857,5902}, {4863,5904}, {5010,5443}, {5046,7693}, {5070,10171}, {5073,5882}, {5076,12645}, {5079,10172}, {5122,7288}, {5128,10593}, {5221,11238}, {5231,5709}, {5259,7688}, {5271,9958}, {5303,6906}, {5439,6899}, {5530,9554}, {5541,11698}, {5584,6883}, {5708,11019}, {5715,6907}, {5719,10386}, {5720,5763}, {5734,7967}, {5759,6846}, {5762,7330}, {5768,9800}, {5787,5878}, {5806,6827}, {5842,6261}, {6214,12698}, {6215,12697}, {6221,8983}, {6244,6918}, {6260,12631}, {6560,7968}, {6561,7969}, {6583,9961}, {6745,10306}, {6763,7701}, {6767,12575}, {6796,11849}, {6836,10531}, {6842,7680}, {6845,11680}, {6882,7681}, {6911,10310}, {6914,11012}, {6925,10532}, {6928,7686}, {6972,7704}, {7502,9591}, {7530,8185}, {7580,10267}, {7745,9620}, {7951,11010}, {7962,9613}, {7970,10723}, {7978,10733}, {7983,10722}, {7984,10721}, {8193,9818}, {8200,12458}, {8207,12459}, {8725,12264}, {8981,9616}, {9655,10106}, {9708,12572}, {9821,12263}, {9904,10264}, {9943,10202}, {9996,12497}, {10039,10895}, {10167,10596}, {10679,11500}, {10680,12114}, {10695,10727}, {10696,10732}, {10697,10725}, {10698,10724}, {10703,10726}, {10796,12197}, {10915,11236}, {10916,11235}, {10942,12703}, {11529,12433}, {11599,12188}, {11699,12383}, {11720,12121}, {11928,12616}, {12163,12259}

X(12699) = midpoint of X(i) and X(j) for these {i,j}: {4,962}, {40,9589}, {382,1482}, {944,3146}, {1320,10728}, {2102,10737}, {2103,10736}, {5691,7982}, {5812,12700}, {5881,11531}, {7970,10723}, {7978,10733}, {7983,10722}, {7984,10721}, {10695,10727}, {10696,10732}, {10697,10725}, {10698,10724}, {10703,10726}
X(12699) = reflection of X(i) in X(j) for these (i,j): (3,946), (20,1385), (40,5), (74,12261), (100,12611), (145,11278), (355,4), (550,5901), (1482,4301), (1657,4297), (1768,1484), (3359,7956), (3416,3818), (3534,551), (3579,9955), (3654,381), (3655,3656), (3679,3845), (5493,6684), (5541,11698), (5690,546), (5691,3627), (5887,9856), (6265,1537), (6361,3579), (6769,5763), (7991,5690), (8725,12264), (9778,11230), (9821,12263), (9904,10264), (11500,12608), (12121,11720), (12163,12259), (12188,11599), (12368,1539), (12383,11699), (12515,11), (12702,10), (12778,113)
X(12699) = isogonal conjugate of X(10623)
X(12699) = anticomplement of X(3579)
X(12699) = complement of X(6361)
X(12699) = X(40)-of-Johnson-triangle
X(12699) = homothetic center of Ehrmann mid-triangle and outer Garcia triangle
X(12699) = X(12702)-of-Ehrmann-mid-triangle
X(12699) = X(12702)-of-outer-Garcia-triangle
X(12699) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,79,10404), (2,6361,3579), (3,946,5886), (4,5758,5777), (7,1058,5045), (10,12702,3654), (20,5603,1385), (40,1699,5), (55,12047,11374), (57,9614,496), (381,12702,10), (962,9812,4), (962,10248,12245), (1699,9589,40), (1836,10404,79), (1836,12701,1), (3058,3649,1), (3434,11415,72), (3579,9955,2), (10165,12512,3)

X(12700) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO 3rd EXTOUCH

Barycentrics a^7-2*(b^2+b*c+c^2)*a^5-(b+c)*(b^2-8*b*c+c^2)*a^4+(b^4+c^4+2*b*c*(b^2-7*b*c+c^2))*a^3+2*(b^2-c^2)*(b-c)*(b^2-3*b*c+c^2)*a^2-(b^2-c^2)^3*(b-c) : :
X(12700) = 3*X(5886)-2*X(11248) = 3*X(11235)-2*X(12616)

The reciprocal orthologic center of these triangles is X(4).

X(12700) lies on these lines: {1,11826}, {3,10624}, {4,8}, {5,1706}, {10,10893}, {11,40}, {12,12703}, {30,12650}, {46,10948}, {65,10947}, {78,1537}, {79,11224}, {390,1385}, {474,5886}, {496,3359}, {515,10912}, {516,8666}, {528,6261}, {550,3254}, {946,1376}, {952,3680}, {1058,9940}, {1158,3813}, {1482,10106}, {1519,5687}, {1621,6940}, {1697,6907}, {1709,6763}, {1836,7982}, {2077,11376}, {2802,12761}, {3579,6926}, {3656,11112}, {3753,10531}, {3880,6256}, {3913,12608}, {4002,6898}, {4187,5250}, {4863,5693}, {5048,7702}, {5119,10523}, {5439,10596}, {5603,6904}, {5657,6919}, {5709,10943}, {5840,12737}, {5881,12679}, {6361,10785}, {6850,9957}, {6891,7743}, {6916,9785}, {6964,9955}, {7991,10826}, {9911,10829}, {10167,10806}, {10679,11374}, {10794,12197}, {10871,12497}, {10919,12697}, {10920,12698}, {10949,12704}, {11235,12616}, {11865,12458}, {11866,12459}, {11903,12696}, {11928,12702}

X(12700) = reflection of X(i) in X(j) for these (i,j): (355,10525), (1158,3813), (3913,12608), (5812,12699), (10306,946)
X(12700) = X(40)-of-inner-Johnson-triangle
X(12700) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,10914,355), (40,9614,6922), (962,3434,12672), (3434,12672,355)

X(12701) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO 3rd EXTOUCH

Barycentrics a^4+(b+c)*a^3-4*b*c*a^2-(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(12701) = 2*X(1210)-3*X(11238) = 3*X(5603)-X(6934) = 3*X(5886)-2*X(6924)

The reciprocal orthologic center of these triangles is X(4).

X(12701) lies on these lines: {1,30}, {3,11376}, {4,1000}, {5,5119}, {7,5558}, {8,3967}, {10,4679}, {11,40}, {12,1697}, {20,1319}, {21,5832}, {35,5886}, {36,11373}, {46,496}, {55,946}, {56,516}, {57,9589}, {63,3813}, {65,497}, {72,4863}, {78,528}, {145,5057}, {149,3869}, {165,5433}, {210,5082}, {226,3303}, {354,1058}, {355,3583}, {381,10039}, {388,5919}, {390,3485}, {498,9955}, {499,3579}, {515,2098}, {517,1479}, {518,1898}, {546,10827}, {550,1387}, {908,3913}, {944,5048}, {950,2099}, {960,3434}, {999,1770}, {1012,10966}, {1038,12652}, {1125,5217}, {1155,3086}, {1191,3914}, {1210,11238}, {1317,6259}, {1385,4302}, {1388,4297}, {1478,9957}, {1482,9668}, {1512,10893}, {1519,11500}, {1537,6261}, {1698,7173}, {1709,10949}, {1737,9669}, {1788,5183}, {1839,2256}, {1864,5758}, {1902,11393}, {2475,3890}, {2478,5836}, {2646,4294}, {2802,12764}, {2886,5250}, {3146,3476}, {3295,12047}, {3296,5551}, {3304,4292}, {3305,9710}, {3333,11246}, {3416,3702}, {3419,3878}, {3421,3893}, {3436,3880}, {3486,11011}, {3487,3748}, {3586,5812}, {3601,5805}, {3612,5901}, {3616,5880}, {3673,5195}, {3698,5084}, {3746,11374}, {3772,3915}, {3868,5180}, {3876,7673}, {3877,5794}, {3885,5080}, {3895,12607}, {3911,5493}, {4305,10595}, {4342,10106}, {4388,4673}, {4640,10527}, {4857,5722}, {4861,11114}, {4870,5703}, {5087,5552}, {5123,5187}, {5221,11019}, {5432,8227}, {5533,12515}, {5657,10591}, {5690,10826}, {5691,7962}, {5727,11531}, {5840,12740}, {6001,12116}, {6734,11235}, {6949,7704}, {6985,11508}, {7288,9778}, {7580,11510}, {7686,10531}, {7741,11010}, {7991,9581}, {8715,11813}, {8727,10957}, {9578,9819}, {9671,11362}, {9779,10588}, {9911,10832}, {10065,12261}, {10087,12611}, {10306,11502}, {10366,10373}, {10698,12743}, {10738,12758}, {10798,12197}, {10806,12675}, {10874,12497}, {10925,12697}, {10926,12698}, {10947,12672}, {10958,12703}, {10959,12704}, {10965,12608}, {11871,12458}, {11872,12459}, {12666,12688}

X(12701) = midpoint of X(962) and X(6836)
X(12701) = reflection of X(i) in X(j) for these (i,j): (40,6922), (46,496), (56,12053), (1837,1479), (3149,946)
X(12701) = X(40)-of-2nd-Johnson-Yff-triangle
X(12701) = inner-Johnson-to-ABC similarity image of X(40)
X(12701) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1836,10404), (1,9579,5434), (1,9580,6284), (1,12699,1836), (4,3057,5252), (40,9614,11), (55,946,11375), (226,12575,3303), (388,9785,5919), (497,962,65), (946,10624,55), (950,4301,2099), (1058,4295,354), (1482,9668,10572), (1697,1699,12), (2099,9670,950), (3086,6361,1155), (3579,7743,499), (3583,5697,355), (9785,9812,388)

X(12702) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO 3rd EXTOUCH

Trilinears    2 r - 3 R cos A : :
Trilinears    2 cos A - cos B - cos C + 1 : :
Trilinears    a^3+2*(b+c)*a^2-(b^2+4*b*c+c^2)*a-2*(b^2-c^2)*(b-c) : :
X(12702) = 2*X(1)-3*X(3) = X(1)-3*X(40) = 5*X(1)-9*X(165) = 5*X(1)-6*X(1385) = 4*X(1)-3*X(1482) = 7*X(1)-9*X(3576) = 5*X(1)-3*X(7982) = 3*X(4)-5*X(3617) = 2*X(4)-3*X(5790) = 5*X(3617)-6*X(5690) = 10*X(3617)-9*X(5790) = 4*X(5690)-3*X(5790)

The reciprocal orthologic center of these triangles is X(4).

X(12702) lies on these lines: {1,3}, {4,3617}, {5,962}, {8,30}, {10,381}, {20,952}, {44,1766}, {45,573}, {63,10914}, {72,3426}, {79,11237}, {100,5730}, {140,5550}, {145,376}, {149,6903}, {219,2173}, {220,5011}, {355,382}, {378,11396}, {390,12433}, {399,12778}, {474,3877}, {495,4295}, {496,1788}, {515,1657}, {519,3534}, {546,5818}, {548,1483}, {549,3616}, {550,944}, {582,595}, {631,5901}, {758,3913}, {946,1656}, {958,3647}, {960,9709}, {984,5492}, {1000,3600}, {1001,3754}, {1125,3656}, {1145,3436}, {1254,7086}, {1351,4663}, {1376,3878}, {1386,12017}, {1387,7288}, {1389,7508}, {1480,4642}, {1511,7978}, {1537,6834}, {1571,5024}, {1572,9605}, {1597,1829}, {1598,1902}, {1698,5055}, {1699,3851}, {1702,6417}, {1703,6418}, {1706,5044}, {1737,9669}, {1759,4513}, {1770,5252}, {1836,9654}, {1837,9668}, {1871,11471}, {2771,5541}, {2775,4730}, {2778,3556}, {2800,11500}, {2802,11256}, {2948,12308}, {3098,3242}, {3218,3885}, {3240,4192}, {3241,8703}, {3244,3655}, {3305,4002}, {3488,10386}, {3522,7967}, {3523,10595}, {3524,3622}, {3526,4301}, {3530,5734}, {3543,4678}, {3555,3895}, {3614,6980}, {3623,10304}, {3633,4880}, {3636,3653}, {3649,10056}, {3651,3871}, {3679,3830}, {3753,5250}, {3817,5079}, {3843,5587}, {3861,10248}, {3869,3940}, {3870,4018}, {3911,11373}, {3928,12629}, {3935,7580}, {3987,4383}, {4188,5330}, {4299,10944}, {4302,10950}, {4313,11041}, {4388,5827}, {4421,4930}, {4816,5881}, {4848,5722}, {5070,8227}, {5072,10175}, {5073,5691}, {5082,6851}, {5180,11681}, {5184,9301}, {5225,6928}, {5229,6923}, {5302,5836}, {5440,11682}, {5534,12565}, {5554,11113}, {5704,6922}, {5714,5758}, {5729,5759}, {5762,6850}, {5763,6825}, {5771,6847}, {5780,12672}, {5812,11929}, {5840,11827}, {5882,12512}, {5884,11495}, {5899,8185}, {6197,7497}, {6221,7969}, {6284,10573}, {6398,7968}, {6407,9583}, {6445,9582}, {6472,9618}, {6759,7973}, {6762,7171}, {6842,10592}, {6882,10593}, {6942,10698}, {6971,7173}, {7354,12647}, {7489,11496}, {7517,9911}, {7983,12042}, {7984,12041}, {8666,10912}, {8715,12635}, {9584,10145}, {9798,12083}, {9905,12316}, {9928,12164}, {10800,12054}, {11230,11522}, {11842,12197}, {11911,12696}, {11916,12697}, {11917,12698}, {11928,12700}

X(12702) = midpoint of X(i) and X(j) for these {i,j}: {8,6361}, {20,12245}, {40,7991}, {1657,12645}
X(12702) = reflection of X(i) in X(j) for these (i,j): (1,3579), (3,40), (4,5690), (355,11362), (381,3654), (382,355), (399,12778), (944,550), (962,5), (1482,3), (1483,548), (3241,8703), (3242,3098), (3830,3679), (4301,6684), (4930,4421), (5073,5691), (5882,12512), (6767,3587), (7973,6759), (7978,1511), (7982,1385), (7983,12042), (7984,12041), (8148,1), (8158,5709), (9301,5184), (10247,165), (10742,1145), (10912,8666), (12164,9928), (12308,2948), (12316,9905), (12635,8715), (12699,10), (12773,12515)
X(12702) = X(40)-of-X3-ABC-reflections-triangle
X(12702) = X(382)-of-1st-circumperp-triangle
X(12702) = X(1657)-of-2nd-circumperp-triangle
X(12702) = Stammler isogonal conjugate of X(3913)
X(12702) = center of circle that is the poristic locus of X(20)
X(12702) = endo-homothetic center of Ehrmann mid-triangle and outer Garcia triangle; the homothetic center is X(12699)
X(12702) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,40,3579), (1,3579,3), (1,8148,1482), (3,1482,10246), (3,8148,1), (3,10247,1385), (3,10306,11849), (40,1697,3587), (40,7982,165), (46,3057,999), (57,9957,7373), (65,5119,3295), (165,7982,1385), (484,5697,56), (942,1697,6767), (942,3587,3), (1385,7982,10247), (1697,2093,942), (3057,5183,46), (3428,11248,3), (7982,10247,1482)

X(12703) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO 3rd EXTOUCH

Trilinears a^6-(3*b^2+4*b*c+3*c^2)*a^4+10*b*c*(b+c)*a^3+(3*b^4+3*c^4+2*b*c*(b^2-9*b*c+c^2))*a^2-10*(b^2-c^2)*(b-c)*b*c*a-(b^2-c^2)^2*(b-c)^2 : :
X(12703) = 4*X(55)-3*X(3576) = 2*X(3434)-3*X(5587)

The reciprocal orthologic center of these triangles is X(4).

X(12703) lies on these lines: {1,3}, {4,10915}, {9,6976}, {10,6898}, {12,12700}, {119,1699}, {145,1158}, {515,3895}, {516,12115}, {528,11372}, {946,5552}, {952,1709}, {962,10528}, {1012,3880}, {1706,6983}, {1836,10956}, {1902,11400}, {2136,5881}, {2800,3870}, {2802,12775}, {3158,6326}, {3434,5587}, {3632,7330}, {3656,6174}, {3871,6261}, {3913,12672}, {5250,5554}, {5555,7160}, {5657,10596}, {5693,6765}, {5812,10955}, {5840,12749}, {6361,10805}, {7966,10860}, {9911,10834}, {10525,10827}, {10803,12197}, {10878,12497}, {10914,11496}, {10929,12697}, {10930,12698}, {10942,12699}, {10958,12701}, {11914,12696}, {12245,12514}

X(12703) = reflection of X(i) in X(j) for these (i,j): (1,10679), (40,5119)
X(12703) = X(40)-of-inner-Yff-tangents-triangle
X(12703) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,165,10269), (1,2077,3576), (40,7982,12704), (65,10965,1), (962,10528,12608), (2136,12705,5881), (3057,11509,1), (5709,11010,40), (11010,11531,5709)

X(12704) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO 3rd EXTOUCH

Trilinears a^6-3*(b^2+c^2)*a^4-2*b*c*(b+c)*a^3+(b^2+c^2)*(3*b^2-2*b*c+3*c^2)*a^2+2*(b^2-c^2)*(b-c)*b*c*a-(b^2-c^2)^2*(b-c)^2 : :
X(12704) = 4*X(56)-3*X(3576) = 2*X(3436)-3*X(5587)

The reciprocal orthologic center of these triangles is X(4).

X(12704) lies on these lines: {1,3}, {4,10916}, {9,6832}, {10,6854}, {11,1728}, {30,10085}, {63,946}, {84,10431}, {191,11522}, {210,6918}, {283,4228}, {411,3873}, {496,5762}, {515,12649}, {516,12116}, {518,3149}, {580,614}, {583,8557}, {956,7686}, {962,1158}, {1068,1435}, {1072,5292}, {1125,6878}, {1329,5705}, {1473,9911}, {1699,6763}, {1708,3086}, {1709,10943}, {1766,2260}, {1768,9589}, {1836,10957}, {1902,11401}, {2270,2323}, {2360,3193}, {2802,12776}, {2829,10864}, {2949,3646}, {2990,10692}, {3306,6684}, {3436,5587}, {3475,6988}, {3555,11500}, {3681,6915}, {3811,6905}, {3868,6261}, {3870,6796}, {3916,11496}, {3928,12705}, {4005,5780}, {4333,5840}, {5231,5715}, {5437,10198}, {5603,12514}, {5657,10597}, {5720,5904}, {5722,11827}, {5735,7701}, {5805,6067}, {5881,6762}, {5905,10530}, {6326,11523}, {6361,10806}, {6907,10404}, {7580,12675}, {7682,12527}, {10526,10826}, {10804,12197}, {10879,12497}, {10884,12005}, {10931,12697}, {10932,12698}, {10949,12700}, {10959,12701}, {11915,12696}

X(12704) = reflection of X(i) in X(j) for these (i,j): (1,10680), (40,46), (11415,946)
X(12704) = X(40)-of-outer-Yff-tangents-triangle
X(12704) = X(46)-of-tangential-of-excentral-triangle
X(12704) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,165,10267), (1,5536,5709), (1,5709,40), (1,11012,3576), (40,3333,3576), (40,7982,12703), (65,10966,1), (354,3338,3333), (962,3218,1158), (1699,6763,7330), (3336,7991,3359), (3359,7991,40), (4860,5584,9940), (5535,7982,40)

X(12705) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HEXYL TO 4th EXTOUCH

Trilinears a^6-(3*b^2-2*b*c+3*c^2)*a^4+4*b*c*(b+c)*a^3+3*(b^2-c^2)^2*a^2-4*(b^2-c^2)*(b-c)*b*c*a-(b^2-c^2)^2*(b+c)^2 : :
X(12705) = 2*X(3)-3*X(4512) = 3*X(3576)-4*X(5248) = 3*X(3576)-2*X(12520) = 2*X(3671)-3*X(5603) = 3*X(4512)-X(12565) = 3*X(5587)-4*X(12617) = 5*X(8227)-4*X(12609) = 5*X(10595)-4*X(12563)

The reciprocal orthologic center of these triangles is X(65).

X(12705) lies on these lines: {1,84}, {3,4512}, {4,9}, {5,3359}, {11,2950}, {12,12679}, {20,5250}, {46,1699}, {55,1490}, {57,946}, {63,962}, {72,6769}, {78,12529}, {90,3577}, {104,7091}, {165,3149}, {191,9589}, {196,1712}, {200,5777}, {226,8803}, {380,5776}, {390,9799}, {495,6259}, {497,6245}, {515,1697}, {517,3927}, {595,990}, {758,6762}, {774,2263}, {936,10310}, {942,3358}, {944,4314}, {960,6282}, {968,4300}, {971,3295}, {1001,8726}, {1056,12246}, {1181,1449}, {1385,7171}, {1389,7285}, {1420,5450}, {1453,3073}, {1486,9914}, {1519,5437}, {1537,1768}, {1621,9961}, {1728,2093}, {1750,11500}, {1765,2257}, {1788,7682}, {1836,5715}, {2077,5438}, {2096,4298}, {2136,5881}, {2800,3340}, {2829,9613}, {3057,12650}, {3062,7160}, {3085,6260}, {3176,7008}, {3303,12680}, {3333,3671}, {3576,5248}, {3601,6261}, {3683,5584}, {3731,8915}, {3870,12528}, {3928,12704}, {5044,6244}, {5119,5691}, {5219,12608}, {5231,5709}, {5285,9911}, {5441,7966}, {5534,10679}, {5687,5927}, {5693,11523}, {5720,11248}, {5768,9948}, {5884,11518}, {5918,8273}, {6256,9578}, {6326,12775}, {6684,7308}, {6767,12684}, {7675,9960}, {7680,9612}, {7967,9845}, {8081,12714}, {8111,12707}, {8112,12708}, {8234,12712}, {8235,12713}, {9581,12616}, {9709,10157}, {9940,10582}, {10042,10058}, {10476,12544}, {10595,12563}

X(12705) = midpoint of X(i) and X(j) for these {i,j}: {20,9800}, {4314,9949}, {12526,12651}
X(12705) = reflection of X(i) in X(j) for these (i,j): (1,11496), (40,12514), (944,4314), (4295,946), (12520,5248), (12565,3)
X(12705) = excentral-to-hexyl similarity image of X(12565)
X(12705) = anticomplement, wrt hexyl triangle, of X(12520)
X(12705) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1709,84), (1,1777,1394), (1,2956,222), (40,5587,1706), (40,11372,4), (55,12688,1490), (946,1158,57), (946,6705,3086), (1001,9943,8726), (1012,12672,1), (1519,6833,8227), (1621,9961,10884), (1768,11522,3338), (4512,12565,3), (5248,12520,3576), (5777,10306,200), (5881,12703,2136), (6212,6213,2270)

X(12706) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO 4th EXTOUCH

Trilinears (b+c)*a^7-(3*b^2+b*c+3*c^2)*a^6+(b+c)*(b^2-4*b*c+c^2)*a^5+(5*b^2+13*b*c+5*c^2)*(b-c)^2*a^4-(b+c)*(5*b^4+5*c^4-2*b*c*(4*b^2+3*b*c+4*c^2))*a^3-(b^2-c^2)^2*(b^2+3*b*c+c^2)*a^2+(b^2-c^2)*(b-c)*(3*b^4+3*c^4+2*b*c*(b^2+5*b*c+c^2))*a-(b^2-c^2)^2*(b-c)*(b^3-c^3) : :
X(12706) = 3*X(7671)-2*X(12560) = 3*X(11038)-4*X(12710)

The reciprocal orthologic center of these triangles is X(65).

X(12706) lies on these lines: {7,9800}, {9,12529}, {390,6001}, {758,7673}, {1445,12565}, {3671,11025}, {7671,12560}, {7675,9960}, {7676,12514}, {7677,12520}, {7678,12609}, {7679,12617}, {8232,12688}, {8236,12709}, {8237,12712}, {8238,12713}, {8385,12707}, {8386,12708}, {8387,12714}, {8389,12716}, {8732,9943}, {9949,10865}, {10889,12548}, {11038,12710}, {11526,12651}

X(12706) = reflection of X(i) in X(j) for these (i,j): (7,12711), (12529,9)
X(12706) = excentral-to-Honsberger similarity image of X(12565)

X(12707) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-HUTSON TO 4th EXTOUCH

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=2*(-a+b+c)*((b+c)*a^4+4*a^3*b*c-2*(b+c)*(b^2-3*b*c+c^2)*a^2+(b^4-c^4)*(b-c))
G(a,b,c)=-2*(a-b+c)*(a^4-2*(b+c)*(b-2*c)*a^2+4*(b^2-c^2)*c*a+(b^2-c^2)*(b+c)^2)*b
H(a,b,c)=a^6+2*(b+c)*a^5-(5*b^2-2*b*c+5*c^2)*a^4-4*(b+c)*(b^2-3*b*c+c^2)*a^3+7*(b^2-c^2)^2*a^2+2*(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a-(3*b^2+2*b*c+3*c^2)*(b^2-c^2)^2

The reciprocal orthologic center of these triangles is X(65).

X(12707) lies on these lines: {363,12565}, {3671,11026}, {5934,12688}, {6001,9836}, {8107,12514}, {8109,12520}, {8111,12705}, {8113,12711}, {8133,12714}, {8377,12609}, {8380,12617}, {8385,12706}, {8390,12709}, {8391,12713}, {9783,9800}, {9943,11854}, {9949,11856}, {9961,11886}, {11039,12710}, {11527,12651}, {11685,12529}, {11892,12548}, {11922,12712}

X(12707) = excentral-to-inner-Hutson similarity image of X(12565)

X(12708) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-HUTSON TO 4th EXTOUCH

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)-H(a,b,c) : :
where F, G, H are as given at X(12707)

The reciprocal orthologic center of these triangles is X(65).

X(12708) lies on these lines: {3671,11027}, {5935,12688}, {6001,9837}, {8108,12514}, {8110,12520}, {8112,12705}, {8114,12711}, {8135,12714}, {8378,12609}, {8381,12617}, {8386,12706}, {8392,12709}, {9943,11855}, {9949,11857}, {9961,11887}, {11040,12710}, {11528,12651}, {11686,12529}, {11893,12548}, {11925,12712}, {11926,12713}

X(12708) = excentral-to-outer-Hutson similarity image of X(12565)

X(12709) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON INTOUCH TO 4th EXTOUCH

Trilinears (a^3-(b+c)*a^2-(b^2-4*b*c+c^2)*a+(b+c)*(b^2+c^2))*(b+c)/(-a+b+c) : :
X(12709) = 3*X(1)-2*X(12710) = 3*X(392)-2*X(12514) = 8*X(3841)-7*X(4002) = X(3893)-4*X(12446) = 2*X(4314)-3*X(5919) = 4*X(12710)-3*X(12711)

The reciprocal orthologic center of these triangles is X(65).

X(12709) lies on these lines: {1,84}, {7,3869}, {10,12}, {11,12617}, {55,12520}, {56,392}, {57,960}, {73,3931}, {145,12529}, {227,4424}, {281,2358}, {354,12563}, {388,517}, {497,5787}, {516,3057}, {518,3340}, {942,3086}, {950,12688}, {971,3486}, {986,1465}, {997,1466}, {1042,1214}, {1062,7986}, {1319,5248}, {1400,4047}, {1420,4512}, {1617,5250}, {1697,12565}, {1788,5044}, {1837,5927}, {1858,5728}, {1864,6738}, {1898,9844}, {2099,3555}, {2646,10167}, {3304,10569}, {3339,5692}, {3476,4294}, {3600,3877}, {3601,9943}, {3666,10571}, {3812,5219}, {3868,5173}, {3873,4323}, {3878,4298}, {3884,4315}, {3890,4308}, {3893,12446}, {3899,4355}, {4313,9961}, {4314,5919}, {4551,4646}, {4870,10199}, {5018,11533}, {5083,12564}, {5252,10914}, {5439,10200}, {5440,11509}, {5693,11529}, {5694,6858}, {5784,6737}, {5836,9578}, {5884,6705}, {7681,12047}, {7686,9612}, {7962,12651}, {8236,12706}, {8239,12712}, {8240,12713}, {8390,12707}, {8392,12708}, {8543,10177}, {9785,9800}, {9949,10866}, {10480,12544}

X(12709) = midpoint of X(145) and X(12529)
X(12709) = reflection of X(i) in X(j) for these (i,j): (65,3671), (3555,12559), (4294,9957), (12526,960), (12711,1)
X(12709) = excentral-to-Hutson-intouch similarity image of X(12565)
X(12709) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (12,65,3753), (65,210,4848), (1042,2292,1214), (3057,8581,10106)

X(12710) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO 4th EXTOUCH

Trilinears (b+c)*a^5-(b^2-4*b*c+c^2)*a^4-2*(b+c)*(b^2+c^2)*a^3+2*(b^2-4*b*c+c^2)*(b+c)^2*a^2+(b^2-c^2)^2*(b+c)*a-(b^4-c^4)*(b^2-c^2) : :
X(12710) = 3*X(1)-X(12709) = X(72)-3*X(4512) = 3*X(354)-X(4295) = 5*X(3616)-X(12529) = 3*X(10167)-X(12565) = 3*X(10178)-2*X(12511) = 3*X(11038)+X(12706) = X(12709)+3*X(12711)

The reciprocal orthologic center of these triangles is X(65).

X(12710) lies on these lines: {1,84}, {40,4326}, {65,3488}, {72,4512}, {354,1058}, {495,12617}, {496,3742}, {516,942}, {517,4314}, {518,3295}, {758,3635}, {774,2293}, {938,7671}, {943,3683}, {946,9942}, {950,5842}, {960,5248}, {962,11020}, {999,12520}, {1056,12680}, {1062,1386}, {1864,3085}, {3333,10167}, {3487,12688}, {3555,12526}, {3616,12529}, {3671,5045}, {3745,6198}, {3812,5722}, {4319,5706}, {5049,12563}, {5173,10122}, {5223,7160}, {5587,9844}, {5603,9848}, {8351,12715}, {9800,11037}, {9940,11019}, {9949,11035}, {9961,11036}, {10178,12511}, {10578,12528}, {10595,10866}, {11038,12706}, {11039,12707}, {11042,12712}, {11043,12713}, {11529,12651}

X(12710) = midpoint of X(i) and X(j) for these {i,j}: {1,12711}, {65,4294}, {3555,12526}, {4326,5728}
X(12710) = reflection of X(i) in X(j) for these (i,j): (942,12564), (960,5248), (3671,5045)
X(12710) = excentral-to-incircle-circles similarity image of X(12565)
X(12710) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,10391,12675), (5572,9943,942), (10122,10624,5173)

X(12711) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INTOUCH TO 4th EXTOUCH

Trilinears ((b+c)*a^4+4*b*c*a^3-2*(b+c)*(b^2-3*b*c+c^2)*a^2+(b^4-c^4)*(b-c))*(-a+b+c) : :
X(12711) = 3*X(354)-2*X(3671) = 3*X(392)-4*X(5248) = 2*X(960)-3*X(4512) = 3*X(10167)-2*X(12520) = X(12709)-4*X(12710)

The reciprocal orthologic center of these triangles is X(65).

X(12711) lies on these lines: {1,84}, {2,12529}, {7,9800}, {8,10394}, {10,1864}, {11,5439}, {12,1898}, {33,5711}, {40,9786}, {55,72}, {56,10167}, {57,9943}, {65,516}, {174,12715}, {226,12688}, {243,1871}, {354,3671}, {380,9119}, {386,9371}, {388,971}, {390,3868}, {392,2646}, {496,10202}, {497,942}, {517,3486}, {518,1697}, {758,3057}, {774,1214}, {912,3295}, {960,3601}, {962,5173}, {1155,12511}, {1284,12713}, {1617,10884}, {1708,5584}, {1837,3753}, {2089,12714}, {2093,10399}, {2098,12559}, {2269,4047}, {2292,2293}, {3085,5777}, {3086,9940}, {3340,12651}, {3485,9856}, {3586,7686}, {3600,11220}, {3812,9581}, {3869,4313}, {3873,9785}, {3874,12575}, {3876,5281}, {3881,4342}, {3925,10395}, {5044,5218}, {5225,5806}, {5250,7675}, {5493,12432}, {5572,10384}, {5722,10525}, {5727,5836}, {5842,10572}, {7288,11227}, {8113,12707}, {8114,12708}, {8243,12712}, {8581,9949}, {10106,12680}, {10157,10588}, {10473,12544}, {10480,11997}, {10502,12570}, {10503,12568}, {10569,10866}, {10914,10950}

X(12711) = midpoint of X(i) and X(j) for these {i,j}: {7,12706}, {9800,9961}
X(12711) = reflection of X(i) in X(j) for these (i,j): (1,12710), (72,12514), (3057,4314), (3671,12564), (4295,942), (12560,5572), (12565,9943), (12672,11496), (12709,1)
X(12711) = complement of X(12529)
X(12711) = excentral-to-intouch similarity image of X(12565)
X(12711) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (12,1898,5927), (55,1858,72), (354,9848,12053), (774,4300,1214), (3671,12564,354), (3753,9844,1837), (4326,12526,1697), (9856,11018,3485), (12715,12716,174)

X(12712) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO 4th EXTOUCH

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

The reciprocal orthologic center of these triangles is X(65).

X(12712) lies on these lines: {516,9808}, {3671,11030}, {6001,7596}, {8224,12514}, {8225,12520}, {8228,12609}, {8230,12617}, {8231,12565}, {8233,12688}, {8234,12705}, {8237,12706}, {8239,12709}, {8243,12711}, {8246,12713}, {9789,9800}, {9943,10858}, {9949,10867}, {9961,10885}, {10891,12548}, {11042,12710}, {11211,12566}, {11532,12651}, {11687,12529}, {11996,12716}

X(12712) = excentral-to-2nd-Pamfilos-Zhou similarity image of X(12565)

X(12713) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO 4th EXTOUCH

Trilinears (b-c)^2*a^7-(b+c)*(b^2+6*b*c+c^2)*a^6-(3*b^4+3*c^4+2*b*c*(b^2+b*c+c^2))*a^5+(b+c)*(3*b^4+3*c^4+4*b*c*(b^2-b*c+c^2))*a^4+(3*b^4+3*c^4-2*b*c*(2*b-c)*(b-2*c))*(b+c)^2*a^3-(b+c)*(3*b^6+3*c^6-(2*b^4+2*c^4+b*c*(7*b^2-4*b*c+7*c^2))*b*c)*a^2-(b^2-c^2)^2*(b^4+c^4-2*b*c*(b^2-3*b*c+c^2))*a+(b^2-c^2)^2*(b+c)*(b^4+c^4) : :
X(12713) = 3*X(11203)-2*X(12567)

The reciprocal orthologic center of these triangles is X(65).

X(12713) lies on these lines: {21,1709}, {516,2292}, {846,12565}, {1284,12711}, {3671,11031}, {4199,12688}, {4220,12514}, {5051,12617}, {6001,9840}, {8229,12609}, {8235,12705}, {8238,12706}, {8240,12709}, {8246,12712}, {8249,12714}, {8391,12707}, {8425,12716}, {8731,9943}, {9791,9800}, {9949,10868}, {10892,12548}, {11043,12710}, {11203,12567}, {11533,12651}, {11688,12529}, {11926,12708}

X(12713) = 2nd-Sharygin-to-1st-Sharygin similarity image of X(13258)
X(12713) = excentral-to-1st-Sharygin similarity image of X(12565)

X(12714) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL-MIDARC TO 4th EXTOUCH

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=((b+c)*a^4+4*a^3*b*c-2*(b+c)*(b^2-3*b*c+c^2)*a^2+(b^4-c^4)*(b-c))*(-a+b+c)^2
G(a,b,c)=-(a^4-2*(b+c)*(b-2*c)*a^2+4*(b^2-c^2)*c*a+(b^2-c^2)*(b+c)^2)*(a-b+c)^2*b
H(a,b,c)=-2*S^2*(a^3-(b+c)*a^2-(b+c)^2*a+(b^2-c^2)*(b-c))
X(12714) = 3*X(11192)-2*X(12568)

The reciprocal orthologic center of these triangles is X(65).

X(12714) lies on these lines: {1,12715},{516,8093}, {2089,12711}, {3671,11032}, {6001,8091}, {8075,12514}, {8077,12520}, {8078,12565}, {8079,12688}, {8081,12705}, {8084,12569}, {8085,12609}, {8087,12617}, {8133,12707}, {8135,12708}, {8241,12709}, {8247,12712}, {8249,12713}, {8387,12706}, {8733,9943}, {9793,9800}, {9961,11888}, {11192,12568}, {11690,12529}, {11894,12548}

X(12714) = reflection of X(8084) in X(12569)
X(12714) = excentral-to-tangential-midarc similarity image of X(12565)
X(12714) = reflection of X(12715) in X(1)

X(12715) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO 4th EXTOUCH

Trilinears -2*(a+b+c)*(a^3-(b+c)*a^2-(b+c)^2*a+(b^2-c^2)*(b-c))*b*c*sin(A/2)-(-a+b+c)*((b+c)*a^4+4*b*c*a^3-2*(b+c)*(b^2-3*b*c+c^2)*a^2+(b^4-c^4)*(b-c)) : :

The reciprocal orthologic center of these triangles is X(65).

X(12715) lies on these lines: {1,12714}, {174,12711}, {258,12565}, {3671,11033}, {7588,12520}, {8083,12564}, {8125,12529}, {8351,12710}, {8734,9943}, {9949,11859}, {11895,12548}, {11899,12651}

X(12715) = excentral-to-2nd-tangential-midarc similarity image of X(12565)
X(12715) = reflection of X(12714) in X(1)

X(12716) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO 4th EXTOUCH

Trilinears 2*(a+b+c)*(a^3-(b+c)*a^2-(b+c)^2*a+(b^2-c^2)*(b-c))*b*c*sin(A/2)-(-a+b+c)*((b+c)*a^4+4*b*c*a^3-2*(b+c)*(b^2-3*b*c+c^2)*a^2+(b^4-c^4)*(b-c)) : :
X(12716) = 3*X(11195)-2*X(12570)

The reciprocal orthologic center of these triangles is X(65).

X(12716) lies on these lines: {174,12711}, {516,12445}, {3671,8083}, {6001,8351}, {7587,12520}, {8126,12529}, {8382,12617}, {8389,12706}, {8425,12713}, {8729,9943}, {9800,11891}, {9949,11860}, {9961,11890}, {11033,12564}, {11195,12570}, {11535,12651}, {11896,12548}, {11996,12712}

X(12716) = excentral-to-Yff-central similarity image of X(12565)

X(12717) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HEXYL TO 5th EXTOUCH

Trilinears a^5+(b+c)*a^4+6*b*c*a^3-2*b*c*(b+c)*a^2-(b^2+c^2)*(b+c)^2*a+(b^2-c^2)*(b-c)*(-c^2-4*b*c-b^2) : :
X(12717) = 2*X(990)-3*X(3576) = 3*X(5587)-4*X(12618) = 5*X(8227)-4*X(12610)

The reciprocal orthologic center of these triangles is X(65).

Let A'B'C' be the hexyl triangle. Let A" be the trilinear pole, wrt A'B'C', of line BC, and define B" and C" cyclically. Let A* be the trilinear pole, wrt A'B'C', of line B"C", and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X(12717). (Randy Hutson, July 21, 2017)

X(12717) lies on these lines: {1,7175}, {3,1721}, {4,9}, {20,2128}, {78,12530}, {84,309}, {515,3886}, {517,1351}, {726,6762}, {894,962}, {946,10436}, {990,3576}, {1490,12689}, {1699,2941}, {1709,1764}, {1757,7991}, {1836,10319}, {2796,3928}, {2961,5709}, {3333,3663}, {3683,9816}, {3821,5437}, {5227,5695}, {6001,10477}, {7675,12718}, {8081,12726}, {8111,12719}, {8112,12720}, {8227,12610}, {8234,12724}, {8235,12725}, {8726,9944}, {9950,10864}, {9962,10884}

X(12717) = midpoint of X(i) and X(j) for these {i,j}: {1,7996}, {20,9801}
X(12717) = reflection of X(i) in X(j) for these (i,j): (40,1766), (1721,3)
X(12717) = X(317)-of-hexyl-triangle
X(12717) = excentral-to-hexyl similarity image of X(1721)
X(12717) = hexyl-isotomic conjugate of X(84)
X(12717) = anticomplement, wrt hexyl triangle, of X(990)
X(12717) = {X(40), X(11372)}-harmonic conjugate of X(6210)

X(12718) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO 5th EXTOUCH

Trilinears (b+c)*a^5-(3*b^2+b*c+3*c^2)*a^4+4*(b^2-c^2)*(b-c)*a^3-2*(2*b^4+2*c^4-b*c*(2*b^2+b*c+2*c^2))*a^2+(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a-(b^4+c^4+b*c*(b-c)^2)*(b-c)^2 : :
X(12718) = 3*X(8236)-2*X(12721) = 3*X(11038)-4*X(12722)

The reciprocal orthologic center of these triangles is X(65).

X(12718) lies on these lines: {7,9801}, {9,12530}, {990,7677}, {1445,1721}, {1766,7676}, {3663,11025}, {4326,7996}, {7675,12717}, {7678,12610}, {7679,12618}, {8232,12689}, {8236,12721}, {8237,12724}, {8238,12725}, {8385,12719}, {8386,12720}, {8387,12726}, {8389,12728}, {8732,9944}, {9950,10865}, {10889,12549}, {11038,12722}, {11526,12652}

X(12718) = reflection of X(i) in X(j) for these (i,j): (7,12723), (12530,9)

X(12718) = X(317)-of-Honsberger-triangle
X(12718) = excentral-to-Honsberger similarity image of X(1721)
X(12718) = Honsberger-isotomic conjugate of X(12669)

X(12719) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-HUTSON TO 5th EXTOUCH

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=2*(b+c)*a^3-2*(b-c)^2*a^2+2*(b^2-c^2)*(b-c)*a-2*(b^2-c^2)^2
G(a,b,c)=2*(a^3-(b-c)*a^2+(b^2-2*b*c-c^2)*a-(b-c)*(b^2+2*b*c-c^2))*b
H(a,b,c)=-a^4-2*(b+c)*a^3+2*(b^2-3*b*c+c^2)*a^2-2*(b+c)*(b^2-4*b*c+c^2)*a+(3*b^2+8*b*c+3*c^2)*(b-c)^2

The reciprocal orthologic center of these triangles is X(65).

X(12719) lies on these lines: {363,1721}, {990,8109}, {1766,8107}, {3663,11026}, {5934,12689}, {7996,8140}, {8111,12717}, {8113,12723}, {8133,12726}, {8377,12610}, {8380,12618}, {8385,12718}, {8390,12721}, {8391,12725}, {9783,9801}, {9944,11854}, {9950,11856}, {9962,11886}, {11039,12722}, {11527,12652}, {11685,12530}, {11892,12549}, {11922,12724}

X(12719) = reflection of X(12720) in X(7996)
X(12719) = X(317)-of-inner-Hutson-triangle
X(12719) = excentral-to-inner-Hutson similarity image of X(1721)
X(12719) = inner-Hutson-isotomic conjugate of X(12673)

X(12720) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-HUTSON TO 5th EXTOUCH

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)-H(a,b,c) : :
where F, G, H are as given at X(12719)

The reciprocal orthologic center of these triangles is X(65).

X(12720) lies on these lines: {990,8110}, {1766,8108}, {3663,11027}, {5935,12689}, {7996,8140}, {8112,12717}, {8114,12723}, {8135,12726}, {8378,12610}, {8381,12618}, {8386,12718}, {8392,12721}, {9944,11855}, {9950,11857}, {9962,11887}, {11040,12722}, {11528,12652}, {11686,12530}, {11893,12549}, {11925,12724}, {11926,12725}

X(12720) = reflection of X(12719) in X(7996)
X(12720) = X(317)-of-outer-Hutson-triangle
X(12720) = excentral-to-outer-Hutson similarity image of X(1721)
X(12720) = outer-Hutson-isotomic conjugate of X(12674)

X(12721) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON INTOUCH TO 5th EXTOUCH

Trilinears (b+c)*a^4-4*b*c*(b^2+c^2)*a-(b^2-c^2)^2*(b+c) : :
X(12721) = 3*X(354)-4*X(4353) = 3*X(392)-2*X(3923) = 3*X(3753)-4*X(3821) = 3*X(8236)-X(12718)

The reciprocal orthologic center of these triangles is X(65).

X(12721) lies on these lines: {1,7175}, {11,12618}, {12,12610}, {38,1824}, {55,990}, {56,1766}, {65,3663}, {72,726}, {145,12530}, {210,3030}, {354,4353}, {392,3923}, {516,3057}, {517,1469}, {518,3875}, {537,4523}, {950,12689}, {960,3729}, {971,3056}, {1362,2823}, {1682,10445}, {1697,1721}, {3601,9944}, {3688,5784}, {3753,3821}, {4313,9962}, {4660,10914}, {7962,12652}, {8236,12718}, {8239,12724}, {8240,12725}, {8241,12726}, {8390,12719}, {8392,12720}, {9785,9801}, {9950,10866}, {10444,10480}, {10544,12680}, {11924,12728}

X(12721) = midpoint of X(145) and X(12530)
X(12721) = reflection of X(i) in X(j) for these (i,j): (65,3663), (3729,960), (10914,4660), (12723,1)
X(12721) = X(317)-of-Hutson-intouch-triangle
X(12721) = excentral-to-Hutson-intouch similarity image of X(1721)
X(12721) = Hutson-intouch-isotomic conjugate of X(12672)

X(12722) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO 5th EXTOUCH

Trilinears (b+c)*a^4+6*b*c*a^3+2*b*c*(b^2+c^2)*a-(b^2-c^2)^2*(b+c) : :
X(12722) = 3*X(3742)-2*X(3821) = 2*X(4353)-3*X(5049) = 3*X(11038)+X(12718) = X(12721)+3*X(12723)

The reciprocal orthologic center of these triangles is X(65).

X(12722) lies on these lines: {1,7175}, {495,12618}, {496,12610}, {516,942}, {518,3923}, {990,999}, {1721,3333}, {1766,3295}, {3487,12689}, {3555,3729}, {3663,5045}, {3742,3821}, {3812,4660}, {4353,5049}, {5255,6211}, {8351,12727}, {9801,11037}, {9950,11035}, {9962,11036}, {11038,12718}, {11039,12719}, {11040,12720}, {11042,12724}, {11043,12725}, {11529,12652}

X(12722) = midpoint of X(i) and X(j) for these {i,j}: {1,12723}, {3555,3729}
X(12722) = reflection of X(i) in X(j) for these (i,j): (3663,5045), (4660,3812)
X(12722) = X(317)-of-incircle-circles-triangle
X(12722) = excentral-to-incircle-circles similarity image of X(1721)
X(12722) = incircle-circles-isotomic conjugate of X(12675)
X(12722) = anticomplement, wrt incircle-circles triangle, of X(4353)

X(12723) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INTOUCH TO 5th EXTOUCH

Trilinears (b+c)*a^3-(b-c)^2*a^2+(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(12723) = 3*X(354)-2*X(3663) = 3*X(3753)-2*X(4660) = X(12721)-4*X(12722)

The reciprocal orthologic center of these triangles is X(65).

X(12723) lies on these lines: {1,7175}, {2,12530}, {4,4008}, {7,9801}, {11,12610}, {12,12618}, {19,6059}, {31,1824}, {33,1460}, {37,2223}, {55,1766}, {56,990}, {57,1721}, {65,516}, {72,3923}, {174,12727}, {181,1864}, {226,12689}, {354,3663}, {517,3056}, {518,3729}, {604,4336}, {726,3555}, {971,1469}, {1108,4516}, {1122,4014}, {1284,12725}, {1359,2823}, {1400,1827}, {1418,3675}, {1742,7146}, {1871,3073}, {1872,3072}, {1876,4331}, {1900,5230}, {2089,12726}, {2171,2293}, {2175,2182}, {2262,3271}, {2285,4319}, {2309,3010}, {2356,8898}, {2805,4852}, {3340,12652}, {3501,4073}, {3753,4660}, {3821,5439}, {3941,8609}, {4523,4672}, {8113,12719}, {8114,12720}, {8243,12724}, {8581,9950}, {10391,10444}

X(12723) = midpoint of X(i) and X(j) for these {i,j}: {7,12718}, {9801,9962}
X(12723) = reflection of X(i) in X(j) for these (i,j): (1,12722), (72,3923), (1721,9944), (4523,4672), (12721,1)
X(12723) = complement of X(12530)
X(12723) = {X(12727), X(12728)}-harmonic conjugate of X(174)
X(12723) = X(317)-of-intouch-triangle
X(12723) = excentral-to-intouch similarity image of X(1721)
X(12723) = intouch-isogonal conjugate of X(222)
X(12723) = intouch-isotomic conjugate of X(1071)
X(12723) = anticomplement, wrt intouch triangle, of X(3663)

X(12724) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO 5th EXTOUCH

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

The reciprocal orthologic center of these triangles is X(65).

X(12724) lies on these lines: {516,9808}, {990,8225}, {1721,8231}, {1766,8224}, {3663,11030}, {3817,8228}, {7996,8244}, {8230,12618}, {8233,12689}, {8234,12717}, {8237,12718}, {8239,12721}, {8243,12723}, {8246,12725}, {8247,12726}, {9789,9801}, {9944,10858}, {9950,10867}, {9962,10885}, {10891,12549}, {11042,12722}, {11532,12652}, {11687,12530}, {11922,12719}, {11925,12720}, {11996,12728}

X(12724) = X(317)-of-2nd-Pamfilos-Zhou-triangle
X(12724) = excentral-to-2nd-Pamfilos-Zhou similarity image of X(1721)
X(12724) = 2nd-Pamfilos-Zhou-isotomic conjugate of X(12681)

X(12725) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO 5th EXTOUCH

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

The reciprocal orthologic center of these triangles is X(65).

X(12725) lies on these lines: {4,240}, {21,990}, {165,846}, {516,2292}, {1284,12723}, {3663,11031}, {4199,12689}, {5051,8582}, {8229,12610}, {8235,12717}, {8238,12718}, {8240,12721}, {8246,12724}, {8249,12726}, {8391,12719}, {8425,12728}, {8731,9944}, {9791,9801}, {10892,12549}, {11043,12722}, {11533,12652}, {11688,12530}, {11926,12720}

X(12725) = X(317)-of-1st-Sharygin-triangle
X(12725) = 2nd-Sharygin-to-1st-Sharygin similarity image of X(13259)
X(12725) = excentral-to-1st-Sharygin similarity image of X(1721)
X(12725) = hexyl-to-1st-Sharygin similarity image of X(12717)
X(12725) = 1st-Sharygin-isotomic conjugate of X(12682)

X(12726) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL-MIDARC TO 5th EXTOUCH

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=-2*(-a+b+c)*((b+c)*a^3-(b-c)^2*a^2+(b^2-c^2)*(b-c)*a-(b^2-c^2)^2)
G(a,b,c)=-2*b*(a-b+c)*(a^3-(b-c)*a^2+(b^2-2*b*c-c^2)*a-(b-c)*(b^2+2*b*c-c^2))
H(a,b,c)=-(-a+b+c)*(a+b-c)*(a-b+c)*(a^2+b^2+c^2)

The reciprocal orthologic center of these triangles is X(65).

X(12726) lies on these lines: {1,12727}, {516,8093}, {990,8077}, {1721,8078}, {1766,8075}, {2089,12723}, {3663,11032}, {7996,8089}, {8079,12689}, {8081,12717}, {8085,12610}, {8087,12618}, {8133,12719}, {8135,12720}, {8241,12721}, {8247,12724}, {8249,12725}, {8387,12718}, {8733,9944}, {9793,9801}, {9962,11888}, {11690,12530}, {11894,12549}

X(12726) = reflection of X(12727) in X(1)
X(12726) = X(317)-of-tangential-midarc-triangle
X(12726) = excentral-to-tangential-midarc similarity image of X(1721)
X(12726) = tangential-midarc-isotomic conjugate of X(8095)

X(12727) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO 5th EXTOUCH

Trilinears -2*(a^2+b^2+c^2)*b*c*sin(A/2)+(b+c)*a^3-(b-c)^2*a^2+(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :

The reciprocal orthologic center of these triangles is X(65).

X(12727) lies on these lines: {1,12726}, {174,12723}, {258,1721}, {990,7588}, {3663,11033}, {8125,12530}, {8351,12722}, {8734,9944}, {9950,11859}, {11895,12549}, {11899,12652}

X(12727) = reflection of X(12726) in X(1)
X(12727) = X(317)-of-2nd-tangential-midarc-triangle
X(12727) = excentral-to-2nd-tangential-midarc similarity image of X(1721)
X(12727) = 2nd-tangential-midarc-isotomic conjugate of X(8096)
X(12727) = {X(174), X(12723)}-harmonic conjugate of X(12728)

X(12728) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO 5th EXTOUCH

Trilinears 2*(a^2+b^2+c^2)*b*c*sin(A/2)+(b+c)*a^3-(b-c)^2*a^2+(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :

The reciprocal orthologic center of these triangles is X(65).

X(12728) lies on these lines: {174,12723}, {516,12445}, {990,7587}, {3663,8083}, {8126,12530}, {8382,12618}, {8389,12718}, {8425,12725}, {8729,9944}, {9801,11891}, {9950,11860}, {9962,11890}, {11535,12652}, {11924,12721}, {11996,12724}

X(12728) = {X(174), X(12723)}-harmonic conjugate of X(12727)
X(12728) = X(317)-of-Yff-central-triangle
X(12728) = excentral-to-Yff-central similarity image of X(1721)
X(12728) = Yff-central-isotomic conjugate of X(12685)

X(12729) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO FUHRMANN

Barycentrics (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(2*a^12-(b+c)*a^11-2*(2*b^2-b*c+2*c^2)*a^10+(b+c)*(2*b^2-b*c+2*c^2)*a^9-(3*b^4+3*c^4+2*(b^2-8*b*c+c^2)*b*c)*a^8+(b+c)*(2*b^4+2*c^4+(b^2-9*b*c+c^2)*b*c)*a^7+(12*b^6+12*c^6-(5*b^4+5*c^4+2*(7*b^2-6*b*c+7*c^2)*b*c)*b*c)*a^6-(b+c)*(8*b^6+8*c^6-(4*b^4+4*c^4+9*b*c*(b^2-b*c+c^2))*b*c)*a^5-2*(b^2-c^2)^2*(4*b^4+4*c^4-(4*b^2-11*b*c+4*c^2)*b*c)*a^4+(b^2-c^2)^2*(b+c)*(7*b^4+7*c^4-b*c*(7*b^2-15*b*c+7*c^2))*a^3-(b^2-c^2)^2*(3*b^4+3*c^4-2*b*c*(5*b^2-4*b*c+5*c^2))*b*c*a^2-(b^2-c^2)^2*(b+c)*(2*b^6+2*c^6-(3*b^4+3*c^4-7*b*c*(b^2-b*c+c^2))*b*c)*a+(b^4-c^4)^2*(b^2-c^2)^2) : :
X(12729) = 3*X(11845)-X(12247) = 3*X(11911)-X(12747)

The reciprocal orthologic center of these triangles is X(3).

X(12729) lies on these lines: {11,11831}, {30,6265}, {80,402}, {100,11900}, {214,1650}, {515,12752}, {952,12438}, {2771,12790}, {2800,12113}, {2802,12626}, {2829,12668}, {4240,6224}, {5840,12696}, {6262,11902}, {6263,11901}, {7972,11910}, {9897,11852}, {9912,11853}, {10057,11912}, {10073,11913}, {11832,12137}, {11839,12198}, {11845,12247}, {11848,12331}, {11863,12460}, {11864,12461}, {11885,12498}, {11903,12737}, {11904,12738}, {11905,12739}, {11906,12740}, {11907,12741}, {11908,12742}, {11909,12743}, {11911,12747}, {11914,12749}, {11915,12750}

X(12729) = midpoint of X(4240) and X(6224)
X(12729) = reflection of X(i) in X(j) for these (i,j): (80,402), (1650,214)
X(12729) = X(80)-of-Gossard-triangle

X(12730) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO FUHRMANN

Barycentrics (5*a^4-12*(b+c)*a^3+11*(b^2+b*c+c^2)*a^2-(b+c)*(6*b^2-5*b*c+6*c^2)*a+2*(b^2-c^2)^2)/(-a+b+c) : :
X(12730) = 2*X(11)-3*X(8236) = 3*X(11038)-4*X(12735)

The reciprocal orthologic center of these triangles is X(8).

X(12730) lies on these lines: {7,528}, {9,12531}, {11,7679}, {80,2346}, {100,2078}, {119,7678}, {145,5856}, {390,952}, {516,7972}, {517,12755}, {1387,6854}, {1445,5541}, {2800,7673}, {2802,7672}, {4326,7993}, {5219,10707}, {5252,8543}, {5854,12630}, {6264,7675}, {8097,8387}, {8098,8388}, {8232,12690}, {8237,12744}, {8238,12746}, {8385,12733}, {8386,12734}, {8389,12748}, {8732,9945}, {9951,10865}, {10889,12550}, {11025,12736}, {11038,12735}, {11526,12653}

X(12730) = reflection of X(i) in X(j) for these (i,j): (7,1317), (1156,390), (12531,9)
X(12730) = X(74)-of-Honsberger-triangle
X(12730) = excentral-to-Honsberger similarity image of X(5541)

X(12731) = ORTHOLOGIC CENTER OF THESE TRIANGLES: FUHRMANN TO HUTSON EXTOUCH

Barycentrics a^7-(b+c)*a^6-(b^2+10*b*c+c^2)*a^5+(b+c)*(b^2+12*b*c+c^2)*a^4-(b^4+c^4+2*b*c*(b^2-19*b*c+c^2))*a^3+(b+c)*(b^4+c^4-2*b*c*(7*b^2+11*b*c+7*c^2))*a^2+(b^2-c^2)^2*(b^2+12*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :

The reciprocal orthologic center of these triangles is X(12732).

X(12731) lies on these lines: {1,12521}, {1158,5493}, {2475,9874}, {3625,6849}, {5082,9953}, {7160,12620}, {9782,9804}

X(12731) = reflection of X(7160) in X(12620)
X(12731) lies on the Jerabek hyperbola of the Fuhrmann triangle.

X(12732) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON EXTOUCH TO FUHRMANN

Barycentrics 4*a^4+2*(b+c)*a^3-(3*b^2+14*b*c+3*c^2)*a^2-2*(b+c)*(b^2-5*b*c+c^2)*a-(b^2-c^2)^2 : :
X(12732) = 7*X(11)-8*X(3634) = 2*X(80)-3*X(1145) = 5*X(80)-6*X(3036) = 7*X(80)-9*X(3679) = 5*X(1145)-4*X(3036) = 7*X(1317)-6*X(3244) = X(1317)-3*X(6154) = 2*X(3244)-7*X(6154) = 4*X(3244)-7*X(10609) = 3*X(3621)+7*X(9963)

The reciprocal orthologic center of these triangles is X(12731).

X(12732) lies on these lines: {9,80}, {11,3634}, {20,952}, {65,1317}, {100,474}, {149,5084}, {214,3748}, {392,9951}, {1320,9945}, {1537,12331}, {1617,2932}, {2094,6224}, {3871,5719}, {3895,11112}, {4304,10914}, {6957,10738}

X(12732) = reflection of X(i) in X(j) for these (i,j): (1145,5541), (1320,9945), (1537,12331), (9802,1387), (10609,6154), (12690,1145)
X(12732) = {X(100), X(9802)}-harmonic conjugate of X(1387)

X(12733) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-HUTSON TO FUHRMANN

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=-2*(2*a-b-c)^2*(a-b+c)*(a+b-c)*b*c
G(a,b,c)=-2*(a-2*b+c)*(-a+b+c)*(a+b-c)*(a^2-a*b+b^2-c^2)*c
H(a,b,c)=a^6-2*(b+c)*a^5-(b^2-15*b*c+c^2)*a^4+(b-4*c)*(4*b-c)*(b+c)*a^3-(b^4+c^4+b*c*(9*b^2-28*b*c+9*c^2))*a^2-(b^2-c^2)*(b-c)*(2*b^2-13*b*c+2*c^2)*a+(b^2-6*b*c+c^2)*(b^2-c^2)^2

The reciprocal orthologic center of these triangles is X(8).

X(12733) lies on these lines: {11,8380}, {100,8109}, {104,8107}, {119,8377}, {363,5541}, {517,12759}, {952,9836}, {1317,8113}, {5854,12633}, {5934,12690}, {6264,8111}, {7993,8140}, {8097,8133}, {8385,12730}, {8391,12746}, {9783,9802}, {9945,11854}, {9951,11856}, {9963,11886}, {11026,12736}, {11039,12735}, {11527,12653}, {11685,12531}, {11892,12550}, {11922,12744}

X(12733) = reflection of X(12734) in X(7993)
X(12733) = X(74)-of-inner-Hutson-triangle
X(12733) = excentral-to-inner-Hutson similarity image of X(5541)

X(12734) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-HUTSON TO FUHRMANN

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)-H(a,b,c) : :
where F, G, H are as given at X(12733)

The reciprocal orthologic center of these triangles is X(8).

X(12734) lies on these lines: {11,8381}, {100,8110}, {104,8108}, {119,8378}, {517,12760}, {952,9837}, {1317,8114}, {5854,12634}, {5935,12690}, {6264,8112}, {7993,8140}, {8097,8135}, {8098,8138}, {8386,12730}, {9945,11855}, {9951,11857}, {9963,11887}, {11027,12736}, {11040,12735}, {11528,12653}, {11686,12531}, {11893,12550}, {11925,12744}, {11926,12746}

X(12734) = reflection of X(12733) in X(7993)
X(12734) = X(74)-of-outer-Hutson-triangle
X(12734) = excentral-to-outer-Hutson similarity image of X(5541)

X(12735) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO FUHRMANN

Barycentrics 6*a^4-6*(b+c)*a^3-(5*b^2-16*b*c+5*c^2)*a^2+6*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(12735) = 3*X(1)-X(11) = 5*X(1)-X(80) = 3*X(1)+X(7972) = 4*X(1)-X(12019) = 5*X(11)-3*X(80) = X(11)+3*X(1317) = X(100)+3*X(3241) = 3*X(7967)+X(10698) = 9*X(7967)-X(12248) = 3*X(10698)+X(12248)

The reciprocal orthologic center of these triangles is X(8).

X(12735) lies on these lines: {1,5}, {30,5048}, {55,10074}, {56,10087}, {100,999}, {104,3295}, {145,1145}, {149,1056}, {153,1058}, {214,3244}, {388,10738}, {390,6938}, {497,10742}, {517,5083}, {519,3035}, {528,5542}, {551,6667}, {631,7317}, {942,2802}, {944,1537}, {1125,3036}, {1319,5844}, {1320,3296}, {1388,5690}, {1479,12763}, {1482,4293}, {1862,1870}, {2098,4302}, {2099,11046}, {2800,9957}, {2829,4342}, {3057,11570}, {3303,10058}, {3304,10090}, {3333,5541}, {3340,10993}, {3476,10247}, {3487,12690}, {3576,8275}, {3616,12531}, {3636,6702}, {3655,7962}, {3890,12532}, {4311,11278}, {4312,12119}, {5045,12736}, {5049,6797}, {5218,10246}, {5556,5734}, {5919,12758}, {6154,11034}, {6198,12138}, {6767,12773}, {7373,12331}, {9802,11037}, {9951,11035}, {9963,11036}, {11011,11551}, {11038,12730}, {11039,12733}, {11040,12734}, {11042,12744}, {11043,12746}, {12053,12611}

X(12735) = midpoint of X(i) and X(j) for these {i,j}: {1,1317}, {11,7972}, {145,1145}, {214,3244}, {944,1537}, {1320,10609}, {3057,11570}, {6154,12653}
X(12735) = reflection of X(i) in X(j) for these (i,j): (1387,1), (3036,1125), (6702,3636), (12019,1387), (12736,5045)
X(12735) = incircle-inverse-of-X(7972)
X(12735) = X(74)-of-incircle-circles-triangle
X(12735) = excentral-to-incircle-circles similarity image of X(5541)
X(12735) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5252,10283), (1,7972,11), (1,10944,5901), (11,1317,7972), (944,4345,9668)

X(12736) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE TO FUHRMANN

Trilinears (b+c)*a^5-(b+c)^2*a^4-(2*b-c)*(b-2*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^2-3*b*c+c^2)*a-(b^2-3*b*c+c^2)*(b^2-c^2)^2 : :
X(12736) = X(80)+3*X(5902) = 3*X(354)-X(1317) = X(1145)-3*X(3753) = 3*X(3873)+X(12531) = 3*X(5587)-X(12665) = 3*X(5902)-X(11570)

The reciprocal orthologic center of these triangles is X(8).

X(12736) lies on these lines: {1,88}, {7,80}, {8,11023}, {11,65}, {46,10058}, {56,11715}, {57,104}, {119,226}, {142,1145}, {149,938}, {354,1317}, {388,12751}, {499,3878}, {517,1387}, {518,3036}, {519,5570}, {528,5572}, {653,1845}, {758,908}, {942,952}, {950,5840}, {954,6594}, {960,6667}, {999,12737}, {1155,5427}, {1156,10398}, {1411,11700}, {1445,2093}, {1768,3339}, {1771,3924}, {1836,12764}, {1837,5884}, {1876,12138}, {1938,10006}, {2099,12740}, {2771,7687}, {2829,4292}, {2840,3937}, {3035,3812}, {3057,10165}, {3333,6264}, {3338,10074}, {3340,10698}, {3486,12119}, {3586,10724}, {3738,10015}, {3873,12531}, {3874,10057}, {3887,11028}, {3918,10039}, {3919,9951}, {4345,5697}, {4654,10711}, {5045,12735}, {5328,5692}, {5587,12665}, {5708,12773}, {5722,10738}, {5728,12690}, {5836,5854}, {6147,11698}, {6326,11529}, {6738,10122}, {7993,10980}, {8083,12748}, {8097,11032}, {9579,10728}, {9802,10580}, {9945,11018}, {9963,11020}, {10404,12763}, {10532,12247}, {10950,12005}, {11021,12550}, {11025,12730}, {11026,12733}, {11027,12734}, {11030,12744}, {11031,12746}

X(12736) = midpoint of X(i) and X(j) for these {i,j}: {11,65}, {80,11570}, {942,6797}
X(12736) = reflection of X(i) in X(j) for these (i,j): (960,6667), (3035,3812), (5083,942), (12735,5045)
X(12736) = incircle-inverse-of-X(106)
X(12736) = X(74)-of-inverse-in-incircle-triangle
X(12736) = X(113)-of-intouch-triangle
X(12736) = complement, wrt intouch triangle, of X(1317)
X(12736) = excentral-to-inverse-in-incircle similarity image of X(5541)
X(12736) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,10090,214), (80,5902,11570), (1320,3306,214), (1737,8068,6702)

X(12737) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO FUHRMANN

Trilinears a^6-2*(b+c)*a^5-(b^2-9*b*c+c^2)*a^4+(b+c)*(4*b^2-11*b*c+4*c^2)*a^3-(b^4+c^4+2*b*c*(3*b^2-8*b*c+3*c^2))*a^2-(b^2-c^2)*(b-c)*(2*b^2-7*b*c+2*c^2)*a+(b^2-3*b*c+c^2)*(b^2-c^2)^2 : :
X(12737) = 5*X(1)-X(5531) = 3*X(1)-X(6326) = 3*X(1)+X(7993) = 4*X(1)-X(12738) = X(153)-3*X(5603) = 2*X(214)-3*X(10246) = 2*X(1320)+X(12515) = 2*X(1537)-3*X(3656) = 3*X(5603)-2*X(12611) = 3*X(10246)-X(12331)

The reciprocal orthologic center of these triangles is X(3).

X(12737) lies on these lines: {1,5}, {3,2802}, {8,12619}, {40,12653}, {65,10074}, {100,1385}, {104,517}, {145,6972}, {149,944}, {153,5603}, {214,1376}, {515,10738}, {519,10265}, {528,3655}, {912,5048}, {946,10742}, {962,12248}, {997,3036}, {999,12736}, {1319,10090}, {1389,6583}, {1482,2800}, {1537,3656}, {1538,10707}, {1768,7982}, {2098,12758}, {2099,11570}, {2646,10087}, {2771,7984}, {2801,10247}, {2827,6095}, {2829,12676}, {2932,10269}, {3057,10058}, {3241,9803}, {3244,12616}, {3434,6224}, {3576,5541}, {3653,6174}, {3898,7489}, {4511,12531}, {5330,5694}, {5731,9802}, {5734,9809}, {5790,6702}, {5840,12700}, {5844,11219}, {6175,10031}, {6262,10920}, {6263,10919}, {6906,10284}, {9912,10829}, {10522,10806}, {10679,12332}, {10794,12198}, {10871,12498}, {10945,12741}, {10946,12742}, {10947,12743}, {11009,11571}, {11014,11826}, {11224,12767}, {11390,12137}, {11865,12460}, {11866,12461}, {11903,12729}, {11928,12747}, {12047,12763}

X(12737) = midpoint of X(i) and X(j) for these {i,j}: {1,6264}, {40,12653}, {104,1320}, {145,12247}, {149,944}, {962,12248}, {1482,12773}, {1768,7982}, {6326,7993}
X(12737) = reflection of X(i) in X(j) for these (i,j): (3,11715), (8,12619), (80,1484), (100,1385), (119,1387), (153,12611), (355,11), (1145,6713), (5660,10283), (6265,1), (7972,1483), (10742,946), (11698,5901), (12331,214), (12515,104), (12738,6265), (12751,5)
X(12737) = hexyl circle-inverse-of-X(7993)
X(12737) = X(80)-of-inner-Johnson-triangle
X(12737) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,80,12740), (1,7972,12739), (1,7993,6326), (11,10944,10057), (119,1387,5886), (153,5603,12611), (7972,10057,10944), (10246,12331,214)

X(12738) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO FUHRMANN

Trilinears a^6-2*(b+c)*a^5-(b^2-b*c+c^2)*a^4+(b+c)*(4*b^2-3*b*c+4*c^2)*a^3-(b^4+c^4+2*b*c*(b^2+c^2))*a^2-(b^2-c^2)*(b-c)*(2*b^2+b*c+2*c^2)*a+(b+c)*(b^2-c^2)*(b^3-c^3) : :
X(12738) = X(1)+3*X(5531) = 5*X(1)-3*X(6264) = 2*X(1)-3*X(6265) = X(1)-3*X(6326) = 7*X(1)-3*X(7993) = 4*X(1)-3*X(12737) = 2*X(5)-3*X(5660) = 3*X(100)-2*X(3579) = 4*X(140)-3*X(11219) = 4*X(3579)-3*X(12515)

The reciprocal orthologic center of these triangles is X(3).

X(12738) lies on these lines: {1,5}, {3,2801}, {35,3652}, {72,74}, {78,10609}, {104,6986}, {140,11219}, {149,12611}, {153,6895}, {200,3654}, {214,958}, {500,5293}, {515,12762}, {517,3935}, {528,3811}, {912,1155}, {943,1156}, {997,3655}, {1259,2932}, {1385,5260}, {1490,5528}, {2800,11500}, {2802,8148}, {2829,12677}, {3035,5791}, {3436,6224}, {3617,10786}, {3634,10265}, {3656,3870}, {4860,6911}, {5204,12757}, {5217,12665}, {5221,11570}, {5694,11491}, {5708,9946}, {5812,5840}, {6262,10922}, {6263,10921}, {6583,6915}, {8167,10246}, {9780,9803}, {9912,10830}, {9955,10707}, {9963,10728}, {10698,11278}, {10742,12437}, {10795,12198}, {10872,12498}, {10951,12741}, {10952,12742}, {10953,12743}, {11391,12137}, {11827,12119}, {11867,12460}, {11868,12461}, {11904,12729}, {11929,12747}

X(12738) = midpoint of X(i) and X(j) for these {i,j}: {5531,6326}, {9963,10728}
X(12738) = reflection of X(i) in X(j) for these (i,j): (80,11698), (149,12611), (6265,6326), (9803,12619), (12515,100), (12737,6265), (12773,214)
X(12738) = X(80)-of-outer-Johnson-triangle

X(12739) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO FUHRMANN

Trilinears (a^4-2*(b+c)*a^3+b*c*a^2+(b+c)*(2*b^2-b*c+2*c^2)*a-(b^3+c^3)*(b+c))*(a-b+c)*(a+b-c) : :
X(12739) = X(80)-4*X(5719)

The reciprocal orthologic center of these triangles is X(3).

X(12739) lies on these lines: {1,5}, {3,10093}, {4,12743}, {21,12532}, {35,11571}, {55,2800}, {56,214}, {59,518}, {65,100}, {78,3035}, {104,943}, {149,3485}, {153,3486}, {388,6224}, {498,12619}, {515,12763}, {517,10087}, {758,5172}, {942,10090}, {950,12764}, {954,2801}, {956,1388}, {1145,3811}, {1320,11011}, {1385,10074}, {1454,12559}, {1464,5018}, {1479,12611}, {1537,6261}, {1768,3601}, {1836,5840}, {2078,4867}, {2099,2802}, {2771,10058}, {2829,12678}, {2932,11509}, {3057,10698}, {3085,12247}, {3295,12758}, {3340,5541}, {3868,4996}, {3870,5854}, {4305,12248}, {4313,9809}, {4321,5856}, {4323,9802}, {4861,11256}, {4870,10707}, {5528,12560}, {5538,5762}, {5703,9803}, {5730,11510}, {5851,7675}, {6001,12775}, {6262,10924}, {6263,10923}, {7354,12119}, {9654,12747}, {9912,10831}, {10404,10609}, {10572,10742}, {10738,12047}, {10797,12198}, {10873,12498}, {11392,12137}, {11501,12331}, {11870,12461}, {11905,12729}, {11930,12741}, {11931,12742}

X(12739) = midpoint of X(i) and X(j) for these {i,j}: {1317,10956}, {7972,10057}
X(12739) = reflection of X(i) in X(j) for these (i,j): (5252,10956), (10057,495)
X(12739) = X(80)-of-1st-Johnson-Yff-triangle
X(12739) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4551,1411), (1,6265,12740), (1,6326,11), (1,7972,12737), (35,11571,12515), (214,5083,56), (495,10944,5252), (1317,10944,7972)

X(12740) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO FUHRMANN

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

The reciprocal orthologic center of these triangles is X(3).

X(12740) lies on these lines: {1,5}, {3,10094}, {33,5151}, {36,12515}, {55,214}, {56,2800}, {65,10698}, {78,5854}, {100,3057}, {104,1319}, {106,10703}, {153,3476}, {497,6224}, {499,12619}, {515,12764}, {517,10090}, {997,1145}, {999,11570}, {1318,1320}, {1385,10058}, {1388,11715}, {1420,1768}, {1470,12332}, {1478,12611}, {1519,12761}, {1537,1836}, {1964,4336}, {2098,2802}, {2099,12736}, {2646,10179}, {2771,10074}, {2829,12679}, {3086,12247}, {3254,6596}, {3304,5083}, {3877,4996}, {4308,9809}, {4345,9802}, {5433,11014}, {5541,7962}, {5563,11571}, {5840,12701}, {6262,10926}, {6263,10925}, {6284,12119}, {6958,10043}, {9669,12747}, {9912,10832}, {9957,10087}, {10106,12763}, {10798,12198}, {10874,12498}, {11256,12531}, {11393,12137}, {11502,12331}, {11871,12460}, {11872,12461}, {11906,12729}, {11932,12741}, {11933,12742}

X(12740) = reflection of X(i) in X(j) for these (i,j): (1837,11), (2932,214), (10073,496)
X(12740) = X(80)-of-2nd-Johnson-Yff-triangle
X(12740) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,80,12737), (1,6265,12739), (1,6326,1317), (497,6224,12743)

X(12741) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO FUHRMANN

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

The reciprocal orthologic center of these triangles is X(3).

X(12741) lies on these lines: {11,11377}, {80,493}, {100,8214}, {214,8222}, {515,12765}, {952,12440}, {2800,9838}, {2802,12636}, {6224,6462}, {6262,8218}, {6263,8216}, {6265,8220}, {6461,12742}, {7972,8210}, {8188,9897}, {8194,9912}, {10057,11951}, {10073,11953}, {10875,12498}, {10945,12737}, {10951,12738}, {11394,12137}, {11503,12331}, {11828,12119}, {11840,12198}, {11846,12247}, {11907,12729}, {11930,12739}, {11932,12740}, {11947,12743}, {11949,12747}, {11955,12749}, {11957,12750}

X(12741) = X(80)-of-Lucas-homothetic-triangle

X(12742) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO FUHRMANN

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

The reciprocal orthologic center of these triangles is X(3).

X(12742) lies on these lines: {11,11378}, {80,494}, {100,8215}, {214,8223}, {515,12766}, {952,12441}, {2800,9839}, {2802,12637}, {6224,6463}, {6262,8219}, {6263,8217}, {6265,8221}, {6461,12741}, {7972,8211}, {8189,9897}, {8195,9912}, {10057,11952}, {10073,11954}, {10876,12498}, {10946,12737}, {10952,12738}, {11395,12137}, {11504,12331}, {11829,12119}, {11841,12198}, {11847,12247}, {11908,12729}, {11931,12739}, {11933,12740}, {11948,12743}, {11950,12747}, {11956,12749}, {11958,12750}

X(12742) = X(80)-of-Lucas(-1)-homothetic-triangle

X(12743) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO FUHRMANN

Barycentrics (-a+b+c)*(2*a^6-(b+c)*a^5-(3*b^2-4*b*c+3*c^2)*a^4+(b+c)*(2*b^2-3*b*c+2*c^2)*a^3-b*c*(b-c)^2*a^2-(b^2-c^2)*(b-c)^3*a+(b^2-c^2)^2*(b-c)^2) : :
X(12743) = 3*X(11114)-X(12532)

The reciprocal orthologic center of these triangles is X(3).

X(12743) lies on these lines: {1,10738}, {3,10073}, {4,12739}, {11,214}, {30,11570}, {33,12137}, {35,12619}, {55,80}, {56,12119}, {65,5840}, {100,1837}, {149,3486}, {355,10087}, {497,6224}, {515,1317}, {952,1898}, {1385,5533}, {1479,6265}, {1697,9897}, {1836,10724}, {2098,7972}, {2800,6284}, {2802,10950}, {2829,12680}, {2932,11502}, {3295,10057}, {3583,12611}, {3586,6326}, {4294,12247}, {4302,12515}, {4304,10265}, {4542,5853}, {5083,7354}, {5432,6702}, {5541,5727}, {5691,12763}, {5722,10090}, {6262,10928}, {6263,10927}, {9912,10833}, {10698,12701}, {10799,12198}, {10877,12498}, {10947,12737}, {10953,12738}, {10965,12749}, {10966,12750}, {11114,12532}, {11873,12460}, {11909,12729}, {11947,12741}, {11948,12742}

X(12743) = reflection of X(i) in X(j) for these (i,j): (11,950), (7354,5083)
X(12743) = X(80)-of-Mandart-incircle-triangle
X(12743) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (497,6224,12740), (3295,12747,10057), (3586,6326,12764)

X(12744) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO FUHRMANN

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

The reciprocal orthologic center of these triangles is X(8).

X(12744) lies on these lines: {11,8230}, {80,7133}, {100,8225}, {104,8224}, {119,8228}, {517,12768}, {952,7596}, {1317,8243}, {1320,7595}, {2800,12681}, {2802,9808}, {5541,8231}, {5854,12638}, {6264,8234}, {7993,8244}, {8097,8247}, {8098,8248}, {8233,12690}, {8237,12730}, {8246,12746}, {9789,9802}, {9945,10858}, {9951,10867}, {9963,10885}, {10891,12550}, {11030,12736}, {11042,12735}, {11532,12653}, {11687,12531}, {11922,12733}, {11925,12734}, {11996,12748}

X(12744) = X(74)-of-2nd-Pamfilos-Zhou-triangle
X(12744) = excentral-to-2nd-Pamfilos-Zhou similarity image of X(5541)

X(12745) = ORTHOLOGIC CENTER OF THESE TRIANGLES: FUHRMANN TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(11604).

X(12745) lies on the Jerabek hyperbola of trhe Furhmann triangle and these lines: {1,6597}, {8,6595}, {191,12342}, {1158,12519}, {2476,9782}, {10266,12623}

X(12745) = midpoint of X(6597) and X(12786)
X(12745) = reflection of X(i) in X(j) for these (i,j): (10266,12623), (12342,191)

X(12746) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO FUHRMANN

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

The reciprocal orthologic center of these triangles is X(8).

X(12746) lies on these lines: {1,3909}, {10,21}, {11,5051}, {104,4220}, {119,8229}, {256,1320}, {517,12770}, {846,5541}, {855,1145}, {952,9840}, {1281,2787}, {1284,1317}, {2292,2802}, {2800,12683}, {4199,12690}, {5854,12642}, {6264,8235}, {7993,8245}, {8097,8249}, {8098,8250}, {8238,12730}, {8246,12744}, {8391,12733}, {8425,12748}, {8731,9945}, {9791,9802}, {9951,10868}, {10892,12550}, {11031,12736}, {11043,12735}, {11533,12653}, {11688,12531}, {11926,12734}

X(12746) = X(74)-of-1st-Sharygin-triangle
X(12746) = 2nd-Sharygin-to-1st-Sharygin similarity image of X(13266)
X(12746) = excentral-to-1st-Sharygin similarity image of X(5541)

X(12747) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS 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-(5*b^4+5*c^4+b*c*(5*b^2-18*b*c+5*c^2))*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) : :
X(12747) = 3*X(3)-2*X(12119) = 3*X(3)-4*X(12619) = 4*X(11)-3*X(10246) = 3*X(80)-X(12119) = 3*X(80)-2*X(12619) = 4*X(214)-5*X(1656) = 5*X(3843)-4*X(12611) = 3*X(11911)-2*X(12729)

The reciprocal orthologic center of these triangles is X(3).

X(12747) lies on these lines: {3,80}, {4,145}, {5,6224}, {11,6980}, {30,12247}, {40,3065}, {100,5790}, {214,1656}, {355,8715}, {382,2800}, {515,12773}, {517,9897}, {528,5779}, {944,1484}, {999,10073}, {1598,12137}, {1657,12515}, {2771,5691}, {2802,12645}, {2829,12684}, {3036,10993}, {3295,10057}, {3526,6702}, {3843,12611}, {5180,5844}, {5727,6797}, {5840,11827}, {6262,11917}, {6263,11916}, {6862,10609}, {6863,12019}, {6892,9945}, {7517,9912}, {7972,10247}, {9301,12498}, {9654,12739}, {9655,11570}, {9668,12758}, {9669,12740}, {10679,12751}, {11842,12198}, {11875,12460}, {11876,12461}, {11911,12729}, {11928,12737}, {11929,12738}, {11949,12741}, {11950,12742}, {12000,12749}, {12001,12750}

X(12747) = reflection of X(i) in X(j) for these (i,j): (3,80), (944,1484), (1482,10738), (1657,12515), (6224,5), (10993,3036), (12119,12619), (12331,355)
X(12747) = X(80)-of-X3-ABC-reflections-triangle
X(12747) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (80,12119,12619), (10057,12743,3295)

X(12748) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO FUHRMANN

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

The reciprocal orthologic center of these triangles is X(8).

X(12748) lies on these lines: {11,8382}, {100,7587}, {174,1317}, {517,12774}, {952,8351}, {2800,12685}, {2802,12445}, {5854,12646}, {7993,8423}, {8083,12736}, {8126,12531}, {8389,12730}, {8425,12746}, {8729,9945}, {9802,11891}, {9951,11860}, {9963,11890}, {11535,12653}, {11896,12550}, {11996,12744}

X(12748) = X(74)-of-Yff-central-triangle
X(12748) = excentral-to-Yff-central similarity image of X(5541)

X(12749) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO FUHRMANN

Barycentrics a^7-2*(b+c)*a^6+11*b*c*a^5+(b+c)*(3*b^2-13*b*c+3*c^2)*a^4-(3*b^2-7*b*c+3*c^2)*(b^2+4*b*c+c^2)*a^3+11*(b^2-c^2)*(b-c)*b*c*a^2+2*(b^2-c^2)^2*(b^2-3*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :
X(12749) = X(80)-4*X(5252)

The reciprocal orthologic center of these triangles is X(3).

X(12749) lies on these lines: {1,5}, {8,10940}, {10,10074}, {36,6735}, {46,1145}, {79,12641}, {100,10915}, {104,10039}, {153,10935}, {214,5552}, {498,11715}, {515,10087}, {517,12763}, {956,5445}, {1320,12047}, {1478,2802}, {1768,11919}, {2098,12611}, {2800,12115}, {2829,5119}, {3057,10742}, {5083,10573}, {5541,9613}, {5697,6256}, {5840,12703}, {5856,9814}, {6224,10528}, {6262,10930}, {6263,10929}, {9612,12653}, {9912,10834}, {9957,12764}, {10090,10106}, {10698,12608}, {10803,12198}, {10805,12247}, {10878,12498}, {10965,12743}, {10970,12767}, {11248,12119}, {11400,12137}, {11509,12331}, {11881,12460}, {11882,12461}, {11914,12729}, {11955,12741}, {11956,12742}, {12000,12747}

X(12749) = reflection of X(i) in X(j) for these (i,j): (1,10956), (80,10057), (10057,5252)
X(12749) = X(80)-of-inner-Yff-tangents-triangle
X(12749) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,12751,80), (80,7972,12750), (355,10073,80), (1317,11729,1), (10942,10944,1)

X(12750) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO FUHRMANN

Barycentrics a^7-2*(b+c)*a^6-b*c*a^5+(b+c)*(3*b^2-b*c+3*c^2)*a^4-3*(b^2+c^2)*(b^2-b*c+c^2)*a^3-(b^2-c^2)*(b-c)*b*c*a^2+2*(b^2-c^2)*(b-c)*(b^3+c^3)*a-(b^2-c^2)^3*(b-c) : :
X(12750) = 3*X(80)-4*X(1837)

The reciprocal orthologic center of these triangles is X(3).

X(12750) lies on these lines: {1,5}, {35,11219}, {46,528}, {79,3254}, {100,10916}, {149,4295}, {214,10527}, {515,12776}, {1478,3892}, {1479,2801}, {1768,11920}, {1898,4857}, {2771,12374}, {2800,12116}, {2802,12649}, {2829,12687}, {4311,10074}, {4314,10058}, {4333,5840}, {5083,11048}, {5086,10031}, {5445,5687}, {6224,10529}, {6262,10932}, {6263,10931}, {9785,9803}, {9912,10835}, {10087,10265}, {10707,12047}, {10804,12198}, {10806,12247}, {10879,12498}, {10966,12743}, {10971,12767}, {11249,12119}, {11401,12137}, {11510,12331}, {11883,12460}, {11884,12461}, {11915,12729}, {11957,12741}, {11958,12742}, {12001,12747}

X(12750) = reflection of X(80) in X(10073)
X(12750) = X(80)-of-outer-Yff-tangents-triangle
X(12750) = {X(80), X(7972)}-harmonic conjugate of X(12749)

X(12751) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO INNER-GARCIA

Barycentrics a^7-2*(b+c)*a^6+9*b*c*a^5+(b+c)*(3*b^2-11*b*c+3*c^2)*a^4-(3*b^4+3*c^4+b*c*(3*b^2-16*b*c+3*c^2))*a^3+9*(b^2-c^2)*(b-c)*b*c*a^2+2*(b^2-c^2)^2*(b^2-3*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :
X(12751) = 3*X(1)-4*X(11729) = 3*X(8)+X(9809) = 2*X(11)-3*X(5587) = 3*X(119)-2*X(11729) = 3*X(153)-X(9809) = 4*X(1387)-5*X(8227) = 5*X(1698)-4*X(6713) = 3*X(1699)-X(12653) = X(1768)-3*X(3679) = X(10698)-3*X(10711)

The reciprocal orthologic center of these triangles is X(40).

X(12751) lies on the cubic K684 and these lines: {1,5}, {2,11715}, {4,2802}, {8,153}, {10,104}, {40,1145}, {65,12763}, {72,12762}, {100,515}, {214,944}, {388,12736}, {516,10728}, {517,10742}, {519,1519}, {528,11372}, {529,5535}, {912,11571}, {946,1320}, {1482,12611}, {1537,5854}, {1698,6713}, {1699,12653}, {1737,5193}, {1768,3359}, {2550,2801}, {2787,9864}, {2806,12784}, {2827,4768}, {2932,12114}, {3035,3576}, {3036,5794}, {3057,12764}, {3419,11525}, {3813,11256}, {3898,6965}, {4413,5790}, {4668,12767}, {4996,6796}, {5086,12531}, {5090,12138}, {5541,5691}, {5552,6224}, {5554,9803}, {5657,12248}, {5687,12332}, {5688,12754}, {5689,12753}, {5690,12515}, {5787,9945}, {5818,6702}, {5847,10759}, {6797,8581}, {8193,9913}, {8197,12462}, {8204,12463}, {8214,12765}, {8215,12766}, {9857,12499}, {10039,10058}, {10087,10572}, {10573,11570}, {10679,12747}, {10707,10863}, {10791,12199}, {10914,12761}, {10915,12775}, {10916,12776}, {11248,12331}, {11362,11684}, {11900,12752}, {12647,12758}

X(12751) = midpoint of X(i) and X(j) for these {i,j}: {8,153}, {5531,9897}, {5541,5691}, {5881,6326}
X(12751) = reflection of X(i) in X(j) for these (i,j): (1,119), (40,1145), (80,355), (104,10), (944,214), (1320,946), (1482,12611), (2077,6735), (5693,12665), (6264,11), (6265,11698), (7972,6265), (7982,1537), (11219,5790), (11256,3813), (12119,100), (12515,5690), (12737,5), (12773,12619)
X(12751) = anticomplement of X(11715)
X(12751) = Fuhrmann circle-inverse-of-X(5881)
X(12751) = X(104)-of-outer-Garcia-triangle
X(12751) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (80,7972,10073), (80,12749,1), (355,5252,5587), (5587,6264,11), (5660,7972,6265), (5790,12773,12619)

X(12752) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO INNER-GARCIA

Barycentrics ((b+c)^2*a^13-(b+c)^3*a^12-4*(b^4-b^2*c^2+c^4)*a^11+2*(2*b^2-3*b*c+2*c^2)*(b+c)^3*a^10+(5*b^6+5*c^6-(15*b^4+15*c^4+4*b*c*(b^2-6*b*c+c^2))*b*c)*a^9-(5*b^4+5*c^4-b*c*(20*b^2-31*b*c+20*c^2))*(b+c)^3*a^8+2*(10*b^6+10*c^6-(11*b^4+11*c^4+b*c*(9*b^2-22*b*c+9*c^2))*b*c)*b*c*a^7-2*(b^2-c^2)^2*(b+c)*b*c*(10*b^2-11*b*c+10*c^2)*a^6-(b^2-c^2)^2*(5*b^6+5*c^6-b^2*c^2*(25*b^2-42*b*c+25*c^2))*a^5+5*(b^2-c^2)^2*(b+c)*(b^6+c^6+(2*b^4+2*c^4-b*c*(5*b^2-8*b*c+5*c^2))*b*c)*a^4+2*(b^2-c^2)^2*(2*b^8+2*c^8-(6*b^6+6*c^6+(3*b^4+3*c^4-b*c*(9*b^2-14*b*c+9*c^2))*b*c)*b*c)*a^3-2*(b^2-c^2)^3*(b-c)*(2*b^6+2*c^6+(3*b^4+3*c^4+b*c*(b^2+8*b*c+c^2))*b*c)*a^2-(b^2-c^2)^4*(b^6+c^6-(5*b^4+5*c^4-2*b*c*(2*b^2-7*b*c+2*c^2))*b*c)*a+(b^2-c^2)^5*(b-c)*(b^4+3*b^2*c^2+c^4))*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(12752) = 2*X(11)-3*X(11897) = X(1768)-3*X(11852) = 2*X(11715)-3*X(11831) = 3*X(11845)-X(12248) = 3*X(11911)-X(12773)

The reciprocal orthologic center of these triangles is X(40).

X(12752) lies on these lines: {11,11897}, {30,100}, {104,402}, {119,1650}, {153,4240}, {515,12729}, {952,11251}, {1317,11909}, {1768,11852}, {2787,12181}, {2800,12438}, {2802,12696}, {2806,12796}, {2829,12113}, {9913,11853}, {10058,11912}, {10074,11913}, {10698,11910}, {11715,11831}, {11832,12138}, {11839,12199}, {11845,12248}, {11848,12332}, {11885,12499}, {11900,12751}, {11901,12753}, {11902,12754}, {11903,12761}, {11904,12762}, {11905,12763}, {11906,12764}, {11907,12765}, {11908,12766}, {11911,12773}, {11914,12775}, {11915,12776}

X(12752) = midpoint of X(153) and X(4240)
X(12752) = X(104)-of-Gossard-triangle
X(12752) = reflection of X(i) in X(j) for these (i,j): (104,402), (1650,119)

X(12753) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO INNER-GARCIA

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

The reciprocal orthologic center of these triangles is X(40).

X(12753) lies on these lines: {6,104}, {11,6202}, {100,11824}, {119,5591}, {153,1271}, {515,6263}, {952,1161}, {1317,10927}, {1768,5589}, {2771,7732}, {2783,6319}, {2787,6227}, {2800,3641}, {2802,12697}, {2806,12805}, {2829,5871}, {5595,9913}, {5605,10698}, {5689,12751}, {6215,10742}, {8198,12462}, {8205,12463}, {8216,12765}, {8217,12766}, {9994,12499}, {10040,10058}, {10048,10074}, {10783,12248}, {10792,12199}, {10919,12761}, {10921,12762}, {10923,12763}, {10925,12764}, {10929,12775}, {10931,12776}, {11370,11715}, {11388,12138}, {11497,12332}, {11901,12752}, {11916,12773}

X(12753) = reflection of X(12754) in X(104)
X(12753) = X(104)-of-inner-Grebe-triangle

X(12754) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO INNER-GARCIA

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

The reciprocal orthologic center of these triangles is X(40).

X(12754) lies on these lines: {6,104}, {11,6201}, {100,11825}, {119,5590}, {153,1270}, {515,6262}, {952,1160}, {1317,10928}, {1768,5588}, {2771,7733}, {2783,6320}, {2787,6226}, {2800,3640}, {2802,12698}, {2806,12806}, {2829,5870}, {5594,9913}, {5604,10698}, {5688,12751}, {6214,10742}, {8199,12462}, {8206,12463}, {8218,12765}, {8219,12766}, {9995,12499}, {10041,10058}, {10049,10074}, {10784,12248}, {10793,12199}, {10920,12761}, {10922,12762}, {10924,12763}, {10926,12764}, {10930,12775}, {10932,12776}, {11371,11715}, {11389,12138}, {11498,12332}, {11902,12752}, {11917,12773}

X(12754) = reflection of X(12753) in X(104)
X(12754) = X(104)-of-outer-Grebe-triangle

X(12755) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO INNER-GARCIA

Trilinears (b+c)*a^7-3*(b^2+b*c+c^2)*a^6+(b+c)*(b^2+b*c+c^2)*a^5+(5*b^4-3*b^2*c^2+5*c^4)*a^4-(b+c)*(5*b^4+5*c^4-b*c*(6*b^2-5*b*c+6*c^2))*a^3-(b^4+c^4+b*c*(b^2-b*c+c^2))*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(3*b^4+3*c^4-b*c*(b^2-3*b*c+c^2))*a-(b^4-c^4)*(b^2-c^2)*(b-c)^2 : :
X(12755) = 3*X(8236)-2*X(12758)

The reciprocal orthologic center of these triangles is X(3869).

X(12755) lies on these lines: {7,80}, {9,12532}, {100,518}, {104,2346}, {390,2800}, {516,11571}, {517,12730}, {952,7672}, {971,10728}, {1156,2771}, {1387,11025}, {1445,6326}, {1768,7675}, {2802,12630}, {2829,12669}, {3868,5856}, {4326,12767}, {5083,11038}, {5809,9809}, {5851,10394}, {6224,7674}, {6264,11526}, {6265,7677}, {7676,12515}, {7678,12611}, {7679,12619}, {8232,12691}, {8236,12758}, {8237,12768}, {8238,12770}, {8385,12759}, {8386,12760}, {8387,12771}, {8389,12774}, {8732,9946}, {9952,10865}, {10889,12551}

X(12755) = reflection of X(i) in X(j) for these (i,j): (7,11570), (1156,5728), (12532,9)
X(12755) = X(265)-of-Honsberger-triangle
X(12755) = excentral-to-Honsberger similarity image of X(6326)

X(12756) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GARCIA TO HUTSON EXTOUCH

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

The reciprocal orthologic center of these triangles is X(12757).

X(12756) lies on these lines: {8,6835}, {40,12757}, {191,9898}, {1728,10059}, {3957,12260}, {11224,12654}

X(12756) = reflection of X(12777) in X(12670)

X(12757) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON EXTOUCH TO INNER-GARCIA

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

The reciprocal orthologic center of these triangles is X(12756).

X(12757) lies on these lines: {9,48}, {20,2800}, {40,12756}, {65,952}, {80,6826}, {442,12675}, {1768,10268}, {3560,6265}, {3754,5881}, {5204,12738}, {5445,5770}, {5554,9803}, {5693,5731}, {6897,12247}, {9940,12619}

X(12757) = midpoint of X(6224) and X(9964)
X(12757) = reflection of X(i) in X(j) for these (i,j): (80,9946), (12665,6326), (12691,214)

X(12758) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON INTOUCH TO INNER-GARCIA

Trilinears (b+c)*a^5-(b^2+4*b*c+c^2)*a^4-(b+c)*(2*b^2-7*b*c+2*c^2)*a^3+(2*b^4+2*c^4+3*(b^2-4*b*c+c^2)*b*c)*a^2+(b^2-c^2)*(b-c)*(b^2-5*b*c+c^2)*a+(b^2-c^2)*(b-c)*(-b^3-c^3) : :
X(12758) = 3*X(1)-2*X(5083) = 3*X(1)-X(11571) = X(100)-3*X(3877) = 3*X(392)-2*X(3035) = 3*X(3899)+X(12653) = 4*X(5083)-3*X(11570) = 3*X(5919)-2*X(12735) = 3*X(8236)-X(12755) = 3*X(11570)-2*X(11571)

The reciprocal orthologic center of these triangles is X(3869).

X(12758) lies on these lines: {1,104}, {3,10094}, {4,10043}, {8,80}, {11,517}, {12,12611}, {35,214}, {40,10090}, {55,6265}, {56,12515}, {65,1387}, {72,5854}, {90,1320}, {100,997}, {119,10039}, {145,12532}, {153,10935}, {355,12764}, {390,2801}, {392,3035}, {497,10051}, {758,2611}, {946,8068}, {950,12691}, {952,1898}, {960,1145}, {1317,2771}, {1537,12047}, {1697,6326}, {2098,12737}, {2829,12672}, {3036,10914}, {3295,12739}, {3476,12248}, {3586,8275}, {3601,9946}, {3612,3890}, {3753,6667}, {3885,12531}, {3899,5223}, {4294,6224}, {4302,12119}, {4313,9964}, {5252,10742}, {5531,9819}, {5533,10265}, {5730,8668}, {5919,12735}, {6264,7962}, {6702,7741}, {8071,12332}, {8236,12755}, {8239,12768}, {8240,12770}, {8241,12771}, {8390,12759}, {8392,12760}, {9668,12747}, {9785,9803}, {9952,10866}, {10284,10950}, {10738,12701}, {11924,12774}, {12647,12751}

X(12758) = midpoint of X(i) and X(j) for these {i,j}: {80,5697}, {145,12532}, {1320,3869}, {3885,12531}
X(12758) = reflection of X(i) in X(j) for these (i,j): (65,1387), (214,3884), (1145,960), (1317,9957), (10914,3036), (11570,1), (11571,5083), (12665,5887)
X(12758) = X(265)-of-Hutson-intouch-triangle
X(12758) = X(12121)-of-intouch-triangle
X(12758) = excentral-to-Hutson-intouch similarity image of X(6326)
X(12758) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1768,10074), (1,11571,5083), (497,12247,10073), (1697,6326,10087), (5083,11571,11570), (10265,12053,5533)

X(12759) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-HUTSON TO INNER-GARCIA

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=-2*(a^2-b^2+b*c-c^2)*((b+c)*a^3-(b^2+c^2)*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2)
G(a,b,c)=-2*(a^2-a*c-b^2+c^2)*(a^3-a^2*b-(b-c)^2*a+b*(b^2-c^2))*b
H(a,b,c)=a^6+2*(b+c)*a^5-(5*b^2+b*c+5*c^2)*a^4-(b+c)*(4*b^2-11*b*c+4*c^2)*a^3+(7*b^2+13*b*c+7*c^2)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(2*b^2-7*b*c+2*c^2)*a-(3*b^2-2*b*c+3*c^2)*(b^2-c^2)^2

The reciprocal orthologic center of these triangles is X(3869).

X(12759) lies on these lines: {363,6326}, {517,12733}, {1387,11026}, {1768,8111}, {2800,9836}, {2802,12633}, {5083,11039}, {5934,12691}, {6264,11527}, {6265,8109}, {8107,12515}, {8113,11570}, {8133,12771}, {8140,12760}, {8377,12611}, {8380,12619}, {8385,12755}, {8390,12758}, {8391,12770}, {9783,9803}, {9946,11854}, {9952,11856}, {9964,11886}, {11685,12532}, {11892,12551}, {11922,12768}

X(12759) = reflection of X(12760) in X(12767)
X(12759) = X(265)-of-inner-Hutson-triangle
X(12759) = excentral-to-inner-Hutson similarity image of X(6326)

X(12760) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-HUTSON TO INNER-GARCIA

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)-H(a,b,c) : :
where F, G, H are as given at X(12759)

The reciprocal orthologic center of these triangles is X(3869).

X(12760) lies on these lines: {517,12734}, {1387,11027}, {1768,8112}, {2800,9837}, {2802,12634}, {5083,11040}, {5935,12691}, {6264,11528}, {6265,8110}, {8108,12515}, {8114,11570}, {8135,12771}, {8140,12759}, {8378,12611}, {8381,12619}, {8386,12755}, {8392,12758}, {9946,11855}, {9952,11857}, {9964,11887}, {11686,12532}, {11893,12551}, {11925,12768}, {11926,12770}

X(12760) = reflection of X(12759) in X(12767)
X(12760) = X(265)-of-outer-Hutson-triangle
X(12760) = excentral-to-outer-Hutson similarity image of X(6326)

X(12761) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO INNER-GARCIA

Barycentrics a^10-2*(b+c)*a^9-(b^2-9*b*c+c^2)*a^8+(b+c)*(4*b^2-11*b*c+4*c^2)*a^7-2*(b^4+c^4+b*c*(4*b^2-11*b*c+4*c^2))*a^6+12*(b^2-c^2)*(b-c)*b*c*a^5+(2*b^4+2*c^4-b*c*(5*b^2+18*b*c+5*c^2))*(b-c)^2*a^4-(b^2-c^2)*(b-c)*(4*b^4+4*c^4-b*c*(b^2+14*b*c+c^2))*a^3+(b^2-c^2)^2*(b^4+c^4+2*b*c*(3*b^2-8*b*c+3*c^2))*a^2+2*(b^2-c^2)^3*(b-c)*(b^2-3*b*c+c^2)*a-(b^2-c^2)^4*(b-c)^2 : :
X(12761) = X(2950)-3*X(5587)

The reciprocal orthologic center of these triangles is X(40).

X(12761) lies on these lines: {4,11}, {12,12775}, {80,6001}, {100,11826}, {119,1376}, {149,12667}, {153,3434}, {355,2800}, {515,10738}, {952,6256}, {1012,8068}, {1158,12619}, {1317,10947}, {1478,1537}, {1519,12740}, {1532,10090}, {1768,10826}, {2787,12182}, {2802,12700}, {2950,5587}, {3035,6850}, {3419,12665}, {3585,10057}, {4996,6932}, {5840,11500}, {5842,10724}, {6265,12608}, {6667,6893}, {6713,6929}, {7971,9897}, {9913,10829}, {10058,10523}, {10074,10948}, {10698,10944}, {10794,12199}, {10871,12499}, {10914,12751}, {10919,12753}, {10920,12754}, {10945,12765}, {10946,12766}, {10949,12776}, {11373,11715}, {11390,12138}, {11865,12462}, {11866,12463}, {11903,12752}, {11928,12773}

X(12761) = midpoint of X(i) and X(j) for these {i,j}: {149,12667}, {7971,9897}
X(12761) = reflection of X(i) in X(j) for these (i,j): (1158,12619), (6265,12608), (12114,11), (12332,119), (12762,10742)
X(12761) = X(104)-of-inner-Johnson-triangle
X(12761) = {X(4), X(104)}-harmonic conjugate of X(12764)

X(12762) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO INNER-GARCIA

Barycentrics a^10-2*(b+c)*a^9-(b^2-9*b*c+c^2)*a^8+(b+c)*(4*b^2-15*b*c+4*c^2)*a^7-2*(b^4+c^4+b*c*(2*b^2-15*b*c+2*c^2))*a^6+4*b*c*(b+c)*(5*b^2-11*b*c+5*c^2)*a^5+(2*b^4+2*c^4-b*c*(13*b^2+38*b*c+13*c^2))*(b-c)^2*a^4-(b^2-c^2)*(b-c)*(4*b^4+4*c^4+b*c*(3*b^2-26*b*c+3*c^2))*a^3+(b^2-c^2)^2*(b^4+c^4+10*b*c*(b-c)^2)*a^2+2*(b^2-c^2)^3*(b-c)*(b^2-3*b*c+c^2)*a-(b^2-c^2)^4*(b-c)^2 : :
X(12762) = 3*X(153)-X(12667)

The reciprocal orthologic center of these triangles is X(40).

X(12762) lies on these lines: {11,10532}, {12,104}, {20,100}, {72,12751}, {80,7686}, {119,958}, {355,2800}, {515,12738}, {952,10526}, {1317,10806}, {1768,10827}, {2787,12183}, {2802,5812}, {2886,6982}, {4295,12247}, {4298,10265}, {4301,10738}, {5220,5690}, {5270,11219}, {5432,12115}, {6253,10728}, {9913,10830}, {10058,10954}, {10074,10523}, {10698,10950}, {10786,12248}, {10795,12199}, {10872,12499}, {10921,12753}, {10922,12754}, {10951,12765}, {10952,12766}, {10955,12775}, {11236,12114}, {11374,11715}, {11391,12138}, {11867,12462}, {11868,12463}, {11904,12752}, {11929,12773}

X(12762) = reflection of X(12761) in X(10742)
X(12762) = X(104)-of-outer-Johnson-triangle

X(12763) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO INNER-GARCIA

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

The reciprocal orthologic center of these triangles is X(40).

X(12763) lies on the Johnson-Yff-inner circle and these lines: {1,10742}, {4,1317}, {5,10074}, {11,153}, {12,104}, {30,10087}, {55,2829}, {56,119}, {65,12751}, {80,942}, {100,7354}, {149,5229}, {355,11570}, {495,10058}, {515,12739}, {952,1478}, {1388,11729}, {1466,9657}, {1479,12735}, {1537,2098}, {1768,9578}, {1836,2802}, {1837,5083}, {2771,10057}, {2800,5252}, {3032,9553}, {3035,3436}, {3045,9653}, {3085,12248}, {3585,7972}, {5434,10711}, {5541,9579}, {5691,12743}, {6264,9612}, {6284,10728}, {6326,9613}, {8068,9654}, {9655,12331}, {9913,10831}, {10039,12515}, {10090,11698}, {10106,12740}, {10404,12736}, {10698,10944}, {10797,12199}, {10827,12619}, {10873,12499}, {10923,12753}, {10924,12754}, {10957,12776}, {11375,11715}, {11392,12138}, {11501,12332}, {11869,12462}, {11870,12463}, {11905,12752}, {11930,12765}, {11931,12766}, {12047,12737}

X(12763) = reflection of X(i) in X(j) for these (i,j): (55,10956), (10058,495)
X(12763) = X(104)-of-1st-Johnson-Yff-triangle
X(12763) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,10742,12764), (153,388,11), (3585,7972,10738), (9654,12773,8068)

X(12764) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO INNER-GARCIA

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

The reciprocal orthologic center of these triangles is X(40).

X(12764) lies on the Johnson-Yff-outer circle and these lines: {1,10742}, {4,11}, {5,10058}, {30,10090}, {55,119}, {80,517}, {100,1329}, {149,3436}, {153,497}, {355,12758}, {377,6667}, {381,8068}, {480,528}, {496,10074}, {515,12740}, {529,10707}, {950,12739}, {952,1479}, {1156,11604}, {1319,1538}, {1320,5080}, {1387,1478}, {1388,6256}, {1532,5172}, {1537,2099}, {1737,12515}, {1768,9581}, {1836,12736}, {1837,2800}, {1898,2771}, {2475,6691}, {2478,3035}, {2787,12185}, {2802,12701}, {2841,10774}, {3032,9554}, {3036,3434}, {3045,9666}, {3057,12751}, {3058,10711}, {3303,10956}, {3586,6326}, {4186,9672}, {4857,7972}, {4996,11114}, {5432,6965}, {5533,9669}, {5541,9580}, {5722,11570}, {5840,6928}, {6264,9614}, {6265,10572}, {6713,6923}, {6840,10724}, {9668,12331}, {9670,10953}, {9913,10832}, {9957,12749}, {10087,11698}, {10698,10950}, {10798,12199}, {10826,12619}, {10874,12499}, {10925,12753}, {10926,12754}, {10958,12775}, {10959,12776}, {11376,11715}, {11393,12138}, {11502,12332}, {11871,12462}, {11872,12463}, {11906,12752}, {11932,12765}, {11933,12766}

X(12764) = midpoint of X(149) and X(3436)
X(12764) = reflection of X(i) in X(j) for these (i,j): (56,11), (100,1329), (10074,496)
X(12764) = X(104)-of-2nd-Johnson-Yff-triangle
X(12764) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,10742,12763), (80,3583,10738), (153,497,1317), (9669,12773,5533)

X(12765) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO INNER-GARCIA

Trilinears (a^10-(b+c)*a^9-(4*b^2-5*b*c+4*c^2)*a^8+4*(b^3+c^3)*a^7+2*(5*b^4+5*c^4-3*(b^2-4*b*c+c^2)*b*c)*a^6-2*(b+c)*(5*b^4+5*c^4-2*b*c*(b^2-6*b*c+c^2))*a^5-4*(3*b^6+3*c^6-(5*b^4+5*c^4-b*c*(3*b^2-11*b*c+3*c^2))*b*c)*a^4+12*(b^3-c^3)*(b^4-c^4)*a^3+(b^2+c^2)*(5*b^4+5*c^4-2*b*c*(4*b^2+9*b*c+4*c^2))*(b-c)^2*a^2-(b^4-c^4)*(b-c)^3*(5*b^2+8*b*c+5*c^2)*a-(b^4-c^4)^2*b*c)*S+a*(a^11-(b+c)*a^10-(5*b^2-3*b*c+5*c^2)*a^9+(b+c)*(5*b^2-2*b*c+5*c^2)*a^8+2*(5*b^4+5*c^4-2*b*c*(3*b^2-4*b*c+3*c^2))*a^7-2*(b+c)*(5*b^4+5*c^4-4*b*c*(b-c)^2)*a^6-2*(5*b^6+5*c^6-(9*b^4+9*c^4-5*(b^2-4*b*c+c^2)*b*c)*b*c)*a^5+2*(b+c)*(5*b^6+5*c^6-(6*b^4+6*c^4-b*c*(5*b^2-16*b*c+5*c^2))*b*c)*a^4+(5*b^8+5*c^8-2*(6*b^6+6*c^6+(4*b^4+4*c^4-b*c*(2*b^2+19*b*c+2*c^2))*b*c)*b*c)*a^3-(b^2-c^2)*(b-c)*(5*b^6+5*c^6+(2*b^4+2*c^4-b*c*(9*b^2+28*b*c+9*c^2))*b*c)*a^2-(b^2-c^2)^2*(b^6+c^6-(3*b^4+3*c^4+b*c*(5*b-c)*(b-5*c))*b*c)*a+(b^2-c^2)^3*(b-c)*((b^2-c^2)^2-4*b^2*c^2)) : :

The reciprocal orthologic center of these triangles is X(40).

X(12765) lies on these lines: {11,8212}, {100,11828}, {104,493}, {119,8222}, {153,6462}, {515,12741}, {952,10669}, {1317,11947}, {1768,8188}, {2787,12186}, {2800,12440}, {2829,9838}, {6461,12766}, {8194,9913}, {8201,12462}, {8208,12463}, {8210,10698}, {8214,12751}, {8216,12753}, {8218,12754}, {8220,10742}, {10058,11951}, {10074,11953}, {10875,12499}, {11377,11715}, {11394,12138}, {11503,12332}, {11840,12199}, {11846,12248}, {11930,12763}, {11932,12764}, {11949,12773}, {11955,12775}, {11957,12776}

X(12765) = X(104)-of-Lucas-homothetic-triangle

X(12766) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO INNER-GARCIA

Trilinears -(a^10-(b+c)*a^9-(4*b^2-5*b*c+4*c^2)*a^8+4*(b^3+c^3)*a^7+2*(5*b^4+5*c^4-3*(b^2-4*b*c+c^2)*b*c)*a^6-2*(b+c)*(5*b^4+5*c^4-2*b*c*(b^2-6*b*c+c^2))*a^5-4*(3*b^6+3*c^6-(5*b^4+5*c^4-b*c*(3*b^2-11*b*c+3*c^2))*b*c)*a^4+12*(b^3-c^3)*(b^4-c^4)*a^3+(b^2+c^2)*(5*b^4+5*c^4-2*b*c*(4*b^2+9*b*c+4*c^2))*(b-c)^2*a^2-(b^4-c^4)*(b-c)^3*(5*b^2+8*b*c+5*c^2)*a-(b^4-c^4)^2*b*c)*S+a*(a^11-(b+c)*a^10-(5*b^2-3*b*c+5*c^2)*a^9+(b+c)*(5*b^2-2*b*c+5*c^2)*a^8+2*(5*b^4+5*c^4-2*b*c*(3*b^2-4*b*c+3*c^2))*a^7-2*(b+c)*(5*b^4+5*c^4-4*b*c*(b-c)^2)*a^6-2*(5*b^6+5*c^6-(9*b^4+9*c^4-5*(b^2-4*b*c+c^2)*b*c)*b*c)*a^5+2*(b+c)*(5*b^6+5*c^6-(6*b^4+6*c^4-b*c*(5*b^2-16*b*c+5*c^2))*b*c)*a^4+(5*b^8+5*c^8-2*(6*b^6+6*c^6+(4*b^4+4*c^4-b*c*(2*b^2+19*b*c+2*c^2))*b*c)*b*c)*a^3-(b^2-c^2)*(b-c)*(5*b^6+5*c^6+(2*b^4+2*c^4-b*c*(9*b^2+28*b*c+9*c^2))*b*c)*a^2-(b^2-c^2)^2*(b^6+c^6-(3*b^4+3*c^4+b*c*(5*b-c)*(b-5*c))*b*c)*a+(b^2-c^2)^3*(b-c)*((b^2-c^2)^2-4*b^2*c^2)) : :

The reciprocal orthologic center of these triangles is X(40).

X(12766) lies on these lines: {11,8213}, {100,11829}, {104,494}, {119,8223}, {153,6463}, {515,12742}, {952,10673}, {1317,11948}, {1768,8189}, {2787,12187}, {2800,12441}, {2829,9839}, {6461,12765}, {8195,9913}, {8202,12462}, {8209,12463}, {8211,10698}, {8215,12751}, {8217,12753}, {8219,12754}, {8221,10742}, {10058,11952}, {10074,11954}, {10876,12499}, {11378,11715}, {11395,12138}, {11504,12332}, {11841,12199}, {11847,12248}, {11931,12763}, {11933,12764}, {11950,12773}, {11956,12775}, {11958,12776}

X(12766) = X(104)-of-Lucas(-1)-homothetic-triangle

X(12767) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR TO INNER-GARCIA

Trilinears a^6+2*(b+c)*a^5-(5*b^2+b*c+5*c^2)*a^4-(b+c)*(4*b^2-11*b*c+4*c^2)*a^3+(7*b^2+13*b*c+7*c^2)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(2*b^2-7*b*c+2*c^2)*a-(3*b^2-2*b*c+3*c^2)*(b^2-c^2)^2 : :
X(12767) = 3*X(1)-4*X(104) = 5*X(1)-4*X(10698) = 7*X(1)-8*X(11715) = 3*X(40)-2*X(12331) = 2*X(104)-3*X(1768) = 5*X(104)-3*X(10698) = 7*X(104)-6*X(11715) = 2*X(153)-3*X(3679) = 5*X(1768)-2*X(10698) = 3*X(5531)-4*X(12331)

The reciprocal orthologic center of these triangles is X(3869).

X(12767) lies on these lines: {1,104}, {10,9809}, {11,3339}, {40,2771}, {80,2093}, {100,3984}, {149,9589}, {153,3679}, {165,6326}, {200,12532}, {484,6001}, {516,9803}, {517,7993}, {952,7991}, {971,3245}, {1145,5223}, {1317,9819}, {1387,10980}, {1537,11219}, {1699,10265}, {1709,3065}, {1750,12691}, {2717,2958}, {2801,2951}, {2802,11519}, {2829,7992}, {3337,12672}, {4326,12755}, {4668,12751}, {4674,9355}, {5010,12332}, {5691,12247}, {5732,9964}, {6264,11531}, {6265,7987}, {7280,7971}, {7972,7990}, {7982,12773}, {7988,12611}, {7989,12619}, {8089,12771}, {8140,12759}, {8244,12768}, {8245,12770}, {8423,12774}, {9946,10857}, {10045,10057}, {10073,10092}, {10970,12749}, {10971,12750}, {11224,12737}, {11280,12114}

X(12767) = midpoint of X(12759) and X(12760)
X(12767) = reflection of X(i) in X(j) for these (i,j): (1,1768), (5531,40), (5691,12247), (6326,12515), (7982,12773), (9589,149), (9809,10), (11531,6264)
X(12767) = X(265)-of-6th-mixtilinear-triangle
X(12767) = excentral-to-6th-mixtilinear similarity image of X(6326)
X(12767) = {X(6326), X(12515)}-harmonic conjugate of X(165)

X(12768) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO INNER-GARCIA

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

The reciprocal orthologic center of these triangles is X(3869).

X(12768) lies on these lines: {80,7595}, {104,7133}, {517,12744}, {952,9808}, {1387,11030}, {1768,8234}, {2771,12490}, {2800,7596}, {2802,12638}, {2829,12681}, {5083,11042}, {6264,11532}, {6265,8225}, {6326,8231}, {8224,12515}, {8228,12611}, {8230,12619}, {8233,12691}, {8237,12755}, {8239,12758}, {8243,11570}, {8244,12767}, {8246,12770}, {9789,9803}, {9946,10858}, {9952,10867}, {9964,10885}, {10265,12610}, {10891,12551}, {11687,12532}, {11996,12774}

X(12768) = X(265)-of-2nd-Pamfilos-Zhou-triangle
X(12768) = excentral-to-2nd-Pamfilos-Zhou similarity image of X(6326)

X(12769) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GARCIA TO 1st SCHIFFLER

Barycentrics 5*a^7-13*(b^2+c^2)*a^5-2*(b^2+b*c+c^2)*(b+c)*a^4+(11*b^4+11*c^4+b*c*(2*b^2+3*b*c+2*c^2))*a^3+2*(b^3+c^3)*(2*b^2+b*c+2*c^2)*a^2-(b^2-c^2)^2*(3*b^2+2*b*c+3*c^2)*a-2*(b^2-c^2)^3*(b-c) : :

The reciprocal orthologic center of these triangles is X(3065).

X(12769) lies on these lines: {8,12535}, {90,6599}, {191,12409}, {1657,5693}, {1836,6763}, {2476,3336}

X(12769) = reflection of X(i) in X(j) for these (i,j): (12409,191), (12786,12682)
X(12769) = X(6595)-of-inner-Garcia-triangle

X(12770) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO INNER-GARCIA

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

The reciprocal orthologic center of these triangles is X(3869).

X(12770) lies on these lines: {5,3120}, {21,104}, {80,256}, {517,12746}, {846,6326}, {952,2292}, {1284,11570}, {1387,11031}, {1768,8235}, {2800,9840}, {2802,12642}, {2829,12683}, {4199,12691}, {4220,12515}, {4425,10265}, {5051,12619}, {5083,11043}, {6264,11533}, {8229,12611}, {8238,12755}, {8240,12758}, {8245,12767}, {8246,12768}, {8249,12771}, {8391,12759}, {8425,12774}, {8731,9946}, {9791,9803}, {9952,10868}, {10892,12551}, {11688,12532}, {11926,12760}

X(12770) = X(265)-of-1st-Sharygin-triangle
X(12770) = 2nd-Sharygin-to-1st-Sharygin similarity image of X(13277)
X(12770) = excentral-to-1st-Sharygin similarity image of X(6326)
X(12770) = hexyl-to-1st-Sharygin similarity image of X(1768)

X(12771) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL-MIDARC TO INNER-GARCIA

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=2*(-a+b+c)*(a^2-b^2+b*c-c^2)*((b+c)*a^3-(b^2+c^2)*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2)
G(a,b,c)=2*(a-b+c)*(a^3-a^2*b-(b-c)^2*a+b*(b^2-c^2))*(a^2-a*c-b^2+c^2)*b
H(a,b,c)=(a+b+c)*(a+b-c)*(-a+b+c)*(a-b+c)*(a^3-(b+c)*a^2-(b^2-3*b*c+c^2)*a+(b^2-c^2)*(b-c))

The reciprocal orthologic center of these triangles is X(3869).

X(12771) lies on these lines: {1,12772}, {517,8097}, {952,8093}, {1387,11032}, {1768,8081}, {2089,11570}, {2771,8099}, {2800,8091}, {2829,8095}, {6265,8077}, {6326,8078}, {8075,12515}, {8079,12691}, {8085,12611}, {8087,12619}, {8089,12767}, {8133,12759}, {8135,12760}, {8241,12758}, {8247,12768}, {8249,12770}, {8387,12755}, {8733,9946}, {9793,9803}, {9964,11888}, {11690,12532}, {11894,12551}

X(12771) = reflection of X(12772) in X(1)
X(12771) = X(265)-of-tangential-midarc-triangle
X(12771) = X(12898)-of-2nd-tangential-midarc-triangle
X(12771) = excentral-to-tangential-midarc similarity image of X(6326)

X(12772) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO INNER-GARCIA

Trilinears 2*(a+b+c)*(a^3-(b+c)*a^2-(b^2-3*b*c+c^2)*a+(b^2-c^2)*(b-c))*b*c*sin(A/2)-(a^2-b^2+b*c-c^2)*((b+c)*a^3-(b^2+c^2)*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(3869).

X(12772) lies on these lines: {1,12771}, {174,11570}, {258,6326}, {1387,11033}, {2802,12644}, {5083,8351}, {6264,11899}, {6265,7588}, {8125,12532}, {8734,9946}, {9952,11859}, {11895,12551}

X(12772) = reflection of X(12771) in X(1)
X(12772) = X(265)-of-2nd-tangential-midarc-triangle
X(12772) = X(12898)-of-tangential-midarc-triangle
X(12772) = excentral-to-2nd-tangential-midarc similarity image of X(6326)

X(12773) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO INNER-GARCIA

Trilinears a^6-(b+c)*a^5-(2*b^2-7*b*c+2*c^2)*a^4+2*(b^2-3*b*c+c^2)*(b+c)*a^3+(b^4+c^4-5*b*c*(b-c)^2)*a^2-(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a-2*(b^2-c^2)^2*b*c : :
X(12773) = 3*X(3)-2*X(100) = 4*X(11)-3*X(381) = X(100)-3*X(104) = 4*X(100)-3*X(12331) = 4*X(104)-X(12331) = 4*X(119)-5*X(1656) = 3*X(119)-4*X(6667) = 3*X(381)-2*X(10742) = 15*X(1656)-16*X(6667) = 3*X(1768)+X(12653) = 3*X(6264)-X(12653)

The reciprocal orthologic center of these triangles is X(40).

X(12773) lies on the Stammler circle and these lines: {1,399}, {2,11698}, {3,8}, {4,1484}, {5,153}, {11,381}, {30,149}, {36,9897}, {40,7993}, {55,7972}, {56,80}, {57,6797}, {119,1656}, {214,958}, {355,10265}, {382,2829}, {515,12747}, {517,1768}, {528,3534}, {993,3655}, {1001,2801}, {1012,10247}, {1317,3295}, {1320,8148}, {1385,5251}, {1387,3485}, {1482,2800}, {1483,6906}, {1537,10941}, {1597,1862}, {1598,12138}, {1657,5840}, {2099,11571}, {2787,12188}, {2802,11256}, {2830,11258}, {3032,9566}, {3035,5054}, {3036,9709}, {3045,9703}, {3243,3358}, {3304,12611}, {3359,11525}, {3428,12119}, {3526,6713}, {3576,5531}, {3579,3893}, {3652,3884}, {3830,10707}, {4413,5790}, {4428,11274}, {5055,10711}, {5073,10724}, {5093,10759}, {5450,11849}, {5533,9669}, {5603,9809}, {5708,12736}, {5730,12532}, {5844,6909}, {6361,9802}, {6767,12735}, {6862,10805}, {6912,10283}, {6913,11729}, {6914,7967}, {6971,10785}, {6980,12115}, {7517,9913}, {7982,12767}, {8068,9654}, {9301,12499}, {10966,12743}, {11492,12461}, {11493,12460}, {11842,12199}, {11875,12462}, {11876,12463}, {11911,12752}, {11916,12753}, {11917,12754}, {11928,12761}, {11929,12762}, {11949,12765}, {11950,12766}, {12000,12775}

X(12773) = midpoint of X(i) and X(j) for these {i,j}: {40,7993}, {149,12248}, {944,9803}, {1768,6264}, {6361,9802}, {7982,12767}
X(12773) = reflection of X(i) in X(j) for these (i,j): (3,104), (4,1484), (153,5), (355,10265), (382,10738), (1482,12737), (3830,10707), (5073,10724), (5541,3579), (5790,11219), (6265,11715), (6326,1385), (8148,1320), (10742,11), (12331,3), (12332,5450), (12702,12515), (12738,214), (12751,12619)
X(12773) = anticomplement of X(11698)
X(12773) = antipode of X(12331) in Stammler circle
X(12773) = X(104)-of-X3-ABC-reflections-triangle
X(12773) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11,10074,999), (11,10742,381), (1317,10058,3295), (5533,12764,9669), (6265,11715,10246), (8068,12763,9654)

X(12774) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO INNER-GARCIA

Trilinears 2*(a+b+c)*(a^3-(b+c)*a^2-(b^2-3*b*c+c^2)*a+(b^2-c^2)*(b-c))*b*c*sin(A/2)+(a^2-b^2+b*c-c^2)*((b+c)*a^3-(b^2+c^2)*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(3869).

X(12774) lies on these lines: {174,11570}, {517,12748}, {952,12445}, {1387,8083}, {2771,12491}, {2800,8351}, {2802,12646}, {2829,12685}, {6264,11535}, {6265,7587}, {8126,12532}, {8382,12619}, {8423,12767}, {8425,12770}, {8729,9946}, {9803,11891}, {9952,11860}, {9964,11890}, {11896,12551}, {11924,12758}, {11996,12768}

X(12774) = X(265)-of-Yff-central-triangle
X(12774) = excentral-to-Yff-central similarity image of X(6326)

X(12775) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO INNER-GARCIA

Trilinears a^9-2*(b+c)*a^8-(2*b^2-7*b*c+2*c^2)*a^7+(b+c)*(6*b^2-7*b*c+6*c^2)*a^6-b*c*(13*b^2-8*b*c+13*c^2)*a^5-(b+c)*(6*b^4+6*c^4-b*c*(13*b^2-10*b*c+13*c^2))*a^4+(2*b^4+2*c^4+b*c*(9*b^2+10*b*c+9*c^2))*(b-c)^2*a^3+(b^2-c^2)*(b-c)^3*(2*b^2+3*b*c+2*c^2)*a^2-(b-c)*(b^3+c^3)*(b^4-c^4)*a-(b^2-c^2)^3*(b-c)*b*c : :
X(12775) = 2*X(4302)+X(10728)

The reciprocal orthologic center of these triangles is X(40).

X(12775) lies on these lines: {1,104}, {3,1537}, {4,100}, {11,6833}, {12,12761}, {35,12608}, {55,2829}, {56,11047}, {149,6847}, {153,10528}, {515,10087}, {516,1519}, {946,10090}, {952,1012}, {962,4996}, {1006,3359}, {1145,10306}, {1317,10965}, {1376,6968}, {1470,5603}, {1512,5537}, {1621,6950}, {2787,12189}, {2802,12703}, {3035,6834}, {3295,10935}, {3560,5554}, {3601,11919}, {3811,12665}, {4302,6256}, {5528,11372}, {6001,12739}, {6265,12672}, {6326,12705}, {6713,6977}, {6831,10738}, {6935,10596}, {9913,10834}, {10073,12616}, {10742,10942}, {10803,12199}, {10805,12248}, {10878,12499}, {10915,12751}, {10929,12753}, {10930,12754}, {10955,12762}, {10958,12764}, {11400,12138}, {11881,12462}, {11882,12463}, {11914,12752}, {11955,12765}, {11956,12766}, {12000,12773}

X(12775) = reflection of X(i) in X(j) for these (i,j): (104,10058), (12115,10956)
X(12775) = X(104)-of-inner-Yff-tangents-triangle
X(12775) = {X(104),X(10698)}-harmonic conjugate of X(12776)
X(12775) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (119,11248,100), (5450,10074,104), (6906,10698,104)

X(12776) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO INNER-GARCIA

Trilinears a^9-2*(b+c)*a^8-(2*b^2-11*b*c+2*c^2)*a^7+(b+c)*(6*b^2-19*b*c+6*c^2)*a^6-b*c*(13*b^2-44*b*c+13*c^2)*a^5-(b+c)*(6*b^4+6*c^4-b*c*(37*b^2-66*b*c+37*c^2))*a^4+(2*b^4+2*c^4-b*c*(3*b^2+38*b*c+3*c^2))*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(2*b^4+2*c^4-b*c*(13*b^2-30*b*c+13*c^2))*a^2-(b^2-c^2)^2*(b^4+c^4-b*c*(9*b^2-14*b*c+9*c^2))*a-(b^2-c^2)^3*(b-c)*b*c : :

The reciprocal orthologic center of these triangles is X(40).

X(12776) lies on these lines: {1,104}, {4,10707}, {11,10532}, {72,6265}, {100,6942}, {119,10527}, {153,10529}, {411,10031}, {515,12750}, {519,6905}, {528,6934}, {952,3149}, {1317,10966}, {1537,10941}, {2787,12190}, {2802,12704}, {2829,12116}, {3058,6938}, {3304,6833}, {3829,6968}, {4848,10090}, {4996,6585}, {5288,5660}, {6326,6762}, {6830,10072}, {6834,12513}, {6941,10711}, {6956,10597}, {9851,10971}, {9913,10835}, {10742,10943}, {10804,12199}, {10806,12248}, {10879,12499}, {10916,12751}, {10931,12753}, {10932,12754}, {10949,12761}, {10957,12763}, {10959,12764}, {11401,12138}, {11510,12332}, {11883,12462}, {11884,12463}, {11915,12752}, {11957,12765}, {11958,12766}

X(12776) = reflection of X(104) in X(10074)
X(12776) = X(104)-of-outer-Yff-tangents-triangle
X(12776) = {X(104),X(10698)}-harmonic conjugate of X(12775)

X(12777) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO HUTSON EXTOUCH

Barycentrics a^7-(b+c)*a^6-(b^2+6*b*c+c^2)*a^5+(b+c)*(b^2+10*b*c+c^2)*a^4-(b^2+6*b*c+c^2)*(b-c)^2*a^3+(b+c)*(b^4+c^4-2*b*c*(6*b^2+5*b*c+6*c^2))*a^2+(b^2-c^2)^2*(b^2+10*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :
X(12777) = 3*X(354)-2*X(12439) = 3*X(3873)-X(12537) = 3*X(5587)-2*X(12599) = 3*X(5657)-X(12249)

The reciprocal orthologic center of these triangles is X(40).

X(12777) lies on these lines: {1,12521}, {2,12260}, {4,5223}, {8,6835}, {10,6601}, {40,4847}, {100,3523}, {354,12439}, {497,10395}, {515,12120}, {518,12692}, {519,8000}, {942,2550}, {1737,10075}, {2551,12019}, {3295,6675}, {3419,12667}, {3434,11684}, {3679,9898}, {3873,12537}, {5090,12139}, {5587,12599}, {5657,12249}, {5687,12333}, {5688,12802}, {5689,12801}, {6737,11525}, {6743,6864}, {8193,12411}, {8197,12464}, {8204,12465}, {9804,11024}, {9857,12500}, {10039,10059}, {10791,12200}, {11900,12789}

X(12777) = midpoint of X(8) and X(9874)
X(12777) = reflection of X(i) in X(j) for these (i,j): (7160,10), (12756,12670)
X(12777) = anticomplement of X(12260)
X(12777) = X(7160)-of-outer-Garcia-triangle

X(12778) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO 1st HYACINTH

Trilinears a^9+(b+c)*a^8-(3*b^2+2*b*c+3*c^2)*a^7-2*(b^3+c^3)*a^6+(3*b^4+3*c^4+b*c*(2*b^2+3*b*c+2*c^2))*a^5-b*c*(2*b-c)*(b-2*c)*(b+c)*a^4-(b^6+c^6-b*c*(2*b-c)*(b-2*c)*(b+c)^2)*a^3+(b^2-c^2)*(b-c)*(2*b^4+2*c^4+b*c*(2*b^2-b*c+2*c^2))*a^2-(b^2-c^2)^2*b*c*(2*b^2-b*c+2*c^2)*a-(b^4-c^4)*(b^2-c^2)^2*(b-c) : :
X(12778) = 3*X(3)-2*X(11709) = 3*X(110)-X(7978) = 3*X(110)-2*X(11699) = 3*X(165)-2*X(12041) = 3*X(3656)-4*X(11723) = 3*X(3679)-X(12407) = 3*X(5587)-2*X(10113) = 2*X(5609)+X(7991) = 3*X(5642)-2*X(11723) = 3*X(5886)-4*X(5972)

The reciprocal orthologic center of these triangles is X(6102).

X(12778) lies on these lines: {1,1511}, {2,12261}, {3,11709}, {8,12383}, {10,265}, {30,12368}, {35,1807}, {40,2940}, {46,3028}, {65,5504}, {72,74}, {110,517}, {113,12699}, {146,6361}, {165,12041}, {399,12702}, {484,4551}, {515,12121}, {516,7728}, {542,3416}, {1155,10081}, {1385,7984}, {1482,11720}, {1770,12373}, {2777,12779}, {2778,9934}, {2836,11579}, {3057,10091}, {3448,5657}, {3656,5642}, {3679,12407}, {5090,12140}, {5183,11670}, {5587,10113}, {5609,7991}, {5687,12334}, {5688,12804}, {5689,12803}, {5690,12785}, {5886,5972}, {7727,11010}, {7968,10820}, {7969,10819}, {8193,12412}, {8197,12466}, {8204,12467}, {9778,12244}, {9857,12501}, {10778,12619}, {10791,12201}, {11900,12790}

X(12778) = midpoint of X(i) and X(j) for these {i,j}: {8,12383}, {40,2948}, {146,6361}, {399,12702}
X(12778) = reflection of X(i) in X(j) for these (i,j): (1,1511), (74,3579), (265,10), (1482,11720), (3656,5642), (7978,11699), (7984,1385), (10778,12619), (12699,113)
X(12778) = anticomplement of X(12261)
X(12778) = X(265)-of-outer-Garcia-triangle

X(12779) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO MIDHEIGHT

Barycentrics a^10-(b+c)*a^9+(b+c)^2*a^8+2*(b^3+c^3)*a^7-2*(4*b^2-7*b*c+4*c^2)*(b+c)^2*a^6+2*(b^2-c^2)*(b-c)*b*c*a^5+2*(b^2-c^2)^2*(4*b^2-b*c+4*c^2)*a^4-2*(b^3+c^3)*(b^2-c^2)^2*a^3-(b^2-c^2)^2*(b^4+c^4-2*b*c*(b^2-5*b*c+c^2))*a^2+(b^4-c^4)*(b^2-c^2)^2*(b-c)*a-(b^4-c^4)*(b^2-c^2)^3 : :
X(12779) = 3*X(154)-2*X(4297) = 3*X(165)-2*X(5894) = X(944)-3*X(5656) = 5*X(1698)-4*X(6696) = 3*X(1699)-4*X(5893) = 3*X(3679)-X(9899) = 3*X(5587)-2*X(6247) = 3*X(5657)-X(12250) = 5*X(7987)-6*X(10192) = 5*X(8567)-6*X(10164)

The reciprocal orthologic center of these triangles is X(4).

X(12779) lies on these lines: {1,2883}, {2,12262}, {4,65}, {8,6225}, {10,64}, {30,9928}, {154,4297}, {165,5894}, {221,950}, {226,1854}, {355,6000}, {440,12520}, {515,1498}, {516,5895}, {517,5878}, {519,7973}, {607,5776}, {944,5656}, {1103,1490}, {1503,3751}, {1698,6696}, {1699,5893}, {1712,8899}, {1737,10076}, {2192,10106}, {2777,12778}, {3197,8804}, {3556,7580}, {3679,9899}, {5090,11381}, {5252,6285}, {5587,6247}, {5657,12250}, {5687,12335}, {5688,6266}, {5689,6267}, {6684,10606}, {7522,12617}, {7987,10192}, {8193,9914}, {8197,12468}, {8204,12469}, {8567,10164}, {9857,12502}, {10039,10060}, {10791,12202}, {11900,12791}

X(12779) = midpoint of X(8) and X(6225)
X(12779) = reflection of X(i) in X(j) for these (i,j): (1,2883), (64,10)
X(12779) = anticomplement of X(12262)
X(12779) = X(64)-of-outer-Garcia-triangle

X(12780) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO INNER-NAPOLEON

Barycentrics -(6*a^3-6*(b+c)*(b^2+c^2))*S+sqrt(3)*(a+b+c)*(a^4+3*(b+c)*a^3-2*(b^2+3*b*c+c^2)*a^2-3*(b^2-c^2)*(b-c)*a+(b^2-c^2)^2) : :
X(12780) = 3*X(5470)-2*X(11599) = 2*X(5479)-3*X(5587)

The reciprocal orthologic center of these triangles is X(3).

X(12780) lies on these lines: {1,619}, {2,11706}, {8,617}, {10,14}, {40,2946}, {515,5474}, {517,5613}, {519,5464}, {530,9881}, {531,3679}, {542,3416}, {1018,1276}, {1698,6670}, {1737,10077}, {5090,12141}, {5470,11599}, {5479,5587}, {5657,6773}, {5687,12336}, {5688,6269}, {5689,6271}, {7975,11711}, {7983,11705}, {8193,9915}, {8197,12470}, {8204,12471}, {9857,9981}, {10039,10061}, {10791,12204}, {11900,12792}

X(12780) = midpoint of X(8) and X(617)
X(12780) = reflection of X(i) in X(j) for these (i,j): (1,619), (14,10), (7975,11711), (7983,11705)
X(12780) = X(14)-of-outer-Garcia-triangle
X(12780) = {X(3416),X(3654)}-harmonic conjugate of X(12781)

X(12781) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO OUTER-NAPOLEON

Barycentrics (6*a^3-6*(b+c)*(b^2+c^2))*S+sqrt(3)*(a+b+c)*(a^4+3*(b+c)*a^3-2*(b^2+3*b*c+c^2)*a^2-3*(b^2-c^2)*(b-c)*a+(b^2-c^2)^2) : :
X(12781) = 3*X(5463)-X(7975) = 3*X(5469)-2*X(11599) = 2*X(5478)-3*X(5587)

The reciprocal orthologic center of these triangles is X(3).

X(12781) lies on these lines: {1,618}, {2,11705}, {8,616}, {10,13}, {40,2945}, {515,5473}, {517,5617}, {519,5463}, {530,3679}, {531,9881}, {542,3416}, {1018,1277}, {1698,6669}, {1737,10078}, {5090,12142}, {5469,11599}, {5478,5587}, {5657,6770}, {5687,12337}, {5688,6268}, {5689,6270}, {7974,11711}, {7983,11706}, {8193,9916}, {8197,12472}, {8204,12473}, {9857,9982}, {10039,10062}, {10791,12205}, {11900,12793}

X(12781) = midpoint of X(8) and X(616)
X(12781) = reflection of X(i) in X(j) for these (i,j): (1,618), (13,10), (7974,11711), (7983,11706)
X(12781) = X(13)-of-outer-Garcia-triangle
X(12781) = {X(3416),X(3654)}-harmonic conjugate of X(12780)

X(12782) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO 1st NEUBERG

Trilinears (b+c)*(b^2+c^2)*a-b^2*c^2 : :
X(12782) = 3*X(165)-2*X(5188) = 3*X(262)-2*X(946) = X(944)-3*X(7709) = 2*X(1385)-3*X(11171) = 5*X(1698)-4*X(3934) = 2*X(4669)+X(11055) = 3*X(5587)-2*X(6248) = 3*X(5657)-X(12251) = 3*X(5886)-4*X(11272)

The reciprocal orthologic center of these triangles is X(3).

X(12782) lies on these lines: {1,39}, {2,12263}, {3,11364}, {6,12194}, {8,194}, {10,75}, {37,4446}, {38,3661}, {40,511}, {99,12195}, {165,5188}, {190,3764}, {192,3778}, {238,3730}, {256,3729}, {262,946}, {274,4476}, {355,2782}, {384,10791}, {515,11257}, {517,3095}, {518,3094}, {519,7757}, {536,4443}, {538,3679}, {712,4424}, {732,3416}, {734,4680}, {736,4769}, {944,7709}, {982,3912}, {985,5280}, {1125,7786}, {1385,11171}, {1469,3503}, {1582,2273}, {1698,3934}, {1700,12021}, {1701,12020}, {1737,10079}, {1740,3688}, {1757,3496}, {2664,4517}, {3122,4664}, {3579,9821}, {3624,6683}, {4642,4712}, {4649,5145}, {4669,11055}, {5007,10789}, {5090,12143}, {5587,6248}, {5657,12251}, {5687,12338}, {5688,6272}, {5689,6273}, {5886,11272}, {5969,9881}, {7697,9956}, {7772,10800}, {8193,9917}, {8197,12474}, {8204,12475}, {8298,8715}, {9857,9983}, {10039,10063}, {11900,12794}

X(12782) = midpoint of X(8) and X(194)
X(12782) = reflection of X(i) in X(j) for these (i,j): (1,39), (76,10), (4443,4735), (9821,3579)
X(12782) = anticomplement of X(12263)
X(12782) = X(76)-of-outer-Garcia-triangle
X(12782) = {X(1), X(3097)}-harmonic conjugate of X(39)

X(12783) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO 2nd NEUBERG

Barycentrics a^5+(b^2+c^2)*a^3-2*(b+c)*(b^2+c^2)*a^2+b^2*c^2*a-(b+c)*(b^2+c^2)^2 : :
X(12783) = 4*X(4745)-X(12156) = 3*X(5587)-2*X(6249) = 3*X(5657)-X(12252)

The reciprocal orthologic center of these triangles is X(3).

X(12783) lies on these lines: {1,6292}, {2,12264}, {8,2896}, {10,82}, {515,12122}, {517,6287}, {519,7977}, {732,3416}, {754,3679}, {1018,3496}, {1698,6704}, {1737,10080}, {3579,8725}, {4745,12156}, {5090,12144}, {5587,6249}, {5657,12252}, {5687,12339}, {5688,6274}, {5689,6275}, {5690,9864}, {6308,11364}, {6684,9751}, {8193,9918}, {8197,12476}, {8204,12477}, {10039,10064}, {10791,12206}, {11900,12795}

X(12783) = midpoint of X(8) and X(2896)
X(12783) = reflection of X(i) in X(j) for these (i,j): (1,6292), (83,10), (8725,3579)
X(12783) = anticomplement of X(12264)
X(12783) = X(83)-of-outer-Garcia-triangle

X(12784) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO 1st ORTHOSYMMEDIAL

Barycentrics a^14-(b+c)*a^13+2*b*c*a^12+2*(b^3+c^3)*a^11-(2*b^4+2*c^4+b*c*(2*b^2-b*c+2*c^2))*a^10-(b+c)*(b^2-b*c+c^2)^2*a^9-(b^4+c^4-b*c*(2*b^2-b*c+2*c^2))*(b+c)^2*a^8+(b^2-c^2)*(b-c)*b^2*c^2*a^7+(b^2-c^2)^2*(b^4-b^2*c^2+c^4)*a^6+(b^6-c^6)*(b^2-c^2)*(b+c)*a^5+(b^2-c^2)^2*(2*b^6+2*c^6-b*c*(2*b^2+b*c+2*c^2)*(b-c)^2)*a^4-(b^2-c^2)^2*(b+c)*(2*b^6+2*c^6-b*c*(2*b^2+b*c+2*c^2)*(b-c)^2)*a^3+2*(b^2-c^2)*(b-c)*b*c*(b^3-c^3)*(b^4-c^4)*a^2+(b^8-c^8)*a*(b^2-c^2)^2*(b-c)+(b^4-c^4)*(b^2-c^2)^3*(-c^4-b^4) : :
X(12784) = 2*X(127)-3*X(5587) = 3*X(3576)-4*X(6720) = 3*X(3679)-X(12408) = 3*X(5657)-X(12253)

The reciprocal orthologic center of these triangles is X(4).

X(12784) lies on these lines: {1,132}, {2,12265}, {8,12384}, {10,1297}, {80,2831}, {112,515}, {127,5587}, {944,11722}, {946,10705}, {1837,3320}, {2794,5691}, {2799,9864}, {2806,12751}, {3576,6720}, {3679,9530}, {5090,12145}, {5252,6020}, {5657,12253}, {5687,12340}, {5688,12806}, {5689,12805}, {8193,12413}, {8197,12478}, {8204,12479}, {9517,12368}, {9857,12503}, {10791,12207}, {11900,12796}

X(12784) = midpoint of X(8) and X(12384)
X(12784) = reflection of X(i) in X(j) for these (i,j): (1,132), (944,11722), (1297,10), (10705,946)
X(12784) = anticomplement of X(12265)
X(12784) = X(1297)-of-outer-Garcia-triangle

X(12785) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO REFLECTION

Barycentrics a^10-(b+c)*a^9-2*(b^2-b*c+c^2)*a^8+2*(b^3+c^3)*a^7+(b^2-b*c+c^2)^2*a^6+b*c*(b+c)*(2*b^2-b*c+2*c^2)*a^5-(b^4+b^2*c^2+c^4)*(b+c)^2*a^4-(b^2-c^2)*(b-c)*(2*b^4+2*c^4+b*c*(2*b^2+3*b*c+2*c^2))*a^3+2*(b^2-c^2)^2*(b^4+c^4+b*c*(b^2+b*c+c^2))*a^2+(b^4-c^4)*(b^2-c^2)^2*(b-c)*a-(b^4-c^4)*(b^2-c^2)^3 : :
X(12785) = X(195)-3*X(5790) = 5*X(1698)-4*X(6689) = 2*X(3574)-3*X(5587) = 3*X(5657)-X(12254)

The reciprocal orthologic center of these triangles is X(4).

X(12785) lies on these lines: {1,1209}, {2,12266}, {8,2888}, {10,54}, {65,2962}, {72,6145}, {80,6286}, {195,5790}, {355,1154}, {515,7691}, {517,6288}, {519,7979}, {539,3679}, {1698,6689}, {1737,10082}, {3468,4551}, {3574,5587}, {3751,5965}, {5090,11576}, {5657,12254}, {5687,12341}, {5688,6276}, {5689,6277}, {5690,12778}, {8193,9920}, {8197,12480}, {8204,12481}, {9857,9985}, {10039,10066}, {10628,12368}, {10791,12208}, {11900,12797}

X(12785) = midpoint of X(8) and X(2888)
X(12785) = reflection of X(i) in X(j) for these (i,j): (1,1209), (54,10)
X(12785) = anticomplement of X(12266)
X(12785) = X(54)-of-outer-Garcia-triangle

X(12786) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO 1st SCHIFFLER

Trilinears a^6+2*(b+c)*a^5-5*(b^2+c^2)*a^4-2*(b+c)*(2*b^2-b*c+2*c^2)*a^3+(7*b^4+7*c^4+b*c*(2*b^2-b*c+2*c^2))*a^2+2*(b+c)*(b^4+c^4-(b^2-4*b*c+c^2)*b*c)*a-(3*b^2+2*b*c+3*c^2)*(b^2-c^2)^2 : :
X(12786) = 3*X(3679)-X(12409) = 3*X(5587)-2*X(12600) = 3*X(5657)-X(12255)

The reciprocal orthologic center of these triangles is X(79).

X(12786) lies on these lines: {1,6597}, {2,12267}, {8,12535}, {10,10266}, {100,191}, {2802,6595}, {3679,12409}, {5090,12146}, {5538,5694}, {5587,12600}, {5657,12255}, {5687,12342}, {5688,12808}, {5689,12807}, {8193,12414}, {8197,12482}, {8204,12483}, {9857,12504}, {10791,12209}, {11024,12543}, {11900,12798}

X(12786) = reflection of X(i) in X(j) for these (i,j): (6597,12745), (10266,10), (12769,12682)
X(12786) = anticomplement of X(12267)
X(12786) = X(10266)-of-outer-Garcia-triangle

X(12787) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO INNER-VECTEN

Barycentrics -2*(a^3-(b+c)*(b^2+c^2))*S+(a+b+c)*(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) : :
X(12787) = 5*X(3617)-X(12221) = 3*X(5587)-2*X(6251) = X(6281)+2*X(11362)

The reciprocal orthologic center of these triangles is X(3).

X(12787) lies on these lines: {1,642}, {2,12268}, {8,487}, {10,486}, {515,12123}, {517,6290}, {519,7980}, {1018,6212}, {1698,6119}, {1737,10083}, {3416,3564}, {3617,12221}, {3679,9906}, {5090,12147}, {5587,6251}, {5657,12256}, {5687,12343}, {5688,6280}, {5689,6281}, {5790,12601}, {8193,9921}, {8197,12484}, {8204,12485}, {9857,9986}, {10039,10067}, {10791,12210}, {11900,12799}

X(12787) = midpoint of X(8) and X(487)
X(12787) = reflection of X(i) in X(j) for these (i,j): (1,642), (486,10)
X(12787) = anticomplement of X(12268)
X(12787) = X(486)-of-outer-Garcia-triangle
X(12787) = {X(3416),X(5690)}-harmonic conjugate of X(12788)

X(12788) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO OUTER-VECTEN

Barycentrics 2*(a^3-(b+c)*(b^2+c^2))*S+(a+b+c)*(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) : :
X(12788) = 5*X(3617)-X(12222) = 3*X(3679)-X(9907) = 3*X(5587)-2*X(6250) = X(6278)+2*X(11362)

The reciprocal orthologic center of these triangles is X(3).

X(12788) lies on these lines: {1,641}, {2,12269}, {8,488}, {10,485}, {515,12124}, {517,6289}, {519,7981}, {1018,6213}, {1698,6118}, {1737,10084}, {3416,3564}, {3617,12222}, {3679,9907}, {5090,12148}, {5587,6250}, {5657,12257}, {5687,12344}, {5688,6278}, {5689,6279}, {5790,12602}, {8193,9922}, {8197,12486}, {8204,12487}, {9857,9987}, {10039,10068}, {10791,12211}, {11900,12800}

X(12788) = midpoint of X(8) and X(488)
X(12788) = reflection of X(i) in X(j) for these (i,j): (1,641), (485,10)
X(12788) = anticomplement of X(12269)
X(12788) = X(485)-of-outer-Garcia-triangle
X(12788) = {X(3416),X(5690)}-harmonic conjugate of X(12787)

X(12789) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO HUTSON EXTOUCH

Barycentrics (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*((b+c)*a^14+4*b*c*a^13-2*(b+c)*(2*b^2-b*c+2*c^2)*a^12-(b^4+c^4+2*(10*b^2-b*c+10*c^2)*b*c)*a^11+(b+c)*(4*b^4+4*c^4-(6*b^2+19*b*c+6*c^2)*b*c)*a^10+(5*b^6+5*c^6+(30*b^4+30*c^4-(5*b^2-8*b*c+5*c^2)*b*c)*b*c)*a^9+(b+c)*(5*b^6+5*c^6+2*(b^4+c^4+(8*b^2+7*b*c+8*c^2)*b*c)*b*c)*a^8-2*(5*b^8+5*c^8+2*(7*b^4+7*c^4+(4*b^2-19*b*c+4*c^2)*b*c)*b^2*c^2)*a^7-(b+c)*(15*b^8+15*c^8-2*(6*b^6+6*c^6+(37*b^4+37*c^4-(8*b^2+71*b*c+8*c^2)*b*c)*b*c)*b*c)*a^6+2*(b^2-c^2)^2*(5*b^6+5*c^6-(20*b^4+20*c^4-(41*b^2+8*b*c+41*c^2)*b*c)*b*c)*a^5+2*(b^2-c^2)^2*(b+c)*(7*b^6+7*c^6-(9*b^4+9*c^4+2*b*c*(18*b^2+5*b*c+18*c^2))*b*c)*a^4-(b^4-c^4)*(b^2-c^2)*(5*b^6+5*c^6-(36*b^4+36*c^4-5*b*c*(7*b^2+8*b*c+7*c^2))*b*c)*a^3-(b^2-c^2)^2*(b+c)*(6*b^8+6*c^8-(10*b^6+10*c^6+(21*b^4+21*c^4+10*b*c*(b^2+9*b*c+c^2))*b*c)*b*c)*a^2+(b^2-c^2)^4*(b^6+c^6-(10*b^4+10*c^4-b*c*(3*b^2-32*b*c+3*c^2))*b*c)*a+(b^2-c^2)^5*(b-c)*(b^4+3*b^2*c^2+c^4)) : :
X(12789) = 3*X(11845)-X(12249) = 3*X(11897)-2*X(12599)

The reciprocal orthologic center of these triangles is X(40).

X(12789) lies on these lines: {30,12120}, {402,7160}, {4240,9874}, {8000,11910}, {9898,11852}, {10059,11912}, {10075,11913}, {11831,12260}, {11832,12139}, {11839,12200}, {11845,12249}, {11848,12333}, {11853,12411}, {11885,12500}, {11897,12599}, {11900,12777}, {11901,12801}, {11902,12802}

X(12789) = midpoint of X(4240) and X(9874)
X(12789) = reflection of X(7160) in X(402)
X(12789) = X(10266)-of-Gossard-triangle

X(12790) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO 1st HYACINTH

Barycentrics (2*S^2+3*SA*(SA-SW))*((SA-SW+3*R^2)*S^2+(9*R^2-2*SW)*SA^2+(18*R^2-5*SW)*(6*R^2-SW)*SA+(4*R^2-SW)*(162*R^4-90*R^2*SW+11*SW^2)) : :
X(12790) = X(3448)-3*X(11845) = 2*X(10113)-3*X(11897) = 3*X(11831)-2*X(12261) = 3*X(11852)-X(12407)

The reciprocal orthologic center of these triangles is X(6102).

X(12790) lies on these lines: {30,110}, {265,402}, {542,12583}, {1511,1650}, {2771,12729}, {2777,12791}, {3448,11845}, {4240,12383}, {5663,12113}, {10088,11905}, {10091,11906}, {10113,11897}, {11831,12261}, {11832,12140}, {11839,12201}, {11848,12334}, {11852,12407}, {11853,12412}, {11885,12501}, {11900,12778}, {11901,12803}, {11902,12804}

X(12790) = midpoint of X(4240) and X(12383)
X(12790) = X(265)-of-Gossard-triangle
X(12790) = reflection of X(i) in X(j) for these (i,j): (265,402), (1650,1511)

X(12791) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO MIDHEIGHT

Barycentrics (2*S^2+3*SA*(SA-SW))*((2*R^2-SA)*S^2+(-14*R^2+3*SW)*SA^2-(8*R^2-3*SW)*(6*R^2-SW)*SA+(4*R^2-SW)*(288*R^4-80*R^2*SW+5*SW^2)) : :
X(12791) = 2*X(6247)-3*X(11897) = X(9899)-3*X(11852) = 3*X(11831)-2*X(12262) = 3*X(11845)-X(12250)

The reciprocal orthologic center of these triangles is X(4).

X(12791) lies on these lines: {30,155}, {64,402}, {1650,2883}, {2777,12790}, {4240,6225}, {5502,12113}, {6000,11251}, {6001,12696}, {6247,11897}, {6266,11902}, {6267,11901}, {7355,11909}, {7973,11910}, {9899,11852}, {9914,11853}, {10060,11912}, {10076,11913}, {11381,11832}, {11831,12262}, {11839,12202}, {11845,12250}, {11848,12335}, {11885,12502}, {11900,12779}

X(12791) = midpoint of X(4240) and X(6225)
X(12791) = reflection of X(i) in X(j) for these (i,j): (64,402), (1650,2883)
X(12791) = X(64)-of-Gossard-triangle

X(12792) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO INNER-NAPOLEON

Barycentrics (S^2-3*SB*SC)*((-2*SW+SA+6*R^2)*S^2+(18*R^2-5*SW)*SA^2+(6*R^2-SW)*SW*SA-(4*R^2-SW)*SW^2-1/3*sqrt(3)*S*(-3*SA^2+6*(6*R^2-SW)*SA+216*R^4-S^2+12*SW^2-102*R^2*SW)) : :

The reciprocal orthologic center of these triangles is X(3).

X(12792) lies on these lines: {30,5464}, {530,12347}, {531,1651}, {542,12583}, {617,4240}, {619,1650}, {5479,11897}, {6269,11902}, {6271,11901}, {6773,11845}, {7974,11910}, {9900,11852}, {9915,11853}, {9981,11885}, {10061,11912}, {10077,11913}, {11706,11831}, {11832,12141}, {11839,12204}, {11848,12336}, {11900,12780}

X(12792) = X(14)-of-Gossard-triangle

X(12793) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO OUTER-NAPOLEON

Barycentrics (S^2-3*SB*SC)*((-2*SW+SA+6*R^2)*S^2+(18*R^2-5*SW)*SA^2+(6*R^2-SW)*SW*SA-(4*R^2-SW)*SW^2+1/3*sqrt(3)*S*(-3*SA^2+6*(6*R^2-SW)*SA+216*R^4-S^2+12*SW^2-102*R^2*SW)) : :

The reciprocal orthologic center of these triangles is X(3).

X(12793) lies on these lines: {30,5463}, {530,1651}, {531,12347}, {542,12583}, {616,4240}, {618,1650}, {5478,11897}, {6268,11902}, {6270,11901}, {6770,11845}, {7975,11910}, {9901,11852}, {9916,11853}, {9982,11885}, {10062,11912}, {10078,11913}, {11705,11831}, {11832,12142}, {11839,12205}, {11848,12337}, {11900,12781}

X(12793) = X(13)-of-Gossard-triangle

X(12794) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO 1st NEUBERG

Barycentrics (S^2-3*SB*SC)*((-SA^2+(12*R^2-4*SW)*SA-76*R^2*SW+9*SW^2+144*R^4)*S^2-SW*((36*R^2-7*SW)*SA^2-4*(6*R^2-SW)*SW*SA+(4*R^2-SW)*SW^2)) : :
X(12794) = 3*X(11831)-2*X(12263) = 3*X(11845)-X(12251)

The reciprocal orthologic center of these triangles is X(3).

X(12794) lies on these lines: {30,3095}, {39,1650}, {76,402}, {194,4240}, {384,11839}, {511,12113}, {538,1651}, {730,12438}, {732,12583}, {2782,11251}, {5969,12347}, {6248,11897}, {6272,11902}, {6273,11901}, {7976,11910}, {9902,11852}, {9917,11853}, {9983,11885}, {10063,11912}, {10079,11913}, {11831,12263}, {11832,12143}, {11845,12251}, {11848,12338}, {11863,12474}, {11864,12475}, {11900,12782}

X(12794) = midpoint of X(194) and X(4240)
X(12794) = X(76)-of-Gossard-triangle
X(12794) = reflection of X(i) in X(j) for these (i,j): (76,402), (1650,39)

X(12795) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO 2nd NEUBERG

Barycentrics (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(2*(b^2+c^2)*a^10-4*b^2*c^2*a^8-2*(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^4-3*b^2*c^2+c^4)*b^2*c^2)*a^4+2*(b^8-c^8)*a^2*(b^2-c^2)-(b^4-c^4)^2*(b^2+c^2)^2) : :
X(12795) = 2*X(6249)-3*X(11897) = 3*X(11845)-X(12252)

The reciprocal orthologic center of these triangles is X(3).

X(12795) lies on these lines: {30,6287}, {83,402}, {732,12583}, {754,1651}, {1650,6292}, {2896,4240}, {6249,11897}, {6274,11902}, {6275,11901}, {7977,11910}, {9903,11852}, {9918,11853}, {10064,11912}, {10080,11913}, {11831,12264}, {11832,12144}, {11839,12206}, {11845,12252}, {11848,12339}, {11863,12476}, {11864,12477}, {11900,12783}

X(12795) = midpoint of X(2896) and X(4240)
X(12795) = reflection of X(i) in X(j) for these (i,j): (83,402), (1650,6292)
X(12795) = X(83)-of-Gossard-triangle

X(12796) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO 1st ORTHOSYMMEDIAL

Barycentrics (S^2-3*SB*SC)*((SW-2*R^2)*S^4+SW*(4*R^2-SW)*(-2*SW+21*R^2)*S^2+(36*R^4-14*R^2*SW+SW^2)*SA*S^2+(-3*R^2+SW)*SA^2*S^2-3*SW^2*(4*R^2-SW)*(6*R^2-SW)*SA+SW^2*(36*R^2-5*SW)*(4*R^2-SW)^2+3*SW*(4*R^2-SW)*(6*R^2-SW)*SA^2) : :
X(12796) = 2*X(127)-3*X(11897) = 3*X(11831)-2*X(12265) = 3*X(11845)-X(12253) = 3*X(11852)-X(12408)

The reciprocal orthologic center of these triangles is X(4).

X(12796) lies on these lines: {30,112}, {127,11897}, {132,1650}, {402,1297}, {1651,9530}, {2799,12181}, {2806,12752}, {3320,11909}, {4240,12384}, {9517,12369}, {11831,12265}, {11832,12145}, {11839,12207}, {11845,12253}, {11848,12340}, {11852,12408}, {11853,12413}, {11885,12503}, {11900,12784}, {11901,12805}, {11902,12806}

X(12796) = midpoint of X(4240) and X(12384)
X(12796) = reflection of X(i) in X(j) for these (i,j): (1297,402), (1650,132)
X(12796) = X(1297)-of-Gossard-triangle

X(12797) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO REFLECTION

Barycentrics (S^2-3*SB*SC)*(SA*S^2+(-3*SW+10*R^2)*S^2+(4*R^2-SW)*(SW^2-19*R^2*SW+36*R^4)-(-3*SW+10*R^2)*(6*R^2-SW)*SA+(23*R^2-6*SW)*SA^2) : :
X(12797) = X(195)-3*X(11911) = 2*X(3574)-3*X(11897) = X(9905)-3*X(11852) = 3*X(11831)-2*X(12266) = 3*X(11845)-X(12254)

The reciprocal orthologic center of these triangles is X(4).

X(12797) lies on these lines: {30,6288}, {54,402}, {195,11911}, {539,1651}, {1154,11251}, {1209,1650}, {2888,4240}, {3574,11897}, {6276,11902}, {6277,11901}, {7979,11910}, {9905,11852}, {9920,11853}, {9985,11885}, {10066,11912}, {10082,11913}, {10628,12369}, {11576,11832}, {11831,12266}, {11839,12208}, {11845,12254}, {11848,12341}, {11900,12785}

X(12797) = midpoint of X(2888) and X(4240)
X(12797) = reflection of X(i) in X(j) for these (i,j): (54,402), (1650,1209)
X(12797) = X(54)-of-Gossard-triangle

X(12798) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO 1st SCHIFFLER

Barycentrics (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(2*a^15-8*(b^2+c^2)*a^13+(7*b^4+7*c^4+2*(2*b^2+15*b*c+2*c^2)*b*c)*a^11-(b+c)*(b^4-10*b^2*c^2+c^4)*a^10+(15*b^6+15*c^6-(12*b^4+12*c^4+(39*b^2-8*b*c+39*c^2)*b*c)*b*c)*a^9+(b+c)*(5*b^6+5*c^6-(2*b^4+2*c^4+(13*b^2-4*b*c+13*c^2)*b*c)*b*c)*a^8-(40*b^8+40*c^8-(8*b^6+8*c^6+(27*b^4+27*c^4-4*(b^2-11*b*c+c^2)*b*c)*b*c)*b*c)*a^7-(b+c)*(10*b^8+10*c^8-(8*b^6+8*c^6-(11*b^4+11*c^4+2*(4*b^2-25*b*c+4*c^2)*b*c)*b*c)*b*c)*a^6+2*(b^2-c^2)^2*(19*b^6+19*c^6+(4*b^4+4*c^4+(29*b^2-10*b*c+29*c^2)*b*c)*b*c)*a^5+2*(b^2-c^2)^2*(b+c)*(5*b^6+5*c^6-2*(3*b^4+3*c^4-2*b*c*(5*b^2-2*b*c+5*c^2))*b*c)*a^4-(b+c)*(b^2-c^2)*(b^3-c^3)*(17*b^6+17*c^6-b*c*(5*b^2-3*b*c+5*c^2)*(b-c)^2)*a^3-(b^2-c^2)*(b+c)^2*(b^3-c^3)*(5*b^6+5*c^6-(13*b^4+13*c^4-b*c*(23*b^2-18*b*c+23*c^2))*b*c)*a^2+(b^2-c^2)^4*(3*b^6+3*c^6+(4*b^4+4*c^4+b*c*(11*b^2+12*b*c+11*c^2))*b*c)*a+(b^2-c^2)^5*(b-c)*(b^2+c^2)^2) : :
X(12798) = 3*X(11831)-2*X(12267) = 3*X(11845)-X(12255) = 3*X(11897)-2*X(12600)

The reciprocal orthologic center of these triangles is X(79).

X(12798) lies on these lines: {402,10266}, {11831,12267}, {11832,12146}, {11839,12209}, {11845,12255}, {11848,12342}, {11852,12409}, {11853,12414}, {11885,12504}, {11897,12600}, {11900,12786}, {11901,12807}, {11902,12808}

X(12798) = reflection of X(10266) in X(402)

X(12799) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO INNER-VECTEN

Barycentrics (S^2-3*SB*SC)*(S*(S^2+2*SA^2-(24*R^2-4*SW)*SA-(18*R^2-5*SW)*(4*R^2-SW))+(6*R^2+SA-2*SW)*S^2+(18*R^2-5*SW)*SA^2+(6*R^2-SW)*SW*SA-(4*R^2-SW)*SW^2) : :

The reciprocal orthologic center of these triangles is X(3).

X(12799) lies on these lines: {30,6290}, {402,486}, {487,4240}, {642,1650}, {3564,12418}, {6251,11897}, {6280,11902}, {6281,11901}, {7980,11910}, {9906,11852}, {9921,11853}, {9986,11885}, {10067,11912}, {10083,11913}, {11831,12268}, {11832,12147}, {11839,12210}, {11845,12256}, {11848,12343}, {11863,12484}, {11864,12485}, {11900,12787}, {11911,12601}

X(12799) = X(486)-of-Gossard-triangle

X(12800) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO OUTER-VECTEN

Barycentrics (S^2-3*SB*SC)*(-S*(S^2+2*SA^2-(24*R^2-4*SW)*SA-(18*R^2-5*SW)*(4*R^2-SW))+(6*R^2+SA-2*SW)*S^2+(18*R^2-5*SW)*SA^2+(6*R^2-SW)*SW*SA-(4*R^2-SW)*SW^2) : :

The reciprocal orthologic center of these triangles is X(3).

X(12800) lies on these lines: {30,6289}, {402,485}, {488,4240}, {641,1650}, {3564,12418}, {6250,11897}, {6278,11902}, {6279,11901}, {7981,11910}, {9907,11852}, {9922,11853}, {9987,11885}, {10068,11912}, {10084,11913}, {11831,12269}, {11832,12148}, {11845,12257}, {11848,12344}, {11863,12486}, {11864,12487}, {11900,12788}, {11911,12602}

X(12800) = X(485)-of-Gossard-triangle

X(12801) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO HUTSON EXTOUCH

Trilinears -(a^2*(a^2-b^2-6*b*c-c^2)^2-(b-c)^2*(a+b+c)^2*(a-b-c)^2)*S+(b+c)*a^7+(b^2+4*b*c+c^2)*a^6-(b+c)*(3*b^2-2*b*c+3*c^2)*a^5-(b^2+c^2)*(3*b^2+26*b*c+3*c^2)*a^4+(b+c)*(3*b^4+3*c^4-2*b*c*(2*b^2+15*b*c+2*c^2))*a^3+(b^2+6*b*c+c^2)*(3*b^4+3*c^4+2*b*c*(3*b^2-b*c+3*c^2))*a^2-(b^2-c^2)^3*(b-c)*a-(b^4-c^4)*(b^2-c^2)*(b+c)^2 : :

The reciprocal orthologic center of these triangles is X(40).

X(12801) lies on these lines: {6,7160}, {1271,9874}, {5589,9898}, {5595,12411}, {5605,8000}, {5689,12777}, {6202,12599}, {8198,12464}, {8205,12465}, {9994,12500}, {10040,10059}, {10048,10075}, {10783,12249}, {10792,12200}, {11370,12260}, {11388,12139}, {11497,12333}, {11824,12120}, {11901,12789}

X(12801) = reflection of X(12802) in X(7160)
X(12801) = X(7160)-of-inner-Grebe-triangle

X(12802) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO HUTSON EXTOUCH

Trilinears (a^2*(a^2-b^2-6*b*c-c^2)^2-(b-c)^2*(a+b+c)^2*(a-b-c)^2)*S+(b+c)*a^7+(b^2+4*b*c+c^2)*a^6-(b+c)*(3*b^2-2*b*c+3*c^2)*a^5-(b^2+c^2)*(3*b^2+26*b*c+3*c^2)*a^4+(b+c)*(3*b^4+3*c^4-2*b*c*(2*b^2+15*b*c+2*c^2))*a^3+(b^2+6*b*c+c^2)*(3*b^4+3*c^4+2*b*c*(3*b^2-b*c+3*c^2))*a^2-(b^2-c^2)^3*(b-c)*a-(b^4-c^4)*(b^2-c^2)*(b+c)^2 : :

The reciprocal orthologic center of these triangles is X(40).

X(12802) lies on these lines: {6,7160}, {1270,9874}, {5588,9898}, {5594,12411}, {5604,8000}, {5688,12777}, {6201,12599}, {8199,12464}, {8206,12465}, {9995,12500}, {10041,10059}, {10049,10075}, {10784,12249}, {10793,12200}, {11371,12260}, {11389,12139}, {11498,12333}, {11825,12120}, {11902,12789}

X(12802) = reflection of X(12801) in X(7160)
X(12802) = X(7160)-of-outer-Grebe-triangle

X(12803) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO 1st HYACINTH

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

The reciprocal orthologic center of these triangles is X(6102).

X(12803) lies on these lines: {6,13}, {30,7725}, {110,6215}, {568,7720}, {1163,1986}, {1271,12383}, {1511,5591}, {2771,6263}, {2777,6267}, {2931,8903}, {3448,10783}, {3581,10814}, {5589,12407}, {5595,12412}, {5663,5871}, {5689,12778}, {5875,6277}, {6202,10113}, {6218,12236}, {8198,12466}, {8205,12467}, {9994,12501}, {10088,10923}, {10091,10925}, {10792,12201}, {11370,12261}, {11388,12140}, {11497,12334}, {11824,12121}, {11901,12790}

X(12803) = reflection of X(12804) in X(265)
X(12803) = X(265)-of-inner-Grebe-triangle

X(12804) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO 1st HYACINTH

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

The reciprocal orthologic center of these triangles is X(6102).

X(12804) lies on these lines: {6,13}, {30,7726}, {110,6214}, {568,7721}, {1162,1986}, {1270,12383}, {1511,5590}, {2771,6262}, {2777,6266}, {2931,8904}, {3448,10784}, {3581,10815}, {5588,12407}, {5594,12412}, {5663,5870}, {5688,12778}, {5874,6276}, {6201,10113}, {6217,12236}, {8199,12466}, {8206,12467}, {9995,12501}, {10088,10924}, {10091,10926}, {10793,12201}, {11371,12261}, {11389,12140}, {11498,12334}, {11825,12121}, {11902,12790}

X(12804) = reflection of X(12803) in X(265)
X(12804) = X(265)-of-outer-Grebe-triangle

X(12805) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO 1st ORTHOSYMMEDIAL

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

The reciprocal orthologic center of these triangles is X(4).

X(12805) lies on these lines: {6,1297}, {112,11824}, {127,6202}, {132,5591}, {1271,12384}, {2781,7732}, {2794,6319}, {2799,6227}, {2806,12753}, {3320,10927}, {5589,12408}, {5595,12413}, {5689,12784}, {5861,9530}, {7725,9517}, {8198,12478}, {8205,12479}, {9994,12503}, {10783,12253}, {10792,12207}, {11370,12265}, {11388,12145}, {11497,12340}, {11901,12796}

X(12805) = X(1297)-of-inner-Grebe-triangle
X(12805) = reflection of X(12806) in X(1297)

X(12806) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO 1st ORTHOSYMMEDIAL

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

The reciprocal orthologic center of these triangles is X(4).

X(12806) lies on these lines: {6,1297}, {112,11825}, {127,6201}, {132,5590}, {1270,12384}, {2781,7733}, {2794,6320}, {2799,6226}, {2806,12754}, {3320,10928}, {5588,12408}, {5594,12413}, {5688,12784}, {5860,9530}, {7726,9517}, {8199,12478}, {8206,12479}, {9995,12503}, {10784,12253}, {10793,12207}, {11371,12265}, {11389,12145}, {11498,12340}, {11902,12796}

X(12806) = X(1297)-of-outer-Grebe-triangle
X(12806) = reflection of X(12805) in X(1297)

X(12807) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(79).

X(12807) lies on these lines: {6,10266}, {5589,12409}, {5595,12414}, {5689,12786}, {6202,12600}, {8198,12482}, {8205,12483}, {9994,12504}, {10783,12255}, {10792,12209}, {11370,12267}, {11388,12146}, {11497,12342}, {11901,12798}

X(12807) = reflection of X(12808) in X(10266)
X(12807) = X(10266)-of-inner-Grebe-triangle

X(12808) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(79).

X(12808) lies on these lines: {6,10266}, {5588,12409}, {5594,12414}, {5688,12786}, {6201,12600}, {8199,12482}, {8206,12483}, {9995,12504}, {10784,12255}, {10793,12209}, {11371,12267}, {11389,12146}, {11498,12342}, {11902,12798}

X(12808) = reflection of X(12807) in X(10266)
X(12808) = X(10266)-of-outer-Grebe-triangle

X(12809) = X(1)X(12810)∩X(65,2089)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25581.

X(12809) lies on the incircle and these lines: {1,12810}, {65,2089}, {177,10505}, {1122,7 371}, {6018,10508}

X(12809) = X(7371)-Ceva conjugate of X(3669)
X(12809) = X(108)-of-intouch-triangle

X(12810) = X(1)X(12809)∩X(3,6585)

Barycentrics a (a^4 b+2 a^3 b^2-2 a b^4-b^5+a^4 c-6 a^3 b c+4 a^2 b^2 c-2 a b^3 c+3 b^4 c+2 a^3 c^2+4 a^2 b c^2-2 b^3 c^2-2 a b c^3-2 b^2 c^3-2 a c^4+3 b c^4-c^5-4 b c ((2 a^3+a^2 b-2 a b^2-b^3+a^2 c-4 a b c+b^2 c-2 a c^2+b c^2-c^3) Sin[A/2]-a ((a^2-b^2+6 a c+c^2) Sin[B/2]+(a^2+6 a b+b^2-c^2) Sin[C/2]))) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25582.

X(12810) lies on these lines: {1,12809}, {3,6585}

X(12811) = MIDPOINT OF X(3) and X(12102)

Barycentrics 11*SB*SC+7*S^2 : :
X(12811) = 27*X(2)-11*X(3), 3*X(2)-11*X(5), 15*X(2)-11*X(140), 5*X(2)+11*X(381), 9*X(2)+11*X(546), 7*X(2)-11*X(547), 19*X(2)-11*X(549), 21*X(2)-11*X(3530), X(4701)+11*X(9955), X(5609)+3*X(11801)

As a point of the Euler line, X(12811) has Shinagawa coefficients (7,11).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25590.

X(12811) lies on these lines: {2,3}, {517,4540}, {3303,10592}, {3304,10593}, {3614,3746}, {4701,5844}, {5418,10147}, {5420,10148}, {5563,7173}, {5609,11801}, {6488,8253}, {6489,8252}, {11695,12046}

X(12811) = midpoint of X(i) and X(j) for these {i,j}: {3,12102}, {4,3530}, {5,3850}, {140,3861}, {381,10109}, {546,3628}, {547,3860}, {3845,10124}, {5066,11737}
X(12811) = reflection of X(i) in X(j) for these (i,j): (3856,3850), (11540,547), (11695,12046), (12108,3628)
X(12811) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,5,547), (4,547,3530), (4,632,12103), (4,3859,3860), (4,5070,8703), (4,5079,632), (5,3627,3090), (5,3845,1656), (140,12101,20), (381,3627,546), (546,3627,3861), (632,3627,8703), (1656,3845,548), (3090,3091,381), (3090,3627,140), (3091,3146,3855), (3525,5076,550), (3628,12102,3), (3843,5056,549), (3850,3861,381), (3861,10109,140), (5055,5076,3525)

X(12812) = MIDPOINT OF X(5) AND X(1656)

Barycentrics 7*SB*SC+11*S^2 : :
X(12812) = 27*X(2)-7*X(3), 3*X(2)+7*X(5), 12*X(2)-7*X(140), 13*X(2)+7*X(381), 2*X(2)-7*X(547), 6*X(2)-X(548), 17*X(2)-7*X(549), 15*X(2)-7*X(631), 9*X(373)+X(5876), 7*X(576)+3*X(3630)

As a point of the Euler line, X(12812) has Shinagawa coefficients (11,7).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25590.

X(12812) lies on these lines: {2,3}, {373,5876}, {576,3630}, {3303,10593}, {3304,10592}, {3614,5563}, {3625,10175}, {3633,5886}, {3635,5901}, {3746,7173}, {4668,5844}, {4691,9956}, {5305,7603}, {5690,7988}, {5943,12046}, {6560,10148}, {6561,10147}, {10095,10170}

X(12812) = midpoint of X(i) and X(j) for these {i,j}: {5,1656}, {140,3859}, {631,3858}, {632,3091}
X(12812) = reflection of X(i) in X(j) for these (i,j): (546,3091), (632,3628), (3522,3530), (3843,3850), (5071,10109)
X(12812) = complement of X(15712)
X(12812) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3627,12108), (3,11541,550), (4,5,11737), (5,3627,5072), (5,3845,5068), (140,546,12103), (546,547,3628), (546,548,3627), (546,3091,3859), (546,3628,140), (546,12103,3853), (1656,3843,2), (3091,5076,3858), (3627,3850,546), (3627,5072,3850), (3627,12108,548), (3628,3856,10303), (3843,5072,3091), (3857,12102,546), (5070,12101,140), (10303,11541,3)

X(12813) = X(164)X(5708)∩X(177,942)

Barycentrics a ((b-c) (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-a b^3 c-2 b^4 c-a^3 c^2+a^2 b c^2+2 a b^2 c^2+2 b^3 c^2-a^2 c^3-a b c^3+2 b^2 c^3+a c^4-2 b c^4)+b (b-c) c (2 a^3+a^2 b-2 a b^2-b^3+a^2 c+2 a b c+5 b^2 c-2 a c^2+5 b c^2-c^3) Sin[A/2]-a (c (a^3 b+2 a^2 b^2-a b^3-2 b^4+a^3 c-a^2 b c+5 a b^2 c+3 b^3 c-a^2 c^2-5 a b c^2+b^2 c^2-a c^3-3 b c^3+c^4) Sin[B/2]- b (a^3 b-a^2 b^2-a b^3+b^4+a^3 c-a^2 b c-5 a b^2 c-3 b^3 c+2 a^2 c^2+5 a b c^2+b^2 c^2-a c^3+3 b c^3-2 c^4) Sin[C/2])) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25592.

X(12813) lies on these lines: {1,10502}, {164,5708}, {177,942}, {5049, 8422}, {5439,11691}, {8083,8091} ,{9957,11191}

X(12814) = X(1)X(7597)∩X(57,3659)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25592.

X(12814) lies on these lines: {1,7597}, {57,3659}, {65,2089}, {174,354}, {177,942}

X(12814) = X(11)-of-intouch triangle

X(12815) = MIDPOINT OF X(17) AND X(18)

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

See Tran Quang Hung and César Lozada, Hyacinthos 25601.

X(12815) lies on these lines: {2,7765}, {4,5206}, {6,17}, {32,5056}, {115,140}, {187,3850}, {532,8260}, {533,8259}, {547,5007}, {550,3054}, {574,3533}, {629,6674}, {630,6673}, {1504,10195}, {1505,10194}, {3090,7753}, {3523,7756}, {3525,11648}, {3628,9698}, {3851,7747}, {5059,8588}, {5067,7772}, {5070,5309}, {5461,7824}, {6292,6722}

X(12815) = midpoint of X(17) and X(18)
X(12815) = reflection of X(i) in X(j) for these (i,j): (629,6674), (630,6673)

X(12816) = X(17,30)∩X(18,381)

Barycentrics 1/(S+3*sqrt(3)*SA) : :
X(12816) = 7*X(17)-4*X(5238) = X(17)+8*X(5350) = X(5238)+14*X(5350)

See Tran Quang Hung and César Lozada, Hyacinthos 25606.

X(12816) = outer Hung-Lozada two-hexagons point, and X(12817) = inner Hung-Lozada two-hexagons point. See Tran Quang Hung and César Lozada, Hyacinthos 25607.

Let A' be the orthocenter of BCX(17), and define B', C' cyclically. X(12816) is the centroid of A'B'C'. (Randy Hutson, July 21, 2017)

X(12816) lies on the Kiepert hyperbola and these lines: {2,10646}, {3,10188}, {5,10187}, {6,12817}, {13,3830}, {14,3845}, {16,5066}, {17,30}, {18,381}, {62,3839}, {98,5470}, {383,7608}, {395,3860}, {531,11122}, {532,5487}, {542,11602}, {671,6778}, {1080,7607}, {2043,10195}, {2044,10194}, {3412,3627}, {5071,5237}, {5485,5863}, {8781,9116}, {10159,11303}

X(12816) = isogonal conjugate of X(10645)

X(12817) = X(18,30)∩X(17,381)

Barycentrics 1/(S-3*sqrt(3)*SA) : :
X(12817) = 7*X(18)-4*X(5237) = X(18)+8*X(5349) = X(5237)+14*X(5349)

See Tran Quang Hung and César Lozada, Hyacinthos 25606.

X(12816) = outer Hung-Lozada two-hexagons point, and X(12817) = inner Hung-Lozada two-hexagons point. See Tran Quang Hung and César Lozada, Hyacinthos 25607.

Let A' be the orthocenter of BCX(18), and define B', C' cyclically. X(12817) is the centroid of A'B'C'. (Randy Hutson, July 21, 2017)

X(12817) lies on the Kiepert hyperbola and these lines: {2,10645}, {3,10187}, {5,10188}, {6,12816}, {13,3845}, {14,3830}, {15,5066}, {17,381}, {18,30}, {61,3839}, {98,5469}, {383,7607}, {396,3860}, {530,11121}, {533,5488}, {542,11603}, {671,6777}, {1080,7608}, {2043,10194}, {2044,10195}, {3411,3627}, {5071,5238}, {5485,5862}, {8781,9114}, {10159,11304}

X(12817) = isogonal conjugate of X(10646)

X(12818) = OUTER HUNG-LOZADA THREE-SQUARES POINT

Barycentrics 1/(S+5*SA) : :

See Tran Quang Hung and César Lozada, Hyacinthos 25607.

X(12818) lies on the Kiepert hyperbola and these lines: {5,6434}, {6,12819}, {372,3591}, {382,485}, {486,546}, {550,10195}, {1131,6561}, {1132,6436}, {1152,11737}, {1327,6470}, {1328,3070} et al

X(12819) = INNER HUNG-LOZADA THREE-SQUARES POINT

Barycentrics 1/(-S+5*SA) : :

See Tran Quang Hung and César Lozada, Hyacinthos 25607.

X(12819) lies on the Kiepert hyperbola and these lines: {5,6433}, {6,12818}, {371,3590}, {382,486}, {485,546}, {550,10194}, {1131,6435}, {1132,6560}, {1151,11737}, {1327,3071}, {1328,6471}

X(12820) = OUTER HUNG-LOZADA THREE-HEXAGONS POINT

Barycentrics 1/(S+5*sqrt(3)*SA) : :

See Tran Quang Hung and César Lozada, Hyacinthos 25607.

X(12820) lies on the Kiepert hyperbola and these lines: {6,12821}, {17,382}, {18,546}, {383,11669}, {550,10188} et al

X(12821) = INNER HUNG-LOZADA THREE-HEXAGONS POINT

Barycentrics 1/(S-5*sqrt(3)*SA) : :

See Tran Quang Hung and César Lozada, Hyacinthos 25607.

X(12821) is the radical center of the de Longchamps circles of the adjunct anti-altimedial triangles. (Randy Hutson, November 2, 2017)

X(12821) lies on the Kiepert hyperbola and these lines: {6,12820}, {17,546}, {18,382}, {550,10187}, {1080,11669 et al

X(12822) = OUTER HUNG-LOZADA TWO-OCTAGONS POINT

Barycentrics 1/((3*(1+sqrt(2)))*SA+S)

See Tran Quang Hung and César Lozada, Hyacinthos 25607.

X(12822) lies on the Kiepert hyperbola and these lines: {6,12823}, {30,3373}, {381,3388} et al

X(12823) = INNER HUNG-LOZADA TWO-OCTAGONS POINT

Barycentrics 1/((3*(1+sqrt(2)))*SA-S) : :

See Tran Quang Hung and César Lozada, Hyacinthos 25607.

X(12823) lies on the Kiepert hyperbola and these lines: {6,12822}, {30,3388}, {381,3373} et al

X(12824) = HATZIPOLAKIS-MOSES NINE-POINT IMAGE OF X(2)

Barycentrics a^2 (a^4-b^4+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) : :
X(12824) = X(110)+2*X(1112), 2*X(113)+X(1986), X(113)+2*X(11557), 2*X(143)+X(5609), X(146)+2*X(974), X(399)+2*X(12236), X(1539)+2*X(11561), X(1986)-4*X(11557), 3*X(5640)-X(9140), X(9143)+3*X(11002)

Let P = p : q : r be barycentrics for a point P in the plane of a triangle ABC. Let
A' = reflection of A in P, and define B' and C' cyclically
Ab = orthogonal projection of A' on AC, and define Bc and Ca cyclically
Ac = orthogonal projection of A' on AB, and define Ba and Cb cyclically
(Na) = nine-point circle of AAbAC, and define (Nb) and (Nc) cyclically
The circles concur in the point Q given by

Q = a^2 (2 a^2 b^2 c^2 p+a^4 c^2 q+b^4 c^2 q-2 a^2 c^4 q-2 b^2 c^4 q+c^6 q+a^4 b^2 r-2 a^2 b^4 r+b^6 r-2 b^4 c^2 r+b^2 c^4 r) (b^2 c^2 p^2+a^2 c^2 p q-c^4 p q+a^2 b^2 p r-b^4 p r+a^4 q r-a^2 b^2 q r-a^2 c^2 q r) : : The point Q = HM(P) is here named the Hatzipolakis-Moses nine-point image of P. The appearance of (i,j) in the following list means that X(j) = HM(X(i)): {1,11570}, {2,12824}, {4,1986}, {5,11557}, {6,5477}, {15,6783}, {16,6782}, {20,12825}, {21,12826}, {22,12827}, {23,3580}, {25,12828}, {32,12829}, {36,1737}, {39,12830}, {55,12831}, {56,12832}, {99,12833}, {110,7471}, {186,403}, {187,230}

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609 and Antreas Hatzipolakis and César Lozada, Hyacinthos 25623

X(12824) lies on these lines:
{2,2781}, {23,6593}, {25,110}, {51,542}, {52,10294}, {74,9818}, {113,403}, {125,5133}, {143,5609}, {146,974}, {265,11818}, {381,5640}, {399,12236}, {511,5642}, {541,9730}, {568,5655}, {1495,11649}, {1511,2070}, {1539,11561}, {1550,11751}, {1992,2854}, {1995,9970}, {3448,7394}, {3796,10117}, {3917,5972}, {5095,8681}, {5422,5622}, {5621,10601}, {5643,12006}, {9517,9979}, {9729,10990}

X(12824) = midpoint of X(i) and X(j) for these {i,j}: {110,3060}, {568,5655}, {5890,10706}
X(12824) = reflection of X(i) in X(j) for these (i,j): (125,5943), (3060,1112), (3917,5972), (9140,12099)
X(12824) = isoconjugate of X(2157) and X(2986)
X(12824) = barycentric product X(i)*X(j) for these {i,j}: {23, 3580}, {316, 3003}
X(12824) = barycentric quotient X(i)/X(j) for these (i,j): (23, 2986), (3003, 67), (8744, 1300), (10317, 5504)
X(12824) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (113,1986,12825), (113,11557,1986), (113,12828,12827), (5640,9140,12099), (12827,12828,3580)

X(12825) = HATZIPOLAKIS-MOSES NINE-POINT IMAGE OF X(20)

Barycentrics a^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) (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) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609.

X(12825) is the radical center of the polar circles of the adjunct anti-altimedial triangles. (Randy Hutson, November 2, 2017)

X(12825) lies on these lines: {2,974}, {3,74}, {22,9934}, {69, 146}, {113,403}, {125,5907} et al

X(12825) = {X(113),X(1986)}-harmonic conjugate of X(12824)

X(12826) = HATZIPOLAKIS-MOSES NINE-POINT IMAGE OF X(21)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609.

X(12826) lies on these lines: {21,2778}, {28,110}, {113,403} et all

X(12827) = HATZIPOLAKIS-MOSES NINE-POINT IMAGE OF X(22)

Barycentrics (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) (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) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609.

X(12827) lies on these lines: {2,98}, {5,12099}, {113,403} et al

X(12827) = {X(113),X(12828)}-harmonic conjugate of X(12824)

X(12828) = HATZIPOLAKIS-MOSES NINE-POINT IMAGE OF X(25)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609.

X(12828) lies on these lines: {4,541}, {25,542}, {51,125}, {107,11005}, {110,6353}, {112,6792} ,{113,403} et al

X(12828) = {X(12824),X(12827)}-harmonic conjugate of X(113)

X(12829) = HATZIPOLAKIS-MOSES NINE-POINT IMAGE OF X(32)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609.

X(12829) lies on these lines: {6,98}, {32,2782}, {39,12042}, {99,3053}, {114,230}, {115,546} et al

X(12829) = {X(114),X(5477)}-harmonic conjugate of X(12830)

X(12830) = HATZIPOLAKIS-MOSES NINE-POINT IMAGE OF X(39)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609.

X(12830) lies on these lines: {6,147}, {30,1569}, {98,3815}, {99,7762}, {114,230}, {115,3850} et al

X(12830) = {X(114),X(5477)}-harmonic conjugate of X(12829)

X(12831) = HATZIPOLAKIS-MOSES NINE-POINT IMAGE OF X(55)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609.

X(12831) lies on these lines: {11,118}, {12,5884}, {57,5660}, {63,3035}, {80,11529}, {100,3474}, {119,912} et al

X(12831) = {X(119),X(11570)}-harmonic conjugate of X(12832)

X(12832) = HATZIPOLAKIS-MOSES NINE-POINT IMAGE OF X(56)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609.

X(12832) lies on these lines: {1,6713}, {10,5083}, {11,65}, {12,5883}, {46,5840}, {56,952}, {57 ,80}, {78,3035}, {100,1788}, {104 ,1470}, {109,6788}, {119,912} et al

X(12832) = {X(119),X(11570)}-harmonic conjugate of X(12831)

X(12833) = HATZIPOLAKIS-MOSES NINE-POINT IMAGE OF X(99)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609. See also Antreas Hatzipolakis and César Lozada, Hyacinthos 25623.

X(12833) lies on these lines: {4,69}, {99,512}, {112,249}, {526,9182}, {924,4590}, {2715,4611}, {2855,9160}, {9181,10411}

X(12833) = reflection of X(99) in its Simson line (line X(114)X(325))

X(12834) = X(2)X(576)∩X(6)X(11451)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25612.

X(12834) lies on these lines: {2,576}, {6,11451}, {22,10541}, {25,5012}, {51,5092}, {110,5943}, {140,1173}, {182,5645}, {184,10545}, {186,5462}, {323,6688}, {373,1994}, {589,8956}, {597,11416}, {694,3108}, {1350,3060}, {1597,10574}, {2979,5644}, {3567,7514}, {3580,11548}, {5020,11422}, {5133,9140}, {5899,10095}, {9781,12083}, {9815,20009}, {10546,11402}

X(12834) = {X(5422), X(5640)}-harmonic conjugate of X(5012)

CENTERS ASSOCIATED WITH THE ELLIPSE IE59: X(12835)-X(12841)

This preamble and centers X(12835)-X(12841) were contributed by Peter Moses, March 29, 2017.

Let IE59 denote the inellipse with perspector X(59). The center of IE59 is X(13006), and IE59 passes through X(i) for these i:

55, 56, 181, 202, 203, 215, 1124, 1335, 1362, 1397, 1672, 1673, 1682, 2007, 2008, 3235, 3236, 3237, 3238, 6056, 7005, 7006, 7066, 10799, 12835, 12836, 12837, 12838, 12839, 12840, 12841

This ellipse IE59 is the locus of the centers of similtude (insimilicenter and exsimilicenter) of the incircle with Tucker circles. Also, IE59 intersects the incircle in X(1362) and three other points, so that the corresponding four Tucker circles are tangent to the incircle. The Tucker circle through X(1362) has the following parameter:

arccos[(t² - s²)/(t² + s²)], where t = r + 4R.

The centers of the other three Tucker circles are the extraversions of X(970), and they lie on the Brocard axis. Not only are these circle internally tangent to the incircle, but they are also externally tangent to the two corresponding excircles. In this section, the names for centers X(12835) to X(12841), the notation "Tucker (X,p)-circle" represents the Tucker circle with center X and parameter p.

Let f(a,b,c,x,y,z) = b⁴c⁴(a - b - c)²(b - c)⁴x² - 2a⁴b²c²(a - b)² (a - b + c)(c - a)²(a + b - c)yz. The ellipse IE59 is given by the barycentric equation f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,z,x,y) = 0.

Possibly the earliest mention of IE59 occurs in TCCT, page 238, in a list of inscribed ellipses; in that list, this ellipse is denoted by W(X₁₁).

X(12835) = EXSIMILICENTER OF THE INCIRCLE AND TUCKER (X(3398),2ω)-CIRCLE

Barycentrics Sin[A]*(1 - Cos[A - 2*ω]) : :
Barycentrics a^2 (a+b-c) (a-b+c) (a^2-b c)^2 : :

The insimilicenter of the incircle and Tucker (X(3398),2ω) circle is X(10799).

X(12835) lies the inellipse IE(59) on these lines: {1, 3398}, {3, 10801}, {4, 10798}, {11, 98}, {12, 83}, {32, 56}, {34, 11380}, {35, 12054}, {36, 2080}, {55, 182}, {57, 10789}, {65, 12194}, {109, 727}, {181, 4279}, {388, 7787}, {499, 10104}, {999, 11842}, {1078, 5433}, {1319, 11364}, {1342, 3237}, {1343, 3238}, {1357, 1412}, {1428, 1691}, {1469, 5332}, {1478, 10796}, {1687, 2007}, {1688, 2008}, {2099, 10800}, {2276, 5038}, {2477, 3203}, {3023, 12176}, {3024, 12192}, {3027, 4027}, {3057, 12197}, {3085, 10359}, {3271, 8852}, {4293, 10788}, {5171, 5204}, {5182, 12350}, {5252, 10791}, {5434, 12150}, {6020, 12207}, {6285, 12202}, {7288, 7793}, {7354, 12110}, {10345, 10873}, {10358, 10895}, {10803, 11490}, {10944, 12195}

X(12835) = isoconjugate of X(j) and X(j) for these (i,j): {291,4518}, {334,7077}, {335,4876}
X(12835) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3398,10799), (388,7787,10797)
X(12835) = barycentric product X(i) X(j) for these {i,j}: {56,4366}, {57,8300}, {109,4375}, {238,1429}, {239,1428}, {593,3027}, {1412,4368}, {1447,1914}, {2210,10030}
X(12835) = barycentric quotient X(i)/X(j) for these (i,j): (1428,335), (1429,334), (1914,4518), (2210,4876), (4366,3596), (6652,4087), (8300,312)

X(12836) = EXSIMILICENTER OF THE INCIRCLE AND TUCKER (X(3395),-2ω)-CIRCLE

Barycentrics Sin[A]*(1 + Cos[A + 2*ω]) : :
Barycentrics a^2 (a-b-c) (b^2-b c+c^2)^2 : :

X(12836) lies the inellipse IE(59) on these lines: {1, 3095}, {3, 10801}, {5, 10063}, {6,10799}, {11, 76}, {12, 262}, {35, 11171}, {36, 9821}, {39, 55}, {56, 511}, {172, 8540}, {194, 497}, {202, 3105}, {203, 3104}, {215, 3202}, {330, 1916}, {384, 10798}, {496, 10079}, {498, 11272}, {538, 11238}, {726, 12053}, {730, 1837}, {982, 3865}, {1124, 3103}, {1335, 3102}, {1362, 10571}, {1479, 2782}, {1670, 1673}, {1671, 1672}, {1689, 3235}, {1690, 3236}, {1697, 3097}, {2053, 3271}, {2275, 3056}, {3058, 7757}, {3086, 12251}, {3106, 7006}, {3107, 7005}, {4294, 7709}, {5188, 5204}, {5432, 7786}, {5969, 12351}, {6194, 7288}, {6248, 10896}, {6272, 10926}, {6273, 10925}, {6284, 11257}, {7697, 7741}, {7976, 10950}, {9581, 9902}, {9917, 10832}, {9983, 10874}, {11152, 12354}, {11376, 12263}, {11393, 12143}, {11502, 12338}

X(12836) = reflection of X(10079) in X(496)
X(12836) = barycentric product X(i) X(j) for these {i,j}: {982,3061}, {1252,3020}, {2275,3705}, {3056,3662}, {3721,3794}
X(12836) = barycentric quotient X(i)/X(j) for these (i,j): (3061,7033), (7032,7132)
X(12836) = orthologic center of these triangles: 2nd Johnson-Yff to 1st Neuberg
X(12836) = X(76)-of-2nd-Johnson-Yff-triangle
X(12836) = {X(1),X(3095)}-harmonic conjugate of X(12837)

X(12837) = INSIMILICENTER OF THE INCIRCLE AND TUCKER (X(3395),-2ω)-CIRCLE

Barycentrics Sin[A]*(1 - Cos[A + 2*ω]) : :
Barycentrics a^2 (a+b-c) (a-b+c) (b^2+b c+c^2)^2 : :

X(12837) lies the inellipse IE(59) on these lines: {1,3095}, {3,10799}, {5,10079}, {6,12835}, {11,262}, {12,76}, {35,9821}, {36,11171}, {39,56}, {55,511}, {57,3097}, {65,12782}, {181,1403}, {192,1916}, {194,388}, {202,3106}, {203,3107}, {226,726}, {371,12839}, {372,12838}, {384,10797}, {495,10063}, {499,11272}, {538,11237}, {730,5252}, {732,12588}, {1124,3102}, {1335,3103}, {1397,5145}, {1469,2276}, {1478,2782}, {1670,1672}, {1671,1673}, {1689,3236}, {1690,3235}, {2175,8852}, {2477,3202}, {3085,12251}, {3104,7005}, {3105,7006}, {3790,7179}, {3864,7146}, {4293,7709}, {5188,5217}, {5218,6194}, {5433,7786}, {5434,7757}, {5969,12350}, {6248,10895}, {6272,10924}, {6273,10923}, {7354,11257}, {7697,7951}, {7976,10944}, {9578,9902}, {9917,10831}, {9983,10873}, {11375,12263}, {11392,12143}, {11501,12338}, {11869,12474}, {11870,12475}, {11905,12794}, {11930,12992}, {11931,12993}

X(12837) = reflection of X(10063) in X(495)
X(12837) = barycentric product X(i) X(j) for these {i,j}: {65,4469}, {226,4476}, {593,7142}, {984,7146}, {1469,3661}, {2276,7179}
X(12837) = barycentric quotient X(i)/X(j) for these (i,j): (869,2344), (4469,314), (4476,333),. (7146,870)
X(12837) = orthologic center of these triangles: 1st Johnson-Yff to 1st Neuberg
X(12837) = X(76)-of-1st-Johnson-Yff-triangle
X(12837) = {X(1),X(3095)}-harmonic conjugate of X(12836)

X(12838) = EXSIMILICENTER OF THE INCIRCLE AND TUCKER (X(1691), π/2 + 2ω)-CIRCLE

Barycentrics Sin[A]*(1 - Sin[A - 2*ω]) : :
Barycentrics a^2 (b c (a^2 b^2+a^2 c^2+b^2 c^2)+(-a^2+b c) (a^2+b c) S) : :

X(12838) lies the inellipse IE(59) on these lines: {1, 1691}, {32, 1124}, {182, 1335}, {1342, 2008}, {1343, 2007}, {1687, 3237}, {1688, 3238}, {3299, 12212}, {3301, 5038}

X(12838) = {X(1),X(1691)}-harmonic conjugate of X(12839)

X(12839) = INSIMILICENTER OF THE INCIRCLE AND TUCKER (X(1691), π/2 + 2ω)-CIRCLE

Barycentrics Sin[A]*(1 + Sin[A - 2*ω]) : :
Barycentrics a^2 (b c (a^2 b^2+a^2 c^2+b^2 c^2)-(-a^2+b c) (a^2+b c) S) : :

X(12839) lies the inellipse IE(59) on these lines: {1, 1691}, {32, 1335}, {182, 1124}, {1342, 2007}, {1343, 2008}, {1687, 3238}, {1688, 3237}, {3299, 5038}, {3301, 12212}

X(12839) = {X(1),X(1691)}-harmonic conjugate of X(12838)

X(12840) = EXSIMILICENTER OF THE INCIRCLE AND TUCKER (X(3094), π/2 - 2ω)-CIRCLE

Barycentrics Sin[A]*(1 - Sin[A + 2*ω]) : :
Barycentrics a^2 (b c (a^2 b^2+a^2 c^2+b^2 c^2)-(b^2-b c+c^2) (b^2+b c+c^2) S) : :

X(12840) lies the inellipse IE(59) on these lines: {1, 3094}, {39, 1124}, {55, 3102}, {56, 3103}, {371, 10799}, {511, 1335}, {1670, 3236}, {1671, 3235}, {1672, 1690}, {1673, 1689}

X(12840) = {X(1),X(3094)}-harmonic conjugate of X(12841)

X(12841) = INSIMILICENTER OF THE INCIRCLE AND TUCKER (X(3094), π/2 - 2ω)-CIRCLE

Barycentrics Sin[A]*(1 + Sin[A + 2*ω]) : :
Barycentrics a^2 (b c (a^2 b^2+a^2 c^2+b^2 c^2)+(b^2-b c+c^2) (b^2+b c+c^2) S) : :

X(12841) lies the inellipse IE(59) on these lines: {1, 3094}, {39, 1335}, {55, 3103}, {56, 3102}, {372, 10799}, {511, 1124}, {1670, 3235}, {1671, 3236}, {1672, 1689}, {1673, 1690}

X(12841) = {X(1),X(3094)}-harmonic conjugate of X(12840)

Orthologic centers: X(12842)-X(13005)

Centers X(12842)-X(13005) were contributed by César Eliud Lozada, April 1, 2017.

X(12842) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HEXYL TO HUTSON EXTOUCH

Trilinears p^5*(p-q)-(q^2+3)*p^4+(q^2-1)*q*p^3+(5*q^2+1)*p^2-(q^2-5)*q*p-2-2*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(12842) = 3*X(3576)-2*X(12521) = 3*X(5587)-4*X(12620) = 5*X(8227)-4*X(12612)

The reciprocal orthologic center of these triangles is X(3555).

X(12842) lies on these lines: {1,5920}, {3,12658}, {20,9804}, {40,6764}, {78,12533}, {84,6361}, {144,962}, {517,12654}, {1490,12692}, {3333,12855}, {3576,12521}, {5587,12620}, {5732,6762}, {5777,8158}, {7675,12846}, {7966,12245}, {8227,12612}, {8273,12333}, {8726,12439}, {9953,10864}, {10884,12537}

X(12842) = midpoint of X(i) and X(j) for these {i,j}: {1,8001}, {20,9804}
X(12842) = reflection of X(i) in X(j) for these (i,j): (40,12516), (12658,3)

X(12843) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HEXYL TO MANDART-EXCIRCLES

Trilinears a^9-(b+c)*a^8-2*(b^2+c^2)*a^7+2*(b+c)*(b^2+3*b*c+c^2)*a^6+4*b*c*(b^2-6*b*c+c^2)*a^5-2*b*c*(b+c)*(3*b^2+2*b*c+3*c^2)*a^4+2*(b^4+c^4-2*b*c*(b^2-5*b*c+c^2))*(b+c)^2*a^3-2*(b+c)*(b^6+c^6-b*c*(b^2+4*b*c+c^2)*(b^2-3*b*c+c^2))*a^2-(b^2-c^2)^2*(b-c)^2*(b^2+6*b*c+c^2)*a+(b^4-c^4)*(b^2-c^2)^2*(b-c) : :
X(12843) = 3*X(3576)-2*X(12522) = 3*X(5587)-4*X(12621) = 5*X(8227)-4*X(12613)

The reciprocal orthologic center of these triangles is X(3555).

X(12843) lies on these lines: {1,12553}, {3,12659}, {20,12542}, {40,12517}, {78,12534}, {517,12655}, {962,4511}, {1490,12693}, {3576,12522}, {5587,12621}, {7675,12847}, {8227,12613}, {8726,12442}, {10864,12449}, {10884,12538}

X(12843) = midpoint of X(20) and X(12542)
X(12843) = reflection of X(i) in X(j) for these (i,j): (40,12517), (12659,3)

X(12844) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HEXYL TO MIDARC

Trilinears a^3-(b+c)*a^2-(b-c)^2*a+(b^2-c^2)*(b-c)+4*b*c*(b+c)*sin(A/2)-4*a*sin(B/2)*b*c-4*a*sin(C/2)*b*c : :
X(12844) = 5*X(8227)-4*X(12614)

The reciprocal orthologic center of these triangles is X(1).

X(12844) lies on these lines: {1,167}, {3,164}, {20,9807}, {40,12518}, {78,11691}, {188,1490}, {517,12656}, {1482,11528}, {3333,5571}, {3576,12523}, {5587,12622}, {5732,9836}, {6765,9837}, {7587,11032}, {7588,8084}, {7670,7675}, {8075,8094}, {8076,8093}, {8227,12614}, {8726,12443}, {10864,12450}, {10884,12539}

X(12844) = midpoint of X(i) and X(j) for these {i,j}: {1,167}, {20,9807}
X(12844) = reflection of X(i) in X(j) for these (i,j): (40,12518), (164,3), (11528,1482)
X(12844) = orthologic center of these triangles: hexyl to 2nd midarc
X(12844) = {X(8081), X(8082)}-harmonic conjugate of X(1)
X(12844) = X(1)-of-hexyl-triangle
X(12844) = X(8)-of-2nd-circumperp-triangle
X(12844) = X(355)-of-excentral-triangle
X(12844) = X(944)-of-1st-circumperp-triangle
X(12844) = excentral-to-hexyl similarity image of X(164)

X(12845) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HEXYL TO 1st SCHIFFLER

Trilinears a^9-3*(b+c)*a^8-2*b*c*a^7+4*(b+c)*(2*b^2-b*c+2*c^2)*a^6-(6*b^4+6*c^4-b*c*(10*b^2+9*b*c+10*c^2))*a^5-(b+c)*(6*b^4+6*c^4-b*c*(8*b^2-13*b*c+8*c^2))*a^4+(8*b^6+8*c^6-(14*b^4+14*c^4+3*b*c*(3*b^2-2*b*c+3*c^2))*b*c)*a^3-(b+c)*(4*b^4+4*c^4-b*c*(9*b^2-14*b*c+9*c^2))*b*c*a^2-3*(b^4-c^4)*(b^2-c^2)*(b-c)^2*a+(b^2-c^2)^4*(b+c) : :
X(12845) = 3*X(3576)-2*X(12524) = 3*X(5587)-4*X(12623) = 5*X(8227)-4*X(12615)

The reciprocal orthologic center of these triangles is X(21).

X(12845) lies on these lines: {1,5180}, {3,12660}, {20,12543}, {40,12519}, {78,12535}, {84,6597}, {411,1768}, {517,12657}, {1490,12695}, {3576,12524}, {5587,12623}, {6599,7491}, {7675,12850}, {8227,12615}, {8726,12444}, {10864,12451}, {10884,12540}

X(12845) = midpoint of X(20) and X(12543)
X(12845) = reflection of X(i) in X(j) for these (i,j): (40,12519), (12660,3)

X(12846) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO HUTSON EXTOUCH

Trilinears ((b+c)*a^8-(4*b^2+3*b*c+4*c^2)*a^7+(b+c)*(4*b^2-19*b*c+4*c^2)*a^6+(4*b^4+4*c^4+b*c*(53*b^2+50*b*c+53*c^2))*a^5-(b+c)*(10*b^4+10*c^4+3*b*c*(9*b^2-26*b*c+9*c^2))*a^4+(4*b^6+4*c^6-11*(3*b^4+3*c^4+2*b*c*(8*b^2+9*b*c+8*c^2))*b*c)*a^3+(b+c)*(4*b^6+4*c^6+(47*b^4+47*c^4+2*b*c*(24*b^2-19*b*c+24*c^2))*b*c)*a^2-(b^2-c^2)^2*(4*b^4+4*c^4+b*c*(17*b^2+6*b*c+17*c^2))*a+(b^3-c^3)*(b^2-c^2)^3)/(-a+b+c) : :
X(12846) = 2*X(5920)-3*X(8236) = 3*X(11038)-4*X(12853)

The reciprocal orthologic center of these triangles is X(3555).

X(12846) lies on these lines: {7,3555}, {9,12533}, {1445,12658}, {2346,7160}, {4326,8001}, {5920,8236}, {7675,12842}, {7676,12516}, {7677,12521}, {7678,12612}, {7679,12620}, {8232,12692}, {8732,12439}, {9953,10865}, {10889,12552}, {11025,12855}, {11038,12853}, {11526,12654}

X(12846) = reflection of X(i) in X(j) for these (i,j): (7,12854), (12533,9)

X(12847) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO MANDART-EXCIRCLES

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

X(12847) lies on these lines: {7,12538}, {9,12534}, {1445,12659}, {7675,12843}, {7676,12517}, {7677,12522}, {7678,12613}, {7679,12621}, {8232,12693}, {8732,12442}, {10865,12449}, {10889,12553}, {11526,12655}

X(12847) = reflection of X(12534) in X(9)

X(12848) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO MIXTILINEAR

Barycentrics (3*a^3-5*(b+c)*a^2+(b+c)^2*a+(b^2-c^2)*(b-c))*(a-b+c)*(a+b-c) : :
X(12848) = 3*X(2)-4*X(8257) = 2*X(3421)-3*X(5686) = 2*X(3586)-3*X(5809) = X(3586)-3*X(10398) = 2*X(7962)-3*X(8236)

The reciprocal orthologic center of these triangles is X(1).

X(12848) lies on the cubic K295 and these lines: {1,5766}, {2,7}, {4,653}, {6,347}, {20,10394}, {44,948}, {56,6068}, {65,452}, {72,3600}, {145,4552}, {190,6604}, {218,279}, {241,4644}, {348,3758}, {388,5220}, {390,517}, {391,1441}, {405,8543}, {516,2093}, {518,3476}, {664,1992}, {954,999}, {971,2096}, {997,4321}, {1020,4253}, {1210,5735}, {1471,4310}, {1490,8544}, {1728,4295}, {1736,3332}, {1737,4312}, {1743,3668}, {1788,5177}, {1864,3474}, {2095,5762}, {2097,5845}, {2182,10402}, {2801,4293}, {3339,12572}, {3421,5686}, {3487,5265}, {3522,10393}, {3672,7961}, {3820,7679}, {3832,10395}, {4294,10399}, {4308,11523}, {4323,5436}, {4326,7994}, {4419,5228}, {4641,7365}, {4848,5175}, {5173,10177}, {5218,8255}, {5223,12573}, {5704,5715}, {5740,5798}, {5779,6826}, {5784,6904}, {5805,6844}, {5812,11662}, {5817,6843}, {5843,6911}, {5924,7682}, {6244,7676}, {6282,7675}, {7678,7956}, {7962,8236}, {8101,8387}, {8102,8388}, {9954,10865}, {10889,12555}

X(12848) = midpoint of X(144) and X(9965)
X(12848) = reflection of X(i) in X(j) for these (i,j): (7,57), (329,9), (5809,10398)
X(12848) = X(25)-of-Honsberger-triangle
X(12848) = excentral-to-Honsberger similarity image of X(57)
X(12848) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7,9,8232), (7,1445,8732), (7,6172,8545), (390,7672,12849), (5728,5759,390)

X(12849) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO 1st SCHIFFLER

Barycentrics 3*a^7+(b+c)*a^6-(9*b^2-4*b*c+9*c^2)*a^5-(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(9*b^4+b^2*c^2+9*c^4)*a^3+(b+c)*(3*b^4+3*c^4-b*c*(4*b^2-9*b*c+4*c^2))*a^2-(b^2-c^2)^2*(3*b^2+4*b*c+3*c^2)*a-(b+c)^3*(b-c)^4 : :
X(12849) = 3*X(2)-4*X(13089) = 5*X(3091)-4*X(12600)

The reciprocal orthologic center of these triangles is X(79)

X(12849) lies on these lines: {2,3467}, {3,12255}, {4,12146}, {5,13126}, {7,6597}, {8,12535}, {10,12409}, {20,5694}, {22,12414}, {100,12342}, {145,13100}, {149,6595}, {153,5690}, {388,12947}, {497,12957}, {1270,12808}, {1271,12807}, {2475,12745}, {2896,12504}, {3085,13128}, {3086,13129}, {3091,12600}, {3434,12927}, {3436,12937}, {3616,12267}, {3648,3988}, {3878,6224}, {4240,12798}, {4309,12877}, {5601,12482}, {5602,12483}, {6462,13000}, {6463,13001}, {7787,12209}, {10528,13130}, {10529,13131}

X(12849) = reflection of X(i) in X(j) for these (i,j): (4,12919), (8,12786), (20,12556), (145,13100), (149,6595), (4240,12798), (10266,13089), (12255,3), (12409,10), (12535,12682), (12543,6597), (13126,5)
X(12849) = anticomplement of X(10266)
X(12849) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10266,13089,2), (12957,13080,497)

X(12850) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO 1st SCHIFFLER

Trilinears 16*p^6+(-32*q^2-56)*p^4-40*q*p^3+(16*q^4+56*q^2+17)*p^2+2*(20*q^2+11)*q*p-11*q^2+12-4*q/p : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(21).

X(12850) lies on these lines: {7,6597}, {9,12535}, {1445,12660}, {2346,10266}, {3889,12701}, {7675,12845}, {7676,12519}, {7677,12524}, {7678,12615}, {7679,12623}, {8232,12695}, {8732,12444}, {10865,12451}, {10889,12557}, {11526,12657}

X(12850) = reflection of X(12535) in X(9)

X(12851) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-HUTSON TO HUTSON EXTOUCH

Trilinears -2*(a^8-6*(b+c)^2*a^6+4*(b+c)*(b^2-5*b*c+c^2)*a^5+8*(b^4+c^4+b*c*(7*b^2+4*b*c+7*c^2))*a^4-8*(b+c)*(b^4+c^4-b*c*(b^2+12*b*c+c^2))*a^3-2*(b^4+c^4+10*b*c*(2*b^2-b*c+2*c^2))*(b+c)^2*a^2+4*(b^2-c^2)^2*(b+c)*(b^2+3*b*c+c^2)*a-(b^2-c^2)^4)*b*c*sin(A/2)+4*(a^3-(b-c)*a^2-(b^2+6*b*c+c^2)*a+(b+c)*(b^2-c^2))*(a^3-(b-c)*a^2-(b^2+8*b*c-c^2)*a+(b^2-c^2)*(b-c))*a*b^2*c*sin(B/2)+4*(a^3+(b-c)*a^2+(b^2-8*b*c-c^2)*a+(b^2-c^2)*(b-c))*(a^3+(b-c)*a^2-(b^2+6*b*c+c^2)*a-(b+c)*(b^2-c^2))*a*b*c^2*sin(C/2)+(a^2-2*(b+c)*a+(b-c)^2)*((b+c)*a^5-(b^2+c^2)*a^4-2*(b+c)^3*a^3+2*(b^4+6*b^2*c^2+c^4)*a^2+(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)*a-(b^4-c^4)*(b^2-c^2))*(a+b+c)^2 : :

The reciprocal orthologic center of these triangles is X(3555).

X(12851) lies on these lines: {363,12658}, {5920,8390}, {5934,12692}, {8001,8140}, {8107,12516}, {8109,12521}, {8377,12612}, {8380,12620}, {9783,9804}, {9953,11856}, {11527,12654}, {11685,12533}, {11854,12439}, {11886,12537}, {11892,12552}

X(12851) = reflection of X(12852) in X(8001)

X(12852) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-HUTSON TO HUTSON EXTOUCH

Trilinears 2*(a+b-c)*(a-b+c)*(a^3-(b+c)*a^2-(b^2+8*b*c+c^2)*a+(b+c)*(b^2+c^2))*((b+c)*a^3-(b-c)^2*a^2-(b^2+6*b*c+c^2)*(b+c)*a+(b^2-c^2)^2)*sin(A/2)-2*(a+b-c)*(-a+b+c)*(a^3-(b-c)*a^2-(b^2+6*b*c+c^2)*a+(b+c)*(b^2-c^2))*(a^3-(b-c)*a^2-(b^2+8*b*c-c^2)*a+(b^2-c^2)*(b-c))*b*sin(B/2)-2*(-a+b+c)*(a-b+c)*(a^3+(b-c)*a^2-(b^2+6*b*c+c^2)*a-(b+c)*(b^2-c^2))*(a^3+(b-c)*a^2+(b^2-8*b*c-c^2)*a+(b^2-c^2)*(b-c))*c*sin(C/2)+a^9+(b+c)*a^8-4*(2*b^2+3*b*c+2*c^2)*a^7-28*b*c*(b+c)*a^6+2*(3*b^2+2*b*c+3*c^2)*(3*b^2+8*b*c+3*c^2)*a^5-2*(b+c)*(3*b^4+3*c^4-2*b*c*(15*b^2+13*b*c+15*c^2))*a^4-4*(4*b^6+4*c^6+(21*b^4+21*c^4+2*b*c*(4*b^2+31*b*c+4*c^2))*b*c)*a^3+4*(b^2-c^2)*(b-c)*(2*b^4+2*c^4-b*c*(b+5*c)*(5*b+c))*a^2+(b^2-c^2)^2*(5*b^4+5*c^4+18*b*c*(2*b^2-b*c+2*c^2))*a-(b^2-c^2)^3*(b-c)*(3*b^2+2*b*c+3*c^2) : :

The reciprocal orthologic center of these triangles is X(3555).

X(12852) lies on these lines: {168,12658}, {5920,8392}, {5935,12692}, {7160,7707}, {8001,8140}, {8108,12516}, {8110,12521}, {8378,12612}, {8381,12620}, {9787,9804}, {9953,11857}, {11528,12654}, {11686,12533}, {11855,12439}, {11887,12537}, {11893,12552}

X(12852) = reflection of X(12851) in X(8001)

X(12853) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO HUTSON EXTOUCH

Trilinears 2*q*p^5-(4*q^2-3)*p^4+(2*q^2-13)*q*p^3+(11*q^2-9)*p^2-(q^2-15)*q*p+2-2*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(12853) = 3*X(1)-X(5920) = 5*X(3616)-X(12533) = X(5920)+3*X(12854) = 3*X(11038)+X(12846)

The reciprocal orthologic center of these triangles is X(3555).

X(12853) lies on these lines: {1,5920}, {495,12620}, {496,12612}, {942,12439}, {999,12521}, {3295,12516}, {3333,12658}, {3487,12692}, {3616,12533}, {4295,12680}, {4326,6766}, {5045,12855}, {5542,9953}, {6764,12777}, {8351,12871}, {9797,9874}, {9804,11037}, {11036,12537}, {11038,12846}, {11042,12865}, {11043,12869}, {11529,12654}

X(12853) = midpoint of X(1) and X(12854)
X(12853) = reflection of X(12855) in X(5045)
X(12853) = {X(1), X(8001)}-harmonic conjugate of X(7160)

X(12854) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INTOUCH TO HUTSON EXTOUCH

Trilinears (a-b+c)*(a+b-c)*(a^3-(b+c)*a^2-(b^2+8*b*c+c^2)*a+(b+c)*(b^2+c^2))*((b+c)*a^3-(b-c)^2*a^2-(b^2+6*b*c+c^2)*(b+c)*a+(b^2-c^2)^2) : :
X(12854) = 3*X(354)-2*X(12855) = X(5920)-4*X(12853)

The reciprocal orthologic center of these triangles is X(3555).

X(12854) lies on these lines: {1,5920}, {2,12533}, {11,12612}, {12,12620}, {55,12516}, {56,12521}, {57,12439}, {72,11526}, {174,12871}, {226,12692}, {354,12855}, {1284,12869}, {2089,12870}, {3340,12654}, {3555,5082}, {5173,12777}, {8243,12865}, {8581,9953}, {12670,12864}, {12731,12859}

X(12854) = midpoint of X(i) and X(j) for these {i,j}: {7,12846}, {9804,12537}
X(12854) = reflection of X(i) in X(j) for these (i,j): (1,12853), (5920,1), (12658,12439), (12670,12864)
X(12854) = complement of X(12533)

X(12855) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE TO HUTSON EXTOUCH

Barycentrics (b+c)*a^6+12*b*c*a^5-(b+c)*(3*b^2+2*b*c+3*c^2)*a^4-4*b*c*(5*b^2+14*b*c+5*c^2)*a^3+3*(b^2-c^2)^2*(b+c)*a^2+8*(b^2-c^2)^2*b*c*a-(b^2-c^2)^3*(b-c) : :
X(12855) = 3*X(354)-X(12854) = 3*X(3873)+X(12533)

The reciprocal orthologic center of these triangles is X(3555).

X(12855) lies on these lines: {1,12521}, {7,40}, {10,5572}, {57,12516}, {65,5920}, {142,3913}, {226,9589}, {354,12854}, {495,12599}, {942,11362}, {946,3295}, {1056,12120}, {1210,12620}, {3085,7308}, {3303,12859}, {3333,12842}, {3339,9898}, {3873,12533}, {3922,12736}, {4866,10398}, {5045,12853}, {5703,9624}, {5728,12692}, {6767,12856}, {8001,10980}, {8083,12873}, {9804,10580}, {9874,11024}, {9953,11019}, {10056,10075}, {10122,12670}, {11018,12439}, {11020,12537}, {11021,12552}, {11025,12846}, {11030,12865}, {11031,12869}, {11032,12870}, {11033,12871}

X(12855) = midpoint of X(i) and X(j) for these {i,j}: {65,5920}, {12658,12777}
X(12855) = reflection of X(12853) in X(5045)

X(12856) = ORTHOLOGIC CENTER OF THESE TRIANGLES: JOHNSON TO HUTSON EXTOUCH

Trilinears 2*p^5*(p-q)-7*p^4-2*(q^2-4)*q*p^3+(2*q^4-7*q^2+10)*p^2+2*(3*q^2-5)*q*p-2+4*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(40).

X(12856) lies on these lines: {1,12859}, {2,12249}, {3,12411}, {4,9874}, {5,7160}, {11,10075}, {12,10059}, {30,12120}, {355,12731}, {381,12599}, {517,12777}, {952,5665}, {1479,12863}, {3652,12516}, {5587,9898}, {5779,12699}, {5805,6601}, {5886,12260}, {6214,12802}, {6215,12801}, {6265,12521}, {6767,12855}, {6864,9957}, {8200,12464}, {8207,12465}, {8220,12861}, {8221,12862}, {9996,12500}, {10796,12200}, {10942,12874}, {10943,12875}, {11499,12333}

X(12856) = midpoint of X(i) and X(j) for these {i,j}: {4,9874}, {12857,12858}
X(12856) = reflection of X(i) in X(j) for these (i,j): (3,12864), (7160,5), (12872,12599)
X(12856) = complement of X(12249)
X(12856) = X(7160)-of-Johnson-triangle
X(12856) = {X(12859),X(12860)}-harmonic conjugate of X(1)

X(12857) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO HUTSON EXTOUCH

Trilinears 2*p^5*(p-q)-7*p^4-2*(q^2-5)*q*p^3+(2*q^4-9*q^2+11)*p^2+2*(3*q^2-7)*q*p-1+5*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(40).

X(12857) lies on these lines: {11,7160}, {12,12874}, {355,12731}, {1376,12333}, {8000,10944}, {9898,10826}, {10059,10523}, {10075,10948}, {10785,12249}, {10794,12200}, {10829,12411}, {10871,12500}, {10893,12599}, {10914,12777}, {10919,12801}, {10920,12802}, {10945,12861}, {10946,12862}, {10947,12863}, {10949,12875}, {11373,12260}, {11390,12139}, {11826,12120}, {11865,12464}, {11866,12465}, {11903,12789}, {11928,12872}

X(12857) = reflection of X(i) in X(j) for these (i,j): (12333,12864), (12858,12856)
X(12857) = X(7160)-of-inner-Johnson-triangle

X(12858) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO HUTSON EXTOUCH

Trilinears 2*p^5*(p-q)-7*p^4-2*(q^2-3)*q*p^3+(2*q^4-5*q^2+9)*p^2+6*(q^2-1)*q*p-3+3*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(40).

X(12858) lies on these lines: {4,5173}, {11,12875}, {12,7160}, {72,12777}, {355,12731}, {946,3295}, {958,12864}, {2886,5791}, {3436,9874}, {5220,5812}, {5572,5805}, {8000,10950}, {9898,10827}, {10059,10954}, {10075,10523}, {10786,12249}, {10795,12200}, {10830,12411}, {10872,12500}, {10894,12599}, {10921,12801}, {10922,12802}, {10951,12861}, {10952,12862}, {10953,12863}, {10955,12874}, {11391,12139}, {11827,12120}, {11867,12464}, {11868,12465}, {11904,12789}, {11929,12872}

X(12858) = reflection of X(12857) in X(12856)
X(12858) = X(7160)-of-outer-Johnson-triangle

X(12859) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO HUTSON EXTOUCH

Trilinears p^6-q*p^5-3*p^4-(q^2-3)*q*p^3+(q^4-3*q^2+3)*p^2+3*(q^2-1)*q*p+2*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(40).

X(12859) lies on these lines: {1,12856}, {4,12863}, {5,10075}, {12,7160}, {56,12864}, {65,12777}, {354,11023}, {388,9874}, {495,10059}, {3085,12249}, {3303,12855}, {4863,5173}, {7354,12120}, {8000,10944}, {9654,12872}, {10797,12200}, {10831,12411}, {10873,12500}, {10895,12599}, {10923,12801}, {10924,12802}, {10956,12874}, {10957,12875}, {11375,12260}, {11392,12139}, {11501,12333}, {11905,12789}, {11930,12861}, {11931,12862}, {12731,12854}

X(12859) = reflection of X(10059) in X(495)
X(12859) = X(7160)-of-1st-Johnson-Yff-triangle
X(12859) = {X(1),X(12856)}-harmonic conjugate of X(12860)

X(12860) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO HUTSON EXTOUCH

Trilinears 2*p^5*(p-q)-7*p^4-2*(q^2-3)*q*p^3+(2*q^4-5*q^2+9)*p^2+6*(q^2-1)*q*p-3+3*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(40).

X(12860) lies on these lines: {1,12856}, {5,10059}, {11,7160}, {55,12864}, {480,12053}, {496,10075}, {497,9874}, {3057,12777}, {3086,12249}, {3601,6154}, {5920,12731}, {6284,12120}, {8000,10950}, {9581,9898}, {9669,12872}, {10798,12200}, {10832,12411}, {10874,12500}, {10896,12599}, {10925,12801}, {10926,12802}, {10958,12874}, {10959,12875}, {11376,12260}, {11393,12139}, {11502,12333}, {11871,12464}, {11872,12465}, {11906,12789}, {11932,12861}, {11933,12862}

X(12860) = reflection of X(10075) in X(496)
X(12860) = X(7160)-of-2nd-Johnson-Yff-triangle
X(12860) = {X(1),X(12856)}-harmonic conjugate of X(12859)

X(12861) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO HUTSON EXTOUCH

Trilinears (a^10-5*(b+c)^2*a^8+4*(b+c)*(b^2+c^2)*a^7+2*(5*b^4+5*c^4+24*b*c*(b^2+b*c+c^2))*a^6-4*(b+c)*(b^2+c^2)*(3*b^2-2*b*c+3*c^2)*a^5-2*(5*b^6+5*c^6+(50*b^4+50*c^4+b*c*(41*b^2+96*b*c+41*c^2))*b*c)*a^4+4*(b+c)*(b^2+c^2)*(3*b^4+3*c^4-2*b*c*(2*b^2+15*b*c+2*c^2))*a^3+(b^2+c^2)*(5*b^6+5*c^6+(64*b^4+64*c^4+5*b*c*(7*b^2-16*b*c+7*c^2))*b*c)*a^2-4*(b^4-c^4)*(b^2-c^2)^2*(b-c)*a-(b^4-c^4)^2*(b+c)^2)*S+a*(a^11-(b+c)*a^10-(5*b^2+14*b*c+5*c^2)*a^9+(b+c)*(5*b^2-2*b*c+5*c^2)*a^8+2*(5*b^4+5*c^4+4*b*c*(7*b^2+6*b*c+7*c^2))*a^7-2*(b+c)*(5*b^4+5*c^4-4*b*c*(b^2+3*b*c+c^2))*a^6-2*(5*b^6+5*c^6+(42*b^4+42*c^4+b*c*(41*b^2+40*b*c+41*c^2))*b*c)*a^5+2*(b+c)*(5*b^6+5*c^6-(6*b^4+6*c^4+35*b*c*(b^2+c^2))*b*c)*a^4+(b^2+c^2)*(5*b^6+5*c^6+(56*b^4+56*c^4+b*c*(35*b^2-128*b*c+35*c^2))*b*c)*a^3-(b+c)*(5*b^8+5*c^8-2*(4*b^6+4*c^6+(24*b^4+24*c^4-b*c*(12*b^2+91*b*c+12*c^2))*b*c)*b*c)*a^2-(b^2-c^2)^2*(b^6+c^6+(14*b^4+14*c^4+b*c*(3*b^2-68*b*c+3*c^2))*b*c)*a+(b^2-c^2)^3*(b-c)*((b^2-c^2)^2-4*b^2*c^2)) : :

The reciprocal orthologic center of these triangles is X(40).

X(12861) lies on these lines: {493,7160}, {6461,12862}, {6462,9874}, {8000,8210}, {8188,9898}, {8194,12411}, {8201,12464}, {8208,12465}, {8212,12599}, {8214,12777}, {8216,12801}, {8218,12802}, {8220,12856}, {8222,12864}, {10059,11951}, {10875,12500}, {11377,12260}, {11394,12139}, {11503,12333}, {11828,12120}, {11840,12200}, {11846,12249}, {11930,12859}, {11932,12860}, {11947,12863}, {11949,12872}, {11955,12874}, {11957,12875}

X(12861) = X(7160)-of-Lucas-homothetic-triangle

X(12862) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO HUTSON EXTOUCH

Trilinears -(a^10-5*(b+c)^2*a^8+4*(b+c)*(b^2+c^2)*a^7+2*(5*b^4+5*c^4+24*b*c*(b^2+b*c+c^2))*a^6-4*(b+c)*(b^2+c^2)*(3*b^2-2*b*c+3*c^2)*a^5-2*(5*b^6+5*c^6+(50*b^4+50*c^4+b*c*(41*b^2+96*b*c+41*c^2))*b*c)*a^4+4*(b+c)*(b^2+c^2)*(3*b^4+3*c^4-2*b*c*(2*b^2+15*b*c+2*c^2))*a^3+(b^2+c^2)*(5*b^6+5*c^6+(64*b^4+64*c^4+5*b*c*(7*b^2-16*b*c+7*c^2))*b*c)*a^2-4*(b^4-c^4)*(b^2-c^2)^2*(b-c)*a-(b^4-c^4)^2*(b+c)^2)*S+a*(a^11-(b+c)*a^10-(5*b^2+14*b*c+5*c^2)*a^9+(b+c)*(5*b^2-2*b*c+5*c^2)*a^8+2*(5*b^4+5*c^4+4*b*c*(7*b^2+6*b*c+7*c^2))*a^7-2*(b+c)*(5*b^4+5*c^4-4*b*c*(b^2+3*b*c+c^2))*a^6-2*(5*b^6+5*c^6+(42*b^4+42*c^4+b*c*(41*b^2+40*b*c+41*c^2))*b*c)*a^5+2*(b+c)*(5*b^6+5*c^6-(6*b^4+6*c^4+35*b*c*(b^2+c^2))*b*c)*a^4+(b^2+c^2)*(5*b^6+5*c^6+(56*b^4+56*c^4+b*c*(35*b^2-128*b*c+35*c^2))*b*c)*a^3-(b+c)*(5*b^8+5*c^8-2*(4*b^6+4*c^6+(24*b^4+24*c^4-b*c*(12*b^2+91*b*c+12*c^2))*b*c)*b*c)*a^2-(b^2-c^2)^2*(b^6+c^6+(14*b^4+14*c^4+b*c*(3*b^2-68*b*c+3*c^2))*b*c)*a+(b^2-c^2)^3*(b-c)*((b^2-c^2)^2-4*b^2*c^2)) : :

The reciprocal orthologic center of these triangles is X(40).

X(12862) lies on these lines: {494,7160}, {6461,12861}, {6463,9874}, {8000,8211}, {8189,9898}, {8195,12411}, {8202,12464}, {8209,12465}, {8213,12599}, {8215,12777}, {8217,12801}, {8219,12802}, {8221,12856}, {8223,12864}, {10059,11952}, {10075,11954}, {10876,12500}, {11378,12260}, {11395,12139}, {11504,12333}, {11829,12120}, {11841,12200}, {11847,12249}, {11931,12859}, {11933,12860}, {11948,12863}, {11950,12872}, {11956,12874}, {11958,12875}

X(12862) = X(7160)-of-Lucas(-1)-homothetic-triangle

X(12863) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO HUTSON EXTOUCH

Trilinears q*p^5-2*(q^2-1)*p^4+(q^2-3)*q*p^3+(2*q^2-3)*p^2-(q^2-1)*q*p+2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(40).

X(12863) lies on these lines: {3,10075}, {4,12859}, {11,12864}, {12,12599}, {33,12139}, {55,84}, {56,12120}, {497,9874}, {1479,12856}, {1697,5223}, {1837,12777}, {2098,8000}, {2646,12260}, {3057,3488}, {3295,10059}, {3601,9850}, {4294,12249}, {5920,10543}, {10799,12200}, {10833,12411}, {10877,12500}, {10927,12801}, {10928,12802}, {10947,12857}, {10953,12858}, {10965,12874}, {10966,12875}, {11873,12464}, {11874,12465}, {11909,12789}, {11947,12861}, {11948,12862}

X(12863) = X(7160)-of-Mandart-incircle-triangle

X(12864) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MEDIAL TO HUTSON EXTOUCH

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

The reciprocal orthologic center of these triangles is X(40).

X(12864) lies on these lines: {1,12521}, {2,7160}, {3,12411}, {4,12120}, {5,12599}, {8,8000}, {9,946}, {11,12863}, {55,12860}, {56,12859}, {83,12200}, {142,5045}, {427,12139}, {442,3555}, {498,10059}, {499,10075}, {631,12249}, {958,12858}, {1125,6600}, {1145,4002}, {1376,12333}, {1650,12789}, {1656,12872}, {1698,9898}, {2886,6260}, {3090,12612}, {3096,12500}, {3333,9776}, {3889,12537}, {5552,12874}, {5590,12802}, {5591,12801}, {5599,12464}, {5600,12465}, {5795,6849}, {6864,9623}, {8222,12861}, {8223,12862}, {9709,12631}, {10527,12875}, {12670,12854}

X(12864) = midpoint of X(i) and X(j) for these {i,j}: {1,12777}, {3,12856}, {4,12120}, {8,8000}, {1650,12789}, {3555,12692}, {7160,9874}, {12333,12857}, {12521,12731}, {12670,12854}
X(12864) = reflection of X(i) in X(j) for these (i,j): (12260,1125), (12439,5045), (12599,5)
X(12864) = complement of X(7160)
X(12864) = {X(2), X(9874)}-harmonic conjugate of X(7160)

X(12865) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO HUTSON EXTOUCH

Trilinears -2*b*c*((b+c)*a^6-2*(b-c)^2*a^5-(b+c)*(b^2+14*b*c+c^2)*a^4+4*(b^2+5*b*c+c^2)*(b-c)^2*a^3-(b^2-18*b*c+c^2)*(b+c)^3*a^2-2*(b^2-c^2)^2*(b^2+8*b*c+c^2)*a+(b^2-c^2)^3*(b-c))*S+(b+c)*a^10-2*(b^2+c^2)*a^9-(b+c)*(b^2+15*b*c+c^2)*a^8+4*(b^4+c^4+b*c*(3*b^2+b*c+3*c^2))*a^7-2*(b+c)*(b^4+c^4-2*b*c*(5*b^2+7*b*c+5*c^2))*a^6-4*b*c*(3*b^2-b*c+3*c^2)*(b-c)^2*a^5+2*(b+c)*(b^2+6*b*c+c^2)*(b^4+c^4-b*c*(3*b^2-8*b*c+3*c^2))*a^4-4*(b^8+c^8+(3*b^6+3*c^6+7*(b^4+c^4+3*b*c*(b^2+c^2))*b*c)*b*c)*a^3+(b^2-c^2)*(b-c)*(b^6+c^6-(10*b^4+10*c^4+b*c*(33*b^2+28*b*c+33*c^2))*b*c)*a^2+2*(b^2-c^2)^2*(b^4+c^4)*(b^2+6*b*c+c^2)*a-(b^3+c^3)*(b^2-c^2)^4 : :

The reciprocal orthologic center of these triangles is X(3555).

X(12865) lies on these lines: {5920,8239}, {8001,8244}, {8224,12516}, {8225,12521}, {8228,12612}, {8230,12620}, {8231,12658}, {8233,12692}, {8234,12842}, {8237,12846}, {8243,12854}, {8246,12869}, {9789,9804}, {9953,10867}, {10858,12439}, {10885,12537}, {10891,12552}, {11030,12855}, {11042,12853}, {11532,12654}, {11687,12533}, {11996,12873}

X(12866) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON EXTOUCH TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(12867).

X(12866) lies on these lines: {9,10266}, {20,5538}, {65,2475}, {6597,12444}, {11024,12543}

X(12866) = reflection of X(i) in X(j) for these (i,j): (6597,12444), (12682,12660), (12695,12639)

X(12867) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SCHIFFLER TO HUTSON EXTOUCH

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

The reciprocal orthologic center of these triangles is X(12866).

X(12867) lies on the Feuerbach hyperbola and these lines: {7,442}, {30,10429}, {84,3651}, {191,3062}, {210,943}, {758,5665}, {3647,7285}, {4900,9898}, {5556,11684}

X(12868) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd SCHIFFLER TO HUTSON EXTOUCH

Trilinears (-a+b+c)/((b+c)*a^3-(b^2-12*b*c+c^2)*a^2-(b+c)*(b^2+4*b*c+c^2)*a+(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(12632).

X(12868) lies on the Feuerbach hyperbola and these lines: {7,12732}, {90,12756}, {100,5558}, {952,10429}, {1000,4423}, {2802,5665}, {6601,8168}

X(12868) = reflection of X(100) in X(12631)

X(12869) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO HUTSON EXTOUCH

Trilinears 8*(q^2-1)*p^6-8*(2*q^2-1)*q*p^5+(8*q^4-44*q^2+26)*p^4+(44*q^2-15)*q*p^3+(-8*q^4+49*q^2-15)*p^2-(25*q^2+2)*q*p+(q^2-13)*q^2+2*q^3/p : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(3555).

X(12869) lies on these lines: {21,3870}, {846,12658}, {1284,12854}, {4199,12692}, {4220,12516}, {5051,12620}, {5920,8240}, {8001,8245}, {8229,12612}, {8235,12842}, {8238,12846}, {8246,12865}, {8249,12870}, {8425,12873}, {8731,12439}, {9953,10868}, {10892,12552}, {11031,12855}, {11043,12853}, {11533,12654}, {11688,12533}

X(12869) = 2nd-Sharygin-to-1st-Sharygin similarity image of X(13286)
X(12869) = excentral-to-1st-Sharygin similarity image of X(12658)
X(12869) = hexyl-to-1st-Sharygin similarity image of X(12842)
X(12869) = Hutson-intouch-to-1st-Sharygin similarity image of X(5920)

X(12870) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL-MIDARC TO HUTSON EXTOUCH

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=2*((b+c)*a^3-(b-c)^2*a^2-(b+c)*(b^2+6*b*c+c^2)*a+(b^2-c^2)^2)*(a^3-(b+c)*a^2-(b^2+8*b*c+c^2)*a+(b+c)*(b^2+c^2))
G(a,b,c)=-2*(a^3-(b-c)*a^2-(b^2+6*b*c+c^2)*a+(b+c)*(b^2-c^2))*(a^3-(b-c)*a^2-(b^2+8*b*c-c^2)*a+(b^2-c^2)*(b-c))*b
H(a,b,c)=(a+b+c)*(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*b*c*(6*b^2+5*b*c+6*c^2))*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)

The reciprocal orthologic center of these triangles is X(3555).

X(12870) lies on these lines: {1,12871}, {2089,12854}, {5920,8241}, {8001,8089}, {8075,12516}, {8077,12521}, {8078,12658}, {8079,12692}, {8081,12842}, {8085,12612}, {8087,12620}, {8247,12865}, {8249,12869}, {8387,12846}, {8733,12439}, {9793,9804}, {11032,12855}, {11690,12533}, {11888,12537}, {11894,12552}

X(12870) = reflection of X(12871) in X(1)

X(12871) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO HUTSON EXTOUCH

Trilinears 2*(a+b+c)*(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*b*c*(6*b^2+5*b*c+6*c^2))*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)*b*c*sin(A/2)-(a+b-c)*(a-b+c)*(a^3-(b+c)*a^2-(b^2+8*b*c+c^2)*a+(b+c)*(b^2+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) : :

The reciprocal orthologic center of these triangles is X(3555).

X(12871) lies on these lines: {1,12870}, {174,12854}, {258,12658}, {7588,12521}, {8125,12533}, {8351,12853}, {8734,12439}, {9953,11859}, {11033,12855}, {11895,12552}, {11899,12654}

X(12871) = reflection of X(12870) in X(1)

X(12872) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO HUTSON EXTOUCH

Trilinears 2*p^6-3*(2*q^2+1)*p^4+2*(2*q^2+1)*q*p^3+(5*q^2+4)*p^2-4*(q^2+2)*q*p+2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(40).

X(12872) lies on these lines: {3,7091}, {5,9874}, {30,12249}, {381,12599}, {517,9898}, {960,1482}, {999,10075}, {1598,12139}, {1656,12864}, {3295,10059}, {5790,12777}, {7517,12411}, {8000,10247}, {9301,12500}, {9654,12859}, {9669,12860}, {10246,12260}, {10679,12631}, {11842,12200}, {11849,12333}, {11875,12464}, {11876,12465}, {11911,12789}, {11916,12801}, {11917,12802}, {11928,12857}, {11929,12858}, {11949,12861}, {11950,12862}, {12000,12874}, {12001,12875}

X(12872) = reflection of X(i) in X(j) for these (i,j): (3,7160), (9874,5), (12856,12599)
X(12872) = X(7160)-of-X3-ABC-reflections-triangle

X(12873) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO HUTSON EXTOUCH

Trilinears -2*(a+b+c)*(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*b*c*(6*b^2+5*b*c+6*c^2))*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)*b*c*sin(A/2)-(a+b-c)*(a-b+c)*(a^3-(b+c)*a^2-(b^2+8*b*c+c^2)*a+(b+c)*(b^2+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) : :

The reciprocal orthologic center of these triangles is X(3555).

X(12873) lies on these lines: {174,12854}, {5920,11924}, {7587,12521}, {8001,8423}, {8083,12855}, {8126,12533}, {8382,12620}, {8389,12846}, {8425,12869}, {8729,12439}, {9804,11891}, {9953,11860}, {11535,12654}, {11890,12537}, {11896,12552}, {11996,12865}

X(12874) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO HUTSON EXTOUCH

Trilinears 2*p^6-2*q*p^5-(2*q^2+7)*p^4+2*(q^2+5)*q*p^3-(q^2-11)*p^2-2*(q^2+7)*q*p-1+q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(40).

X(12874) lies on these lines: {1,5920}, {12,12857}, {5552,12864}, {9874,10528}, {10531,12599}, {10803,12200}, {10805,12249}, {10834,12411}, {10878,12500}, {10915,12777}, {10929,12801}, {10930,12802}, {10942,12856}, {10955,12858}, {10956,12859}, {10958,12860}, {10965,12863}, {11248,12120}, {11400,12139}, {11509,12333}, {11881,12464}, {11882,12465}, {11914,12789}, {11955,12861}, {11956,12862}, {12000,12872}

X(12874) = reflection of X(7160) in X(10059)
X(12874) = X(7160)-of-inner-Yff-tangents-triangle
X(12874) = {X(7160),X(8000)}-harmonic conjugate of X(12875)

X(12875) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO HUTSON EXTOUCH

Trilinears 2*p^6-2*q*p^5-(2*q^2+5)*p^4+2*(q^2+1)*q*p^3+(5*q^2+1)*p^2-2*(q^2-1)*q*p+1-q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(40).

X(12875) lies on these lines: {1,5920}, {2,10941}, {9,6675}, {11,12858}, {56,12687}, {57,6833}, {938,10936}, {1210,12116}, {1445,12704}, {6734,12620}, {6878,11048}, {9874,10529}, {10527,12864}, {10532,12599}, {10804,12200}, {10806,12249}, {10835,12411}, {10879,12500}, {10916,12777}, {10931,12801}, {10932,12802}, {10943,12856}, {10949,12857}, {10957,12859}, {10959,12860}, {10966,12863}, {11249,12120}, {11401,12139}, {11510,12333}, {11883,12464}, {11884,12465}, {11915,12789}, {11957,12861}, {11958,12862}, {12001,12872}

X(12875) = reflection of X(7160) in X(10075)
X(12875) = X(7160)-of-outer-Yff-tangents-triangle
X(12875) = {X(7160),X(8000)}-harmonic conjugate of X(12874)

X(12876) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON INTOUCH TO MANDART-EXCIRCLES

Trilinears a^6-(b+c)*a^5-2*(b^2-b*c+c^2)*a^4+6*b*c*(b+c)*a^3-(b^4+14*b^2*c^2+c^4)*a^2+(b+c)*(b^4+c^4-2*b*c*(3*b^2-7*b*c+3*c^2))*a+(b^2-c^2)*(b-c)*(2*b^3+2*c^3) : :

The reciprocal orthologic center of these triangles is X(3555).

X(12876) lies on these lines: {1,12553}, {11,12621}, {12,12613}, {34,517}, {55,12522}, {56,12517}, {145,12534}, {522,10912}, {950,12693}, {1482,4292}, {1697,12659}, {3601,12442}, {4313,12538}, {7962,12655}, {8236,12847}, {8390,12878}, {8392,12883}, {9785,12542}, {10866,12449}

X(12876) = midpoint of X(145) and X(12534)
X(12876) = reflection of X(12912) in X(1)

X(12877) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON INTOUCH TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(21).

X(12877) lies on these lines: {1,5180}, {8,6597}, {11,21}, {12,12615}, {35,2475}, {55,12524}, {145,12535}, {950,12695}, {1697,12660}, {3057,12682}, {3601,12444}, {3648,4018}, {4294,10043}, {4313,12540}, {5441,12758}, {6872,10051}, {7962,12657}, {8236,12850}, {8390,12882}, {8392,12887}, {9785,12543}, {10866,12451}

X(12877) = midpoint of X(145) and X(12535)
X(12877) = reflection of X(12913) in X(1)

X(12878) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-HUTSON TO MANDART-EXCIRCLES

Trilinears 4*(a^3-(b+c)^2*a+2*b*c*(b+c))*(a^3+(b+c)*a^2+(b^2-4*b*c+c^2)*a+(b^2-c^2)*(b-c))*a*b*c*sin(A/2)-2*(a^3+(b-c)*a^2+(b^2-4*b*c-c^2)*a+(b+c)*(b^2+c^2))*(a^5-(b-c)*a^4-2*b*(b+c)*a^3+4*b^2*c*a^2+(b^2-c^2)*(b-c)^2*a+(b^2-c^2)^2*(b-c))*c*sin(B/2)-2*(a^3-(b-c)*a^2-(b^2+4*b*c-c^2)*a+(b+c)*(b^2+c^2))*(a^5+(b-c)*a^4-2*c*(b+c)*a^3+4*b*c^2*a^2-(b^2-c^2)*(b-c)^2*a-(b^2-c^2)^2*(b-c))*b*sin(C/2)-a^9+(b+c)*a^8+2*(b^2+c^2)*a^7-2*(b+c)*(b^2+5*b*c+c^2)*a^6-8*b*c*(b^2-5*b*c+c^2)*a^5+2*b*c*(3*b+c)*(b+3*c)*(b+c)*a^4-2*(b^6+c^6+b^2*c^2*(11*b^2+40*b*c+11*c^2))*a^3+2*(b+c)*(b^6+c^6+(b^4+c^4-3*b*c*(3*b-c)*(b-3*c))*b*c)*a^2+(b^2-c^2)^2*(b-c)^2*(b^2+10*b*c+c^2)*a-(b^4-c^4)*(b^2-c^2)^2*(b-c) : :

The reciprocal orthologic center of these triangles is X(3555).

X(12878) lies on these lines: {363,12659}, {5934,12693}, {8107,12517}, {8109,12522}, {8111,12843}, {8140,12883}, {8377,12613}, {8380,12621}, {8385,12847}, {8390,12876}, {9783,12542}, {11527,12655}, {11685,12534}, {11854,12442}, {11856,12449}, {11886,12538}, {11892,12553}

X(12879) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-HUTSON TO MIDARC

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

The reciprocal orthologic center of these triangles is X(1).

X(12879) lies on these lines: {1,6724}, {40,164}, {167,8140}, {177,8113}, {1130,11923}, {3577,11528}, {5571,11026}, {5934,11523}, {7670,8385}, {8107,12518}, {8390,8422}, {9783,9807}, {11685,11691}, {11856,12450}, {11886,12539}, {11892,12554}

X(12879) = reflection of X(i) in X(j) for these (i,j): (164,188), (12884,167)
X(12879) = orthologic center of these triangles: inner-Hutson to 2nd midarc
X(12879) = X(1)-of-inner-Hutson-triangle
X(12879) = excentral-to-inner-Hutson similarity image of X(164)
X(12879) = {X(6732),X(8133)}-harmonic conjugate of X(1)

X(12880) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-HUTSON TO MIXTILINEAR

Trilinears -4*(a+b-c)*(a-b+c)*a*b*c*sin(A/2)-2*(a+b-c)*(-a+b+c)*(a^2-2*a*b-b^2+2*b*c-c^2)*c*sin(B/2)-2*(-a+b+c)*(a-b+c)*(a^2-2*a*c-b^2+2*b*c-c^2)*b*sin(C/2)+a^5-3*(b+c)*a^4+2*(b^2+8*b*c+c^2)*a^3+2*(b+c)*(b^2-6*b*c+c^2)*a^2-3*(b^2-c^2)^2*a+(b^2-c^2)*(b-c)^3 : :

The reciprocal orthologic center of these triangles is X(1).

X(12880) lies on these lines: {57,363}, {329,5934}, {999,8109}, {3820,8380}, {6244,8107}, {6282,8111}, {7956,8377}, {7962,8390}, {7994,8140}, {8101,8133}, {8385,12848}, {9954,11856}, {9965,11886}, {11892,12555}

X(12880) = reflection of X(12885) in X(7994)
X(12880) = X(25)-of-inner-Hutson-triangle
X(12880) = excentral-to-inner-Hutson similarity image of X(57)

X(12881) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-HUTSON TO ANTLIA

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=2*(a^2+(b-c)^2)*((b+c)*a^6-2*(b^2+4*b*c+c^2)*a^5+(b+c)*(b^2+16*b*c+c^2)*a^4-12*b*c*(b+c)^2*a^3-(b^2-6*b*c+c^2)*(b+c)^3*a^2+2*(b^2-c^2)^2*(b-c)^2*a-(b^4-c^4)*(b-c)^3)
G(a,b,c)=2*b*(a^2-2*a*c+b^2+c^2)*(a^6-2*(b+3*c)*a^5+(b^2+12*b*c+c^2)*a^4-4*c*(2*b^2-2*b*c+3*c^2)*a^3-(b-c)*(b^3+11*c^3-7*b*c*(b-c))*a^2+2*(b^3+c^3-b*c*(3*b+c))*(b-c)^2*a+(b^2-c^2)*(b-c)^2*(-3*c^2+2*b*c-b^2))
H(a,b,c)=a^9+(b+c)*a^8-4*(b^2+7*b*c+c^2)*a^7+4*(b+c)*(b^2+15*b*c+c^2)*a^6-2*(3*b^4+3*c^4+2*b*c*(21*b^2+16*b*c+21*c^2))*a^5+2*(b+c)*(b^4+c^4+2*b*c*(27*b^2-26*b*c+27*c^2))*a^4+4*(b^6+c^6-(21*b^4+21*c^4-b*c*(b^2+14*b*c+c^2))*b*c)*a^3-4*(b^3-c^3)*(b^2-c^2)*(b^2-8*b*c+c^2)*a^2+(5*b^6+5*c^6-3*(6*b^4+6*c^4+b*c*(3*b^2-4*b*c+3*c^2))*b*c)*(b-c)^2*a-(b^4-c^4)*(b-c)^3*(3*b^2-8*b*c+3*c^2)

The reciprocal orthologic center of these triangles is X(1)

X(12881) lies on these lines: {168,12396}, {5935,12397}, {7707,12406}, {8108,12387}, {8110,12388}, {8112,12398}, {8114,12402}, {8140,12404}, {8378,12393}, {8381,12394}, {8386,12399}, {8392,12400}, {9787,12391}, {11027,12403}, {11040,12401}, {11528,12395}, {11686,12389}, {11855,12385}, {11857,12386}, {11887,12390}, {11893,12392}, {11926,12405}

X(12882) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-HUTSON TO 1st SCHIFFLER

Trilinears 4*(a^3+(b+c)*a^2-(b^2-3*b*c+c^2)*a-(b^2-c^2)*(b-c))*(a^4-(2*b^2+3*b*c+2*c^2)*a^2+b*c*(b+c)*a+(b^2-c^2)^2)*b*c*sin(A/2)-2*(-a+b+c)*(a^3+(3*b-c)*a^2+(3*b^2-b*c-c^2)*a+(b^2-c^2)*(b-c))*(a^4-(2*b^2-b*c+2*c^2)*a^2-b*c*(3*b-c)*a+(b^2-c^2)^2)*c*sin(B/2)-2*(-a+b+c)*(a^3-(b-3*c)*a^2-(b^2+b*c-3*c^2)*a+(b^2-c^2)*(b-c))*(a^4-(2*b^2-b*c+2*c^2)*a^2+b*c*(b-3*c)*a+(b^2-c^2)^2)*b*sin(C/2)+a^9-3*(b+c)*a^8-10*b*c*a^7+8*(b+c)*(b^2+c^2)*a^6-(6*b^4+6*c^4-b*c*(26*b^2+9*b*c+26*c^2))*a^5-(b+c)*(6*b^4+6*c^4-b*c*(4*b^2-5*b*c+4*c^2))*a^4+(2*b^2-7*b*c+2*c^2)*(4*b^4+4*c^4+b*c*(3*b^2+2*b*c+3*c^2))*a^3-(b+c)*(8*b^4+8*c^4-b*c*(9*b^2-10*b*c+9*c^2))*b*c*a^2-3*(b^4-c^4)*(b^2-c^2)*(b-c)^2*a+(b^2-c^2)^3*(b-c)*(b^2+6*b*c+c^2) : :

The reciprocal orthologic center of these triangles is X(21).

X(12882) lies on these lines: {363,12660}, {5934,12695}, {8107,12519}, {8109,12524}, {8111,12845}, {8140,12887}, {8377,12615}, {8380,12623}, {8385,12850}, {8390,12877}, {9783,12543}, {11527,12657}, {11685,12535}, {11854,12444}, {11856,12451}, {11886,12540}, {11892,12557}

X(12883) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-HUTSON TO MANDART-EXCIRCLES

Trilinears 4*a*b*c*(a^3-(b+c)^2*a+2*b*c*(b+c))*(a^3+(b+c)*a^2+(b^2-4*b*c+c^2)*a+(b^2-c^2)*(b-c))*sin(A/2)-2*c*(a^3+(b-c)*a^2+(b^2-4*b*c-c^2)*a+(b+c)*(b^2+c^2))*(a^5-(b-c)*a^4-2*b*(b+c)*a^3+4*a^2*b^2*c+(b^2-c^2)*(b-c)^2*a+(b^2-c^2)^2*(b-c))*sin(B/2)-2*b*(a^3-(b-c)*a^2-(b^2+4*b*c-c^2)*a+(b+c)*(b^2+c^2))*(a^5+(b-c)*a^4-2*c*(b+c)*a^3+4*a^2*b*c^2-(b^2-c^2)*(b-c)^2*a-(b^2-c^2)^2*(b-c))*sin(C/2)+a^9-(b+c)*a^8-2*(b^2+c^2)*a^7+2*(b+c)*(b^2+5*b*c+c^2)*a^6+8*b*c*(b^2-5*b*c+c^2)*a^5-2*b*c*(3*b+c)*(b+3*c)*(b+c)*a^4+2*(b^6+c^6+b^2*c^2*(11*b^2+40*b*c+11*c^2))*a^3-2*(b+c)*(b^6+c^6+(b^4+c^4-3*b*c*(3*b-c)*(b-3*c))*b*c)*a^2-(b^2-c^2)^2*(b-c)^2*(b^2+10*b*c+c^2)*a+(b^4-c^4)*(b^2-c^2)^2*(b-c) : :

The reciprocal orthologic center of these triangles is X(3555).

X(12883) lies on these lines: {5935,12693}, {8108,12517}, {8110,12522}, {8112,12843}, {8140,12878}, {8378,12613}, {8381,12621}, {8386,12847}, {8392,12876}, {11528,12655}, {11686,12534}, {11855,12442}, {11857,12449}, {11887,12538}, {11893,12553}

X(12884) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-HUTSON TO MIDARC

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

The reciprocal orthologic center of these triangles is X(1).

X(12884) lies on these lines: {1,8135}, {9,164}, {177,8114}, {5571,11027}, {7670,8386}, {7982,9837}, {8108,12518}, {8110,12523}, {8140,10233}, {8378,12614}, {8381,12622}, {8392,8422}, {9787,9807}, {11686,11691}, {11855,12443}, {11857,12450}, {11887,12539}, {11893,12554}

X(12884) = reflection of X(i) in X(j) for these (i,j): (11528,9837), (12879,167)
X(12884) = orthologic center of these triangles: outer-Hutson to 2nd midarc
X(12884) = X(1)-of-outer-Hutson-triangle
X(12884) = excentral-to-outer-Hutson similarity image of X(164)
X(12884) = {X(8135),X(8138)}-harmonic conjugate of X(1)

X(12885) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-HUTSON TO MIXTILINEAR

Trilinears -4*(a+b-c)*(a-b+c)*a*b*c*sin(A/2)-2*(a+b-c)*(-a+b+c)*(a^2-2*a*b-b^2+2*b*c-c^2)*c*sin(B/2)-2*(-a+b+c)*(a-b+c)*(a^2-2*a*c-b^2+2*b*c-c^2)*b*sin(C/2)-a^5+3*(b+c)*a^4-2*(b^2+8*b*c+c^2)*a^3-2*(b+c)*(b^2-6*b*c+c^2)*a^2+3*(b^2-c^2)^2*a-(b^2-c^2)*(b-c)^3 : :

The reciprocal orthologic center of these triangles is X(1).

X(12885) lies on these lines: {329,5935}, {999,8110}, {3820,8381}, {6244,8108}, {6282,8112}, {7956,8378}, {7994,8140}, {8101,8135}, {8102,8138}, {8386,12848}, {9954,11857}, {9965,11887}, {11893,12555}

X(12885) = reflection of X(12880) in X(7994)
X(12885) = X(25)-of-outer-Hutson-triangle
X(12885) = excentral-to-outer-Hutson similarity image of X(57)

X(12886) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-HUTSON TO ANTLIA

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)-H(a,b,c)
where F, G, H are given at X(12881)

The reciprocal orthologic center of these triangles is X(1)

X(12886) lies on these lines: {363,12396}, {5934,12397}, {8107,12387}, {8109,12388}, {8111,12398}, {8113,12402}, {8140,12404}, {8377,12393}, {8380,12394}, {8385,12399}, {8390,12400}, {8391,12405}, {9783,12391}, {11026,12403}, {11039,12401}, {11527,12395}, {11685,12389}, {11854,12385}, {11856,12386}, {11886,12390}, {11892,12392}, {11923,12406}

X(12887) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-HUTSON TO 1st SCHIFFLER

Trilinears 4*(a^3+(b+c)*a^2-(b^2-3*b*c+c^2)*a-(b^2-c^2)*(b-c))*(a^4-(2*b^2+3*b*c+2*c^2)*a^2+b*c*(b+c)*a+(b^2-c^2)^2)*b*c*sin(A/2)-2*(-a+b+c)*(a^3+(3*b-c)*a^2+(3*b^2-b*c-c^2)*a+(b^2-c^2)*(b-c))*(a^4-(2*b^2-b*c+2*c^2)*a^2-b*c*(3*b-c)*a+(b^2-c^2)^2)*c*sin(B/2)-2*(-a+b+c)*(a^3-(b-3*c)*a^2-(b^2+b*c-3*c^2)*a+(b^2-c^2)*(b-c))*(a^4-(2*b^2-b*c+2*c^2)*a^2+b*c*(b-3*c)*a+(b^2-c^2)^2)*b*sin(C/2)-a^9+3*(b+c)*a^8+10*a^7*b*c-8*(b+c)*(b^2+c^2)*a^6+(6*b^4+6*c^4-b*c*(26*b^2+9*b*c+26*c^2))*a^5+(b+c)*(6*b^4+6*c^4-b*c*(4*b^2-5*b*c+4*c^2))*a^4-(2*b^2-7*b*c+2*c^2)*(4*b^4+4*c^4+b*c*(3*b^2+2*b*c+3*c^2))*a^3+(b+c)*(8*b^4+8*c^4-b*c*(9*b^2-10*b*c+9*c^2))*b*c*a^2+3*(b^4-c^4)*(b^2-c^2)*(b-c)^2*a-(b^2-c^2)^3*(b-c)*(b^2+6*b*c+c^2) : :

The reciprocal orthologic center of these triangles is X(21).

X(12887) lies on these lines: {5935,12695}, {8108,12519}, {8110,12524}, {8112,12845}, {8140,12882}, {8378,12615}, {8381,12623}, {8386,12850}, {8392,12877}, {11528,12657}, {11686,12535}, {11855,12444}, {11857,12451}, {11887,12540}, {11893,12557}

X(12888) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INTANGENTS TO 1st HYACINTH

Trilinears cos(A)*((4*cos(A)+4*cos(2*A)+5)*cos(B-C)-(2*cos(A)+1)*cos(2*(B-C))-2*cos(2*A)-cos(3*A)-6*cos(A)-2) : :

The reciprocal orthologic center of these triangles is X(10112).

X(12888) lies on the intangents circle and these lines: {1,12896}, {33,113}, {34,12295}, {35,12893}, {36,12901}, {55,2931}, {56,12302}, {74,3100}, {110,6198}, {125,1062}, {146,9539}, {399,3157}, {497,12319}, {1040,6699}, {1250,10664}, {1870,10733}, {2066,12891}, {2948,9577}, {3031,9551}, {3043,9637}, {3047,9638}, {3295,12310}, {3448,9538}, {4354,10065}, {5414,12892}, {5504,10091}, {5663,6285}, {7071,12168}, {8540,12596}, {9576,9904}, {9627,12903}, {9628,12373}, {9629,12374}, {9630,12904}, {9632,10819}, {9633,10817}, {9641,10620}, {9645,10117}, {9817,12900}, {10638,10663}, {11429,12228}, {11436,12236}, {11446,12273}, {11461,12284}

X(12888) = reflection of X(i) in X(j) for these (i,j): (10118,8144), (12661,2931)
X(12888) = antipode of X(10118) in intangents circle

X(12889) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO 1st HYACINTH

Trilinears 64*p^7*(p-q)-32*p^5*(p-2*q)+4*(16*q^2-19)*p^4-4*(8*q^2-3)*q*p^3+4*(8*q^4-21*q^2+15)*p^2+2*(10*q^2-9)*q*p-(4*q^2-3)^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(6102).

X(12889) lies on these lines: {11,265}, {12,12905}, {30,12371}, {110,355}, {542,12586}, {1376,1511}, {2771,7984}, {3434,12383}, {3448,10785}, {5663,12114}, {10088,10944}, {10113,10893}, {10523,12903}, {10794,12201}, {10826,12407}, {10829,12412}, {10871,12501}, {10914,12778}, {10919,12803}, {10920,12804}, {10945,12894}, {10946,12895}, {10947,12896}, {10948,12904}, {10949,12906}, {11373,12261}, {11390,12140}, {11826,12121}, {11903,12790}, {11928,12902}

X(12889) = reflection of X(i) in X(j) for these (i,j): (12334,1511), (12890,110)

X(12890) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO 1st HYACINTH

Trilinears 64*p^7*(p-q)-32*p^5*(5*p-4*q)-4*(16*q^2-37)*p^4+4*(8*q^2-21)*q*p^3-4*(8*q^4-19*q^2+15)*p^2-6*(2*q^2-3)*q*p+(4*q^2-3)^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(6102).

X(12890) lies on these lines: {11,12906}, {12,265}, {30,12372}, {72,74}, {110,355}, {542,12587}, {958,1511}, {3436,12383}, {3448,10786}, {5663,11500}, {6253,7728}, {10091,10950}, {10113,10894}, {10523,12904}, {10795,12201}, {10827,12407}, {10830,12412}, {10872,12501}, {10921,12803}, {10922,12804}, {10951,12894}, {10952,12895}, {10953,12896}, {10954,12903}, {10955,12905}, {11374,12261}, {11391,12140}, {11827,12121}, {11904,12790}, {11929,12902}

X(12890) = reflection of X(12889) in X(110)
X(12890) = X(265)-of-outer-Johnson-triangle

X(12891) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS TO 1st HYACINTH

Barycentrics SA*(SB+SC)*((-9*R^2+2*SW)*S^2+S*(2*SW^2-S^2-SA^2+18*R^4+(-12*SW+3*SA)*R^2)-SA*(9*R^2-2*SW)*(SA-SW)) : :

The reciprocal orthologic center of these triangles is X(10112).

X(12891) lies on these lines: {6,1511}, {74,11417}, {110,10666}, {113,5412}, {125,10897}, {265,6413}, {372,12893}, {1151,12302}, {2066,12888}, {3068,12319}, {3311,12310}, {5410,12168}, {5415,12661}, {5663,11265}, {6699,11513}, {10961,12900}, {11447,12273}, {11462,12284}, {11473,12295}

X(12891) = {X(6),X(2931)}-harmonic conjugate of X(12892)

X(12892) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS TO 1st HYACINTH

Barycentrics SA*(SB+SC)*((-9*R^2+2*SW)*S^2-S*(2*SW^2-S^2-SA^2+18*R^4+(-12*SW+3*SA)*R^2)-SA*(9*R^2-2*SW)*(SA-SW)) : :

The reciprocal orthologic center of these triangles is X(10112).

X(12892) lies on these lines: {6,1511}, {74,11418}, {110,10665}, {113,5413}, {125,10898}, {265,6414}, {371,12893}, {1152,12302}, {3069,12319}, {3312,12310}, {5411,12168}, {5414,12888}, {5416,12661}, {5663,11266}, {6396,12901}, {6699,11514}, {10963,12900}, {11448,12273}, {11463,12284}, {11474,12295}

X(12892) = {X(6),X(2931)}-harmonic conjugate of X(12891)

X(12893) = ORTHOLOGIC CENTER OF THESE TRIANGLES: KOSNITA TO 1st HYACINTH

Barycentrics SA*(SB+SC)*(S^2-54*R^4-(3*SA-29*SW)*R^2-4*SW^2+SA^2) : :
X(12893) = 3*X(3)-X(12302) = 3*X(3)+X(12310) = 3*X(2931)+X(12302) = 3*X(2931)-X(12310) = 2*X(2931)+X(12901)

The reciprocal orthologic center of these triangles is X(10112).

X(12893) lies on these lines: {3,125}, {15,10664}, {16,10663}, {23,10721}, {24,113}, {26,2777}, {35,12888}, {54,5504}, {68,5963}, {74,7488}, {110,186}, {371,12892}, {372,12891}, {378,12295}, {389,11536}, {399,12163}, {541,10117}, {549,11804}, {575,12596}, {578,12236}, {631,12319}, {1147,1511}, {1658,5663}, {2070,7728}, {2935,7387}, {3043,5889}, {3047,11464}, {3448,10298}, {3515,12168}, {3520,10733}, {3564,12584}, {5972,6644}, {6642,12900}, {6723,7514}, {7502,8717}, {7526,7687}, {7556,12244}, {8723,9517}, {8998,9682}, {9590,12368}, {9932,10114}, {10821,12235}, {10902,12661}, {11430,11800}, {11449,12273}

X(12893) = midpoint of X(i) and X(j) for these {i,j}: {3,2931}, {68,12383}, {399,12163}, {2935,7387}, {12302,12310}
X(12893) = reflection of X(i) in X(j) for these (i,j): (265,5449), (1147,1511), (5504,12038), (12596,575), (12901,3)
X(12893) = anticomplement of X(33547)
X(12893) = {X(3), X(12310)}-harmonic conjugate of X(12302)

X(12894) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO 1st HYACINTH

Barycentrics (-6*(-SW^2+3*SA^2)*R^2+(4*SA^2+SA*SW-SW^2)*SW)*S^2-2*S*(S^2*R^2*(5*SA-13*SW+12*R^2)-2*(SB+SC)*(6*(3*R^2-SW)*R^2*SA-S^2*SW))+3*(3*R^2-SW)*(SB+SC)*SA*SW^2 : :

The reciprocal orthologic center of these triangles is X(6102).

X(12894) lies on these lines: {30,12377}, {110,8220}, {265,493}, {542,12590}, {1511,8222}, {2771,12741}, {3448,11846}, {5663,9838}, {6461,12895}, {6462,12383}, {8188,12407}, {8194,12412}, {8210,12898}, {8212,10113}, {8214,12778}, {8216,12803}, {8218,12804}, {10088,11930}, {10091,11932}, {10875,12501}, {10945,12889}, {11377,12261}, {11394,12140}, {11503,12334}, {11828,12121}, {11840,12201}, {11907,12790}, {11947,12896}, {11949,12902}, {11951,12903}, {11953,12904}, {11955,12905}, {11957,12906}

X(12894) = X(265)-of-Lucas-homothetic-triangle

X(12895) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO 1st HYACINTH

Barycentrics (-6*(-SW^2+3*SA^2)*R^2+(4*SA^2+SA*SW-SW^2)*SW)*S^2+2*S*(S^2*R^2*(5*SA-13*SW+12*R^2)-2*(SB+SC)*(6*(3*R^2-SW)*R^2*SA-S^2*SW))+3*(3*R^2-SW)*(SB+SC)*SA*SW^2 : :

The reciprocal orthologic center of these triangles is X(6102).

X(12895) lies on these lines: {30,12378}, {110,8221}, {265,494}, {542,12591}, {1511,8223}, {2771,12742}, {3448,11847}, {5663,9839}, {6461,12894}, {6463,12383}, {8189,12407}, {8195,12412}, {8211,12898}, {8213,10113}, {8215,12778}, {8217,12803}, {8219,12804}, {10088,11931}, {10091,11933}, {10946,12889}, {11378,12261}, {11395,12140}, {11504,12334}, {11829,12121}, {11841,12201}, {11908,12790}, {11948,12896}, {11950,12902}, {11952,12903}, {11954,12904}, {11956,12905}, {11958,12906}

X(12895) = X(265)-of-Lucas(-1)-homothetic-triangle

X(12896) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO 1st HYACINTH

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

The reciprocal orthologic center of these triangles is X(6102).

X(12896) lies on these lines: {1,12888}, {3,12904}, {4,10088}, {11,1511}, {12,10113}, {20,10081}, {30,3028}, {33,12140}, {35,125}, {55,265}, {56,12121}, {74,4302}, {79,6062}, {80,4092}, {110,1479}, {113,3583}, {382,12373}, {399,9668}, {497,10091}, {542,3056}, {1478,10733}, {1697,12407}, {1837,12778}, {2098,12898}, {2646,12261}, {2771,12743}, {2777,7355}, {2948,3586}, {3295,12902}, {3448,4294}, {3585,12295}, {5010,6699}, {5663,6284}, {5972,7741}, {7687,7951}, {10058,10778}, {10086,11005}, {10799,12201}, {10833,12412}, {10877,12501}, {10927,12803}, {10928,12804}, {10947,12889}, {10953,12890}, {10965,12905}, {10966,12906}, {11874,12467}, {11909,12790}, {11947,12894}, {11948,12895}

X(12896) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (399,9668,12374), (497,12383,10091), (3295,12902,12903), (3448,4294,10065)

X(12896) = X(265)-of-Mandart-incircle-triangle

X(12897) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st HYACINTH TO MIDHEIGHT

Barycentrics 2*a^10-4*(b^2+c^2)*a^8+(b^4+12*b^2*c^2+c^4)*a^6+(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^4+((b^2-c^2)^2-4*b^2*c^2)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(12897) = 5*X(4)-X(12278) = 3*X(3543)+X(12289) = X(10575)-3*X(12022)

The reciprocal orthologic center of these triangles is X(7687).

X(12897) lies on these lines: {4,110}, {30,143}, {195,7728}, {235,12038}, {378,5449}, {382,1181}, {539,12162}, {568,10937}, {1209,7527}, {1493,2883}, {1533,10619}, {1593,9927}, {1597,12293}, {1885,12421}, {2777,6102}, {2781,12585}, {3146,11750}, {3543,12289}, {5073,11820}, {5097,8550}, {5663,10112}, {6000,10116}, {6699,11250}, {7687,10224}, {7706,10982}, {10575,12022}, {10628,12899}, {11472,12429}

X(12897) = midpoint of X(3146) and X(11750)
X(12897) = reflection of X(10116) in X(12370)
X(12897) = X(1320)-of-1st-Hyacinth-triangle if ABC is acute

X(12898) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR TO 1st HYACINTH

Trilinears 48*p^6-32*q*p^5+16*(3*q^2-5)*p^4-16*(2*q^2-3)*q*p^3+(16*q^4-48*q^2+39)*p^2+2*(8*q^2-9)*q*p-1/2*(4*q^2-3)^2/2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(12898) = 3*X(1)-X(12407) = 3*X(265)-2*X(12407) = 3*X(381)-4*X(11723) = X(3448)-3*X(7967) = 3*X(3655)-2*X(11709) = 3*X(5603)-2*X(10113) = 3*X(5655)-4*X(11699) = 3*X(5655)-2*X(12368) = 3*X(10247)-X(12902)

The reciprocal orthologic center of these triangles is X(6102).

X(12898) lies on these lines: {1,265}, {8,1511}, {30,6742}, {56,12334}, {110,952}, {125,10246}, {145,12383}, {355,11720}, {381,11723}, {515,7728}, {517,12121}, {519,12778}, {542,3242}, {944,5663}, {1483,7979}, {2098,12896}, {2771,3057}, {2777,7973}, {3448,7967}, {3655,11709}, {5597,12467}, {5598,12466}, {5603,10113}, {5604,12804}, {5605,12803}, {5655,11699}, {5790,5972}, {8192,12412}, {8210,12894}, {8211,12895}, {9997,12501}, {10088,10944}, {10091,10950}, {10247,12902}, {10283,11801}, {10800,12201}, {11396,12140}, {11910,12790}

X(12898) = midpoint of X(145) and X(12383)
X(12898) = reflection of X(i) in X(j) for these (i,j): (8,1511), (265,1), (355,11720), (7984,1483), (12368,11699), (12407,12261)

X(12898) = X(265)-of-5th-mixtilinear-triangle
X(12898) = {X(12905),X(12906)}-harmonic conjugate of X(265)

X(12899) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st HYACINTH TO REFLECTION

Trilinears (4*cos(2*A)+cos(4*A))*cos(B-C)-cos(A)*cos(2*(B-C))+(cos(2*A)+1)*cos(3*(B-C))-2*cos(A)-cos(5*A) : :

The reciprocal orthologic center of these triangles is X(399).

X(12899) lies on these lines: {5,11536}, {195,10255}, {389,539}, {567,3519}, {1154,12370}, {1209,1493}, {1353,9977}, {2888,12161}, {6102,11562}, {10115,12236}, {10628,12897}, {11801,11803}

X(12900) = ORTHOLOGIC CENTER OF THESE TRIANGLES: SUBMEDIAL TO 1st HYACINTH

Trilinears (2*cos(2*A)+4)*cos(B-C)-3*cos(A)*cos(2*(B-C))-cos(A)+cos(3*A) : :
X(12900) = 9*X(2)-X(74) = 3*X(2)+X(113) = 15*X(2)+X(146) = 7*X(2)+X(10706) = 3*X(5)-X(7687) = X(74)+3*X(113) = 5*X(74)+3*X(146) = X(74)-3*X(6699) = 7*X(74)+9*X(10706) = X(1511)-3*X(5972) = 5*X(1511)+3*X(10113) = 5*X(5972)+X(10113)

The reciprocal orthologic center of these triangles is X(10112).

X(12900) lies on these lines: {2,74}, {5,1511}, {10,11723}, {110,569}, {125,399}, {140,2777}, {265,5642}, {486,8998}, {542,3589}, {690,6721}, {974,5892}, {1112,1216}, {1209,2914}, {1568,3581}, {1986,5891}, {2771,6667}, {3619,10752}, {3624,12368}, {3819,11807}, {5448,11438}, {5449,12227}, {5663,9729}, {5943,12236}, {6053,10264}, {7978,9780}, {8253,8994}, {9306,12228}, {9813,12596}, {9817,12888}, {9827,11746}, {10170,11557}, {10546,12140}, {10643,10663}, {10644,10664}, {10961,12891}, {10963,12892}, {11230,11735}, {11451,12273}, {11465,12284}

X(12900) = midpoint of X(i) and X(j) for these {i,j}: {5,5972}, {10,11723}, {113,6699}, {1112,1216}, {1511,7687}, {6053,10264}, {11557,12358}
X(12900) = reflection of X(6723) in X(3628)
X(12900) = complement of X(6699)
X(12900) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,113,6699), (5,1511,7687), (5972,7687,1511)

X(12901) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TRINH TO 1st HYACINTH

Trilinears cos(A)*((18*cos(A)+4*cos(3*A))*cos(B-C)-(2*cos(2*A)+3)*cos(2*(B-C))-9*cos(2*A)-cos(4*A)-8) : :
X(12901) = 3*X(3)-X(2931) = 5*X(3)-X(12310) = X(146)-3*X(5654) = 3*X(376)+X(12319) = X(2931)+3*X(12302) = 5*X(2931)-3*X(12310) = 2*X(2931)-3*X(12893)

The reciprocal orthologic center of these triangles is X(10112).

X(12901) lies on these lines: {3,125}, {24,12295}, {36,12888}, {74,323}, {110,3520}, {113,378}, {146,5654}, {155,10620}, {186,10733}, {376,12319}, {511,12596}, {541,2935}, {1092,7723}, {1147,3357}, {1511,4550}, {2777,12084}, {3043,12270}, {3047,6241}, {3098,9976}, {3448,12118}, {5448,7728}, {5972,7526}, {6101,7689}, {6396,12892}, {6644,7687}, {7688,12661}, {9818,12900}, {10117,12085}, {10539,12292}, {10645,10663}, {10646,10664}, {10721,12086}, {11410,12168}, {11430,12228}, {11438,12236}, {11442,12383}, {11454,12273}, {11468,12284}, {11999,12163}

X(12901) = midpoint of X(i) and X(j) for these {i,j}: {3,12302}, {74,5504}, {155,10620}, {2935,12412}, {3448,12118}, {10117,12085}
X(12901) = reflection of X(i) in X(j) for these (i,j): (110,12038), (7689,12041), (7728,5448), (9927,125), (12893,3)
X(12901) = X(104)-of-Trinh-triangle if ABC is acute

X(12902) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO 1st HYACINTH

Trilinears cos(A)*(6*cos(A)*cos(B-C)-2*cos(2*(B-C))-3*cos(2*A)-5/2) : :
X(12902) = 3*X(2)-4*X(11801) = 3*X(3)-4*X(125) = 7*X(3)-8*X(6699) = 3*X(3)-2*X(12121) = 2*X(110)-3*X(381) = 2*X(125)-3*X(265) = 7*X(125)-6*X(6699) = 7*X(265)-4*X(6699) = 3*X(3448)-X(12244) = 3*X(10620)-2*X(12244)

The reciprocal orthologic center of these triangles is X(6102).

X(12902) lies on these lines: {2,11801}, {3,125}, {4,195}, {5,12383}, {20,10264}, {30,3448}, {68,11559}, {74,1657}, {110,381}, {113,3843}, {146,3627}, {382,5663}, {517,12407}, {542,1351}, {568,11562}, {578,11597}, {999,12904}, {1154,12281}, {1511,1656}, {1539,5076}, {1598,12140}, {1699,11699}, {1986,12173}, {2079,10413}, {2771,5691}, {2777,5073}, {2930,3818}, {2937,12289}, {3028,9655}, {3043,7547}, {3091,10272}, {3146,12317}, {3295,12896}, {3521,10116}, {3534,9140}, {3567,11561}, {3845,9143}, {3851,7687}, {5055,5972}, {5071,11694}, {5790,12778}, {5876,12273}, {5898,6288}, {5899,10117}, {6102,12270}, {6243,10628}, {6407,8994}, {7517,12412}, {7723,11898}, {8976,10819}, {9301,12501}, {9654,10088}, {9669,10091}, {10246,12261}, {10247,12898}, {10255,12118}, {10516,12584}, {10778,12773}, {11744,12315}, {11842,12201}, {11849,12334}, {11850,12358}, {11875,12466}, {11876,12467}, {11911,12790}, {11916,12803}, {11917,12804}, {11928,12889}, {11929,12890}, {11949,12894}, {11950,12895}, {12000,12905}, {12001,12906}

X(12902) = midpoint of X(3146) and X(12317)
X(12902) = reflection of X(i) in X(j) for these (i,j): (3,265), (20,10264), (110,10113), (146,3627), (382,10733), (399,4), (1657,74), (2930,3818), (2931,9927), (3534,9140), (5898,6288), (7728,12295), (7731,10263), (9143,3845), (10620,3448), (11562,11800), (12121,125), (12270,6102), (12273,5876), (12308,7728), (12315,11744), (12383,5), (12773,10778)
X(12902) = anticomplement of X(34153)
X(12902) = X(265)-of-X3-ABC-reflections-triangle
X(12902) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (110,10113,381), (125,12121,3), (265,12121,125), (3830,12308,7728), (7728,12295,3830), (11562,11800,568), (12896,12903,3295)

X(12903) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO 1st HYACINTH

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

The reciprocal orthologic center of these triangles is X(6102).

X(12903) lies on the Johnson-Yff-inner-circle and these lines: {1,265}, {5,10091}, {12,110}, {30,10065}, {35,12121}, {56,125}, {67,1469}, {74,7354}, {113,10895}, {146,5229}, {388,3028}, {399,9654}, {495,10066}, {496,11801}, {498,1511}, {542,611}, {1112,11392}, {1317,10778}, {1388,11735}, {1478,5663}, {1479,10113}, {2771,10057}, {2777,10060}, {2854,12588}, {2931,9659}, {2948,9578}, {3023,11005}, {3031,9552}, {3043,9653}, {3047,9652}, {3085,12383}, {3295,12896}, {3585,7727}, {5204,6699}, {5434,9140}, {6284,10733}, {6285,11744}, {7687,10896}, {7732,10923}, {7733,10924}, {7984,10944}, {9579,9904}, {9627,12888}, {9628,10118}, {9646,10819}, {9649,10817}, {9655,10620}, {9658,10117}, {10037,12412}, {10039,12778}, {10040,12803}, {10041,12804}, {10081,10264}, {10082,11804}, {10272,10592}, {10523,12889}, {10801,12201}, {10831,12310}, {10954,12890}, {11375,11720}, {11398,12140}, {11507,12334}, {11912,12790}, {11951,12894}, {11952,12895}

X(12903) = midpoint of X(265) and X(12905)
X(12903) = reflection of X(i) in X(j) for these (i,j): (10088,495), (12373,1478)
X(12903) = antipode of X(12373) in Johnson-Yff-inner-circle
X(12903) = X(265)-of-inner-Yff-triangle
X(12903) = {X(1), X(265)}-harmonic conjugate of X(12904)

X(12904) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO 1st HYACINTH

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

The reciprocal orthologic center of these triangles is X(6102).

X(12904) lies on the Johnson-Yff-outer-circle and these lines: {1,265}, {3,12896}, {4,3028}, {5,10088}, {11,110}, {30,10081}, {36,12121}, {55,125}, {67,3056}, {74,6284}, {113,10896}, {146,5225}, {399,9669}, {495,11801}, {496,10082}, {499,1511}, {542,613}, {999,12902}, {1112,11393}, {1478,10113}, {1479,5663}, {1737,12778}, {1898,2771}, {2777,10076}, {2854,12589}, {2931,9672}, {2948,9581}, {3027,11005}, {3031,9555}, {3043,9666}, {3047,9667}, {3058,9140}, {3086,12383}, {3583,7728}, {4857,7727}, {5217,6699}, {5504,12428}, {7354,10733}, {7355,11744}, {7687,10895}, {7732,10925}, {7733,10926}, {7743,11699}, {7984,10950}, {9580,9904}, {9629,10118}, {9630,12888}, {9661,10819}, {9662,10817}, {9668,10620}, {9673,10117}, {10046,12412}, {10047,12501}, {10048,12803}, {10049,12804}, {10065,10264}, {10066,11804}, {10272,10593}, {10523,12890}, {10802,12201}, {10832,12310}, {10948,12889}, {11006,12354}, {11376,11720}, {11399,12140}, {11508,12334}, {11879,12466}, {11880,12467}, {11913,12790}, {11953,12894}, {11954,12895}

X(12904) = midpoint of X(265) and X(12906)
X(12904) = reflection of X(i) in X(j) for these (i,j): (10091,496), (12374,1479)
X(12904) = antipode of X(12374) in Johnson-Yff-outer-circle
X(12904) = X(265)-of-outer-Yff-triangle
X(12904) = {X(1), X(265)}-harmonic conjugate of X(12903)

X(12905) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO 1st HYACINTH

Trilinears 64*p^8-64*q*p^7+32*(2*q^2-1)*p^6-64*(q^2-1)*q*p^5+4*(16*q^4-8*q^2-19)*p^4-4*(16*q^4-16*q^2-3)*q*p^3-12*(4*q^2-5)*p^2+2*(16*q^4-8*q^2-9)*q*p-(4*q^2-3)^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(6102).

X(12905) lies on these lines: {1,265}, {12,12889}, {30,12381}, {110,10942}, {542,12594}, {1511,5552}, {2771,12749}, {3448,10805}, {5663,12115}, {6256,7728}, {10088,10956}, {10091,10958}, {10113,10531}, {10528,12383}, {10803,12201}, {10834,12412}, {10878,12501}, {10915,12778}, {10929,12803}, {10930,12804}, {10955,12890}, {10965,12896}, {11248,12121}, {11400,12140}, {11509,12334}, {11882,12467}, {11914,12790}, {11955,12894}, {11956,12895}, {12000,12902}

X(12905) = reflection of X(265) in X(12903)
X(12905) = X(265)-of-inner-Yff-tangents-triangle
X(12905) = {X(265),X(12898)}-harmonic conjugate of X(12906)

X(12906) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO 1st HYACINTH

Trilinears 64*p^8-64*q*p^7+32*(2*q^2-7)*p^6-64*(q^2-3)*q*p^5+4*(16*q^4-56*q^2+61)*p^4-4*(16*q^4-48*q^2+45)*q*p^3-16*(4*q^4-9*q^2+6)*p^2+2*(16*q^4-40*q^2+27)*q*p+(4*q^2-3)^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(6102).

X(12906) lies on these lines: {1,265}, {11,12890}, {30,12382}, {110,10943}, {542,12595}, {1511,10527}, {2771,12374}, {3448,10806}, {5663,12116}, {10088,10957}, {10091,10959}, {10113,10532}, {10529,12383}, {10804,12201}, {10835,12412}, {10879,12501}, {10916,12778}, {10931,12803}, {10932,12804}, {10949,12889}, {10966,12896}, {11249,12121}, {11401,12140}, {11510,12334}, {11883,12466}, {11884,12467}, {11915,12790}, {11957,12894}, {11958,12895}, {12001,12902}

X(12906) = reflection of X(265) in X(12904)
X(12906) = X(265)-of-outer-Yff-tangents-triangle
X(12906) = {X(265),X(12898)}-harmonic conjugate of X(12905)

X(12907) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO MANDART-EXCIRCLES

Trilinears 2*a^6+(b+c)*a^5-(b^2+8*b*c+c^2)*a^4-2*(b^4+c^4-b*c*(3*b^2+4*b*c+3*c^2))*a^2-(b+c)*(b^2+c^2)^2*a+(b^2-c^2)^2*(b+c)^2 : :
X(12907) = 5*X(3616)-X(12534) = X(12876)+3*X(12912)

The reciprocal orthologic center of these triangles is X(3555).

X(12907) lies on these lines: {1,12553}, {3,4319}, {495,12621}, {496,12613}, {522,3159}, {942,12442}, {999,12522}, {1385,10386}, {3295,12517}, {3333,12659}, {3487,12693}, {3616,12534}, {5045,12914}, {11035,12449}, {11036,12538}, {11037,12542}, {11038,12847}, {11039,12878}, {11040,12883}, {11529,12655}

X(12907) = midpoint of X(1) and X(12912)
X(12907) = reflection of X(12914) in X(5045)

X(12908) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO MIDARC

Trilinears -2*(-a+b+c)*a*b*c*sin(A/2)-(a-b+c)*(a^2+4*a*b+b^2-c^2)*c*sin(B/2)-(a+b-c)*(a^2+4*a*c-b^2+c^2)*b*sin(C/2) : :
X(12908) = 5*X(3616)-X(11691)

The reciprocal orthologic center of these triangles is X(1).

X(12908) lies on these lines: {1,167}, {164,3333}, {495,12622}, {942,12443}, {3295,12518}, {3487,12694}, {3616,11691}, {5045,5571}, {11035,12450}, {11529,12656}

X(12908) = midpoint of X(1) and X(177)
X(12908) = reflection of X(5571) in X(5045)
X(12908) = X(1)-of-incircle-circles-triangle
X(12908) = X(5)-of-mid-arc-triangle
X(12908) = X(550)-of-2nd-mid-arc-triangle
X(12908) = excentral-to-incircle-circles similarity image of X(164)
X(12908) = orthologic center of these triangles: incircle-circles to 2nd midarc

X(12909) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO 1st SCHIFFLER

Barycentrics 3*(b+c)*a^6-2*(b^2+c^2)*a^5-(b+c)*(7*b^2-2*b*c+7*c^2)*a^4+4*(b^4+c^4-b*c*(b^2+3*b*c+c^2))*a^3+(b+c)*(5*b^4+5*c^4-b*c*(4*b^2+b*c+4*c^2))*a^2-2*(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :
X(12909) = 3*X(1)-X(12877) = 5*X(3616)-X(12535) = X(12877)+3*X(12913)

The reciprocal orthologic center of these triangles is X(21).

X(12909) lies on these lines: {1,5180}, {495,12623}, {496,12615}, {942,3838}, {999,12524}, {3295,12519}, {3296,6597}, {3487,12695}, {3616,12535}, {3649,5083}, {5045,12917}, {11035,12451}, {11036,12540}, {11037,12543}, {11038,12850}, {11039,12882}, {11040,12887}, {11529,12657}, {11551,11604}

X(12909) = midpoint of X(1) and X(12913)
X(12909) = reflection of X(12917) in X(5045)

X(12910) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INTANGENTS TO LUCAS ANTIPODAL

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

The reciprocal orthologic center of these triangles is X(3).

X(12910) lies on these lines: {33,487}, {34,12296}, {55,12662}, {56,12303}, {486,1040}, {497,12320}, {642,9817}, {1038,12123}, {1062,12601}, {3100,12221}, {3295,12311}, {3564,12911}, {6198,12509}, {7071,12169}, {8540,12597}, {11429,12229}, {11436,12237}, {11446,12274}, {11461,12285}

X(12910) = reflection of X(12662) in X(12978)
X(12910) = orthic-to-intangents similarity image of X(487)

X(12911) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INTANGENTS TO LUCAS(-1) ANTIPODAL

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

The reciprocal orthologic center of these triangles is X(3).

X(12911) lies on these lines: {33,488}, {34,12297}, {55,12663}, {56,12304}, {485,1040}, {497,12321}, {641,9817}, {1038,12124}, {1062,12602}, {3100,12222}, {3295,12312}, {3564,12910}, {6198,12510}, {7071,12170}, {8540,12598}, {11429,12230}, {11436,12238}, {11446,12275}, {11461,12286}

X(12911) = reflection of X(12663) in X(12979)
X(12911) = orthic-to-intangents similarity image of X(488)

X(12912) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INTOUCH TO MANDART-EXCIRCLES

Trilinears (a^3-(b+c)^2*a+2*(b+c)*b*c)*(a^3+(b+c)*a^2+(b^2-4*b*c+c^2)*a+(b^2-c^2)*(b-c)) : :
X(12912) = X(12876)-4*X(12907)

The reciprocal orthologic center of these triangles is X(3555).

X(12912) lies on these lines: {1,12553}, {2,12534}, {3,10624}, {7,12538}, {11,12613}, {12,12621}, {40,1736}, {55,12517}, {56,12522}, {57,12442}, {226,12693}, {354,12914}, {522,3913}, {3295,3663}, {3340,12655}, {4329,6361}, {4890,5711}, {4941,5255}, {8113,12878}, {8114,12883}, {8581,12449}

X(12912) = midpoint of X(i) and X(j) for these {i,j}: {7,12847}, {12538,12542}
X(12912) = reflection of X(i) in X(j) for these (i,j): (1,12907), (12659,12442), (12876,1)
X(12912) = complement of X(12534)

X(12913) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INTOUCH TO 1st SCHIFFLER

Barycentrics (a^4-(2*b^2+3*b*c+2*c^2)*a^2+(b+c)*b*c*a+(b^2-c^2)^2)*(a^3+(b+c)*a^2-(b^2-3*b*c+c^2)*a-(b^2-c^2)*(b-c)) : :
X(12913) = 3*X(354)-2*X(12917) = X(12877)-4*X(12909)

The reciprocal orthologic center of these triangles is X(21).

X(12913) lies on these lines: {1,5180}, {2,10044}, {7,6597}, {11,12615}, {12,12623}, {55,12519}, {56,12524}, {57,12444}, {65,2475}, {79,11570}, {226,12695}, {354,12917}, {1484,6595}, {2771,12600}, {3337,11263}, {3340,12657}, {8113,12882}, {8114,12887}, {8581,12451}, {12745,12947}

X(12913) = midpoint of X(i) and X(j) for these {i,j}: {7,12850}, {12540,12543}
X(12913) = reflection of X(i) in X(j) for these (i,j): (1,12909), (12660,12444), (12877,1)
X(12913) = complement of X(12535)

X(12914) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE TO MANDART-EXCIRCLES

Trilinears 8*q*p^7-4*(2*q^2-1)*p^6+8*(q^2-3)*q*p^5-2*(2*q^2+4*q+1)*(2*q^2-4*q+1)*p^4-4*(4*q^2-5)*q*p^3+2*(2*q^4-4*q^2-3)*p^2+(2*q^2+3)*q*p-q^2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(12914) = 3*X(3873)+X(12534)

The reciprocal orthologic center of these triangles is X(3555).

X(12914) lies on these lines: {1,12522}, {57,12517}, {65,12876}, {226,12613}, {354,12912}, {1210,12621}, {3333,12843}, {3873,12534}, {5045,12907}, {5728,12693}, {10580,12542}, {11018,12442}, {11019,12449}, {11020,12538}, {11021,12553}, {11025,12847}, {11026,12878}, {11027,12883}

X(12914) = midpoint of X(65) and X(12876)
X(12914) = reflection of X(12907) in X(5045)

X(12915) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE TO MIXTILINEAR

Trilinears (b+c)*a^4-2*(b^2+b*c+c^2)*a^3+4*b*c*(b+c)*a^2+2*(b^2-b*c+c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)^3 : :
X(12915) = X(57)-3*X(354) = X(329)+3*X(3873) = 3*X(1699)-2*X(10241)

The reciprocal orthologic center of these triangles is X(1).

X(12915) lies on these lines: {1,3}, {7,10569}, {11,10157}, {226,1538}, {329,3873}, {388,5806}, {392,5273}, {496,5777}, {497,971}, {518,3452}, {527,5572}, {938,3421}, {954,4666}, {1210,3820}, {1699,8581}, {2550,10855}, {2810,11028}, {2823,10271}, {3035,3742}, {3086,5044}, {3697,5704}, {3881,6744}, {5083,10391}, {5218,10156}, {5274,5927}, {5691,9850}, {8101,11032}, {9581,9947}, {9856,12053}, {9965,11020}, {10106,12128}, {11025,12848}, {11026,12880}, {11027,12885}, {12005,12710}

X(12915) = midpoint of X(i) and X(j) for these {i,j}: {65,7962}, {3421,3555}
X(12915) = reflection of X(i) in X(j) for these (i,j): (999,5045), (3359,9940), (9954,3452)
X(12915) = complement of X(17658)
X(12915) = incircle-inverse-of-X(5537)
X(12915) = X(25)-of-inverse-in-incircle-triangle
X(12915) = excentral-to-inverse-in-incircle similarity image of X(57)
X(12915) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,354,11018), (1,942,12916), (57,7962,7994), (57,10388,6244), (65,354,10980), (354,5173,942), (3873,10580,5728), (5806,11035,388), (7994,10980,57), (9957,11227,55)

X(12916) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL-MIDARC TO ANTLIA

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : : , where
F(a,b,c)=(-a+b+c)*(a^2+(b-c)^2)*((b+c)*a^6-2*(b^2+4*b*c+c^2)*a^5+(b+c)*(b^2+16*b*c+c^2)*a^4-12*b*c*(b+c)^2*a^3-(b^2-6*b*c+c^2)*(b+c)^3*a^2+2*(b^2-c^2)^2*(b-c)^2*a-(b^4-c^4)*(b-c)^3) : :
G(a,b,c)=(a-b+c)*(a^2-2*a*c+b^2+c^2)*(a^6-2*(b+3*c)*a^5+(b^2+12*b*c+c^2)*a^4-4*c*(2*b^2-2*b*c+3*c^2)*a^3-(b-c)*(b^3+11*c^3-7*b*c*(b-c))*a^2+2*(b^3+c^3-b*c*(3*b+c))*(b-c)^2*a+(b^2-c^2)*(b-c)^2*(-b^2+2*b*c-3*c^2))*b
H(a,b,c)=2*S^2*(a^6-2*(b+c)*a^5+3*(b+c)^2*a^4-4*(b+c)*(b^2+c^2)*a^3+(3*b^4+3*c^4-2*b*c*(2*b^2-7*b*c+2*c^2))*a^2-2*(b^2-c^2)*(b-c)^3*a+(b-c)^2*(b^2+c^2)^2)

The reciprocal orthologic center of these triangles is X(1)

X(12916) lies on these lines: {177,12406}, {2089,12402}, {8078,12396}, {8079,12397}, {8081,12398}, {8085,12393}, {8087,12394}, {8089,12404}, {8241,12400}, {8249,12405}, {8387,12399}, {8733,12385}, {9793,12391}, {11032,12403}, {11044,12401}, {11534,12395}, {11690,12389}, {11858,12386}, {11888,12390}, {11894,12392}

X(12916) = reflection of X(12475) in X(1)

X(12917) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE TO 1st SCHIFFLER

Trilinears 32*q*p^5-16*(4*q^2-3)*p^4+32*(q^2-3)*q*p^3+8*(8*q^2-7)*p^2-2*(8*q^2-29)*q*p+15+q/p : :
where p=sin(A/2), q=cos((B-C)/2)
X(12917) = 3*X(354)-X(12913) = 3*X(3873)+X(12535)

The reciprocal orthologic center of these triangles is X(21).

X(12917) lies on these lines: {1,6597}, {7,10266}, {57,12519}, {65,12877}, {142,12639}, {226,12615}, {354,12913}, {946,12267}, {1210,12623}, {3333,12845}, {3873,12535}, {5045,12909}, {5728,12695}, {8261,12736}, {10580,12543}, {11018,12444}, {11019,12451}, {11020,12540}, {11021,12557}, {11025,12850}, {11026,12882}, {11027,12887}

X(12917) = midpoint of X(65) and X(12877)
X(12917) = reflection of X(12909) in X(5045)

X(12918) = ORTHOLOGIC CENTER OF THESE TRIANGLES: JOHNSON TO 1st ORTHOSYMMEDIAL

Barycentrics (4*R^2-2*SW)*S^4+((3*SA^2+SA*SW-12*SW^2)*R^2-(SA+SW)*(2*SA-3*SW)*SW)*S^2-3*(4*R^2-SW)*(SA-SW)*SA*SW^2 : :
X(12918) = 3*X(3)-4*X(6720) = 2*X(127)-3*X(381) = 3*X(132)-2*X(6720) = 3*X(5587)-X(12408) = 3*X(5886)-2*X(12265) = X(10749)+2*X(12384)

The reciprocal orthologic center of these triangles is X(4).

X(12918) lies on these lines: {1,12945}, {2,12253}, {3,132}, {4,339}, {5,1297}, {30,112}, {66,265}, {127,133}, {355,12925}, {382,2794}, {517,12784}, {1478,6020}, {1479,3320}, {2799,6033}, {2806,10742}, {2825,10739}, {2831,10738}, {2853,10740}, {3627,10735}, {3845,10718}, {5587,12408}, {5886,12265}, {6214,12806}, {6215,12805}, {7728,9517}, {8200,12478}, {8207,12479}, {9518,10741}, {9523,10743}, {9527,10744}, {9532,10747}, {9996,12503}, {10796,12207}, {11499,12340}

X(12918) = midpoint of X(i) and X(j) for these {i,j}: {4,12384}, {12925,12935}
X(12918) = reflection of X(i) in X(j) for these (i,j): (3,132), (1297,5), (10718,3845), (10735,3627), (10749,4)
X(12918) = complement of X(12253)
X(12918) = X(1297)-of-Johnson-triangle
X(12918) = {X(12945),X(12955)}-harmonic conjugate of X(1)

X(12919) = ORTHOLOGIC CENTER OF THESE TRIANGLES: JOHNSON TO 1st SCHIFFLER

Trilinears 16*p^6-16*(3*q^2-1)*p^4+8*(4*q^2-3)*q*p^3+(40*q^2-27)*p^2-2*(16*q^2-13)*q*p-8*q^2+9/2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(12919) = 3*X(5587)-X(12409)

The reciprocal orthologic center of these triangles is X(79).

X(12919) lies on these lines: {1,12947}, {2,12255}, {3,7701}, {4,12146}, {5,10266}, {30,12798}, {355,12745}, {381,12600}, {517,12786}, {952,6595}, {2475,10742}, {3652,12519}, {5587,12409}, {5886,12267}, {6214,12808}, {6215,12807}, {6265,12524}, {8200,12482}, {8207,12483}, {9996,12504}, {10796,12209}, {11499,12342}, {11849,12660}

X(12919) = midpoint of X(12927) and X(12937)
X(12919) = reflection of X(10266) in X(5)
X(12919) = complement of X(12255)
X(12919) = X(10266)-of-Johnson-triangle

X(12920) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO MIDHEIGHT

Trilinears 2*p^7*(p-q)+(8*q^2-7)*p^6+(-q^2+1)*(-p*q^2*(2*p^2-1)*(p-q)+(8*p^4-9*p^2+3)*p*q+6*p^4-1) : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(4).

X(12920) lies on these lines: {11,64}, {30,12422}, {355,5878}, {1376,2883}, {1498,11826}, {2777,12889}, {3434,6225}, {6000,10525}, {6001,12700}, {6247,10893}, {6266,10920}, {6267,10919}, {7355,10947}, {7973,10944}, {9899,10826}, {9914,10829}, {10060,10523}, {10076,10948}, {10785,12250}, {10794,12202}, {10871,12502}, {10914,12779}, {11373,12262}, {11381,11390}, {11865,12468}, {11866,12469}, {11903,12791}

X(12920) = reflection of X(i) in X(j) for these (i,j): (12335,2883), (12930,5878)
X(12920) = X(64)-of-inner-Johnson-triangle
X(12920) = X(13094)-of-outer-Johnson-triangle

X(12921) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO INNER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(3).

X(12921) lies on these lines: {11,14}, {355,5613}, {530,12348}, {531,11235}, {542,12586}, {617,3434}, {619,1376}, {5474,11826}, {5479,10893}, {6269,10920}, {6271,10919}, {6773,10785}, {7974,10944}, {9900,10826}, {9915,10829}, {9981,10871}, {10061,10523}, {10077,10948}, {10794,12204}, {10914,12780}, {11373,11706}, {11390,12141}, {11865,12470}, {11866,12471}, {11903,12792}

X(12921) = reflection of X(12931) in X(5613)
X(12921) = X(14)-of-inner-Johnson-triangle
X(12921) = X(13104)-of-outer-Johnson-triangle

X(12922) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO OUTER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(3).

X(12922) lies on these lines: {11,13}, {355,5617}, {530,11235}, {531,12348}, {542,12586}, {616,3434}, {618,1376}, {5473,11826}, {5478,10893}, {6268,10920}, {6270,10919}, {6770,10785}, {7975,10944}, {9901,10826}, {9916,10829}, {9982,10871}, {10062,10523}, {10078,10948}, {10794,12205}, {10914,12781}, {11373,11705}, {11390,12142}, {11865,12472}, {11866,12473}, {11903,12793}

X(12922) = reflection of X(12932) in X(5617)
X(12922) = X(13)-of-inner-Johnson-triangle
X(12922) = X(13105)-of-outer-Johnson-triangle

X(12923) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO 1st NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(12923) lies on these lines: {11,76}, {39,1376}, {194,3434}, {355,730}, {384,10794}, {511,12114}, {538,11235}, {732,12586}, {2782,10525}, {5969,12348}, {6248,10893}, {6272,10920}, {6273,10919}, {7976,10944}, {9902,10826}, {9917,10829}, {9983,10871}, {10063,10523}, {10079,10948}, {10785,12251}, {10914,12782}, {11257,11826}, {11373,12263}, {11390,12143}, {11865,12474}, {11866,12475}, {11903,12794}

X(12923) = reflection of X(i) in X(j) for these (i,j): (12338,39), (12933,3095)
X(12923) = X(76)-of-inner-Johnson-triangle
X(12923) = X(13109)-of-outer-Johnson-triangle

X(12924) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(12924) lies on these lines: {11,83}, {355,6287}, {732,12586}, {754,11235}, {1376,6292}, {2896,3434}, {6249,10893}, {6274,10920}, {6275,10919}, {7977,10944}, {9903,10826}, {9918,10829}, {10064,10523}, {10080,10948}, {10785,12252}, {10794,12206}, {10914,12783}, {10943,12182}, {11373,12264}, {11390,12144}, {11826,12122}, {11865,12476}, {11866,12477}, {11903,12795}

X(12924) = reflection of X(i) in X(j) for these (i,j): (12339,6292), (12934,6287)
X(12924) = X(83)-of-inner-Johnson-triangle
X(12924) = X(13112)-of-outer-Johnson-triangle

X(12925) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO 1st ORTHOSYMMEDIAL

Barycentrics a^17-(b+c)*a^16-(b-c)^2*a^15+(b+c)*(b^2+c^2)*a^14-(2*b^4+2*c^4-b*c*(2*b^2-3*b*c+2*c^2))*a^13+(b+c)*(2*b^4+2*c^4-b*c*(4*b^2-3*b*c+4*c^2))*a^12+(b^6+c^6-b*c*(6*b^2+5*b*c+6*c^2)*(b-c)^2)*a^11-(b^2-c^2)*(b-c)*(b^4+c^4-2*b*c*(2*b^2+b*c+2*c^2))*a^10+2*(b^6+c^6-(b^4+c^4+b*c*(5*b^2+4*b*c+5*c^2))*b*c)*(b-c)^2*a^9-2*(b^2-c^2)*(b-c)*(b^6+c^6+(b^3-c^3)*(b-c)*b*c)*a^8+(b^2-c^2)^2*(b^6+c^6+3*(2*b^4+2*c^4+b*c*(b^2+c^2))*b*c)*a^7-(b^2-c^2)^2*(b+c)^3*(b^4+c^4+2*b*c*(b^2-b*c+c^2))*a^6-(b^2-c^2)^2*(2*b^8+2*c^8-(b^2+c^2)*(6*b^4+6*c^4-b*c*(3*b^2-4*b*c+3*c^2))*b*c)*a^5+(b^2-c^2)^2*(b+c)*(2*b^4+b^2*c^2+c^4)*(b^4+b^2*c^2+2*c^4)*a^4-(b^2-c^2)^2*(b^10+c^10+(2*b^8+2*c^8+(b^6+c^6+2*(3*b^4+3*c^4-b*c*(b^2-8*b*c+c^2))*b*c)*b*c)*b*c)*a^3+(b^2-c^2)^5*(b-c)*(b^2+c^2)^2*a^2+(b^8-c^8)*a*(b-c)^2*(b^2-c^2)^3-(b^2-c^2)^5*(b-c)*(b^2+c^2)*(b^4+c^4) : :

The reciprocal orthologic center of these triangles is X(4).

X(12925) lies on these lines: {11,1297}, {112,11826}, {127,10893}, {132,1376}, {355,12918}, {2799,12182}, {2806,12761}, {3320,10947}, {3434,12384}, {9517,12371}, {9530,11235}, {10785,12253}, {10794,12207}, {10826,12408}, {10829,12413}, {10871,12503}, {10914,12784}, {10919,12805}, {10920,12806}, {10944,12945}, {11373,12265}, {11390,12145}, {11865,12478}, {11866,12479}, {11903,12796}

X(12925) = reflection of X(i) in X(j) for these (i,j): (12340,132), (12935,12918)
X(12925) = X(1297)-of-inner-Johnson-triangle
X(12925) = X(13118)-of-outer-Johnson-triangle

X(12926) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO REFLECTION

Trilinears 32*p^7*(p-q)+32*(q^2-2)*p^6-16*(2*q^2-5)*q*p^5+2*(4*q^2-3)*(4*q^2-7)*p^4-2*(16*q^4-40*q^2+33)*q*p^3-(16*q^4-28*q^2+3)*p^2+4*(q^2-1)*(4*q^2-3)*q*p-1+q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(4).

X(12926) lies on these lines: {11,54}, {195,11928}, {355,6288}, {539,11235}, {1154,10525}, {1209,1376}, {2888,3434}, {3574,10893}, {6276,10920}, {6277,10919}, {7691,11826}, {7979,10944}, {9905,10826}, {9920,10829}, {9985,10871}, {10066,10523}, {10082,10948}, {10628,12371}, {10785,12254}, {10794,12208}, {10914,12785}, {10943,12889}, {11373,12266}, {11390,11576}, {11865,12480}, {11866,12481}, {11903,12797}

X(12926) = reflection of X(i) in X(j) for these (i,j): (12341,1209), (12936,6288)
X(12926) = X(54)-of-inner-Johnson-triangle
X(12926) = X(13121)-of-outer-Johnson-triangle

X(12927) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(79).

X(12927) lies on these lines: {11,6595}, {355,12745}, {1376,12342}, {3149,7701}, {10785,12255}, {10794,12209}, {10826,12409}, {10829,12414}, {10871,12504}, {10893,12600}, {10912,11280}, {10914,12786}, {10919,12807}, {10920,12808}, {10944,12947}, {11373,12267}, {11390,12146}, {11865,12482}, {11866,12483}, {11903,12798}

X(12927) = reflection of X(12937) in X(12919)
X(12927) = X(10266)-of-inner-Johnson-triangle
X(12927) = X(13130)-of-outer-Johnson-triangle

X(12928) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO INNER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(12928) lies on these lines: {11,486}, {355,6290}, {487,3434}, {642,1376}, {3564,10943}, {6251,10893}, {6280,10920}, {6281,10919}, {7980,10944}, {9906,10826}, {9921,10829}, {9986,10871}, {10067,10523}, {10083,10948}, {10785,12256}, {10794,12210}, {10914,12787}, {11373,12268}, {11390,12147}, {11826,12123}, {11865,12484}, {11866,12485}, {11903,12799}, {11928,12601}

X(12928) = X(486)-of-inner-Johnson-triangle
X(12928) = X(13132)-of-outer-Johnson-triangle
X(12928) = {X(10943),X(12586)}-harmonic conjugate of X(12929)

X(12929) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO OUTER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(12929) lies on these lines: {11,485}, {355,6289}, {488,3434}, {641,1376}, {3564,10943}, {6250,10893}, {6278,10920}, {6279,10919}, {7981,10944}, {9907,10826}, {9922,10829}, {9987,10871}, {10068,10523}, {10084,10948}, {10785,12257}, {10794,12211}, {10914,12788}, {11373,12269}, {11390,12148}, {11826,12124}, {11865,12486}, {11866,12487}, {11903,12800}, {11928,12602}

X(12929) = X(485)-of-inner-Johnson-triangle
X(12929) = X(13134)-of-outer-Johnson-triangle
X(12929) = {X(10943),X(12586)}-harmonic conjugate of X(12928)

X(12930) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO MIDHEIGHT

Trilinears 2*p^7*(p-q)+(8*q^2-7)*p^6-2*(4*q^2-3)*q*p^5+(2*q^4-12*q^2+9)*p^4-(2*q^4-9*q^2+6)*q*p^3+(1-q^2)*(1-p*q*(q^2-2)+p^2*(q^2-5)) : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(4).

X(12930) lies on these lines: {12,64}, {30,12423}, {72,12779}, {355,5878}, {958,2883}, {1498,11827}, {2777,12890}, {3436,6225}, {5812,6001}, {5895,6253}, {6000,10526}, {6247,10894}, {6266,10922}, {6267,10921}, {7355,10953}, {7973,10950}, {9899,10827}, {9914,10830}, {10060,10954}, {10076,10523}, {10786,12250}, {10795,12202}, {10872,12502}, {11374,12262}, {11381,11391}, {11500,12335}, {11867,12468}, {11868,12469}, {11904,12791}

X(12930) = reflection of X(12920) in X(5878)
X(12930) = X(64)-of-outer-Johnson-triangle
X(12930) = X(13095)-of-inner-Johnson-triangle

X(12931) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO INNER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(3).

X(12931) lies on these lines: {12,14}, {72,12780}, {355,5613}, {530,12349}, {531,11236}, {542,12587}, {617,3436}, {619,958}, {5474,11827}, {5479,10894}, {6269,10922}, {6271,10921}, {6773,10786}, {7974,10950}, {9900,10827}, {9915,10830}, {9981,10872}, {10061,10954}, {10077,10523}, {10795,12204}, {11374,11706}, {11391,12141}, {11500,12336}, {11867,12470}, {11868,12471}, {11904,12792}

X(12931) = reflection of X(12921) in X(5613)
X(12931) = X(14)-of-outer-Johnson-triangle
X(12931) = X(13106)-of-inner-Johnson-triangle

X(12932) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO OUTER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(3)

X(12932) lies on these lines: {12,13}, {72,12781}, {355,5617}, {530,11236}, {531,12349}, {542,12587}, {616,3436}, {618,958}, {5473,11827}, {5478,10894}, {6268,10922}, {6270,10921}, {6770,10786}, {7975,10950}, {9901,10827}, {9916,10830}, {9982,10872}, {10062,10954}, {10078,10523}, {10795,12205}, {11374,11705}, {11391,12142}, {11500,12337}, {11867,12472}, {11868,12473}, {11904,12793}

X(12932) = reflection of X(12922) in X(5617)
X(12932) = X(13)-of-outer-Johnson-triangle
X(12932) = X(13107)-of-inner-Johnson-triangle

X(12933) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO 1st NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(12933) lies on these lines: {12,76}, {39,958}, {72,12782}, {194,3436}, {355,730}, {384,10795}, {511,11500}, {538,11236}, {726,12610}, {732,12587}, {2782,10526}, {5969,12349}, {6248,10894}, {6272,10922}, {6273,10921}, {7976,10950}, {9902,10827}, {9917,10830}, {9983,10872}, {10063,10954}, {10079,10523}, {10786,12251}, {11257,11827}, {11374,12263}, {11391,12143}, {11867,12474}, {11868,12475}, {11904,12794}

X(12933) = reflection of X(12923) in X(3095)
X(12933) = X(76)-of-outer-Johnson-triangle
X(12933) = X(13110)-of-inner-Johnson-triangle

X(12934) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(12934) lies on these lines: {12,83}, {72,12783}, {355,6287}, {732,12587}, {754,11236}, {958,6292}, {2896,3436}, {6249,10894}, {6274,10922}, {6275,10921}, {7977,10950}, {9903,10827}, {9918,10830}, {10064,10954}, {10080,10523}, {10786,12252}, {10795,12206}, {10942,12183}, {11374,12264}, {11391,12144}, {11500,12339}, {11827,12122}, {11867,12476}, {11868,12477}, {11904,12795}

X(12934) = reflection of X(12924) in X(6287)
X(12934) = X(83)-of-outer-Johnson-triangle
X(12934) = X(13113)-of-inner-Johnson-triangle

X(12935) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO 1st ORTHOSYMMEDIAL

Barycentrics a^17-(b+c)*a^16-(b-c)^2*a^15+(b+c)*(b^2-4*b*c+c^2)*a^14-(2*b^4+2*c^4+b*c*(2*b-c)*(b-2*c))*a^13+(b+c)*(2*b^4+2*c^4+b*c*(4*b^2-5*b*c+4*c^2))*a^12+(b^6+c^6-b*c*(b+2*c)*(2*b+c)*(b-c)^2)*a^11-(b+c)*(b^6+c^6-(2*b^4+2*c^4+b*c*(b^2-8*b*c+c^2))*b*c)*a^10+2*(b^2-c^2)^2*(b+c)*(b^3+c^3)*a^9-2*(b^3-c^3)*(b^2-c^2)^3*a^8+(b^4-c^4)^2*(b-c)^2*a^7-(b^2-c^2)^2*(b+c)*(b^6+c^6+b^2*c^2*(3*b^2-4*b*c+3*c^2))*a^6-(b^2-c^2)^2*(2*b^8+2*c^8-(2*b^6+2*c^6-(11*b^4+11*c^4-2*b*c*(b^2-7*b*c+c^2))*b*c)*b*c)*a^5+(b^2-c^2)^3*(b-c)*(2*b^6+2*c^6-(4*b^4+4*c^4-b*c*(b^2-6*b*c+c^2))*b*c)*a^4-(b^2-c^2)^2*(b^10+c^10-(2*b^8+2*c^8+(7*b^6+7*c^6-2*(b^4+c^4-5*b*c*(b^2+c^2))*b*c)*b*c)*b*c)*a^3+(b^4-c^4)*(b^2-c^2)^2*(b-c)*(b^6+c^6+(4*b^4+4*c^4-b*c*(b^2+c^2))*b*c)*a^2+(b^8-c^8)*a*(b-c)^2*(b^2-c^2)^3-(b^2-c^2)^5*(b-c)*(b^2+c^2)*(b^4+c^4) : :

The reciprocal orthologic center of these triangles is X(4).

X(12935) lies on these lines: {12,1297}, {72,12784}, {112,11827}, {127,10894}, {132,958}, {355,12918}, {2799,12183}, {2806,12762}, {3320,10953}, {3436,12384}, {9517,12372}, {9530,11236}, {10786,12253}, {10795,12207}, {10827,12408}, {10830,12413}, {10872,12503}, {10921,12805}, {10922,12806}, {10950,12955}, {11374,12265}, {11391,12145}, {11500,12340}, {11867,12478}, {11868,12479}, {11904,12796}

X(12935) = reflection of X(12925) in X(12918)
X(12935) = X(1297)-of-outer-Johnson-triangle
X(12935) = X(13119)-of-inner-Johnson-triangle

X(12936) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO REFLECTION

Trilinears 32*p^7*(p-q)+32*(q^2-2)*p^6-16*(2*q^2-3)*q*p^5+2*(16*q^4-24*q^2+21)*p^4-2*(4*q^2-3)^2*q*p^3-(16*q^4-12*q^2+11)*p^2+2*(4*q^2-1)*(2*q^2-1)*q*p+1-q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(4).

X(12936) lies on these lines: {12,54}, {72,6145}, {195,11929}, {355,6288}, {539,11236}, {958,1209}, {1154,10526}, {2888,3436}, {3574,10894}, {6276,10922}, {6277,10921}, {7691,11827}, {7979,10950}, {9905,10827}, {9920,10830}, {9985,10872}, {10066,10954}, {10082,10523}, {10628,12372}, {10786,12254}, {10795,12208}, {10942,12890}, {11374,12266}, {11391,11576}, {11500,12341}, {11867,12480}, {11868,12481}, {11904,12797}

X(12936) = reflection of X(12926) in X(6288)
X(12936) = X(54)-of-outer-Johnson-triangle
X(12936) = X(13122)-of-inner-Johnson-triangle

X(12937) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(79).

X(12937) lies on these lines: {12,10266}, {72,12786}, {191,12519}, {355,12745}, {1012,5693}, {2771,12524}, {10786,12255}, {10795,12209}, {10827,12409}, {10830,12414}, {10872,12504}, {10894,12600}, {10921,12807}, {10922,12808}, {10950,12957}, {11374,12267}, {11391,12146}, {11500,12342}, {11867,12482}, {11868,12483}, {11904,12798}

X(12937) = reflection of X(i) in X(j) for these (i,j): (12519,191), (12927,12919)
X(12937) = X(10266)-of-outer-Johnson-triangle
X(12937) = X(13131)-of-inner-Johnson-triangle

X(12938) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO INNER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(12938) lies on these lines: {12,486}, {72,12787}, {355,6290}, {487,3436}, {642,958}, {3564,10942}, {6251,10894}, {6280,10922}, {6281,10921}, {7980,10950}, {9906,10827}, {9921,10830}, {9986,10872}, {10067,10954}, {10083,10523}, {10786,12256}, {10795,12210}, {11374,12268}, {11391,12147}, {11500,12343}, {11827,12123}, {11867,12484}, {11868,12485}, {11904,12799}, {11929,12601}

X(12938) = X(486)-of-outer-Johnson-triangle
X(12938) = X(13133)-of-inner-Johnson-triangle
X(12938) = {X(10942),X(12587)}-harmonic conjugate of X(12939)

X(12939) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO OUTER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(12939) lies on these lines: {12,485}, {72,12788}, {355,6289}, {488,3436}, {641,958}, {3564,10942}, {6250,10894}, {6278,10922}, {6279,10921}, {7981,10950}, {9907,10827}, {9922,10830}, {9987,10872}, {10068,10954}, {10084,10523}, {10786,12257}, {10795,12211}, {11374,12269}, {11391,12148}, {11500,12344}, {11827,12124}, {11867,12486}, {11868,12487}, {11904,12800}, {11929,12602}

X(12939) = X(485)-of-outer-Johnson-triangle
X(12939) = X(13135)-of-inner-Johnson-triangle
X(12939) = {X(10942),X(12587)}-harmonic conjugate of X(12938)

X(12940) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO MIDHEIGHT

Trilinears 2*p^6+2*(4*q^2-3)*p^4+(2*q^4-9*q^2+6)*p^2-(q^2-1)*(q^2-2) : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(4).

X(12940) lies on these lines: {1,5878}, {4,65}, {5,10076}, {12,64}, {30,3157}, {56,2883}, {221,5895}, {388,6225}, {495,10060}, {498,3357}, {607,3330}, {851,3556}, {1056,11189}, {1478,6000}, {1498,7354}, {1854,3649}, {2192,5434}, {2777,4302}, {3028,11744}, {3085,12250}, {4293,5656}, {4299,6759}, {5217,5894}, {5229,12324}, {5432,10606}, {5663,10055}, {5893,10896}, {6247,10895}, {6266,10924}, {6267,10923}, {7973,10944}, {9578,9899}, {9655,12315}, {9833,10483}, {9914,10831}, {10797,12202}, {10873,12502}, {11375,12262}, {11381,11392}, {11501,12335}, {11905,12791}

X(12940) = reflection of X(10060) in X(495)
X(12940) = X(64)-of-1st-Johnson-Yff-triangle
X(12940) = {X(1), X(5878)}-harmonic conjugate of X(12950)

X(12941) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO INNER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(3).

X(12941) lies on these lines: {1,5613}, {5,10077}, {12,14}, {13,3027}, {56,619}, {65,12780}, {114,12952}, {388,617}, {495,10061}, {498,6774}, {531,11237}, {542,10053}, {2782,10062}, {3085,6773}, {5434,5464}, {5474,7354}, {5479,10895}, {6269,10924}, {6271,10923}, {7974,10944}, {9578,9900}, {9915,10831}, {9981,10873}, {10797,12204}, {11375,11706}, {11392,12141}, {11501,12336}, {11869,12470}, {11870,12471}, {11905,12792}

X(12941) = reflection of X(10061) in X(495)
X(12941) = X(14)-of-1st-Johnson-Yff-triangle
X(12941) = {X(10056),X(12588)}-harmonic conjugate of X(12942)

X(12942) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO OUTER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(3).

X(12942) lies on these lines: {1,5617}, {5,10078}, {12,13}, {14,3027}, {56,618}, {65,12781}, {114,12951}, {388,616}, {495,10062}, {498,6771}, {530,11237}, {531,12350}, {542,10053}, {2782,10061}, {3085,6770}, {5434,5463}, {5473,7354}, {5478,10895}, {6268,10924}, {6270,10923}, {6774,10069}, {7975,10944}, {9578,9901}, {9916,10831}, {10797,12205}, {11375,11705}, {11392,12142}, {11501,12337}, {11869,12472}, {11870,12473}, {11905,12793}

X(12942) = reflection of X(10062) in X(495)
X(12942) = X(13)-of-1st-Johnson-Yff-triangle
X(12942) = {X(10056),X(12588)}-harmonic conjugate of X(12941)

X(12943) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO ANTICEVIAN-OF-X(4)

Barycentrics 3*a^4-(b-c)^2*a^2-2*(b^2-c^2)^2 : :
X(12943) = 3*X(55)-4*X(495) = 3*X(55)-2*X(4302) = 5*X(55)-6*X(10056) = 2*X(55)-3*X(11237) = 2*X(495)-3*X(1478) = 3*X(1478)-X(4302)

The reciprocal orthologic center of these triangles is X(4).

X(12943) lies on these lines: {1,382}, {3,3585}, {4,11}, {5,4299}, {12,20}, {30,55}, {34,7221}, {35,1657}, {36,381}, {65,971}, {84,1454}, {100,11236}, {354,3586}, {355,1770}, {376,5432}, {377,3826}, {388,390}, {484,5790}, {485,9647}, {496,3853}, {497,3543}, {498,550}, {499,546}, {515,1836}, {516,5252}, {529,3434}, {535,956}, {548,10592}, {631,3614}, {946,1388}, {950,5542}, {958,2475}, {962,10944}, {999,3583}, {1001,11114}, {1012,5172}, {1056,3058}, {1155,5587}, {1159,11552}, {1317,10724}, {1319,1699}, {1329,4190}, {1357,10730}, {1358,10729}, {1361,10732}, {1362,10725}, {1364,10726}, {1376,5080}, {1479,3304}, {1482,7972}, {1503,12940}, {1539,10091}, {1593,9672}, {1597,10832}, {1614,9653}, {1656,7280}, {1837,4292}, {1885,11392}, {2098,12699}, {2307,5340}, {2477,6759}, {2646,9612}, {2777,10060}, {2794,12945}, {3022,10727}, {3023,10722}, {3027,10723}, {3028,10733}, {3057,9613}, {3085,3529}, {3091,5433}, {3157,12373}, {3295,5073}, {3320,10735}, {3324,10152}, {3325,10734}, {3476,4345}, {3486,3649}, {3487,10543}, {3522,10588}, {3524,5326}, {3534,5010}, {3579,4333}, {3600,5225}, {3679,5183}, {3790,7270}, {3832,7173}, {3839,5298}, {3843,4325}, {3861,10593}, {4056,4089}, {4295,10950}, {4297,11375}, {4305,5714}, {4308,10248}, {4311,11376}, {4323,5556}, {4342,10106}, {4413,11112}, {4423,11113}, {4857,7373}, {4860,5722}, {4872,7223}, {4995,8164}, {4999,6871}, {5056,7294}, {5057,5289}, {5059,5261}, {5064,5322}, {5076,5563}, {5141,5303}, {5187,6691}, {5584,6850}, {5840,12763}, {5842,12115}, {5895,6285}, {5919,9580}, {6240,11398}, {6253,12667}, {6256,11509}, {6851,10953}, {6906,10894}, {6925,12943}, {6938,7680}, {7286,10296}, {7352,12293}, {7387,9659}, {7745,9597}, {7747,9651}, {7756,9650}, {8275,9589}, {8972,9663}, {9539,10149}, {10037,12085}, {10081,10113}, {10310,10526}, {10572,11551}, {11194,11680}, {11499,11698}, {11571,12747}, {12295,12904}

X(12943) = reflection of X(i) in X(j) for these (i,j): (55,1478), (2099,1836), (3428,6923), (4302,495), (6938,7680)
X(12943) = X(20)-of-1st-Johnson-Yff-triangle
X(12943) = outer-Johnson-to-ABC similarity image of X(20)
X(12943) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,9655,9657), (3,3585,10895), (4,56,10896), (4,4293,11), (4,7354,56), (5,4299,5204), (11,4293,56), (11,7354,4293), (12,20,5217), (12,5229,9656), (20,5229,12), (55,1478,11237), (382,9655,1), (382,9657,9670), (495,4302,55), (1478,4302,495), (3585,4316,7951), (3585,10483,3), (4316,7951,3), (5217,9656,12), (7951,10483,4316)

X(12944) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(12944) lies on these lines: {1,6287}, {5,10080}, {12,83}, {35,8725}, {56,6292}, {388,2896}, {495,10053}, {732,12588}, {754,11237}, {3027,11606}, {3085,12252}, {5432,9751}, {6249,10895}, {6274,10924}, {6275,10923}, {7354,12122}, {7977,10944}, {9918,10831}, {10797,12206}, {11375,12264}, {11392,12144}, {11501,12339}, {11869,12476}, {11870,12477}, {11905,12795}

X(12944) = reflection of X(10064) in X(495)

X(12945) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO 1st ORTHOSYMMEDIAL

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

The reciprocal orthologic center of these triangles is X(4).

X(12945) lies on the Johnson-Yff-inner-circle and these lines: {1,12918}, {4,3320}, {12,1297}, {56,132}, {65,12784}, {112,7354}, {127,10895}, {388,6020}, {2781,12903}, {2799,12184}, {2806,12763}, {3085,12253}, {3585,10749}, {5204,6720}, {9517,12373}, {9530,11237}, {9578,12408}, {10797,12207}, {10831,12413}, {10873,12503}, {10923,12805}, {10924,12806}, {10944,12925}, {11375,12265}, {11392,12145}, {11501,12340}, {11905,12796}

X(12945) = X(1297)-of-1st-Johnson-Yff-triangle
X(12945) = {X(1),X(12918)}-harmonic conjugate of X(12955)

X(12946) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO REFLECTION

Trilinears (4*cos(A)-2*cos(2*A))*cos(B-C)+2*cos(A)*cos(2*(B-C))+2*cos(A)-2*cos(2*A)+cos(3*A)+1 : :

The reciprocal orthologic center of these triangles is X(4).

X(12946) lies on these lines: {1,6288}, {5,10082}, {12,54}, {56,1209}, {65,2962}, {73,6145}, {195,9654}, {388,2888}, {495,10066}, {498,10610}, {539,3157}, {1154,1478}, {2917,9659}, {3085,12254}, {3574,10895}, {3585,6286}, {6276,10924}, {6277,10923}, {7354,7691}, {7979,10944}, {8254,10592}, {9578,9905}, {9655,12307}, {9920,10831}, {9985,10873}, {10628,12373}, {10797,12208}, {11375,12266}, {11392,11576}, {11501,12341}, {11905,12797}

X(12946) = reflection of X(10066) in X(495)
X(12946) = X(54)-of-1st-Johnson-Yff-triangle
X(12946) = {X(1),X(6288)}-harmonic conjugate of X(12956)

X(12947) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(79).

X(12947) lies on these lines: {1,12919}, {12,10266}, {65,12786}, {1317,6595}, {3085,12255}, {9578,12409}, {10797,12209}, {10831,12414}, {10873,12504}, {10895,12600}, {10923,12807}, {10924,12808}, {10944,12927}, {11375,12267}, {11392,12146}, {11501,12342}, {11869,12482}, {11870,12483}, {11905,12798}, {12745,12913}

X(12947) = X(10266)-of-1st-Johnson-Yff-triangle
X(12947) = {X(1),X(12919)}-harmonic conjugate of X(12957)

X(12948) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO INNER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(12948) lies on these lines: {1,6290}, {5,10083}, {12,486}, {56,642}, {65,12787}, {388,487}, {495,611}, {3085,12256}, {5229,12296}, {6251,10895}, {6280,10924}, {6281,10923}, {7354,12123}, {7980,10944}, {9578,9906}, {9654,12601}, {9921,10831}, {9986,10873}, {10037,12972}, {10797,12210}, {11375,12268}, {11392,12147}, {11501,12343}, {11869,12484}, {11870,12485}, {11905,12799}

X(12948) = reflection of X(10067) in X(495)
X(12948) = X(486)-of-1st-Johnson-Yff-triangle

X(12949) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO OUTER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(12949) lies on these lines: {1,6289}, {5,10084}, {12,485}, {56,641}, {65,12788}, {388,488}, {495,611}, {3085,12257}, {5229,12297}, {5261,12222}, {6250,10895}, {6278,10924}, {6279,10923}, {7354,12124}, {7981,10944}, {9578,9907}, {9654,12602}, {9922,10831}, {9987,10873}, {10037,12973}, {10797,12211}, {11375,12269}, {11392,12148}, {11501,12344}, {11869,12486}, {11870,12487}, {11905,12800}

X(12949) = reflection of X(10068) in X(495)
X(12949) = X(485)-of-1st-Johnson-Yff-triangle

X(12950) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO MIDHEIGHT

Trilinears 2*p^6+8*(q^2-1)*p^4+(q^2-1)*(2*q^2-9)*p^2-(q^2-1)*(q^2-3) : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(4).

X(12950) lies on these lines: {1,5878}, {4,6285}, {5,10060}, {11,64}, {20,10535}, {30,1069}, {55,2883}, {146,9538}, {221,3058}, {388,11189}, {496,10076}, {497,6225}, {499,3357}, {1479,6000}, {1498,6284}, {2192,5895}, {2777,4299}, {3057,12779}, {3086,12250}, {4294,5656}, {4302,6759}, {5204,5894}, {5225,12324}, {5433,10606}, {5663,10071}, {5893,10895}, {6001,12116}, {6247,10896}, {6266,10926}, {6267,10925}, {7973,10950}, {9581,9899}, {9668,12315}, {9914,10832}, {10798,12202}, {10874,12502}, {11376,12262}, {11381,11393}, {11502,12335}, {11871,12468}, {11872,12469}, {11906,12791}

X(12950) = reflection of X(10076) in X(496)
X(12950) = X(64)-of-2nd-Johnson-Yff-triangle
X(12950) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5878,12940), (2192,5895,7354)

X(12951) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO INNER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(3).

X(12951) lies on these lines: {1,5613}, {5,10061}, {11,14}, {13,3023}, {55,619}, {114,12942}, {496,10077}, {497,617}, {499,6774}, {530,12351}, {531,11238}, {542,10069}, {2782,10078}, {3057,12780}, {3058,5464}, {3086,6773}, {5474,6284}, {5479,10896}, {6269,10926}, {6271,10925}, {6771,10053}, {7974,10950}, {9114,12354}, {9581,9900}, {9915,10832}, {9981,10874}, {10798,12204}, {11376,11706}, {11393,12141}, {11502,12336}, {11871,12470}, {11872,12471}, {11906,12792}

X(12951) = reflection of X(10077) in X(496)
X(12951) = X(14)-of-2nd-Johnson-Yff-triangle
X(12951) = {X(10072),X(12589)}-harmonic conjugate of X(12952)

X(12952) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO OUTER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(3).

X(12952) lies on these lines: {1,5617}, {5,10062}, {11,13}, {14,3023}, {55,618}, {114,12941}, {496,10078}, {497,616}, {499,6771}, {530,11238}, {531,12351}, {542,10069}, {2782,10077}, {3057,12781}, {3058,5463}, {3086,6770}, {5473,6284}, {5478,10896}, {6268,10926}, {6270,10925}, {6774,10053}, {7975,10950}, {9581,9901}, {9916,10832}, {9982,10874}, {10798,12205}, {11376,11705}, {11393,12142}, {11502,12337}, {11871,12472}, {11872,12473}, {11906,12793}

X(12952) = reflection of X(10078) in X(496)
X(12952) = X(13)-of-2nd-Johnson-Yff-triangle
X(12952) = {X(10072),X(12589)}-harmonic conjugate of X(12951)

X(12953) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO ANTICEVIAN-OF-X(4)

Barycentrics 3*a^4-(b+c)^2*a^2-2*(b^2-c^2)^2) : :
X(12953) = 3*X(56)-4*X(496) = 3*X(56)-2*X(4299) = 2*X(56)-3*X(11238) = 2*X(496)-3*X(1479) = 3*X(1479)-X(4299) = 4*X(1479)-3*X(11238) = 3*X(1837)-2*X(4848)

The reciprocal orthologic center of these triangles is X(4).

X(12953) lies on these lines: {1,382}, {3,3583}, {4,12}, {5,4302}, {11,20}, {30,56}, {33,4348}, {34,9627}, {35,381}, {36,1657}, {40,7082}, {65,3586}, {80,12702}, {149,12513}, {215,6759}, {218,5134}, {354,4355}, {376,5433}, {377,4423}, {388,3058}, {390,5229}, {405,3841}, {452,3925}, {485,9660}, {495,3853}, {497,3146}, {498,546}, {499,550}, {515,2098}, {516,1837}, {517,1898}, {528,3436}, {548,10593}, {631,7173}, {938,11246}, {950,1836}, {958,11114}, {962,10950}, {999,4857}, {1001,2475}, {1058,5434}, {1069,12374}, {1155,9581}, {1317,10728}, {1319,9614}, {1361,10726}, {1362,10727}, {1364,10732}, {1376,5046}, {1466,6851}, {1478,3303}, {1486,4214}, {1503,12950}, {1539,10088}, {1593,9659}, {1597,10831}, {1614,9666}, {1656,5010}, {1699,2646}, {1770,5221}, {1885,11393}, {2099,10572}, {2478,4413}, {2777,10076}, {2794,12955}, {2829,12116}, {2886,6872}, {2975,11235}, {3021,10729}, {3022,10725}, {3023,10723}, {3027,10722}, {3028,10721}, {3035,5187}, {3057,5691}, {3086,3529}, {3091,5432}, {3295,3585}, {3318,10731}, {3428,7491}, {3485,10543}, {3486,4323}, {3488,3649}, {3522,10589}, {3524,7294}, {3534,7280}, {3579,10826}, {3612,9955}, {3614,3832}, {3715,12572}, {3746,5076}, {3748,5290}, {3816,4190}, {3839,4995}, {3843,4330}, {3861,10592}, {3871,11236}, {3913,5080}, {3962,12625}, {4292,4860}, {4297,11376}, {4304,11375}, {4313,10248}, {4387,7270}, {4421,11681}, {4680,7206}, {4854,5716}, {4855,5087}, {4863,12527}, {4872,7185}, {5016,5695}, {5056,5326}, {5057,12635}, {5059,5274}, {5064,5310}, {5160,9538}, {5172,6985}, {5175,5698}, {5178,5220}, {5252,10624}, {5261,10385}, {5270,6767}, {5298,11001}, {5339,7127}, {5727,9589}, {5790,11010}, {5840,6928}, {5895,7355}, {5919,9613}, {6018,10730}, {6019,10734}, {6020,10735}, {6154,7080}, {6238,12293}, {6240,11399}, {6256,10965}, {6690,6871}, {6827,11826}, {6840,11502}, {6850,8273}, {6905,10893}, {6934,7681}, {7158,10152}, {7387,9672}, {7727,12902}, {7728,12896}, {7745,9598}, {7747,9664}, {7756,9665}, {8972,9648}, {10046,12085}, {10056,10386}, {10065,10113}, {10738,11249}, {12295,12903}, {12667,12763}

X(12953) = reflection of X(i) in X(j) for these (i,j): (56,1479), (2098,12701), (4299,496), (6934,7681), (10310,6928)
X(12953) = X(20)-of-2nd-Johnson-Yff-triangle
X(12953) = inner-Johnson-to-ABC similarity image of X(20)
X(12953) = homothetic center of intangents triangle and reflection of tangential triangle in X(4)
X(12953) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,382,12943), (1,9668,9670), (1,12943,9657), (3,3583,10896), (4,55,10895), (4,4294,12), (4,6284,55), (5,4302,5217), (11,20,5204), (11,5225,9671), (12,4294,55), (12,6284,4294), (20,5225,11), (56,1479,11238), (382,9668,1), (382,9670,9657), (1479,4299,496), (3583,4324,7741), (4324,7741,3), (5204,9671,11), (9670,12943,1)

X(12954) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(12954) lies on these lines: {1,6287}, {5,10064}, {11,83}, {36,8725}, {55,6292}, {496,10069}, {497,2896}, {732,12589}, {754,11238}, {3023,11606}, {3057,12783}, {3086,12252}, {5433,9751}, {6249,10896}, {6274,10926}, {6275,10925}, {6284,12122}, {7977,10950}, {9581,9903}, {9918,10832}, {10798,12206}, {11376,12264}, {11393,12144}, {11502,12339}, {11871,12476}, {11872,12477}, {11906,12795}

X(12954) = reflection of X(10080) in X(496)
X(12954) = X(83)-of-2nd-Johnson-Yff-triangle
X(12954) = {X(1), X(6287)}-harmonic conjugate of X(12944)

X(12955) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO 1st ORTHOSYMMEDIAL

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

The reciprocal orthologic center of these triangles is X(4).

X(12955) lies on the Johnson-Yff-outer-circle and these lines: {1,12918}, {4,6020}, {11,1297}, {55,132}, {112,6284}, {127,10896}, {497,3320}, {2781,12904}, {2799,12185}, {2806,12764}, {3057,12784}, {3086,12253}, {3583,10749}, {5217,6720}, {9517,12374}, {9530,11238}, {9581,12408}, {10798,12207}, {10832,12413}, {10874,12503}, {10925,12805}, {10926,12806}, {10950,12935}, {11376,12265}, {11393,12145}, {11502,12340}, {11871,12478}, {11872,12479}, {11906,12796}

X(12955) = X(1297)-of-2nd-Johnson-Yff-triangle
X(12955) = {X(1),X(12918)}-harmonic conjugate of X(12945)

X(12956) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO REFLECTION

Trilinears (4*cos(A)+2*cos(2*A))*cos(B-C)-2*cos(A)*cos(2*(B-C))-2*cos(A)-2*cos(2*A)-cos(3*A)+1 : :

The reciprocal orthologic center of these triangles is X(4).

X(12956) lies on these lines: {1,6288}, {5,10066}, {11,54}, {55,1209}, {195,9669}, {496,10082}, {497,2888}, {499,10610}, {539,1069}, {1154,1479}, {2917,9672}, {3057,12785}, {3086,12254}, {3519,4857}, {3574,10896}, {3583,7356}, {6145,9630}, {6276,10926}, {6277,10925}, {6284,7691}, {7979,10950}, {8254,10593}, {9581,9905}, {9668,12307}, {9920,10832}, {9985,10874}, {10628,12374}, {10798,12208}, {11376,12266}, {11393,11576}, {11502,12341}, {11871,12480}, {11872,12481}, {11906,12797}

X(12956) = reflection of X(10082) in X(496)
X(12956) = X(54)-of-2nd-Johnson-Yff-triangle
X(12956) = {X(1),X(6288)}-harmonic conjugate of X(12946)

X(12957) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(79).

X(12957) lies on these lines: {1,12919}, {11,6595}, {3057,12786}, {3086,12255}, {9581,12409}, {10798,12209}, {10832,12414}, {10874,12504}, {10896,12600}, {10925,12807}, {10926,12808}, {10950,12937}, {11376,12267}, {11393,12146}, {11502,12342}, {11871,12482}, {11872,12483}, {11906,12798}, {12745,12877}

X(12957) = X(10266)-of-2nd-Johnson-Yff-triangle
X(12957) = {X(1),X(12919)}-harmonic conjugate of X(12947)

X(12958) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO INNER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(12958) lies on these lines: {1,6290}, {5,10067}, {11,486}, {55,642}, {487,497}, {496,613}, {3057,12787}, {3086,12256}, {5225,12296}, {5274,12221}, {6251,10896}, {6280,10926}, {6281,10925}, {6284,12123}, {7980,10950}, {9581,9906}, {9669,12601}, {9921,10832}, {9986,10874}, {10046,12972}, {10798,12210}, {11376,12268}, {11393,12147}, {11502,12343}, {11871,12484}, {11872,12485}, {11906,12799}

X(12958) = reflection of X(10083) in X(496)
X(12958) = X(486)-of-2nd-Johnson-Yff-triangle

X(12959) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO OUTER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(12959) lies on these lines: {1,6289}, {5,10068}, {11,485}, {55,641}, {488,497}, {496,613}, {3057,12788}, {3086,12257}, {5225,12297}, {5274,12222}, {6250,10896}, {6278,10926}, {6279,10925}, {6284,12124}, {7981,10950}, {9581,9907}, {9669,12602}, {9922,10832}, {9987,10874}, {10046,12973}, {10798,12211}, {11376,12269}, {11393,12148}, {11502,12344}, {11871,12486}, {11872,12487}, {11906,12800}

X(12959) = reflection of X(10084) in X(496)
X(12959) = X(485)-of-2nd-Johnson-Yff-triangle

X(12960) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS TO LUCAS ANTIPODAL

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

The reciprocal orthologic center of these triangles is X(3).

X(12960) lies on these lines: {6,12229}, {371,5254}, {372,12972}, {486,3547}, {487,5412}, {642,10961}, {1151,12303}, {2066,12910}, {3068,12320}, {3311,12311}, {3564,12961}, {5409,5491}, {5410,12169}, {5415,12662}, {10880,12509}, {10897,12601}, {11417,12221}, {11447,12274}, {11462,12285}, {11473,12296}

X(12960) = orthic-to-1st-Kenmotu-diagonals similarity image of X(487)
X(12960) = X(12972)-of-1st-Kenmotu-free-vertices-triangle
X(12960) = {X(6),X(12978)}-harmonic conjugate of X(12966)

X(12961) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS TO LUCAS(-1) ANTIPODAL

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

The reciprocal orthologic center of these triangles is X(3).

X(12961) lies on these lines: {6,12230}, {372,12973}, {485,6643}, {488,5412}, {641,10961}, {1151,12304}, {2066,12911}, {3068,12321}, {3564,12960}, {5410,12170}, {5415,12663}, {10880,12510}, {10897,12602}, {11417,12222}, {11447,12275}, {11462,12286}, {11473,12297}

X(12961) = orthic-to-1st-Kenmotu-diagonals similarity image of X(488)
X(12961) = X(12973)-of-1st-Kenmotu-free-vertices-triangle
X(12961) = {X(6),X(12979)}-harmonic conjugate of X(12967)

X(12962) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS TO LUCAS CENTRAL

Trilinears (a^2-3*b^2-3*c^2-6*S)*a : :
X(12962) = 2*X(5023)-3*X(12963)

The reciprocal orthologic center of these triangles is X(3).

X(12962) lies on these lines: {3,6}, {115,8960}, {315,1991}, {493,9225}, {590,637}, {639,8253}, {2066,6283}, {2067,7362}, {3068,12322}, {5410,12171}, {5412,6291}, {5415,6252}, {6239,10880}, {9823,10961}, {11417,12223}, {11447,12276}, {11462,12287}, {11473,12298}

X(12962) = reflection of X(1151) in X(371)
X(12962) = Kenmotu circle-inverse-of-X(2459)
X(12962) = X(176)-of-1st-Kenmotu-diagonals-triangle if ABC is acute
X(12962) = X(12974)-of-1st-Kenmotu-free-vertices-triangle
X(12962) = orthic-to-1st-Kenmotu-diagonals similarity image of X(6291)
X(12962) = center of inverse-in-1st-Kenmotu-circle-of-circumcircle
X(12962) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,371,12963), (6,1151,12968), (6,5013,12969), (6,6425,3053), (39,6419,6), (371,1504,6), (3311,6422,6), (6417,6421,6)

X(12963) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS TO LUCAS(-1) CENTRAL

Trilinears (3*a^2-b^2-c^2-2*S)*a : :
Trilinears 2 cos A + sin A (1 - cot ω) : :
X(12963) = 4*X(32)-X(12969) = 2*X(371)+X(1152) = 2*X(5023)+X(12962)

The reciprocal orthologic center of these triangles is X(3).

X(12963) lies on these lines: {3,6}, {112,6400}, {172,2066}, {230,3071}, {485,7737}, {590,7388}, {615,637}, {639,8252}, {1501,1599}, {1583,1915}, {1613,10132}, {1914,2067}, {1968,5412}, {2548,5418}, {3068,11294}, {3156,8576}, {3767,6561}, {5286,9541}, {5319,9681}, {5410,8778}, {5415,6404}, {5475,10576}, {6222,6251}, {6459,7735}, {6564,7747}, {6565,7746}, {7749,10577}, {8253,11314}, {9575,9615}, {9582,9593}, {9824,10961}, {10311,11473}, {11417,12224}, {11447,12277}, {11462,12288}

X(12963) = X(175)-of-1st-Kenmotu-diagonals-triangle if ABC is acute
X(12963) = X(12975)-of-1st-Kenmotu-free-vertices-triangle
X(12963) = orthic-to-1st-Kenmotu-diagonals similarity image of X(6406)
X(12963) = center of conic {X(371),PU(1),PU(2)}
X(12963) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1691,12968), (3,6424,6), (6,371,12962), (6,1152,12969), (6,3053,12968), (6,5023,1152), (6,6409,5013), (32,371,6), (32,5017,12968), (372,5058,6), (1384,8375,6), (3311,6423,6), (5062,6419,6)

X(12964) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS TO MIDHEIGHT

Barycentrics (S^2+(4*R^2-SW)*S-2*(4*R^2-SA)*SA)*(SB+SC) : :
X(12964) = 2*X(371)-3*X(11241)

X(12964) lies on these lines: {3,10533}, {4,6}, {30,10665}, {64,1151}, {154,1152}, {184,11474}, {185,5412}, {221,3298}, {371,6000}, {372,6759}, {394,490}, {590,6247}, {1971,12968}, {2066,6285}, {2067,7355}, {2192,3297}, {2777,12375}, {3068,12324}, {3311,12315}, {3312,11242}, {3357,6200}, {5410,12174}, {5415,6254}, {5663,11265}, {5878,6561}, {5907,11513}, {6001,7969}, {6221,13093}, {6225,6459}, {6241,10880}, {6396,10282}, {6409,10606}, {6411,8567}, {6460,11206}, {6502,10535}, {6560,9833}, {8909,12085}, {9541,12250}, {9616,9899}, {9729,10961}, {9934,20124}, {10897,12162}, {11381,11473}, {11417,12111}, {11447,12279}, {11462,12290}, {12305,13055}

X(12964) = {X(4),X(17849)}-harmonic conjugate of X(12970)
X(12964) = {X(6),X(1498)}-harmonic conjugate of X(12970)

X(12965) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS TO REFLECTION

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

The reciprocal orthologic center of these triangles is X(6243).

X(12965) lies on these lines: {6,17}, {54,372}, {371,1154}, {485,2888}, {539,10665}, {615,8254}, {1151,12307}, {1493,6420}, {2066,6286}, {2067,7356}, {3068,12325}, {3070,12375}, {3311,12316}, {3574,6565}, {5410,12175}, {5412,6152}, {5415,6255}, {6200,7691}, {6242,10880}, {6288,6564}, {6396,10610}, {6560,12254}, {9827,10961}, {10897,12606}, {11417,12226}, {11447,12280}, {11462,12291}, {11473,12300}, {11513,12363}

X(12965) = X(79)-of-1st-Kenmotu-diagonals-triangle if ABC is acute
X(12965) = X(54)-of-1st-Kenmotu-free-vertices-triangle
X(12965) = perspector of 1st Kenmotu diagonals triangle and 1st Kenmotu free vertices triangle
X(12965) = {X(6),X(195)}-harmonic conjugate of X(12971)

X(12966) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS TO LUCAS ANTIPODAL

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

The reciprocal orthologic center of these triangles is X(3).

X(12966) lies on these lines: {6,12229}, {371,12972}, {486,6643}, {487,5413}, {642,10963}, {1152,12303}, {3069,12320}, {3312,12311}, {3564,12967}, {5411,12169}, {5414,12910}, {5416,12662}, {10881,12509}, {10898,12601}, {11418,12221}, {11448,12274}, {11463,12285}, {11474,12296}

X(12966) = orthic-to-2nd-Kenmotu-diagonals similarity image of X(487)
X(12966) = X(12972)-of-2nd-Kenmotu-free-vertices-triangle
X(12966) = {X(6),X(12978)}-harmonic conjugate of X(12960)

X(12967) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS TO LUCAS(-1) ANTIPODAL

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

The reciprocal orthologic center of these triangles is X(3).

X(12967) lies on these lines: {6,12230}, {371,12973}, {372,5254}, {485,3547}, {488,5413}, {641,10963}, {1152,12304}, {3069,12321}, {3312,12312}, {3564,12966}, {5408,5490}, {5411,12170}, {5414,12911}, {5416,12663}, {10881,12510}, {10898,12602}, {11418,12222}, {11448,12275}, {11463,12286}, {11474,12297}

X(12967) = orthic-to-2nd-Kenmotu-diagonals similarity image of X(488)
X(12967) = X(12973)-of-2nd-Kenmotu-free-vertices-triangle
X(12967) = {X(6),X(12979)}-harmonic conjugate of X(12961)

X(12968) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS TO LUCAS CENTRAL

Trilinears (3*a^2-b^2-c^2+2*S)*a : :
Trilinears 2 cos A - sin A (1 + cot ω) : :
X(12968) = 2*X(372)+X(1151) = 2*X(5023)+X(12969)

The reciprocal orthologic center of these triangles is X(3).

X(12968) lies on these lines: {3,6}, {112,6239}, {172,5414}, {230,3070}, {486,7737}, {590,638}, {615,7389}, {640,8253}, {1501,1600}, {1584,1915}, {1613,10133}, {1914,6502}, {1968,5413}, {2079,8989}, {2548,5420}, {3069,11293}, {3155,8577}, {3767,6560}, {5411,8778}, {5416,6252}, {5475,10577}, {6250,6399}, {6460,7735}, {6564,7746}, {6565,7747}, {7749,10576}, {8252,11313}, {9823,10963}, {10311,11474}, {11418,12223}, {11448,12276}, {11463,12287}

X(12968) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1691,12963), (3,6423,6), (6,372,12969), (6,1151,12962), (6,3053,12963), (6,5023,1151), (6,6410,5013), (32,372,6), (32,5017,12963), (371,5062,6), (3312,6424,6), (5058,6420,6)
X(12968) = X(176)-of-2nd-Kenmotu-diagonals-triangle if ABC is acute
X(12968) = X(12974)-of-2nd-Kenmotu-free-vertices-triangle
X(12968) = orthic-to-2nd-Kenmotu-diagonals similarity image of X(6291)
X(12968) = center of conic {X(372),PU(1),PU(2)}

X(12969) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS TO LUCAS(-1) CENTRAL

Trilinears (a^2-3*b^2-3*c^2+6*S)*a : :
X(12969) = 4*X(32)-3*X(12963) = 2*X(5023)-3*X(12968)

The reciprocal orthologic center of these triangles is X(3).

X(12969) lies on these lines: {3,6}, {494,9225}, {615,638}, {640,8252}, {3069,12323}, {3071,8982}, {5411,12172}, {5413,6406}, {5416,6404}, {6400,10881}, {6502,7353}, {9824,10963}, {11418,12224}, {11448,12277}, {11463,12288}, {11474,12299}

X(12969) = reflection of X(1152) in X(372)
X(12969) = X(175)-of-2nd-Kenmotu-diagonals-triangle if ABC is acute
X(12969) = X(12975)-of-2nd-Kenmotu-free-vertices-triangle
X(12969) = orthic-to-2nd-Kenmotu-diagonals similarity image of X(6406)
X(12969) = center of inverse-in-2nd-Kenmotu-circle-of-circumcircle
X(12969) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,372,12968), (6,1152,12963), (6,5013,12962), (6,6426,3053), (39,6420,6), (372,1505,6), (3312,6421,6), (5038,9605,12962), (6418,6422,6)

X(12970) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS TO MIDHEIGHT

Barycentrics (S^2-(4*R^2-SW)*S-2*(4*R^2-SA)*SA)*(SB+SC) : :
X(12970) = 2*X(372)-3*X(11242)

X(12970) lies on these lines: {3,10534}, {4,6}, {30,10666}, {64,1152}, {154,1151}, {184,11473}, {185,5413}, {221,3297}, {371,6759}, {372,6000}, {394,489}, {615,6247}, {1971,12963}, {2067,10535}, {2192,3298}, {2777,12376}, {3069,12324}, {3311,11241}, {3312,12315}, {3357,6396}, {5411,12174}, {5414,6285}, {5416,6254}, {5663,11266}, {5878,6560}, {5907,11514}, {6001,7968}, {6200,10282}, {6225,6460}, {6241,10881}, {6398,13093}, {6410,10606}, {6412,8567}, {6459,11206}, {6502,7355}, {6561,9833}, {8991,10192}, {9729,10963}, {9934,20123}, {10898,12162}, {11381,11474}, {11418,12111}, {11448,12279}, {11463,12290}, {12306,13056}

X(12970) = {X(4),X(17849)}-harmonic conjugate of X(12964)
X(12970) = {X(6),X(1498)}-harmonic conjugate of X(12964)

X(12971) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS TO REFLECTION

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

The reciprocal orthologic center of these triangles is X(6243).

X(12971) lies on these lines: {6,17}, {54,371}, {372,1154}, {486,2888}, {539,10666}, {590,8254}, {1152,12307}, {1493,6419}, {3069,12325}, {3071,12376}, {3299,8953}, {3312,12316}, {3574,6564}, {5411,12175}, {5413,6152}, {5414,6286}, {5416,6255}, {6200,10610}, {6242,10881}, {6288,6565}, {6396,7691}, {6502,7356}, {6561,12254}, {9827,10963}, {10898,12606}, {11418,12226}, {11448,12280}, {11463,12291}, {11474,12300}, {11514,12363}

X(12971) = X(79)-of-2nd-Kenmotu-diagonals-triangle if ABC is acute
X(12971) = X(54)-of-2nd-Kenmotu-free-vertices-triangle
X(12971) = perspector of 2nd Kenmotu diagonals triangle and 2nd Kenmotu free vertices triangle
X(12971) = {X(6),X(195)}-harmonic conjugate of X(12965)

X(12972) = ORTHOLOGIC CENTER OF THESE TRIANGLES: KOSNITA TO LUCAS ANTIPODAL

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

The reciprocal orthologic center of these triangles is X(3).

X(12972) lies on these lines: {3,486}, {15,12981}, {16,12980}, {22,12256}, {24,487}, {25,6290}, {26,159}, {35,12910}, {186,12509}, {371,12966}, {372,12960}, {378,12296}, {389,12229}, {575,12597}, {578,12237}, {631,12320}, {642,6642}, {3155,6503}, {3515,12169}, {5413,9732}, {6119,7393}, {6251,9818}, {6281,9714}, {7488,12221}, {10037,12948}, {10046,12958}, {10067,10831}, {10083,10832}, {10902,12662}, {11449,12274}, {11464,12285}

X(12972) = midpoint of X(3) and X(12978)
X(12972) = orthic-to-Kosnita similarity image of X(487)
X(12972) = reflection of X(i) in X(j) for these (i,j): (9921,26), (12597,575), (12984,3)

X(12973) = ORTHOLOGIC CENTER OF THESE TRIANGLES: KOSNITA TO LUCAS(-1) ANTIPODAL

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

The reciprocal orthologic center of these triangles is X(3).

X(12973) lies on these lines: {3,485}, {15,12983}, {16,12982}, {22,12257}, {24,488}, {25,6289}, {26,159}, {35,12911}, {186,12510}, {371,12967}, {372,12961}, {378,12297}, {389,12230}, {511,8909}, {575,12598}, {578,12238}, {631,12321}, {641,6642}, {3156,6503}, {3515,12170}, {5412,9733}, {6118,7393}, {6250,9818}, {6278,9714}, {7488,12222}, {10037,12949}, {10046,12959}, {10068,10831}, {10084,10832}, {10902,12663}, {11449,12275}, {11464,12286}

X(12973) = midpoint of X(3) and X(12979)
X(12973) = orthic-to-Kosnita similarity image of X(488)
X(12973) = reflection of X(i) in X(j) for these (i,j): (9922,26), (12598,575), (12985,3)

X(12974) = ORTHOLOGIC CENTER OF THESE TRIANGLES: KOSNITA TO LUCAS CENTRAL

Trilinears (-4*(-a^2+b^2+c^2)*S+a^4-(b^2+c^2)*a^2-2*b^2*c^2)*a : :
X(12974) = 3*X(3)-X(12305) = 3*X(3)+X(12313) = 5*X(631)-X(12322) = 2*X(1151)+X(7690) = 3*X(1151)+X(12305)

The reciprocal orthologic center of these triangles is X(3).

X(12974) lies on these lines: {3,6}, {24,6291}, {30,6250}, {35,6283}, {36,7362}, {186,6239}, {378,12298}, {488,5965}, {542,12257}, {631,12322}, {642,7761}, {1599,9306}, {3515,12171}, {3819,5407}, {6252,10902}, {6642,9823}, {6759,8155}, {7488,12223}, {11449,12276}, {11464,12287}

X(12974) = midpoint of X(i) and X(j) for these {i,j}: {3,1151}, {12305,12313}
X(12974) = reflection of X(i) in X(j) for these (i,j): (7690,3), (9974,575)
X(12974) = X(176)-of-Kosnita-triangle if ABC is acute
X(12974) = orthic-to-Kosnita similarity image of X(6291)
X(12974) = {X(3),X(182)}-harmonic conjugate of X(12975)

X(12975) = ORTHOLOGIC CENTER OF THESE TRIANGLES: KOSNITA TO LUCAS(-1) CENTRAL

Trilinears (4*(-a^2+b^2+c^2)*S+a^4-(b^2+c^2)*a^2-2*b^2*c^2)*a : :
X(12975) = 3*X(3)-X(12306) = 3*X(3)+X(12314) = 5*X(631)-X(12323) = 2*X(1152)+X(7692) = 3*X(1152)+X(12306)

The reciprocal orthologic center of these triangles is X(3).

X(12975) lies on these lines: {3,6}, {24,6406}, {30,6251}, {35,6405}, {36,7353}, {186,6400}, {378,12299}, {487,5965}, {542,12256}, {631,12323}, {641,7761}, {1600,9306}, {3515,12172}, {3819,5406}, {6404,10902}, {6642,9824}, {6759,8156}, {7488,12224}, {11449,12277}, {11464,12288}

X(12975) = midpoint of X(i) and X(j) for these {i,j}: {3,1152}, {12306,12314}
X(12975) = reflection of X(i) in X(j) for these (i,j): (7692,3), (9975,575)
X(12975) = X(175)-of-Kosnita-triangle if ABC is acute
X(12975) = orthic-to-Kosnita similarity image of X(6406)
X(12975) = {X(3),X(182)}-harmonic conjugate of X(12974)

X(12976) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS ANTIPODAL TO INNER-SQUARES

Barycentrics -8*S^3*(5*a^4-5*(b^2+c^2)*a^2-2*(b^2-c^2)^2)+(-a^2+b^2+c^2)*((b^2+c^2)*a^6-(b^4-26*b^2*c^2+c^4)*a^4-(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4) : :
X(12976) = 2*S*(2*R^2+S)*X(20)+SW*(2*S+SW)*X(485)

The reciprocal orthologic center of these triangles is X(12977).

X(12976) lies on the line {20,485}

X(12977) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-SQUARES TO LUCAS ANTIPODAL

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

The reciprocal orthologic center of these triangles is X(12976).

X(12977) lies on these lines: {371,3167}, {487,590}, {3564,12426}, {11949,12311}

X(12978) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL TO LUCAS ANTIPODAL

Barycentrics (SB+SC)*(2*S^3-4*R^2*S^2-2*(2*R^2-SA+SW)*SA*S+SA*SW^2) : :
X(12978) = 3*X(6)-2*X(12597)

The reciprocal orthologic center of these triangles is X(3).

X(12978) lies on these lines: {2,12320}, {3,486}, {6,12229}, {22,12221}, {24,12509}, {25,487}, {55,12662}, {110,12274}, {372,10673}, {642,5020}, {1593,12296}, {1598,6290}, {1614,12285}, {3564,5596}, {6251,11479}, {8193,9906}, {9909,9921}, {11414,12256}

X(12978) = midpoint of X(i) and X(j) for these {i,j}: {3,12311}, {12662,12910}
X(12978) = reflection of X(3) in X(12972)
X(12978) = complement of X(12320)
X(12978) = orthic-to-tangential similarity image of X(487)
X(12978) = {X(12960),X(12966)}-harmonic conjugate of X(6)
X(12978) = {X(12980),X(12981)}-harmonic conjugate of X(6)

X(12979) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL TO LUCAS(-1) ANTIPODAL

Barycentrics (SB+SC)*(-2*S^3-4*R^2*S^2+2*(2*R^2-SA+SW)*SA*S+SA*SW^2) : :
X(12979) = 3*X(6)-2*X(12598)

The reciprocal orthologic center of these triangles is X(3).

X(12979) lies on these lines: {2,12321}, {3,485}, {6,12230}, {22,12222}, {24,12510}, {25,488}, {55,12663}, {110,12275}, {371,10669}, {641,5020}, {1593,12297}, {1598,6289}, {1614,12286}, {3564,5596}, {6250,11479}, {8193,9907}, {8996,9909}, {11414,12257}

X(12979) = midpoint of X(i) and X(j) for these {i,j}: {3,12312}, {12663,12911}
X(12979) = reflection of X(3) in X(12973)
X(12979) = complement of X(12321)
X(12979) = orthic-to-tangential similarity image of X(488)
X(12979) = {X(12961),X(12967)}-harmonic conjugate of X(6)
X(12979) = {X(12982),X(12983)}-harmonic conjugate of X(6)

X(12980) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL TO LUCAS ANTIPODAL

Barycentrics ((4*R^2-SW)*(2*S-SW)*S*sqrt(3)+6*S^3-12*R^2*S^2-6*(2*R^2-SA+SW)*SA*S+3*SA*SW^2)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(3).

X(12980) lies on these lines: {6,12229}, {16,12972}, {486,11515}, {487,10641}, {642,10643}, {3564,12982}, {10632,12509}, {10634,12601}, {10636,12662}, {10638,12910}, {10645,12984}, {11408,12169}, {11420,12221}, {11452,12274}, {11466,12285}, {11475,12296}, {11480,12303}, {11485,12311}, {11488,12320}

X(12980) = orthic-to-inner-tri-equilateral similarity image of X(487)
X(12980) = {X(6),X(12978)}-harmonic conjugate of X(12981)

X(12981) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL TO LUCAS ANTIPODAL

Barycentrics (-(4*R^2-SW)*(2*S-SW)*S*sqrt(3)+6*S^3-12*R^2*S^2-6*(2*R^2-SA+SW)*SA*S+3*SA*SW^2)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(3).

X(12981) lies on these lines: {6,12229}, {15,12972}, {486,11516}, {487,10642}, {642,10644}, {1250,12910}, {3564,12983}, {10633,12509}, {10635,12601}, {10637,12662}, {10646,12984}, {11409,12169}, {11421,12221}, {11453,12274}, {11467,12285}, {11476,12296}, {11481,12303}, {11486,12311}, {11489,12320}

X(12981) = orthic-to-outer-tri-equilateral similarity image of X(487)
X(12981) = {X(6),X(12978)}-harmonic conjugate of X(12980)

X(12982) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL TO LUCAS(-1) ANTIPODAL

Barycentrics ((4*R^2-SW)*(-2*S-SW)*S*sqrt(3)-6*S^3-12*R^2*S^2+6*(2*R^2-SA+SW)*SA*S+3*SA*SW^2)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(3).

X(12982) lies on these lines: {6,12230}, {16,12973}, {485,11515}, {488,10641}, {641,10643}, {3564,12980}, {10632,12510}, {10634,12602}, {10636,12663}, {10638,12911}, {10645,12985}, {11408,12170}, {11420,12222}, {11452,12275}, {11466,12286}, {11475,12297}, {11480,12304}, {11485,12312}, {11488,12321}

X(12982) = orthic-to-inner-tri-equilateral similarity image of X(488)
X(12982) = {X(6),X(12979)}-harmonic conjugate of X(12983)

X(12983) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL TO LUCAS(-1) ANTIPODAL

Barycentrics (-(4*R^2-SW)*(-2*S-SW)*S*sqrt(3)-6*S^3-12*R^2*S^2+6*(2*R^2-SA+SW)*SA*S+3*SA*SW^2)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(3).

X(12983) lies on these lines: {6,12230}, {15,12973}, {485,11516}, {488,10642}, {641,10644}, {1250,12911}, {3564,12981}, {10633,12510}, {10635,12602}, {10637,12663}, {10646,12985}, {11409,12170}, {11421,12222}, {11453,12275}, {11467,12286}, {11476,12297}, {11481,12304}, {11486,12312}, {11489,12321}

X(12983) = orthic-to-outer-tri-equilateral similarity image of X(488)
X(12983) = {X(6),X(12979)}-harmonic conjugate of X(12982)

X(12984) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TRINH TO LUCAS ANTIPODAL

Barycentrics (SB+SC)*(S^3-2*R^2*S^2-(14*R^2-SA-2*SW)*SA*S+(6*R^2-SW)*SA*SW) : :

The reciprocal orthologic center of these triangles is X(3).

X(12984) lies on these lines: {3,486}, {24,12296}, {30,9921}, {36,12910}, {376,12320}, {378,487}, {511,12597}, {642,9818}, {1593,6290}, {2071,12221}, {3520,12509}, {3564,12084}, {6200,12960}, {6251,6642}, {6396,12966}, {7688,12662}, {9732,11474}, {10645,12980}, {10646,12981}, {11410,12169}, {11413,12256}, {11430,12229}, {11438,12237}, {11454,12274}, {11468,12285}

X(12984) = reflection of X(12972) in X(3)
X(12984) = orthic-to-Trinh similarity image of X(487)

X(12985) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TRINH TO LUCAS(-1) ANTIPODAL

Barycentrics (SB+SC)*(-S^3-2*R^2*S^2+(14*R^2-SA-2*SW)*SA*S+(6*R^2-SW)*SA*SW) : :

The reciprocal orthologic center of these triangles is X(3).

X(12985) lies on these lines: {3,485}, {24,12297}, {30,9922}, {36,12911}, {376,12321}, {378,488}, {511,12598}, {641,9818}, {1593,6289}, {2071,12222}, {3520,12510}, {3564,12084}, {6200,12961}, {6250,6642}, {6396,12967}, {7688,12663}, {9733,11473}, {10645,12982}, {10646,12983}, {11410,12170}, {11413,12257}, {11430,12230}, {11438,12238}, {11454,12275}, {11468,12286}

X(12985) = midpoint of X(3) and X(12304)
X(12985) = reflection of X(12973) in X(3)
X(12985) = orthic-to-Trinh similarity image of X(488)

X(12986) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO MIDHEIGHT

Barycentrics (SB+SC)*((4*(SA+SW)*R^2-(SA+1/2*SW)*SW)*S^2-S*(SA*(64*R^4-4*(3*SA+2*SW)*R^2+SA*SW)-2*(6*R^2-SW)*S^2)-(8*R^2-SA-SW)*SA*SW^2) : :

The reciprocal orthologic center of these triangles is X(4).

X(12986) lies on these lines: {30,12426}, {64,493}, {1498,11828}, {2777,12894}, {2883,8222}, {5878,8220}, {6000,10669}, {6225,6462}, {6247,8212}, {6266,8218}, {6267,8216}, {6461,12987}, {7355,11947}, {7973,8210}, {8188,9899}, {8194,9914}, {8201,12468}, {8214,12779}, {10060,11951}, {10076,11953}, {10875,12502}, {10945,12920}, {10951,12930}, {11377,12262}, {11381,11394}, {11503,12335}, {11840,12202}, {11846,12250}, {11907,12791}, {11930,12940}, {11932,12950}

X(12986) = X(64)-of-Lucas-homothetic-triangle

X(12987) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO MIDHEIGHT

Barycentrics (SB+SC)*((4*(SA+SW)*R^2-(SA+1/2*SW)*SW)*S^2+S*(SA*(64*R^4-4*(3*SA+2*SW)*R^2+SA*SW)-2*(6*R^2-SW)*S^2)-(8*R^2-SA-SW)*SA*SW^2) : :

The reciprocal orthologic center of these triangles is X(4).

X(12987) lies on these lines: {30,12427}, {64,494}, {1498,11829}, {2777,12895}, {2883,8223}, {5878,8221}, {6000,10673}, {6225,6463}, {6247,8213}, {6266,8219}, {6267,8217}, {6461,12986}, {7355,11948}, {7973,8211}, {8189,9899}, {8195,9914}, {8202,12468}, {8209,12469}, {8215,12779}, {10060,11952}, {10076,11954}, {10876,12502}, {10946,12920}, {10952,12930}, {11378,12262}, {11381,11395}, {11504,12335}, {11841,12202}, {11847,12250}, {11908,12791}, {11931,12940}, {11933,12950}

X(12987) = X(64)-of-Lucas(-1)-homothetic-triangle

X(12988) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO INNER-NAPOLEON

Barycentrics 2*(3*SA^2+sqrt(3)*(4*R^2*SA-SA*SW+SW^2)-SW^2)*S^2+((24*R^2*SA-sqrt(3)*(SW+2*SA)*SW)*(SA-SW)+2*(3*SA+4*R^2-3*SW)*S^2)*S+3*(SA-SW)*SA*SW^2 : :

The reciprocal orthologic center of these triangles is X(3).

X(12988) lies on these lines: {14,493}, {530,12352}, {531,12152}, {542,12590}, {617,6462}, {619,8222}, {5474,11828}, {5479,8212}, {5613,8220}, {6269,8218}, {6271,8216}, {6461,12989}, {6773,11846}, {7974,8210}, {8188,9900}, {8194,9915}, {8214,12780}, {9981,10875}, {10061,11951}, {10077,11953}, {10945,12921}, {10951,12931}, {11377,11706}, {11394,12141}, {11503,12336}, {11840,12204}, {11907,12792}, {11930,12941}, {11932,12951}

X(12989) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO INNER-NAPOLEON

Barycentrics 2*(3*SA^2-sqrt(3)*(4*R^2*SA-SA*SW+SW^2)-SW^2)*S^2-((24*R^2*SA+sqrt(3)*(SW+2*SA)*SW)*(SA-SW)+2*(3*SA+4*R^2-3*SW)*S^2)*S+3*(SA-SW)*SA*SW^2 : :

The reciprocal orthologic center of these triangles is X(3).

X(12989) lies on these lines: {14,494}, {530,12353}, {531,12153}, {542,12591}, {617,6463}, {619,8223}, {5474,11829}, {5479,8213}, {5613,8221}, {6269,8219}, {6271,8217}, {6461,12988}, {6773,11847}, {7974,8211}, {8189,9900}, {8195,9915}, {8215,12780}, {9981,10876}, {10061,11952}, {10077,11954}, {10946,12921}, {10952,12931}, {11378,11706}, {11395,12141}, {11504,12336}, {11841,12204}, {11908,12792}, {11931,12941}, {11933,12951}

X(12990) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO OUTER-NAPOLEON

Barycentrics 2*(3*SA^2-sqrt(3)*(4*R^2*SA-SA*SW+SW^2)-SW^2)*S^2+((24*R^2*SA+sqrt(3)*(SW+2*SA)*SW)*(SA-SW)+2*(3*SA+4*R^2-3*SW)*S^2)*S+3*(SA-SW)*SA*SW^2 : :

The reciprocal orthologic center of these triangles is X(3).

X(12990) lies on these lines: {13,493}, {530,12152}, {531,12352}, {542,12590}, {616,6462}, {618,8222}, {5473,11828}, {5478,8212}, {5617,8220}, {6268,8218}, {6270,8216}, {6461,12991}, {6770,11846}, {7975,8210}, {8188,9901}, {8194,9916}, {8214,12781}, {9982,10875}, {10062,11951}, {10078,11953}, {10945,12922}, {10951,12932}, {11377,11705}, {11394,12142}, {11503,12337}, {11840,12205}, {11907,12793}, {11930,12942}, {11932,12952}

X(12991) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO OUTER-NAPOLEON

Barycentrics 2*(3*SA^2+sqrt(3)*(4*R^2*SA-SA*SW+SW^2)-SW^2)*S^2-((24*R^2*SA-sqrt(3)*(SW+2*SA)*SW)*(SA-SW)+2*(3*SA+4*R^2-3*SW)*S^2)*S+3*(SA-SW)*SA*SW^2 : :

The reciprocal orthologic center of these triangles is X(3).

X(12991) lies on these lines: {13,494}, {530,12153}, {531,12353}, {542,12591}, {616,6463}, {618,8223}, {5473,11829}, {5478,8213}, {5617,8221}, {6268,8219}, {6270,8217}, {6461,12990}, {6770,11847}, {7975,8211}, {8189,9901}, {8195,9916}, {8215,12781}, {9982,10876}, {10062,11952}, {10078,11954}, {10946,12922}, {10952,12932}, {11378,11705}, {11395,12142}, {11504,12337}, {11841,12205}, {11908,12793}, {11931,12942}, {11933,12952}

X(12992) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO 1st NEUBERG

Barycentrics SA^2*S^2+S*((8*SA^2-4*SW^2)*R^2+(SA-SW)*SW^2+(4*R^2+SA-SW)*S^2)+(2*SA^2-SW^2)*SW^2 : :

The reciprocal orthologic center of these triangles is X(3).

X(12992) lies on these lines: {39,8222}, {76,493}, {194,6462}, {384,11840}, {511,9838}, {538,12152}, {730,12440}, {732,12590}, {2782,10669}, {3095,8220}, {5969,12352}, {6248,8212}, {6272,8218}, {6273,8216}, {6461,12993}, {7976,8210}, {8188,9902}, {8194,9917}, {8201,12474}, {8208,12475}, {8214,12782}, {9983,10875}, {10063,11951}, {10079,11953}, {10945,12923}, {10951,12933}, {11257,11828}, {11377,12263}, {11394,12143}, {11503,12338}, {11846,12251}, {11907,12794}, {11930,12943}, {11932,12953}

X(12993) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO 1st NEUBERG

Barycentrics SA^2*S^2-S*((8*SA^2-4*SW^2)*R^2+(SA-SW)*SW^2+(4*R^2+SA-SW)*S^2)+(2*SA^2-SW^2)*SW^2 : :

The reciprocal orthologic center of these triangles is X(3).

X(12993) lies on these lines: {39,8223}, {76,494}, {194,6463}, {384,11841}, {511,9839}, {538,12153}, {730,12441}, {732,12591}, {2782,10673}, {3095,8221}, {5969,12353}, {6248,8213}, {6272,8219}, {6273,8217}, {6461,12992}, {7976,8211}, {8189,9902}, {8195,9917}, {8202,12474}, {8209,12475}, {8215,12782}, {9983,10876}, {10063,11952}, {10079,11954}, {10946,12923}, {10952,12933}, {11257,11829}, {11378,12263}, {11395,12143}, {11504,12338}, {11847,12251}, {11908,12794}, {11931,12943}, {11933,12953}

X(12994) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(12994) lies on these lines: {83,493}, {732,12590}, {754,12152}, {1271,2896}, {6249,8212}, {6274,8218}, {6287,8220}, {6292,8222}, {6461,12995}, {7977,8210}, {8188,9903}, {8194,9918}, {8201,12476}, {8208,12477}, {8214,12783}, {10064,11951}, {10080,11953}, {10945,12924}, {10951,12934}, {11377,12264}, {11394,12144}, {11503,12339}, {11828,12122}, {11840,12206}, {11846,12252}, {11907,12795}, {11930,12944}, {11932,12954}

X(12995) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(12995) lies on these lines: {83,494}, {732,12591}, {754,12153}, {1270,2896}, {6249,8213}, {6275,8217}, {6287,8221}, {6292,8223}, {6461,12994}, {7977,8211}, {8189,9903}, {8195,9918}, {8202,12476}, {8209,12477}, {8215,12783}, {10064,11952}, {10080,11954}, {10946,12924}, {10952,12934}, {11378,12264}, {11395,12144}, {11504,12339}, {11829,12122}, {11841,12206}, {11847,12252}, {11908,12795}, {11931,12944}, {11933,12954}

X(12996) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO 1st ORTHOSYMMEDIAL

Barycentrics (-S^2*(-((16*SA+24*SW)*R^2+SA^2-6*SA*SW-6*SW^2)*SW^2+((12*SA+12*SW)*R^2-(4*SA+5*SW)*SW)*S^2)+4*S*(-SA*SW^2*(4*R^2-SW)*(8*R^2-SA)+(-4*R^2+2*SW)*S^4+(16*SW*R^4+(-SA^2-4*SA*SW+4*SW^2)*R^2+(SA^2-2*SW^2)*SW)*S^2)-(4*(4*R^2-SW))*SA*SW^4)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(4).

X(12996) lies on these lines: {112,11828}, {127,8212}, {132,8222}, {493,1297}, {2799,12186}, {2806,12765}, {3320,11947}, {6461,12997}, {6462,12384}, {8188,12408}, {8194,12413}, {8214,12784}, {8216,12805}, {8218,12806}, {8220,12918}, {9517,12377}, {9530,12152}, {10875,12503}, {11377,12265}, {11394,12145}, {11503,12340}, {11840,12207}, {11846,12253}, {11907,12796}, {11930,12945}, {11932,12955}

X(12996) = X(1297)-of-Lucas-homothetic-triangle

X(12997) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO 1st ORTHOSYMMEDIAL

Barycentrics (-S^2*(-((16*SA+24*SW)*R^2+SA^2-6*SA*SW-6*SW^2)*SW^2+((12*SA+12*SW)*R^2-(4*SA+5*SW)*SW)*S^2)-4*S*(-SA*SW^2*(4*R^2-SW)*(8*R^2-SA)+(-4*R^2+2*SW)*S^4+(16*SW*R^4+(-SA^2-4*SA*SW+4*SW^2)*R^2+(SA^2-2*SW^2)*SW)*S^2)-4*(4*R^2-SW)*SA*SW^4)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(4).

X(12997) lies on these lines: {112,11829}, {127,8213}, {132,8223}, {494,1297}, {2799,12187}, {2806,12766}, {3320,11948}, {6461,12996}, {6463,12384}, {8189,12408}, {8195,12413}, {8215,12784}, {8217,12805}, {8219,12806}, {8221,12918}, {9517,12378}, {9530,12153}, {10876,12503}, {11378,12265}, {11395,12145}, {11504,12340}, {11841,12207}, {11847,12253}, {11908,12796}, {11931,12945}, {11933,12955}

X(12997) = X(1297)-of-Lucas(-1)-homothetic-triangle

X(12998) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO REFLECTION

Barycentrics ((-20*(SA+SW)*R^2+(8*SA+7*SW)*SW)*S^2+4*S*(SA*(8*R^4-(3*SA+4*SW)*R^2+2*SA*SW)-4*(3*R^2-SW)*S^2)+(4*R^2+SA-2*SW)*SA*SW^2)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(4).

X(12998) lies on these lines: {54,493}, {195,11949}, {539,12152}, {1154,10669}, {1209,8222}, {2888,6462}, {3574,8212}, {6276,8218}, {6277,8216}, {6288,8220}, {6461,12999}, {7691,11828}, {7979,8210}, {8188,9905}, {8194,9920}, {8201,12480}, {8208,12481}, {8214,12785}, {9985,10875}, {10066,11951}, {10082,11953}, {10628,12377}, {10945,12926}, {10951,12936}, {11377,12266}, {11394,11576}, {11503,12341}, {11840,12208}, {11846,12254}, {11907,12797}, {11930,12946}, {11932,12956}

X(12998) = X(54)-of-Lucas-homothetic-triangle

X(12999) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO REFLECTION

Barycentrics ((-20*(SA+SW)*R^2+(8*SA+7*SW)*SW)*S^2-4*S*(SA*(8*R^4-(3*SA+4*SW)*R^2+2*SA*SW)-4*(3*R^2-SW)*S^2)+(4*R^2+SA-2*SW)*SA*SW^2)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(4).

X(12999) lies on these lines: {54,494}, {195,11950}, {539,12153}, {1154,10673}, {1209,8223}, {2888,6463}, {3574,8213}, {6276,8219}, {6277,8217}, {6288,8221}, {6461,12998}, {7691,11829}, {7979,8211}, {8189,9905}, {8195,9920}, {8202,12480}, {8209,12481}, {8215,12785}, {9985,10876}, {10066,11952}, {10082,11954}, {10628,12378}, {10946,12926}, {10952,12936}, {11378,12266}, {11395,11576}, {11504,12341}, {11841,12208}, {11847,12254}, {11908,12797}, {11931,12946}, {11933,12956}

X(12999) = X(54)-of-Lucas(-1)-homothetic-triangle

X(13000) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO 1st SCHIFFLER

Barycentrics (a^11+(b+c)*a^10+(3*b^2+4*b*c+3*c^2)*a^9-(b+c)*(5*b^2-2*b*c+5*c^2)*a^8-(14*b^4+14*c^4+3*b*c*(4*b^2+13*b*c+4*c^2))*a^7+(b+c)*(10*b^4+10*c^4-b*c*(8*b^2-15*b*c+8*c^2))*a^6+2*(7*b^4+7*c^4+b*c*(14*b^2+15*b*c+14*c^2))*(b^2+c^2)*a^5-2*(b+c)*(5*b^6+5*c^6-2*(3*b^4+3*c^4+4*b*c*(b^2+b*c+c^2))*b*c)*a^4-(3*b^6+3*c^6+4*(5*b^4+5*c^4-b*c*(b^2+6*b*c+c^2))*b*c)*(b^2+c^2)*a^3+(b^3+c^3)*(b^2+c^2)*(5*b^4+5*c^4-3*b*c*(b+c)^2)*a^2-(b^4-c^4)^2*(b^2+c^2)*a-(b^4-c^4)^2*(b^2-c^2)*(b-c))*S-a^2*(a^11-(b+c)*a^10-(5*b^2+4*b*c+5*c^2)*a^9+(b+c)*(5*b^2-2*b*c+5*c^2)*a^8+(10*b^4+10*c^4+b*c*(16*b^2+9*b*c+16*c^2))*a^7-(b+c)*(10*b^4+10*c^4-b*c*(8*b^2-7*b*c+8*c^2))*a^6-2*(5*b^6+5*c^6+2*(6*b^4+6*c^4-b*c*(3*b^2-8*b*c+3*c^2))*b*c)*a^5+2*(b+c)*(5*b^6+5*c^6-2*b*c*(3*b^4+4*b^2*c^2+3*c^4))*a^4+(5*b^8+5*c^8+(16*b^6+16*c^6-(31*b^4+31*c^4+8*b*c*(2*b^2+3*b*c+2*c^2))*b*c)*b*c)*a^3-(b+c)*(5*b^8+5*c^8-(8*b^6+8*c^6+(b^4+c^4+4*b*c*(2*b^2-13*b*c+2*c^2))*b*c)*b*c)*a^2-(b^2-c^2)^2*(b^6+c^6+(4*b^4+4*c^4-b*c*(13*b^2+24*b*c+13*c^2))*b*c)*a+(b^4-c^4)^2*(b^2-c^2)*(b-c)) : :

The reciprocal orthologic center of these triangles is X(79).

X(13000) lies on these lines: {493,10266}, {6461,13001}, {8188,12409}, {8194,12414}, {8212,12600}, {8214,12786}, {8216,12807}, {8218,12808}, {8220,12919}, {10875,12504}, {11377,12267}, {11394,12146}, {11840,12209}, {11846,12255}, {11930,12947}, {11932,12957}

X(13000) = X(10266)-of-Lucas-homothetic-triangle

X(13001) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO 1st SCHIFFLER

Barycentrics -(a^11+(b+c)*a^10+(3*b^2+4*b*c+3*c^2)*a^9-(b+c)*(5*b^2-2*b*c+5*c^2)*a^8-(14*b^4+14*c^4+3*b*c*(4*b^2+13*b*c+4*c^2))*a^7+(b+c)*(10*b^4+10*c^4-b*c*(8*b^2-15*b*c+8*c^2))*a^6+2*(7*b^4+7*c^4+b*c*(14*b^2+15*b*c+14*c^2))*(b^2+c^2)*a^5-2*(b+c)*(5*b^6+5*c^6-2*(3*b^4+3*c^4+4*b*c*(b^2+b*c+c^2))*b*c)*a^4-(3*b^6+3*c^6+4*(5*b^4+5*c^4-b*c*(b^2+6*b*c+c^2))*b*c)*(b^2+c^2)*a^3+(b^3+c^3)*(b^2+c^2)*(5*b^4+5*c^4-3*b*c*(b+c)^2)*a^2-(b^4-c^4)^2*(b^2+c^2)*a-(b^4-c^4)^2*(b^2-c^2)*(b-c))*S-a^2*(a^11-(b+c)*a^10-(5*b^2+4*b*c+5*c^2)*a^9+(b+c)*(5*b^2-2*b*c+5*c^2)*a^8+(10*b^4+10*c^4+b*c*(16*b^2+9*b*c+16*c^2))*a^7-(b+c)*(10*b^4+10*c^4-b*c*(8*b^2-7*b*c+8*c^2))*a^6-2*(5*b^6+5*c^6+2*(6*b^4+6*c^4-b*c*(3*b^2-8*b*c+3*c^2))*b*c)*a^5+2*(b+c)*(5*b^6+5*c^6-2*b*c*(3*b^4+4*b^2*c^2+3*c^4))*a^4+(5*b^8+5*c^8+(16*b^6+16*c^6-(31*b^4+31*c^4+8*b*c*(2*b^2+3*b*c+2*c^2))*b*c)*b*c)*a^3-(b+c)*(5*b^8+5*c^8-(8*b^6+8*c^6+(b^4+c^4+4*b*c*(2*b^2-13*b*c+2*c^2))*b*c)*b*c)*a^2-(b^2-c^2)^2*(b^6+c^6+(4*b^4+4*c^4-b*c*(13*b^2+24*b*c+13*c^2))*b*c)*a+(b^4-c^4)^2*(b^2-c^2)*(b-c)) : :

The reciprocal orthologic center of these triangles is X(79).

X(13001) lies on these lines: {494,10266}, {6461,13000}, {8189,12409}, {8195,12414}, {8213,12600}, {8215,12786}, {8217,12807}, {8219,12808}, {8221,12919}, {10876,12504}, {11378,12267}, {11395,12146}, {11504,12342}, {11841,12209}, {11847,12255}, {11931,12947}, {11933,12957}

X(13001) = X(10266)-of-Lucas(-1)-homothetic-triangle

X(13002) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO INNER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(13002) lies on these lines: {485,487}, {486,493}, {642,8222}, {3564,12426}, {6251,8212}, {6280,8218}, {6281,8216}, {6290,8220}, {6461,13003}, {7980,8210}, {8188,9906}, {8194,9921}, {8201,12484}, {8208,12485}, {8214,12787}, {9838,12976}, {9986,10875}, {10067,11951}, {10083,11953}, {10945,12928}, {10951,12938}, {11377,12268}, {11394,12147}, {11503,12343}, {11828,12123}, {11840,12210}, {11846,12256}, {11907,12799}, {11930,12948}, {11932,12958}, {11949,12601}

X(13003) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO INNER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(13003) lies on these lines: {193,372}, {486,494}, {642,8223}, {3564,12427}, {6251,8213}, {6280,8219}, {6281,8217}, {6290,8221}, {6461,13002}, {7980,8211}, {8189,9906}, {8195,9921}, {8202,12484}, {8209,12485}, {8215,12787}, {9986,10876}, {10067,11952}, {10083,11954}, {10946,12928}, {10952,12938}, {11378,12268}, {11395,12147}, {11504,12343}, {11829,12123}, {11841,12210}, {11847,12256}, {11908,12799}, {11931,12948}, {11933,12958}, {11950,12601}

X(13004) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO OUTER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(13004) lies on these lines: {193,371}, {485,493}, {641,8222}, {3564,12426}, {6250,8212}, {6278,8218}, {6279,8216}, {6289,8220}, {6461,13005}, {7981,8210}, {8188,9907}, {8194,9922}, {8201,12486}, {8208,12487}, {8214,12788}, {9987,10875}, {10068,11951}, {10084,11953}, {10945,12929}, {10951,12939}, {11377,12269}, {11394,12148}, {11503,12344}, {11828,12124}, {11840,12211}, {11846,12257}, {11907,12800}, {11930,12949}, {11932,12959}, {11949,12602}

X(13005) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO OUTER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(13005) lies on these lines: {485,494}, {486,488}, {641,8223}, {3564,12427}, {6250,8213}, {6278,8219}, {6279,8217}, {6289,8221}, {6461,13004}, {7981,8211}, {8189,9907}, {8195,9922}, {8202,12486}, {8209,12487}, {8215,12788}, {9987,10876}, {10068,11952}, {10084,11954}, {10946,12929}, {10952,12939}, {11378,12269}, {11395,12148}, {11504,12344}, {11829,12124}, {11841,12211}, {11847,12257}, {11908,12800}, {11931,12949}, {11933,12959}, {11950,12602}

X(13006) = CROSSSUM OF X(6) AND X(11)

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

If you have The Geometer's Sketchpad, you can view X(13006) as the center of the ellipse IE59.

X(13006) lies on these lines: {1,39}, {3,1415}, {6,906}, {9,216}, {37,570}, {44,3003}, {45,566}, {101,7117}, {115,8068}, {232,1785}, {498,5283}, {572,2197}, {672,3002}, {800,1743}, {952,11998}, {1100,5421}, {1107,10039}, {1506,8070}, {1575,1737}, {1772,3125}, {3767,10320}, {5013,8071}, {5069,9456}, {5254,10523}, {5286,10321}, {5299,10315}, {5393,8962}, {7738,10629}

X(13006) = midpoint of X(2007) and X(2008)
X(13006) = complement of X(34387)
X(13006) = crosspoint of X(2) and X(59)
X(13006) = crosssum of X(6) and X(11)
X(13006) = polar conjugate of isogonal conjugate of X(23198)
X(13006) = center of the inscribed ellipse IE59; see the preamble just before X(12841)
X(13006) = {X(2275),X(2276)}-harmonic conjugate of X(9620)
X(13006) = harmonic center of incircle and Gallatly circle
X(13006) = X(i)-complementary conjugate of X(j) for these (i,j): (59, 2887), (692, 124), (1110, 1329), (1415, 116), (2149, 141), (4564, 626)

Orthologic centers: X(13007)-X(13135)

Centers X(13007)-X(13135) were contributed by César Eliud Lozada, April, 5, 2017.

X(13007) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO LUCAS REFLECTION

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

The reciprocal orthologic center of these triangles is X(10670).

X(13007) lies on these lines: {3,13009}, {4,13023}, {25,13051}, {427,13025}, {1593,12170}, {1993,13015}, {3515,13049}, {3516,13021}, {5410,13045}, {5411,13047}, {7071,13043}, {7395,13039}, {7484,13027}, {7592,13017}, {9777,13013}, {11284,13053}, {11402,13011}, {11403,13019}, {11405,13037}, {11406,13041}, {11408,13057}, {11409,13059}, {11410,13061}

X(13008) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO LUCAS(-1) REFLECTION

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

The reciprocal orthologic center of these triangles is X(10674).

X(13008) lies on these lines: {3,13010}, {4,13024}, {25,13052}, {427,13026}, {1593,12169}, {1993,13016}, {3515,13050}, {3516,13022}, {5410,13046}, {5411,13048}, {7071,13044}, {7395,13040}, {7484,13028}, {7592,13018}, {9777,13014}, {11284,13054}, {11402,13012}, {11403,13020}, {11405,13038}, {11406,13042}, {11408,13058}, {11409,13060}, {11410,13062}

X(13009) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP TO LUCAS REFLECTION

Barycentrics S^2*((2*R^2-SA)*(2*SA-SW)-S^2)-S*((8*R^2+SW)*(SA-SW)*SA+6*S^2*R^2)-(SA-SW)*SA*SW^2 : :

The reciprocal orthologic center of these triangles is X(10670).

X(13009) lies on these lines: {2,13027}, {3,13007}, {4,13039}, {20,6462}, {22,13055}, {1370,13025}, {1975,13033}, {2071,13061}, {2979,13015}, {3060,13013}, {3100,13043}, {3101,13041}, {3146,13019}, {5012,13011}, {7488,13049}, {11412,13017}, {11413,13021}, {11414,13023}, {11416,13037}, {11417,13045}, {11418,13047}, {11420,13057}, {11421,13059}

X(13009) = midpoint of X(11412) and X(13017)
X(13009) = reflection of X(i) in X(j) for these (i,j): (4,13039), (3146,13019), (13035,3)
X(13009) = anticomplement of X(13051)

X(13010) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP TO LUCAS(-1) REFLECTION

Barycentrics S^2*((2*R^2-SA)*(2*SA-SW)-S^2)+S*((8*R^2+SW)*(SA-SW)*SA+6*S^2*R^2)-(SA-SW)*SA*SW^2 : :

The reciprocal orthologic center of these triangles is X(10674).

X(13010) lies on these lines: {2,13028}, {3,13008}, {4,13040}, {20,6463}, {22,13056}, {1370,13026}, {1975,13034}, {2071,13062}, {2979,13016}, {3060,13014}, {3100,13044}, {3101,13042}, {3146,13020}, {5012,13012}, {7488,13050}, {11412,13018}, {11413,13022}, {11414,13024}, {11416,13038}, {11417,13046}, {11418,13048}, {11420,13058}, {11421,13060}

X(13010) = midpoint of X(11412) and X(13018)
X(13010) = reflection of X(i) in X(j) for these (i,j): (4,13040), (3146,13020)
X(13010) = anticomplement of X(13052)

X(13011) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS REFLECTION TO ANTI-CONWAY

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

The reciprocal orthologic center of these triangles is X(10670).

X(13011) lies on these lines: {6,13013}, {54,13035}, {182,13027}, {184,13051}, {389,13049}, {569,13039}, {578,12230}, {5012,13009}, {9306,13053}, {11402,13007}, {11422,13015}, {11423,13017}, {11424,13019}, {11425,13021}, {11426,13023}, {11427,13025}, {11428,13041}, {11429,13043}, {11430,13061}

X(13012) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) REFLECTION TO ANTI-CONWAY

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

The reciprocal orthologic center of these triangles is X(10674).

X(13012) lies on these lines: {6,13014}, {54,13036}, {182,13028}, {184,13052}, {389,13050}, {569,13040}, {578,12229}, {5012,13010}, {9306,13054}, {11402,13008}, {11422,13016}, {11423,13018}, {11424,13020}, {11425,13022}, {11426,13024}, {11427,13026}, {11428,13042}, {11429,13044}, {11430,13062}

X(13013) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CONWAY TO LUCAS REFLECTION

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

The reciprocal orthologic center of these triangles is X(10670).

X(13013) lies on these lines: {6,13011}, {51,13051}, {52,13039}, {185,13019}, {389,12238}, {511,13027}, {578,13049}, {3060,13009}, {3567,13035}, {5640,13015}, {5943,13053}, {9777,13007}, {9781,13017}, {9786,13021}, {11432,13023}, {11433,13025}, {11435,13041}, {11436,13043}, {11438,13061}

X(13014) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CONWAY TO LUCAS(-1) REFLECTION

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

The reciprocal orthologic center of these triangles is X(10674).

X(13014) lies on these lines: {6,13012}, {51,13052}, {52,13040}, {185,13020}, {389,12237}, {511,13028}, {578,13050}, {3060,13010}, {3567,13036}, {5640,13016}, {5943,13054}, {9777,13008}, {9781,13018}, {9786,13022}, {11432,13024}, {11433,13026}, {11435,13042}, {11436,13044}, {11438,13062}

X(13015) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO LUCAS REFLECTION

Barycentrics (S^2*(3*SA^2+2*SA*SW+SW^2+32*R^4+(-16*SA-10*SW)*R^2+S^2)-2*S*(SA*(32*R^4+(-5*SA+6*SW)*R^2-(SA+2*SW)*SW)+(SW-7*R^2)*S^2)-((6*SA+12*SW)*R^2-(3*SA+2*SW)*SW)*SA*SW)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(10670).

X(13015) lies on these lines: {3,13017}, {110,13055}, {1993,13007}, {2979,13009}, {3060,13051}, {5640,13013}, {5889,13035}, {7998,13027}, {11422,13011}, {11439,13019}, {11440,13021}, {11441,13023}, {11442,13025}, {11443,13037}, {11444,13039}, {11445,13041}, {11446,13043}, {11447,13045}, {11448,13047}, {11449,13049}, {11451,13053}, {11452,13057}, {11453,13059}, {11454,13061}, {12111,12275}

X(13016) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO LUCAS(-1) REFLECTION

Barycentrics (S^2*(3*SA^2+2*SA*SW+SW^2+32*R^4+(-16*SA-10*SW)*R^2+S^2)+2*S*(SA*(32*R^4+(-5*SA+6*SW)*R^2-(SA+2*SW)*SW)+(SW-7*R^2)*S^2)-((6*SA+12*SW)*R^2-(3*SA+2*SW)*SW)*SA*SW)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(10674).

X(13016) lies on these lines: {3,13018}, {110,13056}, {1993,13008}, {2979,13010}, {3060,13052}, {5640,13014}, {5889,13036}, {7998,13028}, {11422,13012}, {11439,13020}, {11440,13022}, {11441,13024}, {11442,13026}, {11443,13038}, {11444,13040}, {11445,13042}, {11446,13044}, {11447,13046}, {11448,13048}, {11449,13050}, {11451,13054}, {11452,13058}, {11453,13060}, {11454,13062}, {12111,12274}

X(13017) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO LUCAS REFLECTION

Barycentrics (S^2*(32*R^4-10*(2*SA+SW)*R^2+S^2+3*SA^2+2*SA*SW+SW^2)-2*S*(SA*(28*R^4-5*(SA-2*SW)*R^2-(SA+2*SW)*SW)+(SW-7*R^2)*S^2)+(8*R^4-2*(3*SA+8*SW)*R^2+(3*SA+2*SW)*SW)*SA*SW)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(10670).

X(13017) lies on these lines: {3,13015}, {74,13021}, {1614,13055}, {3567,13051}, {5890,13035}, {6241,12286}, {7592,13007}, {7999,13027}, {9781,13013}, {11412,13009}, {11423,13011}, {11455,13019}, {11456,13023}, {11457,13025}, {11458,13037}, {11459,13039}, {11460,13041}, {11461,13043}, {11462,13045}, {11463,13047}, {11464,13049}, {11465,13053}, {11466,13057}, {11467,13059}, {11468,13061}

X(13018) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO LUCAS(-1) REFLECTION

Barycentrics (S^2*(32*R^4-10*(2*SA+SW)*R^2+S^2+3*SA^2+2*SA*SW+SW^2)+2*S*(SA*(28*R^4-5*(SA-2*SW)*R^2-(SA+2*SW)*SW)+(SW-7*R^2)*S^2)+(8*R^4-2*(3*SA+8*SW)*R^2+(3*SA+2*SW)*SW)*SA*SW)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(10674).

X(13018) lies on these lines: {3,13016}, {74,13022}, {1614,13056}, {3567,13052}, {5890,13036}, {6241,12285}, {7592,13008}, {7999,13028}, {9781,13014}, {11412,13010}, {11423,13012}, {11455,13020}, {11456,13024}, {11457,13026}, {11458,13038}, {11459,13040}, {11460,13042}, {11461,13044}, {11462,13046}, {11463,13048}, {11464,13050}, {11465,13054}, {11466,13058}, {11467,13060}, {11468,13062}

X(13019) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO LUCAS REFLECTION

Barycentrics (2*S^2-(16*R^2+SA-7*SW)*S-SA^2+3*SW^2-8*R^2*SW)*SB*SC : :

The reciprocal orthologic center of these triangles is X(10670).

X(13019) lies on these lines: {4,488}, {20,13027}, {24,13061}, {25,13021}, {30,13039}, {34,13043}, {185,13013}, {378,13049}, {1593,13055}, {1597,13023}, {3070,12231}, {3091,13053}, {3146,13009}, {11403,13007}, {11424,13011}, {11439,13015}, {11455,13017}, {11470,13037}, {11471,13041}, {11473,13045}, {11474,13047}, {11475,13057}, {11476,13059}

X(13019) = midpoint of X(3146) and X(13009)
X(13019) = reflection of X(i) in X(j) for these (i,j): (20,13027), (185,13013), (13051,4)

X(13020) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO LUCAS(-1) REFLECTION

Barycentrics (2*S^2+(16*R^2+SA-7*SW)*S-SA^2+3*SW^2-8*R^2*SW)*SB*SC : :

The reciprocal orthologic center of these triangles is X(10674).

X(13020) lies on these lines: {4,487}, {20,13028}, {24,13062}, {25,13022}, {30,13040}, {34,13044}, {185,13014}, {378,13050}, {1593,13056}, {1597,13024}, {3071,12232}, {3091,13054}, {3146,13010}, {11403,13008}, {11424,13012}, {11439,13016}, {11455,13018}, {11470,13038}, {11471,13042}, {11473,13046}, {11474,13048}, {11475,13058}, {11476,13060}

X(13020) = midpoint of X(3146) and X(13010)
X(13020) = reflection of X(i) in X(j) for these (i,j): (20,13028), (185,13014), (13052,4)

X(13021) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH TO LUCAS REFLECTION

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

The reciprocal orthologic center of these triangles is X(10670).

X(13021) lies on these lines: {3,485}, {20,13025}, {25,13019}, {56,13043}, {74,13017}, {311,1975}, {378,13035}, {1151,13045}, {1152,13047}, {1350,11828}, {1593,13051}, {3516,13007}, {5584,13041}, {9786,13013}, {11413,13009}, {11425,13011}, {11440,13015}, {11477,13037}, {11479,13053}, {11480,13057}, {11481,13059}

X(13021) = midpoint of X(20) and X(13025)
X(13021) = reflection of X(i) in X(j) for these (i,j): (3,13061), (11477,13037), (13055,3)

X(13022) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH TO LUCAS(-1) REFLECTION

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

The reciprocal orthologic center of these triangles is X(10674).

X(13022) lies on these lines: {3,486}, {20,13026}, {25,13020}, {56,13044}, {74,13018}, {311,1975}, {378,13036}, {1151,13046}, {1152,13048}, {1350,11829}, {1593,13052}, {3516,13008}, {5584,13042}, {9786,13014}, {11413,13010}, {11425,13012}, {11440,13016}, {11477,13038}, {11479,13054}, {11480,13058}, {11481,13060}

X(13022) = midpoint of X(20) and X(13026)
X(13022) = reflection of X(i) in X(j) for these (i,j): (3,13062), (11477,13038), (13056,3)

X(13023) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO LUCAS REFLECTION

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

The reciprocal orthologic center of these triangles is X(10670).

X(13023) lies on these lines: {3,485}, {4,13007}, {5,13025}, {25,13035}, {1597,13019}, {1598,13051}, {3295,13043}, {3311,13045}, {3312,13047}, {10306,13041}, {11414,13009}, {11426,13011}, {11432,13013}, {11441,13015}, {11456,13017}, {11482,13037}, {11484,13053}, {11485,13057}, {11486,13059}

X(13023) = reflection of X(i) in X(j) for these (i,j): (3,13055), (13025,5)

X(13024) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO LUCAS(-1) REFLECTION

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

The reciprocal orthologic center of these triangles is X(10674).

X(13024) lies on these lines: {3,486}, {4,13008}, {5,13026}, {25,13036}, {1597,13020}, {1598,13052}, {3295,13044}, {3311,13046}, {3312,13048}, {10306,13042}, {11414,13010}, {11426,13012}, {11432,13014}, {11441,13016}, {11456,13018}, {11482,13038}, {11484,13054}, {11485,13058}, {11486,13060}

X(13024) = reflection of X(i) in X(j) for these (i,j): (3,13056), (13026,5)

X(13025) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO LUCAS REFLECTION

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

The reciprocal orthologic center of these triangles is X(10670).

X(13025) lies on these lines: {2,13055}, {4,488}, {5,13023}, {20,13021}, {376,13061}, {427,13007}, {489,11828}, {497,13043}, {631,13049}, {1370,13009}, {1992,13037}, {2550,13041}, {3068,13045}, {3069,13047}, {6643,13039}, {7386,13027}, {7392,13053}, {11427,13011}, {11433,13013}, {11442,13015}, {11457,13017}, {11488,13057}, {11489,13059}

X(13025) = anticomplement of X(13055)

X(13026) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO LUCAS(-1) REFLECTION

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

The reciprocal orthologic center of these triangles is X(10674).

X(13026) lies on these lines: {2,13056}, {4,487}, {5,13024}, {20,13022}, {376,13062}, {427,13008}, {490,11829}, {497,13044}, {631,13050}, {1370,13010}, {1992,13038}, {2550,13042}, {3068,13046}, {3069,13048}, {6643,13040}, {7386,13028}, {7392,13054}, {11427,13012}, {11433,13014}, {11442,13016}, {11457,13018}, {11488,13058}, {11489,13060}

X(13026) = anticomplement of X(13056)

X(13027) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR TO LUCAS REFLECTION

Barycentrics (2*(2*R^2-SA+SW)*S^2-S*((SA-SW)*(8*R^2-2*SA+SW)-2*S^2)+(SA-SW)^2*SW)*SA : :

The reciprocal orthologic center of these triangles is X(10670).

X(13027) lies on these lines: {2,13009}, {3,485}, {20,13019}, {182,13011}, {511,13013}, {631,13035}, {1040,13043}, {7386,13025}, {7484,13007}, {7998,13015}, {7999,13017}, {9306,13030}, {10319,13041}, {11511,13037}, {11513,13045}, {11514,13047}, {11515,13057}, {11516,13059}

X(13027) = midpoint of X(i) and X(j) for these {i,j}: {3,13039}, {20,13019}
X(13027) = anticomplement of X(13053)
X(13027) = complement of X(13051)

X(13028) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR TO LUCAS(-1) REFLECTION

Barycentrics (2*(2*R^2-SA+SW)*S^2+S*((SA-SW)*(8*R^2-2*SA+SW)-2*S^2)+(SA-SW)^2*SW)*SA : :

The reciprocal orthologic center of these triangles is X(10674).

X(13028) lies on these lines: {2,13010}, {3,486}, {20,13020}, {182,13012}, {511,13014}, {631,13036}, {1040,13044}, {7386,13026}, {7484,13008}, {7998,13016}, {7999,13018}, {8964,9306}, {10319,13042}, {11511,13038}, {11513,13046}, {11514,13048}, {11515,13058}, {11516,13060}

X(13028) = midpoint of X(i) and X(j) for these {i,j}: {3,13040}, {20,13020}
X(13028) = anticomplement of X(13054)
X(13028) = complement of X(13052)

X(13029) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS REFLECTION TO 1st BROCARD

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

The reciprocal orthologic center of these triangles is X(13030).

X(13029) lies on these lines: {2,98}, {511,13063}

X(13030) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD TO LUCAS REFLECTION

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

The reciprocal orthologic center of these triangles is X(13029).

X(13030) lies on these lines: {2,13063}, {182,13065}, {384,13033}, {1152,8374}, {9306,13027}

X(13031) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) REFLECTION TO 1st BROCARD

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

The reciprocal orthologic center of these triangles is X(13032).

X(13031) lies on these lines: {2,98}, {511,13064}

X(13032) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD TO LUCAS(-1) REFLECTION

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

The reciprocal orthologic center of these triangles is X(13031).

X(13032) lies on these lines: {2,13064}, {182,9687}, {384,13034}, {1151,8373}, {8964,9306}

X(13033) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th BROCARD TO LUCAS REFLECTION

Barycentrics S^2*(SW*((8*SA^2-12*SA*SW+2*SW^2)*R^2+(3*SA-2*SW)*SA*SW)+S^4+(6*SW*R^2+2*SA^2-SA*SW+SW^2)*S^2)-2*S*(-SA*SW^2*(4*R^2+SW)*(SA-SW)+(-SW+R^2)*S^4+((2*SA^2+SA*SW-4*SW^2)*R^2-(2*SA-SW)*SA*SW)*S^2)+(SA-SW)*SA*SW^4 : :

The reciprocal orthologic center of these triangles is X(13029).

X(13033) lies on these lines: {3,13063}, {384,13030}, {1975,13009}, {10131,13065}

X(13034) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th BROCARD TO LUCAS(-1) REFLECTION

Barycentrics S^2*(SW*((8*SA^2-12*SA*SW+2*SW^2)*R^2+(3*SA-2*SW)*SA*SW)+S^4+(6*SW*R^2+2*SA^2-SA*SW+SW^2)*S^2)+2*S*(-SA*SW^2*(4*R^2+SW)*(SA-SW)+(-SW+R^2)*S^4+((2*SA^2+SA*SW-4*SW^2)*R^2-(2*SA-SW)*SA*SW)*S^2)+(SA-SW)*SA*SW^4 : :

The reciprocal orthologic center of these triangles is X(13031).

X(13034) lies on these lines: {3,13064}, {384,13032}, {1975,13010}, {10131,13066}

X(13035) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC TO LUCAS REFLECTION

Barycentrics (S^2+(10*R^2+SA-SW)*S+SA^2+2*R^2*SW)*SB*SC : :

The reciprocal orthologic center of these triangles is X(10670).

X(13035) lies on these lines: {2,13039}, {3,13007}, {4,488}, {24,13055}, {25,13023}, {54,13011}, {186,13049}, {378,13021}, {631,13027}, {3090,13053}, {3520,13061}, {3567,13013}, {5889,13015}, {5890,13017}, {6197,13041}, {6198,13043}, {8537,13037}, {10632,13057}, {10633,13059}, {10880,13045}, {10881,13047}

X(13035) = midpoint of X(5889) and X(13015)
X(13035) = reflection of X(i) in X(j) for these (i,j): (4,13051), (13009,3)
X(13035) = anticomplement of X(13039)

X(13036) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC TO LUCAS(-1) REFLECTION

Barycentrics (S^2-(10*R^2+SA-SW)*S+SA^2+2*R^2*SW)*SB*SC : :

The reciprocal orthologic center of these triangles is X(10674).

X(13036) lies on these lines: {2,13040}, {3,13008}, {4,487}, {24,13056}, {25,13024}, {54,13012}, {186,13050}, {378,13022}, {631,13028}, {3090,13054}, {3520,13062}, {3567,13014}, {5889,13016}, {5890,13018}, {6197,13042}, {6198,13044}, {8537,13038}, {10632,13058}, {10633,13060}, {10880,13046}, {10881,13048}

X(13036) = midpoint of X(5889) and X(13016)
X(13036) = reflection of X(i) in X(j) for these (i,j): (4,13052), (13010,3)
X(13036) = anticomplement of X(13040)

X(13037) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EHRMANN TO LUCAS REFLECTION

Barycentrics (S^2*(SW*(4*R^2+SA-3*SW)-3*S^2)+S*(SA*SW*(4*R^2-SA+2*SW)+(6*R^2-7*SW)*S^2)+SA*SW^3)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(10670).

X(13037) lies on these lines: {6,13011}, {511,13061}, {575,13049}, {576,12598}, {1992,13025}, {8537,13035}, {8538,13039}, {8539,13041}, {8540,13043}, {8541,13051}, {9813,13053}, {11405,13007}, {11416,13009}, {11443,13015}, {11458,13017}, {11470,13019}, {11477,13021}, {11482,13023}, {11511,13027}

X(13037) = midpoint of X(11477) and X(13021)
X(13037) = reflection of X(13049) in X(575)

X(13038) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EHRMANN TO LUCAS(-1) REFLECTION

Barycentrics (S^2*(SW*(4*R^2+SA-3*SW)-3*S^2)-S*(SA*SW*(4*R^2-SA+2*SW)+(6*R^2-7*SW)*S^2)+SA*SW^3)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(10674).

X(13038) lies on these lines: {6,13012}, {511,13062}, {575,13050}, {576,12597}, {1992,13026}, {8537,13036}, {8538,13040}, {8539,13042}, {8540,13044}, {8541,13052}, {9813,13054}, {11405,13008}, {11416,13010}, {11443,13016}, {11458,13018}, {11470,13020}, {11477,13022}, {11482,13024}, {11511,13028}

X(13038) = midpoint of X(11477) and X(13022)

X(13039) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EULER TO LUCAS REFLECTION

Barycentrics ((4*R^2-SA+SW)*S^2-S*((SA-SW)*(10*R^2-2*SA-SW)-2*S^2)-(2*R^2-SA)*(SA-SW)*SW)*SA : :

The reciprocal orthologic center of these triangles is X(10670).

X(13039) lies on these lines: {2,13035}, {3,485}, {4,13009}, {5,13051}, {30,13019}, {52,13013}, {569,13011}, {1062,13043}, {1656,13053}, {6643,13025}, {7395,13007}, {8251,13041}, {8538,13037}, {10634,13057}, {10635,13059}, {10897,13045}, {10898,13047}, {11444,13015}, {11459,13017}

X(13039) = reflection of X(52) in X(13013)
X(13039) = complement of X(13035)

X(13040) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EULER TO LUCAS(-1) REFLECTION

Barycentrics ((4*R^2-SA+SW)*S^2+S*((SA-SW)*(10*R^2-2*SA-SW)-2*S^2)-(2*R^2-SA)*(SA-SW)*SW)*SA : :

The reciprocal orthologic center of these triangles is X(10674).

X(13040) lies on these lines: {2,13036}, {3,486}, {4,13010}, {5,13052}, {30,13020}, {52,13014}, {569,13012}, {1062,13044}, {1656,13054}, {5254,8961}, {6643,13026}, {7395,13008}, {8251,13042}, {8538,13038}, {10634,13058}, {10635,13060}, {10897,13046}, {10898,13048}, {11444,13016}, {11459,13018}

X(13040) = reflection of X(i) in X(j) for these (i,j): (52,13014), (13052,5)
X(13040) = complement of X(13036)

X(13041) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXTANGENTS TO LUCAS REFLECTION

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

The reciprocal orthologic center of these triangles is X(10670).

X(13041) lies on these lines: {19,13051}, {40,9907}, {55,13043}, {2550,13025}, {3101,13009}, {5415,13045}, {5416,13047}, {5584,13021}, {6197,13035}, {7688,13061}, {8251,13039}, {8539,13037}, {9816,13053}, {10306,13023}, {10319,13027}, {10636,13057}, {10637,13059}, {10902,13049}, {11406,13007}, {11428,13011}, {11435,13013}, {11445,13015}, {11460,13017}, {11471,13019}

X(13042) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXTANGENTS TO LUCAS(-1) REFLECTION

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

The reciprocal orthologic center of these triangles is X(10674).

X(13042) lies on these lines: {19,13052}, {40,9906}, {55,13044}, {2550,13026}, {3101,13010}, {5415,13046}, {5416,13048}, {5584,13022}, {6197,13036}, {7688,13062}, {8251,13040}, {8539,13038}, {9816,13054}, {10306,13024}, {10319,13028}, {10636,13058}, {10637,13060}, {10902,13050}, {11406,13008}, {11428,13012}, {11435,13014}, {11445,13016}, {11460,13018}, {11471,13020}

X(13043) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INTANGENTS TO LUCAS REFLECTION

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

The reciprocal orthologic center of these triangles is X(10670).

X(13043) lies on these lines: {1,12911}, {33,13051}, {34,13019}, {35,13049}, {36,13061}, {55,13041}, {56,13021}, {497,13025}, {1040,13027}, {1062,13039}, {1250,13059}, {2066,13045}, {3100,13009}, {3295,13023}, {5414,13047}, {6198,13035}, {7071,13007}, {8540,13037}, {9817,13053}, {10638,13057}, {11429,13011}, {11436,13013}, {11446,13015}, {11461,13017}

X(13044) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INTANGENTS TO LUCAS(-1) REFLECTION

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

The reciprocal orthologic center of these triangles is X(10674).

X(13044) lies on these lines: {1,12910}, {33,13052}, {34,13020}, {35,13050}, {36,13062}, {55,13042}, {56,13022}, {497,13026}, {1040,13028}, {1062,13040}, {1250,13060}, {2066,13046}, {3100,13010}, {3295,13024}, {5414,13048}, {6198,13036}, {7071,13008}, {8540,13038}, {9817,13054}, {10638,13058}, {11429,13012}, {11436,13014}, {11446,13016}, {11461,13018}

X(13045) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS TO LUCAS REFLECTION

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

The reciprocal orthologic center of these triangles is X(10670).

X(13045) lies on these lines: {6,13011}, {371,12961}, {372,13049}, {1151,13021}, {2066,13043}, {3068,13025}, {3311,13023}, {5410,13007}, {5412,13051}, {5415,13041}, {6200,13061}, {6413,8408}, {10880,13035}, {10897,13039}, {10961,13053}, {11417,13009}, {11447,13015}, {11462,13017}, {11473,13019}, {11513,13027}

X(13046) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS TO LUCAS(-1) REFLECTION

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

The reciprocal orthologic center of these triangles is X(10674).

X(13046) lies on these lines: {6,13012}, {371,5254}, {372,13050}, {1151,13022}, {2066,13044}, {3053,11829}, {3068,13026}, {3311,13024}, {5410,13008}, {5412,13052}, {5415,13042}, {6200,13062}, {10880,13036}, {10897,13040}, {10961,13054}, {11417,13010}, {11447,13016}, {11462,13018}, {11473,13020}, {11513,13028}

X(13047) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS TO LUCAS REFLECTION

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

The reciprocal orthologic center of these triangles is X(10670).

X(13047) lies on these lines: {6,13011}, {371,13049}, {372,5254}, {1152,13021}, {3053,11828}, {3069,13025}, {3312,13023}, {5411,13007}, {5413,13051}, {5414,13043}, {5416,13041}, {6396,13061}, {10881,13035}, {10898,13039}, {10963,13053}, {11418,13009}, {11448,13015}, {11463,13017}, {11474,13019}, {11514,13027}

X(13048) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS TO LUCAS(-1) REFLECTION

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

The reciprocal orthologic center of these triangles is X(10674).

X(13048) lies on these lines: {6,13012}, {371,13050}, {372,12966}, {1152,13022}, {3069,13026}, {3312,13024}, {5411,13008}, {5413,13052}, {5414,13044}, {5416,13042}, {6396,13062}, {6414,8420}, {10881,13036}, {10898,13040}, {10963,13054}, {11418,13010}, {11448,13016}, {11463,13018}, {11474,13020}, {11514,13028}

X(13049) = ORTHOLOGIC CENTER OF THESE TRIANGLES: KOSNITA TO LUCAS REFLECTION

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

The reciprocal orthologic center of these triangles is X(10670).

X(13049) lies on these lines: {3,485}, {15,13059}, {16,13057}, {24,13051}, {35,13043}, {186,13035}, {371,13047}, {372,13045}, {378,13019}, {389,13011}, {575,13037}, {578,13013}, {631,13025}, {3515,13007}, {6642,13053}, {7488,13009}, {10902,13041}, {11449,13015}, {11464,13017}

X(13049) = midpoint of X(3) and X(13055)
X(13049) = reflection of X(13061) in X(3)

X(13050) = ORTHOLOGIC CENTER OF THESE TRIANGLES: KOSNITA TO LUCAS(-1) REFLECTION

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

The reciprocal orthologic center of these triangles is X(10674).

X(13050) lies on these lines: {3,486}, {15,13060}, {16,13058}, {24,13052}, {35,13044}, {186,13036}, {371,13048}, {372,13046}, {378,13020}, {389,13012}, {575,13038}, {578,13014}, {631,13026}, {3515,13008}, {6642,13054}, {7488,13010}, {10902,13042}, {11449,13016}, {11464,13018}

X(13050) = midpoint of X(3) and X(13056)
X(13050) = reflection of X(13062) in X(3)

X(13051) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ORTHIC TO LUCAS REFLECTION

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

The reciprocal orthologic center of these triangles is X(10670).

X(13051) lies on these lines: {2,13009}, {4,488}, {5,13039}, {19,13041}, {24,13049}, {25,13007}, {33,13043}, {51,13013}, {184,13011}, {378,13061}, {1321,1899}, {1593,13021}, {1598,13023}, {1907,13052}, {3060,13015}, {3567,13017}, {5412,13045}, {5413,13047}, {8541,13037}, {10641,13057}, {10642,13059}

X(13051) = midpoint of X(4) and X(13035)
X(13051) = reflection of X(i) in X(j) for these (i,j): (13019,4), (13039,5)
X(13051) = anticomplement of X(13027)
X(13051) = complement of X(13009)

X(13052) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ORTHIC TO LUCAS(-1) REFLECTION

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

The reciprocal orthologic center of these triangles is X(10674).

X(13052) lies on these lines: {2,13010}, {4,487}, {5,13040}, {19,13042}, {24,13050}, {25,13008}, {33,13044}, {51,13014}, {184,13012}, {378,13062}, {1322,1899}, {1593,13022}, {1598,13024}, {1907,13051}, {3060,13016}, {3567,13018}, {5412,13046}, {5413,13048}, {8541,13038}, {10641,13058}, {10642,13060}

X(13052) = midpoint of X(4) and X(13036)
X(13052) = reflection of X(i) in X(j) for these (i,j): (13020,4), (13040,5)
X(13052) = anticomplement of X(13028)
X(13052) = complement of X(13010)

X(13053) = ORTHOLOGIC CENTER OF THESE TRIANGLES: SUBMEDIAL TO LUCAS REFLECTION

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

The reciprocal orthologic center of these triangles is X(10670).

X(13053) lies on these lines: {2,13009}, {5,641}, {1656,13039}, {3070,8944}, {3090,13035}, {3091,13019}, {5020,13055}, {5943,13013}, {6642,13049}, {7392,13025}, {9306,13011}, {9813,13037}, {9816,13041}, {9817,13043}, {9818,13061}, {10643,13057}, {10644,13059}, {10961,13045}, {10963,13047}, {11284,13007}, {11451,13015}, {11465,13017}, {11479,13021}, {11484,13023}

X(13053) = complement of X(13027)

X(13054) = ORTHOLOGIC CENTER OF THESE TRIANGLES: SUBMEDIAL TO LUCAS(-1) REFLECTION

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

The reciprocal orthologic center of these triangles is X(10674).

X(13054) lies on these lines: {2,13010}, {5,642}, {1656,13040}, {3071,8940}, {3090,13036}, {3091,13020}, {5020,13056}, {5943,13014}, {6642,13050}, {7392,13026}, {9306,13012}, {9813,13038}, {9816,13042}, {9817,13044}, {9818,13062}, {10643,13058}, {10644,13060}, {10961,13046}, {10963,13048}, {11284,13008}, {11451,13016}, {11465,13018}, {11479,13022}, {11484,13024}

X(13054) = complement of X(13028)

X(13055) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL TO LUCAS REFLECTION

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

The reciprocal orthologic center of these triangles is X(10670).

X(13055) lies on these lines: {2,13025}, {3,485}, {6,13011}, {22,13009}, {24,13035}, {25,13007}, {55,13041}, {110,13015}, {511,10670}, {1151,11828}, {1152,8374}, {1593,13019}, {1614,13017}, {5020,13053}, {8266,10323}

X(13055) = midpoint of X(3) and X(13023)
X(13055) = reflection of X(i) in X(j) for these (i,j): (3,13049), (13021,3)
X(13055) = complement of X(13025)

X(13056) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL TO LUCAS(-1) REFLECTION

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

The reciprocal orthologic center of these triangles is X(10674).

X(13056) lies on these lines: {2,13026}, {3,486}, {6,13012}, {22,13010}, {24,13036}, {25,13008}, {55,13042}, {110,13016}, {511,10674}, {1151,8373}, {1152,11829}, {1593,13020}, {1614,13018}, {5020,13054}, {8266,10323}

X(13056) = midpoint of X(3) and X(13024)
X(13056) = reflection of X(i) in X(j) for these (i,j): (3,13050), (13022,3)
X(13056) = complement of X(13026)

X(13057) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL TO LUCAS REFLECTION

Trilinears ((4-2*sqrt(3))*S^3+(-3*SA+4*R^2+2*SW+sqrt(3)*(SA-2*SW))*S^2+(3-sqrt(3))*((SA^2-2*SA*(SW+2*R^2)+1/3*sqrt(3)*(-SW+2*R^2)*SW)*S-SA*SW^2))*a : :

The reciprocal orthologic center of these triangles is X(10670).

X(13057) lies on these lines: {6,13011}, {15,12982}, {16,13049}, {10632,13035}, {10634,13039}, {10636,13041}, {10638,13043}, {10641,13051}, {10643,13053}, {10645,13061}, {11408,13007}, {11420,13009}, {11452,13015}, {11466,13017}, {11475,13019}, {11480,13021}, {11485,13023}, {11488,13025}, {11515,13027}

X(13058) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL TO LUCAS(-1) REFLECTION

Trilinears (-(2*sqrt(3)+4)*S^3+(-3*SA+4*R^2+2*SW-sqrt(3)*(SA-2*SW))*S^2+(3+sqrt(3))*(-(SA^2-2*SA*(SW+2*R^2)-1/3*sqrt(3)*(-SW+2*R^2)*SW)*S-SA*SW^2))*a : :

X(13058) lies on these lines: {6,13012}, {15,12980}, {16,13050}, {10632,13036}, {10634,13040}, {10636,13042}, {10638,13044}, {10641,13052}, {10643,13054}, {10645,13062}, {11408,13008}, {11420,13010}, {11452,13016}, {11466,13018}, {11475,13020}, {11480,13022}, {11485,13024}, {11488,13026}, {11515,13028}

X(13059) = ORTHOLOGIC CENTER OF THESE TRIANGLES:OUTER TRI-EQUILATERAL TO LUCAS REFLECTION

Trilinears ((2*sqrt(3)+4)*S^3+(-3*SA+4*R^2+2*SW-sqrt(3)*(SA-2*SW))*S^2+(3+sqrt(3))*((SA^2-2*SA*(SW+2*R^2)-1/3*sqrt(3)*(-SW+2*R^2)*SW)*S-SA*SW^2))*a : :

X(13059) lies on these lines: {6,13011}, {15,13049}, {16,12983}, {1250,13043}, {10633,13035}, {10635,13039}, {10637,13041}, {10642,13051}, {10644,13053}, {10646,13061}, {11409,13007}, {11421,13009}, {11453,13015}, {11467,13017}, {11476,13019}, {11481,13021}, {11486,13023}, {11489,13025}, {11516,13027}

X(13060) = ORTHOLOGIC CENTER OF THESE TRIANGLES:OUTER TRI-EQUILATERAL TO LUCAS(-1) REFLECTION

Trilinears (-(4-2*sqrt(3))*S^3+(-3*SA+4*R^2+2*SW+sqrt(3)*(SA-2*SW))*S^2+(3-sqrt(3))*(-(SA^2-2*SA*(SW+2*R^2)+1/3*sqrt(3)*(-SW+2*R^2)*SW)*S-SA*SW^2))*a : :

X(13060) lies on these lines: {6,13012}, {15,13050}, {16,12981}, {1250,13044}, {10633,13036}, {10635,13040}, {10637,13042}, {10642,13052}, {10644,13054}, {10646,13062}, {11409,13008}, {11421,13010}, {11453,13016}, {11467,13018}, {11476,13020}, {11481,13022}, {11486,13024}, {11489,13026}, {11516,13028}

X(13061) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TRINH TO LUCAS REFLECTION

Barycentrics (2*(SA+R^2)*S^2-S*(SA*(10*R^2-SA-4*SW)-S^2)-2*(3*R^2-SW)*SA*SW)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(10670).

X(13061) lies on these lines: {3,485}, {24,13019}, {36,13043}, {376,13025}, {378,13051}, {511,13037}, {2071,13009}, {3520,13035}, {6200,13045}, {6396,13047}, {7688,13041}, {9818,13053}, {10645,13057}, {10646,13059}, {11410,13007}, {11430,13011}, {11438,13013}, {11454,13015}, {11468,13017}

X(13061) = midpoint of X(3) and X(13021)
X(13061) = reflection of X(13049) in X(3)

X(13062) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TRINH TO LUCAS(-1) REFLECTION

Barycentrics (2*(SA+R^2)*S^2+S*(SA*(10*R^2-SA-4*SW)-S^2)-2*(3*R^2-SW)*SA*SW)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(10674).

X(13062) lies on these lines: {3,486}, {24,13020}, {36,13044}, {376,13026}, {378,13052}, {511,13038}, {2071,13010}, {3520,13036}, {6200,13046}, {6396,13048}, {7688,13042}, {9818,13054}, {10645,13058}, {10646,13060}, {11410,13008}, {11430,13012}, {11438,13014}, {11454,13016}, {11468,13018}

X(13062) = midpoint of X(3) and X(13022)
X(13062) = reflection of X(13050) in X(3)

X(13063) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-BROCARD TO LUCAS REFLECTION

Barycentrics S^2*(SW*((12*SA^2-16*SA*SW+2*SW^2)*R^2+SA^2*SW)+S^4+SW*(6*R^2+SA+SW)*S^2)-2*S*(-SA*SW^2*(4*R^2+SW)*(SA-SW)+(-SW+R^2)*S^4+SW*((3*SA-4*SW)*R^2-SA*SW)*S^2)+(SA-SW)*SA*SW^4 : :

The reciprocal orthologic center of these triangles is X(13029).

X(13063) lies on these lines: {2,13030}, {3,13033}, {22,13009}, {511,13029}, {4027,13065}

X(13064) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-BROCARD TO LUCAS(-1) REFLECTION

Barycentrics S^2*(SW*((12*SA^2-16*SA*SW+2*SW^2)*R^2+SA^2*SW)+S^4+SW*(6*R^2+SA+SW)*S^2)+2*S*(-SA*SW^2*(4*R^2+SW)*(SA-SW)+(-SW+R^2)*S^4+SW*((3*SA-4*SW)*R^2-SA*SW)*S^2)+(SA-SW)*SA*SW^4 : :

The reciprocal orthologic center of these triangles is X(13031).

X(13064) lies on these lines: {2,13032}, {3,13034}, {22,13010}, {511,13031}, {4027,13066}

X(13065) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-BROCARD TO LUCAS REFLECTION

Barycentrics (S^2*(SW^2*((20*SA-4*SW)*R^2-2*SA^2-2*SA*SW+SW^2)-5*S^4+((12*SA+4*SW)*R^2-2*SA^2-3*SA*SW-4*SW^2)*S^2)+S*(SA*SW^3*(8*R^2-SA+2*SW)+(2*R^2-7*SW)*S^4+SW*((16*SA-6*SW)*R^2-SA^2-6*SA*SW+SW^2)*S^2)+SA*SW^5)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(13029).

X(13065) lies on these lines: {182,13030}, {4027,13063}, {10131,13033}

X(13066) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-BROCARD TO LUCAS(-1) REFLECTION

Barycentrics (S^2*(SW^2*((20*SA-4*SW)*R^2-2*SA^2-2*SA*SW+SW^2)-5*S^4+((12*SA+4*SW)*R^2-2*SA^2-3*SA*SW-4*SW^2)*S^2)-S*(SA*SW^3*(8*R^2-SA+2*SW)+(2*R^2-7*SW)*S^4+SW*((16*SA-6*SW)*R^2-SA^2-6*SA*SW+SW^2)*S^2)+SA*SW^5)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(13031).

X(13066) lies on these lines: {182,9687}, {4027,13064}, {10131,13034}

X(13067) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) ANTIPODAL TO OUTER-SQUARES

Barycentrics 2*(-SW+4*R^2)*S^2-S*((8*SA+SW)*(SA-SW)+4*S^2)+(16*R^2-SW)*(SA-SW)*SA : :

The reciprocal orthologic center of these triangles is X(13068).

X(13067) lies on these lines: {20,486}, {9722,12976}, {9839,13005}

X(13068) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-SQUARES TO LUCAS(-1) ANTIPODAL

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

The reciprocal orthologic center of these triangles is X(13067).

X(13068) lies on these lines: {372,3167}, {488,615}, {3564,12427}, {9723,12977}, {11950,12312}

X(13069) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR TO MANDART-EXCIRCLES

Trilinears 2*p^8-4*q*p^7+4*(q^2-1)*p^6-(4*q^2-13)*q*p^5+(2*q^4-7*q^2-5)*p^4+(7*q^2-8)*q*p^3-(q^4-q^2-8)*p^2-5*q*p+q^2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(13069) = 3*X(165)-4*X(12517) = 3*X(165)-2*X(12659) = 5*X(7987)-4*X(12522) = 9*X(7988)-8*X(12613) = 7*X(7989)-8*X(12621)

The reciprocal orthologic center of these triangles is X(3555).

X(13069) lies on these lines: {1,12553}, {165,12517}, {200,12534}, {516,12542}, {1750,12693}, {3062,12449}, {4326,12847}, {5732,12538}, {6769,9589}, {7987,12522}, {7988,12613}, {7989,12621}, {8089,13072}, {8140,12878}, {8244,13070}, {8245,13071}, {8423,13074}, {10857,12442}, {10980,12914}, {11531,12655}

X(13069) = midpoint of X(12878) and X(12883)
X(13069) = reflection of X(i) in X(j) for these (i,j): (1,12843), (11531,12655), (12659,12517)

X(13070) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO MANDART-EXCIRCLES

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

The reciprocal orthologic center of these triangles is X(3555).

X(13070) lies on these lines: {8224,12517}, {8225,12522}, {8228,12613}, {8230,12621}, {8231,12659}, {8233,12693}, {8234,12843}, {8237,12847}, {8239,12876}, {8243,12912}, {8244,13069}, {8246,13071}, {9789,12542}, {10858,12442}, {10867,12449}, {10885,12538}, {10891,12553}, {11030,12914}, {11042,12907}, {11532,12655}, {11687,12534}, {11996,13074}

X(13071) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO MANDART-EXCIRCLES

Trilinears (b+c)*a^11+(b^2+c^2)*a^10-(b+c)*(2*b^2+3*b*c+2*c^2)*a^9-(2*b^4+2*c^4+b*c*(3*b^2-8*b*c+3*c^2))*a^8+4*b*c*(b+c)*(b^2+b*c+c^2)*a^7-4*b^2*c^2*(2*b^2+b*c+2*c^2)*a^6+2*(b+c)*(b^6+c^6-b*c*(b^2+b*c+c^2)*(3*b^2-8*b*c+3*c^2))*a^5+2*(b^6+c^6-(b^4+c^4-b*c*(b^2-4*b*c+c^2))*b*c)*(b+c)^2*a^4-(b+c)*(b^8+c^8-2*(2*b^6+2*c^6-(2*b^4+2*c^4+b*c*(12*b^2-13*b*c+12*c^2))*b*c)*b*c)*a^3-(b^8+c^8-2*(b^2-b*c+c^2)*(b^4+c^4-b*c*(b^2-5*b*c+c^2))*b*c)*(b+c)^2*a^2+(b^2-c^2)^2*(b+c)*(b^4+c^4-2*b*c*(4*b^2-5*b*c+4*c^2))*b*c*a+(b^4-c^4)*(b^2-c^2)^3*b*c : :

The reciprocal orthologic center of these triangles is X(3555).

X(13071) lies on these lines: {21,12522}, {846,12659}, {1284,12912}, {4199,12693}, {4220,12517}, {5051,12621}, {8229,12613}, {8235,12843}, {8238,12847}, {8240,12876}, {8245,13069}, {8246,13070}, {8249,13072}, {8391,12878}, {8425,13074}, {8731,12442}, {9791,12542}, {10868,12449}, {10892,12553}, {11031,12914}, {11043,12907}, {11533,12655}, {11688,12534}, {11926,12883}

X(13071) = 2nd-Sharygin-to-1st-Sharygin similarity image of X(13300)
X(13071) = excentral-to-1st-Sharygin similarity image of X(12659)
X(13071) = hexyl-to-1st-Sharygin similarity image of X(12843)
X(13071) = intouch-to-1st-Sharygin similarity image of X(12912)

X(13072) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL-MIDARC TO MANDART-EXCIRCLES

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=2*a*b*c*(-a+b+c)*(a^3-(b+c)^2*a+2*b*c*(b+c))*(a^3+(b+c)*a^2+(b^2-4*b*c+c^2)*a+(b^2-c^2)*(b-c))
G(a,b,c)=-c*(a-b+c)*(a^5-(b-c)*a^4-2*b*(b+c)*a^3+4*a^2*b^2*c+(b^2-c^2)*(b-c)^2*a+(b^2-c^2)^2*(b-c))*(a^3+(b-c)*a^2+(b^2-4*b*c-c^2)*a+(b+c)*(b^2+c^2))
H(a,b,c)=2*S^2*(a^6-(b+c)^2*a^4+2*b*c*(b+c)*a^3-(b^2+c^2)*(b-c)^2*a^2-2*(b^2-c^2)*(b-c)*b*c*a+(b^4-c^4)*(b^2-c^2))

The reciprocal orthologic center of these triangles is X(3555).

X(13072) lies on these lines: {1,13073}, {2089,12912}, {8075,12517}, {8077,12522}, {8078,12659}, {8079,12693}, {8081,12843}, {8085,12613}, {8087,12621}, {8089,13069}, {8241,12876}, {8247,13070}, {8249,13071}, {8387,12847}, {8733,12442}, {9793,12542}, {11032,12914}, {11690,12534}, {11888,12538}, {11894,12553}

X(13072) = reflection of X(13073) in X(1)

X(13073) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO MANDART-EXCIRCLES

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

The reciprocal orthologic center of these triangles is X(3555).

X(13073) lies on these lines: {1,13072}, {174,12912}, {258,12659}, {7588,12522}, {8125,12534}, {8351,12907}, {8734,12442}, {11033,12914}, {11859,12449}, {11895,12553}, {11899,12655}

X(13073) = reflection of X(13072) in X(1)

X(13074) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO MANDART-EXCIRCLES

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

The reciprocal orthologic center of these triangles is X(3555).

X(13074) lies on these lines: {174,12912}, {7587,12522}, {8083,12914}, {8126,12534}, {8382,12621}, {8389,12847}, {8423,13069}, {8425,13071}, {8729,12442}, {11535,12655}, {11860,12449}, {11890,12538}, {11891,12542}, {11896,12553}, {11924,12876}, {11996,13070}

X(13075) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO INNER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(3).

X(13075) lies on these lines: {3,10077}, {4,12941}, {11,619}, {12,5479}, {14,55}, {33,12141}, {35,6774}, {56,5474}, {99,12952}, {115,10638}, {497,617}, {530,12354}, {531,3058}, {542,3056}, {1250,5471}, {1479,5613}, {1697,9900}, {1837,12780}, {2098,7974}, {2646,11706}, {3295,10061}, {4092,7043}, {4294,6773}, {4995,5460}, {5432,6670}, {5464,11238}, {6269,10928}, {6271,10927}, {6321,10062}, {9114,12351}, {9915,10833}, {9981,10877}, {10799,12204}, {10947,12921}, {10953,12931}, {10965,13104}, {10966,13106}, {11873,12470}, {11874,12471}, {11909,12792}, {11947,12988}, {11948,12989}

X(13075) = X(14)-of-Mandart-incircle-triangle

X(13076) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO OUTER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(3).

X(13076) lies on these lines: {3,10078}, {4,12942}, {11,618}, {12,5478}, {13,55}, {33,12142}, {35,6771}, {56,5473}, {99,12951}, {115,1250}, {497,616}, {530,3058}, {531,12354}, {542,3056}, {1479,5617}, {1697,9901}, {1837,12781}, {2098,7975}, {2646,11705}, {3295,10062}, {4092,7026}, {4294,6770}, {4995,5459}, {5432,6669}, {5463,11238}, {5472,10638}, {5613,10086}, {6268,10928}, {6270,10927}, {6321,10061}, {9116,12351}, {9982,10877}, {10799,12205}, {10947,12922}, {10953,12932}, {10965,13105}, {10966,13107}, {11873,12472}, {11874,12473}, {11909,12793}, {11947,12990}, {11948,12991}

X(13076) = X(13)-of-Mandart-incircle-triangle

X(13077) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO 1st NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(13077) lies on these lines: {1,2782}, {3,10079}, {4,12837}, {6,10798}, {11,39}, {12,6248}, {33,12143}, {55,76}, {56,11257}, {194,497}, {262,10896}, {384,10799}, {498,7697}, {499,11171}, {511,6284}, {538,3058}, {726,950}, {730,3057}, {732,3056}, {1479,3095}, {1697,9902}, {1837,12782}, {2098,7976}, {2646,12263}, {3086,7709}, {3094,9598}, {3097,9581}, {3202,9667}, {3295,10063}, {3934,5432}, {4294,12251}, {4302,9821}, {4995,9466}, {5969,12354}, {6272,10928}, {6273,10927}, {7741,11272}, {7757,11238}, {9917,10833}, {9983,10877}, {10947,12923}, {10953,12933}, {10965,13109}, {10966,13110}, {11152,12351}, {11873,12474}, {11874,12475}, {11909,12794}, {11947,12992}, {11948,12993}

X(13077) = X(76)-of-Mandart-incircle-triangle
X(13077) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (194,497,12836), (3295,13108,10063)

X(13078) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(13078) lies on these lines: {3,10080}, {4,12944}, {11,6292}, {12,6249}, {33,12144}, {55,83}, {56,12122}, {497,2896}, {732,3056}, {754,3058}, {1479,6287}, {1697,9903}, {1837,12783}, {2098,7977}, {2646,12264}, {3295,10064}, {4294,12252}, {4302,8725}, {5217,9751}, {5432,6704}, {6022,6026}, {6274,10928}, {6275,10927}, {9918,10833}, {10947,12924}, {10953,12934}, {10965,13112}, {10966,13113}, {11873,12476}, {11909,12795}, {11947,12994}, {11948,12995}

X(13078) = X(83)-of-Mandart-incircle-triangle
X(13078) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (497,2896,12954), (3295,13111,10064)

X(13079) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO REFLECTION

Trilinears ((b-c)^2*a^6-(3*b^4+3*c^4-2*b*c*(b^2-b*c+c^2))*a^4+(3*b^4+3*c^4-b*c*(4*b^2-5*b*c+4*c^2))*(b+c)^2*a^2-(b^2-c^2)^2*(b^4+c^4+b*c*(2*b^2+b*c+2*c^2)))*a : :
X(13079) = 3*X(1)-X(7356) = 3*X(6286)+X(7356)

The reciprocal orthologic center of these triangles is X(4).

X(13079) lies on these lines: {1,1154}, {3,10082}, {4,12946}, {11,1209}, {12,3574}, {33,11576}, {35,10610}, {54,55}, {56,7691}, {195,3295}, {497,2888}, {539,3058}, {999,12307}, {1058,12325}, {1062,12363}, {1479,6288}, {1493,3746}, {1697,9905}, {1837,12785}, {1858,9957}, {1870,12300}, {2098,7979}, {2646,12266}, {3028,10628}, {3270,10619}, {4294,12254}, {5432,6689}, {6152,6198}, {6276,10928}, {6277,10927}, {6767,12316}, {7159,12896}, {9538,12226}, {9670,11446}, {9920,10833}, {9985,10877}, {10799,12208}, {10947,12926}, {10953,12936}, {10965,13121}, {10966,13122}, {11873,12480}, {11874,12481}, {11909,12797}, {11947,12998}, {11948,12999}

X(13079) = midpoint of X(1) and X(6286)
X(13079) = X(54)-of-Mandart-incircle-triangle
X(13079) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (195,3295,10066), (497,2888,12956)

X(13080) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO 1st SCHIFFLER

Trilinears 16*p^5*(p-q)-8*(q^2+2)*p^4+24*q*p^3+(8*q^4-4*q^2+9)*p^2-(4*q^2+5)*q*p+(q^2-9)/2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(79).

X(13080) lies on these lines: {4,12947}, {11,13089}, {12,12600}, {33,12146}, {55,10266}, {1317,5441}, {1479,12919}, {1697,12409}, {1837,12786}, {2098,13100}, {2646,12267}, {4294,12255}, {10543,12877}, {10799,12209}, {10833,12414}, {10877,12504}, {10927,12807}, {10928,12808}, {10947,12927}, {10953,12937}, {11874,12483}, {11909,12798}, {11947,13000}, {11948,13001}

X(13080) = X(10266)-of-Mandart-incircle-triangle

X(13081) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO INNER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(13081) lies on these lines: {1,12910}, {3,10083}, {4,12948}, {11,642}, {12,6251}, {33,12147}, {55,486}, {56,12123}, {388,12296}, {390,12221}, {487,497}, {1058,12509}, {1479,6290}, {1697,9906}, {1837,12787}, {2098,7980}, {2646,12268}, {3056,3564}, {3295,10067}, {4294,12256}, {5432,6119}, {6280,10928}, {6281,9670}, {6337,12959}, {9921,10833}, {9986,10877}, {10799,12210}, {10947,12928}, {10953,12938}, {11873,12484}, {11874,12485}, {11909,12799}, {11947,13002}, {11948,13003}

X(13081) = X(486)-of-Mandart-incircle-triangle
X(13081) = {X(3056),X(15171)}-harmonic conjugate of X(13082)

X(13082) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO OUTER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(13082) lies on these lines: {1,12911}, {3,10084}, {4,12949}, {11,641}, {12,6250}, {30,6283}, {33,12148}, {55,485}, {56,12124}, {388,12297}, {390,12222}, {488,497}, {1058,12510}, {1479,6289}, {1697,9907}, {1837,12788}, {2098,7981}, {2646,12269}, {3056,3564}, {3295,10068}, {4294,12257}, {5432,6118}, {6278,9670}, {6279,10927}, {6337,12958}, {9922,10833}, {9987,10877}, {10799,12211}, {10947,12929}, {10953,12939}, {11873,12486}, {11874,12487}, {11909,12800}, {11947,13004}, {11948,13005}

X(13082) = X(485)-of-Mandart-incircle-triangle
X(13082) = {X(3056),X(15171)}-harmonic conjugate of X(13081)

X(13083) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MCCAY TO INNER-NAPOLEON

Barycentrics (3*a^6+6*(b^2+c^2)*a^4-6*(2*b^4+3*b^2*c^2+2*c^4)*a^2+3*(b^4-c^4)*(b^2-c^2)-2*sqrt(3)*S*(7*a^4-7*(b^2+c^2)*a^2-4*b^2*c^2+c^4+b^4)) : :
X(13083) = 2*X(619)+X(6295)

The reciprocal orthologic center of these triangles is X(9885).

X(13083) lies on these lines: {2,14}, {3,530}, {30,9735}, {182,524}, {299,5463}, {303,10645}, {396,574}, {532,3524}, {533,5054}, {543,6771}, {599,618}, {624,11295}, {630,11306}, {5215,10613}, {5238,11304}, {5459,6772}, {6672,11485}, {6774,7619}, {9762,9774}, {9886,11171}, {11151,12155}

X(13083) = midpoint of X(3) and X(9763)
X(13083) = reflection of X(13084) in X(549)
X(13083) = X(13)-of-McCay-triangle

X(13084) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MCCAY TO OUTER-NAPOLEON

Barycentrics (3*a^6+6*(b^2+c^2)*a^4-6*(2*b^4+3*b^2*c^2+2*c^4)*a^2+3*(b^4-c^4)*(b^2-c^2)+2*sqrt(3)*S*(7*a^4-7*(b^2+c^2)*a^2-4*b^2*c^2+c^4+b^4)) : :
X(13084) = 4*X(618)-X(6298)

The reciprocal orthologic center of these triangles is X(9886).

X(13084) lies on these lines: {2,13}, {3,531}, {30,9736}, {182,524}, {298,5464}, {302,10646}, {395,574}, {532,5054}, {533,3524}, {543,6774}, {599,619}, {623,11296}, {629,11305}, {5215,10614}, {5237,11303}, {5460,6775}, {6671,11486}, {6771,7619}, {9760,9774}, {9885,11171}, {11151,12154}

X(13084) = midpoint of X(3) and X(9761)
X(13084) = reflection of X(13083) in X(549)
X(13084) = X(14)-of-McCay-triangle

X(13085) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MCCAY TO 1st NEUBERG

Barycentrics 2*(b^2+c^2)*a^6-(b^2+2*c^2)*(2*b^2+c^2)*a^4+2*(b^4-c^4)*(b^2-c^2)*a^2-b^2*c^2*(b^2+c^2)^2 : :
X(13085) = 4*X(39)-X(6309) = 3*X(262)-2*X(7775) = 2*X(6248)-3*X(7615) = 5*X(7786)-2*X(8149) = 3*X(11184)-4*X(11272)

The reciprocal orthologic center of these triangles is X(9888).

X(13085) lies on these lines: {2,39}, {3,5969}, {262,736}, {524,3095}, {543,9774}, {698,11171}, {1916,7833}, {2023,11318}, {3094,8359}, {5188,8182}, {6248,7615}, {7840,9983}, {8667,10104}, {11184,11272}

X(13085) = {X(6294),X(6581)}-harmonic conjugate of X(76)

X(13086) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MCCAY TO 2nd NEUBERG

Barycentrics a^8+5*(b^2+c^2)*a^6-(2*b^2-b*c+2*c^2)*(2*b^2+b*c+2*c^2)*a^4-(b^2+c^2)*(b^4+12*b^2*c^2+c^4)*a^2+(b^4-3*b^2*c^2+c^4)*(b^2+c^2)^2 : :
X(13086) = X(2896)+2*X(8150) = 2*X(6287)+3*X(8182) = 2*X(6292)+X(6308) = 3*X(7615)+2*X(12122)

The reciprocal orthologic center of these triangles is X(9890).

X(13086) lies on these lines: {2,32}, {3,9890}, {732,11171}, {3523,8350}, {5569,9774}, {6287,8182}, {7615,12122}, {7841,9478}

X(13087) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MCCAY TO INNER-VECTEN

Barycentrics a^6+2*(b^2+c^2)*a^4-2*(2*b^4+3*b^2*c^2+2*c^4)*a^2+(b^4-c^4)*(b^2-c^2)-2*S*(4*a^4-4*(b^2+c^2)*a^2-4*b^2*c^2+c^4+b^4) : :
X(13087) = 4*X(642)-X(6315) = 2*X(642)+X(6316) = X(6315)+2*X(6316)

The reciprocal orthologic center of these triangles is X(9892).

X(13087) lies on these lines: {2,371}, {3,9892}, {13088, 14645

X(13088) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MCCAY TO OUTER-VECTEN

Barycentrics a^6+2*(b^2+c^2)*a^4-2*(2*b^4+3*b^2*c^2+2*c^4)*a^2+(b^4-c^4)*(b^2-c^2)+2*S*(4*a^4-4*(b^2+c^2)*a^2-4*b^2*c^2+c^4+b^4) : :
X(13088) = 4*X(641)-X(6311) = 2*X(641)+X(6312) = X(6311)+2*X(6312)

The reciprocal orthologic center of these triangles is X(9894).

X(13088) lies on these lines: {2,372}, {3,9894}, {13087, 14645}

X(13089) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MEDIAL TO 1st SCHIFFLER

Trilinears (a^3+(b+c)*a^2-(b^2-3*b*c+c^2)*a-(b^2-c^2)*(b-c))*(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+(b+c)^3) : :
X(13089) = 5*X(631)-X(12255) = 2*X(11263)+X(12682)

The reciprocal orthologic center of these triangles is X(79).

X(13089) lies on these lines: {1,6597}, {2,3467}, {3,7701}, {5,12600}, {8,13100}, {9,12519}, {10,191}, {11,13080}, {21,214}, {55,12957}, {56,12947}, {83,12209}, {100,6595}, {119,3652}, {142,12444}, {427,12146}, {442,1749}, {499,12543}, {631,12255}, {958,12937}, {1125,12267}, {1376,12342}, {1650,12798}, {1698,12409}, {1768,6853}, {2476,3336}, {3096,12504}, {3337,11263}, {5590,12808}, {5591,12807}, {5599,12482}, {5600,12483}, {5696,6600}, {6763,12535}, {7280,12845}, {8222,13000}, {8223,13001}

X(13089) = midpoint of X(i) and X(j) for these {i,j}: {1,12786}, {3,12919}, {8,13100}, {100,6595}, {1650,12798}, {6597,12660}, {12342,12927}, {12524,12745}, {12682,12913}
X(13089) = reflection of X(i) in X(j) for these (i,j): (12267,1125), (12600,5), (12913,11263)
X(13089) = complement of X(10266)
X(13089) = X(10266)-of-medial-triangle
X(13089) = {X(191), X(2475)}-harmonic conjugate of X(484)

X(13090) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO MIDARC

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2) : :
where
F(a,b,c)=-((2*b+2*c)*S+(-a+b+c)*((b+c)*a+(b-c)^2))*b*c
G(a,b,c)=(2*b*(b-c)*S+(a-b+c)*(a^3+(b-c)*a^2-c*(b-c)*a+(b^2-c^2)*c))*c

The reciprocal orthologic center of these triangles is X(1).

X(13090) lies on these lines: {1,8247}, {164,8231}, {177,8243}, {5571,11030}, {7670,8237}, {8224,12518}, {8225,12523}, {8230,12622}, {8233,12694}, {8239,8422}, {8246,13091}, {9789,9807}, {10858,12443}, {10867,12450}, {10885,12539}, {10891,12554}, {11042,12908}, {11532,12656}, {11687,11691}

X(13090) = X(1)-of-2nd-Pamfilos-Zhou-triangle
X(13090) = excentral-to-2nd-Pamfilos-Zhou similarity image of X(164)

X(13091) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO MIDARC

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

The reciprocal orthologic center of these triangles is X(1).

X(13091) lies on these lines: {1,8249}, {21,12523}, {164,846}, {167,8245}, {177,1284}, {5571,11031}, {7670,8238}, {8240,8422}, {9791,9807}, {11043,12908}, {11533,12656}, {11688,11691}

X(13091) = X(1)-of-1st-Sharygin-triangle
X(13091) = 2nd-Sharygin-to-1st-Sharygin similarity image of X(13301)
X(13091) = excentral-to-1st-Sharygin similarity image of X(164)
X(13091) = hexyl-to-1st-Sharygin similarity image of X(12844)
X(13091) = intouch-to-1st-Sharygin similarity image of X(177)

X(13092) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO MIDARC

Trilinears (a+b-c)*(a-b+c)*(4*c*sin(B/2)+4*b*sin(C/2)-a+b+c) : :
Trilinears 1 + 2 sec(A/2) (cos(B/2) + cos(C/2)) : :

The reciprocal orthologic center of these triangles is X(1).

X(13092) lies on these lines: {1,167}, {7,10489}

X(13092) = X(1)-of-Yff-central-triangle
X(13092) = excentral-to-Yff-central similarity image of X(164)

X(13093) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO MIDHEIGHT

Trilinears (a^8+2*(b^2+c^2)*a^6-4*(3*b^4-5*b^2*c^2+3*c^4)*a^4+14*(-c^4+b^4)*(b^2-c^2)*a^2-(5*b^4+18*b^2*c^2+5*c^4)*(b^2-c^2)^2)*a : :
X(13093) = 7*X(3)-6*X(154) = 3*X(3)-2*X(1498) = 3*X(3)-4*X(3357) = 9*X(3)-10*X(8567) = 9*X(3)-8*X(10282) = 13*X(3)-12*X(11202) = 11*X(3)-12*X(11204) = 4*X(140)-3*X(5656) = 3*X(381)-2*X(5878) = 9*X(381)-8*X(5893) = 3*X(381)-4*X(6247) = 3*X(5878)-4*X(5893) = 2*X(5893)-3*X(6247)

The reciprocal orthologic center of these triangles is X(4).

X(13093) lies on these lines: {3,64}, {4,3426}, {5,6225}, {25,12290}, {30,11411}, {74,3515}, {140,5656}, {185,1597}, {378,12174}, {381,5878}, {517,9899}, {548,11206}, {999,6285}, {1159,1854}, {1204,3517}, {1351,8549}, {1503,1657}, {1593,6241}, {1598,10605}, {1614,11410}, {1656,2883}, {1853,3843}, {2070,11999}, {2777,5073}, {2935,12308}, {3167,12084}, {3295,7355}, {3516,11456}, {3526,6696}, {3527,5890}, {3534,5894}, {3830,5895}, {5198,11455}, {5643,10574}, {5663,12085}, {5790,12779}, {6001,12702}, {6266,11917}, {6267,11916}, {6447,11241}, {6448,11242}, {6455,10533}, {6456,10534}, {6804,11469}, {7517,9914}, {7689,9909}, {7691,9920}, {7973,10247}, {9301,12502}, {9654,12940}, {9669,12950}, {9715,11440}, {9935,12307}, {9968,10249}, {10246,12262}, {11472,11479}, {11842,12202}, {11849,12335}, {11875,12468}, {11876,12469}, {11911,12791}, {11928,12920}, {11929,12930}, {11949,12986}, {11950,12987}, {12000,13094}, {12001,13095}

X(13093) = midpoint of X(12250) and X(12324)
X(13093) = reflection of X(i) in X(j) for these (i,j): (3,64), (1498,3357), (5878,6247), (6225,5), (9833,5894), (9919,10620), (12164,12085), (12308,2935), (12315,3)
X(13093) = X(64)-of-X3-ABC-reflections-triangle
X(13093) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (185,1597,11432), (1498,3357,3), (1498,8567,10282), (3357,10282,8567), (5878,6247,381), (5890,11403,3527), (5894,9833,3534), (6285,10076,999), (6759,10606,3), (7355,10060,3295), (8567,10282,3), (10605,11381,1598), (11414,12279,11820)

X(13094) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO MIDHEIGHT

Trilinears 2*p^8-2*q*p^7+(6*q^2-7)*p^6-2*(3*q^2-4)*q*p^5-(5*q^2-6)*p^4+(8*q^2-9)*q*p^3-3*(q^2-1)*q*p-1+q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(4).

X(13094) lies on these lines: {1,64}, {12,12920}, {30,12430}, {154,2077}, {1470,2192}, {1498,11248}, {2777,12905}, {2883,5552}, {5878,10942}, {5895,6256}, {6000,10679}, {6001,12703}, {6225,10528}, {6247,10531}, {6266,10930}, {6267,10929}, {6285,11509}, {7355,10965}, {9914,10834}, {10269,10606}, {10803,12202}, {10805,12250}, {10878,12502}, {10915,12779}, {10955,12930}, {10956,12940}, {10958,12950}, {11381,11400}, {11881,12468}, {11882,12469}, {11914,12791}, {11955,12986}, {11956,12987}, {12000,13093}

X(13094) = reflection of X(64) in X(10060)
X(13094) = X(64)-of-inner-Yff-tangents-triangle
X(13094) = {X(64),X(7973)}-harmonic conjugate of X(13095)

X(13095) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO MIDHEIGHT

Trilinears 2*p^8-2*q*p^7+(6*q^2-5)*p^6-2*(3*q^2-2)*q*p^5-(7*q^2-6)*p^4+(4*q^2-3)*q*p^3+4*(q^2-1)*p^2-(q^2-1)*q*p+1-q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(4).

X(13095) lies on these lines: {1,64}, {11,12930}, {30,12431}, {154,11012}, {1498,11249}, {2777,12906}, {2883,10527}, {5878,10943}, {6000,10680}, {6001,12704}, {6225,10529}, {6247,10532}, {6266,10932}, {6267,10931}, {7355,10966}, {8567,10902}, {9914,10835}, {10267,10606}, {10804,12202}, {10806,12250}, {10879,12502}, {10916,12779}, {10949,12920}, {10957,12940}, {10959,12950}, {11381,11401}, {11510,12335}, {11883,12468}, {11884,12469}, {11915,12791}, {11957,12986}, {11958,12987}, {12001,13093}

X(13095) = reflection of X(64) in X(10076)
X(13095) = X(64)-of-outer-Yff-tangents-triangle
X(13095) = {X(64),X(7973)}-harmonic conjugate of X(13094)

X(13096) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO MIXTILINEAR

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

The reciprocal orthologic center of these triangles is X(1).

X(13096) lies on these lines: {57,8231}, {329,8233}, {517,7596}, {527,1991}, {999,8225}, {3452,12610}, {3820,8230}, {6244,8224}, {6282,8234}, {7956,8228}, {7962,8239}, {7994,8244}, {8101,8247}, {8102,8248}, {8237,12848}, {8246,13097}, {9954,10867}, {9965,10885}, {10891,12555}, {11030,12915}, {11922,12880}, {11925,12885}, {11996,13098}

X(13096) = X(25)-of-2nd-Pamfilos-Zhou-triangle
X(13096) = excentral-to-2nd-Pamfilos-Zhou similarity image of X(57)

X(13097) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO MIXTILINEAR

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

The reciprocal orthologic center of these triangles is X(1).

X(13097) lies on these lines: {1,8421}, {21,999}, {57,846}, {329,4199}, {517,2292}, {3452,4425}, {3820,5051}, {4220,6244}, {6282,8235}, {7956,8229}, {7962,8240}, {7994,8245}, {8101,8249}, {8102,8250}, {8238,12848}, {8246,13096}, {8391,12880}, {8425,13098}, {9954,10868}, {10892,12555}, {11031,12915}, {11926,12885}

X(13097) = X(25)-of-1st-Sharygin-triangle
X(13097) = excentral-to-1st-Sharygin similarity image of X(57)
X(13097) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (846,1284,8731), (9791,11688,4199)

X(13098) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO MIXTILINEAR

Barycentrics (a^3-(b+c)*a^2-(b^2-6*b*c+c^2)*a+(b^2-c^2)*(b-c))*sin(A/2)+(a+b-c)*(a-b+c)*a : :

The reciprocal orthologic center of these triangles is X(1).

X(13098) lies on these lines: {517,8130}, {999,7587}, {3820,8382}, {7962,11535}, {7994,8423}, {8083,12915}, {8389,12848}, {8425,13097}, {9954,11860}, {11896,12555}

X(13098) = X(25)-of-Yff-central-triangle
X(13098) = excentral-to-Yff-central similarity image of X(57)

X(13099) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR TO 1st ORTHOSYMMEDIAL

Trilinears 8*p^12*(2*p^2-4*p*q-5)-16*(2*q^2-7)*q*p^11-(16*q^4-32*q^2-17)*p^10-2*(16*q^4-56*q^2+73)*q*p^9+(56*q^4-97*q^2+31)*p^8-4*(8*q^6-28*q^4+36*q^2-21)*q*p^7+(16*q^6-89*q^4+107*q^2-33)*p^6+2*(24*q^6-53*q^4+37*q^2-9)*q*p^5+(-q^2+1)*((23*q^4-38*q^2+9)*p^4+4*(5*q^2-3)*q^3*p^3-3*(3*q^2-2)*q^2*p^2-2*q^5*p+q^4) : :
where p=sin(A/2), q=cos((B-C)/2)
X(13099) = 3*X(1)-2*X(12265) = 3*X(1)-X(12408) = 2*X(127)-3*X(5603) = 3*X(1297)-4*X(12265) = 3*X(1297)-2*X(12408) = 3*X(5657)-4*X(6720) = 3*X(7967)-X(12253) = 3*X(10247)-X(13115)

The reciprocal orthologic center of these triangles is X(4).

X(13099) lies on these lines: {1,1297}, {8,132}, {40,11722}, {56,12340}, {112,517}, {127,5603}, {145,12384}, {519,12784}, {952,12918}, {962,2794}, {1320,2831}, {1482,10705}, {2098,3320}, {2099,6020}, {2781,7984}, {2799,7970}, {2806,10698}, {2825,10695}, {2853,10696}, {3241,9530}, {3656,10718}, {5604,12806}, {5605,12805}, {5657,6720}, {7967,12253}, {7978,9517}, {8192,12413}, {8210,12996}, {8211,12997}, {9518,10697}, {9523,10699}, {9527,10700}, {9532,10703}, {9997,12503}, {10247,13115}, {10735,12699}, {10800,12207}, {10944,12925}, {10950,12935}, {11396,12145}, {11910,12796}

X(13099) = midpoint of X(145) and X(12384)
X(13099) = X(1297)-of-5th-mixtilinear-triangle
X(13099) = {X(13118),X(13119)}-harmonic conjugate of X(1297)
X(13099) = reflection of X(i) in X(j) for these (i,j): (8,132), (40,11722), (1297,1), (10705,1482), (10718,3656), (10735,12699), (12408,12265)

X(13100) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR TO 1st SCHIFFLER

Barycentrics 3*a^7-3*(b+c)*a^6-(5*b^2-4*b*c+5*c^2)*a^5+(b+c)*(5*b^2-2*b*c+5*c^2)*a^4+(b^4+9*b^2*c^2+c^4)*a^3-(b+c)*(b^4-b^2*c^2+c^4)*a^2+(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :
X(13100) = 3*X(1)-X(12409) = 3*X(7967)-X(12255) = 3*X(10266)-2*X(12409)

The reciprocal orthologic center of these triangles is X(79).

X(13100) lies on these lines: {1,5180}, {8,13089}, {12,100}, {56,12342}, {519,12786}, {758,12535}, {952,6595}, {2098,13080}, {2136,12660}, {3243,12657}, {3648,4757}, {5597,12483}, {5598,12482}, {5603,12600}, {5604,12808}, {5605,12807}, {6872,12917}, {7701,9803}, {7967,12255}, {8192,12414}, {8210,13000}, {8211,13001}, {9997,12504}, {10800,12209}, {10944,12927}, {10950,12937}, {11396,12146}, {11910,12798}

X(13100) = reflection of X(i) in X(j) for these (i,j): (8,13089), (10266,1), (12409,12267)

X(13101) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR TO 1st SCHIFFLER

Trilinears a^9-3*(b+c)*a^8-10*b*c*a^7+8*(b+c)*(b^2+c^2)*a^6-(6*b^4+6*c^4-b*c*(26*b^2+9*b*c+26*c^2))*a^5-(b+c)*(6*b^4+6*c^4-b*c*(4*b^2-5*b*c+4*c^2))*a^4+(2*b^2-7*b*c+2*c^2)*(4*b^4+4*c^4+b*c*(3*b^2+2*b*c+3*c^2))*a^3-(b+c)*(8*b^4+8*c^4-b*c*(9*b^2-10*b*c+9*c^2))*b*c*a^2-3*(b^4-c^4)*(b^2-c^2)*(b-c)^2*a+(b^2-c^2)^3*(b-c)*(b^2+6*b*c+c^2) : :
X(13101) = 3*X(165)-4*X(12519) = 3*X(165)-2*X(12660) = 5*X(7987)-4*X(12524) = 9*X(7988)-8*X(12615) = 7*X(7989)-8*X(12623)

The reciprocal orthologic center of these triangles is X(21).

X(13101) lies on these lines: {1,5180}, {165,12519}, {200,12535}, {516,12543}, {1750,12695}, {3062,6597}, {4326,12850}, {5732,12540}, {7987,12524}, {7988,12615}, {7989,12623}, {8089,13124}, {8140,12882}, {8244,13120}, {8245,13123}, {9961,12767}, {10857,12444}, {10980,12917}, {11531,12657}

X(13101) = midpoint of X(12882) and X(12887)
X(13101) = reflection of X(i) in X(j) for these (i,j): (1,12845), (11531,12657), (12660,12519)

X(13102) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO INNER-NAPOLEON

Barycentrics sqrt(3)*a^2*(a^4-4*(b^2+c^2)*a^2-2*b^2*c^2+3*c^4+3*b^4)+2*S*(5*a^4-(b^2+c^2)*a^2-4*(b^2-c^2)^2) : :
X(13102) = 3*X(3)-2*X(5474) = 3*X(381)-2*X(5613) = 7*X(3526)-8*X(6670) = 3*X(5054)-4*X(5460) = 3*X(5790)-2*X(12780)

The reciprocal orthologic center of these triangles is X(3).

X(13102) lies on these lines: {3,14}, {4,3180}, {5,617}, {30,5615}, {115,11485}, {381,531}, {382,5868}, {517,9900}, {530,12355}, {542,1351}, {619,1656}, {999,10077}, {1080,7777}, {1598,12141}, {3295,10061}, {3526,6670}, {5054,5460}, {5055,5464}, {5469,6771}, {5471,11486}, {5790,12780}, {6269,11917}, {6271,11916}, {7517,9915}, {7974,10247}, {9301,9981}, {9654,12941}, {9669,12951}, {10246,11706}, {10796,11296}, {11842,12204}, {11849,12336}, {11875,12470}, {11876,12471}, {11911,12792}, {11928,12921}, {11929,12931}, {11949,12988}, {11950,12989}, {12000,13104}, {12001,13106}

X(13102) = reflection of X(i) in X(j) for these (i,j): (3,14), (617,5), (5474,6774), (5613,5479), (13103,6321)

X(13103) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO OUTER-NAPOLEON

Barycentrics sqrt(3)*a^2*(a^4-4*(b^2+c^2)*a^2-2*b^2*c^2+3*c^4+3*b^4)-2*S*(5*a^4-(b^2+c^2)*a^2-4*(b^2-c^2)^2) : :
X(13103) = 3*X(3)-2*X(5473) = 3*X(3)-4*X(6771) = 3*X(381)-4*X(5478) = 3*X(381)-2*X(5617) = 4*X(618)-5*X(1656) = 7*X(3526)-8*X(6669) = 3*X(5054)-4*X(5459) = 3*X(5055)-2*X(5463) = 3*X(5790)-2*X(12781) = 2*X(7975)-3*X(10247)

The reciprocal orthologic center of these triangles is X(3).

X(13103) lies on these lines: {3,13}, {4,3181}, {5,616}, {30,5611}, {115,11486}, {381,530}, {382,5869}, {383,7777}, {517,9901}, {531,12355}, {542,1351}, {618,1656}, {999,10078}, {1598,12142}, {3295,10062}, {3526,6669}, {5054,5459}, {5055,5463}, {5470,6774}, {5472,11485}, {5790,12781}, {6268,11917}, {6270,11916}, {7517,9916}, {7975,10247}, {9301,9982}, {9654,12942}, {9669,12952}, {10246,11705}, {10796,11295}, {11842,12205}, {11849,12337}, {11875,12472}, {11876,12473}, {11911,12793}, {11928,12922}, {11929,12932}, {11949,12990}, {11950,12991}, {12000,13105}, {12001,13107}

X(13103) = reflection of X(i) in X(j) for these (i,j): (3,13), (616,5), (5473,6771), (5617,5478), (13102,6321)

X(13104) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO INNER-NAPOLEON

Barycentrics -2*sqrt(3)*a^2*(a^4-2*(b^2+b*c+c^2)*a^2+2*b*c*(b+c)*a+b^4+c^4-4*b^3*c-2*b^2*c^2-4*b*c^3)*S+(a+b+c)*(a^7-(b+c)*a^6+16*b*c*a^5-14*b*c*(b+c)*a^4-(b^2+4*b*c-3*c^2)*(3*b^2-4*b*c-c^2)*a^3+(b^2-c^2)*(b-c)*(3*b^2+16*b*c+3*c^2)*a^2+2*(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-2*(b^2-c^2)^3*(b-c)) : :

The reciprocal orthologic center of these triangles is X(3).

X(13104) lies on these lines: {1,14}, {12,12921}, {530,12356}, {531,11239}, {542,12594}, {617,10528}, {619,5552}, {5474,11248}, {5479,10531}, {5613,10942}, {6269,10930}, {6271,10929}, {6773,10805}, {9915,10834}, {9981,10878}, {10803,12204}, {10915,12780}, {10955,12931}, {10956,12941}, {10958,12951}, {10965,13075}, {11400,12141}, {11509,12336}, {11881,12470}, {11882,12471}, {11914,12792}, {11955,12988}, {11956,12989}, {12000,13102}

X(13105) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO OUTER-NAPOLEON

Barycentrics 2*sqrt(3)*a^2*(a^4-2*(b^2+b*c+c^2)*a^2+2*b*c*(b+c)*a+b^4+c^4-4*b^3*c-2*b^2*c^2-4*b*c^3)*S+(a+b+c)*(a^7-(b+c)*a^6+16*b*c*a^5-14*b*c*(b+c)*a^4-(b^2+4*b*c-3*c^2)*(3*b^2-4*b*c-c^2)*a^3+(b^2-c^2)*(b-c)*(3*b^2+16*b*c+3*c^2)*a^2+2*(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-2*(b^2-c^2)^3*(b-c)) : :

The reciprocal orthologic center of these triangles is X(3).

X(13105) lies on these lines: {1,13}, {12,12922}, {530,11239}, {531,12356}, {542,12594}, {616,10528}, {618,5552}, {5473,11248}, {5478,10531}, {5617,10942}, {6268,10930}, {6270,10929}, {6770,10805}, {9916,10834}, {9982,10878}, {10803,12205}, {10915,12781}, {10955,12932}, {10956,12942}, {10958,12952}, {10965,13076}, {11400,12142}, {11509,12337}, {11881,12472}, {11882,12473}, {11914,12793}, {11955,12990}, {11956,12991}, {12000,13103}

X(13106) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO INNER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(3).

X(13106) lies on these lines: {1,14}, {11,12931}, {530,12357}, {531,11240}, {542,12595}, {617,10529}, {619,10527}, {5474,11249}, {5479,10532}, {5613,10943}, {6269,10932}, {6271,10931}, {6773,10806}, {9915,10835}, {9981,10879}, {10804,12204}, {10916,12780}, {10949,12921}, {10957,12941}, {10959,12951}, {10966,13075}, {11401,12141}, {11510,12336}, {11883,12470}, {11884,12471}, {11915,12792}, {11957,12988}, {11958,12989}, {12001,13102}

X(13107) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO OUTER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(3).

X(13107) lies on these lines: {1,13}, {11,12932}, {530,11240}, {531,12357}, {542,12595}, {616,10529}, {618,10527}, {5473,11249}, {5478,10532}, {5617,10943}, {6268,10932}, {6270,10931}, {6770,10806}, {9916,10835}, {9982,10879}, {10804,12205}, {10916,12781}, {10949,12922}, {10957,12942}, {10959,12952}, {10966,13076}, {11401,12142}, {11510,12337}, {11883,12472}, {11884,12473}, {11915,12793}, {11957,12990}, {11958,12991}, {12001,13103}

X(13108) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO 1st NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(13108) lies on these lines: {3,76}, {4,7779}, {5,194}, {30,9863}, {39,1656}, {140,7709}, {262,3851}, {355,726}, {381,538}, {382,511}, {384,11842}, {517,9902}, {550,6194}, {698,1352}, {730,1482}, {732,1351}, {999,10079}, {1569,7746}, {1598,12143}, {1657,9821}, {2080,7751}, {2937,5938}, {3097,9956}, {3104,5339}, {3105,5340}, {3295,10063}, {3398,3734}, {3526,3934}, {3534,5188}, {5054,9466}, {5055,7757}, {5070,7786}, {5790,12782}, {5965,7747}, {5969,12355}, {6036,7863}, {6272,11917}, {6273,11916}, {6309,10983}, {7517,9917}, {7748,11646}, {7760,10796}, {7770,10334}, {7798,10358}, {7801,11632}, {7976,10247}, {9301,9983}, {9654,12837}, {9669,12836}, {10246,12263}, {11849,12338}, {11875,12474}, {11876,12475}, {11911,12794}, {11928,12923}, {11929,12933}, {11949,12992}, {11950,12993}, {12000,13109}, {12001,13110}

X(13108) = reflection of X(i) in X(j) for these (i,j): (3,76), (194,5), (1657,9821), (3095,6248)
X(13108) = anticomplement of X(32448)
X(13108) = X(76)-of-X3-ABC-reflections-triangle
X(13108) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,7697,1656), (99,10104,3), (3095,6248,381), (3934,11171,3526), (10063,13077,3295)

X(13109) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO 1st NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(13109) lies on these lines: {1,76}, {12,12923}, {39,5552}, {119,262}, {194,10528}, {384,10803}, {511,12115}, {538,11239}, {732,12594}, {2782,10679}, {3095,10942}, {5969,12356}, {6248,10531}, {6272,10930}, {6273,10929}, {9917,10834}, {9983,10878}, {10805,12251}, {10915,12782}, {10955,12933}, {10956,12837}, {10958,12836}, {10965,13077}, {11248,11257}, {11400,12143}, {11509,12338}, {11881,12474}, {11882,12475}, {11914,12794}, {11955,12992}, {11956,12993}, {12000,13108}

X(13109) = reflection of X(76) in X(10063)
X(13109) = {X(76), X(7976)}-harmonic conjugate of X(13110)
X(13109) = X(76)-of-inner-Yff-tangents-triangle

X(13110) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO 1st NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(13110) lies on these lines: {1,76}, {11,12933}, {39,10527}, {194,10529}, {384,10804}, {511,12116}, {538,11240}, {732,12595}, {2782,10680}, {3095,10943}, {5969,12357}, {6248,10532}, {6272,10932}, {6273,10931}, {9917,10835}, {9983,10879}, {10806,12251}, {10916,12782}, {10949,12923}, {10957,12837}, {10959,12836}, {10966,13077}, {11249,11257}, {11401,12143}, {11510,12338}, {11883,12474}, {11884,12475}, {11915,12794}, {11957,12992}, {11958,12993}, {12001,13108}

X(13110) = reflection of X(76) in X(10079)
X(13110) = X(76)-of-outer-Yff-tangents-triangle
X(13110) = {X(76), X(7976)}-harmonic conjugate of X(13109)

X(13111) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(13111) lies on these lines: {3,83}, {4,5984}, {5,2896}, {6,382}, {30,12252}, {183,3851}, {381,754}, {385,546}, {517,9903}, {550,3329}, {732,1351}, {999,10080}, {1598,12144}, {1656,6292}, {1657,8725}, {2080,8150}, {3095,7781}, {3295,10064}, {3526,6704}, {3830,12156}, {5790,12783}, {6274,11917}, {6275,11916}, {7517,9918}, {7977,10247}, {9478,9753}, {9654,12944}, {9669,12954}, {9821,10358}, {10246,12264}, {11842,12206}, {11849,12339}, {11875,12476}, {11876,12477}, {11911,12795}, {11928,12924}, {11929,12934}, {11949,12994}, {11950,12995}, {12000,13112}, {12001,13113}

X(13111) = reflection of X(i) in X(j) for these (i,j): (3,83), (1657,8725), (2896,5), (6287,6249)
X(13111) = X(83)-of-X3-ABC-reflections-triangle
X(13111) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6249,6287,381), (10064,13078,3295)

X(13112) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(13112) lies on these lines: {1,83}, {12,12924}, {732,12594}, {754,11239}, {2896,10528}, {5552,6292}, {6249,10531}, {6274,10930}, {6287,10942}, {9751,10269}, {9918,10834}, {10803,12206}, {10805,12252}, {10915,12783}, {10955,12934}, {10956,12944}, {10958,12954}, {10965,13078}, {11248,12122}, {11400,12144}, {11509,12339}, {11881,12476}, {11882,12477}, {11914,12795}, {11955,12994}, {11956,12995}, {12000,13111}

X(13112) = reflection of X(83) in X(10064)
X(13112) = X(83)-of-inner-Yff-tangents-triangle
X(13112) = {X(83),X(7977)}-harmonic conjugate of X(13113)

X(13113) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(13113) lies on these lines: {1,83}, {11,12934}, {732,12595}, {754,11240}, {2896,10529}, {6249,10532}, {6274,10932}, {6275,10931}, {6287,10943}, {6292,10527}, {9751,10267}, {9918,10835}, {10804,12206}, {10806,12252}, {10916,12783}, {10949,12924}, {10957,12944}, {10959,12954}, {10966,13078}, {11249,12122}, {11401,12144}, {11510,12339}, {11883,12476}, {11884,12477}, {11915,12795}, {11957,12994}, {11958,12995}, {12001,13111}

X(13113) = reflection of X(83) in X(10080)
X(13113) = X(83)-of-outer-Yff-tangents-triangle
X(13113) = {X(83),X(7977)}-harmonic conjugate of X(13112)

X(13114) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO 1st ORTHOSYMMEDIAL

Barycentrics (3*S^4+3*(12*(SA+SW)*R^2-3*SA^2-4*S^2+4*SB*SC-3*SW^2)*S^2-(48*R^2+SA-12*SW)*SA*SW^2)*(SB^2-SC^2) : :

The reciprocal orthologic center of these triangles is X(6).

X(13114) lies on the Parry circle and these lines: {2,1637}, {23,647}, {110,112}, {111,1297}, {351,2881}, {684,2492}, {686,10766}, {2781,9138}, {2794,9147}, {2806,9978}, {2831,9980}, {2848,9123}, {9210,11673}, {9998,12503}

X(13114) = reflection of X(9157) in X(351)
X(13114) = antipode of X(9157) in Parry circle
X(13114) = X(1297)-of-1st-Parry-triangle
X(13114) = X(112)-of-2nd-Parry-triangle

X(13115) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO 1st ORTHOSYMMEDIAL

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

The reciprocal orthologic center of these triangles is X(4).

X(13115) lies on the Stammler circle and these lines: {3,112}, {5,12384}, {30,12253}, {127,133}, {132,1656}, {159,399}, {382,10749}, {517,12408}, {999,6020}, {1350,1625}, {1598,12145}, {1657,2794}, {2799,12188}, {2806,12773}, {2831,12331}, {2937,9821}, {3295,3320}, {3830,10718}, {5054,6720}, {5073,10735}, {5790,12784}, {7517,12413}, {8148,10705}, {9301,12503}, {9517,10620}, {9654,12945}, {9669,12955}, {10246,12265}, {10247,13099}, {11842,12207}, {11849,12340}, {11875,12478}, {11876,12479}, {11911,12796}, {11916,12805}, {11917,12806}, {11928,12925}, {11929,12935}, {11949,12996}, {11950,12997}, {12000,13118}, {12001,13119}

X(13115) = reflection of X(i) in X(j) for these (i,j): (3,1297), (382,10749), (3830,10718), (5073,10735), (8148,10705), (12384,5), (12918,127)
X(13115) = X(1297)-of-X3-ABC-reflections-triangle
X(13115) = {X(127), X(12918)}-harmonic conjugate of X(381)

X(13116) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO 1st ORTHOSYMMEDIAL

Trilinears a*(S^4+(2*(3*R^2-SW)*c*b+(SA-SW)^2+(8*R^2-3*SW)*SW)*S^2-2*(4*R^2-SW)*(b*c+2*SA)*SW^2) : :

The reciprocal orthologic center of these triangles is X(4).

X(13116) lies on these lines: {1,1297}, {3,6020}, {5,12955}, {12,12918}, {35,112}, {127,1479}, {132,498}, {388,12253}, {495,12945}, {2781,10088}, {2794,4302}, {2799,10053}, {2806,10058}, {2831,10087}, {3085,12384}, {3295,3320}, {3612,11722}, {5697,10705}, {6284,10749}, {7298,9157}, {9517,10065}, {9530,10056}, {10037,12413}, {10038,12503}, {10039,12784}, {10040,12805}, {10041,12806}, {10523,12925}, {10801,12207}, {10954,12935}, {11398,12145}, {11507,12340}, {11877,12478}, {11878,12479}, {11912,12796}, {11951,12996}, {11952,12997}

X(13116) = reflection of X(12945) in X(495)
X(13116) = X(1297)-of-inner-Yff-triangle
X(13116) = X(1),X(1297)}-harmonic conjugate of X(13117)

X(13117) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO 1st ORTHOSYMMEDIAL

Trilinears a*(S^4+((-6*R^2+2*SW)*b*c+(SA-SW)^2+(8*R^2-3*SW)*SW)*S^2-2*(4*R^2-SW)*(-b*c+2*SA)*SW^2) : :

The reciprocal orthologic center of these triangles is X(4).

X(13117) lies on these lines: {1,1297}, {3,3320}, {5,12945}, {11,12918}, {36,112}, {127,1478}, {132,499}, {496,12955}, {497,12253}, {999,6020}, {1737,12784}, {1795,9532}, {2781,10091}, {2794,4299}, {2799,10069}, {2806,10074}, {2831,10090}, {3086,12384}, {5345,9157}, {7354,10749}, {9517,10081}, {9530,10072}, {10046,12413}, {10047,12503}, {10048,12805}, {10049,12806}, {10483,10735}, {10523,12935}, {10802,12207}, {10948,12925}, {11399,12145}, {11508,12340}, {11879,12478}, {11880,12479}, {11913,12796}, {11953,12996}, {11954,12997}

X(13117) = reflection of X(12955) in X(496)
X(13117) = X(1297)-of-outer-Yff-triangle
X(13117) = X(1),X(1297)}-harmonic conjugate of X(13116)

X(13118) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO 1st ORTHOSYMMEDIAL

Trilinears 8*R*S^4*(-4*R^2*SW^2-S^2*SW+3*R^2*S^2+SW^3)-(R-r)*((SA-SC)*S^2-2*(SA*SB-SC^2)*SA)*((SA-SB)*S^2-2*(SA*SC-SB^2)*SA)*a*(a+b+c) : :

The reciprocal orthologic center of these triangles is X(4).

X(13118) lies on these lines: {1,1297}, {12,12925}, {112,11248}, {127,10531}, {132,5552}, {2799,12189}, {2806,12775}, {3320,10965}, {6020,11509}, {9517,12381}, {9530,11239}, {10528,12384}, {10803,12207}, {10805,12253}, {10834,12413}, {10878,12503}, {10915,12784}, {10929,12805}, {10930,12806}, {10942,12918}, {10955,12935}, {10956,12945}, {10958,12955}, {11400,12145}, {11881,12478}, {11882,12479}, {11914,12796}, {11955,12996}, {11956,12997}, {12000,13115}

X(13118) = reflection of X(1297) in X(13116)
X(13118) = X(1297)-of-inner-Yff-tangents-triangle
X(13118) = {X(1297),X(13099)}-harmonic conjugate of X(13119)

X(13119) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO 1st ORTHOSYMMEDIAL

Trilinears 8*R*S^4*(-4*R^2*SW^2-S^2*SW+3*R^2*S^2+SW^3)-2*(R+r)*((SA-SC)*S^2-2*(SA*SB-SC^2)*SA)*((SA-SB)*S^2-2*(SA*SC-SB^2)*SA)*a*s : :

The reciprocal orthologic center of these triangles is X(4).

X(13119) lies on these lines: {1,1297}, {11,12935}, {112,11249}, {127,10532}, {132,10527}, {2799,12190}, {2806,12776}, {3320,10966}, {9517,12382}, {9530,11240}, {10529,12384}, {10804,12207}, {10806,12253}, {10835,12413}, {10879,12503}, {10916,12784}, {10931,12805}, {10932,12806}, {10943,12918}, {10949,12925}, {10957,12945}, {10959,12955}, {11401,12145}, {11510,12340}, {11883,12478}, {11884,12479}, {11915,12796}, {11957,12996}, {11958,12997}, {12001,13115}

X(13119) = reflection of X(1297) in X(13117)
X(13119) = X(1297)-of-outer-Yff-tangents-triangle
X(13119) = {X(1297),X(13099)}-harmonic conjugate of X(13118)

X(13120) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO 1st SCHIFFLER

Trilinears (-a+b+c)*((2*(b+c)*a^8+8*b*c*a^7-4*(b+c)*(2*b^2-3*b*c+2*c^2)*a^6-4*b*c*(5*b^2-2*b*c+5*c^2)*a^5+2*(b+c)*(6*b^4+6*c^4-b*c*(16*b^2-13*b*c+16*c^2))*a^4+4*(4*b^4+4*c^4-b*c*(3*b^2+4*b*c+3*c^2))*b*c*a^3-2*(b^2-c^2)*(b-c)*(4*b^4+4*c^4-b*c*(6*b^2+5*b*c+6*c^2))*a^2-4*(b^2-c^2)^2*(b-c)^2*b*c*a+2*(b^2-c^2)^3*(b-c)^3)*S-(a+b-c)*(a-b+c)*(2*a^9+3*(b+c)*a^8-4*(b^2+b*c+c^2)*a^7-2*(b+c)*(4*b^2+b*c+4*c^2)*a^6+2*b*c*(b^2-9*b*c+c^2)*a^5+(b+c)*(6*b^4+6*c^4-b*c*(4*b^2-7*b*c+4*c^2))*a^4+4*(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^3+(b+c)*(6*b^4+6*c^4+b*c*(3*b^2-14*b*c+3*c^2))*b*c*a^2-2*(b^2-c^2)^2*(b-c)*(b^3-c^3)*a-(b^2-c^2)^4*(b+c)))/a : :

The reciprocal orthologic center of these triangles is X(21).

X(13120) lies on these lines: {8224,12519}, {8225,12524}, {8228,12615}, {8230,12623}, {8231,12660}, {8233,12695}, {8234,12845}, {8237,12850}, {8239,12877}, {8243,12913}, {8244,13101}, {8246,13123}, {9789,12543}, {10858,12444}, {10867,12451}, {10885,12540}, {10891,12557}, {11030,12917}, {11042,12909}, {11532,12657}, {11687,12535}, {11996,13127}

X(13121) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO REFLECTION

Trilinears 8*R*S^4*(-2*SW+5*R^2)+(R-r)*(S^2+SA*SC)*(S^2+SA*SB)*a*(a+b+c) : :

The reciprocal orthologic center of these triangles is X(4).

X(13121) lies on these lines: {1,54}, {12,12926}, {195,12000}, {539,11239}, {1154,10679}, {1209,5552}, {2888,10528}, {3574,10531}, {6276,10930}, {6277,10929}, {6288,10942}, {7691,11248}, {9920,10834}, {9985,10878}, {10628,12381}, {10803,12208}, {10805,12254}, {10915,12785}, {10955,12936}, {10956,12946}, {10958,12956}, {10965,13079}, {11400,11576}, {11509,12341}, {11881,12480}, {11882,12481}, {11914,12797}, {11955,12998}, {11956,12999}

X(13121) = reflection of X(54) in X(10066)
X(13121) = X(54)-of-inner-Yff-tangents-triangle
X(13121) = {X(54),X(7979)}-harmonic conjugate of X(13122)

X(13122) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO REFLECTION

Trilinears 8*R*S^4*(-2*SW+5*R^2)+(R+r)*(S^2+SA*SC)*(S^2+SA*SB)*a*(a+b+c) : :

The reciprocal orthologic center of these triangles is X(4).

X(13122) lies on these lines: {1,54}, {11,12936}, {195,12001}, {539,11240}, {1154,10680}, {1209,10527}, {2888,10529}, {3574,10532}, {6276,10932}, {6277,10931}, {6288,10943}, {7691,11249}, {9920,10835}, {9985,10879}, {10628,12382}, {10804,12208}, {10806,12254}, {10916,12785}, {10949,12926}, {10957,12946}, {10959,12956}, {10966,13079}, {11401,11576}, {11510,12341}, {11883,12480}, {11884,12481}, {11915,12797}, {11957,12998}, {11958,12999}

X(13122) = reflection of X(54) in X(10082)
X(13122) = X(54)-of-outer-Yff-tangents-triangle
X(13122) = {X(54),X(7979)}-harmonic conjugate of X(13121)

X(13123) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO 1st SCHIFFLER

Barycentrics a*S*(3*R+2*r)*(7*S*s+2*r^3-4*R*SW-6*SW*r)*(a+c)*(a+b)*(-a+b+c)+2*(2*r^3+4*R*SW-S*s+2*SW*r)*((2*a-2*b+2*c)*SB+(R+2*r)*S)*((2*b-2*c+2*a)*SC+(R+2*r)*S)*(-a+b+c) : :

The reciprocal orthologic center of these triangles is X(21).

X(13123) lies on these lines: {21,10266}, {256,6597}, {846,12660}, {1284,12913}, {4199,12695}, {4220,12519}, {5051,12623}, {8229,12615}, {8235,12845}, {8238,12850}, {8240,12877}, {8245,13101}, {8246,13120}, {8249,13124}, {8391,12882}, {8425,13127}, {8731,12444}, {9791,12543}, {10868,12451}, {10892,12557}, {11043,12909}, {11533,12657}, {11688,12535}, {11926,12887}

X(13124) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL-MIDARC TO 1st SCHIFFLER

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=2*(-a+b+c)*b*c*(a^3+(b+c)*a^2-(b^2-3*b*c+c^2)*a-(b^2-c^2)*(b-c))*(a^4-(2*b^2+3*b*c+2*c^2)*a^2+b*c*(b+c)*a+(b^2-c^2)^2)
G(a,b,c)=-(-a+b+c)*c*(a-b+c)*(a^3+(3*b-c)*a^2+(3*b^2-b*c-c^2)*a+(b^2-c^2)*(b-c))*(a^4-(2*b^2-b*c+2*c^2)*a^2-b*c*(3*b-c)*a+(b^2-c^2)^2)
H(a,b,c)=-2*S^2*(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+(b^2-c^2)*(b-c))^2

The reciprocal orthologic center of these triangles is X(21).

X(13124) lies on these lines: {1,13125}, {2089,12913}, {8075,12519}, {8077,12524}, {8078,12660}, {8079,12695}, {8081,12845}, {8085,12615}, {8087,12623}, {8089,13101}, {8241,12877}, {8249,13123}, {8733,12444}, {9793,12543}, {11032,12917}, {11690,12535}, {11888,12540}, {11894,12557}

X(13124) = reflection of X(13125) in X(1)

X(13125) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(21).

X(13125) lies on these lines: {1,13124}, {174,12913}, {258,12660}, {7588,12524}, {8125,12535}, {8351,12909}, {8734,12444}, {11033,12917}, {11859,12451}, {11895,12557}, {11899,12657}

X(13125) = reflection of X(13124) in X(1)

X(13126) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO 1st SCHIFFLER

Trilinears 48*p^6-32*q*p^5-16*(3*q^2+1)*p^4+24*q*p^3+(32*q^4+8*q^2-9)*p^2-16*(q^2-1)*q*p+2*q^2-9/2 : : , where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(79).

X(13126) lies on these lines: {3,10266}, {30,12255}, {381,12600}, {517,12409}, {999,13129}, {1598,12146}, {1656,13089}, {3295,13080}, {5790,12786}, {7517,12414}, {9301,12504}, {9654,12947}, {9669,12957}, {10246,12267}, {10247,13100}, {11681,12682}, {11842,12209}, {11849,12342}, {11875,12482}, {11876,12483}, {11911,12798}, {11916,12807}, {11917,12808}, {11928,12927}, {11929,12937}, {11949,13000}, {11950,13001}, {12000,13130}, {12001,13131}

X(13126) = reflection of X(i) in X(j) for these (i,j): (3,10266), (12919,12600)
X(13126) = X(10266)-of-X3-ABC-reflections-triangle

X(13127) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(21).

X(13127) lies on these lines: {174,12913}, {7587,12524}, {8083,12917}, {8126,12535}, {8382,12623}, {8389,12850}, {8423,13101}, {8425,13123}, {8729,12444}, {11535,12657}, {11860,12451}, {11890,12540}, {11891,12543}, {11896,12557}, {11924,12877}, {11996,13120}

X(13128) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO 1st SCHIFFLER

Trilinears 8*p^6-8*(2*q^2-1)*p^4-12*q*p^3+1/2*(16*q^4+16*q^2-27)*p^2-(4*q^2-13)*q*p+9/4+1/2*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(79).

X(13128) lies on these lines: {1,5180}, {5,12957}, {12,12919}, {388,12255}, {495,12947}, {498,13089}, {1479,12600}, {5719,10058}, {6595,8068}, {10037,12414}, {10038,12504}, {10039,12786}, {10040,12807}, {10041,12808}, {10523,12927}, {10801,12209}, {10954,12937}, {11398,12146}, {11912,12798}, {11952,13001}

X(13128) = midpoint of X(10266) and X(13130)
X(13128) = reflection of X(12947) in X(495)
X(13128) = X(10266)-of-inner-Yff-triangle
X(13128) = {X(1),X(10266)}-harmonic conjugate of X(13129)

X(13129) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO 1st SCHIFFLER

Trilinears 16*p^6-32*q^2*p^4+8*q*p^3+(16*q^4-3)*p^2-2*(4*q^2-1)*q*p+q^2-9/2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal orthologic center of these triangles is X(79).

X(13129) lies on these lines: {1,5180}, {3,13080}, {5,12947}, {11,12919}, {496,12957}, {497,12255}, {499,12543}, {999,13126}, {1478,12600}, {1737,12786}, {1749,12535}, {5437,12660}, {5533,6595}, {10046,12414}, {10047,12504}, {10048,12807}, {10049,12808}, {10523,12937}, {10948,12927}, {11399,12146}, {11508,12342}, {11879,12482}, {11913,12798}, {11954,13001}

X(13129) = midpoint of X(10266) and X(13131)
X(13129) = reflection of X(12957) in X(496)
X(13129) = X(10266)-of-outer-Yff-triangle
X(13129) = {X(1),X(10266)}-harmonic conjugate of X(13128)

X(13130) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO 1st SCHIFFLER

Trilinears (a^10-5*(b-c)^2*a^8-8*b*c*(b+c)*a^7+(10*b^4+10*c^4-b*c*(20*b^2-17*b*c+20*c^2))*a^6+2*b*c*(b+c)*(6*b^2-b*c+6*c^2)*a^5-2*(5*b^6+5*c^6-(6*b^4+6*c^4+b*c*(b^2+9*b*c+c^2))*b*c)*a^4-2*b^2*c^2*(3*b^2-5*b*c+3*c^2)*(b+c)*a^3+(b^2-c^2)^2*(5*b^4+5*c^4-b*c*(4*b^2+7*b*c+4*c^2))*a^2-4*(b^2-c^2)^3*(b-c)*b*c*a-(b^2-c^2)^4*(b-c)^2)/a : :

The reciprocal orthologic center of these triangles is X(79).

X(13130) lies on these lines: {1,5180}, {12,12927}, {119,6595}, {5552,13089}, {10531,12600}, {10803,12209}, {10805,12255}, {10834,12414}, {10878,12504}, {10915,12786}, {10929,12807}, {10930,12808}, {10942,12919}, {10955,12937}, {10956,12947}, {10958,12957}, {10965,13080}, {11400,12146}, {11509,12342}, {11881,12482}, {11882,12483}, {11914,12798}, {11955,13000}, {11956,13001}, {12000,13126}

X(13130) = reflection of X(10266) in X(13128)
X(13130) = X(10266)-of-inner-Yff-tangents-triangle

X(13131) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(79).

X(13131) lies on these lines: {1,5180}, {11,12937}, {3649,12524}, {10527,13089}, {10532,12600}, {10804,12209}, {10806,12255}, {10835,12414}, {10879,12504}, {10916,12786}, {10931,12807}, {10932,12808}, {10943,12919}, {10949,12927}, {10957,12947}, {10959,12957}, {10966,13080}, {11401,12146}, {11510,12342}, {11883,12482}, {11884,12483}, {11915,12798}, {11957,13000}, {11958,13001}, {12001,13126}

X(13131) = reflection of X(10266) in X(13129)
X(13131) = X(10266)-of-outer-Yff-tangents-triangle

X(13132) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO INNER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(13132) lies on these lines: {1,486}, {12,12928}, {487,10528}, {642,5552}, {3564,12430}, {6251,10531}, {6280,10930}, {6281,10929}, {6290,10942}, {9921,10834}, {9986,10878}, {10803,12210}, {10805,12256}, {10915,12787}, {10955,12938}, {10956,12948}, {10958,12958}, {10965,13081}, {11248,12123}, {11400,12147}, {11509,12343}, {11881,12484}, {11882,12485}, {11914,12799}, {11955,13002}, {11956,13003}, {12000,12601}

X(13133) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO INNER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(13133) lies on these lines: {1,486}, {11,12938}, {487,10529}, {642,10527}, {3564,12431}, {6251,10532}, {6280,10932}, {6281,10931}, {6290,10943}, {9921,10835}, {9986,10879}, {10804,12210}, {10806,12256}, {10916,12787}, {10949,12928}, {10957,12948}, {10959,12958}, {10966,13081}, {11249,12123}, {11401,12147}, {11510,12343}, {11883,12484}, {11884,12485}, {11915,12799}, {11957,13002}, {11958,13003}, {12001,12601}

X(13134) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO OUTER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(13134) lies on these lines: {1,485}, {12,12929}, {488,10528}, {641,5552}, {3564,12430}, {6250,10531}, {6278,10930}, {6279,10929}, {6289,10942}, {9922,10834}, {9987,10878}, {10803,12211}, {10805,12257}, {10915,12788}, {10955,12939}, {10956,12949}, {10958,12959}, {10965,13082}, {11248,12124}, {11400,12148}, {11509,12344}, {11881,12486}, {11882,12487}, {11914,12800}, {11955,13004}, {11956,13005}, {12000,12602}

X(13135) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO OUTER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3).

X(13135) lies on these lines: {1,485}, {11,12939}, {488,10529}, {641,10527}, {3564,12431}, {6250,10532}, {6278,10932}, {6279,10931}, {6289,10943}, {9922,10835}, {9987,10879}, {10804,12211}, {10806,12257}, {10916,12788}, {10949,12929}, {10957,12949}, {10959,12959}, {10966,13082}, {11249,12124}, {11510,12344}, {11883,12486}, {11884,12487}, {11915,12800}, {11957,13004}, {11958,13005}, {12001,12602}

X(13136) = TRILINEAR POLE OF X(3)X(8)

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

X(13136) lies on the MacBeath circumconic and these lines: {104, 898}, {110, 1309}, {190, 1813}, {320, 908}, {645, 4558}, {646, 1016}, {651, 4391}, {655, 3904}, {666, 1814}, {765, 1331}, {895, 5380}, {1275, 4554}, {2250, 4584}, {2397, 2401}, {2399, 2406}, {2423, 5381}, {2720, 8707}, {4563, 4601}

X(13136) = isogonal conjugate of X(3310)
X(13136) = isotomic conjugate of X(10015)
X(13136) = X(i)-cross conjugate of X(j) for these (i,j): (2423, 104), (2427, 100), (4358, 1016), (4511, 4998), (5548, 6079)
X(13136) = isoconjugate of X(j) and X(j) for these (i,j): {1, 3310}, {6, 1769}, {19, 8677}, {31, 10015}, {92,23220}, {244, 2427}, {513, 2183}, {517, 649}, {604, 2804}, {650, 1457}, {652, 1875}, {661, 859}, {663, 1465}, {667, 908}, {1919, 3262}, {2397, 3248}
X(13136) = cevapoint of X(i) and X(j) for these (i,j): {2, 3904}, {6, 900}, {100, 2427}, {104, 2423}, {190, 4585}, {519, 3239}, {525, 3936}, {651, 2406} X(13136) = X(645)-beth conjugate of X(4564)
X(13136) = X(i)-zayin conjugate of X(j) for these (i,j): (1, 3310), (1054, 2183)
X(13136) = barycentric product X(i)*X(j) for these {i,j}: {69, 1309}, {104, 668}, {799, 2250}, {909, 1978}, {1016, 2401}, {2342, 4572}, {2720, 3596}
X(13136) = barycentric quotient X(i)/X(j) for these {i,j}: (1,1769), (2,10015), (3,8677), (6,3310), (8,2804), (100,517), (101,2183), (104,513), (108,1875), (109,1457), (110,859), (190,908), (651,1465), (668,3262), (900,3259), (909,649), (1016,2397), (1252,2427), (1309,4), (1795,1459), (1809,521), (1897,1785), (2250,661), (2342,663), (2401,1086), (2423,1015), (2720,56), (2804,3326), (3699,6735), (4242,1845), (5379,4246)

X(13137) = POINT BEID 23

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25623.

X(13137) on the cubic K289 and these lines:
{4,512}, {23,1976}, {98,385}, {232,1692}, {895,11653}, {5640,5967}

X(13137) = reflection of X(98) in its Simson line, X(115)X(523)

X(13138) = TRILINEAR POLE OF X(3)X(9)

Trilinears    1/[cos B cot(C/2) - cos C cot(B/2)] : :
Trilinears    1/{(b - c)[a^3 - b^3 - c^3 + (a - b)(a - c)(b + c)]} : :
Barycentrics    1/(b*(b - c)*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)) : :

The line X(3)X(9) is perpendicular to the trilinear polar of X(9).

X(13138) lies on the MacBeath circumconic and these lines: {1, 271}, {2, 7358}, {78, 3341}, {84, 1320}, {100, 1813}, {108, 521}, {145, 280}, {189, 1814}, {282, 1998}, {285, 3193}, {287, 1999}, {643, 4558}, {644, 1331}, {651, 1897}, {895, 1903}, {1120, 1413}, {1256, 6765}, {1259, 8886}, {1280, 1422}, {1332, 3699}, {1815, 3870}, {2208, 8851}, {2988, 7020}, {2989, 3187}, {2990, 12649}, {2991, 7151}, {3913, 9376}, {4025, 4617}, {4563, 7257}, {7003, 8759}

X(13138) = isogonal conjugate of X(6129)
X(13138) = isotomic conjugate of X(17896)
X(13138) = anticomplement X(7358)
X(13138) = cevapoint of X(i) and X(j) for these (i,j): {1, 521}, {6, 3900}, {109, 8059}, {522, 1210}
X(13138) = X(i)-cross conjugate of X(j) for these (i,j): {109, 100}, {521, 271}, {1783, 651}, {6765, 765}
X(13138) = trilinear pole of X(3)X(9)
X(13138) = isoconjugate of X(j) and X(j) for these (i,j): {1, 6129}, {40, 513}, {56, 8058}, {102, 6087}, {196, 652}, {198, 514}, {208, 521}, {221, 522}, {223, 650}, {227, 3737}, {278, 10397}, {322, 667}, {329, 649}, {342, 1946}, {347, 663}, {512, 8822}, {523, 2360}, {656, 3194}, {661, 1817}, {693, 2187}, {905, 2331}, {1459, 7952}, {1461, 5514}, {2199, 4391}, {2324, 3669}, {3064, 7011}, {3195, 4025}, {3209, 6332}, {3239, 6611}, {3318, 8059}, {3676, 7074}, {7078, 7649}, {7152, 8063}
X(13138) = barycentric product X(i)*(X(j) for these {i,j}: {84, 190}, {99, 1903}, {100, 189}, {101, 309}, {271, 653}, {280, 651}, {282, 664}, {285, 4552}, {312, 8059}, {643, 8808}, {644, 1440}, {646, 1413}, {668, 1436}, {799, 2357}, {1422, 3699}, {1433, 6335}, {1813, 7020}, {1978, 2208}, {2192, 4554}, {4561, 7129}, {4569, 7367}, {4572, 7118}, {6516, 7003}
X(13138) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6129}, {9, 8058}, {84, 514}, {100, 329}, {101, 40}, {108, 196}, {109, 223}, {110, 1817}, {112, 3194}, {163, 2360}, {189, 693}, {190, 322}, {212, 10397}, {268, 521}, {271, 6332}, {280, 4391}, {282, 522}, {285, 4560}, {309, 3261}, {644, 7080}, {651, 347}, {653, 342}, {662, 8822}, {692, 198}, {906, 7078}, {1413, 3669}, {1415, 221}, {1422, 3676}, {1433, 905}, {1436, 513}, {1490, 8063}, {1783, 7952}, {1813, 7013}, {1903, 523}, {2182, 6087}, {2188, 652}, {2192, 650}, {2208, 649}, {2357, 661}, {3900, 5514}, {3939, 2324}, {4559, 227}, {7008, 3064}, {7118, 663}, {7129, 7649}, {7151, 6591}, {7367, 3900}, {8059, 57}, {8064, 3345}, {8750, 2331}, {8808, 4077}

X(13139) = POINT BEID 24

Barycentrics 1/(2*(b^2+c^2)*a^6-3*(2*b^4+b^ 2*c^2+2*c^4)*a^4+(b^2+c^2)*(6* b^4-11*b^2*c^2+6*c^4)*a^2-2*( b^4-b^2*c^2+c^4)*(b^2-c^2)^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25625.

X(13139) lies on this line: {6748,7577}

X(13139) = isogonal conjugate of X(13321)

X(13140) = POINT BEID 25

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25630.

X(13140) lies on these lines: {67,524} et al

X(13140) = complementary conjugate of X(111)

X(13141) = POINT BEID 26

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25650.

Let P=X(36). The nine-point circles of ABC, BCP, CAP, ABP concur in X(13141). (Randy Hutson, July 21, 2017)

X(13141) lies on the nine-point circle and this line: {137,8286}

X(13141) = crosssum of circumcircle intercepts of line X(3)X(80)
X(13141) = orthopole of line X(3)X(80)
X(13141) = Kirikami-six-circles image of X(36)
X(13141) = center of hyperbola {{A,B,C,X(4),X(36)}}

X(13142) = X(4)X(193)∩X(30)X(52)

Barycentrics 2*a^10-7*(b^2+c^2)*a^8+2*(5*b^ 4+8*b^2*c^2+5*c^4)*a^6-8*(c^6+ b^6)*a^4+4*(b^2-c^2)^2*(b^4-b^ 2*c^2+c^4)*a^2-(b^4-c^4)*(b^2- c^2)^3 : :
X(13142) = X(20)-3*X(11245) = 3*X(3060)-X(3575) = 9*X(3060)-X(12278) = 3*X(3575)-X(12278)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25658.

X(13142) lies on these lines: {3,11433}, {4,193}, {5,394}, {6,6823}, {20,11245}, {30,52}, {51,9825}, {68,1595}, {69,11479}, {155,1596}, {195,11799}, {235,1993}, {343,11424}, {382,6225}, {511,12241}, {524,5907}, {576,12233}, {578,6676}, {1092,6677}, {1181,1353}, {1503,10112}, {1593,6515}, {1597,11411}, {1598,6193}, {1885,5889}, {1906,11441}, {1907,11442}, {2883,3629}, {3060,3575}, {3089,3167}, {3527,7401}, {3547,11426}, {5050,7400}, {5446,6756}, {6815,9777}, {10095,10127}

X(13142) = midpoint of X(i) and X(j) for these {i,j}: {1885,5889}, {10263,12370}
X(13142) = reflection of X(i) in X(j) for these (i,j): (6756,5446), (12362,12241)
X(13142) = {X(4), X(193)}-harmonic conjugate of X(12164)

X(13143) = POINT BEID 27

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25666.

X(13143) lies on the Feuerbach hyperbola and these lines: {1,6797}, {3,11279}, {4,9897}, {8,11524}, {11,5559}, {21,2802}, {79,952}, {80,5844}, {104,484}, {498,7320}, {517,3065}, {519, 11604}, {528,3255}

X(13143) = reflection of X(5559) in X(11)

X(13144) = POINT BEID 28

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+12 a^4 b c-22 a^3 b^2 c-6 a^2 b^3 c+24 a b^4 c-6 b^5 c-a^4 c^2-22 a^3 b c^2+67 a^2 b^2 c^2-34 a b^3 c^2-b^4 c^2+4 a^3 c^3-6 a^2 b c^3-34 a b^2 c^3+12 b^3 c^3-a^2 c^4+24 a b c^4-b^2 c^4-2 a c^5-6 b c^5+c^6) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25666.

X(13144) lies on the Jerabek hyperbola of the excentral triangle and on these lines: {40,7993}, {191,2802}

X(13145) = MIDPOINT OF X(65) AND X(3579)

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+3 a^3 b^2 c-2 a^2 b^3 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^3-2 a^2 b 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(13145) = 3 X[1385] - X[3057], 2 X[942] - 3 X[5885], 3 X[3] + X[5903], 4 X[942] - 3 X[6583], 3 X[10202] - X[10222], 3 X[10246] - X[10284], 3 X[354] - X[11278], 3 X[354] - 4 X[12009], X[11278] - 4 X[12009].

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25666.

X(13145) lies on these lines: {1,3}, {10,2771}, {30,3754}, {140,2800}, {355,6951}, {500, 4642}, {549,3878}

X(13145) = midpoint of X(i) and X(j) for these {i,j}: {{65, 3579}, {5690,5884}
X(13145) = reflection of X(i) in X(j) for these (i,j): (6583, 5885), (9955,3812)
X(13145) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,484,3579), (1385,3579,35)

X(13146) = POINT BEID 29

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25666.

X(13146) lies on the Jerabek hyperbola of the excentral triangle and these lines: {1,149}, {9,1030}, {21,5506}, {30,5538}, {40,2771}, {80,3925}, {100,191}, {214,5284}

X(13146) = reflection of X(i) in X(j) for these (i,j): (149, 11263), (191, 100), (1768,3651)

X(13147) = POINT BEID 30

Barycentrics (2 a^4-3 a^2 b^2+b^4-3 a^2 c^2-2 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^2-2 a^6 b^4 c^2-2 a^4 b^6 c^2-9 a^2 b^8 c^2+7 b^10 c^2+9 a^8 c^4-2 a^6 b^2 c^4+6 a^4 b^4 c^4+6 a^2 b^6 c^4-19 b^8 c^4-6 a^6 c^6-2 a^4 b^2 c^6+6 a^2 b^4 c^6+26 b^6 c^6-a^4 c^8-9 a^2 b^2 c^8-19 b^4 c^8+3 a^2 c^10+7 b^2 c^10-c^12) : :

See Tran Quang Hung and Peter Moses, Hyacinthos 25668.

X(13147) lies on this line: {2,3}

X(13147) = orthocenter of cevian triangle of cyclocevian conjugate of X(3)

X(13148) = POINT BEID 31

Barycentrics SB*SC*(SB+SC)*((18*R^2+6*SA-5* SW)*S^2+(36*R^2-11*SW)*SA^2) : :
X(13148) = 2*X(4)-3*X(1112) = X(4)-3*X(1986) = X(4)+3*X(7722) = 4*X(4)-3*X(12133) = 5*X(4)-3*X(12292) = 4*X(140)-3*X(12358) = 3*X(185)-X(10990) = 4*X(389)-3*X(12099) = X(1112)+2*X(7722) = 5*X(1656)-3*X(7723)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25674.

X(13148) lies on these lines: {4,94}, {24,5609}, {74,3516}, {110,3515}, {113,12359}, {125,12233}, {140,12358}, {185,1205}, {389,12099}, {399,3517}, {541,1885}, {542,3575}, {974,10628}, {1154,10295}, {1593,11482}, {1656,7723}, {2777,6146}, {2914,10610}, {3523,12219}, {3542,5655}, {5094,5890}, {7507,9140}, {10018,11561}, {10019,11557}, {10294,11591}

X(13148) = midpoint of X(1986) and X(7722)
X(13148) = reflection of X(i) in X(j) for these (i,j): (1112,1986), (7723,9826), (12133,1112)

X(13149) = TRILINEAR POLE OF X(4)X(7)

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

X(13149) lies on the circumconic {{A,B,C,X(107), X(648)}} and these lines: {7, 1364}, {77, 8764}, {92, 1088}, {107, 934}, {108, 927}, {196, 7056}, {279, 331}, {281, 1996}, {286, 3668}, {648, 4569}, {653, 658}, {664, 1897}, {1847, 6336}, {1857, 2898}, {4554, 6335}

X(13149) = trilinear pole of X(4)X(7)
X(13149) = cevapoint of X(i) and X(j) for these (i,j): {7,905}, {92,693}, {514,3668}, {650,1836}, {1851,6591}
X(13149) = X(i)-cross conjugate of X(j) for these (i,j): {514,286}, {693,1088}, {905,7}, {934,4569}
X(13149) = polar conjugate of X(3900)
X(13149) = isoconjugate of X(j) and X(j) for these (i,j): {3,657}, {9,1946}, {41,521}, {48,3900}, {55,652}, {63,8641}, {78,3063}, {101,3270}, {184,3239}, {212,650}, {219,663}, {220,1459}, {222,4105}, {228,1021}, {283,3709}, {512,2327}, {513,1802}, {520,2332}, {603,4130}, {647,2328}, {649,1260}, {667,3692}, {798,1792}, {810,2287}, {822,4183}, {905,1253}, {906,2310}, {1043,3049}, {1265,1919}, {1437,4171}, {1783,2638}, {1790,4524}, {1803,6607}, {1813,3022}, {2175,6332}, {2192,10397}, {2193,4041}, {2194,8611}, {2200,7253}, {2318,7252}, {3064,6056}, {3271,4587}, {3939,7117}, {4091,7071}, {4397,9247}, {4477,7116}, {8606,9404}
X(13149) = barycentric product X(i)*X(j) for these {i,j}: {4,4569}, {34,4572}, {85,653}, {92,658}, {108,6063}, {190,1847}, {225,4625}, {264,934}, {273,664}, {278,4554}, {279,6335}, {286,4566}, {318,4626}, {331,651}, {648,1446}, {668,1119}, {670,1426}, {811,3668}, {1088,1897}, {1398,6386}, {1427,6331}, {1435,1978}, {1439,6528}, {1461,1969}, {1826,4635}, {3261,7128}, {4617,7017}
X(13149) = barycentric quotient X(i)/X(j) for these {i,j}: {4,3900}, {7,521}, {19,657}, {25,8641}, {27,1021}, {33,4105}, {34,663}, {56,1946}, {57,652}, {85,6332}, {92,3239}, {99,1792}, {100,1260}, {101,1802}, {107,4183}, {108,55}, {109,212}, {162,2328}, {190,3692}, {223,10397}, {225,4041}, {226,8611}, {264,4397}, {269,1459}, {273,522}, {278,650}, {279,905}, {281,4130}, {286,7253}, {318,4163}, {331,4391}, {342,8058}, {513,3270}, {608,3063}, {648,2287}, {651,219}, {653,9}, {658,63}, {662,2327}, {664,78}, {668,1265}, {693,2968}, {811,1043}, {823,2322}, {934,3}, {1020,71}, {1042,810}, {1088,4025}, {1119,513}, {1262,906}, {1275,1332}, {1396,7252}, {1398,667}, {1414,283}, {1426,512}, {1427,647}, {1435,649}, {1439,520}, {1446,525}, {1459,2638}, {1461,48}, {1783,220}, {1813,2289}, {1824,4524}, {1826,4171}, {1827,6607}, {1847,514}, {1876,926}, {1877,4895}, {1880,3709}, {1897,200}, {3064,3119}, {3668,656}, {3669,7117}, {3676,7004}, {4551,2318}, {4552,3694}, {4554,345}, {4564,4587}, {4565,2193}, {4566,72}, {4569,69}, {4572,3718}, {4573,1812}, {4605,3949}, {4616,1444}, {4617,222}, {4625,332}, {4626,77}, {4637,1790}, {4998,4571}, {6335,346}, {6516,1259}, {6614,603}, {7009,4477}, {7012,3939}, {7045,1331}, {7056,4131}, {7103,8678}, {7128,101}, {7177,4091}, {7365,2522}, {7649,2310}, {8059,2188}, {8750,1253}

X(13150) = POINT BEID 32

Barycentrics S^4+(SW^2-2*(4*R^2+SA)*SW+13* R^4+2*SA^2)*S^2+3*(2*R^2-SW)*( SA-SW)*R^2*SA : :

See Tran Quang Hung and César Lozada, Hyacinthos 25683.

X(13150) lies on this line: {2,3}

X(13151) = POINT BEID 33

Barycentrics a(2*a^6-3*(b+c)*a^5-(3*b^2+2*b*c +3*c^2)*a^4+6*(b^3+c^3)*a^3+4* b*c*(b^2+b*c+c^2)*a^2-3*(b^4- c^4)*(b-c)*a+(b^2-c^2)^2*(b-c) ^2) : :
X(13151) = 3*X(1006)-X(3219) = 6*X(1385)-X(3748)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25689.

X(13151) lies on these lines: {1,3}, {30,5249}, {214,5745}, {443,3897}, {500,1104}, {515,6881}, {549,5440}, {912,1006}, {944,6989}, {968,7986}, {971,7489}, {1125,6841}, {2320,9776}, {2771,3683}, {2772,11709}, {3305,6883}, {3534,6173}, {3560,10884}, {3616,6851}, {3740,12738}, {3916,5428}, {5226,6827}, {5731,6826}, {5787,6861}, {6914,10167}

X(13151) = midpoint of X(1) and X(7688)
X(13151) = {X(5709), X(7987)}-harmonic conjugate of X(3)
X(13151) = {X(1),X(3)}-harmonic conjugate of X(37585)

X(13152) = POINT BEID 34

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25690.

X(13152) is the infinite point of the orthic axes of the following homothetic triangles: polar triangle of nine-point circle, orthoanticevian triangle of X(2), orthic axes triangle (see X(2501)), Yiu tangents triangle (see X(7495)). (Randy Hutson, August 19, 2019)

X(13152) lies on this line: {30,511}

X(13152) = isogonal conjugate of X(33639)
X(13152) = complementary conjugate of complement of X(33639)
X(13152) = anticomplementary conjugate of anticomplement of X(33639)
X(13152) = crossdifference of every pair of points on line X(6)X(22462)
X(13152) = (ABC-X3 reflections)-isogonal conjugate of-X(33643)

X(13153) = POINT BEID 35

Barycentrics a^2 (a-b) (a+b) (a-c) (a+c) (2 a^18-6 a^16 b^2+16 a^12 b^6-12 a^10 b^8-12 a^8 b^10+16 a^6 b^12-6 a^2 b^16+2 b^18-11 a^16 c^2+6 a^14 b^2 c^2+31 a^12 b^4 c^2-2 a^10 b^6 c^2-48 a^8 b^8 c^2-2 a^6 b^10 c^2+31 a^4 b^12 c^2+6 a^2 b^14 c^2-11 b^16 c^2+23 a^14 c^4+23 a^12 b^2 c^4-34 a^10 b^4 c^4-12 a^8 b^6 c^4-12 a^6 b^8 c^4-34 a^4 b^10 c^4+23 a^2 b^12 c^4+23 b^14 c^4-22 a^12 c^6-27 a^10 b^2 c^6+54 a^8 b^4 c^6+105 a^6 b^6 c^6+54 a^4 b^8 c^6-27 a^2 b^10 c^6-22 b^12 c^6+13 a^10 c^8-15 a^8 b^2 c^8-137 a^6 b^4 c^8-137 a^4 b^6 c^8-15 a^2 b^8 c^8+13 b^10 c^8-22 a^8 c^10-4 a^6 b^2 c^10+55 a^4 b^4 c^10-4 a^2 b^6 c^10-22 b^8 c^10+41 a^6 c^12+69 a^4 b^2 c^12+69 a^2 b^4 c^12+41 b^6 c^12-38 a^4 c^14-63 a^2 b^2 c^14-38 b^4 c^14+17 a^2 c^16+17 b^2 c^16-3 c^18) (2 a^18-11 a^16 b^2+23 a^14 b^4-22 a^12 b^6+13 a^10 b^8-22 a^8 b^10+41 a^6 b^12-38 a^4 b^14+17 a^2 b^16-3 b^18-6 a^16 c^2+6 a^14 b^2 c^2+23 a^12 b^4 c^2-27 a^10 b^6 c^2-15 a^8 b^8 c^2-4 a^6 b^10 c^2+69 a^4 b^12 c^2-63 a^2 b^14 c^2+17 b^16 c^2+31 a^12 b^2 c^4-34 a^10 b^4 c^4+54 a^8 b^6 c^4-137 a^6 b^8 c^4+55 a^4 b^10 c^4+69 a^2 b^12 c^4-38 b^14 c^4+16 a^12 c^6-2 a^10 b^2 c^6-12 a^8 b^4 c^6+105 a^6 b^6 c^6-137 a^4 b^8 c^6-4 a^2 b^10 c^6+41 b^12 c^6-12 a^10 c^8-48 a^8 b^2 c^8-12 a^6 b^4 c^8+54 a^4 b^6 c^8-15 a^2 b^8 c^8-22 b^10 c^8-12 a^8 c^10-2 a^6 b^2 c^10-34 a^4 b^4 c^10-27 a^2 b^6 c^10+13 b^8 c^10+16 a^6 c^12+31 a^4 b^2 c^12+23 a^2 b^4 c^12-22 b^6 c^12+6 a^2 b^2 c^14+23 b^4 c^14-6 a^2 c^16-11 b^2 c^16+2 c^18) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25695.

X(13153) lies on the circumcircle and these lines: {}

X(13154) = POINT BEID 36

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

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 25698.

X(13154) lies on these lines: {2,3}, {141,9925}, {156,5085}, {394,1493}, {569,5650}, {1173,2979}, {1350,10095}, {6101,10601}

X(13155) = POINT BEID 37

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25701.

X(13155) lies on this line: {2,3}

X(13155) = {X(20), X(6616)}-harmonic conjugate of X(3)

X(13156) = POINT BEID 38

Barycentrics a*((b+c)*a-(b-c)^2)/a/(a^3+(b+c) *a^2-(b+c)^2*a-(b^2-c^2)*(b-c)) : :
X(13156) = 3*(r+2*R)*X(2)-(r+6*R)*X(77)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25701.

X(13156) lies on these lines: {2,77}, {84,516}

X(13156) = {X(189), X(1440)}-harmonic conjugate of X(282)

X(13157) = POINT BEID 39

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25701.

X(13157) lies on the cubic K671 and these lines: {2,253}, {5,8798}, {30,64}, {381,6526}, {8703,11589}

X(13157) = polar conjugate of X(38808)

X(13158) = POINT BEID 409

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25724.

X(13158) lies on this line: {8674, 33593}

X(13159) = POINT BEID 41

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25724.

X(13159) lies on these lines: {7,79}, {9,6701}, {11,553}, {30, 5542}

X(13159) = midpoint of X(7) and X(79)

X(13160) = POINT BEID 42

Barycentrics 2*(3*R^2-SW)*S^2-(4*R^2-SW)*(S B+SC)*SA : :

As a point of the Euler line, X(13160) hasw Shinagawa coefficients (E + 2F, 2F).

See Tran Quang Hung and César Lozada, Hyacinthos 25728.

In the plane of a triangle ABC, let
DEF = circummedial triangle,
H_aH_bH_c = orthic triangle,
Γ_a = circumcircle of EFH_a,
Γ_b = circumcircle of FDH_b,
Γ_c = circumcircle of DEH_b,
U = circle tangent to and encompassing Γ_a, Γ_b, Γ_c.
Then X(13160) = center of U. See X(13160) (Angel Montesdeoca, March 25, 2020)

X(13160) lies on these lines: {2,3}, {12,9630}, {68,7592}, {125,9729}, {141,11444}, {343,5889}, {389,3580}, {511,3574}, {567,12370}, {569,9927}, {1176,1503}, {1181,11442}, {1209,11802}, {1352,11441}, {1568,11793}, {1614,12134}, {2888,3564}, {4348,7741}, {5012,6146}, {5448,5891}, {5449,9730}, {5890,12359}, {6800,9833}, {7221,7951}, {7699,7999}, {9019,11743}, {11402,12429}

X(13160) = midpoint of X(4) and X(7512)
X(13160) = reflection of X(i) in X(j) for these (i,j): (3,7568), (5576,5)
X(13160) = nine-point-circle-inverse of X(3153)
X(13160) = orthocentroidal circle-inverse-of-X(7503)
X(13160) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,7503), (2,3091,6816), (3,4,12225), (4,5,5133), (4,3090,7404), (4,3091,7566), (4,3547,22), (4,7494,20), (4,7558,3), (4,12088,7553), (5,235,3091), (5,10024,403), (5,11563,3850), (381,7387,4), (546,7553,4), (1656,10254,5), (3146,5169,1595), (3542,7401,1995), (3575,6676,7488), (3832,7500,4), (5133,7495,858)

X(13161) = X(1)X(4)∩X(10)X(75)

Barycentrics (b+c)*a^3+(b^2+c^2)*a^2+(b+c)* (b^2+c^2)*a+(b^2-c^2)^2 : :
X(13161) = SW*X(1) - 2*r^2*X(4) = 2*(r^2 - s^2)*X(10) + (2*s^2 - SW)*X(75)

See Tran Quang Hung and César Lozada, Hyacinthos 25735.

Let A'B'C' be the Gergonne line extraversion triangle, as defined at X(10180). Let La be the reflection of line BC in line B'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(13161). (Randy Hutson, July 21, 2017)

For another construction see: Antreas Hatzipolakis and Peter Moses, Euclid 2531 .

X(13161) lies on these lines: {1,4}, {2,988}, {7,989}, {8,3914}, {9,5286}, {10,75}, {12,3666}, {19,1773}, {21,3011}, {30,5266}, {37,5254}, {38,6734}, {40,1423}, {63,5230}, {65,3782}, {141,3714}, {171,4292}, {238,12572}, {315,3879}, {377,612}, {387,3751}, {443,5268}, {495,3931}, {516,5255}, {518,1834}, {519,5015}, {527,1046}, {536,3704}, {611,5706}, {614,2478}, {908,1193}, {938,4310}, {940,10404}, {958,3772}, {960,4415}, {978,3452}, {982,1210}, {1074,3085}, {1076,4293}, {1086,3812}, {1100,7745}, {1329,3752}, {1330,5847}, {1469,10441}, {1722,2551}, {1737,3670}, {1770,5264}, {1836,5710}, {2475,3920}, {2782,5988}, {3120,10459}, {3146,4339}, {3339,4862}, {3361,7397}, {3664,4911}, {3672,5261}, {3674,4920}, {3677,9581}, {3701,4202}, {3744,6284}, {3749,4294}, {3891,5016}, {3912,6656}, {3947,7377}, {3976,11019}, {4104,9534}, {4187,5121}, {4201,7081}, {4298,6996}, {4352,7179}, {4416,7754}, {4642,6735}, {4646,12607}, {4696,4972}, {4850,11681}, {4851,7784}, {4902,5586}, {4968,5051}, {5080,5262}, {5084,5272}, {5247,12527}, {5269,9579}, {5393,7389}, {5405,7388}, {5725,9654}, {7383,10320}, {7395,8071}, {7399,10523}, {7613,11024}, {8069,11414}

X(13161) = isogonal conjugate, wrt the Gergonne line extraversion triangle, of X(1)
X(13161) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3944,946), (10,3663,986), (12,3666,5530), (225,5307,1838), (341,4429,10), (2551,4000,1722)

X(13162) = POINT BEID 42

Barycentrics (3*SA-SW)*(6*SA^2-(9*R^2+2*SW) *SA+6*S^2-2*SW^2+3*R^2*SW) : :

See Tran Quang Hung and César Lozada, Hyacinthos 25737.

X(13162) lies on these lines: {2,99}, {5,5099}, {3580,10413}

X(13163) = 9TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics 2 a^10-3 a^8 b^2-2 a^6 b^4+4 a^4 b^6-b^10-3 a^8 c^2+2 a^6 b^2 c^2-9 a^4 b^4 c^2+7 a^2 b^6 c^2+3 b^8 c^2-2 a^6 c^4-9 a^4 b^2 c^4-14 a^2 b^4 c^4-2 b^6 c^4+4 a^4 c^6+7 a^2 b^2 c^6-2 b^4 c^6+3 b^2 c^8-c^10 : :
X(13163) = 3 X[428] + X[548], 5 X[546] - X[1885], X[3575] + 3 X[5066], 7 X[3857] + X[6240], 5 X[632] + 3 X[7540], 3 X[140] + X[7553], 5 X[5] + 3 X[7576], 7 X[140] - 3 X[7667], 7 X[7553] + 9 X[7667], X[3530] - 3 X[10127], 3 X[547] + X[11819], 9 X[5] - X[12225], 3 X[10109] - X[12362]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25749.

X(13163) lies on these lines: {2,3}, {143,6153}, {3574,10272} ,{7693,12254}

X(13163) = midpoint of X(3628) and X(6756)
X(13163) = {X(5),X(2070)}-harmonic conjugate of X(140)

X(13164) = HUNG-MOSES-EULER POINT

Barycentrics 2 a^16-5 a^14 b^2+a^12 b^4+5 a^10 b^6-5 a^8 b^8+13 a^6 b^10-25 a^4 b^12+19 a^2 b^14-5 b^16-5 a^14 c^2+10 a^12 b^2 c^2-3 a^10 b^4 c^2-10 a^8 b^6 c^2-3 a^6 b^8 c^2+54 a^4 b^10 c^2-69 a^2 b^12 c^2+26 b^14 c^2+a^12 c^4-3 a^10 b^2 c^4+16 a^8 b^4 c^4-7 a^6 b^6 c^4-44 a^4 b^8 c^4+93 a^2 b^10 c^4-56 b^12 c^4+5 a^10 c^6-10 a^8 b^2 c^6-7 a^6 b^4 c^6+30 a^4 b^6 c^6-43 a^2 b^8 c^6+70 b^10 c^6-5 a^8 c^8-3 a^6 b^2 c^8-44 a^4 b^4 c^8-43 a^2 b^6 c^8-70 b^8 c^8+13 a^6 c^10+54 a^4 b^2 c^10+93 a^2 b^4 c^10+70 b^6 c^10-25 a^4 c^12-69 a^2 b^2 c^12-56 b^4 c^12+19 a^2 c^14+26 b^2 c^14-5 c^16 : :

See Tran Quang Hung and Peter Moses, Hyacinthos 25750.

X(13164) lies on these lines: {2, 3}, {137, 11801}

Parallelogic centers: X(13165)-X(13320)

Centers X(13165)-X(13320) were contributed by César Eliud Lozada, April, 9, 2017.

X(13165) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd SHARYGIN TO ANDROMEDA

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

The reciprocal parallelogic center of these triangles is X(1).

X(13165) lies on these lines: {659,3900}, {2254,3667}

X(13165) = 1st-Sharygin-to-2nd-Sharygin similarity image of X(9852)
X(13165) = excentral-to-2nd-Sharygin similarity image of X(4882)
X(13165) = intouch-to-2nd-Sharygin similarity image of X(9850)

X(13166) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO 1st ORTHOSYMMEDIAL

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

The reciprocal parallelogic center of these triangles is X(6).

X(13166) lies on these lines: {4,339}, {25,111}, {33,6020}, {34,3320}, {127,427}, {132,235}, {468,6720}, {1112,9517}, {1297,1593}, {1597,13115}, {1843,2781}, {1862,2806}, {2794,3575}, {2799,5186}, {2831,12138}, {5064,10718}, {5140,10151}, {5185,9518}, {7487,13200}, {10705,11396}, {10735,12173}, {10766,12167}, {11363,11722}, {11380,13195}, {11383,13206}

X(13166) = reflection of X(12145) in X(4)
X(13166) = X(112)-of-anti-Ara-triangle

X(13167) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-ARTZT TO 4th ANTI-BROCARD

Trilinears a*((3*SA+SW)*b^2*c^2*(3*(9*R^2-SW)*S^2-SW^3)-(SA-SC)*(SA-SB)*(SW^2*(SW+2*SA)-(7*S^2+6*SA^2)*SW+9*R^2*S^2)*SW) : :

The reciprocal parallelogic center of these triangles is X(13168).

X(13167) lies on these lines: {2,13191}, {6,110}, {183,9869}, {511,13168}, {599,12149}, {2780,13169}, {6088,11162}, {9027,9870}

X(13167) = reflection of X(12149) in X(599)
X(13167) = X(6236)-of-anti-Artzt-triangle

X(13168) = PARALLELOGIC CENTER OF THESE TRIANGLES: 4th ANTI-BROCARD TO ANTI-ARTZT

Barycentrics a^10+(2*b^4-25*b^2*c^2+2*c^4)*a^6+10*b^2*c^2*(b^2+c^2)*a^4-(b^2-c^2)^2*(3*b^4-19*b^2*c^2+3*c^4)*a^2-2*(b^4-c^4)*(b^2-c^2)*b^2*c^2 : :

The reciprocal parallelogic center of these triangles is X(13167).

X(13168) lies on these lines: {2,3}, {511,13167}, {524,9871}, {1499,13192}, {2782,10787}, {9870,11258}, {10753,12112}

X(13168) = reflection of X(9870) in X(11258)
X(13168) = X(6236)-of-4th-anti-Brocard-triangle

X(13169) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-ARTZT TO 4th BROCARD

Barycentrics 5*a^8-6*(b^2+c^2)*a^6-(b^2-3*b*c-c^2)*(b^2+3*b*c-c^2)*a^4+(3*b^2-2*c^2)*(2*b^2-3*c^2)*(b^2+c^2)*a^2-4*(b^4-c^4)^2 : :
X(13169) = 4*X(67)-X(895)

The reciprocal parallelogic center of these triangles is X(4).

X(13169) lies on these lines: {2,9769}, {67,524}, {69,74}, {110,599}, {125,1992}, {183,9759}, {381,10752}, {541,11180}, {690,11161}, {1352,10706}, {2780,13167}, {2781,11188}, {2854,2979}, {3448,11160}, {5181,9143}, {5476,7577}, {5505,10989}, {5642,11061}, {5648,6030}, {9970,11178}

X(13169) = midpoint of X(3448) and X(11160)
X(13169) = reflection of X(i) in X(j) for these (i,j): (110,599), (895,9140), (1992,125), (9140,67), (9143,5181), (9970,11178), (10706,1352), (10752,381), (11061,5642)
X(13169) = X(1296)-of-anti-Artzt-triangle
X(13169) = 4th-Brocard-to-anti-Artzt similarity image of X(4)

X(13170) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-ARTZT TO 3rd PARRY

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

The reciprocal parallelogic center of these triangles is X(9147).

X(13170) lies on the anti-Artzt circle and these lines: {99,512}, {110,11186}, {183,12434}, {511,8597}, {599,12157}

X(13170) = reflection of X(12157) in X(599)
X(13170) = circumsymmedial-to-anti-Artzt similarity image of X(805)

X(13171) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO 1st HYACINTH

Trilinears (a^10-(b^2+c^2)*a^8-(2*b^2-c^2)*(b^2-2*c^2)*a^6+2*(-c^4+b^4)*(b^2-c^2)*a^4+(b^6-c^6)*(b^2-c^2)*a^2-(-c^4+b^4)^2*(b^2+c^2))*a : :
X(13171) = 4*X(74)+X(12174) = 3*X(11402)-4*X(13198)

The reciprocal parallelogic center of these triangles is X(10116).

X(13171) lies on these lines: {3,74}, {4,9919}, {6,1205}, {22,3448}, {25,125}, {26,10264}, {67,159}, {113,7395}, {146,7503}, {265,7387}, {378,12244}, {427,13203}, {1112,5622}, {1181,10628}, {1204,7729}, {1593,2777}, {1597,10721}, {1993,13201}, {2781,11402}, {2931,9715}, {2935,3516}, {2937,11457}, {3028,10831}, {3047,3167}, {3556,10693}, {5198,7687}, {5504,9908}, {5972,7484}, {6723,11284}, {7071,10118}, {7512,12317}, {7516,10272}, {7530,11801}, {7592,7731}, {7728,9818}, {7984,12410}, {9140,9909}, {9861,11005}, {9876,11006}, {9914,11744}, {10119,11406}, {10323,12383}, {10681,11408}, {10682,11409}, {10833,12904}, {10982,11807}, {11403,13202}, {12083,12902}, {12164,12219}

X(13171) = reflection of X(i) in X(j) for these (i,j): (12165,1181), (12168,3)
X(13171) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22,3448,12310), (125,10117,25), (5621,10117,125)
X(13171) = X(100)-of-anti-Ascella-triangle if ABC is acute

X(13172) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO 1st ANTI-BROCARD

Barycentrics 3*a^8-7*(b^2+c^2)*a^6+(6*b^4+7*b^2*c^2+6*c^4)*a^4-(b^2+c^2)^3*a^2-(b^4-3*b^2*c^2+c^4)*(b^2-c^2)^2 : :
X(13172) = 3*X(4)-4*X(114) = 3*X(4)-2*X(10723) = X(4)-4*X(10992) = 3*X(20)-X(5984) = 2*X(98)-3*X(376) = X(98)-3*X(12117) = 4*X(98)-3*X(12243) = 3*X(99)-2*X(114) = 3*X(99)-X(10723) = X(147)-3*X(8591) = 3*X(8591)-2*X(13188)

The reciprocal parallelogic center of these triangles is X(385).

X(13172) lies on these lines: {2,6321}, {3,148}, {4,99}, {20,2782}, {24,13175}, {30,147}, {98,376}, {115,631}, {388,10086}, {497,10089}, {515,13174}, {542,11001}, {549,12355}, {550,12188}, {620,3090}, {671,3524}, {690,12383}, {1569,7737}, {1916,7709}, {2482,3545}, {2783,12248}, {2787,13199}, {2794,3529}, {2797,5667}, {2799,13200}, {3023,4294}, {3027,4293}, {3085,13182}, {3086,13183}, {3095,6658}, {3146,6033}, {3522,12042}, {3533,6722}, {3534,11177}, {3543,8724}, {3576,11599}, {4027,10788}, {5071,6721}, {5186,7487}, {5473,6773}, {5474,6770}, {5603,11711}, {5657,13178}, {5969,6776}, {5989,11676}, {6319,10783}, {6320,10784}, {7738,10359}, {7756,10357}, {7967,7983}, {8596,10304}, {9861,12082}, {10353,10796}, {10519,11646}, {10785,13180}, {10786,13181}, {10805,13189}, {10806,13190}, {11257,12252}, {11491,13173}, {11843,13176}, {11844,13177}, {11845,13179}, {11846,13184}, {11847,13185}

X(13172) = reflection of X(i) in X(j) for these (i,j): (4,99), (99,10992), (147,13188), (148,3), (376,12117), (3146,6033), (3543,8724), (6770,5474), (6773,5473), (8596,11632), (9862,20), (10723,114), (11177,3534), (12188,550), (12243,376), (12355,549)
X(13172) = anticomplement of X(6321)
X(13172) = X(99)-of-anti-Euler-triangle
X(13172) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (99,10723,114), (114,10723,4), (147,8591,13188), (8596,10304,11632)

X(13173) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 1st ANTI-BROCARD

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

The reciprocal parallelogic center of these triangles is X(385).

X(13173) lies on these lines: {3,11710}, {35,13174}, {55,99}, {56,7983}, {98,10310}, {100,148}, {114,11496}, {115,1376}, {197,13175}, {542,12327}, {543,4421}, {620,1001}, {690,13204}, {1012,9864}, {2482,4428}, {2782,11248}, {2783,12332}, {2787,13205}, {2794,12340}, {2799,13206}, {3027,11509}, {3295,11711}, {4027,11490}, {5186,11383}, {5537,9860}, {5687,13178}, {5969,12329}, {6319,11497}, {6320,11498}, {6321,11499}, {8782,11494}, {10086,11507}, {10089,11508}, {11491,13172}, {11492,13176}, {11493,13177}, {11500,13181}, {11501,13182}, {11502,13183}, {11503,13184}, {11504,13185}, {11510,13190}, {11848,13179}, {11849,13188}

X(13173) = reflection of X(i) in X(j) for these (i,j): (12178,11248), (12326,4421), (13180,115)
X(13173) = X(99)-of-anti-Mandart-incircle-triangle

X(13174) = PARALLELOGIC CENTER OF THESE TRIANGLES: AQUILA TO 1st ANTI-BROCARD

Barycentrics a^5+2*(b+c)*a^4-(b^2+c^2)*a^3-2*(b+c)*(b^2+c^2)*a^2-(b^4-3*b^2*c^2+c^4)*a+2*b^2*c^2*(b+c) : :
X(13174) = 3*X(1)-2*X(7983) = 2*X(98)-3*X(165) = 3*X(99)-X(7983) = 3*X(99)-2*X(11711) = 4*X(114)-3*X(1699) = 4*X(115)-5*X(1698) = 8*X(620)-7*X(3624) = 2*X(1916)-3*X(3097) = 3*X(2482)-2*X(11725) = 3*X(3679)-2*X(13178)

The reciprocal parallelogic center of these triangles is X(385).

X(13174) lies on the Bevan circle, and the bianticevian conic of X(1) and X(2), and on these lines: {1,99}, {2,846}, {10,148}, {20,2784}, {35,13173}, {40,2782}, {43,3029}, {57,3027}, {98,165}, {114,1699}, {115,1571}, {147,516}, {190,2640}, {191,2795}, {194,1046}, {291,3571}, {515,13172}, {517,13188}, {519,8591}, {538,5184}, {542,9904}, {543,3679}, {620,3624}, {690,2948}, {726,2959}, {1282,2786}, {1569,1572}, {1697,3023}, {1764,1768}, {1916,3097}, {2023,9574}, {2482,11725}, {2787,5541}, {2794,12408}, {2938,6194}, {3044,9587}, {3509,4037}, {3579,12188}, {3751,5969}, {4027,10789}, {4654,12350}, {5186,7713}, {5537,12178}, {5587,6321}, {5588,6320}, {5589,6319}, {5691,9864}, {5984,9778}, {7970,11531}, {7987,11710}, {8185,13175}, {8187,13177}, {8188,13184}, {8189,13185}, {9578,13182}, {9579,12184}, {9580,12185}, {9581,13183}, {9900,12781}, {9901,12780}, {9903,12782}, {10826,13180}, {10827,13181}, {11852,13179}

X(13174) = reflection of X(i) in X(j) for these (i,j): (1,99), (148,10), (3679,9881), (5691,9864), (7983,11711), (9860,40), (9875,3679), (9900,12781), (9901,12780), (11531,7970), (12188,3579)
X(13174) = anticomplement of X(11599)
X(13174) = anticomplementary conjugate of X(20558)
X(13174) = anticomplementary isotomic conjugate of X(20536)
X(13174) = antipode of X(9860) in Bevan circle
X(13174) = X(99)-of-Aquila-triangle
X(13174) = excentral isogonal conjugate of X(511)
X(13174) = X(188)-aleph conjugate of X(511)
X(13174) = intersection, other than excenters, of the Bevan circle and the bianticevian conic of X(1) and X(2)
X(13174) = antipode of X(1) in the bianticevian conic of X(1) and X(2)
X(13174) = Bevan circle antipode of X(9860)
X(13174) = X(3027) of tangential triangle of excentral triangle
X(13174) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (99,7983,11711), (7983,11711,1)

X(13175) = PARALLELOGIC CENTER OF THESE TRIANGLES: ARA TO 1st ANTI-BROCARD

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

The reciprocal parallelogic center of these triangles is X(385).

X(13175) lies on these lines: {3,115}, {22,148}, {24,13172}, {25,99}, {98,11414}, {114,1598}, {159,5969}, {197,13173}, {542,9919}, {543,9876}, {620,5020}, {690,12310}, {1593,10723}, {2782,7387}, {2783,9913}, {2794,12413}, {2799,11641}, {2971,4558}, {3023,10833}, {3517,10992}, {4027,10790}, {5594,6320}, {5595,6319}, {5984,12087}, {6721,11484}, {7517,13188}, {7983,8192}, {8185,13174}, {8190,13176}, {8191,13177}, {8193,13178}, {8194,13184}, {8195,13185}, {8782,10828}, {9862,12082}, {10037,10086}, {10046,10089}, {10829,13180}, {10830,13181}, {10831,13182}, {10832,13183}, {10834,13189}, {10835,13190}, {11365,11711}, {11853,13179}, {12083,12188}

X(13175) = reflection of X(i) in X(j) for these (i,j): (9861,7387), (9876,9909)
X(13175) = X(99)-of-Ara-triangle

X(13176) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO 1st ANTI-BROCARD

Barycentrics (b+c)*(a^4-(b^2+c^2)*a^2-(b^2-c^2)*(b-c)*a+b^2*c^2)*D-a^2*(a+b+c)*(a-b-c)*(a^4-(b^2+c^2)*a^2-b^2*c^2+c^4+b^4) : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal parallelogic center of these triangles is X(385).

X(13176) lies on these lines: {55,13177}, {98,11822}, {99,5597}, {114,8196}, {115,5599}, {148,5601}, {517,12180}, {519,12346}, {542,12365}, {543,11207}, {2782,11252}, {2783,12462}, {2794,12478}, {3023,11873}, {4027,11837}, {5186,11384}, {5598,7983}, {5969,12452}, {6319,8198}, {6320,8199}, {6321,8200}, {8190,13175}, {8197,13178}, {8201,13184}, {8202,13185}, {8782,11861}, {10086,11877}, {10089,11879}, {11366,11711}, {11492,13173}, {11843,13172}, {11865,13180}, {11867,13181}, {11869,13182}, {11871,13183}, {11875,13188}, {11881,13189}, {11883,13190}

X(13176) = X(99)-of-1st-Auriga-triangle
X(13176) = X(7983)-of-2nd-Auriga-triangle

X(13177) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO 1st ANTI-BROCARD

Barycentrics (b+c)*(a^4-(b^2+c^2)*a^2-(b^2-c^2)*(b-c)*a+b^2*c^2)*D+a^2*(a-b-c)*(a+b+c)*(a^4-(b^2+c^2)*a^2-b^2*c^2+c^4+b^4) : :
where D=4*S*sqrt(R*(4*R+r))

The reciprocal parallelogic center of these triangles is X(385).

X(13177) lies on these lines: {55,13176}, {98,11823}, {99,5598}, {114,8203}, {115,5600}, {148,5602}, {517,12179}, {519,12345}, {542,12366}, {543,11208}, {2782,11253}, {2783,12463}, {2794,12479}, {3023,11874}, {4027,11838}, {5186,11385}, {5597,7983}, {5969,12453}, {6319,8205}, {6320,8206}, {6321,8207}, {8187,13174}, {8191,13175}, {8204,13178}, {8208,13184}, {8209,13185}, {8782,11862}, {10086,11878}, {10089,11880}, {11367,11711}, {11493,13173}, {11844,13172}, {11866,13180}, {11868,13181}, {11870,13182}, {11872,13183}, {11876,13188}, {11882,13189}, {11884,13190}

X(13177) = X(99)-of-2nd-Auriga-triangle
X(13177) = X(7983)-of-1st-Auriga-triangle

X(13178) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO 1st ANTI-BROCARD

Barycentrics a^5-(b^2+c^2)*a^3+b^2*c^2*a-(b^2-c^2)^2*(b+c) : :
X(13178) = 3*X(1)-4*X(11725) = 2*X(114)-3*X(5587) = 3*X(115)-2*X(11725) = 4*X(620)-5*X(1698) = 3*X(671)-X(7983) = 2*X(1386)-3*X(6034) = 2*X(1569)-3*X(3097) = 3*X(3679)-X(13174) = 3*X(5470)-X(7974) = 3*X(5657)-X(13172)

The reciprocal parallelogic center of these triangles is X(385).

X(13178) lies on these lines: {1,115}, {2,11711}, {8,148}, {10,99}, {30,5184}, {65,13182}, {72,13181}, {80,291}, {98,515}, {114,5587}, {355,2782}, {516,10723}, {517,6321}, {518,11646}, {519,671}, {542,3751}, {543,3679}, {551,9166}, {620,1698}, {730,1916}, {944,11710}, {946,7970}, {1012,12178}, {1386,6034}, {1569,3097}, {1737,10089}, {1837,3023}, {2640,4092}, {2783,12751}, {2794,5691}, {2796,4669}, {2802,10769}, {3027,5252}, {3057,13183}, {3241,12258}, {3416,5969}, {3576,6036}, {3624,6722}, {4027,10791}, {4769,9902}, {5090,5186}, {5469,7975}, {5470,7974}, {5657,13172}, {5687,13173}, {5688,6320}, {5689,6319}, {5790,13188}, {5847,10754}, {8193,13175}, {8197,13176}, {8204,13177}, {8214,13184}, {8215,13185}, {8227,11724}, {8782,9857}, {8980,9583}, {10039,10086}, {10053,10572}, {10914,13180}, {10915,13189}, {10916,13190}, {11900,13179}

X(13178) = midpoint of X(i) and X(j) for these {i,j}: {8,148}, {3679,9875}, {5691,9860}, {9900,9901}
X(13178) = reflection of X(i) in X(j) for these (i,j): (1,115), (99,10), (944,11710), (3241,12258), (7970,946), (7974,11705), (7975,11706), (7983,11599), (9864,355), (9881,3679), (9884,551)
X(13178) = anticomplement of X(11711)
X(13178) = X(99)-of-outer-Garcia-triangle
X(13178) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (671,7983,11599), (9166,9884,551)

X(13179) = PARALLELOGIC CENTER OF THESE TRIANGLES: GOSSARD TO 1st ANTI-BROCARD

Barycentrics (4*S^2-(3*SA+SW)*(SB+SC))*(S^2-3*SB*SC)*(S^4-6*(6*R^2-SW)*SA*S^2+SA^2*SW^2) : :
X(13179) = 2*X(114)-3*X(11897) = 2*X(11711)-3*X(11831) = 3*X(11845)-X(13172) = 3*X(11852)-X(13174)

The reciprocal parallelogic center of these triangles is X(385).

X(13179) lies on these lines: {30,98}, {99,402}, {114,11897}, {115,1650}, {148,4240}, {542,12369}, {543,1651}, {2782,11251}, {2783,12752}, {2794,12796}, {3023,11909}, {4027,11839}, {5186,11832}, {5969,12583}, {6319,11901}, {6320,11902}, {7983,11910}, {8782,11885}, {9166,11049}, {10086,11912}, {10089,11913}, {11711,11831}, {11845,13172}, {11848,13173}, {11852,13174}, {11853,13175}, {11863,13176}, {11864,13177}, {11900,13178}, {11903,13180}, {11904,13181}, {11905,13182}, {11906,13183}, {11907,13184}, {11908,13185}, {11911,13188}, {11914,13189}, {11915,13190}

X(13179) = midpoint of X(148) and X(4240)
X(13179) = X(99)-of-Gossard-triangle
X(13179) = reflection of X(i) in X(j) for these (i,j): (99,402), (1650,115), (12181,11251), (12347,1651)

X(13180) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO 1st ANTI-BROCARD

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

The reciprocal parallelogic center of these triangles is X(385).

X(13180) lies on these lines: {11,99}, {12,13189}, {98,11826}, {114,10893}, {115,1376}, {148,3434}, {355,6321}, {542,12371}, {543,11235}, {2782,10525}, {2783,12761}, {2794,12925}, {3023,10947}, {4027,10794}, {5186,11390}, {5969,12586}, {6319,10919}, {6320,10920}, {7983,10944}, {8782,10871}, {10086,10523}, {10089,10948}, {10785,13172}, {10826,13174}, {10829,13175}, {10914,13178}, {10945,13184}, {10946,13185}, {10949,13190}, {11373,11711}, {11865,13176}, {11866,13177}, {11903,13179}, {11928,13188}

X(13180) = reflection of X(i) in X(j) for these (i,j): (12182,10525), (12348,11235), (13173,115), (13181,6321)
X(13180) = X(99)-of-inner-Johnson-triangle
X(13180) = {X(99), X(10769)}-harmonic conjugate of X(13183)

X(13181) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO 1st ANTI-BROCARD

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

The reciprocal parallelogic center of these triangles is X(385).

X(13181) lies on these lines: {11,13190}, {12,99}, {72,13178}, {98,11827}, {114,10894}, {115,958}, {148,3436}, {355,6321}, {542,12372}, {543,11236}, {2782,10526}, {2783,12762}, {2794,12935}, {3023,10953}, {4027,10795}, {5186,11391}, {5969,12587}, {6253,10723}, {6319,10921}, {6320,10922}, {7983,10950}, {8782,10872}, {10086,10954}, {10089,10523}, {10786,13172}, {10827,13174}, {10830,13175}, {10951,13184}, {10952,13185}, {10955,13189}, {11374,11711}, {11500,13173}, {11608,12527}, {11867,13176}, {11868,13177}, {11904,13179}, {11929,13188}

X(13181) = reflection of X(i) in X(j) for these (i,j): (12183,10526), (12349,11236), (13180,6321)
X(13181) = X(99)-of-outer-Johnson-triangle

X(13182) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO 1st ANTI-BROCARD

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

The reciprocal parallelogic center of these triangles is X(385).

X(13182) lies on the Johnson-Yff-inner-circle and these lines: {1,6321}, {4,3023}, {5,10089}, {12,99}, {30,10053}, {56,115}, {65,13178}, {98,7354}, {114,10895}, {147,5229}, {148,388}, {381,12351}, {495,10086}, {542,12373}, {543,11237}, {671,5434}, {690,12903}, {1317,10769}, {1388,11725}, {1428,6034}, {1469,11646}, {1478,2782}, {1569,9650}, {2023,9597}, {2783,12763}, {2794,12943}, {3029,9552}, {3044,9652}, {3085,13172}, {3585,6033}, {4027,10797}, {4299,12042}, {4654,9875}, {4995,12117}, {5186,11392}, {5204,6036}, {5298,9166}, {5478,12951}, {5479,12952}, {5969,12588}, {6284,10723}, {6319,10923}, {6320,10924}, {7983,10944}, {8782,10873}, {9578,13174}, {9579,9860}, {9654,13188}, {9655,12188}, {9880,11238}, {10106,11599}, {10831,13175}, {10956,13189}, {10957,13190}, {11375,11711}, {11501,13173}, {11869,13176}, {11870,13177}, {11905,13179}, {11930,13184}, {11931,13185}

X(13182) = reflection of X(i) in X(j) for these (i,j): (10086,495), (12184,1478), (12350,11237)
X(13182) = antipode of X(12184) in Johnson-Yff-inner-circle
X(13182) = X(99)-of-1st-Johnson-Yff-triangle
X(13182) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,6321,13183), (4,3023,12185), (148,388,3027)

X(13183) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO 1st ANTI-BROCARD

Barycentrics (-a+b+c)*(a^6-(b^2+c^2)*a^4+b^2*c^2*a^2+(b^2-c^2)^2*(b-c)^2) : :
X(13183) = X(10069)-3*X(10070)

The reciprocal parallelogic center of these triangles is X(385).

X(13183) lies on the Johnson-Yff-outer-circle and these lines: {1,6321}, {2,12354}, {4,3027}, {5,10086}, {11,99}, {13,13075}, {14,13076}, {30,10069}, {55,115}, {98,6284}, {114,10896}, {147,5225}, {148,497}, {496,10089}, {542,12374}, {543,11238}, {671,3058}, {690,12904}, {950,11599}, {1479,2782}, {1569,9665}, {1916,13077}, {2023,9598}, {2330,6034}, {2783,12764}, {2794,12953}, {3029,9555}, {3044,9667}, {3056,11646}, {3057,13178}, {3086,13172}, {3583,6033}, {4027,10798}, {4302,12042}, {4995,9166}, {5186,11393}, {5217,6036}, {5298,12117}, {5479,12942}, {5969,12589}, {6319,10925}, {6320,10926}, {7354,10723}, {7983,10950}, {8782,10874}, {9580,9860}, {9581,13174}, {9668,12188}, {9669,13188}, {10832,13175}, {10958,13189}, {10959,13190}, {11376,11711}, {11502,13173}, {11606,13078}, {11871,13176}, {11872,13177}, {11906,13179}, {11932,13184}, {11933,13185}

X(13183) = reflection of X(i) in X(j) for these (i,j): (10089,496), (12185,1479), (12351,11238)
X(13183) = antipode of X(12185) in Johnson-Yff-outer-circle
X(13183) = X(99)-of-2nd-Johnson-Yff-triangle
X(13183) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,6321,13182), (4,3027,12184), (148,497,3023)

X(13184) = PARALLELOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO 1st ANTI-BROCARD

Barycentrics (-3*SA^2+4*SW^2)*S^2+S*(4*(6*SA^2-SW^2-4*SA*SW)*R^2+(SA-SW)*SW^2+(20*R^2-3*SA+3*SW)*S^2)+(4*SA^2-2*SA*SW-SW^2)*SW^2 : :

The reciprocal parallelogic center of these triangles is X(385).

X(13184) lies on these lines: {98,11828}, {99,493}, {114,8212}, {115,8222}, {148,1131}, {542,12377}, {543,12152}, {2782,10669}, {2783,12765}, {2794,12996}, {3023,11947}, {4027,11840}, {5186,11394}, {5969,12590}, {6319,8216}, {6320,8218}, {6321,8220}, {6461,13185}, {7983,8210}, {8188,13174}, {8194,13175}, {8201,13176}, {8208,13177}, {8214,13178}, {8782,10875}, {10086,11951}, {10089,11953}, {10945,13180}, {10951,13181}, {10981,12187}, {11377,11711}, {11503,13173}, {11846,13172}, {11907,13179}, {11930,13182}, {11932,13183}, {11949,13188}, {11955,13189}, {11957,13190}

X(13184) = X(99)-of-Lucas-homothetic-triangle

X(13185) = PARALLELOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO 1st ANTI-BROCARD

Barycentrics (-3*SA^2+4*SW^2)*S^2-S*(4*(6*SA^2-SW^2-4*SA*SW)*R^2+(SA-SW)*SW^2+(20*R^2-3*SA+3*SW)*S^2)+(4*SA^2-2*SA*SW-SW^2)*SW^2 : :

The reciprocal parallelogic center of these triangles is X(385).

X(13185) lies on these lines: {98,11829}, {99,494}, {114,8213}, {115,8223}, {148,1132}, {542,12378}, {543,12153}, {690,13216}, {2782,10673}, {2783,12766}, {2794,12997}, {3023,11948}, {4027,11841}, {5186,11395}, {5969,12591}, {6319,8217}, {6320,8219}, {6321,8221}, {6461,13184}, {7983,8211}, {8189,13174}, {8195,13175}, {8202,13176}, {8209,13177}, {8215,13178}, {8782,10876}, {10086,11952}, {10089,11954}, {10946,13180}, {10952,13181}, {10981,12186}, {11378,11711}, {11504,13173}, {11847,13172}, {11908,13179}, {11931,13182}, {11933,13183}, {11950,13188}, {11956,13189}, {11958,13190}

X(13185) = X(99)-of-Lucas(-1)-homothetic-triangle

X(13186) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st ANTI-BROCARD TO X-PARABOLA-TANGENTIAL

Barycentrics ((b^2+c^2)*a^6-2*(2*b^4-b^2*c^2+2*c^4)*a^4+3*b^2*c^2*(b^2+c^2)*a^2+b^8+c^8-b^6*c^2-2*b^4*c^4-b^2*c^6)*(b^2-c^2) : :
X(13186) = 8*X(230)-9*X(9189)

The reciprocal parallelogic center of these triangles is X(13187).

X(13186) lies on these lines: {2,13232}, {3,13237}, {230,231}, {690,7779}, {4027,13197}

X(13186) = X(9293)-of-1st-anti-Brocard-triangle

X(13187) = PARALLELOGIC CENTER OF THESE TRIANGLES: X-PARABOLA-TANGENTIAL TO 1st ANTI-BROCARD

Barycentrics (a^8-2*(b^2+c^2)*a^6-(b^4-8*b^2*c^2+c^4)*a^4+2*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-b^8+2*b^6*c^2-b^4*c^4+2*b^2*c^6-c^8)*(b^2-c^2) : :
X(13187) = 2*X(115)-3*X(10278) = X(148)-3*X(8029) = 4*X(620)-3*X(10190) = X(9293)-3*X(10278)

The reciprocal parallelogic center of these triangles is X(13186).

X(13187) lies on these lines: {99,523}, {115,9293}, {148,8029}, {543,12076}, {620,10190}

X(13187) = reflection of X(9293) in X(115)
X(13187) = {X(9293), X(10278)}-harmonic conjugate of X(115)

X(13188) = PARALLELOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO 1st ANTI-BROCARD

Barycentrics a^8-4*(b^2+c^2)*a^6+(4*b^4+5*b^2*c^2+4*c^4)*a^4-(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^2+2*(b^2-c^2)^2*b^2*c^2 : :
X(13188) = 3*X(3)-2*X(98) = 5*X(3)-4*X(12042) = X(98)-3*X(99) = 5*X(98)-6*X(12042) = 4*X(98)-3*X(12188) = 5*X(99)-2*X(12042) = 4*X(114)-3*X(381) = 8*X(114)-3*X(12355) = X(147)+3*X(8591) = 3*X(381)-2*X(6321) = X(6321)-3*X(8724) = 3*X(8591)-X(13172)

The reciprocal parallelogic center of these triangles is X(385

X(13188) lies on the Stammler circle and these lines: {3,76}, {5,148}, {6,1569}, {20,9990}, {22,5987}, {30,147}, {114,381}, {115,1656}, {194,10788}, {376,5984}, {382,6033}, {384,10353}, {399,690}, {517,13174}, {538,2080}, {542,1350}, {549,12243}, {550,9862}, {574,7697}, {620,3526}, {671,5055}, {999,3027}, {1003,4027}, {1351,5969}, {1384,12829}, {1597,12131}, {1598,5186}, {1634,2453}, {1657,2794}, {1916,11170}, {2023,5024}, {2070,2936}, {2482,5054}, {2783,12773}, {2787,12331}, {2796,3656}, {3023,3295}, {3029,9567}, {3044,9704}, {3095,7781}, {3398,7816}, {3545,8596}, {3579,9860}, {3734,11171}, {3830,6054}, {5026,5050}, {5073,10722}, {5093,10754}, {5613,13103}, {5617,13102}, {5790,13178}, {5886,11599}, {5965,6781}, {6248,8178}, {6319,11916}, {6320,11917}, {7517,13175}, {7709,8290}, {7737,12830}, {7757,10796}, {7970,8148}, {7983,10247}, {8289,9755}, {8703,11177}, {8782,9301}, {9654,13182}, {9655,12184}, {9668,12185}, {9669,13183}, {9861,12083}, {10246,11711}, {10352,11286}, {11005,12902}, {11849,13173}, {11875,13176}, {11876,13177}, {11911,13179}, {11928,13180}, {11929,13181}, {11949,13184}, {11950,13185}, {12000,13189}, {12001,13190}

X(13188) = midpoint of X(147) and X(13172)
X(13188) = reflection of X(i) in X(j) for these (i,j): (3,99), (148,5), (381,8724), (382,6033), (1351,12177), (3830,6054), (5073,10722), (6321,114), (8148,7970), (9301,11676), (9860,3579), (9862,550), (11177,8703), (11632,2482), (12188,3), (12243,549), (12355,381), (12902,11005), (13102,5617), (13103,5613)
X(13188) = antipode of X(12188) in Stammler circle
X(13188) = X(99)-of-X3-ABC-reflections-triangle
X(13188) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (114,6321,381), (147,8591,13172), (2482,11632,5054), (3023,10086,3295), (3027,10089,999), (6321,8724,114), (7782,10104,3)

X(13189) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO 1st ANTI-BROCARD

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

The reciprocal parallelogic center of these triangles is X(385).

X(13189) lies on these lines: {1,99}, {12,13180}, {98,11248}, {114,10531}, {115,5552}, {119,10769}, {148,10528}, {542,12381}, {543,11239}, {690,13217}, {2782,10679}, {2783,12775}, {2794,13118}, {3023,10965}, {3027,11509}, {4027,10803}, {5186,11400}, {5969,12594}, {6256,10723}, {6319,10929}, {6320,10930}, {6321,10942}, {8782,10878}, {10805,13172}, {10834,13175}, {10915,13178}, {10955,13181}, {10956,13182}, {10958,13183}, {11881,13176}, {11882,13177}, {11914,13179}, {11955,13184}, {11956,13185}, {12000,13188}

X(13189) = reflection of X(i) in X(j) for these (i,j): (99,10086), (12189,10679), (12356,11239)
X(13189) = X(99)-of-inner-Yff-tangents-triangle
X(13189) = {X(99),X(7983)}-harmonic conjugate of X(13190)

X(13190) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO 1st ANTI-BROCARD

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

The reciprocal parallelogic center of these triangles is X(385).

X(13190) lies on these lines: {1,99}, {11,13181}, {98,11249}, {114,10532}, {115,10527}, {148,10529}, {542,12382}, {543,11240}, {690,13218}, {2782,10680}, {2783,12776}, {2794,13119}, {3023,10966}, {4027,10804}, {5186,11401}, {5969,12595}, {6319,10931}, {6320,10932}, {6321,10943}, {8782,10879}, {10806,13172}, {10835,13175}, {10916,13178}, {10949,13180}, {10957,13182}, {10959,13183}, {11510,13173}, {11883,13176}, {11884,13177}, {11915,13179}, {11957,13184}, {11958,13185}, {12001,13188}

X(13190) = reflection of X(i) in X(j) for these (i,j): (99,10089), (12190,10680), (12357,11240)
X(13190) = X(99)-of-outer-Yff-tangents-triangle
X(13190) = {X(99),X(7983)}-harmonic conjugate of X(13189)

X(13191) = PARALLELOGIC CENTER OF THESE TRIANGLES: ARTZT TO 4th ANTI-BROCARD

Trilinears ((2*b^2+c^2)*(b^2+2*c^2)*a^10+(b^2+c^2)*(4*b^4-35*b^2*c^2+4*c^4)*a^8-b^2*c^2*(7*b^4-95*b^2*c^2+7*c^4)*a^6-(2*b-c)*(b+2*c)*(b-2*c)*(2*b+c)*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^4-(b^2-c^2)^2*(2*b^8+2*c^8+b^2*c^2*(2*b^4-63*b^2*c^2+2*c^4))*a^2+2*(-c^4+b^4)*(b^2-c^2)*b^2*c^2*((b^2+c^2)^2-9*b^2*c^2))*a : :

The reciprocal parallelogic center of these triangles is X(13168).

X(13191) lies on these lines: {2,13167}, {111,351}, {183,12149}, {2780,9769}, {2854,9759}, {7610,9869}

X(13191) = reflection of X(9869) in X(7610)
X(13191) = X(6236)-of-Artzt-triangle

X(13192) = PARALLELOGIC CENTER OF THESE TRIANGLES: 4th ANTI-BROCARD TO CIRCUMSYMMEDIAL

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

The reciprocal parallelogic center of these triangles is X(111).

X(13192) lies on these lines: {3,7708}, {6,23}, {30,6792}, {110,5107}, {111,352}, {187,9218}, {237,5024}, {323,2502}, {524,9870}, {574,5640}, {1499,13168}, {1648,10989}, {2393,10765}, {3124,5104}, {3569,10561}, {3981,5354}, {5166,9019}, {5969,5971}, {6088,9212}, {6323,13233}, {6791,11647}, {7998,8585}, {8430,9138}, {9463,11173}, {9830,10787}, {9871,11258}

X(13192) = reflection of X(i) in X(j) for these (i,j): (352,111), (9871,11258)
X(13192) = X(110)-of-4th-anti-Brocard-triangle
X(13192) = X(323)-of-circumsymmedial-triangle
X(13192) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2502,8586,323), (3124,5104,11580)

X(13193) = PARALLELOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO ANTI-ORTHOCENTROIDAL

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

The reciprocal parallelogic center of these triangles is X(323).

X(13193) lies on these lines: {32,110}, {74,182}, {83,125}, {98,113}, {265,10796}, {399,11842}, {542,12150}, {690,4027}, {895,5039}, {1078,5972}, {1112,11380}, {1511,2080}, {1691,6593}, {2771,12199}, {2781,12207}, {2854,12212}, {2948,10789}, {3028,12835}, {3031,4279}, {3047,3203}, {3398,5663}, {3448,7787}, {5182,11006}, {5640,11060}, {7732,10792}, {7733,10793}, {7984,10800}, {9517,13195}, {10088,10801}, {10091,10802}, {10788,12383}, {10790,12310}, {10791,13211}, {10794,13213}, {10795,13214}, {10797,12903}, {10798,12904}, {10803,13217}, {10804,13218}, {11364,11720}, {11490,13204}, {11837,13208}, {11838,13209}, {11839,13212}, {11840,13215}, {11841,13216}, {12041,12054}, {12201,12208}

X(13193) = reflection of X(12192) in X(3398)
X(13193) = X(110)-of-5th-anti-Brocard-triangle
X(13193) = orthologic center of these triangles: 5th anti-Brocard to orthocentroidal

X(13194) = PARALLELOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO INNER-GARCIA

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

The reciprocal parallelogic center of these triangles is X(1).

X(13194) lies on these lines: {11,83}, {32,100}, {80,10791}, {98,119}, {104,182}, {149,7787}, {214,11364}, {528,12150}, {952,3398}, {1078,3035}, {1317,12835}, {1320,10800}, {1862,11380}, {2771,12192}, {2783,12176}, {2787,4027}, {2800,12197}, {2802,12194}, {2806,13195}, {2831,12207}, {3032,4279}, {3045,3203}, {5039,10755}, {5541,10789}, {5840,12110}, {9024,12212}, {10087,10801}, {10090,10802}, {10738,10796}, {10788,13199}, {10790,13222}, {11490,13205}, {11837,13228}, {11838,13230}, {11842,12331}

X(13194) = reflection of X(12199) in X(3398)
X(13194) = X(100)-of-5th-anti-Brocard-triangle

X(13195) = PARALLELOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO 1st ORTHOSYMMEDIAL

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

The reciprocal parallelogic center of these triangles is X(6).

X(13195) lies on these lines: {4,32}, {83,127}, {125,251}, {182,1297}, {1078,6720}, {2781,12192}, {2799,4027}, {2806,13194}, {2831,12199}, {3320,12835}, {3398,12207}, {5039,10766}, {6020,10799}, {6793,10313}, {7787,13219}, {9517,13193}, {10705,10800}, {10749,10796}, {10789,13221}, {10790,11641}, {11364,11722}, {11380,13166}, {11490,13206}, {11837,13229}, {11838,13231}

X(13195) = reflection of X(12207) in X(3398)
X(13195) = X(112)-of-5th-anti-Brocard-triangle
X(13195) = {X(112), X(11610)}-harmonic conjugate of X(13236)

X(13196) = PARALLELOGIC CENTER OF THESE TRIANGLES: 6th ANTI-BROCARD TO SCHROETER

Barycentrics 2*a^6-2*(b^2+c^2)*a^4+(b^2-c^2)^2*a^2+b^2*c^2*(b^2+c^2) : :
X(13196) = X(1691)-3*X(5182) = X(1691)+3*X(12151) = X(3629)+2*X(6390)

The reciprocal parallelogic center of these triangles is X(115).

X(13196) lies on these lines: {6,194}, {30,12213}, {32,3629}, {83,6329}, {99,5111}, {140,141}, {230,10352}, {249,524}, {325,4027}, {385,10353}, {511,5026}, {542,5031}, {597,5034}, {620,5965}, {732,1692}, {1078,3631}, {1503,2456}, {1570,5969}, {2482,10631}, {3398,7789}, {3589,5038}, {3630,5033}, {5039,8584}, {5097,7816}, {5102,10788}, {5254,10349}, {7750,10131}, {7792,10334}

X(13196) = midpoint of X(i) and X(j) for these {i,j}: {6,12215}, {99,5111}, {2456,12177}, {5182,12151}, {12213,12214}
X(13196) = orthologic center of these triangles: 6th anti-Brocard to Steiner
X(13196) = X(115)-of-6th-anti-Brocard-triangle

X(13197) = PARALLELOGIC CENTER OF THESE TRIANGLES: 6th ANTI-BROCARD TO X-PARABOLA-TANGENTIAL

Barycentrics ((96*R^2+7*SA-17*SW)*S^4+2*(8*(3*SA+SW)*R^2+6*SA^2-11*SA*SW-SW^2)*SW*S^2-(4*SA^2-3*SA*SW+SW^2)*SW^3)*(SB-SC) : :

The reciprocal parallelogic center of these triangles is X(13187).

X(13197) lies on these lines: {182,13232}, {826,2451}, {4027,13186}, {10131,13237}

X(13198) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-CONWAY TO 1st HYACINTH

Trilinears (-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+(-c^4+b^4)*(b^2-c^2)*a^2+(b^2-c^2)^2*b^2*c^2)*a : :
X(13198) = X(74)+2*X(12227) = 3*X(11402)+X(13171)

The reciprocal parallelogic center of these triangles is X(10116).

X(13198) lies on these lines: {2,98}, {3,974}, {4,9934}, {6,1112}, {25,11746}, {26,12236}, {54,74}, {68,10111}, {113,569}, {154,12099}, {155,12358}, {265,6146}, {567,7728}, {578,2777}, {895,1176}, {1147,6699}, {1181,5663}, {1498,12133}, {1885,11744}, {1986,7592}, {2781,11402}, {2935,11425}, {3516,11598}, {5157,5181}, {6759,7687}, {7503,12825}, {7731,11423}, {8547,10602}, {9833,12140}, {9919,11426}, {10118,11429}, {10119,11428}, {10605,12041}, {11422,13201}, {11424,13202}, {11427,13203}, {11456,12292}, {11806,12893}, {12134,12419}

X(13198) = Brocard circle-inverse-of-X(1899)
X(13198) = X(100)-of-anti-Conway-triangle if ABC is acute
X(13198) = X(11)-of-2nd-anti-extouch-triangle if ABC is acute
X(13198) = crosspoint of X(15460) and X(15461)
X(13198) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3047,110), (6,10117,1112), (110,5622,125), (125,184,110), (5622,6776,11579), (6776,11003,184)

X(13199) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO INNER-GARCIA

Trilinears 6*sin(3*A/2)*cos((B-C)/2)+(-2*cos(A)+1)*cos(B-C)-2*sin(A/2)*cos(3*(B-C)/2)-5*cos(A)+3*cos(2*A)-1 : :
X(13199) = 3*X(3)-2*X(1484) = 3*X(4)-4*X(119) = 3*X(4)-2*X(10724) = X(4)-4*X(10993) = 4*X(11)-5*X(631) = 2*X(80)-3*X(5657) = 3*X(100)-2*X(119) = 3*X(100)-X(10724) = 3*X(149)-4*X(1484) = 4*X(214)-3*X(5603)

The reciprocal parallelogic center of these triangles is X(1).

X(13199) lies on these lines: {2,10738}, {3,149}, {4,100}, {11,631}, {20,952}, {24,13222}, {30,153}, {35,6853}, {40,12247}, {55,6951}, {80,4302}, {104,376}, {165,10265}, {214,5603}, {382,11698}, {388,10087}, {390,1387}, {497,10090}, {515,2950}, {516,5528}, {517,6224}, {550,12773}, {944,2802}, {962,6265}, {1006,7676}, {1145,6938}, {1317,4293}, {1320,6948}, {1376,6965}, {1537,9945}, {1788,10073}, {1862,7487}, {2475,11849}, {2771,9961}, {2783,9862}, {2787,13172}, {2800,6361}, {2803,5667}, {2806,13200}, {2829,3529}, {2831,12253}, {2932,6905}, {3035,3090}, {3146,10742}, {3434,4996}, {3474,11570}, {3488,12736}, {3524,6713}, {3533,6667}, {3545,6174}, {3600,12735}, {4190,10595}, {4295,12739}, {4297,6264}, {4299,7972}, {4324,9897}, {4330,6684}, {5218,8068}, {5533,7288}, {5690,12747}, {5731,9802}, {5759,12691}, {5817,6594}, {5842,12332}, {5882,12653}, {6284,6902}, {6776,9024}, {6875,10058}, {6885,11729}, {6900,11496}, {6903,10310}, {6934,10609}, {6936,12019}, {6949,10525}, {6963,9668}, {6987,12690}, {9778,9803}, {9809,12738}, {9812,12611}, {9913,12082}, {10788,13194}, {11491,11826}, {11843,13228}, {11844,13230}

X(13199) = reflection of X(i) in X(j) for these (i,j): (4,100), (100,10993), (149,3), (153,12331), (382,11698), (944,12119), (962,6265), (1537,9945), (3146,10742), (6264,4297), (9802,12737), (9803,12515), (9809,12738), (9897,11362), (10698,10609), (10724,119), (12247,40), (12248,20), (12653,5882), (12747,5690), (12773,550)
X(13199) = anticomplement of X(10738)
X(13199) = X(100)-of-anti-Euler-triangle
X(13199) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (100,10724,119), (119,10724,4), (5731,9802,12737), (9778,9803,12515)

X(13200) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO 1st ORTHOSYMMEDIAL

Barycentrics SB*SC*((9*R^2+SA-2*SW)*S^2-(4*R^2*SW-SA^2+SA*SW-SW^2)*SW) : :
X(13200) = 3*X(4)-4*X(132) = 3*X(4)-2*X(10735) = 3*X(112)-2*X(132) = 3*X(112)-X(10735) = 4*X(127)-5*X(631) = 3*X(376)-2*X(1297) = 3*X(5603)-4*X(11722)

The reciprocal parallelogic center of these triangles is X(6).

X(13200) lies on these lines: {2,10749}, {3,13219}, {4,32}, {20,12253}, {24,11641}, {30,12384}, {127,631}, {147,4235}, {376,1297}, {515,13221}, {550,13115}, {2781,6776}, {2799,13172}, {2806,13199}, {2831,12248}, {2848,5667}, {3090,6720}, {3146,12918}, {3320,4293}, {3524,10718}, {4294,6020}, {5603,11722}, {6353,9157}, {7487,13166}, {7967,10705}, {9517,12383}, {9530,11001}, {11491,13206}, {11843,13229}, {11844,13231}, {12082,12413}

X(13200) = reflection of X(i) in X(j) for these (i,j): (4,112), (3146,12918), (10735,132), (12253,20), (13115,550), (13219,3)
X(13200) = anticomplement of X(10749)
X(13200) = X(112)-of-anti-Euler-triangles
X(13200) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (112,10735,132), (132,10735,4)

X(13201) = PARALLELOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO 1st HYACINTH

Barycentrics ((11*R^2-2*SW)*S^2-((3*SA+22*SW)*R^2-2*(SA+2*SW)*SW)*SA)*(SB+SC) : :
X(13201) = 5*X(3)-4*X(11561) = 2*X(110)-3*X(2979) = 4*X(113)-5*X(11444) = 4*X(125)-3*X(3060) = 3*X(376)-2*X(11562) = 5*X(631)-4*X(11557) = 8*X(1112)-9*X(5640) = 5*X(3091)-4*X(11807) = 3*X(5890)-4*X(12041) = 5*X(7731)-8*X(11561)

The reciprocal parallelogic center of these triangles is X(10116).

X(13201) lies on these lines: {3,7731}, {20,10628}, {22,110}, {30,12281}, {74,5889}, {113,11444}, {125,3060}, {146,2889}, {193,1205}, {376,11562}, {399,6101}, {511,3448}, {631,11557}, {1112,5094}, {1154,10620}, {1657,5663}, {1986,10574}, {1993,13171}, {2777,12111}, {2935,11440}, {3091,11807}, {3313,11061}, {5890,12041}, {5972,7998}, {6243,10264}, {6723,11451}, {7525,11597}, {7723,10721}, {7728,11459}, {9919,11441}, {10118,11446}, {10119,11445}, {10625,12383}, {10681,11452}, {10682,11453}, {11422,13198}, {11439,13202}, {11442,13203}

X(13201) = reflection of X(i) in X(j) for these (i,j): (146,5562), (193,1205), (399,6101), (5889,74), (6243,10264), (7731,3), (10721,7723), (11061,3313), (12111,12219), (12270,20), (12273,11412), (12284,10620), (12383,10625)
X(13201) = X(100)-of-3rd-anti-Euler-triangle if ABC is acute

X(13202) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO 1st HYACINTH

Barycentrics (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(2*a^6-(b^2+c^2)*a^4-4*(b^2-c^2)^2*a^2+3*(b^4-c^4)*(b^2-c^2)) : :
X(13202) = 3*X(3)-4*X(12900) = 3*X(4)-X(74) = 3*X(4)-2*X(7687) = 4*X(4)-X(10990) = 5*X(4)-X(12244) = 2*X(74)-3*X(125) = 3*X(113)-2*X(1511) = 4*X(113)-3*X(5642) = 13*X(113)-9*X(11693) = 17*X(113)-12*X(11694) = X(1511)-3*X(1539) = 8*X(1511)-9*X(5642)

Let Oa be the circle centered at A and tangent to the Euler line; define Ob and Oc cyclically. Let La be the polar of X(4) wrt Oa, and define Lb and Lc cyclically. (Note: X(4) is the perspector of any circle centered at a vertex of ABC.) Let A' = Lb∩Lc, and define B' and C' cyclically. Triangle A'B'C' is the reflection of ABC in X(5972), which is the radical center of Oa, Ob, Oc. A'B'C' is homothetic to the medial triangle at X(125) and to the anticomplementary triangle at X(110). X(13202) = X(20)-of-A'B'C'. (Randy Hutson, July 21, 2017)

The reciprocal parallelogic center of these triangles is X(10116).

X(13202) lies on the curve Q107 and these lines: {3,12900}, {4,74}, {6,1562}, {20,5972}, {25,2935}, {30,113}, {34,10118}, {51,974}, {52,3627}, {110,3146}, {146,148}, {155,382}, {184,9934}, {185,1112}, {247,5191}, {265,541}, {381,6699}, {511,12825}, {546,12041}, {1146,1839}, {1162,7726}, {1163,7725}, {1503,5095}, {1593,10117}, {1597,9919}, {1650,3184}, {1699,11735}, {1829,2778}, {1843,2781}, {1885,3574}, {1986,6000}, {2771,12690}, {2904,11456}, {2914,12112}, {3091,6723}, {3531,10293}, {3853,10113}, {5073,12121}, {5076,10620}, {6053,10706}, {6103,9412}, {6126,12896}, {6152,10628}, {6564,8994}, {6793,9408}, {7517,12893}, {7722,12290}, {7731,11455}, {7984,9812}, {8972,10817}, {10119,11471}, {10151,11598}, {10681,11475}, {10682,11476}, {11403,13171}, {11424,13198}, {11439,13201}, {11801,12102}, {12373,12953}, {12374,12943}

X(13202) = midpoint of X(i) and X(j) for these {i,j}: {4,10721}, {110,3146}, {146,10733}, {382,7728}, {5073,12121}, {7722,12290}
X(13202) = reflection of X(i) in X(j) for these (i,j): (20,5972), (74,7687), (113,1539), (125,4), (185,1112), (1495,1514), (1986,11807), (10113,3853), (10990,125), (11801,12102), (12041,546), (12295,3627), (12383,6053)
X(13202) = polar circle-inverse-of-X(10152)
X(13202) = X(100)-of-anti-excenter-reflections-triangle if ABC is acute
X(13202) = crosssum of X(3) and X(74)
X(13202) = crosspoint of X(4) and X(30)
X(13202) = orthic-triangle-syngonal conjugate of X(4)
X(13202) = crossdifference of every pair of points on line X(1636)X(2433)
X(13202) = orthic-isogonal conjugate of X(10151)
X(13202) = X(1320)-of-orthic-triangle if ABC is acute
X(13202) = antipode of X(185) in Hatzipolakis-Lozada hyperbola
X(13202) = orthopole of line X(4)X(523)
X(13202) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,74,7687), (74,7687,125), (146,3543,10733)

X(13203) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO 1st HYACINTH

Barycentrics a^12-3*(b^4-b^2*c^2+c^4)*a^8+b^2*c^2*(b^2+c^2)*a^6+(b^2-c^2)^2*(3*b^4+5*b^2*c^2+3*c^4)*a^4-5*(b^4-c^4)*(b^2-c^2)*b^2*c^2*a^2-(b^4-c^4)^2*(b^2-c^2)^2 : :
X(13203) = 4*X(110)-3*X(11206) = 4*X(1177)-5*X(3618)

The reciprocal parallelogic center of these triangles is X(10116).

X(13203) lies on these lines: {2,10117}, {4,74}, {5,9919}, {20,2917}, {30,12310}, {69,2892}, {110,1370}, {113,6643}, {146,6225}, {323,1503}, {427,13171}, {497,10118}, {962,2778}, {1112,11433}, {1177,3618}, {2550,10119}, {2781,3448}, {5621,7378}, {5663,12319}, {5972,7386}, {6403,11550}, {6699,7401}, {6723,7392}, {7731,11457}, {10628,12284}, {10681,11488}, {10682,11489}, {11427,13198}, {11442,13201}

X(13203) = reflection of X(i) in X(j) for these (i,j): (20,2935), (69,2892), (6225,146), (9919,5)
X(13203) = anticomplement of X(10117)
X(13203) = anticomplementary circle-inverse-of-X(107)
X(13203) = X(100)-of-anti-inverse-in-incircle-triangle if ABC is acute

X(13204) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO ANTI-ORTHOCENTROIDAL

Trilinears 32*p^7*(p-q)+32*(q^2-2)*p^6-16*(2*q^2-5)*q*p^5-6*(8*q^2-7)*p^4+6*(8*q^2-11)*q*p^3+(16*q^2-9)*p^2-2*(8*q^2-9)*q*p-q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal parallelogic center of these triangles is X(323).

X(13204) lies on these lines: {3,11709}, {35,2948}, {55,110}, {56,7984}, {74,10310}, {78,10693}, {100,3448}, {113,11496}, {125,1376}, {165,2836}, {197,12310}, {265,11499}, {399,11849}, {542,4421}, {690,13173}, {1001,5972}, {1012,12368}, {1112,11383}, {1158,2771}, {1511,10267}, {2781,12340}, {2854,12329}, {3028,11509}, {3295,11720}, {4428,5642}, {5537,9904}, {5663,11248}, {5687,13211}, {6911,12261}, {7732,11497}, {7733,11498}, {9517,13206}, {10088,11507}, {10091,11508}, {11490,13193}, {11491,12383}, {11492,13208}, {11493,13209}, {11494,13210}, {11500,13214}, {11501,12903}, {11502,12904}, {11503,13215}, {11504,13216}, {11510,13218}, {11848,13212}, {12334,12341}

X(13204) = reflection of X(i) in X(j) for these (i,j): (12327,11248), (13213,125)
X(13204) = X(110)-of-anti-Mandart-incircle-triangle
X(13204) = orthologic center of these triangles: anti-Mandart-incircle to orthocentroidal

X(13205) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO INNER-GARCIA

Trilinears (a^4-2*(b+c)*a^3+7*b*c*a^2+(b+c)*(2*b^2-7*b*c+2*c^2)*a-b^4+4*b^2*c^2-c^4)*a : :
X(13205) = 2*X(100)-3*X(4421) = 3*X(3158)-X(5531) = 2*X(10742)-3*X(11236)

The reciprocal parallelogic center of these triangles is X(1).

X(13205) lies on these lines: {1,2932}, {2,11}, {3,2802}, {35,5541}, {36,12653}, {56,1320}, {80,5687}, {104,5854}, {119,11496}, {153,12607}, {197,13222}, {214,3295}, {355,8715}, {480,1156}, {518,1768}, {519,12773}, {529,12248}, {952,3913}, {958,1145}, {1012,12751}, {1317,11509}, {1709,3158}, {1862,11383}, {2077,3880}, {2136,7993}, {2771,3811}, {2783,12178}, {2787,13173}, {2800,10306}, {2806,13206}, {2831,12340}, {3189,9803}, {3746,10179}, {3871,6224}, {4996,5217}, {5289,12758}, {5840,11500}, {5851,10307}, {5856,11495}, {6265,10679}, {6326,12672}, {6702,9709}, {6796,12700}, {9024,12329}, {10087,10609}, {10090,11508}, {10738,10893}, {10742,11236}, {11490,13194}, {11491,11826}, {11492,13228}, {11493,13230}, {11494,13235}, {11517,12690}, {11523,12767}

X(13205) = midpoint of X(i) and X(j) for these {i,j}: {2136,7993}, {3189,9803}, {11523,12767}
X(13205) = reflection of X(i) in X(j) for these (i,j): (153,12607), (10912,12737), (12331,8715), (12332,11248), (12513,104)
X(13205) = X(100)-of-anti-Mandart-incircle-triangle
X(13205) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11,100,1376), (1145,10058,958), (8668,10310,12513)

X(13206) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 1st ORTHOSYMMEDIAL

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

The reciprocal parallelogic center of these triangles is X(6).

X(13206) lies on these lines: {3,12265}, {35,13221}, {55,112}, {56,10705}, {100,13219}, {127,1376}, {132,11496}, {197,11641}, {1001,6720}, {1012,12784}, {1297,10310}, {2781,12327}, {2794,11500}, {2799,13173}, {2806,13205}, {2831,12332}, {3295,11722}, {3320,11509}, {5537,12408}, {9517,13204}, {10749,11499}, {11248,12340}, {11383,13166}, {11490,13195}, {11491,13200}, {11492,13229}, {11493,13231}, {11494,13236}

X(13206) = reflection of X(12340) in X(11248)
X(13206) = X(112)-of-anti-Mandart-incircle-triangle

X(13207) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-MCCAY TO CIRCUMSYMMEDIAL

Trilinears ((2*b^4+b^2*c^2+2*c^4)*a^4-2*(b^6+c^6)*a^2-b^2*c^2*(2*b^4-3*b^2*c^2+2*c^4))*a : :
X(13207) = 3*X(8859)-2*X(11673)

The reciprocal parallelogic center of these triangles is X(98).

X(13207) lies on these lines: {6,23}, {98,385}, {323,9149}, {512,8597}, {671,9879}, {2387,6787}, {3329,5640}, {6784,8859}, {6786,7925}

X(13207) = reflection of X(i) in X(j) for these (i,j): (9879,671), (11673,6784)
X(13207) = X(6233)-of-anti-McCay-triangle
X(13207) = {X(6784), X(11673)}-harmonic conjugate of X(8859)

X(13208) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO ANTI-ORTHOCENTROIDAL

Trilinears (b+c)*(a^5-(b^2+c^2)*a^3+(b^2-c^2)*(b-c)*a^2+b^2*c^2*a-(b^4-c^4)*(b-c))*D-a*(a-b-c)*(a+b+c)*(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)) : :
where D=*S*sqrt(R*(4*R+r))

The reciprocal parallelogic center of these triangles is X(323).

X(13208) lies on these lines: {55,13209}, {74,11822}, {110,5597}, {113,8196}, {125,5599}, {265,8200}, {399,11875}, {517,12366}, {542,11207}, {690,13176}, {952,12467}, {1112,11384}, {2771,12462}, {2781,12478}, {2854,12452}, {3448,5601}, {5598,7984}, {5663,11252}, {7732,8198}, {7733,8199}, {8190,12310}, {8197,13211}, {8201,13215}, {8202,13216}, {9517,13229}, {10088,11877}, {10091,11879}, {11366,11720}, {11492,13204}, {11837,13193}, {11843,12383}, {11861,13210}, {11865,13213}, {11867,13214}, {11869,12903}, {11871,12904}, {11881,13217}, {11883,13218}

X(13208) = reflection of X(13209) in X(55)
X(13208) = X(110)-of-1st-Auriga-triangle
X(13208) = X(7984)-of-2nd-Auriga-triangle

X(13209) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO ANTI-ORTHOCENTROIDAL

Trilinears (b+c)*(a^5-(b^2+c^2)*a^3+(b^2-c^2)*(b-c)*a^2+b^2*c^2*a-(b^4-c^4)*(b-c))*D+a*(a-b-c)*(a+b+c)*(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)) : :
where D=*S*sqrt(R*(4*R+r))

The reciprocal parallelogic center of these triangles is X(323).

X(13209) lies on these lines: {55,13208}, {74,11823}, {110,5598}, {113,8203}, {125,5600}, {265,8207}, {399,11876}, {517,12365}, {542,11208}, {690,13177}, {952,12466}, {1112,11385}, {2771,12463}, {2781,12479}, {2854,12453}, {2948,8187}, {3448,5602}, {5597,7984}, {5663,11253}, {7732,8205}, {7733,8206}, {8191,12310}, {8204,13211}, {8208,13215}, {8209,13216}, {9517,13231}, {10088,11878}, {10091,11880}, {11367,11720}, {11493,13204}, {11838,13193}, {11844,12383}, {11862,13210}, {11866,13213}, {11868,13214}, {11870,12903}, {11872,12904}, {11882,13217}, {11884,13218}

X(13209) = reflection of X(13208) in X(55)
X(13209) = X(110)-of-2nd-Auriga-triangle
X(13209) = X(7984)-of-1st-Auriga-triangle

X(13210) = PARALLELOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO ANTI-ORTHOCENTROIDAL

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

The reciprocal parallelogic center of these triangles is X(323).

X(13210) lies on these lines: {32,110}, {69,74}, {113,9993}, {125,3096}, {265,9996}, {323,2387}, {399,3511}, {526,887}, {690,8782}, {1112,11386}, {2076,2930}, {2771,12499}, {2781,12503}, {2854,3094}, {2896,3448}, {2948,3099}, {5027,9138}, {5663,9821}, {5972,7846}, {6787,9149}, {7732,9994}, {7733,9995}, {7753,10545}, {7865,9140}, {7984,9997}, {9517,13236}, {9857,13211}, {9985,12501}, {10038,10088}, {10047,10091}, {10828,12310}, {10871,13213}, {10872,13214}, {10873,12903}, {10874,12904}, {10875,13215}, {10876,13216}, {10878,13217}, {10879,13218}, {11368,11720}, {11494,13204}, {11861,13208}, {11862,13209}, {11885,13212}

X(13210) = reflection of X(9984) in X(9821)
X(13210) = X(110)-of-5th-Brocard-triangle
X(13210) = X(76)-of-anti-orthocentroidal-triangle
X(13210) = 4th-Brocard-to-circumsymmedial similarity image of X(76)

X(13211) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO ANTI-ORTHOCENTROIDAL

Barycentrics a^7-(b^2+c^2)*a^5+b^2*c^2*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^4-c^4)*(b^2-c^2)*(-b-c) : :
X(13211) = 2*X(113)-3*X(5587) = 3*X(125)-2*X(11735) = X(399)-3*X(5790) = 5*X(1698)-4*X(5972) = 3*X(1699)-4*X(7687) = X(2948)-3*X(3679) = 3*X(5657)-X(12383) = 5*X(8227)-4*X(11723)

The reciprocal parallelogic center of these triangles is X(323).

X(13211) lies on these lines: {1,125}, {2,11720}, {3,12334}, {8,3448}, {10,110}, {40,12407}, {65,12903}, {67,518}, {72,13214}, {74,515}, {80,7727}, {113,5587}, {265,517}, {355,5663}, {399,5790}, {519,7984}, {542,2948}, {690,13178}, {895,5847}, {944,11709}, {946,7978}, {952,10264}, {1112,5090}, {1385,12898}, {1482,12261}, {1698,5972}, {1699,7687}, {1737,10091}, {2771,12751}, {2802,10778}, {2854,3416}, {3028,5252}, {3057,12904}, {3576,6699}, {3579,12121}, {5119,12896}, {5657,12383}, {5687,13204}, {5688,7733}, {5689,7732}, {5690,12778}, {5691,9904}, {8185,10117}, {8193,12310}, {8197,13208}, {8204,13209}, {8214,13215}, {8215,13216}, {8227,11723}, {8994,9583}, {9798,13171}, {9857,13210}, {9956,11699}, {10039,10088}, {10065,10572}, {10113,12699}, {10791,13193}, {10914,13213}, {10915,13217}, {10916,13218}, {11670,12373}, {11900,13212}, {12702,12902}

X(13211) = midpoint of X(i) and X(j) for these {i,j}: {8,3448}, {40,12407}, {5691,9904}, {12702,12902}
X(13211) = reflection of X(i) in X(j) for these (i,j): (1,125), (110,10), (944,11709), (1482,12261), (7978,946), (11699,9956), (12121,3579), (12368,355), (12699,10113), (12778,5690), (12898,1385)
X(13211) = anticomplement of X(11720)
X(13211) = X(110)-of-outer-Garcia-triangle

X(13212) = PARALLELOGIC CENTER OF THESE TRIANGLES: GOSSARD TO ANTI-ORTHOCENTROIDAL

Barycentrics SA*((SW+3*SA)*(SA-SW)+4*S^2)*(3*SA*(SA-SW)+2*S^2)*(SA*(3*SA-18*R^2+2*SW)+S^2) : :
X(13212) = 2*X(113)-3*X(11897) = X(399)-3*X(11911) = X(2948)-3*X(11852) = 3*X(11845)-X(12383)

The reciprocal parallelogic center of these triangles is X(323).

X(13212) lies on these lines: {30,74}, {110,402}, {113,11897}, {122,125}, {399,11911}, {542,1651}, {690,13179}, {1112,11832}, {2771,12752}, {2781,12796}, {2854,12583}, {2948,11852}, {3448,4240}, {5663,11251}, {7732,11901}, {7733,11902}, {7984,11910}, {10088,11912}, {10091,11913}, {11720,11831}, {11839,13193}, {11845,12383}, {11848,13204}, {11853,12310}, {11863,13208}, {11864,13209}, {11885,13210}, {11900,13211}, {11903,13213}, {11904,13214}, {11905,12903}, {11906,12904}, {11907,13215}, {11908,13216}, {11914,13217}, {11915,13218}, {12790,12797}

X(13212) = midpoint of X(3448) and X(4240)
X(13212) = reflection of X(i) in X(j) for these (i,j): (110,402), (1650,125), (12369,11251)
X(13212) = X(110)-of-Gossard-triangle

X(13213) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO ANTI-ORTHOCENTROIDAL

Trilinears 32*p^7*(p-q)+32*(q^2-2)*p^6-16*(2*q^2-5)*q*p^5+2*(4*q^2-3)*(4*q^2-7)*p^4-2*(16*q^4-40*q^2+33)*q*p^3-(16*q^4-32*q^2+9)*p^2+2*(8*q^4-16*q^2+9)*q*p-q^2 : : , where p=sin(A/2), q=cos((B-C)/2)

The reciprocal parallelogic center of these triangles is X(323).

X(13213) lies on these lines: {11,110}, {12,13217}, {74,11826}, {113,10893}, {125,1376}, {265,355}, {399,11928}, {542,11235}, {690,13180}, {1112,11390}, {2771,12761}, {2781,12925}, {2836,5927}, {2854,12586}, {2948,10826}, {3434,3448}, {5663,10525}, {7732,10919}, {7733,10920}, {7984,10944}, {10088,10523}, {10091,10948}, {10785,12383}, {10794,13193}, {10829,12310}, {10871,13210}, {10914,13211}, {10943,12889}, {10945,13215}, {10946,13216}, {10949,13218}, {11373,11720}, {11865,13208}, {11866,13209}, {11903,13212}, {12737,12898}

X(13213) = reflection of X(i) in X(j) for these (i,j): (12371,10525), (12889,10943), (13204,125), (13214,265)
X(13212) = X(110)-of-inner-Johnson-triangle

X(13214) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO ANTI-ORTHOCENTROIDAL

Trilinears 32*p^7*(p-q)+32*(q^2-2)*p^6-16*(2*q^2-3)*q*p^5+2*(16*q^4-24*q^2+21)*p^4-2*(4*q^2-3)^2*q*p^3-(16*q^4-16*q^2+9)*p^2+16*(q^2-1)*q^3*p+q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal parallelogic center of these triangles is X(323).

X(13214) lies on these lines: {11,13218}, {12,110}, {72,13211}, {74,11827}, {113,10894}, {125,958}, {265,355}, {399,11929}, {542,11236}, {690,13181}, {1112,11391}, {2771,12762}, {2781,12935}, {2854,12587}, {2948,10827}, {3436,3448}, {5663,10526}, {6253,10733}, {7732,10921}, {7733,10922}, {7984,10950}, {10088,10954}, {10091,10523}, {10786,12383}, {10795,13193}, {10830,12310}, {10872,13210}, {10942,12890}, {10951,13215}, {10952,13216}, {10955,13217}, {11374,11720}, {11500,13204}, {11867,13208}, {11868,13209}, {11904,13212}

X(13214) = reflection of X(i) in X(j) for these (i,j): (12372,10526), (12890,10942), (13213,265)
X(13214) = X(110)-of-outer-Johnson-triangle

X(13215) = PARALLELOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO ANTI-ORTHOCENTROIDAL

Barycentrics SA*((36*R^2-17*SW)*S^2+4*S*((3*SA+4*SW)*R^2-4*S^2-4*SA^2+2*SA*SW)-(9*SA^2-6*SA*SW-2*SW^2)*SW)*(SB+SC) : :

The reciprocal parallelogic center of these triangles is X(323).

X(13215) lies on these lines: {74,11828}, {110,493}, {113,8212}, {125,8222}, {265,8220}, {399,11949}, {542,12152}, {690,13184}, {1112,11394}, {2771,12765}, {2781,12996}, {2854,12590}, {2948,8188}, {3448,6462}, {5663,10669}, {6461,13216}, {7732,8216}, {7733,8218}, {7984,8210}, {8194,12310}, {8201,13208}, {8208,13209}, {8214,13211}, {10088,11951}, {10091,11953}, {10875,13210}, {10945,13213}, {10951,13214}, {10981,12378}, {11377,11720}, {11503,13204}, {11840,13193}, {11846,12383}, {11907,13212}, {11930,12903}, {11932,12904}, {11955,13217}, {11957,13218}, {12894,12998}

X(13215) = X(110)-of-Lucas-homothetic-triangle

X(13216) = PARALLELOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO ANTI-ORTHOCENTROIDAL

Barycentrics SA*((36*R^2-17*SW)*S^2-4*S*((3*SA+4*SW)*R^2-4*S^2-4*SA^2+2*SA*SW)-(9*SA^2-6*SA*SW-2*SW^2)*SW)*(SB+SC) : :

The reciprocal parallelogic center of these triangles is X(323).

X(13216) lies on these lines: {74,11829}, {110,494}, {113,8213}, {125,8223}, {265,8221}, {399,11950}, {542,12153}, {690,13185}, {1112,11395}, {2771,12766}, {2781,12997}, {2854,12591}, {2948,8189}, {3448,6463}, {5663,10673}, {6461,13215}, {7732,8217}, {7733,8219}, {7984,8211}, {8195,12310}, {8202,13208}, {8209,13209}, {8215,13211}, {10088,11952}, {10091,11954}, {10876,13210}, {10946,13213}, {10952,13214}, {10981,12377}, {11378,11720}, {11504,13204}, {11841,13193}, {11847,12383}, {11908,13212}, {11931,12903}, {11933,12904}, {11956,13217}, {11958,13218}, {12895,12999}

X(13216) = X(110)-of-Lucas(-1)-homothetic-triangle

X(13217) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO ANTI-ORTHOCENTROIDAL

Trilinears 32*p^8-32*q*p^7-64*p^6+80*q*p^5-2*(16*q^2-21)*p^4+2*(16*q^2-33)*q*p^3+(14*q^2-9)*p^2-2*(7*q^2-9)*q*p-q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal parallelogic center of these triangles is X(323).

X(13217) lies on these lines: {1,60}, {12,13213}, {74,11248}, {113,10531}, {119,10778}, {125,5552}, {265,10942}, {399,12000}, {542,11239}, {690,13189}, {1112,11400}, {2771,12775}, {2781,13118}, {2854,12594}, {3028,11509}, {3448,10528}, {5663,10679}, {6256,10733}, {7732,10929}, {7733,10930}, {10803,13193}, {10805,12383}, {10834,12310}, {10878,13210}, {10915,13211}, {10955,13214}, {10956,12903}, {10958,12904}, {11881,13208}, {11882,13209}, {11914,13212}, {11955,13215}, {11956,13216}, {12905,13121}

X(13217) = reflection of X(i) in X(j) for these (i,j): (110,10088), (12381,10679)
X(13217) = X(110)-of-inner-Yff-tangents-triangle
X(13217) = {X(110),X(7984)}-harmonic conjugate of X(13218)

X(13218) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO ANTI-ORTHOCENTROIDAL

Trilinears 32*p^8-32*q*p^7-32*p^6+16*q*p^5+2*(16*q^2-3)*p^4-2*(16*q^2-15)*q*p^3-9*(2*q^2-1)*p^2+18*(q^2-1)*q*p+q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal parallelogic center of these triangles is X(323).

X(13218) lies on these lines: {1,60}, {11,13214}, {74,11249}, {113,10532}, {125,10527}, {265,10943}, {399,12001}, {542,11240}, {690,13190}, {1112,11401}, {2771,12776}, {2781,13119}, {2854,12595}, {3448,10529}, {5663,10680}, {7732,10931}, {7733,10932}, {10804,13193}, {10806,12383}, {10835,12310}, {10879,13210}, {10916,13211}, {10949,13213}, {10957,12903}, {10959,12904}, {11510,13204}, {11883,13208}, {11884,13209}, {11915,13212}, {11957,13215}, {11958,13216}, {12906,13122}

X(13218) = reflection of X(i) in X(j) for these (i,j): (110,10091), (12382,10680)
X(13218) = X(110)-of-outer-Yff-tangents-triangle
X(13218) = {X(110),X(7984)}-harmonic conjugate of X(13217)

X(13219) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO 1st ORTHOSYMMEDIAL

Barycentrics a^10-(b^2+c^2)*a^8-(2*b^2-c^2)*(b^2-2*c^2)*a^6+2*(b^4-c^4)*(b^2-c^2)*a^4+(b^2-c^2)^2*(b^4-b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(-b^4-c^4) : :
X(13219) = 3*X(2)-4*X(127) = 3*X(4)-2*X(12918) = 3*X(112)-4*X(6720) = X(112)-3*X(10718) = 3*X(127)-2*X(6720) = 2*X(127)-3*X(10718) = 4*X(132)-5*X(3091) = 4*X(10749)-X(12384) = 3*X(10749)-X(12918) = 3*X(12384)-4*X(12918)

The reciprocal parallelogic center of these triangles is X(6).

X(13219) lies on the anticomplementary circle and these lines: {2,112}, {3,13200}, {4,339}, {10,13221}, {20,99}, {22,1369}, {30,12253}, {69,146}, {100,13206}, {132,3091}, {145,10705}, {148,2799}, {149,2806}, {150,9518}, {151,2893}, {152,1330}, {153,322}, {193,10766}, {253,317}, {316,3153}, {325,2071}, {388,3320}, {497,6020}, {516,12408}, {754,10313}, {858,5971}, {2838,5300}, {2848,3268}, {2896,13236}, {3146,10735}, {3164,7898}, {3448,9517}, {3616,11722}, {4293,13117}, {4294,13116}, {5225,12955}, {5229,12945}, {5601,13229}, {5602,13231}, {5731,12265}, {6031,7493}, {7391,11605}, {7488,7750}, {7664,9157}, {7776,11413}, {7787,13195}

X(13219) = reflection of X(i) in X(j) for these (i,j): (2,10718), (4,10749), (20,1297), (112,127), (145,10705), (149,10780), (193,10766), (3146,10735), (7391,11605), (12253,13115), (12384,4), (13200,3), (13221,10)
X(13219) = isogonal conjugate of X(34190)
X(13219) = anticomplement of X(112)
X(13219) = antipode of X(12384) in anticomplementary circle
X(13219) = polar circle-inverse-of-X(13166)
X(13219) = X(112)-of-anticomplementary-triangle
X(13219) = inverse-in-de-Longchamps-circle of X(99)
X(13219) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(2373)
X(13219) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (112,127,2), (112,10718,127)

X(13220) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd SHARYGIN TO ANTLIA

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

The reciprocal parallelogic center of these triangles is X(1).

X(13220) lies on the line {659,905}

X(13220) = 1st-Sharygin-to-2nd-Sharygin similarity image of X(12405)

X(13221) = PARALLELOGIC CENTER OF THESE TRIANGLES: AQUILA TO 1st ORTHOSYMMEDIAL

Trilinears a^10+2*(b+c)*a^9-(b^2+c^2)*a^8-2*(b^2+c^2)*(b+c)*a^7-(2*b^2-c^2)*(b^2-2*c^2)*a^6-2*(b^4-3*b^2*c^2+c^4)*(b+c)*a^5+2*(b^4-c^4)*(b^2-c^2)*a^4+2*(b^4-c^4)*(b^2-c^2)*(b+c)*a^3+(b^2-c^2)^2*(b^4-b^2*c^2+c^4)*a^2-2*(b^2-c^2)^2*(b+c)*b^2*c^2*a+(b^4-c^4)*(b^2-c^2)*(-b^4-c^4) : :
X(13221) = 3*X(1)-2*X(10705) = 3*X(112)-X(10705) = 4*X(127)-5*X(1698) = 4*X(132)-3*X(1699) = 3*X(165)-2*X(1297) = 3*X(5587)-2*X(10749) = 5*X(7987)-4*X(12265)

The reciprocal parallelogic center of these triangles is X(6).

X(13221) lies on the Bevan circle and these lines: {1,112}, {10,13219}, {35,13206}, {40,12408}, {57,3320}, {127,1698}, {132,1699}, {165,1297}, {515,13200}, {516,12384}, {1046,2825}, {1054,2844}, {1282,9518}, {1697,6020}, {1724,2838}, {1754,1768}, {2781,3751}, {2794,5691}, {2799,13174}, {2806,5541}, {2948,9517}, {3099,13236}, {3579,13115}, {3624,6720}, {5537,12340}, {5587,10749}, {7713,13166}, {7987,12265}, {8185,11641}, {8187,13231}, {9579,12945}, {9580,12955}, {10789,13195}, {11531,13099}

X(13221) = reflection of X(i) in X(j) for these (i,j): (1,112), (5691,12784), (10705,11722), (11531,13099), (12408,40), (13115,3579), (13219,10)
X(13221) = antipode of X(12408) in Bevan circle
X(13221) = X(112)-of-Aquila-triangle

X(13222) = PARALLELOGIC CENTER OF THESE TRIANGLES: ARA TO INNER-GARCIA

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

The reciprocal parallelogic center of these triangles is X(1).

X(13222) lies on these lines: {3,11}, {22,149}, {24,13199}, {25,100}, {80,8193}, {104,11414}, {119,1598}, {159,9024}, {197,13205}, {214,11365}, {528,9909}, {659,2804}, {952,7387}, {1320,8192}, {1593,10724}, {2771,9919}, {2783,9861}, {2787,13175}, {2800,9911}, {2802,9798}, {2806,11641}, {2831,12413}, {3035,5020}, {3517,10993}, {5541,8185}, {7517,12331}, {7530,11698}, {8190,13228}, {8191,13230}, {10037,10087}, {10778,13171}, {10790,13194}, {10828,13235}, {12082,12248}, {12083,12773}

X(13222) = reflection of X(9913) in X(7387)
X(13222) = X(100)-of-Ara-triangle

X(13223) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO ARIES

Barycentrics (3*(4*R^2-SW)*SA^2+(3*S^2-(16*R^2-3*SW)*SW)*SA-6*(S^2-SW^2)*R^2-SW^3)*(SB-SC) : :

The reciprocal parallelogic center of these triangles is X(9833).

X(13223) lies on these lines: {351,13224}, {924,9131}, {3566,9135}, {9138,9979}

X(13223) = reflection of X(13224) in X(351)
X(13223) = X(68)-of-1st-Parry-triangle

X(13224) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO ARIES

Barycentrics (3*(4*R^2-SW)*SA^2-(3*S^2+(16*R^2-5*SW)*SW)*SA+(6*S^2+2*SW^2)*R^2-SW^3)*(SB-SC) : :

The reciprocal parallelogic center of these triangles is X(9833).

X(13224) lies on these lines: {110,925}, {351,13223}, {924,9979}, {1899,9134}, {2450,3566}

X(13224) = reflection of X(13223) in X(351)
X(13224) = X(68)-of-2nd-Parry-triangle

X(13225) = PARALLELOGIC CENTER OF THESE TRIANGLES: ARTZT TO 3rd PARRY

Barycentrics (27*S^6*(S^2+SA^2)+18*(SA^2-(12*R^2-SW)*SA+(8*R^2-29/18*SW)*SW)*SW^2*S^4-(9*SA^2+4*(18*R^2-5*SW)*SA+2*SW^2)*SW^4*S^2+2*SA*SW^7)*(SB+SC) : :

The reciprocal parallelogic center of these triangles is X(9147).

X(13225) lies on the Artzt circle and these lines: {2,13170}, {98,385}, {111,2080}, {183,12157}, {512,9877}, {2698,9080}, {7610,12434}

X(13225) = reflection of X(12434) in X(7610)
X(13225) = circumsymmedial-to-Artzt similarity image of X(805)

X(13226) = PARALLELOGIC CENTER OF THESE TRIANGLES: ASCELLA TO FUHRMANN

Barycentrics 2*a^6-2*(b+c)*a^5-(5*b^2-12*b*c+5*c^2)*a^4+6*(b^2-c^2)*(b-c)*a^3+2*(b^2-3*b*c+c^2)*(b-c)^2*a^2-4*(b^3+c^3)*(b-c)^2*a+(b^2-c^2)^2*(b-c)^2 : :
X(13226) = 3*X(2)+X(13243) = 5*X(11)-3*X(1699) = X(11)-3*X(11219) = 5*X(104)-X(944) = 3*X(104)+X(12247) = 3*X(165)-X(6154) = 3*X(944)+5*X(12247) = 3*X(1699)+5*X(1768) = X(1699)-5*X(11219) = X(1768)+3*X(11219)

The reciprocal parallelogic center of these triangles is X(4).

X(13226) lies on these lines: {2,5779}, {3,8}, {5,3306}, {11,57}, {20,12690}, {28,12138}, {80,5787}, {119,8728}, {142,5851}, {153,443}, {165,4863}, {381,2096}, {496,1158}, {515,5122}, {658,1565}, {908,5843}, {942,1387}, {971,3911}, {1156,8732}, {1317,3601}, {1466,12832}, {1537,5708}, {1862,4219}, {2771,5972}, {2801,3035}, {2829,6245}, {3218,5762}, {3522,9963}, {3653,6265}, {3927,6926}, {5083,11018}, {5531,6174}, {5658,5825}, {5660,11407}, {5704,12246}, {5709,12515}, {5763,6890}, {6147,6833}, {6264,6282}, {6326,8726}, {6692,10157}, {6702,12436}, {6826,10742}, {6848,12684}, {6851,10738}, {6906,12433}, {6935,10698}, {7483,9964}, {8103,8733}, {8729,13267}, {8731,13265}, {9588,9845}, {9776,9809}, {10202,11729}, {10855,13227}, {10856,13244}, {10858,13262}, {11518,13253}, {11715,12735}, {11854,13260}, {11855,13261}

X(13226) = midpoint of X(i) and X(j) for these {i,j}: {11,1768}, {20,12690}, {9803,10609}, {13243,13257}
X(13226) = reflection of X(i) in X(j) for these (i,j): (9945,3), (9946,9940), (12019,10265), (12735,11715)
X(13226) = complement of X(13257)
X(13226) = X(110)-of-Ascella-triangle
X(13226) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,13243,13257), (1768,11219,11)

X(13227) = PARALLELOGIC CENTER OF THESE TRIANGLES: ATIK TO FUHRMANN

Trilinears 4*q*p^5-4*(2*q^2-1)*p^4+(4*q^2-3)*q*p^3+4*q^2*p^2-(5*q^2-1)*q*p-2+3*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal parallelogic center of these triangles is X(4).

X(13227) lies on these lines: {8,153}, {11,118}, {100,10860}, {104,8583}, {119,8582}, {952,9856}, {971,6745}, {1156,10865}, {1317,10866}, {1387,11035}, {1768,8580}, {2771,9947}, {3062,3174}, {3870,11372}, {5587,5884}, {6259,12059}, {6326,10864}, {10855,13226}, {10861,13243}, {10862,13244}, {10867,13262}, {10868,13265}, {11519,13253}, {11856,13260}, {11857,13261}, {11860,13267}

X(13227) = reflection of X(i) in X(j) for these (i,j): (9951,9856), (9952,9947)
X(13227) = X(110)-of-Atik-triangle
X(13227) = {X(5927), X(8581)}-harmonic conjugate of X(10863)

X(13228) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO INNER-GARCIA

Trilinears (-(b+c)*a^2+4*b*c*a+(b+c)*(b^2-3*b*c+c^2))*D+a*(a-b-c)*(a+b+c)*(a^3-(b+c)*a^2-(b^2-3*b*c+c^2)*a+(b^2-c^2)*(b-c)) : :
where D=*S*sqrt(R*(4*R+r))

The reciprocal parallelogic center of these triangles is X(1).

X(13228) lies on these lines: {11,5599}, {55,2802}, {80,8197}, {100,5597}, {104,11822}, {119,8196}, {149,5601}, {214,11366}, {517,12463}, {519,12461}, {528,11207}, {1145,5600}, {1320,5598}, {1862,11384}, {2771,12365}, {2783,12179}, {2787,13176}, {2800,12457}, {2806,13229}, {2831,12478}, {5854,12455}, {8187,12653}, {8190,13222}, {8200,10738}, {9024,12452}, {10087,11877}, {10090,11879}, {11492,13205}, {11837,13194}, {11843,13199}, {11861,13235}, {11875,12331}

X(13228) = reflection of X(13230) in X(55)
X(13228) = X(100)-of-1st-Auriga-triangle

X(13229) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO 1st ORTHOSYMMEDIAL

Trilinears -(b+c)*(a^9-(b^2+c^2)*a^7-(b^2-c^2)*(b-c)*a^6-(b^4-3*b^2*c^2+c^4)*a^5+(b^4-c^4)*(b-c)*a^4+(b^4-c^4)*(b^2-c^2)*a^3+(b^2-c^2)*(b-c)*(b^4+c^4)*a^2-(b^2-c^2)^2*b^2*c^2*a-(b^8-c^8)*(b-c))*D+a*(a+b+c)*(a-b-c)*(a^10-(b^2+c^2)*a^8+b^2*c^2*a^6-(b^6-c^6)*(b^2-c^2)*a^2+(b^8-c^8)*(b^2-c^2)) : :
where D=*S*sqrt(R*(4*R+r))

The reciprocal parallelogic center of these triangles is X(6).

X(13229) lies on these lines: {55,13231}, {112,5597}, {127,5599}, {132,8196}, {517,12479}, {1297,11822}, {2781,12365}, {2799,13176}, {2806,13228}, {2831,12462}, {5598,10705}, {5601,13219}, {6020,11873}, {8190,11641}, {8200,10749}, {9517,13208}, {11252,12478}, {11366,11722}, {11492,13206}, {11837,13195}, {11843,13200}, {11861,13236}

X(13229) = reflection of X(13231) in X(55)
X(13229) = X(112)-of-1st-Auriga-triangle

X(13230) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO INNER-GARCIA

Trilinears ((b+c)*a^2-4*b*c*a-(b+c)*(b^2-3*b*c+c^2))*D+a*(a+b+c)*(a-b-c)*(a^3-(b+c)*a^2-(b^2-3*b*c+c^2)*a+(b^2-c^2)*(b-c)) : :
where D=*S*sqrt(R*(4*R+r))

The reciprocal parallelogic center of these triangles is X(1).

X(13230) lies on these lines: {11,5600}, {55,2802}, {80,8204}, {100,5598}, {104,11823}, {119,8203}, {149,5602}, {214,11367}, {517,12462}, {519,12460}, {528,11208}, {1145,5599}, {1320,5597}, {1862,11385}, {2771,12366}, {2783,12180}, {2787,13177}, {2800,12456}, {2806,13231}, {2831,12479}, {5541,8187}, {5854,12454}, {8191,13222}, {8207,10738}, {9024,12453}, {10087,11878}, {10090,11880}, {11493,13205}, {11838,13194}, {11844,13199}, {11862,13235}, {11876,12331}

X(13230) = reflection of X(13228) in X(55)
X(13230) = X(100)-of-2nd-Auriga-triangle

X(13231) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO 1st ORTHOSYMMEDIAL

Trilinears (b+c)*(a^9-(b^2+c^2)*a^7-(b^2-c^2)*(b-c)*a^6-(b^4-3*b^2*c^2+c^4)*a^5+(b^4-c^4)*(b-c)*a^4+(b^4-c^4)*(b^2-c^2)*a^3+(b^2-c^2)*(b-c)*(b^4+c^4)*a^2-(b^2-c^2)^2*b^2*c^2*a-(b^8-c^8)*(b-c))*D+a*(a+b+c)*(a-b-c)*(a^10-(b^2+c^2)*a^8+b^2*c^2*a^6-(b^6-c^6)*(b^2-c^2)*a^2+(b^8-c^8)*(b^2-c^2)) : :
where D=*S*sqrt(R*(4*R+r))

The reciprocal parallelogic center of these triangles is X(6).

X(13231) lies on these lines: {55,13229}, {112,5598}, {127,5600}, {132,8203}, {517,12478}, {1297,11823}, {2781,12366}, {2799,13177}, {2806,13230}, {2831,12463}, {5597,10705}, {5602,13219}, {6020,11874}, {8187,13221}, {8191,11641}, {8207,10749}, {9517,13209}, {11253,12479}, {11367,11722}, {11493,13206}, {11838,13195}, {11844,13200}, {11862,13236}

X(13231) = reflection of X(13229) in X(55)
X(13231) = X(112)-of-2nd-Auriga-triangle

X(13232) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st BROCARD TO X-PARABOLA-TANGENTIAL

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

The reciprocal parallelogic center of these triangles is X(13187).

X(13232) lies on these lines: {2,13186}, {69,690}, {182,13197}, {384,13237}, {523,3589}

X(13232) = X(9293)-of-1st-Brocard-triangle

X(13233) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st EHRMANN TO 2nd BROCARD

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

The reciprocal parallelogic center of these triangles is X(3).

X(13233) lies on these lines: {3,543}, {23,671}, {98,2696}, {99,7496}, {111,9176}, {115,1995}, {148,7492}, {2930,11646}, {2936,5461}, {6323,13192}, {8546,9830}

X(13233) = reflection of X(9966) in X(8546)
X(13233) = X(1380)-of-1st-Ehrmann-triangle

X(13234) = PARALLELOGIC CENTER OF THESE TRIANGLES: 5th EULER TO 2nd BROCARD

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

The reciprocal parallelogic center of these triangles is X(6322).

X(13234) lies on the nine-points circle and these lines: {2,6323}, {4,6233}, {5,12494}, {115,597}, {353,7790}, {2793,11569}, {4045,10166}, {5099,8705}

X(13234) = midpoint of X(4) and X(6233)
X(13234) = reflection of X(12494) in X(5)
X(13234) = complementary conjugate of X(3849)
X(13234) = complement of X(6323)
X(13234) = antipode of X(12494) in nine-points circle
X(13234) = orthoptic circle of Steiner inellipse-inverse-of-X(9100)

X(13235) = PARALLELOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO INNER-GARCIA

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

The reciprocal parallelogic center of these triangles is X(1).

X(13235) lies on these lines: {11,3096}, {32,100}, {80,9857}, {104,3098}, {119,9993}, {149,2896}, {214,995}, {528,7811}, {952,9821}, {1320,9997}, {1862,11386}, {2771,9984}, {2783,9862}, {2787,8782}, {2800,12497}, {2802,9941}, {2806,13236}, {2831,12503}, {3035,7846}, {3094,9024}, {3099,5541}, {5840,9873}, {7865,10707}, {9301,12331}, {9978,9998}, {9980,9999}, {9996,10738}, {10038,10087}, {10047,10090}, {10828,13222}, {11494,13205}, {11861,13228}, {11862,13230}

X(13235) = reflection of X(12499) in X(9821)
X(13235) = X(100)-of-5th-Brocard-triangle

X(13236) = PARALLELOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO 1st ORTHOSYMMEDIAL

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

The reciprocal parallelogic center of these triangles is X(6).

X(13236) lies on these lines: {4,32}, {127,3096}, {1297,3098}, {2781,3094}, {2799,8782}, {2806,13235}, {2831,12499}, {2881,9420}, {2896,13219}, {3099,13221}, {6020,10877}, {6720,7846}, {7865,10718}, {9157,9998}, {9517,13210}, {9821,12503}, {9996,10749}, {9997,10705}, {9999,13114}, {10828,11641}, {11368,11722}, {11386,13166}, {11494,13206}, {11861,13229}, {11862,13231}

X(13236) = reflection of X(12503) in X(9821)
X(13236) = X(112)-of-5th-Brocard-triangle
X(13236) = {X(112), X(11610)}-harmonic conjugate of X(13195)

X(13237) = PARALLELOGIC CENTER OF THESE TRIANGLES: 6th BROCARD TO X-PARABOLA-TANGENTIAL

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

The reciprocal parallelogic center of these triangles is X(13187).

X(13237) lies on these lines: {3,13186}, {384,13232}, {690,7893}, {882,7927}, {10131,13197}

X(13237) = X(9293)-of-6th-Brocard-triangle

X(13238) = PARALLELOGIC CENTER OF THESE TRIANGLES: CIRCUMMEDIAL TO 2nd ORTHOSYMMEDIAL

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

The reciprocal parallelogic center of these triangles is X(13239).

X(13238) lies on the circumcircle and these lines: {2,13249}, {3,12507}, {4,12624}, {935,8705}, {2781,6236}, {6325,9517}

X(13238) = reflection of X(i) in X(j) for these (i,j): (4,12624), (12507,3)
X(13238) = anticomplement of X(13249)
X(13238) = antipode of X(12507) in circumcircle
X(13238) = 2nd-orthosymmedial-to-circummedial similarity image of X(13239)

X(13239) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd ORTHOSYMMEDIAL TO CIRCUMMEDIAL

Barycentrics (S^2+SW^2)*((9*R^2-3*SW)*S^2-SW^3)*SA^2+(-(18*R^2+SW)*(3*R^2-SW)*S^4-(R^2-2*SW)*SW^3*S^2+SW^6)*SA+2*((3*R^2-SW)*S^4+(27*R^4-7*R^2*SW-SW^2)*SW*S^2-2*R^2*SW^4)*S^2 : :

The reciprocal parallelogic center of these triangles is X(13238).

X(13239) lies on the orthosymmedial circle and these lines: {1316,6322}, {5480,12508}

X(13239) = reflection of X(12508) in X(5480)
X(13239) = circummedial-to-2nd-orthosymmedial similarity image of X(13238)

X(13240) = PARALLELOGIC CENTER OF THESE TRIANGLES: MCCAY TO CIRCUMSYMMEDIAL

Trilinears ((2*b^4-b^2*c^2+2*c^4)*a^4-(b^2+c^2)*(2*b^4-7*b^2*c^2+2*c^4)*a^2-4*(b^2-c^2)^2*b^2*c^2)*a : :

The reciprocal parallelogic center of these triangles is X(98).

X(13240) lies on the McCay circles and these lines: {2,13207}, {6,6784}, {230,263}, {373,2393}, {381,512}, {511,7610}, {1624,3066}, {2387,5055}, {2871,5640}, {3111,11159}, {6324,7708}, {7606,8705}, {7617,12525}, {8859,11002}, {8860,11673}

X(13240) = reflection of X(12525) in X(7617)
X(13240) = antipode of X(12525) in McCay circles
X(13240) = X(6233)-of-McCay-triangle

X(13241) = PARALLELOGIC CENTER OF THESE TRIANGLES: CIRCUMSYMMEDIAL TO 3rd PARRY

Trilinears a/(a^6-9*(b^2+c^2)*a^4+3*(2*b^4+b^2*c^2+2*c^4)*a^2-(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2))/(b^2-c^2) : :

The reciprocal parallelogic center of these triangles is X(13242).

X(13241) lies on the circumcircle and these lines: {3,9831}, {32,9136}, {98,3849}, {110,11186}, {111,2080}, {182,843}, {511,6323}, {512,6233}, {842,8722}, {2770,9829}, {5970,11842}

X(13241) = reflection of X(9831) in X(3)
X(13241) = antipode of X(9831) in circumcircle
X(13241) = X(98)-of-circummedial-triangle
X(13241) = X(2698)-of-circumsymmedial-triangle
X(13241) = reflection of X(6233) in the Brocard axis

X(13242) = PARALLELOGIC CENTER OF THESE TRIANGLES: 3rd PARRY TO CIRCUMSYMMEDIAL

Trilinears (4*a^8-8*(b^2+c^2)*a^6-3*(6*b^4+7*b^2*c^2+6*c^4)*a^4+(b^2+c^2)*(22*b^4-25*b^2*c^2+22*c^4)*a^2-8*b^8+10*b^6*c^2+27*b^4*c^4+10*b^2*c^6-8*c^8)*(b^2-c^2)*a : :

The reciprocal parallelogic center of these triangles is X(13241).

X(13242) lies on the Parry circle and these lines: {23,9871}, {110,6233}, {111,6323}, {351,353}, {8705,9213}, {9135,9999}, {9147,9830}

X(13242) = reflection of X(353) in X(351)
X(13242) = antipode of X(353) in Parry circle
X(13242) = X(2698)-of-3rd-Parry-triangle

X(13243) = PARALLELOGIC CENTER OF THESE TRIANGLES: CONWAY TO FUHRMANN

Trilinears a^5-(4*b^2-5*b*c+4*c^2)*a^3+(b+c)*(2*b^2-3*b*c+2*c^2)*a^2+(3*b^2+b*c+3*c^2)*(b-c)^2*a+(b^2-c^2)*(b-c)*(-2*c^2-b*c-2*b^2) : :
X(13243) = 3*X(2)-4*X(13226) = 5*X(100)-6*X(165) = 3*X(100)-2*X(5531) = 5*X(104)-4*X(1385) = 3*X(104)-2*X(6265) = 3*X(165)-5*X(1768) = 9*X(165)-5*X(5531) = 6*X(1385)-5*X(6265) = 3*X(1768)-X(5531) = 9*X(9779)-5*X(9809)

The reciprocal parallelogic center of these triangles is X(4).

X(13243) lies on these lines: {2,5779}, {7,11}, {20,952}, {21,104}, {63,100}, {80,4292}, {84,1320}, {88,5400}, {119,4197}, {149,9965}, {153,377}, {244,9355}, {404,5720}, {528,10430}, {651,7004}, {912,6909}, {971,3218}, {1012,10247}, {1158,3871}, {1317,4313}, {1387,11036}, {1709,3873}, {1776,7677}, {2095,3543}, {2096,12247}, {2802,11519}, {2808,3937}, {2829,9799}, {2950,12658}, {3035,5273}, {3065,10122}, {3146,12684}, {3219,10167}, {3522,3927}, {3748,10391}, {3832,5708}, {3869,10085}, {3889,12705}, {3951,9841}, {4198,12138}, {4297,11684}, {4304,7972}, {4661,6244}, {5047,7330}, {5083,11020}, {5249,10171}, {5768,11114}, {5770,6932}, {6154,9778}, {6326,10884}, {6839,10742}, {7701,12005}, {8103,11888}, {8104,11889}, {10444,13244}, {10711,12619}, {10861,13227}, {10885,13262}, {11520,13253}, {11886,13260}, {11887,13261}, {11890,13267}, {12246,12649}

X(13243) = reflection of X(i) in X(j) for these (i,j): (100,1768), (3146,12690), (9809,11), (9963,20), (10698,12773), (12528,12691), (13257,13226)
X(13243) = anticomplement of X(13257)
X(13243) = X(110)-of-Conway-triangle
X(13243) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (63,11220,7411), (13226,13257,2)

X(13244) = PARALLELOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO FUHRMANN

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

The reciprocal parallelogic center of these triangles is X(4).

X(13244) lies on these lines: {1,5}, {100,10434}, {104,10882}, {149,1999}, {1156,10889}, {1764,1768}, {2771,10441}, {2800,12435}, {2801,10439}, {5083,11021}, {5851,10442}, {8103,11894}, {9803,10449}, {9809,10446}, {10265,10479}, {10444,13243}, {10856,13226}, {10862,13227}, {10888,13257}, {10891,13262}, {10892,13265}, {11521,13253}, {11892,13260}, {11893,13261}, {11896,13267}

X(13244) = reflection of X(i) in X(j) for these (i,j): (12550,1), (12551,10441)
X(13244) = Conway circle-inverse-of-X(11)
X(13244) = X(110)-of-3rd-Conway-triangle

X(13245) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd SHARYGIN TO 4th CONWAY

Trilinears (a^5-(b+c)*a^4-(b+c)^2*a^3+(b^3+c^3)*a^2+b*c*(b^2-b*c+c^2)*a+b^2*c^2*(b+c))*(b-c) : :
X(13245) = 3*X(1635)-X(13258)

The reciprocal parallelogic center of these triangles is X(1).

X(13245) lies on these lines: {514,659}, {521,9508}, {1021,1635}, {2254,6003}

X(13245) = X(578)-of-2nd-Sharygin-triangle
X(13245) = 1st-Sharygin-to-2nd-Sharygin similarity image of X(12567)

X(13246) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd SHARYGIN TO 5th CONWAY

Barycentrics (3*a^3-b*c*a-b^3-c^3)*(b-c) : :
X(13246) = X(659)+3*X(4809) = X(4088)-3*X(10196) = X(4458)-3*X(4809)

The reciprocal parallelogic center of these triangles is X(1).

X(13246) lies on these lines: {514,659}, {522,4874}, {676,812}, {900,6667}, {1635,4765}, {1960,2785}, {2254,3667}, {2488,6003}, {2786,3716}, {2789,10015}, {4088,10196}, {4707,5592}

X(13246) = midpoint of X(i) and X(j) for these {i,j}: {659,4458}, {667,4142}, {4707,5592}
X(13246) = {X(659), X(4809)}-harmonic conjugate of X(4458)
X(13246) = X(389)-of-2nd-Sharygin-triangle
X(13246) = 1st-Sharygin-to-2nd-Sharygin similarity image of X(12579)

X(13247) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st EHRMANN TO 2nd ORTHOSYMMEDIAL

Barycentrics (3*(3*R^2-SW)*S^4-(3*R^2-SW)*(12*(3*SA-2*SW)*R^2-6*SA^2-SA*SW+6*SW^2)*S^2-(4*R^2-SW)*(3*SA-5*SW)*SA*SW^2)*(SB+SC) : :

The reciprocal parallelogic center of these triangles is X(4).

X(13247) lies on the line {8546,12593}

X(13247) = reflection of X(12593) in X(8546)
X(13247) = 2nd-orthosymmedial-to-1st-Ehrmann similarity image of X(4)
X(13247) = circummedial-to-1st-Ehrmann similarity image of X(1297)

X(13248) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd EHRMANN TO 1st HYACINTH

Barycentrics ((108*R^4-56*R^2*SW+7*SW^2)*S^2-(2*(3*SA-5*SW)*R^2-(SA-2*SW)*SW)*SA*SW)*(SB+SC) : :
X(13248) = 3*X(1992)+X(13203) = 3*X(10249)-2*X(12041)

The reciprocal parallelogic center of these triangles is X(10116).

X(13248) lies on these lines: {6,1112}, {66,193}, {74,8537}, {110,2393}, {113,8538}, {125,8541}, {159,6593}, {576,2777}, {1351,2781}, {1503,7728}, {1992,13203}, {2935,11477}, {5663,8549}, {5972,11511}, {6723,9813}, {6776,10721}, {7722,10752}, {7731,11458}, {8539,10119}, {8540,10118}, {9919,11482}, {9976,10628}, {10249,12041}, {11405,13171}, {11443,13201}, {11470,13202}

X(13248) = midpoint of X(i) and X(j) for these {i,j}: {193,2892}, {2935,11477}
X(13248) = reflection of X(i) in X(j) for these (i,j): (159,6593), (1177,6), (12596,11255)
X(13248) = X(100)-of-2nd-Ehrmann-triangle if ABC is acute
X(13248) = orthic-to-2nd-Ehrmann similarity image of X(125)

X(13249) = PARALLELOGIC CENTER OF THESE TRIANGLES: 5th EULER TO 2nd ORTHOSYMMEDIAL

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

The reciprocal parallelogic center of these triangles is X(13239).

X(13249) lies on the nine-points circle and these lines: {2,13238}, {4,12507}, {5,12624}, {115,9971}

X(13249) = midpoint of X(4) and X(12507)
X(13249) = reflection of X(12624) in X(5)
X(13249) = complement of X(13238)
X(13249) = antipode of X(12624) in nine-points circle

X(13250) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO EXCENTERS-MIDPOINTS

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

The reciprocal parallelogic center of these triangles is X(10).

X(13250) lies on these lines: {351,900}, {513,9131}, {522,9811}, {2827,13263}, {3667,9123}, {4926,9185}

X(13250) = reflection of X(i) in X(j) for these (i,j): (9979,9811), (13251,351)
X(13250) = X(8)-of-1st-Parry-triangle
X(13250) = X(944)-of-2nd-Parry-triangle

X(13251) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO EXCENTERS-MIDPOINTS

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

The reciprocal parallelogic center of these triangles is X(10).

X(13251) lies on these lines: {351,900}, {513,9979}, {522,9131}, {2827,13264}, {3667,9185}, {4926,9123}

X(13251) = reflection of X(i) in X(j) for these (i,j): (9131,9810), (13250,351)
X(13251) = X(8)-of-2nd-Parry-triangle
X(13251) = X(944)-of-1st-Parry-triangle

X(13252) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd SHARYGIN TO EXCENTERS-MIDPOINTS

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

The reciprocal parallelogic center of these triangles is X(1).

X(13252) lies on these lines: {513,13256}, {659,3667}, {900,13205}, {3309,4498}

X(13252) = 1st-Sharygin-to-2nd-Sharygin similarity image of X(12642)
X(13252) = excentral-to-2nd-Sharygin similarity image of X(2136)
X(13252) = X(64)-of-2nd-Sharygin-triangle

X(13253) = PARALLELOGIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS TO FUHRMANN

Trilinears (30*sin(A/2)-8*sin(3*A/2))*cos((B-C)/2)+(6*cos(A)-4)*cos(B-C)+11*cos(A)-cos(2*A)-12 : :
X(13253) = 3*X(1)-2*X(104) = 5*X(1)-4*X(11715) = 3*X(1)-X(12767) = 4*X(11)-5*X(11522) = 2*X(80)-3*X(1699) = 4*X(104)-3*X(1768) = X(104)-3*X(10698) = 5*X(104)-6*X(11715) = 4*X(1537)-3*X(1699) = 8*X(9952)-15*X(11522)

The reciprocal parallelogic center of these triangles is X(4).

X(13253) lies on these lines: {1,104}, {4,9897}, {11,3340}, {36,12332}, {40,6265}, {57,12740}, {80,1537}, {100,7991}, {119,3679}, {145,9809}, {149,4301}, {153,519}, {165,214}, {191,11014}, {516,6224}, {517,3689}, {946,12247}, {952,3627}, {1145,5660}, {1156,11526}, {1317,7962}, {1320,2801}, {1387,11219}, {1482,2771}, {1484,3656}, {1697,12739}, {1709,12737}, {2093,10090}, {2783,12550}, {2802,5531}, {2827,4895}, {2829,7971}, {2932,5537}, {3035,9588}, {3243,5851}, {3576,12515}, {3624,11729}, {3632,12751}, {4677,10711}, {5587,12611}, {5603,10265}, {5727,12764}, {5840,9589}, {5854,11523}, {5881,10742}, {5882,12248}, {6282,9946}, {6702,7988}, {8187,12462}, {8227,12619}, {9580,12743}, {9612,10057}, {9614,10073}, {11009,12672}, {11278,12688}, {11518,13226}, {11519,13227}, {11520,13243}, {11521,13244}, {11524,12645}, {11527,13260}, {11528,13261}, {11532,13262}, {11533,13265}, {11535,13267}

X(13253) = midpoint of X(i) and X(j) for these {i,j}: {145,9809}, {5531,11531}
X(13253) = reflection of X(i) in X(j) for these (i,j): (1,10698), (40,6265), (80,1537), (149,4301), (1768,1), (3632,12751), (4677,10711), (5541,6326), (5881,10742), (6264,1482), (7991,100), (7993,1320), (9897,4), (12247,946), (12248,5882), (12653,7982), (12767,104)
X(13253) = X(110)-of-excenters-reflections-triangle
X(13253) = X(10733)-of-excentral-triangle
X(13253) = X(265)-of-hexyl-triangle
X(13253) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,12767,104), (80,1537,1699), (104,12767,1768), (7993,11224,1320)

X(13254) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO EXTOUCH

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

The reciprocal parallelogic center of these triangles is X(40).

X(13254) lies on these lines: {351,3900}, {521,9810}, {522,9811}, {2804,13263}, {8058,9131}

X(13254) = reflection of X(13255) in X(351)
X(13254) = X(84)-of-1st-Parry-triangle
X(13254) = X(1490)-of-2nd-Parry-triangle

X(13255) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO EXTOUCH

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

The reciprocal parallelogic center of these triangles is X(40).

X(13255) lies on these lines: {351,3900}, {521,9811}, {522,9131}, {2804,13264}, {8058,9979}

X(13255) = reflection of X(13254) in X(351)
X(13255) = X(84)-of-2nd-Parry-triangle
X(13255) = X(1490)-of-1st-Parry-triangle

X(13256) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd SHARYGIN TO EXTOUCH

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

The reciprocal parallelogic center of these triangles is X(72).

X(13256) lies on these lines: {513,13252}, {521,659}, {522,693}, {926,3033}, {2804,13277}, {3738,13266}, {3900,9508}, {3907,13259}

X(13256) = X(68)-of-2nd-Sharygin-triangle
X(13256) = 1st-Sharygin-to-2nd-Sharygin similarity image of X(12683)

X(13257) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH TO FUHRMANN

Barycentrics 2*(b+c)*a^5-(3*b^2+2*b*c+3*c^2)*a^4-2*(b+c)*(b^2-3*b*c+c^2)*a^3+4*(b^2-c^2)^2*a^2-6*(b^2-c^2)*(b-c)*b*c*a-(b^2-c^2)^2*(b-c)^2 : :
X(13257) = 5*X(11)-6*X(3817) = 5*X(100)-3*X(9778) = 5*X(119)-4*X(9956) = 3*X(119)-2*X(12619) = 5*X(1145)-4*X(11362) = X(1768)-3*X(5660) = 2*X(3035)-3*X(5660) = 3*X(9778)+5*X(9809) = 6*X(9956)-5*X(12619) = 3*X(10742)-X(12747)

The reciprocal parallelogic center of these triangles is X(4).

X(13257) lies on these lines: {2,5779}, {4,145}, {5,9964}, {9,1768}, {11,118}, {20,9945}, {72,1145}, {78,6259}, {80,5665}, {100,329}, {104,405}, {119,125}, {214,12572}, {355,7700}, {440,2972}, {516,3689}, {528,1750}, {908,971}, {912,1532}, {950,1317}, {997,12678}, {1086,5400}, {1156,8232}, {1387,3487}, {1490,2829}, {1699,3243}, {2826,4120}, {3091,6147}, {3146,5763}, {3218,5843}, {3419,11525}, {3452,10167}, {3488,12735}, {3586,7972}, {3655,6265}, {3811,12679}, {3873,7956}, {3927,6838}, {3940,6925}, {4199,13265}, {5175,12531}, {5177,9952}, {5249,10157}, {5536,5852}, {5708,6953}, {5714,9803}, {5719,6912}, {5720,11112}, {5730,12667}, {5758,12732}, {5780,6897}, {5789,6933}, {5812,5840}, {5854,11523}, {5934,13260}, {5935,13261}, {6667,11219}, {6890,12684}, {6907,11698}, {6913,11729}, {6976,10246}, {6987,12248}, {7330,7483}, {8079,8103}, {8080,8104}, {8233,13262}, {10393,12739}, {10711,12247}, {10888,13244}, {11517,12332}

X(13257) = midpoint of X(i) and X(j) for these {i,j}: {100,9809}, {3146,9963}, {9964,12528}
X(13257) = reflection of X(i) in X(j) for these (i,j): (20,9945), (1768,3035), (9803,12019), (10609,6326), (12690,4), (12691,5777), (12773,11729), (13243,13226)
X(13257) = anticomplement of X(13226)
X(13257) = complement of X(13243)
X(13257) = X(110)-of-2nd-extouch-triangle
X(13257) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,13243,13226), (226,5927,8226), (329,5658,7580), (1768,5660,3035)

X(13258) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd SHARYGIN TO 4th EXTOUCH

Trilinears (b-c)*((b+c)*a^4-2*(b+c)^2*a^3-2*b*c*(b+c)*a^2+2*(b^4+c^4+b*c*(b^2+3*b*c+c^2))*a-(b+c)*(b^4+c^4)) : :
X(13258) = 3*X(1635)-2*X(13245)

The reciprocal parallelogic center of these triangles is X(65).

X(13258) lies on these lines: {514,1734}, {521,659}, {656,3776}, {1021,1635}

X(13258) = 1st-Sharygin-to-2nd-Sharygin similarity image of X(12713)

X(13259) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd SHARYGIN TO 5th EXTOUCH

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

The reciprocal parallelogic center of these triangles is X(65).

X(13259) lies on these lines: {291,2401}, {514,1734}, {659,3803}, {984,1769}, {1027,3887}, {1282,2812}, {1635,4160}, {3738,3751}, {3907,13256}, {4147,4458}, {9373,9511}

X(13259) = X(317)-of-2nd-Sharygin-triangle
X(13259) = 1st-Sharygin-to-2nd-Sharygin similarity image of X(12725)
X(13259) = excentral-to-2nd-Sharygin similarity image of X(1721)
X(13259) = hexyl-to-2nd-Sharygin similarity image of X(12717)

X(13260) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-HUTSON TO FUHRMANN

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=-2*(b-c)^2*(-a+b+c)*b*c
G(a,b,c)=2*c*(a-c)*(a-b+c)*(a^2+(b-2*c)*a+c^2-b^2)
H(a,b,c)=a^5-3*(b+c)*a^4+(b+2*c)*(2*b+c)*a^3+2*(b+c)*(b^2-3*b*c+c^2)*a^2-(3*b^2+5*b*c+3*c^2)*(b-c)^2*a+(b^2-c^2)^2*(b+c)

The reciprocal parallelogic center of these triangles is X(4).

X(13260) lies on these lines: {11,8113}, {100,8107}, {104,8109}, {119,8380}, {363,1768}, {952,9836}, {1156,8385}, {1317,8390}, {1387,11039}, {2771,12488}, {2800,9805}, {2801,11222}, {5083,11026}, {5531,8140}, {5934,13257}, {6326,8111}, {6732,8104}, {8103,8133}, {8391,13265}, {9783,9809}, {11527,13253}, {11854,13226}, {11856,13227}, {11886,13243}, {11892,13244}, {11922,13262}, {11923,13267}

X(13260) = reflection of X(i) in X(j) for these (i,j): (12733,9836), (12759,12488), (13261,5531)
X(13260) = X(110)-of-inner-Hutson-triangle

X(13261) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-HUTSON TO FUHRMANN

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)-H(a,b,c) : :
where F, G, H are as given at X(13260)

The reciprocal parallelogic center of these triangles is X(4).

X(13261) lies on these lines: {11,8114}, {100,8108}, {104,8110}, {119,8381}, {168,1768}, {952,9837}, {1156,8386}, {1317,8392}, {1387,11040}, {2771,12489}, {2800,9806}, {2801,11223}, {5083,11027}, {5531,8140}, {5935,13257}, {6326,8112}, {7707,13267}, {8103,8135}, {8104,8138}, {9787,9809}, {11528,13253}, {11855,13226}, {11857,13227}, {11887,13243}, {11893,13244}, {11925,13262}, {11926,13265}

X(13261) = reflection of X(i) in X(j) for these (i,j): (12734,9837), (12760,12489), (13260,5531)
X(13261) = X(110)-of-outer-Hutson-triangle

X(13262) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO FUHRMANN

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

The reciprocal parallelogic center of these triangles is X(4).

X(13262) lies on these lines: {11,8228}, {100,8224}, {104,8225}, {119,8230}, {952,7596}, {1156,8237}, {1317,8239}, {1387,11042}, {1768,8231}, {2771,12490}, {2800,9808}, {2801,11211}, {5083,11030}, {5531,8244}, {6326,8234}, {8103,8247}, {8104,8248}, {8233,13257}, {8246,13265}, {9789,9809}, {10858,13226}, {10867,13227}, {10885,13243}, {10891,13244}, {11532,13253}, {11922,13260}, {11925,13261}, {11996,13267}

X(13262) = reflection of X(12768) in X(12490)
X(13262) = X(110)-of-2nd-Pamfilos-Zhou-triangle

X(13263) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO FUHRMANN

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

The reciprocal parallelogic center of these triangles is X(3).

X(13263) lies on these lines: {351,13264}, {522,9980}, {900,9810}, {2804,13254}, {2827,13250}, {3738,9131}

X(13263) = reflection of X(13264) in X(351)
X(13263) = X(80)-of-1st-Parry-triangle
X(13263) = X(12119)-of-2nd-Parry-triangle

X(13264) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO FUHRMANN

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

The reciprocal parallelogic center of these triangles is X(3).

X(13264) lies on these lines: {351,13263}, {522,9978}, {900,9811}, {2804,13255}, {2827,13251}, {3035,3700}, {3738,9979}

X(13264) = reflection of X(13263) in X(351)
X(13264) = X(80)-of-2nd-Parry-triangle
X(13264) = X(12119)-of-1st-Parry-triangle

X(13265) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO FUHRMANN

Trilinears 16*q*p^7-16*q^2*p^6-16*q*p^5+4*(6*q^2-1)*p^4-q*p^3-(7*q^2-3)*p^2+q^3*p-(q^2-1)*q^2 : : , where p=sin(A/2), q=cos((B-C)/2)

The reciprocal parallelogic center of these triangles is X(4).

X(13265) lies on these lines: {3,4427}, {11,1284}, {21,104}, {98,100}, {119,5051}, {846,1768}, {952,9840}, {1156,8238}, {1283,13266}, {1317,8240}, {1387,11043}, {2292,2800}, {2801,11203}, {2826,9147}, {2831,3744}, {4199,13257}, {5083,11031}, {5492,6906}, {5531,8245}, {6326,8235}, {8103,8249}, {8104,8250}, {8246,13262}, {8391,13260}, {8425,13267}, {8731,13226}, {9791,9809}, {10868,13227}, {10892,13244}, {11533,13253}, {11926,13261}

X(13265) = reflection of X(i) in X(j) for these (i,j): (12746,9840), (12770,9959)
X(13265) = X(110)-of-1st-Sharygin-triangle
X(13265) = 2nd-Sharygin-to-1st-Sharygin similarity image of X(100)
X(13265) = excentral-to-1st-Sharygin similarity image of X(1768)
X(13265) = hexyl-to-1st-Sharygin similarity image of X(6326)

X(13266) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd SHARYGIN TO FUHRMANN

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

The reciprocal parallelogic center of these triangles is X(8).

X(13266) lies on these lines: {9,1635}, {100,190}, {104,105}, {513,3218}, {812,9318}, {891,1320}, {1054,1768}, {1281,2787}, {1282,3887}, {1283,13265}, {3738,13256}, {11689,12531}

X(13266) = reflection of X(100) in X(659)
X(13266) = X(74)-of-2nd-Sharygin-triangle
X(13266) = 1st-Sharygin-to-2nd-Sharygin similarity image of X(12746)
X(13266) = excentral-to-2nd-Sharygin similarity image of X(5541)
X(13266) = hexyl-to-2nd-Sharygin similarity image of X(6264)

X(13267) = PARALLELOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO FUHRMANN

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

The reciprocal parallelogic center of these triangles is X(4).

X(13267) lies on these lines: {1,8097}, {11,174}, {100,7589}, {104,7587}, {119,8382}, {173,1768}, {177,8103}, {236,3035}, {942,12772}, {952,8351}, {1156,8389}, {1317,11924}, {1387,8092}, {2771,12491}, {2800,12445}, {2801,11195}, {3320,10758}, {3662,4735}, {3931,8817}, {4003,8728}, {5083,8083}, {5129,8380}, {5531,8423}, {5840,8130}, {6326,7590}, {6667,7028}, {6713,8129}, {7593,13257}, {7707,13261}, {8094,12736}, {8425,13265}, {8729,13226}, {9809,11891}, {11535,13253}, {11860,13227}, {11890,13243}, {11896,13244}, {11923,13260}, {11996,13262}

X(13267) = reflection of X(i) in X(j) for these (i,j): (12748,8351), (12774,12491)
X(13267) = {X(11), X(174)}-harmonic conjugate of X(8104)
X(13267) = X(110)-of-Yff-central-triangle
X(13267) = excentral-to-Yff-central similarity image of X(1768)
X(13267) = hexyl-to-Yff-central similarity image of X(6326)

X(13268) = PARALLELOGIC CENTER OF THESE TRIANGLES: GOSSARD TO INNER-GARCIA

Trilinears (-cos(B-C)+2*cos(A))*(-16*p^8+16*q*p^7-8*(2*q^2-5)*p^6+8*(2*q^2-5)*q*p^5+(20*q^2-33)*p^4-(16*q^2-33)*q*p^3-9*(q^2-1)*p^2+(5*q^2-9)*q*p+q^2)*(-q^2+1) : :
where p=sin(A/2), q=cos((B-C)/2)
X(13268) = 2*X(119)-3*X(11897) = X(5541)-3*X(11852) = 3*X(11845)-X(13199)

The reciprocal parallelogic center of these triangles is X(1).

X(13268) lies on these lines: {11,1650}, {30,104}, {80,11900}, {100,402}, {119,11897}, {149,4240}, {214,11831}, {528,1651}, {952,11251}, {1320,11910}, {1862,11832}, {2771,12369}, {2783,12181}, {2787,13179}, {2800,12696}, {2802,12438}, {2806,13281}, {2831,12796}, {5541,11852}, {5840,12113}, {9024,12583}, {10087,11912}, {10090,11913}, {11839,13194}, {11845,13199}, {11848,13205}, {11853,13222}, {11863,13228}, {11864,13230}, {11885,13235}, {11901,13269}, {11902,13270}, {11903,13271}, {11904,13272}, {11905,13273}, {11906,13274}, {11907,13275}, {11908,13276}, {11911,12331}, {11914,13278}, {11915,13279}

X(13268) = midpoint of X(149) and X(4240)
X(13268) = reflection of X(i) in X(j) for these (i,j): (100,402), (1650,11), (12752,11251)
X(13268) = X(100)-of-Gossard-triangle

X(13269) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO INNER-GARCIA

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

The reciprocal parallelogic center of these triangles is X(1).

X(13269) lies on these lines: {6,100}, {11,5591}, {80,5689}, {104,11824}, {119,6202}, {149,1271}, {214,11370}, {528,5861}, {952,1161}, {1320,5605}, {1862,11388}, {2771,7725}, {2783,6227}, {2787,6319}, {2800,12697}, {2802,3641}, {2806,13282}, {2831,12805}, {5541,5589}, {5595,13222}, {5840,5871}, {6215,10738}, {8198,13228}, {8205,13230}, {8216,13275}, {8217,13276}, {9994,13235}, {10040,10087}, {10048,10090}, {10783,13199}, {10792,13194}, {10919,13271}, {10921,13272}, {10923,13273}, {10925,13274}, {10929,13278}, {10931,13279}, {11497,13205}, {11901,13268}, {11916,12331}

X(13269) = reflection of X(i) in X(j) for these (i,j): (12753,1161), (13270,100)
X(13269) = X(100)-of-inner-Grebe-triangle

X(13270) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO INNER-GARCIA

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

The reciprocal parallelogic center of these triangles is X(1).

X(13270) lies on these lines: {6,100}, {11,5590}, {80,5688}, {104,11825}, {119,6201}, {149,1270}, {214,11371}, {528,5860}, {952,1160}, {1320,5604}, {1862,11389}, {2771,7726}, {2783,6226}, {2787,6320}, {2800,12698}, {2802,3640}, {2806,13283}, {2831,12806}, {5541,5588}, {5594,13222}, {5840,5870}, {6214,10738}, {8199,13228}, {8206,13230}, {8218,13275}, {8219,13276}, {9995,13235}, {10041,10087}, {10049,10090}, {10784,13199}, {10793,13194}, {10920,13271}, {10922,13272}, {10924,13273}, {10926,13274}, {10930,13278}, {10932,13279}, {11498,13205}, {11902,13268}, {11917,12331}

X(13270) = reflection of X(i) in X(j) for these (i,j): (12754,1160), (13269,100)
X(13270) = X(100)-of-outer-Grebe-triangle

X(13271) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO INNER-GARCIA

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

The reciprocal parallelogic center of these triangles is X(1).

X(13271) lies on these lines: {2,11}, {4,5854}, {8,12764}, {12,13278}, {80,10914}, {104,3813}, {119,3913}, {145,12763}, {214,11373}, {355,2802}, {519,10742}, {529,10728}, {952,6256}, {1145,1479}, {1320,10944}, {1537,12635}, {1862,11390}, {2771,12371}, {2783,12182}, {2787,13180}, {2800,12700}, {2806,13294}, {2829,12513}, {2831,12925}, {2932,5533}, {3036,5082}, {3419,12758}, {3811,12611}, {3880,12751}, {5541,10826}, {5840,12114}, {5851,6601}, {5882,12737}, {9024,12586}, {10087,10523}, {10090,10948}, {10785,13199}, {10794,13194}, {10829,13222}, {10871,13235}, {10916,12515}, {10919,13269}, {10920,13270}, {10945,13275}, {10946,13276}, {10949,13279}, {11865,13228}, {11866,13230}, {11903,13268}, {11928,12331}, {12625,13253}

X(13271) = midpoint of X(12625) and X(13253)
X(13271) = reflection of X(i) in X(j) for these (i,j): (104,3813), (3811,12611), (3913,119), (12515,10916), (12635,1537), (12761,10525), (13205,11), (13272,10738)
X(13271) = X(100)-of-inner-Johnson-triangle
X(13271) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (100,149,13274), (149,3434,11), (3434,10947,1376), (11235,13205,11)

X(13272) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO INNER-GARCIA

Barycentrics (-a+b+c)*(a^6-(b^2-b*c+c^2)*a^4-b*c*(b+c)*a^3-(b^4-4*b^2*c^2+c^4)*a^2+(b^2-c^2)^2*(b-c)^2) : :
X(13272) = 2*X(3813)-3*X(10707) = 3*X(4421)-2*X(10993)

The reciprocal parallelogic center of these triangles is X(1).

X(13272) lies on these lines: {4,528}, {11,958}, {12,100}, {72,80}, {104,6903}, {119,10894}, {149,3436}, {214,11374}, {355,2802}, {377,6174}, {443,3035}, {529,6840}, {952,10526}, {1320,10950}, {1478,10609}, {1479,10912}, {1862,11391}, {2551,6068}, {2771,12372}, {2783,12183}, {2787,13181}, {2800,5812}, {2801,5787}, {2806,13295}, {2829,6851}, {2831,12935}, {3555,12750}, {3583,3880}, {3813,5046}, {3829,6965}, {4421,6923}, {5220,5856}, {5541,10827}, {5791,6702}, {5840,11500}, {6224,12763}, {6253,10724}, {6827,11194}, {6928,12513}, {6929,11235}, {8668,12953}, {9024,12587}, {10087,10954}, {10090,10523}, {10742,12437}, {10786,13199}, {10795,13194}, {10830,13222}, {10872,13235}, {10921,13269}, {10922,13270}, {10951,13275}, {10952,13276}, {10955,13278}, {11867,13228}, {11868,13230}, {11904,13268}, {11929,12331}

X(13272) = reflection of X(i) in X(j) for these (i,j): (12762,10526), (13271,10738)
X(13272) = X(100)-of-outer-Johnson-triangle

X(13273) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO INNER-GARCIA

Barycentrics (a^5-(b+c)*a^4+b*c*a^3-(b^2-c^2)^2*a+(b^2-c^2)^2*(b+c))*(a-b+c)*(a+b-c) : :
X(13273) = 2*X(11)+X(12943)

The reciprocal parallelogic center of these triangles is X(1).

X(13273) lies on the Johnson-Yff-inner-circle and these lines: {1,10738}, {3,8068}, {4,11}, {5,10090}, {12,100}, {30,5172}, {46,12619}, {55,5840}, {65,79}, {115,1415}, {119,6917}, {149,388}, {153,5229}, {214,11375}, {226,12739}, {355,12665}, {377,3035}, {381,1470}, {495,10087}, {517,10057}, {528,10956}, {942,10073}, {946,12740}, {952,1478}, {999,5533}, {1319,3583}, {1320,10944}, {1357,10774}, {1358,10773}, {1361,10777}, {1362,10770}, {1364,10771}, {1387,1388}, {1411,3120}, {1454,1768}, {1484,10074}, {1617,3830}, {1770,12515}, {1836,2800}, {1837,5884}, {1862,11392}, {2098,10525}, {2476,4996}, {2478,6667}, {2550,6068}, {2646,12119}, {2783,12184}, {2787,13182}, {2802,5252}, {2806,13296}, {2831,12945}, {3022,10772}, {3023,10768}, {3027,10769}, {3028,10778}, {3032,9552}, {3036,3436}, {3045,9652}, {3085,13199}, {3254,8581}, {3303,10629}, {3320,10780}, {3324,10775}, {3325,10779}, {3340,9897}, {3434,5854}, {3485,6224}, {3614,6901}, {3649,11604}, {4185,9658}, {4292,10265}, {4295,12247}, {5046,5303}, {5083,10404}, {5204,6713}, {5217,6850}, {5221,12019}, {5270,7972}, {5432,6951}, {5434,10707}, {5541,9578}, {6264,9613}, {6265,12047}, {6284,10724}, {6326,9612}, {7680,12775}, {9024,12588}, {9654,12331}, {9655,12773}, {10797,13194}, {10831,13222}, {10873,13235}, {10894,12332}, {10923,13269}, {10924,13270}, {10957,13279}, {11501,13205}, {11510,12953}, {11869,13228}, {11870,13230}, {11905,13268}, {11930,13275}, {11931,13276}, {12699,12758}

X(13273) = reflection of X(i) in X(j) for these (i,j): (10087,495), (12739,226), (12763,1478), (12775,7680)
X(13273) = antipode of X(12763) in Johnson-Yff-inner-circle
X(13273) = X(100)-of-1st-Johnson-Yff-triangle
X(13273) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,10738,13274), (4,11,12764), (11,7354,104), (79,80,11571), (80,3585,10742), (149,388,1317)

X(13274) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO INNER-GARCIA

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

The reciprocal parallelogic center of these triangles is X(1).

X(13274) lies on the Johnson-Yff-outer-circle and these lines: {1,10738}, {2,11}, {3,5533}, {4,1317}, {5,10087}, {30,10074}, {56,5840}, {80,3057}, {104,6284}, {119,10896}, {153,5225}, {214,11376}, {496,10090}, {517,10073}, {944,12761}, {946,12739}, {952,1479}, {1319,12119}, {1320,10950}, {1361,10771}, {1362,10772}, {1364,10777}, {1388,10525}, {1478,12735}, {1484,10058}, {1768,9580}, {1836,5083}, {1837,2802}, {1862,11393}, {2478,3036}, {2771,12374}, {2783,12185}, {2787,13183}, {2800,12701}, {2806,13297}, {2829,12116}, {2831,12955}, {3022,10770}, {3023,10769}, {3027,10768}, {3032,9555}, {3045,9667}, {3086,13199}, {3254,3255}, {3295,8068}, {3318,10776}, {3583,5048}, {3586,6264}, {5046,12531}, {5119,12619}, {5217,6713}, {5541,9581}, {5727,12653}, {6018,10774}, {6019,10779}, {6020,10780}, {6068,6601}, {6326,9614}, {6595,13080}, {7158,10775}, {7354,10724}, {7962,9897}, {9024,12589}, {9668,12773}, {9669,12331}, {9670,10966}, {9957,10057}, {10265,10624}, {10531,10895}, {10543,11604}, {10572,12737}, {10798,13194}, {10832,13222}, {10874,13235}, {10925,13269}, {10926,13270}, {10958,13278}, {10959,13279}, {11570,12699}, {11871,13228}, {11872,13230}, {11906,13268}, {11932,13275}, {11933,13276}, {12053,12740}

X(13274) = reflection of X(i) in X(j) for these (i,j): (10090,496), (12740,12053), (12764,1479)
X(13274) = antipode of X(12764) in Johnson-Yff-outer-circle
X(13274) = X(100)-of-2nd-Johnson-Yff-triangle
X(13274) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,10738,13273), (4,1317,12763), (100,149,13271), (149,497,11), (497,10947,55)

X(13275) = PARALLELOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO INNER-GARCIA

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

The reciprocal parallelogic center of these triangles is X(1).

X(13275) lies on these lines: {11,8222}, {80,8214}, {100,493}, {104,11828}, {119,8212}, {149,6462}, {214,11377}, {528,12152}, {952,10669}, {1320,8210}, {1862,11394}, {2771,12377}, {2783,12186}, {2787,13184}, {2802,12440}, {2806,13298}, {2831,12996}, {5541,8188}, {5840,9838}, {6461,13276}, {8194,13222}, {8201,13228}, {8208,13230}, {8216,13269}, {8218,13270}, {8220,10738}, {9024,12590}, {10087,11951}, {10090,11953}, {10875,13235}, {10945,13271}, {10951,13272}, {10981,12766}, {11503,13205}, {11840,13194}, {11846,13199}, {11907,13268}, {11930,13273}, {11932,13274}, {11949,12331}, {11955,13278}, {11957,13279}

X(13275) = X(100)-of-Lucas-homothetic-triangle

X(13276) = PARALLELOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO INNER-GARCIA

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

The reciprocal parallelogic center of these triangles is X(1).

X(13276) lies on these lines: {11,8223}, {80,8215}, {100,494}, {104,11829}, {119,8213}, {149,6463}, {214,11378}, {528,12153}, {952,10673}, {1320,8211}, {1862,11395}, {2771,12378}, {2783,12187}, {2787,13185}, {2802,12441}, {2806,13299}, {2831,12997}, {5541,8189}, {5840,9839}, {6461,13275}, {8195,13222}, {8202,13228}, {8209,13230}, {8217,13269}, {8219,13270}, {8221,10738}, {9024,12591}, {10087,11952}, {10090,11954}, {10876,13235}, {10946,13271}, {10952,13272}, {10981,12765}, {11504,13205}, {11841,13194}, {11847,13199}, {11908,13268}, {11931,13273}, {11933,13274}, {11950,12331}, {11956,13278}, {11958,13279}

X(13276) = X(100)-of-Lucas(-1)-homothetic-triangle

X(13277) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd SHARYGIN TO INNER-GARCIA

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

The reciprocal parallelogic center of these triangles is X(3869).

X(13277) lies on these lines: {11,244}, {80,291}, {100,110}, {214,3126}, {513,3218}, {659,3738}, {690,4736}, {952,4922}, {2775,12515}, {2802,4730}, {2804,13256}, {2827,13252}

X(13277) = reflection of X(i) in X(j) for these (i,j): (100,9508), (4010,11)
X(13277) = X(265)-of-2nd-Sharygin-triangle
X(13277) = 1st-Sharygin-to-2nd-Sharygin similarity image of X(12770)
X(13277) = excentral-to-2nd-Sharygin similarity image of X(6326)
X(13277) = hexyl-to-2nd-Sharygin similarity image of X(1768)

X(13278) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO INNER-GARCIA

Trilinears (-a+b+c)*(a^5-(2*b^2+b*c+2*c^2)*a^3+3*b*c*(b+c)*a^2+(b^4+c^4+b*c*(3*b-c)*(b-3*c))*a-(b^2-c^2)*(b-c)*b*c) : :
X(13278) = X(1320)+2*X(3895)

The reciprocal parallelogic center of these triangles is X(1).

X(13278) lies on these lines: {1,88}, {8,4571}, {11,3913}, {12,13271}, {21,12641}, {55,5854}, {80,10915}, {104,145}, {119,149}, {519,10058}, {528,10956}, {952,1012}, {1145,3295}, {1317,11509}, {1387,5687}, {1470,3241}, {1862,11400}, {2771,12381}, {2783,12189}, {2787,13189}, {2800,3870}, {2806,13313}, {2831,13118}, {2932,12735}, {2950,12658}, {3035,3303}, {3244,10074}, {3256,10031}, {3555,12515}, {3811,12758}, {5840,12115}, {5853,6735}, {6256,10724}, {6713,10529}, {6918,11729}, {9024,12594}, {10738,10942}, {10803,13194}, {10805,13199}, {10834,13222}, {10878,13235}, {10929,13269}, {10930,13270}, {10955,13272}, {10958,13274}, {11881,13228}, {11882,13230}, {11914,13268}, {11955,13275}, {11956,13276}, {12607,12764}

X(13278) = reflection of X(i) in X(j) for these (i,j): (100,10087), (12775,10679)
X(13278) = X(100)-of-inner-Yff-tangents-triangle
X(13278) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (100,1320,13279), (149,10528,119), (1320,3871,100), (3913,10965,5552), (8715,10090,100)

X(13279) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO INNER-GARCIA

Trilinears (-a+b+c)*(a^5-(2*b^2-3*b*c+2*c^2)*a^3-b*c*(b+c)*a^2+(b^4+c^4-5*b*c*(b-c)^2)*a-(b^2-c^2)*(b-c)*b*c) : :
X(13279) = 2*X(1479)-3*X(10707)

The reciprocal parallelogic center of these triangles is X(1).

X(13279) lies on these lines: {1,88}, {11,958}, {20,104}, {21,3254}, {56,528}, {80,5288}, {119,6953}, {145,11499}, {952,3149}, {956,12019}, {999,1004}, {1145,9709}, {1479,2975}, {1862,11401}, {2771,12382}, {2783,12190}, {2787,13190}, {2800,12704}, {2806,13314}, {2831,13119}, {3555,12738}, {3825,5260}, {4308,6224}, {4311,10074}, {4996,9785}, {5267,10058}, {6911,12648}, {9024,12595}, {10269,10993}, {10738,10943}, {10804,13194}, {10806,13199}, {10835,13222}, {10879,13235}, {10931,13269}, {10932,13270}, {10949,13271}, {10957,13273}, {10959,13274}, {11510,13205}, {11883,13228}, {11884,13230}, {11915,13268}, {11957,13275}, {11958,13276}, {12001,12331}, {12690,12773}

X(13279) = reflection of X(i) in X(j) for these (i,j): (100,10090), (12776,10680)
X(13279) = X(100)-of-outer-Yff-tangents-triangle
X(13279) = {X(100), X(1320)}-harmonic conjugate of X(13278)

X(13280) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO 1st ORTHOSYMMEDIAL

Barycentrics a^11-(b^2+c^2)*a^9-(b^4-3*b^2*c^2+c^4)*a^7-(b^2-c^2)^2*(b+c)*a^6+(b^4-c^4)*(b^2-c^2)*a^5+(b^4-c^4)*(b^2-c^2)*(b+c)*a^4-(b^2-c^2)^2*b^2*c^2*a^3+(b^2-c^2)^2*(b+c)*(b^4+c^4)*a^2-(b^8-c^8)*(b+c)*(b^2-c^2) : :
X(13280) = 2*X(132)-3*X(5587) = 3*X(3679)-X(13221) = 3*X(5657)-X(13200) = 3*X(5790)-X(13310) = X(10705)-3*X(10718)

The reciprocal parallelogic center of these triangles is X(6).

X(13280) lies on these lines: {1,127}, {2,11722}, {10,112}, {40,2794}, {65,13296}, {72,13295}, {80,2806}, {132,5587}, {355,12784}, {515,1297}, {516,10735}, {517,10749}, {519,10705}, {944,12265}, {946,13099}, {1012,12340}, {1698,6720}, {1737,13312}, {1837,6020}, {2781,3416}, {2799,13178}, {2802,10780}, {2831,12751}, {3057,13297}, {3320,5252}, {3679,13221}, {5090,13166}, {5657,13200}, {5687,13206}, {5688,13283}, {5689,13282}, {5691,12408}, {5790,13310}, {5847,10766}, {8193,11641}, {8197,13229}, {8204,13231}, {8214,13298}, {8215,13299}, {9517,13211}, {9857,13236}, {10039,13311}, {10572,13116}, {10791,13195}, {10914,13294}, {10915,13313}, {10916,13314}, {11900,13281}

X(13280) = midpoint of X(i) and X(j) for these {i,j}: {8,13219}, {5691,12408}
X(13280) = reflection of X(i) in X(j) for these (i,j): (1,127), (112,10), (944,12265), (12784,355), (13099,946)
X(13280) = anticomplement of X(11722)
X(13280) = X(112)-of-outer-Garcia-triangle

X(13281) = PARALLELOGIC CENTER OF THESE TRIANGLES: GOSSARD TO 1st ORTHOSYMMEDIAL

Barycentrics SA*(4*S^2+4*SA^2-(SA+SW)^2)*(2*S^2+3*SA^2-3*SA*SW)*((36*R^2*SW+S^2-9*SW^2)*SA^2-4*(4*R^2-SW)*SW^2*SA+(48*R^2*(3*R^2-SW)+S^2+3*SW^2)*S^2) : :
X(13281) = 2*X(132)-3*X(11897) = 3*X(11845)-X(13200) = 3*X(11852)-X(13221) = 3*X(11911)-X(13310)

The reciprocal parallelogic center of these triangles is X(6).

X(13281) lies on these lines: {30,935}, {112,402}, {127,1650}, {132,11897}, {2781,12369}, {2794,12113}, {2799,13179}, {2806,13268}, {2831,12752}, {4240,13219}, {6020,11909}, {9517,13212}, {10705,11910}, {11251,12796}, {11641,11853}, {11722,11831}, {11832,13166}, {11839,13195}, {11845,13200}, {11848,13206}, {11852,13221}, {11863,13229}, {11864,13231}, {11885,13236}, {11900,13280}, {11901,13282}, {11902,13283}, {11903,13294}, {11904,13295}, {11905,13296}, {11906,13297}, {11907,13298}, {11908,13299}, {11911,13310}, {11912,13311}, {11913,13312}, {11914,13313}, {11915,13314}

X(13281) = midpoint of X(4240) and X(13219)
X(13281) = reflection of X(i) in X(j) for these (i,j): (112,402), (1650,127), (12796,11251)
X(13281) = X(112)-of-Gossard-triangle

X(13282) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO 1st ORTHOSYMMEDIAL

Barycentrics (2*(6*R^2-SW)*S^2+S*((12*SA-8*SW)*R^2-S^2-SA^2-2*SA*SW+2*SW^2)-2*(12*R^2-SA-2*SW)*SA*SW)*(SB+SC) : :

The reciprocal parallelogic center of these triangles is X(6).

X(13282) lies on these lines: {6,74}, {127,5591}, {132,6202}, {1161,12805}, {1271,13219}, {1297,11824}, {2794,5871}, {2799,6319}, {2806,13269}, {2831,12753}, {5589,13221}, {5595,11641}, {5605,10705}, {5689,13280}, {6020,10927}, {6215,10749}, {7732,9517}, {8198,13229}, {8205,13231}, {8216,13298}, {8217,13299}, {9994,13236}, {10040,13311}, {10048,13312}, {10783,13200}, {10792,13195}, {10919,13294}, {10921,13295}, {10923,13296}, {10925,13297}, {10929,13313}, {10931,13314}, {11370,11722}, {11388,13166}, {11497,13206}, {11901,13281}, {11916,13310}

X(13282) = reflection of X(13283) in X(112)
X(13282) = X(112)-of-inner-Grebe-triangle

X(13283) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO 1st ORTHOSYMMEDIAL

Barycentrics (2*(6*R^2-SW)*S^2-S*(-S^2+12*R^2*SA-8*R^2*SW-SA^2-2*SA*SW+2*SW^2)-2*(12*R^2-SA-2*SW)*SA*SW)*(SB+SC) : :

The reciprocal parallelogic center of these triangles is X(6).

X(13283) lies on these lines: {6,74}, {127,5590}, {132,6201}, {1160,12806}, {1270,13219}, {1297,11825}, {2794,5870}, {2799,6320}, {2806,13270}, {2831,12754}, {5588,13221}, {5594,11641}, {5604,10705}, {5688,13280}, {6020,10928}, {6214,10749}, {7733,9517}, {8199,13229}, {8206,13231}, {8218,13298}, {8219,13299}, {9995,13236}, {10041,13311}, {10049,13312}, {10793,13195}, {10920,13294}, {10922,13295}, {10924,13296}, {10926,13297}, {10930,13313}, {10932,13314}, {11371,11722}, {11389,13166}, {11498,13206}, {11902,13281}, {11917,13310}

X(13283) = reflection of X(13282) in X(112)
X(13283) = X(112)-of-outer-Grebe-triangle

X(13284) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO HUTSON EXTOUCH

Trilinears (a^10-4*(b+c)*a^9+2*(b^2+5*b*c+c^2)*a^8+(b+c)*(9*b^2+20*b*c+9*c^2)*a^7-(9*b^4+9*c^4+b*c*(42*b^2+89*b*c+42*c^2))*a^6-(b+c)*(7*b^4+7*c^4+2*b*c*(17*b^2+12*b*c+17*c^2))*a^5+(9*b^6+9*c^6+2*b*c*(24*b^2+55*b*c+24*c^2)*(b^2-b*c+c^2))*a^4+(b+c)*(3*b^6+3*c^6+(16*b^4+16*c^4+b*c*(9*b^2+16*b*c+9*c^2))*b*c)*a^3-(4*b^8+4*c^8+(18*b^6+18*c^6+(37*b^4+37*c^4-14*b*c*(b^2+3*b*c+c^2))*b*c)*b*c)*a^2-(b^2-c^2)^2*(b+c)*(b^4+c^4+2*b*c*(b-c)^2)*a+(b^2-c^2)^2*(b+c)^2*(b^4-b^2*c^2+c^4))*(b-c) : :

The reciprocal parallelogic center of these triangles is X(40).

X(13284) lies on the line {351,13285}

X(13284) = reflection of X(13285) in X(351)
X(13284) = X(7160)-of-1st-Parry-triangle
X(13284) = X(12120)-of-2nd-Parry-triangle

X(13285) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO HUTSON EXTOUCH

Trilinears (a^10-6*(b^2+b*c+c^2)*a^8+(b+c)*(3*b^2-20*b*c+3*c^2)*a^7+(11*b^4+11*c^4+15*b*c*(2*b^2+b*c+2*c^2))*a^6-(b+c)*(9*b^4+9*c^4-2*b*c*(17*b^2+10*b*c+17*c^2))*a^5-(7*b^6+7*c^6+2*(16*b^4+16*c^4-b*c*(7*b^2+9*b*c+7*c^2))*b*c)*a^4+(b+c)*(9*b^6+9*c^6-(16*b^4+16*c^4+b*c*(13*b^2+16*b*c+13*c^2))*b*c)*a^3+(6*b^6+6*c^6-(37*b^4+37*c^4+2*b*c*(5*b^2-17*b*c+5*c^2))*b*c)*b*c*a^2-(b^2-c^2)^2*(b+c)*(3*b^4+3*c^4-2*b*c*(b^2+c^2))*a+(b^2-c^2)^2*(b+c)^2*(b^4-b^2*c^2+c^4))*(b-c) : :

The reciprocal parallelogic center of these triangles is X(40).

X(13285) lies on the line {351,13284}

X(13285) = reflection of X(13284) in X(351)
X(13285) = X(7160)-of-2nd-Parry-triangle
X(13285) = X(12120)-of-1st-Parry-triangle

X(13286) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd SHARYGIN TO HUTSON EXTOUCH

Trilinears ((b+c)*a^7-(3*b^2+2*b*c+3*c^2)*a^6+(b+c)*(b^2-16*b*c+c^2)*a^5+(5*b^4+5*c^4+8*b*c*(4*b^2-b*c+4*c^2))*a^4-(b+c)*(5*b^2+16*b*c+5*c^2)*(b^2-4*b*c+c^2)*a^3-(b^6+c^6+(30*b^4+30*c^4+b*c*(63*b^2-20*b*c+63*c^2))*b*c)*a^2+(b+c)*(3*b^6+3*c^6+(12*b^4+12*c^4-b*c*(15*b^2-16*b*c+15*c^2))*b*c)*a-(b^4+c^4)*(b^2-c^2)^2)*(b-c) : :

The reciprocal parallelogic center of these triangles is X(3555).

X(13286) lies on these lines: {}

X(13286) = 1st-Sharygin-to-2nd-Sharygin similarity image of X(12869)
X(13286) = excentral-to-2nd-Sharygin similarity image of X(12658)
X(13286) = hexyl-to-2nd-Sharygin similarity image of X(12842)

X(13287) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS TO 1st HYACINTH

Barycentrics ((5*R^2-SW)*S^2+(4*R^2-SW)*(9*R^2-2*SW)*S+(2*(3*SA-5*SW)*R^2-(SA-2*SW)*SW)*SA)*(SB+SC) : :

The reciprocal parallelogic center of these triangles is X(10116).

X(13287) lies on these lines: {6,1112}, {26,12892}, {74,10880}, {110,10533}, {113,10897}, {125,5412}, {371,2777}, {372,13289}, {1151,2935}, {2066,10118}, {2781,11241}, {3068,13203}, {3311,9919}, {5415,10119}, {5663,11265}, {5972,11513}, {6200,13293}, {6723,10961}, {6759,12376}, {7731,11462}, {10628,12375}, {11447,13201}, {11473,13202}

X(13287) = X(100)-of-1st-Kenmotu-diagonals-triangle if ABC is acute
X(13287) = orthic-to-1st-Kenmotu-diagonals similarity image of X(125)
X(13287) = {X(6),X(10117)}-harmonic conjugate of X(13288)

X(13288) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS TO 1st HYACINTH

Barycentrics ((5*R^2-SW)*S^2-(4*R^2-SW)*(9*R^2-2*SW)*S+(2*(3*SA-5*SW)*R^2-(SA-2*SW)*SW)*SA)*(SB+SC) : :

The reciprocal parallelogic center of these triangles is X(10116).

X(13288) lies on these lines: {6,1112}, {26,12891}, {74,10881}, {110,10534}, {113,10898}, {125,5413}, {371,13289}, {372,2777}, {1152,2935}, {2781,11242}, {3069,13203}, {3312,9919}, {5411,13171}, {5414,10118}, {5416,10119}, {5663,11266}, {5972,11514}, {6396,13293}, {6723,10963}, {6759,12375}, {7731,11463}, {10628,12376}, {11448,13201}, {11474,13202}

X(13288) = X(100)-of-2nd-Kenmotu-diagonals-triangle if ABC is acute
X(13288) = orthic-to-2nd-Kenmotu-diagonals similarity image of X(125)
X(13288) = {X(6),X(10117)}-harmonic conjugate of X(13287)

X(13289) = PARALLELOGIC CENTER OF THESE TRIANGLES: KOSNITA TO 1st HYACINTH

Barycentrics ((5*R^2-SW)*S^2+(36*R^4+(6*SA-27*SW)*R^2-(SA-4*SW)*SW)*SA)*(SB+SC) : :
Barycentrics a^2[a^14 - 3a^12(b^2 + c^2) + a^10(b^4 + 9b^2c^2 + c^4) + a^8(5b^6 - 9b^4c^2 - 9b^2c^4 + 5c^6) - a^6(5b^8 - b^6c^2 - 10b^4c^4 - b^2c^6 + 5 c^8) - a^4(b^2 - c^2)^2(b^6 - 4b^4c^2 - 4b^2c^4 + c^6) + 3a^2(b^2 - c^2)^2(b^8 + c^8) - (b^2 - c^2)^4(b^6 + 2b^4c^2 + 2b^2c^4 + c^6)] : :
X(13289) = 3*X(3)-X(2935) = 3*X(154)-X(399) = 5*X(631)-X(13203) = 2*X(1511)-3*X(11202) = 3*X(2917)-X(5898) = 2*X(2935)-3*X(13293) = 3*X(11204)-2*X(11598) = 3*X(11206)+X(12317)

The reciprocal parallelogic center of these triangles is X(10116).

X(13289) lies on these lines: {3,113}, {15,10682}, {16,10681}, {23,10733}, {24,125}, {25,7687}, {30,12901}, {35,10118}, {74,186}, {110,5562}, {146,10298}, {154,399}, {159,542}, {182,9826}, {184,1986}, {206,1511}, {265,2070}, {371,13288}, {372,13287}, {378,13202}, {389,13198}, {511,1177}, {575,13248}, {578,1112}, {631,13203}, {974,11438}, {1498,10620}, {1503,7575}, {1531,2071}, {1614,7722}, {1658,5663}, {1843,5622}, {2393,9976}, {2778,3579}, {2917,5898}, {2937,12121}, {3043,7731}, {3047,5889}, {3357,12041}, {3448,9833}, {3515,13171}, {3520,10721}, {3818,6644}, {4550,11204}, {5480,11566}, {5878,12244}, {5944,10274}, {6053,12168}, {6642,6723}, {7387,12302}, {7514,12900}, {7517,12295}, {7556,12383}, {7723,10539}, {7724,10536}, {7727,10535}, {8994,9682}, {9306,12358}, {9590,13211}, {10119,10902}, {10192,10272}, {10533,12375}, {10534,12376}, {11206,12317}, {11430,11807}, {11449,13201}, {11557,12228}

X(13289) = midpoint of X(i) and X(j) for these {i,j}: {3,10117}, {74,9934}, {1498,10620}, {2931,12412}, {2935,9919}, {3448,9833}, {5878,12244}, {7387,12302}
X(13289) = reflection of X(i) in X(j) for these (i,j): (110,10282), (3357,12041), (12893,1658), (13248,575), (13293,3)
X(13289) = anticomplement of X(32743)
X(13289) = circumcircle-inverse-of-X(10745)
X(13289) = X(100)-of-Kosnita-triangle if ABC is acute
X(13289) = orthic-to-Kosnita similarity image of X(125)
X(13289) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (184,1986,12227), (7731,11464,3043)

X(13290) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO 1st HYACINTH

Barycentrics (3*(6*R^2-SA-SW)*S^2-3*(9*SA^2-12*SA*SW+5*SW^2)*R^2+3*(2*SA^2+SW^2)*SW-7*SW^2*SA)*(SB-SC) : :

The reciprocal parallelogic center of these triangles is X(6102).

X(13290) lies on these lines: {110,930}, {351,13291}, {523,9138}, {526,9131}, {690,8030}, {9033,13302}, {11419,12219}

X(13290) = reflection of X(13291) in X(351)
X(13290) = X(265)-of-1st-Parry-triangle
X(13290) = X(12121)-of-2nd-Parry-triangle

X(13291) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO 1st HYACINTH

Barycentrics (2*a^10-4*(b^2+c^2)*a^8+2*(b^4+4*b^2*c^2+c^4)*a^6-(b^2+c^2)^3*a^4+2*(b^4-b^2*c^2+c^4)^2*a^2-(b^6+c^6)*(b^2-c^2)^2)*(b^2-c^2) : :
X(13291) = 2*X(125)-3*X(8371) = 3*X(1649)-4*X(5972) = X(3448)-3*X(5466) = X(3448)-4*X(12064)

The reciprocal parallelogic center of these triangles is X(6102).

Let A* be the intersection of the lies through X(110) perpendicular to BC, and define B* and C* cyclically. Then X(13291) is the centroid of the degenerate triangle A*B*C*. (Angel Montesdeoca, November 1, 2021). See X(13291).

See Antreas Hatzipolakis and Angel Montesdeoca, euclid 2948.

X(13291) lies on these lines: {110,476}, {115,125}, {351,13290}, {525,5653}, {526,9979}, {542,8029}, {1499,7728}, {1649,5972}, {3448,5466}, {5642,11123}, {9033,13303}, {9138,13318}, {9140,10278}

X(13291) = reflection of X(i) in X(j) for these (i,j): (9140,10278), (11123,5642), (13290,351)
X(13291) = X(265)-of-2nd-Parry-triangle
X(13291) = X(12121)-of-1st-Parry-triangle

X(13292) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st HYACINTH TO SCHROETER

Barycentrics 2*a^10-7*(b^2+c^2)*a^8+2*(5*b^4+4*b^2*c^2+5*c^4)*a^6-4*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^4+4*(b^2-c^2)^2*(b^4+c^4)*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(13292) = X(3)-3*X(11245) = 3*X(51)-X(12134) = 3*X(52)+X(11750) = X(389)-3*X(11225) = 3*X(3060)-X(7553) = X(5889)+3*X(12022) = 3*X(6146)-X(11750) = X(10112)+3*X(11225) = 2*X(10627)-3*X(10691)

The reciprocal parallelogic center of these triangles is X(125).

X(13292) lies on these lines: {3,6515}, {5,6}, {30,52}, {49,468}, {51,12134}, {54,3580}, {69,7393}, {140,343}, {143,6756}, {193,6643}, {195,2072}, {389,10112}, {524,1216}, {539,5462}, {542,10110}, {568,3575}, {578,12359}, {973,12236}, {1112,10111}, {1154,12362}, {1209,1493}, {1503,5446}, {1594,1994}, {1993,11585}, {2917,7575}, {3060,7553}, {3284,10600}, {3549,11402}, {3627,5878}, {3861,10113}, {5422,7405}, {5889,12022}, {5946,9825}, {5965,11793}, {6101,9967}, {6193,6642}, {6644,9937}, {6776,7387}, {7403,11442}, {7528,9777}, {7550,12325}, {8263,9925}, {9545,10018}, {9786,12118}, {9818,11411}, {9927,12233}, {10114,11800}, {10115,11262}, {10627,10691}, {11432,12429}, {11436,12428}

X(13292) = midpoint of X(i) and X(j) for these {i,j}: {52,6146}, {143,11264}, {155,12421}, {389,10112}, {1112,10111}, {1493,12899}, {5446,10116}, {5889,12605}, {6102,12370}, {10114,11800}
X(13292) = reflection of X(6756) in X(143)
X(13292) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,1353,12161), (6,68,5), (54,3580,7542), (343,569,140), (5446,11232,10116), (5889,12022,12605), (10112,11225,389)
X(13292) = X(11)-of-1st-Hyacinth-triangle if ABC is acute
X(13292) = orthic-to-1st-Hyacinth similarity image of X(1112)

X(13293) = PARALLELOGIC CENTER OF THESE TRIANGLES: TRINH TO 1st HYACINTH

Barycentrics ((-5*R^2+SW)*S^2+(108*R^4+(-6*SA-41*SW)*R^2+(SA+4*SW)*SW)*SA)*(SB+SC) : :
X(13293) = 3*X(376)+X(13203) = 3*X(1853)-X(12902) = 2*X(2935)+X(13289) = 3*X(11204)-2*X(12041)

The reciprocal parallelogic center of these triangles is X(10116).

X(13293) lies on the Trihn circle and these lines: {3,113}, {24,13202}, {30,12893}, {36,10118}, {54,74}, {64,399}, {110,2071}, {125,378}, {182,2781}, {186,10721}, {376,13203}, {511,13248}, {542,12302}, {578,974}, {1092,12825}, {1112,11438}, {1147,3357}, {1177,5092}, {1204,1986}, {1385,2778}, {1503,12584}, {1511,6759}, {1593,7687}, {1853,12902}, {2771,12262}, {2883,10272}, {2931,12085}, {3043,6241}, {5894,10226}, {6200,13287}, {6396,13288}, {6696,10264}, {6699,7526}, {6723,9818}, {7688,10119}, {7731,11468}, {8717,11202}, {8718,9934}, {8907,11413}, {9932,12084}, {10606,10620}, {10645,10681}, {10646,10682}, {10733,12086}, {11410,13171}, {11440,12219}, {11454,13201}

X(13293) = midpoint of X(i) and X(j) for these {i,j}: {3,2935}, {64,399}, {2931,12085}
X(13293) = reflection of X(i) in X(j) for these (i,j): (1177,5092), (2883,10272), (3357,11598), (6759,1511), (9934,10282), (10264,6696), (12901,11250), (13289,3)
X(13293) = X(100)-of-Trinh-triangle if ABC is acute

X(13294) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO 1st ORTHOSYMMEDIAL

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

The reciprocal parallelogic center of these triangles is X(6).

X(13294) lies on these lines: {11,112}, {12,13313}, {127,1376}, {132,10893}, {355,10749}, {1297,11826}, {2781,12371}, {2794,12114}, {2799,13180}, {2806,13271}, {2831,12761}, {3434,13219}, {6020,10947}, {9517,13213}, {10523,13311}, {10525,12925}, {10705,10944}, {10785,13200}, {10794,13195}, {10826,13221}, {10829,11641}, {10871,13236}, {10914,13280}, {10919,13282}, {10920,13283}, {10945,13298}, {10946,13299}, {10948,13312}, {10949,13314}, {11373,11722}, {11390,13166}, {11865,13229}, {11866,13231}, {11903,13281}, {11928,13310}

X(13294) = reflection of X(i) in X(j) for these (i,j): (12925,10525), (13206,127), (13295,10749)
X(13294) = X(112)-of-inner-Johnson-triangle

X(13295) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO 1st ORTHOSYMMEDIAL

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

The reciprocal parallelogic center of these triangles is X(6).

X(13295) lies on these lines: {11,13314}, {12,112}, {72,13280}, {127,958}, {132,10894}, {355,10749}, {1297,11827}, {2781,12372}, {2794,11500}, {2799,13181}, {2806,13272}, {2831,12762}, {3436,13219}, {6020,10953}, {6253,10735}, {9517,13214}, {10523,13312}, {10526,12935}, {10705,10950}, {10786,13200}, {10795,13195}, {10827,13221}, {10830,11641}, {10872,13236}, {10921,13282}, {10922,13283}, {10951,13298}, {10952,13299}, {10954,13311}, {10955,13313}, {11374,11722}, {11391,13166}, {11867,13229}, {11868,13231}, {11904,13281}, {11929,13310}

X(13295) = reflection of X(i) in X(j) for these (i,j): (12935,10526), (13294,10749)
X(13295) = X(112)-of-outer-Johnson-triangle

X(13296) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO 1st ORTHOSYMMEDIAL

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

The reciprocal parallelogic center of these triangles is X(6).

X(13296) lies on the Johnson-Yff-inner-circle and these lines: {1,10749}, {4,6020}, {5,13312}, {12,112}, {30,13116}, {55,2794}, {56,127}, {65,13280}, {132,10895}, {388,3320}, {495,13311}, {1297,7354}, {1478,12945}, {2781,12373}, {2799,13182}, {2806,13273}, {2831,12763}, {3085,13200}, {3585,12918}, {5229,12384}, {5434,10718}, {6284,10735}, {9517,12903}, {9578,13221}, {9579,12408}, {9654,13310}, {9655,13115}, {10705,10944}, {10797,13195}, {10831,11641}, {10873,13236}, {10923,13282}, {10924,13283}, {10956,13313}, {10957,13314}, {11375,11722}, {11392,13166}, {11501,13206}, {11905,13281}, {11930,13298}, {11931,13299}

X(13296) = reflection of X(i) in X(j) for these (i,j): (12945,1478), (13311,495)
X(13296) = Johnson-Yff-inner-circle-antipode of X(12945)
X(13296) = X(112)-of-1st-Johnson-Yff-triangle
X(13296) = {X(1),X(10749)}-harmonic conjugate of X(13297)

X(13297) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO 1st ORTHOSYMMEDIAL

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

The reciprocal parallelogic center of these triangles is X(6).

X(13297) lies on the Johnson-Yff-outer-circle and these lines: {1,10749}, {4,3320}, {5,13311}, {11,112}, {30,13117}, {55,127}, {56,2794}, {132,10896}, {496,13312}, {497,6020}, {1297,6284}, {1479,12955}, {2781,12374}, {2799,13183}, {2806,13274}, {2831,12764}, {3057,13280}, {3058,10718}, {3086,13200}, {3583,12918}, {5225,12384}, {7354,10735}, {9517,12904}, {9580,12408}, {9581,13221}, {9668,13115}, {9669,13310}, {10705,10950}, {10798,13195}, {10832,11641}, {10874,13236}, {10925,13282}, {10926,13283}, {10958,13313}, {10959,13314}, {11376,11722}, {11393,13166}, {11502,13206}, {11871,13229}, {11872,13231}, {11906,13281}, {11932,13298}, {11933,13299}

X(13297) = reflection of X(i) in X(j) for these (i,j): (12955,1479), (13312,496)
X(13297) = Johnson-Yff-outer-circle-antipode of X(12955)
X(13297) = X(112)-of-2nd-Johnson-Yff-triangle
X(13297) = {X(1),X(10749)}-harmonic conjugate of X(13296)

X(13298) = PARALLELOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO 1st ORTHOSYMMEDIAL

Barycentrics SA*(S^2*((12*R^2-7*SW)*S^2-(3*SA^2-8*SW^2+4*(7*R^2-SA)*SW)*SW)-4*S*(2*S^4+(24*R^4+2*SA^2-2*SW^2-(5*R^2+SW)*SA)*S^2+(SW^2+2*(2*R^2-SW)*SA)*SA*SW)+2*(SA-SW)*SA*SW^3)*(SB+SC) : :

The reciprocal parallelogic center of these triangles is X(6).

X(13298) lies on these lines: {112,493}, {127,8222}, {132,8212}, {1297,11828}, {2781,12377}, {2794,9838}, {2799,13184}, {2806,13275}, {2831,12765}, {6020,11947}, {6461,13299}, {6462,13219}, {8188,13221}, {8194,11641}, {8201,13229}, {8208,13231}, {8214,13280}, {8216,13282}, {8218,13283}, {8220,10749}, {9517,13215}, {10669,12996}, {10875,13236}, {10945,13294}, {10951,13295}, {10981,12997}, {11377,11722}, {11394,13166}, {11503,13206}, {11840,13195}, {11846,13200}, {11907,13281}, {11930,13296}, {11932,13297}, {11949,13310}, {11951,13311}, {11953,13312}, {11955,13313}, {11957,13314}

X(13298) = X(112)-of-Lucas-homothetic-triangle

X(13299) = PARALLELOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO 1st ORTHOSYMMEDIAL

Barycentrics SA*(S^2*((12*R^2-7*SW)*S^2-(3*SA^2-8*SW^2+4*(7*R^2-SA)*SW)*SW)+4*S*(2*S^4+(24*R^4+2*SA^2-2*SW^2-(5*R^2+SW)*SA)*S^2+(SW^2+2*(2*R^2-SW)*SA)*SA*SW)+2*(SA-SW)*SA*SW^3)*(SB+SC) : :

The reciprocal parallelogic center of these triangles is X(6).

X(13299) lies on these lines: {112,494}, {127,8223}, {132,8213}, {1297,11829}, {2781,12378}, {2794,9839}, {2799,13185}, {2806,13276}, {2831,12766}, {6020,11948}, {6461,13298}, {6463,13219}, {8189,13221}, {8195,11641}, {8202,13229}, {8209,13231}, {8211,10705}, {8215,13280}, {8217,13282}, {8219,13283}, {8221,10749}, {9517,13216}, {10673,12997}, {10876,13236}, {10946,13294}, {10952,13295}, {10981,12996}, {11378,11722}, {11395,13166}, {11504,13206}, {11841,13195}, {11847,13200}, {11908,13281}, {11931,13296}, {11933,13297}, {11950,13310}, {11952,13311}, {11954,13312}, {11956,13313}, {11958,13314}

X(13299) = X(112)-of-Lucas(-1)-homothetic-triangle

X(13300) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd SHARYGIN TO MANDART-EXCIRCLES

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

The reciprocal parallelogic center of these triangles is X(3555).

X(13300) lies on these lines: {}

X(13300) = 1st-Sharygin-to-2nd-Sharygin similarity image of X(13071)
X(13300) = excentral-to-2nd-Sharygin similarity image of X(12659)
X(13300) = hexyl-to-2nd-Sharygin similarity image of X(12843)

X(13301) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd SHARYGIN TO MIDARC

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)-G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=(b-c)*(-a+b+c)*(a^4-4*(b+c)*a^3+3*(b^2-b*c+c^2)*a^2+b*c*(b+c)^2)
G(a,b,c)=(a-b+c)*((3*b-2*c)*a^4-(4*b^2+3*b*c-3*c^2)*a^3+(b^3+c^3-b*c*(4*b-7*c))*a^2+(b^3-3*c^3+b*c*(b-c))*c*a+c*(b-c)*(2*b^3+b*c^2-c^3))
H(a,b,c)=-4*S^2*(b-c)*(2*a-b-c)

The reciprocal parallelogic center of these triangles is X(1).

X(13301) lies on these lines: pending)

X(13301) = X(1)-of-2nd-Sharygin-triangle
X(13301) = 1st-Sharygin-to-2nd-Sharygin similarity image of X(13091)
X(13301) = excentral-to-2nd-Sharygin similarity image of X(164)
X(13301) = hexyl-to-2nd-Sharygin similarity image of X(12844)
X(13301) = intouch-to-2nd-Sharygin similarity image of X(177)

X(13302) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO MIDHEIGHT

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

The reciprocal parallelogic center of these triangles is X(4).

X(13302) lies on these lines: {297,525}, {351,520}, {521,9811}, {523,13223}, {8057,9131}, {9033,13290}

X(13302) = reflection of X(13303) in X(351)
X(13302) = X(64)-of-1st-Parry-triangle
X(13302) = X(1498)-of-2nd-Parry-triangle

X(13303) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO MIDHEIGHT

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

The reciprocal parallelogic center of these triangles is X(4).

X(13303) lies on these lines: {110,1301}, {351,520}, {521,9810}, {523,13224}, {525,9131}, {1636,3167}, {2444,9517}, {3569,6753}, {8057,9979}, {9033,13291}

X(13303) = reflection of X(i) in X(j) for these (i,j): (1636,3167), (13302,351)
X(13303) = X(64)-of-2nd-Parry-triangle
X(13303) = X(1498)-of-1st-Parry-triangle

X(13304) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO INNER-NAPOLEON

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

The reciprocal parallelogic center of these triangles is X(3).

X(13304) lies on these lines: {351,9200}, {690,8030}, {804,9205}, {5917,9215}, {8595,9485}, {9158,9162}

X(13304) = reflection of X(9200) in X(351)
X(13304) = X(14)-of-1st-Parry-triangle
X(13304) = X(5474)-of-2nd-Parry-triangle

X(13305) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO OUTER-NAPOLEON

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

The reciprocal parallelogic center of these triangles is X(3).

X(13305) lies on these lines: {351,9201}, {690,8030}, {804,9204}, {5916,9215}, {8594,9485}, {9158,9163}

X(13305) = reflection of X(9201) in X(351)
X(13305) = X(13)-of-1st-Parry-triangle
X(13305) = X(5473)-of-2nd-Parry-triangle

X(13306) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO 1st NEUBERG

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

The reciprocal parallelogic center of these triangles is X(3).

X(13306) lies on these lines: {2,351}, {512,9131}, {2780,7833}, {5025,11615}, {9135,13308}

X(13306) = reflection of X(13307) in X(351)
X(13306) = X(76)-of-1st-Parry-triangle
X(13306) = X(11257)-of-2nd-Parry-triangle

X(13307) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO 1st NEUBERG

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

The reciprocal parallelogic center of these triangles is X(3).

X(13307) lies on these lines: {2,351}, {512,9979}, {850,5027}, {3569,13309}

X(13307) = reflection of X(13306) in X(351)
X(13307) = X(76)-of-2nd-Parry-triangle
X(13307) = X(11257)-of-1st-Parry-triangle

X(13308) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO 2nd NEUBERG

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

The reciprocal parallelogic center of these triangles is X(3).

X(13308) lies on these lines: {351,13309}, {826,5027}, {9135,13306}, {9147,9479}

X(13308) = reflection of X(13309) in X(351)
X(13308) = X(83)-of-1st-Parry-triangle
X(13308) = X(12122)-of-2nd-Parry-triangle

X(13309) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO 2nd NEUBERG

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

The reciprocal parallelogic center of these triangles is X(3).

X(13309) lies on these lines: {2,9479}, {351,13308}, {826,9208}, {3569,13307}, {5466,9302}

X(13309) = reflection of X(13308) in X(351)
X(13309) = X(83)-of-2nd-Parry-triangle
X(13309) = X(12122)-of-1st-Parry-triangle

X(13310) = PARALLELOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO 1st ORTHOSYMMEDIAL

Barycentrics (S^4-((9*SA-8*SW)*R^2-(SA+2*SW)*(SA-SW))*S^2-(4*R^2-SW)*SA*SW^2)*(SB+SC) : :
X(13310) = 3*X(3)-2*X(1297) = 3*X(112)-X(1297) = 4*X(112)-X(13115) = 4*X(127)-5*X(1656) = 4*X(132)-3*X(381) = 3*X(381)-2*X(10749) = 4*X(1297)-3*X(13115) = 7*X(3526)-8*X(6720) = 3*X(3830)-2*X(10735) = 3*X(5055)-2*X(10718)

The reciprocal parallelogic center of these triangles is X(6).

X(13310) lies on the Stammler circle and these lines: {3,112}, {5,13219}, {30,12384}, {127,1656}, {132,381}, {382,2794}, {399,9517}, {517,13221}, {550,12253}, {999,3320}, {1351,2781}, {1597,12145}, {1598,13166}, {2070,2080}, {2799,13188}, {2806,12331}, {2831,12773}, {3295,6020}, {3526,6720}, {3534,9530}, {3579,12408}, {3830,10735}, {5055,10718}, {5093,10766}, {5790,13280}, {7517,11641}, {8148,13099}, {9301,13236}, {9654,13296}, {9655,12945}, {9668,12955}, {9669,13297}, {10246,11722}, {10247,10705}, {11842,13195}, {11849,13206}, {11875,13229}, {11876,13231}, {11911,13281}, {11916,13282}, {11917,13283}, {11928,13294}, {11929,13295}, {11949,13298}, {11950,13299}, {12000,13313}, {12001,13314}, {12083,12413}

X(13310) = midpoint of X(12384) and X(13200)
X(13310) = reflection of X(i) in X(j) for these (i,j): (3,112), (382,12918), (8148,13099), (10749,132), (12253,550), (12408,3579), (13115,3), (13219,5)
X(13310) = antipode of X(13115) in Stammler circle
X(13310) = {X(132), X(10749)}-harmonic conjugate of X(381)
X(13310) = X(112)-of-X3-ABC-reflections-triangle

X(13311) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO 1st ORTHOSYMMEDIAL

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

The reciprocal parallelogic center of these triangles is X(6).

X(13311) lies on these lines: {1,112}, {3,3320}, {5,13297}, {12,10749}, {30,12945}, {35,1297}, {55,13116}, {127,498}, {132,1479}, {388,13200}, {495,13296}, {499,6720}, {611,2781}, {1478,2794}, {2799,10086}, {2806,10087}, {2831,10058}, {3085,13219}, {3295,6020}, {3584,10718}, {3585,10735}, {3612,12265}, {4294,12384}, {5697,13099}, {6284,12918}, {8068,10780}, {9517,10088}, {10037,11641}, {10038,13236}, {10039,13280}, {10523,13294}, {10572,12784}, {10801,13195}, {10954,13295}, {11398,13166}, {11507,13206}, {11877,13229}, {11878,13231}, {11912,13281}, {11951,13298}, {11952,13299}

X(13311) = midpoint of X(112) and X(13313)
X(13311) = reflection of X(i) in X(j) for these (i,j): (13116,55), (13296,495)
X(13311) = X(112)-of-inner-Yff-triangle
X(13311) = {X(1), X(112)}-harmonic conjugate of X(13312)

X(13312) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO 1st ORTHOSYMMEDIAL

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

The reciprocal parallelogic center of these triangles is X(6).

X(13312) lies on these lines: {1,112}, {3,6020}, {5,13296}, {11,10749}, {30,12955}, {36,1297}, {56,13117}, {127,499}, {132,1478}, {496,13297}, {497,13200}, {498,6720}, {613,2781}, {999,3320}, {1479,2794}, {1737,13280}, {1795,2853}, {2799,10089}, {2806,10090}, {2831,10074}, {3086,13219}, {3582,10718}, {3583,10735}, {4293,12384}, {5533,10780}, {7354,12918}, {9517,10091}, {10046,11641}, {10047,13236}, {10048,13282}, {10049,13283}, {10523,13295}, {10802,13195}, {10948,13294}, {11399,13166}, {11508,13206}, {11879,13229}, {11880,13231}, {11913,13281}, {11953,13298}, {11954,13299}

X(13312) = midpoint of X(112) and X(13314)
X(13312) = reflection of X(i) in X(j) for these (i,j): (13117,56), (13297,496)
X(13312) = X(112)-of-outer-Yff-triangle
X(13312) = {X(1), X(112)}-harmonic conjugate of X(13311)

X(13313) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO 1st ORTHOSYMMEDIAL

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

The reciprocal parallelogic center of these triangles is X(6).

X(13313) lies on these lines: {1,112}, {12,13294}, {119,10780}, {127,5552}, {132,10531}, {1297,11248}, {2781,12381}, {2794,12115}, {2799,13189}, {2806,13278}, {2831,12775}, {3320,11509}, {6020,10965}, {6256,10735}, {9517,13217}, {10528,13219}, {10679,13118}, {10749,10942}, {10803,13195}, {10805,13200}, {10834,11641}, {10878,13236}, {10915,13280}, {10929,13282}, {10930,13283}, {10955,13295}, {10956,13296}, {10958,13297}, {11400,13166}, {11881,13229}, {11882,13231}, {11914,13281}, {11955,13298}, {11956,13299}, {12000,13310}

X(13313) = reflection of X(i) in X(j) for these (i,j): (112,13311), (13118,10679)
X(13313) = X(112)-of-inner-Yff-tangents-triangle
X(13313) = {X(112),X(10705)}-harmonic conjugte of X(13314)

X(13314) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO 1st ORTHOSYMMEDIAL

Trilinears a*(8*R*S^2*(-SW*(S^2-SW^2)+(-4*SW^2+3*S^2)*R^2)+(R+r)*SB*(SA-SB)*SC*(SA-SC)*a*(a+b+c)) : :

The reciprocal parallelogic center of these triangles is X(6).

X(13314) lies on these lines: {1,112}, {11,13295}, {127,10527}, {132,10532}, {1297,11249}, {2794,12116}, {2799,13190}, {2806,13279}, {2831,12776}, {6020,10966}, {9517,13218}, {10529,13219}, {10680,13119}, {10749,10943}, {10804,13195}, {10806,13200}, {10835,11641}, {10879,13236}, {10916,13280}, {10931,13282}, {10932,13283}, {10949,13294}, {10957,13296}, {10959,13297}, {11401,13166}, {11510,13206}, {11883,13229}, {11884,13231}, {11915,13281}, {11957,13298}, {11958,13299}, {12001,13310}

X(13314) = reflection of X(i) in X(j) for these (i,j): (112,13312), (13119,10680)
X(13314) = X(112)-of-outer-Yff-tangents-triangle
X(13314) = {X(112),X(10705)}-harmonic conjugte of X(13313)

X(13315) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO REFLECTION

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

The reciprocal parallelogic center of these triangles is X(4).

X(13315) lies on these lines: {110,930}, {351,1510}, {6368,9131}, {9123,13223}

X(13315) = reflection of X(13318) in X(351)
X(13315) = X(54)-of-1st-Parry-triangle
X(13315) = X(7691)-of-2nd-Parry-triangle

X(13316) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO INNER-VECTEN

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

The reciprocal parallelogic center of these triangles is X(3).

X(13316) lies on these lines: {351,13319}, {3566,9135}

X(13316) = reflection of X(13319) in X(351)
X(13316) = X(486)-of-1st-Parry-triangle
X(13316) = X(12123)-of-2nd-Parry-triangle

X(13317) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO OUTER-VECTEN

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

The reciprocal parallelogic center of these triangles is X(3).

X(13317) lies on these lines: {351,13320}, {3566,9135}

X(13317) = reflection of X(13320) in X(351)
X(13317) = X(485)-of-1st-Parry-triangle
X(13317) = X(12124)-of-2nd-Parry-triangle

X(13318) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO REFLECTION

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

The reciprocal parallelogic center of these triangles is X(4).

X(13318) lies on these lines: {110,933}, {351,1510}, {2081,5012}, {6368,9979}, {9138,13291}, {9185,13224}

X(13318) = reflection of X(13315) in X(351)
X(13318) = X(54)-of-2nd-Parry-triangle
X(13318) = X(7691)-of-1st-Parry-triangle

X(13319) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO INNER-VECTEN

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

The reciprocal parallelogic center of these triangles is X(3).

X(13319) lies on these lines: {351,13316}, {2450,3566}

X(13319) = reflection of X(13316) in X(351)
X(13319) = X(486)-of-2nd-Parry-triangle
X(13319) = X(12123)-of-1st-Parry-triangle

X(13320) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO OUTER-VECTEN

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

The reciprocal parallelogic center of these triangles is X(3).

X(13320) lies on these lines: {351,13317}, {2450,3566}

X(13320) = reflection of X(13317) in X(351)
X(13320) = X(485)-of-2nd-Parry-triangle
X(13320) = X(12124)-of-1st-Parry-triangle

X(13321) = ISOGONAL CONJUGATE OF X(13139)

Barycentrics a^2 (2 a^6 b^2-6 a^4 b^4+6 a^2 b^6-2 b^8+2 a^6 c^2-3 a^4 b^2 c^2-5 a^2 b^4 c^2+6 b^6 c^2-6 a^4 c^4-5 a^2 b^2 c^4-8 b^4 c^4+6 a^2 c^6+6 b^2 c^6-2 c^8) : :
X(13321) = X[3] + 8 X[143] = 4 X[51] - X[381] = 2 X[51] + X[568]

X(13321) lies on these lines: {3,143}, {6,2070}, {30,11002}, {51,381}, {52,1656}, {185,5076}, {195,973} et al

X(13321) = reflection of X(5050) in X(5640)
X(13321) = isogonal conjugate of X(13139)
X(13321) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (51,568,381), (143,3567,3), (143,5946,3060), (3060,3567,5946), (3060,5946,3)

X(13322) = X(2)X(3)∩X(52)X(6751)

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

Let P = X(68) in the construction given at X(3146); then P' = X(13322).

X(13322) lies on these lines: {2,3}, {52,6751}, {2055,8884}

X(13322) = X(10441)-of-orthic-triangle if ABC is acute
X(13322) = {X(4),X(418)}-harmonic conjugate of X(5)

Centers on X(3)X(6) represented by Tucker parameter: X(13323)-X(13357)

A Tucker parameter is a function p = p(a,b,c) symmetric and homogeneous of degree zero in a,b,c. A point P with barycentric coordinates (sin A)[cos(A - arccot(p))] lies on the Brocard axis, X(3)X(6) and has combo X(3) + ((cot ω)/p)*X(6). Contributed by Peter Moses, April 14, 2017.

X(13323) = CROSSSUM OF X(1685) AND X(1686)

Trilinears    (r^2 - s^2) cos A - 2rs sin A : :
Trilinears    a[a^5 + a^4(b + c) - a^3(b - c)^2 - a^2(b + c)(b^2 + c^2) - 2abc(b^2 + bc + c^2) - 2b^2c^2(b + c)] : :
Barycentrics    (sin A)[cos(A - arccot(p))] : : , where p = (s^2-r^2)/(2 r s)
X(13323) = X(3) + ((cot ω)/p)X(6)

X(13323) lies on these lines: {1,987}, {3,6}, {4,1798}, {5,6703}, {21,184}, {30,5799}, {51,11337}, {54,6875}, {60,5320}, {81,10441}, {84,2344}, {140,5743}, {283,1011}, {405,1437}, {411,11424}, {712,7781}, {912,960}, {940,1408}, {952,5835}, {988,1428}, {1006,1092}, {2049,5788}, {3560,6759}, {4189,5012}, {5047,5651}, {6906,10984}, {7489,10539}

X(13323) = isogonal conjugate of X(3597)
X(13323) = crosssum of X(1685) and X(1686)
X(13323) = inverse-in-Brocard-circle of X(970)
X(13323) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(2092)
X(13323) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,970), (58,572,3), (371,372,2092), (405,1437,9306)

X(13324) = INVERSE-IN-BROCARD CIRCLE OF X(2012)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = (e-Cot[w])/(1-e Cot[w])
X(13324) = X(3) + ((cot ω)/p)X(6)

X(13324) lies on this line: {3,6}

X(13324) = inverse-in-Brocard-circle of X(2012)
X(13324) = {X(3),X(6)}-harmonic conjugate of X(2012)

X(13325) = REFLECTION OF X(3558) IN X(39)

Trilinears cos A + e cos(A + ω) : :
Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = -Cot[w]-Csc[w]/e
X(13325) = X(3) + ((cot ω)/p)X(6) = 2 X[39] - 3 X[1340] = 3 X[1340] - X[3558]

X(13325) lies on these lines: {3,6}, {76,3414}, {1916,6178}, {2040,5025}, {2564,3238}, {2565,3237}, {2566,5403}, {2567,5404}, {3413,11257}

X(13325) = reflection X(3558) in X(39)
X(13325) = reflection of X(13326) in X(3)
X(13325) = inverse-in Brocard-circle of X(2558)
X(13325) = inverse-in-second-Brocard-circle of X(1380)
X(13325) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,2558), (1340,3558,39), (1670,1671,1380), (3102,3103,1341)

X(13326) = REFLECTION OF X(3557) IN X(39)

Trilinears cos A - e cos(A + ω) : :
Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = -Cot[w]+Csc[w]/e
X(13326) = X(3) + ((cot ω)/p)X(6) = 2 X[39] - 3 X[1341] = 3 X[1341] - X[3557]

X(13326) lies on these lines: {3,6}, {76,3413}, {1916,6177}, {2039,5025}, {2564,3237}, {2565,3238}, {2566,5404}, {2567,5403}, {3414,11257}

X(13326) = reflection X(3557) in X(39)
X(13326) = reflection of X(13325) in X(3)
X(13326) = inverse-in Brocard-circle of X(2559)
X(13326) = inverse-in-second-Brocard-circle of X(1379)
X(13326) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,2559), (1341,3557,39), (1670,1671,1379), (3102,3103,1340)

X(13327) = X(3)X(6)∩X(2564)X(3236)

Trilinears cos A + e sin (A - ω) : :
Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = Sec[w]/e-Tan[w]
X(13327) = X(3) + ((cot ω)/p)X(6)

X(13327) lies on these lines: {3,6}, {2564,3236}, {2565,3235}

X(13327) = inverse-in Brocard-circle of X(2563)
X(13327) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,2563), (3,32,13328), (1342,1343,1668)

X(13328) = X(3)X(6)∩X(2564)X(3235)

Trilinears cos A - e sin (A - ω) : :
Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = -(Sec[w]/e)-Tan[w]
X(13328) = X(3) + ((cot ω)/p)X(6)

X(13328) lies on these lines: {3,6}, {2564,3235}, {2565,3236}

X(13328) = inverse-in-Brocard-circle of X(2562)
X(13328) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,2562), (3,32,13327), (1342,1343,1669)

X(13329) = X(3)X(6)∩X(36)X(59)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = -((r+4 R)/s)
X(13329) = X(3) + ((cot ω)/p)X(6)

X(13329) lies on these lines: {1,1170}, {2,1754}, {3,6}, {9,990}, {20,1724}, {31,165}, {35,2293}, {36,59}, {40,595}, {44,971}, {46,2263}, {57,212}, {101,7193}, {103,672}, {106,2742}, {109,1155}, {140,3019}, {171,4349}, {184,4191}, {238,516}, {244,5536}, {255,269}, {283,404}, {411,1780}, {484,1772}, {517,1279}, {518,3939}, {603,1419}, {605,9616}, {631,4648}, {656,3737}, {748,1699}, {896,1768}, {902,5537}, {954,5228}, {995,3428}, {1006,4653}, {1040,1708}, {1064,7688}, {1086,5762}, {1203,4300}, {1284,5091}, {1293,2382}, {1331,3218}, {1428,2223}, {1451,3601}, {1468,7987}, {1496,3361}, {1616,8158}, {1617,7074}, {1714,6836}, {1730,4224}, {1736,3100}, {1743,5732}, {1757,2801}, {1790,4210}, {1818,2323}, {1936,3911}, {1955,2636}, {2051,7413}, {2183,3220}, {2191,12704}, {2299,4219}, {3052,6244}, {3072,6684}, {3074,4292}, {3523,3945}, {3550,7220}, {3796,11350}, {3915,7991}, {4000,5759}, {4297,5247}, {4306,7078}, {4344,5264}, {4383,7580}, {4641,10167}, {4859,5735}, {5122,6610}, {5131,6149}, {5292,6865}, {5542,9440}, {5713,6989}

X(13329) = midpoint of X(238) and X(9441)
X(13329) = crosssum of X(11) and X(2254)
X(13329) = crossdifference of every pair of points on line {523,2294}
X(13329) = inverse-in-Brocard-circle of X(991)
X(13329) = inverse-in-Schoute-circle of X(5030)
X(13329) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(4253)
X(13329) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,991), (3,182,572), (3,580,58), (3,582,580), (3,970,3430), (15,16,5030), (40,602,595), (371,372,4253), (1155,2361,109), (1253,1471,1), (4210,5012,1790), (5132,5135,284)

X(13330) = REFLECTION OF X(6) IN X(5052)

Trilinears    2 sin A - sin(A + 2ω) : :
Trilinears    a - R sin(A + 2ω) : :
Barycentrics    (sin A)[cos(A - arccot(p))] : : , where p = Sin[2 w]/(-2+Cos[2 w])
Barycentrics    a^2 (2 a^2 b^2-b^4+2 a^2 c^2+b^2 c^2-c^4) : :
X(13330) = X(3) + ((cot ω)/p)X(6) = 3 X[6] - 2 X[39] = X[194] - 3 X[1992] = 4 X[39] - 3 X[3094]

X(13330) lies on the cubic K382 and these lines: {3,6}, {4,11646}, {25,2056}, {51,1613}, {69,7785}, {76,524}, {112,8537}, {141,7752}, {172,8540}, {181,2162}, {193,732}, {194,1992}, {230,262}, {263,694}, {538,11159}, {542,7747}, {597,7786}, {599,3934}, {698,3629}, {881,9005}, {1003,12151}, {1153,6683}, {1383,9716}, {1501,1994}, {1914,12837}, {1915,1993}, {1916,5939}, {1968,8541}, {1995,9225}, {2023,7735}, {2176,3271}, {2211,6403}, {2493,9419}, {2782,7737}, {3051,3060}, {3124,9463}, {3231,5640}, {3299,12841}, {3301,12840}, {3552,5026}, {3589,7857}, {3618,10007}, {3763,7862}, {3767,6034}, {3787,5943}, {4663,12782}, {5475,7697}, {5476,7746}, {5976,7774}, {6194,7736}, {7749,11261}, {7757,8584}, {7760,10754}, {7837,9865}, {7838,8149}, {8550,11257}, {8627,11003}, {8778,11405}, {10311,11470}, {10753,12110}

X(13330) = midpoint of X(j) and X(j) for these (i,j): {1,1742}, {20,10446}
X(13330) = reflection of X(i) in X(j) for these (i,j): (6, 5052), (3094, 6), (3095, 576), (7757, 8584), (11152, 8787), (11257, 8550), (12782, 4663)
X(13330) = inverse-in-Brocard-circle of X(5038)
X(13330) = inverse-in-second-Brocard-circle of X(574)
X(13330) = inverse-in-circle-{{X(4),X(194),X(3557),X(3558)}} of X(39)
X(13330) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(2080)
X(13330) = 2nd-Lemoine-circle-inverse of X(35377)
X(13330) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,5038), (6,187,10485), (6,1351,5111), (6,2076,182), (6,3094,13331), (6,5017,1691), (6,5116,5034), (6,11173,5104), (32,576,6), (187,1692,8590), (371,372,2080), (1351,3311,9975), (1351,3312,9974), (1384,3311,6424), (1384,3312,6423), (1664,1665,5116), (1670,1671,574), (1689,1690,576), (1692,5097,6), (3051,3060,3981), (3094,10485,11171), (3098,5034,5116), (3104,3105,3095), (3557,3558,39), (5028,5039,6), (5038,5104,3), (5111,12212,6), (6423,9975,6), (6424,9974,6), (9463,11002,3124), (12963,12968,1384)

X(13331) = X(2)X(732)∩X(3)X(6)

Trilinears    2 sin A + sin(A + 2ω) : :
Trilinears    a + R sin(A + 2ω) : :
Barycentrics    (sin A)[cos(A - arccot(p))] : : , where p = Sin[2 w]/(2+Cos[2 w])
Barycentrics    a^2 (2 a^2 b^2+b^4+2 a^2 c^2+3 b^2 c^2+c^4) : :
X(13331) = X(3) + ((cot ω)/p)X(6) = X[6] + 2 X[39] = X[1691] - 4 X[2024] = 4 X[39] - X[3094] = 2 X[6] + X[3094]

X(13331) lies on these lines: {2,732}, {3,6}, (6,3094,13330), {69,10007}, {76,3589}, {83,4048}, {141,7786}, {147,2023}, {194,3618}, {262,1503}, {373,1194}, {597,698}, {694,9155}, {1180,3981}, {1352,11272}, {1386,12782}, {1428,12837}, {1613,5650}, {1915,6800}, {1916,5026}, {2056,6090}, {2330,12836}, {2782,6034}, {3051,7998}, {3108,5012}, {3299,12840}, {3301,12841}, {3329,10334}, {3763,6683}, {5031,7777}, {5103,7790}, {5309,7697}, {5480,9607}, {5965,11261}, {6309,7819}, {7760,8177}, {7829,8149}, {7875,9865}, {10347,12216}

X(13331) = inverse-in-Brocard-circle of X(12212)
X(13331) = inverse-in-second-Brocard-circle of X(7772)
X(13331) = centroid of X(6)PU(1)
X(13331) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(12054)
X(13331) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,12212), (6,39,3094), (6,2076,5039), (6,3094,13330), (6,5013,5017), (6,5024,5104), (6,5116,32), (39,2021,5024), (39,7772,3095), (182,7772,6), (371,372,12054), (574,5039,2076), (1670,1671,7772), (1689,1690,3098), (3106,3107,3095), (12055,12212,3)

X(13332) = INVERSE-IN-BROCARD-CIRCLE OF X(1686)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = (s-r)/(s+r)
Barycentrics a^2 (-(a+b) (a-b-c) (a+b-c) (a+c) (a-b+c)+Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)] (-a^3+a b^2-a b c+b^2 c+a c^2+b c^2)) : :
X(13332) = X(3) + ((cot ω)/p)X(6)

X(13332) lies on these lines: {3,6}, {485,2048}, {486,2047}, {1124,1397}

X(13332) = inverse-in-Brocard-circle of X(1686)
X(13332) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(1685)
X(13332) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,1686), (371,372,1685), (3371,3372,386), (3385,3386,573)

X(13333) = INVERSE-IN-BROCARD-CIRCLE OF X(1685)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = cot[w]+2 tan[w]
Barycentrics a^2 (-(a+b) (a-b-c) (a+b-c) (a+c) (a-b+c)+Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)] (-a^3+a b^2-a b c+b^2 c+a c^2+b c^2)) : :
X(13333) = X(3) + ((cot ω)/p)X(6)

X(13333) lies on these lines: {3,6}, {1335,1397}

X(13333) = inverse-in-Brocard-circle of X(1685)
X(13333) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(1686)
X(13333) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,1685), (371,372,1686), (3371,3372,573), (3385,3386,386)

X(13334) = MIDPOINT OF X(3) AND X(39)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = (s-r)/(s+r)
Barycentrics a^2 (3 a^4 b^2-4 a^2 b^4+b^6+3 a^4 c^2-6 a^2 b^2 c^2-3 b^4 c^2-4 a^2 c^4-3 b^2 c^4+c^6) : :
X(13334) = X(3) + ((cot ω)/p)X(6) = 3 X[39] - X[3095] = 3 X[3] + X[3095]

X(13334) lies on these lines:
{2,6248}, {3,6}, {4,7786}, {5,4045}, {20,262}, {30,11272}, {76,631}, {83,11676}, {98,7824}, {114,6656}, {140,620}, {160,9822}, {194,3523}, {237,5943}, {538,549}, {542,8359}, {730,6684}, {1503,10007}, {1656,7913}, {3097,7987}, {3329,12110}, {3524,7757}, {3526,7697}, {3576,12782}, {3818,8721}, {3972,10359}, {5054,9466}, {5204,12837}, {5217,12836}, {5418,8992}, {5657,7976}, {5965,7767}, {6688,11328}, {6721,8361}, {7791,9744}, {7878,10788}, {8369,10168}, {10165,12263}, {10358,11174}

X(13334) = midpoint of X(i) and X(j) for these {i,j}: {3, 39}, {3095, 5188}, {6248, 11257}
X(13334) = reflection of X(i) in X(j) for these (i,j): (5,6683) (3934,140)
X(13334) = complement of X[6248]
X(13334) = inverse-in-Brocard-circle of X(5171)
X(13334) = inverse-in-second-Brocard-circle of X(1351)
X(13334) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(5034)
X(13334) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,5171), (3,3095,5188), (3,3398,187), (3,5013,9737), (3,5050,3053), (3,10983,1350), (3,11171,39), (39,5188,3095), (371,372,5034), (1670,1671,1351), (1689,1690,5013), (8160,8161,182), (2,11257,6248), (631,7709,76)

X(13335) = MIDPOINT OF X(3) AND X(32)

Trilinears    2 cos(A - 2ω) + cos(A + 2ω) - cos A : :
Barycentrics    (sin A)[cos(A - arccot(p))] : : , where p = Cot[w]-2 Tan[w]
Barycentrics    a^2 (2 a^6-3 a^4 b^2+2 a^2 b^4-b^6-3 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(13335) = X(3) + ((cot ω)/p)X(6) = X[315] - 5 X[631] = 3 X[5054] - X[7818] = 7 X[3526] - 5 X[7867] = X[20] + 3 X[9753]

X(13335) lies on these lines:
{3,6}, {4,3972}, {5,2794}, {20,7797}, {30,7817}, {98,384}, {114,7807}, {140,626}, {157,9822}, {262,7787}, {315,631}, {376,7827}, {542,8369}, {549,754}, {682,3491}, {736,7780}, {760,1385}, {1975,9755}, {2386,6644}, {2782,7816}, {3148,5943}, {3526,7867}, {3552,11257}, {3564,7789}, {3933,5965}, {3934,10104}, {5026,8550}, {5054,7818}, {5152,12176}, {5999,12110}, {6055,8370}, {6179,12251}, {7709,7782}, {7786,10359}, {7824,10350}, {7892,9863}, {8359,10168}, {11676,12203}

X(13335) = midpoint of X(3) and X(32)
X(13335) = complement of complement of X(36998)
X(13335) = inverse-in-Brocard-circle of X(9737)
X(13335) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(5028)
X(13335) = harmonic center of circles {{X(3102),X(3103),PU(1)}} and {{X(2459),X(2460),PU(2)}}
X(13335) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,9737), (3,2080,5188), (3,3053,5171), (3,3398,39), (3,5013,9734), (3,5050,5013), (3,11842,3095), (39,3398,575), (371,372,5028), (1342,1343,2456), (3095,5007,5097), (3095,11842,5007), (5023,5085,3), (98,384,6248)

X(13336) = INVERSE-IN-BROCARD-CIRCLE OF X(10625)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = Cot[w]+1/2 Csc[A] Csc[B] Csc[C]
Barycentrics a^2 (a^8-3 a^6 b^2+3 a^4 b^4-a^2 b^6-3 a^6 c^2-2 a^4 b^2 c^2+7 a^2 b^4 c^2-2 b^6 c^2+3 a^4 c^4+7 a^2 b^2 c^4+4 b^4 c^4-a^2 c^6-2 b^2 c^6) : :
X(13336) = X(3) + ((cot ω)/p)X(6)

X(13336) lies on these lines:
{2, 1614}, {3, 6}, {5, 10984}, {22, 5462}, {24, 5892}, {49, 5054}, {54, 3523}, {68, 7383}, {110, 3525}, {113, 6816}, {140, 184}, {155, 7484}, {156, 632}, {185, 7514}, {323, 11423}, {549, 1092}, {550, 11424}, {631, 1147}, {1176, 7404}, {1181, 5891}, {1199, 2979}, {1209, 1899}, {1216, 7485}, {1503, 7405}, {1588, 9687}, {1595, 1974}, {1656, 1853}, {1993, 5447}, {3146, 8717}, {3526, 9306}, {3567, 6636}, {3589, 7403}, {3796, 6642}, {3832, 8718}, {3917, 12161}, {4550, 6241}, {5070, 10540}, {5326, 9652}, {5422, 5446}, {5449, 7558}, {5562, 7516}, {5640, 12088}, {5943, 7517}, {5946, 7525}, {7294, 9667}, {7387, 10601}, {7395, 12162}, {7496, 7999}, {7499, 12359}, {7502, 12006}, {7506, 11695}, {7550, 12111}, {7689, 10574}, {7706, 12225}, {9818, 10575}, {10110, 12083}, {10170, 11441}, {10303, 11003}, {10610, 11577} }

X(13336) = inverse-in-Brocard-circle of X(10625)
X(13336) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(5421)
X(13336) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,10625), (3,182,569), (3,6243,3098), (3,11425,10564), (156,632,5651), (371,372,5421), (389,5092,3), (631,5012,1147), (1181,7393,5891), (5422,10323,5446), (7485,7592,1216)

X(13337) = CROSSSUM OF X(6) AND X(5054)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = 4/(4 Cot[w]+3 Csc[A] Csc[B] Csc[C])
Barycentrics a^2 (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2-7 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6),b^2 (a^6-2 a^4 b^2+a^2 b^4-a^4 c^2-7 a^2 b^2 c^2+b^4 c^2-a^2 c^4-2 b^2 c^4+c^6) : :
X(13337) = X(3) + ((cot ω)/p)X(6)

X(13337) lies on these lines:
{3, 6}, {231, 5355}, {1180, 9300}, {1989, 7739}, {2493, 7736}, {3815, 9465}, {5254, 9220}

X(13337) = crosssum of X(6) and X(5054)
X(13337) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,13338), (6,39,566), (5063,7772,6)

X(13338) = X(3)X(6)∩X(51)X(7669)

Barycentrics    (sin A)[cos(A - arccot(p))] : : , where p = -(4/(4 Cot[w]+3 Csc[A] Csc[B] Csc[C]))
Barycentrics    a^4 (a^4-2 a^2 b^2+b^4-2 a^2 c^2-5 b^2 c^2+c^4) : :
Barycentrics    Sin[A]^2 (3+4 Sin[A]^2) : :
X(13338) = X(3) + ((cot ω)/p)X(6)

X(13338) lies on these lines:
{3, 6}, {51, 7669}, {112, 6749}, {251, 1989}, {1627, 9300}, {1990, 10312}, {2493, 5354}, {3767, 9220}, {4558, 8584}, {5304, 7519}

X(13338) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,13337), (6,32,50), (6,11063,39), (32,50,2965), (3003,5007,6)

X(13339) = X(3)X(6)∩X(110)X(140)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = Cot[w]+3/4 Csc[A] Csc[B] Csc[C]
Barycentrics a^2 (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+8 a^2 b^4 c^2-2 b^6 c^2+3 a^4 c^4+8 a^2 b^2 c^4+4 b^4 c^4-a^2 c^6-2 b^2 c^6) : :
X(13339) = X(3) + ((cot ω)/p)X(6)

X(13339) lies on these lines:
{2, 10540}, {3, 6}, {4, 7605}, {49, 631}, {54, 3530}, {110, 140}, {156, 3525}, {184, 5054}, {195, 5447}, {373, 7545}, {382, 8717}, {399, 10170}, {549, 5012}, {632, 1614}, {1092, 11935}, {1199, 10627}, {1493, 11592}, {1656, 10984}, {2070, 5892}, {3066, 7517}, {3526, 5651}, {3850, 8718}, {5070, 6759}, {5544, 7529}, {5663, 7550}, {5899, 5943}, {5907, 12308}, {5946, 6636}, {6699, 11597}, {7512, 12006}, {10601, 12083}, {11561, 12041}

X(13339) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,13340),. (3,182,567), (3,9730,3581), (5092,9730,3)

X(13340) = X(3)X(6)∩X(140)X(5640)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = -Cot[w]-3/4 Csc[A] Csc[B] Csc[C]
Barycentrics a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2-7 a^4 b^2 c^2+4 a^2 b^4 c^2+2 b^6 c^2-3 a^4 c^4+4 a^2 b^2 c^4-2 b^4 c^4+3 a^2 c^6+2 b^2 c^6-c^8) : :
X(13340) = X(3) + ((cot ω)/p)X(6)

X(13340) lies on these lines:
{3, 6}, {4, 10627}, {5, 7998}, {20, 5663}, {30, 2979}, {49, 6800}, {51, 5054}, {140, 5640}, {143, 3523}, {195, 10984}, {373, 3526}, {376, 1154}, {381, 3917}, {382, 1216}, {394, 10540}, {546, 7999}, {548, 5889}, {549, 3060}, {550, 11412}, {631, 10263}, {632, 9781}, {1092, 2937}, {1511, 7556}, {1656, 5447}, {1657, 5562}, {3146, 11591}, {3522, 6102}, {3524, 5946}, {3525, 10095}, {3529, 5876}, {3530, 3567}, {3627, 11444}, {3819, 5055}, {3830, 5891}, {3843, 11793}, {5070, 10110}, {5073, 5907}, {5651, 7545}, {5890, 8703}, {5899, 9306}, {6090, 7387}, {6241, 12103}, {6759, 9919}, {7555, 11464}, {7689, 12302}, {7691, 11250}, {10303, 11592}, {11451, 11539}

X(13340) = reflection of X(i) in X(j) for these (i,j): (381,3917), (568,3), (3060,549), (3830,5891), (5890,8703), (6243,568)
X(13340) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,13339), (394,12083,13354)

X(13341) = X(3)X(6)∩X(53)X(7765)

Barycentrics    (sin A)[cos(A - arccot(p))] : : , where p = 1/(Cot[w]+2 Csc[A] Csc[B] Csc[C])
Barycentrics    a^2 (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2-12 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6) : :
Barycentrics    (sin^2 A)(2 + sin^2 B + sin^2 C) : :
X(13341) = X(3) + ((cot ω)/p)X(6)

X(13341) lies on these lines:
{3, 6}, {53, 7765}, {232, 7714}, {393, 7739}, {1015, 3553}, {1194, 7398}, {1196, 7736}, {1500, 3554}, {3087, 3199}, {6748, 7753}, {7586, 8962}, {7603, 9722}, {9300, 10128}

X(13341) = crosssum of X(i) and X(j) for these (i,j): {2,10601}, {6,3523}
X(13341) = barycentric product X(3)*X(1907)
X(13341) = barycentric quotient X(i)/X(j) for these (i,j): (1907,264), (3442,5312)
X(13341) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,13342), (6,39,800), (6,577,5007), (6,5421,216), (216,5421,39)

X(13342) = X(3)X(6)∩X(53)X(5309)

Barycentrics    (sin A)[cos(A - arccot(p))] : : , where p = -(1/(Cot[w]+2 Csc[A] Csc[B] Csc[C]))
Barycentrics    a^4 (a^4-2 a^2 b^2+b^4-2 a^2 c^2-10 b^2 c^2+c^4) : :
Barycentrics    Sin[A]^2 (2+Sin[A]^2) : :
X(13342) = X(3) + ((cot ω)/p)X(6)

X(13342) lies on these lines:
{3, 6}, {53, 5309}, {393, 5319}, {1249, 7714}, {2241, 3553}, {2242, 3554}, {5304, 7398}, {5306, 10128}

X(13342) = barycentric product X(3)*X(5198)
X(13342) = barycentric quotient X(5198)/X(264)
X(13342) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,13341), (6,32,5065), (6,216,7772), (6,8573,39)

X(13343) = POINT BEID 44

Trilinears    e cos A + sin A : :
Barycentrics    (sin A)[cos(A - arccot(p))] : : , where p = e
Barycentrics    a^2 (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)] Sqrt[a^2 b^2+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]) : :
X(13343) = X(3) + ((cot ω)/p)X(6)

X(13343) lies on this line: {3, 6}

X(13343) = radical center of Lucas(2/e) circles
X(13343) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(2560)
X(13343) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,13344), (371,372,2560), (1689,1690,2026)

X(13344) = POINT BEID 45

Trilinears    e cos A - sin A : :
Barycentrics    (sin A)[cos(A - arccot(p))] : : , where p = -e
Barycentrics    a^2 (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)] Sqrt[a^2 b^2+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]) : :
X(13344) = X(3) + ((cot ω)/p)X(6)

X(13344) lies on this line: {3, 6}

X(13344) = radical center of Lucas(-2/e) circles
X(13344) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(2561)
X(13344) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,13343), (371,372,2561), (1689,1690,2027)

X(13345) = X(3)X(6)∩X(51)X(157)

Barycentrics    (sin A)[cos(A - arccot(p))] : : , where p = -(2/(2 Cot[w]+Csc[A] Csc[B] Csc[C]))
Barycentrics    a^4 (a^4-2 a^2 b^2+b^4-2 a^2 c^2-4 b^2 c^2+c^4) : :
Barycentrics    Sin[A]^2+2 Sin[A]^4 : :
X(13345) = X(3) + ((cot ω)/p)X(6)

X(13345) lies on these lines:
{3, 6}, {51, 157}, {53, 428}, {112, 3087}, {233, 7753}, {251, 2165}, {393, 1179}, {609, 3554}, {1249, 8882}, {1627, 7736}, {1879, 3767}, {1907, 1968}, {3553, 7031}, {5304, 7500}, {7745, 9722}

X(13345) = inverse-in-Brocard-circle of X(5421)
X(13345) = crosspoint of X(5422) and X(10594)
X(13345) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(10625)
X(13345) = barycentric product X(i)X(j) for these {i,j}: {3,10594}, {6,5422}
X(13345) = barycentric quotient X(i)/X(j) for these (i,j): (5422,76), (10594,264)
X(13345) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,5421), (6,32,571), (6,50,5065), (6,571,5063), (6,1609,570), (6,2965,577), (6,8553,39), (6,8573,3003), (32,577,2965), (32,5007,10316), (216,5007,6), (371,372,10625), (577,2965,571), (1182,5037,6)

X(13346) = X(3)X(6)∩X(20)X(184)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = Cot[w]-2 Csc[A] Csc[B] Csc[C]
Barycentrics a^2 (a^8-3 a^6 b^2+3 a^4 b^4-a^2 b^6-3 a^6 c^2+8 a^4 b^2 c^2-3 a^2 b^4 c^2-2 b^6 c^2+3 a^4 c^4-3 a^2 b^2 c^4+4 b^4 c^4-a^2 c^6-2 b^2 c^6) :
X(13346) = X(3) + ((cot ω)/p)X(6)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung); then X(13347) = X(20)-of-A'B'C'. (Randy Hutson, July 21, 2017)

X(13346) lies on these lines:
{2, 11424}, {3, 6}, {4, 801}, {20, 184}, {23, 11449}, {26, 11202}, {30, 156}, {40, 3955}, {49, 1657}, {54, 376}, {64, 12164}, {84, 7193}, {110, 3146}, {155, 6000}, {185, 1993}, {235, 11064}, {323, 12086}, {378, 5562}, {382, 10539}, {394, 1593}, {517, 4347}, {524, 6696}, {542, 2892}, {1038, 11429}, {1154, 7689}, {1204, 2071}, {1216, 7526}, {1352, 3088}, {1368, 12241}, {1437, 7580}, {1498, 3167}, {1595, 3818}, {1614, 3529}, {1853, 12429}, {1899, 10112}, {1935, 6056}, {1936, 7335}, {1941, 2052}, {1968, 3289}, {1994, 10574}, {2777, 5504}, {3068, 9686}, {3091, 5651}, {3292, 11381}, {3357, 9938}, {3520, 11412}, {3522, 5012}, {3542, 5972}, {3564, 6247}, {3819, 7395}, {3917, 7503}, {4296, 9637}, {4550, 11591}, {5059, 9544}, {5073, 10540}, {5446, 6644}, {5447, 7514}, {5480, 9825}, {5943, 10982}, {5965, 11411}, {6090, 11403}, {6241, 7464}, {6642, 10110}, {7387, 10282}, {7527, 11444}, {8548, 10250}, {8549, 8681}, {8717, 12103}, {8718, 11001}, {9707, 12082}, {9818, 11793}, {10605, 12160}, {10628, 12302}, {10996, 11427}, {11464, 12088}, {12123, 12229}, {12124, 12230}

X(13346) = midpoint of X(i) and X(j) for these {i,j}: {64,12164}, {155,12085}, {32614, 32615}
X(13346) = reflection of X(i) in X(j) for these (i,j)}: (26,12038), (3357,12084), (6759,1147), (7387,10282), (7689,11250)
X(13346) = inverse-in-Brocard-circle of X(9729)
X(13346) = X(20)-of-Trinh-triangle
X(13346) = inverse-in-circle-{X(371),X(372),PU(1),PU(39)} of X(800)
X(13346) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,9729), (3,52,11438), (3,182,13347), (3,578,182), (3,1351,9786), (4,1092,9306), (26,12038,11202), (52,10564,3), (54,376,10984), (323,12086,12111), (371,372,800), (394,1593,5907), (1660,2883,6759), (1993,11413,185), (2071,5889,1204), (3292,11381,11441), (7689,11250,11204)

X(13347) = X(3)X(6)∩X(20)X(9815)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = Cot[w]+2 Csc[A] Csc[B] Csc[C]
Barycentrics a^2 (a^8-3 a^6 b^2+3 a^4 b^4-a^2 b^6-3 a^6 c^2-8 a^4 b^2 c^2+13 a^2 b^4 c^2-2 b^6 c^2+3 a^4 c^4+13 a^2 b^2 c^4+4 b^4 c^4-a^2 c^6-2 b^2 c^6) : :
X(13347) = X(3) + ((cot ω)/p)X(6)

X(13347) lies on these lines:
{3,6}, {20,9815}, {54,10299}, {140,6247}, {156,12108}, {184,3523}, {185,7485}, {206,6696}, {546,8717}, {550,11745}, {631,1614} et al

X(13347) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,182,13346), (3,389,3098), (3,12017,11425), (631,10984,9306)

X(13348) = X(3)X(6)∩X(20)X(3917)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = -Cot[w]-2 Csc[A] Csc[B] Csc[C]
Barycentrics a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2-12 a^4 b^2 c^2+9 a^2 b^4 c^2+2 b^6 c^2-3 a^4 c^4+9 a^2 b^2 c^4-2 b^4 c^4+3 a^2 c^6+2 b^2 c^6-c^8) : :
X(13348) = X(3) + ((cot ω)/p)X(6)

X(13348) lies on these lines:
{3, 6}, {4, 3819}, {20, 3917}, {30, 5447}, {40, 3784}, {51, 3523}, {84, 3781}, {140, 6688}, {143, 12100}, {185, 2979}, {373, 10303}, {376, 5562}, {517, 4298}, {548, 10627}, {549, 5446}, {550, 1216}, {631, 5943}, {632, 12045}, {1092, 9707}, {1657, 5891}, {2393, 6696}, {2807, 12512}, {3088, 11387}, {3091, 5650}, {3146, 7998}, {3526, 10219}, {3528, 11412}, {3529, 7999}, {3530, 5462}, {3534, 12162}, {3567, 10299}, {3627, 10170}, {3628, 11592}, {5889, 10304}, {5892, 10263}, {6101, 8703}, {6759, 9914}, {6916, 10441}, {7485, 11424}, {7492, 11449}, {7525, 12038}, {9052, 12675}, {9306, 11414}, {9940, 12109}, {10095, 12108}, {10295, 12300}, {10691, 12241}, {11591, 12103}, {11645, 12134}

X(13348) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,578,5092), (3,10625,389), (20,3917,5907), (20,11444,11381), (140,10110,6688), (549,5446,11695), (2979,3522,185), (3917,11381,11444), (11381,11444,5907)

X(13349) = MIDPOINT OF X(3) AND X(16)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = -2 Sqrt[3]+Cot[w]
Barycentrics a^2 (a^4-a^2 b^2-a^2 c^2-2 b^2 c^2+Sqrt[3] Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)] (-a^2+b^2+c^2)) : :
X(13349) = X(3) + ((cot ω)/p)X(6) = X[622] - 5 X[631] = 3 X[16] - X[5615] = 3 X[3] + X[5615]

X(13349) lies on these lines:
{3,6}, {5,6672}, {30,5460}, {110,11145}, {140,624}, {397,10616}, {517,11708}, {530,549}, {622,631}, {2381,9203}, {3131,5943}, {3292,11130}, {5617,11300}, {5650,11131}, {6036,6108}

X(13349) = midpoint of X(3) and X(16)
X(13349) = reflection of X(i) in X(j) for these (i,j): (5, 6672), (624, 140)
X(13349) = inverse-in-circumcircle of X(5611)
X(13349) = inverse-in-Brocard-circle of X(9735)
X(13349) = vertex conjugate of X(512) and X(5611)
X(13349) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,9735), (3,182,13350), (3,5050,11480), (3,11481,9736), (62,5611,5097), (1379,1380,5611)

X(13350) = MIDPOINT OF X(3) AND X(15)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = 2 Sqrt[3]+Cot[w]
Barycentrics a^2 (a^4-a^2 b^2-a^2 c^2-2 b^2 c^2-Sqrt[3] Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)] (-a^2+b^2+c^2)) : :
X(13350) = X(3) + ((cot ω)/p)X(6) = X[621] - 5 X[631] = 3 X[15] - X[5611] = 3 X[3] + X[5611]

X(13350) lies on these lines:
{3,6}, {5,6671}, {30,5459}, {110,11146}, {140,623}, {398,10617}, {517,11707}, {531,549}, {621,631}, {2380,9202}, {3132,5943}, {3292,11131}, {5613,11299}, {5650,11130}, {6036,6109}

X(13350) = midpoint of X(3) and X(15)
X(13350) = reflection of X(i) in X(j) for these (i,j): (5, 6671), (623, 140)
X(13350) = inverse-in-circumcircle of X(5615)
X(13350) = inverse-in-Brocard-circle of X(9736)
X(13350) = vertex conjugate of X(512) and X(5615)
X(13350) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,9736), (3,182,13349), (3,5050,11481), (3,11480,9735), (61,5615,5097), (1379,1380,5615)

X(13351) = CROSSSUM OF X(6) AND X(1656)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = -(4/(-4 Cot[w]+Csc[A] Csc[B] Csc[C]))
Barycentrics a^2 (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2-3 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6) : :
X(13351) = X(3) + ((cot ω)/p)X(6)

In the plane of a triangle ABC, let
DEF = circum-orthic triangle;
Ab = center of circle (BFO), and define Bc and Ca cyclically;
Ac = center of circle (CEO), and define Ba and Cb cyclically;
A' = BAc∩CAb, and define B' and C' cyclically;
T = the affine transformation that carries ABC onto A'B'C'.
Then X(1335 1) = the finite fixed point of T. (Angel Montesdeoca, March 4, 2024)

X(13351) lies on these lines:
{2,1225}, {3,6}, {53,9606}, {141,1238}, {160,9971}, {230,1180}, {232,5064}, {233,7765}, {378,8746}, {1506,1879}, {2549,11818}, {3054,9465}, {3087,11062}, {3815,5133}, {6748,7576}, {7391,7736}, {7544,7738}, {8253,8962}

X(13351) = inverse-in-Brocard-circle of X(2965)
X(13351) = crosssum of X(6) and X(1656)
X(13351) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,2965), (6,570,566), (39,216,5421), (39,570,6), (216,5421,6), (570,5421,216), (1609,9605,6)

X(13352) = X(3)X(6)∩X(4)X(110)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = Cot[w]-3/2 Csc[A] Csc[B] Csc[C]
Barycentrics a^2 (a^8-3 a^6 b^2+3 a^4 b^4-a^2 b^6-3 a^6 c^2+6 a^4 b^2 c^2-a^2 b^4 c^2-2 b^6 c^2+3 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(13352) = X(3) + ((cot ω)/p)X(6)

X(13352) lies on these lines:
{3,6}, {4,110}, {5,1092}, {20,54}, {23,11464}, {24,5446}, {30,184}, {49,382}, {51,6644}, {68,3541}, {74,11004}, {143,9826}, {155,1593}, {156,1514}, {185,12084}, {186,3060}, {195,2935}, {215,12943}, {235,9820}, {249,6785}, {323,4550}, {376,5012}, {378,1993}, {381,9306}, {394,5891}, {539,11442}, {550,10984}, {858,12022}, {974,1204}, {1060,11429}, {1181,10575}, {1199,10574}, {1216,7503}, {1437,6985}, {1495,7530}, {1511,12106}, {1594,9927}, {1595,12134}, {1597,3167}, {1614,3146}, {1658,10263}, {1870,9637}, {1986,12901}, {1992,5622}, {1994,2071}, {2070,11202}, {2477,12953}, {2914,12270}, {3044,10722}, {3045,10728}, {3046,10727}, {3047,10721}, {3066,6642}, {3088,5921}, {3093,8909}, {3431,7556}, {3516,12160}, {3518,11449}, {3520,5889}, {3543,9544}, {3830,9703}, {3917,7514}, {5059,8718}, {5073,9704}, {5422,5892}, {5447,7509}, {5562,7526}, {6241,12086}, {6284,9653}, {6564,9676}, {6640,6723}, {6689,7558}, {6800,12082}, {7354,9666}, {7464,11422}, {7506,10110}, {7507,12293}, {7517,10282}, {7525,10610}, {7550,7998}, {7592,11413}, {7752,10411}, {7760,12192}, {8981,9686}, {9541,9687}, {10116,11457}, {10224,11801}, {10661,11476}, {10662,11475}, {10665,11474}, {10666,11473}, {11562,12302}, {11585,12241}, {12111,12364}

X(13352) = midpoint of X(378) and X(1993)
X(13352) = reflection of X(3) in X(11430)
X(13352) = inverse-in-Brocard-circle of X(9730)
X(13352) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(3003)
X(13352) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,9730), (3,567,182), (3,568,11438), (3,578,569), (4,1147,10539), (4,5654,113), (49,382,6759), (155,1593,12162), (182,567,569), (182,578,567), (323,7527,11459), (371,372,3003), (394,9818,5891), (576,11438,568), (1092,11424,5), (1181,12085,10575), (1994,2071,5890), (2935,7729,3357), (3146,9545,1614), (3516,12160,12163), (3520,5889,7689), (3830,9703,10540), (5446,12038,24), (5448,12897,4), (5504,5654,1147), (6102,11250,1204), (7527,11459,4550), (9730,10564,3), (12084,12161,185)

X(13353) = X(2)X(49)∩X(3)X(6)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = Cot[w]-1/4 Csc[A] Csc[B] Csc[C]
Barycentrics a^2 (a^8-3 a^6 b^2+3 a^4 b^4-a^2 b^6-3 a^6 c^2+a^4 b^2 c^2+4 a^2 b^4 c^2-2 b^6 c^2+3 a^4 c^4+4 a^2 b^2 c^4+4 b^4 c^4-a^2 c^6-2 b^2 c^6) : :
X(13353) = X(3) + ((cot ω)/p)X(6)

X(13353) lies on these lines:
{2,49}, {3,6}, {5,1614}, {17,11134}, {18,11137}, {23,10095}, {26,5422}, {51,2937}, {54,140}, {110,3628}, {125,6689}, {143,7512}, {156,3090}, {184,1656}, {186,6152}, {195,1216}, {323,1493}, {381,11572}, {382,10984}, {597,7540}, {1092,5054}, {1147,3526}, {1154,1199}, {1176,7553}, {1511,10821}, {1657,11424}, {1993,7516}, {1994,6101}, {2070,2918}, {2888,11264}, {3060,7525}, {3525,9545}, {3567,7502}, {3574,7574}, {3580,7568}, {3589,12134}, {3618,7528}, {3796,7517}, {3851,6759}, {3853,8718}, {5055,10539}, {5067,9544}, {5070,9306}, {5899,10110}, {5946,7488}, {6146,6288}, {6636,10263}, {7393,11402}, {7506,10601}, {7509,12161}, {7514,7592}, {7550,11591}, {7746,9604}, {7999,11422}, {9818,12174}, {10277,12026}, {10574,11468}, {10982,12083}, {11423,11444}

X(13353) = inverse-in-circumcircle of X(11811)
X(13353) = inverse-in-Brocard-circle of X(6243)
X(13353) = vertex conjugate of X(512) and X(11811)
X(13353) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6243), (3,389,3581), (3,569,567), (182,569,3), (1379,1380,11811), (3090,11003,156), (5070,9704,9306), (5092,10625,3), (10610,12006,186), (11426,12017,3)

X(13354) = INVERSE-IN-MOSES-CIRCLE OF X(2022)

Trilinears    sin ω cos(A + ω) - cot ω cos A : :
Barycentrics    (sin A)[cos(A - arccot(p))] : : , where p = Cot[w]^3
Barycentrics    a^2 (3 a^6 b^2-3 a^4 b^4+a^2 b^6-b^8+3 a^6 c^2-2 a^4 b^2 c^2-5 a^2 b^4 c^2-3 a^4 c^4-5 a^2 b^2 c^4-6 b^4 c^4+a^2 c^6-c^8) : :
X(13354) = X(3) + ((cot ω)/p)X(6) = X[3095] - 3 X[5050] = X[3094] - 3 X[5085] = 3 X[11171] - 5 X[12017]

X(13354) lies on these lines:
{3,6}, {51,7467}, {69,9744}, {76,6776}, {114,141}, {193,6194}, {262,3618}, {538,11179}, {542,9466}, {732,8550}, {1352,3934}, {1503,6248}, {5149,12177}, {7763,10519}, {10352,10753}

X(13354) = midpoint of X(i) and X(j) for these {i,j}: {76, 6776}, {1351, 9821}, {5052, 5188}
X(13354) = reflection of X(i) in X(j) for these (i,j): (39, 182), (1352, 3934)
X(13354) = inverse-in-Moses-circle of X(2022)
X(13354) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2024,3094,39), (2028,2029,2022), (3098,5092,8589)

X(13355) = X(3)X(6)∩X(69)X(98)

Trilinears    2 cos(A - ω) [b sin(B - ω) + c sin(C - ω)] + sin(A - ω) [a cos(A - ω) - b cos(B - ω) - c cos(C - ω)] : :
Barycentrics    (sin A)[cos(A - arccot(p))] : : , where p = -Cot[w]^3
Barycentrics    a^2 (a^8-a^6 b^2+3 a^4 b^4-3 a^2 b^6-a^6 c^2+2 a^4 b^2 c^2-a^2 b^4 c^2-4 b^6 c^2+3 a^4 c^4-a^2 b^2 c^4-3 a^2 c^6-4 b^2 c^6) : :
X(13355) = X(3) + ((cot ω)/p)X(6) = 5 X[3618] - 3 X[9753] = X[5017]-3 X[5085]

X(13355) lies on these lines:
{3,6}, {20,10350}, {69,98}, {184,7467}, {315,6776}, {376,5182}, {542,7818}, {626,1352}, {754,11179}, {895,12192}, {1078,10519}, {2794,12177}, {3618,9753}, {3751,12197}, {4027,10753}, {5480,10358}, {10754,12176}, {10755,12199}, {10766,12207}, {11257,12215}, {12216,12252}

X(13355) = midpoint of X(315) and X(6776)
X(13355) = reflection of X(i) in X(j) for these (i,j): (32, 182), (1352, 626)
X(13355) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(13357)
X(13355) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,2456,182), (182,5039,3398), (182,5171,1691), (371,372,13357), (1350,1691,5171), (1350,5085,5023), (1351,3398,5039), (2458,5028,32), (3098,5092,8588), (5038,5085,182), (5050,12054,182)

X(13356) = X(3)X(6)∩X(98)X(5286)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = Tan[w]^3
Barycentrics a^2 (a^6+3 a^2 b^4+4 a^2 b^2 c^2+2 b^4 c^2+3 a^2 c^4+2 b^2 c^4) : :
X(13356) = X(3) + ((cot ω)/p)X(6)

X(13356) lies on these lines:
{3,6}, {83,7736}, {98,5286}, {114,2548}, {230,7815}, {1078,7735}, {1186,3289}, {1194,3148}, {2909,3203}, {3117,9306}, {3407,7783}, {3734,8149}, {3788,3815}, {5280,10802}, {5299,10801}, {5304,7793}, {5305,10104}, {5306,8359}, {7738,12203}, {7774,10350}, {7787,10352}, {7796,10347}, {7836,10345}, {7906,10333}, {8369,9300}, {9575,12194}, {9593,12197}, {9744,12110}, {10334,10346}

X(13356) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,13357), (32,39,182), (32,5034,3398), (32,8722,3053), (3053,12212,32), (3398,9605,5034), (5039,5171,32), (12048,12049,6)

X(13357) = MIDPOINT OF X(32) AND X(39)

Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = -Tan[w]^3
Barycentrics a^2 (3 a^4 b^2+b^6+3 a^4 c^2+2 a^2 b^2 c^2+b^4 c^2+b^2 c^4+c^6) : :
X(13357) = X(3) + ((cot ω)/p)X(6) = X[315] - 5 X[7786] = 3 X[9753] + X[11257]

X(13357) lies on these lines:
{3,6}, {76,7735}, {194,5304}, {230,736}, {237,1194}, {315,7736}, {538,5306}, {626,3815}, {732,7789}, {754,8359}, {1196,11328}, {1569,5368}, {2023,2794}, {2782,5305}, {3329,10350}, {3767,6248}, {3819,8623}, {5286,9753}, {7892,9983}, {8569,8570}, {9475,11326}

X(13357) = midpoint of X(i) and X(j) for these {i,j}: {32,39}, {2021,2022}
X(13357) = reflection of X(i) in X(j) for these (i,j): (626,6683), (3934,6680)
X(13357) = crosssum of X(6) and X(7467)
X(13357) = centroid of PU(1)PU(39)
X(13357) = harmonic center of circles O(15,16) (Shoute circle) and O(61,62)
X(13357) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(13355)
X(13357) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,13356), (371,372,13355)

X(13358) = X(4)X(94)∩X(30)X(11800)

Barycentrics a^2 (a^12 b^2-4 a^10 b^4+5 a^8 b^6-5 a^4 b^10+4 a^2 b^12-b^14+a^12 c^2-4 a^10 b^2 c^2+7 a^8 b^4 c^2-11 a^6 b^6 c^2+16 a^4 b^8 c^2-13 a^2 b^10 c^2+4 b^12 c^2-4 a^10 c^4+7 a^8 b^2 c^4+2 a^6 b^4 c^4-8 a^4 b^6 c^4+9 a^2 b^8 c^4-6 b^10 c^4+5 a^8 c^6-11 a^6 b^2 c^6-8 a^4 b^4 c^6+3 b^8 c^6+16 a^4 b^2 c^8+9 a^2 b^4 c^8+3 b^6 c^8-5 a^4 c^10-13 a^2 b^2 c^10-6 b^4 c^10+4 a^2 c^12+4 b^2 c^12-c^14) : :
X(13358) = 3 X[143] - 2 X[1112], 3 X[568] + X[3448], X[399] - 5 X[3567], X[110] - 3 X[5946], 3 X[6102] - X[7722], 3 X[265] + X[7722], X[1112] - 3 X[12236], 3 X[381] + X[12284], 3 X[10113] - X[12292]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25752.

X(13358) lies on these lines:
{4,94}, {30,11800}, {52,10264}, {54,1511}, {74,10263}, {110, 5946}, {113,10095}, {125,1154}, {182,12893}, {381,12284}, {389, 6153}, {399,3567}, {974,11565}, {1199,11597}, {1493,3043}

X(13358) = midpoint of X(i) and X(j) for these {i,j}: {52, 10264}, {74, 10263}, {265, 6102}, {11800, 11806}
X(13358) = reflection of X(i) in X(j) for these (i,j): (113,10095), (143,12236), ( 1511,12006), (10272,5462), ( 10627,6699), (11561,389)

X(13359) = POINT BEID 46

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

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 25755.

X(13359) lies on these lines: {1, 6}, {4662, 7090}, {6212, 9943}

X(13359) = reflection of X(13360) in X(9)

X(13360) = POINT BEID 47

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

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 25755.

X(13360) lies on these lines: {1, 6}, {3812, 7090}, {6213, 9943}

X(13360) = reflection of X(13359) in X(9)

X(13361) = EULER LINE INTERCEPT OF LINE X(1503)X(10219)

Barycentrics 2 a^6+a^4 (b^2+c^2)-2 a^2 (b^4-16 b^2 c^2+c^4) -(b^2-c^2)^2 (b^2+c^2) : :
X(13361) = X(2) + X(10128)

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25759.

X(13361) lies on these lines: {2,3}, {1503,10219}, {3564,6688}

X(13361) = midpoint of X(2) and X(10128)

X(13362) = INTERSECTION OF EULER LINE AND X(137)X(10095)

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

See Tran Quang Hung and César Lozada, Hyacinthos 25762.

X(13362) lies on these lines: {2,3}, {137,10095}

X(13362) =

X(13363) = MIDPOINT OF X(5) AND X(9730)

Barycentrics a^4(b^2 + c^2 - 6 R^2) - a^2[(b^4 + c^4) + 9(b^2 + c^2) R^2] - 3(b^2 - c^2)^2 R^2 : :

See Antreas Hatzipolakis and Randy Hutson, Hyacinthos 25769.

X(13363) lies on these lines:
{2,568}, {3,5640}, {4,7693}, {5,113}, {30,5892}, {51,549}, {52,632}, {140,143}, {381,11451}, {546,9729}, {547,6688}, {548,10110}, {631,10263}

X(13363) = midpoint of X(5) and X(9730)
X(13363) = complement of complement of X(568)
X(13363) = X(10272)-of-orthocentroidal-triangle
X(13363) = nine-point center of triangle formed by the centroids of the three altimedial triangles
X(13363) = centroid of triangle formed by the nine-point centers of the three altimedial triangles

X(13364) = MIDPOINT OF X(5) AND X(51)

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

See Antreas Hatzipolakis and Randy Hutson, Hyacinthos 25769.

X(13364) lies on these lines:
{3,11451}, {4,12006}, {5,51}, {30,5892}, {140,6688}, {381,5640}, {3526,11592}

X(13364) = midpoint of X(i) and X(j) for these ({i,j}: {5,51}, {381,5946}
X(13364) = X(140)-of-orthocentroidal-triangle
X(13364) = X(549)-of-orthic-triangle

X(13365) = POINT BEID 48

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25773.

X(13365) lies on these lines: {5,51}, {195,5640}, {5943,6153} et al

X(13365) = midpoint of X(143) and X(1209)
X(13365) = complement of the complement of X(32196)
X(13365) = X(5)-of-pedal-triangle-of-X(5)

X(13366) = CROSSSUM OF X(2) AND X(5)

Trilinears (sin^2 A)(cos A + 2 sin B sin C) : :
Barycentrics a^2 (2 a^4-3 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4) : :
X(13366) = (3 X[5012] - X[6636] = 3 X[1994] + X[6636]
X(13366) = 6 SW X[6] - R^2 (3+J^2) X[25]

X(13366) lies on these lines: {2,575}, {4,11423}, {5,11264}, {6,25}, {22,576}, {49,5462}, {52,7502}, {54,186}, {61,3131}, {62,3132}, {110,5943}, {125,11245}, {140,1493}, {182,1993}, {185,378}, {195,1216}, {228,2317}, {237,5007}, {288,933}, {323,3819}, {343,1353}, {373,5422}, {394,5050}, {418,3284}, {427,8550}, {511,1994}, {524,7499}, {542,5133}, {569,5562}, {570,8603}, {1181,1597}, {1204,11410}, {1351,3796}, {1370,11179}, {1501,5052}, {1594,12242}, {1614,10110}, {1627,2030}, {1692,3051}, {1899,8889}, {1976,3108}, {1992,7494}, {2003,3937}, {2056,3291}, {2323,3690}, {2979,5092}, {3060,5097}, {3066,8780}, {3148,7772}, {3155,6419}, {3156,6420}, {3167,5544}, {3270,11429}, {3311,10133}, {3312,10132}, {3564,11548}, {3567,10282}, {3574,6146}, {3575,10619}, {3580,11225}, {3611,11428}, {5032,10565}, {5066,5609}, {5111,10329}, {5158,6641}, {5310,8540}, {5446,5899}, {5476,7394}, {5576,10116}, {5640,9544}, {5642,6677}, {5702,6618}, {5890,11430}, {6749,6755}, {6776,7378}, {7571,11178}, {8877,10558}, {9820,12421}, {9909,11482}, {10151,12241}

X(13366) = midpoint of X(1994) and X(5012)
X(13366) = isogonal conjugate of isotomic conjugate of X(140)
X(13366) = isogonal conjugate of polar conjugate of X(6748)
X(13366) = X(933)-Ceva conjugate of X(647)
X(13366) = X(75)-isoconjugate of X(1173)
X(13366) = X(92)-isoconjugate of X(31626)
X(13366) = crosssum of X(i) and X(j) for these (i,j): {2,5}, {3,1994}, {302,303}
X(13366) = crosspoint of X(i) and X(j) for these (i,j): {4,2963}, {6,54}, {140,6748}
X(13366) = polar conjugate of isotomic conjugate of X(22052)
X(13366) = crossdifference of every pair of points on line X(525)X(15340)
X(13366) = intersection of tangents to Moses-Jerabek conic at X(6) and X(185)
X(13366) = barycentric product X(i)*X(j) for these {i,j}: {3,6748}, {6,140}, {32,1232}, {54,233}, {1493,2963}
X(13366) = barycentric quotient X(i)/X(j) for these (i,j): (32,1173), (140,76), (233,311), (1232,1502), (1493,7769), (6748,264)
X(13366) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,154,9777), (6,184,51), (6,11402,184), (51,184,1495), (54,1199,389), (182,1993,3917), (569,12161,5562), (575,11422,3292), (578,7592,185), (1181,11424,11381), (1181,11426,11424), (3167,10601,5651), (3574,6146,11572), (5422,9306,373), (8603,8604,570)

X(13367) = CROSSSUM OF X(4) AND X(5)

Barycentrics a^2 (a^2-b^2-c^2) (2 a^6-3 a^4 b^2+b^6-3 a^4 c^2-b^4 c^2-b^2 c^4+c^6) : :
X(13367) =(1+J^2) X[24] - 3X[51]

X(13367) lies on these lines:
{2,11449}, {3,49}, {4,1495}, {6,3515}, {15,8839}, {16,8837}, {24,51}, {25,11424}, {30,5944}, {35,3270}, {36,1425}, {39,8779}, {52,1658}, {54,186}, {64,11410}, {110,5907}, {125,128}, {143,7575}, {154,1593}, {156,12162}, {182,6467}, {187,217}, {195,3581}, {206,12294}, {232,1970}, {235,10192}, {287,7824}, {323,7691}, {373,6642}, {378,6759}, {468,12241}, {511,7488}, {550,10564}, {567,5462}, {569,6644}, {631,1899}, {1498,3516}, {1531,5448}, {1568,9820}, {1594,11572}, {1614,3520}, {2070,5446}, {2931,12235}, {3043,10628}, {3357,11456}, {3517,10982}, {3518,10110}, {3523,3620}, {3541,9833}, {3549,12118}, {3574,3575}, {3580,10112}, {3611,10902}, {5012,9729}, {5059,7712}, {5068,10546}, {5622,12584}, {5650,7509}, {5651,7395}, {5889,9545}, {6143,12254}, {6457,8908}, {6639,9927}, {6800,11413}, {7464,8718}, {7502,10625}, {7503,9306}, {7526,10539}, {7577,12289}, {7592,11438}, {8542,10541}, {9544,12111}, {9786,11402}, {10018,10182}, {10020,12370}, {10263,12107}, {10533,11474}, {10534,11473}, {10574,11003}, {10575,11250}, {10606,12174}, {11064,12362}, {11577,12006}, {11799,12897}

X(13367) = midpoint of X(i) and X(j) for these {i,j}: {3, 49}, {1614, 3520}
X(13367) = reflection of X(11572) in X(1594)
X(13367) = crosssum of X(i) and X(j) for these (i,j): {3,5}, {4,5}
X(13367) = crosspoint of X(3) and X(54)
X(13367) = barycentric product X(i)*X(j) for these {i,j}: {97,3574}, {394,3575}, {3917,10548}
X(13367) = barycentric quotient X(i)/X(j) for these {(i,j): (3574,324), (3575,2052)
X(13367) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,184,185), (3,1092,3917), (3,1147,5562), (3,1181,1204), (4,10282,1495), (4,11464,10282), (24,578,51), (25,11425,11424), (54,186,389), (125,10619,6146), (140,6146,125), (184,1204,1181), (378,6759,11381), (378,9707,6759), (578,11202,24), (1147,5562,3292), (1181,1204,185), (1511,10610,140), (3431,11464,11430), (3541,9833,11550), (9545,10298,5889), (9820,12605,1568), (10282,11430,4), (11430,11464,1495)

X(13368) = POINT BEID 49

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25773.

X(13368) lies on these lines: {4, 93}, {5, 6153}, {54, 5946}, {110, 143}, {182, 9977}, {973, 1493}, {1209, 10224}, {1511, 6746}, {2917, 5944}, {2937, 10203}, {3060, 12316}, {5448, 11808}, {6102, 11562}, {6640, 12363}, {10255, 12606}, {12006, 12291}

X(13368) = X(4)-of-reflection-triangle-of-X(5)

X(13369) = MIDPOINT OF X(3) AND X(1071)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25778.

X(13369) lies on these lines:
{1,1406}, {3,63}, {4,10202}, {5,142}, {7,6851}, {30,553}, {36,1858}, {40,4880}, {57,6985,10399}, {65,4299}, {84,3560}

X(13369) = midpoint of X(i) and X(j) for these {i,j}: {3,1071}, {355,12680}, {3555, 12702}, {4297,5884}, {5787, 12671}, {9943,12675}, {10202, 11220}
X(13369) = reflection of X(i) in X(j) for these (i,j): (5,9940), (3627,5806), (5777, 140), (7686,5885), (9856,5901)

X(13370) = POINT BEID 50

Barycentrics a^2 (a^5-a^4 (b+c)+a^3 (-2 b^2+7 b c-2 c^2)+a (b-c)^2 (b^2-5 b c+c^2)-(b-c)^2 (b^3+c^3)+a^2 (2 b^3-b^2 c-b c^2+2 c^3)) : :
X(13370) = (r - 4 R) R X(1) + 2 r (2 R - r) X(3)

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 25789.

X(13370) lies on these lines: {1,3} et al

X(13370) =

X(13371) = INTERSECTION OF EULER LINE AND X(52)X(125)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25817.

Let A'B'C' be the complement of the tangential triangle, as at X(11585). Then X(13371) = X(3)-of-A'B'C'. (Randy Hutson, July 21, 2017)

Let A' be the reflection in BC of the A-vertex of the tangential triangle. Let Oa be the circumcenter of AB'C', and define Ob and Oc cyclically. Let Oa' be the circumcenter of A'BC, and define Ob' and Oc' cyclically. The lines OaOa', ObOb', OcOc' concur in X(13371). (Randy Hutson, July 21, 2017)

X(13371) lies on these lines:
{2, 3}, {11, 8144}, {50, 9722}, {52, 125}, {70, 1993}, {113, 11381}, {141, 12061}, {155, 1853}, {156, 1503}, {343, 6101}, {496, 9630}, {511, 5449}, {524, 11255}, {590, 11265}, {615, 11266}, {1154, 12359}, {1209, 3917}, {1236, 3933}, {1568, 12162}, {1899, 12161}, {3574, 9730}, {3580, 6243}, {3925, 8141}, {5448, 6000}, {5480, 10095}, {5504, 6145}, {5663, 6247}, {7703, 11444}, {10539, 11550}, {11064, 12134}

X(13371) = midpoint of X(4) and X(12084)
X(13371) = reflection of X(i) in X(j) for these (i,j): (5,10224), (26,10020), (156,9820), (550,10226), (1658,140), (12107,10125)
X(13371) = complement of X(26)
X(13371) = anticomplement of X(10020)
X(13371) = complementary conjugate of X(34116)
X(13371) = center of inverse-in-first-Droz-Farny-circle-of-circumcircle
X(13371) = inverse-in-first-Droz-Farny-circle of X(186)
X(13371) = X(5)-of-AAOA-triangle

X(13372) = MIDPOINT OF X(3) AND X(128)

Barycentrics 2 a^12-8 a^10 b^2+14 a^8 b^4-15 a^6 b^6+11 a^4 b^8-5 a^2 b^10+b^12-8 a^10 c^2+16 a^8 b^2 c^2-9 a^6 b^4 c^2-4 a^4 b^6 c^2+8 a^2 b^8 c^2-3 b^10 c^2+14 a^8 c^4-9 a^6 b^2 c^4+4 a^4 b^4 c^4-3 a^2 b^6 c^4+3 b^8 c^4-15 a^6 c^6-4 a^4 b^2 c^6-3 a^2 b^4 c^6-2 b^6 c^6+11 a^4 c^8+8 a^2 b^2 c^8+3 b^4 c^8-5 a^2 c^10-3 b^2 c^10+c^12 : :
X(13372) = 3 X[2] + X[930] = 5 X[631] - X[1141] = 5 X[632] - X[1263] = 9 X[2] - X[11671] = 3 X[137] - X[11671] = 3 X[930] + X[11671] = 3 X[140] - X[12026] = 3 X[6592] + X[12026]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25824.

X(13372) lies on the nine-point circle of the medial triangle, and on these lines:
{2,137}, {3,128}, {140,6592}, {631,1141}, {632,1263}

X(13372) = midpoint of X(i) and X(j) for these {i,j}: {3, 128}, {137, 930}, {140, 6592}
X(13372) = complement of X(137)
X(13372) = {X(2),X(930)}-harmonic conjugate of X(137)

X(13373) = MIDPOINT OF X(5) AND X(12675)

Barycentrics a (a^5 (b+c)-a^4 (b^2-4 b c+c^2)-2 a^3 (b^3+b^2 c+b c^2+c^3)+2 a^2 (b^4-3 b^3 c+2 b^2 c^2-3 b c^3+c^4)+a (b-c)^2 (b+c)^3-(b-c)^4 (b+c)^2) : :
X(13373) = (r+3 R) X(1) + (R-r) X(3)

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25830.

X(13373) lies on these lines:
{1,3}, {5,3742}, {72,6878}, {119,9947}, {140,518}, {143,9037}, {210,3526}, {355,5439}, {381,12680}, {496,10391}, {551,5884}, {575,9004} et al

X(13373) = midpoint of X(i) and X(j) for these {i,j}: {5,12675}, {942,1385}, {1125,12005}, {1483,5836}, {3881,6684}, {5045,9940}, {5083,6713}
X(13373) = excentral-to-incircle-circles similarity image of X(11249)

X(13374) = MIDPOINT OF X(1) AND X(7686)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25831.

X(13374) lies on these lines:
{1,227}, {3,3742}, {4,354}, {5,518}, {7,10309}, {10,12864}, {11,12691}, {40,5439}, {57,11496}, {65,3086}, {72,5231,8227}, {84,10980}, {140,517}, {210,3090}, {226,7681}, {388,5804}, {392,9624}, {405,12704}, {496,942}, {497,12710}, et al

X(13374) = midpoint of X(i) and X(j) for these {i,j}: {1,7686}, {4,12675}, {942,946}, {1482,5836}, {3874,5777}, {5045,5806}, {5173,7680}, {5572,5805}, {5884,9856}, {6583,9955}, {7682,12915}, {9943,12699}

X(13375) = MIDPOINT OF X(5559) AND X(5903)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25833.

X(13375) lies on these lines: {1, 1389}, {7, 5559}, {8, 10941}, {65, 5844}, {145, 3881}, {388, 10052}, {517, 3649}, {758, 12913}, {942, 1317}, {999, 10094}, {1056, 10044}, {1537, 12047}, {1737, 10957}, {2800, 5270}, {3057, 5719}, {3754, 4861}, {5761, 10056}, {10106, 11570}

X(13375) = midpoint of X(5559) and X(5903)

X(13376) = MIDPOINT OF X(5) AND X(11692)

Barycentrics a^2 (a^12 b^2-4 a^10 b^4+5 a^8 b^6-5 a^4 b^10+4 a^2 b^12-b^14+a^12 c^2-6 a^10 b^2 c^2+6 a^8 b^4 c^2+a^6 b^6 c^2+6 a^4 b^8 c^2-15 a^2 b^10 c^2+7 b^12 c^2-4 a^10 c^4+6 a^8 b^2 c^4-4 a^6 b^4 c^4-5 a^4 b^6 c^4+22 a^2 b^8 c^4-15 b^10 c^4+5 a^8 c^6+a^6 b^2 c^6-5 a^4 b^4 c^6-22 a^2 b^6 c^6+9 b^8 c^6+6 a^4 b^2 c^8+22 a^2 b^4 c^8+9 b^6 c^8-5 a^4 c^10-15 a^2 b^2 c^10-15 b^4 c^10+4 a^2 c^12+7 b^2 c^12-c^14) : :
X(13376) = 3 X[51] + X[3153] = X[186] - 3 X[5943]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25843.

X(13376) lies on these lines:
{5,6153}, {30,5462}, {51,3153}, {182,5899}, {186,5943}, {511, 2072} et al

X(13376) = midpoint of X(i) and X(j) for these {i,j}: {5,11692}, {1568,11800}

X(13377) = REFLECTION OF X(6322) IN X(2)

Barycentrics 4 a^8+10 a^6 (b^2+c^2)+33 a^4 b^2 c^2+a^2 (4 b^6+6 b^4 c^2+6 b^2 c^4+4 c^6)-4 (2 b^8+2 b^6 c^2-9 b^4 c^4+2 b^2 c^6+2 c^8) : :
X(13377) = 2 X(2) - X(6322)

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25862.

X(13377) lies on these lines:
{2,6322}, {381,8704}, {542,6232}, {599,3734}, {5094,10162}, {6032,11163}, {7840,9464}, {9829,10130}

X(13377) = reflection of X(6322) in X(2)
X(13377) = isogonal conjugate of X(353)

X(13378) = POINT BEID 51

Barycentrics 4 a^6+9 a^4 (b^2+c^2)+a^2 (9 b^4+6 b^2 c^2+9 c^4)-2 (7 b^6-9 b^4 c^2-9 b^2 c^4+7 c^6) : :
X(13378) = X(10166) - 2 X(10173)

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25862.

X(13378) lies on these lines:
{2,1495}, {125,3363}, {9830,10162}, {10166,10173}

X(13379) = POINT BEID 52

Barycentrics a^4 (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) (a^12 b^4-6 a^10 b^6+15 a^8 b^8-20 a^6 b^10+15 a^4 b^12-6 a^2 b^14+b^16-a^10 b^4 c^2+10 a^6 b^8 c^2-20 a^4 b^10 c^2+15 a^2 b^12 c^2-4 b^14 c^2+a^12 c^4-a^10 b^2 c^4-2 a^8 b^4 c^4+9 a^4 b^8 c^4-11 a^2 b^10 c^4+4 b^12 c^4-6 a^10 c^6-4 a^4 b^6 c^6+2 a^2 b^8 c^6+4 b^10 c^6+15 a^8 c^8+10 a^6 b^2 c^8+9 a^4 b^4 c^8+2 a^2 b^6 c^8-10 b^8 c^8-20 a^6 c^10-20 a^4 b^2 c^10-11 a^2 b^4 c^10+4 b^6 c^10+15 a^4 c^12+15 a^2 b^2 c^12+4 b^4 c^12-6 a^2 c^14-4 b^2 c^14+c^16) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25870.

X(13379) lies on the line {185,1986}

X(13380) = X(3)X(801)∩X(4)X(800)

Barycentrics 1/(a^8-2 b^2 c^2 (b^2-c^2)^2-3 a^6 (b^2+c^2)-a^2 (b^2+c^2)^3+a^4 (3 b^4+8 b^2 c^2+3 c^4)) : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25871.

X(13380) lies on these lines: {2, 185}, {3, 801}, {4, 800}, {76, 6823}, {83, 11479}, {96, 11456}, {98, 1498} et al

X(13381) = POINT BEID 53

Barycentrics 1/(a^8-3 a^6 (b^2+c^2)+3 a^4 (b^4+3 b^2 c^2+c^4)-a^2 (b^2+c^2)^3-3 b^2 c^2 (b^2-c^2)^2) : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25871.

X(13381) lies on these lines: {140,185}, {800,6748} et al

X(13382) = POINT BEID 54

Barycentrics a^2 (3 a^6 (b^2+c^2)+a^4 (-9 b^4+4 b^2 c^2-9 c^4)+9 a^2 (b^2-c^2)^2 (b^2+c^2)-(b^2-c^2)^2 (3 b^4+4 b^2 c^2+3 c^4)) : :
X(13382) = X(185) + X(389)

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25871.

X(13382) lies on these lines: {4,51}, {6,3357}, {30,11565}, {52,1657}, {64,11432}, {74,1199}, {140,9729}, {182,12163}, {511,550} et al

X(13382) = midpoint of X(185) and X(389)
X(13382) = reflection of X(i) in X(j) for these {i,j}: (5907,11695), (10110,389), (11793,9729)

X(13383) = MIDPOINT OF X(5) AND X(26)

Barycentrics 2 a^10-5 a^8 b^2+2 a^6 b^4+4 a^4 b^6-4 a^2 b^8+b^10-5 a^8 c^2+8 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4-8 a^2 b^4 c^4+2 b^6 c^4+4 a^4 c^6+8 a^2 b^2 c^6+2 b^4 c^6-4 a^2 c^8-3 b^2 c^8+c^10 : :
X(13383) = 3 (3 J^2 - 7) X[2] + (13 - J^2) X[4], where J = |OH|/R

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25879.

X(13383) lies on these lines: {2,3}, {68,154}, {156,206}, {343,10539}, {498,9645}, {511, 9820} et al

X(13383) = midpoint of X(i) and X(j) for these {i,j}: {5,26}, {2883,7689}, {6759, 12359}, {10154,10201}
X(13383) = reflection of X(i) in X(j) for these (i,j)}: (140,10020), (11250,3530)
X(13383) = anticomplement of X(32144)

X(13384) = POINT BEID

Barycentrics a (a-b-c) (5 a^2+2 a b-3 b^2+2 a c+6 b c-3 c^2) : :
X(13384) = 3 X[5219] - 2 X[10590] = (3/2 + R/r) X[1] + X[3]

See Tran Quang Hung and Peter Moses, Hyacinthos 25881.

X(13384) lies on these lines: {1,3}, {2,5727}, {9,2320}, {78, 3897}, {200,4711}, {226,5731}, {390,6173}, {392,10391}, {495, 3655}, {497,551}, {498,5881}, {515,5219}, {519,5218}, {944, 6956}, {946,4305}, {950,3616}, {991,1457}, {993,3929}, {997, 7308}, {1125,3486}, {1149,2293}, {1317,9952}, {1419,1455}, {1479, 9624}, {1698,5326}, {1706,4855}, {1837,3624}, {2136,4861}, {2268, 3247}, {2975,11523}, {3085,5882} ,{3158,3872}, {3241,5281}, {3306,4881}, {3475,4315}, {3485, 4297}, {3487,4311}, {3522,4323}, {3523,4848}, {3524,11041}, {3577,6905}, {3586,5886}, {3622, 4313}, {3636,4314}, {3679,5432}, {3680,3871}, {3689,4915}, {4189, 11682}, {4293,4654}, {4304,5603} ,{4342,10385}, {4512,5289}, {4845,11712}, {4853,8168}, {4870, 12943}, {5270,10953}, {5284, 5436}, {5332,9575}, {5424,7284}, {5440,9623}, {5587,6859}, {5691, 11375}, {5703,10106}, {5901, 9614}, {6284,11522}, {6738,7288} ,{6867,8227}, {6906,7971}, {7675,10384}, {8167,8583}, {9613, 11374}, {10527,12625}, {10543, 11376}, {10595,10624}

X(13384) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3,3340), (1,35,7982), (1,36, 11529), (1,55,7962), (1,56,11518), (1,165,2099), (1,1385, 1420), (1,2646,3601), (1,3576, 57), (1,3601,1697), (1,3612,40), (1,7987,65), (1,7991,11011), (1, 9819,5048), (3,3340,5128), (36, 11529,57), (55,5048,9819), (55, 7962,1697), (1125,3486,9581), ( 3485,4297,9579), (3576,11529, 36), (3601,7962,55), (3622,4313, 12053), (4304,5603,9580), (5048, 9819,7962), (5217,11011,7991)

X(13385) = POINT BEID 56

Barycentrics a ((b-c) (a+b-c) (a-b+c) (a^2-b^2-4 b c-c^2)+2 b (b-c) c (5 a^2-2 a b-3 b^2-2 a c-10 b c-3 c^2) Sin[A/2]+2 c (a-b+c) (2 a^3+6 a^2 b+3 a b^2+b^3-9 a b c-2 a c^2-b c^2) Sin[B/2]-2 b (a+b-c) (2 a^3-2 a b^2+6 a^2 c-9 a b c-b^2 c+3 a c^2+c^3) Sin[C/2]) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25925.

X(13385) lies on these lines: {1,167}, {517,12814}, {3616,7057}, {5571,10503}, {6728,10492}, {11033,11192}

X(13385) = X(10215)-Ceva conjugate of X(177)
X(13385) = incircle-inverse of X(177)
X(13385) = Conway-circle-inverse of X(12554)
X(13385) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,2089,177), (1,8091,8422), (1, 8241,11234), (1,11044,11191)

X(13386) = X(2)X(175)∩X(4)X(8)

Barycentrics sec A + tan A : :
Barycentrics (b c + S) SB SC : :
X(13386) = 2 r X[4] - (r + 2 R - s) X[8]

The appearance of (i,j) in the following list means that X(j) = {X(13386),X(13387)}-harmonic conjugate of X(i): (4,92), (8,329), (321,3436), (3869,5739).

See Tran Quang Hung and Peter Moses, AdvGeom 3776.

X(13386) lies on the cubics K170 and K200 and these lines: {2,175}, {4,8}, {63,488}, {77,3083}, {81,1124}, {242,5200}, {278,1585}, {281,1586}, {1267,1444}

X(13386) = isogonal conjugate of X(34121)
X(13386) = isotomic conjugate of X(13387)
X(13386) = X(1124)-cross conjugate of X(1267)
X(13386) = anticomplement of X(13388)
X(13386) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (6,175), (2066,20), (6502,347)
X(13386) = polar conjugate of X(1123)
X(13386) = isoconjugate of X(j) and X(j) for these (i,j): {4,606}, {6,6213}, {19,1335}, {25,3084}, {48,1123}, {1459,6135}, {1973,5391}, {2067,7133}, {2362,5414}, {6365,8750}
X(13386) = barycentric product X(i)*X(j) for these {i,j}: {4,1267}, {69,1336}, {75,6212}, {92,3083}, {264,1124}, {605,1969}, {6335,6364}
X(13386) = barycentric quotient X(i)/X(j) for these (i,j): (1,6213), (3,1335), (4,1123), (48,606), (63,3084), (69,5391), (605,48), (905,6365), (1124,3), (1267,69), (1336,4), (1783,6135), (1806,1805), (2066,5414), (3083,63), (6136,1783), (6212,1), (6364,905), (6502,2067)

X(13387) = X(2)X(176)∩X(4)X(8)

Barycentrics sec A - tan A : :
Barycentrics (b c - S) SB SC : :
X(13387) = 2 r X[4] - (r + 2 R + s) X[8]

The appearance of (i,j) in the following list means that X(j) = {X(13386),X(13387)}-harmonic conjugate of X(i): (4,92), (8,329), (321,3436), (3869,5739).

See Tran Quang Hung and Peter Moses, AdvGeom 3776.

X(13387) lies on the cubics K170, K200, and these lines: {2,176}, {4,8}, {63,487}, {77,3084}, {81,1123}, {193,7133}, {278,1586}, {281,1585}, {1444,5391}, {5200,7009}, {8048,9789}

X(13387) = isogonal conjugate of X(34125)
X(13387) = isotomic conjugate of X(13386)
X(13387) = cevapoint of X(1335) and X(34121)
X(13387) = X(1335)-cross conjugate of X(5391)
X(13387) = anticomplement of X(13389)
X(13387) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (6,176), (1659,3434), (2067,347), (2362,7), (5414,20), (7090,69), (7133,8)
X(13387) = polar conjugate of X(1336)
X(13387) = isoconjugate of X(j) and X(j) for these (i,j): {4,605}, {6,6212}, {19,1124}, {25,3083}, {48,1336}, {1267,1973}, {1459,6136}, {6364,8750}
X(13387) = barycentric product X(i)*X(j) for these {i,j}: {4,5391}, {69,1123}, {75,6213}, {92,3084}, {264,1335}, {606,1969}, {6335,6365}
X(13387) = barycentric quotient X(i)/X(j) for these (i,j): (1,6212), (3,1124), (4,1336), (48,605), (63,3083), (69,1267), (606,48), (905,6364), (1123,4), (1335,3), (1783,6136), (1805,1806), (2067,6502), (3084,63), (5391,69), (5414,2066), (6135,1783), (6213,1), (6365,905)

X(13388) = COMPLEMENT OF X(13386)

Trilinears    1 - tan(A/2) : :
Trilinears    Ra - s : Rb - s : Rc - s, where Ra, Rb, Rc are the exradii
Barycentrics    cos A + sin A - 1 : :

See Tran Quang Hung and Peter Moses, AdvGeom 3776.

The appearance of (i,j) in the following list means that X(j) = {X(13388),X(13389)}-harmonic conjugate of X(i): {1,57}, {3,1214}, {55,241}, {56,3666}, {65,940}, {354,5228}, {980,1402}, {982,1429}, {1038,10319}. (Randy Hutson, July 21, 2017)

X(13388) lies on the cubic K168 and K363, and on these lines: {1,3}, {2,175}, {6,6203}, {7,1659}, {12,10911}, {37,6204}, {63,2067}, {69,5391}, {77,5414}, {81,1805}, {174,558}, {189,7090}, {200,3640}, {222,1335}, {226,481}, {371,1708}, {478,8231}, {482,553}, {914,11090}, {1211,10908}, {1372,5219}, {1374,4654}, {1407,3298}, {1445,2066}, {1589,6350}, {1590,6349}, {2003,3301}, {3083,3306}, {3751,8941}, {3752,7968}, {3911,5405}, {5256,6502}, {5408,6505}, {7248,7353}

X(13388) = complement of X(13386)
X(13388) = X(651)-Ceva conjugate of X(6365)
X(13388) = X(6365)-cross conjugate of X(651)
X(13388) = cevapoint of X(2067) and X(5414)
X(13388) = trilinear pole of line {905,6365}
X(13388) = X(i)-complementary conjugate of X(j) for these (i,j): (606, 3), (3084, 1368). (6213, 141). (8750, 6365)
X(13388) = isoconjugate of X(j) and X(j) for these (i,j): {4,2066}, {281,6502}, {1336,5414}, {1806,1826}, {6212,7133}
X(13388) = {X(481),X(5393)}-harmonic conjugate of X(226)
X(13388) = barycentric product X(i)*X(j) for these {i,j}: {63,1659}, {69,2362}, {75,2067}, {77,7090}, {85,5414}, {348,7133}, {1441,1805}
X(13388) = barycentric quotient X(i)/X(j) for these (i,j): (48,2066), (603,6502), (606,5414), (1437,1806), (1659,92), (1805,21), (2067,1), (2362,4), (5414,9), (6213,7090), (6502,6212), (7090,318), (7133,281)

X(13389) = COMPLEMENT OF X(13387)

Trilinears    1 + tan(A/2) : :
Trilinears    Ra + s : Rb + s : Rc + s, where Ra, Rb, Rc are the exradii
Barycentrics    cos A - sin A - 1 : :

See Tran Quang Hung and Peter Moses, AdvGeom 3776.

X(13389) lies on the cubics K168 and K363, and on these lines: {1,3}, {2,176}, {6,6204}, {12,10910}, {37,6203}, {63,3083}, {69,1267}, {77,2066}, {81,1806}, {174,557}, {200,3641}, {222,1124}, {226,482}, {371,8978}, {372,1708}, {481,553}, {914,11091}, {1211,10907}, {1371,5219}, {1373,4654}, {1407,3297}, {1445,5414}, {1587,8957}, {1589,6349}, {1590,6350}, {2003,3299}, {2067,5256}, {3084,3306}, {3751,8945}, {3752,7969}, {3911,5393}, {5409,6505}, {6200,8973}, {6213,8965}, {7248,7362}

X(13389) = isogonal conjugate of X(7133)
X(13389) = complement of X(13387)
X(13389) = X(651)-Ceva conjugate of X(6364)
X(13389) = X(6364)-cross conjugate of X(651)
X(13389) = X(i)-complementary conjugate of X(j) for these (i,j): (605,3), (3083,1368), (6212,141), (8750,6364)
X(13389) = cevapoint of X(2066) and X(6502)
X(13389) = trilinear pole of line {905,6364}
X(13389) = barycentric product X(i)*X(j) for these {i,j}: {75,6502}, {85,2066}, {1267,2362}, {1441,1806}, {1659,3083}
X(13389) = barycentric quotient X(i)/X(j) for these (i,j): {1,7090), (6,7133), (48,5414), (56,2362), (57,1659), (603,2067), (605,2066), (1437,1805), (1806,21), (2066,9), (2067,6213), (2362,1123), (6502,1)
X(13389) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,176,1659), (482,5405,226)
X(13389) = isoconjugate of X(j) and X(j) for these {i,j}: {1,7133}, {4,5414}, {6,7090}, {9,2362}, {55,1659}, {281,2067}, {1123,2066}, {1805,1826}

X(13390) = X(1)X(4)∩X(2)X(175)

Barycentrics (sin A)/(1 + cos A - sin A) : :
Barycentrics 1/(1 - cot A - csc A) : :

X(13390) lies on the cubics K070a, K332, the circumconic {{A,B,C(X(20, X(7)}}, and these lines: 1,4}, {2,175}, {19,3069}, {27,6502}, {56,10911}, {57,481}, {75,492}, {92,1586}, {281,3536}, {482,4654}, {553,1374}, {638,1943}, {908,3084}, {940,10908}, {1372,3911}, {1373,3982}, {1465,2048}, {1738,8945}, {1826,6351}, {1851,5200}, {3071,6354}, {3083,5249}, {3128,7102}, {3640,4847}, {3672,8243}, {3772,7968}, {5219,5393}, {6204,6352}, {10905,11347}

X(13390) = isogonal conjugate of (5414)
X(13390) = polar conjugate of X(7090)
X(13390) = X(92)-Ceva conjugate of X(1659)
X(13390) = X(i)-cross conjugate of X(j) for these (i,j): (57,1659), (481,7), (5405,2)
X(13390) = isoconjugate of X(j) and X(j) for these {i,j}: {1,5414}, {3,7133}, {9,2067}, {37,1805}, {48,7090}, {212,1659}, {219,2362}, {2066,6213}
X(13390) = cevapoint of X(1) and X(6203)
X(13390) = crosssum of X(i) and X(j) for these (i,j): {48,606}, {6502,8833}
X(13390) = {X(1),X(226)}-harmonic conjugate of X(1659)
X(13390) = {X(4),X(278)}-harmonic conjugate of X(1659)
X(13390) = X(481)-cross conjugate of X(7)
X(13390) = barycentric product X(i)*X(j) for these {i,j}: {264,6502}, {331,2066}
X(13390) = barycentric quotient X(i)/X(j) for these (i,j): (4,7090), (6,5414), (19,7133), (34,2362), (56,2067), (58,1805), (278,1659), (1806,283), (2066,219), (2067,1335), (2362,6213), (6502,3)

X(13391) = X(3)X(143)∩X(30)X(511)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25820 and Angel Montesdeoca, Triángulos paralelógicos y cónicas asociadas

X(13391) lies on these lines: {3,143}, {4,2889}, {5,3917}, {15,2058}, {16,2059}, {20,6102}, {30,511}, {36,500}, {49,12088}, {51,549}, {52,550}, {54,6030}, {110,5899}, {128,11583}, {140,5446}, {156,3167}, {186,1112}, {265,5900}, {323,10540}, {373,11539}, {376,568}, {381,2979}, {382,5876}, {389,548}, {394,7530}, {546,1216}, {547,3819}, {567,6636}, {578,7525}, {1319,5453}, {1350,7514}, {1511,2070}, {1568,11563}, {1597,6403}, {1657,5889}, {1993,12083}, {2071,3581}, {2077,6097}, {2937,5944}, {3153,10113}, {3520,6746}, {3521,12226}, {3524,11002}, {3526,9781}, {3530,5462}, {3534,5890}, {3627,5562}, {3628,5447}, {3830,11459}, {3843,11444}, {3845,5891}, {3850,11793}, {3851,7999}, {3853,5907}, {5054,5640}, {5055,7998}, {5066,10170}, {5073,12111}, {5892,12100}, {5972,10096}, {6688,10124}, {6699,11692}, {7512,10610}, {7516,10982}, {8567,12084}, {8703,9730}, {9826,10564}, {10272,11807}, {11414,12161}, {11660,12225}, {11695,12108}, {12038,12107}

X(13391) = isogonal conjugate of X(13597)

X(13392) = MIDPOINT OF X(110)X(140)

Barycentrics (2a^4-(b^2-c^2)^2-a^2(b^2+c^2))(3a^6-7a^4(b^2+c^2)+a^2(5b^4+b^2c^2+5c^4)-(b^2-c^2)^2(b^2+c^2)) : :
X(13392) = X[265] - 3 X[547] = X[399] + 3 X[549] = X[113] + 3 X[1511] = 7 X[113] - 3 X[1539] = 7 X[1511] + X[1539] = 5 X[632] - X[3448] = X[113] - 9 X[5642]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25820 and Angel Montesdeoca, Triángulos paralelógicos y cónicas asociadas

X(13392) lies on these lines: {5,12383}, {30,113}, {74,12100}, {110,140}, {146,8703}, {265,547}, {399,549}, {468,3043}, {542,10124}, {546,10733}, {632,3448}, {3292,11702}, {3523,12308}, {3530,5663}, {3580,11597}, {3628,5972}, {3853,12121}, {4995,7343}, {5054,12317}, {5298,6126}, {5609,12108}, {5844,11720}, {6677,12228}, {6699,11812}, {7687,11737}, {7728,12103}, {9143,11539}, {10109,12900}, {10113,12811}

X(13392) = midpoint of X(i) and X(j) for these {i,j}: {{110, 140}, {1511, 10272}, {3853, 12121}, {5642, 11694}, {7728, 12103}
X(13392) = reflection of X(i) in X(j) for these (i,j): (3628, 5972), (10113, 12811)
X(13392) = X(3471)-Ceva conjugate of X(30)
X(13392) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1511,5642,10272), (10272,11694,1511)

X(13393) = X(30)X(6070)∩X(110)X(140)

Barycentrics 2a^10-11a^8(b^2+c^2)+ 2a^6(13b^4-7b^2c^2+13c^4)+ a^4(-32b^6+23b^4c^2+23b^2c^4-32c^6)+ a^2(b^2-c^2)^2(20b^4+27b^2c^2+20c^4)-5(b^2-c^2)^4(b^2+c^2) : :
X(13393) = 3 X[110] - 5 X[140] = X[550] + 3 X[3448] = X[546] - 3 X[9140] = X[110] - 5 X[10264] = X[140] - 3 X[10264]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25820 and Angel Montesdeoca, Triángulos paralelógicos y cónicas asociadas

X(13393) lies on these lines: {30,6070}, {110,140}, {541,12102}, {542,3530}, {546,9140}, {550,3448}, {1656,12317}, {3850,5462}, {3859,10706}, {5056,12308}, {5655,12812}

X(13394) = MIDPOINT OF X(2) AND X(6800)

Barycentrics (a^2-b^2-c^2) (4 a^4-(b^2-c^2)^2+a^2 (b^2+c^2)) : :
X(13394) = 2 X[184] + X[343] = X[343] - 4 X[6676] = X[184] + 2 X[6676] = 5 X[631] + X[11456]

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 25910.

X(13394) lies on these lines: {2,154}, {3,4549}, {5,1495}, {6,7493}, {23,5480}, {51,10154}, {67,110}, {140,5651}, {156,7568}, {182,468}, {184,343}, {394,7494}, {549,5642}, {597,5640}, {631,11456}, {1995,3589}, {2502,3054}, {3066,3618}, {3233,11007}, {3549,6146}, {3580,8550}, {3629,11422}, {5092,5972}, {5169,7712}, {5647,12012}, {6353,10601}, {6689,7403}, {7399,10282}, {7488,12233}, {7499,9306}, {7552,12022}, {7558,9707}, {10565,11427}

X(13394) = midpoint of X(2) and X(6800)
X(13394) = X(22)-of-X(2)-Brocard-triangle
X(13394) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (110,7495,141), (184,6676,343), (3580,11003,8550), (3618,4232,3066)

X(13395) = TRILINEAR POLE OF LINE X(6)X(1214)

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

Suppose that P = u : v : w (barycentrics). The point U(P) = a^2 (b^4 w (-u+w)+(a^2-c^2) v (c^2 (u-v)+a^2 w)+b^2 (a^2 (u-v-w) w+c^2 (u^2+2 v w-u (v+w)))) : : is constructed in Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25820 and Angel Montesdeoca, Triángulos paralelógicos y cónicas asociadas

The point U(P) is also the trilinear pole of the line X(6)X, where X = crossum of X(6) and P. (Peter Moses, May 14, 2017).

X(13395) = U(X(19)). See also X(13396-X(13398).

X(13395) lies on the circumcircle and these lines: {103,10884}, {105,3485}, {108,4566}, {110,6516}, {112,651}, {1305,1633}

X(13395) = trilinear pole of line X(6)X(1214)
X(13395) = trilinear pole, wrt circumtangential triangle, of line X(3)X(19)
X(13395) = Ψ(X(6), X(1214))
X(13395) = isoconjugate of X(j) and X(j) for these (i,j): {377, 663}, {1448, 3900}
X(13395) = barycentric quotient X(i)/X(j) for these (i,j): (651, 377), (1461, 1448)

X(13396) = TRILINEAR POLE OF LINE X(6)X(2243)

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

X(13396) = U(X(45)); see X(13395 for the mapping U(P).

X(13396) lies on the circumcircle and these lines: {1,753}, {86,759}, {101,4585}, {664,2222}, {668,9059}, {743,1015}, {761,1001}

X(13396) = anticomplement of X(38963)
X(13396) = trilinear pole of line X(6)X(2243)
X(13396) = X(21)-beth conjugate of X(753)
X(13396) = Ψ(X(6), X(2243))
X(13396) = isoconjugate of X(663) and X(5252)
X(13396) = barycentric quotient X(651)/X(5252)br>

X(13397) = TRILINEAR POLE OF LINE X(6)X(169)

Trilinears 1/[(sin B)(1 + cos A + cos B) - (sin C)(1 + cos C + cos A)] : :
Barycentrics a (a-b) (a-c) (a^3+a^2 b+a b^2+b^3-a^2 c-2 a b c-b^2 c-a c^2-b c^2+c^3) (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(13397) = U(X(63)); see X(13395 for the mapping U(P).

X(13397) lies on the circumcircle and these lines: {2,5521}, {3,915}, {20,104}, {22,105}, {107,3658}, {110,1633}, {112,4236}, {513,6099}, {759,4278}, {840,3100}, {858,2752}, {917,7411}, {935,7475}, {1289,4238}, {1295,11413}, {1299,7414}, {1300,3651}, {1301,4246}, {1304,7477}, {1305,6516}, {2071,2687}, {2249,4269}, {2374,4239}, {2757,10538}, {3563,4220}, {7465,9085}, {7476,10423}, {7493,9061}

X(13397) = reflection of X(915) in X(3)
X(13397) = trilinear pole of line X(6)X(169)
X(13397) = anticomplement of X(5521)
X(13397) = cevapoint of X(3) and X(513)
X(13397) = X(i)-cross conjugate of X(j) for these (i,j): (906, 651), (6591, 2), (7742, 59)
X(13397) = DeLongchamps-circle-inverse of X(149)
X(13397) = reflection of X(6099) in the line X(1)X(3)
X(13397) = Ψ(X(6), X(169))
X(13397) = Λ(trilinear polar of X(7040))
X(13397) = isoconjugate of X(j) and X(j) for these (i,j): {513, 3811}, {514, 2911}, {523, 1780}, {650, 1708}, {1331, 5521}, {3064, 3173}, {3900, 4341}, {7649, 11517}
X(13397) = barycentric quotient X(i)/X(j) for these (i,j): (101, 3811), (109, 1708), (163, 1780), (692, 2911), (906, 11517), (1461, 4341), (6591, 5521)

X(13398) = TRILINEAR POLE OF LINE X(6)X(1147)

Trilinears a/(cot B cos 2B - cot C cos 2C) : :
Barycentrics a^2 (a-b) (a+b) (a-c) (a+c) (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(13398) = U(X(68)); see X(13395 for the mapping U(P).

X(13398) lies on the circumcircle and these lines: {2,135}, {3,1299}, {20,254}, {22,3563}, {74,9938}, {98,1370}, {759,921}, {925,4558}, {1141,8800}, {1289,4226}, {2374,7493}, {2383,7488}, {2713,9218}, {7468,10423}

X(13398) = reflection of X(1299) in X(3)
X(13398) = trilinear pole of line X(6)X(1147)
X(13398) = anticomplement of X(135)
X(13398) = cevapoint of X(i) and X(j) for these {i, j}: {3, 924}, {512, 577}, {523, 11585}
X(13398) = X(i)-cross conjugate of X(j) for these (i,j): (6562, 251), (6753, 2), (7387, 250)
X(13398) = circumcircle-antipode of X(1299)
X(13398) = Λ(trilinear polar of X(6515))
X(13398) = X(108)-of-dual-of-orthic-triangle if ABC is acute
X(13398) = isoconjugate of X(j) and X(j) for these {i,j}: {523, 920}, {656, 3542}, {661, 6515}, {1577, 1609}, {2618, 8883}
X(13398) = barycentric product X(i)*X(j) for these {i,j}: {110, 6504}, {254, 4558}, {662, 921}
X(13398) = barycentric quotient X(i)/X(j) for these (i,j): (110, 6515), (112, 3542), (163, 920), (921, 1577), (1576, 1609), (6504, 850), (6753, 135)

X(13399) = X(30)X(6070)∩X(74)X(10421)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25933.

X(13399) lies on these lines: {30,6070}, {74,10421}, {125,403}, {185,427}, {542,2071}, {1533,10264}, {1568,5663}, {1596,11381}, {3357,11457}, {3520,10619}, {5642,10257}, {6241,7577}, {6353,12324}, {6699,10540}, {10112,12086}, {10193,11464}, {10605,11550}

X(13400) = POINT BEID 57

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25939.

X(13400) lies on this line: {230,231}

X(13400) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2501,6753,6587)
X(13400) = barycentric product X(i)*X(j) for these {i,j}: {523, 3089}, {2501, 11433}
X(13400) = barycentric quotient X(i)/X(j) for these {i,j}: {3089, 99}, {8573, 4558}, {11433, 4563}

X(13401) = POINT BEID 58

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25945.

X(13401) lies on these lines: {44,513}, {521,4976}, {2488, 11934}, {3738,4765}, {4131,4762} ,{8774,11068}

X(13401) = reflection of X(11934) in X(2488)
X(13401) = X(42)-complementary conjugate of X(5522)
X(13401) = crosspoint of X(651) and X(3296)
X(13401) = crossdifference of every pair of points on line {1, 6883}
X(13401) = crosssum of X(650) and X(3295)
X(13401) = barycentric product X(i)*X(j) for these {i,j}: {513, 10527}, {522, 3338}, {3737, 12609}
X(13401) = barycentric quotient X(i)/X(j) for these {i,j}: {663, 7162}, {3338, 664}, {10527, 668}
X(13401) = {X(650),X(4790)}-harmonic conjugate of X(654)

X(13402) = POINT BEID 59

Barycentrics a^2 (3 a^8 b^2-6 a^6 b^4+6 a^2 b^8-3 b^10+3 a^8 c^2+10 a^6 b^2 c^2-7 a^4 b^4 c^2-9 a^2 b^6 c^2+3 b^8 c^2-6 a^6 c^4-7 a^4 b^2 c^4+16 a^2 b^4 c^4-9 a^2 b^2 c^6+6 a^2 c^8+3 b^2 c^8-3 c^10) : :
X(13402) = 8 X[6723] - 9 X[10219] = 3 X[9729] - X[10620] = 3 X[389] + X[12308]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25952.

X(13402) lies on these lines: {23, 110}, {389, 12308}, {5663, 12811}, {6723, 10219}, {9729, 10620}

X(13402) =

X(13403) = POINT BEID 60

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25958.

Let A'B'C' be the midheight triangle. Let A_B, A_C be the orthogonal projections of A' on CA, AB, resp. Define B_C, B_A, C_A, C_B cyclically. 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(13403) = X(382)-of-A"B"C". (Randy Hutson, March 29, 2020)

X(13403) lies on these lines: {3,2929}, {4,54}, {5,1511}, {6,382}, {20,11438}, {30,143}, {49,113}, {51,6240}, {115,1970}, {125,3520}

X(13403) = midpoint of X(i) and X(j) for these {i,j}: {382,11750}, {1885,6146}
X(13403) = reflection of X(i) in X(j) for these {i,j}: {389,12241}, {3575,10110}, {101 12,12370}
X(13403) = crosssum of X(3) and X(6102)
X(13403) = X(3878)-of-orthic-triangle if ABC is acute
X(13403) = {X(5),X(12038)}-harmonic conjugate of X(5972)

X(13404) = POINT BEID 61

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25964.

X(13404) lies on the hyperbola {{A,B,C,X(1),X(6)}} and these lines:

X(13404) = cevapoint of X(6) and X(2293)
X(13404) = X(2488)-cross conjugate of X(101)

X(13405) = POINT BEID 62

Barycentrics 2 a^3-3 a^2 b+b^3-3 a^2 c-b^2 c-b c^2+c^3 : :
X(13405) = 3 X[226] - X[1836] = 3 X[55] + X[1836] = 3 X[2] + X[3870] = 5 X[3616] - X[3872] = X[1] + 3 X[10056] = X[10] - 3 X[10197]
X(13405) = 5 X[3616] + 3 X[11239] = X[3872] + 3 X[11239] = 9 X[10056] - X[12647] = 3 X[1] + X[12647] = 7 X[3622] + X[12648]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25964.

X(13405) lies on these lines:
{1,2}, {3,4298}, {4,3947}, {7,165}, {11,3748}, {12,950}, {20,5290}, {21,12527}, {35,4292}, {37,800}, {40,3487}, {55,226}, {56,12577}, {57,3475}, {63,5850}, {65,12563}, {100,5249}, et al.

X(13405) = midpoint of X(i) and X(j) for these {i,j}: {55, 226}, {3870, 4847}, {4028, 4362}
X(13405) = complement X(4847)
X(13405) = X(6606)-Ceva conjugate of X(514)
X(13405) = crosssum of X(6) and X(2293)
X(13405) = complement of X(4847)
X(13405) = X(i)-complementary conjugate of X(j) for these (i,j): (1170, 141), (1174, 3452), (2346, 1329)
X(13405) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,2,11019), ( 1,10,6738), ( 1,498,1210), (1,1210,6744), (1,1698,938), (1,3085,10), (1,3584,1737), (2,200,10), (2,3870,4847), (2,10578,1), (7,5281,165), (10,3811,6743), (35,4292,12512), (40,3487,3671), (57,3475,5542), (57,5218,10164), (498,1210,3634), (3475,5218,57), (3616,11239,3872), (3634,6744,1210), (3811,10198,10), (3947,4314,4), (5542,10164,57)

X(13406) = POINT BEID 63

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25967.

Let La be the polar of X(4) wrt the circle centered at A and passing through X(5). Define Lb and Lc cyclically. (Note: X(4) is the perspector of any circle centered at a vertex of ABC.) Let A' = Lb∩Lc, and define B' and C' cyclically. X(13406) = X(3)-of-A'B'C'. (Randy Hutson, July 21, 2017)

See Anopolis #436, Angel Montesdeoca, 6/18/2013.

X(13406) lies on these lines: {2,3}, {113,5876}, {156,9927}, {265,1614}, {1154,5448}, {1568,6101}, {5449,5663}, {5476,11255}, {5944,10113}, {6241,10264}, {6564,11266}, {6565,11265}, {7728,11440}, {7951,8144}, {11459,11805}, {11804,12254}

X(13406) = midpoint of X(i) and X(j) for these {i,j}: {4,1658}, {156,9927}
X(13406) = reflection of X(i) on X(j) for these {i,j}: {3,10125}, {10125,12010}, {10224,5}, {10226,140}, {11250,5498}
X(13406) = complement of X(11250)
X(13406) = X(26286)-of-orthic-triangle if ABC is acute

X(13407) = POINT BEID 64

Barycentrics a^3 b+a^2 b^2-a b^3-b^4+a^3 c+4 a^2 b c+a b^2 c+a^2 c^2+a b c^2+2 b^2 c^2-a c^3-c^4 : :
X(13407) = X[5178] - 3 X[6175]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25968.

X(13407) lies on these lines: {1,4}, {2,3338}, {3,10404}, {5, 354}, {7,46}, {8,12559}, {10, 3681}, {11,5045}, {12,942}, {35,4292}, {36,4298}, {40,4654}, {55, 1770}, {56,6883}, {57,498}, {58, 3011}, {63,10198}, {65,495}, {79, 516}, {80,6738}, {142,1698}, {191,527}, {210,8728}, {355, 11237}, {377,3811}, {442,518}, {484,3982}, {499,3333}, {517, 3649}, {519,5178}, {529,11281}, {551,3897}, {553,3336}, {726, 3178}, {908,1125}, {938,10590}, {999,11375}, {1071,7680}, {1089, 3912}, {1103,4328}, {1145,10107} ,{1210,3947}, {1329,5439}, {1330,3757}, {1385,5434}, {1714, 3751}, {1718,5262}, {1738,3293}, {1788,8164}, {1836,3295}, {1837, 9654}, {1892,11398}, {2476,3873} ,{2646,5719}, {2801,10122}, {2886,3555}, {3086,5226}, {3090, 3296}, {3091,11038}, {3303, 12699}, {3304,5886}, {3337,3911} ,{3340,12647}, {3452,3624}, {3576,4317}, {3579,11246}, {3600, 6992}, {3601,4299}, {3612,4293}, {3634,5557}, {3636,11813}, {3670,5530}, {3671,5903}, {3697, 3826}, {3742,4187}, {3753,12607} ,{3754,6735}, {3782,3931}, {3822,3874}, {3824,3925}, {3889, 11680}, {3916,6690}, {3936,4968} ,{4004,8256}, {4294,10578}, {4295,5119}, {4302,9579}, {4304, 10483}, {4309,10389}, {4338, 6361}, {4415,6051}, {4870,5901}, {5049,9955}, {5083,8068}, {5252, 12645}, {5259,12572}, {5261, 6993}, {5425,12563}, {5443, 12577}, {5586,9588}, {5587, 11518}, {5687,5880}, {5722, 10895}, {5745,6763}, {5905, 12514}, {6745,12436}, {6767, 12701}, {6825,12704}, {6831, 12675}, {6847,10085}, {6890, 7284}, {7373,11376}, {7741, 11019}, {7958,10157}, {8227, 10072}, {8727,12680}, {9578, 10573}, {10052,10075}, {10580, 10591}, {10587,11415}, {11010, 11552}, {12528,12617}

X(13407) = midpoint of X(i) and X(j) for these {i,j}: {1, 5270}, {79, 3746}
X(13407) = crosspoint of X(92) and X(1268)
X(13407) = crosssum of X(48) and X(2308)
X(13407) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,226,12047), (1,1478,10572), ( 1,3585,950), (1,5290,1478), (1, 9612,1479), (4,3475,1), (7,3085, 46), (12,942,1737), (65,495, 10039), (65,6147,11551), (388, 3487,1), (388,10629,1478), (495, 6147,65), (553,6684,3336), (938, 10590,10826), (1056,3485,1), ( 1125,12527,5251), (1210,3947, 7951), (2476,3873,10916), (3333, 5219,499), (3336,3584,6684), ( 3681,4197,10), (3822,3874,6734), (3947,5542,1210), (4293,5703, 3612), (5226,11037,3086), (9578, 11529,10573), (10039,11551,65)

X(13408) = POINT BEID 65

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25968.

X(13408) lies on these lines: {1,30}, {3,5713}, {4,81}, {5,1724}, {58,6841}, {155,5706}, {186,229}, {225,6357}, {283,442}, {323,2475}, {355,3564}, {381, 5292}, {382,5733}, {407,1437}, {540,10916}, {580,6881}, {582, 8728}, {942,1835}, {1478,3157}, {2003,3585}, {3109,11657}, {3332, 6850}, {4340,6851}, {5230,9958}, {5712,6869}, {9840,11249}

X(13409) = POINT BEID 66

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25977.

X(13409) lies on these lines: {2,1972}, {3,54}, {5,6662}, {6,426}, {51,852}, {216,3289}, {264,11197}, {389,417}, {394,6641}, {408,970}, {418,511}

X(13409) = crosspoint of X(3) and X(264)
X(13409) = crosssum of X(4) and X(184)
X(13409) = X(9251)-anticomplementary conjugate of X(2888)
X(13409) = barycentric product X(i)*X(j) for these {i,j}: {76,6752}, {394,6747}
X(13409) = {X(51),X(6509)}-harmonic conjugate of X(852)

X(13410) = POINT BEID 67

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25977.

X(13410) lies on these lines: {6,110}, {23,5038}, {39,51}, {18 2,8627}, {251,12834}, {373,3231}

X(13410) = crosspoint of X(6) and X(598)
X(13410) = crosssum of X(2) and X(574)
X(13410) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6,5640,3124}, {373,5052,3231}

X(13411) = POINT BEID 68

Barycentrics 2 a^4-a^3 b-3 a^2 b^2+a b^3+b^4-a^3 c-2 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(13411) = X[1] + 3 X[3584] = 5 X[3616] - X[4861] = 3 X[3584] - X[10039]
X(13411) = (r + 2 R) X[1] + 3 r X[2] = r X[3] + (r + R) X[226] = r X[4] + (3 r + 2 R) X[3601]
X(13411) = 2 r X[5] + R X[950] = (r + 4 R) X[7] + 7 r X[3523] = (r+4 R) X[9] - (5 r + 6 R) X[6857] = r X[20] + (7 r+4 R) X[5226]

See Antreas Hatzipolakis, Angel Montesdeoca, and Peter Moses, Hyacinthos 25986.

X(13411) lies on these lines: {1,2}, {3,226}, {4,3601}, {5, 950}, {7,3523}, {9,6857}, {12, 515}, {20,5226}, {21,908}, {29, 1785}, {35,411}, {36,4298}, {37, 216}, {40,3485}, {46,3671}, {55, 946}, {56,10165}, {57,631}, {58, 3074}, {63,6910}, {65,5432}, {72, 5745}, {84,6935}, {86,1167}, {140,942}, {142,474}, {165,4295}, {255,307}, {354,5433}, {376, 5714}, {377,4855}, {388,3576}, {390,9614}, {404,5249}, {405, 3452}, {442,5440}, {443,5438}, {452,5748}, {495,1385}, {496, 11230}, {497,6864}, {518,4999}, {527,3916}, {549,553}, {750,1771} ,{940,7078}, {943,6905}, {944, 6956}, {960,6690}, {962,5281}, {965,5257}, {991,1745}, {993, 12527}, {1012,6260}, {1056,1420} ,{1058,10389}, {1071,6705}, {1150,4101}, {1155,3649}, {1323, 1446}, {1335,8983}, {1445,3338}, {1478,3612}, {1479,3817}, {1490, 6847}, {1656,5722}, {1697,5603}, {1699,4294}, {1728,3305}, {1735, 2292}, {1770,5010}, {1788,11529} ,{1836,5217}, {1837,10175}, {1838,7513}, {1892,3515}, {2099, 11362}, {3035,3812}, {3075,7572} ,{3090,3488}, {3091,3586}, {3158,5082}, {3295,5886}, {3303, 11376}, {3306,6921}, {3333,3475} ,{3336,11551}, {3340,5657}, {3419,12437}, {3486,5587}, {3524, 4654}, {3525,11518}, {3530,3982} ,{3583,6894}, {3585,6895}, {3614,10543}, {3628,12433}, {3686,5742}, {3742,6691}, {3746, 5443}, {3772,4255}, {3822,10523} ,{3940,5791}, {4293,5290}, {4301,5119}, {4305,5691}, {4652, 5905}, {4870,4995}, {4909,5740}, {5044,6675}, {5054,5708}, {5083, 6713}, {5084,5436}, {5129,5328}, {5135,9028}, {5204,10404}, {5248,8069}, {5252,5882}, {5261, 5731}, {5265,11037}, {5294, 11031}, {5316,11108}, {5425, 5445}, {5435,10303}, {5439,6692} ,{5444,5563}, {5482,11573}, {5570,6681}, {5709,5761}, {5717, 5718}, {5720,6824}, {5727,5818}, {5728,6666}, {5730,5837}, {5732, 8232}, {5735,5766}, {5747,8804}, {5750,5830}, {5777,10391}, {5850,6763}, {5901,9957}, {5902, 12563}, {6198,7537}, {6245,6833} ,{6282,6908}, {6710,11028}, {6718,12016}, {6767,11373}, {6828,7951}, {6834,7682}, {6837, 7675}, {6846,10382}, {6860, 10827}, {6861,10395}, {6878, 11048}, {6890,10884}, {6892, 7330}, {6926,8726}, {6966,8545}, {6991,7741}, {7308,10396}, {7498,7952}, {7742,12573}, {7962, 10595}, {7988,10591}, {10086, 11599}, {10265,12739}, {10399, 11020}, {10956,11715}, {11507, 12609}

X(13411) = midpoint of X(i) and X(j) for these {i,j}: {1, 10039}, {12, 2646}, {35, 12047}, {4870, 4995}
X(13411) = complement X(6734)
X(13411) = X(i)-complementary conjugate of X(j) for these (i,j): {943, 1329}, {1175, 960}, {2259, 3452}, {2982, 141}
X(13411) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,2,1210), (1,498,10), (1,499, 11019), (1,1737,6738), (1,3584, 10039), (1,3624,3086), (1,12647, 3244), (2,8,5705), (2,78,10), (2, 5703,1), (3,226,4292), (3,11374, 226), (4,3601,4304), (20,5226, 9612), (21,908,12572), (55,946, 10624), (55,11375,946), (65, 5432,6684), (72,7483,5745), ( 140,942,3911), (140,5719,942), ( 307,5736,3664), (376,5714,9579) ,(388,3576,4311), (404,5249, 12436), (495,1385,10106), (631, 3487,57), (944,8164,9578), ( 1125,6700,2), (1478,3612,4297), (1770,5010,12512), (3035,11281, 3812), (3090,3488,9581), (3091, 4313,3586), (3295,5886,12053), ( 3475,7288,3333), (3485,5218,40) ,(3486,10588,5587), (3601,5219, 4), (3634,6738,1737), (3671, 10164,46), (3817,4314,1479), ( 3947,4297,1478), (4305,10590, 5691), (5261,5731,9613), (5290, 7987,4293), (5761,6954,5709), ( 10303,11036,5435)

X(13412) = POINT BEID 69

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26001.

X(13412) lies on this line: {5,3629}

X(13413) = 10TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics 2 a^8 b^2-4 a^6 b^4+4 a^2 b^8-2 b^10+2 a^8 c^2-2 a^6 b^2 c^2-3 a^4 b^4 c^2-3 a^2 b^6 c^2+6 b^8 c^2-4 a^6 c^4-3 a^4 b^2 c^4-2 a^2 b^4 c^4-4 b^6 c^4-3 a^2 b^2 c^6-4 b^4 c^6+4 a^2 c^8+6 b^2 c^8-2 c^10 : :
X(13413) = 3 X[5] + X[427] = X[378] + 7 X[3851] = X[22] - 9 X[5055] = 3 X[547] - X[6676] = 15 X[5071] + X[7391] = 5 X[1656] - X[7502] = 7 X[3090] - X[7555] = 3 (2 J^2 - 7) X[2] - (2 J^2 - 9) X[3], where J = |OH|/R

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26006.

X(13413) lies on these lines: {2,3}, {1216,11808}, {5943,11557}, {6146,8254}, {10610,11572}, {11264,12242}

X(13413) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,1594,140), (5,2072,547), (5,3 845,10254), (5,5133,5066), (5,55 76,546), (5,10224,3628), (140,54 6,3575), (546,547,10096), (5169, 10254,3845), (7574,7579,1594)

X(13414) = POINT BEID 70

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

X(13414) and X(13415) were contributed by Peter Moses, May 22, 2017.

X(182) = midpoint of X(13414) and X(13415)
X(i) = {X(13414),X(13415)}-harmonic conjugate of X(j) for these {i,j}: {2,110}, {125,184}, {1899,13198}, {3448,5012}, {5622,6776}, {5642,5651}, {5972,9306}, {9140,11003}, {9744,11653}, {11179,11579}

X(13414) lies on the Brocard circle, the cubics K019, K048, K223, K417, K418, and these lines: {2,98}, {3,2575}, {6,1344}, {511,1113}, {1114,1495}, {1312,11064}, {1313,1503}, {1347,3818}, {3292,8115}, {10719,11645}

X(13414) = midpoint of X(1113) and X(8116)
X(13414) = X(1113)-of-1st-Brocard-triangle
X(13414) = X(9513)-Ceva conjugate of X(13415)
X(13414) = X(i)-line conjugate of X(j) for these (i,j): {3, 2575}, {6, 8105}, {1344, 8105}, {2574, 8105}, {9173, 8105}, {10287, 2575}
X(13414) = crossdifference of every pair of points on line X(2575)X(3569)
X(13414) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,110,13415), (125,184,13415), (1899,13198,13415), (3448,5012,13415), (5622,6776,13415), (5642,5651,13415), (5972,9306,13415), (9140,11003,13415), (9744,11653,13415), (11179,11579,13415)

X(13415) = POINT BEID 71

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

See X(13414).

X(13415) lies on the Brocard circle, the cubics K019, K048, K223, K417, K418, and these lines: {2,98}, {3,2574}, {6,1345}, {511,1114}, {1113,1495}, {1312,1503}, {1313,11064}, {1346,3818}, {3292,8116}, {10720,11645}}

X(13415) = midpoint of X(1114) and X(8115)
X(13415) = X(1114)-of-1st-Brocard-triangle
X(13415) = X(9513)-Ceva conjugate of X(13414)
X(13415) = X(i)-line conjugate of X(j) for these (i,j): {3, 2574}, {6, 8106}, {1345, 8106}, {2575, 8106}, {9174, 8106}, {10288, 2574}
X(13415) = crossdifference of every pair of points on line X(2575)X(3569)
X(13415) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,110,13414), (125,184,13414), (1899,13198,13414), (3448,5012,13414), (5622,6776,13414), (5642,5651,13414), (5972,9306,13414), (9140,11003,13414), (9744,11653,13414), (11179,11579,13414)

X(13416) = POINT BEID 72

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26038.

X(13416) lies on these lines: {2,1112}, {3,74}, {20,12133}, {1 13,6823}, {125,343}, {140,9826}, {143,6640}, {146,10996}, {265,66 43}, {339,4576}, {376,12292}, {39 4,13198}, {511,5159}, {542,10691 }, {631,1986}

X(13416) = midpoint of X(i) and X(j) for these {i,j}: {3,12358}, {20,12133}, {974,556 2}, {1216,6699}, {2979,12099}, {6101,12236}, {12219,13148}
X(13416) = reflection of X(i) in X(j) for these {i,j}: {9826, 140}, {11746, 6723}
X(13416) = complement X(1112)

X(13417) = POINT BEID 73

Barycentrics a^2 (a^8 b^2-2 a^6 b^4+2 a^2 b^8-b^10+a^8 c^2+a^4 b^4 c^2-3 a^2 b^6 c^2+b^8 c^2-2 a^6 c^4+a^4 b^2 c^4+2 a^2 b^4 c^4-3 a^2 b^2 c^6+2 a^2 c^8+b^2 c^8-c^10) : :
X(13417) = 3 X[51] - 2 X[125] = 3 X[51] - 4 X[1112] = 9 X[373] - 8 X[6723] = 3 X[568] - X[10620] = 9 X[51] - 8 X[11746] = 3 X[125] - 4 X[11746]
X(13417) = 3 X[1112] - 2 X[11746] = 3 X[568] - 2 X[11806] = X[7731] + 2 X[11807] = 10 X[11746] - 9 X[12099] = 5 X[125] - 6 X[12099] = 5 X[51] - 4 X[12099] = 5 X[1112] - 3 X[12099] = 3 X[4] - X[12281]
X(13417) = 6 X[11807] - X[12281] = 3 X[7731] + X[12281] = 5 X[110] - 8 X[13402]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26039.

X(13417) lies on these lines: {2,13201}, {3,11557}, {4,7730}, {6,1205}, {23,110}, {30,11562}, {51,125}, {52,3627}, {67,9969}, {74,389}, {113,5562}, {143,10264} ,{146,5889}, {184,10117}, {185, 1986}, {265,5446}, {373,6723}, {399,6243}, {550,11561}, {568, 10620}, {974,10990}, {1181,9919} ,{1204,2935}, {1498,12165}

X(13417) = midpoint of X(i) and X(j) for these {i,j}: {4,7731}, {146,5889}, {399, 6243}, {3146,12270}, {7722, 10721}
X(13417) = reflection of X(i) in X(j) for these {i,j}: {3,11557}, {4,11807}, {67,9969} ,{74,389}, {125,1112}, {185, 1986}, {265,5446}, {550,11561}, {1205,6}, {3313,6593}, {3448,11800}, {3917,12824}, {5562,113} ,{6101,10272}, {6467,5095}, {10264,143}, {10620,11806}, {10625,1511}, {10990,974}, {11381,13202}, {12162,1539}, {12219,5907}
X(13417) = complement X(13201)
X(13417) = crosssum of X(3) and X(3448)
X(13417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (125,1112,51), (568,10620, 11806)

X(13418) = POINT BEID 74

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26047.

X(13418) lies on the Jerabek hyperbola and these lines: {3, 12325}, {74, 12254}, {265, 2888}, {1154, 3521}, {1176, 5965}, {9706, 11271}

X(13418) = pedal antipodal perspector of X(5)

X(13419) = POINT BEID 75

Barycentrics 2 a^10-4 a^8 b^2+a^6 b^4+a^4 b^6+a^2 b^8-b^10-4 a^8 c^2-a^4 b^4 c^2+2 a^2 b^6 c^2+3 b^8 c^2+a^6 c^4-a^4 b^2 c^4-6 a^2 b^4 c^4-2 b^6 c^4+a^4 c^6+2 a^2 b^2 c^6-2 b^4 c^6+a^2 c^8+3 b^2 c^8-c^10 : :
X(13419) = 3 X[428] - X[6146] = X[52] - 3 X[7540] = X[185] - 3 X[7576] = 3 X[428] - 2 X[10110] = 4 X[143] - 3 X[11225]
X(13419) = 2 X[10116] - 3 X[11225] = 3 X[389] - 4 X[11745] = 3 X[6756] - 2 X[11745] = 3 X[381] - X[11750] = 3 X[3543] + X[12278] = 5 X[4] - X[12289]
X(13419) = 4 X[13163] - 3 X[13363] = 2 X[12289] - 5 X[13403]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26047.

Let P be a point on the circumcircle and D a point on the line BC. Let U and V be the reflections of D in PB and PC, respectively. The envelope of lines UV when D moves on BC is a parabola. Let P be its focus and d_a its directrix. The envelope of d_a as P moves on circumcircle is a circle, O_a. Define (O_b) and (O_c) cyclically. The radical center of the circles (O_a), (O_b), (O_c) is X(13419); see Euclid #611. (Angel Montesdeoca, February 6, 2020)

X(13419) lies on these lines: {3,2916}, {4,54}, {24,11550}, {30,1216}, {52,542}, {125,3518}, {143,10116}, {182,7528}, {185, 7576}, {211,2794}, {235,7687}, {381,11750}, {389,1503}, {403, 11572}, {427,10282}, {428,6146}, {511,7553}, {539,10263}, {1092, 7391}, {1112,10114}, {1209,2937} ,{1495,1594}, {1539,3627}, {1595,11430}, {1598,1619}, {1853, 3517}, {2777,6240}, {3426,5925}, {3541,11202}, {3543,12278}, {3575,6000}, {3853,12897}, {5092, 7405}, {5446,10112}, {5899,6288} ,{5900,12244}, {5965,6243}, {5972,13371}, {6145,10117}, {6242,7731}, {6800,7566}, {7487, 11438}, {7530,9927}, {7544, 10984}, {9729,11645}, {13163, 13363}

X(13419) = midpoint of X(i) and X(j) for these {i,j}: {6240, 11381}, {7553, 12134}
X(13419) = reflection of X(i) in X(j) for these {i,j}: {389, 6756}, {6146, 10110}, {10112, 5446}, {10114, 1112}, {10116, 143}, {12897, 3853}, {13403, 4}
X(13419) = crosspoint of X(4) and X(11816)
X(13419) = crosssum of X(3) and X(6101)
X(13419) = X(3874)-of-orthic-triangle if ABC is acute
X(13419) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1614,3574), (4,1629,6750), ( 4,9833,578), (143,10116,11225), (428,6146,10110)

X(13420) = POINT BEID 76

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26047.

X(13420) lies on the Feuerbach hyperbola of the orthic triangle and these lines: {4,12175}, {52,11271}, {185, 12254}, {1986,6242}, {3518,5898}

X(13420) = X(4)-Ceva conjugate of X(6143)

X(13421) = POINT BEID 77

Barycentrics a^2 (3 a^6 b^2-9 a^4 b^4+9 a^2 b^6-3 b^8+3 a^6 c^2-10 a^4 b^2 c^2+7 b^6 c^2-9 a^4 c^4-8 b^4 c^4+9 a^2 c^6+7 b^2 c^6-3 c^8) : :
X(13421) = 2 X[140] - 3 X[143] = 3 X[52] - X[550] = 5 X[1656] - 9 X[3060] = 9 X[568] - 5 X[3522] = 2 X[3850] - 3 X[5446] = 7 X[140] - 6 X[5447] = 7 X[143] - 4 X[5447] = 5 X[5447] - 7 X[5462] = 5 X[140] - 6 X[5462]
X(13421) = 5 X[143] - 4 X[5462] = 5 X[3858] - 3 X[5562] = 5 X[4] - 3 X[5876] = X[5073] + 3 X[5889] = 7 X[3523] - 9 X[5946] = 5 X[1656] - 3 X[6101]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26047.

X(13421) lies on these lines: {4,93}, {26,11477}, {30,10112}, {52,550}, {140,143}, {546,12002} ,{568,3522}, {576,7525}, {1176, 1351}, {1216,13364}, {1493,2937} ,{1656,3060}, {1657,6102}, {3523,5946}, {3532,12084}, {3850, 5446}, {3851,11412}, {3858,5562} ,{5073,5889}, {10299,13340}, {10625,12006}

X(13421) = midpoint of X(6243) and X(10263)
X(13421) = reflection of X(i) in X(j) for these {i,j}: {6101, 10095}, {10625, 12006}, {10627, 143}, {11591, 5446}
X(13421) = 2{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (143,10627,13363), (3060,6101, 10095)

X(13421) =

X(13422) = POINT BEID 78

Barycentrics 7 a^16-38 a^14 b^2+78 a^12 b^4-62 a^10 b^6-20 a^8 b^8+78 a^6 b^10-62 a^4 b^12+22 a^2 b^14-3 b^16-38 a^14 c^2+134 a^12 b^2 c^2-160 a^10 b^4 c^2+84 a^8 b^6 c^2-74 a^6 b^8 c^2+106 a^4 b^10 c^2-64 a^2 b^12 c^2+12 b^14 c^2+78 a^12 c^4-160 a^10 b^2 c^4+79 a^8 b^4 c^4-4 a^6 b^6 c^4-41 a^4 b^8 c^4+60 a^2 b^10 c^4-12 b^12 c^4-62 a^10 c^6+84 a^8 b^2 c^6-4 a^6 b^4 c^6-6 a^4 b^6 c^6-18 a^2 b^8 c^6-12 b^10 c^6-20 a^8 c^8-74 a^6 b^2 c^8-41 a^4 b^4 c^8-18 a^2 b^6 c^8+30 b^8 c^8+78 a^6 c^10+106 a^4 b^2 c^10+60 a^2 b^4 c^10-12 b^6 c^10-62 a^4 c^12-64 a^2 b^2 c^12-12 b^4 c^12+22 a^2 c^14+12 b^2 c^14-3 c^16 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26047.

X(13422) lies on this line: {11558,12316}

X(13422) =

X(13423) = POINT BEID 79

Barycentrics a^2 (a^12 b^2-4 a^10 b^4+5 a^8 b^6-5 a^4 b^10+4 a^2 b^12-b^14+a^12 c^2-7 a^10 b^2 c^2+11 a^8 b^4 c^2-5 a^6 b^6 c^2+4 a^4 b^8 c^2-8 a^2 b^10 c^2+4 b^12 c^2-4 a^10 c^4+11 a^8 b^2 c^4-5 a^6 b^4 c^4+a^4 b^6 c^4+3 a^2 b^8 c^4-6 b^10 c^4+5 a^8 c^6-5 a^6 b^2 c^6+a^4 b^4 c^6+2 a^2 b^6 c^6+3 b^8 c^6+4 a^4 b^2 c^8+3 a^2 b^4 c^8+3 b^6 c^8-5 a^4 c^10-8 a^2 b^2 c^10-6 b^4 c^10+4 a^2 c^12+4 b^2 c^12-c^14) : :
X(13423) = 3 X[54] - 4 X[973] = 2 X[195] - 3 X[3060] = 16 X[973] - 15 X[3567] = 4 X[54] - 5 X[3567] = 5 X[3567] - 8 X[6152] = 2 X[973] - 3 X[6152] = 3 X[2] - 4 X[6153] = X[6241] - 4 X[6242] = 8 X[973] - 9 X[7730]
X(13423) = 5 X[3567] - 6 X[7730] = 2 X[54] - 3 X[7730] = 4 X[6152] - 3 X[7730] = 8 X[1209] - 7 X[7999] = 9 X[5640] - 8 X[8254] = 4 X[6288] - 3 X[11459] = 16 X[6689] - 17 X[11465]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26047.

Let A' be the inverse of A in the circumcircle of the A-adjunct anti-altimedial triangle, and define B', C' cyclically. Triangle A'B'C' is perspective to the reflection triangle at X(13423). (Randy Hutson, November 2, 2017)

X(13423) lies on these lines: {2,6153}, {3,13368}, {6,24}, {22,10203}, {143,9706}, {156,195} ,{382,1154}, {399,10263}, {511, 12325}, {1209,7999}, {1614,9920} ,{2888,11412}, {5640,8254}, {5890,12254}, {6241,6242}, {6288, 11459}, {6689,11465}, {7512, 11649}, {7691,12084}, {9512, 11816}, {9781,11808}, {10628, 12290}, {10938,12289}, {11061, 11271}, {11451,13365}

X(13423) = reflection of X(i) in X(j) for these {i,j}: {3, 13368}, {54, 6152}, {11412, 2888}, {12226, 1209}, {12291, 54}, {12316, 10263
X(13423) = X(4)-of-reflection-triangle
X(13423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (54,2917,11464), (54,6152,7730) ,(54,7730,3567), (54,12380, 2917), (6152,12291,3567), (7730, 12291,54)

X(13424) = BARYCENTRIC SQUARE OF X(13386)

Barycentrics (1 + sin A)/(1 - sin A) : :

X(13424) lies on these lines: {2, 585}, {158, 7080}, {6136, 9098}, {13386, 13389}

X(13424) = barycentric square of X(13386)
X(13424) = X(606)-isoconjugate of X(1123)
X(13424) = barycentric product X(i)*X(j) for these {i,j}: {1267, 1336}, {13386, 13386}
X(13424) = barycentric quotient X(i)/X(j) for these {i,j}: {605, 606}, {1124, 1335}, {1267, 5391}, {1336, 1123}, {3083, 3084}, {6136, 6135}, {6212, 6213}, {6364, 6365}, {13386, 13387}

X(13425) = POINT BEID 80

Barycentrics (1 + sin A)/(1 - cos A) : :
Barycentrics (a - b - c) (bc + S) : :

X(13425) lies on these lines: {8, 210}, {69, 13386}, {326, 1267}, {5391, 6348}

X(13425) = isoconjugate of X(j) and X(j) for these (i,j): {604, 1123}, {606, 1118}, {608, 6213}, {1395, 13387}, {3084, 7337}
X(13425) = barycentric product X(i)*X(j) for these {i,j}: {8, 1267}, {312, 3083}, {345, 13386}, {646, 6364}, {1124, 3596}, {1264, 1336}, {3718, 6212}
X(13425) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 1123}, {78, 6213}, {345, 13387}, {605, 604}, {644, 6135}, {1124, 56}, {1259, 1335}, {1264, 5391}, {1267, 7}, {1336, 1118}, {2289, 606}, {3083, 57}, {3719, 3084}, {6212, 34}, {6364, 3669}, {13386, 278}

X(13426) = POINT BEID 81

Barycentrics (1 + cos A)/(1 - sin A) : :
Barycentrics (a - b - c) (ab - S) (ac - S) : :

X(13426) lies on the Feuerbach hyperbola and these lines: {1, 1336}, {7, 13386}, {84, 6212}, {104, 6136}, {210, 1857}, {281, 7133}

X(13426) = isoconjugate of X(j) and X(j) for these (i,j): {7, 606}, {56, 3084}, {57, 1335}, {109, 6365}, {222, 6213}, {603, 13387}, {604, 5391}, {1123, 7125}, {2067, 13388}
X(13426) = barycentric product X(i)*X(j) for these {i,j}: {8, 1336}, {281, 13386}, {318, 6212}, {1267, 1857}, {4391, 6136}
X(13426) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 5391}, {9, 3084}, {33, 6213}, {41, 606}, {55, 1335}, {281, 13387}, {605, 7125}, {650, 6365}, {1124, 1804}, {1267, 7055}, {1336, 7}, {1857, 1123}, {3083, 7183}, {6136, 651}, {6212, 77}, {13386, 348}

X(13427) = POINT BEID 82

Barycentrics (1 + cos A)/(1 - csc A) : :
Barycentrics a(a - b - c)(ab - S) (ac - S) : :

X(13427) lies on these lines: {6, 9043}, {19, 5200}, {57, 481}, {497, 7347}, {1334, 1857}, {1776, 7348}, {2291, 6136}

X(13427) = isoconjugate of X(j) and X(j) for these {i,j}: {7, 1335}, {56, 5391}, {57, 3084}, {77, 6213}, {85, 606}, {222, 13387}, {651, 6365}, {1123, 1804} 1336}, {281, 13386}, {318, 6212}, {1267, 1857}, {4391, 6136}
X(13427) = barycentric product X(i)*X(j) for these {i,j}: {9, 1336}, {33, 13386}, {281, 6212}, {522, 6136}, {1857, 3083}
X(13427) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 5391}, {33, 13387}, {41, 1335}, {55, 3084}, {605, 1804}, {607, 6213}, {663, 6365}, {1124, 7183}, {1336, 85}, {2175, 606}, {3083, 7055}, {6136, 664}, {6212, 348}, {13386, 7182}

X(13428) = POINT BEID 83

Barycentrics (1 + tan A)/(1 - tan A) : :
Barycentrics (a^2-b^2-c^2-2 S) (a^2+b^2-c^2-2 S) (a^2-b^2+c^2-2 S) : :

X(13428) lies on the cubics K170 and K267 and on these lines: {2, 371}, {4, 52}, {97, 1590}, {343, 3071}, {485, 5417}, {489, 1600}, {492, 1599}, {571, 3069}, {588, 2165}, {591, 5406}, {1585, 1993}, {1586, 3580}, {2994, 13387}, {5422, 7389}, {5870, 7500}, {5905, 13386}, {6414, 11418}, {6561, 11090}, {7386, 12603}, {11433, 12239}

X(13428) = anticomplement of X(5409)
X(13428) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {19, 487}, {486, 4329}, {8576, 6360}
X(13428) = X(5408)-cross conjugate of X(2)
X(13428) = {X(486),X11091)}-harmonic conjjugate of X(2)
X(13428) = isoconjugate of X(j) and X(j) for these {i,j}: {6, 3377}, {19, 10665}
X(13428) = barycentric product X(i)*X(j) for these {i,j}: {75, 3378}, {264, 10666}, {486, 492}, {1585, 11091}, {1599, 5392}
X(13427) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3377}, {3, 10665}, {371, 372}, {486, 485}, {492, 491}, {1585, 1586}, {1599, 1993}, {1993, 1600}, {3378, 1}, {5408, 5409}, {5413, 5412}, {6414, 6413}, {8576, 8577}, {8940, 8944}, {10666, 3}, {11091, 11090}

X(13429) = POINT BEID 84

Barycentrics (1 + tan A)/(1 - cot A) : :
Barycentrics (a^2+b^2-c^2) (a^2-b^2+c^2) (a^2-b^2-c^2-2 S) (a^2+b^2-c^2-2 S) (a^2-b^2+c^2-2 S) : :

X(13429) lies on these lines: {2, 8961}, {4, 372}, {54, 1588}, {393, 847}, {1068, 1336}, {1123, 7040}, {1585, 1993}, {3535, 11091}, {8576, 10880}

X(13429) = isogonal conjugate of X(10665)
X(13429) = X(371)-cross conjugate of X(4)
X(13429) = X(1)-zayin conjugate of X(10665)
X(13429) = isoconjugate of X(j) and X(j) for these {i,j}: {1, 10665}, {3, 3377}, {1600, 1820}
X(13429) = barycentric product X(i)*X(j) for these {i,j}: {92, 3378}, {486, 1585}, {847, 1599}, {2052, 10666}
X(13429) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 10665}, {19, 3377}, {24, 1600}, {371, 5409}, {486, 11090}, {1585, 491}, {1599, 9723}, {3378, 63}, {5413, 372}, {8576, 6413}, {10666, 394}

X(13430) = POINT BEID 85

Barycentrics (1 + cot A)/(1 - tan A) : :
Barycentrics (a^2-b^2-c^2) (a^2-b^2-c^2-2 S) (a^2+b^2-c^2-2 S) (a^2-b^2+c^2-2 S) : :

X(13430) lies on these lines: {2,311}, {68, 488}, {69, 1590}, {486, 641}, {487, 1147}, {492, 1599}, {591, 10607}

X(13430) = isoconjugate of X(j) and X(j) for these {i,j}: {25, 3377}, {1096, 10665}
X(13430) = barycentric product X(i)*X(j) for these {i,j}: {76, 10666}, {304, 3378}, {492, 11091}
X(13430) = barycentric quotient X(i)/X(j) for these {i,j}: {63, 3377}, {371, 5412}, {394, 10665}, {492, 1586}, {1599, 24}, {3378, 19}, {5408, 372}, {6414, 8577}, {9723, 1600}, {10666, 6}, {11091, 485}

X(13431) = POINT BEID 86

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^2-2 a^2 b^2 c^2+b^4 c^2+4 a^2 c^4+b^2 c^4-c^6) : :
X(13431) = 2 X[140] - 3 X[1493] = 9 X[1209] - 10 X[1656] = 9 X[195] - 5 X[1656] = 5 X[1656] - 3 X[3519] = 3 X[1209] - 2 X[3519] = 3 X[195] - X[3519] = 9 X[54] - 7 X[3523] = 9 X[3574] - 8 X[3850] = 9 X[2888] - 13 X[5068] = 17 X[3533] - 18 X[6689] = 2 X[550] - 3 X[10619] = X[4] + 3 X[11271] = 3 X[3574] - 4 X[11803] = 2 X[3850] - 3 X[11803] = 5 X[1656] - 6 X[12242], 3 X[1209] - 4 X[12242], 3 X[195] - 2 X[12242], X[5073] - 9 X[12316], 17 X[3533] - 9 X[12325]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26050.

X(13431) lies on the Feuerbach hyperbola of the orthic triangle and on these lines: {4,539}, {6,17}, {54,3523}, {140,1493}, {185,550}, {193,10539}, {1858,9957}, {2888,5068}, {3533,6689}, {3574,3850}, {5073,5895}, {5446,11817}, {8550,12363}

X(13431) = reflection of X(i) in X(j) for these {i,j}: {1209, 195}, {3519, 12242}, {10625, 11577}, {12325, 6689}
X(13431) = X(4)-Ceva conjugate of X(140)
X(13431) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (195,3519,12242), (3519,12242,1209)

X(13432) = POINT BEID 87

Barycentrics 5 a^10-20 a^8 b^2+32 a^6 b^4-26 a^4 b^6+11 a^2 b^8-2 b^10-20 a^8 c^2+29 a^6 b^2 c^2-a^4 b^4 c^2-14 a^2 b^6 c^2+6 b^8 c^2+32 a^6 c^4-a^4 b^2 c^4+6 a^2 b^4 c^4-4 b^6 c^4-26 a^4 c^6-14 a^2 b^2 c^6-4 b^4 c^6+11 a^2 c^8+6 b^2 c^8-2 c^10 : :
X(13432) = 5 X[195] - 4 X[1209] = 6 X[195] - 5 X[1656] = 6 X[1209] - 5 X[3519] = 5 X[1656] - 4 X[3519] = 3 X[195] - 2 X[3519] = 8 X[1493] - 7 X[3526] = 6 X[2888] - 7 X[3851] = 7 X[3851] - 8 X[11803] = 3 X[2888] - 4 X[11803] = 15 X[1656] - 16 X[12242] = 9 X[1209] - 10 X[12242] = 9 X[195] - 8 X[12242] = 3 X[3519] - 4 X[12242] = 4 X[10619] - 3 X[12307] = 2 X[4] - 3 X[12316] = 4 X[140] - 3 X[12325]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26050.

X(13432) lies on the cubic K119 and these lines: {3,2889}, {4,12175}, {6,17}, {140,12325}, {382,539}, {1154,1657}, {1493,3526}, {2888,3851}, {7517,11061}, {10605,10619}

X(13432) = reflection of X(3) in X(11271)
X(13432) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (195,3519,1656), (2888,11803,3851)

X(13433) = POINT BEID 88

Barycentrics a^2 (3 a^12 b^2-12 a^10 b^4+15 a^8 b^6-15 a^4 b^10+12 a^2 b^12-3 b^14+3 a^12 c^2-22 a^10 b^2 c^2+34 a^8 b^4 c^2-13 a^6 b^6 c^2+10 a^4 b^8 c^2-25 a^2 b^10 c^2+13 b^12 c^2-12 a^10 c^4+34 a^8 b^2 c^4-16 a^6 b^4 c^4+5 a^4 b^6 c^4+10 a^2 b^8 c^4-21 b^10 c^4+15 a^8 c^6-13 a^6 b^2 c^6+5 a^4 b^4 c^6+6 a^2 b^6 c^6+11 b^8 c^6+10 a^4 b^2 c^8+10 a^2 b^4 c^8+11 b^6 c^8-15 a^4 c^10-25 a^2 b^2 c^10-21 b^4 c^10+12 a^2 c^12+13 b^2 c^12-3 c^14) : :
X(13433) = 2 X[140] - 3 X[6153] = 3 X[389] - 2 X[10619] = 3 X[6152] - X[10619] = 3 X[5446] - 2 X[11803] = 3 X[11808] - 2 X[12242] = 3 X[51] - X[12291] = X[550] - 3 X[13368]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26050.

X(13433) lies on these lines: {51,12291}, {140,6153}, {389,6152}, {511,3519}, {539,11819}, {550,13368}, {5446,11803}, {6000,6242}, {6759,12175}, {9935,12234}, {9969,11808}, {11793,12226}, {12380,13367}

X(13433) = reflection of X(i) in X(j) for these {i,j}: {389, 6152}, {12226, 11793}

X(13434) = X(1)X(59)∩X(3)X(143)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26069.

X(13434) lies on these lines: {1,59}, {2,578}, {3,143}, {4,569}, {5,49}, {6,5889}, {20,182}, {22,10982}, {23,10110}, {24,5640}, {26,9781}, {30,13353}, {51,7488}, {52,7691}, {60,2617}, {97,2055}, {155,11422}, {156,3851}, {184,3091}, {185,575}, {186,5462}, {195,11591}, {215,3614}, {323,11793}, {376,13336}, {378,10574}, {381,1614}, {548,13339}, {631,13352}, {1147,3090}, {1176,5480}, {1216,7550}, {1437,6915}, {1568,12242}, {1593,5050}, {1598,6800}, {1993,7395}, {1994,5562}, {2070,10095}, {2071,9729}, {2477,7173}, {2888,10112}, {2979,7509}, {3146,10984}, {3153,3574}, {3520,9730}, {3521,10721}, {3523,13346}, {3527,9715}, {3545,10539}, {3618,6815}, {3627,8718}, {3832,6759}, {3850,10540}, {5056,9306}, {5068,9544}, {5072,9704}, {5079,9703}, {5133,6146}, {5446,7512}, {5609,11017}, {5643,12038}, {5651,7486}, {5890,7526}, {5907,13366}, {5943,13367}, {5944,13364}, {6030,12088}, {6642,11449}, {6816,11427}, {7393,7998}, {7394,9833}, {7404,11442}, {7499,13142}, {7506,10545}, {7514,11412}, {7529,9707}, {7539,12429}, {7565,11572}, {7592,9818}, {8548,12282}, {10304,13347}, {10601,11425}, {10733,13403}, {11402,11441}, {11439,11456}, {11459,12161}, {12241,13160}

X(13434) = X(i)-aleph conjugate of X(j) for these (i,j): {21, 2940}, {6727, 1048}
X(13434) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,569,5012), (5,54,110), (5,567,54), (6,7503,5889), (54,110,9706), (182,11424,20), (1993,7395,11444), (3832,11003,6759), (5056,9545,9306), (5890,7526,11440), (7395,11426,1993), (7592,9818,12111), (10095,10610,2070), (11402,11479,11441), (11449,11451,6642)
X(13434) = SS(cos A → cos(B-C)) of X(2) (trilinear substitution)
X(13434) = barycentric product X(249)*X(8902)
X(13434) = barycentric quotient X(8902)/X(338)

X(13435) = POINT BEID 89

Barycentrics (1 - sin A)/(1 + sin A) : :
Barycentrics (bc - S)(ab + S)(ac + S) : :

X(13435) lies on these lines: {2,586}, {158,7080}, {6135,9099}, {13387,13388}

X(13435) = isotomic conjugate of X(13424)
X(13435) = isoconjugate of X(j) and X(j) for these (i,j): {31, 13424}, {605, 1336}
X(13435) = barycentric product X(i)*X(j) for these {i,j}: {1123, 5391}, {13387, 13387}
X(13435) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 13424}, {606, 605}, {1123, 1336}, {1335, 1124}, {3084, 3083}, {5391, 1267}, {6135, 6136}, {6213, 6212}, {6365, 6364}, {13387, 13386}

X(13436) = POINT BEID 90

Barycentrics (1 - sin A)/(1 + cos A) : :
Barycentrics (a + b - c)(a - b + c)(bc - S) : :

X(13436) lies on these lines: {7,8}, {175,490}, {226,5490}, {481,9312}, {5391,7183}, {6203,7131}, {7056,13387}

X(13436) = isotomic conjugate of X(13426)
X(13436) = X(99)-beth conjugate of X(175)
X(13436) = X(3084)-cross conjugate of X(5391)
X(13436) = isoconjugate of X(j) and X(j) for these (i,j): {6, 13427}, {31, 13426}, {41, 1336}, {605, 1857}, {607, 6212}, {663, 6136}, {2212, 13386}, {3083, 6059}
X(13436) = barycentric product X(i)*X(j) for these {i,j}: {7, 5391}, {85, 3084}, {348, 13387}, {1123, 7055}, {1335, 6063}, {4554, 6365}, {6213, 7182}
X(13436) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 13427}, {2, 13426}, {7, 1336}, {77, 6212}, {348, 13386}, {606, 41}, {651, 6136}, {1123, 1857}, {1335, 55}, {1804, 1124}, {3084, 9}, {3926, 13425}, {5391, 8}, {6213, 33}, {6365, 650}, {7055, 1267}, {7125, 605}, {7183, 3083}, {13387, 281}
X(13436) = {X(7),X(85)}-harmonic conjugate of X(13453)

X(13437) = POINT BEID 91

Barycentrics (1 - cos A)/(1 + sin A) : :
Barycentrics (a + b - c)(a - b + c)(ab + S)(ac + S : :

X(13437) lies on these lines: {4,65}, {7,13387}, {57,482}, {226,7090}, {278,2362}, {653,1585}

X(13437) = isotomic conjugate of X(13425)
X(13437) = isoconjugate of X(j) and X(j) for these (i,j): {8, 605}, {9, 1124}, {31, 13425}, {41, 1267}, {55, 3083}, {212, 13386}, {219, 6212}, {255, 13426}, {394, 13427}, {1336, 2289}, {3939, 6364}
X(13437) = barycentric product X(i)*X(j) for these {i,j}: {7, 1123}, {273, 6213}, {278, 13387}, {1118, 5391}, {1659, 1659}
X(13437) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 13425}, {7, 1267}, {34, 6212}, {56, 1124}, {57, 3083}, {278, 13386}, {393, 13426}, {604, 605}, {606, 2289}, {1096, 13427}, {1118, 1336}, {1123, 8}, {1335, 1259}, {3084, 3719}, {3669, 6364}, {5391, 1264}, {6135, 644}, {6213, 78}, {13387, 345}

X(13438) = POINT BEID 92

Barycentrics (1 - cos A)/(1 + csc A) : :
Barycentrics a(a + b - c)(a - b + c)(ab + S)(ac + S : :

X(13438) lies on these lines: {19,208}, {56,2362}, {57,482}, {65,7133}, {388,6203}, {1477,6135}, {6204,7098}

X(13438) = isoconjugate of X(j) and X(j) for these (i,j): {6, 13425}, {8, 1124}, {9, 3083}, {55, 1267}, {78, 6212}, {219, 13386}, {312, 605}, {326, 13427}, {394, 13426}, {644, 6364}, {1259, 1336}
X(13438) = barycentric product X(i)*X(j) for these {i,j}: {34, 13387}, {57, 1123}, {278, 6213}, {1118, 3084}, {1659, 2362}, {3676, 6135}
X(13438) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 13425}, {34, 13386}, {56, 3083}, {57, 1267}, {604, 1124}, {606, 1259}, {608, 6212}, {1096, 13426}, {1123, 312}, {1335, 3719}, {1397, 605}, {2207, 13427}, {3084, 1264}, {6135, 3699}, {6213, 345}, {13387, 3718}

X(13439) = POINT BEID 93

Barycentrics (1 - tan A)/(1 + tan A) : :
Barycentrics (a^2-b^2-c^2+2 S) (a^2+b^2-c^2+2 S) (a^2-b^2+c^2+2 S) : :

X(13439) lies on the cubics on K170 and K267 and on these lines: {2,372}, {4,52}, {97,1589}, {343,3070}, {486,5419}, {490,1599}, {491,1600}, {571,3068}, {589,2165}, {1585,3580}, {1586,1993}, {1991,5407}, {2351,8982}, {2994,13386}, {5422,7388}, {5871,7500}, {5905,13387}, {6413,11417}, {6560,11091}, {7386,12604}, {11433,12240}

X(13439) = isotomic conjugate of X(13428)
X(13439) = anticomplement X(5408)
X(13439) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {19, 488}, {485, 4329}, {8577, 6360}
X(13439) = X(5409)-cross conjugate of X(2)
X(13439) = isoconjugate of X(j) and X(j) for these (i,j): {6, 3378}, {19, 10666}, {31, 13428}, {48, 13429}, {1973, 13430}
X(13439) = cevapoint of X(6) and X(8996)
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,6515,13428), (68,5392,13428), (485,11090,2)
X(13439) = barycentric product X(i)*X(j) for these {i,j}: {75, 3377}, {264, 10665}, {485, 491}, {1586, 11090}, {1600, 5392}
X(13439) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3378}, {2, 13428}, {3, 10666}, {4, 13429}, {69, 13430}, {372, 371}, {485, 486}, {491, 492}, {1586, 1585}, {1600, 1993}, {1993, 1599}, {3377, 1}, {5409, 5408}, {5412, 5413}, {6413, 6414}, {8577, 8576}, {8944, 8940}, {10665, 3}, {11090, 11091}

X(13440) = POINT BEID 94

Barycentrics (1 - tan A)/(1 + cot A) : :
Barycentrics (a^2+b^2-c^2) (a^2-b^2+c^2) (a^2-b^2-c^2+2 S) (a^2+b^2-c^2+2 S) (a^2-b^2+c^2+2 S) : :

X(13440) lies on these lines: {4,371}, {54,1587}, {393,847}, {1068,1123}, {1336,7040}, {1586,1993}, {3536,11090}, {8577,10881}

X(13440) = isogonal conjugate of X(10666)
X(13440) = isotomic conjugate of X(13430)
X(13440) = X(372)-cross conjugate of X(4)
X(13440) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (393,3542,13429), (847,2165,13429)
X(13440) = isoconjugate of X(j) and X(j) for these (i,j): {1, 10666}, {3, 3378}, {31, 13430}, {48, 13428}, {255, 13429}, {1599, 1820}
X(13440) = barycentric product X(i)*X(j) for these {i,j}: {92, 3377}, {485, 1586}, {847, 1600}, {2052, 10665}
X(13440) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 13430}, {4, 13428}, {6, 10666}, {19, 3378}, {24, 1599}, {372, 5408}, {393, 13429}, {485, 11091}, {1586, 492}, {1600, 9723}, {3377, 63}, {5412, 371}, {8577, 6414}, {10665, 394}

X(13441) = POINT BEID 95

Barycentrics (1 - cot A)/(1 + tan A) : :
Barycentrics (a^2-b^2-c^2) (a^2-b^2-c^2+2 S) (a^2+b^2-c^2+2 S) (a^2-b^2+c^2+2 S) : :

X(13441) lies on these lines: {2,311}, {68,487}, {69,1589}, {485,642}, {488,1147}, {491,1600}, {1991,10607}

X(13441) = isotomic conjugate of X(13429)
X(13441) = isoconjugate of X(j) and X(j) for these (i,j): {25, 3378}, {31, 13429}, {1096, 10666}, {1973, 13428}
X(13441) = barycentric product X(i)*X(j) for these {i,j}: {76, 10665}, {304, 3377}, {491, 11090}
X(13441) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 13429}, {63, 3378}, {69, 13428}, {372, 5413}, {394, 10666}, {491, 1585}, {1600, 24}, {3377, 19}, {3926, 13430}, {5409, 371}, {6413, 8576}, {9723, 1599}, {10665, 6}, {11090, 486}

X(13442) = EULER LINE INTERCEPT OF THE LINE X(72)X(511)

Barycentrics 2 a^7-a^5 b^2+3 a^4 b^3-2 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+4 a^3 b^2 c^2+2 a^2 b^3 c^2+a b^4 c^2+b^5 c^2+3 a^4 c^3+2 a^3 b c^3+2 a^2 b^2 c^3+4 a b^3 c^3+b^4 c^3+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 : :
X(13442) = 3(r² + 2rR - s²)*X(2) - 2(2r² + 4rR - s²)*X(3)

Let I be the incenter of a triangle ABC. Let X = AI∩BC and let D be the midpoint of segment AX. Define E and F cyclically. Then X(13442) = X(4)-of-DEF. See Navneel Singhal, Tsihong Lau et al, AdvGeom3801 and AdvGeom3802, based on a problem stated in Art of Problem Solving, May 30, 2017..

X(13442) lies on these lines: {1,1503}, {2,3}, {72,511}, {226,3429}, {355,9958}, {524,11523}, {950,3666}, {1043,2893}, {1211,3430}, {1214,1891}, {1333,1901}, {1709,12779}, {1724,5480}, {1754,5799}, {1842,6708}, {2352,7354}, {2794,5988}, {4296,6356}, {4657,5436}, {5453,5483}, {5928,10393}, {6000,12672}

X(13442) = reflection of X(355) in X(9958)
X(13442) = X(4)-of-Gergonne-line-extraversion-triangle
X(13442) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,7413,442), (4,7567,1532), (20,7379,1010), (1314,1315,447)

X(13443) = CENTER OF 1st DAO-APOLLONIUS CIRCLE

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

Let A'B'C' be the intouch triangle of a triangle ABC. There exists a circle U, here named the 1st Dao-Apollonius circle, that is tangent to each of the four circles (AB'C'), (BC'A'), (CA'B'), (ABC). The center of U is X(13443), and the tangency point is X(13444). If, instead, A'B'C' is the orthic triangle, there exists a circle V, here named the 2nd Dao-Apollonius circle, tangents to each of the circles (AB'C'), (BC'A'), (CA'B'), (ABC); the center of V is X(7952) and the tangent point is X(108). (Dao Thanh Oai, Peter Moses, June 1, 2017).

X(13443) lies on the cubic K259 and these lines: {1,167}, {57,1130}

X(13443) = X(21)-beth conjugate of X(8082)
X(13443) = isoconjugate of X(260) and X(8372)

X(13444) = TANGENCY POINT OF THE 1st DAO-APOLLONIUS CIRCLE

Trilinears    sin(A/2)/(cos(B/2) - cos(C/2)) : :
Barycentrics    a^2 Sec[A/2] (Cos[B/2]+Cos[C/2])/((b-c) (b+c-a)) : :
Barycentrics    a^2/(sec(B/2) - sec(C/2)) : :

Let A'B'C' be the intouch triangle of a triangle ABC. There exists a circle U that is tangent to each of the four circles (AB'C'), (BC'A'), (CA'B'), (ABC). The center of U is X(13443), and the tangency point is X(13444); see A(13443).

X(13444) lies on the circumcircle and these lines: {1, 7597}, {56, 12809}, {101, 6733}, {104, 177}, {174, 10497}

X(13444) = trilinear pole of line X(6)X(266)
X(13444) = crosssum of X(3900) and X(6730)
X(13444) = incircle-inverse of X(12814)
X(13444) = X(21)-beth conjugate of X(7597)
X(13444) = X(164)-zayin conjugate of X(10495)
X(13444) = isoconjugate of X(j) and X(j) for these (i,j): {9, 10492}, {188, 10495}, {260, 522}
X(13444) = barycentric product X(i)*X(j) for these {i,j}: {177, 651}, {234, 6733}
X(13444) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 10492}, {177, 4391}, {1415, 260}

X(13445) = REFLECTION OF X(110) IN X(2071)

Barycentrics a^2 (a^8-a^6 b^2-3 a^4 b^4+5 a^2 b^6-2 b^8-a^6 c^2+11 a^4 b^2 c^2-7 a^2 b^4 c^2-3 b^6 c^2-3 a^4 c^4-7 a^2 b^2 c^4+10 b^4 c^4+5 a^2 c^6-3 b^2 c^6-2 c^8) : :
X(13445) = 5 X[74] - 2 X[3581] = X[5189] + 2 X[10990]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26078.

X(13445) lies on these lines: {3,6030}, {20,2888}, {22,10606}, {26,11468}, {30,74}, {64,394}, {110,2071}, {146,1568}, {185,1994}, {378,5012}, {539,12317}, {1154,10620}, {1204,3146}, {1498,11449}, {1593,5422}, {1597,5640}, {1614,11250}, {2070,12041}, {2777,3153}, {2781,11416}, {2935,12270}, {3060,10605}, {3098,11180}, {3426,10546}, {3448,13399}, {3520,10575}, {3524,4550}, {3529,7689}, {3543,11438}, {3839,10545}, {5189,10990}, {5889,12085}, {5894,12225}, {5897,6080}, {6241,12084}, {6644,11455}, {6800,11410}, {9730,12834}, {10060,11446}, {10298,11204}, {11064,12379}, {11441,13093}

X(13445) = reflection of X(i) in X(j) for these {i,j}: {110,2071}, {146,1568}, {2070, 12041}, {3448,13399}
X(13445) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (20,3357,11440), (20,11440, 7691), (22,10606,11454), (64, 11413,12111)}
X(13445) = X(775)-anticomplementary conjugate of X(146)}
X(13445) = crosssum of X(1562) and X(9409)}

X(13446) = POINT BEID 96

Barycentrics a^2 (a^12 (b^2+c^2) - 2 a^10 (2 b^4+3 b^2 c^2+2 c^4) + a^8 (5 b^6+6 b^4 c^2+6 b^2 c^4+5 c^6) + a^6 b^2 c^2 (b^4-20 b^2 c^2+c^4) + a^4 (-5 b^10+6 b^8 c^2+3 b^6 c^4+3 b^4 c^6+6 b^2 c^8-5 c^10) + a^2 (b^2-c^2)^2 (4 b^8-7 b^6 c^2+12 b^4 c^4-7 b^2 c^6+4 c^8) - (b^2-c^2)^4 (b^6-3 b^4 c^2-3 b^2 c^4+c^6) ) : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 26077.

X(13446) lies on these lines: {30, 5462}, {143, 11558}, {389, 3521}, {403, 511}, {569, 5899}, {2070, 11430}, {2071, 5943}, {5446, 11563}, {6000, 10113}, {6688, 10257}, {12900, 13391}

X(13446) = midpoint of X(i) and X(j) for these {i,j}: {143,11558}, {5446,11563}}
X(13446) = reflection of X(13376) in X(10110)}

X(13447) = POINT BEID 97

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

Let A'B'C' be the orthic triangle and I the incenter of a triangle ABC, and let
Ma = midpoint of AA", and define Mb and Mc cyclically
M1 = orthogonal projection of Ma on AI, and define M2 and M3 cyclically
. Then X(13447) is the A'B'C'-to-M1M2M3 orthology center, and X(9729) is the M1M2M-to-A'B'C' orthology center.

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 26077.

X(13447) lies on these lines: {}

X(13448) = 11th HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics a^10 b^2-a^8 b^4-4 a^4 b^8+7 a^2 b^10-3 b^12+a^10 c^2-4 a^8 b^2 c^2+3 a^6 b^4 c^2+6 a^4 b^6 c^2-16 a^2 b^8 c^2+10 b^10 c^2-a^8 c^4+3 a^6 b^2 c^4-6 a^4 b^4 c^4+9 a^2 b^6 c^4-13 b^8 c^4+6 a^4 b^2 c^6+9 a^2 b^4 c^6+12 b^6 c^6-4 a^4 c^8-16 a^2 b^2 c^8-13 b^4 c^8+7 a^2 c^10+10 b^2 c^10-3 c^12 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26084.

X(13448) lies on this line: {2,3}

X(13448) = centroid of (degenerate) cross-triangle of medial and orthic triangles

X(13449) = POINT BEID 98

Barycentrics 2 a^8-3 a^6 b^2+3 a^2 b^6-2 b^8-3 a^6 c^2-a^2 b^4 c^2+6 b^6 c^2-a^2 b^2 c^4-8 b^4 c^4+3 a^2 c^6+6 b^2 c^6-2 c^8 : :
X(13449) = 2 X[1353] - 3 X[1570] = 3 X[381] - X[2080] = 4 X[547] - 3 X[5215] = X[5184] - 3 X[5587] = 5 X[3843] - X[9301] = X[6033] + 3 X[10242] = X[5104] - 3 X[10516] = 3 X[7809] + X[10723] = 3 X[6787] - X[11674] = 3 X[7799] - X[13172]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26086.

X(13449) lies on these lines: {3,625}, {4,69}, {5,187}, {30, 114}, {182,7841}, {297,5972}, {381,2080}, {524,9880}, {538, 6321}, {542,8352}, {547,5215}, {575,7790}, {598,8590}, {631, 7910}, {842,10296}, {1353,1570}, {1478,5194}, {1479,5148}, {1691, 10358}, {1692,7745}, {2021,5475} ,{2031,3767}, {2076,10356}, {2459,10577}, {2460,10576}, {3095,7843}, {3398,7861}, {3564, 5107}, {3843,9301}, {5025,13335} ,{5097,7812}, {5104,10516}, {5184,5587}, {5613,8594}, {5617, 8595}, {5999,10722}, {6033, 10242}, {6054,8597}, {6390, 10992}, {6655,13334}, {7574, 10748}, {7773,9737}, {7799, 13172}, {7809,10723}, {7817, 11842}, {7918,10359}, {9775, 10989}, {10104,10631}, {11178, 11317}, {11303,13349}, {11304, 13350}, {11645,12177}

X(13449) = midpoint of X(i) and X(j) for these {i,j}: {4, 316}, {842, 10296}, {5999, 10722}, {6054, 8597}
X(13449) = reflection of X(i) in X(j) for these {i,j}: {3, 625}, {187, 5}, {10992, 6390}
X(13449) = polar-circle-inverse of X(6403)
X(13449) = {X(33517), X(33518)}-harmonic conjugate of X(5107)

X(13450) = POINT BEID 99

Trilinears sec^2 A cos(B - C) : :
Barycentrics b^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^2-2 b^2 c^2+c^4) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26090.

The trilinear polar of X(13450) passes through X(12077). (Randy Hutson, July 21, 2017)

X(13450) lies on the conic {{A, B, C, X(4), X(5)}} and these lines: {4,51}, {5,324}, {54,436}, {93, 1487}, {107,1141}, {235,2970}, {264,3090}, {327,1235}, {393,847} ,{467,8800}, {1173,4994}, {1594, 3613}, {1629,11816}

X(13450) = isogonal conjugate of X(19210)
X(13450) = polar conjugate of X(97)
X(13450) = X(2052)-Ceva conjugate of X(53)
X(13450) = X(i)-cross conjugate of X(j) for these (i,j): {53, 324}, {6750, 4}
X(13450) = trilinear product of vertices of Euler triangle
X(13450) = perspector of ABC and orthoanticevian triangle of X(324)
X(13450) = isoconjugate of X(j) and X(j) for these (i,j): {3, 2169}, {48, 97}, {54, 255}, {275, 4100}, {394, 2148}, {577, 2167}, {1092, 2190}, {6507, 8882}
X(13450) = barycentric product X(i)*X(j) for these {i,j}: {4, 324}, {5, 2052}, {53, 264}, {311, 393}, {343, 1093}, {467, 847}, {823, 2618}, {1969, 2181}, {6528, 12077}
X(13450) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 97}, {5, 394}, {19, 2169}, {51, 577}, {53, 3}, {158, 2167}, {216, 1092}, {311, 3926}, {324, 69}, {343, 3964}, {393, 54}, {467, 9723}, {1093, 275}, {1096, 2148}, {1393, 7125}, {1953, 255}, {2052, 95}, {2181, 48}, {3199, 184}, {6520, 2190}, {6524, 8882}, {6529, 933}, {7069, 2289}, {12077, 520}
X(13450) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1075,5890), (4,3168,3567), ( 51,8887,4), (107,8884,3518), ( 1093,2052,4)

X(13451) = POINT BEID 100

Barycentrics a^2 (2 a^6 (b^2+c^2)-6 a^4 (b^4+b^2 c^2+c^4)+a^2 (6 b^6-5 b^4 c^2-5 b^2 c^4+6 c^6)-(b^2-c^2)^2 (2 b^4-5 b^2 c^2+2 c^4)) : :
X(13451) = X(5) + X(3060)

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 26100.

X(13451) lies on these lines: {4, 13321}, {5, 3060}, {26, 3527}, {30, 51}, {52, 3850}, {140, 5446}, {143, 546}, {185, 12102}, {373, 10124}, {381, 11002}, {389, 3853}, {511, 547}, {548, 5462}, {549, 5640}, {568, 3845}, {1112, 11801}, {1154, 5066}, {1216, 12812}, {1658, 10982}, {1994, 7545}, {3567, 3627}, {3628, 3917}, {3856, 5876}, {3858, 5889}, {3859, 5907}, {3861, 6102}, {5447, 10219}, {5562, 12811}, {5891, 11737}, {6000, 12101}, {6030, 13353}, {7530, 9777}, {8254, 13383}, {9729, 12002}, {9969, 10272}, {11451, 11539}, {11793, 13421}, {12006, 12103}, {12100, 13363}, {12834, 13339}

X(13451) = midpoint of X(i) and X(j) for these {i,j}: {5,3060}, {568,3845}, {3917,10263}, {5446,5943}
X(13451) = reflection of X(i) in X(j) for these {i,j}: {140, 5943}, {547, 13364}, {3917, 3628}, {5447, 10219}, {5891, 11737}, {5943, 10095}, {12100, 13363}
=

X(13452) = POINT BEID 101

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26104.

X(13452) lies on the Jerabek hyperbola and these lines: {20,3519}, {54,3357}, { 64,3518}, {68,3529}, {186,3532}, {265,3146}, {3090,4846}, {3091, 3521}, {3426,10594}, {3431,6241} ,{3531,11403}, {3542,10293}, { 5365,11138}, {5366,11139}, { 6000,11270}, {6415,6447}, {6416, 6448}, {8717,11440}, {10990, 11564}, {11381,11738}, {11744, 12250}

X(13452) = isogonal conjugate of X(1657)}
X(13452) = X(12290)-cross conjugate of X(4)}
X(13452) = X(i)-vertex conjugate of X(j) for these (i,j): {3, 11270}, {4, 3532}
X(13452) = barycentric quotient X(6)/X(1657)}

X(13453) = POINT BEID 102

Barycentrics (1 + sin A)/(1 + cos A) : :

X(13453) lies on these lines: {7, 8}, {176, 489}, {226, 5491}, {482, 9312}, {1267, 7183}, {6204, 7131}, {7055, 13425}, {7056, 13386}

X(13453) = {X(7),X(85)}-harmonic conjugate of X(13436)
X(13453) = (3083)-cross conjugate of X(1267)
X(13453) = isoconjugate of X(j) and X(j) for these (i,j): {41, 1123}, {220, 13438}, {606, 1857}, {607, 6213}, {663, 6135}, {1253, 13437}, {2212, 13387}, {3084, 6059}
X(13453) = barycentric product X(i)X(j) for these {i,j}: {7, 1267}, {85, 3083}, {279, 13425}, {348, 13386}, {1124, 6063}, {1336, 7055}, {4554, 6364}, {6212, 7182}, {13424, 13436}
X(13453) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 1123}, {77, 6213}, {269, 13438}, {279, 13437}, {348, 13387}, {605, 41}, {651, 6135}, {1124, 55}, {1267, 8}, {1336, 1857}, {1804, 1335}, {3083, 9}, {6212, 33}, {6364, 650}, {7055, 5391}, {7125, 606}, {7183, 3084}, {13386, 281}, {13389, 7133}, {13424, 13426}, {13425, 346}, {13436, 13435}
X(13453) = {X(7),X(85)}-harmonic conjugate of X(13436)

X(13454) = POINT BEID 103

Barycentrics (1 + cos A)/(1 + sin A) : :

X(13454) lies on the Feuerbach hyperbola and these lines: {1, 1123}, {7, 13387}, {84, 6213}, {104, 6135}, {210, 1857}, {7091, 13438}

X(13455) = POINT BEID 104

Barycentrics (1 + cos A)/(1 + cot A) : :

X(13455) lies on these lines: {37, 5414}, {41, 7069}, {226, 481}, {1826, 7133}, {1903, 6413}

X(13455) = isoconjugate of X(j) and X(j) for these (i,j): {7, 371}, {56, 492}, {222, 1585}, {278, 5408}, {331, 8911}, {348, 5413}
X(13455) = barycentric product X(i)X(j) for these {i,j}: {9, 485}, {33, 11090}, {312, 8577}, {318, 6413}
X(13455) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 492}, {33, 1585}, {41, 371}, {212, 5408}, {485, 85}, {2212, 5413}, {6413, 77}, {8577, 57}, {11090, 7182}

X(13456) = POINT BEID 105

Barycentrics (1 + cos A)/(1 + csc A) : :

X(13456) lies on these lines:
{6, 7133}, {57, 482}, {497, 7090}, {1334, 1857}, {1776, 7347}, {2291, 6135}

X(13456) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1123,6213,13438)
X(13456) = isoconjugate of X(j) and X(j) for these (i,j): {7, 1124}, {56, 1267}, {57, 3083}, {77, 6212}, {85, 605}, {222, 13386}, {651, 6364}, {1336, 1804}, {1407, 13425}
X(13456) = barycentric product X(i)X(j) for these {i,j}: {9, 1123}, {33, 13387}, {200, 13437}, {281, 6213}, {346, 13438}, {522, 6135}, {1857, 3084}, {7090, 7133}, {13427, 13435}
X(13456) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 1267}, {33, 13386}, {41, 1124}, {55, 3083}, {200, 13425}, {606, 1804}, {607, 6212}, {663, 6364}, {1123, 85}, {1335, 7183}, {2175, 605}, {3084, 7055}, {6135, 664}, {6213, 348}, {13387, 7182}, {13427, 13424}, {13437, 1088}, {13438, 279}

X(13457) = POINT BEID 106

Barycentrics (1 + cot A)/(1 + csc A) : :

X(13457) lies on these lines:
{2, 13437}, {63, 13438}, {81, 1123}, {92, 1586}, {1585, 1748}

X(13457) = isoconjugate of X(j) and X(j) for these (i,j): {485, 605}, {3083, 8577}, {6212, 6413}
X(13457) = barycentric product X(i)X(j) for these {i,j}: {492, 1123}, {1585, 13387}
X(13457) = barycentric quotient X(i)/X(j) for these {i,j}: {371, 1124}, {492, 1267}, {1123, 485}, {1585, 13386}, {13387, 11090}

X(13458) = POINT BEID 107

Barycentrics (1 - sin A)/(1 - cos A) : :
Barycentrics (S - bc)(s - a) : :

X(13458) lies on these lines:
{8, 210}, {69, 13387}, {326, 3084}, {1267, 6347}, {7055, 13436}

X(13458) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,312,13425)
X(13458) = isoconjugate of X(j) and X(j) for these (i,j): {604, 1336}, {605, 1118}, {608, 6212}, {1106, 13426}, {1395, 13386}, {1407, 13427}, {3083, 7337}
X(13458) = barycentric product X(i)X(j) for these {i,j}: {8, 5391}, {312, 3084}, {345, 13387}, {346, 13436}, {646, 6365}, {1123, 1264}, {1335, 3596}, {3718, 6213}, {13425, 13435}
X(13458) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 1336}, {78, 6212}, {200, 13427}, {345, 13386}, {346, 13426}, {606, 604}, {644, 6136}, {1123, 1118}, {1259, 1124}, {1264, 1267}, {1335, 56}, {2289, 605}, {3084, 57}, {3719, 3083}, {5391, 7}, {6213, 34}, {6365, 3669}, {13387, 278}, {13425, 13424}, {13435, 13437}, {13436, 279}

X(13459) = POINT BEID 108

Barycentrics (1 - cos A)/(1 - sin A) : :

X(13459) lies on these lines:
{4, 65}, {7, 13386}, {57, 481}, {653, 1586}

X(13459) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,196,13437)
X(13459) = isoconjugate of X(j) and X(j) for these (i,j): {8, 606}, {9, 1335}, {41, 5391}, {55, 3084}, {212, 13387}, {219, 6213}, {1123, 2289}, {1253, 13436}, {3939, 6365}
X(13459) = barycentric product X(i)X(j) for these {i,j}: {7, 1336}, {273, 6212}, {278, 13386}, {279, 13426}, {1088, 13427}, {1118, 1267}, {13390, 13390}, {13424, 13437}
X(13459) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 5391}, {34, 6213}, {56, 1335}, {57, 3084}, {278, 13387}, {279, 13436}, {604, 606}, {605, 2289}, {1118, 1123}, {1124, 1259}, {1267, 1264}, {1336, 8}, {3083, 3719}, {3669, 6365}, {6136, 644}, {6212, 78}, {13386, 345}, {13424, 13425}, {13426, 346}, {13427, 200}, {13437, 13435}

X(13460) = POINT BEID 109

Barycentrics (1 - cos A)/(1 - csc A) : :

X(13460) lies on these lines:
{19, 208}, {56, 7968}, {57, 481}, {388, 6204}, {1477, 6136}, {1875, 2362}, {6203, 7098}, {7091, 13426}

X(13460) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (19,208,13438), (1336,6212,13427)
X(13460) = isoconjugate of X(j) and X(j) for these (i,j): {8, 1335}, {9, 3084}, {55, 5391}, {78, 6213}, {219, 13387}, {220, 13436}, {312, 606}, {644, 6365}, {1123, 1259}
X(13460) = barycentric product X(i)X(j) for these {i,j}: {34, 13386}, {57, 1336}, {269, 13426}, {278, 6212}, {279, 13427}, {1118, 3083}, {3676, 6136}, {13424, 13438}
X(13460) = barycentric quotient X(i)/X(j) for these {i,j}: {34, 13387}, {56, 3084}, {57, 5391}, {269, 13436}, {604, 1335}, {605, 1259}, {608, 6213}, {1124, 3719}, {1336, 312}, {1397, 606}, {3083, 1264}, {6136, 3699}, {6212, 345}, {13386, 3718}, {13426, 341}, {13427, 346}, {13438, 13435}

X(13461) = POINT BEID 110

Barycentrics (1 - tan A)/(1 - sec A) : :

X(13461) lies on these lines:
{8, 21}, {190, 13386}, {312, 7090}, {346, 13425}, {4417, 13387}

X(13461) = isoconjugate of X(j) and X(j) for these (i,j): {34, 6414}, {57, 8576}, {486, 604}, {1395, 11091}
X(13461) = barycentric product X(i)X(j) for these {i,j}: {8, 491}, {345, 1586}, {372, 3596}, {5409, 7017}
X(13461) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 486}, {55, 8576}, {219, 6414}, {345, 11091}, {372, 56}, {491, 7}, {1586, 278}, {5409, 222}, {5412, 608}

X(13462) = POINT BEID 111

Barycentrics a(5a - b - c)(a + b - c)(a - b + c) : :
X(13462) = (4R - r)*X(1) - 4r*X(3)

See Tran Quang Hung and Peter Moses, Hyacinthos 26107.

X(13462) lies on these lines:
{1,3}, {2,4315}, {7,551}, {9,11194}, {10,4308}, {101,604}, {104,3062}, {106,269}

X(13462) = isogonal conjugate of X(4900)
X(13462) = crosssum of X(i) and X(j) for these (i,j): {2170, 4814}
X(13462) = X(21)-beth conjugate of X(165)
X(13462) = isoconjugate of X(j) and X(j) for these (i,j): {1, 4900}, {522, 6014}
X(13462) = barycentric product X(i)X(j) for these {i,j}: {57, 3241}, {651, 6006}, {1014, 4029}, {4554, 8656}
X(13462) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 4900}, {1415, 6014}, {3241, 312}, {4029, 3701}, {4982, 3702}, {6006, 4391}, {8656, 650}
X(13462) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,36,165), (1,46,11531), (1,56, 3361), (1,165,9819), (1,2093, 11224), (1,3361,3339), (3,56, 13370), (56,1319,57), (56,1420, 1), (56,1617,36), (57,1319,1), ( 57,1420,1319), (354,13384,1), ( 999,3576,1), (999,5126,3576), ( 1385,3333,1), (1388,3340,1), ( 1470,2078,5010), (3304,3601,1), (4308,5265,10), (10246,11529,1)

X(13463) = POINT BEID 112

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

See Alexander Skutin and Angel Montesdeoca, Hyacinthos 26115.

Let A' be the orthocenter of BCX(1), and define B' and C' cyclically. A'B'C' is also the anticevian triangle, wrt intouch triangle, of X(1), and also the reflection of the 2nd Schiffler triangle in X(11). X(13463) is the nine-point center of A'B'C'; see also X(1699) and X(3680). (Randy Hutson, July 21, 2017)

X(13463) lies on these lines:
{1, 528}, {4, 10912}, {5, 2802}, {8, 10896}, {10, 3829}, {11, 8256}, {12, 3885}, {145, 5229}, {149, 10950}, {355, 5854}, {392, 9710}, {404, 13205}, {496, 6797}, {516, 11260}, {517, 3813}, {518, 4301}, {519, 3845}, {529, 12699}, {908, 3893}, {946, 3880}, {962, 12513}, {1001, 9785}, {1145, 7741}, {1320, 10944,}, {1329, 10914}, {1537, 5881}, {1697, 6690}, {1699, 3680}

X(13463) = midpoint of X(i) and X(j) for these {i,j}: {4,10912}, {962,12513}, {1320,13271}, {3161,12512}
X(13463) = reflection of X(i) in X(j) for these {i,j}: {8715,5901}, {10915,9955}, {12607,946}
X(13463) = anticomplement of X(32157)

X(13464) = POINT BEID 113

Trilinears 3 r + 2 R cos B cos C : :
Barycentrics 2 a^4-3 a^3 b-3 a^2 b^2+3 a b^3+b^4-3 a^3 c+6 a^2 b c-3 a b^2 c-3 a^2 c^2-3 a b c^2-2 b^2 c^2+3 a c^3+c^4 : :
X(13464) = X(13464) = 3 X[1] + X[4] =X[3] - 3 X[551] = 5 X[1] - X[944] = 5 X[4] + 3 X[944] = X[4] - 3 X[946] = X[944] + 5 X[946] = 2 X[140] - 3 X[1125] = X[550] - 3 X[1385] = 3 X[10] - 5 X[1656]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26117.

X(13464) lies on these lines:
{1,4}, {2,5734}, {3,551}, {5,519}, {8,5056}, {10,1482}, { 11,11011}, {12,5048}, {40,3306}, {84,5558}, {104,5557}, {140,517} ,{145,5068}, {165,10299}, {214, 10993}, {354,5884}, {355,3244}, { 376,9589}, {382,3655}, {495, 7681}, {496,6738}, {499,4848}, { 516,550}, {527,3560}, {547,4745} ,{553,5563}, {631,7991}

X(13464) = midpoint of X(i) and X(j) for these {i,j}: {1, 946}, {3, 4301}, {4, 5882}, {5, 10222}, {10, 1482}, {355, 3244}, {551, 3656}, {3817, 10247}, {5884, 12672}, {7982, 11362}
X(13464) = reflection of X(i) in X(j) for these {i,j}: {1125, 5901}, {1385, 3636}, {4745, 547}, {6684, 1125}, {10172, 5886}
X(13464) = complement X(11362)
X(13464) = crosssum of X(55) and X(2317)
X(13464) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4,5882), (1,1699,944), (1, 5603,946), (1,5691,7967), (1, 9612,3476), (1,9614,3486), (1, 11522,4), (1,12047,10106), (2, 5734,7982), (2,7982,11362), (3, 3656,4301), (4,5603,11522), (4, 11522,946), (8,8227,10175), (40, 3616,10165), (354,12672,5884), ( 355,10247,3244), (551,4301,3), ( 946,5882,4), (1125,3754,6692), ( 1385,10283,3636), (1482,5886, 10), (3244,3817,355), (5603, 10595,1), (5734,9624,11362), ( 7982,9624,2), (10531,10597, 1478), (10532,10596,1479)

X(13465) = POINT BEID 114

Barycentrics a (a^6+a^5 (b+c)-a^4 (4 b^2+b c+4 c^2)-2 a^3 (b-c)^2 (b+c)+a^2 (5 b^4+b^3 c-2 b^2 c^2+b c^3+5 c^4)+a (b-c)^4 (b+c)-2 (b^2-c^2)^2 (b^2+c^2)) : :
X(13465) =

See Le Viet An and Angel Montesdeoca, Hyacinthos 26119.

X(13465) lies on these lines:
{3, 191}, {8, 30}, {21, 10246}, {40, 12786}, {56, 1749}, {79, 10895}, {153, 5690}, {355, 12745}, {499, 3649}, {517, 7701}, {758, 1482}, {1046, 5492}, {1656, 11263}, {2095, 5789}, {2475, 5790}, {3065, 5697}, {3467, 5902}, {3577, 6597}, {3579, 4005}, {3678, 12515}, {3811, 11849}, {3878, 12773}, {4880, 9955}, {5055, 5221}, {12331, 12342}

X(13465) = reflection of X(3) in X(191)
X(13465) = anticomplement of X(33668)

X(13466) = POINT BEID 115

Barycentrics (ab + ac - 2bc)² : :
X(13466) = 2 X[668] + X[1015] = 3 X[1015] - 2 X[3227] = 3 X[668] + X[3227] = 5 X[3227] - 3 X[9263] = 5 X[1015] - 2 X[9263] = 5 X[2] - X[9263] = 5 X[668] + X[9263]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26123.

X(13466) lies on the Steiner inellipse and these lines:
{2,668}, {10,537}, {76, 4740}, {115,1211}, {536,6381}, { 599,2810}, {812,4370}, {891, 4728}, {1017,3570}, {1084,4755}, {1146,3452}, {1500,4033}, {2482, 2787}, {3679,3789}, {3762,6184}, {4482,8649}

X(13466) = midpoint of X(2) and X(668)
X(13466) = reflection X(1015) in X(2)
X(13466) = complement of X(3227)
X(13466) = X(8031)-cross conjugate of X(536)
X(13466) = crosspoint of X(2) and X(536)
X(13466) = crossdifference of every pair of points on line {739, 890}
X(13466) = crosssum of X(6) and X(739)
X(13466) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 4871}, {31, 536}, {101, 891}, {536, 2887}, {692, 4763}, {890, 1086}, {891, 116}, {899, 141}, {1918, 2229}, {1919, 1646}, {3230, 10}, {3768, 11}, {4526, 124}, {6381, 626}
X(13466) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 536}, {536, 8031}, {668, 891}
X(13466) = Steiner-inellipse-antipode of X(1015)
X(13466) = barycentric product X(i)X(j) for these {i,j}: {536, 536}, {899, 6381}, {3227, 8031}
X(13466) = barycentric quotient X(i)/X(j) for these {i,j}: {536, 3227}, {3230, 739}, {8031, 536}

X(13467) = POINT BEID 116

Trilinears cos(B-C)*(2*(2*cos(A)-cos(3*A) )*cos(B-C)+cos(2*(B-C))-cos(2* A)-2*cos(4*A)+3/2) : :
Barycentrics (S^2+SB*SC)*(-5*R^4+(-7*SA+5* SW)*R^2+5*S^2+2*SA^2-4*SB*SC- SW^2) : :
X(13467) = (9*R^4-6*SW*R^2+S^2+SW^2)*X(140)-R^4*X(389)

See Le Viet An and César Lozada, Hyacinthos 26128.

X(13467) lies on this line: {140,389}

X(13468) = POINT BEID 117

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26131.

X(13468) is the centroid of BaCaCbAbAcBc used in construction of the 9th Lozada circle; see X(9737). (Randy Hutson, July 21, 2017)

X(13468) lies on these lines:
{2,6}, {5,754}, {30,5171}, {140,7751}, {157,9909}, {523,11 633}, {538,549}, {543,8703}, {547 ,7775}, {632,7764}

X(13468) = complement of X(9766)
X(13468) = {X(591),X(1991)}-harmonic conjugate of X(69)

X(13469) = POINT BEID 118

Barycentrics 2 a^16 - 5 a^14 (b^2+c^2) - a^12 (17 b^4+26 b^2 c^2+17 c^4) + a^10 (91 b^6+115 b^4 c^2+115 b^2 c^4+91 c^6) -3 a^8 (55 b^8+28 b^6 c^2+28 b^4 c^4+28 b^2 c^6+55 c^8) +3 a^6 (b^2+c^2) (51 b^8-92 b^6 c^2+79 b^4 c^4-92 b^2 c^6+51 c^8) - a^4 (b^2-c^2)^2 (75 b^8-68 b^6 c^2-93 b^4 c^4-68 b^2 c^6+75 c^8) + a^2 (b^2-c^2)^4 (b^2+c^2) (17 b^4-64 b^2 c^2+17 c^4) - (b^2-c^2)^6 (b^4-14 b^2 c^2+c^4) : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 26135.

X(13469) lies on these lines: {2,3}

X(13469) = midpoint of X(5501) and X(10289)

X(13470) = POINT BEID 119

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26141.

X(13470) lies on these lines: {3,12278}, {4,13353}, {5, 1495}, {20,3581}, {30,143}, {54,7574}, {265,7512}, {1154,6146}, {1594,10610}, {2072,5944}, {3521,10296}, {3575,12006}, {3850,13419 }, {5073,9777}, {5663,12605}, {6102,12225}, {6689,13413}, {6756,13364}, {7525,9927}, {10095,11819}, {10263,12022}, {11591,12362}, {12370,13391}

X(13470) = midpoint of X(i) and X(j) for these {i,j}: {5, 11750}, {6102, 12225}
X(13470) = reflection of X(i) in X(j) for these {i,j}: {3575, 12006}, {6146, 11565}, {11264, 6146}, {11591, 12362}, {11819, 10095}, {13419, 3850}

X(13471) = POINT BEID 120

Barycentrics 8*S^4+(918*R^4-360*R^2*SW+12* SA^2-12*SW*SA+32*SW^2)*S^2+( 63*R^2-16*SW)*(27*R^2-4*SW)*( SA-SW)*SA : :
X(13471) = 3*X(548) - 2*X(7471)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26145.

X(13471) lies on this line: {2,3}

X(13472) = POINT BEID 121

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26152.

X(13472) lies on the Jerabek hyperbola and these lines: {2,1493}, {3,1199}, {4,11423}, {6,3518}, {54,11202}, {64,7592}, {68,3090}, {69,575}, {70,11427}, {74,578}, {184,1173}, {248,7772}, {265,3091}, {323,13154}, {389, 3431}, {568,12226}, {576,1176}, {879,7950}, {3146,3521}, {3147, 5486}, {3426,11426}, {3520,3532} ,{3527,10594}, {3529,4846}, {3531,5198}, {3628,9716}, {5890, 11270}, {6241,11738}, {6391, 9925}, {6413,6420}, {6414,6419}, {6415,6428}, {6416,6427}, {10261,10783}, {10262,10784}, {11004,13353}

X(13472) = isogonal conjugate of X(1656)
X(13472) = cevapoint of X(i) and X(j) for these (i,j): {6, 11402}, {184, 13345}
X(13472) = X(i)-cross conjugate of X(j) for these (i,j): {3567, 4}, {13351, 2}
X(13472) = isoconjugate of X(j) and X(j) for these (i,j): {1, 1656}, {92, 10979}
X(13472) = trilinear pole of line {647, 1510}
X(13472) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 1656}, {184, 10979}, {8882, 4994}

X(13473) = 12th HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics (a^2+b^2-c^2) (a^2-b^2+c^2) (6 a^6-7 a^4 b^2-4 a^2 b^4+5 b^6-7 a^4 c^2+16 a^2 b^2 c^2-5 b^4 c^2-4 a^2 c^4-5 b^2 c^4+5 c^6) : :
X(13473) = 5 X[4] - X[186], 3 X[186] - 5 X[403], 3 X[4] - X[403], 4 X[186] - 5 X[468], 4 X[403] - 3 X[468], 4 X[4] - X[468], X[2071] + 3 X[3543], X[3146] + 2 X[5159], 2 X[186] - 5 X[10151], 2 X[403] - 3 X[10151], 7 X[186] - 5 X[10295], 7 X[468] - 4 X[10295], 7 X[403] - 3 X[10295], 7 X[10151] - 2 X[10295], 7 X[4] - X[10295], 2 X[3627] + X[10297], 10 X[12102] - X[12105], 3 X[13202] + X[13399]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26156.

X(13473) lies on these lines: {2,3}, {34,10149}, {1112, 6000}, {1552,10421}, {10152,12079}, {12828,13202}

X(13473) = midpoint of X(382) & X(2072)
X(13473) = reflection of X(i) in X(j) for these {i,j}: {468, 10151}, {10151, 4}
X(13473) = polar-circle-inverse of X(3146)
X(13473) = X(3535)-Hirst inverse of X(3536)
X(13473) = orthoptic-circle-of-Steiner-inellipse-inverse of X(38282)
X(13473) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,378,3845), (4,382,235), (4, 1594,3861), (4,3543,25), (4, 3627,3575), (4,10736,1313), (4, 10737,1312), (4,12173,1906), ( 3575,10297,468)

X(13474) = POINT BEID 122

Barycentrics a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2+8 a^4 b^2 c^2-3 a^2 b^4 c^2-6 b^6 c^2-3 a^4 c^4-3 a^2 b^2 c^4+14 b^4 c^4+3 a^2 c^6-6 b^2 c^6-c^8) : :
X(13443) = 5 X[4] - 3 X[51] = 9 X[51] - 5 X[185] = 3 X[4] - X[185] = 6 X[51] - 5 X[389] = 2 X[185] - 3 X[389] = 9 X[389] - 10 X[3567] = 9 X[4] - 5 X[3567] = 3 X[185] - 5 X[3567] = 2 X[550] - 3 X[3819] = X[52] - 3 X[3830] = 9 X[373] - 11 X[3855] = X[3529] - 3 X[3917] = 3 X[3845] - 2 X[5462] = 7 X[3528] - 9 X[5650] = 7 X[185] - 9 X[5890] = 7 X[389] - 6 X[5890] = 7 X[51] - 5 X[5890] = 7 X[4] - 3 X[5890] = X[1657] - 3 X[5891] = 4 X[3850] - 3 X[5892] = 2 X[1216] - 3 X[5907] = 4 X[546] - 3 X[5943] = 15 X[5890] - 7 X[6241]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26156.

X(13474) lies on these lines: {4,51}, {6,12315}, {20,7998}, {25,3357}, {30,1216}, {52, 3830}, {54,12112}, {64,1598}, {140,11017}, {143,12102}, {373,3855}, {378,10282}, {381,9729}, {382,511}, {546,5943}, {548, 10170}, {550,3819}, {578,1498}, {1154,13433}, {1181,11403}, {1204,10594}, {1495,3520}, {1503, 13403}, {1593,6759}, {1595,2883} ,{1596,6247}, {1657,5891}, {1872,2818}, {1885,11577}, {2777, 3575}, {3091,11695}, {3146,5562}, {3515,11204}, {3516,11202}, {3517,10606}, {3528,5650}, {3529, 3917}, {3543,12111}, {3627, 10263}, {3839,10574}, {3843, 9730}, {3845,5462}, {3850,5892}, {3851,6688}, {3853,5446}, {3856, 13363}, {5059,11444}, {5066, 12046}, {5073,10625}, {5079, 10219}, {5198,10605}, {6152, 10628}, {6995,12250}, {7387, 11472}, {7516,8717}, {7530,7689} ,{7999,11001}, {9306,12085}, {9786,13093}, {10982,12174}, {11424,11456}, {11574,12605}, {11645,11750}

X(13474) = midpoint of X(i) and X(j) for these {i,j}: {4, 11381}, {185, 12290}, {382, 12162}, {3146, 5562}, {5073, 10625}, {12292, 13202}
X(13474) = reflection of X(i) in X(j) for these {i,j}: {20, 11793}, {143, 12102}, {185, 10110}, {389, 4}, {1657, 13348}, {5446, 3853}, {6241, 13382}, {10575, 9729}
X(13474) = crosssum of X(3) and X(550)
X(13474) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,185,10110), (4,6241,51), (4, 11455,11381), (4,12290,185), ( 51,6241,13382), (51,13382,389), (64,1598,11438), (185,10110, 389), (185,11381,12290), (381, 10575,9729), (1498,1597,578), ( 1593,6759,11430), (1598,3426, 64), (1657,5891,13348)

X(13475) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO ANTLIA

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)-H(a,b,c) : :
where F, G, H are as defined at X(12916)

The reciprocal orthologic center of these triangles is X(1)

X(13475) lies on these lines: {1,12916}, {174,12402}, {258,12396}, {6732,12886}, {7588,12388}, {8076,12387}, {8080,12397}, {8082,12398}, {8086,12393}, {8088,12394}, {8090,12404}, {8125,12389}, {8138,12881}, {8242,12400}, {8250,12405}, {8351,12401}, {8388,12399}, {8734,12385}, {9795,12391}, {11033,12403}, {11859,12386}, {11889,12390}, {11895,12392}, {11899,12395}

X(13475) = reflection of X(12916) in X(1)

Wolfram´s triangle conics perspectors: X(13476)-X(13482)

This preamble and centers X(13476)-X(13482) were contributed by César Eliud Lozada, June 13, 2017.

The appearance of (ℭ, n) in the following list means that the perspector of the conic ℭ is X(n):

(Brocard inellipse, 6), (De Longchamps ellipse, 13476), (dual of Yff parabola, 514), (Evans conic, 13477), (excentral-hexyl ellipse, 13478), (Feuerbach hyperbola, 650), (Jerabek hyperbola, 647), (Johnson circumconic, 216), (Kiepert hyperbola, 523), (Kiepert parabola, 99), (Lemoine inellipse, 598), (MacBeath circumconic, 3), (MacBeath inconic, 264), (Mandart inellipse, 8), (orthic inconic, 4), (Stammler hyperbola,*), (Steiner circumellipse, 2), (Steiner inellipse, 2), (Thomson-Gibert-Moses hyperbola, 13480), (Yff hyperbola, 13481), (Yff parabola, 190).

*: The polar triangle of ABC with respect to the Stammler hyperbola is ABC, i.e., ABC is self-polar with respect to the Stammler hyperbola.

For definitions of these conics, see Wolfram's Triangle Conics. For Thomson-Gibert-Moses hyperbola, see X(5642).

X(13476) = PERSPECTOR OF THE DE LONGCHAMPS ELLIPSE

Trilinears 1/(a^2-(b+c)*a-b*c) : :
X(13476) = X(37)-3*X(354) = X(75)+3*X(3873) = X(3555)+X(3696) = 3*X(3681)-7*X(4751) = 3*X(3742)-2*X(4698) = 3*X(4430)+5*X(4699)

X(13476) lies on these lines: {10,141}, {37,38}, {65,1418}, {75,3873}, {81,82}, {225,1876}, {244,872}, {596,740}, {674,3664}, {692,3449}, {876,4132}, {1002,4000}, {1037,5228}, {1468,2218}, {1486,3423}, {3271,7277}, {3446,5091}, {3555,3696}, {3668,5173}, {3681,4751}, {3686,9038}, {3742,4698}, {3779,4675}, {4032,5083}, {4430,4699}, {4644,9309}, {4674,5902}, {6007,7228}, {6665,9055}

X(13476) = midpoint of X(3555) and X(3696)
X(13476) = isogonal conjugate of X(1621)
X(13476) = trilinear pole of the line {661,665}

X(13477) = PERSPECTOR OF THE EVANS CONIC

Barycentrics 1/(3*a^8-9*(b^2+c^2)*a^6+(9*(b^2+c^2)^2-b^2*c^2)*a^4-(b^2+c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^2-7*(b^2-c^2)^2*b^2*c^2) : :

X(13477) lies on the line {632,9730}

X(13477) = isogonal conjugate of X(13482)

X(13478) = PERSPECTOR OF THE EXCENTRAL-HEXYL ELLIPSE

Barycentrics 1/((b+c)*a^2-b*c*a-b^3-c^3) : :
X(13478) = (r^2-s^2)*X(3)+(s^2+2*R*r+r^2)*X(10) = (s^2+r^2)*X(4)+(2*r^2-SW)*X(58)

X(13478) lies on the Kiepert hyperbola, cubic K321 and these lines: {2,572}, {3,10}, {4,58}, {5,6703}, {6,2050}, {11,1397}, {21,10454}, {27,2052}, {57,5307}, {63,321}, {76,6996}, {81,10478}, {83,7377}, {222,226}, {262,5145}, {275,469}, {295,2801}, {333,573}, {386,3597}, {485,2048}, {517,5769}, {527,4052}, {758,10441}, {867,11550}, {946,5707}, {991,7413}, {1446,7177}, {1503,5138}, {1719,1768}, {1796,6539}, {1797,4080}, {2067,5393}, {2792,11599}, {2996,7406}, {3452,5778}, {5397,6830}, {5405,6502}, {6504,7381}, {6625,7384}, {6994,8796}

X(13478) = isogonal conjugate of X(573)
X(13478) = isotomic conjugate of X(4417)
X(13478) = polar conjugate of X(17555)
X(13478) = trilinear pole of the line {523,1459}
X(13478) = Cundy-Parry Phi transform of X(10)
X(13478) = Cundy-Parry Psi transform of X(58)
X(13478) = perspector of ABC and tangential triangle wrt excentral triangle of the excentral-hexyl ellipse
X(13478) = pole of antiorthic axis wrt excentral-hexyl ellipse
X(13478) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,5788,10), (6,2050,2051)

X(13479) = {98,648} ∩ {99,895}

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

X(13479) lies on these lines: {39,11169}, {98,648}, {99,895}, {107,8749}, {125,1494}, {523,9139}, {3269,10762}, {5095,9862}, {8541,10788}, {9166,9214}

X(13479) = isogonal conjugate of X(13480)
X(13479) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (98,2452,648), (98,11596,2452)

X(13480) = PERSPECTOR OF THE THOMSON-GIBERT-MOSES HYPERBOLA

Trilinears a/(a^8-(b^2+c^2)*a^6-(5*b^4-11*b^2*c^2+5*c^4)*a^4+5*(b^4-c^4)*(b^2-c^2)*a^2-(b^2-c^2)^2*b^2*c^2) : :
Trilinears 1/((2*cos(2*A)-1)*cos(B-C)+(8*cos(A)-10*cos(3*A))*cos(2*(B-C))-cos(3*(B-C))+cos(5*A)-7*cos(A)+8*cos(3*A)) : :

The Thomson-Gibert-Moses hyperbola is defined at X(5642) .

X(13480) = isogonal conjugate of X(13479)

X(13481) = PERSPECTOR OF THE YFF HYPERBOLA

Barycentrics 1/(2*a^4-2*(b^2+c^2)*a^2+b^2*c^2) : :

X(13481) lies on these lines: {157,3447}, {250,2453}, {338,9307}, {382,511}, {1485,7669}

X(13481) = isogonal conjugate of X(9544)
X(13481) = isotomic conjugate of X(7782)

X(13482) = ISOGONAL CONJUGATE OF X(13477)

Trilinears a*(3*a^8-9*(b^2+c^2)*a^6+(3*b^2-b*c+3*c^2)*(3*b^2+b*c+3*c^2)*a^4-(b^2+c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^2-7*(b^2-c^2)^2*b^2*c^2) : :
X(13482) = 4*X(54)-X(8718)

X(13482) lies on these lines: {2,13352}, {3,13421}, {4,9705}, {30,54}, {74,1994}, {110,3845}, {376,578}, {549,13434}, {567,8703}, {569,10304}, {1092,5071}, {1147,3839}, {1614,3543}, {3200,12817}, {3201,12816}, {3524,13346}, {3534,5012}, {3545,11424}, {3627,9706}, {5622,8584}, {7464,13366}, {10540,12101}, {11423,12085}

X(13482) = isogonal conjugate of X(13477)

X(13483) = CYCLOCEVIAN CONJUGATE OF X(13)

Barycentrics 1/ (b^2 c^2 (3 a^2 + 4 Sqrt[3] S)+4 (-a^2+b^2+c^2) S^2) : :

X(13483) lies on the curves K129b and Q066, and on these lines: {13,8446}, {15,8172}, {61,8014}, {396,11063}

X(13483) = X(i)-cross conjugate of X(j) for these (i,j): {2981, 2}, {11139, 4}
X(13483) = X(463)-vertex conjugate of X(11146)
X(13483) = cyclocevian conjugate of X(13)
X(13483) = barycentric quotient X(i)/X(j) for these {i,j}: {3489, 8437}, {8495, 627}

X(13484) = CYCLOCEVIAN CONJUGATE OF X(14)

Barycentrics 1/ (b^2 c^2 (3 a^2 - 4 Sqrt[3] S)+4 (-a^2+b^2+c^2) S^2) : :

X(13484) lies on the curves K129b and Q066, and on these lines: {14,8456}, {16,8173}, {62,8015}, {395,11063}

X(13484) = X(i)-cross conjugate of X(j) for these (i,j): {6151, 2}, {11138, 4}
X(13484) = X(462)-vertex conjugate of X(11145)
X(13484) = cyclocevian conjugate of X(14)
X(13484) = barycentric quotient X(i)/X(j) for these {i,j}: {3490, 8438}, {8496, 628}

X(13485) = CYCLOCEVIAN CONJUGATE OF X(99)

Barycentrics (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(13485) is the perspector of the conic through X(4), X(8), and the extraversions of X(8). This conic is a rectangular hyperbola centered at X(3448). (Randy Hutson, July 21, 2017)

X(13485) lies on these lines: {2,9514}, {23,325}, {297,323}, {315,6328}, {850,3448}, {5641,9143}, {6563,13219}

X(13485) = isogonal conjugate of X(7669)
X(13485) = isotomic conjugate of X(3448)
X(13485) = anticomplement of X(36830)
X(13485) = X(3447)-anticomplementary conjugate of X(4560)
X(13485) = X(110)-cross conjugate of X(2)
X(13485) = cyclocevian conjugate of X(99)
X(13485) = isoconjugate of X(j) and X(j) for these (i,j): {1, 7669}, {31, 3448}, {662, 8574}
X(13485) = cevapoint of X(i) and X(j) for these (i,j): {6, 11641}, {524, 5099}
X(13485) = barycentric product X(i)X(j) for these {i,j}: {76, 3447}, {4590, 6328}
X(13485) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3448}, {6, 7669}, {512, 8574}, {3447, 6}, {6328, 115}

X(13486) = KIEPERT IMAGE OF X(1)

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

Suppose that P = p : q : r (barycentrics) is a point in the plane of a triangle ABC. The Kiepert image of P is introduced here as the point

K(P) = 1/(c^2 p^2 q^2+2 c^2 p q^3+2 c^2 p q^2 r+(a^2-b^2+c^2) q^3 r-b^2 p^2 r^2-2 b^2 p q r^2+(-b^2+c^2) q^2 r^2-2 b^2 p r^3+(-a^2-b^2+c^2) q r^3) : :

Let K be the Kiepert hyperbola of the cevian triangle A'B'C' of P. Let A'' be the point, other than A', in which K meets line BC, and define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in K(P). (Peter Moses, June 15, 2017; see also X(13610).)

X(13486) lies on these lines: {21,7100}, {58,79}, {81,7073}, {109,476}, {110,9811}

X(13486) = cevapoint of X(i) and X(j) for these (i,j): {58, 513}, {650, 7073}
X(13486) = X(i)-cross conjugate of X(j) for these (i,j): {513, 79}, {650, 81}
X(13486) = X(163)-vertex conjugate of X(6742)
X(13486) = X(13486) = trilinear pole of line X(284)X(501)
X(13486) = (i)-zayin conjugate of X(j) for these (i,j): {1717, 656}, {1781, 9404}, {2940, 513}
X(13486) = isoconjugate of X(j) and X(j) for these (i,j): {6, 7265}, {10, 2605}, {35, 523}, {42, 4467}, {80, 526}, {100, 2611}, {101, 8287}, {109, 6741}, {226, 9404}, {319, 512}, {513, 3678}, {521, 1825}, {522, 2594}, {649, 3969}, {656, 6198}, {661, 3219}, {1018, 7202}, {1399, 4086}, {1442, 4041}, {1577, 2174}, {2003, 3700}, {3268, 6187}, {3733, 7206}, {4017, 4420}
X(13486) = barycentric product X(i)X(j) for these {i,j}: {79, 662}, {81, 6742}, {99, 2160}, {476, 3218}, {648, 7100}, {651, 3615}, {653, 1789}, {799, 6186}, {1414, 7110}, {4556, 6757}, {4573, 7073}
X(13486) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 7265}, {79, 1577}, {81, 4467}, {100, 3969}, {101, 3678}, {110, 3219}, {112, 6198}, {163, 35}, {513, 8287}, {649, 2611}, {650, 6741}, {662, 319}, {1018, 7206}, {1333, 2605}, {1415, 2594}, {1576, 2174}, {1789, 6332}, {2160, 523}, {2194, 9404}, {3218, 3268}, {3615, 4391}, {3733, 7202}, {4565, 1442}, {5546, 4420}, {6186, 661}, {6742, 321}, {7073, 3700}, {7100, 525}, {7110, 4086}, {7113, 526}, {8606, 8611}, {8818, 4036}

X(13487) = 13th HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics 2 a^10+a^8 b^2-10 a^6 b^4+4 a^4 b^6+8 a^2 b^8-5 b^10+a^8 c^2+16 a^6 b^2 c^2+4 a^4 b^4 c^2-36 a^2 b^6 c^2+15 b^8 c^2-10 a^6 c^4+4 a^4 b^2 c^4+56 a^2 b^4 c^4-10 b^6 c^4+4 a^4 c^6-36 a^2 b^2 c^6-10 b^4 c^6+8 a^2 c^8+15 b^2 c^8-5 c^10 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26162.

X(13487) lies on this line: {2,3}

X(13487) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,3845,12084), (381,7528,3858) , (546,5066,10224), (546,6677,4), (3091,7392,3851)

X(13488) = 14th HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics (a^2+b^2-c^2) (a^2-b^2+c^2) (2 a^6-3 a^4 b^2+b^6-3 a^4 c^2+12 a^2 b^2 c^2-b^4 c^2-b^2 c^4+c^6) : :
X(13488) = 5 X[4] - 3 X[428] = 3 X[428] + 5 X[1885] = 9 X[428] - 5 X[3575] = 3 X[4] - X[3575] = 3 X[1885] + X[3575] = 5 X[3575] - 3 X[6240] = 5 X[4] - X[6240] = 3 X[428] - X[6240] = 5 X[1885] + X[6240] = 6 X[428] - 5 X[6756] = 2 X[6240] - 5 X[6756] = 2 X[3575] - 3 X[6756] = 2 X[1885] + X[6756] = 3 X[3830] - X[7553] = 7 X[6240] - 15 X[7576] = 7 X[3575] - 9 X[7576] = 7 X[6756] - 6 X[7576] = 7 X[428] - 5 X[7576] = 7 X[4] - 3 X[7576] = 7 X[1885] + 3 X[7576] = X[3529] - 3 X[7667] = 5 X[3] - 6 X[7734] = 3 X[381] - 2 X[9825] = 4 X[3850] - 3 X[10127] = 7 X[3851] - 6 X[10128] = 2 X[550] - 3 X[10691] = X[6241] - 3 X[11245] = 3 X[3543] + X[12225] = 3 X[12022] + X[12290] = 13 X[5079] - 12 X[13361]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26162.

X(13488) lies on these lines: {2,3}, {6,5878}, {53,5065}, {68,11472}, {578,2883}, { 800,6748}, {952,1902}, {1503,13403}, {1876,12433}, {1968,5305}, {1990,7765}, {2777,10110}, {3426,12324}, {3564,12162}, {5663,13292}, {5894,11438}, {6000,12241}, {6146,11381}, {6241,11245}, {6776,12315}, {7583,11473}, {7584,11474}, { 8550,9968}, {11433,12250}, { 11475,11542}, {11476,11543}, { 12022,12290}

X(13488) = midpoint of X(i) and X(j) for these {i,j}: {4, 1885}, {382, 12605}, {6146, 11381}
X(13488) = reflection of X(6756) in X(4)
X(13488) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,4,1596), (4,20,1598), (4,24,1906), (4,378,235), (4,427,546), (4,1593,5), (4,1594,10151), (4,1597,1595), (4,3088,381), (4,3529,6995), (4,6240,428), (4,7507,3845), (20,6804,3), (235,378,140), (381,3548,5), (468,3520,3530), (1594,10151,3850), (3516,3542,549), (7577,10019,12811)

X(13489) = ISOGONAL CONJUGATE OF X(13491)

Barycentrics (a^8+a^6 b^2-4 a^4 b^4+a^2 b^6+b^8-3 a^6 c^2+4 a^4 b^2 c^2+4 a^2 b^4 c^2-3 b^6 c^2+3 a^4 c^4-4 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+4 a^4 b^2 c^2-4 a^2 b^4 c^2-b^6 c^2-4 a^4 c^4+4 a^2 b^2 c^4+3 b^4 c^4+a^2 c^6-3 b^2 c^6+c^8) : :
Barycentrics 1/((b^2+c^2)*a^6-(3*b^4-4*b^2* c^2+3*c^4)*a^4+(b^2+c^2)*(3*b^ 4-7*b^2*c^2+3*c^4)*a^2-(b^4+3* b^2*c^2+c^4)*(b^2-c^2)^2) : :

See Antreas Hatzipolakis, Peter Moses, and César Lozada, Hyacinthos 26166 and Hyacinthos 26167.

X(13489) lies on these lines: {30, 18350}, {1990, 35491}, {2452, 22949}, {3260, 18354}, {5627, 20299}

X(13489) = isogonal conjugate of X(13491)
X(13489) = isogonal conjugate of the complement of X(18439)

X(13490) = MIDPOINT OF X(2) AND X(7540)

Barycentrics 2 a^10-3 a^8 b^2-2 a^6 b^4+4 a^4 b^6-b^10-3 a^8 c^2-6 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^4-12 a^2 b^4 c^4-2 b^6 c^4+4 a^4 c^6+6 a^2 b^2 c^6-2 b^4 c^6+3 b^2 c^8-c^10 : :
Trilinears (3*cos(2*A)-2)*cos(B-C)-cos(A) *cos(2*(B-C))-cos(3*A) : :
X(13490) = (J^2 - 11) X[2] - (J^2 - 7) X[3]
X(13490) = 2 X[546] + X[3575] = X[1885] - 4 X[3861] = 5 X[3843] + X[6240] = X[5] + 2 X[6756] = 2 X[140] + X[7553] = 5 X[632] - 4 X[7734] = X[550] - 4 X[9825] = X[6146] - 4 X[10095] = 2 X[10691] - 3 X[11539] = X[6102] - 4 X[11745] = 4 X[6756] - X[11819] = 2 X[5] + X[11819] = 2 X[143] + X[12134] = 7 X[3851] - X[12225] = 5 X[5] - 2 X[12362] = 5 X[6756] + X[12362] = 5 X[11819] + 4 X[12362] = 4 X[10110] - X[12370] = 4 X[3850] - X[12605] = X[7667] - 8 X[13163] = X[140] - 4 X[13163] = X[7553] + 8 X[13163] = 2 X[5462] + X[13419]

As a point of the Euler line, X(13490) has Shinagawa coefficients (E - 4F, 9E + 12F)

See Antreas Hatzipolakis, Peter Moses, and César Lozada, Hyacinthos 26166 and Hyacinthos 26167.

X(13490) lies on these lines: {2, 3}, {143, 12134}, {206, 5476}, {524, 41714}, {539, 11808}, {542, 9969}, {1503, 5946}, {3564, 9971}, {3567, 43588}, {3654, 34657}, {5462, 13419}, {5892, 29012}, {6102, 11745}, {6146, 10095}, {10110, 12370}, {12824, 13451}, {13470, 18874}, {13491, 16621}, {13567, 34514}, {13630, 16655}, {14848, 19125}, {16657, 30522}, {16659, 37481}, {17814, 31815}, {18374, 18583}, {18474, 34417}, {18475, 19130}, {19139, 20423}

X(13490) = midpoint of X(i) and X(j) for these {i,j}: {2, 7540}, {3, 34603}, {4, 38321}, {381, 7576}, {3534, 34613}, {3654, 34657}, {3830, 38323}, {3845, 38322}, {7553, 7667}, {14269, 38320}
X(13490) = reflection of X(i) in X(j) for these {i,j}: {2, 23410}, {549, 10127}, {7667, 140}, {34614, 15690}, {34664, 5066}, {38321, 31830}
X(13490) = 1st- Droz-Farny circle-inverse of X(5189)
X(13490) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 7506, 13371}, {4, 13595, 2072}, {4, 13861, 5}, {5, 26, 7568}, {5, 6756, 11819}, {5, 7715, 37440}, {5, 37936, 6676}, {23, 37347, 25337}, {25, 381, 10201}, {25, 403, 21841}, {25, 11818, 5}, {26, 7528, 5}, {26, 13861, 7506}, {143, 12134, 32358}, {381, 7426, 547}, {381, 10201, 5}, {546, 10020, 5576}, {546, 10096, 13413}, {546, 21841, 5}, {2070, 5133, 140}, {3518, 5576, 10020}, {3542, 7564, 5}, {3830, 7529, 16072}, {3830, 16072, 18569}, {3845, 16532, 33332}, {6676, 21841, 10096}, {6756, 21841, 3575}, {6995, 18420, 7530}, {6997, 7514, 5}, {7516, 31305, 550}, {7528, 37122, 26}, {7529, 18569, 5}, {10201, 11818, 381}, {12362, 23411, 5}, {18586, 18587, 37444}

X(13491) = X(3)X(74)∩X(30)X(52)

Trilinears a*((b^2+c^2)*a^6-(3*b^4-4*b^2* c^2+3*c^4)*a^4+(b^2+c^2)*(3*b^ 4-7*b^2*c^2+3*c^4)*a^2-(b^4+3* b^2*c^2+c^4)*(b^2-c^2)^2) : :
Trilinears (2*cos(2*A)+3)*cos(B-C)-6*cos( A) : :
X(13491) = 11*X(3)-9*X(7998) = 9*X(3)-7*X(7999) = 7*X(3)-5*X(11444) = 5*X(3)-3*X(11459) = 3*X(3)-2*X(11591) = 9*X(3)-8*X(11592) = 2*X(4)-3*X(5946) = 3*X(4)-4*X(10095) = 5*X(5)-6*X(5892) = 11*X(5)-12*X(6688) = 3*X(5)-4*X(9729) = 7*X(5)-8*X(11695) = 11*X(5892)-10*X(6688) = 9*X(5892)-10*X(9729) = 21*X(5892)-20*X(11695) = 9*X(5946)-8*X(10095) = 9*X(6688)-11*X(9729)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26167.

X(13491) lies on these lines: {3,74}, {4,3521}, {5,2883}, {20,1154}, {26,10605}, {30,52}, {49,2071}, {51,3853}, {54,13445}, {64,7526}, {125,13406}, {140,12162}, {143,382}, {182,9968}, {184,11250}, {373,12811}, {376,10627}, {381,10574}, {389,3627}, {546,1514}, {548,5562}, {549,5907}, {550,6101}, {568,3146}, {974,10113}, {1181,12084}, {1204,1658}, {1216,8703}, {1350,9925}, {1493,13352}, {1498,6644}, {1657,5889}, {1986,11565}, {2772,5694}, {2888,12317}, {2935,11702}, {2937,8718}, {3060,5073}, {3091,13363}, {3357,10274}, {3529,6243}, {3530,5891}, {3534,11412}, {3567,3830}, {3581,12088}, {3843,11455}, {3845,5462}, {3851,11439}, {3858,5943}, {4846,6145}, {5055,11017}, {5059,13421}, {5076,9781}, {5446,13382}, {5449,10264}, {6642,12315}, {7502,7689}, {7527,13353}, {7530,9786}, {7728,11561}, {9818,13093}, {10226,13367}, {10625,12103}, {10733,13358}, {11003,12300}, {11465,12046}, {12085,12161}

X(13491) = midpoint of X(i) and X(j) for these {i,j}: {3,6241}, {185,10575}, {382,12279}, {1657,5889}, {3529,6243}, {10620,12270}
X(13491) = reflection of X(i) in X(j) for these (i,j): (382,143), (3627,389), (5446,13382), (5562,548), (5876,3), (6101,550), (6102,185), (7728,11561), (10113,974), (10263,6102), (10625,12103), (10733,13358), (11381,546), (11455,13364), (12111,11591), (12162,140), (13474,5462)
X(13491) = isogonal conjugate of X(13489)
X(13491) = X(5)-of-X(4)-anti-altimedial-triangle
X(13491) = X(20)-of-A*B*C*, where A*B*C is defined at X(5694)
X(13491) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,156,1511), (3,7999,11592), (3,10620,11440), (3,11456,156), (3,12111,11591), (381,10574,12006), (382,5890,143), (5462,13474,3845), (5890,12279,382), (5944,12041,3), (9730,11381,546), (10574,12290,381), (11591,11592,7999), (11591,12111,5876)

Cyclologic centers: X(13492)-X(13560)

This preamble and centers X(13492)-X(13560) were contributed by César Eliud Lozada, June 15, 2017.

X(13492) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-BROCARD TO ABC

Trilinears a*(5*b^2-c^2-a^2)*(5*c^2-a^2-b^2)*((a^2+b^2+c^2)^2-9*b^2*c^2) : :
X(13492) = 2*X(187)-3*X(9136)

The reciprocal cyclologic center of these triangles is X(1296)

X(13492) lies on the cubics K273, K792 and these lines: {2,2418}, {6,9871}, {23,10355}, {187,1296}, {895,10630}, {10354,11580}, {11186,13192}

X(13492) = isogonal conjugate of X(34581)
X(13492) = antigonal conjugate of X(39157)
X(13492) = cevapoint of X(6) and X(10355)
X(13492) = trilinear pole of the line {6088,10354}
X(13492) = X(112)-of-4th-anti-Brocard-triangle

X(13493) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 1st EHRMANN TO ABC

Trilinears a*(a^4-b^4+4*b^2*c^2-c^4)/((a^2+b^2+c^2)^2-9*b^2*c^2) : :

The reciprocal cyclologic center of these triangles is X(1296)

X(13493) lies on the cubic K108 and these lines: {6,10355}, {25,2930}, {6082,9084}, {8787,9966}

X(13493) = isogonal conjugate of X(39157)
X(13493) = isogonal conjugate of isotomic conjugate of X(34166)
X(13493) = circumcircle-inverse of X(34581)
X(13493) = circummedial-to-1st-Ehrmann similarity image of X(9084)
X(13493) = trilinear product X(31)*X(34166)

X(13494) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ABC TO GOSSARD

Trilinears a*(SA^2-SB^2)*(SA^2-SC^2)/(S^4+(SA^2-124*SW*R^2+288*R^4+13*SW^2)*S^2+3*SW*SA^2*(4*R^2-SW)) : :

The reciprocal cyclologic center of these triangles is X(13495)

X(13494) lies on the circumcircle and these lines: {74,12113}, {402,13495}, {842,13202}, {2706,12825}

X(13494) = reflection of X(13495) in X(402)
X(13494) = X(13495)-of-Gossard-triangle

X(13495) = CYCLOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO ABC

Barycentrics
(b^2-c^2)^2*(-a^2+b^2+c^2)^2*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)/(a^14-2*(b^2+c^2)*a^12-3*(b^4-4*b^2*c^2+c^4)*a^10+(b^2+c^2)*(11*b^4-24*b^2*c^2+11*c^4)*a^8-(9*b^8+9*c^8+b^2*c^2*(9*b^4-37*b^2*c^2+9*c^4))*a^6+20*(b^4-c^4)*(b^2-c^2)*b^2*c^2*a^4+(b^2-c^2)^2*(3*b^8+3*c^8-b^2*c^2*(3*b^4+13*b^2*c^2+3*c^4))*a^2-(b^4-c^4)^3*(b^2-c^2)) : :

The reciprocal cyclologic center of these triangles is X(13494)

X(13495) lies on the cubic K025 and these lines: {4,12369}, {402,13494}

X(13495) = reflection of X(13494) in X(402)
X(13495) = antigonal conjugate of X(1650)
X(13495) = X(13494)-of-Gossard-triangle

X(13496) = CYCLOLOGIC CENTER OF THESE TRIANGLES: TRINH TO ABC-X3 REFLECTIONS

Barycentrics SA*(54*R^4-(3*SA+25*SW)*R^2+SA^2+3*SW^2)*(SB+SC) : :

The reciprocal cyclologic center of these triangles is X(74)

X(13496) lies on these lines: {3,125}, {93,930}, {136,378}, {477,925}

X(13496) = reflection of X(5961) in X(3)
X(13496) = circumcircle-inverse-of-X(12121)
X(13496) = X(109)-of-Trinh-triangle if ABC is acute

X(13497) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO FUHRMANN

Trilinears 128*p^7*(p-4*q)+8*(84*q^2+13)*p^6-4*(88*q^2+39)*q*p^5+2*(32*q^4+12*q^2-15)*p^4+(28*q^2+79)*q*p^3-(8*q^4+44*q^2+9)*p^2+(7*q^2+6)*q*p-q^2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(13497) = 3*X(1)-X(13542)

The reciprocal cyclologic center of these triangles is X(13498)

X(13497) lies on the line {1,13542}

X(13498) = CYCLOLOGIC CENTER OF THESE TRIANGLES: FUHRMANN TO ANTI-AQUILA

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

The reciprocal cyclologic center of these triangles is X(13497)

X(13498) lies on the Fuhrmann circle and the line {8,1392}

X(13499) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 5th EULER TO 1st ANTI-BROCARD

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

The reciprocal cyclologic center of these triangles is X(13500)

X(13499) lies on the nine-points circle and these lines: {2,13510}, {83,115}

X(13500) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-BROCARD TO 5th EULER

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

The reciprocal cyclologic center of these triangles is X(13499)

X(13500) lies on the line {2,13511}

X(13501) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-BROCARD TO INNER-GREBE

Trilinears
a*(25*a^16-110*(b^2+c^2)*a^14+(50*b^4+39*b^2*c^2+50*c^4)*a^12+2*(b^2+c^2)*(141*b^4+790*b^2*c^2+141*c^4)*a^10+(692*b^8+692*c^8-b^2*c^2*(7853*b^4+3622*b^2*c^2+7853*c^4))*a^8-2*(b^2+c^2)*(549*b^8+549*c^8-2*b^2*c^2*(1666*b^4+1053*b^2*c^2+1666*c^4))*a^6-(898*b^12+898*c^12-(10133*b^8+10133*c^8-2*b^2*c^2*(20429*b^4-19103*b^2*c^2+20429*c^4))*b^2*c^2)*a^4+2*(b^2+c^2)*(335*b^12+335*c^12-(3146*b^8+3146*c^8-b^2*c^2*(10545*b^4-13132*b^2*c^2+10545*c^4))*b^2*c^2)*a^2-5*(5*b^8+5*c^8+(9*b^6+9*c^6-(14*b^4+14*c^4+b*c*(9*b^2-34*b*c+9*c^2))*b*c)*b*c)*(5*b^8+5*c^8-(9*b^6+9*c^6+(14*b^4+14*c^4-b*c*(9*b^2+34*b*c+9*c^2))*b*c)*b*c)) : :

The reciprocal cyclologic center of these triangles is X(13502)

X(13501) lies on the circumcircle of the 4th anti-Broard triangle and on these lines: (pending)

X(13501) = cyclologic center of these triangles: 4th anti-Brocard to outer-Grebe

X(13502) = CYCLOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO 4th ANTI-BROCARD

Trilinears
(25*(b^2+c^2)*a^16-10*(19*b^4+36*b^2*c^2+19*c^4)*a^14+(b^2+c^2)*(350*b^4+1509*b^2*c^2+350*c^4)*a^12+2*(13*b^8+13*c^8-b^2*c^2*(1507*b^4+3908*b^2*c^2+1507*c^4))*a^10-b^2*c^2*(b^2+c^2)*(653*b^4-13554*b^2*c^2+653*c^4)*a^8-2*(13*b^12+13*c^12-(34*b^8+34*c^8+5*b^2*c^2*(269*b^4-3068*b^2*c^2+269*c^4))*b^2*c^2)*a^6-(b^2+c^2)*(350*b^12+350*c^12-(3515*b^8+3515*c^8-2*b^2*c^2*(7179*b^4-14849*b^2*c^2+7179*c^4))*b^2*c^2)*a^4+2*(95*b^16+95*c^16-(763*b^12+763*c^12-(1954*b^8+1954*c^8-b^2*c^2*(1029*b^4+2690*b^2*c^2+1029*c^4))*b^2*c^2)*b^2*c^2)*a^2-(b^2+c^2)*(25*b^16+25*c^16-(205*b^12+205*c^12-(538*b^8+538*c^8-9*b^2*c^2*(43*b^4+22*b^2*c^2+43*c^4))*b^2*c^2)*b^2*c^2)-S*(a^2-b^2)*(a^2-c^2)*(5*a^6-(19*b^2+13*c^2)*a^4-(b^4-66*b^2*c^2+13*c^4)*a^2+5*c^6-b^6-b^4*c^2-19*b^2*c^4)*(5*a^6-(13*b^2+19*c^2)*a^4-(13*b^4-66*b^2*c^2+c^4)*a^2-c^6+5*b^6-19*b^4*c^2-b^2*c^4))*a : :

The reciprocal cyclologic center of these triangles is X(13501)

X(13502) lies on the line {6,13503}

X(13503) = CYCLOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO 4th ANTI-BROCARD

Trilinears
(25*(b^2+c^2)*a^16-10*(19*b^4+36*b^2*c^2+19*c^4)*a^14+(b^2+c^2)*(350*b^4+1509*b^2*c^2+350*c^4)*a^12+2*(13*b^8+13*c^8-(1507*b^4+3908*b^2*c^2+1507*c^4)*b^2*c^2)*a^10-b^2*c^2*(b^2+c^2)*(653*b^4-13554*b^2*c^2+653*c^4)*a^8-2*(13*b^12+13*c^12-(34*b^8+34*c^8+5*(269*b^4-3068*b^2*c^2+269*c^4)*b^2*c^2)*b^2*c^2)*a^6-(b^2+c^2)*(350*b^12+350*c^12-(3515*b^8+3515*c^8-2*(7179*b^4-14849*b^2*c^2+7179*c^4)*b^2*c^2)*b^2*c^2)*a^4+2*(95*b^16+95*c^16-(763*b^12+763*c^12-(1954*b^8+1954*c^8-b^2*c^2*(1029*b^4+2690*b^2*c^2+1029*c^4))*b^2*c^2)*b^2*c^2)*a^2-(b^2+c^2)*(25*b^16+25*c^16-(205*b^12+205*c^12-(538*b^8+538*c^8-9*b^2*c^2*(43*b^4+22*b^2*c^2+43*c^4))*b^2*c^2)*b^2*c^2)+S*(a^2-b^2)*(a^2-c^2)*(5*a^6-(19*b^2+13*c^2)*a^4-(b^4-66*b^2*c^2+13*c^4)*a^2+5*c^6-b^6-b^4*c^2-19*b^2*c^4)*(5*a^6-(13*b^2+19*c^2)*a^4-(13*b^4-66*b^2*c^2+c^4)*a^2-c^6+5*b^6-19*b^4*c^2-b^2*c^4))*a : :

The reciprocal cyclologic center of these triangles is X(13501)

X(13503) lies on the line {6,13502}

X(13504) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO 4th ANTI-EULER

Barycentrics ((35*R^4-4*(5*SA+6*SW)*R^2+4*SA^2+4*SA*SW+4*SW^2)*S^2-((3*SA+22*SW)*R^4-4*SW^2*(5*R^2-SW))*SA)*(SB+SC) : :
X(13504) = 4*X(128)-5*X(11444) = 4*X(137)-3*X(3060) = 3*X(568)-4*X(12026) = 2*X(930)-3*X(2979)

The reciprocal cyclologic center of these triangles is X(13505)

X(13504) lies on these lines: {3,13505}, {128,11444}, {137,3060}, {511,11671}, {568,12026}, {930,1298}, {1141,5889}, {1263,6243}, {6101,13512}, {12273,12281}

X(13504) = reflection of X(i) in X(j) for these (i,j): (5889,1141), (6243,1263), (13505,3), (13512,6101)

X(13504) = X(110)-of-3rd-anti-Euler-triangle
X(13504) = X(265)-of-4th-anti-Euler-triangle

X(13505) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO 3rd ANTI-EULER

Barycentrics ((35*R^4-4*(7*SA+6*SW)*R^2+4*SA^2+4*SA*SW+4*SW^2)*S^2+(36*R^6-3*(SA+18*SW)*R^4+4*SW^2*(7*R^2-SW))*SA)*(SB+SC) : :
X(13505) = 4*X(128)-3*X(11459) = 4*X(137)-5*X(3567) = 3*X(568)-2*X(1263) = 2*X(1141)-3*X(5890)

The reciprocal cyclologic center of these triangles is X(13504)

X(13505) lies on these lines: {3,13504}, {52,11671}, {128,11459}, {137,3567}, {568,1263}, {930,11412}, {1141,1303}, {1154,13512}, {12270,12278}

X(13505) = reflection of X(i) in X(j) for these (i,j): (11412,930), (11671,52), (13504,3)
X(13505) = X(265)-of-3rd-anti-Euler-triangle
X(13505) = X(110)-of-4th-anti-Euler-triangle

X(13506) = CYCLOLOGIC CENTER OF THESE TRIANGLES: REFLECTION TO 4th ANTI-EULER

Barycentrics ((-7*R^2+2*SW)*S^4+(7*R^2-2*SW)*(32*R^4+20*(SA-SW)*R^2-3*SA^2-2*SA*SW+3*SW^2)*S^2-(4*R^2-SW)*(4*(3*SA+4*SW)*R^4-(SA+14*SW)*SW*R^2-2*(SA-2*SW)*SW^2)*SA)*(SB+SC) : :

The reciprocal cyclologic center of these triangles is X(7731)

X(13506) lies on the reflection circle and these lines: {184,933}, {1614,8157}, {3567,10214}, {6241,10628}

X(13506) = X(5620)-of-4th-anti-Euler-triangle if ABC is acute
X(13506) = orthic-to-4th-anti-Euler similarity image of X(10214)

X(13507) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO ANTI-INVERSE-IN-INCIRCLE

Barycentrics (3*S^4+(2*R^4-2*(SA+2*SW)*R^2+3*SA^2-4*SA*SW-SW^2)*S^2+(36*R^6-3*(4*SA-9*SW)*R^4-2*(4*SA-3*SW)*SW*R^2-SA*SW^2)*SA)*(SB+SC) : :

The reciprocal cyclologic center of these triangles is X(13508)

X(13507) lies on these lines: {5,13508}, {160,11641}, {1598,11792}, {11411,12310}

X(13507) = reflection of X(13508) in X(5)
X(13507) = X(8701)-of-anti-incircle-circles-triangle if ABC is acute
X(13507) = orthic-to-anti-incircle-circles similarity image of X(11792)

X(13508) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO ANTI-INCIRCLE-CIRCLES

Barycentrics ((7*R^2-4*SW)*S^2+5*(2*R^2+SW)*R^2*SW)*SA^2+((7*R^4-5*R^2*SW+4*SW^2)*S^2-5*(2*R^2+SW)*R^2*SW^2)*SA-2*((-5*R^2+2*SW)*S^2+(2*R^2+SW)*(3*R^2-2*SW)*R^2)*S^2 : :

The reciprocal cyclologic center of these triangles is X(13507)

X(13508) lies on these lines: {4,11792}, {5,13507}, {12164,12319}

X(13508) = reflection of X(13507) in X(5)
X(13508) = X(8701)-of-anti-inverse-in-incircle-triangle if ABC is acute
X(13508) = orthic-to-anti-inverse-in-incircle similarity image of X(11792)

X(13509) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL TO CIRCUMSYMMEDIAL

Trilinears a*(a^8-2*(b^2+c^2)*a^6+(2*b^4-b^2*c^2+2*c^4)*a^4-2*(b^4-c^4)*(b^2-c^2)*a^2+(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(13509) = 2*X(187)-3*X(2715)

The reciprocal cyclologic center of these triangles is X(74)

Let L1 be the line that is the polar conjugate, wrt the 1st Ehrmann circumscribing triangle, of the Johnson circle. Let L2 be the line that is the polar conjugate, wrt the 2nd Ehrmann circumscribing triangle, of the Johnson circle. Then X(13509) = L1∩L2. (Randy Hutson, June 27, 2018)

X(13509) lies on the cubics K854, K890, the circle {{X(4),X(15),X(16),X(186),X(3484)}}, and on these lines: {4,6}, {32,6241}, {39,1614}, {74,187}, {112,6000}, {185,10312}, {186,1971}, {323,401}, {353,3148}, {574,11464}, {577,11459}, {1504,11462}, {1505,11463}, {1968,12290}, {2241,11461}, {2275,9638}, {2393,10766}, {3172,12315}, {3767,11457}, {5013,9707}, {5028,12283}, {5206,11468}, {5663,10317}, {5890,10311}, {7748,12289}, {8778,13093}, {10316,12111}, {10986,11438}

X(13509) = reflection of X(112) in X(8779)
X(13509) = isogonal conjugate of X(34579)
X(13509) = polar circle-inverse-of-X(53)
X(13509) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,3331,8744), (1971,3269,186), (8744,12112,3331)
X(13509) = crossdifference of every pair of points on line X(51)X(520)
X(13509) = intersection, other than X(4), of van Aubel line and circle {{X(4),X(15),X(16),X(186),X(3484)}}

X(13510) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO 2nd BROCARD

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

The reciprocal cyclologic center of these triangles is X(13511)

X(13510) lies on the anticomplementary circle and the line {2,13499}

X(13511) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 2nd BROCARD TO ANTICOMPLEMENTARY

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

The reciprocal cyclologic center of these triangles is X(13510)

X(13511) lies on the Brocard circle, cubic K548 and these lines: {2,13500}, {6,6655}

X(13511) = anticomplement of X(33665)

X(13512) = CYCLOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO ANTICOMPLEMENTARY

Barycentrics (3*R^2*(3*R^2-2*SW)+8*S^2)*SA^2-(2*R^2*(5*S^2-3*SW^2)+(9*R^4+4*S^2)*SW)*SA+2*(3*R^2*(2*R^2-SW)+2*S^2)*S^2 : :
X(13512) = 3*X(2)-4*X(6592) = 3*X(3)-2*X(1141) = 4*X(128)-3*X(381) = 4*X(137)-5*X(1656) = 5*X(631)-4*X(12026) = 3*X(930)-X(1141)

The reciprocal cyclologic center of these triangles is X(11671)

X(13512) lies on the Stammler circle and these lines: {2,1263}, {3,252}, {5,11671}, {20,10620}, {128,381}, {137,1656}, {399,6069}, {631,12026}, {999,7159}, {1154,13505}, {2925,2926}, {3295,3327}, {6101,13504}

X(13512) = reflection of X(i) in X(j) for these (i,j): (3,930), (1263,6592), (11671,5), (13504,6101)
X(13512) = anticomplement of X(1263)
X(13512) = {X(1263), X(6592)}-harmonic conjugate of X(2)

X(13513) = CYCLOLOGIC CENTER OF THESE TRIANGLES: AQUILA TO FUHRMANN

Trilinears 64*p^8+128*q*p^7-80*p^6-16*(8*q^2+3)*q*p^5-4*(16*q^4-12*q^2-3)*p^4+16*(11*q^2-7)*q*p^3+(32*q^4-28*q^2+9)*p^2-(64*q^2-57)*q*p-2*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(13513) = 3*X(165)-2*X(5951) = 5*X(1698)-4*X(5952) = 3*X(1699)-4*X(5950)

The reciprocal cyclologic center of these triangles is X(13514)

X(13513) lies on the Bevan circle, cubic K800 and these lines: {1,5606}, {3,9904}, {5,1768}, {165,5951}, {267,3336}, {583,5540}, {1698,5952}, {1699,5950}, {5127,5131}

X(13513) = reflection of X(1) in X(5606)

X(13514) = CYCLOLOGIC CENTER OF THESE TRIANGLES: FUHRMANN TO AQUILA

Trilinears 16*p^6+16*q*p^5+8*(2*q^2-3)*p^4-16*q*p^3+(16*q^4-32*q^2+9)*p^2-(4*q^2-3)*q*p-16*(q^2-1)*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal cyclologic center of these triangles is X(13513)

X(13514) lies on the Fuhrmann circle and these lines: {1,502}, {8,12535}

X(13514) = reflection of X(1) in X(502)

X(13515) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD TO CIRCUMMEDIAL

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

The reciprocal cyclologic center of these triangles is X(737)

X(13515) lies on the Brocard circle and these lines: {}

X(13516) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD TO 5th EULER

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

The reciprocal cyclologic center of these triangles is X(13517)

X(13516) lies on the Brocard circle and the line {858,6795}

X(13517) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 5th EULER TO 1st BROCARD

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

The reciprocal cyclologic center of these triangles is X(13516)

X(13517) lies on the nine-points circle and these lines: {115,182}, {127,9967}, {136,458}

X(13518) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 1st NEUBERG TO 2nd BROCARD

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

The reciprocal cyclologic center of these triangles is X(3)

X(13518) lies on the cubics K509, K794 and these lines: {2,694}, {83,689}, {125,626}, {670,2086}, {858,5103}, {1316,5149}, {3231,4563}, {3978,4609}, {5652,6033}

X(13518) = complement of X(9998)
X(13518) = orthoptic circle of Steiner inellipse-inverse-of-X(5976)

X(13519) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 2nd NEUBERG TO 2nd BROCARD

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

The reciprocal cyclologic center of these triangles is X(3)

X(13519) lies on the Neuberg 2nd circle, cubics K509, K796 and these lines: {2,4048}, {76,689}

X(13519) = orthoptic circle of Steiner inellipse-inverse-of-X(9478)

X(13520) = CYCLOLOGIC CENTER OF THESE TRIANGLES: INNER-VECTEN TO 2nd BROCARD

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

The reciprocal cyclologic center of these triangles is X(3)

X(13520) lies on the Vecten-inner circle, cubic K509 and these lines: {2,7599}, {485,925}

X(13520) = anticomplement of X(32500)

X(13521) = CYCLOLOGIC CENTER OF THESE TRIANGLES: OUTER-VECTEN TO 2nd BROCARD

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

The reciprocal cyclologic center of these triangles is X(3)

X(13521) lies on the Vecten-outer circle, cubic K509 and these lines: {2,7598}, {486,925}

X(13521) = anticomplement of X(32501)

X(13522) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 4th BROCARD TO INNER-GREBE

Barycentrics
-16*(a^2+b^2+c^2)*(a^8-14*(b^2+c^2)*a^6-4*(b^4+b^2*c^2+c^4)*a^4+18*(b^4-c^4)*(b^2-c^2)*a^2-(b^4-10*b^2*c^2+c^4)*(b^2-c^2)^2)*S+15*a^12+28*(b^2+c^2)*a^10-(285*b^4+382*b^2*c^2+285*c^4)*a^8-8*(b^2+c^2)*(3*b^4+52*b^2*c^2+3*c^4)*a^6+(317*b^8+317*c^8-2*b^2*c^2*(78*b^4+233*b^2*c^2+78*c^4))*a^4-4*(b^4-c^4)*(b^2-c^2)*(9*b^4-154*b^2*c^2+9*c^4)*a^2-5*(b^2-c^2)^4*(3*b^2+c^2)*(b^2+3*c^2) : :

X(13522) lies on the orthocentroidal circle and these lines: {}

X(13523) = CYCLOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO 4th BROCARD

Barycentrics
(2*S^2*(-4*SW*(27*(9*SA^2-6*SA*SW+8*SW^2)*R^4-(135*SA^2+57*SA*SW+64*SW^2)*SW*R^2+2*(14*SA^2+7*SA*SW+2*SW^2)*SW^2)+(5*SW-42*R^2)*S^4+(5832*R^6-3996*R^4*SW+8*(21*SA+80*SW)*SW*R^2-(5*SA^2+34*SA*SW+32*SW^2)*SW)*S^2)+S*(-16*SW^3*((27*SA^2+33*SA*SW-8*SW^2)*R^2-(11*SA^2+4*SA*SW-2*SW^2)*SW)+(108*R^4-4*(12*SA-83*SW)*R^2+SA^2+10*SA*SW-50*SW^2)*S^4+(108*(9*SA^2-6*SA*SW+8*SW^2)*R^4-12*(39*SA^2+49*SA*SW+4*SW^2)*SW*R^2+8*(12*SA+17*SW)*SA*SW^2)*S^2+S^6)+96*((3*SA+SW)*R^2-SA*SW)*SA*SW^4)*(SB+SC) : :

The reciprocal cyclologic center of these triangles is X(13522)

X(13523) lies on these lines: {}

X(13524) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 4th BROCARD TO OUTER-GREBE

Barycentrics
16*(a^2+b^2+c^2)*(a^8-14*(b^2+c^2)*a^6-4*(b^4+b^2*c^2+c^4)*a^4+18*(b^4-c^4)*(b^2-c^2)*a^2-(b^4-10*b^2*c^2+c^4)*(b^2-c^2)^2)*S+15*a^12+28*(b^2+c^2)*a^10-(285*b^4+382*b^2*c^2+285*c^4)*a^8-8*(b^2+c^2)*(3*b^4+52*b^2*c^2+3*c^4)*a^6+(317*b^8+317*c^8-2*b^2*c^2*(78*b^4+233*b^2*c^2+78*c^4))*a^4-4*(b^4-c^4)*(b^2-c^2)*(9*b^4-154*b^2*c^2+9*c^4)*a^2-5*(b^2-c^2)^4*(3*b^2+c^2)*(b^2+3*c^2) : :

The reciprocal cyclologic center of these triangles is X(0)

X(13524) lies on the orthocentroidal circle and these lines: {}

X(13525) = CYCLOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO 4th BROCARD

Barycentrics
(2*S^2*(-4*SW*(27*(9*SA^2-6*SA*SW+8*SW^2)*R^4-(135*SA^2+57*SA*SW+64*SW^2)*SW*R^2+2*(14*SA^2+7*SA*SW+2*SW^2)*SW^2)+(5*SW-42*R^2)*S^4+(5832*R^6-3996*R^4*SW+8*(21*SA+80*SW)*SW*R^2-(5*SA^2+34*SA*SW+32*SW^2)*SW)*S^2)-S*(-16*SW^3*((27*SA^2+33*SA*SW-8*SW^2)*R^2-(11*SA^2+4*SA*SW-2*SW^2)*SW)+(108*R^4-4*(12*SA-83*SW)*R^2+SA^2+10*SA*SW-50*SW^2)*S^4+(108*(9*SA^2-6*SA*SW+8*SW^2)*R^4-12*(39*SA^2+49*SA*SW+4*SW^2)*SW*R^2+8*(12*SA+17*SW)*SA*SW^2)*S^2+S^6)+96*((3*SA+SW)*R^2-SA*SW)*SA*SW^4)*(SB+SC) : :

The reciprocal cyclologic center of these triangles is X(13524)

X(13525) lies on these lines: {}

X(13526) = CYCLOLOGIC CENTER OF THESE TRIANGLES: MIDHEIGHT TO 4th BROCARD

Barycentrics
2*a^16-2*(b^2+c^2)*a^14-(3*b^4-8*b^2*c^2+3*c^4)*a^12-2*(b^4-c^4)*(b^2-c^2)*a^10+(b^2-c^2)^2*(9*b^4+22*b^2*c^2+9*c^4)*a^8+2*(b^4-c^4)*(b^2-c^2)*((b^2-c^2)^2-16*b^2*c^2)*a^6-(b^2-c^2)^2*(9*b^8+9*c^8-2*b^2*c^2*(3*b^4+19*b^2*c^2+3*c^4))*a^4+2*(b^4-c^4)*(b^2-c^2)^3*(b^4+6*b^2*c^2+c^4)*a^2+(b^2+c^2)^2*(b^2-c^2)^6 : :

The reciprocal cyclologic center of these triangles is X(4)

X(13526) lies on the half-altitude circle and these lines: {6,1562}, {1593,10229}

X(13527) = CYCLOLOGIC CENTER OF THESE TRIANGLES: REFLECTION TO 4th BROCARD

Barycentrics
a^16-4*(b^2+c^2)*a^14+2*(3*b^4+5*b^2*c^2+3*c^4)*a^12-4*(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^10+b^2*c^2*(b^2+2*c^2)*(2*b^2+c^2)*a^8+4*(b^8-c^8)*a^6*(b^2-c^2)-(b^2-c^2)^2*(6*b^8+5*b^4*c^4+6*c^8)*a^4+4*(b^8-c^8)*a^2*(b^2-c^2)^3-(b^2+c^2)^2*(b^2-c^2)^6 : :

The reciprocal cyclologic center of these triangles is X(4)

X(13527) lies on the reflection circle and the line {6,2914}

X(13528) = CYCLOLOGIC CENTER OF THESE TRIANGLES: EXTOUCH TO 1st CIRCUMPERP

Trilinears 2*a^6-(b+c)*a^5-(5*b^2-2*b*c+5*c^2)*a^4+2*(b+c)^3*a^3+4*(b^4+c^4-b*c*(b+c)^2)*a^2-(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2 : :
X(13528) = X(36)-3*X(165) = X(1155)-4*X(3579) = 3*X(1155)-4*X(10225) = 3*X(3579)-X(10225) = X(3689)+2*X(12515) = 2*X(5087)+X(6361) = 2*X(6681)-3*X(10164)

The reciprocal cyclologic center of these triangles is X(40)

X(13528) lies on these lines: {1,3}, {4,5123}, {10,11826}, {20,5176}, {100,2745}, {102,2743}, {104,3880}, {499,12700}, {515,1145}, {516,1532}, {601,4646}, {901,1295}, {912,3689}, {962,6921}, {972,2742}, {1158,5687}, {1293,2716}, {1519,3035}, {1753,4186}, {2800,5440}, {2829,6735}, {3560,3698}, {3683,6929}, {3871,12675}, {3916,11362}, {4187,6684}, {4640,5657}, {4863,5770}, {5057,6838}, {5080,6925}, {5087,6361}, {5180,6962}, {5252,6948}, {5450,10914}, {5493,11813}, {5836,6906}, {6681,10164}, {6891,12701}, {6959,12699}, {6961,11376}, {7701,9947}, {9943,11491}, {11499,12688}

X(13528) = midpoint of X(i) and X(j) for these {i,j}: {20,5176}, {40,2077}, {484,5537}, {3245,5538}, {5493,11813}
X(13528) = isogonal conjugate of X(34256)
X(13528) = reflection of X(i) in X(j) for these (i,j): (4,5123), (1319,3), (1519,3035)
X(13528) = circumcircle-inverse-of-X(10310)
X(13528) = extouch-isogonal conjugate of X(12665)
X(13528) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,10306,11508), (40,165,3428), (165,5119,3)

X(13529) = CYCLOLOGIC CENTER OF THESE TRIANGLES: INTANGENTS TO 2nd CIRCUMPERP

Trilinears (2*p^5*(4*p^3+4*p^2*q-5*p-4*q)-(8*q^4-14*q^2+1)*p^4-2*(4*(q^2-1)^2-q^2)*q*p^3+(12*q^4-21*q^2+8)*p^2+2*(q^2-1)*(3*q^2-4)*q*p-4*(1-q^2)^2)*(-p^2+1) : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal cyclologic center of these triangles is X(103)

X(13529) lies on the intangents circle and these lines: {1012,11700}, {6326,9611}

X(13530) = CYCLOLOGIC CENTER OF THESE TRIANGLES: CIRCUMSYMMEDIAL TO ORTHOCENTROIDAL

Barycentrics 1/(2*(b^2+c^2)*a^6-2*(3*b^4+b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(6*b^4-7*b^2*c^2+6*c^4)*a^2-(2*b^4+3*b^2*c^2+2*c^4)*(b^2-c^2)^2) : :

The reciprocal cyclologic center of these triangles is X(13531)

X(13530) lies on the circumcircle and these lines: {110,381}, {476,7575}, {925,10298}, {6325,7418}, {10296,10420}, {10788,11636}

X(13530) = orthocentroidal circle-inverse-of-X(7699)

X(13531) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ORTHOCENTROIDAL TO CIRCUMSYMMEDIAL

Barycentrics ((-27*R^2+8*SW)*SA^2+4*(3*R^2*(9*R^2-4*SW)+S^2+SW^2)*SA+72*SW*R^4+(-45*S^2-34*SW^2)*R^2+4*(3*S^2+SW^2)*SW)*(SB+SC) : :

The reciprocal cyclologic center of these triangles is X(13530)

X(13531) lies on the orthocentroidal circle and these lines: {187,6785}, {237,6324}, {11005,11564}

X(13532) = CYCLOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO INNER-CONWAY

Trilinears 4*p^4-4*q*p^3+(4*q^2-3)*p^2-4*(q^2-1)*q*p+(q^2-1)*q/p : :
where p=sin(A/2), q=cos((B-C)/2)
X(13532) = 3*X(1)-4*X(11734) = 2*X(117)-3*X(5587) = 3*X(124)-2*X(11734) = 5*X(1698)-4*X(6718) = 3*X(3576)-4*X(6711) = X(10703)-3*X(10716)

The reciprocal cyclologic center of these triangles is X(100)

X(13532) lies on these lines: {1,124}, {2,11700}, {4,2817}, {8,153}, {10,109}, {80,3738}, {102,515}, {117,5587}, {355,2818}, {516,10732}, {517,10747}, {519,10703}, {944,11713}, {946,10696}, {1361,5252}, {1364,1837}, {1478,1845}, {1698,6718}, {1737,1795}, {2349,2816}, {2773,13211}, {2779,5086}, {2785,13178}, {2792,3732}, {2802,10777}, {2807,3419}, {2829,2968}, {2835,3421}, {2849,4768}, {2853,13280}, {3042,5794}, {3153,5080}, {3576,6711}, {3699,7270}, {5847,10764}, {8227,11727}, {9532,12784}

X(13532) = reflection of X(i) in X(j) for these (i,j): (1,124), (109,10), (944,11713), (10696,946)
X(13532) = anticomplement of X(11700)

X(13533) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 2nd CONWAY TO INNER-GARCIA

Barycentrics
a^12+2*(b+c)*a^11-(4*b^2+5*b*c+4*c^2)*a^10-(b+c)*(10*b^2+11*b*c+10*c^2)*a^9+(5*b^4+5*c^4+b*c*(32*b^2-35*b*c+32*c^2))*a^8+4*(b+c)*(5*b^4+5*c^4+2*b*c*(b^2+12*b*c+c^2))*a^7-(50*b^4+50*c^4-b*c*(3*b^2+131*b*c+3*c^2))*b*c*a^6-(b+c)*(20*b^6+20*c^6-(26*b^4+26*c^4-b*c*(128*b^2+7*b*c+128*c^2))*b*c)*a^5-(5*b^8+5*c^8-(24*b^6+24*c^6+(49*b^4+49*c^4-b*c*(61*b^2+110*b*c+61*c^2))*b*c)*b*c)*a^4+(b+c)*(10*b^8+10*c^8-(32*b^6+32*c^6-(32*b^4+32*c^4+b*c*(7*b^2+110*b*c+7*c^2))*b*c)*b*c)*a^3+(b^2-c^2)^2*(4*b^6+4*c^6-(b^4+c^4+b*c*(11*b^2+68*b*c+11*c^2))*b*c)*a^2-(b^2-c^2)^3*(b-c)*(2*b^4+2*c^4-b*c*(5*b^2+18*b*c+5*c^2))*a-(b^2-c^2)^6 : :

The reciprocal cyclologic center of these triangles is X(13534)

X(13533) lies on these lines: {}

X(13534) = CYCLOLOGIC CENTER OF THESE TRIANGLES: INNER-GARCIA TO 2nd CONWAY

Barycentrics
a^10+2*(b+c)*a^9-2*(b^2+3*b*c+c^2)*a^8-2*(b+c)*(3*b^2+7*b*c+3*c^2)*a^7+b*c*(2*b^2-3*b*c+2*c^2)*a^6+3*(2*b^2+3*b*c+2*c^2)*(b+c)^3*a^5+(2*b^6+2*c^6+b*c*(b^2+3*b*c+c^2)*(3*b^2+22*b*c+3*c^2))*a^4-(b^2-c^2)*(b-c)*(2*b^4+2*c^4+b*c*(16*b^2+45*b*c+16*c^2))*a^3-(b^2-c^2)^2*(b^4+28*b^2*c^2+c^4)*a^2+(b^2-c^2)^2*(b+c)*b*c*(5*b^2-9*b*c+5*c^2)*a+(b^2-c^2)^4*b*c : :

The reciprocal cyclologic center of these triangles is X(13533)

X(13534) lies on these lines: {}

X(13535) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 1st EHRMANN TO 2nd EHRMANN

Barycentrics
(3*S^2*(3*R^2-SW)*(SW*(4*(18*SA^2-3*SA*SW-4*SW^2)*R^2-(30*SA^2-13*SA*SW-6*SW^2)*SW)+3*(144*R^4-24*R^2*SW-7*SW^2)*S^2)-(4*(6*SA-SW)*R^2-3*(3*SA-SW)*SW)*SA*SW^4)*(SB+SC) : :

The reciprocal cyclologic center of these triangles is X(13536)

X(13535) lies on these lines: {}

X(13536) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 2nd EHRMANN TO 1st EHRMANN

Barycentrics
((144*S^2*R^4-2*(42*S^2+5*SW^2)*SW*R^2+3*(4*S^2+SW^2)*SW^2)*SW*SA^2+2*(432*S^2*R^6-4*(105*S^2+4*SW^2)*SW*R^4+(128*S^2+15*SW^2)*SW^2*R^2-3*(4*S^2+SW^2)*SW^3)*SW*SA-(5184*S^2*R^6-4*(1044*S^2+55*SW^2)*SW*R^4+4*(267*S^2+34*SW^2)*SW^2*R^2-21*(4*S^2+SW^2)*SW^3)*S^2)*(SB+SC) : :

The reciprocal cyclologic center of these triangles is X(13535)

X(13536) lies on these lines: {}

X(13537) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 2nd EHRMANN TO KOSNITA

Barycentrics
(((1944*S^4-144*S^2*SW^2-6*SW^4)*R^4-(864*S^4+6*S^2*SW^2-11*SW^4)*SW*R^2+3*(30*S^4+5*S^2*SW^2-SW^4)*SW^2)*SW*SA^2-2*(432*S^2*SW*R^6+(1512*S^4-534*S^2*SW^2-13*SW^4)*R^4-(684*S^4-129*S^2*SW^2-14*SW^4)*SW*R^2+3*(24*S^4-S^2*SW^2-SW^4)*SW^2)*SW^2*SA-((54*S^2-SW^2)*R^2-(9*S^2+SW^2)*SW)*((648*S^2-36*SW^2)*R^4-4*(99*S^2-5*SW^2)*SW*R^2+3*(20*S^2-SW^2)*SW^2)*S^2)*(SB+SC) : :

The reciprocal cyclologic center of these triangles is X(13538)

X(13537) lies on the line {575,13538}

X(13537) = reflection of X(13538) in X(575)

X(13538) = CYCLOLOGIC CENTER OF THESE TRIANGLES: KOSNITA TO 2nd EHRMANN

Barycentrics
(((1944*S^4-144*S^2*SW^2-6*SW^4)*R^4-(864*S^4+6*S^2*SW^2-11*SW^4)*SW*R^2+3*(30*S^4+5*S^2*SW^2-SW^4)*SW^2)*SA^2+(36*(18*S^2-SW^2)^2*R^6-3*(3852*S^4-352*S^2*SW^2+5*SW^4)*SW*R^4+4*(846*S^4-45*S^2*SW^2-2*SW^4)*SW^2*R^2-3*(11*S^2-SW^2)*(9*S^2+SW^2)*SW^3)*SA+(72*(18*S^2+SW^2)*SW*R^6+(1620*S^4-1188*S^2*SW^2-139*SW^4)*R^4-(756*S^4-240*S^2*SW^2-77*SW^4)*SW*R^2+3*(27*S^4-S^2*SW^2-4*SW^4)*SW^2)*S^2)*(SB+SC) : :

The reciprocal cyclologic center of these triangles is X(13537)

X(13538) lies on the line {575,13537}

X(13538) = reflection of X(13537) in X(575)

X(13539) = CYCLOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-MIDPOINTS TO EXTOUCH

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

The reciprocal cyclologic center of these triangles is X(1145)

X(13539) lies on the cubic K338 and the line {1,104}

X(13540) = CYCLOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-MIDPOINTS TO HUTSON INTOUCH

Trilinears (-a+b+c)*(2*a^5-5*(b+c)*a^4+2*(b^2+7*b*c+c^2)*a^3+(b+c)*(4*b^2-15*b*c+4*c^2)*a^2-4*(b^4+c^4-b*c*(2*b^2-b*c+2*c^2))*a+(b^3+c^3)*(b-c)^2) : :

The reciprocal cyclologic center of these triangles is X(3021)

X(13540) lies on these lines: {9,644}, {1477,2137}

X(13541) = CYCLOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS TO EXCENTRAL

Trilinears a^3-4*(b+c)*a^2-(2*b^2-17*b*c+2*c^2)*a+3*(b+c)*(b^2-3*b*c+c^2) : :
X(13541) = 3*X(1)-2*X(106) = 4*X(106)-3*X(1054) = X(106)-3*X(10700) = 4*X(121)-3*X(3679) = X(1054)-4*X(10700) = 5*X(1054)-8*X(11717) = 7*X(3624)-8*X(11731) = 4*X(5510)-5*X(11522) = 5*X(10700)-2*X(11717)

The reciprocal cyclologic center of these triangles is X(1054)

X(13541) lies on these lines: {1,88}, {6,4919}, {8,11814}, {121,3679}, {537,1120}, {644,3973}, {1293,7991}, {1357,3340}, {1721,9519}, {2163,6095}, {2170,3731}, {2796,3241}, {2810,3022}, {2827,4895}, {3624,11731}, {4677,10713}, {5510,11522}, {5881,10744}, {7993,9355}, {9897,10774}

X(13541) = reflection of X(i) in X(j) for these (i,j): (1,10700), (8,11814), (1054,1), (4677,10713), (5881,10744), (7991,1293), (9897,10774)

X(13542) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR TO FUHRMANN

Trilinears 32*p^7*(2*p-11*q)+4*(168*q^2+37)*p^6-4*(128*q^2+135)*q*p^5+(128*q^4+624*q^2+57)*p^4-4*(64*q^2+31)*q*p^3+(32*q^4+65*q^2+9)*p^2-2*(5*q^2+3)*q*p+q^2 : :
where p=sin(A/2), q=cos((B-C)/2)
X(13542) = 3*X(1)-2*X(13497)

The reciprocal cyclologic center of these triangles is X(13498)

X(13542) lies on these lines: {1,13497}, {3,10700}, {5881,10698}

X(13543) = CYCLOLOGIC CENTER OF THESE TRIANGLES: FUHRMANN TO INNER-YFF

Trilinears
p*(64*p^11-192*q*p^10+64*(3*q^2-2)*p^9-32*(2*q^2-11)*q*p^8+32*(2*q^4-11*q^2+2)*p^7-16*(12*q^4-14*q^2+11)*q*p^6+4*(48*q^6-64*q^4+56*q^2+1)*p^5-4*(16*q^6-72*q^4+56*q^2-5)*q*p^4-4*(40*q^6-44*q^4+17*q^2-1)*p^3+4*(8*q^6-20*q^4+15*q^2-2)*q*p^2+(16*q^6-16*q^4+12*q^2-7)*p-(4*q^2-3)*q) : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal cyclologic center of these triangles is X(13544)

X(13543) lies on the Fuhrmann circle and these lines: {}

X(13544) = CYCLOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO FUHRMANN

Trilinears
256*p^13*(p-3*q)+64*(8*q^2-9)*p^12+128*(4*q^2+13)*q*p^11-32*(24*q^4+34*q^2-13)*p^10+32*(8*q^4-28*q^2-39)*q*p^9+16*(76*q^4+62*q^2-7)*p^8-16*(16*q^4-16*q^2-29)*q*p^7-4*(16*q^6+136*q^4+144*q^2-7)*p^6+8*(12*q^4+24*q^2-17)*q*p^5+2*(16*q^6+24*q^4+96*q^2-1)*p^4-4*(4*q^4+22*q^2-3)*q*p^3+(4*q^4-14*q^2-7)*p^2+4*(q^2+2)*q*p-q^2-1/2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal cyclologic center of these triangles is X(13543)

X(13544) lies on these lines: {}

X(13545) = CYCLOLOGIC CENTER OF THESE TRIANGLES: FUHRMANN TO OUTER-YFF

Trilinears
p*(64*p^11-448*q*p^10+64*(19*q^2+2)*p^9-32*(58*q^2+15)*q*p^8+32*(58*q^4+31*q^2-4)*p^7-16*(76*q^4+102*q^2-33)*q*p^6+4*(112*q^6+448*q^4-136*q^2-47)*p^5-4*(16*q^6+264*q^4+24*q^2-117)*q*p^4+4*(72*q^6+92*q^4-81*q^2-13)*p^3-4*(8*q^6+36*q^4-7*q^2-18)*q*p^2+(16*q^6+16*q^4-12*q^2-7)*p-(4*q^2-3)*q) : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal cyclologic center of these triangles is X(13546)

X(13545) lies on the Fuhrmann circle and these lines: {}

X(13546) = CYCLOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO FUHRMANN

Trilinears
512*p^13*(p-7*q)+128*(72*q^2+11)*p^12-256*(44*q^2+25)*q*p^11+64*(52*q^2+5)*(2*q^2+3)*p^10-64*(24*q^4+116*q^2+41)*q*p^9+32*(44*q^4+42*q^2+5)*p^8+32*(16*q^4+64*q^2+5)*q*p^7-8*(16*q^6+312*q^4+176*q^2+5)*p^6+16*(52*q^4+112*q^2+19)*q*p^5-4*(16*q^6+200*q^4+144*q^2-5)*p^4+8*(12*q^4+50*q^2-1)*q*p^3-2*(44*q^4+18*q^2-5)*p^2+8*(3*q^2-1)*q*p+1-2*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal cyclologic center of these triangles is X(13545)

X(13546) lies on these lines: {}

X(13547) = CYCLOLOGIC CENTER OF THESE TRIANGLES: FUHRMANN TO INNER-YFF TANGENTS

Barycentrics
(-a+b+c)*(a^15+(b+c)*a^14-3*(3*b^2-2*b*c+3*c^2)*a^13-(b+c)*(b^2+6*b*c+c^2)*a^12+(29*b^4+29*c^4-2*b*c*(6*b^2-11*b*c+6*c^2))*a^11-(b+c)*(11*b^4+11*c^4-2*b*c*(12*b^2-5*b*c+12*c^2))*a^10-(45*b^6+45*c^6+(14*b^4+14*c^4+3*b*c*(b^2-4*b*c+c^2))*b*c)*a^9+(b^2-c^2)*(b-c)*(35*b^4+35*c^4+2*b*c*(16*b^2+9*b*c+16*c^2))*a^8+(35*b^8+35*c^8+2*(28*b^6+28*c^6-(26*b^4+26*c^4+b*c*(52*b^2-105*b*c+52*c^2))*b*c)*b*c)*a^7-(b+c)*(45*b^8+45*c^8-2*(16*b^6+16*c^6+(46*b^4+46*c^4-b*c*(136*b^2-185*b*c+136*c^2))*b*c)*b*c)*a^6-(11*b^8+11*c^8+2*(38*b^6+38*c^6+(34*b^4+34*c^4-7*b*c*(2*b^2+b*c+2*c^2))*b*c)*b*c)*(b-c)^2*a^5+(b^2-c^2)*(b-c)*(29*b^8+29*c^8+2*(20*b^6+20*c^6-(30*b^4+30*c^4-b*c*(28*b^2+39*b*c+28*c^2))*b*c)*b*c)*a^4-(b^2-c^2)^2*(b^8+c^8-2*(10*b^6+10*c^6-(18*b^4+18*c^4+b*c*(2*b-c)*(b-2*c))*b*c)*b*c)*a^3-(b^2-c^2)^3*(b-c)*(9*b^6+9*c^6+(10*b^4+10*c^4-b*c*(17*b^2-12*b*c+17*c^2))*b*c)*a^2+(b^2-c^2)^4*(b-c)^2*(b^4+6*b^2*c^2+c^4)*a+(b^2-c^2)^7*(b-c)) : :

The reciprocal cyclologic center of these triangles is X(13548)

X(13547) lies on the Fuhrmann circle and these lines: {}

X(13548) = CYCLOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO FUHRMANN

Trilinears
128*p^16-256*q*p^15-32*(20*q^2-3)*p^14+32*(80*q^2-31)*q*p^13-16*(200*q^4-258*q^2+45)*p^12+32*(56*q^4-265*q^2+106)*q*p^11-16*(24*q^6-570*q^4+447*q^2-25)*p^10-16*(306*q^4-574*q^2+117)*q*p^9+8*(124*q^6-966*q^4+404*q^2+39)*p^8+8*(4*q^6+480*q^4-344*q^2-105)*q*p^7-2*(392*q^6-728*q^4-436*q^2+59)*p^6-2*(16*q^6+312*q^4+312*q^2-225)*q*p^5+(160*q^6+368*q^4-458*q^2-75)*p^4-2*(44*q^4-61*q^2-71)*q*p^3-(4*q^4+63*q^2+9)*p^2+2*(4*q^2+3)*q*p-q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal cyclologic center of these triangles is X(13547)

X(13548) lies on these lines: {}

X(13549) = CYCLOLOGIC CENTER OF THESE TRIANGLES: FUHRMANN TO OUTER-YFF TANGENTS

Barycentrics
(-a+b+c)*(a^15-7*(b+c)*a^14+(7*b^2+54*b*c+7*c^2)*a^13+(b+c)*(23*b^2-126*b*c+23*c^2)*a^12-(43*b^4+43*c^4+2*b*c*(10*b^2-219*b*c+10*c^2))*a^11-(b+c)*(11*b^4+11*c^4-2*b*c*(164*b^2-437*b*c+164*c^2))*a^10+(75*b^6+75*c^6-(390*b^4+390*c^4+b*c*(219*b^2-1532*b*c+219*c^2))*b*c)*a^9-(b+c)*(45*b^6+45*c^6+(62*b^4+62*c^4-b*c*(1637*b^2-3284*b*c+1637*c^2))*b*c)*a^8-(45*b^8+45*c^8-2*(340*b^6+340*c^6-(922*b^4+922*c^4+b*c*(92*b^2-1497*b*c+92*c^2))*b*c)*b*c)*a^7+(b+c)*(75*b^8+75*c^8-2*(256*b^6+256*c^6-(46*b^4+46*c^4+b*c*(1944*b^2-3551*b*c+1944*c^2))*b*c)*b*c)*a^6-(11*b^8+11*c^8+2*(166*b^6+166*c^6-(862*b^4+862*c^4-b*c*(146*b^2+1209*b*c+146*c^2))*b*c)*b*c)*(b-c)^2*a^5-(b^2-c^2)*(b-c)*(43*b^8+43*c^8-2*(216*b^6+216*c^6-(390*b^4+390*c^4+b*c*(600*b^2-1631*b*c+600*c^2))*b*c)*b*c)*a^4+(b^2-c^2)^2*(23*b^8+23*c^8-2*(42*b^6+42*c^6+(266*b^4+266*c^4-b*c*(1250*b^2-1923*b*c+1250*c^2))*b*c)*b*c)*a^3+(b^2-c^2)^3*(b-c)*(7*b^6+7*c^6-b*c*(2*b-c)*(b-2*c)*(61*b^2-120*b*c+61*c^2))*a^2-(b^2-c^2)^4*(b-c)^2*(7*b^4+7*c^4-2*b*c*(28*b^2-45*b*c+28*c^2))*a+(b^2-c^2)^5*(b-c)^3*(b^2-6*b*c+c^2)) : :

The reciprocal cyclologic center of these triangles is X(13550)

X(13549) lies on the Fuhrmann circle and these lines: {}

X(13550) = CYCLOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO FUHRMANN

Trilinears
(a^20-9*(b+c)*a^19+(19*b^2+89*b*c+19*c^2)*a^18+(b+c)*(29*b^2-294*b*c+29*c^2)*a^17-(140*b^4+140*c^4-(43*b^2+1328*b*c+43*c^2)*b*c)*a^16+2*(b+c)*(22*b^4+22*c^4+(582*b^2-1751*b*c+582*c^2)*b*c)*a^15+2*(182*b^6+182*c^6-(916*b^4+916*c^4+(911*b^2-5110*b*c+911*c^2)*b*c)*b*c)*a^14-2*(b+c)*(182*b^6+182*c^6+(438*b^4+438*c^4-(6291*b^2-12662*b*c+6291*c^2)*b*c)*b*c)*a^13-2*(217*b^8+217*c^8-(2552*b^6+2552*c^6-(4941*b^4+4941*c^4+4*(2119*b^2-6467*b*c+2119*c^2)*b*c)*b*c)*b*c)*a^12+2*(b+c)*(385*b^8+385*c^8-(1362*b^6+1362*c^6+(5851*b^4+5851*c^4-2*(17715*b^2-29906*b*c+17715*c^2)*b*c)*b*c)*b*c)*a^11+2*(77*b^10+77*c^10-(2887*b^8+2887*c^8-(14460*b^6+14460*c^6-(15570*b^4+15570*c^4+b*c*(36937*b^2-88122*b*c+36937*c^2))*b*c)*b*c)*b*c)*a^10-2*(b+c)*(413*b^10+413*c^10-(3300*b^8+3300*c^8-(4840*b^6+4840*c^6+3*(9010*b^4+9010*c^4-b*c*(39391*b^2-60596*b*c+39391*c^2))*b*c)*b*c)*b*c)*a^9+2*(98*b^12+98*c^12+(1103*b^10+1103*c^10-(13977*b^8+13977*c^8-(44075*b^6+44075*c^6-2*(16251*b^4+16251*c^4+b*c*(45905*b^2-96029*b*c+45905*c^2))*b*c)*b*c)*b*c)*b*c)*a^8+2*(b+c)*(238*b^12+238*c^12-(2958*b^10+2958*c^10-(12523*b^8+12523*c^8-2*(6655*b^6+6655*c^6+(29465*b^4+29465*c^4-b*c*(113726*b^2-165903*b*c+113726*c^2))*b*c)*b*c)*b*c)*b*c)*a^7-2*(130*b^14+130*c^14-(536*b^12+536*c^12+(4031*b^10+4031*c^10-(31134*b^8+31134*c^8-(75591*b^6+75591*c^6-4*(11446*b^4+11446*c^4+b*c*(30565*b^2-59847*b*c+30565*c^2))*b*c)*b*c)*b*c)*b*c)*b*c)*a^6-2*(b+c)*(62*b^14+62*c^14-(1158*b^12+1158*c^12-(8019*b^10+8019*c^10-(25518*b^8+25518*c^8-(28955*b^6+28955*c^6+2*(24167*b^4+24167*c^4-6*b*c*(17501*b^2-25225*b*c+17501*c^2))*b*c)*b*c)*b*c)*b*c)*b*c)*a^5+(b^2-c^2)^2*(121*b^12+121*c^12-(1208*b^10+1208*c^10-(3204*b^8+3204*c^8+(5344*b^6+5344*c^6-3*(17451*b^4+17451*c^4-8*b*c*(5703*b^2-7681*b*c+5703*c^2))*b*c)*b*c)*b*c)*b*c)*a^4-(b^2-c^2)^3*(b-c)*(b^10+c^10+(206*b^8+206*c^8-(2545*b^6+2545*c^6-4*(2897*b^4+2897*c^4-b*c*(6632*b^2-8743*b*c+6632*c^2))*b*c)*b*c)*b*c)*a^3-(b^2-c^2)^4*(b-c)^2*(21*b^8+21*c^8-(259*b^6+259*c^6-(1188*b^4+1188*c^4-b*c*(2781*b^2-3454*b*c+2781*c^2))*b*c)*b*c)*a^2+(b^2-c^2)^5*(b-c)^3*(5*b^6+5*c^6-(46*b^4+46*c^4-b*c*(113*b^2-96*b*c+113*c^2))*b*c)*a-(b^2-c^2)^6*(b-c)^4*b*c*(b^2-6*b*c+c^2))*a : :

The reciprocal cyclologic center of these triangles is X(13549)

X(13550) lies on these lines: {}

X(13551) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 1st HYACINTH TO 2nd HYACINTH

Barycentrics
(16*R^6*(4*R^2-11*SW)-(88*S^2-149*SW^2)*R^4+4*(13*S^2-12*SW^2)*SW*R^2-(7*S^2-5*SW^2)*SW^2)*SA^2+(-64*SW*R^8-4*(42*S^2-44*SW^2)*R^6+(269*S^2-149*SW^2)*SW*R^4+(24*S^4-116*S^2*SW^2+48*SW^4)*R^2-(5*S^4-14*S^2*SW^2+5*SW^4)*SW)*SA+(8*R^6*(16*R^2-21*SW)-(84*S^2-83*SW^2)*R^4+16*(2*S^2-SW^2)*SW*R^2-(3*S^2-SW^2)*SW^2)*S^2 : :

The reciprocal cyclologic center of these triangles is X(13552)

X(13551) lies on these lines: {}

X(13552) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 2nd HYACINTH TO 1st HYACINTH

Barycentrics
(-(4*R^2-SW)*(96*R^8-44*SW*R^6+(75*S^2-20*SW^2)*R^4-10*(3*S^2-SW^2)*SW*R^2+(3*S^2-SW^2)*SW^2)*SA^2+2*(4*R^2-SW)*(5*R^2-SW)*(16*SW*R^6+(115*S^2-18*SW^2)*R^4-(43*S^2-3*SW^2)*SW*R^2+4*S^2*SW^2)*SA+(1504*R^10-1916*SW*R^8+(600*S^2+888*SW^2)*R^6-(365*S^2+198*SW^2)*SW*R^4+2*(37*S^2+11*SW^2)*SW^2*R^2-(5*S^2+SW^2)*SW^3)*S^2)*(SB+SC) : :

The reciprocal cyclologic center of these triangles is X(13551)

X(13552) lies on these lines: {}

X(13553) = CYCLOLOGIC CENTER OF THESE TRIANGLES: MIDHEIGHT TO 2nd HYACINTH

Barycentrics (SA*(3*SW-14*R^2)*(6*R^2+SA-2*SW)-12*R^4*(24*R^2-17*SW)-4*(11*SW^2+5*S^2)*R^2+(4*S^2+3*SW^2)*SW)*SB*SC : :

The reciprocal cyclologic center of these triangles is X(974)

X(13553) lies on these lines: {4,110}, {6000,12004}

X(13554) = CYCLOLOGIC CENTER OF THESE TRIANGLES: INTANGENTS TO 2nd SCHIFFLER

Trilinears 64*p^7*(2*p-5*q)+8*(48*q^2-19)*p^6-40*(8*q^2-9)*q*p^5+2*(64*q^4-116*q^2-13)*p^4+60*(2*q^2-1)*q*p^3-(80*q^4-24*q^2-47)*p^2+(1-q^2)*(4*q^2-50*p*q+3) : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal cyclologic center of these triangles is X(13555)

X(13554) lies on the intangents circle and these lines: {}

X(13555) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 2nd SCHIFFLER TO INTANGENTS

Trilinears 16*p^6-56*q*p^5+(104*q^2-31)*p^4-6*(8*q^2-1)*q*p^3-(13*q^2-22)*p^2+18*(q^2-1)*q*p+2-2*q^2 : :
where p=sin(A/2), q=cos((B-C)/2)

The reciprocal cyclologic center of these triangles is X(13554)

X(13555) lies on the line {1768,7971}

X(13556) = CYCLOLOGIC CENTER OF THESE TRIANGLES: JOHNSON TO YIU

Barycentrics (12*R^2*(3*R^2-2*SW)-5*S^2+5*SW^2)*SA^2+(-4*R^2*(9*SW*R^2-2*S^2-6*SW^2)+(3*S^2-5*SW^2)*SW)*SA+2*(8*R^2*(3*R^2-2*SW)-2*S^2+3*SW^2)*S^2 : :
X(13556) = 2*X(131)-3*X(381)

The reciprocal cyclologic center of these triangles is X(13557)

X(13556) lies on these lines: {3,136}, {5,925}, {30,1300}, {131,381}, {155,382}, {2070,5961}, {7517,13558}, {11641,12918}

X(13556) = reflection of X(i) in X(j) for these (i,j): (3,136), (925,5)

X(13557) = CYCLOLOGIC CENTER OF THESE TRIANGLES: YIU TO JOHNSON

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

The reciprocal cyclologic center of these triangles is X(13556)

X(13557) is the perspector of ABC and the reflection of the 2nd extouch triangle in line X(924)X(6753). Line X(924)X(6753) is the trilinear polar of X(24) and the perspectrix of ABC and the 2nd extouch triangle. (Randy Hutson, August 19, 2019)

X(13557) lies on the Yiu circle, cubic K725, and these lines: {3,49}, {5,5962}, {30,1300}, {131,539}, {924,10540}

X(13557) = reflection of X(i) in X(j) for these (i,j): (3,12095), (5962,5)
X(13557) = circumcircle-inverse-of-X(1147)
X(13557) = X(5962)-of-Johnson-triangle
X(13557) = Stammler circle-inverse-of-X(12164)

X(13558) = CYCLOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL TO X3-ABC REFLECTIONS

Trilinears (a^8-(b^2+c^2)*a^6+b^2*c^2*a^4-(b^4-c^4)*(b^2-c^2)*a^2+(b^4-b^2*c^2+c^4)*(b^2-c^2)^2)*(-a^2+b^2+c^2)*a : :
X(13558) = 3*X(3)-2*X(13496) = 3*X(5961)-X(13496)

The reciprocal cyclologic center of these triangles is X(399)

X(13558) lies on the tangential circle and these lines: {3,125}, {4,11587}, {22,98}, {24,107}, {25,132}, {186,6761}, {1112,1576}, {1609,6103}, {1637,2079}, {2080,6660}, {3124,8429}, {3129,7684}, {3130,7685}, {6793,8573}, {7517,13556}, {7669,10117}

X(13558) = reflection of X(3) in X(5961)
X(13558) = circumcircle-inverse-of-X(125)
X(13558) = Dao-Moses-Telv circle-inverse-of-X(2079)
X(13558) = X(109)-of-tangential-triangle if ABC is acute
X(13558) = Stammler circle-inverse-of-X(12902)

X(13559) = CYCLOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL-MIDARC TO 2nd TANGENTIAL-MIDARC

Trilinears F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=2*b*c*(-a+b+c)*(a^3-2*(b+c)*a^2+(b^2-c^2)*(b-c))
G(a,b,c)=-2*b*c*a*(a-b+c)*(2*a^2+(b-6*c)*a-(3*b+4*c)*(b-c))
H(a,b,c)=-3*b*c*a*(a+b-c)*(a-b+c)*(-a+b+c)

The reciprocal cyclologic center of these triangles is X(13560)

X(13559) lies on these lines: {1,13560}, {2089,8084}, {3659,8075}, {5919,8241}, {7597,8077}, {8091,8094}, {11032,12814}, {11044,11217}

X(13559) = reflection of X(13560) in X(1)

X(13560) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO TANGENTIAL-MIDARC

Barycentrics F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : :
where
F(a,b,c)=-2*(-a+b+c)*(a^3+2*(b+c)*a^2-2*(b-c)^2*a-(b^2-c^2)*(b-c))
G(a,b,c)=-2*a*(a-b+c)*((b-2*c)*a-(b+2*c)*(b-c))
H(a,b,c)=3*(a+b-c)*(a-b+c)*(-a+b+c)*a

The reciprocal cyclologic center of these triangles is X(13559)

X(13560) lies on these lines: {1,13559}, {174,354}, {3659,8076}, {7588,7597}, {8084,8092}, {8242,10506}

X(13560) = reflection of X(13559) in X(1)

X(13561) = X(2)X(156)∩X(5)X(113)

Barycentrics a^6 b^4-3 a^4 b^6+3 a^2 b^8-b^10-2 a^6 b^2 c^2+2 a^4 b^4 c^2-3 a^2 b^6 c^2+3 b^8 c^2+a^6 c^4+2 a^4 b^2 c^4-2 b^6 c^4-3 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 : :
X(13561) = X[26] + 3 X[1853]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26170.

X(13561) lies on these lines: {2,156}, {3,12278}, {5, 113}, {26,1853}, {30,5449} et al

X(13561) = midpoint of X(i) and X(j) for these {i,j}: {9927,11250}, {12359,13371}
X(13561) = reflection of X(i) in X(j) for these (i,j): (10282,10125), (12038, 5498)
X(13561) = complement of X(156)
X(13561) = circumcenter of nine-point centers of BCX(3), CAX(3), ABX(3)
X(13561) = X(5)-of-A'B'C' as defined at X(11585)
X(13561) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,10264,185), (3448,6143,49)

X(13562) = POINT BEID 123

Barycentrics 2*a^8-(b^2+c^2)*a^6-(b^2-c^2)^ 2*a^4+(b^2+c^2)^3*a^2-(b^4-c^4 )^2 : :
X(13562) = 5*X(3618) - 3*X(11245)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26172.

X(13562) lies on these lines: {3,5596}, {5,6}, {25,69}, {30,3313}, {66,1368}, {140,5157}, {141,206}, {193,6997}, {311,460}, {343,1974}, {511,6756}, {524,9969}, {599,10154}, {1176,7499}, {1351,7528}, {1503,5907}, {3589,11548}, {3618,7539}, {3620,7493}, {3818,3867}, {5159,6697}, {5480,13142}, {5921,6816}, {5972,6698}, {6248,6748}, {6776,7395}, {7529,11898}, {7819,10547}, {9715,10519}, {9967,12134}

X(13562) = midpoint of X(9967) and X(12134)
X(13562) = reflection of X(13142) in X(5480)
X(13562) = {X(141), X(206)}-harmonic conjugate of X(6676)

X(13563) = POINT BEID 124

Barycentrics (b+c)*a^7-(3*b^2+2*b*c+3*c^2)* a^6+(b+c)*(2*b^2-b*c+2*c^2)*a^ 5+(2*b^4+2*c^4+b*c*(b+c)^2)*a^ 4-3*(b^2-c^2)^2*(b+c)*a^3+(b^2 +c^2)*(b^2+4*b*c+c^2)*(b-c)^2* a^2+(b^2-c^2)*(b-c)^3*b*c*a-(b ^2-c^2)^2*(b-c)^2*b*c : :
X(13563) =

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26172.

X(13563) lies on these lines: {5572, 28850}, {8680, 29957}

X(13564) = EULER LINE INTERCEPT OF X(195)X(511)

Trilinears (a^8-2*(b^2+c^2)*a^6-3*b^2*c^ 2*a^4+(b^2+c^2)*(2*b^4+b^2*c^2 +2*c^4)*a^2-(b^4+c^4)*(b^2-c^2 )^2)*a : :
X(13564) = X(54)-3*X(6030) = 3*X(381)-4*X(13160) = 5*X(1656)-4*X(5576) = 7*X(3526)-8*X(7568) = 7*X(3851)-6*X(7565)

As a point on the Euler line, X(13564) has Shinagawa coefficients (5E + 8F, -9E - 8*F).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26172.

X(13564) lies on these lines: {2,3}, {49,10625}, {54,6030}, {110,10627}, {156,2979}, {195,511}, {399,2918}, {524,13432}, {542,3519}, {568,10984}, {1147,13340}, {1204,8717}, {1216,10540}, {1385,9591}, {1495,5447}, {1503,9920}, {1614,6101}, {2883,9919}, {2889,9143}, {2917,6000}, {3053,9700}, {3098,10539}, {3311,9683}, {3579,9626}, {5012,10263}, {5446,13353}, {5462,13339}, {5663,7691}, {6455,9682}, {6800,9704}, {7755,11063}, {8193,12645}, {10117,10282}, {10575,10620}, {11255,12220}, {12310,12359}

X(13564) = midpoint of X(7691) and X(8718)
X((13564) = anticomplement of X(33332)
X(13564) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,22,2937), (3,2937,2070), (3,3830,7503), (3,3843,7514), (3,5073,7526), (3,5899,5), (3,7387,381), (3,7517,1656), (3,9909,7506), (3,12083,382), (4,7492,7525), (4,7525,3), (5,12088,5899), (20,7502,3), (22,10323,26), (26,10323,3), (550,7555,7488), (1656,7517,7545), (6636,12088,5), (7509,7530,3851), (7526,12082,5073)

X(13565) = POINT BEID 125

Barycentrics ((b^2+c^2)*a^2-(b^2-c^2)^2)*(a ^6-3*(b^4+b^2*c^2+c^4)*a^2+2*( b^4-c^4)*(b^2-c^2)) : :
X(13565) = = 3*X(2)+X(6288) = 9*X(2)-X(12254) = 3*X(5)-X(3574) = X(54)-5*X(1656) = X(195)-9*X(5055) = 3*X(381)+X(7691) = 3*X(547)-X(8254) = 3*X(1209)+X(3574) = 3*X(6288)+X(12254) = 3*X(10610)-X(12254)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26172.

X(13565) lies on these lines: {2,6288}, {3,7703}, {5,51}, {54,1656}, {140,13470}, {195,5055}, {252,12060}, {381,7691}, {403,11017}, {498,12956}, {499,12946}, {539,547}, {858,11592}, {1216,11808}, {1493,2888}, {1594,11576}, {2072,12363}, {2917,7514}, {3519,5056}, {3628,5972}, {3851,12307}, {5071,12325}, {5449,13363}, {5576,13391}, {5663,13160}, {5790,7979}, {5886,12785}, {5907,11802}, {5943,10115}, {5965,12812}, {6145,9833}, {6152,7577}, {6153,10170}, {7393,9920}, {7579,7999}, {7730,11444}, {7741,13079}, {9777,12316}, {9977,11178}, {10255,12606}, {11230,12266}

X(13565) = midpoint of X(i) and X(j) for these {i,j}: {5,1209}, {1216,11808}, {1493,2888}, {5907,11802}, {6288,10610}, {12606,13368}
X(13565) = reflection of X(i) in X(j) for these (i,j): (973,13365), (6689,3628)
X(13565) = complement of X(10610)
X(13565) = {X(2), X(6288)}-harmonic conjugate of X(10610)

X(13566) = POINT BEID 126

Barycentrics (13*cos(2*A)-9/2)*cos(B-C)-4*cos(A)*cos(2*(B-C))-10*cos(A)-2 *cos(3*A) : :
X(13566) =

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26172.

X(13566) lies on these lines: {4,12006}, {548,3589}

X(13567) = X(2)X(6)∩X(4)X(64)

Barycentrics (b^2+c^2)*a^4-2*(b^2-c^2)^2*a^ 2+(b^4-c^4)*(b^2-c^2) : :
Barycentrics cos^2 B + cos^2 C : :
X(13567) = 3*X(2) + X(6515)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26172.

Let P be the trilinear pole of the tangent to the MacBeath circumconic at a point Q. (i.e., P = X(3)-cross conjugate of a point Q on the MacBeath circumconic.) The locus of the polar conjugate of P as Q varies is an inconic centered at X(13567), and passing through X(338), X(1146), and X(3269). The Brianchon point (perspector) of this inconic is X(2052). (Randy Hutson, November 2, 2017)

X(13567) lies on these lines: {2,6}, {3,12241}, {4,64}, {5,389}, {11,11436}, {15,465}, {16,466}, {19,5928}, {20,1192}, {24,161}, {25,1503}, {30,11438}, {32,441}, {34,10361}, {51,125}, {52,11585}, {53,2052}, {54,10018}, {68,6642}, {92,1146}, {140,578}, {143,13371}, {154,6353}, {182,6676}, {184,468}, {185,235}, {186,12022}, {189,7365}, {226,6708}, {275,6749}, {287,1915}, {297,3981}, {306,3965}, {324,338}, {329,6554}, {397,470}, {398,471}, {403,5890}, {406,5706}, {428,11550}, {429,5799}, {440,573}, {458,7745}, {461,3332}, {472,5321}, {473,5318}, {511,1368}, {541,1539}, {549,11430}, {550,13403}, {568,2072}, {569,7542}, {572,7536}, {576,5159}, {580,7515}, {631,11425}, {800,6509}, {858,3060}, {860,5721}, {1147,13292}, {1151,1589}, {1152,1590}, {1181,3542}, {1196,6388}, {1204,1885}, {1209,7405}, {1350,7386}, {1352,5020}, {1353,5972}, {1498,3089}, {1583,11090}, {1584,11091}, {1585,3070}, {1586,3071}, {1587,3535}, {1588,3536}, {1591,12239}, {1592,12240}, {1593,6696}, {1594,3567}, {1595,10110}, {1596,6000}, {1620,3522}, {1656,11432}, {1848,2262}, {1861,1864}, {1906,11381}, {1990,11547}, {1995,11442}, {3066,6997}, {3098,10691}, {3168,6530}, {3517,9833}, {3526,11426}, {3541,10982}, {3564,6677}, {3796,7493}, {3832,11469}, {3867,9969}, {3925,11435}, {4232,11206}, {5012,13394}, {5067,11431}, {5085,7494}, {5094,9777}, {5097,6723}, {5133,5640}, {5432,11429}, {5713,7532}, {5810,7535}, {5816,7522}, {6389,6617}, {6619,10002}, {6823,9729}, {7392,10516}, {7505,7592}, {7506,12134}, {7583,8968}, {7715,13419}, {8254,12234}, {8263,8681}, {8991,11473}, {9820,12161}, {10257,13352}, {10272,12227}, {10594,11457}, {11402,12007}, {13142,13346}

X(13567) = midpoint of X(i) and X(j) for these {i,j}: {4,10605}, {25,1899}, {125,12828}, {394,6515}
X(13567) = reflection of X(9306) in X(6677)
X(13567) = isotomic conjugate of X(801)
X(13567) = complement of X(394)
X(13567) = complementary conjugate of X(6389)
X(13567) = polar conjugate of X(1105)
X(13567) = pole wrt polar circle of trilinear polar of X(1105) (line X(450)X(2451))
X(13567) = crosssum of X(6) and X(577)
X(13567) = crosspoint of X(2) and X(2052)
X(13567) = crosspoint of X(6) and X(2929) wrt excentral triangle
X(13567) = crosspoint of X(6) and X(2929) wrt tangential triangle
X(13567) = inverse-in-Jerabek-hyperbola of X(12294)
X(13567) = midpoint of polar conjugates of PU(17)
X(13567) = trilinear pole of the polar, wrt circle O(3,4), of the perspector of circle O(3,4)
X(13567) = trilinear pole of the polar, wrt the second Droz-Farny circle, of the perspector of the second Droz-Farny circle
X(13567) = perspector of the midheight triangle and the polar triangle of the Yiu-Hutson conic
X(13567) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,343,141), (2,1993,11064), (2,3580,343), (2,6515,394), (2,10601,3589), (2,11433,6), (5,389,12233), (51,125,427), (51,427,5480), (184,468,10192), (184,11245,8550), (185,235,2883), (468,11245,184), (1146,6354,92), (1204,1885,5894), (5449,5462,5), (6353,6776,154), (8550,10192,184)

X(13568) = X(4)X(64)∩X(6)X(20)

Barycentrics 2*a^10-(b^2+c^2)*a^8-8*(b^4-b^ 2*c^2+c^4)*a^6+10*(b^4-c^4)*(b ^2-c^2)*a^4-2*(b^4-c^4)^2*a^2- (b^4-c^4)*(b^2-c^2)^3 : :
X(13568) = 3*X(51)-X(1885) = 3*X(389)-X(13403) = 3*X(428)-X(11381) = 3*X(12241)-2*X(13403)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26172.

X(13568) lies on these lines: {2,1192}, {3,12233}, {4,64}, {5,4550}, {6,20}, {25,2883}, {30,143}, {51,1885}, {54,10295}, {141,6815}, {185,1503}, {235,5893}, {376,11425}, {427,1204}, {428,11381}, {516,12432}, {524,5889}, {548,11430}, {550,578}, {973,974}, {1350,10996}, {1498,7487}, {1593,5480}, {1595,3357}, {1598,5878}, {1620,3523}, {1657,11432}, {1890,12688}, {1899,12173}, {2777,10110}

X(13568) = midpoint of X(185) and X(3575)
X(13568) = reflection of X(12241) in X(389)
X(13568) = reflection of X(4) in X(11745)
X(13568) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,10605,6247), (427,1204,6696), (5480,5894,1593), (7689,7706,5)
>

X(13569) = POINT BEID 127

Barycentrics ((b+c)*a^6-(b+c)*(2*b^2-b*c+2* c^2)*a^4-4*b*c*(b^2+3*b*c+c^2) *a^3+(b^2-c^2)*(b-c)*(b^2+4*b* c+c^2)*a^2+(b^2-c^2)^2*(b+c)*b *c)/(-a+b+c) : :
X(13569) =

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26172.

X(13569) lies on this line: {57,4008}

X(13570) = POINT BEID 128

Barycentrics a^2*((b^2+c^2)*a^6-3*(b^4+4*b^ 2*c^2+c^4)*a^4+(b^2-3*c^2)*(3* b^2-c^2)*(b^2+c^2)*a^2-(b^4-16 *b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(13570) = 2*X(4)+X(9729) = 4*X(5)-X(13348) = X(51)+3*X(3839) = X(143)+5*X(546) = 2*X(143)-5*X(10110) = 3*X(373)+X(3543) = 5*X(381)-X(5891) = X(389)+5*X(3843) = 2*X(546)+X(10110) = 3*X(546)+X(13451)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26172.

X(13570) lies on these lines: {3,10219}, {4,5943}, {5,13348}, {30,6688}, {51,3839}, {143,546}, {373,3543}, {381,511}, {389,3843}, {3060,5907}, {5480,8681}

X(13570) = midpoint of X(i) and X(j) for these {i,j}: {4,5943}, {3060,5907}
X(13570) = reflection of X(3) in X(10219)

X(13571) = POINT BEID 129

Barycentrics a^4+3 a^2 b^2-b^4+3 a^2 c^2-b^2 c^2-c^4 : :
X(13571) = 3X(2) - 4X(9698)

X(13571) lies on these lines: {2,3108}, {3,7837}, {4,147}, {6,7836}, {32,10353}, {39,2896}, {61,618}, {62,619}, {69,10007}, {83,7813}, {99,7838}, {140,385}, {182,193}, {325,7797}, {524,7824}, {538,7858}, {550,7762}, {574,7877}, {633,3107}, {634,3106}, {1078,7890}, {1654,3216}, {1656,7754}, {1657,7823}, {1975,7921}, {2549,7900}, {3096,7916}, {3314,9605}, {3329,3933}, {3788,7894}, {3926,7787}, {4045,7917}, {5007,7799}, {5013,7893}, {5024,7904}, {5025,9766}, {5041,7832}, {5068,6392}, {5189,8878}, {5254,7941}, {5286,7912}, {5305,7925}, {5309,7814}, {5346,7940}, {5355,7899}, {6655,7757}, {6656,7840}, {6658,7781}, {7533,8267}, {7738,7898}, {7739,7933}, {7748,7926}, {7752,7798}, {7761,7949}, {7763,7766}, {7765,7809}, {7769,7805}, {7776,7864}, {7778,7920}, {7786,7855}, {7788,7876}, {7790,7903}, {7791,7946}, {7792,7947}, {7801,7878}, {7803,7897}, {7821,7827}, {7834,7871}, {7845,7847}, {7846,7908}, {7859,7895}, {7863,12150}, {7875,7881}, {7924,9607}, {9863,10983}

X(13571) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,7836,10583), (6,7906,7836), (39,7779,2896), (39,7882,7831), (39,7905,7779), (194,7774,7785), (194,7785,148), (325,7839,7797), (6658,7781,8591), (7757,7759,6655), (7760,7764,2), (7772,7796,2), (7781,7812,6658), (7829,7909,2), (7856,7888,2)

X(13572) = POINT BEID 130

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26172.

X(13572) lies on this line: {2809,12573}

X(13573) = CYCLOCEVIAN CONJUGATE OF X(648)

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

The appearance of {i,j} in the following list means that X(i) and X(j) are a pair of cyclocevian conjugates:
{1,1029}, {2,4}, {6,1031}, {7,7}, {8,189}, {20,1032}, {66,2998}, {69,253}, {75,8044}, {80,8046}, {290,9473}, {329,1034}, {330,7357}, {648,13573}, {671,13574}, {668,8047}, {2113,6650}, {2994,7219}, {2996,13576}, {3346,6504}, {4373,8048}, {6625,8049}, {6630,8050}, {7319,8051}, {9510,13577}, {10405,13578}

X(13573) lies on these lines: {1503,2071}, {3267,13219}, {3448,8057}

X(13573) = isogonal conjugate of X(10117)
X(13573) = isotomic conjugate of X(13219)
X(13573) = cevapoint of de Longchamps circle intercepts of Euler line
X(13573) = X(112)-cross conjugate of X(2)
X(13573) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 13219}, {6, 10117}

X(13574) = CYCLOCEVIAN CONJUGATE OF X(671)

Barycentrics (a^6-3 a^4 b^2-3 a^2 b^4+b^6+a^4 c^2+5 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^6-3 a^4 c^2+5 a^2 b^2 c^2-b^4 c^2-3 a^2 c^4+b^2 c^4+c^6) : :
X(13574) = 4X(468) - 3X(11580)

X(13574) lies on the curves K008, K535, Q066, and these lines: {2,8877}, {23,524}, {316,3266}, {468,8744}, {523,10561}, {4062,5525}, {4232,10424}, {5099,10415}

X(13574) = reflection of X[10630] in X(5099)
X(13574) = isogonal conjugate of X(2930)
X(13574) = cyclocevian conjugate of X(671)
X(13574) = antigonal image of X(10630)
X(13574) = complement of the anticomplementary conjugate of X(32255)
X(13574) = X(i)-cross conjugate of X(j) for these (i,j): {67, 4}, {111, 2}
X(13574) = X(i)-vertex conjugate of X(j) for these (i,j): {2, 3447}, {23, 468}
X(13574) = cevapoint of X(523) and X(5099)
X(13574) = trilinear pole of line {690, 2492}

X(13575) = CYCLOCEVIAN CONJUGATE OF X(2996)

Barycentrics (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) : :

X(13575) lies on the curves K169, Q066, and these lines: {2,2138}, {22,69}, {253,6995}, {264,6997}, {287,6515}, {305,315}, {306,1763}, {307,8270}, {858,6340}, {1799,7493}, {2373,6353}, {2419,6563}, {4329,4463}, {6330,11547}, {7391,11605}

X(13575) = isogonal conjugate of X(159)
X(13575) = isotomic conjugate of X(1370)
X(13575) = anticomplement of X(3162)
X(13575) = cyclocevian conjugate of X(2996)
X(13575) = X(i)-cross conjugate of X(j) for these (i,j): {25, 2}, {66, 4}
X(13575) = isoconjugate of X(j) and X(j) for these (i,j): {1, 159}, {19,23115}, {31, 1370}, {38, 8793}, {63, 3162}
X(13575) = X(2)-vertex conjugate of X(22)
X(13575) = cevapoint of X(i) and X(j) for these (i,j): {2, 7500}, {127, 523}
X(13575) = trilinear pole of line {525, 2485}
X(13575) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 1370}, {6, 159}, {25, 3162}, {251, 8793}, {3162, 455}

X(13576) = CYCLOCEVIAN CONJUGATE OF X(9510)

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

X(13576) lies on the Kiepert hyperbola, the cubic K299, and these lines: {1,2140}, {2,11}, {4,218}, {7,1002}, {8,76}, {10,1018}, {42,226}, {43,1699}, {65,1446}, {80,885}, {98,919}, {210,321}, {291,812}, {388,2334}, {516,672}, {544,1478}, {666,671}, {740,3930}, {962,10822}, {1111,2809}, {1282,9318}, {1416,11269}, {1462,6817}, {1751,2195}, {1754,13478}, {1814,5800}, {1824,1893}, {1857,2052}, {1916,5992}, {2357,8808}, {2475,6625}, {2486,4557}, {2551,6559}, {2996,3436}, {3030,3038}, {3416,4863}, {3421,5485}, {4049,4674}, {4052,4685}, {4080,4442}, {4419,4492}, {4516,4552}, {4589,4645}, {4648,10013}, {4733,6539}, {5091,5773}, {5377,11604}

X(13576) = anticomplement of X(8299)
X(13576) = reflection of X(i) in X(j) for these {i,j}: {1018, 10}
X(13576) = isogonal conjugate of X(3286)
X(13576) = isotomic conjugate of X(30941)
X(13576) = cyclocevian conjugate of X(9510)
X(13576) = X(8)-beth conjugate of X(1018)
X(13576) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 3286}, {6, 672}, {3737, 2254}
X(13576) = X(673)-daleth conjugate of X(2)
X(13576) = X(i)-cross conjugate of X(j) for these (i,j): {2238, 2}, {4010, 3952}
X(13576) = isoconjugate of X(j) and X(j) for these (i,j): {1, 3286}, {21, 1458}, {28, 1818}, {58, 518}, {81, 672}, {86, 2223}, {110, 2254}, {163, 918}, {241, 284}, {274, 9454}, {283, 1876}, {310, 9455}, {593, 3930}, {662, 665}, {741, 8299}, {849, 3932}, {926, 1414}, {1014, 2340}, {1019, 2284}, {1025, 7252}, {1026, 3733}, {1178, 4447}, {1333, 3912}, {1408, 3717}, {1412, 3693}, {1437, 1861}, {1444, 2356}, {1459, 4238}, {1790, 5089}, {2193, 5236}, {2194, 9436}, {2206, 3263}, {2283, 3737}, {3675, 4570}, {4625, 8638}
X(13576) = cevapoint of X(i) and X(j) for these (i,j): {10, 740}, {115, 4155}
X(13576) = trilinear pole of line {37, 523}
X(13576) = crosssum of X(i) and X(j) for these (i,j): {21, 8849}, {672, 2223}
X(13576) = polar conjugate of X(15149)
X(13576) = barycentric product X(i)X(j) for these {i,j}: {10, 673}, {37, 2481}, {105, 321}, {294, 1441}, {313, 1438}, {349, 2195}, {523, 666}, {850, 919}, {885, 4552}, {927, 3700}, {1027, 4033}, {1462, 3701}, {3668, 6559}, {3932, 6185}, {6335, 10099}
X(13576) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 3286}, {10, 3912}, {37, 518}, {42, 672}, {65, 241}, {71, 1818}, {105, 81}, {210, 3693}, {213, 2223}, {225, 5236}, {226, 9436}, {294, 21}, {321, 3263}, {512, 665}, {523, 918}, {594, 3932}, {661, 2254}, {666, 99}, {673, 86}, {756, 3930}, {884, 7252}, {885, 4560}, {919, 110}, {927, 4573}, {1018, 1026}, {1024, 3737}, {1027, 1019}, {1213, 4966}, {1334, 2340}, {1400, 1458}, {1416, 1412}, {1438, 58}, {1462, 1014}, {1783, 4238}, {1814, 1444}, {1824, 5089}, {1826, 1861}, {1880, 1876}, {1918, 9454}, {2195, 284}, {2205, 9455}, {2238, 8299}, {2295, 4447}, {2321, 3717}, {2333, 2356}, {2481, 274}, {3125, 3675}, {3709, 926}, {3930, 4712}, {3932, 4437}, {3950, 4899}, {4024, 4088}, {4551, 1025}, {4552, 883}, {4557, 2284}, {4559, 2283}, {5257, 4684}, {5377, 4567}, {6559, 1043}, {8751, 28}, {10099, 905}

X(13577) = CYCLOCEVIAN CONJUGATE OF X(10405)

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

Let A₂₉B₂₉C₂₉ and A₃₀B₃₀C₃₀ be Gemini triangles 29 and 30, resp. Let A' be the intersection of the tangents to the {Gemini 29, Gemini 30}-circumconic at A₂₉ and A₃₀. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(13577). (Randy Hutson, January 15, 2019)

X(13577 lies on these lines: {2,1814}, {63,3730}, {69,3263}, {77,3870}, {100,8817}, {497,693}, {1444,3433}, {3434,6063}, {3873,6604}

X(13577) = anticomplement X(5452)
X(13577) = isogonal conjugate of X(1486)
X(13577) = isotomic conjugate of X(3434)
X(13577) = cyclocevian conjugate of X(10405)
X(13577) = X(3433)-anticomplementary conjugate of X(3177)
X(13577) = X(55)-cross conjugate of X(2)
X(13577) = polar conjugate of X(17905)
X(13577) = isoconjugate of X(j) and X(j) for these (i,j): {1, 1486}, {6, 169}, {31, 3434}, {42, 4228}, {57, 5452}, {109, 11934}
X(13577) = cevapoint of X(i) and X(j) for these (i,j): {1, 7289}, {116, 522}, {513, 4904}
X(13577) = trilinear pole of line {905, 918}
X(13577) = barycentric product X(76)X(3433)
X(13577) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 169}, {2, 3434}, {6, 1486}, {55, 5452}, {81, 4228}, {650, 11934}, {3433, 6}, {4904, 5511}
X(13577) = center of {Gemini 29, Gemini 30}-circumconic
X(13577) = perspector of {Gemini 29, Gemini 30}-circumconic

X(13578) = KIEPERT IMAGE OF X(6)

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

The Kiepert image of a point is defined at X(13486).

X(13578) lies on no line X(i)X(j) for 1 ≤ i < j ≤ 13577.
X(13578) = cevapoint of X(512) and X(10329)
X(13578) = trilinear pole of line X(5007)X(5943)

X(13579) = CYCLOCEVIAN CONJUGATE OF X(254)

Barycentrics    (a^6-a^4 b^2-a^2 b^4+b^6-3 a^4 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+3 b^4 c^2-a^2 c^4-3 b^2 c^4+c^6) : :
Barycentrics    1/(- cos 2A + cos 2B + cos 2C) : :
Barycentrics    1/(SA - R^2) : :

The trilinear polar of X(13579) meets the line at infinity at X(523). (Randy Hutson, July 21, 2017)

X(13579) lies on the Kiepert hyperbola and these lines: {2,9609}, {4,1994}, {69,11140}, {94,6515}, {98,7391}, {262,7394}, {323,6504}, {1370,7612}, {7392,10155}, {7578,11427}

X(13579) = isogonal conjugate of X(8553)
X(13579) = polar conjugate of X(7505)
X(13579) = cyclocevian conjugate of X(254)
X(13579) = X(i)-cross conjugate of X(j) for these (i,j): {1993, 2}, {13371, 264}
X(13579) = isoconjugate of X(j) and X(j) for these (i,j): {1, 8553}, {48, 7505}, {1820, 2904}
X(13579) = cevapoint of X(i) and X(j) for these (i,j): {6, 7517}, {115, 924}
X(13579) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 7505}, {6, 8553}, {24, 2904}

X(13580) = CYCLOCEVIAN CONJUGATE OF X(1113)

Barycentrics 1/(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)+(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) J) : :

X(13580) lies on the Kiepert hyperbola and this line: {262,2552}, {323,13581}

X(13580) = cyclocevian conjugate of X(1113)
X(13580) = X(8115)-cross conjugate of X(2)
X(13580) = cevapoint of X(115) and X(2575)

X(13581) = CYCLOCEVIAN CONJUGATE OF X(1114)

Barycentrics 1/(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)-(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) J) : :

X(13581) lies on the Kiepert hyperbola and this line: {262,2553}, {323,13580}

X(13581) = cyclocevian conjugate of X(1114)
X(13581) = X(8116)-cross conjugate of X(2)
X(13581) = cevapoint of X(115) and X(2574)

X(13582) = CYCLOCEVIAN CONJUGATE OF X(1138)

Barycentrics (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) : :

X(13582) lies on the Kiepert hyperbola, the curve Q110, and these lines: {4,195}, {13,5612}, {14,5616}, {17,11130}, {18,11131}, {94,11071}, {98,1291}, {140,10277}, {262,7533}, {265,1117}, {338,11140}, {340,9381}, {1994,11538}, {5671,10264}, {11639,11669}

X(13582) = isogonal conjugate of X(11063)
X(13862) = anticomplement of X(37450)
X(13582) = X(11071)-anticomplementary conjugate of X(8)
X(13582) = cyclocevian conjugate of X(1138)
X(13582) = X(i)-cross conjugate of X(j) for these (i,j): {323, 2}, {10413, 523}, {11600, 2993}, {11601, 2992}
X(13582) = isoconjugate of X(j) and X(j) for these (i,j): {1, 11063}, {6, 1749}, {662, 6140}, {1101, 10413}, {1157, 1953}, {2153, 5616}, {2154, 5612}, {2159, 10272}, {2173, 3470}
X(13582) = cevapoint of X(i) and X(j) for these (i,j): {6, 5899}, {115, 526}, {233, 1154}, {523, 10413}, {524, 13162}, {758, 5949}, {1117, 11071}
X(13582) = trilinear pole of line {140, 523} (the line through the nine-point centers of the 1st and 2nd Ehrmann inscribed triangles)
X(13582) = barycentric product X(i)*X(j) for these {i,j}: {95, 1263}, {850, 1291}, {1494, 3471}, {7799, 11071}
X(13582) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1749}, {6, 11063}, {15, 5616}, {16, 5612}, {30, 10272}, {54, 1157}, {74, 3470}, {115, 10413}, {186, 2914}, {512, 6140}, {526, 8562}, {1263, 5}, {1291, 110}, {3459, 11584}, {3471, 30}, {11071, 1989}

X(13583) = CYCLOCEVIAN CONJUGATE OF X(2184)

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

X(13583) lies on the Kiepert hyperbola and these lines: {226,2475}, {1751,5046}, {2051,6894}, {6895,13478}

X(13583) = cyclocevian conjugate of X(2184)
X(13583) = X(i)-cross conjugate of X(j) for these (i,j): {2287, 2}, {6598, 8}
X(13583) = cevapoint of X(i) and X(j) for these (i,j): {115, 3900}, {522, 8286}

X(13584) = CYCLOCEVIAN CONJUGATE OF X(3223)

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

X(13584) lies on the Kiepert hyperbola and not on any line X(i)X(j) for 1 ≤ i < j ≤ 13583.

X(13584) = cevapoint of X(115) and X(4083)
X(13584) = cyclocevian conjugate of X(3223)

X(13585) = CYCLOCEVIAN CONJUGATE OF X(3459)

Barycentrics (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) : :

X(13585) lies on the Kiepert hyperbola and these lines: {6,11538}, {96,1157}, {5189,7607}, {6997,10155}, {7391,7612}, {7533,7608}

X(13585) = X(1994)-cross conjugate of X(2)
X(13585) = cevapoint of X(115) and X(1510)
X(13585) = cyclocevian conjugate of X(3459)

X(13586) = EULER LINE INTERCEPT OF THE LINE X(99)X(187)

Barycentrics 3a⁴ - 2a²b² - 2a²c² + b²c² : :

X(13586) lies on these lines: {2,3}, {32,7757}, {35,6645}, {36,4366}, {39,12150}, {76,5206}, {99,187}, {148,230}, {183,5210}, {194,3053}, {262,9734}, {315,7891}, {316,620}, {511,5182}, {524,2076}, {532,8595}, {533,8594}, {543,5152}, {574,3329}, {597,5116}, {598,7622}, {599,4048}, {754,2482}, {1078,7816}, {1384,7766}, {1691,5969}, {1916,2021}, {1975,5023}, {1992,5017}, {2030,10754}, {2080,4027}, {2549,7806}, {2896,7789}, {3111,13207}, {3734,7771}, {3788,7802}, {3849,5149}, {3926,7893}, {5013,7787}, {5026,5104}, {5184,11711}, {5215,9166}, {5913,7665}, {5989,8591}, {6179,7781}, {6337,7906}, {6390,7779}, {6680,7847}, {7610,11164}, {7737,7777}, {7738,7920}, {7747,7769}, {7748,7857}, {7750,7836}, {7756,7828}, {7761,7835}, {7763,7823}, {7768,7863}, {7778,7898}, {7784,7945}, {7788,9939}, {7795,7904}, {7801,7811}, {7804,8589}, {7818,7870}, {7820,7831}, {7825,7940}, {7830,7832}, {7842,7899}, {7850,7908}, {7860,7888}, {7865,10000}, {7867,7910}, {7869,7936}, {7872,7942}, {7873,7909}, {7874,7911}, {7880,7883}, {7881,7929}, {7930,7935}, {8291,11153}, {8292,11154}, {9751,11155}, {9770,11147}

X(13586) = midpoint of X(2) and X(33265)
X(13586) = reflection of X(14041) in X(2)
X(13586) = anticomplement of X(33228)
X(13586) = crossdifference of every pair of points on line {647, 6375}
X(13586) = barycentric product X(99)*X(11176)
X(13586) = barycentric quotient X(11176)/X(523)
X(13586) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,33273), (2,20,33017), (2,376,7833), (2,1003,384), (2,3552,1003), (2,7833,7924), (2,8598,9855), (2,9855,8597), (3,384,7824), (3,1003,2), (3,3552,384), (3,11676,5999), (32,7782,7783), (32,7783,7839), (99,187,385), (315,7891,7947), (316,620,7925), (548,8368,8354), (549,8370,2), (550,7807,6655), (574,3972,3329), (620,6781,316), (1003,8598,11676), (1003,10997,9855), (1975,5023,7793), (3534,11288,7841), (3552,10997,11676), (3734,8588,7771), (3788,7802,7885), (6655,7807,7901), (6656,8368,2), (6661,8359,2), (6680,7847,7923), (7750,7836,7939), (7761,7835,7931), (7763,7823,7941), (7791,7892,7948), (7830,7832,7928), (7841,11288,2), (7870,11057,7818), (8354,8368,6656), (8356,8369,2), (8369,8703,8356)

X(13587) = EULER LINE INTERCEPT OF THE LINE X(36)X(100)

Barycentrics a(3a³ - 3ab² - 3ac² + b²c + bc² + abc) : :
X(13587) = 2 X[36] + X[100] = 2 X[214] + X[484] = 4 X[1319] - X[1320] = X[20] + 2 X[1532] = 2 X[3814] + X[4316] = 2 X[1155] + X[4511] = 4 X[3035] - X[5080] = X[3218] - 4 X[5122] = 2 X[5122] + X[5440] = X[3218] + 2 X[5440] = X[3583] - 4 X[6681] = 7 X[3523] - X[6840] = 5 X[631] - 2 X[6882] = 2 X[3] + X[6905] = 4 X[3] - X[6909] = 2 X[6905] + X[6909] = 8 X[3911] + X[9963] = X[6265] + 2 X[10225]

X(13587) lies on these lines: {1,9352}, {2,3}, {8,5204}, {10,5303}, {35,551}, {36,100}, {40,5330}, {55,4345}, {56,3241}, {78,3928}, {81,4256}, {165,3877}, {214,484}, {517,4881}, {524,5096}, {528,5172}, {529,4996}, {574,5276}, {597,4265}, {644,1055}, {758,5131}, {896,5529}, {1155,4511}, {1210,11015}, {1319,1320}, {1420,3885}, {1470,11239}, {1621,5010}, {2078,13279}, {2975,3679}, {3035,5080}, {3218,5122}, {3361,3889}, {3582,10090}, {3583,6681}, {3616,4428}, {3655,11491}, {3814,4316}, {3825,4324}, {3828,5260}, {3829,5433}, {3833,5426}, {3868,4855}, {3876,3929}, {3895,13462}, {3897,7987}, {3911,9963}, {4299,11681}, {4677,8666}, {4745,5258}, {5193,13278}, {5251,9342}, {5362,10646}, {5367,10645}, {5691,7705}, {6265,10225}, {9782,11281}

X(13587) = reflection of X(10707) in X(3582)
X(13587) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,376,11114), (2,11112,6175), (3,404,21), (3,474,4189), (3,4188,404), (3,4191,4216), (3,6905,6909), (3,6911,6950), (3,6924,6906), (3,6940,6986), (3,6942,411), (20,6921,4193), (56,4421,3241), (140,2475,7504), (404,5047,474), (474,4189,5047), (549,11112,2), (631,4190,2476), (3241,4421,3871), (4189,5047,21), (4203,9909,8654), (4652,5438,3876), (5122,5440,3218), (6880,6948,6932), (6885,6977,6828), (6904,6910,4197), (6906,6924,6915), (6911,6950,6912), (6934,6961,6943), (6938,6970,6945)

X(13588) = EULER LINE INTERCEPT OF THE LINE X(55)X(86)

Barycentrics a(a + b) (a + c)(a²b + a²c - ab² - ac² - abc + b²c + bc²) : :

X(13588) lies on these lines: {2,3}, {10,4278}, {31,1582}, {36,3741}, {42,81}, {43,58}, {55,86}, {56,1043}, {57,5208}, {63,3786}, {75,2352}, {99,9082}, {108,1947}, {228,894}, {274,2223}, {314,1402}, {332,1014}, {333,1376}, {662,2194}, {672,2287}, {750,10458}, {1333,1575}, {1444,7224}, {1621,5333}, {1778,2238}, {1790,7350}, {1792,5323}, {1812,2651}, {2276,2303}, {3072,3193}, {3720,5253}, {3724,4418}, {3871,5711}, {4255,5331}, {4276,6685}, {4658,8715}, {6516,7196}, {10434,10455}

X(13588) = reflection of X(10707) in X(3582)
X(13588) = isoconjugate of X(37) and X(3500)
X(13588) = cevapoint of X(3) and X(1740)
X(13588) = X(3063)-zayin conjugate of X(661)
X(13588) = barycentric product X(86)*X(3501)
X(13588) = barycentric quotient X(i)/X(j) for these {i,j}: {58, 3500}, {3501, 10}
X(13588) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,2475,3136), (2,4184,21), (2,4188,4191), (2,4190,6817), (2,11322,4203), (3,1010,21), (3,11358,2), (171,3736,81), (404,4203,2), (859,4234,21), (1376,3286,333), (3724,4418,11688), (4188,11115,4225), (4225,11115,21)

X(13589) = EULER LINE INTERCEPT OF THE LINE X(100)X(190)

Barycentrics a(a - b) (a - c)(a³ + b³ + c³ - b²c - bc² - abc) : :

X(13589) lies on these lines: {2,3}, {36,1647}, {99,9070}, {100,190}, {108,13397}, {110,6011}, {643,3909}, {675,2481}, {833,835}, {901,2222}, {1283,3120}, {1292,9058}, {1623,10707}, {1626,11680}, {1754,3060}, {2948,13146}, {3573,6161}, {3938,5697}, {3961,11010}, {4597,13396}, {6012,9059}

X(13589) =anticomplement of X(867)
X(13589) = X(100)-aleph conjugate of X(2948)
X(13589) = X(765)-zayin conjugate of X(901)
X(13589) = crosspoint of X(100) and X(1290)
X(13589) = crossdifference of every pair of points on line {647, 1015}
X(13589) = crosssum of X(i) and X(j) for these (i,j): {512, 2610}, {513, 8674}, {661, 4895}
X(13589) = barycentric product X(i)*X(j) for these {i,j}: {664, 1731}, {1332, 5146}
X(13589) = barycentric quotient X(1731)/X(522)
X(13589) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2475,3145,11101), (3658,4246,4243), (4236,7475,11634), (5004,5005,7427), (7437,7451,3658), (7451,7461,7450)

X(13590) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CIRCUMTANGENTIAL TO 1st MORLEY ADJUNCT

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

The reciprocal orthologic triangle of these triangles is X(356)

X(13590) lies on these lines: {3,3276}, {3278,3605}

X(13590) = reflection of X(3281) in X(3)

X(13591) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CIRCUMTANGENTIAL TO 2nd MORLEY ADJUNCT

Trilinears F(A-2*Pi, B-2*Pi, C-2*Pi, a, b, c) : : , where F(A, B, C, a, b, c) = X(13590)

The reciprocal orthologic triangle of these triangles is X(3276)

X(13591) lies on these lines: {3,3277}, {3280,3606}

X(13591) = reflection of X(3283) in X(3)

X(13592) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CIRCUMTANGENTIAL TO 3rd MORLEY ADJUNCT

Trilinears F(A-4*Pi, B-4*Pi, C-4*Pi, a, b, c) : : , where F(A, B, C, a, b, c) = X(13590)

The reciporcal orthologic center of these triangles is X(3277)

X(13592) lies on these lines: {3,356}, {3282,3607}

X(13592) = reflection of X(3279) in X(3)

X(13593) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CIRCUMTANGENTIAL TO INNER-NAPOLEON

Barycentrics Sin[A]/(Sin[B] Sin[A/3-B/3]+Sin[C] Sin[A/3-C/3]) : : (Peter Moses, March 8, 2018)

The reciprocal orthologic center of these triangles is X(8011)

X(13593) lies on the circumcircle and the line {3,8008}

X(13593) = midpoint of X(3) and X(8008)
X(13593) = reflection of X(13594) in X(3)
X(13593) = antipode of X(13594) in circumcircle
X(13593) = parallelogic center of these triangles: circumnormal to inner-Napoleon

X(13594) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CIRCUMNORMAL TO INNER-NAPOLEON

Barycentrics Sin[A]/(Sin[B] Cos[A/3-B/3]-Sin[C] Cos[A/3-C/3]) : : (Peter Moses, March 8, 2018)

The reciprocal orthologic center of these triangles is X(8011)

X(13594) lies on the circumcircle and the line {3,8008}

X(13594) = midpoint of X(3) and X(8009)
X(13594) = reflection of X(13593) in X(3)
X(13594) = antipode of X(13593) in circumcircle
X(13594) = parallelogic center of these triangles: circumtangential to inner-Napoleon

X(13595) = EULER LINE INTERCEPT OF THE LINE X(51)X(110)

Barycentrics a²(a⁴ - b⁴ - c⁴ + 3b²c²) : :
X(13595) = X(3) + 5X(7545)

X(13595) lies on these lines: {2,3}, {6,9544}, {49,10095}, {51,110}, {107,324}, {111,251}, {145,11365}, {148,2936}, {154,3066}, {156,1199}, {159,10169}, {182,11451}, {184,5640}, {305,5971}, {323,3060}, {567,13364}, {575,12834}, {669,10278}, {1141,1302}, {1147,9781}, {1173,9705}, {1194,5041}, {1204,11439}, {1287,2770}, {1383,8770}, {1495,5012}, {1614,5462}, {1627,3291}, {1843,11416}, {1915,3124}, {1974,9813}, {1993,5102}, {2056,2502}, {2930,8584}, {2979,5651}, {3047,11746}, {3167,11004}, {3410,3580}, {3455,9166}, {3567,10539}, {3616,8185}, {3622,9798}, {3634,9591}, {3796,7712}, {3817,9590}, {4678,12410}, {5092,6030}, {5218,9673}, {5281,10833}, {5297,5310}, {5322,7292}, {5946,10540}, {6800,10601}, {7288,9658}, {7605,13394}, {7693,10192}, {8780,9777}, {9625,10175}, {10282,13434}, {10313,10985}, {10643,11421}, {10644,11420}, {10961,11418}, {10963,11417}, {11424,11449}, {11465,13336}

X(13595) = orthoptic-circle-of-Steiner-inellipe inverse of X(11563)
X(13595) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,23,6636), (2,25,23), (2,5189,1368), (2,6636,7496), (2,6995,7391), (2,7394,5169), (2,7492,7485), (2,7519,1370), (2,7533,5133), (3,3845,13596), (4,6644,2071), (5,3518,7488), (5,7576,3153), (22,1995,5020), (22,5020,2), (24,7529,3091), (24,9818,10298), (25,1995,2), (25,5020,22), (25,11284,9909), (51,110,1994), (111,251,1196), (154,3066,5422), (154,5422,11003), (186,381,7527), (251,1196,5354), (381,12106,186), (428,6677,858), (468,5133,2), (858,6677,2), (1370,7714,7519), (1495,5943,5012), (2072,13490,4), (3060,9306,323), (3060,10546,9306), (3091,10298,9818), (4232,7398,2), (5004,5005,550), (5012,10545,5943), (5055,7502,7550), (5071,7556,7514), (6353,6997,2), (6642,10594,20), (7392,7493,2), (7485,9909,7492), (7485,11284,2), (7577,11818,7565), (9909,11284,7485)

X(13596) = EULER LINE INTERCEPT OF THE LINE X(51)X(74)

Trilinears 4 cos A + 5 sec A : :
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+7 b^2 c^2+c^4) : :

X(13596) lies on these lines: {2,3}, {6,11738}, {51,74}, {54,11381}, {112,5008}, {154,3431}, {184,11455}, {569,12279}, {578,12290}, {1147,11439}, {1192,11270}, {1199,6241}, {1204,9781}, {1614,13474}, {1993,11472}, {1994,5663}, {2979,4550}, {3043,12133}, {3357,3567}, {3426,11402}, {3455,10722}, {5097,7722}, {5412,6480}, {5413,6481}, {5446,11440}, {5480,5621}, {6000,13366}, {9730,12834}, {10575,13434}

X(13596) = Stammler-circles-radical-circle inverse of X(23)
X(13596) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,3845,13595), (4,378,186), (4,3520,3518), (4,6143,235), (24,378,11410), (24,11403,4), (184,11455,12112), (186,378,3520), (376,9818,7550), (378,1597,4), (1593,1597,378), (1907,6240,4), (3146,7526,7512), (6241,11424,1199)

X(13597) = POINT BEID 131

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 26178 and Antreas Hatipolakis and Peter Moses, Hyacinthos 26850.

X(13597) lies on the circumcircle and these lines: {4,11792},{25,13507},{30, 11703},{99,1232},{110,140},{ 112,6748},{476,5899},{953, 5957},{2687,5959},{2699,5958} X(13597) = reflection of X(4) in X(11792)
X(13597) = isogonal conjugate of X(13391)
X(13597) = Collings transform of X(11792)

X(13598) = X(4)X(69)∩X(20)X(51)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 26178.

X(13598) lies on the circumcircle and these lines: {2, 13348}, {3, 5943}, {4, 69}, {5, 3819}, {20, 51}, {22, 11424}, {23, 13367}, {25, 13346}, {26, 11430}, {30, 143}, {52, 382}, {182, 10790}, {185, 3060}, {186, 13446}, {343, 1907}, {373, 3523}, {376, 9781}, {381, 10625}, {394, 5198}, {517, 12527}, {546, 1216}, {548, 5892}, {550, 5462}, {568, 5073}, {569, 12083}, {575, 10984}, {576, 1181}, {578, 7387}, {631, 6688}, {970, 1012}, {1071, 12109}, {1092, 10594}, {1147, 7530}, {1154, 3853}, {1199, 8718}, {1350, 9822}, {1351, 1498}, {1503, 10112}, {1539, 13433}, {1598, 9306}, {1629, 1941}, {1657, 9730}, {2071, 13376}, {2393, 2883}, {2777, 11800}, {2979, 3832}, {3091, 3917}, {3098, 7395}, {3522, 5640}, {3525, 10219}, {3529, 3567}, {3530, 13364}, {3533, 12045}, {3543, 5889}, {3627, 10263}, {3830, 6243}, {3839, 11444}, {3843, 5891}, {3845, 6101}, {3850, 10170}, {3851, 13340}, {3855, 7999}, {3861, 11591}, {5012, 12087}, {5056, 5650}, {5059, 10574}, {5068, 7998}, {5092, 10323}, {5097, 7592}, {5480, 6823}, {5806, 11573}, {7517, 10282}, {8549, 9914}, {8681, 11477}, {9714, 11202}, {9815, 10996}, {9909, 11425}, {9969, 12362}, {10299, 11465}, {10601, 13347}, {10628, 12295}, {10733, 13417}, {11438, 12085}, {11807, 11819}, {12006, 12103} X(13598) = midpoint of X(i) and X(j) for these {i, j}: {52,382}, {185,3146}, {3627,10263}, {5073,10575}, {5889,11381}, {6243,12162}, {10733,13417}
X(13598) = reflection of X(i) in X(j) for these {i, j}: {3,10110}, {20,9729}, {186,13446}, {389,5446}, {548,10095}, {550,5462}, {1071,12109}, {1216,546}, {1350,9822}, {2071,13376}, {5462,12002}, {5907,4}, {10112,13142}, {10575,13382}, {10625,11793}, {10627,3850}, {11573,5806}, {11574,5480}, {11591,3861}, {12103,12006}, {13474,3627}

X(13599) = X(2)X(389)∩X(3)X(275)

Trilineaers    1/(cos A cos 2A - cos(B-C)) : :
Trilineaers    csc(A - T) : : , where T is as at X(389)
Barycentrics    1/(a^8-3*(b^2+c^2)*a^6+(3*b^4+ 4*b^2*c^2+3*c^4)*a^4-(-c^4+b^ 4)*(b^2-c^2)*a^2-2*(b^2-c^2)^ 2*b^2*c^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26181.

X(13599) lies on the Kiepert hyperbola and these lines: {2,389}, {3,275}, {4,216}, {5,2052}, {76,7399}, {83,7395}, {96,7592}, {98,1181}, {485,6810}, {486,6809}, {1751,7567}, {3091,8796}, {5392,13160}, {6504,6815} X(13599) = reflection of X(4) in X(8799)
X(13599) = isogonal conjugate of X(578)

X(13600) = X(1)X(3)∩X(5)X(6736)

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-8 a^4 b c+12 a^3 b^2 c+6 a^2 b^3 c-13 a b^4 c+2 b^5 c-a^4 c^2+12 a^3 b c^2-24 a^2 b^2 c^2+12 a b^3 c^2+b^4 c^2-2 a^3 c^3+6 a^2 b c^3+12 a b^2 c^3-4 b^3 c^3+2 a^2 c^4-13 a b c^4+b^2 c^4+a c^5+2 b c^5-c^6) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26194.

X(13600) lies on these lines: {1,3}, {5,6736}, {8,6939}, {145,12528}, {388,12700}, {392,12245}, {519,5777}, {946,3880}, {952,9856}, {1071,3241} X(13600) = midpoint of X(i) and X(j) for these {i,j}: {145, 12672}, {3057, 7982}, {10284, 11278}
X(13600) = reflection of X(i) in X(j) for these {i,j}: {942,10222}, {5836,13464}, {12645,9947}, {12675,3635}
X(13600) = {X(1),X(10310)}-harmonic conjugate of X(1385)

X(13601) = X(1)X(3)∩X(8)X(12709)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26194.

X(13601) lies on these lines: {1,3}, {8,12709}, {144,7672}, {226,5836}, {388,10914}, {392,1788}, {960,4848}, {971,10950} X(13601) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,13370,1319), (65,2099,942), ( 65,3057,57), (65,3340,5173), ( 1466,2099,1)

X(13602) = X(1)X(3530)∩X(8)X(4540)

Barycentrics (a^2 - 7ab + b^2 - c^2)(a^2 - 7ac - b^2 + c^2) : :
Barycentrics (sin A)/(7 - 2 cos A) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26201.

X(13602) lies on the Feuerbach hyperbola and these lines: {1, 3530}, {8, 4540}, {21, 3635}, {79, 9957}, {80, 5919}, et al X(13602) = isoconjugate of X(58) and X(3968)
X(13602) = barycentric quotient X(37)/X(3968)

X(13603) = ISOGONAL CONJUGATE OF X(8703)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26201.

X(13603) lies on the Jerabek hyperbola and these lines: {3,5888}, {6,11455}, {54, 12112}, {67,10721}, et al X(13603) = isogonal conjugate of X(8703)
X(13603) = X(54)-vertex conjugate of X(3431)

X(13604) = X(10)X(21)∩X(11)X(5620)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 26203.

X(13604) lies on these lines: {10,21}, {11,5620}
X(13604) = reflection of X(5620) in X(11)

X(13605) = X(10)X(125)∩X(74)X(516)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 26203.

X(13605) lies on these lines:
{1, 3448}, {2, 2948}, {5, 2771}, {10, 125}, {67, 5847}, {74, 516}, {80, 5620}, {110, 1125}, {113, 3817}, {141, 2836}, {146, 1699}, {226, 3028}, {265, 515}, {399, 5886}, {517, 10264}, {519, 7984}, {542, 551}, {690, 4049}, {758, 10693}, {944, 12407}, {946, 5663}, {950, 12904}, {962, 9904}, {1511, 10165}, {2778, 6247}, {2784, 11005}, {2796, 11006}, {2842, 11814}, {3120, 6788}, {3576, 12383}, {4292, 10081}, {4297, 11709}, {4304, 12896}, {4466, 6326}, {5603, 12317}, {5901, 11699}, {6684, 12778}, {6699, 10164}, {10065, 10624}, {10088, 13411}, {10106, 12903}, {10272, 11230}, {10620, 12699} X(13605 = X(2948)-excentral-isogonal conjugate X(30)
X(13605) = midpoint of X(i) and X(j) for these {i, j}: {1,3448}, {944,12407}, {962,9904}, {7984,13211}, {10620,12699}
X(13605) = reflection of X(i) in X(j) for these {i, j}: {10,125}, {110,1125}, {946,12261}, {4297,11709}, {11699,5901}, {11720,11735}, {12778,6684}

X(13606) = X(1)X(549)∩X(7)X(5697)

Barycentrics (a^2-5 a b+b^2-c^2) (a^2-b^2-5 a c+c^2) : :
Barycentrics (sin A)/(5 - 2 cos A) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26206.

X(13606) lies on the Feuerbach hyperbola and these lines:
{1,549}, {7,5697}, {8,3898}, {9,3633}, {21,3244}, {35,1476}, {79,3057}, {80,9957}, {104,3746} X(13606) = X(58)-isoconjugate of X(3918)
X(13606) = trilinear pole of line X(650)X(4949)
X(13606) = barycentric quotient X(37)/X(3918)

X(13607) = X(1)X(4)∩X(3)X(3244)

Trilinears 5 r - 2 R cos B cos C : :
Barycentrics 6 a^4-5 a^3 b-5 a^2 b^2+5 a b^3-b^4-5 a^3 c+10 a^2 b c-5 a b^2 c-5 a^2 c^2-5 a b c^2+2 b^2 c^2+5 a c^3-c^4 : :
X(13607) = 5 X[1] - X[4] = 3 X[1] + X[944] = 3 X[4] + 5 X[944] = 3 X[4] - 5 X[946] = 3 X[1] - X[946] = 11 X[4] - 15 X[1699] = 11 X[946] - 9 X[1699] = 11 X[1] - 3 X[1699] = 11 X[944] + 9 X[1699] = X[40] + 3 X[3241] = 5 X[10] - 7 X[3526] = 7 X[4] - 15 X[5603] = 7 X[1699] - 11 X[5603] = 7 X[946] - 9 X[5603] = 7 X[1] - 3 X[5603] = 7 X[944] + 9 X[5603] = 9 X[4] - 5 X[5691] = 9 X[1] - X[5691] = 3 X[946] - X[5691] = 3 X[944] + X[5691] = X[944] - 3 X[5882] = X[946] + 3 X[5882] = X[4] + 5 X[5882] = 3 X[5603] + 7 X[5882] = X[5691] + 9 X[5882] = 3 X[1699] + 11 X[5882] = X[944] - 9 X[7967] = X[5882] - 3 X[7967] = X[1] + 3 X[7967] = X[5603] + 7 X[7967] = X[946] + 9 X[7967] = X[1699] + 11 X[7967] = X[4] + 15 X[7967] = X[8] - 3 X[10165] = 7 X[3526] - 15 X[10246] = X[10] - 3 X[10246] = 5 X[8] - 13 X[10303]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26206.

X(13607) lies on these lines:
{1,4}, {3,3244}, {5,3636}, {8,10165}, {10,3526}, {40,3241}, {104,3746} et al X(13607) = midpoint of X(i) and X(j) for these {i,j}: {1,5882}, {3,3244}, {145,11362}, {550,11278}, {944,946}, {1317,11715}, {1385,1483}, {1482,4297}, {3057,5884}, {5493,8148}, {7972,10265}, {9957,12675}
X(13607) = reflection of X(i) in X(j) for these {i,j}: {5,3636}, {3626,140}, {3754, 13373}, {6684,1385}, {13464,1}
X(13607) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,944,946), (1,5691,10595), (1, 7967,5882), (944,10595,5691), ( 946,5882,944), (5691,10595,946)

X(13608) = X(3)X(524)∩X(4)X(111)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26212.

Let P be a point in the plane of ABC. Let (Oa) be the circumcircle of BCP, and define (Ob) and (Oc) cyclically. Let A' be the intersection, other than P, of (Oa) and AP, and define B' and C' cyclically. Let A" be the antipode of A' in (Oa), and define B" and C" cyclically. The lines AA", BB" CC" concur for all P. When P = X(2), the lines AA", BB" CC" concur in X(13608). (Randy Hutson, July 21, 2017) X(13608) lies on the cubics K009, K043 and these lines: {2,12505}, {3,524}, {4,111}
X(13608) = isogonal conjugate of X(14262)
X(13608) = complement of X(34165)
X(13608) = Cundy-Parry Phi transform of X(524)
X(13608) = Cundy-Parry Psi transform of X(111)

X(13609) = COMPLEMENT OF X(658)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26212.

X(13609) lies on these lines:
{2,658}, {9,1768}, {11,1146}, {124,1566}, {1212,5316}, {1615,5658}, {2632,6587}, {2968,3239}, {5328,6554} X(13609) = complement X(658)
X(13609) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 7658}, {7, 3900}, {4081, 1146}, {10405, 522}X(13609)
X(13609) = crosspoint of X(i) and X(j) for these (i,j): {2, 3239}, {514, 10307}, {522, 10405}X(13609)
X(13609) = crosssum of X(i) and X(j) for these (i,j): {6, 1461}, {101, 6244}, {109, 3207}
X(13609) = isoconjugate of X(j) and X(j) for these (i,j): {1262, 3062}, {7045, 11051}
X(13609) = {X(11),X(3119)}-harmonic conjugate of X(1146)
X(13609) = barycentric product X(i)X(j) for these {i,j}: {144, 1146}, {3160, 4081}, {3239, 7658}
X(13609) = barycentric quotient X(i)/X(j) for these {i,j}: {144, 1275}, {165, 7045}, {1146, 10405}, {2310, 3062}, {3207, 1262}, {7658, 658}
X(13609) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 3900}, {31, 7658}, {32, 6129}, {41, 522}, {55, 4885}, {200, 3835}, {213, 656}, {220, 513}, {294, 926}, {607, 521}, {649, 11019}, {650, 2886}, {657, 10}, {663, 142}, {667, 4000}, {798, 1834}, {949, 6182}, {1021, 3741}, {1170, 6607}, {1253, 514}, {2175, 905}, {2287, 512}, {2310, 116}, {2328, 4369}, {2332, 8062}, {3063, 1}, {3119, 124}, {3239, 2887}, {3271, 4904}, {3709, 442}, {3900, 141}, {4105, 3452}, {4130, 1329}, {4171, 3454}, {4397, 626}, {4524, 1211}, {6602, 4521}, {7118, 8058}, {7252, 3742}, {8641, 2}, {9447, 6589}

X(13610) = STEINER IMAGE OF X(1)

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

Suppose that P = p : q : r (barycentrics) is a point in the plane of a triangle ABC. The Steiner image of P is introduced here as the point

S(P) = p/(q² + r² - p² + qr + rp + pq) : :

Let S be the Steiner circumellipse of the cevian triangle A'B'C' of P. Let A'' be the point, other than A', in which S meets line BC, and define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in S(P). (Contributed by Peter Moses, June 23, 2017; see also X(13486).

X(13610) lies on the cubics K132 and K328, and on these lines:
{1,1326}, {6,2640}, {10,894}, {37,171}, {58,1247}, {65,4649}, {75,8033}, {192,13174}, {757,2643}, {759,5429}, {1178,4128}, {3017,5620}, {4418,6535}

X(13610) = isogonal conjugate of X(846)
X(13610) = cevapoint of X(i) and X(j) for these {i,j}: {513, 2643}, {649, 4128}
X(13610) = trilinear pole of line {661, 4367}
X(13610) = X(9278)-he conjugate of X(9)
X(13610) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 846}, {2653, 43}
X(13610) = X(i)-cross conjugate of X(j) for these (i,j): {81, 1}, {256, 87}, {3512, 9499}
X(13610) = trilinear pole, wrt medial triangle, of line X(4)X(9)
X(13610) = isoconjugate of X(j) and X(j) for these (i,j): {1, 846}, {3, 4213}, {6, 1654}, {42, 6626}, {71, 2905}, {249, 6627}, {2664, 8937}
X(13610) = barycentric product X(i)X(j) for these {i,j}: {1, 6625}, {75, 2248}
X(13610) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1654}, {6, 846}, {19, 4213}, {28, 2905}, {81, 6626}, {2248, 1}, {2643, 6627}, {6625, 75}

X(13611) = POINT BEID 132

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26214.

X(13611) lies on these lines: {3,6723}, {4,3184}, {122,125}, {136,3154}
X(13611) = {X(125), X(1650)}-harmonic conjugate of X(122)

X(13612) = POINT BEID 133

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26214.

X(13612) lies on the nine-point circle and these lines: {117,6260}, {124,7358}, {7952,10271}

X(13613) = POINT BEID 134

Trilinears sin(2*A)*sin(B-C)^2*(3*cos(A)- cos(B-C))^2*((2*cos(2*A)+6)* cos(B-C)-9*cos(A)+cos(3*A)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26214.

X(13613) lies on the nine-point circle and these lines: {122,1562}, {132,2883}, {133,1249}
X(13613) = intersection, other than X(122), of nine-point circle and cevian circle of X(20)

X(13614) = ISOGONAL CONJUGATE OF X(8811)

Barycentrics a (a+b) (a-b-c) (a+c) (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 b^4 c+2 b^5 c-a^4 c^2+2 a^2 b^2 c^2-b^4 c^2+4 a^3 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) : :
X(13614) = 3 R (2 r R+4 R^2-s^2) X[2] + 2 r (r+2 R)^2 X[3]

X(13614) lies on these lines: {2,3}, {78,7070}, {271,282}, {936,2328}, {1034,1792}, {1035,5932}, {1259,8805}, {1819,3347}, {3176,8885}

X(13614) = isogonal conjugate of X(8811)
X(13614) = X(1792)-Ceva conjugate of X(21)
X(13614) = X(8885)-cross conjugate of X(21)
X(13614) = isoconjugate of X(j) and X(j) for these (i,j): {1, 8811}, {56, 8806}, {65, 3345}, {73, 7149}, {226, 7152}, {1034, 1042}, {1439, 7007}, {3668, 7037}
X(13614) = barycentric product X(i)*X(j) for these {i,j}: {314, 3197}, {333, 1490}, {345, 8885}, {1812, 3176}, {2287, 5932}
X(13614) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8811}, {9, 8806}, {284, 3345}, {1035, 1427}, {1172, 7149}, {1490, 226}, {2194, 7152}, {2287, 1034}, {2332, 7007}, {3197, 65}, {3341, 8808}, {5932, 1446}, {8885, 278}
X(13614) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,4183,21), (21,1816,1817)

X(13615) = ISOGONAL CONJUGATE OF X(8814)

a^2 (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(13615) = 6 R^2 X[2] + r (r+4 R) X[3]

X(13615) lies on these lines: {2,3}, {6,2328}, {9,55}, {35,8580}, {42,218}, {56,5436}, {63,5728}, {72,3295}, {154,572}, {169,3198}, {191,10399}, {197,8053}, {220,3190}, {226,1001}, {268,7008}, {329,954}, {950,958}, {956,3488}, {960,10393}, {993,11019}, {999,4666}, {1259,9844}, {1486,8804}, {1708,4640}, {1728,11507}, {2975,10580}, {3303,11523}, {3419,9708}, {3586,5251}, {3877,3957}, {3897,7373}, {5175,5260}, {5248,12572}, {5259,7742}, {5273,5809}, {5777,10267}, {8273,8583}

X(13615) = isogonal conjugate of X(8814)
X(13615) = isoconjugate of X(j) and X(j) for these (i,j): {1, 8814}, {19, 8813}
X(13615) = crossdifference of every pair of points on line {647, 3669}
X(13615) = crosssum of X(i) and X(j) for these (i,j): {7, 5933}
X(13615) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 8813}, {6, 8814}
X(13615) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,1005,7580), (2,6872,10431), (3,405,37244), (3,11108,16410), (9,55,1260), (9,2900,210), (21,452,405), (21,11344,3), (25,1011,3), (25,11284,7453), (55,210,6600), (329,1621,954), (405,442,11108), (405,7580,2), (405,11113,6913), (1006,3651,3524), (1864,3683,9), (1995,4184,11350), (4184,11350,3)

X(13616) = ISOGONAL CONJUGATE OF X(10261)

Barycentrics a^2 (a^4-b^4-b^2 c^2-c^4-a^2 S+b^2 S+c^2 S) : :
X(13616) = 3 a^2 b^2 c^2 X[2]-4 (a^2+b^2+c^2-S) S^2 X[3]

X(13616) lies on these lines: {2,3}, {589,8406}, {1161,1994}, {1993,11824}, {2979,10133}, {3796,12306}, {5012,9732}, {5409,6800}

X(13616) = isogonal conjugate of X(10261)
X(13616) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,22,1600), (3,3155,7485), (3,6636,13617)

X(13617) = ISOGONAL CONJUGATE OF X(10262)

Barycentrics a^2 (a^4-b^4-b^2 c^2-c^4-(-a^2+b^2+c^2) S) : :
X(13617) = 3a^2 b^2 c^2 X[2] - 4 (a^2+b^2+c^2+S) S^2 X[3]

X(13617) lies on these lines: {2,3}, {588,8414}, {1160,1994}, {1993,11825}, {2979,10132}, {3796,12305}, {5012,9733}, {5408,6800}

X(13617) = isogonal conjugate of X(10262)
X(13617) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,22,1599), (3,3156,7485), (3,6636,13616)

X(13618) = ISOGONAL CONJUGATE OF X(10378)

Barycentrics a (a+b) (a+c) (a^10+3 a^8 b^2-14 a^6 b^4+14 a^4 b^6-3 a^2 b^8-b^10+6 a^8 b c-16 a^6 b^3 c+12 a^4 b^5 c-2 b^9 c+3 a^8 c^2-4 a^6 b^2 c^2+2 a^4 b^4 c^2-4 a^2 b^6 c^2+3 b^8 c^2-16 a^6 b c^3+8 a^4 b^3 c^3+8 b^7 c^3-14 a^6 c^4+2 a^4 b^2 c^4+14 a^2 b^4 c^4-2 b^6 c^4+12 a^4 b c^5-12 b^5 c^5+14 a^4 c^6-4 a^2 b^2 c^6-2 b^4 c^6+8 b^3 c^7-3 a^2 c^8+3 b^2 c^8-2 b c^9-c^10) : : X(13618) lies on these lines: {2,3}, {255,610}, {2360,7070}, {7011,8885} X(13618) = isogonal conjugate of X(10378)

X(13619) = ISOGONAL CONJUGATE OF X(11559)

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+5 a^2 b^2 c^2-b^4 c^2+a^2 c^4-b^2 c^4+c^6) : :
X(13619) = 3 X[4] - 4 X[403] = 3 X[186] - 2 X[403] = 5 X[4] - 8 X[468] = 5 X[403] - 6 X[468] = 5 X[186] - 4 X[468] = 3 X[376] - 2 X[2071] = 5 X[631] - 4 X[2072]
X(13619) = 3 (-2+J^2)X[2]-2 (-3+2 J^2)X[3]

X(13619) lies on these lines: {2,3}, {74,10421}, {112,6781}, {146,10540}, {185,12254}, {340,1272}, {477,933}, {562,930}, {1154,7722}, {1199,13568}, {1204,12289}, {1291,1300}, {1870,4316}, {1986,13391}, {3521,5944}, {4294,10149}, {4324,6198}, {6000,12244}, {7689,12278}, {7756,10312}, {9076,10098}, {9730,11692}, {12270,12273}

X(13619) = midpoint of X(i) and X(j) for these {i,j}: {1657, 5899}
X(13619) = reflection of X(i) in X(j) for these {i,j}: {4, 186}, {146, 10540}, {186, 10295}, {382, 11563}, {3153, 3}, {3627, 10096}, {10296, 2072}
X(13619) = isogonal conjugate of X(11559)
X(13619) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,4,6143), (3,382,10224), (376,3529,1370), (376,7494,3528), (468,10989,8889), (550,6240,3520), (550,6636,376), (1113,1114,3520), (3146,7505,4), (3520,6240,4), (15160,15161,4)
X(13619) = circumcircfle-inverse of X(3520)
X(13619) = polar-circle inverse of X(546)
X(13619) = X(523)-vertex conjugate of X(3520)

X(13620) = ISOGONAL CONJUGATE OF X(12023)

Barycentrics a^2 (5 a^8-10 a^6 b^2+10 a^2 b^6-5 b^8-10 a^6 c^2+9 a^4 b^2 c^2-4 a^2 b^4 c^2+5 b^6 c^2-4 a^2 b^2 c^4+10 a^2 c^6+5 b^2 c^6-5 c^8) : : X(13620) lies on these lines: {2,3}, {323,11202}, {7712,10605} X(13620) = isogonal conjugate of X(12023)
X(13620) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,24,5056), (3,2070,3845), (3,11001,2071), (186,7488,6636), (3543,10298,3)

X(13621) = ISOGONAL CONJUGATE OF X(13418)

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+4 b^6 c^2-3 a^2 b^2 c^4-6 b^4 c^4+2 a^2 c^6+4 b^2 c^6-c^8) : :
X(13621) = 12X[2]+(-8+J^2)X[3]

X(13621) lies on these lines: {2,3}, {6,3205}, {49,51}, {52,12316}, {54,10095}, {110,143}, {156,3567}, {389,10540}, {399,6102}, {567,9920}, {568,10539}, {669,10279}, {827,5966}, {1141,11815}, {1147,12310}, {1157,10228}, {1173,9706}, {1493,9705}, {1495,5462}, {1614,5946}, {1624,3432}, {2079,7747}, {2929,7728}, {2931,5448}, {3357,9919}, {3581,5907}, {5943,13353}, {5944,13364}, {6199,8276}, {6243,9306}, {6395,8277}, {6767,10046}, {7373,10037}, {8185,10246}, {9590,9955}, {9591,11231}, {9625,9956}, {9626,11230}, {10247,11365}, {10546,11412}, {11591,12307}, {11597,11808}, {11695,13339}, {12121,12897}, {12161,13321}

X(13621) = inverse in the circumcircle of X(10096)
X(13621) = isogonal conjugate of X(13418)
X(13621) = X(11538)-Ceva conjugate of X(6)
X(13621) = X(523)-vertex conjugate of X(10096)
X(13621) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,2937,3), (3,1598,3830), (3,5020,5070), (3,7529,3851), (5,2070,3), (5,3518,2070), (22,3526,3), (23,140,13564), (24,381,3), (25,6642,7517), (25,7506,3), (26,1656,3), (26,1995,1656), (110,143,195), (140,13564,3), (382,6644,3), (1113,1114,10096), (3153,7488,11413), (3517,7529,3), (5020,9714,3), (5944,13364,13434), (6353,7528,6639), (6642,7517,3), (6644,10594,382), (7506,7517,6642), (7545,12106,3), (7715,11585,7540), (10096,13163,5)

X(13622) = ISOGONAL CONJUGATE OF X(13595)

Barycentrics (a^4-3 a^2 b^2+b^4-c^4) (a^4-b^4-3 a^2 c^2+c^4) : :
X(13622) = 8 X[3628] - 5 X[9972] = 5 X[54] - 4 X[12007]

X(13622) lies on the Jerabek hyperbola and these lines: {3,5965}, {4,9973}, {54,12007}, {67,6467}, {141,895}, {265,5891}, {511,3521}, {524,1176}, {542,11559}, {549,5504}, {599,6391}, {826,10097}, {1154,4846}, {3532,10619}, {3628,9972}, {6776,11270}, {7748,13481}, {10293,10628}, {12254,13452}

X(13622) = isogonal conjugate of X(13595)

X(13623) = ISOGONAL CONJUGATE OF X(13596)

Barycentrics (a^2-b^2-c^2) (a^4+7 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+7 a^2 c^2-2 b^2 c^2+c^4) : :
X(13623) = 6 X[549] - 5 X[5888] = 3 X[4] - 5 X[7693] = 12 X[5066] - 5 X[13603]

X(13623) lies on the Jerabek hyperbola and these lines: {2,11738}, {4,7693}, {6,3534}, {54,548}, {64,3526}, {74,549}, {895,12121}, {1531,3521}, {3426,5055}, {3431,10304}, {5066,13603}, {6699,11559}, {10303,13452}

X(13622) = isogonal conjugate of X(13596)

X(13624) = X(1)X(3)∩X(30)X(1125)

Trilinears    4*a^3-(b+c)*a^2-2*(2*b^2-b*c+ 2*c^2)*a+(b^2-c^2)*(b-c) : :
Trilinears    4 cos A + cos B + cos C - 1 : :
Trilinears    r + 3 R cos B cos C : :
X(13624) = X(1)+3*X(3) = 5*X(1)+3*X(40) = 7*X(1)+9*X(165) = X(1)-3*X(1385) = 7*X(1)-3*X(1482) = X(1)-9*X(3576) = 11*X(1)-3*X(7982) = 3*X(4)-11*X(5550) = X(4)-3*X(11230) = X(5)-3*X(10165) = X(4297)+3*X(10165)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26229.

Let DEF be the cevian triangle of I=X(1) and A"B"C" the 2nd circumperp triangle. Let D_b and D_c be the circumcenters of ABD and ACD. The lines through X(3) parallel to DD_b and DD_c intersects the lines through I perpendicular to IB and IC at A_b and A_c respectively. Let (A") be the circle with center centered at A" and tangent to line A_bA_c, and define (B") and (C") cyclically. X(13624) is the radical center of (A"), (B"), (C"). (Angel Montesdeoca, November 7, 2019)

X(13624) lies on these lines: {1,3}, {4,5550}, {5,4297}, {8,3524}, {10,549}, {20,5886}, {21,4881}, {30,1125}, {44,572}, {72,3431}, {74,11699}, {79,4870}, {104,6986}, {140,515}, {145,3654}, {182,4663}, {186,1829}, {214,960}, {229,4221}, {355,631}, {376,3616}, {378,11363}, {381,3624}, {382,8227}, {392,4189}, {495,4311}, {496,4304}, {500,1193}, {518,5092}, {582,1468}, {912,12038}, {944,3617}, {952,3626}, {956,4855}, {993,5044}, {1000,6049}, {1480,1616}, {1483,11362}, {2975,4420}, {3488,5265}, {3585,5444}, {3621,5657}, {3622,3656}, {3625,5690}, {3689,5288}, {3811,11194}, {3916,4511}, {4004,9352}, {4292,11544}, {4293,11374}, {4298,5719}, {4299,11375}, {4301,10283}, {4302,11376}, {4316,5443}, {4652,5730}, {4816,9588}, {5219,9655}, {5258,12773}, {5298,10543}, {5433,10572}, {5438,9708}, {5844,13607}, {5887,10167}, {6051,8143}, {6284,7743}, {6713,12019}, {6734,10609}, {6759,12262}, {8546,9004}, {8715,11260}, {9778,10595}, {10610,12675}, {11711,12042}

X(13624 = midpoint of X(i) and X(j) for these {i,j}: {1,3579}, {3,1385}, {5,4297}, {74,11699}, {548,5901}, {550,946}, {551,8703}, {960,13369}, {1386,3098}, {1483,11362}, {1511,11709}, {5690,5882}, {5731,11231}, {6759,12262}, {8715,11260}, {11278,12702}, {11711,12042}, {11720,12041}, {12512,13464}
X(13624 = reflection of X(i) in X(j) for these (i,j): (3828,11812), (3853,12571), (5885,9940), (6583,13373), (6684,3530), (9955,1125), (9956,140)
X(13624) = X(140)-of-2nd-circumperp-triangle
X(13624) = X(3850)-of-excentral-triangle
X(13624) = X(548)-of-1st-circumperp-triangle
X(13624) = radical center of circles centered at A, B, C with respective radii 1/2*sqrt(bc), 1/2*sqrt(ca), 1/2*sqrt(ab)
X(13624) = QA-P32 (Centroid of the Circumcenter Quadrangle) of quadrangle ABCX(1)
X(13624 = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3,3579), (1,40,8148), (1,12702,11278), (3,1482,165), (3,3576,1385), (3,10246,40), (3,10680,5584), (3,13151,942), (35,1319,9957), (36,2646,942), (36,3576,13151), (65,7280,5122), (214,5267,960), (355,631,11231), (631,5731,355), (1385,3579,1), (2646,13151,1385), (3576,7987,3), (3579,11278,12702), (4297,10165,5), (8148,10246,1), (37524,37525,1)

X(13625) = REFLECTION OF X(6789) IN X(13607)

Trilinears 4*p^5*(4*p-17*q)+4*(17*q^2+12) *p^4-(28*q^2+73)*q*p^3+(4*q^4+ 52*q^2-9)*p^2-(17*q^2-6)*q*p+( 2*q^2-1)*q^2 : : , where p = sin(A/2) and q = cos(B/2 - C/2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26234.

X(13625) lies on these lines: {3,519}, {952,11717}, {3244,3667}, {6789,13607}

X(13625) = reflection of X(6789) in X(13607)

X(13626) = MIDPOINT OF X(381) AND X(1113)

Barycentrics 3*SB*SC - S^2 + K*(2*S)/(3*R) : : , where K = 2*S*|OH|
X(13626) = (-3*R*S-2*K)*X(3) + (3*R*S-K)*X( 4)

As a point of the Euler line, X(13626) has Shinagawa coefficients (-3RS + 2K, 9RS).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26240.

X(13626) lies on this line: {2,3}

X(13626) = midpoint of X(i) and X(j) for these {i,j}: {3,10720}, {381,1113}
X(13626) = reflection of X(i) in X(j) for these (i,j): (1313,547), (13627,2)

X(13627) = MIDPOINT OF X(381) AND X(1114)

Barycentrics 3*SB*SC - S^2 - K*(2*S)/(3*R) : : , where K = 2*S*|OH|
X(13627) = (-3*R*S-2*K)*X(3) - (3*R*S-K)*X( 4)

As a point of the Euler line, X(13626) has Shinagawa coefficients (-3RS - 2K, 9RS).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26240.

X(13627) lies on this line: {2,3}

X(13627) = midpoint of X(i) and X(j) for these {i,j}: {3,10719}, {381,1114}
X(13627) = reflection of X(i) in X(j) for these (i,j): (1312,547), (13626,2)

X(13628) = POINT BEID 135

Barycentrics 3*(27*R^2-8*SW)*S^2 + 81*R^2*SB* SC+8*SW^3 : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26240.

X(13628) lies on this line: {2,3}

X(13629) = POINT BEID 136

Barycentrics (117*R^2-32*SW)*S^2-27*R^2*SB* SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26240.

As a point of the Euler line, X(13629) has Shinagawa coefficients (11E + 128F, 27E).

X(13629) lies on this line: {2,3}

X(13630) = X(3)X(54)∩X(30)X(143)

Barycentrics a^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^2-4 a^2 b^4 c^2+b^6 c^2-3 a^4 c^4-4 a^2 b^2 c^4+3 a^2 c^6+b^2 c^6-c^8) : :
X(13630) = 7 X[5] - 9 X[373] = 7 X[185] + 9 X[373] = X[20] + 3 X[568] = 7 X[3] - 3 X[2979] = X[1657] + 3 X[3060] = X[382] - 5 X[3567] = 3 X[51] - X[3627]

See Telv Cohl and Peter Moses, Hyacinthos 26257.

Let P be a point on the circumcircle. Let Pa be the orthogonal projection of P on the A-altitude, and define Pb and Pc cyclically. The locus of the nine-point center of PaPbPc as P varies is an ellipse centered at X(13630); see also X(185), X(5884), X(8550), X(9730). (Randy Hutson, July 21, 2017)

Let Ha, Hb, Hc be the orthocenters of the A-, B-, and C-altimedial triangles. Then X(13630) is the nine-point center of HaHbHc. (Randy Hutson, July 21, 2017)

Let Na, Nb, Nc be the nine-point centers of the A-, B-, and C-altimedial triangles. Then X(13630) is the orthocenter of NaNbNc. (Randy Hutson, July 21, 2017)

Let (A) be the circle centered at A that cuts off a segment of line BC equal to |BC|. Define (B) and (C) cyclically. X(13630) is the radical center of circles (A), (B), (C). (Randy Hutson, July 21, 2017)

X(13630) lies on these lines: {2,5876}, {3,54}, {4,3521}, {5, 113}, {6,12084}, {20,568}, {26, 9786}, {30,143}, {49,1511}, {51, 3627}, {52,550}, {74,13434}, { 140,9729}, {156,1181}, {182, 7689}, {186,5944}, {376,6243}, { 381,6241}, {382,3567}, {511,548} ,{546,5462}, {547,11695}, {549, 5562}, {567,1986}, {569,1204}, { 575,2781}, {578,11250}, {632, 5891}, {973,11750}, {1199,2071}, {1216,3530}, {1656,12111}, { 1657,3060}, {1658,11438}, {1853, 7564}, {2779,5885}, {2807,5901}, {3448,6288}, {3526,11459}, { 3528,13340}, {3581,7512}, {3628, 5892}, {3819,12108}, {3830,9781} ,{3843,5640}, {3845,11381}, { 3850,5943}, {3853,10110}, {3861, 13474}, {3917,11592}, {5054, 11444}, {5072,11451}, {5073, 13321}, {5079,11465}, {5447, 12100}, {5498,6699}, {5878,7729} ,{5899,8718}, {6143,7722}, { 6240,6746}, {6688,12812}, {6759, 12106}, {7502,10984}, {7506, 11456}, {7514,12163}, {7526, 10605}, {7529,12174}, {8254, 10628}, {8548,12301}, {8703, 10625}, {9704,11449}, {9705, 12284}, {10116,11802}, {10226, 11430}, {11245,12370}, {11432, 12085}, {12022,12236}, {12233, 13371}, {12254,13368}

X(13630) = midpoint of X(i) and X(j) for these {i,j}: {3, 6102}, {4, 13491}, {5, 185}, {20, 10263}, {52, 550}, {1986, 12041}, {3627, 10575}, {5889, 6101}, {9729, 13382}, {10264, 11562}, {12254, 13368}, {14374,14375}
X(13630) = reflection of X(i) in X(j) for these {i,j}: {4, 10095}, {5, 12006}, {140, 9729}, {143, 389}, {546, 5462}, {1216, 3530}, {3853, 10110}, {5907, 3628}, {10627, 3}, {11591, 140}, {13363, 9730}, {13421, 52}, {13474, 3861}
X(13630) = complement X(5876)
X(13630) = crosssum of X(3) and X(5448)
X(13630) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,5012,10610), (3,5889,6101), ( 3,5890,6102), (4,5946,10095), ( 5,373,12046), (5,9730,12006), ( 5,12006,13363), (20,568,10263), (51,10575,3627), (185,9730,5), ( 546,5462,13364), (1181,6644, 156), (5640,12290,3843), (5890, 10574,3), (5892,5907,3628), ( 5946,13491,4), (6101,6102,5889) ,(9781,12279,3830)

X(13631) = EULER LINE INTERCEPT OF X(143)X(1263)

Barycentrics (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (3 a^12-10 a^10 b^2+9 a^8 b^4+4 a^6 b^6-11 a^4 b^8+6 a^2 b^10-b^12-10 a^10 c^2+16 a^8 b^2 c^2-6 a^6 b^4 c^2+10 a^4 b^6 c^2-16 a^2 b^8 c^2+6 b^10 c^2+9 a^8 c^4-6 a^6 b^2 c^4+5 a^4 b^4 c^4+10 a^2 b^6 c^4-15 b^8 c^4+4 a^6 c^6+10 a^4 b^2 c^6+10 a^2 b^4 c^6+20 b^6 c^6-11 a^4 c^8-16 a^2 b^2 c^8-15 b^4 c^8+6 a^2 c^10+6 b^2 c^10-c^12) : :

See Telv Cohl and Peter Moses, Hyacinthos 26257.

X(13631) lies on these lines: {2,3}, {143,1263}

X(13631) = reflection of X(5) in X(13362)

X(13632) = EULER LINE INTERCEPT OF X(39)X(3017)

Barycentrics a^6+3 a^5 b-a^4 b^2-3 a^3 b^3-a^2 b^4+b^6+3 a^5 c+3 a^4 b c-3 a^3 b^2 c-3 a^2 b^3 c-a^4 c^2-3 a^3 b c^2-6 a^2 b^2 c^2-b^4 c^2-3 a^3 c^3-3 a^2 b c^3-a^2 c^4-b^2 c^4+c^6 : :
X(13632) = (a² + b² + c²)*X(2) + (bc + ca + ab)*X(3)
X(13632) = 2 X[3] + X[36707] = 4 X[140] - X[13727]

Steiner coordinates: (x,x), where x = (-a^2 + b^2)*(b^2 - c^2)*(-a^2 + c^2)*(a*b + a*c + b*c) / (a^2 + a*b + b^2 + a*c + b*c + c^2)

X(13632) lies on these lines: {2,3}, {39,3017}, {542,572}, {573,5476}, {2223,3584}, {10168,13329}

X(13632) lies on these lines: {2, 3}, {39, 3017}, {355, 48809}, {542, 572}, {573, 5476}, {597, 37510}, {599, 37474}, {991, 50977}, {1482, 48830}, {2223, 3584}, {3579, 29633}, {3582, 37575}, {3654, 50287}, {3655, 50311}, {3656, 48822}, {5337, 49744}, {9301, 37678}, {10056, 37590}, {10168, 13329}, {13624, 29637}, {17264, 51045}, {17399, 51044}, {17754, 37584}, {18480, 19856}, {19130, 37508}, {20423, 48875}, {20430, 41312}, {24512, 45923}, {26446, 28850}, {34718, 50282}, {37499, 38072}, {37676, 51340}, {48802, 50798}

X(13632) = reflection of X(13634) in X(549)
X(13632) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 13633}, {3, 36530, 36716}, {140, 36654, 36527}, {2043, 2044, 36685}, {15765, 18585, 6998}, {35303, 35304, 22351}, {37340, 37341, 16060}

X(13633) = EULER LINE INTERCEPT OF X(542)X(13329)

Barycentrics a^6-3 a^5 b-a^4 b^2+3 a^3 b^3-a^2 b^4+b^6-3 a^5 c-3 a^4 b c+3 a^3 b^2 c+3 a^2 b^3 c-a^4 c^2+3 a^3 b c^2-6 a^2 b^2 c^2-b^4 c^2+3 a^3 c^3+3 a^2 b c^3-a^2 c^4-b^2 c^4+c^6 : :
X(13633) = (a² + b² + c²)*X(2) - (bc + ca + ab)*X(3)
X(13633) = 2 X[3] + X[36707] = 4 X[140] - X[13727]

Steiner coordinates: (x,x), where x = (a^2 - b^2)*(b^2 - c^2)*(c^2 - a^2)*(a*b + a*c + b*c) / (a^2 + a*b + b^2 + a*c + b*c + c^2)

X(13633) lies on these lines: {2, 3}, {524, 37510}, {542, 13329}, {572, 10168}, {573, 50977}, {991, 5476}, {2223, 3582}, {3579, 29637}, {3584, 37575}, {3653, 48822}, {3654, 50311}, {3655, 50287}, {8299, 35000}, {9301, 37686}, {10072, 37590}, {10246, 48830}, {13624, 29633}, {16569, 18528}, {17320, 51045}, {17342, 51044}, {18527, 37589}, {20423, 48908}, {20430, 41313}, {26446, 48809}, {29010, 37756}, {34718, 50316}, {37474, 47352}, {38066, 48802}, {38072, 50677}

X(13633) = reflection of X(13634) in X(549)
X(13633) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 13633}, {3, 36530, 36716}, {140, 36654, 36527}, {2043, 2044, 36685}, {15765, 18585, 6998}, {35303, 35304, 22351}, {37340, 37341, 16060}

X(13634) = EULER LINE INTERCEPT OF X(86)X(3098)

Barycentrics 3 a^6+a^5 b-2 a^3 b^3-3 a^2 b^4+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-2 a^3 b c^2-6 a^2 b^2 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^3-3 a^2 c^4+a b c^4+a c^5+b c^5 : :
X(13634) = (bc + ca + ab)*X(2) + (a² + b² + c²)*X(3)
X(13634) = 2 X[3] + X[13727] = 4 X[140] - X[36707]

Steiner coordinates: (x,x), where x = (-a^2 + b^2)*(b^2 - c^2)*(-a^2 + c^2)*(a^2 + b^2 + c^2) / (a^2 + a*b + b^2 + a*c + b*c + c^2)

X(13634) lies on these lines: {2, 3}, {40, 48854}, {86, 3098}, {165, 44430}, {511, 46922}, {542, 17271}, {551, 48932}, {944, 48849}, {990, 51044}, {1447, 24929}, {1654, 48906}, {3576, 9746}, {3579, 16830}, {3654, 50286}, {3655, 50310}, {3818, 17307}, {4297, 48853}, {4664, 46475}, {5092, 17277}, {5224, 46264}, {5232, 39874}, {5306, 18755}, {5337, 50182}, {5434, 17798}, {6210, 50300}, {6211, 50094}, {7788, 17206}, {9300, 33863}, {9441, 50291}, {10056, 37576}, {11179, 17346}, {12017, 17349}, {12512, 39605}, {13624, 16823}, {17238, 18440}, {17251, 43273}, {17297, 50977}, {17327, 48905}, {17343, 39899}, {17379, 33878}, {17381, 31670}, {17398, 48881}, {22712, 47040}, {24257, 50086}, {24808, 26446}, {31162, 48900}, {34638, 48925}, {35242, 39586}, {37677, 44456}, {48851, 50811}, {48856, 50810}

X(13634) = reflection of X(13632) in X(549). X(13634) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 13635}, {2, 13635, 21554}, {3, 6998, 21554}, {3, 36477, 6996}, {6998, 13635, 2}, {21869, 21898, 16060}

X(13635) = EULER LINE INTERCEPT OF X(86)X(5092)

Barycentrics 3 a^6+a^5 b-2 a^3 b^3-3 a^2 b^4+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-2 a^3 b c^2-6 a^2 b^2 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^3-3 a^2 c^4+a b c^4+a c^5+b c^5 : :
X(13635) = (bc + ca + ab)*X(2) - (a² + b² + c²)*X(3)
X(13635) = 2 X[3] + X[6996], X[4] - 4 X[19512], X[20] + 2 X[36654], 4 X[140] - X[36716], 7 X[3523] - X[6999], 4 X[12100] - X[19703], 2 X[15977] + X[50424]

Steiner coordinates: (x,x), where x = (-a^2 + b^2)*(b^2 - c^2)*(-a^2 + c^2)*(a^2 + b^2 + c^2)/(a^2 - a*b + b^2 - a*c - b*c + c^2)

+X(13635) lies on these lines: {2, 3}, {86, 5092}, {182, 46922}, {515, 24808}, {519, 19589}, {537, 6211}, {542, 17297}, {1447, 5122}, {1766, 51044}, {3098, 17277}, {3576, 44430}, {3579, 16823}, {3654, 50310}, {3655, 50286}, {3818, 17283}, {4869, 39874}, {5298, 17798}, {5306, 33863}, {5657, 48849}, {6361, 16020}, {9300, 18755}, {9441, 28194}, {10072, 37576}, {10164, 48853}, {11179, 17378}, {12017, 17379}, {13624, 16830}, {17206, 37671}, {17232, 18440}, {17234, 46264}, {17265, 48905}, {17271, 50977}, {17300, 48906}, {17313, 43273}, {17337, 48881}, {17349, 33878}, {17352, 31670}, {17375, 39899}, {18524, 31073}, {19883, 48932}, {22712, 47039}, {26241, 35238}, {37654, 50967}, {38021, 48900}, {46475, 51488}

X(13635) = reflection of X(13633) in X(549)
X(13635) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 13634}, {2, 13634, 6998}, {3, 21554, 6998}, {13634, 21554, 2}, {21869, 21898, 16061}

X(13636) = TRIPOLAR CENTROID OF X(3413)

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

Let U be the circle through X(13) and X(14) with center on the minor axis of the Steiner inellipse. The center of U is X(13636).

See Benedetto Scimemi and César Lozada, Hyacinthos 26259.

X(13636) lies on the Hutson-Parry circle, the cubics K219 and K237, and these lines: {2, 1340}, {115, 125}, {476, 1380}, {892, 6189}, {2039, 5996}, {2395, 5638}, {3413, 5466}, {6142, 6795}

X(13636) = tripolar centroid of X(3413)
X(13636) = reflection of X(13722) in X(8371)
X(13636) = Hutson-Parry-circle antipode of X(13722)
X(13636) = {X(115),X(1648)}-harmonic conjugate of X(13722)
X(13636) = {X(125),X(868)}-harmonic conjugate of X(13722)
X(13636) = barycentric product X(115)*X(6189)
X(13636) = exsimilicenter of circles {{X(13),X(14),X(16)}} and {{X(13),X(14),X(15)}}; the insimilicenter is X(13722)

Tri-squares triangles and related centers: X(13637)-X(13721)

This preamble and centers X(13637) - X(13721) were contributed by César Eliud Lozada, July 1, 2017.

Corrected on July 7, 2017. Thanks to Peter Moses.

Inscribe three squares into a triangle ABC such that each square has two vertices on two distinct sides of ABC and the other vertices of the three squares coincide at the vertices of another triangle A'B'C'.

Let the squares be σ_a=B'C'A_cA_b, σ_b=C'A'B_aB_c and σ_c=A'B'C_bC_a with B_a, C_a on BC, C_b, A_b on CA, A_c, B_c on AB and centers A_o, B_o, C_o, respectively. This construction has four solutions. For each case, the triangle A'B'C' will be named here the tri-squares triangle of ABC and the triangle A_oB_oC_o will be referred here as tri-squares-central triangle of ABC.

Barycentric coordinates of the first points on each square are shown in the following table:

1st tri-squares 2nd tri-squares 3rd tri-squares 4th tri-squares

A' = S : SA+2*SC+S : SA+2*SB+S

A_b = SB+2*SC+3*S : 0 : 2*SA+SB
A_c = 2*SB+SC+3*S : 2*SA+SC : 0

A_o = 3*(SB+SC)+4*S : 2*SA+SC+S : 2*SA+SB+S

The squares are internal to ABC and outwards A'B'C' A' = -S : SA+2*SC-S : SA+2*SB-S

A_b = SB+2*SC-3*S : 0 : 2*SA+SB
A_c = 2*SB+SC-3*S : 2*SA+SC

A_o = 3*(SB+SC)-4*S : 2*SA+SC-S : 2*SA+SB-S

The squares are external to ABC and inwards A'B'C' A' = 2*(SB+SC+S) : SA+S : SA+S

A_b = SB+2*SC+3*S : 0 : 2*SA+SB+S
A_c = 2*SB+SC+3*S : 2*SA+SC+S : 0

A_o = SB+SC+2*S : SA+SC+S : SA+SB+S

The squares are internal to ABC and inwards A'B'C' A' = 2*(SB+SC-S) : SA-S : SA-S

A_b = SB+2*SC-3*S : 0 : 2*SA+SB-S
A_c = 2*SB+SC-3*S : 2*SA+SC-S : 0 : 0

A_o = SB+SC-2*S : SA+SC-S : SA+SB-S

The squares are external to ABC and outwards A'B'C'

1st tri-squares	2nd tri-squares	3rd tri-squares	4th tri-squares
A' = S : SA+2SC+S : SA+2SB+S A_b = SB+2SC+3S : 0 : 2SA+SB A_c = 2SB+SC+3S : 2SA+SC : 0 A_o = 3(SB+SC)+4S : 2SA+SC+S : 2SA+SB+S The squares are internal to ABC and outwards A'B'C'	A' = -S : SA+2SC-S : SA+2SB-S A_b = SB+2SC-3S : 0 : 2SA+SB A_c = 2SB+SC-3S : 2SA+SC A_o = 3(SB+SC)-4S : 2SA+SC-S : 2SA+SB-S The squares are external to ABC and inwards A'B'C'	A' = 2(SB+SC+S) : SA+S : SA+S A_b = SB+2SC+3S : 0 : 2SA+SB+S A_c = 2SB+SC+3S : 2SA+SC+S : 0 A_o = SB+SC+2S : SA+SC+S : SA+SB+S The squares are internal to ABC and inwards A'B'C'	A' = 2(SB+SC-S) : SA-S : SA-S A_b = SB+2SC-3S : 0 : 2SA+SB-S A_c = 2SB+SC-3S : 2SA+SC-S : 0 : 0 A_o = SB+SC-2S : SA+SC-S : SA+SB-S The squares are external to ABC and outwards A'B'C'

The following results correspond to the 1st tri-squares triangles only:

A'B'C' is:
directly homothetic to these triangles: anti-Artzt, Artzt
directly similar to the circumsymmedial triangle
inversely similar to these triangles: 4th anti-Brocard, anti-McCay, 4th Brocard, McCay, 3rd Parry

The appearance of (T, n) in the following list means that A'B'C' and triangle T are perspective with perspector X(n): (anti-Artzt*, 13637), (Artzt*, 13638).
An asterisk means that both triangles are homothetic.

The appearance of (T, m, n) in the following list means that A'B'C' and triangle T are orthologic with centers X(m) and X(n):
(ABC, 3068, 2), (ABC-X3 reflections, 3068, 376), (anti-Aquila, 3068, 551), (anti-Ara, 3068, 428), (anti-Artzt, 13639, 1992), (1st anti-Brocard, 13640, 7840), (4th anti-Brocard, 13641, 9870), (5th anti-Brocard, 3068, 12150), (6th anti-Brocard, 13640, 12151), (anti-Euler, 3068, 376), (anti-Mandart-incircle, 3068, 4421), (anti-McCay, 13642, 385), (anticomplementary, 3068, 2), (Aquila, 3068, 3679), (Ara, 3068, 9909), (Artzt, 13639, 9770), (1st Auriga, 3068, 11207), (2nd Auriga, 3068, 11208), (1st Brocard, 13640, 599), (4th Brocard, 13643, 2), (5th Brocard, 3068, 7811), (6th Brocard, 13640, 9939), (circummedial, 13644, 2), (Euler, 3068, 381), (5th Euler, 13644, 2), (outer-Garcia, 3068, 3679), (Gossard, 3068, 1651), (inner-Grebe, 3068, 5861), (outer-Grebe, 3068, 5860), (Johnson, 3068, 381), (inner-Johnson, 3068, 11235), (outer-Johnson, 3068, 11236), (1st Johnson-Yff, 3068, 11237), (2nd Johnson-Yff, 3068, 11238), (Lucas homothetic, 3068, 12152), (Lucas(-1) homothetic, 3068, 12153), (Mandart-incircle, 3068, 3058), (McCay, 13642, 7610), (medial, 3068, 2), (5th mixtilinear, 3068, 3241), (inner-Napoleon, 13645, 9761), (outer-Napoleon, 13646, 9763), (1st Neuberg, 13647, 8667), (2nd Neuberg, 13648, 9766), (3rd Parry, 13649, 2), (inner-Vecten, 13650, 591), (outer-Vecten, 13651, 1991), (X3-ABC reflections, 3068, 381), (inner-Yff, 3068, 10056), (outer-Yff, 3068, 10072), (inner-Yff tangents, 3068, 11239), (outer-Yff tangents, 3068, 11240)

The appearance of (T, m, n) in the following list means that A'B'C' and triangle T are parallelogic with centers X(m) and X(n):
(4th anti-Brocard, 13652, 13168), (anti-McCay, 13653, 8597), (4th Brocard, 13654, 4), (McCay, 13653, 381), (1st Parry, 3068, 9123), (2nd Parry, 3068, 9185), (3rd Parry, 13655, 9147)

The appearance of (I, J) in the following list means that X(I)-of-A'B'C' = X(J) :
(2,3068), (3,13663), (4,13639), (5,13664), (6,13644), (13,13646), (14,13645), (98,13642), (99,13653), (111,13643), (376,2), (485,13662), (671,13640), (1296,13654), (1327,13651), (1328,13650), (5860,9541), (6236,13652), (6325,13641), (9741,7374), (9831,13655), (11001,1270)

A_oB_oC_o and A'B'C' are perspective with perspector X(13662).

The appearance of (T, m, n) in the following list means that A_oB_oC_o and triangle T are orthologic with centers X(m) and X(n):
(ABC, 13665, 1327), (ABC-X3 reflections, 13665, 13666), (anti-Aquila, 13665, 13667), (anti-Ara, 13665, 13668), (anti-Artzt, 13663, 13669), (1st anti-Brocard, 13670, 13671), (5th anti-Brocard, 13665, 13672), (6th anti-Brocard, 13670, 13673), (anti-Euler, 13665, 13674), (anti-Mandart-incircle, 13665, 13675), (anti-McCay, 13676, 13677), (anticomplementary, 13665, 13678), (Aquila, 13665, 13679), (Ara, 13665, 13680), (Artzt, 13663, 13681), (1st Auriga, 13665, 13682), (2nd Auriga, 13665, 13683), (1st Brocard, 13670, 13684), (5th Brocard, 13665, 13685), (6th Brocard, 13670, 13686), (Euler, 13665, 13687), (outer-Garcia, 13665, 13688), (Gossard, 13665, 13689), (inner-Grebe, 13665, 13690), (outer-Grebe, 13665, 13691), (Johnson, 13665, 13692), (inner-Johnson, 13665, 13693), (outer-Johnson, 13665, 13694), (1st Johnson-Yff, 13665, 13695), (2nd Johnson-Yff, 13665, 13696), (Lucas homothetic, 13665, 13697), (Lucas(-1) homothetic, 13665, 13698), (Mandart-incircle, 13665, 13699), (McCay, 13676, 13700), (medial, 13665, 13701), (5th mixtilinear, 13665, 13702), (inner-Napoleon, 13703, 13704), (outer-Napoleon, 13705, 13706), (1st Neuberg, 13707, 13708), (2nd Neuberg, 13709, 13710), (tri-squares, 13663, 13662), (inner-Vecten, 13711, 13712), (outer-Vecten, 3068, 2), (X3-ABC reflections, 13665, 13713), (inner-Yff, 13665, 13714), (outer-Yff, 13665, 13715), (inner-Yff tangents, 13665, 13716), (outer-Yff tangents, 13665, 13717)

The appearance of (T, m, n) in the following list means that A_oB_oC_o and triangle T are parallelogic with centers X(m) and X(n): (1st Parry, 13665, 13718), (2nd Parry, 13665, 13719)

The appearance of (I, J) in the following list means that X(I)-of-A_oB_oC_o = X(J) :
(2,3068), (3,13720), (4,13662), (5,13721), (13,13705), (14,13703), (486,13663), (487,13639), (642,13664), (1328,13665), (5861,13651)

X(13637) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES AND ANTI-ARTZT

Barycentrics 3*(S-SA)+2*SW : :
X(13637) = X(492)-4*X(590)

X(13637) lies on these lines: {2,6}, {99,13642}, {110,13643}, {485,489}, {490,9540}, {542,6811}, {637,8976}, {638,8981}, {6222,10783}, {7388,7812}, {7389,8960}, {11055,13647}, {11159,13644}, {11161,13653}, {12149,13641}, {12154,13645}, {12155,13646}, {12156,13648}, {12157,13649}, {12158,13650}, {13167,13652}, {13169,13654}, {13170,13655}

X(13637) = reflection of X(i) in X(j) for these (i,j): (2,590), (492,2)
X(13637) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,5032,3069), (2,13639,1992), (1991,8667,5861)

X(13638) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES AND ARTZT

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

X(13638) lies on these lines: {2,6}, {32,7388}, {98,485}, {371,6813}, {381,13644}, {490,12968}, {638,6423}, {1194,8963}, {1586,10311}, {3103,5418}, {3767,7389}, {5254,11293}, {5286,11291}, {6054,13640}, {6459,7000}, {6561,9993}, {8854,8969}, {9605,11316}, {9759,13643}, {9760,13645}, {9762,13646}, {9764,13647}, {9765,13648}, {9767,13650}, {9768,13651}, {9769,13654}, {9869,13641}, {9877,13642}, {12434,13649}, {13191,13652}, {13225,13655}

X(13638) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,385,492), (2,5304,3069), (2,7585,7736), (2,8974,3068), (2,13639,9770), (230,590,2)

X(13639) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO ANTI-ARTZT

Barycentrics 5*a^2-b^2-c^2+3*S : :
X(13639) = 3*X(2)-4*X(13663) = 3*X(2)-8*X(13664) = X(1270)-4*X(3068) = 3*X(1270)-8*X(13663) = 3*X(1270)-16*X(13664) = 3*X(3068)-2*X(13663)

The reciprocal orthologic center of these triangles is X(1992)

X(13639) lies on these lines: {2,6}, {542,7374}, {543,13640}, {671,1131}, {6462,8591}, {11148,13669}

X(13639) = reflection of X(i) in X(j) for these (i,j): (2,3068), (1270,2), (13663,13664)
X(13639) = orthologic center of these triangles: 1st tri-squares to Artzt
X(13639) = X(4)-of-1st-tri-squares-triangle
X(13639) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,5032,7586), (193,8972,3593), (1992,13637,2), (9770,13638,2)

X(13640) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO 1st ANTI-BROCARD

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

The reciprocal orthologic center of these triangles is X(7840)

X(13640) lies on these lines: {2,5477}, {115,7585}, {147,8974}, {491,5182}, {524,13642}, {530,13645}, {531,13646}, {542,3068}, {543,13639}, {620,1271}, {1991,5026}, {2782,13644}, {5969,13647}, {6054,13638}, {6306,9113}, {6307,9112}, {6561,10722}

X(13640) = reflection of X(13653) in X(3068)
X(13640) = orthologic center of the 1st tri-squares triangle to these triangles: 6th anti-Brocard, 1st Brocard, 6th Brocard
X(13640) = X(671)-of-1st-tri-squares-triangle
X(13640) = Artzt-to-1st-tri-squares similarity image of X(6054)
X(13640) = anti-Artzt-to-1st-tri-squares similarity image of X(8593)
X(13640) = McCay-to-1st-tri-squares similarity image of X(3)

X(13641) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO 4th ANTI-BROCARD

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

The reciprocal orthologic center of these triangles is X(9870)

X(13641) lies on these lines: {2,13167}, {2780,13643}, {2854,13654}, {9869,13638}, {12149,13637}, {13652,13663}

X(13641) = X(6325)-of-1st-tri-squares-triangle
X(13641) = 4th-anti-Brocard-to-1st-tri-squares similarity image of X(9870)
X(13641) = Artzt-to-1st-tri-squares similarity image of X(9869)
X(13641) = anti-Artzt-to-1st-tri-squares similarity image of X(12149)

X(13642) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO ANTI-MCCAY

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

The reciprocal orthologic center of these triangles is X(385)

X(13642) lies on these lines: {2,98}, {99,13637}, {511,13649}, {512,13655}, {524,13640}, {543,3068}, {690,13643}, {5591,9167}, {9830,13653}, {9877,13638}, {11159,13665}

X(13642) = reflection of X(13653) in X(13663)
X(13642) = orthologic center of these triangles: 1st tri-squares to McCay
X(13642) = X(98)-of-1st-tri-squares-triangle

X(13643) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO 4th BROCARD

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

The reciprocal orthologic center of these triangles is X(2)

X(13643) lies on these lines: {2,9769}, {110,13637}, {542,3068}, {690,13642}, {2780,13641}, {2854,13652}, {9759,13638}, {13654,13663}

X(13643) = X(111)-of-1st-tri-squares-triangle

X(13644) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO CIRCUMMEDIAL

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

The reciprocal orthologic center of these triangles is X(2)

X(13644) lies on these lines: {4,8974}, {30,3068}, {32,11313}, {381,13638}, {491,11286}, {543,13645}, {590,7737}, {1161,1587}, {1991,3734}, {2548,11316}, {2782,13640}, {3053,11315}, {3849,13663}, {7583,12313}, {7745,11314}, {9605,11293}, {11159,13637}, {13647,13648}, {13650,13651}, {13662,13664}

X(13644) = orthologic center of these triangles: 1st tri-squares to 5th euler
X(13644) = X(6)-of-1st-tri-squares-triangle

X(13645) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO INNER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(9761)

X(13645) lies on these lines: {2,9113}, {530,13640}, {531,3068}, {543,13644}, {9760,13638}, {12154,13637}

X(13645) = X(14)-of-1st-tri-squares-triangle

X(13646) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO OUTER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(9763)

X(13646) lies on these lines: {2,9112}, {530,3068}, {531,13640}, {543,13644}, {9762,13638}, {12155,13637}

X(13646) = X(13)-of-1st-tri-squares-triangle

X(13647) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO 1st NEUBERG

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

The reciprocal orthologic center of these triangles is X(8667)

X(13647) lies on these lines: {39,5591}, {491,7757}, {538,3068}, {5969,13640}, {9764,13638}, {11055,13637}, {13644,13648}

X(13648) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(9766)

X(13648) lies on these lines: {83,491}, {754,3068}, {9765,13638}, {13644,13647}

X(13649) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO 3rd PARRY

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

The reciprocal orthologic center of these triangles is X(2)

X(13649) lies on these lines: {2,13170}, {511,13642}, {512,13653}, {12157,13637}, {12434,13638}, {13655,13663}

X(13649) = 3rd-Parry-to-1st-tri-squares similarity image of X(2)
X(13649) = circumsymmedial-to-1st-tri-squares similarity image of X(2698)
X(13649) = X(13241)-of-1st-tri-squares-triangle

X(13650) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO INNER-VECTEN

Barycentrics 27*a^6-30*(b^2+c^2)*a^4+(23*b^4-22*b^2*c^2+23*c^4)*a^2-4*(b^4-c^4)*(b^2-c^2)+2*(8*a^4-15*(b^2+c^2)*a^2+2*b^2*c^2+7*c^4+7*b^4)*S : :

The reciprocal orthologic center of these triangles is X(591)

X(13650) lies on these lines: {371,6281}, {486,590}, {487,7585}, {491,7926}, {642,3069}, {6560,9732}, {9767,13638}, {12158,13637}, {13644,13651}

X(13650) = X(1328)-of-1st-tri-squares-triangle
X(13650) = 4th-anti-Brocard-to-1st-tri-squares similarity image of X(11836)

X(13651) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO OUTER-VECTEN

Barycentrics 17*a^6-56*(b^2+c^2)*a^4+(17*b^4-74*b^2*c^2+17*c^4)*a^2+6*(b^4-c^4)*(b^2-c^2)-2*S*(20*a^4+29*(b^2+c^2)*a^2-9*(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(1991)

X(13651) lies on these lines: {5,6}, {488,8972}, {524,13662}, {5490,11008}, {6199,12602}, {6200,12124}, {6304,11489}, {6305,11488}, {9768,13638}, {11149,12159}, {13644,13650}

X(13651) = X(1327)-of-1st-tri-squares-triangle
X(13651) = 4th-anti-Brocard-to-1st-tri-squares similarity image of X(11835)

X(13652) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO 4th ANTI-BROCARD

Trilinears
(6*b^2*c^2*(a^6-3*(b^2+c^2)*a^4-3*(b^4-5*b^2*c^2+c^4)*a^2+(b^2+c^2)*(b^4-4*b^2*c^2+c^4))*S-(b^2-c^2)^2*(a^2+b^2+c^2)*(2*a^6-(2*b^4-13*b^2*c^2+2*c^4)*a^2-b^2*c^2*(b^2+c^2)))*a : :

The reciprocal parallelogic center of these triangles is X(13168)

X(13652) lies on these lines: {2,9869}, {2780,13654}, {2854,13643}, {13191,13638}, {13641,13663}

X(13652) = X(6236)-of-1st-tri-squares-triangle

X(13653) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO ANTI-MCCAY

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

The reciprocal parallelogic center of these triangles is X(8597)

X(13653) lies on these lines: {2,99}, {98,485}, {492,10754}, {511,13655}, {512,13649}, {542,3068}, {590,11646}, {615,6034}, {690,13654}, {2794,7374}, {5477,7585}, {6036,12975}, {8974,11177}, {9830,13642}, {11161,13637}

X(13653) = reflection of X(i) in X(j) for these (i,j): (13640,3068), (13642,13663)
X(13653) = parallelogic center of these triangles: 1st tri-squares to McCay
X(13653) = X(99)-of-1st-tri-squares-triangle

X(13654) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO 4th BROCARD

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

The reciprocal parallelogic center of these triangles is X(4)

X(13654) lies on these lines: {2,98}, {67,590}, {74,6811}, {492,895}, {690,13653}, {2780,13652}, {2854,13641}, {5095,7585}, {5181,5590}, {9769,13638}, {13169,13637}, {13643,13663}

X(13654) = X(2696)-of-1st-tri-squares-triangle

X(13655) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO 3rd PARRY

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

The reciprocal parallelogic center of these triangles is X(9147)

X(13655) lies on these lines: {2,12157}, {511,13653}, {512,13642}, {13170,13637}, {13225,13638}, {13649,13663}

X(13655) = X(9831)-of-1st-tri-squares-triangle

X(13656) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: 1st TRI-SQUARES AND 4th ANTI-BROCARD

Trilinears
a*(-6*((b^2+c^2)*a^4+26*b^2*c^2*a^2-((b^2-c^2)^2-4*b^2*c^2)*(b^2+c^2))*S+a^8-3*(b^2+c^2)*a^6+((b^2-c^2)^2-36*b^2*c^2)*a^4+(b^2+c^2)*(3*b^4-80*b^2*c^2+3*c^4)*a^2-2*(b^4-25*b^2*c^2+c^4)*(b^2-c^2)^2) : :

X(13656) lies on these lines: {}

X(13657) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: 1st TRI-SQUARES AND ANTI-MCCAY

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

X(13657) lies on these lines: {2,5062}, {148,3068}, {597,20221}

X(13658) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: 1st TRI-SQUARES AND 4th BROCARD

Barycentrics
2*(10*a^8-6*(b^2+c^2)*a^6-(11*b^4-16*b^2*c^2+11*c^4)*a^4+2*(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^2+(b^4-c^4)^2)*S+5*a^10+6*(b^2+c^2)*a^8-(23*b^4+2*b^2*c^2+23*c^4)*a^6+(b^2+c^2)*(b^4+c^4)*a^4+2*(b^2-c^2)^2*(9*b^4+5*b^2*c^2+9*c^4)*a^2-7*(b^4-c^4)*(b^2-c^2)^3 : :

X(13658) lies on these lines: {}

X(13659) = CENTER OF SIMILITUDE OF THESE TRIANGLES: 1st TRI-SQUARES AND CIRCUMSYMMEDIAL

Barycentrics (9*(2*S+SW)*SA^2-3*(3*S^2+2*(9*R^2-SW)*S+SW^2)*SA+(15*S^2+3*(18*R^2+SW)*S+(36*R^2-5*SW)*SW)*S)*(SB+SC) : :

X(13659) lies on the line {6200,7464}

X(13660) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: 1st TRI-SQUARES AND MCCAY

Barycentrics S^2*(33*S^2+18*SA^2-18*SA*SW+SW^2)-S*(3*S^2-SW^2)*(3*SA-5*SW)+2*(SA-SW)*SW^3 : :

X(13660) lies on the line {3068,7615}

X(13661) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: 1st TRI-SQUARES AND 3rd PARRY

Barycentrics
(SB+SC)*(9*S^3*(S^2+SA^2)*(-S^2+54*R^4)-9*S^3*(S^3+(22*R^2+SA)*S^2+12*R^2*SA*S+24*(3*R^2+SA)*R^2*SA)*SW+9*S^3*(3*S^2+(8*R^2+SA)*S+2*(9*R^2+2*SA)*SA)*SW^2-(3*S^2+SA*S+2*SA^2)*SW^5+2*S*(-2*S^3+5*(-SA+3*R^2)*S^2-3*(6*R^2-SA)*SA*S-18*R^2*SA^2)*SW^3+(-S+SA)*SW^6+S*(-5*S^2+10*SA*S+(18*R^2+5*SA)*SA)*SW^4) : :

X(13661) lies on these lines: {}

X(13662) = RADICAL CENTER OF THE 1st TRI-SQUARES CIRCLES

Barycentrics 18*S^2+(4*SA+19*SB+19*SC)*S-3*(SA-2*SB-2*SC)*(SB+SC) : :

X(13662) lies on these lines: {99,13637}, {485,542}, {524,13651}, {13644,13664}

X(13662) = reflection of X(1327) in X(13665)
X(13662) = perspector of these triangles: 1st tri-squares and 1st tri-squares central
X(13662) = reflection of X(13720) in X(13721)
X(13662) = X(4)-of-1st-tri-squares-central-triangle
X(13662) = X(485)-of-1st-tri-squares-triangle
X(13662) = orthologic center of these triangles: 1st tri-squares to 1st tri-squares central

X(13663) = X(3) OF THE 1st TRI-SQUARES TRIANGLE

Barycentrics 3*SA-6*S-5*SW : :
X(13663) = 5*X(2)-X(1270) = 3*X(2)+X(13639) = 3*X(2)+2*X(13664) = X(1270)+5*X(3068) = 3*X(1270)+5*X(13639) = 3*X(1270)+10*X(13664) = 3*X(3068)-X(13639)

X(13663) lies on these lines: {2,6}, {371,5461}, {3311,6118}, {3849,13644}, {7817,11313}, {8180,13088}, {8960,11315}, {9830,13642}, {11157,13669}, {13641,13652}, {13643,13654}, {13649,13655}

X(13663) = midpoint of X(i) and X(j) for these {i,j}: {2,3068}, {13641,13652}, {13642,13653}, {13643,13654}, {13649,13655}
X(13663) = reflection of X(13639) in X(13664)
X(13663) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,13637,599), (2,13638,7610), (599,13637,1991)
X(13663) = orthologic center of 1st tri-squares central triangle to these triangles: {1st tri-squares, Artzt, anti-Artzt}

X(13664) = X(5) OF THE 1st TRI-SQUARES TRIANGLE

Barycentrics 15*SA-12*S-13*SW : :
X(13664) = 13*X(2)-5*X(1270) = X(2)-5*X(3068) = 3*X(2)+5*X(13639) = 3*X(2)-5*X(13663) = X(1270)-13*X(3068) = 3*X(1270)+13*X(13639) = 3*X(1270)-13*X(13663) = 3*X(3068)+X(13639)

X(13664) lies on these lines: {2,6}, {13644,13662}

X(13664) = midpoint of X(13639) and X(13663)

X(13665) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES-CENTRAL TO ABC

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

The reciprocal orthologic center of these triangles is X(1327)

X(13665) lies on these lines: {2,6398}, {3,485}, {4,1131}, {5,1587}, {6,13}, {20,6449}, {30,3068}, {140,6450}, {262,6569}, {371,382}, {372,1656}, {376,6451}, {403,5411}, {486,3851}, {490,11315}, {546,1588}, {548,6496}, {549,6452}, {550,6455}, {615,5055}, {631,6456}, {632,6522}, {638,11313}, {1124,9669}, {1151,1657}, {1152,3526}, {1335,9654}, {1703,9956}, {2043,11542}, {2044,11543}, {2066,9668}, {2067,9655}, {3071,3843}, {3091,6428}, {3103,13108}, {3146,6447}, {3299,10896}, {3301,10895}, {3316,3523}, {3364,5339}, {3389,5340}, {3529,6519}, {3534,6200}, {3545,7586}, {3590,10299}, {3592,5076}, {3594,5079}, {3627,6459}, {3628,6448}, {3832,7582}, {3854,6499}, {3856,6498}, {5054,6396}, {5070,5420}, {5072,6420}, {5094,13654}, {6811,10846}, {8725,8993}, {8988,12515}, {8992,9821}, {8998,12121}, {9894,11165}, {10665,12429}, {10880,12173}, {11159,13642}

X(13665) = midpoint of X(1327) and X(13662)
X(13665) = reflection of X(6221) in X(3068)
X(13665) = X(1991)-of-1st-tri-squares-triangle
X(13665) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,7583,3311), (5,1587,3312), (6,6564,381), (485,3070,3), (550,9540,6455), (3091,7581,7584), (3843,6417,3071), (3851,6418,486), (7581,7584,6428)

X(13666) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO 1st TRI-SQUARES-CENTRAL

Barycentrics 10*S^3-3*(SB+SC)*((7*S+2*SW)*SA-S^2) : :
X(13666) = 3*X(3576)-2*X(13667) = 5*X(12124)-2*X(12305)

The reciprocal orthologic center of these triangles is X(13665)

X(13666) lies on these lines: {3,1327}, {30,6289}, {182,13672}, {1328,9739}, {1593,13668}, {3534,12123}, {3576,13667}, {11001,11825}

X(13666) = reflection of X(1327) in X(3)
X(13666) = X(1327)-of-ABC-X3-reflections-triangle

X(13667) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO 1st TRI-SQUARES-CENTRAL

Barycentrics (13*a+b+c)*S^2-3*((a+b+c)*SA-(3*a+b+c)*SW)*S+9*(a+b+c)*SB*SC : :
X(13667) = 3*X(3576)-X(13666)

The reciprocal orthologic center of these triangles is X(13665)

X(13667) lies on these lines: {1,1327}, {30,12269}, {3576,13666}, {11363,13668}, {11364,13672}

X(13667) = midpoint of X(1) and X(1327)
X(13667) = X(1327)-of-anti-Aquila-triangle

X(13668) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO 1st TRI-SQUARES-CENTRAL

Barycentrics ((36*R^2+5*SA-15*SW)*S+3*SA^2-3*SW^2)*SB*SC : :
X(13668) = 2*X(6291)-5*X(12148)

The reciprocal orthologic center of these triangles is X(13665)

X(13668) lies on these lines: {25,1327}, {30,6291}, {1593,13666}, {3543,8948}, {7576,12147}, {11363,13667}, {11380,13672}

X(13668) = X(1327)-of-anti-Ara-triangle

X(13669) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARTZT TO 1st TRI-SQUARES-CENTRAL

Barycentrics 6*S^2-(3*SA-5*SW)*S+(3*SA-SW)*(3*SA-2*SW) : :

The reciprocal orthologic center of these triangles is X(13663)

X(13669) lies on these lines: {2,1327}, {6,12158}, {99,13637}, {524,12159}, {597,2549}, {11148,13639}, {11157,13663}

X(13669) = X(485)-of-anti-Artzt-triangle

X(13670) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES-CENTRAL TO 1st ANTI-BROCARD

Barycentrics 6*S^4+(-3*SA+7*SW)*S^3+(3*SA-SW)*SW*S^2+(SA-SW)*SW*(3*SA*SW+3*S*SA+2*S*SW) : :

The reciprocal orthologic center of these triangles is X(13671)

X(13670) lies on these lines: {115,6561}, {542,3068}, {3564,6231}, {6055,7735}, {6230,8980}

X(13670) = orthologic center of these triangles: 1st tri-squares-central to 1st Brocard

X(13671) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-BROCARD TO 1st TRI-SQUARES-CENTRAL

Barycentrics 6*S^4+9*SW*S^3-(3*SA-5*SW)*SW*S^2+3*(3*SA-4*SW)*SA*SW*S+9*(SA-SW)*SA*SW^2 : :

The reciprocal orthologic center of these triangles is X(13670)

X(13671) lies on these lines: {2,1327}, {542,9868}

X(13672) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO 1st TRI-SQUARES-CENTRAL

Barycentrics 10*S^4+(21*SA^2-33*SA*SW+22*SW^2)*S^2+3*(SB+SC)*(-SA*SW*(2*S+3*SW)+S*(S^2+3*SW^2)) : :

The reciprocal orthologic center of these triangles is X(13665)

X(13672) lies on these lines: {30,12211}, {32,1327}, {182,13666}, {11364,13667}, {11380,13668}

X(13672) = X(1327)-of-5th-anti-Brocard-triangle

X(13673) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-BROCARD TO 1st TRI-SQUARES-CENTRAL

Barycentrics 6*S^5+(-27*SA+23*SW)*S^4-3*(SB+SC)*(3*SA-5*SW)*S^3-(15*SA^2-12*SA*SW+SW^2)*SW*S^2-3*(SA^2+SW^2)*SW^2*S-9*(SB+SC)*SA*SW^3 : :

The reciprocal orthologic center of these triangles is X(13670)

X(13673) lies on these lines: {182,13684}, {542,12218}, {4027,13671}, {10131,13686}

X(13674) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO 1st TRI-SQUARES-CENTRAL

Barycentrics 3*SA*(SB+SC)*(S-SW)-S^2*(-3*SA+4*S) : :
X(13674) = 3*X(4)-4*X(13687) = 3*X(376)-2*X(13666) = 3*X(1327)-2*X(13687) = 3*X(3545)+2*X(13690) = 3*X(5603)-4*X(13667)

The reciprocal orthologic center of these triangles is X(13665)

X(13674) lies on these lines: {2,6290}, {3,13678}, {4,1327}, {24,13680}, {30,12257}, {376,13666}, {381,6776}, {515,13679}, {3068,9862}, {3085,13695}, {3086,13696}, {3524,8982}, {3545,10783}, {5603,13667}, {5657,13688}, {5861,12251}, {7487,13668}, {10784,13691}, {10785,13693}, {10786,13694}, {10788,13672}, {11491,13675}, {11843,13682}, {11844,13683}, {11845,13689}

X(13674) = reflection of X(i) in X(j) for these (i,j): (4,1327), (13678,3)
X(13674) = anticomplement of X(13692)
X(13674) = X(1327)-of-anti-Euler-triangle

X(13675) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 1st TRI-SQUARES-CENTRAL

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

The reciprocal orthologic center of these triangles is X(13665)

X(13675) lies on these lines: {30,12344}, {35,13679}, {55,1327}, {100,13678}, {197,13680}, {1376,13693}, {3295,13667}, {5687,13688}, {10310,13666}, {11383,13668}, {11490,13672}, {11491,13674}, {11492,13682}, {11493,13683}, {11496,13687}, {11497,13690}, {11498,13691}, {11499,13692}, {11500,13694}, {11501,13695}, {11502,13696}, {11848,13689}

X(13675) = X(1327)-of-anti-Mandart-incircle-triangle

X(13676) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES-CENTRAL TO ANTI-MCCAY

Barycentrics 18*(b^2-c^2)^2*S+10*a^6-12*(b^2+c^2)*a^4+3*(5*b^4-4*b^2*c^2+5*c^4)*a^2+(b^2+c^2)*(b^4-4*b^2*c^2+c^4) : :

The reciprocal orthologic center of these triangles is X(13677)

X(13676) lies on these lines: {115,1992}, {543,3068}, {671,13640}

X(13676) = midpoint of X(671) and X(13640)
X(13676) = orthologic center of these triangles: 1st tri-squares to McCay
X(13676) = X(6230)-of-1st-tri-squares-triangle

X(13677) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MCCAY TO 1st TRI-SQUARES-CENTRAL

Barycentrics (9*SA^2-9*SA*SW+SW^2)*SW-3*(9*SA-SW)*(SB+SC)*S-3*(3*SA-2*SW)*S^2+12*S^3 : :

The reciprocal orthologic center of these triangles is X(13676)

X(13677) lies on these lines: {2,1327}, {543,9893}

X(13678) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO 1st TRI-SQUARES-CENTRAL

Barycentrics 5*S^2+3*SA*S-9*SB*SC : :
X(13678) = 5*X(488)-2*X(12322) = 5*X(3091)-4*X(13687) = 5*X(3616)-4*X(13667)

The reciprocal orthologic center of these triangles is X(13665)

X(13678) lies on these lines: {2,1327}, {3,13674}, {4,13668}, {8,13688}, {10,13679}, {20,13666}, {22,13680}, {30,488}, {69,3534}, {100,13675}, {376,487}, {388,13695}, {497,13696}, {549,12323}, {638,10304}, {1270,13691}, {1271,13690}, {2896,13685}, {3091,13687}, {3434,13693}, {3436,13694}, {3616,13667}, {4240,13689}, {5601,13682}, {5602,13683}, {7787,13672}

X(13678) = reflection of X(i) in X(j) for these (i,j): (4,13692), (8,13688), (20,13666), (4240,13689), (13674,3), (13679,10)
X(13678) = anticomplement of X(1327)
X(13678) = X(1327)-of-anticomplementary-triangle

X(13679) = ORTHOLOGIC CENTER OF THESE TRIANGLES: AQUILA TO 1st TRI-SQUARES-CENTRAL

Barycentrics 2*(2*a+5*b+5*c)*S^2-3*((a+b+c)*SA-(b+c)*SW)*S-9*(a+b+c)*(SB+SC)*SA : :
X(13679) = 3*X(1)-4*X(13667) = 3*X(165)-2*X(13666) = 3*X(1327)-2*X(13667) = 3*X(3679)-2*X(13688) = 3*X(5587)-2*X(13692)

The reciprocal orthologic center of these triangles is X(13665)

X(13679) lies on these lines: {1,1327}, {10,13678}, {30,9907}, {35,13675}, {165,13666}, {515,13674}, {1699,13687}, {3099,13685}, {3679,13688}, {5587,13692}, {7713,13668}, {10789,13672}

X(13679) = reflection of X(i) in X(j) for these (i,j): (1,1327), (13678,10)
X(13679) = X(1327)-of-Aquila-triangle

X(13680) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARA TO 1st TRI-SQUARES-CENTRAL

Barycentrics (3*SA*SW*(SB+SC)-SA*(SA-6*SW+18*R^2)*S+(6*R^2-3*SW)*S^2-S^3)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(13665)

X(13680) lies on these lines: {3,13692}, {22,13678}, {24,13674}, {25,1327}, {30,9922}, {197,13675}, {1598,13687}

X(13680) = X(1327)-of-Ara-triangle

X(13681) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARTZT TO 1st TRI-SQUARES-CENTRAL

Barycentrics 12*S^3+12*SW*S^2-(3*SA-SW)*SW*S-9*(SB+SC)*SA*SW : :

The reciprocal orthologic center of these triangles is X(13663)

X(13681) lies on these lines: {2,1327}, {30,9757}, {381,7618}, {524,9768}, {7610,13692}, {9767,11184}, {13638,13662}

X(13681) = X(485)-of-Artzt-triangle

X(13682) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO 1st TRI-SQUARES-CENTRAL

Barycentrics ((2*a-10*b-10*c)*S^2+3*((a+b+c)*SA-(-a+b+c)*SW)*S+9*(a+b+c)*(SB+SC)*SA)*K-12*(b*c+SA)*(SB+SC)*(2*S+SW)*S : : , where K=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(13665)

X(13682) lies on these lines: {30,12486}, {55,13683}, {1327,5597}, {5598,13702}, {5599,13701}, {5601,13678}, {8186,13679}, {8190,13680}, {8196,13687}, {8197,13688}, {8198,13690}, {8199,13691}, {8200,13692}, {8201,13697}, {8202,13698}, {11366,13667}, {11384,13668}, {11492,13675}, {11822,13666}, {11837,13672}, {11843,13674}, {11861,13685}, {11863,13689}, {11865,13693}, {11867,13694}, {11869,13695}, {11871,13696}, {11873,13699}, {11875,13713}, {11877,13714}, {11879,13715}, {11881,13716}, {11883,13717}

X(13682) = reflection of X(13683) in X(55)
X(13682) = X(1327)-of-1st-Auriga-triangle
X(13682) = X(13702)-of-2nd-Auriga-triangle

X(13683) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO 1st TRI-SQUARES-CENTRAL

Barycentrics ((2*a-10*b-10*c)*S^2+3*((a+b+c)*SA-(-a+b+c)*SW)*S+9*(a+b+c)*(SB+SC)*SA)*K+12*(b*c+SA)*(SB+SC)*(2*S+SW)*S : : , where K=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(13665)

X(13683) lies on these lines: {30,12487}, {55,13682}, {1327,5598}, {5597,13702}, {5600,13701}, {5602,13678}, {8187,13679}, {8191,13680}, {8203,13687}, {8204,13688}, {8205,13690}, {8206,13691}, {8207,13692}, {8208,13697}, {8209,13698}, {11367,13667}, {11385,13668}, {11493,13675}, {11823,13666}, {11838,13672}, {11844,13674}, {11862,13685}, {11864,13689}, {11866,13693}, {11868,13694}, {11870,13695}, {11872,13696}, {11874,13699}, {11876,13713}, {11878,13714}, {11880,13715}, {11882,13716}, {11884,13717}

X(13683) = reflection of X(13682) in X(55)
X(13683) = X(1327)-of-2nd-Auriga-triangle
X(13683) = X(13702)-of-1st-Auriga-triangle

X(13684) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD TO 1st TRI-SQUARES-CENTRAL

Barycentrics 6*S^3-(3*SA+SW)*S^2+3*(-4*SA*SW-SW^2+3*SA^2)*S-9*(SB+SC)*SA*SW : :

The reciprocal orthologic center of these triangles is X(13670)

X(13684) lies on these lines: {2,1327}, {182,13673}, {384,13686}, {542,6228}, {7697,13692}, {10000,13685}

X(13685) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO 1st TRI-SQUARES-CENTRAL

Barycentrics S^2*(10*S^2+3*(7*SA+3*SW)*(SA-2*SW))+3*(SB+SC)*(S^3-S*SW*(2*SA+SW)+9*SA*SW^2) : :

The reciprocal orthologic center of these triangles is X(13665)

X(13685) lies on these lines: {30,9987}, {32,1327}, {2896,13678}, {3068,9862}, {3098,13666}, {3099,13679}, {9857,13688}, {9993,13687}, {9994,13690}, {9995,13691}, {9996,13692}, {10000,13684}, {10873,13695}, {10874,13696}, {11368,13667}, {11386,13668}

X(13685) = X(1327)-of-5th-Brocard-triangle

X(13686) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th BROCARD TO 1st TRI-SQUARES-CENTRAL

Barycentrics 6*S^4+9*SW*S^3+(12*SA^2-15*SA*SW+5*SW^2)*S^2+3*(5*SA-6*SW)*SA*SW*S-9*(SB+SC)*SA*SW^2 : :

The reciprocal orthologic center of these triangles is X(13670)

X(13686) lies on these lines: {384,13684}, {2896,13678}

X(13687) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EULER TO 1st TRI-SQUARES-CENTRAL

Barycentrics 22*S^3-3*(SA-3*SW)*S^2-3*SA*(SB+SC)*(7*S+2*SW) : :
X(13687) = 3*X(4)+X(13674) = 3*X(381)-X(13692) = 3*X(381)+X(13713) = 3*X(1327)-X(13674) = 5*X(3091)-X(13678) = 3*X(3545)-X(13712) = 3*X(5587)-X(13688) = 3*X(5603)-X(13702)

The reciprocal orthologic center of these triangles is X(13665)

X(13687) lies on these lines: {2,13666}, {4,1327}, {5,13701}, {12,13699}, {30,6250}, {98,13672}, {235,13668}, {381,13692}, {515,13667}, {1478,13715}, {1479,13714}, {1598,13680}, {1699,13679}, {3091,13678}, {3545,13712}, {3845,6251}, {5587,13688}, {5603,13702}, {6201,13691}, {6202,13690}, {9993,13685}, {10895,13695}, {10896,13696}

X(13687) = midpoint of X(i) and X(j) for these {i,j}: {4,1327}, {13692,13713}
X(13687) = reflection of X(13701) in X(5)
X(13687) = complement of X(13666)
X(13687) = X(1327)-of-Euler-triangle
X(13687) = {X(381), X(13713)}-harmonic conjugate of X(13692)

X(13688) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO 1st TRI-SQUARES-CENTRAL

Barycentrics 2*(5*a-b-c)*S^2-3*((a+b+c)*SA+SW*(-a+b+c))*S-9*(a+b+c)*(SB+SC)*SA : :
X(13688) = 3*X(3679)-X(13679) = 3*X(5587)-2*X(13687) = 3*X(5790)-X(13713) = X(13702)-3*X(13712)

The reciprocal orthologic center of these triangles is X(13665)

X(13688) lies on these lines: {1,13701}, {2,13667}, {8,13678}, {10,1327}, {30,12788}, {65,13695}, {515,13666}, {517,13692}, {519,13702}, {1737,13715}, {1837,13699}, {3057,13696}, {3654,12787}, {3679,13679}, {5090,13668}, {5587,13687}, {5657,13674}, {5688,13691}, {5689,13690}, {5790,13713}, {9857,13685}, {10039,13714}, {10791,13672}

X(13688) = midpoint of X(8) and X(13678)
X(13688) = reflection of X(i) in X(j) for these (i,j): (1,13701), (1327,10)
X(13688) = anticomplement of X(13667)
X(13688) = X(1327)-of-outer-Garcia-triangle

X(13689) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO 1st TRI-SQUARES-CENTRAL

Barycentrics (S^3-3*(6*R^2+SA-2*SW)*S^2-3*(216*R^4+(24*SA-98*SW)*R^2-2*SA^2-4*SA*SW+11*SW^2)*S-6*(3*SA+2*SW)*(3*SA-SW)*R^2+3*(5*SA^2+SA*SW-SW^2)*SW)*(S^2-3*SB*SC) : :

The reciprocal orthologic center of these triangles is X(13665)

X(13689) lies on these lines: {30,6289}, {402,1327}, {1650,13701}, {4240,13678}, {11831,13667}, {11832,13668}, {11839,13672}, {11845,13674}, {11848,13675}, {11852,13679}, {11853,13680}, {11863,13682}, {11864,13683}, {11885,13685}, {11897,13687}, {11900,13688}, {11901,13690}, {11902,13691}, {11903,13693}, {11904,13694}, {11905,13695}, {11906,13696}, {11907,13697}, {11908,13698}, {11909,13699}, {11910,13702}, {11911,13713}, {11913,13715}, {11914,13716}, {11915,13717}

X(13689) = midpoint of X(4240) and X(13678)
X(13689) = reflection of X(i) in X(j) for these (i,j): (1327,402), (1650,13701)
X(13689) = X(1327)-of-Gossard-triangle

X(13690) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO 1st TRI-SQUARES-CENTRAL

Barycentrics 5*S^2*(2*S-3*SA-SW)-9*(SB+SC)*SA*(S-2*SW) : :
X(13690) = 6*X(3)-5*X(13712) = 5*X(1327)-4*X(3845) = 3*X(3545)-5*X(13674) = 8*X(3845)-5*X(13691) = 6*X(11539)-5*X(13692)

The reciprocal orthologic center of these triangles is X(13665)

X(13690) lies on these lines: {3,6281}, {6,1327}, {30,6279}, {547,10514}, {1271,13678}, {3543,5871}, {3545,10783}, {5589,13679}, {5591,13701}, {5605,13702}, {5689,13688}, {5861,11001}, {6215,11539}, {9994,13685}, {10792,13672}, {11370,13667}, {11388,13668}

X(13690) = reflection of X(13691) in X(1327)
X(13690) = X(1327)-of-inner-Grebe-triangle

X(13691) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO 1st TRI-SQUARES-CENTRAL

Barycentrics S^2*(10*S+9*SA+11*SW)-9*(SB+SC)*SA*(S+2*SW) : :
X(13691) = 3*X(1327)-4*X(3845) = 2*X(3534)-3*X(13712) = 5*X(3830)-3*X(13713) = 8*X(3845)-3*X(13690) = 2*X(8703)-3*X(13692) = 2*X(11001)-3*X(13666)

The reciprocal orthologic center of these triangles is X(13665)

X(13691) lies on these lines: {2,5870}, {6,1327}, {30,6278}, {1270,13678}, {3534,13712}, {3830,6280}, {5590,13701}, {5688,13688}, {6214,8703}, {9995,13685}, {10793,13672}, {11001,11825}, {11371,13667}, {11389,13668}

X(13691) = reflection of X(13690) in X(1327)
X(13691) = X(1327)-of-outer-Grebe-triangle

X(13692) = ORTHOLOGIC CENTER OF THESE TRIANGLES: JOHNSON TO 1st TRI-SQUARES-CENTRAL

Barycentrics S^2*(2*S+3*SA+3*SW)+3*SA*(SB+SC)*(S-SW) : :
X(13692) = 3*X(381)-2*X(13687) = 3*X(381)-X(13713) = 3*X(5587)-X(13679) = 2*X(8703)+X(13691) = 6*X(11539)-X(13690) = X(13666)-3*X(13712)

The reciprocal orthologic center of these triangles is X(13665)

X(13692) lies on these lines: {1,13695}, {2,6290}, {4,13668}, {5,1327}, {11,13715}, {12,13714}, {30,6289}, {381,13687}, {517,13688}, {549,1352}, {952,13702}, {1479,13699}, {5587,13679}, {5886,13667}, {6214,8703}, {6215,11539}, {6287,13708}, {7610,13681}, {9996,13685}, {10796,13672}

X(13692) = midpoint of X(i) and X(j) for these {i,j}: {4,13678}, {13693,13694}
X(13692) = reflection of X(i) in X(j) for these (i,j): (3,13701), (1327,5), (13713,13687)
X(13692) = complement of X(13674)
X(13692) = X(1327)-of-Johnson-triangle
X(13692) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381,13713,13687), (13695,13696,1)

X(13693) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO 1st TRI-SQUARES-CENTRAL

Barycentrics
6*(a^5-(b+c)*a^4+2*b*c*a^3+(b^2+c^2)*(b-c)^2*a-(b^4-c^4)*(b-c))*S+4*a^7-4*(b+c)*a^6-(9*b^2-28*b*c+9*c^2)*a^5+(b+c)*(9*b^2-20*b*c+9*c^2)*a^4+2*(3*b^4+3*c^4-2*b*c*(5*b^2-8*b*c+5*c^2))*a^3-2*(b^2-c^2)*(b-c)*(3*b^2-5*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2+8*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

The reciprocal orthologic center of these triangles is X(13665)

X(13693) lies on these lines: {11,1327}, {12,13716}, {30,12929}, {355,13692}, {1376,13675}, {3434,13678}, {10523,13714}, {10785,13674}, {10794,13672}, {10826,13679}, {10829,13680}, {10871,13685}, {10893,13687}, {10914,13688}, {10919,13690}, {10920,13691}, {10944,13695}, {10945,13697}, {10946,13698}, {10947,13699}, {10948,13715}, {10949,13717}, {11373,13667}, {11390,13668}, {11826,13666}, {11865,13682}, {11866,13683}, {11903,13689}, {11928,13713}

X(13693) = reflection of X(i) in X(j) for these (i,j): (13675,13701), (13694,13692)
X(13693) = X(1327)-of-inner-Johnson-triangle

X(13694) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO 1st TRI-SQUARES-CENTRAL

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

The reciprocal orthologic center of these triangles is X(13665)

X(13694) lies on these lines: {11,13717}, {12,1327}, {30,12939}, {72,13688}, {355,13692}, {958,13701}, {3436,13678}, {10523,13715}, {10786,13674}, {10795,13672}, {10827,13679}, {10830,13680}, {10872,13685}, {10894,13687}, {10921,13690}, {10922,13691}, {10950,13696}, {10951,13697}, {10952,13698}, {10953,13699}, {10954,13714}, {10955,13716}, {11374,13667}, {11391,13668}, {11500,13675}, {11827,13666}, {11867,13682}, {11868,13683}, {11904,13689}, {11929,13713}

X(13694) = reflection of X(13693) in X(13692)
X(13694) = X(1327)-of-outer-Johnson-triangle

X(13695) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO 1st TRI-SQUARES-CENTRAL

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

The reciprocal orthologic center of these triangles is X(13665)

X(13695) lies on these lines: {1,13692}, {12,1327}, {56,13701}, {65,13688}, {388,13678}, {3085,13674}, {5434,13712}, {10873,13685}, {11375,13667}, {11392,13668}

X(13695) = reflection of X(13714) in X(495)
X(13695) = X(1327)-of-1st-Johnson-Yff-triangle

X(13696) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO 1st TRI-SQUARES-CENTRAL

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

The reciprocal orthologic center of these triangles is X(13665)

X(13696) lies on these lines: {1,13692}, {5,13714}, {11,1327}, {30,12959}, {55,13701}, {496,13715}, {497,13678}, {3057,13688}, {3058,13712}, {3086,13674}, {6284,13666}, {9581,13679}, {9669,13713}, {10072,12958}, {10798,13672}, {10874,13685}, {10896,13687}, {11376,13667}, {11393,13668}

X(13696) = reflection of X(13715) in X(496)
X(13696) = X(1327)-of-2nd-Johnson-Yff-triangle

X(13697) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO 1st TRI-SQUARES-CENTRAL

Barycentrics 4*(3*SA+8*R^2-3*SW)*S^3+2*(6*SA^2+(-12*R^2+3*SW)*SA-4*SW^2)*S^2-3*(SB+SC)*(S*SW^2+(24*S*R^2+2*S*SW+3*SW^2)*SA) : :

The reciprocal orthologic center of these triangles is X(13665)

X(13697) lies on these lines: {30,13004}, {493,1327}, {6461,13698}, {6462,13678}, {8188,13679}, {8194,13680}, {8201,13682}, {8208,13683}, {8210,13702}, {8212,13687}, {8214,13688}, {8216,13690}, {8218,13691}, {8220,13692}, {8222,13701}, {10875,13685}, {10945,13693}, {10951,13694}, {11377,13667}, {11394,13668}, {11503,13675}, {11828,13666}, {11840,13672}, {11846,13674}, {11907,13689}, {11930,13695}, {11932,13696}, {11947,13699}, {11949,13713}, {11951,13714}, {11953,13715}, {11955,13716}, {11957,13717}

X(13697) = X(1327)-of-Lucas-homothetic-triangle

X(13698) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO 1st TRI-SQUARES-CENTRAL

Barycentrics 4*(3*SA+8*R^2-3*SW)*S^3-2*(6*SA^2+(12*R^2-3*SW)*SA+2*SW^2)*S^2-3*(SB+SC)*(-S*SW^2+(24*S*R^2-2*S*SW-3*SW^2)*SA) : :

The reciprocal orthologic center of these triangles is X(13665)

X(13698) lies on these lines: {30,13005}, {494,1327}, {6461,13697}, {6463,13678}, {8189,13679}, {8195,13680}, {8202,13682}, {8209,13683}, {8211,13702}, {8213,13687}, {8215,13688}, {8217,13690}, {8219,13691}, {8221,13692}, {8223,13701}, {10876,13685}, {10946,13693}, {10952,13694}, {11378,13667}, {11395,13668}, {11504,13675}, {11829,13666}, {11841,13672}, {11847,13674}, {11908,13689}, {11931,13695}, {11933,13696}, {11948,13699}, {11950,13713}, {11952,13714}, {11954,13715}, {11956,13716}, {11958,13717}

X(13698) = X(1327)-of-Lucas(-1)-homothetic-triangle

X(13699) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO 1st TRI-SQUARES-CENTRAL

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

The reciprocal orthologic center of these triangles is X(13665)

X(13699) lies on these lines: {11,13701}, {30,6283}, {33,13668}, {55,1327}, {497,13678}, {1837,13688}, {2646,13667}, {4294,13674}, {10799,13672}, {10877,13685}

X(13699) = X(1327)-of-Mandart-incircle-triangle

X(13700) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MCCAY TO 1st TRI-SQUARES-CENTRAL

Barycentrics 6*S^3+3*(3*SA+SW)*S^2-(27*SA^2-30*SW*SA+SW^2)*S+9*(SB+SC)*SA*SW : :
X(13700) = 2*X(13701)+X(13708)

The reciprocal orthologic center of these triangles is X(13676)

X(13700) lies on these lines: {2,1327}, {543,13088}, {5569,13087}

X(13701) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MEDIAL TO 1st TRI-SQUARES-CENTRAL

Barycentrics (2*S+3*SB+3*SC)*(S+3*SA) : :
X(13701) = 3*X(2)+X(13678) = X(1327)+3*X(13712) = 5*X(1656)-X(13713) = 5*X(1698)-X(13679) = X(13678)-3*X(13712) = 3*X(13700)-X(13708) = 3*X(13700)+X(13710)

The reciprocal orthologic center of these triangles is X(13665)

X(13701) lies on these lines: {1,13688}, {2,1327}, {4,13666}, {5,13687}, {8,13702}, {11,13699}, {30,641}, {55,13696}, {56,13695}, {83,13672}, {141,12100}, {376,639}, {427,13668}, {498,13714}, {499,13715}, {549,642}, {631,13674}, {640,5054}, {1125,13667}, {1656,13713}, {1698,13679}, {3096,13685}, {5590,13691}, {5591,13690}

X(13701) = midpoint of X(i) and X(j) for these {i,j}: {1,13688}, {2,13712}, {3,13692}, {4,13666}, {8,13702}, {1327,13678}, {1650,13689}, {13675,13693}, {13704,13706}, {13708,13710}
X(13701) = reflection of X(i) in X(j) for these (i,j): (13667,1125), (13687,5)
X(13701) = complement of X(1327)
X(13701) = X(1327)-of-medial-triangle
X(13701) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,13678,1327), (1327,13712,13678), (13700,13710,13708)

X(13702) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR TO 1st TRI-SQUARES-CENTRAL

Barycentrics 6*(a^2-(b+c)*a+2*b^2+2*c^2)*a*S-(a+b+c)*(16*a^4-12*(b+c)*a^3-(11*b^2-24*b*c+11*c^2)*a^2+12*(b^2-c^2)*(b-c)*a-5*(b^2-c^2)^2) : :
X(13702) = 3*X(1)-2*X(13667) = 3*X(1)-X(13679) = 3*X(1327)-4*X(13667) = 3*X(5603)-2*X(13687) = 3*X(7967)-X(13674) = 3*X(10247)-X(13713) = 2*X(13688)-3*X(13712)

The reciprocal orthologic center of these triangles is X(13665)

X(13702) lies on these lines: {1,1327}, {8,13701}, {30,7981}, {145,13678}, {517,13666}, {519,13688}, {952,13692}, {5603,13687}, {5604,13691}, {5605,13690}, {7967,13674}, {9997,13685}, {10247,13713}, {10800,13672}, {11396,13668}

X(13702) = midpoint of X(145) and X(13678)
X(13702) = reflection of X(i) in X(j) for these (i,j): (8,13701), (1327,1)

X(13703) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES-CENTRAL TO INNER-NAPOLEON

Barycentrics (2*S^2+3*(SB+SC)*S-2*(SB-SC)^2)*sqrt(3)+9*S^2-2*SW*S-6*SW^2+6*SA^2+3*SB*SC : :

The reciprocal orthologic center of these triangles is X(13704)

X(13703) lies on these lines: {115,13705}, {395,6303}, {531,3068}

X(13704) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-NAPOLEON TO 1st TRI-SQUARES-CENTRAL

Barycentrics (2*S+3*SB+3*SC)*(S+3*SA)*sqrt(3)+S*(2*SA-SB-SC)+3*S^2-9*SB*SC : :

The reciprocal orthologic center of these triangles is X(13704)

X(13704) lies on these lines: {2,1327}, {531,6305}, {6301,13084}, {6302,9885}

X(13704) = reflection of X(13706) in X(13701)
X(13704) = anticomplement of X(33487)

X(13705) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES-CENTRAL TO OUTER-NAPOLEON

Barycentrics -(2*S^2+3*(SB+SC)*S-2*(SB-SC)^2)*sqrt(3)+9*S^2-2*SW*S-6*SW^2+6*SA^2+3*SB*SC : :

The reciprocal orthologic center of these triangles is X(13706)

X(13705) lies on these lines: {115,13703}, {396,6302}, {530,3068}

X(13706) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-NAPOLEON TO 1st TRI-SQUARES-CENTRAL

Barycentrics -(2*S+3*SB+3*SC)*(S+3*SA)*sqrt(3)+S*(2*SA-SB-SC)+3*S^2-9*SB*SC : :

The reciprocal orthologic center of these triangles is X(13705)

X(13706) lies on these lines: {2,1327}, {6300,13083}, {6303,9886}

X(13706) = reflection of X(13704) in X(13701)
X(13706) = anticomplement of X(33486)

X(13707) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES-CENTRAL TO 1st NEUBERG

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

The reciprocal orthologic center of these triangles is X(13708)

X(13707) lies on these lines: {69,5475}, {538,3068}, {3734,9675}, {3934,5590}, {6314,8992}

X(13708) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st NEUBERG TO 1st TRI-SQUARES-CENTRAL

Barycentrics 2*S^3-(9*SA^2-12*SA*SW+SW^2)*S-3*(SB+SC)*(S^2-3*SA*SW) : :
X(13708) = X(6312)+2*X(7690) = 3*X(13700)-2*X(13701) = 3*X(13700)-X(13710)

The reciprocal orthologic center of these triangles is X(13707)

X(13708) lies on these lines: {2,1327}, {30,13088}, {538,6312}, {6222,12305}, {6287,13692}

X(13708) = reflection of X(13710) in X(13701)

X(13709) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES-CENTRAL TO 2nd NEUBERG

Barycentrics 4*S^3-2*SA*S^2+(3*SA^2-6*SA*SW+11*SW^2)*S-(3*SA^2+4*SA*SW-9*SW^2)*SW : :

The reciprocal orthologic center of these triangles is X(13710)

X(13709) lies on these lines: {754,3068}, {3618,5355}, {6275,6704}, {6313,8993}

X(13710) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd NEUBERG TO 1st TRI-SQUARES-CENTRAL

Barycentrics 2*S^3+3*(SA+3*SW)*S^2-(-6*SA*SW-SW^2+9*SA^2)*S-9*(SB+SC)*SA*SW : :
X(13710) = 3*X(13700)-4*X(13701) = 3*X(13700)-2*X(13708)

The reciprocal orthologic center of these triangles is X(13709)

X(13710) lies on these lines: {2,1327}, {754,6311}

X(13710) = reflection of X(13708) in X(13701)

X(13711) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES-CENTRAL TO INNER-VECTEN

Barycentrics 6*S^2+(SB+SC)*(S-5*SA-2*SB-2*SC) : :

The reciprocal orthologic center of these triangles is X(13712)

X(13711) lies on these lines: {5,6}, {115,6561}, {230,6560}, {487,8972}, {642,7375}, {3068,13650}, {3619,5491}, {5254,5418}, {5420,7746}, {6200,12123}, {6300,11488}, {6301,11489}, {6564,7735}

X(13711) = reflection of X(13650) in X(3068)

X(13712) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-VECTEN TO 1st TRI-SQUARES-CENTRAL

Barycentrics 2*S^2-(4*SA+SB+SC)*S-9*(SB+SC)*SA : :
X(13712) = 6*X(3)-X(13690) = X(1327)+2*X(13678) = X(1327)-4*X(13701) = 2*X(3534)+X(13691) = 3*X(3545)-2*X(13687) = 3*X(5055)-X(13713) = X(6279)-4*X(12974) = X(13666)+2*X(13692) = X(13678)+2*X(13701) = 2*X(13688)+X(13702)

The reciprocal orthologic center of these triangles is X(13711)

X(13712) lies on these lines: {2,1327}, {3,6281}, {30,6289}, {490,5418}, {519,13688}, {599,8703}, {1991,8182}, {3058,13696}, {3102,7757}, {3534,13691}, {3545,13687}, {3582,13715}, {3584,13714}, {5055,13713}, {5064,13668}, {5434,13695}, {5860,9741}, {5861,6200}, {6279,12974}, {7865,13685}, {11238,13699}

X(13712) = midpoint of X(i) and X(j) for these {i,j}: {2,13678}, {13671,13677}
X(13712) = reflection of X(i) in X(j) for these (i,j): (2,13701), (1327,2), (6560,13669)
X(13712) = complement of X(33456)

X(13713) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO 1st TRI-SQUARES-CENTRAL

Barycentrics 20*S^3+3*(SB+SC)*(2*S^2-8*S*SA-SA*SW) : :
X(13713) = 3*X(3)-2*X(13666) = 3*X(381)-4*X(13687) = 3*X(381)-2*X(13692) = 3*X(1327)-X(13666) = 5*X(1656)-4*X(13701) = 5*X(3830)-2*X(13691) = 3*X(5055)-2*X(13712) = 3*X(5790)-2*X(13688) = 3*X(10246)-4*X(13667) = 2*X(12313)-5*X(12602)

The reciprocal orthologic center of these triangles is X(13665)

X(13713) lies on these lines: {3,1327}, {5,13678}, {30,12257}, {381,13687}, {517,13679}, {999,13715}, {1351,3543}, {1598,13668}, {1656,13701}, {3830,6280}, {3845,12314}, {5055,13712}, {5790,13688}, {9301,13685}, {9669,13696}, {10246,13667}, {10247,13702}, {11842,13672}

X(13713) = reflection of X(i) in X(j) for these (i,j): (3,1327), (13678,5), (13692,13687)
X(13713) = X(1327)-of-X3-ABC-reflections-triangle

X(13714) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO 1st TRI-SQUARES-CENTRAL

Barycentrics 10*S^3+3*(SB+SC)*(S*(S-3*SA)+(2*S+SW)*b*c) : :

The reciprocal orthologic center of these triangles is X(13665)

X(13714) lies on these lines: {1,1327}, {5,13696}, {12,13692}, {30,10068}, {35,13666}, {388,13674}, {498,13701}, {1479,13687}, {3085,13678}, {3584,13712}, {10039,13688}, {10067,11237}, {10801,13672}, {11398,13668}

X(13714) = midpoint of X(1327) and X(13716)
X(13714) = reflection of X(13695) in X(495)
X(13714) = X(1327)-of-inner-Yff-triangle
X(13714) = {X(1),X(1327)}-harmonic conjugate of X(13715)

X(13715) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO 1st TRI-SQUARES-CENTRAL

Barycentrics 10*S^3+3*(SB+SC)*(S*(S-3*SA)-(2*S+SW)*b*c) : :

The reciprocal orthologic center of these triangles is X(13665)

X(13715) lies on these lines: {1,1327}, {11,13692}, {30,10084}, {36,13666}, {496,13696}, {497,13674}, {499,13701}, {999,13713}, {1478,13687}, {1737,13688}, {3086,13678}, {3582,13712}, {10047,13685}, {10802,13672}, {11399,13668}

X(13715) = midpoint of X(1327) and X(13717)
X(13715) = reflection of X(13696) in X(496)
X(13715) = X(1327)-of-outer-Yff-triangle
X(13715) = {X(1),X(1327)}-harmonic conjugate of X(13714)

X(13716) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO 1st TRI-SQUARES-CENTRAL

Barycentrics
6*a^2*(a^4-2*(b^2+b*c+c^2)*a^2+2*b*c*(b+c)*a+b^4+c^4-4*b^3*c-2*b^2*c^2-4*b*c^3)*S+(a+b+c)*(4*a^7-4*(b+c)*a^6-(3*b^2-40*b*c+3*c^2)*a^5+(b+c)*(3*b^2-32*b*c+3*c^2)*a^4-2*(3*b^4+3*c^4+2*b*c*(5*b^2-14*b*c+5*c^2))*a^3+2*(b^2-c^2)*(b-c)*(3*b^2+17*b*c+3*c^2)*a^2+5*(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-5*(b^2-c^2)^3*(b-c)) : :

The reciprocal orthologic center of these triangles is X(13665)

X(13716) lies on these lines: {1,1327}, {12,13693}, {30,13134}, {5552,13701}, {10528,13678}, {10531,13687}, {10803,13672}, {10805,13674}, {10834,13680}, {10915,13688}, {10929,13690}, {10930,13691}, {10942,13692}, {10956,13695}, {10958,13696}, {11248,13666}, {11400,13668}, {11509,13675}, {11881,13682}, {11882,13683}, {11914,13689}, {11955,13697}, {11956,13698}, {12000,13713}

X(13716) = reflection of X(1327) in X(13714)
X(13716) = {X(1327), X(13702)}-Harmonic conjugate of X(13717)
X(13716) = X(1327)-of-inner-Yff-tangents-triangle

X(13717) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO 1st TRI-SQUARES-CENTRAL

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

The reciprocal orthologic center of these triangles is X(13665)

X(13717) lies on these lines: {1,1327}, {11,13694}, {30,13135}, {10527,13701}, {10529,13678}, {10532,13687}, {10804,13672}, {10806,13674}, {10835,13680}, {10879,13685}, {10916,13688}, {10931,13690}, {10932,13691}, {10943,13692}, {10949,13693}, {10957,13695}, {10959,13696}, {10966,13699}, {11249,13666}, {11401,13668}, {11510,13675}, {11883,13682}, {11884,13683}, {11915,13689}, {11957,13697}, {11958,13698}, {12001,13713}

X(13717) = reflection of X(1327) in X(13715)
X(13717) = X(1327)-of-outer-Yff-tangents-triangle
X(13717) = {X(1327), X(13702)}-harmonic conjugate of X(13716)

X(13718) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO 1st TRI-SQUARES-CENTRAL

Barycentrics (SB-SC)*(15*S^3+9*(SB+SC)*S^2-(-24*SA*SW+18*SA^2+11*SW^2)*S+3*(3*SA-2*SW)*(SB+SC)*SW) : :

The reciprocal parallelogic center of these triangles is X(13665)

X(13718) lies on these lines: {351,13719}, {523,13317}

X(13718) = reflection of X(13719) in X(351)
X(13718) = X(1327)-of-1st-Parry-triangle
X(13718) = X(13666)-of-2nd-Parry-triangle

X(13719) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO 1st TRI-SQUARES-CENTRAL

Barycentrics (SB-SC)*(15*S^3+9*(SB+SC)*S^2+(18*SA^2-24*SA*SW+SW^2)*S-9*(SB+SC)*SA*SW) : :

The reciprocal parallelogic center of these triangles is X(13665)

X(13719) lies on these lines: {351,13718}, {523,13320}

X(13719) = reflection of X(13718) in X(351)
X(13719) = X(1327)-of-2nd-Parry-triangle
X(13719) = X(13666)-of-1st-Parry-triangle

X(13720) = X(3) OF THE 1st TRI-SQUARES-CENTRAL TRIANGLE

Barycentrics -14*(4*a^2+b^2+c^2)*S+6*a^4-39*(b^2+c^2)*a^2+9*(b^2-c^2)^2 : :

X(13720) lies on these lines: {1327,6476}, {3068,13721}

X(13720) = midpoint of X(1327) and X(9541)
X(13720) = reflection of X(13662) in X(13721)
X(13720) = X(641)-of-1st-tri-squares-triangle

X(13721) = X(5) OF THE 1st TRI-SQUARES-CENTRAL TRIANGLE

Barycentrics 2*(62*a^2+11*b^2+11*c^2)*S+6*a^4+69*(b^2+c^2)*a^2-27*(b^2-c^2)^2 : :

X(13721) lies on the line {3068,13720}

X(13721) = midpoint of X(13662) and X(13720)
X(13721) = X(6118)-of-1st-tri-squares-triangle

X(13722) = TRIPOLAR CENTROID OF X(3414)

Barycentrics (SB - SC)^2*(SA^2 - SB*SC - Sqrt[-3*S^2 + SW^2]*SA) : :

Contributed by Randy Hutson and Peter Moses, independently, July 2, 2017. See also X(13636), the tripolar centroid of X(3413).

Let V be the circle through X(13) and X(14) with center on the major axis of the Steiner inellipse. The center of V is X(13722). (Randy Hutson and Peter Moses, independently, July 2, 2017)

X(13722) lies on the Hutson-Parry circle, the circle {{X(2), X(13), X(14)}}, the cubics K219 and K237, and these lines: {2,1341}, {115,125}, {476,1379}, {892,6190}, {2040,5996}, {2395,5639}, {3414,5466}, {6141,6795}

X(13722) = reflection of X(13636) in X(8371)
X(13722) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 13636}, {2029, 8287}
X(13722) = crosspoint of X(i) and X(j) for these (i,j): {523, 3413}, {3414, 6190}
X(13722) = crossdifference of every pair of points on line {110, 1380}
X(13722) = crosssum of X(i) and X(j) for these (i,j): {110, 1379}, {1380, 5638}
X(13722) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 13636}, {3413, 523}, {6190, 3414}
X(13722) = crosspoint of X(i) and X(j) for these (i,j): {523, 3413}, {3414, 6190}
X(13722) = crossdifference of every pair of points on line {110, 1380}
X(13722) = crosssum of X(i) and X(j) for these (i,j): {110, 1379}, {1380, 5638}
X(13722) = X(8029)-cross conjugate of X(13636)
X(13722) = isoconjugate of X(j) and X(j) for these (i,j): {163, 6189}, {662, 1380}, {1101, 3413}
X(13722) = X(690)-Hirst inverse of X(13636)
X(13722) = X(2395)-line conjugate of X(5639)
X(13722) = tripolar centroid of X(3414)
X(13722) = Hutson-Parry-circle antipode of X(13636)
X(13722) = insimilicenter of circles {{X(13),X(14),X(16)}} and {{X(13),X(14),X(15)}}; the exsimilicenter is X(13636)
X(13722) = barycentric product X(i)*X(j) for these {i,j}: {115, 6190}, {338, 1379}, {523, 3414}, {850, 5639}
X(13722) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 3413}, {512, 1380}, {523, 6189}, {1379, 249}, {2029, 1379}, {3124, 5638}, {3414, 99}, {5639, 110}, {6190, 4590}, {8029, 13636}
X(13722) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115,1648,13636), (125,868,13636), (1637,1640,13636), (9148,11182,13636), (9200,9201,13636)

X(13723) = EULER LINE INTERCEPT OF X(1)X(32)

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

X(13723) lies on these lines: {1, 32}, {2, 3}, {10, 4112}, {39, 1724}, {41, 72}, {56, 5244}, {58, 980}, {63, 5320}, {238, 2277}, {284, 10477}, {518, 584}, {958, 1146}, {984, 1582}, {986, 1580}, {988, 1453}, {993, 4124}, {1001, 2178}, {1178, 5145}, {1214, 1395}, {1386, 4275}, {1468, 1472}, {1631, 4026}, {2223, 5248}, {5266, 5311}

X(13724) = EULER LINE INTERCEPT OF X(1)X(51)

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

X(13724) lies on these lines: {1, 51}, {2, 3}, {37, 1953}, {73, 1104}, {184, 1724}, {205, 2112}, {228, 950}, {1145, 3695}, {1214, 1828}, {1284, 3924}, {1837, 3185}, {2218, 6187}, {3330, 4268}

X(13725) = EULER LINE INTERCEPT OF X(1)X(69)

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

X(13725) lies on these lines: {1, 69}, {2, 3}, {8, 3896}, {10, 345}, {86, 4340}, {333, 387}, {348, 1448}, {388, 1402}, {958, 4026}, {960, 4259}, {966, 2092}, {1043, 5224}, {1056, 5484}, {1104, 4657}, {1125, 4138}, {1330, 5712}, {1724, 3618}, {1834, 5737}, {2339, 7713}, {2345, 7283}, {3616, 4388}, {3923, 12579}, {4252, 6703}, {4255, 5743}, {4292, 10436}, {4294, 5263}, {5955, 9780}

X(13726) = EULER LINE INTERCEPT OF X(1)X(71)

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

X(13726) lies on these lines: {1, 71}, {2, 3}, {35, 1714}, {55, 387}, {284, 1724}, {345, 10449}, {386, 1453}, {943, 7085}, {1104, 4261}, {2267, 3074}, {2646, 11435}, {2947, 7987}, {3191, 3730}, {3286, 4340}, {4294, 8053}, {5248, 5285}, {5745, 10479}, {6254, 12262}

X(13727) = EULER LINE INTERCEPT OF X(1)X(85)

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

X(13727) lies on these lines: {1, 85}, {2, 3}, {10, 9441}, {69, 3332}, {72, 10025}, {75, 990}, {76, 1043}, {86, 991}, {200, 4385}, {333, 1754}, {516, 4357}, {894, 971}, {1088, 1448}, {1220, 5691}, {1350, 10446}, {1441, 3100}, {1944, 5784}, {2271, 5286}, {3662, 5805}, {3693, 7283}, {4259, 5327}, {4292, 4911}, {4672, 9355}, {4847, 5015}, {5732, 10436}, {5762, 6646}, {13161, 13405}

X(13728) = EULER LINE INTERCEPT OF X(1)X(141)

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

X(13728) lies on these lines: {1, 141}, {2, 3}, {10, 3666}, {72, 4260}, {386, 1211}, {1125, 2887}, {1213, 4261}, {1714, 5737}, {1724, 3589}, {1764, 5799}, {1834, 10479}, {2221, 5711}, {3216, 5743}, {3454, 5718}, {4292, 5750}, {4424, 5835}, {5224, 9534}, {5256, 5814}

X(13729) = EULER LINE INTERCEPT OF X(1)X(153)

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

X(13729) lies on these lines: {1, 153}, {2, 3}, {12, 12764}, {145, 10531}, {149, 355}, {497, 10944}, {946, 4861}, {1478, 4308}, {1479, 9785}, {2551, 8256}, {2829, 5253}, {2975, 7681}, {3583, 10039}, {3585, 3817}, {3616, 6256}, {3622, 12115}, {3623, 10596}, {3648, 5535}, {5057, 7686}, {5225, 10953}, {5274, 10629}, {5722, 12528}, {5804, 5905}, {5811, 12649}, {5818, 10525}, {5884, 9809}, {5901, 10742}, {7173, 13273}, {7701, 10265}, {10893, 11680}, {11496, 11681}, {12433, 13257}

X(13730) = EULER LINE INTERCEPT OF X(1)X(159)

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

X(13730) lies on these lines: {1, 159}, {2, 3}, {35, 197}, {55, 8190}, {56, 1448}, {154, 1437}, {345, 5687}, {387, 5324}, {942, 1473}, {956, 12410}, {958, 8193}, {991, 2360}, {1040, 7713}, {1060, 11363}, {1062, 1829}, {1074, 1842}, {1602, 4293}, {1610, 4305}, {1612, 1617}, {1626, 7742}, {1633, 4295}, {2204, 10316}, {3295, 7291}, {3428, 9911}, {5172, 9658}, {8069, 10037}, {8071, 10046}

X(13731) = EULER LINE INTERCEPT OF X(1)X(181)

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

X(13731) lies on these lines: {1, 181}, {2, 3}, {56, 5718}, {198, 5742}, {228, 6734}, {501, 572}, {517, 6051}, {519, 9568}, {551, 9569}, {573, 10441}, {942, 1400}, {958, 5743}, {978, 3576}, {986, 1284}, {1001, 5799}, {1125, 2051}, {1193, 1385}, {1402, 5530}, {1698, 10434}, {1724, 13323}, {1834, 5132}, {2277, 5336}, {2975, 5741}, {3074, 3955}, {3216, 10470}, {3624, 10882}, {3831, 6684}, {5230, 10267}, {5400, 7987}, {5754, 10246}

X(13732) = EULER LINE INTERCEPT OF X(1)X(182)

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

X(13732) lies on these lines: {1, 182}, {2, 3}, {40, 8616}, {511, 1724}, {517, 582}, {572, 5283}, {580, 10441}, {1935, 3784}, {2933, 3035}, {9041, 12513}

X(13733) = EULER LINE INTERCEPT OF X(1)X(184)

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

X(13733) lies on these lines: {1, 184}, {2, 3}, {31, 65}, {37, 48}, {51, 1724}, {56, 6354}, {283, 10441}, {386, 759}, {498, 1324}, {672, 4426}, {958, 7085}, {1626, 7354}, {2285, 5019}, {2933, 5432}, {3465, 3612}

X(13734) = EULER LINE INTERCEPT OF X(1)X(185)

Trilinears (sec A + sec B + sec C) cos A + (cos A + cos B + cos C) sec A : :
Barycentrics a (a^8 b-3 a^6 b^3+3 a^4 b^5-a^2 b^7+a^8 c-2 a^7 b c+2 a^5 b^3 c-2 a^4 b^4 c+2 a^3 b^5 c-2 a b^7 c+b^8 c+3 a^4 b^3 c^2-2 a^2 b^5 c^2-b^7 c^2-3 a^6 c^3+2 a^5 b c^3+3 a^4 b^2 c^3-4 a^3 b^3 c^3+3 a^2 b^4 c^3+2 a b^5 c^3-3 b^6 c^3-2 a^4 b c^4+3 a^2 b^3 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+2 a b^3 c^5+3 b^4 c^5-3 b^3 c^6-a^2 c^7-2 a b c^7-b^2 c^7+b c^8) : :

X(13734) lies on these lines: {1, 185}, {2, 3}, {65, 774}, {73, 820}, {228, 515}, {374, 1212}, {1724, 11424}, {1745, 3612}, {2278, 3330}

X(13734) = {X(3),X(4)}-harmonic conjugate of X(851)

X(13735) = EULER LINE INTERCEPT OF X(1)X(190)

Barycentrics 3 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(13735) lies on these lines: {1, 190}, {2, 3}, {519, 595}, {536, 1104}, {1043, 1724}, {1220, 5248}, {3749, 4737}, {4302, 4429}, {5251, 5263}

X(13736) = EULER LINE INTERCEPT OF X(1)X(193)

Barycentrics 3 a^4-2 a^3 b-6 a^2 b^2-2 a b^3-b^4-2 a^3 c-10 a^2 b c-10 a b^2 c-2 b^3 c-6 a^2 c^2-10 a b c^2-2 b^2 c^2-2 a c^3-2 b c^3-c^4 : :

X(13736) lies on these lines: {1, 193}, {2, 3}, {8, 968}, {45, 1265}, {391, 941}, {966, 1043}, {4313, 5296}, {4357, 5436}, {6646, 11036}

X(13736) = anticomplement of X(37153)

X(13737) = EULER LINE INTERCEPT OF X(1)X(198)

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

X(13737) lies on these lines: {1, 198}, {2, 3}, {6, 2360}, {34, 7011}, {56, 223}, {154, 580}, {208, 1465}, {228, 3295}, {515, 1622}, {610, 10396}, {963, 10864}, {1104, 2178}, {1214, 7713}, {1437, 5320}, {1439, 4350}, {1617, 5930}, {1724, 5120}, {1728, 2182}, {1730, 5706}, {2183, 7078}, {3182, 3220}, {5909, 11249}, {7742, 8185}, {9708, 10367}

X(13738) = EULER LINE INTERCEPT OF X(1)X(228)

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

X(13738) lies on these lines: {1, 228}, {2, 3}, {6, 41}, {36, 978}, {51, 581}, {55, 2654}, {58, 5320}, {65, 3185}, {78, 10477}, {184, 580}, {197, 5230}, {223, 1410}, {283, 9306}, {610, 1713}, {940, 4267}, {959, 2982}, {992, 3330}, {1104, 2352}, {1214, 1829}, {1398, 7011}, {1425, 10571}, {1426, 1465}, {1437, 5398}, {1617, 8192}, {1790, 13323}, {1867, 6708}, {2053, 8615}, {2176, 2198}, {2635, 5204}, {2933, 5172}, {2975, 5278}, {3556, 7355}, {3724, 3924}, {7742, 9798}, {8583, 10862}

X(13738) = complement of X(37191)

X(13739) = EULER LINE INTERCEPT OF X(1)X(270)

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

X(13739) lies on these lines: {1, 270}, {2, 3}, {7, 229}, {19, 2326}, {34, 162}, {63, 1098}, {107, 158}, {110, 3868}, {163, 1729}, {224, 662}, {1001, 2905}, {1304, 12030}, {1474, 2327}, {2203, 5208}, {3869, 6061}, {9275, 10122}

X(13740) = EULER LINE INTERCEPT OF X(1)X(312)

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

X(13740) lies on these lines: {1, 312}, {2, 3}, {6, 10449}, {10, 82}, {76, 86}, {141, 1330}, {171, 3831}, {239, 5295}, {264, 8747}, {315, 5224}, {321, 5262}, {333, 1724}, {386, 1043}, {387, 3618}, {894, 942}, {938, 5749}, {986, 3923}, {996, 1222}, {1046, 4672}, {1125, 13161}, {1191, 5793}, {1213, 7745}, {1386, 3714}, {1453, 11679}, {1654, 7762}, {1698, 3550}, {1834, 3589}, {2322, 8743}, {2901, 4360}, {3293, 3996}, {3661, 5814}, {3666, 7283}, {3673, 10436}, {3685, 3931}, {3701, 3920}, {3741, 5247}, {3912, 5717}, {4357, 4911}, {4383, 9534}, {4658, 7760}, {4676, 12514}, {4968, 7191}, {5266, 7081}, {5294, 6734}

X(13740) = complement of X(4201)
X(13740) = orthocentroidal-circle-inverse of X(16062)
X(13740) = {X(2),X(4)}-harmonic conjugate of X(16062)

X(13741) = EULER LINE INTERCEPT OF X(1)X(341)

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

X(13741) lies on these lines: {1, 341}, {2, 3}, {10, 4514}, {46, 4676}, {106, 1125}, {190, 3670}, {238, 3831}, {614, 4385}, {894, 5439}, {986, 4011}, {1043, 3216}, {1479, 4429}, {1698, 5263}, {3701, 7191}, {3752, 7283}, {3840, 5247}, {4358, 5262}, {4383, 10449}, {4968, 7292}, {5205, 5266}

X(13742) = EULER LINE INTERCEPT OF X(1)X(344)

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

X(13742) lies on these lines: {1, 344}, {2, 3}, {10, 3749}, {69, 1724}, {1398, 8816}, {1453, 3912}, {4000, 7283}, {4294, 4429}, {4295, 4676}

X(13743) = EULER LINE INTERCEPT OF X(1)X(399)

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

X(13743) lies on these lines: {1, 399}, {2, 3}, {12, 10058}, {36, 9955}, {55, 5441}, {56, 79}, {65, 1727}, {104, 5606}, {191, 517}, {265, 759}, {355, 8715}, {500, 4653}, {758, 1482}, {946, 12600}, {956, 8148}, {958, 3647}, {993, 12699}, {999, 3649}, {1158, 8261}, {1385, 5426}, {1437, 10540}, {1484, 13100}, {1749, 5903}, {1768, 5885}, {2077, 9956}, {2320, 10308}, {2795, 13188}, {2975, 3648}, {3295, 10043}, {3579, 3698}, {3585, 5172}, {3656, 8666}, {3754, 12515}, {3818, 4265}, {5259, 13624}, {5288, 11278}, {5436, 7171}, {5443, 12611}, {5450, 5886}, {5789, 6598}, {5790, 11248}, {6265, 12524}, {7173, 10090}, {7330, 11523}, {8069, 9654}, {8070, 12764}, {8071, 9669}, {8256, 9708}, {10246, 12114}, {10679, 12645}

X(13744) = EULER LINE INTERCEPT OF X(1)X(513)

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

X(13744) lies on these lines: {1, 513}, {2, 3}, {1388, 1464}, {2098, 10544}, {4427, 12746}

X(13745) = EULER LINE INTERCEPT OF X(1)X(524)

Barycentrics 2 a^4-2 a^3 b-5 a^2 b^2-2 a b^3-b^4-2 a^3 c-8 a^2 b c-8 a b^2 c-2 b^3 c-5 a^2 c^2-8 a b c^2-2 b^2 c^2-2 a c^3-2 b c^3-c^4 : :

X(13745) lies on these lines: {1, 524}, {2, 3}, {10, 3712}, {392, 511}, {519, 3743}, {540, 551}, {597, 1724}, {952, 9978}, {1125, 4892}, {1211, 4653}, {2796, 12579}, {3241, 3578}, {3679, 3704}, {4026, 5251}, {4256, 5241}, {4277, 5283}, {4304, 5257}

X(13746) = EULER LINE INTERCEPT OF X(1)X(564)

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

X(13746) lies on these lines: {1, 564}, {2, 3}, {60, 1837}, {110, 355}, {229, 1478}, {759, 7741}, {1789, 10483}, {1793, 7951}, {3580, 13408}

X(13747) = EULER LINE INTERCEPT OF X(1)X(1145)

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

X(13747) lies on these lines: {1, 1145}, {2, 3}, {10, 1319}, {35, 3816}, {36, 1329}, {51, 5482}, {55, 10200}, {72, 3911}, {100, 496}, {191, 5442}, {214, 10950}, {355, 6713}, {392, 6684}, {495, 5253}, {499, 1376}, {946, 13528}, {956, 7288}, {999, 5552}, {1125, 3057}, {1210, 5440}, {1213, 4268}, {1698, 4999}, {1770, 5087}, {1788, 5730}, {1837, 10609}, {2077, 7681}, {2975, 3820}, {3086, 5687}, {3306, 11374}, {3419, 5438}, {3421, 5265}, {3452, 3916}, {3555, 6745}, {3582, 3813}, {3583, 3847}, {3624, 5119}, {3814, 7354}, {3815, 5277}, {3825, 6284}, {3913, 10072}, {3925, 7294}, {4317, 11236}, {4413, 11510}, {4855, 5722}, {5258, 9711}, {5298, 8666}, {5439, 6692}, {5554, 10246}, {5563, 12607}, {5927, 6705}, {6174, 8715}, {6667, 7741}, {6767, 10586}, {7373, 10528}, {8582, 10165}, {9581, 12690}, {9669, 10584}, {9709, 10527}, {10090, 10523}, {10198, 10966}, {12528, 13226}

X(13747) = {X(5),X(404)}-harmonic conjugate of X(11112)

X(13748) = X(3)X(639)∩X(4)X(6)

Barycentrics 2 a^6-a^4 b^2-b^6-a^4 c^2+b^4 c^2+b^2 c^4-c^6+2 (a^2+b^2-c^2) (a^2-b^2+c^2) S : :
X(13748) = 3 X[4] - X[5871] = 3 X[5870] + X[5871] = 3 X[591] - 2 X[9733]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26287.

Let BBaCaC be the external square on side BC, and define CCbAbA and AAcBcB cyclically, as at X(1327). Let Oa be the circumcenter of AAbAc, and define Ob and Oc cyclically. Triangle OaObOc is homothetic to ABC at X(6), and X(13748) is the orthocenter of OaObOc. (Randy Hutson, July 21, 2017)

X(13748) lies on these lines: {3,639}, {4,6}, {20,492}, {30,591}, {154,1585}, {185,6291}, {325, 489}, {372,2794}, {382,12601}, {4 85,7694}, {486,6399}, {590,8414} ,{637,1350}, {1132,3424}, {1151, 6811}, {1513,12963}, {1586,1853} ,{3535,10192}, {5085,7389}, { 5200,13567}, {5921,12323}, {6033 ,6230}, {6459,7374}, {6467,12298 }, {6561,8721}, {6813,9756}, { 7388,10516}, {11381,12299}

X(13748) = midpoint of X(4) and X(5870)
X(13748) = crosssum of X(3) and X(9733)
X(13748) = reflection of X(13749) in X(4)
X(13748) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1588,5480), (4,6776,3070), (4 ,7582,6201), (4,10784,1587), (20 ,492,12305), (1587,10784,8550)

X(13749) = X(3)X(640)∩X(4)X(6)

Barycentrics 2 a^6-a^4 b^2-b^6-a^4 c^2+b^4 c^2+b^2 c^4-c^6-2 (a^2+b^2-c^2) (a^2-b^2+c^2) S : :
X(13749) = 3 X[4] - X[5870] = 3 X[5871] + X[5870] = 3 X[1991] - 2 X[9732]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26287.

Let BBaCaC be the internal square on side BC, and define CCbAbA and AAcBcB cyclically, as at X(1328). Let Oa be the circumcenter of AAbAc, and define Ob and Oc cyclically. Triangle OaObOc is homothetic to ABC at X(6), and X(13749) is the orthocenter of OaObOc. (Randy Hutson, July 21, 2017)

X(13749) lies on these lines: {3,640}, {4,6}, {20,491}, {30,1991}, {154,1586}, {185,6406 }, {325,490}, {371,2794}, {382, 12602}, {485,6222}, {486,7694}, { 615,8406}, {638,1350}, {1131, 3424}, {1152,6813}, {1513,12968} ,{1585,1853}, {3536,10192}, { 5085,7388}, {5921,12322}, {6033, 6231}, {6460,7000}, {6467,12299} ,{6560,8721}, {6811,9756}, { 7374,13638}, {7389,10516}, { 11381,12298}

X(13749) = midpoint of X(4) and X(5871)
X(13749) = crosssum of X(3) and X(9732)
X(13749) = reflection of X(13748) in X(4)
X(13749) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1587,5480), (4,6776,3071), (4 ,7581,6202), (4,10783,1588), (20 ,491,12306), (1588,10783,8550), (6813,8982,1152)

X(13750) = X(1)X(3)∩X(10)X(343)

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^2 b^3 c-a b^4 c+2 b^5 c-a^4 c^2-4 a^2 b^2 c^2+b^4 c^2-2 a^3 c^3-2 a^2 b c^3-4 b^3 c^3+2 a^2 c^4-a b c^4+b^2 c^4+a c^5+2 b c^5-c^6) : :
X(13750) = X[35] + 3 X[5902] = 3 X[354] - X[11011]

X(10023) = X(13750)-of-excentral-triangle

See Peter Moses, Hyacinthos 23283.

X(13750) lies on these lines: {1,3}, {5,1858}, {7,10629}, {10, 343}, {12,912}, {72,498}, {90, 6913}, {226,5884}, {377,5086}, { 381,1898}, {405,920}, {407,1785} ,{442,1737}, {499,5439}, {518, 10039}, {758,13411}, {960,7483}, {971,3585}, {974,2779}, {1006, 7098}, {1046,3074}, {1064,1393}, {1071,1478}, {1210,5883}, {1254, 4303}, {1479,12711}, {1770,9943} , {1776,6920}, {1781,2182}, { 1788,6889}, {1837,6917}, {1864, 10826}, {1905,4185}, {2252,2294} ,{2771,8068}, {3085,3868}, { 3485,6833}, {3486,6934}, {3487, 10321}, {3555,12647}, {3583, 5806}, {3753,5794}, {3827,5135}, {3869,6910}, {4295,6836}, {4299, 10167}, {4333,5918}, {5219,5693}, {5530,10974}, {5691,12671}, {5728,5880}, {5777,7951}, {5887, 6862}, {6001,6831}, {6738,10122}, {7354,13369}, {7686,10391}, { 9612,12664}, {10043,11045}, { 10106,12005}, {10107,10609}, {10320,11374}, {10590,12528}, {10948,11019}, {11013,11032}, {11015,11020}, {11570,13407}

X(13750) = midpoint of X(65) and X(2646)
X(13750) = X(10023)-of-orthic-triangle if ABC is acute
X(13750) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,46,11507),(1,57,8071),(1, 942,5570),(46,3612,165),(46, 5902,65),(942,9957,6583),( 7686,10391,10572),(10122, 12736,6738)

X(13751) = POINT BEID 137

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

X(13751) = (4*R^2-3*r^2-5*s^2+5*SW)*X(1) + (9*r^2-s^2+SW)*X(3)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 23298.

X(13751) lies on these lines: {1,3}, {7,10266}, {11,12005}, {12,5083}, {244,2594}, {1317,3754}, {1421,8614}, {2801,7173}, {3678,7294}, {3874,5433}, {5253,12739}, {5883,10944}, {5901,11570}

X(13751) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (942,1319,65), (942,5045,5425), (5570,13373,2646)

X(13752) = POINT BEID 138

Trilinears (b+c)*a^5-2*(b^2+c^2)*a^4-(b+ c)*(3*b^2-8*b*c+3*c^2)*a^3+(3* b^4+3*c^4-2*b*c*(b^2+3*b*c+c^ 2))*a^2+(b+c)*(2*b^4+2*c^4-b* c*(8*b^2-13*b*c+8*c^2))*a-(b^ 2-c^2)^2*(b-c)^2 : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 23298.

X(13752) lies on these lines: {1,901}, {513,3754}, {517,548}

X(13753) = POINT BEID 139

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26298.

X(13753) lies on these lines: {1,953}, {513,12005}, {517,1125}

X(13754) = X(3)X(49)∩X(4)X(52)

Trilinears (6*R^2-SA-SW)*SA*a : :
Trilinears cos(A)*(2*cos(A)*cos(B-C)-1) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 23787.

X(13754) lies on the curves K039, K114, K339, Q097 and these lines: {1,6238}, {2,5654}, {3,49}, {4,52}, {5,389}, {6,4550}, {20,6193}, {26,6759), (30,511}, {40,6237}, {51,381}, {55,500}, {56,1069}, {69,4846}, {74,323}, {110,186), (113,403}, {125,1568}, {131,1516}, {140,9729}, {143,546}, {146,7731}, {156,1658), (161,1498}, {182,7514}, {232,1625}, {265,1531}, {373,5055}, {376,2979}, {378,1993), (382,6243}, {399,1495}, {547,6688}, {549,3819}, {550,6101}, {569,7503}, {576,8548), (578,7526}, {944,9933}, {974,6699}, {1151,8909}, {1350,8717}, {1351,1597}, {1352,7706}, {1478,20019}, {1614,7488}, {1994,7527}, {3091,3567}, {3153,3448}, {3193,7414}, {3269,3289}, {3357,9938}, {3426,6391}, {3519,3521}, {3523,7999}, {3524,7998}, {3545,5640}, {3574,5576}, {3818,9969}, {3832,9781}, {4549,6776}, {5054,5650}, {5167,6033}, {5609,7575}, {5691,9896}, {5752,6985}, {5870,9930}, {5871,9929}, {5921,6403}, {5986,5999}, {6030,7512}, {6153,6288}, {6285,9931}, {6407,8912}, {6642,9786}, {6644,9306}, {8549,9926}, {9873,9923}

X(13754) = complementary conjugate of X(131)
X(13754) = isogonal conjugate of X(1300)

X(13755) = POINT BEID 140

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26299.

X(13755) lies on this line: {52,517}

X(13756) = POINT BEID 141

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26299.

For another constructions see: Antreas Hatzipolakis and César Lozada, Hyacinthos 28940.

Antreas Hatzipolakis and Peter Moses, Hyacinthos 29110.

X(13756) lies on the incircle and these lines: {1,3025}, {11,517}, {12,3259}, { 55,953}, {56,901}, {513,1317}, { 1155,5577}, {1319,1357}, {1364, 5048}, {3028,4017}, {3057,3326}, {3328,5919}, {3878,7144}

X(13756) = reflection of X(3025) in X(1)
X(13756) = reflection of X(1317) in the line X(1)X(3)
X(13756) = X(477)-of-intouch-triangle
X(13756) = X(953)-of-Mandart-incircle-triangle
X(13756) = intouch-anticomplement of X(33645)

Centers related to the 2nd tri-squares triangles: X(13757)-X(13850)

This preamble and centers X(13757)-X(13850) were contributed by César Eliud Lozada, July 10, 2017.

Tri-squares triangles were defined in the preamble of X(13637). In this section, A'B'C' is the 2nd tri-squares-triangle and A_oB_oC_o is de the 2nd tri-squares-central triangle.

A'B'C' and the circumsymmedial triangle are directly similar with center X(13779).

The appearance of (T, n) in the following list means that triangles A'B'C and T are inversely similar with center X(n):
(4th anti-Brocard, 13776), (anti-McCay, 13777), (4th Brocard, 13778), (McCay, 13780), (3rd Parry, 13781)

The appearance of (T, n) in the following list means that A'B'C' and triangle T are perspective with perspector X(n): (anti-Artzt*, 13757), (Artzt*, 13758), (2nd tri-squares-central, 13782), (1st tri-squares*, 2)
An asterisk means that both triangles are homothetic.

The appearance of (T, m, n) in the following list means that triangles A'B'C' and T are orthologic with centers X(m) and X(n):
(ABC, 3069, 2), (ABC-X3 reflections, 3069, 376), (anti-Aquila, 3069, 551), (anti-Ara, 3069, 428), (anti-Artzt, 13759, 1992), (1st anti-Brocard, 13760, 7840), (4th anti-Brocard, 13845, 9870), (5th anti-Brocard, 3069, 12150), (6th anti-Brocard, 13760, 12151), (anti-Euler, 3069, 376), (anti-Mandart-incircle, 3069, 4421), (anti-McCay, 13761, 385), (anticomplementary, 3069, 2), (Aquila, 3069, 3679), (Ara, 3069, 9909), (Artzt, 13759, 9770), (1st Auriga, 3069, 11207), (2nd Auriga, 3069, 11208), (1st Brocard, 13760, 599), (4th Brocard, 13762, 2), (5th Brocard, 3069, 7811), (6th Brocard, 13760, 9939), (circummedial, 13763, 2), (Euler, 3069, 381), (5th Euler, 13763, 2), (outer-Garcia, 3069, 3679), (Gossard, 3069, 1651), (inner-Grebe, 3069, 5861), (outer-Grebe, 3069, 5860), (Johnson, 3069, 381), (inner-Johnson, 3069, 11235), (outer-Johnson, 3069, 11236), (1st Johnson-Yff, 3069, 11237), (2nd Johnson-Yff, 3069, 11238), (Lucas homothetic, 3069, 12152), (Lucas(-1) homothetic, 3069, 12153), (Mandart-incircle, 3069, 3058), (McCay, 13761, 7610), (medial, 3069, 2), (5th mixtilinear, 3069, 3241), (inner-Napoleon, 13764, 9761), (outer-Napoleon, 13765, 9763), (1st Neuberg, 13766, 8667), (2nd Neuberg, 13767, 9766), (3rd Parry, 13768, 2), (1st tri-squares-central, 13769, 13663), (2nd tri-squares-central, 13782, 13783), (3rd tri-squares-central, 3069, 13846), (4th tri-squares-central, 3069, 13847), (1st tri-squares, 13759, 13639), (3rd tri-squares, 13771, 2), (4th tri-squares, 13770, 2), (inner-Vecten, 13770, 591), (outer-Vecten, 13771, 1991), (X3-ABC reflections, 3069, 381), (inner-Yff, 3069, 10056), (outer-Yff, 3069, 10072), (inner-Yff tangents, 3069, 11239), (outer-Yff tangents, 3069, 11240)

The appearance of (T, m, n) in the following list means that triangles A'B'C' and T are parallelogic with centers X(m) and X(n):
(4th anti-Brocard, 13772, 13168), (anti-McCay, 13773, 8597), (4th Brocard, 13774, 4), (McCay, 13773, 381), (1st Parry, 3069, 9123), (2nd Parry, 3069, 9185), (3rd Parry, 13775, 9147)

The appearance of (I, J) in the following list means that X(I)-of-A'B'C' = X(J) :
(2,3069),(3,13783),(4,13759),(5,13784),(6,13763),(13,13765),(14,13764),(98,13761),(99,13773),(111,13762),(376,2),(485,13769),(486,13782),(488,13831),(642,13843),(671,13760),(1296,13774),(1327,13771),(1328,13770),(1991,6398)

A_oB_oC_o and the 2nd tri-squares triangle are perspective with perspector X(13782).

The appearance of (T, m, n) in the following list means that triangles A_oB_oC_o and T are orthologic with centers X(m) and X(n):
(ABC, 13785, 1328), (ABC-X3 reflections, 13785, 13786), (anti-Aquila, 13785, 13787), (anti-Ara, 13785, 13788), (anti-Artzt, 13783, 13789), (1st anti-Brocard, 13790, 13791), (5th anti-Brocard, 13785, 13792), (6th anti-Brocard, 13790, 13793), (anti-Euler, 13785, 13794), (anti-Mandart-incircle, 13785, 13795), (anti-McCay, 13796, 13797), (anticomplementary, 13785, 13798), (Aquila, 13785, 13799), (Ara, 13785, 13800), (Artzt, 13783, 13801), (1st Auriga, 13785, 13802), (2nd Auriga, 13785, 13803), (1st Brocard, 13790, 13804), (5th Brocard, 13785, 13805), (6th Brocard, 13790, 13806), (Euler, 13785, 13807), (outer-Garcia, 13785, 13808), (Gossard, 13785, 13809), (inner-Grebe, 13785, 13810), (outer-Grebe, 13785, 13811), (Johnson, 13785, 13812), (inner-Johnson, 13785, 13813), (outer-Johnson, 13785, 13814), (1st Johnson-Yff, 13785, 13815), (2nd Johnson-Yff, 13785, 13816), (Lucas homothetic, 13785, 13817), (Lucas(-1) homothetic, 13785, 13818), (Mandart-incircle, 13785, 13819), (McCay, 13796, 13820), (medial, 13785, 13821), (5th mixtilinear, 13785, 13822), (inner-Napoleon, 13823, 13824), (outer-Napoleon, 13825, 13826), (1st Neuberg, 13827, 13828), (2nd Neuberg, 13829, 13830), (1st tri-squares-central, 13831, 13832), (3rd tri-squares-central, 13785, 13848), (4th tri-squares-central, 13785, 13849), (1st tri-squares, 13783, 13833), (2nd tri-squares, 13783, 13782), (3rd tri-squares, 13834, 13850), (4th tri-squares, 3069, 13847), (inner-Vecten, 3069, 2), (outer-Vecten, 13834, 13835), (X3-ABC reflections, 13785, 13836), (inner-Yff, 13785, 13837), (outer-Yff, 13785, 13838), (inner-Yff tangents, 13785, 13839), (outer-Yff tangents, 13785, 13840)

The appearance of (T, m, n) in the following list means that triangles A_oB_oC_o and T are parallelogic with centers X(m) and X(n): (1st Parry, 13785, 13841), (2nd Parry, 13785, 13842)

The appearance of (I, J) in the following list means that X(I)-of-A_oB_oC_o = X(J) :
(2,3069),(3,13843),(4,13782),(5,13844),(13,13823),(14,13825),(486,13783),(487,13759),(642,13784),(1132,13831),(1328,13785)

X(13757) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES AND ANTI-ARTZT

Barycentrics 3*(S+SA)-2*SW : :
X(13757) = X(491)-4*X(615)

X(13757) lies on these lines: {2,6}, {99,13761}, {110,13762}, {486,490}, {542,6813}, {598,13669}, {1505,5461}, {6399,10784}, {7389,7812}, {8593,13760}, {11055,13766}, {11149,12158}, {11159,13763}, {11161,13773}, {12154,13764}, {12155,13765}, {12156,13767}, {12157,13768}, {12159,13771}, {13167,13772}, {13169,13774}, {13170,13775}

X(13757) = reflection of X(i) in X(j) for these (i,j): (2,615), (491,2), (13637,8860)
X(13757) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,1992,13637), (2,5032,3068), (2,13759,1992), (597,11163,13637), (599,13783,2)

X(13758) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES AND ARTZT

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

X(13758) lies on these lines: {2,6}, {32,7389}, {98,486}, {372,6811}, {381,13763}, {489,12963}, {637,6424}, {1585,10311}, {3053,11293}, {3102,5420}, {3767,7388}, {5254,11294}, {5286,11292}, {6054,13760}, {6460,7374}, {6560,9993}, {7000,13748}, {9605,11315}, {9759,13762}, {9760,13764}, {9762,13765}, {9764,13766}, {9765,13767}, {9767,13770}, {9768,13771}, {9769,13774}, {9877,13761}, {12434,13768}, {13191,13772}, {13225,13775}, {13681,13769}

X(13758) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,385,491), (2,5304,3068), (2,7586,7736), (2,7735,13638), (2,13759,9770), (6,615,492), (183,230,13638), (230,615,2), (7610,13783,2)

X(13759) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO ARTZT

Barycentrics 5*a^2-b^2-c^2-3*S : :
X(13759) = 3*X(2)-4*X(13783) = 3*X(2)-8*X(13784) = X(1271)-4*X(3069) = 3*X(1271)-8*X(13783) = 3*X(1271)-16*X(13784) = 3*X(3069)-2*X(13783)

The reciprocal orthologic center of these triangles is X(9770)

X(13759) lies on these lines: {2,6}, {542,7000}, {543,13760}, {671,1132}, {6463,8591}

X(13759) = reflection of X(i) in X(j) for these (i,j): (2,3069), (1271,2), (13783,13784)
X(13759) = orthologic center of these triangles: 2nd tri-squares to Artzt
X(13759) = orthologic center of these triangles: 2nd tri-squares to 1st tri-squares
X(13759) = X(4)-of-2nd-tri-squares-triangle
X(13759) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,1992,13639), (2,5032,7585), (1992,13757,2), (9770,13758,2), (13783,13784,3069)

X(13760) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO 1st ANTI-BROCARD

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

The reciprocal orthologic center of these triangles is X(7840)

X(13760) lies on these lines: {2,5477}, {6,13653}, {115,7586}, {492,5182}, {524,13761}, {530,13764}, {531,13765}, {542,3069}, {543,13759}, {620,1270}, {2782,13763}, {5969,13766}, {6054,13758}, {6302,9113}, {6303,9112}, {6560,10722}, {8593,13757}, {8787,13642}

X(13760) = reflection of X(13773) in X(3069)
X(13760) = orthologic center of 2nd tri-squares triangle to these triangles: 6th anti-Brocard, 1st Brocard, 6th Brocard
X(13760) = {X(2), X(5477)}-harmonic conjugate of X(13640)

X(13761) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO ANTI-MCCAY

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

The reciprocal orthologic center of these triangles is X(385)

X(13761) lies on these lines: {2,98}, {99,13757}, {511,13768}, {512,13775}, {524,13760}, {543,3069}, {597,13653}, {690,13762}, {5590,9167}, {8787,13640}, {9830,13773}, {9877,13758}, {11159,13785}

X(13761) = reflection of X(13773) in X(13783)
X(13761) = orthologic center of these triangles: 2nd tri-squares to McCay

X(13762) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO 4th BROCARD

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

The reciprocal orthologic center of these triangles is X(2)

X(13762) lies on these lines: {2,9769}, {110,13757}, {542,3069}, {597,13654}, {690,13761}, {2854,13772}, {9759,13758}, {13774,13783}

X(13762) = reflection of X(13774) in X(13783)

X(13763) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO CIRCUMMEDIAL

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

The reciprocal orthologic center of these triangles is X(2)

X(13763) lies on these lines: {30,3069}, {32,11314}, {381,13758}, {492,11286}, {543,13764}, {615,7737}, {1160,1588}, {2548,11315}, {2782,13760}, {3053,11316}, {3849,13783}, {7584,12314}, {7745,11313}, {9605,11294}, {11159,13757}, {13766,13767}, {13769,13782}, {13770,13771}

X(13763) = orthologic center of these triangles: 2nd tri-squares to circummedial

X(13764) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO INNER-NAPOLEON

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

X(13764) lies on these lines: {2,9113}, {530,13760}, {531,3069}, {543,13763}, {9760,13758}, {12154,13757}

X(13764) = The reciprocal orthologic center of these triangles is X(9761)

X(13765) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO OUTER-NAPOLEON

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

he reciprocal orthologic center of these triangles is X(9763)

X(13765) lies on these lines: {2,9112}, {530,3069}, {531,13760}, {543,13763}, {9762,13758}, {12155,13757}

X(13766) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO 1st NEUBERG

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

The reciprocal orthologic center of these triangles is X(8667)

X(13766) lies on these lines: {2,13647}, {39,5590}, {492,7757}, {538,3069}, {5969,13760}, {9764,13758}, {11055,13757}, {13763,13767}

X(13767) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(9766)

X(13767) lies on these lines: {2,13648}, {83,492}, {754,3069}, {9765,13758}, {12156,13757}, {13763,13766}

X(13768) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO 3rd PARRY

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

The reciprocal orthologic center of these triangles is X(2)

X(13768) lies on these lines: {2,13170}, {511,13761}, {512,13773}, {597,13655}, {12157,13757}, {12434,13758}, {13775,13783}

X(13768) = reflection of X(13775) in X(13783)

X(13769) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO 1st SQUARES-CENTRAL

Barycentrics 6*S^2+(15*SA-7*SW)*S+3*(3*SA-2*SW)*(SB+SC) : :
X(13769) = 3*X(3069)-X(13831)

The reciprocal orthologic center of these triangles is X(13663)

X(13769) lies on these lines: {2,13662}, {524,13771}, {598,13669}, {1327,5066}, {3069,13831}, {13681,13758}, {13763,13782}

X(13770) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO INNER-VECTEN

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

The reciprocal orthologic center of these triangles is X(591)

X(13770) lies on these lines: {2,13650}, {5,6}, {524,13782}, {5420,9675}, {5491,11008}, {6300,11489}, {6301,11488}, {6395,12601}, {6396,12123}, {9767,13758}, {12158,13757}, {13763,13771}

X(13770) = {X(6), X(486)}-harmonic conjugate of X(13711)

X(13771) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO OUTER-VECTEN

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

The reciprocal orthologic center of these triangles is X(1991)

X(13771) lies on these lines: {2,13651}, {372,6278}, {485,615}, {488,7586}, {492,7926}, {524,13769}, {641,3068}, {3069,13834}, {6144,13650}, {6561,9733}, {9768,13758}, {12159,13757}, {13763,13770}

X(13771) = reflection of X(13834) in X(3069)

X(13772) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO 4th ANTI-BROCARD

Trilinears
(-6*b^2*c^2*(a^6-3*(b^2+c^2)*a^4-3*(b^4-5*b^2*c^2+c^4)*a^2+(b^2+c^2)*(b^4-4*b^2*c^2+c^4))*S-(b^2-c^2)^2*(a^2+b^2+c^2)*(2*a^6-(2*b^4-13*b^2*c^2+2*c^4)*a^2-b^2*c^2*(b^2+c^2)))*a : :

The reciprocal parallelogic center of these triangles is X(13168)

X(13772) lies on these lines: {2,9869}, {597,13641}, {2780,13774}, {2854,13762}, {13167,13757}, {13191,13758}

X(13773) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO ANTI-MCCAY

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

The reciprocal parallelogic center of these triangles is X(8597)

X(13773) lies on these lines: {2,99}, {6,13640}, {98,486}, {491,10754}, {511,13775}, {512,13768}, {542,3069}, {590,6034}, {597,13642}, {615,11646}, {690,13774}, {2794,7000}, {5477,7586}, {6036,12974}, {8997,9994}, {9830,13761}, {11161,13757}

X(13773) = reflection of X(i) in X(j) for these (i,j): (13760,3069), (13761,13783)
X(13773) = parallelogic center of these triangles: 2nd tri-squares to McCay
X(13773) = {X(2), X(115)}-harmonic conjugate of X(13653)

X(13774) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO 4th BROCARD

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

The reciprocal parallelogic center of these triangles is X(4)

X(13774) lies on these lines: {2,98}, {67,615}, {74,6813}, {491,895}, {597,13643}, {690,13773}, {2777,7000}, {2780,13772}, {5094,13785}, {5095,7586}, {5181,5591}, {7374,7687}, {9769,13758}, {13169,13757}, {13762,13783}

X(13774) = reflection of X(13762) in X(13783)
X(13774) = {X(2), X(125)}-harmonic conjugate of X(13654)

X(13775) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO 3rd PARRY

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

The reciprocal parallelogic center of these triangles is X(9147)

X(13775) lies on these lines: {2,12157}, {511,13773}, {512,13761}, {597,13649}, {13170,13757}, {13225,13758}, {13768,13783}

X(13775) = reflection of X(13768) in X(13783)

X(13776) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: 2nd TRI-SQUARES AND 4th ANTI-BROCARD

Trilinears
a*(6*((b^2+c^2)*a^4+26*b^2*c^2*a^2-((b^2-c^2)^2-4*b^2*c^2)*(b^2+c^2))*S+a^8-3*(b^2+c^2)*a^6+((b^2-c^2)^2-36*b^2*c^2)*a^4+(b^2+c^2)*(3*b^4-80*b^2*c^2+3*c^4)*a^2-2*(b^4-25*b^2*c^2+c^4)*(b^2-c^2)^2) : :

X(13776) lies on these lines: {}

X(13777) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: 2nd TRI-SQUARES AND ANTI-MCCAY

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

X(13777) lies on these lines: {2,5058}, {148,3069}, {597,13657}

X(13778) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: 2nd TRI-SQUARES AND 4th BROCARD

Barycentrics
-2*(10*a^8-6*(b^2+c^2)*a^6-(11*b^4-16*b^2*c^2+11*c^4)*a^4+2*(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^2+(b^4-c^4)^2)*S+5*a^10+6*(b^2+c^2)*a^8-(23*b^4+2*b^2*c^2+23*c^4)*a^6+(b^2+c^2)*(b^4+c^4)*a^4+2*(b^2-c^2)^2*(9*b^4+5*b^2*c^2+9*c^4)*a^2-7*(b^4-c^4)*(b^2-c^2)^3 : :

X(13778) lies on these lines: {}

X(13779) = CENTER OF SIMILITUDE OF THESE TRIANGLES: 2nd TRI-SQUARES AND CIRCUMSYMMEDIAL

Barycentrics (9*(-2*S+SW)*SA^2-3*(3*S^2-2*(9*R^2-SW)*S+SW^2)*SA-(15*S^2-3*(18*R^2+SW)*S+(36*R^2-5*SW)*SW)*S)*(SB+SC) : :

X(13779) lies on the line {6396,7464}

X(13780) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: 2nd TRI-SQUARES AND MCCAY

Barycentrics S^2*(33*S^2+18*SA^2-18*SA*SW+SW^2)+S*(3*S^2-SW^2)*(3*SA-5*SW)+2*(SA-SW)*SW^3 : :

X(13780) lies on these lines: {597,13660}, {3069,7615}

X(13781) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: 2nd TRI-SQUARES AND 3rd PARRY

Barycentrics
(SB+SC)*(-9*S^3*(S^2+SA^2)*(-S^2+54*R^4)+9*S^3*(-S^3+(22*R^2+SA)*S^2-12*R^2*SA*S+24*(3*R^2+SA)*R^2*SA)*SW-9*S^3*(3*S^2-(8*R^2+SA)*S+2*(9*R^2+2*SA)*SA)*SW^2-(3*S^2-SA*S+2*SA^2)*SW^5-2*S*(2*S^3+5*(-SA+3*R^2)*S^2+3*(6*R^2-SA)*SA*S-18*R^2*SA^2)*SW^3+(S+SA)*SW^6-S*(-5*S^2-10*SA*S+(18*R^2+5*SA)*SA)*SW^4) : :

X(13781) lies on these lines: {}

X(13782) = RADICAL CENTER OF THE 2nd TRI-SQUARES CIRCLES

Barycentrics 18*S^2-(4*SA+19*SB+19*SC)*S-3*(SA-2*SB-2*SC)*(SB+SC) : :
X(13782) = 3*X(3069)-2*X(13843) = 3*X(3069)-4*X(13844)

X(13782) lies on these lines: {2,13833}, {99,13757}, {486,542}, {524,13770}, {1328,3830}, {3069,13843}, {13758,13801}, {13763,13769}

X(13782) = reflection of X(i) in X(j) for these (i,j): (1328,13785), (13843,13844)
X(13782) = perspector of these triangles: 2nd tri-squares and 2nd tri-squares-central

X(13783) = X(3) OF THE 2nd TRI-SQUARES TRIANGLE

Barycentrics 3*SA+6*S-5*SW : :
X(13783) = 5*X(2)-X(1271) = 3*X(2)+X(13759) = 3*X(2)+2*X(13784) = X(1271)+5*X(3069) = 3*X(1271)+5*X(13759) = 3*X(1271)+10*X(13784) = 3*X(3069)-X(13759)

X(13783) lies on these lines: {2,6}, {372,5461}, {3363,13669}, {3849,13763}, {7817,11314}, {8184,13087}, {9830,13761}, {11158,13789}, {13762,13774}, {13768,13775}

X(13783) = midpoint of X(i) and X(j) for these {i,j}: {2,3069}, {13761,13773}, {13762,13774}, {13768,13775}
X(13783) = reflection of X(13759) in X(13784)
X(13783) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,597,13663), (2,13757,599), (2,13758,7610), (3069,13759,13784)

X(13784) = X(5) OF THE 2nd TRI-SQUARES TRIANGLE

Barycentrics 15*SA+12*S-13*SW : :
X(13784) = 13*X(2)-5*X(1271) = X(2)-5*X(3069) = 3*X(2)+5*X(13759) = 3*X(2)-5*X(13783) = X(1271)-13*X(3069) = 3*X(1271)+13*X(13759) = 3*X(1271)-13*X(13783)

X(13784) lies on these lines: {2,6}, {13763,13769}

X(13784) = midpoint of X(13759) and X(13783)
X(13784) = {X(3069), X(13759)}-harmonic conjugate of X(13783)

X(13785) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES-CENTRAL TO ABC

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

The reciprocal parallelogic center of these triangles is X(1328)

X(13785) lies on these lines: {2,6221}, {3,486}, {4,1132}, {5,1588}, {6,13}, {20,6450}, {30,3069}, {140,6449}, {262,6568}, {371,1656}, {372,382}, {376,6452}, {403,5410}, {485,3851}, {489,11316}, {546,1587}, {548,6497}, {549,6451}, {550,6456}, {590,5055}, {631,6455}, {632,6519}, {637,11314}, {1124,9654}, {1151,3526}, {1152,1657}, {1328,3830}, {1335,9669}, {1702,9956}, {2043,11543}, {2044,11542}, {3070,3843}, {3090,8981}, {3091,6427}, {3102,13108}, {3146,6448}, {3299,10895}, {3301,10896}, {3317,3523}, {3365,5339}, {3390,5340}, {3529,6522}, {3534,6396}, {3545,7585}, {3591,10299}, {3592,5079}, {3594,5076}, {3627,6460}, {3628,6447}, {3832,7581}, {3854,6498}, {3856,6499}, {5054,6200}, {5070,5418}, {5071,8972}, {5072,6419}, {5094,13774}, {5414,9668}, {6251,13749}, {6431,8960}, {6502,9655}, {6813,10845}, {9583,11230}, {9616,11231}, {9680,9691}, {9892,11165}, {10666,12429}, {10881,12173}, {11159,13761}

X(13785) = midpoint of X(1328) and X(13782)
X(13785) = reflection of X(6398) in X(3069)
X(13785) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,7584,3312), (5,1588,3311), (6,381,13665), (6,6565,381), (486,3071,3), (549,9541,6451), (3091,7582,7583), (3843,6418,3070), (3851,6417,485), (7582,7583,6427)

X(13786) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO 2nd TRI-SQUARES-CENTRAL

Barycentrics -10*S^3-3*(SB+SC)*((-7*S+2*SW)*SA-S^2) : :
X(13786) = 3*X(3)-X(13836) = 3*X(165)-X(13799) = 3*X(376)-X(13794) = 3*X(1328)-2*X(13836) = 2*X(11001)+X(13810) = 5*X(12123)-2*X(12306) = 2*X(13812)-3*X(13835)

The reciprocal orthologic center of these triangles is X(13785)

X(13786) lies on these lines: {2,13807}, {3,1328}, {4,13821}, {20,13798}, {30,6290}, {35,13837}, {36,13838}, {56,13819}, {165,13799}, {182,13792}, {376,13794}, {515,13808}, {517,13822}, {1327,9738}, {1350,13666}, {1593,13788}, {3098,13805}, {3534,12124}, {3576,13787}, {6284,13816}, {7354,13815}, {10310,13795}, {11001,11824}, {11248,13839}, {11249,13840}, {11414,13800}, {11825,13811}, {11826,13813}, {11827,13814}, {11828,13817}, {11829,13818}

X(13786) = midpoint of X(20) and X(13798)
X(13786) = reflection of X(i) in X(j) for these (i,j): (4,13821), (1328,3)
X(13786) = anticomplement of X(13807)

X(13787) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO 2nd TRI-SQUARES-CENTRAL

Barycentrics (13*a+b+c)*S^2+3*((a+b+c)*SA-(3*a+b+c)*SW)*S+9*(a+b+c)*SB*SC : :
X(13787) = 3*X(1)+X(13799) = 3*X(1)-X(13822) = 3*X(1328)-X(13799) = 3*X(1328)+X(13822) = 5*X(3616)-X(13798) = 3*X(5603)+X(13794) = 3*X(5886)-X(13812) = 3*X(10246)+X(13836) = 3*X(11831)-X(13809)

The reciprocal orthologic center of these triangles is X(13785)

X(13787) lies on these lines: {1,1328}, {2,13808}, {30,12268}, {515,13807}, {1125,13821}, {1386,13667}, {2646,13819}, {3295,13795}, {3576,13786}, {3616,13798}, {5603,13794}, {5886,13812}, {10246,13836}, {11363,13788}, {11364,13792}, {11365,13800}, {11368,13805}, {11370,13810}, {11371,13811}, {11373,13813}, {11374,13814}, {11375,13815}, {11376,13816}, {11377,13817}, {11378,13818}, {11831,13809}

X(13787) = midpoint of X(i) and X(j) for these {i,j}: {1,1328}, {13799,13822}
X(13787) = reflection of X(13821) in X(1125)
X(13787) = complement of X(13808)
X(13787) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,13799,13822), (1328,13822,13799)

X(13788) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO 2nd TRI-SQUARES-CENTRAL

Barycentrics (-(36*R^2+5*SA-15*SW)*S+3*SA^2-3*SW^2)*SB*SC : :
X(13788) = 2*X(6406)-5*X(12147)

The reciprocal orthologic center of these triangles is X(13785)

X(13788) lies on these lines: {4,13798}, {25,1328}, {30,6406}, {33,13819}, {235,13807}, {427,13821}, {1593,13786}, {1598,13836}, {1843,13668}, {3543,8946}, {5064,13835}, {5090,13808}, {7487,13794}, {7576,12148}, {7713,13799}, {11363,13787}, {11380,13792}, {11383,13795}, {11386,13805}, {11388,13810}, {11389,13811}, {11390,13813}, {11391,13814}, {11392,13815}, {11393,13816}, {11394,13817}, {11395,13818}, {11396,13822}, {11398,13837}, {11399,13838}, {11400,13839}, {11401,13840}, {11832,13809}

X(13789) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARTZT TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 6*S^2+(3*SA-5*SW)*S+(3*SA-SW)*(3*SA-2*SW) : :

The reciprocal orthologic center of these triangles is X(13783)

X(13789) lies on these lines: {2,1328}, {6,12159}, {99,13757}, {524,12158}, {597,2549}, {598,13637}, {3363,13663}, {11148,13759}, {11158,13783}

X(13789) = midpoint of X(6561) and X(13835)
X(13789) = {X(597), X(11159)}-harmonic conjugate of X(13669)

X(13790) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES-CENTRAL TO 1st ANTI-BROCARD

Barycentrics 6*S^4-(-3*SA+7*SW)*S^3+(3*SA-SW)*SW*S^2+(SA-SW)*SW*(3*SW*SA-3*S*SA-2*S*SW) : :

The reciprocal orthologic center of these triangles is X(13791)

X(13790) lies on these lines: {115,6560}, {542,3069}, {3564,6230}, {6055,7735}

X(13791) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-BROCARD TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 6*S^4-9*SW*S^3-(3*SA-5*SW)*SW*S^2-3*(3*SA-4*SW)*SA*SW*S+9*(SA-SW)*SA*SW^2 : :

The reciprocal orthologic center of these triangles is X(13790)

X(13791) lies on these lines: {2,1328}, {3,13806}, {316,13671}, {542,9867}, {4027,13793}

X(13791) = reflection of X(13797) in X(13835)

X(13792) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 10*S^4+(21*SA^2-33*SW*SA+22*SW^2)*S^2+3*(SB+SC)*(-SA*SW*(-2*S+3*SW)-S*(S^2+3*SW^2)) : :

10*S^4+(21*SA^2-33*SW*SA+22*SW^2)*S^2+3*(SB+SC)*(-SA*SW*(-2*S+3*SW)-S*(S^2+3*SW^2))

X(13792) lies on these lines: {30,12210}, {32,1328}, {83,13821}, {98,13807}, {182,13786}, {7787,13798}, {10788,13794}, {10789,13799}, {10790,13800}, {10791,13808}, {10792,13810}, {10793,13811}, {10794,13813}, {10795,13814}, {10796,13812}, {10797,13815}, {10798,13816}, {10799,13819}, {10800,13822}, {10801,13837}, {10802,13838}, {10803,13839}, {10804,13840}, {11364,13787}, {11380,13788}, {11490,13795}, {11839,13809}, {11840,13817}, {11841,13818}, {11842,13836}, {12212,13672}

X(13793) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-BROCARD TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 6*S^5-(-27*SA+23*SW)*S^4-3*(SB+SC)*(3*SA-5*SW)*S^3+(15*SA^2-12*SW*SA+SW^2)*SW*S^2-3*(SA^2+SW^2)*SW^2*S+9*(SB+SC)*SA*SW^3 : :

The reciprocal orthologic center of these triangles is X(13790)

X(13793) lies on these lines: {182,13804}, {542,12217}, {4027,13791}, {10131,13806}

X(13794) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 3*SA*(SB+SC)*(S+SW)-S^2*(3*SA+4*S) : :
X(13794) = 3*X(4)-4*X(13807) = 3*X(376)-2*X(13786) = 5*X(631)-4*X(13821) = 3*X(1328)-2*X(13807) = 3*X(3524)-2*X(13835) = 3*X(3545)+2*X(13811) = 3*X(5603)-4*X(13787) = 3*X(5657)-2*X(13808) = 8*X(6222)-5*X(12257) = 3*X(7967)-2*X(13822)

The reciprocal orthologic center of these triangles is X(13785)

X(13794) lies on these lines: {2,6222}, {3,13798}, {4,1328}, {24,13800}, {30,12256}, {376,13786}, {381,6776}, {388,13837}, {497,13838}, {515,13799}, {631,13821}, {3069,9862}, {3085,13815}, {3086,13816}, {3524,13835}, {3545,10784}, {4294,13819}, {5603,13787}, {5657,13808}, {5860,12251}, {7487,13788}, {7967,13822}, {10783,13810}, {10785,13813}, {10786,13814}, {10788,13792}, {10805,13839}, {10806,13840}, {11491,13795}, {11845,13809}, {11846,13817}, {11847,13818}

X(13794) = reflection of X(i) in X(j) for these (i,j): (4,1328), (13798,3)
X(13794) = anticomplement of X(13812)
X(13794) = {X(381), X(6776)}-harmonic conjugate of X(13674)

X(13795) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 2nd TRI-SQUARES-CENTRAL

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

The reciprocal orthologic center of these triangles is X(13785)

X(13795) lies on these lines: {30,12343}, {35,13799}, {55,1328}, {56,13822}, {100,13798}, {197,13800}, {1376,13813}, {3295,13787}, {5687,13808}, {10310,13786}, {11383,13788}, {11490,13792}, {11491,13794}, {11494,13805}, {11496,13807}, {11497,13810}, {11498,13811}, {11499,13812}, {11500,13814}, {11501,13815}, {11502,13816}, {11503,13817}, {11504,13818}, {11507,13837}, {11508,13838}, {11509,13839}, {11510,13840}, {11848,13809}, {11849,13836}, {12329,13675}

X(13795) = reflection of X(13813) in X(13821)

X(13796) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES-CENTRAL TO ANTI-MCCAY

Barycentrics -18*(b^2-c^2)^2*S+10*a^6-12*(b^2+c^2)*a^4+3*(5*b^4-4*b^2*c^2+5*c^4)*a^2+(b^2+c^2)*(b^4-4*b^2*c^2+c^4) : :

The reciprocal orthologic center of these triangles is X(13797)

X(13796) lies on these lines: {115,1992}, {543,3069}, {671,13760}

X(13796) = midpoint of X(671) and X(13760)

X(13797) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MCCAY TO 2nd TRI-SQUARES-CENTRAL

Barycentrics (9*SA^2-9*SW*SA+SW^2)*SW+3*(9*SA-SW)*(SB+SC)*S-3*(3*SA-2*SW)*S^2-12*S^3 : :

The reciprocal orthologic center of these triangles is X(13796)

X(13797) lies on these lines: {2,1328}, {543,9891}, {6781,13677}

X(13797) = reflection of X(13791) in X(13835)

X(13798) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 5*S^2-3*SA*S-9*SB*SC : :
X(13798) = 3*X(2)-4*X(13821) = 5*X(487)-2*X(12323) = X(1328)-3*X(13835) = 5*X(3091)-4*X(13807) = 5*X(3616)-4*X(13787) = 2*X(13821)-3*X(13835)

The reciprocal orthologic center of these triangles is X(13785)

X(13798) lies on these lines: {2,1328}, {3,13794}, {4,13788}, {5,13836}, {8,13808}, {10,13799}, {20,13786}, {22,13800}, {30,487}, {69,3534}, {100,13795}, {145,13822}, {376,488}, {388,13815}, {497,13816}, {549,12322}, {637,10304}, {1270,13811}, {1271,13810}, {2896,13805}, {3085,13837}, {3086,13838}, {3091,13807}, {3434,13813}, {3436,13814}, {3616,13787}, {4240,13809}, {6462,13817}, {6463,13818}, {7787,13792}, {10528,13839}, {10529,13840}

X(13798) = reflection of X(i) in X(j) for these (i,j): (2,13835), (4,13812), (8,13808), (20,13786), (145,13822), (1328,13821), (4240,13809), (13794,3), (13799,10), (13836,5)
X(13798) = anticomplement of X(1328)
X(13798) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (69,3534,13678), (1328,13821,2), (1328,13835,13821), (13816,13819,497)

X(13799) = ORTHOLOGIC CENTER OF THESE TRIANGLES: AQUILA TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 2*(2*a+5*b+5*c)*S^2+3*((a+b+c)*SA-(b+c)*SW)*S-9*(a+b+c)*(SB+SC)*SA : :
X(13799) = 3*X(1)-4*X(13787) = 3*X(1)-2*X(13822) = 3*X(165)-2*X(13786) = 3*X(1328)-2*X(13787) = 3*X(1328)-X(13822) = 5*X(1698)-4*X(13821) = 3*X(1699)-4*X(13807) = 3*X(3679)-2*X(13808) = 3*X(5587)-2*X(13812) = 3*X(11852)-2*X(13809)

The reciprocal orthologic center of these triangles is X(13785)

X(13799) lies on these lines: {1,1328}, {10,13798}, {30,9906}, {35,13795}, {165,13786}, {515,13794}, {517,13836}, {1697,13819}, {1698,13821}, {1699,13807}, {3099,13805}, {3679,13808}, {3751,13679}, {5587,13812}, {5588,13811}, {5589,13810}, {7713,13788}, {8185,13800}, {8188,13817}, {8189,13818}, {9578,13815}, {9581,13816}, {10789,13792}, {10826,13813}, {10827,13814}, {11852,13809}

X(13799) = reflection of X(i) in X(j) for these (i,j): (1,1328), (13798,10), (13822,13787)
X(13799) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1328,13822,13787), (13787,13822,1)

X(13800) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARA TO 2nd TRI-SQUARES-CENTRAL

Barycentrics (3*SA*SW*(SB+SC)+SA*(SA-6*SW+18*R^2)*S+(6*R^2-3*SW)*S^2+S^3)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(13785)

X(13800) lies on these lines: {3,13812}, {22,13798}, {24,13794}, {25,1328}, {30,9921}, {159,13680}, {197,13795}, {1598,13807}, {5594,13811}, {5595,13810}, {7517,13836}, {8185,13799}, {8192,13822}, {8193,13808}, {8194,13817}, {8195,13818}, {10037,13837}, {10046,13838}, {10790,13792}, {10828,13805}, {10829,13813}, {10830,13814}, {10831,13815}, {10832,13816}, {10833,13819}, {10834,13839}, {10835,13840}, {11365,13787}, {11414,13786}, {11853,13809}

X(13801) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARTZT TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 12*S^3-12*S^2*SW-(3*SA-SW)*SW*S+9*SA*SW*(SB+SC) : :
X(13801) = X(1328)+2*X(13830)

The reciprocal orthologic center of these triangles is X(13783)

X(13801) lies on these lines: {2,1328}, {30,9758}, {381,7618}, {524,9767}, {6054,6813}, {7610,13812}, {9768,11184}, {13638,13833}, {13758,13782}

X(13801) = {X(381), X(9771)}-harmonic conjugate of X(13681)

X(13802) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO 2nd TRI-SQUARES-CENTRAL

Barycentrics ((2*a-10*b-10*c)*S^2-3*((a+b+c)*SA-(-a+b+c)*SW)*S+9*(a+b+c)*(SB+SC)*SA)*K-12*(b*c+SA)*(SB+SC)*(-2*S+SW)*S : : , where K=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(13785)

X(13802) lies on these lines: {}

X(13803) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO 2nd TRI-SQUARES-CENTRAL

Barycentrics ((2*a-10*b-10*c)*S^2-3*((a+b+c)*SA-(-a+b+c)*SW)*S+9*(a+b+c)*(SB+SC)*SA)*K+12*(b*c+SA)*(SB+SC)*(-2*S+SW)*S : : , where K=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(13785)

X(13803) lies on these lines: {}

X(13804) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 6*S^3+(SW+3*SA)*S^2+3*(-4*SW*SA-SW^2+3*SA^2)*S+9*(SB+SC)*SA*SW : :

The reciprocal orthologic center of these triangles is X(13790)

X(13804) lies on these lines: {2,1328}, {182,13793}, {384,13806}, {542,6229}, {7697,13812}, {7761,13684}, {10000,13805}

X(13805) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO 2nd TRI-SQUARES-CENTRAL

Barycentrics S^2*(10*S^2+3*(7*SA+3*SW)*(SA-2*SW))+3*(SB+SC)*(-S^3+S*SW*(2*SA+SW)+9*SW^2*SA) : :

The reciprocal orthologic center of these triangles is X(13785)

X(13805) lies on these lines: {30,9986}, {32,1328}, {2896,13798}, {3069,9862}, {3094,13685}, {3096,13821}, {3098,13786}, {3099,13799}, {7865,13835}, {9301,13836}, {9857,13808}, {9993,13807}, {9994,13810}, {9995,13811}, {9996,13812}, {9997,13822}, {10000,13804}, {10038,13837}, {10047,13838}, {10828,13800}, {10871,13813}, {10872,13814}, {10873,13815}, {10874,13816}, {10875,13817}, {10876,13818}, {10877,13819}, {10878,13839}, {10879,13840}, {11368,13787}, {11386,13788}, {11494,13795}, {11885,13809}

X(13806) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th BROCARD TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 6*S^4-9*SW*S^3+(12*SA^2-15*SW*SA+5*SW^2)*S^2-3*(5*SA-6*SW)*SA*SW*S-9*(SB+SC)*SA*SW^2 : :

The reciprocal orthologic center of these triangles is X(13790)

X(13806) lies on these lines: {3,13791}, {384,13804}, {542,9991}, {2896,13798}, {7802,13686}, {10131,13793}

X(13807) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EULER TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 22*S^3+3*(SA-3*SW)*S^2+3*SA*(SB+SC)*(-7*S+2*SW) : :
X(13807) = 3*X(4)+X(13794) = 3*X(381)-X(13812) = 3*X(381)+X(13836) = 3*X(1328)-X(13794) = 3*X(1699)+X(13799) = 5*X(3091)-X(13798) = 3*X(3545)-X(13835) = 3*X(5587)-X(13808) = 3*X(5603)-X(13822) = 3*X(11897)-X(13809)

The reciprocal orthologic center of these triangles is X(13785)

X(13807) lies on these lines: {2,13786}, {4,1328}, {5,13821}, {12,13819}, {30,6251}, {98,13792}, {235,13788}, {381,13812}, {515,13787}, {1478,13838}, {1479,13837}, {1598,13800}, {1699,13799}, {3091,13798}, {3545,13835}, {3845,6250}, {5480,13687}, {5587,13808}, {5603,13822}, {6201,13811}, {6202,13810}, {8212,13817}, {8213,13818}, {9993,13805}, {10531,13839}, {10532,13840}, {10893,13813}, {10894,13814}, {10895,13815}, {10896,13816}, {11496,13795}, {11897,13809}

X(13807) = midpoint of X(i) and X(j) for these {i,j}: {4,1328}, {13812,13836}
X(13807) = reflection of X(13821) in X(5)
X(13807) = complement of X(13786)
X(13807) = {X(381), X(13836)}-harmonic conjugate of X(13812)

X(13808) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 2*(5*a-b-c)*S^2+3*((a+b+c)*SA+SW*(-a+b+c))*S-9*(a+b+c)*(SB+SC)*SA : :
X(13808) = 3*X(3679)-X(13799) = 3*X(5587)-2*X(13807) = 3*X(5657)-X(13794) = 3*X(5790)-X(13836) = X(13822)-3*X(13835)

The reciprocal orthologic center of these triangles is X(13785)

X(13808) lies on these lines: {1,13821}, {2,13787}, {8,13798}, {10,1328}, {30,12787}, {65,13815}, {72,13814}, {515,13786}, {517,13812}, {519,13822}, {1737,13838}, {1837,13819}, {3057,13816}, {3416,13688}, {3654,12788}, {3679,13799}, {5090,13788}, {5587,13807}, {5657,13794}, {5687,13795}, {5688,13811}, {5689,13810}, {5790,13836}, {8193,13800}, {8214,13817}, {8215,13818}, {9857,13805}, {10039,13837}, {10791,13792}, {10914,13813}, {10915,13839}, {10916,13840}, {11900,13809}

X(13808) = midpoint of X(8) and X(13798)
X(13808) = reflection of X(i) in X(j) for these (i,j): (1,13821), (1328,10)
X(13808) = anticomplement of X(13787)

X(13809) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO 2nd TRI-SQUARES-CENTRAL

Barycentrics (-S^3-3*(6*R^2+SA-2*SW)*S^2+3*(216*R^4+(24*SA-98*SW)*R^2-2*SA^2-4*SW*SA+11*SW^2)*S-6*(3*SA+2*SW)*(3*SA-SW)*R^2+3*(5*SA^2+SW*SA-SW^2)*SW)*(S^2-3*SB*SC) : :
X(13809) = 3*X(11831)-2*X(13787) = 3*X(11845)-X(13794) = 3*X(11852)-X(13799) = 3*X(11897)-2*X(13807) = 3*X(11911)-X(13836)

The reciprocal orthologic center of these triangles is X(13785)

X(13809) lies on these lines: {30,6290}, {402,1328}, {1650,13821}, {4240,13798}, {11831,13787}, {11832,13788}, {11839,13792}, {11845,13794}, {11848,13795}, {11852,13799}, {11853,13800}, {11885,13805}, {11897,13807}, {11900,13808}, {11901,13810}, {11902,13811}, {11903,13813}, {11904,13814}, {11905,13815}, {11906,13816}, {11907,13817}, {11908,13818}, {11909,13819}, {11910,13822}, {11911,13836}, {11912,13837}, {11913,13838}, {11914,13839}, {11915,13840}, {12583,13689}

X(13809) = midpoint of X(4240) and X(13798)
X(13809) = reflection of X(i) in X(j) for these (i,j): (1328,402), (1650,13821)

X(13810) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO 2nd TRI-SQUARES-CENTRAL

Barycentrics S^2*(-10*S+9*SA+11*SW)-9*(SB+SC)*SA*(-S+2*SW) : :
X(13810) = 3*X(1328)-4*X(3845) = 2*X(3534)-3*X(13835) = 5*X(3830)-3*X(13836) = 8*X(3845)-3*X(13811) = 2*X(8703)-3*X(13812) = 2*X(11001)-3*X(13786)

The reciprocal orthologic center of these triangles is X(13785)

X(13810) lies on these lines: {2,5871}, {6,1327}, {30,6281}, {1271,13798}, {3534,13835}, {3830,6279}, {5589,13799}, {5591,13821}, {5595,13800}, {5605,13822}, {5689,13808}, {6202,13807}, {6215,8703}, {8216,13817}, {8217,13818}, {9994,13805}, {10040,13837}, {10048,13838}, {10783,13794}, {10792,13792}, {10919,13813}, {10921,13814}, {10923,13815}, {10925,13816}, {10927,13819}, {10929,13839}, {10931,13840}, {11001,11824}, {11370,13787}, {11388,13788}, {11497,13795}, {11901,13809}

X(13810) = reflection of X(13811) in X(1328)

X(13811) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 5*S^2*(-2*S-3*SA-SW)-9*(SB+SC)*SA*(-S-2*SW) : :
X(13811) = 6*X(3)-5*X(13835) = 5*X(1328)-4*X(3845) = 3*X(3545)-5*X(13794) = 8*X(3845)-5*X(13810) = 6*X(11539)-5*X(13812)

The reciprocal orthologic center of these triangles is X(13785)

X(13811) lies on these lines: {3,6278}, {6,1327}, {30,6280}, {547,10515}, {1270,13798}, {3543,5870}, {3545,10784}, {5588,13799}, {5590,13821}, {5594,13800}, {5604,13822}, {5688,13808}, {5860,11001}, {6201,13807}, {6214,11539}, {8218,13817}, {8219,13818}, {9995,13805}, {10041,13837}, {10049,13838}, {10793,13792}, {10920,13813}, {10922,13814}, {10924,13815}, {10926,13816}, {10928,13819}, {10930,13839}, {10932,13840}, {11371,13787}, {11389,13788}, {11498,13795}, {11825,13786}, {11902,13809}, {11917,13836}

X(13811) = reflection of X(13810) in X(1328)

X(13812) = ORTHOLOGIC CENTER OF THESE TRIANGLES: JOHNSON TO 2nd TRI-SQUARES-CENTRAL

Barycentrics S^2*(-2*S+3*SA+3*SW)+3*SA*(SB+SC)*(-S-SW) : :
X(13812) = 3*X(381)-2*X(13807) = 3*X(381)-X(13836) = X(1991)+2*X(13830) = 3*X(5587)-X(13799) = 3*X(5886)-2*X(13787) = 2*X(8703)+X(13810) = 6*X(11539)-X(13811) = X(13786)-3*X(13835)

The reciprocal orthologic center of these triangles is X(13785)

X(13812) lies on these lines: {1,13815}, {2,6222}, {3,13800}, {4,13788}, {5,1328}, {11,13838}, {12,13837}, {30,6290}, {355,13813}, {381,13807}, {517,13808}, {549,1352}, {952,13822}, {1479,13819}, {1991,3095}, {5587,13799}, {5613,13826}, {5617,13824}, {5886,13787}, {6214,11539}, {6215,8703}, {6230,8724}, {6287,13828}, {7610,13801}, {7697,13804}, {8220,13817}, {8221,13818}, {9996,13805}, {10796,13792}, {10942,13839}, {10943,13840}, {11499,13795}

X(13812) = midpoint of X(i) and X(j) for these {i,j}: {4,13798}, {13813,13814}
X(13812) = reflection of X(i) in X(j) for these (i,j): (3,13821), (1328,5), (13836,13807)
X(13812) = complement of X(13794)
X(13812) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381,13836,13807), (549,1352,13692), (13815,13816,1)

X(13813) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO 2nd TRI-SQUARES-CENTRAL

Barycentrics
-6*(a^5-(b+c)*a^4+2*a^3*b*c+(b^2+c^2)*(b-c)^2*a-(b^4-c^4)*(b-c))*S+4*a^7-4*(b+c)*a^6-(9*b^2-28*b*c+9*c^2)*a^5+(b+c)*(9*b^2-20*b*c+9*c^2)*a^4+2*(3*b^4+3*c^4-2*b*c*(5*b^2-8*b*c+5*c^2))*a^3-2*(b^2-c^2)*(b-c)*(3*b^2-5*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2+8*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

The reciprocal orthologic center of these triangles is X(13785)

X(13813) lies on these lines: {11,1328}, {12,13839}, {30,12928}, {355,13812}, {1376,13795}, {3434,13798}, {10523,13837}, {10785,13794}, {10794,13792}, {10826,13799}, {10829,13800}, {10871,13805}, {10893,13807}, {10914,13808}, {10919,13810}, {10920,13811}, {10944,13815}, {10945,13817}, {10946,13818}, {10947,13819}, {10948,13838}, {10949,13840}, {11373,13787}, {11390,13788}, {11826,13786}, {11903,13809}, {11928,13836}, {12586,13693}

X(13813) = reflection of X(i) in X(j) for these (i,j): (13795,13821), (13814,13812)

X(13814) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO 2nd TRI-SQUARES-CENTRAL

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

The reciprocal orthologic center of these triangles is X(13785)

X(13814) lies on these lines: {11,13840}, {12,1328}, {30,12938}, {72,13808}, {355,13812}, {958,13821}, {3436,13798}, {10523,13838}, {10786,13794}, {10795,13792}, {10827,13799}, {10830,13800}, {10872,13805}, {10894,13807}, {10921,13810}, {10922,13811}, {10950,13816}, {10951,13817}, {10952,13818}, {10953,13819}, {10954,13837}, {10955,13839}, {11374,13787}, {11391,13788}, {11500,13795}, {11827,13786}, {11904,13809}, {11929,13836}, {12587,13694}

X(13814) = reflection of X(13813) in X(13812)

X(13815) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO 2nd TRI-SQUARES-CENTRAL

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

The reciprocal orthologic center of these triangles is X(13785)

X(13815) lies on these lines: {1,13812}, {4,13819}, {5,13838}, {12,1328}, {65,13808}, {388,13798}, {495,13837}, {3085,13794}, {5434,13835}, {7354,13786}, {9578,13799}, {9654,13836}, {10056,12949}, {10797,13792}, {10831,13800}, {10873,13805}, {10895,13807}, {10923,13810}, {10924,13811}, {10944,13813}, {10956,13839}, {10957,13840}, {11375,13787}, {11392,13788}, {11501,13795}, {11905,13809}, {11930,13817}, {11931,13818}, {12588,13695}

X(13815) = reflection of X(13837) in X(495)
X(13815) = {X(1), X(13812)}-harmonic conjugate of X(13816)

X(13816) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO 2nd TRI-SQUARES-CENTRAL

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

The reciprocal orthologic center of these triangles is X(13785)

X(13816) lies on these lines: {1,13812}, {5,13837}, {11,1328}, {30,12958}, {55,13821}, {496,13838}, {497,13798}, {3057,13808}, {3058,13835}, {3086,13794}, {6284,13786}, {9581,13799}, {9669,13836}, {10072,12959}, {10798,13792}, {10832,13800}, {10874,13805}, {10896,13807}, {10925,13810}, {10926,13811}, {10950,13814}, {10958,13839}, {10959,13840}, {11376,13787}, {11393,13788}, {11502,13795}, {11906,13809}, {11932,13817}, {11933,13818}, {12589,13696}

X(13816) = reflection of X(13838) in X(496)
X(13816) = {X(1), X(13812)}-harmonic conjugate of X(13815)

X(13817) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 4*(3*SA+8*R^2-3*SW)*S^3+2*(6*SA^2+(12*R^2-3*SW)*SA+2*SW^2)*S^2+3*(SB+SC)*(SW^2*S+(-24*S*R^2+2*S*SW-3*SW^2)*SA) : :

The reciprocal orthologic center of these triangles is X(13785)

X(13817) lies on these lines: {30,13002}, {493,1328}, {6461,13818}, {6462,13798}, {8188,13799}, {8194,13800}, {8210,13822}, {8212,13807}, {8214,13808}, {8216,13810}, {8218,13811}, {8220,13812}, {8222,13821}, {10875,13805}, {10945,13813}, {10951,13814}, {11377,13787}, {11394,13788}, {11503,13795}, {11828,13786}, {11840,13792}, {11846,13794}, {11907,13809}, {11930,13815}, {11932,13816}, {11947,13819}, {11949,13836}, {11951,13837}, {11953,13838}, {11955,13839}, {11957,13840}, {12590,13697}

X(13818) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 4*(3*SA+8*R^2-3*SW)*S^3-2*(6*SA^2+(-12*R^2+3*SW)*SA-4*SW^2)*S^2+3*(SB+SC)*(-SW^2*S+(-24*S*R^2-2*S*SW+3*SW^2)*SA) : :

The reciprocal orthologic center of these triangles is X(13785)

X(13818) lies on these lines: {30,13003}, {494,1328}, {6461,13817}, {6463,13798}, {8189,13799}, {8195,13800}, {8211,13822}, {8213,13807}, {8215,13808}, {8217,13810}, {8219,13811}, {8221,13812}, {8223,13821}, {10876,13805}, {10946,13813}, {10952,13814}, {11378,13787}, {11395,13788}, {11504,13795}, {11829,13786}, {11841,13792}, {11847,13794}, {11908,13809}, {11931,13815}, {11933,13816}, {11948,13819}, {11950,13836}, {11952,13837}, {11954,13838}, {11956,13839}, {11958,13840}, {12591,13698}

X(13819) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO 2nd TRI-SQUARES-CENTRAL

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

The reciprocal orthologic center of these triangles is X(13785)

X(13819) lies on these lines: {3,13838}, {4,13815}, {11,13821}, {12,13807}, {33,13788}, {55,1328}, {56,13786}, {497,13798}, {1479,13812}, {1697,13799}, {1837,13808}, {2098,13822}, {2646,13787}, {3056,13699}, {3295,13836}, {4294,13794}, {10799,13792}, {10833,13800}, {10877,13805}, {10927,13810}, {10928,13811}, {10947,13813}, {10953,13814}, {10965,13839}, {10966,13840}, {11238,13835}, {11909,13809}, {11947,13817}, {11948,13818}

X(13819) = {X(3295), X(13836)}-harmonic conjugate of X(13837)

X(13820) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MCCAY TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 6*S^3-3*(SW+3*SA)*S^2-(27*SA^2-30*SW*SA+SW^2)*S-9*(SB+SC)*SA*SW : :
X(13820) = 2*X(13821)+X(13828) = 4*X(13821)-X(13830) = 2*X(13828)+X(13830)

The reciprocal orthologic center of these triangles is X(13796)

X(13820) lies on these lines: {2,1328}, {543,13087}, {5569,13088}

X(13821) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MEDIAL TO 2nd TRI-SQUARES-CENTRAL

Barycentrics (-2*S+3*SB+3*SC)*(-S+3*SA) : :
X(13821) = 3*X(2)+X(13798) = 5*X(631)-X(13794) = X(1328)+3*X(13835) = 5*X(1656)-X(13836) = 5*X(1698)-X(13799) = 3*X(11147)-X(13712) = X(13798)-3*X(13835) = 3*X(13820)-X(13828) = 3*X(13820)+X(13830)

The reciprocal orthologic center of these triangles is X(13785)

X(13821) lies on these lines: {1,13808}, {2,1328}, {3,13800}, {4,13786}, {5,13807}, {8,13822}, {11,13819}, {30,642}, {55,13816}, {83,13792}, {141,12100}, {376,640}, {427,13788}, {498,13837}, {499,13838}, {549,641}, {631,13794}, {639,5054}, {958,13814}, {1125,13787}, {1376,13795}, {1650,13809}, {1656,13836}, {1698,13799}, {1991,11165}, {3096,13805}, {5552,13839}, {5590,13811}, {5591,13810}, {8222,13817}, {8223,13818}, {10527,13840}, {11147,13712}

X(13821) = midpoint of X(i) and X(j) for these {i,j}: {1,13808}, {2,13835}, {3,13812}, {4,13786}, {8,13822}, {1328,13798}, {1650,13809}, {13795,13813}, {13824,13826}, {13828,13830}
X(13821) = reflection of X(i) in X(j) for these (i,j): (13787,1125), (13807,5)
X(13821) = complement of X(1328)
X(13821) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,13798,1328), (141,12100,13701), (1328,13835,13798), (13820,13830,13828)

X(13822) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR TO 2nd TRI-SQUARES-CENTRAL

Barycentrics -6*(a^2-(b+c)*a+2*b^2+2*c^2)*a*S-(a+b+c)*(16*a^4-12*(b+c)*a^3-(11*b^2-24*b*c+11*c^2)*a^2+12*(b^2-c^2)*(b-c)*a-5*(b^2-c^2)^2) : :
X(13822) = 3*X(1)-2*X(13787) = 3*X(1)-X(13799) = 3*X(1328)-4*X(13787) = 3*X(1328)-2*X(13799) = 3*X(5603)-2*X(13807) = 3*X(7967)-X(13794) = 3*X(10247)-X(13836)

The reciprocal orthologic center of these triangles is X(13785)

X(13822) lies on these lines: {1,1328}, {8,13821}, {30,7980}, {56,13795}, {145,13798}, {517,13786}, {519,13808}, {952,13812}, {2098,13819}, {3242,13702}, {5603,13807}, {5604,13811}, {5605,13810}, {7967,13794}, {8192,13800}, {8210,13817}, {8211,13818}, {9997,13805}, {10247,13836}, {10800,13792}, {10944,13813}, {10950,13814}, {11396,13788}, {11910,13809}

X(13822) = midpoint of X(145) and X(13798)
X(13822) = reflection of X(i) in X(j) for these (i,j): (8,13821), (1328,1), (13799,13787)
X(13822) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,13799,13787), (13787,13799,1328), (13839,13840,1328)

X(13823) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES-CENTRAL TO INNER-NAPOLEON

Barycentrics -(2*S^2-3*(SB+SC)*S-2*(SB-SC)^2)*sqrt(3)+9*S^2+2*S*SW-6*SW^2+6*SA^2+3*SB*SC : :

The reciprocal orthologic center of these triangles is X(13824)

X(13823) lies on these lines: {115,13825}, {395,6307}, {531,3069}, {5460,13703}

X(13824) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-NAPOLEON TO 2nd TRI-SQUARES-CENTRAL

Barycentrics (-2*S+3*SB+3*SC)*(-S+3*SA)*sqrt(3)+S*(2*SA-SB-SC)-3*S^2+9*SB*SC : :

The reciprocal orthologic center of these triangles is X(13823)

X(13824) lies on these lines: {2,1328}, {531,6301}, {3643,13704}, {6305,13084}, {6306,9885}

X(13824) = reflection of X(13826) in X(13821)
X(13824) = anticomplement of X(33489)

X(13825) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES-CENTRAL TO OUTER-NAPOLEON

Barycentrics (2*S^2-3*(SB+SC)*S-2*(SB-SC)^2)*sqrt(3)+9*S^2+2*S*SW-6*SW^2+6*SA^2+3*SB*SC : :

The reciprocal orthologic center of these triangles is X(13826)

X(13825) lies on these lines: {115,13823}, {396,6306}, {530,3069}, {5459,13705}

X(13826) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-NAPOLEON TO 2nd TRI-SQUARES-CENTRAL

Barycentrics (-2*S+3*SB+3*SC)*(-S+3*SA)*sqrt(3)-S*(2*SA-SB-SC)+3*S^2-9*SB*SC : :

The reciprocal orthologic center of these triangles is X(13825)

X(13826) lies on these lines: {2,1328}, {530,6300}, {3642,13706}, {5613,13812}, {6304,13083}, {6307,9886}

X(13826) = reflection of X(13824) in X(13821)
X(13826) = anticomplement of X(33488)

X(13827) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES-CENTRAL TO 1st NEUBERG

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

The reciprocal orthologic center of these triangles is X(13828)

X(13827) lies on these lines: {69,5475}, {538,3069}, {3934,5591}

X(13828) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st NEUBERG TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 2*S^3-(9*SA^2-12*SW*SA+SW^2)*S+3*(SB+SC)*(S^2-3*SW*SA) : :
X(13828) = X(6316)+2*X(7692) = 3*X(13820)-2*X(13821) = 3*X(13820)-X(13830)

The reciprocal orthologic center of these triangles is X(13827)

X(13828) lies on these lines: {2,1328}, {30,13087}, {538,6316}, {1991,9892}, {3098,13708}, {6287,13812}, {6399,12306}

X(13828) = reflection of X(13830) in X(13821)

X(13829) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES-CENTRAL TO 2nd NEUBERG

Barycentrics 4*S^3+2*S^2*SA+(3*SA^2-6*SW*SA+11*SW^2)*S+(3*SA^2+4*SW*SA-9*SW^2)*SW : :

The reciprocal orthologic center of these triangles is X(13830)

X(13829) lies on these lines: {754,3069}, {3618,5355}, {6274,6704}

X(13830) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd NEUBERG TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 2*S^3-3*(SA+3*SW)*S^2-(-6*SW*SA-SW^2+9*SA^2)*S+9*(SB+SC)*SA*SW : :
X(13830) = X(1328)-3*X(13801) = X(1991)-3*X(13812) = 3*X(13820)-4*X(13821) = 3*X(13820)-2*X(13828)

The reciprocal orthologic center of these triangles is X(13829)

X(13830) lies on these lines: {2,1328}, {754,6315}, {1991,3095}, {3818,13710}

X(13830) = reflection of X(13828) in X(13821)

X(13831) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES-CENTRAL TO 1st TRI-SQUARES-CENTRAL

Barycentrics (5*a^2-b^2-c^2)*S+6*(b^2-c^2)^2 : :
X(13831) = 3*X(3069)-2*X(13769)

The reciprocal orthologic center of these triangles is X(13832)

X(13831) lies on these lines: {2,9600}, {115,1992}, {1327,6436}, {3069,13769}

X(13831) = {X(115), X(1992)}-harmonic conjugate of X(13832)

X(13832) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES-CENTRAL TO 2nd TRI-SQUARES-CENTRAL

Barycentrics -(5*a^2-b^2-c^2)*S+6*(b^2-c^2)^2 : :
X(13832) = 3*X(3068)-2*X(13833)

The reciprocal orthologic center of these triangles is X(13831)

X(13832) lies on these lines: {115,1992}, {1328,6435}, {3068,13833}

X(13832) = {X(115), X(1992)}-harmonic conjugate of X(13831)

X(13833) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 6*S^2-(15*SA-7*SW)*S+3*(3*SA-2*SW)*(SB+SC) : :
X(13833) = 3*X(3068)-X(13832)

The reciprocal orthologic center of these triangles is X(13783)

X(13833) lies on these lines: {2,13782}, {524,13650}, {598,13637}, {1328,5066}, {3068,13832}, {13638,13801}, {13644,13662}

X(13834) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES-CENTRAL TO OUTER-VECTEN

Barycentrics 6*S^2+(SB+SC)*(-S-5*SA-2*SB-2*SC) : :

The reciprocal orthologic center of these triangles is X(13835)

X(13834) lies on these lines: {5,6}, {115,6560}, {230,6561}, {637,8972}, {641,7376}, {3054,9600}, {3069,13771}, {3619,5490}, {5254,5420}, {5286,10577}, {5418,7746}, {6304,11488}, {6305,11489}, {6395,12602}, {6396,12124}, {6565,7735}

X(13834) = reflection of X(13771) in X(3069)
X(13834) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,485,13651), (485,486,6278)

X(13835) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-VECTEN TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 2*S^2+(4*SA+SB+SC)*S-9*(SB+SC)*SA : :
X(13835) = 6*X(3)-X(13811) = X(1328)+2*X(13798) = X(1328)-4*X(13821) = 3*X(3524)-X(13794) = 2*X(3534)+X(13810) = 3*X(3545)-2*X(13807) = 3*X(5055)-X(13836) = X(6280)-4*X(12975) = X(13786)+2*X(13812) = X(13798)+2*X(13821)

The reciprocal orthologic center of these triangles is X(13834)

X(13835) lies on these lines: {2,1328}, {3,6278}, {30,6290}, {489,5420}, {519,13808}, {599,8703}, {1991,6560}, {3058,13816}, {3103,7757}, {3524,13794}, {3534,13810}, {3545,13807}, {3582,13838}, {3584,13837}, {5055,13836}, {5064,13788}, {5434,13815}, {5860,6396}, {5861,9741}, {6280,12975}, {7389,9680}, {7865,13805}, {11147,13701}, {11238,13819}

X(13835) = midpoint of X(i) and X(j) for these {i,j}: {2,13798}, {13791,13797}
X(13835) = reflection of X(i) in X(j) for these (i,j): (2,13821), (1328,2), (6561,13789)
X(13835) = complement of X(33457)
X(13835) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (599,8703,13712), (13798,13821,1328), (13804,13828,2)

X(13836) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 20*S^3-3*(SB+SC)*(2*S^2+8*S*SA-SW*SA) : :
X(13836) = 3*X(3)-2*X(13786) = 3*X(381)-4*X(13807) = 3*X(381)-2*X(13812) = 3*X(1328)-X(13786) = 5*X(1656)-4*X(13821) = 5*X(3830)-2*X(13810) = 3*X(5055)-2*X(13835) = 3*X(5790)-2*X(13808) = 3*X(10246)-4*X(13787) = 2*X(12314)-5*X(12601)

The reciprocal orthologic center of these triangles is X(13785)

X(13836) lies on these lines: {3,1328}, {5,13798}, {30,12256}, {381,13807}, {517,13799}, {999,13838}, {1351,3543}, {1598,13788}, {1656,13821}, {3295,13819}, {3830,6279}, {3845,12313}, {5055,13835}, {5790,13808}, {7517,13800}, {9301,13805}, {9654,13815}, {9669,13816}, {10246,13787}, {10247,13822}, {11842,13792}, {11849,13795}, {11911,13809}, {11917,13811}, {11928,13813}, {11929,13814}, {11949,13817}, {11950,13818}, {12000,13839}, {12001,13840}

X(13836) = reflection of X(i) in X(j) for these (i,j): (3,1328), (13798,5), (13812,13807)
X(13836) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1351,3543,13713), (13807,13812,381), (13819,13837,3295)

X(13837) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 10*S^3-3*(SB+SC)*(-S*(-S-3*SA)+(-2*S+SW)*b*c) : :

The reciprocal orthologic center of these triangles is X(13785)

X(13837) lies on these lines: {1,1328}, {5,13816}, {12,13812}, {30,10067}, {35,13786}, {388,13794}, {495,13815}, {498,13821}, {611,13714}, {1479,13807}, {3085,13798}, {3295,13819}, {3298,13715}, {3584,13835}, {10037,13800}, {10038,13805}, {10039,13808}, {10040,13810}, {10041,13811}, {10068,11237}, {10523,13813}, {10801,13792}, {10954,13814}, {11398,13788}, {11507,13795}, {11912,13809}, {11951,13817}, {11952,13818}

X(13837) = midpoint of X(1328) and X(13839)
X(13837) = reflection of X(13815) in X(495)
X(13837) = {X(3295), X(13836)}-harmonic conjugate of X(13819)

X(13838) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 10*S^3-3*(SB+SC)*(-S*(-S-3*SA)-(-2*S+SW)*b*c) : :

The reciprocal orthologic center of these triangles is X(13785)

X(13838) lies on these lines: {1,1328}, {3,13819}, {5,13815}, {11,13812}, {30,10083}, {36,13786}, {496,13816}, {497,13794}, {499,13821}, {613,13715}, {999,13836}, {1478,13807}, {1737,13808}, {3086,13798}, {3297,13714}, {3582,13835}, {10046,13800}, {10047,13805}, {10048,13810}, {10049,13811}, {10084,11238}, {10523,13814}, {10802,13792}, {10948,13813}, {11399,13788}, {11508,13795}, {11913,13809}, {11953,13817}, {11954,13818}

X(13838) = midpoint of X(1328) and X(13840)
X(13838) = reflection of X(13816) in X(496)

X(13839) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO 2nd TRI-SQUARES-CENTRAL

Barycentrics
-6*a^2*(a^4-2*(b^2+b*c+c^2)*a^2+2*b*c*(b+c)*a+b^4+c^4-4*b^3*c-2*b^2*c^2-4*b*c^3)*S+(a+b+c)*(4*a^7-4*a^6*(b+c)-(3*b^2-40*b*c+3*c^2)*a^5+(b+c)*(3*b^2-32*b*c+3*c^2)*a^4-2*(3*b^4+3*c^4+2*b*c*(5*b^2-14*b*c+5*c^2))*a^3+2*(b^2-c^2)*(b-c)*(3*b^2+17*b*c+3*c^2)*a^2+5*(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-5*(b^2-c^2)^3*(b-c)) : :

The reciprocal orthologic center of these triangles is X(13785)

X(13839) lies on these lines: {1,1328}, {12,13813}, {30,13132}, {5552,13821}, {10528,13798}, {10531,13807}, {10803,13792}, {10805,13794}, {10834,13800}, {10878,13805}, {10915,13808}, {10929,13810}, {10930,13811}, {10942,13812}, {10955,13814}, {10956,13815}, {10958,13816}, {10965,13819}, {11248,13786}, {11400,13788}, {11509,13795}, {11914,13809}, {11955,13817}, {11956,13818}, {12000,13836}, {12594,13716}

X(13839) = reflection of X(1328) in X(13837)

X(13840) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO 2nd TRI-SQUARES-CENTRAL

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

The reciprocal orthologic center of these triangles is X(13785)

X(13840) lies on these lines: {1,1328}, {11,13814}, {30,13133}, {10527,13821}, {10529,13798}, {10532,13807}, {10804,13792}, {10806,13794}, {10835,13800}, {10879,13805}, {10916,13808}, {10931,13810}, {10932,13811}, {10943,13812}, {10949,13813}, {10957,13815}, {10959,13816}, {10966,13819}, {11249,13786}, {11401,13788}, {11510,13795}, {11915,13809}, {11957,13817}, {11958,13818}, {12001,13836}, {12595,13717}

X(13840) = reflection of X(1328) in X(13838)

X(13841) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO 2nd TRI-SQUARES-CENTRAL

Barycentrics (SB-SC)*(-15*S^3+9*(SB+SC)*S^2+(-24*SW*SA+18*SA^2+11*SW^2)*S+3*(3*SA-2*SW)*(SB+SC)*SW) : :

The reciprocal parallelogic center of these triangles is X(13785)

X(13841) lies on these lines: {351,13842}, {523,13316}, {9135,13718}

X(13841) = reflection of X(13842) in X(351)

X(13842) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO 2nd TRI-SQUARES-CENTRAL

Barycentrics (SB-SC)*(-15*S^3+9*(SB+SC)*S^2-(18*SA^2-24*SW*SA+SW^2)*S-9*(SB+SC)*SA*SW) : :

The reciprocal parallelogic center of these triangles is X(13785)

X(13842) lies on these lines: {351,13841}, {523,13319}, {3569,13719}

X(13842) = reflection of X(13841) in X(351)

X(13843) = X(3) OF THE 2nd TRI-SQUARES-CENTRAL TRIANGLE

Barycentrics (3*SA-9*S+4*SW)*(3*SA+2*S-3*SW) : :
X(13843) = 3*X(3069)-X(13782) = 3*X(3069)-2*X(13844)

X(13843) lies on these lines: {597,13720}, {1328,6477}, {3069,13782}

X(13843) = reflection of X(13782) in X(13844)

X(13844) = X(5) OF THE 2nd TRI-SQUARES-CENTRAL TRIANGLE

Barycentrics 54*S^2+(51*SA-73*SW)*S-3*(SB+SC)*(3*SA-8*SW) : :
X(13844) = 3*X(3069)+X(13782) = 3*X(3069)-X(13843)

X(13844) lies on the line {3069,13782}

X(13844) = midpoint of X(13782) and X(13843)

X(13845) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO 4th ANTI-BROCARD

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

The reciprocal orthologic center of these triangles is X(9870)

X(13845) lies on these lines: {2,13167}, {597,13652}, {2780,13762}, {2854,13774}, {9869,13758}, {12149,13757}, {13772,13783}

X(13845) = reflection of X(13772) in X(13783)

X(13846) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO 2nd TRI-SQUARES

Barycentrics 6*a^4-3*(b^2+c^2)*a^2+3*(b^2-c^2)^2-(5*a^2-4*b^2-4*c^2)*S : :
X(13846) = 2*X(485)+X(1151) = 3*X(485)-X(1327) = X(485)+2*X(8981) = 3*X(1151)+2*X(1327) = X(1151)-4*X(8981) = X(1327)+6*X(8981) = X(12602)+2*X(12974)

The reciprocal orthologic center of these triangles is X(3069)

X(13846) lies on these lines: {2,6}, {3,8960}, {4,6425}, {5,3592}, {30,485}, {45,5393}, {140,3594}, {371,381}, {372,5054}, {376,3070}, {382,6453}, {486,547}, {493,1989}, {519,8983}, {530,10668}, {531,10667}, {532,6305}, {533,6304}, {538,8992}, {539,8909}, {541,8994}, {542,8980}, {543,8997}, {549,1152}, {550,9680}, {566,8962}, {631,6426}, {754,8993}, {1124,3582}, {1328,5066}, {1335,3584}, {1587,3524}, {1588,3316}, {1599,11063}, {1656,6419}, {1853,11241}, {1990,3535}, {2043,5340}, {2044,5339}, {2066,11238}, {2067,11237}, {2549,13835}, {3071,3545}, {3155,7669}, {3297,9661}, {3298,9646}, {3311,5055}, {3526,6420}, {3529,10147}, {3534,6200}, {3536,6749}, {3543,6429}, {3590,3832}, {3627,9681}, {3628,10195}, {3679,7969}, {3830,6221}, {3839,6459}, {3843,6447}, {3845,6437}, {5023,13678}, {5064,5412}, {5070,6427}, {5073,6519}, {5309,6422}, {5341,6204}, {5420,6432}, {5475,8375}, {6199,6565}, {6411,6560}, {6412,12100}, {6417,10577}, {6424,7753}, {6433,11001}, {6438,11812}, {6439,9542}, {6442,11540}, {6468,9541}, {7297,7347}, {8553,8939}, {8963,13351}, {8966,8969}, {12602,12974}, {12968,13701}, {13586,13657}, {13662,13712}, {13798,13832}

X(13846) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,1991,599), (2,13637,1991), (2,13639,5860), (6,590,8253), (6,8253,8252), (395,396,3068), (590,3068,6), (615,7585,6), (1991,13663,2), (3068,8972,590), (3070,9540,6409), (5418,7583,1152), (13637,13663,599)

X(13847) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 2nd TRI-SQUARES

Barycentrics -(13*a^2+4*b^2+4*c^2)*S+9*(b^2+c^2)*a^2-3*(b^2-c^2)^2 : :
X(13847) = 2*X(486)+X(1152) = 3*X(486)-X(1328) = 3*X(1152)+2*X(1328) = X(12601)+2*X(12975)

The reciprocal orthologic center of these triangles is X(3069)

X(13847) lies on these lines: {2,6}, {4,6426}, {5,3594}, {30,486}, {45,5405}, {140,3592}, {371,5054}, {372,381}, {376,3071}, {382,6454}, {485,547}, {494,1989}, {530,10672}, {531,10671}, {532,6301}, {533,6300}, {549,1151}, {631,6425}, {1124,3584}, {1327,5066}, {1335,3582}, {1587,3317}, {1588,3524}, {1600,11063}, {1656,6420}, {1853,11242}, {1990,3536}, {2043,5339}, {2044,5340}, {2362,4870}, {2549,13712}, {3070,3545}, {3156,7669}, {3297,10056}, {3298,10072}, {3312,5055}, {3526,6419}, {3529,10148}, {3534,6396}, {3535,6749}, {3543,6430}, {3591,3832}, {3628,10194}, {3679,7968}, {3830,6398}, {3839,6460}, {3843,6448}, {3845,6438}, {5023,13798}, {5064,5413}, {5070,6428}, {5073,6522}, {5309,6421}, {5341,6203}, {5414,11238}, {5418,6431}, {5475,8376}, {6395,6564}, {6411,12100}, {6412,6561}, {6418,10576}, {6423,7753}, {6434,11001}, {6437,11812}, {6441,11540}, {6470,9540}, {6502,11237}, {7297,7348}, {8553,8943}, {8962,13337}, {8981,10124}, {12601,12975}, {12963,13821}, {13586,13777}, {13678,13831}, {13782,13835}

X(13847) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6,13846), (2,13759,5861), (2,13846,8253), (6,615,8252), (6,8252,8253), (395,396,3069), (590,7586,6), (615,3069,6), (5420,7584,1151), (8252,13846,2), (13757,13783,599)

X(13848) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 22*S^2-3*(3*SA-SW)*S-3*(3*SA+2*SW)*(SB+SC) : :
Barycentrics -6*(2*a^2-b^2-c^2)*S+8*a^4-7*(b^2+c^2)*a^2+11*(b^2-c^2)^2 : :

The reciprocal orthologic center of these triangles is X(13785)

X(13848) lies on these lines: {371,13807}, {590,13821}, {1328,3068}, {8972,13798}, {8974,13810}, {8975,13811}, {8976,13812}, {9540,13786}, {13720,13846}, {13832,13835}

X(13849) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 22*S^2+3*(5*SA-7*SW)*S-3*(3*SA-2*SW)*(SB+SC) : :
Barycentrics -6*(6*a^2+b^2+c^2)*S+4*a^4+19*(b^2+c^2)*a^2-11*(b^2-c^2)^2 : :

The reciprocal orthologic center of these triangles is X(13785)

X(13849) lies on these lines: {2,13782}, {6,13848}, {115,13823}, {372,13807}, {615,13821}, {1328,3069}

X(13850) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 14*S^2-(3*SA-SW)*S-3*(3*SA+SW)*(SB+SC)
Barycentrics 2*(-2*a^2+b^2+c^2)*S+a^4-2*(b^2+c^2)*a^2+7*(b^2-c^2)^2 : :
X(13850) = 3*X(485)+X(1328) = X(6222)+2*X(6250)

The reciprocal orthologic center of these triangles is X(13834)

X(13850) lies on these lines: {2,13832}, {115,1991}, {381,485}, {590,13835}, {6222,6250}, {12969,13847}, {13720,13846}

X(13850) = reflection of X(13846) in X(13848)

X(13851) = X(4)X(51)∩X(30)X(125)

Trilinears cos(A)*(3*cos(A)*cos(B-C)-cos( 2*(B-C))-cos(2*A)-1) : :
Barycentrics (-a^2+b^2+c^2)*(2*a^8-(b^2+c^ 2)*a^6-2*(b^2-c^2)^2*a^4-(b^4- c^4)*(b^2-c^2)*a^2+2*(b^2-c^2) ^4) : :
X(13851) = 2*X(265)+X(1531) = X(3292)-4*X(10297) = X(13399)+2*X(13473)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26302.

X(13851) lies on these lines: {4,51}, {5,13367}, {30,125}, {115,8779}, {184,381}, {265,1531}, {382,1204}, {403,1495}, {511,3153}, {546,6146}, {578,7547}, {1092,12293}, {1181,3843}, {1425,3585}, {1503,10151}, {1568,3292}, {1594,13403}, {1650,12096}, {2071,10733}, {3270,3583}, {3410,5907}, {3574,12241}, {3818,6467}, {3830,10605}, {3839,5476}, {3850,8254}, {5562,9927}, {5622,11645}, {7507,11424}, {7577,11430}, {10255,12038}, {10282,12289}, {11017,11577}, {12022,13366}, {12828,13202}

X(13851) = midpoint of X(i) and X(j) for these {i,j}: {2071,10733}, {13202,13399}
X(13851) = reflection of X(i) in X(j) for these (i,j): (403,7687), (1495,403), (1568,10297), (3292,1568), (13202,13473)
X(13851) = X(4511)-of-orthic-triangle if ABC is acute
X(13851) = Ehrmann-side-to-orthic similarity image of X(10540)

X(13852) = POINT BEID 142

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

See Tran Quang Hung and Peter Moses, Hyacinthos 26315.

X(13852) lies on this line: {2,3}

X(13853) = POINT BEID 143

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26317.

X(13853) lies on these lines: {2,7367}, {4,6611}, {11,1435}, {84,5715}, {225,1427}, {226,1439}, {1422,2006}, {1436,7490}, {1440,6612}, {2184,5514}, {3772,7129}, {6356,6358}

X(13853) = {X(226), X(8808)}-harmonic conjugate of X(1903)

X(13854) = X(4)X(251)∩X(6)X(66)

Trilinears    (tan A)(sin A)/(sin 2A - tan ω) : :
Barycentrics    (S^2-2*SA*SC+SB^2)*(S^2-2*SA* SB+SC^2)*SB*SC : :
Barycentrics    1/[(b^2 + c^2 - a^2)(b^4 + c^4 - a^4)] : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26317.

Let A'B'C' be the tangential triangle of the Kiepert hyperbola (i.e., the Schroeter triangle). Let A" be the intersection, other than X(127), of the nine-point circle and line A'X(127); define B" and C" cyclically. The lines AA", BB", CC" concur in X(13854). (Randy Hutson, July 21, 2017)

The trilinear polar of X(13854) meets the line at infinity at X(512). (Randy Hutson, July 21, 2017)

X(13854) lies on the cubic K701, the hyperbola {{A,B,C,X(2),X(6)}}, and these lines: {2,1235}, {4,251}, {6,66}, {22,5523}, {25,2353}, {111,1289}, {112,7391}, {127,13575}, {232,2165}, {468,8770}, {1383,6995}, {1400,2156}, {2395,6753}, {2987,6515}, {3172,5064}, {5133,8743}, {7735,8882}

X(13854) = isogonal conjugate of X(20806)
X(13854) = isotomic conjugate of X(34254)
X(13854) = polar conjugate of X(315)
X(13854) = X(22)-isoconjugate of X(63)
X(13854) = X(1974)-cross conjugate of X(4)

X(13855) = ISOGONAL CONJUGATE OF X(1075)

Trilinears cos(A)/((2*cos(A)+2*cos(3*A))* cos(B-C)+cos(2*(B-C))-3*cos(2* A)-2) : :
Barycentrics (S^2-SB*SC)/(16*R^4-8*(SB+SC)* R^2-S^2-2*SA^2+SW^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26323.

X(13855) lies on the cubic K003 and these lines: {3,1075}, {577,6759}, {1092,2055}

X(13855) = isogonal conjugate of X(1075)
X(13855) = X(4)-cross conjugate of X(3)

X(13856) = POINT BEID 144

Barycentrics (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^12-6 a^10 b^2+14 a^8 b^4-16 a^6 b^6+9 a^4 b^8-2 a^2 b^10-6 a^10 c^2+18 a^8 b^2 c^2-14 a^6 b^4 c^2-3 a^4 b^6 c^2+6 a^2 b^8 c^2-b^10 c^2+14 a^8 c^4-14 a^6 b^2 c^4-3 a^4 b^4 c^4-4 a^2 b^6 c^4+4 b^8 c^4-16 a^6 c^6-3 a^4 b^2 c^6-4 a^2 b^4 c^6-6 b^6 c^6+9 a^4 c^8+6 a^2 b^2 c^8+4 b^4 c^8-2 a^2 c^10-b^2 c^10) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26322.

X(13856) lies on and these lines: {2,11016},{5,128},{140,389},{ 195,252},{930,7604},{1487, 1656}

X(13857) = X(2)X(51)∩X((30)X(113)

Barycentrics (a^2-2 b^2-2 c^2) (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) : :
X(13857) = 2 X[858] + X[3292] = X[3581] - 3 X[5054] = X[1531] + 2 X[10564] = X[1495] - 4 X[11064]

Contributed by Peter Moses, July 11, 2017.

X(13857) lies on and these lines: {2, 51}, {30, 113}, {110, 10989}, {125, 524}, {323, 9140}, {381, 5651}, {401, 12117}, {542, 858}, {599, 5094}, {625, 5108}, {868, 1641}, {1092, 11572}, {1368, 13366}, {1648, 5107}, {1650, 3284}, {2393, 5648}, {3260, 9214}, {3291, 6034}, {3581, 5054}, {3849, 9181}, {3906, 4141}, {5461, 9127}, {5972, 7426}, {7464, 10706}, {8703, 13394}, {10719, 13415}, {10720, 13414}

X(13857) = midpoint of X(i) and X(j) for these {i,j}: {110, 10989}, {323, 9140}, {599, 10510}, {7464, 10706}
X(13857) = reflection of X(i) in X(j) for these {i,j}: {1495, 5642}, {5642, 11064}, {7426, 5972}
X(13857) = crossdifference of every pair of points on line {1383, 2433}
X(13857) = X(2),X(5476)}-harmonic conjugate of X(373)
X(13857) = isoconjugate of X(j) and X(j) for these {i,j}: {598, 2159}, {1383, 2349}
X(13857) = crossdifference of every pair of points on line {1383, 2433}
X(13857) = barycentric product X(i)*X(j) for these {i,j}: {30, 599}, {574, 3260}, {1495, 9464}, {1637, 9146}, {2407, 3906}, {5094, 11064}
X(13857) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 598}, {574, 74}, {599, 1494}, {1495, 1383}, {1637, 8599}, {2420, 11636}, {3906, 2394}, {8288, 12079}, {8541, 8749}

X(13858) = CIRCUMCIRCLE-INVERSE OF X(5463)

Barycentrics (SB+SC)*(3*(3*R^2-SW)*S^2+sqrt(3)*(9*R^2-2*SW)*SA*S-(3*SA-2*SW)*SA*SW) : :
X(13858) = (SW^2+sqrt(3)*(3*R^2-SW)*S)*X(3)-3*SW*R^2*X(67)

Contributed by César Eliud Lozada, July 13, 2017. See cubic K912.

X(13858) lies on the cubic K912 and these lines: {3,67}, {13,13233}, {14,1995}, {15,110}, {16,2854}, {23,531}, {61,6593}, {62,895}, {99,11612}, {617,7492}, {619,7496}, {5981,5987}

X(13858) = reflection of X(10658) in X(110)
X(13858) = circumcircle-inverse of X(5463)
X(13858) = Thomson-isogonal conjugate of X(34313)

X(13859) = CIRCUMCIRCLE-INVERSE OF X(5464)

Barycentrics (SB+SC)*(3*(3*R^2-SW)*S^2-sqrt(3)*(9*R^2-2*SW)*SA*S-(3*SA-2*SW)*SA*SW) : :
X(13859) = (SW^2-sqrt(3)*(3*R^2-SW)*S)*X(3)-3*SW*R^2*X(67)

Contributed by César Eliud Lozada, July 13, 2017. See cubic K912.

X(13859) lies on the cubic K912 and these lines: {3,67}, {13,1995}, {14,13233}, {15,2854}, {16,110}, {23,530}, {61,895}, {62,6593}, {99,11613}, {616,7492}, {618,7496}, {5980,5987}

X(13859) = reflection of X(10657) in X(110)
X(13859) = circumcircle-inverse of X(5464)
X(13859) = Thomson-isogonal conjugate of X(34314)

X(13860) = X(2)X(3)∩X(6)X(98)

Barycentrics a^8-a^6 b^2+3 a^4 b^4-3 a^2 b^6-a^6 c^2+4 a^4 b^2 c^2+3 a^2 b^4 c^2-2 b^6 c^2+3 a^4 c^4+3 a^2 b^2 c^4+4 b^4 c^4-3 a^2 c^6-2 b^2 c^6 : :
Barycentrics (SB*SC - (SB + SC)^2)(S^2 + SA*SW) + 2(S^2 + SB*SW)(S^2 + SC*SW) : :
X(13860) = X(13860) = 2 X[3363] - 3 X[3545]

X(13860) lies on these lines: {2, 3}, {6, 98}, {114, 3818}, {147, 7777}, {157, 3425}, {182, 9418}, {183, 511}, {194, 10983}, {230, 5017}, {325, 1352}, {385, 1351}, {399, 5987}, {542, 11163}, {574, 10837}, {842, 2453}, {1007, 12215}, {1184, 10982}, {1384, 10788}, {1503, 3815}, {1975, 6248}, {2076, 9993}, {2794, 5475}, {2967, 9308}, {3053, 12110}, {3054, 9754}, {3095, 7754}, {3311, 10845}, {3312, 10846}, {3329, 5050}, {3564, 7774}, {5013, 11257}, {5024, 7709}, {5031, 7778}, {5093, 7766}, {5188, 7815}, {5309, 11623}, {5476, 6055}, {5939, 12177}, {6032, 6232}, {6054, 9830}, {6199, 10847}, {6221, 10839}, {6395, 10848}, {6398, 10840}, {6468, 10841}, {6469, 10842}, {6776, 7736}, {6785, 13240}, {7612, 9748}, {7753, 10991}, {7779, 11898}, {7785, 9863}, {7786, 12203}, {8550, 9300}, {8667, 11477}, {8719, 9743}, {9769, 10752}, {9770, 11180}, {9772, 13188}, {9865, 13108}, {10358, 13335}, {10796, 12042}

X(13860) = midpoint of X(10837) and X(10838)
X(13860) = reflection of X(i) in X(j) for these {i,j}: {9744, 3815}, {11317, 381}
X(13860) = complement of X(37182)
X(13860) = anticomplement of X(37451)
X(13860) = orthocentroidal-circle inverse of X(1513)
X(13860) = orthoptic-circle-of-Steiner-inellipse inverse of X(5112)
X(13860) = Thomson-isogonal conjugate of X(34099)
X(13860) = pole, wrt orthoptic circle of Steiner inellipse, of trilinear polar of X(262) (line X(523)X(3569))
X(13860) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,1513),(2,5999,3),(3,5,7770),(5,8361,3090),(6,98,9755),(6,9756,98),(20,7824,3),(98,262,6),(230,5480,9753),(262,9756,9755),(383,1080,381),(631,7470,3),(1344,1345,7418),(2043,2044,8370),(6039,6040,11676),(6248,9737,1975),(6811,6813,5),(6998,7380,2049),(7000,7374,3832)

X(13861) = X(2)X(3)∩X(6)X(156)

Barycentrics a^2 (a^8-2 a^6 b^2+2 a^2 b^6-b^8-2 a^6 c^2-4 a^2 b^4 c^2+6 b^6 c^2-4 a^2 b^2 c^4-10 b^4 c^4+2 a^2 c^6+6 b^2 c^6-c^8) : :
X(13861) = X[3] + 3 X[1598] = X[3] - 3 X[6642] = 7 X[3090] - 3 X[6643] = 5 X[3091] + 3 X[7487] = X[6643] + 3 X[7714] = 7 X[3090] + 9 X[7714]

X(13861) lies on these lines: {2, 3}, {6, 156}, {51, 10539}, {110, 9781}, {115, 9608}, {143, 155}, {154, 13364}, {206, 575}, {373, 13336}, {394, 10263}, {495, 10046}, {496, 10037}, {498, 9673}, {499, 9658}, {567, 9707}, {568, 11441}, {569, 1495}, {576, 9925}, {952, 11365}, {1147, 10110}, {1173, 11422}, {1181, 5946}, {1192, 11472}, {1498, 13630}, {1506, 9609}, {1614, 5640}, {1843, 8538}, {3167, 13451}, {3527, 8780}, {3814, 9712}, {3818, 5449}, {5446, 9306}, {5448, 9932}, {5462, 6759}, {5480, 9820}, {5609, 12236}, {5651, 10625}, {5663, 9786}, {5886, 8185}, {5901, 9798}, {6800, 13353}, {7592, 10540}, {7603, 9700}, {7988, 9626}, {7989, 9625}, {8192, 10283}, {8253, 9683}, {8254, 9920}, {10272, 12310}, {10316, 10985}, {10592, 10831}, {10593, 10832}, {11801, 12412}

X(13861) = midpoint of X(i) and X(j) for these {i,j}: {5, 7715}, {1598, 6642}
X(13861) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12088, 3), (3, 3628, 13154), (3, 5198, 3627), (3, 7517, 12088), (3, 7545, 10594), (3, 10594, 7530), (3, 11284, 632), (4, 6644, 12084), (4, 7506, 6644), (4, 13595, 7506), (5, 25, 26), (5, 26, 7514), (5, 7502, 7395), (5, 10154, 140), (5, 11818, 7564), (5, 13490, 4), (22, 1656, 7516), (23, 3090, 3), (24, 381, 7526), (25, 1598, 7715), (25, 5020, 10154), (25, 7395, 9714), (25, 7529, 5), (51, 10539, 12161), (156, 10095, 6), (381, 13621, 24), (546, 12106, 3), (547, 7525, 7393), (1658, 3850, 9818), (1995, 7545, 7530), (1995, 10594, 3), (2070, 3851, 7503), (2937, 5055, 7509), (3091, 3518, 3), (3517, 9818, 1658), (3526, 5899, 10323), (3542, 7528, 5), (3549, 6997, 5), (3861, 11250, 1597), (5020, 7387, 140), (5070, 13564, 7485), (6642, 7715, 26), (7387, 10154, 26), (7393, 9909, 7525), (7393, 11484, 547), (7394, 7505, 5576), (7395, 9714, 7502), (7502, 9714, 26), (7506, 13490, 26), (9909, 11484, 7393)

X(13862) = X(2)X(3)∩X(6)X(147)

Barycentrics 2 a^6 b^2-a^4 b^4-b^8+2 a^6 c^2+3 a^4 b^2 c^2+2 a^2 b^4 c^2+b^6 c^2-a^4 c^4+2 a^2 b^2 c^4+b^2 c^6-c^8 : :
X(13862) = X[3314] +2 X[9993]

X(13862) lies on these lines: {2, 3}, {6, 147}, {32, 9863}, {98, 3407}, {114, 262}, {132, 264}, {141, 6194}, {182, 7875}, {183, 5207}, {193, 9748}, {265, 5987}, {325, 5480}, {385, 1352}, {511, 3314}, {576, 7837}, {1350, 7868}, {1351, 7779}, {1503, 7792}, {2080, 9996}, {2456, 10334}, {2794, 3972}, {3095, 7906}, {3096, 5188}, {3106, 6115}, {3107, 6114}, {3329, 9744}, {3564, 7766}, {3618, 7710}, {5103, 7778}, {5171, 7904}, {5304, 5921}, {5359, 11441}, {5476, 6054}, {5984, 9755}, {6033, 10796}, {6055, 10033}, {6287, 10104}, {7616, 8556}, {7764, 9764}, {7788, 11477}, {7803, 8721}, {7823, 12110}, {7831, 8722}, {7834, 12203}, {7864, 11257}, {7891, 9737}, {8176, 9877}, {9478, 9756}, {9774, 10168}, {9866, 13111}, {9873, 13335}

X(13862) = orthocentroidal-circle inverse of X(5999)
X(13862) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,5999),(4,5,5025),(5,1513,2),(114,262,7777),(381,11317,3839),(1352,9753,385),(5025,7892,7876)

X(13863) = POINT BEID 145

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26330.

X(13863) lies on these lines: {1141,13619} et al

X(13863) = isoconjugate of X(656) and X(10096)
X(13863) = barycentric quotient X(112)/X(10096)

X(13864) = POINT BEID 146

Barycentrics a^8 (b-c)^2 - 2 a^7 (b^3+5 b^2 c+5 b c^2+c^3) - a^6 (2 b^4+11 b^3 c+30 b^2 c^2+11 b c^3+2 c^4) + a^5 (6 b^5+45 b^4 c+41 b^3 c^2+41 b^2 c^3+45 b c^4+6 c^5) + 2 a^4 b c (15 b^4+94 b^3 c-14 b^2 c^2+94 b c^3+15 c^4) - 6 a^3 (b^7+10 b^6 c-11 b^4 c^3-11 b^3 c^4+10 b c^6+c^7) + a^2 (b^2-c^2)^2 (2 b^4-19 b^3 c-158 b^2 c^2-19 b c^3+2 c^4) + a (b-c)^4 (b+c)^3 (2 b^2+27 b c+2 c^2) - (b-c)^6 (b+c)^4 : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 26332.

X(13864) lies on these lines: {6841, 11227}, {10124, 13865}

X(13865) = POINT BEID 147

Barycentrics a^5 (b^2-10 b c+c^2) - a^4 (b^3+11 b^2 c+11 b c^2+c^3) - 2 a^3 (b-c)^2 (b^2+6 b c+c^2) + 2 a^2 (b-c)^2 (b^3+7 b^2 c+7 b c^2+c^3) + a (b^2-c^2)^2 (b^2+18 b c+c^2) - (b-c)^4 (b+c)^3 : :
X(13865) = 2(4 + 10R)*X(5) -5R(X(40)

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 26332.

X(13865) lies on these lines: {5,40}, {946,12680}

X(13866) = POINT BEID 148

Barycentrics a (2 a^9 - 15 a^8 (b+c) + 10 a^7 (b^2+12 b c+c^2) + 2 a^6 (21 b^3-61 b^2 c-61 b c^2+21 c^3) - 2 a^5 (21 b^4+70 b^3 c-62 b^2 c^2+70 b c^3+21 c^4) - 4 a^4 (9 b^5-67 b^4 c+22 b^3 c^2+22 b^2 c^3-67 b c^4+9 c^5) + 2 a^3 (b-c)^2 (23 b^4+6 b^3 c-74 b^2 c^2+6 b c^3+23 c^4) + 2 a^2 (b-c)^2 (3 b^5-49 b^4 c-2 b^3 c^2-2 b^2 c^3-49 b c^4+3 c^5) - 4 a (b^2-c^2)^2 (4 b^4-25 b^3 c+10 b^2 c^2-25 b c^3+4 c^4) + 3 (b-c)^4 (b+c)^3 (b^2-6 b c+c^2)) : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 26354.

X(13866) lies on this line: {1,5806}

X(13867) = POINT BEID 149

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 26354.

X(13867) lies on these lines:
{1,5806}, {8,3740}, {65,390}, {354,4297}, {950,8581}, {2136,3303}, {2951,11518}, {3057,6738}, {3488,12675}, {3698,10389}, {3893,4423}, {4321,5665}, {5836,8236}, {8275,9957}, {9848,12672}

X(13868) = POINT BEID 150

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26360.

X(13868) lies on these lines:
{104, 5663}, {109, 3028}, {513, 3109}, {6789, 10176}

X(13868) = reflection of X(3109) in the line X(1)X(3)

X(13869) = POINT BEID 151

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26360.

X(13869) lies on these lines:
{1, 523}, {30, 944}, {265, 952}, {405, 2452}, {6741, 11735}

X(13869) = midpoint X(6742) and X(7984)
X(13869) = reflection of X(i) in X(j) for these (i,j): (3109, 1), (6741, 11735)

X(13870) = TOUCHPOINT OF THE NINE-POINTS CIRCLE AND THE AIYAR-CIRCLE-OF-THE-LEMOINE INELLIPSE

Barycentrics ((SW+6*SA)*S^2+3*(SA^2-SB*SC-SW^2)*SA)*K-3*SW*(S^2+SB*SC); : : where K=3*R*SW/(OH*sqrt(SW^2-3*S^2))
X(13870) = (K-1)*X(5)-K*X(597)

Theorem: If any conic C be inscribed in a given triangle and a confocal to it pass through the circumcenter, then the Aiyar-circle-of-C through the intersections of these two confocals touches the nine-points circle of the triangle. Reference: Ramaswami Aiyar: A General Theorem on the Nine-points Circle, Proceedings of the Edinburgh Mathematical Society, Volume 15, February 1896, pp. 74-75.

Notes: The touchpoint of the nine-points circle and the Aiyar-circle-of-the-Steiner inellipse is X(1313). For the Brocard inellipse and the McBeath inconic, the confocal conics through X(3) are degenerated.

Centers X(13870) to X(13872) were contributed by César Eliud Lozada, July 17, 2017.

X(13870) lies on the nine-points circle and these lines: {5,542}, {1312,3414}, {1313,3413}, {1348,13414}, {1349,13415}, {2039,2574}, {2040,2575}

X(13871) = TOUCHPOINT OF THE NINE-POINTS CIRCLE AND THE AIYAR-CIRCLE-OF-THE-MANDART INELLIPSE

Barycentrics ((-8*b*c+12*SA)*S^2+4*(SB*a*b+SB*a*c+SB*b*c+SC*a*b+SC*a*c+SC*b*c+SA^2-SB*SC-SW^2)*SA)*K+8*r*(4*R+r)*(2*S^2-(SB+SC)*SA) : :
where K = (4*R+r)*R/sqrt((4*R+r)^2*R^2-2*(R+r)*S*s)
X(13871) = (K-1)*X(5)-K*X(9)

See X(13870).

X(13871) lies on the nine-points circle and the line {5,9}

X(13872) = TOUCHPOINT OF THE NINE-POINTS CIRCLE AND THE AIYAR-CIRCLE-OF-THE-ORTHIC INCONIC

Barycentrics a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4 + (a*b*c*(a^2 - b^2 - c^2)^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))/(Sqrt[a^10 + b^10 + c^10 - 8*a^2*b^2*c^2*(a^2*b^2 + a^2*c^2 + b^2*c^2) + 9*a^2*b^2*c^2*(a^4 + b^4 + c^4) + 2*(a^6*b^4 + b^6*c^4 + a^4*c^6) + 2*(a^4*b^6 + a^6*c^4 + b^4*c^6) - 3*(a^8*b^2 + b^8*c^2 + a^2*c^8) - 3*(a^2*b^8 + a^8*c^2 + b^2*c^8)]*Abs[(a^2 - b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)]) : :

See X(13870)

X(13872) lies on the nine-points circle and the line {5,6}

3rd & 4th tri-squares triangles and related centers: X(13873)-X(13993)

This preamble and centers X(13873)-X(13993) were contributed by César Eliud Lozada, July 15, 2017.

Tri-squares triangles were defined in the preamble of X(13637). Centers related to the 3rd and 4th tri-squares triangles are showed in the following tables:

3rd tri-squares triangle
In this cell, A'B'C' is the 3rd tri-squares triangle of ABC. A'B'C' is directly similar to the Lucas-tangents triangle. List of triangles perspective to A'B'C' and ETC index of perspector: (Note: An asterisk * means that both triangles are homothetic.) (ABC, 2), (anticomplementary, 2), (4th Brocard, 2), (circummedial, 2), (5th Euler, 2), (medial, 2), (3rd Parry, 2), (3rd tri-squares-central, 485), (4th tri-squares, 2), (outer-Vecten*, 590) List of triangles orthologic to A'B'C' with ETC indexes of orthologic centers: (ABC, 485, 485), (ABC-X3 reflections, 485, 12124), (anti-Aquila, 485, 12269), (anti-Ara, 485, 12148), (anti-Artzt, 2, 12159), (1st anti-Brocard, 13873, 9868), (5th anti-Brocard, 485, 12211), (6th anti-Brocard, 13873, 12218), (anti-Euler, 485, 12257), (anti-Mandart-incircle, 485, 12344), (anti-McCay, 13874, 9893), (anticomplementary, 485, 488), (Aquila, 485, 9907), (Ara, 485, 9922), (Artzt, 2, 9768), (1st Auriga, 485, 12486), (2nd Auriga, 485, 12487), (1st Brocard, 13873, 6228), (5th Brocard, 485, 9987), (6th Brocard, 13873, 9992), (Euler, 485, 6250), (outer-Garcia, 485, 12788), (Gossard, 485, 12800), (inner-Grebe, 485, 6279), (outer-Grebe, 485, 6278), (Johnson, 485, 6289), (inner-Johnson, 485, 12929), (outer-Johnson, 485, 12939), (1st Johnson-Yff, 485, 12949), (2nd Johnson-Yff, 485, 12959), (Lucas homothetic, 485, 13004), (Lucas(-1) homothetic, 485, 13005), (Mandart-incircle, 485, 13082), (McCay, 13874, 13088), (medial, 485, 641), (5th mixtilinear, 485, 7981), (inner-Napoleon, 13875, 6305), (outer-Napoleon, 13876, 6304), (1st Neuberg, 13877, 6312), (2nd Neuberg, 13878, 6311), (inner-squares, 1151, 485), (1st tri-squares-central, 13846, 3068), (2nd tri-squares-central, 13850, 13834), (3rd tri-squares-central, 485, 13879), (4th tri-squares-central, 485, 13880), (1st tri-squares, 2, 13651), (2nd tri-squares, 2, 13771), (4th tri-squares, 13881, 13881), (inner-Vecten, 13881, 488), (outer-Vecten, 13882, 485), (X3-ABC reflections, 485, 12602), (inner-Yff, 485, 10068), (outer-Yff, 485, 10084), (inner-Yff tangents, 485, 13134), (outer-Yff tangents, 485, 13135) List of triangles parallelogic to A'B'C' with ETC indexes of parallelogic centers: (ABC, 2), (anticomplementary, 2), (4th Brocard, 2), (circummedial, 2), (5th Euler, 2), (medial, 2), (3rd Parry, 2), ((1st Parry, 485, 13317), (2nd Parry, 485, 13320) The appearance of (I, J) in the following list means that X(I)-of-A'B'C'=X(J):⁽¹⁾ (2, 13846), (3, 13879), (4, 13882), (5, 13924), (13, 13876), (14, 13875), (486, 485), (487, 1151), (642, 8981), (1132, 13881), (1328, 2) The radical center of the 3rd tri-squares circles is X(485).

3rd tri-squares triangle

In this cell, A'B'C' is the 3rd tri-squares triangle of ABC.

A'B'C' is directly similar to the Lucas-tangents triangle.

List of triangles perspective to A'B'C' and ETC index of perspector: (Note: An asterisk * means that both triangles are homothetic.)

(ABC, 2), (anticomplementary, 2), (4th Brocard, 2), (circummedial, 2), (5th Euler, 2), (medial, 2), (3rd Parry, 2), (3rd tri-squares-central, 485), (4th tri-squares, 2), (outer-Vecten*, 590)

List of triangles orthologic to A'B'C' with ETC indexes of orthologic centers:

(ABC, 485, 485), (ABC-X3 reflections, 485, 12124), (anti-Aquila, 485, 12269), (anti-Ara, 485, 12148), (anti-Artzt, 2, 12159), (1st anti-Brocard, 13873, 9868), (5th anti-Brocard, 485, 12211), (6th anti-Brocard, 13873, 12218), (anti-Euler, 485, 12257), (anti-Mandart-incircle, 485, 12344), (anti-McCay, 13874, 9893), (anticomplementary, 485, 488), (Aquila, 485, 9907), (Ara, 485, 9922), (Artzt, 2, 9768), (1st Auriga, 485, 12486), (2nd Auriga, 485, 12487), (1st Brocard, 13873, 6228), (5th Brocard, 485, 9987), (6th Brocard, 13873, 9992), (Euler, 485, 6250), (outer-Garcia, 485, 12788), (Gossard, 485, 12800), (inner-Grebe, 485, 6279), (outer-Grebe, 485, 6278), (Johnson, 485, 6289), (inner-Johnson, 485, 12929), (outer-Johnson, 485, 12939), (1st Johnson-Yff, 485, 12949), (2nd Johnson-Yff, 485, 12959), (Lucas homothetic, 485, 13004), (Lucas(-1) homothetic, 485, 13005), (Mandart-incircle, 485, 13082), (McCay, 13874, 13088), (medial, 485, 641), (5th mixtilinear, 485, 7981), (inner-Napoleon, 13875, 6305), (outer-Napoleon, 13876, 6304), (1st Neuberg, 13877, 6312), (2nd Neuberg, 13878, 6311), (inner-squares, 1151, 485), (1st tri-squares-central, 13846, 3068), (2nd tri-squares-central, 13850, 13834), (3rd tri-squares-central, 485, 13879), (4th tri-squares-central, 485, 13880), (1st tri-squares, 2, 13651), (2nd tri-squares, 2, 13771), (4th tri-squares, 13881, 13881), (inner-Vecten, 13881, 488), (outer-Vecten, 13882, 485), (X3-ABC reflections, 485, 12602), (inner-Yff, 485, 10068), (outer-Yff, 485, 10084), (inner-Yff tangents, 485, 13134), (outer-Yff tangents, 485, 13135)

List of triangles parallelogic to A'B'C' with ETC indexes of parallelogic centers:

(ABC, 2), (anticomplementary, 2), (4th Brocard, 2), (circummedial, 2), (5th Euler, 2), (medial, 2), (3rd Parry, 2), ((1st Parry, 485, 13317), (2nd Parry, 485, 13320)

The appearance of (I, J) in the following list means that X(I)-of-A'B'C'=X(J):⁽¹⁾

(2, 13846), (3, 13879), (4, 13882), (5, 13924), (13, 13876), (14, 13875), (486, 485), (487, 1151), (642, 8981), (1132, 13881), (1328, 2)

The radical center of the 3rd tri-squares circles is X(485).

3rd tri-squares-central triangle
In this cell, A'B'C' is the 3rd tri-squares-central triangle of ABC. Triangles directly similar to A'B'C': 1st Parry, 2nd Parry Triangles inversely similar to A'B'C': 1st anti-Brocard, 6th anti-Brocard, anti-orthocentroidal, 1st Brocard, 6th Brocard, inner-Garcia, orthocentroidal, orthosymmedial List of triangles perspective to A'B'C' and ETC index of perspector: (Note: An asterisk * means that both triangles are homothetic.) (ABC, 3068), (ABC-X3 reflections, 9540), (anti-Aquila, 13883), (anti-Ara, 13884), (5th anti-Brocard, 13885), (anti-Euler, 13886), (anti-Mandart-incircle, 13887), (anticomplementary, 8972), (Aquila, 13888), (Ara, 13889), (1st Auriga, 13890), (2nd Auriga, 13891), (5th Brocard, 13892), (Euler, 371), (outer-Garcia, 13893), (Gossard, 13894), (inner-Grebe, 8974), (outer-Grebe, 8975), (Johnson, 8976), (inner-Johnson, 13895), (outer-Johnson, 13896), (1st Johnson-Yff, 13897), (2nd Johnson-Yff, 13898), (Lucas homothetic, 13899), (Lucas(-1) homothetic, 13900), (Mandart-incircle, 13901), (medial, 590), (5th mixtilinear, 13902), (inner-squares, 8966), (4th tri-squares-central, 6), (3rd tri-squares, 485), (X3-ABC reflections, 13903), (inner-Yff, 13904), (outer-Yff, 13905), (inner-Yff tangents, 13906), (outer-Yff tangents, 13907) List of triangles orthologic to A'B'C' with ETC indexes of orthologic centers: (ABC, 485, 4), (ABC-X3 reflections, 485, 20), (anti-Aquila, 485, 946), (anti-Ara, 485, 3575), (anti-Artzt, 13846, 2), (anti-Ascella, 8981, 1593), (1st anti-Brocard, 8980, 5999), (5th anti-Brocard, 485, 12110), (6th anti-Brocard, 8980, 2456), (1st anti-circumperp, 8981, 20), (anti-Conway, 8981, 578), (2nd anti-Conway, 8981, 389), (anti-Euler, 485, 4), (3rd anti-Euler, 8981, 12111), (4th anti-Euler, 8981, 6241), (anti-excenters-reflections, 8981, 4), (anti-Hutson intouch, 8981, 3), (anti-incircle-circles, 8981, 3), (anti-inverse-in-incircle, 8981, 4), (anti-Mandart-incircle, 485, 11500), (anti-McCay, 13908, 9855), (6th anti-mixtilinear, 8981, 3), (anti-orthocentroidal, 8994, 12112), (anticomplementary, 485, 20), (Aquila, 485, 5691), (Ara, 485, 3), (Aries, 13909, 9833), (Artzt, 13846, 2), (Ascella, 8983, 3), (Atik, 8983, 9856), (1st Auriga, 485, 9834), (2nd Auriga, 485, 9835), (1st Brocard, 8980, 3), (5th Brocard, 485, 9873), (6th Brocard, 8980, 20), (circumorthic, 8981, 4), (1st circumperp, 8983, 3), (2nd circumperp, 8983, 3), (inner-Conway, 8983, 8), (Conway, 8983, 20), (2nd Conway, 8983, 962), (3rd Conway, 8983, 1), (1st Ehrmann, 13910, 3), (2nd Ehrmann, 8981, 576), (Euler, 485, 4), (2nd Euler, 8981, 3), (3rd Euler, 8983, 5), (4th Euler, 8983, 5), (excenters-midpoints, 13911, 10), (excenters-reflections, 8983, 7982), (excentral, 8983, 40), (extangents, 8981, 40), (extouch, 8987, 40), (2nd extouch, 8983, 4), (3rd extouch, 13912, 4), (Fuhrmann, 8988, 3), (inner-Garcia, 13913, 40), (outer-Garcia, 485, 40), (Gossard, 485, 12113), (inner-Grebe, 485, 5871), (outer-Grebe, 485, 5870), (hexyl, 8983, 1), (Honsberger, 8983, 390), (Hutson extouch, 13914, 40), (inner-Hutson, 8983, 9836), (Hutson intouch, 8983, 1), (outer-Hutson, 8983, 9837), (1st Hyacinth, 13915, 6102), (2nd Hyacinth, 13909, 6146), (incircle-circles, 8983, 1), (intangents, 8981, 1), (intouch, 8983, 1), (inverse-in-incircle, 8983, 942), (Johnson, 485, 3), (inner-Johnson, 485, 12114), (outer-Johnson, 485, 11500), (1st Johnson-Yff, 485, 55), (2nd Johnson-Yff, 485, 56), (1st Kenmotu diagonals, 8981, 371), (2nd Kenmotu diagonals, 8981, 372), (Kosnita, 8981, 3), (Lucas homothetic, 485, 9838), (Lucas(-1) homothetic, 485, 9839), (Mandart-incircle, 485, 6284), (McCay, 13908, 3), (medial, 485, 3), (midheight, 8991, 4), (5th mixtilinear, 485, 944), (6th mixtilinear, 8983, 1), (inner-Napoleon, 13916, 3), (outer-Napoleon, 13917, 3), (1st Neuberg, 8992, 3), (2nd Neuberg, 8993, 3), (orthic, 8981, 4), (orthocentroidal, 8994, 4), (1st orthosymmedial, 13918, 4), (2nd Pamfilos-Zhou, 8983, 7596), (reflection, 8995, 4), (1st Schiffler, 13919, 79), (2nd Schiffler, 13911, 80), (1st Sharygin, 8983, 9840), (submedial, 8981, 5), (tangential, 8981, 3), (tangential-midarc, 8983, 8091), (2nd tangential-midarc, 8983, 8092), (inner tri-equilateral, 8981, 15), (outer tri-equilateral, 8981, 16), (1st tri-squares-central, 13920, 13665), (2nd tri-squares-central, 13848, 13785), (4th tri-squares-central, 485, 486), (1st tri-squares, 13846, 3068), (2nd tri-squares, 13846, 3069), (3rd tri-squares, 13879, 485), (4th tri-squares, 13921, 486), (Trinh, 8981, 3), (inner-Vecten, 13921, 3), (outer-Vecten, 13879, 3), (X3-ABC reflections, 485, 382), (Yff central, 8983, 8351), (inner-Yff, 485, 1478), (outer-Yff, 485, 1479), (inner-Yff tangents, 485, 12115), (outer-Yff tangents, 485, 12116) List of triangles parallelogic to A'B'C' with ETC indexes of parallelogic centers: (1st anti-Brocard, 8997, 385), (6th anti-Brocard, 8997, 1691), (anti-orthocentroidal, 8998, 323), (1st Brocard, 8997, 6), (6th Brocard, 8997, 194), (inner-Garcia, 13922, 1), (orthocentroidal, 8998, 2), (1st orthosymmedial, 13923, 6), (1st Parry, 485, 9131), (2nd Parry, 485, 9979), (2nd Sharygin, 8983, 659) The appearance of (I, J) in the following list means that X(I)-of-A'B'C'=X(J):⁽¹⁾ (1, 8983), (2, 13846), (3, 8981), (4, 485), (5, 13925), (6, 13910), (8, 13911), (13, 13917), (14, 13916), (20, 1151), (40, 13912), (54, 8995), (64, 8991), (68, 13909), (74, 8994), (76, 8992), (80, 8988), (83, 8993), (84, 8987), (98, 8980), (99, 8997), (100, 13922), (104, 13913), (110, 8998), (112, 13923), (193, 6), (265, 13915), (485, 13879), (486, 13921), (488, 13882), (492, 590), (641, 13924), (671, 13908), (1297, 13918), (1327, 13920), (1328, 13848)

3rd tri-squares-central triangle

In this cell, A'B'C' is the 3rd tri-squares-central triangle of ABC.

Triangles directly similar to A'B'C': 1st Parry, 2nd Parry

Triangles inversely similar to A'B'C': 1st anti-Brocard, 6th anti-Brocard, anti-orthocentroidal, 1st Brocard, 6th Brocard, inner-Garcia, orthocentroidal, orthosymmedial

List of triangles perspective to A'B'C' and ETC index of perspector: (Note: An asterisk * means that both triangles are homothetic.)

(ABC*, 3068), (ABC-X3 reflections*, 9540), (anti-Aquila*, 13883), (anti-Ara*, 13884), (5th anti-Brocard*, 13885), (anti-Euler*, 13886), (anti-Mandart-incircle*, 13887), (anticomplementary*, 8972), (Aquila*, 13888), (Ara*, 13889), (1st Auriga*, 13890), (2nd Auriga*, 13891), (5th Brocard*, 13892), (Euler*, 371), (outer-Garcia*, 13893), (Gossard*, 13894), (inner-Grebe*, 8974), (outer-Grebe*, 8975), (Johnson*, 8976), (inner-Johnson*, 13895), (outer-Johnson*, 13896), (1st Johnson-Yff*, 13897), (2nd Johnson-Yff*, 13898), (Lucas homothetic*, 13899), (Lucas(-1) homothetic*, 13900), (Mandart-incircle*, 13901), (medial*, 590), (5th mixtilinear*, 13902), (inner-squares, 8966), (4th tri-squares-central*, 6), (3rd tri-squares, 485), (X3-ABC reflections*, 13903), (inner-Yff*, 13904), (outer-Yff*, 13905), (inner-Yff tangents*, 13906), (outer-Yff tangents*, 13907)

List of triangles orthologic to A'B'C' with ETC indexes of orthologic centers:

(ABC, 485, 4), (ABC-X3 reflections, 485, 20), (anti-Aquila, 485, 946), (anti-Ara, 485, 3575), (anti-Artzt, 13846, 2), (anti-Ascella, 8981, 1593), (1st anti-Brocard, 8980, 5999), (5th anti-Brocard, 485, 12110), (6th anti-Brocard, 8980, 2456), (1st anti-circumperp, 8981, 20), (anti-Conway, 8981, 578), (2nd anti-Conway, 8981, 389), (anti-Euler, 485, 4), (3rd anti-Euler, 8981, 12111), (4th anti-Euler, 8981, 6241), (anti-excenters-reflections, 8981, 4), (anti-Hutson intouch, 8981, 3), (anti-incircle-circles, 8981, 3), (anti-inverse-in-incircle, 8981, 4), (anti-Mandart-incircle, 485, 11500), (anti-McCay, 13908, 9855), (6th anti-mixtilinear, 8981, 3), (anti-orthocentroidal, 8994, 12112), (anticomplementary, 485, 20), (Aquila, 485, 5691), (Ara, 485, 3), (Aries, 13909, 9833), (Artzt, 13846, 2), (Ascella, 8983, 3), (Atik, 8983, 9856), (1st Auriga, 485, 9834), (2nd Auriga, 485, 9835), (1st Brocard, 8980, 3), (5th Brocard, 485, 9873), (6th Brocard, 8980, 20), (circumorthic, 8981, 4), (1st circumperp, 8983, 3), (2nd circumperp, 8983, 3), (inner-Conway, 8983, 8), (Conway, 8983, 20), (2nd Conway, 8983, 962), (3rd Conway, 8983, 1), (1st Ehrmann, 13910, 3), (2nd Ehrmann, 8981, 576), (Euler, 485, 4), (2nd Euler, 8981, 3), (3rd Euler, 8983, 5), (4th Euler, 8983, 5), (excenters-midpoints, 13911, 10), (excenters-reflections, 8983, 7982), (excentral, 8983, 40), (extangents, 8981, 40), (extouch, 8987, 40), (2nd extouch, 8983, 4), (3rd extouch, 13912, 4), (Fuhrmann, 8988, 3), (inner-Garcia, 13913, 40), (outer-Garcia, 485, 40), (Gossard, 485, 12113), (inner-Grebe, 485, 5871), (outer-Grebe, 485, 5870), (hexyl, 8983, 1), (Honsberger, 8983, 390), (Hutson extouch, 13914, 40), (inner-Hutson, 8983, 9836), (Hutson intouch, 8983, 1), (outer-Hutson, 8983, 9837), (1st Hyacinth, 13915, 6102), (2nd Hyacinth, 13909, 6146), (incircle-circles, 8983, 1), (intangents, 8981, 1), (intouch, 8983, 1), (inverse-in-incircle, 8983, 942), (Johnson, 485, 3), (inner-Johnson, 485, 12114), (outer-Johnson, 485, 11500), (1st Johnson-Yff, 485, 55), (2nd Johnson-Yff, 485, 56), (1st Kenmotu diagonals, 8981, 371), (2nd Kenmotu diagonals, 8981, 372), (Kosnita, 8981, 3), (Lucas homothetic, 485, 9838), (Lucas(-1) homothetic, 485, 9839), (Mandart-incircle, 485, 6284), (McCay, 13908, 3), (medial, 485, 3), (midheight, 8991, 4), (5th mixtilinear, 485, 944), (6th mixtilinear, 8983, 1), (inner-Napoleon, 13916, 3), (outer-Napoleon, 13917, 3), (1st Neuberg, 8992, 3), (2nd Neuberg, 8993, 3), (orthic, 8981, 4), (orthocentroidal, 8994, 4), (1st orthosymmedial, 13918, 4), (2nd Pamfilos-Zhou, 8983, 7596), (reflection, 8995, 4), (1st Schiffler, 13919, 79), (2nd Schiffler, 13911, 80), (1st Sharygin, 8983, 9840), (submedial, 8981, 5), (tangential, 8981, 3), (tangential-midarc, 8983, 8091), (2nd tangential-midarc, 8983, 8092), (inner tri-equilateral, 8981, 15), (outer tri-equilateral, 8981, 16), (1st tri-squares-central, 13920, 13665), (2nd tri-squares-central, 13848, 13785), (4th tri-squares-central, 485, 486), (1st tri-squares, 13846, 3068), (2nd tri-squares, 13846, 3069), (3rd tri-squares, 13879, 485), (4th tri-squares, 13921, 486), (Trinh, 8981, 3), (inner-Vecten, 13921, 3), (outer-Vecten, 13879, 3), (X3-ABC reflections, 485, 382), (Yff central, 8983, 8351), (inner-Yff, 485, 1478), (outer-Yff, 485, 1479), (inner-Yff tangents, 485, 12115), (outer-Yff tangents, 485, 12116)

List of triangles parallelogic to A'B'C' with ETC indexes of parallelogic centers:

(1st anti-Brocard, 8997, 385), (6th anti-Brocard, 8997, 1691), (anti-orthocentroidal, 8998, 323), (1st Brocard, 8997, 6), (6th Brocard, 8997, 194), (inner-Garcia, 13922, 1), (orthocentroidal, 8998, 2), (1st orthosymmedial, 13923, 6), (1st Parry, 485, 9131), (2nd Parry, 485, 9979), (2nd Sharygin, 8983, 659)

The appearance of (I, J) in the following list means that X(I)-of-A'B'C'=X(J):⁽¹⁾

(1, 8983), (2, 13846), (3, 8981), (4, 485), (5, 13925), (6, 13910), (8, 13911), (13, 13917), (14, 13916), (20, 1151), (40, 13912), (54, 8995), (64, 8991), (68, 13909), (74, 8994), (76, 8992), (80, 8988), (83, 8993), (84, 8987), (98, 8980), (99, 8997), (100, 13922), (104, 13913), (110, 8998), (112, 13923), (193, 6), (265, 13915), (485, 13879), (486, 13921), (488, 13882), (492, 590), (641, 13924), (671, 13908), (1297, 13918), (1327, 13920), (1328, 13848)

4th tri-squares triangle
In this cell, A'B'C' is the 4th tri-squares triangle of ABC. A'B'C' is directly similar to the Lucas(-1) tangents triangle List of triangles perspective to A'B'C' and ETC index of perspector: (Note: An asterisk * means that both triangles are homothetic.) (ABC, 3068), (ABC-X3 reflections, 9540), (anti-Aquila, 13883), (anti-Ara, 13884), (5th anti-Brocard, 13885), (anti-Euler, 13886), (anti-Mandart-incircle, 13887), (anticomplementary, 8972), (Aquila, 13888), (Ara, 13889), (1st Auriga, 13890), (2nd Auriga, 13891), (5th Brocard, 13892), (Euler, 371), (outer-Garcia, 13893), (Gossard, 13894), (inner-Grebe, 8974), (outer-Grebe, 8975), (Johnson, 8976), (inner-Johnson, 13895), (outer-Johnson, 13896), (1st Johnson-Yff, 13897), (2nd Johnson-Yff, 13898), (Lucas homothetic, 13899), (Lucas(-1) homothetic, 13900), (Mandart-incircle, 13901), (medial, 590), (5th mixtilinear, 13902), (inner-squares, 8966), (4th tri-squares-central, 6), (3rd tri-squares, 485), (X3-ABC reflections, 13903), (inner-Yff, 13904), (outer-Yff, 13905), (inner-Yff tangents, 13906), (outer-Yff tangents, 13907) List of triangles orthologic to A'B'C' with ETC indexes of orthologic centers: (ABC, 485, 4), (ABC-X3 reflections, 485, 20), (anti-Aquila, 485, 946), (anti-Ara, 485, 3575), (anti-Artzt, 13846, 2), (anti-Ascella, 8981, 1593), (1st anti-Brocard, 8980, 5999), (5th anti-Brocard, 485, 12110), (6th anti-Brocard, 8980, 2456), (1st anti-circumperp, 8981, 20), (anti-Conway, 8981, 578), (2nd anti-Conway, 8981, 389), (anti-Euler, 485, 4), (3rd anti-Euler, 8981, 12111), (4th anti-Euler, 8981, 6241), (anti-excenters-reflections, 8981, 4), (anti-Hutson intouch, 8981, 3), (anti-incircle-circles, 8981, 3), (anti-inverse-in-incircle, 8981, 4), (anti-Mandart-incircle, 485, 11500), (anti-McCay, 13908, 9855), (6th anti-mixtilinear, 8981, 3), (anti-orthocentroidal, 8994, 12112), (anticomplementary, 485, 20), (Aquila, 485, 5691), (Ara, 485, 3), (Aries, 13909, 9833), (Artzt, 13846, 2), (Ascella, 8983, 3), (Atik, 8983, 9856), (1st Auriga, 485, 9834), (2nd Auriga, 485, 9835), (1st Brocard, 8980, 3), (5th Brocard, 485, 9873), (6th Brocard, 8980, 20), (circumorthic, 8981, 4), (1st circumperp, 8983, 3), (2nd circumperp, 8983, 3), (inner-Conway, 8983, 8), (Conway, 8983, 20), (2nd Conway, 8983, 962), (3rd Conway, 8983, 1), (1st Ehrmann, 13910, 3), (2nd Ehrmann, 8981, 576), (Euler, 485, 4), (2nd Euler, 8981, 3), (3rd Euler, 8983, 5), (4th Euler, 8983, 5), (excenters-midpoints, 13911, 10), (excenters-reflections, 8983, 7982), (excentral, 8983, 40), (extangents, 8981, 40), (extouch, 8987, 40), (2nd extouch, 8983, 4), (3rd extouch, 13912, 4), (Fuhrmann, 8988, 3), (inner-Garcia, 13913, 40), (outer-Garcia, 485, 40), (Gossard, 485, 12113), (inner-Grebe, 485, 5871), (outer-Grebe, 485, 5870), (hexyl, 8983, 1), (Honsberger, 8983, 390), (Hutson extouch, 13914, 40), (inner-Hutson, 8983, 9836), (Hutson intouch, 8983, 1), (outer-Hutson, 8983, 9837), (1st Hyacinth, 13915, 6102), (2nd Hyacinth, 13909, 6146), (incircle-circles, 8983, 1), (intangents, 8981, 1), (intouch, 8983, 1), (inverse-in-incircle, 8983, 942), (Johnson, 485, 3), (inner-Johnson, 485, 12114), (outer-Johnson, 485, 11500), (1st Johnson-Yff, 485, 55), (2nd Johnson-Yff, 485, 56), (1st Kenmotu diagonals, 8981, 371), (2nd Kenmotu diagonals, 8981, 372), (Kosnita, 8981, 3), (Lucas homothetic, 485, 9838), (Lucas(-1) homothetic, 485, 9839), (Mandart-incircle, 485, 6284), (McCay, 13908, 3), (medial, 485, 3), (midheight, 8991, 4), (5th mixtilinear, 485, 944), (6th mixtilinear, 8983, 1), (inner-Napoleon, 13916, 3), (outer-Napoleon, 13917, 3), (1st Neuberg, 8992, 3), (2nd Neuberg, 8993, 3), (orthic, 8981, 4), (orthocentroidal, 8994, 4), (1st orthosymmedial, 13918, 4), (2nd Pamfilos-Zhou, 8983, 7596), (reflection, 8995, 4), (1st Schiffler, 13919, 79), (2nd Schiffler, 13911, 80), (1st Sharygin, 8983, 9840), (submedial, 8981, 5), (tangential, 8981, 3), (tangential-midarc, 8983, 8091), (2nd tangential-midarc, 8983, 8092), (inner tri-equilateral, 8981, 15), (outer tri-equilateral, 8981, 16), (1st tri-squares-central, 13920, 13665), (2nd tri-squares-central, 13848, 13785), (4th tri-squares-central, 485, 486), (1st tri-squares, 13846, 3068), (2nd tri-squares, 13846, 3069), (3rd tri-squares, 13879, 485), (4th tri-squares, 13921, 486), (Trinh, 8981, 3), (inner-Vecten, 13921, 3), (outer-Vecten, 13879, 3), (X3-ABC reflections, 485, 382), (Yff central, 8983, 8351), (inner-Yff, 485, 1478), (outer-Yff, 485, 1479), (inner-Yff tangents, 485, 12115), (outer-Yff tangents, 485, 12116) List of triangles parallelogic to A'B'C' with ETC indexes of parallelogic centers: (1st Parry, 486, 13316), (2nd Parry, 486, 13319) The appearance of (I, J) in the following list means that X(I)-of-A'B'C'=X(J):⁽¹⁾ (2, 13847), (3, 13933), (4, 13934), (13, 13928), (14, 13929), (486, 486), (487, 1152), (642, 13966), (1132, 13881), (1328, 2) The radical center of the 3rd tri-squares circles is X(486).

4th tri-squares triangle

In this cell, A'B'C' is the 4th tri-squares triangle of ABC.

A'B'C' is directly similar to the Lucas(-1) tangents triangle

List of triangles perspective to A'B'C' and ETC index of perspector: (Note: An asterisk * means that both triangles are homothetic.)

List of triangles orthologic to A'B'C' with ETC indexes of orthologic centers:

List of triangles parallelogic to A'B'C' with ETC indexes of parallelogic centers: (1st Parry, 486, 13316), (2nd Parry, 486, 13319)

The appearance of (I, J) in the following list means that X(I)-of-A'B'C'=X(J):⁽¹⁾

(2, 13847), (3, 13933), (4, 13934), (13, 13928), (14, 13929), (486, 486), (487, 1152), (642, 13966), (1132, 13881), (1328, 2)

The radical center of the 3rd tri-squares circles is X(486).

4th tri-squares-central triangle
In this cell, A'B'C' is the 4th tri-squares-central triangle of ABC. Triangles directly similar to A'B'C': 1st Parry, 2nd Parry Triangles inversely similar to A'B'C': 1st anti-Brocard, 6th anti-Brocard, anti-orthocentroidal, 1st Brocard, 6th Brocard, inner-Garcia, orthocentroidal, orthosymmedial List of triangles perspective to A'B'C' and ETC index of perspector: (Note: An asterisk * means that both triangles are homothetic.) (ABC, 3069), (ABC-X3 reflections, 13935), (anti-Aquila, 13936), (anti-Ara, 13937), (5th anti-Brocard, 13938), (anti-Euler, 13939), (anti-Mandart-incircle, 13940), (anticomplementary, 13941), (Aquila, 13942), (Ara, 13943), (1st Auriga, 13944), (2nd Auriga, 13945), (5th Brocard, 13946), (Euler, 372), (outer-Garcia, 13947), (Gossard, 13948), (inner-Grebe, 13949), (outer-Grebe, 13950), (Johnson, 13951), (inner-Johnson, 13952), (outer-Johnson, 13953), (1st Johnson-Yff, 13954), (2nd Johnson-Yff, 13955), (Lucas homothetic, 13956), (Lucas(-1) homothetic, 13957), (Mandart-incircle, 13958), (medial, 615), (5th mixtilinear, 13959), (outer-squares, 13960), (3rd tri-squares-central, 6), (4th tri-squares, 486), (X3-ABC reflections, 13961), (inner-Yff, 13962), (outer-Yff, 13963), (inner-Yff tangents, 13964), (outer-Yff tangents, 13965) List of triangles orthologic to A'B'C' with ETC indexes of orthologic centers: (ABC, 486, 4), (ABC-X3 reflections, 486, 20), (anti-Aquila, 486, 946), (anti-Ara, 486, 3575), (anti-Artzt, 13847, 2), (anti-Ascella, 13966, 1593), (1st anti-Brocard, 13967, 5999), (5th anti-Brocard, 486, 12110), (6th anti-Brocard, 13967, 2456), (1st anti-circumperp, 13966, 20), (anti-Conway, 13966, 578), (2nd anti-Conway, 13966, 389), (anti-Euler, 486, 4), (3rd anti-Euler, 13966, 12111), (4th anti-Euler, 13966, 6241), (anti-excenters-reflections, 13966, 4), (anti-Hutson intouch, 13966, 3), (anti-incircle-circles, 13966, 3), (anti-inverse-in-incircle, 13966, 4), (anti-Mandart-incircle, 486, 11500), (anti-McCay, 13968, 9855), (6th anti-mixtilinear, 13966, 3), (anti-orthocentroidal, 13969, 12112), (anticomplementary, 486, 20), (Aquila, 486, 5691), (Ara, 486, 3), (Aries, 13970, 9833), (Artzt, 13847, 2), (Ascella, 13971, 3), (Atik, 13971, 9856), (1st Auriga, 486, 9834), (2nd Auriga, 486, 9835), (1st Brocard, 13967, 3), (5th Brocard, 486, 9873), (6th Brocard, 13967, 20), (circumorthic, 13966, 4), (1st circumperp, 13971, 3), (2nd circumperp, 13971, 3), (inner-Conway, 13971, 8), (Conway, 13971, 20), (2nd Conway, 13971, 962), (3rd Conway, 13971, 1), (1st Ehrmann, 13972, 3), (2nd Ehrmann, 13966, 576), (Euler, 486, 4), (2nd Euler, 13966, 3), (3rd Euler, 13971, 5), (4th Euler, 13971, 5), (excenters-midpoints, 13973, 10), (excenters-reflections, 13971, 7982), (excentral, 13971, 40), (extangents, 13966, 40), (extouch, 13974, 40), (2nd extouch, 13971, 4), (3rd extouch, 13975, 4), (Fuhrmann, 13976, 3), (inner-Garcia, 13977, 40), (outer-Garcia, 486, 40), (Gossard, 486, 12113), (inner-Grebe, 486, 5871), (outer-Grebe, 486, 5870), (hexyl, 13971, 1), (Honsberger, 13971, 390), (Hutson extouch, 13978, 40), (inner-Hutson, 13971, 9836), (Hutson intouch, 13971, 1), (outer-Hutson, 13971, 9837), (1st Hyacinth, 13979, 6102), (2nd Hyacinth, 13970, 6146), (incircle-circles, 13971, 1), (intangents, 13966, 1), (intouch, 13971, 1), (inverse-in-incircle, 13971, 942), (Johnson, 486, 3), (inner-Johnson, 486, 12114), (outer-Johnson, 486, 11500), (1st Johnson-Yff, 486, 55), (2nd Johnson-Yff, 486, 56), (1st Kenmotu diagonals, 13966, 371), (2nd Kenmotu diagonals, 13966, 372), (Kosnita, 13966, 3), (Lucas homothetic, 486, 9838), (Lucas(-1) homothetic, 486, 9839), (Mandart-incircle, 486, 6284), (McCay, 13968, 3), (medial, 486, 3), (midheight, 13980, 4), (5th mixtilinear, 486, 944), (6th mixtilinear, 13971, 1), (inner-Napoleon, 13981, 3), (outer-Napoleon, 13982, 3), (1st Neuberg, 13983, 3), (2nd Neuberg, 13984, 3), (orthic, 13966, 4), (orthocentroidal, 13969, 4), (1st orthosymmedial, 13985, 4), (2nd Pamfilos-Zhou, 13971, 7596), (reflection, 13986, 4), (1st Schiffler, 13987, 79), (2nd Schiffler, 13973, 80), (1st Sharygin, 13971, 9840), (submedial, 13966, 5), (tangential, 13966, 3), (tangential-midarc, 13971, 8091), (2nd tangential-midarc, 13971, 8092), (inner tri-equilateral, 13966, 15), (outer tri-equilateral, 13966, 16), (1st tri-squares-central, 13988, 13665), (2nd tri-squares-central, 13849, 13785), (3rd tri-squares-central, 486, 485), (1st tri-squares, 13847, 3068), (2nd tri-squares, 13847, 3069), (3rd tri-squares, 13880, 485), (4th tri-squares, 13933, 486), (Trinh, 13966, 3), (inner-Vecten, 13933, 3), (outer-Vecten, 13880, 3), (X3-ABC reflections, 486, 382), (Yff central, 13971, 8351), (inner-Yff, 486, 1478), (outer-Yff, 486, 1479), (inner-Yff tangents, 486, 12115), (outer-Yff tangents, 486, 12116) List of triangles parallelogic to A'B'C' with ETC indexes of parallelogic centers: (1st anti-Brocard, 13989, 385), (6th anti-Brocard, 13989, 1691), (anti-orthocentroidal, 13990, 323), (1st Brocard, 13989, 6), (6th Brocard, 13989, 194), (inner-Garcia, 13991, 1), (orthocentroidal, 13990, 2), (1st orthosymmedial, 13992, 6), (1st Parry, 486, 9131), (2nd Parry, 486, 9979), (2nd Sharygin, 13971, 659) The appearance of (I, J) in the following list means that X(I)-of-A'B'C'=X(J):⁽¹⁾ (1, 13971), (2, 13847), (3, 13966), (4, 486), (5, 13993), (6, 13972), (8, 13973), (13, 13982), (14, 13981), (20, 1152), (40, 13975), (54, 13986), (64, 13980), (68, 13970), (74, 13969), (76, 13983), (80, 13976), (83, 13984), (84, 13974), (98, 13967), (99, 13989), (100, 13991), (104, 13977), (110, 13990), (112, 13992), (193, 6), (265, 13979), (485, 13880), (486, 13933), (487, 13934), (491, 615), (671, 13968), (1297, 13985), (1327, 13988), (1328, 13849)

4th tri-squares-central triangle

In this cell, A'B'C' is the 4th tri-squares-central triangle of ABC.

Triangles directly similar to A'B'C': 1st Parry, 2nd Parry

Triangles inversely similar to A'B'C': 1st anti-Brocard, 6th anti-Brocard, anti-orthocentroidal, 1st Brocard, 6th Brocard, inner-Garcia, orthocentroidal, orthosymmedial

List of triangles perspective to A'B'C' and ETC index of perspector: (Note: An asterisk * means that both triangles are homothetic.)

(ABC*, 3069), (ABC-X3 reflections*, 13935), (anti-Aquila*, 13936), (anti-Ara*, 13937), (5th anti-Brocard*, 13938), (anti-Euler*, 13939), (anti-Mandart-incircle*, 13940), (anticomplementary*, 13941), (Aquila*, 13942), (Ara*, 13943), (1st Auriga*, 13944), (2nd Auriga*, 13945), (5th Brocard*, 13946), (Euler*, 372), (outer-Garcia*, 13947), (Gossard*, 13948), (inner-Grebe*, 13949), (outer-Grebe*, 13950), (Johnson*, 13951), (inner-Johnson*, 13952), (outer-Johnson*, 13953), (1st Johnson-Yff*, 13954), (2nd Johnson-Yff*, 13955), (Lucas homothetic*, 13956), (Lucas(-1) homothetic*, 13957), (Mandart-incircle*, 13958), (medial*, 615), (5th mixtilinear*, 13959), (outer-squares, 13960), (3rd tri-squares-central*, 6), (4th tri-squares, 486), (X3-ABC reflections*, 13961), (inner-Yff*, 13962), (outer-Yff*, 13963), (inner-Yff tangents*, 13964), (outer-Yff tangents*, 13965)

List of triangles orthologic to A'B'C' with ETC indexes of orthologic centers:

(ABC, 486, 4), (ABC-X3 reflections, 486, 20), (anti-Aquila, 486, 946), (anti-Ara, 486, 3575), (anti-Artzt, 13847, 2), (anti-Ascella, 13966, 1593), (1st anti-Brocard, 13967, 5999), (5th anti-Brocard, 486, 12110), (6th anti-Brocard, 13967, 2456), (1st anti-circumperp, 13966, 20), (anti-Conway, 13966, 578), (2nd anti-Conway, 13966, 389), (anti-Euler, 486, 4), (3rd anti-Euler, 13966, 12111), (4th anti-Euler, 13966, 6241), (anti-excenters-reflections, 13966, 4), (anti-Hutson intouch, 13966, 3), (anti-incircle-circles, 13966, 3), (anti-inverse-in-incircle, 13966, 4), (anti-Mandart-incircle, 486, 11500), (anti-McCay, 13968, 9855), (6th anti-mixtilinear, 13966, 3), (anti-orthocentroidal, 13969, 12112), (anticomplementary, 486, 20), (Aquila, 486, 5691), (Ara, 486, 3), (Aries, 13970, 9833), (Artzt, 13847, 2), (Ascella, 13971, 3), (Atik, 13971, 9856), (1st Auriga, 486, 9834), (2nd Auriga, 486, 9835), (1st Brocard, 13967, 3), (5th Brocard, 486, 9873), (6th Brocard, 13967, 20), (circumorthic, 13966, 4), (1st circumperp, 13971, 3), (2nd circumperp, 13971, 3), (inner-Conway, 13971, 8), (Conway, 13971, 20), (2nd Conway, 13971, 962), (3rd Conway, 13971, 1), (1st Ehrmann, 13972, 3), (2nd Ehrmann, 13966, 576), (Euler, 486, 4), (2nd Euler, 13966, 3), (3rd Euler, 13971, 5), (4th Euler, 13971, 5), (excenters-midpoints, 13973, 10), (excenters-reflections, 13971, 7982), (excentral, 13971, 40), (extangents, 13966, 40), (extouch, 13974, 40), (2nd extouch, 13971, 4), (3rd extouch, 13975, 4), (Fuhrmann, 13976, 3), (inner-Garcia, 13977, 40), (outer-Garcia, 486, 40), (Gossard, 486, 12113), (inner-Grebe, 486, 5871), (outer-Grebe, 486, 5870), (hexyl, 13971, 1), (Honsberger, 13971, 390), (Hutson extouch, 13978, 40), (inner-Hutson, 13971, 9836), (Hutson intouch, 13971, 1), (outer-Hutson, 13971, 9837), (1st Hyacinth, 13979, 6102), (2nd Hyacinth, 13970, 6146), (incircle-circles, 13971, 1), (intangents, 13966, 1), (intouch, 13971, 1), (inverse-in-incircle, 13971, 942), (Johnson, 486, 3), (inner-Johnson, 486, 12114), (outer-Johnson, 486, 11500), (1st Johnson-Yff, 486, 55), (2nd Johnson-Yff, 486, 56), (1st Kenmotu diagonals, 13966, 371), (2nd Kenmotu diagonals, 13966, 372), (Kosnita, 13966, 3), (Lucas homothetic, 486, 9838), (Lucas(-1) homothetic, 486, 9839), (Mandart-incircle, 486, 6284), (McCay, 13968, 3), (medial, 486, 3), (midheight, 13980, 4), (5th mixtilinear, 486, 944), (6th mixtilinear, 13971, 1), (inner-Napoleon, 13981, 3), (outer-Napoleon, 13982, 3), (1st Neuberg, 13983, 3), (2nd Neuberg, 13984, 3), (orthic, 13966, 4), (orthocentroidal, 13969, 4), (1st orthosymmedial, 13985, 4), (2nd Pamfilos-Zhou, 13971, 7596), (reflection, 13986, 4), (1st Schiffler, 13987, 79), (2nd Schiffler, 13973, 80), (1st Sharygin, 13971, 9840), (submedial, 13966, 5), (tangential, 13966, 3), (tangential-midarc, 13971, 8091), (2nd tangential-midarc, 13971, 8092), (inner tri-equilateral, 13966, 15), (outer tri-equilateral, 13966, 16), (1st tri-squares-central, 13988, 13665), (2nd tri-squares-central, 13849, 13785), (3rd tri-squares-central, 486, 485), (1st tri-squares, 13847, 3068), (2nd tri-squares, 13847, 3069), (3rd tri-squares, 13880, 485), (4th tri-squares, 13933, 486), (Trinh, 13966, 3), (inner-Vecten, 13933, 3), (outer-Vecten, 13880, 3), (X3-ABC reflections, 486, 382), (Yff central, 13971, 8351), (inner-Yff, 486, 1478), (outer-Yff, 486, 1479), (inner-Yff tangents, 486, 12115), (outer-Yff tangents, 486, 12116)

List of triangles parallelogic to A'B'C' with ETC indexes of parallelogic centers:

(1st anti-Brocard, 13989, 385), (6th anti-Brocard, 13989, 1691), (anti-orthocentroidal, 13990, 323), (1st Brocard, 13989, 6), (6th Brocard, 13989, 194), (inner-Garcia, 13991, 1), (orthocentroidal, 13990, 2), (1st orthosymmedial, 13992, 6), (1st Parry, 486, 9131), (2nd Parry, 486, 9979), (2nd Sharygin, 13971, 659)

The appearance of (I, J) in the following list means that X(I)-of-A'B'C'=X(J):⁽¹⁾

(1, 13971), (2, 13847), (3, 13966), (4, 486), (5, 13993), (6, 13972), (8, 13973), (13, 13982), (14, 13981), (20, 1152), (40, 13975), (54, 13986), (64, 13980), (68, 13970), (74, 13969), (76, 13983), (80, 13976), (83, 13984), (84, 13974), (98, 13967), (99, 13989), (100, 13991), (104, 13977), (110, 13990), (112, 13992), (193, 6), (265, 13979), (485, 13880), (486, 13933), (487, 13934), (491, 615), (671, 13968), (1297, 13985), (1327, 13988), (1328, 13849)

⁽¹⁾: Centers calculated for 1 ≤ I ≤ 2000.

X(13873) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES TO 1st ANTI-BROCARD

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

The reciprocal orthologic center of these triangles is X(9868)

X(13873) lies on these lines: {2,7598}, {3,115}, {98,485}, {114,371}, {183,6228}, {230,2459}, {385,9868}, {486,8781}, {542,8980}, {590,6230}, {620,11315}, {641,3102}, {1513,2460}, {2023,3103}, {3564,6231}, {6055,13850}, {6108,13875}, {6109,13876}, {6118,9478}, {6561,9757}, {6564,9756}, {6722,11313}, {7793,9992}, {7806,8317}, {8305,13088}, {12602,12968}

X(13873) = midpoint of X(385) and X(9868)
X(13873) = complement of X(33341)
X(13873) = orthoptic-circle-of-Steiner-inellipse-inverse-of-X(13521)
X(13873) = orthologic center of these triangles: 3rd tri-squares to 6th anti-Brocard
X(13873) = orthologic center of these triangles: 3rd tri-squares to 1st Brocard
X(13873) = {X(115), X(6036)}-harmonic conjugate of X(13926)

X(13874) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES TO ANTI-MCCAY

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

The reciprocal orthologic center of these triangles is X(9893)

X(13874) lies on these lines: {115,1991}, {485,489}, {543,8997}, {590,9894}, {599,626}, {2482,13663}, {6222,12602}, {8593,13651}, {8859,9893}, {8860,13088}, {9883,13640}, {9892,13676}

X(13875) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES TO INNER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(6305)

X(13875) lies on these lines: {14,485}, {115,590}, {395,6303}, {531,10667}, {619,6304}, {3643,6301}, {5460,13850}, {6108,13873}, {9113,13651}

X(13876) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES TO OUTER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(6304)

X(13876) lies on these lines: {13,485}, {115,590}, {396,6302}, {530,10668}, {618,6305}, {3642,6300}, {5459,13850}, {6109,13873}, {9112,13651}

X(13877) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES TO 1st NEUBERG

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

The reciprocal orthologic center of these triangles is X(6312)

X(13877) lies on these lines: {3,6222}, {39,13882}, {76,485}, {183,6312}, {488,6194}, {511,6289}, {538,8992}, {590,6314}, {639,6393}, {641,3102}, {3094,3763}, {6318,13707}

X(13877) = reflection of X(3102) in X(641)

X(13878) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(6311)

X(13878) lies on these lines: {83,485}, {590,6313}, {754,8993}, {6118,9478}, {6222,6287}, {6292,13882}, {6311,11174}, {6317,13709}, {6704,7834}

X(13879) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO 3rd TRI-SQUARES

Barycentrics (SA-2*S-SW)*(SA-3*S-2*SW) : :
X(13879) = 3*X(13846)-X(13882) = 3*X(13846)-2*X(13924)

The reciprocal orthologic center of these triangles is X(485)

X(13879) lies on these lines: {4,371}, {6,6118}, {30,13920}, {488,8972}, {590,641}, {3564,13909}, {6278,8975}, {6279,8974}, {6289,8976}, {6329,13933}, {7583,9739}, {7981,13902}, {8970,11209}, {8981,12974}, {9540,12124}, {9907,13888}, {9922,13889}, {9987,13892}, {10068,13904}, {12148,13884}, {12211,13885}, {12344,13887}, {12602,13903}, {12788,13893}, {12800,13894}, {12929,13895}, {12939,13896}, {12949,13897}, {12959,13898}, {12968,13701}, {13005,13900}, {13082,13901}, {13134,13906}, {13135,13907}

X(13879) = reflection of X(i) in X(j) for these (i,j): (641,8180), (13882,13924)
X(13879) = {X(13910), X(13925)}-harmonic conjugate of X(13921)

X(13880) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 3rd TRI-SQUARES

Barycentrics 6*S^2+(3*SA-SW)*S-(SA+2*SW)*(SB+SC) : :
X(13880) = 5*X(13847)+X(13850) = 3*X(13847)+X(13881) = 3*X(13850)-5*X(13881)

The reciprocal orthologic center of these triangles is X(485)

X(13880) lies on these lines: {6,6118}, {30,13988}, {39,615}, {372,6250}, {485,3069}, {488,13834}, {524,7862}, {3564,13933}, {3767,6228}, {6278,13950}, {6279,13949}, {6289,13951}, {7981,13959}, {8252,13882}, {8253,13924}, {9907,13942}, {9922,13943}, {9987,13946}, {10068,13962}, {10084,13963}, {12124,13935}, {12148,13937}, {12211,13938}, {12222,13771}, {12257,13939}, {12269,13936}, {12344,13940}, {12602,13961}, {12788,13947}, {12800,13948}, {12929,13952}, {12939,13953}, {12949,13954}, {12959,13955}, {12969,13847}, {13004,13956}, {13005,13957}, {13082,13958}, {13134,13964}, {13135,13965}

X(13880) = {X(13972), X(13993)}-harmonic conjugate of X(13933)

X(13881) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES TO 4th TRI-SQUARES

Barycentrics a^4-(b^2+c^2)*a^2+2*(b^2-c^2)^2 : :
Barycentrics (cos B) (cot C + cot A - cot B) + (cos C) (cot A + cot B - cot C) : :
X(13881) = 3*X(2)+X(2996) = X(4)+3*X(7612) = 11*X(5056)-3*X(9742) = X(13651)-7*X(13711) = 3*X(13846)-4*X(13921) = X(13846)+2*X(13932) = X(13847)+2*X(13850) = 3*X(13847)-4*X(13880) = 3*X(13850)+2*X(13880) = 2*X(13921)+3*X(13932)

The reciprocal orthologic center of these triangles is X(13881).

Let DEF be the orthic triangle. Let Ab be the point in which the line through B perpendicular to AB meets the perpendicular bisector of segment BD . Let Ac be the point in which the line through C perpendicular to AC meets the perpendicular bisector of segment CD. Define Bc, Ba, Ca, Cb cyclically. The six ponts Ab, Ac, Bc, Ba, Ca, Cb lie on a conic with center X(13881). A barycentric equation for this conic follows:

0 = cyclic sum of (b^2+c^2-a^2)(5 a^4-12 a^2 (b^2+c^2)+7 b^4+2 b^2 c^2+7 c^4)x^2 - 2 (11a^6-13 a^4 (b^2+c^2)+a^2 (b^2-c^2)^2+(b^2-c^2)^2 (b^2+c^2))y z. (Angel Montesdeoca, Decemberr 23, 2018)

X(13881) lies on these lines: {2,1975}, {3,115}, {4,230}, {5,6}, {20,5210}, {30,5023}, {32,381}, {39,1656}, {45,3634}, {53,3089}, {76,2023}, {140,2549}, {141,5490}, {148,7907}, {172,10895}, {183,2896}, {187,382}, {220,1329}, {315,8667}, {371,12601}, {372,12602}, {385,7773}, {393,6622}, {403,2207}, {427,1611}, {487,590}, {488,615}, {538,7862}, {546,7737}, {547,7739}, {548,5585}, {569,9604}, {574,3526}, {599,626}, {625,5111}, {631,3054}, {641,6119}, {642,6118}, {671,7782}, {1003,7857}, {1007,6392}, {1078,7610}, {1180,7571}, {1184,5133}, {1194,7539}, {1384,3843}, {1407,7363}, {1504,8976}, {1505,13951}, {1506,5055}, {1570,11898}, {1571,11231}, {1572,9955}, {1598,1609}, {1640,6587}, {1657,5206}, {1853,2450}, {1914,10896}, {1995,9608}, {2241,9669}, {2242,9654}, {2422,3224}, {2476,5275}, {2485,8574}, {2963,7393}, {3003,3199}, {3055,5067}, {3068,12221}, {3069,12222}, {3070,12256}, {3071,12257}, {3090,3815}, {3091,7735}, {3094,3763}, {3172,6103}, {3291,5094}, {3297,10068}, {3298,10067}, {3413,6177}, {3414,6178}, {3545,5306}, {3614,9596}, {3642,6300}, {3643,6301}, {3734,7886}, {3785,13468}, {3788,6722}, {3851,5475}, {5007,5072}, {5024,5070}, {5038,7617}, {5054,11648}, {5056,7736}, {5058,13785}, {5068,5304}, {5071,9300}, {5073,6781}, {5079,7603}, {5141,5276}, {5346,7753}, {5432,9598}, {5433,9597}, {5523,7505}, {6144,7759}, {6222,6251}, {6250,6399}, {6409,12123}, {6410,12124}, {6421,10577}, {6422,10576}, {6423,6564}, {6424,6565}, {6459,9602}, {6673,11311}, {6674,11312}, {6680,11286}, {6683,7902}, {6704,7834}, {7173,9599}, {7387,8553}, {7486,9606}, {7507,10311}, {7509,9609}, {7530,11063}, {7547,10312}, {7615,8369}, {7752,7754}, {7775,7805}, {7780,7825}, {7787,10807}, {7788,7912}, {7790,11285}, {7795,8361}, {7797,11174}, {7800,8556}, {7807,11185}, {7815,7861}, {7816,11288}, {7833,8860}, {7839,11163}, {7867,9466}, {7868,7901}, {7879,7934}, {7968,9907}, {7969,9906}, {7988,9575}, {8588,11742}, {8719,9754}, {9306,9603}, {9619,11230}, {9620,9956}, {9670,10987}, {9743,11257}, {11623,11646}, {12510,13935}, {12962,13846}, {12969,13847}, {13087,13804}, {13088,13684}

X(13881) = midpoint of X(i) and X(j) for these {i,j}: {485,486}, {2996,6337}, {22591,22592}
X(13881) = reflection of X(i) in X(j) for these (i,j): (641,6119), (642,6118), (10008,141)
X(13881) = complement of X(6337)
X(13881) = orthologic center of these triangles: 3rd tri-squares to inner-Vecten
X(13881) = harmonic center of nine-point and 2nd Lemoine circles
X(13881) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,2996,6337), (2,5254,5013), (4,230,3053), (5,3767,6), (5,5305,2548), (76,7887,7778), (76,7899,7881), (115,7746,3), (115,7749,7748), (183,5025,7784), (625,7751,7776), (1384,3843,7747), (2165,9722,6), (2548,3767,5305), (2548,5305,6), (5490,5491,10008), (7746,7748,7749), (7748,7749,3), (7881,7887,7899), (7881,7899,7778)

X(13882) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES TO OUTER-VECTEN

Barycentrics (SA+2*S+SW)*(SA-S-SW) : :
X(13882) = 3*X(13846)-2*X(13879) = 3*X(13846)-4*X(13924)

The reciprocal orthologic center of these triangles is X(485)

X(13882) lies on these lines: {2,493}, {3,485}, {6,641}, {39,13877}, {114,371}, {488,3068}, {618,6305}, {619,6304}, {642,6118}, {1504,7888}, {1691,6311}, {2066,12959}, {2067,12949}, {2459,8960}, {2482,13663}, {3094,11316}, {3564,8909}, {3589,6387}, {5870,6811}, {6292,13878}, {6421,7807}, {6503,9725}, {7389,9600}, {7969,12788}, {8252,13880}, {8299,9661}, {8854,12967}, {8972,12222}, {9646,10068}, {9757,10839}, {9907,13893}, {12510,13886}, {12968,13701}, {13821,13850}

X(13882) = midpoint of X(5490) and X(6462)
X(13882) = reflection of X(13879) in X(13924)
X(13882) = complement of X(5490)
X(13882) = {X(2), X(6462)}-harmonic conjugate of X(5490)

X(13883) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND ANTI-AQUILA

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

X(13883) lies on these lines: {1,1336}, {2,13893}, {3,13912}, {4,1702}, {6,10}, {8,7585}, {11,8988}, {20,9616}, {40,1587}, {81,6348}, {165,6460}, {214,13922}, {355,3311}, {371,515}, {372,6684}, {376,9582}, {485,946}, {486,10175}, {516,3070}, {517,7583}, {519,7969}, {551,13846}, {590,1125}, {605,1724}, {606,5264}, {615,3634}, {944,9583}, {950,2066}, {1124,1210}, {1131,9812}, {1151,4297}, {1152,10164}, {1385,8981}, {1386,13910}, {1588,5587}, {1698,3069}, {1703,5657}, {1737,3299}, {1743,7090}, {2067,10106}, {2362,4848}, {2646,13901}, {3295,13887}, {3297,11019}, {3301,10039}, {3312,13975}, {3452,5393}, {3576,9540}, {3616,8972}, {3624,13959}, {3911,6502}, {5090,5410}, {5418,10165}, {5603,13886}, {5691,6459}, {5731,9615}, {5790,6417}, {5818,7582}, {5886,8976}, {5901,13925}, {7584,9956}, {7586,9780}, {8960,13464}, {8974,11370}, {8975,11371}, {8980,11710}, {8987,12114}, {8991,12262}, {8992,12263}, {8993,12264}, {8994,11709}, {8995,12266}, {8997,11711}, {8998,11720}, {9646,13411}, {10172,10577}, {10246,13903}, {11108,13940}, {11231,13966}, {11363,13884}, {11364,13885}, {11365,13889}, {11366,13890}, {11368,13892}, {11373,13895}, {11374,13896}, {11375,13897}, {11376,13898}, {11377,13899}, {11378,13900}, {11705,13917}, {11706,13916}, {11715,13913}, {11722,13923}, {11831,13894}, {12258,13908}, {12259,13909}, {12260,13914}, {12261,13915}, {12265,13918}, {12267,13919}, {12268,13921}, {12699,13665}, {13667,13920}, {13787,13848}

X(13883) = reflection of X(5688) in X(10)
X(13883) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3068,8983), (1,13888,13902), (6,10,13936), (6,13911,10), (590,7968,1125), (3068,13902,13888), (5657,7581,1703), (7586,9780,13947), (13888,13902,8983)

X(13884) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND ANTI-ARA

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

X(13884) lies on these lines: {2,5410}, {4,3590}, {5,10880}, {6,468}, {24,7583}, {25,3068}, {33,13901}, {235,371}, {427,590}, {428,13846}, {485,3575}, {1112,8998}, {1151,1885}, {1368,11417}, {1587,3515}, {1593,9540}, {1598,13903}, {1829,8983}, {1843,13910}, {1862,13922}, {1902,13912}, {1906,11473}, {3093,5418}, {3147,3312}, {3311,3542}, {3536,3595}, {3580,11447}, {5090,13893}, {5186,8997}, {5411,6353}, {6561,10151}, {6756,13925}, {7487,13886}, {7505,7584}, {7713,13888}, {8974,11388}, {8975,11389}, {8980,12131}, {8987,12136}, {8988,12137}, {8991,11381}, {8992,12143}, {8993,12144}, {8994,12133}, {8995,11576}, {10018,13966}, {10154,11418}, {11265,11585}, {11363,13883}, {11380,13885}, {11383,13887}, {11384,13890}, {11385,13891}, {11386,13892}, {11391,13896}, {11392,13897}, {11393,13898}, {11394,13899}, {11395,13900}, {11396,13902}, {11398,13904}, {11399,13905}, {11400,13906}, {11401,13907}, {11832,13894}, {12132,13908}, {12134,13909}, {12138,13913}, {12139,13914}, {12140,13915}, {12141,13916}, {12142,13917}, {12145,13918}, {12146,13919}, {12147,13921}, {12148,13879}, {13166,13923}, {13668,13920}, {13788,13848}

X(13885) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND 5th ANTI-BROCARD

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

X(13885) lies on these lines: {6,1078}, {32,638}, {83,590}, {98,371}, {182,9540}, {384,8992}, {485,12110}, {1587,5171}, {2080,7583}, {3069,7815}, {3311,10104}, {3398,8981}, {4027,8997}, {6460,8722}, {7585,7793}, {7787,8972}, {8974,10792}, {8976,10796}, {8980,12176}, {8983,12194}, {8987,12196}, {8988,12198}, {8991,12202}, {8994,12192}, {8995,12208}, {8998,13193}, {10788,13886}, {10789,13888}, {10790,13889}, {10791,13893}, {10794,13895}, {10795,13896}, {10797,13897}, {10798,13898}, {10799,13901}, {10800,13902}, {10801,13904}, {10802,13905}, {10803,13906}, {10804,13907}, {11364,13883}, {11380,13884}, {11490,13887}, {11837,13890}, {11838,13891}, {11839,13894}, {11840,13899}, {11841,13900}, {11842,13903}, {12150,13846}, {12191,13908}, {12193,13909}, {12197,13912}, {12199,13913}, {12200,13914}, {12201,13915}, {12204,13916}, {12205,13917}, {12207,13918}, {12209,13919}, {12210,13921}, {12211,13879}, {12212,13910}, {13194,13922}, {13195,13923}, {13672,13920}, {13792,13848}

X(13885) = {X(6), X(1078)}-harmonic conjugate of X(13938)

X(13886) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND ANTI-EULER

Barycentrics 3*S^2-(SB+SC)*(SA-2*S) : :
X(13886) = 3*X(1131)+2*X(6407) = X(1131)+2*X(13903) = X(6407)-3*X(13903)

X(13886) lies on these lines: {2,3312}, {3,8972}, {4,371}, {5,6417}, {6,3090}, {20,6449}, {24,13889}, {30,1131}, {186,8276}, {372,3525}, {376,3070}, {388,13904}, {486,5071}, {487,13637}, {491,7375}, {497,13905}, {515,13888}, {546,6199}, {590,631}, {632,6395}, {639,5861}, {944,8983}, {1132,3851}, {1151,3529}, {1271,11313}, {1335,8164}, {1588,3545}, {1656,3317}, {3069,5067}, {3071,3855}, {3085,13897}, {3086,13898}, {3091,3311}, {3146,6221}, {3299,10589}, {3301,10588}, {3365,11488}, {3390,11489}, {3448,13915}, {3522,6496}, {3523,6456}, {3524,5418}, {3528,6560}, {3533,8253}, {3544,6419}, {3628,6418}, {3854,6494}, {4294,13901}, {5056,7584}, {5059,9542}, {5068,13785}, {5073,6472}, {5079,6500}, {5603,13883}, {5657,13893}, {6361,13912}, {6392,7389}, {6398,10303}, {6429,9541}, {6446,12108}, {6453,11541}, {6478,9681}, {6484,11001}, {6770,13917}, {6773,13916}, {6776,13910}, {6811,8974}, {7486,13951}, {7487,13884}, {7967,13902}, {8854,10881}, {8975,10784}, {8980,9862}, {8987,12246}, {8988,12247}, {8991,12250}, {8992,12251}, {8993,12252}, {8994,12244}, {8995,12254}, {8997,13172}, {8998,12383}, {9694,12085}, {10138,11812}, {10785,13895}, {10786,13896}, {10788,13885}, {10805,13906}, {10806,13907}, {11411,13909}, {11491,13887}, {11843,13890}, {11844,13891}, {11845,13894}, {11846,13899}, {11847,13900}, {12124,13924}, {12243,13908}, {12245,13911}, {12248,13913}, {12249,13914}, {12253,13918}, {12255,13919}, {12256,13921}, {12510,13882}, {13199,13922}, {13200,13923}, {13674,13920}, {13794,13848}

X(13886) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,7583,7581), (6,3090,13939), (1656,6501,13993), (3070,9540,376), (3316,7581,2), (7583,8976,2)

X(13887) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND ANTI-MANDART-INCIRCLE

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

X(13887) lies on these lines: {1,6}, {3,8983}, {35,13888}, {55,3068}, {56,13902}, {100,8972}, {197,13889}, {371,11496}, {485,11500}, {590,1376}, {606,940}, {615,8167}, {1012,9583}, {1621,7585}, {3069,4423}, {3295,13883}, {3913,13911}, {4421,13846}, {5120,8225}, {5284,7586}, {5687,13893}, {6460,8273}, {7583,10267}, {8974,11497}, {8975,11498}, {8976,11499}, {8980,12178}, {8981,11248}, {8987,12330}, {8988,12331}, {8991,12335}, {8992,12338}, {8993,12339}, {8994,12327}, {8995,12341}, {8997,13173}, {8998,13204}, {9540,10310}, {10306,13912}, {11108,13936}, {11383,13884}, {11490,13885}, {11491,13886}, {11492,13890}, {11493,13891}, {11494,13892}, {11501,13897}, {11502,13898}, {11504,13900}, {11507,13904}, {11508,13905}, {11509,13906}, {11510,13907}, {11848,13894}, {11849,13903}, {12326,13908}, {12328,13909}, {12329,13910}, {12332,13913}, {12334,13915}, {12336,13916}, {12337,13917}, {12340,13918}, {12342,13919}, {12343,13921}, {12344,13879}, {13205,13922}, {13206,13923}, {13795,13848}

X(13888) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND AQUILA

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

X(13888) lies on these lines: {1,1336}, {6,3624}, {10,8972}, {35,13887}, {40,8981}, {165,9540}, {355,13925}, {371,1699}, {485,5691}, {515,13886}, {517,13903}, {590,1698}, {1125,7585}, {1587,7987}, {1588,7988}, {1697,13901}, {1702,11522}, {1703,5418}, {1768,13913}, {2067,5290}, {2948,8998}, {3070,9615}, {3311,8227}, {3316,10175}, {3576,7583}, {4347,9634}, {5273,5393}, {5541,13922}, {5587,8976}, {6199,9955}, {6417,11230}, {7581,10165}, {7713,13884}, {7992,8987}, {8185,13889}, {8186,13890}, {8187,13891}, {8189,13900}, {8276,9590}, {8980,9860}, {8988,9897}, {8991,9899}, {8992,9902}, {8994,9904}, {8995,9905}, {8997,13174}, {9541,9585}, {9578,13897}, {9581,13898}, {9589,9616}, {9875,13908}, {9896,13909}, {9898,13914}, {9900,13916}, {9901,13917}, {9906,13921}, {9907,13879}, {10789,13885}, {10827,13896}, {12407,13915}, {12408,13918}, {12409,13919}, {12788,13924}, {13679,13920}, {13799,13848}

X(13888) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,3624,13942), (485,9583,5691), (3068,8983,1), (3068,13902,13883), (8983,13883,13902), (13883,13902,1)

X(13889) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND ARA

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

X(13889) lies on these lines: {3,485}, {6,1196}, {22,8972}, {24,13886}, {25,3068}, {26,13925}, {159,13910}, {197,13887}, {371,1598}, {486,11484}, {1131,11413}, {1584,9723}, {1597,6564}, {1995,7585}, {3069,11284}, {3146,9694}, {3311,7529}, {3316,7509}, {3517,8960}, {5198,6459}, {5594,8975}, {5595,8974}, {6422,10960}, {6642,7583}, {6767,9632}, {7387,8981}, {7517,13903}, {8185,13888}, {8190,13890}, {8191,13891}, {8192,13902}, {8193,13893}, {8194,13899}, {8195,13900}, {8253,8280}, {8909,12309}, {8980,9861}, {8983,9798}, {8987,9910}, {8988,9912}, {8991,9914}, {8992,9917}, {8993,9918}, {8994,9919}, {8995,9920}, {8997,13175}, {8998,12310}, {9540,11414}, {9876,13908}, {9908,13909}, {9909,13846}, {9911,13912}, {9913,13913}, {9915,13916}, {9916,13917}, {9921,13921}, {9922,13879}, {10046,13905}, {10790,13885}, {10828,13892}, {10829,13895}, {10830,13896}, {10831,13897}, {10832,13898}, {10833,13901}, {10834,13906}, {10835,13907}, {11365,13883}, {11513,12590}, {11641,13923}, {11853,13894}, {12164,12239}, {12410,13911}, {12411,13914}, {12412,13915}, {12413,13918}, {12414,13919}, {13222,13922}, {13680,13920}, {13800,13848}

X(13890) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND 1st AURIGA

Barycentrics ((2*a+b+c)*K-a^2*(a+b+c)*(-a+b+c))*S+K*(a+b+c)*a^2 : : , where K=4*S*sqrt(R*(4*R+r))

X(13890) lies on these lines: {6,13944}, {55,8983}, {371,8196}, {485,9834}, {590,5599}, {1125,13945}, {3068,3361}, {5600,7969}, {5601,8972}, {8186,13888}, {8190,13889}, {8197,13893}, {8198,8974}, {8199,8975}, {8200,8976}, {8981,11252}, {8987,12456}, {8988,12460}, {8992,12474}, {8993,12476}, {8997,13176}, {9540,11822}, {11207,13846}, {11366,13883}, {11384,13884}, {11492,13887}, {11837,13885}, {11843,13886}, {11861,13892}, {11865,13895}, {11867,13896}, {11869,13897}, {11871,13898}, {11873,13901}, {11875,13903}, {11877,13904}, {11879,13905}, {11881,13906}, {11883,13907}, {12345,13908}, {12454,13911}, {12458,13912}, {12462,13913}, {13228,13922}

X(13891) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND 2nd AURIGA

Barycentrics (-(2*a+b+c)*K-a^2*(a+b+c)*(-a+b+c))*S-K*(a+b+c)*a^2 : : , where K=4*S*sqrt(R*(4*R+r))

X(13891) lies on these lines: {6,13945}, {55,8983}, {371,8203}, {485,9835}, {590,5600}, {1125,13944}, {3068,5598}, {3361,13902}, {5599,7969}, {5602,8972}, {8187,13888}, {8191,13889}, {8204,13893}, {8205,8974}, {8206,8975}, {8207,8976}, {8981,11253}, {8987,12457}, {8988,12461}, {8992,12475}, {8993,12477}, {8997,13177}, {8998,13209}, {9540,11823}, {11208,13846}, {11385,13884}, {11493,13887}, {11838,13885}, {11844,13886}, {11862,13892}, {11866,13895}, {11868,13896}, {11870,13897}, {11872,13898}, {11876,13903}, {11878,13904}, {11882,13906}, {12346,13908}, {12455,13911}, {12459,13912}, {12463,13913}, {13230,13922}

X(13892) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND 5th BROCARD

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

X(13892) lies on these lines: {6,7796}, {32,638}, {371,9993}, {485,9873}, {590,3096}, {2896,8972}, {3098,9540}, {8782,8997}, {8976,9996}, {8980,9862}, {8981,9821}, {8987,12496}, {8988,12498}, {8991,12502}, {8992,9983}, {8994,9984}, {8995,9985}, {8998,13210}, {9301,13903}, {9878,13908}, {9923,13909}, {9981,13916}, {9982,13917}, {9986,13921}, {9987,13879}, {10828,13889}, {10871,13895}, {10873,13897}, {10874,13898}, {10876,13900}, {10877,13901}, {11386,13884}, {11494,13887}, {11861,13890}, {11862,13891}, {11885,13894}, {12497,13912}, {12499,13913}, {12500,13914}, {12501,13915}, {12503,13918}, {12504,13919}, {13235,13922}, {13685,13920}, {13805,13848}

X(13892) = {X(6), X(7846)}-harmonic conjugate of X(13946)

X(13893) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND OUTER-GARCIA

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

X(13893) lies on these lines: {1,590}, {2,13883}, {4,9616}, {5,1702}, {6,1698}, {8,8972}, {10,3068}, {30,9582}, {40,485}, {65,13897}, {72,13896}, {80,13922}, {100,8988}, {165,3070}, {355,8981}, {371,5587}, {515,9540}, {517,8976}, {519,13902}, {1131,9778}, {1151,5691}, {1378,5705}, {1587,6684}, {1588,10175}, {1703,7583}, {1737,13905}, {1837,13901}, {2066,9581}, {2067,9578}, {3057,13898}, {3069,3634}, {3071,7989}, {3311,9956}, {3312,11231}, {3416,13910}, {3579,13665}, {3624,7968}, {3679,7969}, {5090,13884}, {5657,13886}, {5687,13887}, {5688,8975}, {5689,8974}, {5690,13925}, {6460,10164}, {7581,13975}, {7585,9780}, {8193,13889}, {8197,13890}, {8204,13891}, {8214,13899}, {8215,13900}, {8227,10576}, {8252,13942}, {8909,9896}, {8980,9864}, {8987,12667}, {8991,12779}, {8993,12783}, {8994,12368}, {8995,12785}, {8997,13178}, {8998,13211}, {9618,9681}, {9857,13892}, {9881,13908}, {9907,13882}, {9928,13909}, {10039,13904}, {10791,13885}, {10914,13895}, {10915,13906}, {10916,13907}, {11900,13894}, {12751,13913}, {12777,13914}, {12778,13915}, {12780,13916}, {12781,13917}, {12784,13918}, {12786,13919}, {12787,13921}, {12788,13879}, {13280,13923}, {13688,13920}, {13808,13848}

X(13893) = reflection of X(9615) in X(9540)
X(13893) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,13912,9616), (6,1698,13947), (8,8972,8983), (590,13911,1), (3679,13888,7969), (7585,9780,13936), (7968,8253,3624)

X(13894) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND GOSSARD

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

X(13894) lies on these lines: {6,13948}, {30,6449}, {371,11897}, {402,3068}, {485,12113}, {590,1650}, {1651,13846}, {4240,8972}, {8974,11901}, {8975,11902}, {8980,12181}, {8981,11251}, {8983,12438}, {8987,12668}, {8988,12729}, {8991,12791}, {8992,12794}, {8993,12795}, {8994,12369}, {8995,12797}, {8997,13179}, {11831,13883}, {11832,13884}, {11839,13885}, {11845,13886}, {11848,13887}, {11853,13889}, {11885,13892}, {11900,13893}, {11903,13895}, {11904,13896}, {11905,13897}, {11906,13898}, {11907,13899}, {11908,13900}, {11909,13901}, {11910,13902}, {11911,13903}, {11912,13904}, {11913,13905}, {11914,13906}, {11915,13907}, {12347,13908}, {12418,13909}, {12583,13910}, {12626,13911}, {12696,13912}, {12752,13913}, {12789,13914}, {12790,13915}, {12792,13916}, {12793,13917}, {12796,13918}, {12798,13919}, {12799,13921}, {12800,13879}, {13268,13922}, {13281,13923}, {13689,13920}, {13809,13848}

X(13895) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND INNER-JOHNSON

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

X(13895) lies on these lines: {6,13952}, {11,3068}, {12,13906}, {355,8976}, {371,10893}, {485,12114}, {590,1376}, {3434,8972}, {7585,10584}, {8974,10919}, {8975,10920}, {8980,12182}, {8981,10525}, {8987,12676}, {8988,12737}, {8991,12920}, {8992,12923}, {8993,12924}, {8994,12371}, {8995,12926}, {8997,13180}, {8998,13213}, {9540,11826}, {10523,13904}, {10785,13886}, {10794,13885}, {10829,13889}, {10871,13892}, {10912,13911}, {10914,13893}, {10943,13925}, {10944,13897}, {10945,13899}, {10946,13900}, {10947,13901}, {10948,13905}, {10949,13907}, {11235,13846}, {11373,13883}, {11865,13890}, {11866,13891}, {11903,13894}, {11928,13903}, {12348,13908}, {12422,13909}, {12586,13910}, {12700,13912}, {12761,13913}, {12857,13914}, {12889,13915}, {12921,13916}, {12922,13917}, {12925,13918}, {12927,13919}, {12928,13921}, {12929,13879}, {13271,13922}, {13813,13848}

X(13896) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND OUTER-JOHNSON

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

X(13896) lies on these lines: {6,13953}, {11,13907}, {12,3068}, {72,13893}, {355,8976}, {371,10894}, {485,11500}, {590,958}, {3436,8972}, {5812,13912}, {7585,10585}, {8974,10921}, {8975,10922}, {8980,12183}, {8981,10526}, {8987,12677}, {8988,12738}, {8991,12930}, {8992,12933}, {8994,12372}, {8995,12936}, {8997,13181}, {8998,13214}, {9540,11827}, {10523,13905}, {10786,13886}, {10795,13885}, {10827,13888}, {10830,13889}, {10942,13925}, {10950,13898}, {10951,13899}, {10952,13900}, {10953,13901}, {10955,13906}, {11236,13846}, {11374,13883}, {11391,13884}, {11867,13890}, {11868,13891}, {11904,13894}, {12349,13908}, {12423,13909}, {12587,13910}, {12635,13911}, {12762,13913}, {12890,13915}, {12931,13916}, {12932,13917}, {12935,13918}, {12937,13919}, {12938,13921}, {12939,13879}, {13272,13922}, {13295,13923}, {13814,13848}

X(13897) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND 1st JOHNSON-YFF

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

X(13897) lies on these lines: {1,8976}, {4,13901}, {5,13905}, {6,13954}, {12,3068}, {35,13665}, {55,485}, {56,590}, {65,13893}, {371,10895}, {388,8972}, {498,7583}, {1124,10576}, {1151,12943}, {1335,8960}, {1478,8981}, {1587,5432}, {1588,3614}, {1656,3299}, {1836,13912}, {2066,10896}, {2067,11237}, {2099,13911}, {3070,5217}, {3085,13886}, {3086,3316}, {3304,9661}, {3311,7951}, {3585,6221}, {3628,13962}, {4316,6455}, {5204,5418}, {5252,8983}, {5326,13935}, {6449,10483}, {6502,8253}, {6564,12953}, {7354,9540}, {7581,13958}, {7585,10588}, {8276,9659}, {8974,10923}, {8975,10924}, {8980,12184}, {8987,12678}, {8988,12739}, {8991,12940}, {8992,12837}, {8993,12944}, {8994,12373}, {8995,12946}, {8997,13182}, {8998,12903}, {9541,9648}, {9578,13888}, {9654,13903}, {10088,13915}, {10797,13885}, {10831,13889}, {10873,13892}, {10944,13895}, {10956,13906}, {10957,13907}, {11375,13883}, {11392,13884}, {11501,13887}, {11869,13890}, {11870,13891}, {11905,13894}, {11930,13899}, {11931,13900}, {12350,13908}, {12588,13910}, {12763,13913}, {12859,13914}, {12941,13916}, {12942,13917}, {12945,13918}, {12947,13919}, {12948,13921}, {12949,13879}, {13273,13922}, {13296,13923}, {13695,13920}, {13815,13848}, {13904,13925}

X(13898) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND 2nd JOHNSON-YFF

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

X(13898) lies on these lines: {1,8976}, {2,13958}, {5,13904}, {6,13955}, {11,3068}, {36,13665}, {55,590}, {56,485}, {371,10896}, {496,13905}, {497,8972}, {499,7583}, {1069,13909}, {1124,8960}, {1151,12953}, {1335,10576}, {1479,8981}, {1587,5433}, {1588,7173}, {1656,3301}, {1837,8983}, {2066,11238}, {2067,10895}, {2098,13911}, {3057,13893}, {3070,5204}, {3085,3316}, {3086,13886}, {3303,9646}, {3311,7741}, {3583,6221}, {3628,13963}, {4324,6455}, {5217,5418}, {5414,8253}, {6284,9540}, {6564,12943}, {7294,13935}, {7585,10589}, {8276,9672}, {8974,10925}, {8975,10926}, {8980,12185}, {8987,12679}, {8988,12740}, {8991,12950}, {8992,12836}, {8993,12954}, {8994,12374}, {8995,12956}, {8997,13183}, {8998,12904}, {9541,9663}, {9581,13888}, {9669,13903}, {10091,13915}, {10798,13885}, {10832,13889}, {10874,13892}, {10950,13896}, {10958,13906}, {10959,13907}, {11376,13883}, {11393,13884}, {11502,13887}, {11871,13890}, {11872,13891}, {11906,13894}, {11932,13899}, {11933,13900}, {12351,13908}, {12589,13910}, {12701,13912}, {12764,13913}, {12860,13914}, {12951,13916}, {12952,13917}, {12955,13918}, {12957,13919}, {12958,13921}, {12959,13879}, {13274,13922}, {13297,13923}, {13696,13920}, {13816,13848}

X(13899) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND LUCAS HOMOTHETIC

Barycentrics (12*R^2+SB+SC)*S^2-(-8*(SB+SC)*R^2+SA^2-2*SW^2)*S+(SB+SC)*SW^2 : :

X(13899) lies on these lines: {6,13956}, {371,8212}, {393,493}, {485,9838}, {590,8222}, {6461,13900}, {8194,13889}, {8220,8976}, {8980,12186}, {8981,10669}, {8988,12741}, {8991,12986}, {8992,12992}, {8994,12377}, {8995,12998}, {8997,13184}, {8998,13215}, {9540,11828}, {10945,13895}, {10951,13896}, {11394,13884}, {11840,13885}, {11846,13886}, {11907,13894}, {11930,13897}, {11932,13898}, {12352,13908}, {12426,13909}, {12765,13913}, {12988,13916}, {12990,13917}, {13002,13921}, {13275,13922}, {13697,13920}, {13817,13848}

X(13900) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND LUCAS(-1) HOMOTHETIC

Barycentrics (12*R^2+SB+SC)*S^2+(8*(SB+SC)*R^2+SA^2-2*SW^2)*S-(SB+SC)*SW^2 : :

X(13900) lies on these lines: {6,13957}, {371,8213}, {485,9839}, {494,3068}, {590,8223}, {6461,13899}, {6463,8972}, {8189,13888}, {8195,13889}, {8211,13902}, {8215,13893}, {8217,8974}, {8219,8975}, {8221,8976}, {8980,12187}, {8981,10673}, {8983,12441}, {8988,12742}, {8991,12987}, {8992,12993}, {8993,12995}, {8994,12378}, {8995,12999}, {8997,13185}, {8998,13216}, {9540,11829}, {10876,13892}, {10946,13895}, {10952,13896}, {11378,13883}, {11395,13884}, {11504,13887}, {11841,13885}, {11847,13886}, {11908,13894}, {11931,13897}, {11933,13898}, {11948,13901}, {11950,13903}, {11952,13904}, {11954,13905}, {11956,13906}, {11958,13907}, {12153,13846}, {12353,13908}, {12427,13909}, {12591,13910}, {12637,13911}, {12766,13913}, {12862,13914}, {12895,13915}, {12989,13916}, {12991,13917}, {12997,13918}, {13001,13919}, {13003,13921}, {13005,13879}, {13276,13922}, {13299,13923}, {13698,13920}, {13818,13848}

X(13901) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND MANDART-INCIRCLE

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

X(13901) lies on these lines: {1,8981}, {2,13955}, {3,13905}, {4,13897}, {6,5432}, {11,590}, {12,371}, {33,13884}, {35,7583}, {55,3068}, {56,9540}, {65,13912}, {140,3299}, {485,6284}, {497,8972}, {498,3311}, {615,5326}, {1124,5418}, {1151,7354}, {1317,13913}, {1478,6221}, {1479,8976}, {1587,5217}, {1697,13888}, {1702,11375}, {1836,9616}, {1837,13893}, {2098,13902}, {2646,13883}, {3023,8997}, {3027,8980}, {3028,8994}, {3056,13910}, {3057,8983}, {3058,13846}, {3071,3614}, {3295,13903}, {3316,10591}, {3320,13918}, {3526,13962}, {4294,13886}, {4299,6449}, {4302,13665}, {4995,5414}, {5218,7585}, {5252,9583}, {5434,9663}, {6020,13923}, {6407,9655}, {6417,13963}, {6453,9647}, {6459,10895}, {6564,9660}, {7173,10576}, {7355,8991}, {7582,13954}, {8974,10927}, {8975,10928}, {8987,12680}, {8992,13077}, {8993,13078}, {8995,13079}, {9541,12943}, {9632,10149}, {10799,13885}, {10833,13889}, {10877,13892}, {10947,13895}, {10950,13911}, {10953,13896}, {11909,13894}, {11948,13900}, {12354,13908}, {12428,13909}, {13075,13916}, {13076,13917}, {13081,13921}, {13699,13920}, {13819,13848}

X(13902) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND 5th MIXTILINEAR

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

X(13902) lies on these lines: {1,1336}, {2,7969}, {6,3616}, {8,590}, {56,13887}, {145,8972}, {371,5603}, {485,944}, {516,9615}, {517,9540}, {519,13893}, {615,5550}, {946,6459}, {952,8976}, {962,1151}, {1125,3069}, {1320,13922}, {1385,1587}, {1482,8981}, {1483,13925}, {1588,5886}, {1702,13464}, {1703,10165}, {2067,3485}, {2098,13901}, {2362,7288}, {3070,5731}, {3241,13846}, {3242,13910}, {3298,5703}, {3311,5901}, {3361,13891}, {3576,6460}, {3622,7585}, {3624,13936}, {4301,9616}, {5418,5657}, {5604,8975}, {5605,8974}, {5818,10576}, {6200,6361}, {6409,9778}, {7583,10246}, {7967,13886}, {7970,8980}, {7971,8987}, {7972,8988}, {7973,8991}, {7974,13916}, {7976,8992}, {7977,8993}, {7978,8994}, {7979,8995}, {7980,13921}, {7981,13879}, {7982,13912}, {7983,8997}, {7984,8998}, {8000,13914}, {8192,13889}, {8210,13899}, {8211,13900}, {8253,9780}, {9541,12699}, {9884,13908}, {9933,13909}, {9997,13892}, {10247,13903}, {10698,13913}, {10705,13923}, {10800,13885}, {10944,13895}, {10950,13896}, {11396,13884}, {11910,13894}, {12898,13915}, {13099,13918}, {13702,13920}, {13822,13848}

X(13902) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,8983,3068), (1,13888,13883), (6,3616,13959), (145,8972,13911), (3622,7585,7968), (8983,13883,13888), (13883,13888,3068), (13906,13907,3068)

X(13903) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND X3-ABC REFLECTIONS

Barycentrics 3*a^4-5*(b^2+c^2)*a^2+2*(b^2-c^2)^2-8*S*a^2 : :
X(13903) = 2*X(1131)+3*X(6407) = X(1131)-3*X(13886) = X(6407)+2*X(13886)

X(13903) lies on these lines: {2,6417}, {3,1587}, {4,13925}, {5,1132}, {6,3411}, {20,6445}, {30,1131}, {140,6418}, {195,8995}, {371,381}, {382,485}, {399,8998}, {486,590}, {517,13888}, {549,7581}, {615,6427}, {631,6395}, {632,7586}, {639,13663}, {999,13905}, {1151,1657}, {1351,13910}, {1482,8983}, {1588,5055}, {1598,13884}, {2070,8276}, {3069,6500}, {3070,3534}, {3071,5072}, {3295,13901}, {3312,5054}, {3523,6408}, {3529,9542}, {3530,6446}, {3590,3858}, {3592,5079}, {3628,7582}, {3843,6459}, {5059,10137}, {5070,7584}, {5073,9541}, {5076,6425}, {5420,6428}, {5790,13893}, {6289,13924}, {6419,8253}, {6431,10577}, {6447,6561}, {6451,9680}, {6455,6560}, {6501,13966}, {7517,13889}, {7747,9602}, {7756,9601}, {8974,11916}, {8975,11917}, {8980,12188}, {8987,12684}, {8988,12747}, {8991,13093}, {8992,13108}, {8994,10620}, {8997,13188}, {9301,13892}, {9654,13897}, {9669,13898}, {9694,12084}, {10246,13883}, {10247,13902}, {11842,13885}, {11849,13887}, {11875,13890}, {11876,13891}, {11911,13894}, {11928,13895}, {11949,13899}, {11950,13900}, {12001,13907}, {12331,13922}, {12355,13908}, {12429,13909}, {12601,13921}, {12602,13879}, {12702,13912}, {12773,13913}, {12902,13915}, {13102,13916}, {13103,13917}, {13115,13918}, {13126,13919}, {13836,13848}

X(13903) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,3526,13961), (485,6221,382), (1151,13665,1657), (3068,8981,3), (3070,6449,3534), (3312,5418,5054), (5073,9691,9541), (7583,9540,3), (13901,13904,3295)

X(13904) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND INNER-YFF

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

X(13904) lies on these lines: {1,1336}, {2,3301}, {5,13898}, {6,499}, {11,3311}, {12,8976}, {35,9540}, {36,1587}, {55,8981}, {56,7583}, {63,5393}, {89,3300}, {371,1479}, {388,13886}, {485,1478}, {495,13897}, {498,590}, {611,13910}, {1124,10072}, {1151,4302}, {1588,7741}, {1709,8987}, {3070,4299}, {3085,8972}, {3086,3299}, {3295,13901}, {3298,9646}, {3312,5433}, {3316,10588}, {3526,13958}, {3583,6459}, {3628,13954}, {5058,9599}, {5119,13912}, {5414,5418}, {6199,9669}, {6221,6284}, {6425,9660}, {6449,9663}, {6460,7280}, {7173,13785}, {7288,7581}, {7354,13665}, {7582,10589}, {7969,10573}, {8974,10040}, {8975,10041}, {8980,10053}, {8988,10057}, {8991,10060}, {8992,10063}, {8993,10064}, {8994,10065}, {8995,10066}, {8997,10086}, {8998,10088}, {9583,10572}, {10037,13889}, {10038,13892}, {10039,13893}, {10054,13908}, {10055,13909}, {10058,13913}, {10061,13916}, {10062,13917}, {10067,13921}, {10068,13879}, {10087,13922}, {10523,13895}, {10801,13885}, {10880,11393}, {11398,13884}, {11507,13887}, {11877,13890}, {11878,13891}, {11912,13894}, {11951,13899}, {11952,13900}, {12647,13911}, {13116,13918}, {13128,13919}, {13714,13920}, {13837,13848}

X(13904) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3301,13963), (6,499,13962), (6,9661,499), (485,2067,1478), (3295,13903,13901), (3298,9646,10056)

X(13905) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND OUTER-YFF

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

X(13905) lies on these lines: {1,1336}, {2,3299}, {3,13901}, {5,13897}, {6,498}, {11,8976}, {12,3311}, {35,1587}, {36,9540}, {46,13912}, {55,7583}, {56,8981}, {371,1478}, {485,1479}, {496,13898}, {497,13886}, {499,590}, {613,13910}, {999,13903}, {1151,4299}, {1335,10056}, {1588,7951}, {1702,12047}, {1737,13893}, {1770,9616}, {3070,4302}, {3085,3301}, {3297,9661}, {3305,5393}, {3312,5432}, {3316,10589}, {3585,6459}, {3614,13785}, {3628,13955}, {5010,6460}, {5058,9596}, {5218,7581}, {5418,6502}, {6199,9654}, {6221,7354}, {6284,13665}, {6418,13958}, {6425,9647}, {6449,9648}, {7582,10588}, {8980,10069}, {8987,10085}, {8988,10073}, {8991,10076}, {8992,10079}, {8994,10081}, {8995,10082}, {8997,10089}, {8998,10091}, {9541,10483}, {10046,13889}, {10070,13908}, {10071,13909}, {10074,13913}, {10077,13916}, {10078,13917}, {10083,13921}, {10090,13922}, {10523,13896}, {10802,13885}, {10880,11392}, {10948,13895}, {11399,13884}, {11913,13894}, {11954,13900}, {13312,13923}

X(13906) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND INNER-YFF TANGENTS

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

X(13906) lies on these lines: {1,1336}, {6,13964}, {12,13895}, {371,10531}, {485,12115}, {590,5552}, {1587,10269}, {8976,10942}, {8980,12189}, {8981,10679}, {8987,12686}, {8988,12749}, {8991,13094}, {8992,13109}, {8994,12381}, {8995,13121}, {8997,13189}, {8998,13217}, {9540,11248}, {10803,13885}, {10805,13886}, {10834,13889}, {10955,13896}, {10956,13897}, {10958,13898}, {10965,13901}, {11400,13884}, {11509,13887}, {11881,13890}, {11882,13891}, {11956,13900}, {12000,13903}, {12356,13908}, {12430,13909}, {12775,13913}, {12874,13914}, {12905,13915}, {13104,13916}, {13105,13917}, {13118,13918}, {13130,13919}, {13132,13921}, {13134,13879}, {13278,13922}, {13313,13923}, {13716,13920}, {13839,13848}

X(13907) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND OUTER-YFF TANGENTS

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

X(13907) lies on these lines: {1,1336}, {6,13965}, {11,13896}, {371,10532}, {485,12116}, {590,10527}, {6460,10902}, {8976,10943}, {8980,12190}, {8981,10680}, {8987,12687}, {8988,12750}, {8991,13095}, {8992,13110}, {8994,12382}, {8995,13122}, {8997,13190}, {8998,13218}, {9540,11249}, {10804,13885}, {10806,13886}, {10835,13889}, {10949,13895}, {10957,13897}, {10959,13898}, {10966,13901}, {11401,13884}, {11510,13887}, {11883,13890}, {11884,13891}, {11915,13894}, {11958,13900}, {12001,13903}, {12357,13908}, {12431,13909}, {12704,13912}, {12776,13913}, {12875,13914}, {12906,13915}, {13106,13916}, {13107,13917}, {13119,13918}, {13135,13879}, {13279,13922}, {13314,13923}, {13717,13920}, {13840,13848}

X(13908) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO ANTI-MCCAY

Barycentrics S*(12*S^2+3*SA^2+6*SB*SC-5*SW^2)+3*(3*S^2-SW^2)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(9855)

X(13908) lies on these lines: {2,13989}, {6,5461}, {30,8980}, {115,13703}, {485,542}, {530,13916}, {531,13917}, {543,8997}, {590,2482}, {671,3068}, {5969,8992}, {6722,13847}, {8724,8976}, {8974,9882}, {8975,9883}, {9540,12117}, {9830,13910}, {9875,13888}, {9876,13889}, {9878,13892}, {9881,13893}, {9884,13902}, {9892,13669}, {9894,13676}, {10054,13904}, {10070,13905}, {12132,13884}, {12191,13885}, {12243,13886}, {12258,13883}, {12326,13887}, {12345,13890}, {12346,13891}, {12347,13894}, {12348,13895}, {12349,13896}, {12350,13897}, {12351,13898}, {12352,13899}, {12353,13900}, {12354,13901}, {12355,13903}, {12356,13906}, {12357,13907}

X(13908) = reflection of X(8997) in X(13846)
X(13908) = orthologic center of these triangles: 3rd tri-squares-central to Mccay
X(13908) = {X(6), X(5461)}-harmonic conjugate of X(13968)

X(13909) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO ARIES

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

The reciprocal orthologic center of these triangles is X(9833)

X(13909) lies on these lines: {5,8969}, {6,5449}, {30,8991}, {68,3068}, {155,8976}, {371,9927}, {485,12239}, {539,8909}, {590,1147}, {1069,13898}, {1151,8994}, {3070,7689}, {3448,11462}, {3564,13879}, {5418,12038}, {6193,8972}, {6221,12293}, {7583,12359}, {8960,10665}, {8974,9929}, {8975,9930}, {9540,12118}, {9896,13888}, {9908,13889}, {9923,13892}, {9928,13893}, {9933,13902}, {10055,13904}, {10071,13905}, {10576,10666}, {11411,13886}, {12134,13884}, {12163,13665}, {12193,13885}, {12259,13883}, {12328,13887}, {12418,13894}, {12422,13895}, {12423,13896}, {12426,13899}, {12427,13900}, {12428,13901}, {12429,13903}, {12430,13906}, {12431,13907}

X(13909) = orthologic center of these triangles: 3rd tri-squares-central to 2nd Hyacinth
X(13909) = {X(6), X(5449)}-harmonic conjugate of X(13970)

X(13910) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO 1st EHRMANN

Barycentrics (2*a^2+b^2+c^2)*S+a^2*(a^2+b^2+c^2) : :
X(13910) = X(6)+3*X(13846)

The reciprocal orthologic center of these triangles is X(3)

X(13910) lies on these lines: {2,6}, {159,13889}, {182,7583}, {206,8969}, {371,5480}, {485,1503}, {511,8981}, {518,8983}, {542,13915}, {611,13904}, {613,13905}, {732,8992}, {1350,9540}, {1351,13903}, {1352,8976}, {1386,13883}, {1587,5085}, {1843,13884}, {2781,8994}, {2854,8998}, {3056,13901}, {3242,13902}, {3416,13893}, {3564,13879}, {3751,13888}, {3867,5412}, {5846,13911}, {5969,8997}, {6561,13644}, {6776,13886}, {8180,13924}, {8550,8960}, {8854,10192}, {9024,13922}, {9830,13908}, {12212,13885}, {12329,13887}, {12583,13894}, {12586,13895}, {12587,13896}, {12588,13897}, {12589,13898}, {12590,13899}, {12591,13900}, {12594,13906}, {12595,13907}, {13848,13920}

X(13910) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,590,141), (6,3589,13972), (3068,13638,590), (3618,7585,6), (8974,8975,3068)

X(13911) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO EXCENTERS-MIDPOINTS

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

The reciprocal orthologic center of these triangles is X(10)

X(13911) lies on these lines: {1,590}, {2,7968}, {6,10}, {8,3068}, {40,3070}, {44,7090}, {145,8972}, {355,371}, {485,517}, {486,9956}, {515,1151}, {518,10068}, {519,8983}, {615,1698}, {940,6348}, {944,9540}, {952,8981}, {1124,1737}, {1125,8253}, {1152,6684}, {1210,3297}, {1335,10039}, {1385,5418}, {1482,8976}, {1587,5657}, {1588,5818}, {1702,3071}, {1837,2066}, {1904,7133}, {2067,5252}, {2098,13898}, {2099,13897}, {2802,8988}, {3311,5790}, {3316,10595}, {3579,6560}, {3594,13975}, {3913,13887}, {4297,6409}, {5090,5412}, {5420,11231}, {5690,7583}, {5691,9616}, {5844,13925}, {5881,9583}, {5886,10576}, {6410,10164}, {6564,12699}, {9620,12787}, {10912,13895}, {10950,13901}, {12245,13886}, {12410,13889}, {12454,13890}, {12455,13891}, {12626,13894}, {12635,13896}, {12637,13900}, {12647,13904}, {12702,13665}

X(13911) = reflection of X(1151) in X(13912)
X(13911) = orthologic center of these triangles: 3rd tri-squares-central to 2nd Schiffler
X(13911) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,13893,590), (6,10,13973), (8,3068,7969), (10,13883,6), (145,8972,13902), (1702,5587,3071)

X(13912) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO 3rd EXTOUCH

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

The reciprocal orthologic center of these triangles is X(4)

X(13912) lies on these lines: {1,9540}, {2,1702}, {3,13883}, {4,9616}, {6,6684}, {8,9583}, {10,371}, {20,9582}, {40,3068}, {46,13905}, {65,13901}, {140,13971}, {165,1587}, {226,9646}, {355,6221}, {372,10164}, {485,516}, {486,3634}, {515,1151}, {517,8981}, {590,946}, {944,9615}, {962,8972}, {1124,3911}, {1125,5418}, {1210,2066}, {1378,5745}, {1588,1698}, {1685,6685}, {1703,7585}, {1836,13897}, {1902,13884}, {2800,13922}, {2802,13913}, {3071,10175}, {3311,13936}, {3579,7583}, {3592,13973}, {3817,10576}, {4297,6200}, {5119,13904}, {5393,6212}, {5493,8960}, {5587,6459}, {5691,9541}, {5795,9678}, {5812,13896}, {5840,8988}, {6001,8991}, {6361,13886}, {6560,12512}, {7582,13947}, {7584,11231}, {7968,10165}, {7969,11362}, {7982,13902}, {8974,12697}, {8975,12698}, {8976,12699}, {9648,10950}, {9661,12053}, {9663,10944}, {9911,13889}, {10306,13887}, {12197,13885}, {12458,13890}, {12459,13891}, {12497,13892}, {12696,13894}, {12700,13895}, {12701,13898}, {12702,13903}, {12704,13907}

X(13912) = midpoint of X(1151) and X(13911)
X(13912) = reflection of X(8983) in X(8981)
X(13912) = {X(6), X(6684)}-harmonic conjugate of X(13975)

X(13913) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO INNER-GARCIA

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

The reciprocal orthologic center of these triangles is X(40)

X(13913) lies on these lines: {6,6713}, {11,371}, {80,9583}, {100,9540}, {104,3068}, {119,590}, {140,13991}, {153,8972}, {485,2829}, {486,6667}, {515,8988}, {952,8981}, {1151,5840}, {1317,13901}, {1377,3035}, {1768,13888}, {2771,8998}, {2783,8997}, {2787,8980}, {2800,8983}, {2802,13912}, {2806,13918}, {2831,13923}, {6221,10738}, {8974,12753}, {8975,12754}, {8976,10742}, {9541,10724}, {9646,10956}, {9913,13889}, {10058,13904}, {10074,13905}, {10698,13902}, {11715,13883}, {12138,13884}, {12199,13885}, {12248,13886}, {12332,13887}, {12463,13891}, {12499,13892}, {12751,13893}, {12752,13894}, {12761,13895}, {12762,13896}, {12763,13897}, {12764,13898}, {12765,13899}, {12766,13900}, {12773,13903}, {12775,13906}, {12776,13907}

X(13913) = reflection of X(13922) in X(8981)

X(13914) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO HUTSON EXTOUCH

Barycentrics
(2*a^7-(b+c)*a^6-6*(b^2+4*b*c+c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+2*(3*b^2+14*b*c+3*c^2)*(b+c)^2*a^3-(b+c)*(3*b^4+3*c^4-2*b*c*(2*b^2+15*b*c+2*c^2))*a^2-2*(b^2-c^2)^2*(b^2+8*b*c+c^2)*a+(b^2-c^2)^3*(b-c))*S+a^2*(a+b+c)*(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*b*c*(6*b^2+5*b*c+6*c^2))*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) : :

The reciprocal orthologic center of these triangles is X(40)

X(13914) lies on these lines: {6,13978}, {371,12599}, {590,12864}, {3068,7160}, {8000,13902}, {8972,9874}, {8974,12801}, {8975,12802}, {8976,12856}, {9540,12120}, {9898,13888}, {10075,13905}, {12139,13884}, {12200,13885}, {12249,13886}, {12260,13883}, {12333,13887}, {12411,13889}, {12500,13892}, {12777,13893}, {12789,13894}, {12857,13895}, {12859,13897}, {12860,13898}, {12861,13899}, {12862,13900}, {12863,13901}, {12874,13906}, {12875,13907}

X(13915) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO 1st HYACINTH

Barycentrics (30*R^2-SA-7*SW)*S^2-(SB+SC)*(SW*(4*S-3*SA)+9*(SA-2*S)*R^2) : :
X(13915) = X(10819)-3*X(13846)

The reciprocal orthologic center of these triangles is X(6102)

X(13915) lies on these lines: {6,13979}, {30,8994}, {74,13665}, {110,8976}, {125,7583}, {265,3068}, {371,10113}, {485,5663}, {542,13910}, {590,1511}, {1131,12244}, {1539,6564}, {2771,8988}, {2777,8991}, {3070,12041}, {3448,13886}, {3628,13990}, {6221,10733}, {6723,13966}, {8253,10820}, {8972,12383}, {8974,12803}, {8975,12804}, {8995,8998}, {9540,12121}, {10088,13897}, {10091,13898}, {10819,13846}, {12140,13884}, {12201,13885}, {12261,13883}, {12334,13887}, {12407,13888}, {12412,13889}, {12501,13892}, {12778,13893}, {12790,13894}, {12889,13895}, {12890,13896}, {12894,13899}, {12895,13900}, {12896,13901}, {12898,13902}, {12902,13903}, {12903,13904}, {12905,13906}, {12906,13907}

X(13915) = reflection of X(8998) in X(13925)

X(13916) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO INNER-NAPOLEON

Barycentrics -(9*S^2+(9*SW-7*SA)*S+(SB+2*SC)*(2*SB+SC))*sqrt(3)+S^2+3*(5*SW-3*SA)*S+6*SW^2-6*SA^2+9*SB*SC : :

The reciprocal orthologic center of these triangles is X(3)

X(13916) lies on these lines: {6,6670}, {14,3068}, {371,5479}, {530,13908}, {531,10667}, {542,13910}, {590,619}, {617,8972}, {5474,9540}, {5613,8976}, {6269,8975}, {6271,8974}, {6307,13703}, {6722,13982}, {6773,13886}, {6774,7583}, {7974,13902}, {9900,13888}, {9915,13889}, {9981,13892}, {10061,13904}, {10077,13905}, {11706,13883}, {12141,13884}, {12204,13885}, {12336,13887}, {12780,13893}, {12792,13894}, {12921,13895}, {12931,13896}, {12941,13897}, {12951,13898}, {12988,13899}, {12989,13900}, {13075,13901}, {13102,13903}, {13104,13906}, {13106,13907}

X(13917) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO OUTER-NAPOLEON

Barycentrics (9*S^2+(9*SW-7*SA)*S+(SB+2*SC)*(2*SB+SC))*sqrt(3)+S^2+3*(5*SW-3*SA)*S+6*SW^2-6*SA^2+9*SB*SC : :

The reciprocal orthologic center of these triangles is X(3)

X(13917) lies on these lines: {6,6669}, {13,3068}, {371,5478}, {530,10668}, {531,13908}, {542,13910}, {590,618}, {616,8972}, {5473,9540}, {5617,8976}, {6270,8974}, {6306,13705}, {6722,13981}, {6770,13886}, {6771,7583}, {9901,13888}, {9916,13889}, {9982,13892}, {10062,13904}, {10078,13905}, {11705,13883}, {12142,13884}, {12205,13885}, {12337,13887}, {12781,13893}, {12793,13894}, {12922,13895}, {12932,13896}, {12942,13897}, {12952,13898}, {12991,13900}, {13076,13901}, {13103,13903}, {13105,13906}, {13107,13907}

X(13918) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO 1st ORTHOSYMMEDIAL

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

The reciprocal orthologic center of these triangles is X(4)

X(13918) lies on these lines: {6,13985}, {112,9540}, {127,371}, {132,590}, {140,13992}, {1151,2794}, {1297,3068}, {2781,8998}, {2799,8980}, {2806,13913}, {3320,13901}, {5418,6720}, {8972,12384}, {8974,12805}, {8975,12806}, {8976,12918}, {8981,13923}, {9530,13846}, {9541,10735}, {9583,13280}, {12145,13884}, {12207,13885}, {12253,13886}, {12265,13883}, {12340,13887}, {12408,13888}, {12413,13889}, {12503,13892}, {12784,13893}, {12796,13894}, {12925,13895}, {12935,13896}, {12945,13897}, {12955,13898}, {12997,13900}, {13099,13902}, {13115,13903}, {13116,13904}, {13118,13906}, {13119,13907}

X(13918) = reflection of X(13923) in X(8981)

X(13919) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(79)

X(13919) lies on these lines: {6,13987}, {371,12600}, {590,13089}, {3068,10266}, {8972,12849}, {8975,12808}, {8976,12919}, {9540,12556}, {12146,13884}, {12255,13886}, {12267,13883}, {12342,13887}, {12409,13888}, {12414,13889}, {12504,13892}, {12786,13893}, {12798,13894}, {12927,13895}, {12937,13896}, {12947,13897}, {12957,13898}, {13001,13900}, {13130,13906}

X(13920) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO 1st TRI-SQUARES-CENTRAL

Barycentrics 19*S^2+3*(-5*SA+7*SW)*S+3*(SB+2*SC)*(2*SB+SC) : :

The reciprocal orthologic center of these triangles is X(13665)

X(13920) lies on these lines: {2,13662}, {6,13988}, {30,13879}, {115,13703}, {371,13687}, {590,13701}, {1327,3068}, {6329,13849}, {8974,13690}, {8975,13691}, {8976,13692}, {9540,13666}, {13667,13883}, {13668,13884}, {13672,13885}, {13674,13886}, {13675,13887}, {13679,13888}, {13680,13889}, {13685,13892}, {13688,13893}, {13689,13894}, {13693,13895}, {13695,13897}, {13696,13898}, {13697,13899}, {13698,13900}, {13699,13901}, {13702,13902}, {13714,13904}, {13715,13905}, {13716,13906}, {13717,13907}, {13848,13910}

X(13921) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO 4th TRI-SQUARES

Barycentrics 6*S^2+(-3*SA+SW)*S-(SA+2*SW)*(SB+SC) : :
X(13921) = 3*X(13846)+X(13881) = 5*X(13846)+X(13932) = 5*X(13881)-3*X(13932)

The reciprocal orthologic center of these triangles is X(486)

X(13921) lies on these lines: {6,6119}, {30,13848}, {39,590}, {486,3068}, {487,8972}, {524,7862}, {3564,13879}, {3767,6229}, {6280,8975}, {6281,8974}, {6290,8976}, {7980,13902}, {8253,13934}, {8960,13638}, {8967,8969}, {9540,12123}, {9906,13888}, {9921,13889}, {9986,13892}, {10067,13904}, {10083,13905}, {12147,13884}, {12210,13885}, {12221,13650}, {12256,13886}, {12268,13883}, {12343,13887}, {12601,13903}, {12787,13893}, {12799,13894}, {12928,13895}, {12938,13896}, {12948,13897}, {12958,13898}, {13002,13899}, {13003,13900}, {13081,13901}, {13132,13906}, {13846,13881}

X(13921) = orthologic center of these triangles: 3rd tri-squares-central to inner-Vecten
X(13921) = {X(13910), X(13925)}-harmonic conjugate of X(13879)

X(13922) = PARALLELOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO INNER-GARCIA

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

The reciprocal parallelogic center of these triangles is X(1)

X(13922) lies on these lines: {6,3035}, {11,590}, {80,13893}, {100,3068}, {104,9540}, {119,371}, {140,13977}, {149,8972}, {214,13883}, {485,5840}, {528,13846}, {952,8981}, {1145,7969}, {1151,2829}, {1320,13902}, {1862,13884}, {2067,10956}, {2771,8994}, {2783,8980}, {2787,8997}, {2800,13912}, {2802,8983}, {2806,13923}, {3634,13976}, {5418,6713}, {5541,13888}, {6221,10742}, {6667,8253}, {8960,10993}, {8974,13269}, {8975,13270}, {8976,10738}, {9024,13910}, {9541,10728}, {9583,12751}, {10087,13904}, {10090,13905}, {12331,13903}, {13194,13885}, {13199,13886}, {13205,13887}, {13222,13889}, {13228,13890}, {13230,13891}, {13235,13892}, {13268,13894}, {13271,13895}, {13272,13896}, {13273,13897}, {13274,13898}, {13275,13899}, {13276,13900}, {13278,13906}, {13279,13907}

X(13922) = reflection of X(13913) in X(8981)
X(13922) = {X(6), X(3035)}-harmonic conjugate of X(13991)

X(13923) = PARALLELOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO 1st ORTHOSYMMEDIAL

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

The reciprocal parallelogic center of these triangles is X(6)

X(13923) lies on these lines: {6,6720}, {112,3068}, {127,590}, {132,371}, {140,13985}, {485,2794}, {1297,9540}, {2781,8994}, {2799,8997}, {2806,13922}, {2831,13913}, {6020,13901}, {6221,12918}, {8972,13219}, {8974,13282}, {8976,10749}, {8981,13918}, {8998,9517}, {9583,12784}, {10705,13902}, {11641,13889}, {11722,13883}, {13166,13884}, {13195,13885}, {13200,13886}, {13206,13887}, {13280,13893}, {13281,13894}, {13294,13895}, {13295,13896}, {13296,13897}, {13297,13898}, {13299,13900}, {13312,13905}, {13313,13906}, {13314,13907}

X(13923) = reflection of X(13918) in X(8981)
X(13923) = {X(6), X(6720)}-harmonic conjugate of X(13992)

X(13924) = X(5) OF THE 3rd TRI-SQUARES TRIANGLE

Barycentrics 10*S^2+(-7*SA+13*SW)*S+(SA+4*SW)*(SB+SC) : :
X(13924) = 3*X(13846)-X(13879) = 3*X(13846)+X(13882)

X(13924) lies on these lines: {20,485}, {590,639}, {641,3068}, {6250,8976}, {6289,13903}, {8180,13910}, {8253,13880}, {12124,13886}, {12788,13888}, {12968,13701}

X(13924) = midpoint of X(13879) and X(13882)

X(13925) = X(5) OF THE 3rd TRI-SQUARES-CENTRAL TRIANGLE

Barycentrics 6*S^2+(SB+SC)*(4*S-SA) : :
X(13925) = 3*X(485)+X(1151) = 11*X(485)-3*X(1327) = X(485)+3*X(13846) = 11*X(1151)+9*X(1327) = X(1151)-3*X(8981) = X(1151)-9*X(13846) = 3*X(1327)+11*X(8981)

X(13925) lies on these lines: {2,6418}, {3,8972}, {4,13903}, {5,1588}, {6,3628}, {26,13889}, {30,485}, {140,372}, {355,13888}, {371,546}, {382,9691}, {395,3392}, {396,3367}, {495,13897}, {496,13898}, {524,6118}, {547,7584}, {548,3070}, {549,1587}, {550,6455}, {631,6408}, {632,3312}, {952,8983}, {1131,1657}, {1152,12108}, {1483,13902}, {1656,3316}, {1658,8276}, {3071,5066}, {3090,6417}, {3091,6199}, {3146,6407}, {3364,11543}, {3389,11542}, {3525,6395}, {3526,7581}, {3529,6445}, {3530,5418}, {3564,13879}, {3590,3851}, {3592,12811}, {3627,6221}, {3830,6474}, {3845,6459}, {3853,6564}, {3861,6561}, {5055,7582}, {5070,7586}, {5073,10145}, {5420,6471}, {5690,13893}, {5844,13911}, {5874,8975}, {5875,8974}, {5901,13883}, {6119,6329}, {6200,12103}, {6419,12812}, {6425,12102}, {6430,11812}, {6432,8253}, {6460,6497}, {6501,13941}, {6756,13884}, {8252,10195}, {8854,11266}, {8980,8993}, {8995,8998}, {9542,10137}, {9663,10483}, {10942,13896}, {10943,13895}, {11539,13935}

X(13925) = midpoint of X(i) and X(j) for these {i,j}: {485,8981}, {8998,13915}
X(13925) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,3628,13993), (1656,6500,13939), (3068,8976,5), (9540,13665,550), (13898,13905,496)

X(13926) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES TO 1st ANTI-BROCARD

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

The reciprocal orthologic center of these triangles is X(9867)

X(13926) lies on these lines: {2,7599}, {3,115}, {98,486}, {114,372}, {183,6229}, {230,2460}, {385,9867}, {485,8781}, {542,13847}, {615,6231}, {620,11316}, {642,3103}, {1513,2459}, {2023,3102}, {3564,6230}, {6055,13932}, {6108,13928}, {6109,13929}, {6119,9478}, {6560,9758}, {6565,9756}, {6722,11314}, {7793,9991}, {7806,8316}, {8304,13087}, {12601,12963}

X(13926) = midpoint of X(385) and X(9867)
X(13926) = complement of X(33340)
X(13926) = orthoptic-circle-of-Steiner-inellipse-inverse-of-X(13520)
X(13926) = orthologic center of these triangles: 4th tri-squares to 1st Brocard
X(13926) = orthologic center of these triangles: 4th tri-squares to 6th Brocard
X(13926) = {X(115), X(6036)}-harmonic conjugate of X(13873)

X(13927) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES TO ANTI-MCCAY

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

The reciprocal orthologic center of these triangles is X(9891)

X(13927) lies on these lines: {486,490}, {543,13847}, {599,626}, {615,9892}, {2482,13783}, {6399,12601}, {8593,13770}, {8859,9891}, {8860,13087}, {9882,13760}, {9894,13796}

X(13927) = orthologic center of these triangles: 4th tri-squares to Mccay

X(13928) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES TO INNER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(6301)

X(13928) lies on these lines: {14,486}, {115,615}, {395,6307}, {531,10671}, {619,6300}, {3643,6301}, {5460,13932}, {6108,13926}, {9113,13770}

X(13929) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES TO OUTER-NAPOLEON

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

The reciprocal orthologic center of these triangles is X(6300)

X(13929) lies on these lines: {13,486}, {115,615}, {396,6306}, {530,10672}, {618,6301}, {3642,6300}, {5459,13932}, {6109,13926}, {9112,13770}

X(13930) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES TO 1st NEUBERG

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

The reciprocal orthologic center of these triangles is X(6316)

X(13930) lies on these lines: {3,6399}, {39,13934}, {76,486}, {183,6316}, {487,6194}, {511,6290}, {538,13847}, {615,6318}, {640,6393}, {642,3103}, {3094,3763}, {6314,13827}, {9466,13932}

X(13930) = reflection of X(3103) in X(642)

X(13931) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(6315)

X(13931) lies on these lines: {83,486}, {615,6317}, {754,13847}, {6119,9478}, {6287,6399}, {6292,13934}, {6313,13829}, {6315,11174}, {6704,7834}

X(13932) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES TO 1st TRI-SQUARES-CENTRAL

Barycentrics 14*S^2+(3*SA-SW)*S-3*(SW+3*SA)*(SB+SC) : :
X(13932) = 2*X(6251)+X(6399) = X(13846)-3*X(13881) = 5*X(13846)-6*X(13921) = 5*X(13881)-2*X(13921)

The reciprocal orthologic center of these triangles is X(13711)

X(13932) lies on these lines: {2,9600}, {381,486}, {524,13850}, {615,13712}, {5459,13929}, {5460,13928}, {6055,13926}, {6251,6399}, {9466,13930}, {12962,13846}, {13701,13934}, {13843,13847}

X(13932) = reflection of X(13847) in X(13988)

X(13933) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 4th TRI-SQUARES

Barycentrics (SA+2*S-SW)*(SA+3*S-2*SW) : :
X(13933) = 3*X(13847)-X(13934)

The reciprocal orthologic center of these triangles is X(486)

X(13933) lies on these lines: {4,372}, {6,6119}, {30,13849}, {487,13770}, {615,642}, {3564,13880}, {6280,13950}, {6281,13949}, {6290,13951}, {6329,13879}, {7980,13959}, {9906,13942}, {9921,13943}, {9986,13946}, {10067,13962}, {10083,13963}, {12123,13935}, {12147,13937}, {12210,13938}, {12268,13936}, {12343,13940}, {12601,13961}, {12787,13947}, {12799,13948}, {12928,13952}, {12938,13953}, {12948,13954}, {12958,13955}, {13002,13956}, {13003,13957}, {13081,13958}, {13132,13964}, {13133,13965}, {13821,13847}

X(13933) = reflection of X(642) in X(8184)
X(13933) = {X(13972), X(13993)}-harmonic conjugate of X(13880)

X(13934) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES TO INNER-VECTEN

Barycentrics (SA-2*S+SW)*(SA+S-SW) : :
X(13934) = 3*X(2)+X(6463) = 3*X(13847)-2*X(13933)

The reciprocal orthologic center of these triangles is X(486)

X(13934) lies on these lines: {2,494}, {3,486}, {6,642}, {39,13930}, {114,372}, {487,3069}, {618,6301}, {619,6300}, {641,6119}, {1505,7888}, {1691,6315}, {2482,13783}, {3094,11315}, {3564,13966}, {3589,6387}, {5414,12958}, {5871,6813}, {6292,13931}, {6422,7807}, {6502,12948}, {6503,9726}, {7968,12787}, {8253,13921}, {8299,10083}, {8855,12960}, {8940,11210}, {9758,10840}, {9906,13947}, {12221,13941}, {12509,13939}, {12963,13821}, {13701,13932}

X(13934) = midpoint of X(5491) and X(6463)
X(13934) = complement of X(5491)
X(13934) = {X(2), X(6463)}-harmonic conjugate of X(5491)

X(13935) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND ABC-X3 REFLECTIONS

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

X(13935) lies on these lines: {1,13975}, {2,372}, {3,1588}, {4,615}, {5,6398}, {6,631}, {20,486}, {22,8277}, {30,6450}, {35,13962}, {36,13963}, {40,13971}, {56,13958}, {74,13990}, {98,13989}, {99,13967}, {100,13977}, {104,13991}, {110,13969}, {112,13985}, {140,3068}, {165,13942}, {182,13938}, {371,3523}, {376,3071}, {381,6408}, {382,6446}, {487,7793}, {489,13757}, {492,3785}, {515,13947}, {517,13959}, {546,6522}, {548,6452}, {549,3311}, {550,6456}, {590,3525}, {632,8976}, {639,3593}, {642,1271}, {944,13973}, {1124,5218}, {1125,1703}, {1131,7486}, {1151,3524}, {1297,13992}, {1327,6479}, {1335,7288}, {1350,13972}, {1490,13974}, {1498,13980}, {1578,7494}, {1593,13937}, {1702,10164}, {2041,11489}, {2042,11488}, {2459,11293}, {3070,3090}, {3085,6502}, {3086,5414}, {3088,5413}, {3089,11474}, {3091,6454}, {3093,6353}, {3098,13946}, {3102,6194}, {3146,6565}, {3448,10820}, {3522,6561}, {3526,6395}, {3528,6412}, {3530,6221}, {3533,8253}, {3541,10881}, {3543,6485}, {3545,6430}, {3546,10898}, {3576,13936}, {3591,12819}, {3628,6448}, {3832,6481}, {3855,6469}, {5054,6418}, {5056,6564}, {5059,6487}, {5067,6438}, {5068,10194}, {5210,12509}, {5326,13897}, {5406,6806}, {5418,6420}, {5473,13982}, {5474,13981}, {5591,11316}, {5657,7968}, {5871,6813}, {6200,9543}, {6201,6811}, {6284,13955}, {6409,10299}, {6421,7735}, {6423,7736}, {6427,12108}, {6433,9693}, {6455,12100}, {6471,13846}, {6489,11541}, {6497,8703}, {7294,13898}, {7354,13954}, {7396,8281}, {7691,13986}, {8416,11315}, {8855,10565}, {9733,13758}, {10310,13940}, {11248,13964}, {11249,13965}, {11257,13983}, {11412,12240}, {11414,13943}, {11539,13925}, {11822,13944}, {11823,13945}, {11824,13949}, {11825,13950}, {11826,13952}, {11827,13953}, {11828,13956}, {11829,13957}, {12117,13968}, {12118,13970}, {12119,13976}, {12120,13978}, {12121,13979}, {12122,13984}, {12123,13933}, {12124,13880}, {12510,13881}, {12556,13987}, {13666,13988}, {13786,13849}

X(13935) = midpoint of X(6450) and X(13951)
X(13935) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,372,1587), (3,1588,9541), (3,13961,7584), (6,631,9540), (372,5420,2), (376,13939,3071), (486,6396,20), (550,13993,13785), (3071,6410,376), (6456,13785,550), (7584,13966,13961)

X(13936) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND ANTI-AQUILA

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

X(13936) lies on these lines: {1,1123}, {2,8983}, {3,13975}, {4,1703}, {6,10}, {8,7586}, {11,13976}, {40,1588}, {81,6347}, {165,6459}, {214,13991}, {226,2362}, {355,3312}, {371,6684}, {372,515}, {485,10175}, {486,946}, {516,3071}, {517,7584}, {519,7968}, {551,13847}, {590,3634}, {605,5264}, {606,1724}, {615,1125}, {631,9583}, {950,5414}, {1132,9812}, {1151,10164}, {1152,4297}, {1210,1335}, {1385,13966}, {1386,13972}, {1587,5587}, {1698,3068}, {1702,5657}, {1737,3301}, {1771,3077}, {2067,3911}, {2646,13958}, {3295,13940}, {3298,11019}, {3299,10039}, {3311,13912}, {3452,5405}, {3523,9615}, {3576,13935}, {3616,13941}, {3624,13902}, {5090,5411}, {5420,10165}, {5603,13939}, {5691,6460}, {5790,6418}, {5818,7581}, {5886,13951}, {5901,13993}, {6502,10106}, {7583,9956}, {7585,9780}, {8981,11231}, {10172,10576}, {10246,13961}, {11108,13887}, {11363,13937}, {11364,13938}, {11365,13943}, {11366,13944}, {11367,13945}, {11368,13946}, {11370,13949}, {11371,13950}, {11373,13952}, {11374,13953}, {11375,13954}, {11376,13955}, {11377,13956}, {11378,13957}, {11705,13982}, {11706,13981}, {11709,13969}, {11710,13967}, {11711,13989}, {11715,13977}, {11720,13990}, {11722,13992}, {11831,13948}, {12114,13974}, {12258,13968}, {12259,13970}, {12260,13978}, {12261,13979}, {12262,13980}, {12263,13983}, {12264,13984}, {12265,13985}, {12266,13986}, {12267,13987}, {12268,13933}, {12269,13880}, {12699,13785}, {13667,13988}, {13787,13849}

X(13936) = reflection of X(5689) in X(10)
X(13936) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3069,13971), (1,13942,13959), (6,10,13883), (6,13973,10), (615,7969,1125), (3069,13959,13942), (5657,7582,1702), (7585,9780,13893), (13942,13959,13971)

X(13937) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND ANTI-ARA

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

X(13937) lies on these lines: {2,5411}, {4,3591}, {5,10881}, {6,468}, {24,7584}, {25,3069}, {33,13958}, {235,372}, {427,615}, {428,13847}, {486,3575}, {1112,13990}, {1152,1885}, {1368,11418}, {1588,3515}, {1593,13935}, {1598,13961}, {1829,13971}, {1843,13972}, {1862,13991}, {1902,13975}, {1906,11474}, {3092,5420}, {3147,3311}, {3312,3542}, {3535,3593}, {3580,11448}, {5090,13947}, {5186,13989}, {5410,6353}, {6560,10151}, {6756,13993}, {7487,13939}, {7505,7583}, {7713,13942}, {8981,10018}, {10154,11417}, {11266,11585}, {11363,13936}, {11380,13938}, {11381,13980}, {11383,13940}, {11384,13944}, {11385,13945}, {11386,13946}, {11388,13949}, {11389,13950}, {11390,13952}, {11391,13953}, {11392,13954}, {11393,13955}, {11394,13956}, {11395,13957}, {11396,13959}, {11398,13962}, {11399,13963}, {11400,13964}, {11401,13965}, {11576,13986}, {11832,13948}, {12131,13967}, {12132,13968}, {12133,13969}, {12134,13970}, {12135,13973}, {12136,13974}, {12137,13976}, {12138,13977}, {12139,13978}, {12140,13979}, {12141,13981}, {12142,13982}, {12143,13983}, {12144,13984}, {12145,13985}, {12146,13987}, {12147,13933}, {12148,13880}, {13166,13992}, {13668,13988}, {13788,13849}

X(13938) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND 5th ANTI-BROCARD

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

X(13938) lies on these lines: {6,1078}, {32,637}, {83,615}, {98,372}, {182,13935}, {384,13983}, {486,12110}, {1588,5171}, {2080,7584}, {3068,7815}, {3312,10104}, {3398,13966}, {4027,13989}, {6459,8722}, {7787,13941}, {10788,13939}, {10789,13942}, {10790,13943}, {10791,13947}, {10792,13949}, {10793,13950}, {10794,13952}, {10795,13953}, {10796,13951}, {10797,13954}, {10798,13955}, {10799,13958}, {10800,13959}, {10801,13962}, {10802,13963}, {10803,13964}, {10804,13965}, {11364,13936}, {11380,13937}, {11490,13940}, {11837,13944}, {11838,13945}, {11839,13948}, {11840,13956}, {11841,13957}, {11842,13961}, {12150,13847}, {12176,13967}, {12191,13968}, {12192,13969}, {12193,13970}, {12194,13971}, {12195,13973}, {12196,13974}, {12197,13975}, {12198,13976}, {12199,13977}, {12200,13978}, {12201,13979}, {12202,13980}, {12204,13981}, {12205,13982}, {12206,13984}, {12207,13985}, {12208,13986}, {12209,13987}, {12210,13933}, {12211,13880}, {12212,13972}, {13193,13990}, {13194,13991}, {13195,13992}, {13672,13988}, {13792,13849}

X(13938) = {X(6), X(1078)}-harmonic conjugate of X(13885)

X(13939) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND ANTI-EULER

Barycentrics 3*S^2-(SB+SC)*(SA+2*S) : :
X(13939) = 3*X(1132)+2*X(6408) = X(1132)+2*X(13961) = X(6408)-3*X(13961)

X(13939) lies on these lines: {2,3311}, {3,13941}, {4,372}, {5,6418}, {6,3090}, {20,6450}, {24,13943}, {30,1132}, {186,8277}, {371,3525}, {376,3071}, {388,13962}, {485,5071}, {488,13757}, {492,7376}, {497,13963}, {515,13942}, {546,6395}, {615,631}, {632,6199}, {640,5860}, {944,13971}, {1124,8164}, {1131,3851}, {1152,3529}, {1270,11314}, {1587,3545}, {1656,3316}, {3068,5067}, {3070,3855}, {3085,13954}, {3086,13955}, {3091,3312}, {3146,6398}, {3299,10588}, {3301,10589}, {3364,11488}, {3389,11489}, {3448,13979}, {3522,6497}, {3523,6455}, {3524,5420}, {3528,6561}, {3533,8252}, {3544,6420}, {3628,6417}, {3854,6495}, {4294,13958}, {5054,9691}, {5056,7583}, {5068,13665}, {5073,6473}, {5079,6501}, {5603,13936}, {5657,13947}, {6221,10303}, {6361,13975}, {6392,7388}, {6445,12108}, {6454,11541}, {6485,11001}, {6770,13982}, {6773,13981}, {6776,13972}, {6813,10784}, {7486,8976}, {7487,13937}, {7967,13959}, {8855,10880}, {9541,10299}, {9862,13946}, {10137,11812}, {10783,13949}, {10785,13952}, {10786,13953}, {10788,13938}, {10805,13964}, {10806,13965}, {11411,13970}, {11491,13940}, {11843,13944}, {11844,13945}, {11845,13948}, {11846,13956}, {11847,13957}, {12243,13968}, {12244,13969}, {12245,13973}, {12246,13974}, {12247,13976}, {12248,13977}, {12249,13978}, {12250,13980}, {12251,13983}, {12252,13984}, {12253,13985}, {12254,13986}, {12255,13987}, {12257,13880}, {12383,13990}, {12509,13934}, {13172,13989}, {13199,13991}, {13200,13992}, {13674,13988}, {13794,13849}

X(13939) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,7584,7582), (6,3090,13886), (1656,6500,13925), (3071,13935,376), (3317,7582,2), (7584,13951,2), (13785,13966,20)

X(13940) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND ANTI-MANDART-INCIRCLE

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

X(13940) lies on these lines: {1,6}, {3,13971}, {35,13942}, {55,3069}, {56,13959}, {100,13941}, {197,13943}, {372,11496}, {486,11500}, {590,8167}, {605,940}, {615,1376}, {1621,7586}, {3068,4423}, {3295,13936}, {3913,13973}, {4254,8225}, {4421,13847}, {5284,7585}, {5687,13947}, {6459,8273}, {7584,10267}, {10306,13975}, {10310,13935}, {11108,13883}, {11248,13966}, {11383,13937}, {11490,13938}, {11491,13939}, {11492,13944}, {11493,13945}, {11494,13946}, {11497,13949}, {11498,13950}, {11499,13951}, {11501,13954}, {11502,13955}, {11503,13956}, {11504,13957}, {11507,13962}, {11508,13963}, {11509,13964}, {11510,13965}, {11848,13948}, {11849,13961}, {12178,13967}, {12326,13968}, {12327,13969}, {12328,13970}, {12329,13972}, {12330,13974}, {12331,13976}, {12332,13977}, {12333,13978}, {12334,13979}, {12335,13980}, {12336,13981}, {12337,13982}, {12338,13983}, {12339,13984}, {12340,13985}, {12341,13986}, {12342,13987}, {12343,13933}, {12344,13880}, {13173,13989}, {13204,13990}, {13205,13991}, {13206,13992}, {13675,13988}, {13795,13849}

X(13941) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND ANTICOMPLEMENTARY

Barycentrics 2*a^2-3*S : :

X(13941) lies on these lines: {2,6}, {3,13939}, {4,3591}, {5,1131}, {8,13947}, {10,13942}, {20,486}, {22,13943}, {30,6446}, {100,13940}, {140,6199}, {145,13959}, {146,13969}, {147,13967}, {148,13989}, {149,13991}, {153,13977}, {194,13983}, {371,10303}, {372,3091}, {376,6452}, {388,13954}, {485,7486}, {487,13770}, {488,13834}, {497,13955}, {549,6445}, {616,13982}, {617,13981}, {631,6221}, {632,6417}, {638,13711}, {962,13975}, {1152,2672}, {1384,11291}, {1587,5056}, {1588,3523}, {1656,7581}, {2888,13986}, {2896,13946}, {3070,5068}, {3071,3522}, {3085,13962}, {3086,13963}, {3090,3312}, {3311,3525}, {3316,5070}, {3434,13952}, {3436,13953}, {3448,13990}, {3524,6451}, {3529,6450}, {3533,8981}, {3543,6481}, {3590,6436}, {3616,13936}, {3617,7968}, {3627,6408}, {3628,6418}, {3723,6351}, {3731,5405}, {3832,6438}, {3839,6560}, {4240,13948}, {5024,11292}, {5059,6434}, {5067,7583}, {5071,13665}, {5261,6502}, {5274,5414}, {5411,8889}, {5413,7378}, {5601,13944}, {5602,13945}, {6193,13970}, {6223,13974}, {6224,13976}, {6225,13980}, {6407,12108}, {6411,6459}, {6433,9543}, {6462,13956}, {6463,13957}, {6501,13925}, {6561,10304}, {6808,11456}, {7396,11418}, {7488,8277}, {7787,13938}, {8591,13968}, {8855,11417}, {9874,13978}, {10528,13964}, {10529,13965}, {10818,13202}, {12221,13934}, {12383,13979}, {12384,13985}, {12849,13987}, {13219,13992}, {13678,13988}, {13798,13849}

X(13941) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6,8972), (2,193,3595), (2,3069,7586), (2,7586,7585), (6,8972,7585), (615,3069,2), (615,13847,3069), (3068,8252,2), (3069,13758,13949), (7586,8972,6), (11488,11489,615)

X(13942) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND AQUILA

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

X(13942) lies on these lines: {1,1123}, {6,3624}, {10,13941}, {35,13940}, {40,13966}, {165,13935}, {355,13993}, {372,1699}, {486,5691}, {515,13939}, {517,13961}, {615,1698}, {1125,7586}, {1587,7988}, {1588,7987}, {1697,13958}, {1702,5420}, {1703,11522}, {1768,13977}, {2948,13990}, {3099,13946}, {3312,8227}, {3317,10175}, {3576,7584}, {3632,13973}, {3679,7968}, {3751,13972}, {5273,5405}, {5290,6502}, {5541,13991}, {5587,13951}, {5588,13950}, {5589,13949}, {6395,9955}, {6418,11230}, {7582,10165}, {7713,13937}, {7991,13975}, {7992,13974}, {8185,13943}, {8186,13944}, {8187,13945}, {8188,13956}, {8189,13957}, {8252,13893}, {8277,9590}, {9578,13954}, {9581,13955}, {9860,13967}, {9875,13968}, {9896,13970}, {9897,13976}, {9898,13978}, {9899,13980}, {9900,13981}, {9901,13982}, {9902,13983}, {9903,13984}, {9904,13969}, {9905,13986}, {9906,13933}, {9907,13880}, {10789,13938}, {10826,13952}, {10827,13953}, {11852,13948}, {12407,13979}, {12408,13985}, {12409,13987}, {13174,13989}, {13221,13992}, {13679,13988}, {13799,13849}

X(13942) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,3624,13888), (3069,13959,13936), (3069,13971,1), (7968,13947,3679), (13936,13959,1), (13936,13971,13959)

X(13943) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND ARA

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

X(13943) lies on these lines: {3,486}, {6,1196}, {22,13941}, {24,13939}, {25,3069}, {26,13993}, {159,13972}, {197,13940}, {372,1598}, {485,11484}, {494,6414}, {631,9695}, {1132,11413}, {1583,9723}, {1597,6565}, {1995,7586}, {3068,11284}, {3312,7529}, {3317,7509}, {5198,6460}, {5594,13950}, {5595,13949}, {6421,10962}, {6642,7584}, {7387,13966}, {7517,13961}, {8185,13942}, {8190,13944}, {8191,13945}, {8192,13959}, {8193,13947}, {8194,13956}, {8195,13957}, {8252,8281}, {9798,13971}, {9861,13967}, {9876,13968}, {9908,13970}, {9909,13847}, {9910,13974}, {9911,13975}, {9912,13976}, {9913,13977}, {9914,13980}, {9915,13981}, {9916,13982}, {9917,13983}, {9918,13984}, {9919,13969}, {9920,13986}, {9921,13933}, {9922,13880}, {10037,13962}, {10046,13963}, {10790,13938}, {10828,13946}, {10829,13952}, {10830,13953}, {10831,13954}, {10832,13955}, {10833,13958}, {10834,13964}, {10835,13965}, {11365,13936}, {11414,13935}, {11514,12591}, {11641,13992}, {11853,13948}, {12164,12240}, {12310,13990}, {12410,13973}, {12411,13978}, {12412,13979}, {12413,13985}, {12414,13987}, {13175,13989}, {13222,13991}, {13680,13988}, {13800,13849}

X(13944) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND 1st AURIGA

Barycentrics -((b+2*a+c)*K-a^2*(c-a+b)*(a+b+c))*S+K*(a+b+c)*a^2 : : , where K=4*S*sqrt(R*(4*R+r))

X(13944) lies on these lines: {6,13890}, {55,13945}, {372,8196}, {486,9834}, {615,5599}, {1125,13891}, {3069,5597}, {5598,13959}, {5600,7968}, {5601,13941}, {8186,13942}, {8190,13943}, {8197,13947}, {8198,13949}, {8199,13950}, {8200,13951}, {11207,13847}, {11252,13966}, {11366,13936}, {11384,13937}, {11492,13940}, {11822,13935}, {11837,13938}, {11843,13939}, {11861,13946}, {11865,13952}, {11867,13953}, {11869,13954}, {11871,13955}, {11873,13958}, {11875,13961}, {11877,13962}, {11879,13963}, {11881,13964}, {11883,13965}, {12345,13968}, {12452,13972}, {12454,13973}, {12456,13974}, {12458,13975}, {12460,13976}, {12462,13977}, {12474,13983}, {12476,13984}, {13176,13989}, {13208,13990}, {13228,13991}

X(13945) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND 2nd AURIGA

Barycentrics -(-(b+2*a+c)*K-a^2*(c-a+b)*(a+b+c))*S-K*(a+b+c)*a^2 : : , where K = 4*S*sqrt(R*(4*R+r))

X(13945) lies on these lines: {6,13891}, {55,13944}, {372,8203}, {486,9835}, {615,5600}, {1125,13890}, {3069,5598}, {5597,13959}, {5599,7968}, {5602,13941}, {8187,13942}, {8191,13943}, {8204,13947}, {8205,13949}, {8206,13950}, {8207,13951}, {11208,13847}, {11253,13966}, {11367,13936}, {11385,13937}, {11493,13940}, {11823,13935}, {11838,13938}, {11844,13939}, {11862,13946}, {11866,13952}, {11868,13953}, {11870,13954}, {11872,13955}, {11874,13958}, {11876,13961}, {11878,13962}, {11880,13963}, {11882,13964}, {11884,13965}, {12346,13968}, {12453,13972}, {12455,13973}, {12457,13974}, {12459,13975}, {12461,13976}, {12463,13977}, {12475,13983}, {12477,13984}, {13177,13989}, {13209,13990}, {13230,13991}

X(13946) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND 5th BROCARD

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

X(13946) lies on these lines: {6,7796}, {32,637}, {372,9993}, {486,9873}, {615,3096}, {2896,13941}, {3094,13972}, {3098,13935}, {3099,13942}, {7586,10583}, {7811,13847}, {8782,13989}, {9301,13961}, {9821,13966}, {9857,13947}, {9862,13939}, {9878,13968}, {9923,13970}, {9941,13971}, {9981,13981}, {9982,13982}, {9983,13983}, {9984,13969}, {9985,13986}, {9986,13933}, {9987,13880}, {9994,13949}, {9995,13950}, {9996,13951}, {9997,13959}, {10038,13962}, {10047,13963}, {10828,13943}, {10871,13952}, {10872,13953}, {10873,13954}, {10874,13955}, {10875,13956}, {10876,13957}, {10877,13958}, {10878,13964}, {10879,13965}, {11368,13936}, {11386,13937}, {11494,13940}, {11861,13944}, {11862,13945}, {11885,13948}, {12495,13973}, {12496,13974}, {12497,13975}, {12498,13976}, {12499,13977}, {12500,13978}, {12501,13979}, {12502,13980}, {12503,13985}, {12504,13987}, {13210,13990}, {13235,13991}, {13236,13992}, {13685,13988}, {13805,13849}

X(13946) = {X(6), X(7846)}-harmonic conjugate of X(13892)

X(13947) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND OUTER-GARCIA

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

X(13947) lies on these lines: {1,615}, {2,8983}, {4,13975}, {5,1703}, {6,1698}, {8,13941}, {10,3069}, {40,486}, {65,13954}, {72,13953}, {80,13991}, {100,13976}, {140,9583}, {165,3071}, {355,13966}, {372,5587}, {515,13935}, {517,13951}, {519,13959}, {631,9615}, {1132,9778}, {1152,5691}, {1377,5705}, {1587,10175}, {1588,6684}, {1702,7584}, {1737,13963}, {1837,13958}, {2362,5219}, {3057,13955}, {3068,3634}, {3070,7989}, {3311,11231}, {3312,9956}, {3317,5603}, {3416,13972}, {3576,5420}, {3579,13785}, {3624,7969}, {3679,7968}, {5090,13937}, {5405,7090}, {5414,9581}, {5657,13939}, {5687,13940}, {5688,13950}, {5689,13949}, {5690,13993}, {5790,13961}, {6459,10164}, {6502,9578}, {7582,13912}, {7586,9780}, {8193,13943}, {8197,13944}, {8204,13945}, {8214,13956}, {8215,13957}, {8227,10577}, {8253,13888}, {9857,13946}, {9864,13967}, {9881,13968}, {9906,13934}, {9928,13970}, {10039,13962}, {10791,13938}, {10820,12407}, {10914,13952}, {10915,13964}, {10916,13965}, {11900,13948}, {12368,13969}, {12667,13974}, {12751,13977}, {12777,13978}, {12778,13979}, {12779,13980}, {12780,13981}, {12781,13982}, {12782,13983}, {12783,13984}, {12784,13985}, {12785,13986}, {12786,13987}, {12787,13933}, {12788,13880}, {13178,13989}, {13211,13990}, {13280,13992}, {13688,13988}, {13808,13849}

X(13947) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,1698,13893), (8,13941,13971), (615,13973,1), (1588,6684,9616), (3679,13942,7968), (7586,9780,13883), (7969,8252,3624)

X(13948) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND GOSSARD

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

X(13948) lies on these lines: {6,13894}, {30,6450}, {372,11897}, {402,3069}, {486,12113}, {615,1650}, {1651,13847}, {4240,13941}, {11251,13966}, {11831,13936}, {11832,13937}, {11839,13938}, {11845,13939}, {11848,13940}, {11852,13942}, {11853,13943}, {11885,13946}, {11900,13947}, {11901,13949}, {11902,13950}, {11903,13952}, {11904,13953}, {11905,13954}, {11906,13955}, {11907,13956}, {11908,13957}, {11909,13958}, {11910,13959}, {11911,13961}, {11912,13962}, {11913,13963}, {11914,13964}, {11915,13965}, {12181,13967}, {12347,13968}, {12369,13969}, {12418,13970}, {12438,13971}, {12583,13972}, {12626,13973}, {12668,13974}, {12696,13975}, {12729,13976}, {12752,13977}, {12789,13978}, {12790,13979}, {12791,13980}, {12792,13981}, {12793,13982}, {12794,13983}, {12795,13984}, {12796,13985}, {12797,13986}, {12798,13987}, {12799,13933}, {12800,13880}, {13179,13989}, {13212,13990}, {13268,13991}, {13281,13992}, {13689,13988}, {13809,13849}

X(13949) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND INNER-GREBE

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

X(13949) lies on these lines: {2,6}, {372,6202}, {486,5871}, {631,8396}, {1161,13966}, {3128,5411}, {3641,13971}, {5589,13942}, {5595,13943}, {5605,13959}, {5689,13947}, {5875,13993}, {6215,13951}, {6227,13967}, {6258,13974}, {6263,13976}, {6267,13980}, {6270,13982}, {6271,13981}, {6273,13983}, {6275,13984}, {6277,13986}, {6279,13880}, {6281,13933}, {6319,13989}, {7725,13969}, {7732,13990}, {8198,13944}, {8205,13945}, {8216,13956}, {8217,13957}, {9882,13968}, {9929,13970}, {9994,13946}, {10040,13962}, {10048,13963}, {10783,13939}, {10792,13938}, {10919,13952}, {10921,13953}, {10923,13954}, {10925,13955}, {10927,13958}, {10929,13964}, {10931,13965}, {11370,13936}, {11388,13937}, {11497,13940}, {11824,13935}, {11901,13948}, {11916,13961}, {12627,13973}, {12697,13975}, {12753,13977}, {12801,13978}, {12803,13979}, {12805,13985}, {12807,13987}, {13269,13991}, {13282,13992}, {13690,13988}, {13810,13849}

X(13949) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,615,5591), (3069,13758,13941), (3069,13972,13950)

X(13950) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND OUTER-GREBE

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

X(13950) lies on these lines: {2,6}, {4,13763}, {25,1165}, {147,13760}, {372,6201}, {486,3424}, {1160,13966}, {3640,13971}, {5305,7376}, {5588,13942}, {5594,13943}, {5604,13959}, {5688,13947}, {5874,13993}, {6214,13951}, {6226,13967}, {6257,13974}, {6262,13976}, {6266,13980}, {6268,13982}, {6269,13981}, {6272,13983}, {6274,13984}, {6276,13986}, {6278,13880}, {6280,13933}, {6320,13989}, {6811,8416}, {6813,10784}, {7726,13969}, {7733,13990}, {8199,13944}, {8206,13945}, {8218,13956}, {8219,13957}, {9883,13968}, {9930,13970}, {9995,13946}, {10041,13962}, {10049,13963}, {10793,13938}, {10920,13952}, {10922,13953}, {10924,13954}, {10926,13955}, {10928,13958}, {10930,13964}, {10932,13965}, {11177,13773}, {11371,13936}, {11389,13937}, {11498,13940}, {11825,13935}, {11902,13948}, {11917,13961}, {12628,13973}, {12698,13975}, {12754,13977}, {12802,13978}, {12804,13979}, {12806,13985}, {12808,13987}, {13270,13991}, {13283,13992}, {13691,13988}, {13811,13849}

X(13950) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,5304,8974), (6,615,5590), (615,7735,2), (3069,13758,2), (3069,13972,13949)

X(13951) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND JOHNSON

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

X(13951) lies on these lines: {1,13954}, {2,3311}, {3,486}, {4,3591}, {5,1587}, {6,17}, {11,13963}, {12,13962}, {20,6456}, {30,6450}, {110,13979}, {140,1588}, {155,13970}, {265,13990}, {355,13952}, {371,3526}, {372,381}, {376,1132}, {382,1152}, {485,5055}, {492,11314}, {517,13947}, {546,6448}, {549,6455}, {550,6452}, {590,5070}, {631,6449}, {632,9540}, {637,11316}, {638,13757}, {641,11165}, {952,13959}, {1131,3544}, {1151,5054}, {1352,13972}, {1479,13958}, {1482,13973}, {1505,13881}, {1594,5411}, {1657,6396}, {1702,11231}, {1703,9955}, {2045,11543}, {2046,11542}, {3068,3628}, {3070,3851}, {3090,6428}, {3095,13983}, {3523,6451}, {3525,6447}, {3530,6496}, {3534,6410}, {3593,7376}, {3594,5072}, {3627,6522}, {3830,6408}, {3843,6560}, {5056,7581}, {5067,7585}, {5073,6446}, {5076,6454}, {5079,6420}, {5410,7505}, {5414,9669}, {5418,6199}, {5587,13942}, {5613,13981}, {5617,13982}, {5790,7968}, {5878,13980}, {5886,13936}, {6033,13967}, {6214,13950}, {6215,13949}, {6259,13974}, {6265,13976}, {6287,13984}, {6288,13986}, {6289,13880}, {6290,13933}, {6321,13989}, {6419,8253}, {6502,9654}, {6519,10303}, {6813,10846}, {7486,13886}, {7507,10881}, {7728,13969}, {8200,13944}, {8207,13945}, {8220,13956}, {8221,13957}, {8724,13968}, {8855,10897}, {9996,13946}, {10738,13991}, {10742,13977}, {10749,13992}, {10796,13938}, {10820,12902}, {10942,13964}, {10943,13965}, {11499,13940}, {12699,13975}, {12856,13978}, {12918,13985}, {12919,13987}, {13692,13988}, {13812,13849}

X(13951) = reflection of X(6450) in X(13935)
X(13951) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,7582,8981), (2,7584,3311), (2,13939,7584), (3,486,13785), (4,13966,6398), (5,3312,13665), (6,1656,8976), (486,5420,3071), (3071,5420,3), (3317,13939,2)

X(13952) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND INNER-JOHNSON

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

X(13952) lies on these lines: {6,13895}, {11,3069}, {12,13964}, {355,13951}, {372,10893}, {486,12114}, {615,1376}, {3434,13941}, {7586,10584}, {10523,13962}, {10525,13966}, {10785,13939}, {10794,13938}, {10826,13942}, {10829,13943}, {10871,13946}, {10912,13973}, {10914,13947}, {10919,13949}, {10920,13950}, {10943,13993}, {10944,13954}, {10945,13956}, {10946,13957}, {10947,13958}, {10948,13963}, {10949,13965}, {11235,13847}, {11373,13936}, {11390,13937}, {11826,13935}, {11865,13944}, {11866,13945}, {11903,13948}, {11928,13961}, {12182,13967}, {12348,13968}, {12371,13969}, {12422,13970}, {12586,13972}, {12676,13974}, {12700,13975}, {12737,13976}, {12761,13977}, {12857,13978}, {12889,13979}, {12920,13980}, {12921,13981}, {12922,13982}, {12923,13983}, {12924,13984}, {12925,13985}, {12926,13986}, {12927,13987}, {12928,13933}, {12929,13880}, {13180,13989}, {13213,13990}, {13271,13991}, {13294,13992}, {13693,13988}, {13813,13849}

X(13953) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND OUTER-JOHNSON

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

X(13953) lies on these lines: {6,13896}, {11,13965}, {12,3069}, {72,13947}, {355,13951}, {372,10894}, {486,11500}, {615,958}, {3436,13941}, {5812,13975}, {7586,10585}, {10523,13963}, {10526,13966}, {10786,13939}, {10795,13938}, {10827,13942}, {10830,13943}, {10872,13946}, {10921,13949}, {10922,13950}, {10942,13993}, {10950,13955}, {10951,13956}, {10952,13957}, {10953,13958}, {10954,13962}, {10955,13964}, {11236,13847}, {11374,13936}, {11391,13937}, {11827,13935}, {11867,13944}, {11868,13945}, {11904,13948}, {11929,13961}, {12183,13967}, {12349,13968}, {12372,13969}, {12423,13970}, {12587,13972}, {12635,13973}, {12677,13974}, {12738,13976}, {12762,13977}, {12858,13978}, {12890,13979}, {12930,13980}, {12931,13981}, {12932,13982}, {12933,13983}, {12934,13984}, {12935,13985}, {12936,13986}, {12937,13987}, {12938,13933}, {12939,13880}, {13181,13989}, {13214,13990}, {13272,13991}, {13295,13992}, {13694,13988}, {13814,13849}

X(13954) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND 1st JOHNSON-YFF

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

X(13954) lies on these lines: {1,13951}, {4,13958}, {5,13963}, {6,13897}, {12,3069}, {35,13785}, {55,486}, {56,615}, {65,13947}, {372,10895}, {388,13941}, {495,13962}, {498,7584}, {1152,12943}, {1335,10577}, {1478,13966}, {1587,3614}, {1588,5432}, {1656,3301}, {1836,13975}, {2067,8252}, {2099,13973}, {3071,5217}, {3085,13939}, {3086,3317}, {3157,13970}, {3312,7951}, {3585,6398}, {3628,13904}, {4316,6456}, {5204,5420}, {5252,13971}, {5326,9540}, {5414,10896}, {6450,10483}, {6502,11237}, {6565,12953}, {7354,13935}, {7582,13901}, {7586,10588}, {8277,9659}, {9578,13942}, {9654,13961}, {10088,13979}, {10797,13938}, {10831,13943}, {10873,13946}, {10923,13949}, {10924,13950}, {10944,13952}, {10956,13964}, {10957,13965}, {11375,13936}, {11392,13937}, {11501,13940}, {11869,13944}, {11870,13945}, {11905,13948}, {11930,13956}, {11931,13957}, {12184,13967}, {12350,13968}, {12373,13969}, {12588,13972}, {12678,13974}, {12739,13976}, {12763,13977}, {12837,13983}, {12859,13978}, {12903,13990}, {12940,13980}, {12941,13981}, {12942,13982}, {12944,13984}, {12945,13985}, {12946,13986}, {12947,13987}, {12948,13933}, {12949,13880}, {13182,13989}, {13273,13991}, {13296,13992}, {13695,13988}, {13815,13849}

X(13954) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (495,13993,13962), (1656,3301,13898)

X(13955) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND 2nd JOHNSON-YFF

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

X(13955) lies on these lines: {1,13951}, {2,13901}, {5,13962}, {6,13898}, {11,3069}, {36,13785}, {55,615}, {56,486}, {372,10896}, {496,13963}, {497,13941}, {499,7584}, {1069,13970}, {1124,10577}, {1152,12953}, {1479,13966}, {1587,7173}, {1588,5433}, {1656,3299}, {1837,13971}, {2066,8252}, {2098,13973}, {3057,13947}, {3071,5204}, {3085,3317}, {3086,13939}, {3312,7741}, {3583,6398}, {3628,13905}, {4324,6456}, {5217,5420}, {5405,8243}, {5414,11238}, {6284,13935}, {6502,10895}, {6565,12943}, {7294,9540}, {7586,10589}, {8277,9672}, {9581,13942}, {9669,13961}, {10091,13979}, {10798,13938}, {10832,13943}, {10874,13946}, {10925,13949}, {10926,13950}, {10950,13953}, {10958,13964}, {10959,13965}, {11376,13936}, {11393,13937}, {11502,13940}, {11871,13944}, {11872,13945}, {11906,13948}, {11932,13956}, {11933,13957}, {12185,13967}, {12351,13968}, {12374,13969}, {12589,13972}, {12679,13974}, {12701,13975}, {12740,13976}, {12764,13977}, {12836,13983}, {12860,13978}, {12904,13990}, {12950,13980}, {12951,13981}, {12952,13982}, {12954,13984}, {12955,13985}, {12956,13986}, {12957,13987}, {12958,13933}, {12959,13880}, {13183,13989}, {13274,13991}, {13297,13992}, {13696,13988}, {13816,13849}

X(13955) = {X(496), X(13993)}-harmonic conjugate of X(13963)

X(13956) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND LUCAS HOMOTHETIC

Barycentrics (12*R^2+SB+SC)*S^2-(8*(SB+SC)*R^2+SA^2-2*SW^2)*S-(SB+SC)*SW^2 : :

X(13956) lies on these lines: {6,13899}, {372,8212}, {486,9838}, {493,3069}, {615,8222}, {6461,13957}, {6462,13941}, {8188,13942}, {8194,13943}, {8210,13959}, {8214,13947}, {8216,13949}, {8218,13950}, {8220,13951}, {10669,13966}, {10875,13946}, {10945,13952}, {10951,13953}, {11377,13936}, {11394,13937}, {11503,13940}, {11828,13935}, {11840,13938}, {11846,13939}, {11907,13948}, {11930,13954}, {11932,13955}, {11947,13958}, {11949,13961}, {11951,13962}, {11953,13963}, {11955,13964}, {11957,13965}, {12152,13847}, {12186,13967}, {12352,13968}, {12377,13969}, {12426,13970}, {12440,13971}, {12590,13972}, {12636,13973}, {12741,13976}, {12765,13977}, {12861,13978}, {12894,13979}, {12986,13980}, {12988,13981}, {12990,13982}, {12992,13983}, {12994,13984}, {12996,13985}, {12998,13986}, {13000,13987}, {13002,13933}, {13004,13880}, {13184,13989}, {13215,13990}, {13275,13991}, {13298,13992}, {13697,13988}, {13817,13849}

X(13957) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND LUCAS(-1) HOMOTHETIC

Barycentrics (12*R^2+SB+SC)*S^2+(-8*(SB+SC)*R^2+SA^2-2*SW^2)*S+(SB+SC)*SW^2 : :

X(13957) lies on these lines: {6,13900}, {372,8213}, {393,494}, {486,9839}, {615,8223}, {6461,13956}, {6463,13941}, {8189,13942}, {8195,13943}, {8211,13959}, {8215,13947}, {8217,13949}, {8219,13950}, {8221,13951}, {10673,13966}, {10876,13946}, {10946,13952}, {10952,13953}, {11378,13936}, {11395,13937}, {11504,13940}, {11829,13935}, {11841,13938}, {11847,13939}, {11908,13948}, {11931,13954}, {11933,13955}, {11948,13958}, {11950,13961}, {11952,13962}, {11954,13963}, {11956,13964}, {11958,13965}, {12153,13847}, {12187,13967}, {12353,13968}, {12378,13969}, {12427,13970}, {12441,13971}, {12591,13972}, {12637,13973}, {12742,13976}, {12766,13977}, {12862,13978}, {12895,13979}, {12987,13980}, {12989,13981}, {12991,13982}, {12993,13983}, {12995,13984}, {12997,13985}, {12999,13986}, {13001,13987}, {13003,13933}, {13005,13880}, {13185,13989}, {13216,13990}, {13276,13991}, {13299,13992}, {13698,13988}, {13818,13849}

X(13958) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND MANDART-INCIRCLE

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

X(13958) lies on these lines: {1,13966}, {2,13898}, {3,13963}, {4,13954}, {6,5432}, {11,615}, {12,372}, {33,13937}, {35,7584}, {55,3069}, {56,13935}, {65,13975}, {140,3301}, {486,6284}, {497,13941}, {498,3312}, {590,5326}, {1152,7354}, {1317,13977}, {1335,5420}, {1478,6398}, {1479,13951}, {1588,5217}, {1697,13942}, {1703,11375}, {1837,13947}, {2066,4995}, {2098,13959}, {2646,13936}, {3023,13989}, {3027,13967}, {3028,13969}, {3056,13972}, {3057,13971}, {3058,13847}, {3070,3614}, {3295,13961}, {3317,10591}, {3320,13985}, {3526,13904}, {3592,9648}, {4294,13939}, {4299,6450}, {4302,13785}, {5218,7586}, {6020,13992}, {6408,9655}, {6418,13905}, {6420,9646}, {6460,10895}, {7173,10577}, {7294,9661}, {7355,13980}, {7581,13897}, {10799,13938}, {10833,13943}, {10877,13946}, {10927,13949}, {10928,13950}, {10947,13952}, {10950,13973}, {10953,13953}, {10965,13964}, {10966,13965}, {11873,13944}, {11874,13945}, {11909,13948}, {11947,13956}, {11948,13957}, {12354,13968}, {12428,13970}, {12680,13974}, {12743,13976}, {12863,13978}, {12896,13979}, {13075,13981}, {13076,13982}, {13077,13983}, {13078,13984}, {13079,13986}, {13080,13987}, {13081,13933}, {13082,13880}, {13699,13988}, {13819,13849}

X(13958) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,5432,13901), (1335,5420,5433), (3295,13961,13962)

X(13959) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND 5th MIXTILINEAR

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

X(13959) lies on these lines: {1,1123}, {2,7968}, {6,3616}, {8,615}, {56,13940}, {145,13941}, {372,5603}, {486,944}, {517,13935}, {519,13947}, {590,5550}, {946,6460}, {952,13951}, {962,1152}, {1125,3068}, {1320,13991}, {1385,1588}, {1482,13966}, {1483,13993}, {1587,5886}, {1702,10165}, {1703,13464}, {2098,13958}, {3071,5731}, {3241,13847}, {3242,13972}, {3297,5703}, {3312,5901}, {3485,6502}, {3576,6459}, {3622,7586}, {3624,13883}, {5420,5657}, {5597,13945}, {5598,13944}, {5604,13950}, {5605,13949}, {5818,10577}, {6361,6396}, {6410,9778}, {7584,10246}, {7967,13939}, {7970,13967}, {7971,13974}, {7972,13976}, {7973,13980}, {7974,13981}, {7975,13982}, {7976,13983}, {7977,13984}, {7978,13969}, {7979,13986}, {7980,13933}, {7981,13880}, {7982,13975}, {7983,13989}, {7984,13990}, {8000,13978}, {8192,13943}, {8210,13956}, {8211,13957}, {8252,9780}, {9541,13624}, {9884,13968}, {9933,13970}, {9997,13946}, {10247,13961}, {10698,13977}, {10705,13992}, {10800,13938}, {10944,13952}, {10950,13953}, {11396,13937}, {11910,13948}, {12898,13979}, {13099,13985}, {13100,13987}, {13702,13988}, {13822,13849}

X(13959) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,13942,13936), (1,13971,3069), (6,3616,13902), (145,13941,13973), (3622,7586,7969), (13936,13942,3069), (13936,13971,13942), (13964,13965,3069)

X(13960) = PERSPECTOR OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND OUTER-SQUARES

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

X(13960) lies on these lines: {3,486}, {6,8969}, {372,6750}, {393,494}, {3156,5593}

X(13961) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND X3-ABC REFLECTIONS

Barycentrics 3*a^4-5*(b^2+c^2)*a^2+2*(b^2-c^2)^2+8*S*a^2 : :
X(13961) = 2*X(1132)+3*X(6408) = X(1132)-3*X(13939) = X(6408)+2*X(13939)

X(13961) lies on these lines: {2,6418}, {3,1588}, {4,13993}, {5,1131}, {6,3411}, {20,6446}, {30,1132}, {140,6417}, {195,13986}, {372,381}, {382,486}, {399,13990}, {485,615}, {517,13942}, {549,7582}, {590,6428}, {631,6199}, {632,7585}, {640,13783}, {999,13963}, {1152,1657}, {1351,13972}, {1482,13971}, {1587,5055}, {1598,13937}, {2070,8277}, {3068,6501}, {3070,5072}, {3071,3534}, {3295,13958}, {3311,5054}, {3523,6407}, {3524,9543}, {3530,6445}, {3591,3858}, {3594,5079}, {3628,7581}, {3843,6460}, {5059,10138}, {5070,7583}, {5076,6426}, {5418,6427}, {5790,13947}, {6420,8252}, {6432,10576}, {6448,6560}, {6456,6561}, {6500,8981}, {7517,13943}, {9301,13946}, {9654,13954}, {9669,13955}, {10246,13936}, {10247,13959}, {10620,13969}, {11842,13938}, {11849,13940}, {11875,13944}, {11876,13945}, {11911,13948}, {11916,13949}, {11917,13950}, {11928,13952}, {11929,13953}, {11949,13956}, {11950,13957}, {12000,13964}, {12001,13965}, {12188,13967}, {12331,13991}, {12355,13968}, {12429,13970}, {12601,13933}, {12602,13880}, {12645,13973}, {12684,13974}, {12702,13975}, {12747,13976}, {12773,13977}, {12872,13978}, {12902,13979}, {13093,13980}, {13102,13981}, {13103,13982}, {13108,13983}, {13111,13984}, {13115,13985}, {13126,13987}, {13188,13989}, {13310,13992}, {13713,13988}, {13836,13849}

X(13961) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,3526,13903), (486,6398,382), (1152,13785,1657), (3071,6450,3534), (3311,5420,5054), (7584,13935,3), (7584,13966,13935), (13958,13962,3295)

X(13962) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND INNER-YFF

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

X(13962) lies on these lines: {1,1123}, {2,3299}, {5,13955}, {6,499}, {11,3312}, {12,13951}, {35,13935}, {36,1588}, {55,13966}, {56,7584}, {63,5405}, {89,3302}, {372,1479}, {388,13939}, {486,1478}, {495,13954}, {498,615}, {611,13972}, {1152,4302}, {1335,10072}, {1587,7741}, {1709,13974}, {2066,5420}, {3071,4299}, {3085,13941}, {3086,3301}, {3295,13958}, {3297,10056}, {3317,10588}, {3526,13901}, {3583,6460}, {3628,13897}, {5062,9599}, {5119,13975}, {6284,6398}, {6395,9669}, {6410,9660}, {6451,9662}, {6459,7280}, {7173,13665}, {7288,7582}, {7354,13785}, {7581,10589}, {7968,10573}, {8252,9646}, {10037,13943}, {10038,13946}, {10039,13947}, {10040,13949}, {10041,13950}, {10053,13967}, {10054,13968}, {10055,13970}, {10057,13976}, {10058,13977}, {10059,13978}, {10060,13980}, {10061,13981}, {10062,13982}, {10063,13983}, {10064,13984}, {10065,13969}, {10066,13986}, {10067,13933}, {10068,13880}, {10086,13989}, {10087,13991}, {10088,13990}, {10523,13952}, {10801,13938}, {10881,11393}, {10954,13953}, {11398,13937}, {11507,13940}, {11877,13944}, {11878,13945}, {11912,13948}, {11951,13956}, {11952,13957}, {12647,13973}, {12903,13979}, {13116,13985}, {13128,13987}, {13311,13992}, {13714,13988}, {13837,13849}

X(13962) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3299,13905), (6,499,13904), (486,6502,1478), (3295,13961,13958)

X(13963) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND OUTER-YFF

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

X(13963) lies on these lines: {1,1123}, {2,3301}, {3,13958}, {5,13954}, {6,498}, {11,13951}, {12,3312}, {35,1588}, {36,13935}, {46,13975}, {55,7584}, {56,13966}, {372,1478}, {486,1479}, {496,13955}, {497,13939}, {499,615}, {613,13972}, {999,13961}, {1124,10056}, {1152,4299}, {1587,7951}, {1703,12047}, {1737,13947}, {2067,5420}, {3071,4302}, {3085,3299}, {3086,13941}, {3298,10072}, {3305,5405}, {3311,5432}, {3317,10589}, {3585,6460}, {3614,13665}, {3628,13898}, {5010,6459}, {5062,9596}, {5218,7582}, {6284,13785}, {6395,9654}, {6398,7354}, {6410,9647}, {6417,13901}, {6451,9649}, {7581,10588}, {7968,12647}, {8252,9661}, {10046,13943}, {10047,13946}, {10048,13949}, {10049,13950}, {10069,13967}, {10070,13968}, {10071,13970}, {10073,13976}, {10074,13977}, {10075,13978}, {10076,13980}, {10077,13981}, {10078,13982}, {10079,13983}, {10080,13984}, {10081,13969}, {10082,13986}, {10083,13933}, {10084,13880}, {10085,13974}, {10089,13989}, {10090,13991}, {10091,13990}, {10523,13953}, {10573,13973}, {10802,13938}, {10881,11392}, {10948,13952}, {11399,13937}, {11508,13940}, {11879,13944}, {11880,13945}, {11913,13948}, {11953,13956}, {11954,13957}, {12904,13979}, {13117,13985}, {13129,13987}, {13312,13992}, {13715,13988}, {13838,13849}

X(13963) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3301, 13904), (6, 498, 13905), (486, 5414, 1479), (496, 13993, 13955)

X(13964) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND INNER-YFF TANGENTS

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

X(13964) lies on these lines: {1,1123}, {6,13906}, {12,13952}, {372,10531}, {486,12115}, {615,5552}, {1588,10269}, {3068,10200}, {5554,7968}, {7586,10586}, {10528,13941}, {10679,13966}, {10803,13938}, {10805,13939}, {10834,13943}, {10878,13946}, {10915,13947}, {10929,13949}, {10930,13950}, {10942,13951}, {10955,13953}, {10956,13954}, {10958,13955}, {10965,13958}, {11239,13847}, {11248,13935}, {11400,13937}, {11509,13940}, {11881,13944}, {11882,13945}, {11914,13948}, {11955,13956}, {11956,13957}, {12000,13961}, {12189,13967}, {12356,13968}, {12381,13969}, {12430,13970}, {12594,13972}, {12648,13973}, {12686,13974}, {12703,13975}, {12749,13976}, {12775,13977}, {12874,13978}, {12905,13979}, {13094,13980}, {13104,13981}, {13105,13982}, {13109,13983}, {13112,13984}, {13118,13985}, {13121,13986}, {13130,13987}, {13132,13933}, {13134,13880}, {13189,13989}, {13217,13990}, {13278,13991}, {13313,13992}, {13716,13988}, {13839,13849}

X(13965) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND OUTER-YFF TANGENTS

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

X(13965) lies on these lines: {1,1123}, {6,13907}, {11,13953}, {372,10532}, {486,12116}, {615,10527}, {1588,10267}, {3068,10198}, {6459,10902}, {7586,10587}, {10529,13941}, {10680,13966}, {10804,13938}, {10806,13939}, {10835,13943}, {10879,13946}, {10916,13947}, {10931,13949}, {10932,13950}, {10943,13951}, {10949,13952}, {10957,13954}, {10959,13955}, {10966,13958}, {11240,13847}, {11249,13935}, {11401,13937}, {11510,13940}, {11883,13944}, {11884,13945}, {11915,13948}, {11957,13956}, {11958,13957}, {12001,13961}, {12190,13967}, {12357,13968}, {12382,13969}, {12431,13970}, {12595,13972}, {12649,13973}, {12687,13974}, {12704,13975}, {12750,13976}, {12776,13977}, {12875,13978}, {12906,13979}, {13095,13980}, {13106,13981}, {13107,13982}, {13110,13983}, {13113,13984}, {13119,13985}, {13122,13986}, {13131,13987}, {13133,13933}, {13135,13880}, {13190,13989}, {13218,13990}, {13279,13991}, {13314,13992}, {13717,13988}, {13840,13849}

X(13966) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO ANTI-ASCELLA

Barycentrics 2*a^4-3*(b^2+c^2)*a^2+(b^2-c^2)^2+4*S*a^2 : :
X(13966) = 7*X(486)-3*X(1328) = X(486)-3*X(13847) = 7*X(1152)+3*X(1328) = X(1152)+3*X(13847) = X(1152)+2*X(13993) = X(1328)-7*X(13847)

The reciprocal orthologic center of these triangles is X(1593)

X(13966) lies on these lines: {1,13958}, {2,3312}, {3,1588}, {4,3591}, {5,372}, {6,140}, {20,6450}, {26,8277}, {30,486}, {40,13942}, {55,13962}, {56,13963}, {69,11316}, {230,1505}, {355,13947}, {371,549}, {376,6456}, {381,6460}, {382,6408}, {395,3365}, {396,3390}, {427,10881}, {485,3594}, {488,11314}, {492,7767}, {495,6502}, {496,5414}, {511,13972}, {517,13971}, {524,642}, {546,6426}, {548,6410}, {550,3071}, {590,632}, {631,3311}, {637,13757}, {639,754}, {640,7886}, {641,3589}, {952,13973}, {971,13974}, {1131,5071}, {1132,3529}, {1151,3530}, {1154,12240}, {1160,13950}, {1161,13949}, {1327,11737}, {1368,10898}, {1385,13936}, {1478,13954}, {1479,13955}, {1482,13959}, {1587,1656}, {1595,5413}, {1596,11474}, {1657,6446}, {1703,5886}, {2045,11485}, {2046,11486}, {2782,13967}, {3068,3526}, {3090,13665}, {3091,3317}, {3146,6522}, {3147,5410}, {3299,5432}, {3301,5433}, {3398,13938}, {3522,6452}, {3523,6221}, {3524,6449}, {3525,6428}, {3528,6497}, {3533,8972}, {3541,5411}, {3564,13934}, {3592,12108}, {3593,7375}, {3618,11315}, {3627,6454}, {3815,5062}, {3850,6438}, {3853,6430}, {5054,6417}, {5305,6421}, {5663,13969}, {5690,7968}, {6000,13980}, {6409,12100}, {6427,10303}, {6431,11812}, {6432,8253}, {6442,10195}, {6451,10299}, {6501,13903}, {6723,13915}, {7387,13943}, {8855,10154}, {9687,13347}, {9821,13946}, {10018,13884}, {10124,13846}, {10525,13952}, {10526,13953}, {10669,13956}, {10673,13957}, {10679,13964}, {10680,13965}, {10721,10818}, {10734,11834}, {11231,13883}, {11248,13940}, {11251,13948}, {11252,13944}, {11253,13945}, {12006,12239}, {12975,13933}, {13985,13992}

X(13966) = midpoint of X(i) and X(j) for these {i,j}: {486,1152}, {13967,13989}, {13969,13990}, {13971,13975}, {13977,13991}, {13985,13992}
X(13966) = reflection of X(486) in X(13993)
X(13966) = orthologic center of the 4th tri-squares-central triangle to these triangles: 1st anti-circumperp, anti-Conway, 2nd anti-Conway, 3rd anti-Euler, 4th anti-Euler, anti-excenters-reflections, anti-Hutson intouch, anti-incircle-circles, anti-inverse-in-incircle, 6th anti-mixtilinear, circumorthic, 2nd Ehrmann, 2nd Euler, extangents, intangents, 1st Kenmotu diagonals, 2nd Kenmotu diagonals, Kosnita, orthic, submedial, tangential, inner tri-equilateral, outer tri-equilateral, Trinh
X(13966) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3312, 7583), (2, 7581, 8976), (4, 13941, 13951), (6, 140, 8981), (6, 5420, 140), (20, 13939, 13785), (1656, 6395, 1587), (3071, 6396, 550), (6450, 13785, 20), (13935, 13961, 7584)

X(13967) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 1st ANTI-BROCARD

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

The reciprocal orthologic center of these triangles is X(5999)

X(13967) lies on these lines: {6,6036}, {30,13968}, {98,3069}, {99,13935}, {114,615}, {115,372}, {140,8997}, {147,13941}, {485,6722}, {486,2794}, {542,13847}, {620,5420}, {690,13969}, {2459,6781}, {2782,13966}, {2783,13991}, {2787,13977}, {2799,13985}, {3027,13958}, {6033,13951}, {6226,13950}, {6227,13949}, {6231,13790}, {6321,6398}, {6721,8252}, {7584,12042}, {7970,13959}, {9860,13942}, {9861,13943}, {9862,13939}, {9864,13947}, {10053,13962}, {10069,13963}, {11710,13936}, {12131,13937}, {12176,13938}, {12178,13940}, {12181,13948}, {12182,13952}, {12183,13953}, {12184,13954}, {12185,13955}, {12186,13956}, {12187,13957}, {12188,13961}, {12189,13964}, {12190,13965}, {13984,13993}

X(13967) = reflection of X(13989) in X(13966)
X(13967) = orthologic center of the 4th tri-squares-central triangle to these triangles: 6th anti-Brocard, 1st Brocard, 6th Brocard
X(13967) = {X(6), X(6036)}-harmonic conjugate of X(8980)

X(13968) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO ANTI-MCCAY

Barycentrics -S*(12*S^2+3*SA^2+6*SB*SC-5*SW^2)+3*(3*S^2-SW^2)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(9855)

X(13968) lies on these lines: {2,8997}, {6,5461}, {30,13967}, {115,13823}, {372,9880}, {486,542}, {530,13981}, {531,13982}, {543,13847}, {615,2482}, {671,3069}, {5969,13983}, {6722,13846}, {8591,13941}, {8724,13951}, {9830,13972}, {9875,13942}, {9876,13943}, {9878,13946}, {9881,13947}, {9882,13949}, {9883,13950}, {9884,13959}, {9892,13796}, {9894,13789}, {10054,13962}, {10070,13963}, {12117,13935}, {12132,13937}, {12191,13938}, {12243,13939}, {12258,13936}, {12326,13940}, {12345,13944}, {12346,13945}, {12347,13948}, {12348,13952}, {12349,13953}, {12350,13954}, {12351,13955}, {12352,13956}, {12353,13957}, {12354,13958}, {12355,13961}, {12356,13964}, {12357,13965}

X(13968) = reflection of X(13989) in X(13847)
X(13968) = orthologic center of these triangles: 4th tri-squares-central to Mccay
X(13968) = {X(6), X(5461)}-harmonic conjugate of X(13908)

X(13969) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO ANTI-ORTHOCENTROIDAL

Barycentrics (12*R^2-SA-3*SW)*S^2-(SB+SC)*(SW*(3*SA-4*S)-18*(SA-S)*R^2) : :

The reciprocal orthologic center of these triangles is X(12112)

X(13969) lies on these lines: {6,6699}, {30,13979}, {74,3069}, {110,13935}, {113,615}, {125,372}, {140,8998}, {146,13941}, {265,6398}, {485,6723}, {486,2777}, {541,13847}, {542,10820}, {690,13967}, {1152,13970}, {2771,13991}, {2781,13972}, {3028,13958}, {5420,5972}, {5642,12375}, {5663,13966}, {6408,12902}, {6450,12121}, {6560,7687}, {6565,13202}, {7584,12041}, {7725,13949}, {7726,13950}, {7728,13951}, {7978,13959}, {8252,12900}, {8277,10117}, {9517,13985}, {9904,13942}, {9919,13943}, {9984,13946}, {10065,13962}, {10081,13963}, {10620,13961}, {10628,13986}, {10733,10818}, {11709,13936}, {12133,13937}, {12192,13938}, {12244,13939}, {12327,13940}, {12368,13947}, {12369,13948}, {12371,13952}, {12372,13953}, {12373,13954}, {12374,13955}, {12377,13956}, {12378,13957}, {12381,13964}, {12382,13965}

X(13969) = reflection of X(13990) in X(13966)
X(13969) = orthologic center of these triangles: 4th tri-squares-central to orthocentroidal
X(13969) = {X(6), X(6699)}-harmonic conjugate of X(8994)

X(13970) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO ARIES

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

The reciprocal orthologic center of these triangles is X(9833)

X(13970) lies on these lines: {6,5449}, {30,13980}, {68,3069}, {155,13951}, {372,9927}, {486,12240}, {539,13847}, {615,1147}, {1069,13955}, {1152,13969}, {3071,7689}, {3157,13954}, {3448,11463}, {3564,13880}, {5420,12038}, {6193,13941}, {6398,12293}, {7584,12359}, {8252,8909}, {9896,13942}, {9908,13943}, {9923,13946}, {9928,13947}, {9929,13949}, {9930,13950}, {9933,13959}, {10055,13962}, {10071,13963}, {10577,10665}, {11411,13939}, {12118,13935}, {12134,13937}, {12163,13785}, {12193,13938}, {12259,13936}, {12328,13940}, {12418,13948}, {12422,13952}, {12423,13953}, {12426,13956}, {12427,13957}, {12428,13958}, {12429,13961}, {12430,13964}, {12431,13965}

X(13970) = rthologic center of the 4th tri-squares-central triangle to these triangles: 2nd Hyacinth
X(13970) = {X(6), X(5449)}-harmonic conjugate of X(13909)

X(13971) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO ASCELLA

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

The reciprocal orthologic center of these triangles is X(3)

X(13971) lies on these lines: {1,1123}, {2,13883}, {3,13940}, {6,1125}, {8,13941}, {10,615}, {40,13935}, {55,13944}, {140,13912}, {226,6502}, {355,13951}, {371,10165}, {372,946}, {486,515}, {516,1152}, {517,13966}, {518,13972}, {519,13847}, {551,7969}, {631,1702}, {730,13983}, {944,13939}, {952,13976}, {1124,13411}, {1378,6700}, {1385,7584}, {1482,13961}, {1587,8227}, {1588,3576}, {1699,6460}, {1703,5603}, {1829,13937}, {1837,13955}, {2800,13977}, {2802,13991}, {3057,13958}, {3068,3624}, {3070,3817}, {3071,4297}, {3297,13405}, {3312,5886}, {3317,5818}, {3523,9616}, {3524,9582}, {3616,7586}, {3634,8252}, {3640,13950}, {3641,13949}, {5252,13954}, {5405,5745}, {5414,12053}, {5420,6684}, {5550,7585}, {6001,13974}, {6398,12699}, {6410,12512}, {6459,7987}, {6667,8988}, {7490,13390}, {7582,9583}, {7583,11230}, {9798,13943}, {9941,13946}, {10175,10577}, {12194,13938}, {12438,13948}, {12440,13956}, {12441,13957}

X(13971) = reflection of X(13975) in X(13966)
X(13971) = orthologic center of the 4th tri-squares-central triangle to these triangles: 1st circumperp, 2nd circumperp, inner-Conway, Conway, 2nd Conway, 3rd Conway, 3rd Euler, 4th Euler, excenters-reflections, excentral, 2nd extouch, hexyl, Honsberger, inner-Hutson, Hutson intouch, outer-Hutson, incircle-circles, intouch, inverse-in-incircle, 6th mixtilinear, 2nd Pamfilos-Zhou, 1st Sharygin, tangential-midarc, 2nd tangential-midarc, Yff central
X(13971) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3069, 13936), (1, 13942, 3069), (6, 1125, 8983), (8, 13941, 13947), (615, 7968, 10), (3069, 13959, 1), (8252, 13911, 3634), (13942, 13959, 13936), (13952, 13953, 13951)

X(13972) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 1st EHRMANN

Barycentrics -(2*a^2+b^2+c^2)*S+a^2*(a^2+b^2+c^2) : :
X(13972) = X(6)+3*X(13847)

The reciprocal orthologic center of these triangles is X(3)

X(13972) lies on these lines: {2,6}, {159,13943}, {182,7584}, {372,5480}, {486,1503}, {511,13966}, {518,13971}, {542,13979}, {611,13962}, {613,13963}, {732,13983}, {1350,13935}, {1351,13961}, {1352,13951}, {1386,13936}, {1588,5085}, {1843,13937}, {2781,13969}, {2854,13990}, {3056,13958}, {3094,13946}, {3242,13959}, {3416,13947}, {3564,13880}, {3751,13942}, {3867,5413}, {5846,13973}, {5969,13989}, {6560,13763}, {6776,13939}, {8855,10192}, {9024,13991}, {9830,13968}, {12212,13938}, {12329,13940}, {12452,13944}, {12453,13945}, {12583,13948}, {12586,13952}, {12587,13953}, {12588,13954}, {12589,13955}, {12590,13956}, {12591,13957}, {12594,13964}, {12595,13965}, {13849,13988}

X(13972) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 615, 141), (6, 3589, 13910), (3618, 7586, 6), (13949, 13950, 3069)

X(13973) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO EXCENTERS-MIDPOINTS

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

The reciprocal orthologic center of these triangles is X(10)

X(13973) lies on these lines: {1,615}, {2,7969}, {6,10}, {8,3069}, {12,2362}, {37,7090}, {40,3071}, {145,13941}, {355,372}, {485,9956}, {486,517}, {515,1152}, {518,10067}, {519,13847}, {590,1698}, {940,6347}, {944,13935}, {952,13966}, {1124,10039}, {1125,8252}, {1151,6684}, {1210,3298}, {1335,1737}, {1385,5420}, {1482,13951}, {1587,5818}, {1588,5657}, {1703,3070}, {1837,5414}, {2098,13955}, {2099,13954}, {2802,13976}, {3068,9780}, {3312,5790}, {3317,10595}, {3579,6561}, {3592,13912}, {3617,7586}, {3632,13942}, {3634,8253}, {3828,13846}, {3913,13940}, {4297,6410}, {4383,6348}, {5090,5413}, {5252,6502}, {5418,11231}, {5690,7584}, {5844,13993}, {5846,13972}, {5886,10577}, {6409,10164}, {6565,12699}, {9588,9616}, {9620,12788}, {10573,13963}, {10912,13952}, {10950,13958}, {12135,13937}, {12195,13938}, {12245,13939}, {12410,13943}, {12454,13944}, {12455,13945}, {12495,13946}, {12626,13948}, {12627,13949}, {12628,13950}, {12635,13953}, {12636,13956}, {12637,13957}, {12645,13961}, {12647,13962}, {12648,13964}, {12649,13965}, {12702,13785}

X(13973) = reflection of X(1152) in X(13975)
X(13973) = orthologic center of these triangles: 4th tri-squares-central to 2nd Schiffler
X(13973) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 13947, 615), (6, 10, 13911), (8, 3069, 7968), (10, 13936, 6), (145, 13941, 13959), (1703, 5587, 3070), (3634, 8983, 8253)

X(13974) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO EXTOUCH

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

The reciprocal orthologic center of these triangles is X(40)

X(13974) lies on these lines: {6,6705}, {84,3069}, {372,6245}, {515,1152}, {615,6260}, {971,13966}, {1490,13935}, {1709,13962}, {2829,13976}, {5787,6398}, {6001,13971}, {6223,13941}, {6257,13950}, {6258,13949}, {6259,13951}, {7971,13959}, {7992,13942}, {9910,13943}, {10085,13963}, {12114,13936}, {12136,13937}, {12196,13938}, {12246,13939}, {12330,13940}, {12456,13944}, {12457,13945}, {12496,13946}, {12667,13947}, {12668,13948}, {12676,13952}, {12677,13953}, {12678,13954}, {12679,13955}, {12680,13958}, {12684,13961}, {12686,13964}, {12687,13965}

X(13975) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 3rd EXTOUCH

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

The reciprocal orthologic center of these triangles is X(4)

X(13975) lies on these lines: {1,13935}, {2,1703}, {3,13936}, {4,13947}, {6,6684}, {10,372}, {40,3069}, {46,13963}, {65,13958}, {140,8983}, {165,1588}, {355,6398}, {371,10164}, {485,3634}, {486,516}, {515,1152}, {517,13966}, {615,946}, {962,13941}, {1125,5420}, {1210,5414}, {1335,3911}, {1377,5745}, {1587,1698}, {1686,6685}, {1702,7586}, {1836,13954}, {1902,13937}, {2362,13411}, {2800,13991}, {2802,13977}, {3070,10175}, {3312,13883}, {3523,9583}, {3524,9615}, {3579,7584}, {3594,13911}, {3817,10577}, {4297,6396}, {5119,13962}, {5405,6213}, {5587,6460}, {5812,13953}, {5840,13976}, {6001,13980}, {6361,13939}, {6561,12512}, {7581,13893}, {7582,9616}, {7583,11231}, {7968,11362}, {7969,10165}, {7982,13959}, {7991,13942}, {9911,13943}, {10306,13940}, {12197,13938}, {12458,13944}, {12459,13945}, {12497,13946}, {12696,13948}, {12697,13949}, {12698,13950}, {12699,13951}, {12700,13952}, {12701,13955}, {12702,13961}, {12703,13964}, {12704,13965}

X(13975) = midpoint of X(1152) and X(13973)
X(13975) = reflection of X(13971) in X(13966)
X(13975) = {X(6), X(6684)}-harmonic conjugate of X(13912)

X(13976) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO FUHRMANN

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

The reciprocal orthologic center of these triangles is X(3)

X(13976) lies on these lines: {6,6702}, {11,13936}, {80,3069}, {100,13947}, {214,615}, {486,2800}, {515,13977}, {952,13971}, {2771,13979}, {2802,13973}, {2829,13974}, {3634,13922}, {5840,13975}, {6224,13941}, {6262,13950}, {6263,13949}, {6265,13951}, {6667,8983}, {7584,12619}, {7972,13959}, {9897,13942}, {9912,13943}, {10057,13962}, {10073,13963}, {12119,13935}, {12137,13937}, {12198,13938}, {12247,13939}, {12331,13940}, {12460,13944}, {12461,13945}, {12498,13946}, {12515,13785}, {12729,13948}, {12737,13952}, {12738,13953}, {12739,13954}, {12740,13955}, {12741,13956}, {12742,13957}, {12743,13958}, {12747,13961}, {12749,13964}, {12750,13965}

X(13976) = {X(6), X(6702)}-harmonic conjugate of X(8988)

X(13977) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO INNER-GARCIA

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

The reciprocal orthologic center of these triangles is X(40)

X(13977) lies on these lines: {6,6713}, {11,372}, {100,13935}, {104,3069}, {119,615}, {140,13922}, {153,13941}, {485,6667}, {486,2829}, {515,13976}, {952,13966}, {1152,5840}, {1317,13958}, {1378,3035}, {1768,13942}, {2771,13990}, {2783,13989}, {2787,13967}, {2800,13971}, {2802,13975}, {2806,13985}, {2831,13992}, {6398,10738}, {9913,13943}, {10058,13962}, {10074,13963}, {10698,13959}, {10742,13951}, {11715,13936}, {12138,13937}, {12199,13938}, {12248,13939}, {12332,13940}, {12499,13946}, {12751,13947}, {12752,13948}, {12753,13949}, {12754,13950}, {12761,13952}, {12762,13953}, {12763,13954}, {12764,13955}, {12765,13956}, {12766,13957}, {12773,13961}, {12775,13964}, {12776,13965}

X(13977) = reflection of X(13991) in X(13966)

X(13978) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO HUTSON EXTOUCH

Barycentrics
(-2*a^7+(b+c)*a^6+6*(b^2+4*b*c+c^2)*a^5-(b+c)*(3*b^2-2*b*c+3*c^2)*a^4-2*(3*b^2+14*b*c+3*c^2)*(b+c)^2*a^3+(b+c)*(3*b^4+3*c^4-2*b*c*(2*b^2+15*b*c+2*c^2))*a^2+2*(b^2-c^2)^2*(b^2+8*b*c+c^2)*a-(b^2-c^2)^3*(b-c))*S+a^2*(a+b+c)*(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*b*c*(6*b^2+5*b*c+6*c^2))*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) : :

The reciprocal orthologic center of these triangles is X(40)

X(13978) lies on these lines: {6,13914}, {372,12599}, {615,12864}, {3069,7160}, {8000,13959}, {9874,13941}, {9898,13942}, {10059,13962}, {10075,13963}, {12120,13935}, {12139,13937}, {12200,13938}, {12249,13939}, {12260,13936}, {12333,13940}, {12411,13943}, {12500,13946}, {12777,13947}, {12789,13948}, {12801,13949}, {12802,13950}, {12856,13951}, {12857,13952}, {12858,13953}, {12859,13954}, {12860,13955}, {12861,13956}, {12862,13957}, {12863,13958}, {12872,13961}, {12874,13964}, {12875,13965}

X(13979) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 1st HYACINTH

Barycentrics (30*R^2-SA-7*SW)*S^2-(SB+SC)*(SW*(-4*S-3*SA)+9*(SA+2*S)*R^2) : :
X(13979) = X(10820)-3*X(13847)

The reciprocal orthologic center of these triangles is X(6102)

X(13979) lies on these lines: {6,13915}, {30,13969}, {74,13785}, {110,13951}, {125,7584}, {265,3069}, {372,10113}, {486,5663}, {542,13972}, {615,1511}, {1132,12244}, {1539,6565}, {2771,13976}, {2777,13980}, {3071,12041}, {3448,13939}, {3628,8998}, {6398,10733}, {6723,8981}, {8252,10819}, {10088,13954}, {10091,13955}, {10820,13847}, {12121,13935}, {12140,13937}, {12201,13938}, {12261,13936}, {12334,13940}, {12383,13941}, {12407,13942}, {12412,13943}, {12501,13946}, {12778,13947}, {12790,13948}, {12803,13949}, {12804,13950}, {12889,13952}, {12890,13953}, {12894,13956}, {12895,13957}, {12896,13958}, {12898,13959}, {12902,13961}, {12903,13962}, {12904,13963}, {12905,13964}, {12906,13965}, {13986,13990}

X(13979) = reflection of X(13990) in X(13993)

X(13980) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO MIDHEIGHT

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

The reciprocal orthologic center of these triangles is X(4)

X(13980) lies on these lines: {6,6696}, {30,13970}, {64,3069}, {372,6247}, {615,2883}, {1152,1503}, {1498,13935}, {1588,10606}, {1853,6460}, {2777,13979}, {2781,12240}, {3071,5894}, {3357,7584}, {5878,13951}, {6000,13966}, {6001,13975}, {6225,13941}, {6266,13950}, {6267,13949}, {6450,9833}, {6459,8567}, {7355,13958}, {7973,13959}, {9899,13942}, {9914,13943}, {10060,13962}, {10076,13963}, {10192,12964}, {11381,13937}, {11474,13567}, {12202,13938}, {12250,13939}, {12262,13936}, {12335,13940}, {12502,13946}, {12779,13947}, {12791,13948}, {12920,13952}, {12930,13953}, {12940,13954}, {12950,13955}, {12986,13956}, {12987,13957}, {13093,13961}, {13094,13964}, {13095,13965}

X(13981) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO INNER-NAPOLEON

Barycentrics (9*S^2-(9*SW-7*SA)*S+(SB+2*SC)*(2*SB+SC))*sqrt(3)+S^2-3*(5*SW-3*SA)*S+6*SW^2-6*SA^2+9*SB*SC : :

The reciprocal orthologic center of these triangles is X(3)

X(13981) lies on these lines: {6,6670}, {14,3069}, {372,5479}, {530,13968}, {531,10671}, {542,13972}, {615,619}, {617,13941}, {5474,13935}, {5613,13951}, {6269,13950}, {6271,13949}, {6303,13823}, {6722,13917}, {6773,13939}, {6774,7584}, {7974,13959}, {9900,13942}, {9915,13943}, {9981,13946}, {10061,13962}, {10077,13963}, {11706,13936}, {12141,13937}, {12204,13938}, {12336,13940}, {12780,13947}, {12792,13948}, {12921,13952}, {12931,13953}, {12941,13954}, {12951,13955}, {12988,13956}, {12989,13957}, {13075,13958}, {13102,13961}, {13104,13964}, {13106,13965}

X(13982) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO OUTER-NAPOLEON

Barycentrics -(9*S^2-(9*SW-7*SA)*S+(SB+2*SC)*(2*SB+SC))*sqrt(3)+S^2-3*(5*SW-3*SA)*S+6*SW^2-6*SA^2+9*SB*SC : :

The reciprocal orthologic center of these triangles is X(3)

X(13982) lies on these lines: {6,6669}, {13,3069}, {372,5478}, {530,10672}, {531,13968}, {542,13972}, {615,618}, {616,13941}, {5473,13935}, {5617,13951}, {6268,13950}, {6270,13949}, {6302,13825}, {6722,13916}, {6770,13939}, {6771,7584}, {7975,13959}, {9901,13942}, {9916,13943}, {9982,13946}, {10062,13962}, {10078,13963}, {11705,13936}, {12142,13937}, {12205,13938}, {12337,13940}, {12781,13947}, {12793,13948}, {12922,13952}, {12932,13953}, {12942,13954}, {12952,13955}, {12990,13956}, {12991,13957}, {13076,13958}, {13103,13961}, {13105,13964}, {13107,13965}

X(13983) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 1st NEUBERG

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

The reciprocal orthologic center of these triangles is X(3)

X(13983) lies on these lines: {6,3934}, {39,615}, {76,3069}, {194,13941}, {372,6248}, {384,13938}, {486,511}, {538,13847}, {639,1506}, {730,13971}, {732,13972}, {2782,13966}, {3071,5188}, {3095,13951}, {3103,10577}, {3312,7697}, {5420,13334}, {5969,13968}, {6272,13950}, {6273,13949}, {6318,13827}, {6683,8252}, {7976,13959}, {9821,13785}, {9902,13942}, {9917,13943}, {9983,13946}, {10063,13962}, {10079,13963}, {11257,13935}, {12143,13937}, {12251,13939}, {12263,13936}, {12338,13940}, {12474,13944}, {12475,13945}, {12782,13947}, {12794,13948}, {12836,13955}, {12837,13954}, {12923,13952}, {12933,13953}, {12992,13956}, {12993,13957}, {13077,13958}, {13108,13961}, {13109,13964}, {13110,13965}

X(13983) = {X(6), X(3934)}-harmonic conjugate of X(8992)

X(13984) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 2nd NEUBERG

Barycentrics (SA^2-2*SB*SC-7*SW^2)*S+(S^2+5*SW^2)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(3)

X(13984) lies on these lines: {6,6704}, {83,3069}, {372,6249}, {615,6292}, {732,13972}, {754,13847}, {1588,9751}, {2896,13941}, {6274,13950}, {6275,13949}, {6287,13951}, {6317,13829}, {7977,13959}, {8725,13785}, {9903,13942}, {9918,13943}, {10064,13962}, {10080,13963}, {12122,13935}, {12144,13937}, {12206,13938}, {12252,13939}, {12264,13936}, {12339,13940}, {12476,13944}, {12477,13945}, {12783,13947}, {12795,13948}, {12924,13952}, {12934,13953}, {12944,13954}, {12954,13955}, {12994,13956}, {12995,13957}, {13078,13958}, {13111,13961}, {13112,13964}, {13113,13965}, {13967,13993}

X(13984) = {X(6), X(6704)}-harmonic conjugate of X(8993)

X(13985) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 1st ORTHOSYMMEDIAL

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

The reciprocal orthologic center of these triangles is X(4)

X(13985) lies on these lines: {6,13918}, {112,13935}, {127,372}, {132,615}, {140,13923}, {1152,2794}, {1297,3069}, {2781,13990}, {2799,13967}, {2806,13977}, {2831,13991}, {3320,13958}, {5420,6720}, {6398,10749}, {9517,13969}, {9530,13847}, {12145,13937}, {12207,13938}, {12253,13939}, {12265,13936}, {12340,13940}, {12384,13941}, {12408,13942}, {12413,13943}, {12503,13946}, {12784,13947}, {12796,13948}, {12805,13949}, {12806,13950}, {12918,13951}, {12925,13952}, {12935,13953}, {12945,13954}, {12955,13955}, {12996,13956}, {12997,13957}, {13099,13959}, {13115,13961}, {13116,13962}, {13117,13963}, {13118,13964}, {13119,13965}, {13966,13992}

X(13985) = reflection of X(13992) in X(13966)

X(13986) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO REFLECTION

Barycentrics (12*R^2+SA-5*SW)*S^2+(SB+SC)*(SW*(4*S-SA)-(10*S-2*SA)*R^2) : :

The reciprocal orthologic center of these triangles is X(4)

X(13986) lies on these lines: {6,6689}, {54,3069}, {195,13961}, {372,3574}, {539,13847}, {615,1209}, {1154,12240}, {2888,13941}, {2917,8277}, {6276,13950}, {6277,13949}, {6288,13951}, {7584,10610}, {7691,13935}, {7979,13959}, {9905,13942}, {9920,13943}, {9985,13946}, {10066,13962}, {10082,13963}, {10628,13969}, {11576,13937}, {12208,13938}, {12254,13939}, {12266,13936}, {12341,13940}, {12785,13947}, {12797,13948}, {12926,13952}, {12936,13953}, {12946,13954}, {12956,13955}, {12998,13956}, {12999,13957}, {13079,13958}, {13121,13964}, {13122,13965}, {13979,13990}

X(13986) = {X(6), X(6689)}-harmonic conjugate of X(8995)

X(13987) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(79)

X(13987) lies on these lines: {6,13919}, {372,12600}, {615,13089}, {3069,10266}, {12146,13937}, {12209,13938}, {12255,13939}, {12267,13936}, {12342,13940}, {12409,13942}, {12414,13943}, {12504,13946}, {12556,13935}, {12786,13947}, {12798,13948}, {12807,13949}, {12808,13950}, {12849,13941}, {12919,13951}, {12927,13952}, {12937,13953}, {12947,13954}, {12957,13955}, {13000,13956}, {13001,13957}, {13080,13958}, {13100,13959}, {13126,13961}, {13128,13962}, {13129,13963}, {13130,13964}, {13131,13965}

X(13988) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 1st TRI-SQUARES-CENTRAL

Barycentrics 22*S^2+3*(3*SA-SW)*S-3*(3*SA+2*SW)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(13665)

X(13988) lies on these lines: {6,13920}, {30,13880}, {372,13687}, {486,13794}, {524,13848}, {615,13701}, {1327,3069}, {13666,13935}, {13667,13936}, {13668,13937}, {13672,13938}, {13674,13939}, {13675,13940}, {13678,13941}, {13679,13942}, {13680,13943}, {13685,13946}, {13688,13947}, {13689,13948}, {13690,13949}, {13691,13950}, {13692,13951}, {13693,13952}, {13694,13953}, {13695,13954}, {13696,13955}, {13697,13956}, {13698,13957}, {13699,13958}, {13702,13959}, {13712,13831}, {13713,13961}, {13714,13962}, {13715,13963}, {13716,13964}, {13717,13965}, {13843,13847}, {13849,13972}

X(13988) = midpoint of X(13847) and X(13932)

X(13989) = PARALLELOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 1st ANTI-BROCARD

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

The reciprocal parallelogic center of these triangles is X(385)

X(13989) lies on these lines: {2,13908}, {6,620}, {98,13935}, {99,3069}, {114,372}, {115,615}, {140,8980}, {148,13941}, {485,6721}, {486,9739}, {542,10820}, {543,13847}, {642,5062}, {690,13990}, {1152,2794}, {2782,13966}, {2783,13977}, {2787,13991}, {2799,13992}, {3023,13958}, {4027,13938}, {5186,13937}, {5420,6036}, {5969,13972}, {6033,6398}, {6319,13949}, {6320,13950}, {6321,13951}, {6722,8252}, {7983,13959}, {8782,13946}, {9995,13653}, {10086,13962}, {10089,13963}, {11711,13936}, {13172,13939}, {13173,13940}, {13174,13942}, {13175,13943}, {13176,13944}, {13177,13945}, {13178,13947}, {13179,13948}, {13180,13952}, {13181,13953}, {13182,13954}, {13183,13955}, {13184,13956}, {13185,13957}, {13188,13961}, {13189,13964}, {13190,13965}

X(13989) = reflection of X(i) in X(j) for these (i,j): (13967,13966), (13968,13847)
X(13989) = parallelogic center of the 4th tri-squares-central triangle these triangles: 6th anti-Brocard, 1st Brocard, 6th Brocard
X(13989) = {X(6), X(620)}-harmonic conjugate of X(8997)

X(13990) = PARALLELOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO ANTI-ORTHOCENTROIDAL

Barycentrics (24*R^2+SA-5*SW)*S^2-(SB+SC)*(18*S*R^2-4*S*SW+SA*SW) : :

The reciprocal parallelogic center of these triangles is X(323)

X(13990) lies on these lines: {6,5181}, {74,13935}, {110,3069}, {113,372}, {125,615}, {140,8994}, {265,13951}, {399,13961}, {485,12900}, {486,10820}, {542,13847}, {690,13989}, {1112,13937}, {1152,2777}, {1511,7584}, {2771,13977}, {2781,13985}, {2854,13972}, {2931,8277}, {2948,13942}, {3448,13941}, {3628,13915}, {5420,6699}, {5663,13966}, {6398,7728}, {6565,12295}, {6723,8252}, {7732,13949}, {7733,13950}, {7984,13959}, {9517,13992}, {9826,12239}, {10088,13962}, {10091,13963}, {11720,13936}, {12121,13785}, {12310,13943}, {12383,13939}, {12903,13954}, {12904,13955}, {13193,13938}, {13204,13940}, {13208,13944}, {13209,13945}, {13210,13946}, {13211,13947}, {13212,13948}, {13213,13952}, {13214,13953}, {13215,13956}, {13216,13957}, {13217,13964}, {13218,13965}, {13979,13986}

X(13990) = midpoint of X(486) and X(10820)
X(13990) = reflection of X(i) in X(j) for these (i,j): (13969,13966), (13979,13993)
X(13990) = parallelogic center of these triangles: 4th tri-squares-central to orthocentroidal
X(13990) = {X(6), X(5972)}-harmonic conjugate of X(8998)

X(13991) = PARALLELOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO INNER-GARCIA

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

The reciprocal parallelogic center of these triangles is X(1)

X(13991) lies on these lines: {6,3035}, {11,615}, {80,13947}, {100,3069}, {104,13935}, {119,372}, {140,13913}, {149,13941}, {214,13936}, {486,5840}, {528,13847}, {952,13966}, {1145,7968}, {1152,2829}, {1320,13959}, {1862,13937}, {2771,13969}, {2783,13967}, {2787,13989}, {2800,13975}, {2802,13971}, {2806,13992}, {2831,13985}, {3634,8988}, {5420,6713}, {5541,13942}, {6398,10742}, {6502,10956}, {6667,8252}, {9024,13972}, {10087,13962}, {10090,13963}, {10738,13951}, {12331,13961}, {13194,13938}, {13199,13939}, {13205,13940}, {13222,13943}, {13228,13944}, {13230,13945}, {13235,13946}, {13268,13948}, {13269,13949}, {13270,13950}, {13271,13952}, {13272,13953}, {13273,13954}, {13274,13955}, {13275,13956}, {13276,13957}, {13278,13964}, {13279,13965}

X(13991) = reflection of X(13977) in X(13966)
X(13991) = {X(6), X(3035)}-harmonic conjugate of X(13922)

X(13992) = PARALLELOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 1st ORTHOSYMMEDIAL

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

The reciprocal parallelogic center of these triangles is X(6)

X(13992) lies on these lines: {6,6720}, {112,3069}, {127,615}, {132,372}, {140,13918}, {486,2794}, {1297,13935}, {2781,13969}, {2799,13989}, {2806,13991}, {2831,13977}, {6020,13958}, {6398,12918}, {9517,13990}, {10705,13959}, {10749,13951}, {11641,13943}, {11722,13936}, {13166,13937}, {13195,13938}, {13200,13939}, {13206,13940}, {13219,13941}, {13221,13942}, {13236,13946}, {13280,13947}, {13281,13948}, {13282,13949}, {13283,13950}, {13294,13952}, {13295,13953}, {13296,13954}, {13297,13955}, {13298,13956}, {13299,13957}, {13310,13961}, {13311,13962}, {13312,13963}, {13313,13964}, {13314,13965}, {13966,13985}

X(13992) = reflection of X(13985) in X(13966)
X(13992) = {X(6), X(6720)}-harmonic conjugate of X(13923)

X(13993) = X(5) OF THE 4th TRI-SQUARES-CENTRAL TRIANGLE

Barycentrics 6*S^2-(SB+SC)*(4*S+SA) : :
X(13993) = 3*X(486)+X(1152) = 11*X(486)-3*X(1328) = X(486)+3*X(13847) = 11*X(1152)+9*X(1328) = X(1152)-9*X(13847) = X(1152)-3*X(13966) = X(1328)+11*X(13847)

X(13993) lies on these lines: {2,6417}, {3,13939}, {4,13961}, {5,1587}, {6,3628}, {26,13943}, {30,486}, {140,371}, {355,13942}, {372,546}, {395,3391}, {396,3366}, {495,13954}, {496,13955}, {524,6119}, {547,7583}, {548,3071}, {549,1588}, {550,6456}, {631,6407}, {632,3311}, {952,13971}, {1132,1657}, {1151,12108}, {1483,13959}, {1656,3317}, {1658,8277}, {3070,5066}, {3090,6418}, {3091,6395}, {3146,6408}, {3365,11543}, {3390,11542}, {3525,6199}, {3526,7582}, {3529,6446}, {3530,5420}, {3564,13880}, {3591,3851}, {3594,12811}, {3627,6398}, {3830,6475}, {3845,6460}, {3853,6565}, {3861,6560}, {5055,7581}, {5070,7585}, {5073,10146}, {5418,6470}, {5690,13947}, {5844,13973}, {5874,13950}, {5875,13949}, {5901,13936}, {6118,6329}, {6396,12103}, {6420,12812}, {6426,12102}, {6429,11812}, {6431,8252}, {6459,6496}, {6500,8972}, {6756,13937}, {8253,10194}, {8855,11265}, {9540,11539}, {10942,13953}, {10943,13952}, {13967,13984}, {13979,13986}

X(13993) = midpoint of X(i) and X(j) for these {i,j}: {486,13966}, {13979,13990}
X(13993) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 3628, 13925), (1656, 6501, 13886), (3069, 13951, 5), (13785, 13935, 550), (13955, 13963, 496)

X(13994) = POINT BEID 152

Barycentrics (SB-SC)^2*(3*S^2-9*SB*SC-2*SW^ 2)*(3*S^2-(3*SA-2*SW)*(3*SA+2* SW)) : :

See Le Viet An and César Lozada, Hyacinthos 26344.

X(13994) lies on the nine-point circle and these lines: {4,11568}, {5,11569}, {114,6032}, {126,9771}, {543,13234}, {2793,12494}, {3849,9127}, {6094,13377}

X(13994) = midpoint of X(i) and X(j) for these {i,j}: {4,11568}, {6094,13377}
X(13994) = reflection of X(11569) in X(5)
X(13994) = nine-point-circle-antipode of X(11569)

X(13995) = POINT BEID 153

Barycentrics 2 a^6 b-a^5 b^2-5 a^4 b^3+2 a^3 b^4+4 a^2 b^5-a b^6-b^7+2 a^6 c+6 a^5 b c-5 a^4 b^2 c-7 a^3 b^3 c+2 a^2 b^4 c+a b^5 c+b^6 c-a^5 c^2-5 a^4 b c^2-10 a^3 b^2 c^2-6 a^2 b^3 c^2+a b^4 c^2+3 b^5 c^2-5 a^4 c^3-7 a^3 b c^3-6 a^2 b^2 c^3-2 a b^3 c^3-3 b^4 c^3+2 a^3 c^4+2 a^2 b c^4+a b^2 c^4-3 b^3 c^4+4 a^2 c^5+a b c^5+3 b^2 c^5-a c^6+b c^6-c^7 : :
X(13995) = (|OI|^2 - 8 R^2) *X(1) + |OI|^2 X(79)

See Le Viet An and Peter Moses, Hyacinthos 26346.

X(13995) lies on these lines: {1, 30}, {442, 3833}, {758, 12267} et al

X(13996) = POINT BEID 154

Barycentrics 4 a^3 (b+c)-a^2 (b^2+14 b c+c^2)-4 a (b^3-2 b^2 c-2 b c^2+c^3)+(b^2-c^2)^2 : :
X(13996) = 4 X(10) - 3 X(11)

See Le Viet An and Angel Montesdeoca, Hyacinthos 26363.

X(13996) lies on these lines: {1, 6174}, {8, 190}, {10, 11}, {12, 10129}, {40, 550}, {56, 100}, {65, 10427}, {80, 4668}, {149, 2551}, {214, 3635}, {519, 1155}, {1000, 4413}, {1018, 4534}, {1320, 3035}, {1376, 13279}, {1387, 3624}, {2183, 3943}, {2254, 6366}, {2800, 3962}, {2829, 6361}, {3434, 13272}, {3617, 10707}, {3649, 10956}, {3679, 4679}, {3885, 8256}, {3893, 11362}, {3922, 12736}, {4701, 12732}, {4746, 12572}, {5082, 10953}, {5087, 6735}, {5252, 5856}, {5433, 10912}, {5434, 12648}, {5660, 11531}, {7091, 12641}, {7173, 13463}, {7972, 9945}, {7993, 13226}, {11500, 12245}, {12247, 12249}

X(13996) = reflection of X(i) in X(j) for these {i, j}: {11,1145}, {149,3036}, {1317,100}, {1320,3035}, {5183,4394}, {6154,5541}, {7972,9945}, {7993,13226}, {12653,1387}

X(13997) = POINT BEID 155

Barycentrics a^2 (a^26-4 a^24 b^2+21 a^20 b^6-23 a^18 b^8-45 a^16 b^10+120 a^14 b^12-78 a^12 b^14-45 a^10 b^16+98 a^8 b^18-56 a^6 b^20+9 a^4 b^22+3 a^2 b^24-b^26-4 a^24 c^2+28 a^22 b^2 c^2-47 a^20 b^4 c^2-68 a^18 b^6 c^2+291 a^16 b^8 c^2-219 a^14 b^10 c^2-273 a^12 b^12 c^2+585 a^10 b^14 c^2-357 a^8 b^16 c^2+19 a^6 b^18 c^2+68 a^4 b^20 c^2-25 a^2 b^22 c^2+2 b^24 c^2-47 a^20 b^2 c^4+235 a^18 b^4 c^4-260 a^16 b^6 c^4-479 a^14 b^8 c^4+1318 a^12 b^10 c^4-839 a^10 b^12 c^4-350 a^8 b^14 c^4+655 a^6 b^16 c^4-251 a^4 b^18 c^4+12 a^2 b^20 c^4+6 b^22 c^4+21 a^20 c^6-68 a^18 b^2 c^6-260 a^16 b^4 c^6+1162 a^14 b^6 c^6-967 a^12 b^8 c^6-1127 a^10 b^10 c^6+2196 a^8 b^12 c^6-964 a^6 b^14 c^6-128 a^4 b^16 c^6+149 a^2 b^18 c^6-14 b^20 c^6-23 a^18 c^8+291 a^16 b^2 c^8-479 a^14 b^4 c^8-967 a^12 b^6 c^8+2852 a^10 b^8 c^8-1587 a^8 b^10 c^8-943 a^6 b^12 c^8+1075 a^4 b^14 c^8-199 a^2 b^16 c^8-20 b^18 c^8-45 a^16 c^10-219 a^14 b^2 c^10+1318 a^12 b^4 c^10-1127 a^10 b^6 c^10-1587 a^8 b^8 c^10+2578 a^6 b^10 c^10-773 a^4 b^12 c^10-220 a^2 b^14 c^10+75 b^16 c^10+120 a^14 c^12-273 a^12 b^2 c^12-839 a^10 b^4 c^12+2196 a^8 b^6 c^12-943 a^6 b^8 c^12-773 a^4 b^10 c^12+560 a^2 b^12 c^12-48 b^14 c^12-78 a^12 c^14+585 a^10 b^2 c^14-350 a^8 b^4 c^14-964 a^6 b^6 c^14+1075 a^4 b^8 c^14-220 a^2 b^10 c^14-48 b^12 c^14-45 a^10 c^16-357 a^8 b^2 c^16+655 a^6 b^4 c^16-128 a^4 b^6 c^16-199 a^2 b^8 c^16+75 b^10 c^16+98 a^8 c^18+19 a^6 b^2 c^18-251 a^4 b^4 c^18+149 a^2 b^6 c^18-20 b^8 c^18-56 a^6 c^20+68 a^4 b^2 c^20+12 a^2 b^4 c^20-14 b^6 c^20+9 a^4 c^22-25 a^2 b^2 c^22+6 b^4 c^22+3 a^2 c^24+2 b^2 c^24-c^26) :

See Le Viet An, César Lozada, and Peter Moses, Hyacinthos 26366 and Hyacinthos 26367.

X(13997) lies on these lines: {74, 186}, {520, 13293}

X(13998) = POINT BEID 156

Barycentrics (a+b-c) (a-b+c) (a^2+b^2-c^2) (a^2-b^2+c^2) (2 a^12 b-2 a^11 b^2-6 a^10 b^3+7 a^9 b^4+4 a^8 b^5-8 a^7 b^6+4 a^6 b^7+2 a^5 b^8-6 a^4 b^9+2 a^3 b^10+2 a^2 b^11-a b^12+2 a^12 c-8 a^11 b c+4 a^10 b^2 c+16 a^9 b^3 c-19 a^8 b^4 c-2 a^7 b^5 c+14 a^6 b^6 c-10 a^5 b^7 c+2 a^4 b^8 c+2 a^3 b^9 c-2 a^2 b^10 c+2 a b^11 c-b^12 c-2 a^11 c^2+4 a^10 b c^2-2 a^9 b^2 c^2-a^8 b^3 c^2+8 a^7 b^4 c^2-11 a^6 b^5 c^2-2 a^5 b^6 c^2+9 a^4 b^7 c^2-2 a^3 b^8 c^2-a^2 b^9 c^2-6 a^10 c^3+16 a^9 b c^3-a^8 b^2 c^3-28 a^7 b^3 c^3+17 a^6 b^4 c^3+10 a^5 b^5 c^3-11 a^4 b^6 c^3+8 a^3 b^7 c^3-3 a^2 b^8 c^3-6 a b^9 c^3+4 b^10 c^3+7 a^9 c^4-19 a^8 b c^4+8 a^7 b^2 c^4+17 a^6 b^3 c^4-16 a^5 b^4 c^4+6 a^4 b^5 c^4-11 a^2 b^7 c^4+9 a b^8 c^4-b^9 c^4+4 a^8 c^5-2 a^7 b c^5-11 a^6 b^2 c^5+10 a^5 b^3 c^5+6 a^4 b^4 c^5-20 a^3 b^5 c^5+15 a^2 b^6 c^5+4 a b^7 c^5-6 b^8 c^5-8 a^7 c^6+14 a^6 b c^6-2 a^5 b^2 c^6-11 a^4 b^3 c^6+15 a^2 b^5 c^6-16 a b^6 c^6+4 b^7 c^6+4 a^6 c^7-10 a^5 b c^7+9 a^4 b^2 c^7+8 a^3 b^3 c^7-11 a^2 b^4 c^7+4 a b^5 c^7+4 b^6 c^7+2 a^5 c^8+2 a^4 b c^8-2 a^3 b^2 c^8-3 a^2 b^3 c^8+9 a b^4 c^8-6 b^5 c^8-6 a^4 c^9+2 a^3 b c^9-a^2 b^2 c^9-6 a b^3 c^9-b^4 c^9+2 a^3 c^10-2 a^2 b c^10+4 b^3 c^10+2 a^2 c^11+2 a b c^11-a c^12-b c^12) : :

See Le Viet An and Peter Moses, Hyacinthos 26373.

X(13998) lies on these lines: {4,109}, {65,7649}

X(13998) =

X(13999) = POINT BEID 157

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

See Le Viet An and Peter Moses, Hyacinthos 26373.

X(13999) lies on the nine-point circle and these lines: {4, 2222}, {11, 7649}, {117, 1737}, {119, 1877}, {650, 5190}, {867, 10017}, {5089, 5513}

X(13999) = polar-circle-inverse of X(2222)
X(13999) = Stevanovic-circle-inverse of X(5190)

X(14000) = POINT BEID 158

Barycentrics a (4 a^6-a^5 b-13 a^4 b^2+2 a^3 b^3+14 a^2 b^4-a b^5-5 b^6-a^5 c+8 a^4 b c+11 a^3 b^2 c-16 a^2 b^3 c-10 a b^4 c+8 b^5 c-13 a^4 c^2+11 a^3 b c^2-6 a^2 b^2 c^2+11 a b^3 c^2+5 b^4 c^2+2 a^3 c^3-16 a^2 b c^3+11 a b^2 c^3-16 b^3 c^3+14 a^2 c^4-10 a b c^4+5 b^2 c^4-a c^5+8 b c^5-5 c^6 : :
X(14000) = (9 r (r + R) - s^2) X[1] + 9 r (R - 2 r) X[3]

See Le Viet An, Angel Montesdeoca, and Peter Moses, Hyacinthos 26369 and Hyacinthos 26370

X(14000) lies on these lines: {1, 3}, {88, 12515}
X(14000) = center of the circumcircle of the cevian triangle of X(80)

This is the end of PART 7: Centers X(12001) - X(14000)

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)