## 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)

### 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) = inverse-in-Schoute-circle 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) =  CIRCUMCENTER OF TRINH TRIANGLE

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(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 .

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 (SA-S (Csc[w]+Tan[w])) : :

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

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

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 .

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 .

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:

1. 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
2. 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).
3. Its directrix and the Euler line of ABC are parallel, therefore its axis and the Euler line of ABC are perpendicular.
4. Its 4th intersection with the circumcircles of ABC and A'B'C' is X(476)
5. The focus and vertex are X(12064) and X(12065), respectively.
6. Its axis is the line {523, 5972}, trilinear polar of X(12066).
7. Its directrix is the line {30, 10279}, trilinear polar of X(12067).
8. The perspector is X(115) and the center is X(523).
9. 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 ta, tb, tc be the tangents to the X-parabola at A, B, C, respectively; the triangle AtBtCt bounded by these tangents is here named the X-parabola-tangential triangle of ABC. Barycentric coordinates of A-vertex are:
At = -(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 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) = 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)

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.

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.

### 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) = 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(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(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) = 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) = 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(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) = 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) = {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) = {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(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

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(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).

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) = 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) = 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) = 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) = {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(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(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)

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), (1,496,5901), (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

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(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(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) = {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

Trilinears    -a*sin(A/2)+(a+c)*sin(B/2)+(a+b)*sin(C/2) : :
X(12523) = X(167)-5*X(7987)

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

X(12523) lies on the cubic K838 and these lines: {1,164}, {3,12443}, {55,8422}, {56,177}, {167,7987}, {2975,11691}, {3616,9807}, {7670,7677}, {8583,12450}

X(12523) = midpoint of X(1) and X(164)
X(12523) = reflection of X(12518) in X(3)
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(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(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) = {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

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(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(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(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(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)

### 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)

### 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) = 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) = 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(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 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): (11,5777), (6797,9947), (11570,119), (12757,6326), (12758,5887)

### 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 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(12666) lies on the Feuerbach hyperbola of the inner Garcia triangle

### 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) = 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) = {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(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(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) = {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), (12