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This is PART 20: Centers X(38001) - X(40000)

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

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Stationary points of Hutson right hyperbolas: X(38001), X(38002)

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This preamble is based on notes from Randy Hutson and Peter Moses, contributed by Clark Kimberling, April 26, 2020.

The following note is copied from X(37904). "Let P and P' be circumcircle antipodes. Let Q be the midpoint of X(2) and P. Let Q' be the midpoint of X(2) and P'. The rectangular hyperbola passing through P, P', Q, Q' has center X(37904) for all P. (Randy Hutson, April 24, 2020)".

There are two stationary points on the family of Hutson rectangular hyperbolas, and on the Euler line, as can be seen by dragging P around the circumcircle in this GeoGebra diagram: Hutson Right Hyperbolas. (Peter Moses, April 26, 2020)

Let HRH(P) denote the Hutson right hyperbola of P.
HRH(X(74) passes through X(i) for these i: 74, 110, 2574, 2575, 2930, 5642, 5646, 38001, 38002.
HRH(X(98)) passes through X(i) for these i: 98, 99, 2482, 6055, 38001, 38002.
HRH(X(111)) passes through X(i) for these i: 111, 1296, 9172, 33900, 38001, 38002.

For P = p : q : r (barycentrics) on the circumcircle, let

h(a,b,c,p,q,r,x,y,z) = (a^2 - 5*b^2 - 5*c^2)*(2*a^2*b^2*c^2*p*(-q + r) - c^2*(a^2 + b^2 - c^2)*(a^2 + c^2)*q^2 - 2*b^2*(b^2 - c^2)*c^2*q*r + b^2*(a^2 + b^2)*(a^2 - b^2 + c^2)*r^2)*x^2 - (a^4 - b^4 + 10*b^2*c^2 - c^4)*((b^4 - c^4)*p^2 + 2*a^2*p*(b^2*q - c^2*r) + a^2*(a^2 + c^2)*q^2 - a^2*(a^2 + b^2)*r^2)*y*z

An equation for HRH(P) is

h(a,b,c,p,q,r,x,y,z) + H(b,c,a,q,r,p,y,z,x) + H(c,a,b,r,p,q,z,x,y) = 0.

The family of Hutson right hyperbolas can be generalized by replacing X(2) in the definition of HRH(P) by U = u : v : w. Let

H(a,b,c,u,v,w,p,q,r,x,y,z) = (c^2*(a^2 + b^2 - c^2)*(a^2 + c^2)*q^2 + 2*a^2*b^2*c^2*p*(q - r) + 2*b^2*c^2*(b^2 - c^2)*q*r - b^2*(a^2 + b^2)*(a^2 - b^2 + c^2)*r^2)*(c^2*u*v + 2*c^2*v^2 + b^2*u*w - (a^2 - 2*b^2 - 2*c^2)*v*w + 2*b^2*w^2)*x^2 - ((b^4 - c^4)*p^2 + a^2*(a^2 + c^2)*q^2 - a^2*(a^2 + b^2)*r^2 + 2*a^2*p*(b^2*q - c^2*r))*(4*b^2*c^2*u^2 + c^2*(a^2 + 3*b^2 - c^2)*u*v + b^2*(a^2 - b^2 + 3*c^2)*u*w + a^2*(a^2 - b^2 - c^2)*v*w)*y*z

Then the general right hyperbola is given by

H(a,b,c,u,v,w,p,q,r,x,y,z) + h(b,c,a,v,w,u,q,r,p,y,z,x) + h(c,a,b,w,u,v,r,p,q,z,x,y) = 0,

with the following point, independent of P, as center:

4*a^2*b^2*c^2*u^3 + c^2*(4*a^4 + 3*a^2*b^2 + b^4 - 5*a^2*c^2 - 2*b^2*c^2 + c^4)*u^2*v + a^2*c^2*(3*a^2 + b^2 - 3*c^2)*u*v^2 + b^2*(4*a^4 - 5*a^2*b^2 + b^4 + 3*a^2*c^2 - 2*b^2*c^2 + c^4)*u^2*w + 2*a^2*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 - c^4)*u*v*w + 3*a^4*(a^2 - b^2 - c^2)*v^2*w + a^2*b^2*(3*a^2 - 3*b^2 + c^2)*u*w^2 + 3*a^4*(a^2 - b^2 - c^2)*v*w^2 : :

and two stationary points on the line UX(3). The midpoint of the two points is the center of the hyperbola. As an example, for U = X(4), the center is X(468) and the stationary points are X(5000) and X(5001), and if P = X(74), then the hyperbola is the Walsmith rectangular hyperbola. (Peter Moses, April 26, 2020)

The general hyperbola is denoted by (U,P)-MHRH and here named the (U,P)-Moses-Hutson right hyperbola. For further examples, see X(38010)-X(38014).

X(38001) =  1ST STATIONARY POINT OF HUTSON FAMILY OF RIGHT HYPERBOLAS

Barycentrics    a*(2*a*(9*(a^4 - b^4 + b^2*c^2 - c^4) + b^2*c^2*J^2)*S - b*c*Sqrt[(a^2 + b^2 + c^2)*(81 - J^2)]*(3*a^2*(a^2 - b^2 - c^2) + 4*S^2)) : : , where J = |OH|/R

X(38001) lies on this line: {2,3}

X(38001) = reflection of X(38002) in X(37904)
X(38001) = circumcircle-inverse of X(38002)
X(38001) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3524, 38002}, {3, 11284, 38002}, {23, 37946, 38002}, {1113, 1114, 38002}

X(38002) =  2ND STATIONARY POINT OF HUTSON FAMILY OF RIGHT HYPERBOLAS

Barycentrics    a*(2*a*(9*(a^4 - b^4 + b^2*c^2 - c^4) + b^2*c^2*J^2)*S + b*c*Sqrt[(a^2 + b^2 + c^2)*(81 - J^2)]*(3*a^2*(a^2 - b^2 - c^2) + 4*S^2)) : : , where J = |OH|/R

X(38002) lies on this line: {2,3}

X(38002) = reflection of X(38001) in X(37904)
X(38002) = circumcircle-inverse of X(38001)
X(38002) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3524, 38001}, {3, 11284, 38001}, {23, 37946, 38001}, {1113, 1114, 38001}

X(38003) =  PERSPECTOR OF THESE TRIANGLES: PAASCHE-HUTSON AND INCENTRAL

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

The Paasche-Hutson triangle is introduced in the preamble just before X(37994).

X(38003) lies on these lines: {1, 1123}, {394, 1124}, {836, 3086}, {1336, 1422}, {37861, 37884}

X(38003) = barycentric product X(i)*X(j) for these {i,j}: {1267, 3554}, {3083, 3086}, {6212, 26871}, {13453, 30223}
X(38003) = barycentric quotient X(i)/X(j) for these {i,j}: {3554, 1123}, {30223, 13454}

X(38004) =  PERSPECTOR OF THESE TRIANGLES: PAASCHE-HUTSON AND EXCENTRAL

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

The Paasche-Hutson triangle is introduced in the preamble just before X(37994).

X(38004) lies on the Jerabek circumhyperbola of the excentral triangle and these lines: {{1, 1123}, {2, 77}, {9, 13389}, {40, 30556}, {380, 7348}, {610, 31438}, {1045, 8945}, {1490, 31562}, {2270, 16432}, {2324, 3083}, {3646, 30557}

X(38004) = X(i)-Ceva conjugate of X(j) for these (i,j): {3083, 1}, {30412, 9}
X(38004) = perspector of these triangles: Paasche-Hutson and cevian triangle of X(3083)
X(38004) = barycentric product X(8)*X(34494)
X(38004) = barycentric quotient X(34494)/X(7)
X(38004) = {X(3083),X(30412)}-harmonic conjugate of X(2324)

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Vu pedal translations: X(38005)-X(38009)

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This preamble is based on notes from Vu Thanh Tung, April 25, 2020.

Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC, but not on the circumcircle, and not the incenter or any of the three excenters. Let

P' = isogonal conjugate of P
A'B'C' = pedal triangle of ABC
V = vector PP'
A1B1C1 = V(A'B'C')

Then A1B1C1 is perspective to ABC, and the perspector, here named the Vu pedal translation of P, is the point

V(P) = (a^4 q r + (b^2 - c^2) p (c^2 q + b^2 r) + a^2 (c^2 q (p - r) + b^2 r (3 p + 3 q + 2 r)))*(a^4 q r - (b^2 - c^2) p (c^2 q + b^2 r) + a^2 (b^2 (p - q) r + c^2 q (3 p + 2 q + 3 r))) : :

See Vu Pedal Translation.

The appearance of (i,j) in the following list means that V(X(i)) = X(j):
(2,38005), (3,4), (4,4846), (5,38006), (6,18842), (7,38007), (8,38008), (9,38009)

See the preamble just before X(38305) for Vu antipedal translation.

X(38005) =  VU PEDAL TRANSLATION OF X(2)

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

X(38005) lies on these lines:

X(38006) =  VU PEDAL TRANSLATION OF X(5)

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

X(38006) lies on these lines:

X(38007) =  VU PEDAL TRANSLATION OF X(7)

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

X(38007) lies on these lines:

X(38008) =  VU PEDAL TRANSLATION OF X(8)

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

X(38008) lies on these lines:

X(38009) =  VU PEDAL TRANSLATION OF X(9)

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

X(38009) lies on these lines:

X(38010) =  CENTER OF THE HYPERBOLAS (X(6),P)-MHRH

Barycentrics    a^2*(8*a^4 + a^2*b^2 - b^4 + a^2*c^2 - 8*b^2*c^2 - c^4) : :
X(38110) = X[6] + 3 X[187], 13 X[6] - 9 X[1570], X[6] - 9 X[1691], 5 X[6] - 9 X[1692], X[6] - 3 X[2030], 7 X[6] + 9 X[2076], 5 X[6] + 3 X[5104], 7 X[6] - 3 X[5107], 17 X[6] - 9 X[5111], 11 X[6] - 3 X[8586], X[69] - 9 X[26613], 13 X[187] + 3 X[1570], X[187] + 3 X[1691], 5 X[187] + 3 X[1692], 7 X[187] - 3 X[2076], 5 X[187] - X[5104], 7 X[187] + X[5107], 17 X[187] + 3 X[5111]

See the preamble just before X(38001).

X(38010) lies on these lines: {3, 6}, {69, 26613}, {111, 32237}, {230, 11645}, {316, 33230}, {373, 1383}, {549, 20194}, {1495, 11580}, {3589, 3849}, {3619, 33197}, {3763, 5215}, {5031, 33211}, {5184, 16491}, {5207, 32952}, {5354, 22352}, {5943, 7708}, {8365, 34573}, {10168, 18907}, {14693, 18358}, {14928, 22329}, {16317, 32267}, {25561, 37637}

X(38010) = midpoint of X(187) and X(2030)
X(38010) = circumcircle-inverse of X(21309)
X(38010) = crossdifference of every pair of points on line {523, 21358}
X(38010) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 15655, 3098}, {187, 1691, 2030}, {187, 1692, 5104}, {187, 5107, 2076}, {1379, 1380, 21309}
.

X(38011) =  1ST STATIONARY POINT OF THE HYPERBOLAS (X(6),P)-MHRH

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

X(38011) lies on this line: {3,6}

X(38012) =  2ND STATIONARY POINT OF THE HYPERBOLAS (X(6),P)-MHRH

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

X(38012) lies on this line: {3,6}

X(38013) =  1ST STATIONARY POINT OF THE HYPERBOLAS (X(1),P)-MHRH

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

X(38013) lies on this line: {1,3}

X(38013) = reflection of X(38014) in X(5126)

X(38014) =  2ND STATIONARY POINT OF THE HYPERBOLAS (X(1),P)-MHRH

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

X(38014) lies on this line: {1,3}
X(38014) = reflection of X(38013) in X(5126)

X(38015) =  PERSPECTOR OF THESE TRIANGLES: PAASCHE-HUTSON AND MEDIAL

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

The Paasche-Hutson triangle, defined in the preamble just before of X(37994), is perspective to the medial triangle of ABC, and the perspector is X(38015). (Dasari Naga Vijay Krishna, April 23, 2020)

See X(38015). (Dasari Naga Vijay Krishna)

X(38015) lies on conics {{A, B, C, X(37), X(836)}}, {{A, B, C, X(77), X(7952)}} and on these lines: {2,77}, {4,15849}, {9,1158}, {19,1528}, {37,158}, {198,5514}, {219,7358}, {836,3086}, {1213,37154}, {1604,12667}, {2270,6848}, {2324,7080}, {2550,37160}, {2551,37320}, {3161,5552}, {4370,34524}, {15836,19843}, {16593,30809}, {17755,27509}, {27382,27522}, {27481,27547}, {27524,27535}, {27539,28830}

X(38015) = complement of X(1440)
X(38015) = crosspoint of X(2) and X(7080)
X(38015) = crosssum of X(6) and X(1413)
X(38015) = X(2)-Ceva conjugate of-X(3086)
X(38015) = X(i)-complementary conjugate of-X(j) for these (i,j): (9, 21239), (31, 3086), (40, 2886), (41, 57)
X(38015) = X(604)-isoconjugate-of-X(34413)
X(38015) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (8, 34413), (329, 34401)
X(38015) = barycentric product X(322)*X(30223)
X(38015) = barycentric quotient X(i)/X(j) for these (i, j): (8, 34413), (329, 34401)
X(38015) = trilinear product X(i)*X(j) for these {i, j}: {329, 30223}, {2324, 3086}
X(38015) = trilinear quotient X(i)/X(j) for these (i, j): (312, 34413), (322, 34401)

X(38016) = PERSPECTOR OF THESE TRIANGLES: ABC AND MEDIAL-OF-3RD-VIJAY-PAASCHE-HUTSON

Barycentrics    4 R^2 + 4 R a - S : :

See X(38016). (Dasari Naga Vijay Krishna)

X(38016) lies on these lines: {1,2}, {390,1600}, {496,6805}, {999,6806}, {1056,1592}, {1058,1584}, {1599,5265}, {5261,15233}, {7373,15235}, {37881,37994}

X(38016) = {X(3083), X(3086)}-harmonic conjugate of X(2)

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Foci of circumparabolas: X(38017) - X(38020) and X(38233)

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This preamble is based on notes from Peter Moses and Randy Hutson, April 28-29, 2020.

Suppose that one focus of a circumparabola is P = p : q : r on the line at infinity. Then the other focus lies on the circumparabolas foci quintic, Q077 and is given by

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

The circumparabola, given by

p(q + r) y z + q( r + p ) z x + r( p + q) = 0

is the isogonal conjugate of the line tangent to the circumcircle at the isogonal conjugate of P.

The appearance of (i,j) in the following list means that X(i) is on the line at infinity and X(j) is the corresponding focus of a circumparabola: (30,38246), (512,38017), (513,38018), (514,38019), (523,12064), (524,38020), (525,38233). (Peter Moses, April 29, 2020)

X(38017) = X(99)X(23610)∩X(512)X(620)

Barycentrics    a^2*(b^2 - c^2)*(a^8*b^4 - a^6*b^6 + a^4*b^8 - 3*a^8*b^2*c^2 + 3*a^6*b^4*c^2 - 4*a^4*b^6*c^2 + a^8*c^4 + 3*a^6*b^2*c^4 + 2*a^2*b^6*c^4 - a^6*c^6 - 4*a^4*b^2*c^6 + 2*a^2*b^4*c^6 - b^6*c^6 + a^4*c^8) : :
X(38017) = X[99] + 3 X[23610]

X(38017) is the focus of the circumparabola given by

a^4*c^4*x*y - 2*a^2*b^2*c^4*x*y + b^4*c^4*x*y + a^4*b^4*x*z - 2*a^2*b^4*c^2*x*z + b^4*c^4*x*z + a^4*b^4*y*z - 2*a^4*b^2*c^2*y*z + a^4*c^4*y*z = 0,

which passes through the points X(i) for these i: 512, 669, 805, 875, 881, 886, 15630, 32729. (Peter Moses, April 28, 2020)

This circumparabola is the isogonal conjugate of line X(99)X(670) (the tangent to the circumcircle at X(99)), and the isotomic conjugate of line X(670)X(888) (the tangent to the Steiner circumellipse at X(670)). (Randy Hutson, April, 28, 2020)

X(38017) lies on the nine-point circle of the cevian triangle of X(512), and on the curve Q077, and on these lines: {99, 23610}, {512, 620}

X(38017) = perspector of ABC and orthic triangle of cevian triangle of X(512)

X(38018) = X(100)X(8027)∩X(513)X(3035)

Barycentrics    a^2*(b - c)*(a^4*b^2 - a^3*b^3 - a^2*b^4 + a*b^5 - 3*a^4*b*c + 3*a^3*b^2*c + 4*a^2*b^3*c - 3*a*b^4*c + a^4*c^2 + 3*a^3*b*c^2 - 12*a^2*b^2*c^2 + 4*a*b^3*c^2 - a^3*c^3 + 4*a^2*b*c^3 + 4*a*b^2*c^3 - b^3*c^3 - a^2*c^4 - 3*a*b*c^4 + a*c^5) : :
X(38018) = X[100] + 3 X[8027], 9 X[14474] - 5 X[31272]

X(38018) is the focus of the circumparabola given by

a^2*c^2*x*y - 2*a*b*c^2*x*y + b^2*c^2*x*y + a^2*b^2*x*z - 2*a*b^2*c*x*z + b^2*c^2*x*z + a^2*b^2*y*z - 2*a^2*b*c*y*z + a^2*c^2*y*z = 0,

which passes through the points X(i) for these i: 513, 649, 660, 889, 901, 3572, 3733, 4581, 7192, 15635, 17929, 17940, 23345, 23836, 32735, 35365. (Peter Moses, April 28, 2020)

This circumparabola is the isogonal conjugate of line X(100)X(190) (the tangent to the circumcircle at X(100)), and the isotomic conjugate of line X(668)X(891) (the tangent to the Steiner circumellipse at X(668)). (Randy Hutson, April, 28, 2020)

X(38018) lies on the nine-point circle of the cevian triangle of X(513), and on the curve Q077, and on these lines: {100, 8027}, {513, 3035}, {3271, 6164}, {4083, 5083}, {14474, 31272}

X(38018) = perspector of ABC and orthic triangle of cevian triangle of X(513)

X(38019) = X(101)X(6545)∩X(116)X(21204)

Barycentrics    (b - c)*(-a^6 + 2*a^5*b - a^3*b^3 - a*b^5 + b^6 + 2*a^5*c - 6*a^4*b*c + 3*a^3*b^2*c + a^2*b^3*c + 3*a*b^4*c - 3*b^5*c + 3*a^3*b*c^2 - 3*a^2*b^2*c^2 - 2*a*b^3*c^2 + 3*b^4*c^2 - a^3*c^3 + a^2*b*c^3 - 2*a*b^2*c^3 - 2*b^3*c^3 + 3*a*b*c^4 + 3*b^2*c^4 - a*c^5 - 3*b*c^5 + c^6) : :
X(38019) = X[101] + 3 X[6545], X[116] - 3 X[21204], X[150] - 9 X[6548], 9 X[14475] - 5 X[31273]

X(38019) is the focus of the circumparabola given by

a^2*x*y - 2*a*b*x*y + b^2*x*y + a^2*x*z - 2*a*c*x*z + c^2*x*z + b^2*y*z - 2*b*c*y*z + c^2*y*z = 0,

which passes through the points X(i) for these i: 514, 693, 927, 3676, 4444, 4555, 4583, 4608, 4817, 6548, 6549, 7192, 15634, 17925, 17930, 37143. (Peter Moses, April 28, 2020)

This circumparabola is the isogonal conjugate of line X(101)X(692) (the tangent to the circumcircle at X(101)), and the isotomic conjugate of line X(100)X(190) (the tangent to the Steiner circumellipse at X(190)). (Randy Hutson, April, 28, 2020)

X(38019) lies on the nine-point circle of the cevian triangle of X(514), and on the curve Q077, and on these lines: {101, 6545}, {116, 21204}, {150, 6548}, {514, 6710}, {1358, 21201}, {3676, 24201}, {14475, 31273}

X(38019) = perspector of ABC and orthic triangle of cevian triangle of X(514)

X(38020) = X(111)X(8030)∩X(126)X(8787)

Barycentrics    (2*a^2 - b^2 - c^2)*(5*a^10 - 18*a^8*b^2 - 2*a^6*b^4 + 17*a^4*b^6 - 3*a^2*b^8 + b^10 - 18*a^8*c^2 + 98*a^6*b^2*c^2 - 75*a^4*b^4*c^2 - 3*a^2*b^6*c^2 - b^8*c^2 - 2*a^6*c^4 - 75*a^4*b^2*c^4 + 81*a^2*b^4*c^4 - 8*b^6*c^4 + 17*a^4*c^6 - 3*a^2*b^2*c^6 - 8*b^4*c^6 - 3*a^2*c^8 - b^2*c^8 + c^10) : :
X(38020) = X[111] + 3 X[8030], 9 X[5468] - X[14360]

X(38020) is the focus of the circumparabola given by

a^4*x*y + 2*a^2*b^2*x*y + b^4*x*y - 4*a^2*c^2*x*y - 4*b^2*c^2*x*y + 4*c^4*x*y + a^4*x*z - 4*a^2*b^2*x*z + 4*b^4*x*z + 2*a^2*c^2*x*z - 4*b^2*c^2*x*z + c^4*x*z + 4*a^4*y*z - 4*a^2*b^2*y*z + b^4*y*z - 4*a^2*c^2*y*z + 2*b^2*c^2*y*z + c^4*y*z = 0,

which passes through the points X(i) for these i: 524, 892, 5468, 6082, 9141, 17708. (Peter Moses, April 28, 2020)

This circumparabola is the isogonal conjugate of line X(111)X(351) (the tangent to the circumcircle at X(111)), and the isotomic conjugate of line X(671)X(690) (the tangent to the Steiner circumellipse at X(671)). (Randy Hutson, April, 28, 2020)

X(38020) lies on the nine-point circle of the cevian triangle of X(524), and on the curve Q077, and on these lines: {111, 8030}, {126, 8787}, {524, 6719}, {5468, 9143}

X(38020) = perspector of ABC and orthic triangle of cevian triangle of X(524)

leftri

Centroids of triangles with central vertices: X(38021)-X(38232)

rightri

This preamble and centers X(38021)-X(38232) were contributed by César Eliud Lozada, April 28, 2020.

This section includes centroids of triangles {X(i), X(j), X(k)}, with i < j< k ≤ 12 and some triangles with vertices X(13) to X(18).

The following list shows centroids of triangles in the above range but not given in this section. The appearance of (i, j, k, n) in this list means that the centroid of trinagle {X(i), X(j), X(k)} is X(n):
(1, 2, 3, 3653), (1, 2, 8, 19875), (1, 2, 10, 19883), (1, 3, 4, 5886), (1, 3, 8, 26446), (1, 3, 10, 10165), (1, 4, 8, 5587), (1, 4, 10, 3817), (1, 4, 20, 3576), (1, 5, 10, 11230), (1, 8, 10, 10), (1, 8, 11, 34122), (1, 8, 20, 165), (1, 10, 11, 32557), (2, 3, 4, 5055), (2, 3, 5, 11539), (2, 3, 20, 15688), (2, 4, 20, 3524), (2, 13, 14, 9166), (2, 15, 16, 26613), (3, 4, 5, 5), (3, 4, 6, 14561), (3, 4, 8, 5790), (3, 4, 10, 10175), (3, 4, 11, 23513), (3, 4, 20, 3), (3, 5, 10, 11231), (3, 5, 11, 34126), (3, 5, 20, 8703), (4, 5, 20, 549), (4, 6, 20, 5085), (4, 7, 20, 21151), (4, 8, 20, 5657), (4, 9, 20, 21153), (4, 10, 20, 10164), (4, 11, 20, 21154), (4, 12, 20, 21155), (4, 13, 14, 14639), (4, 13, 20, 21156), (4, 14, 20, 21157), (4, 15, 20, 21158), (4, 16, 20, 21159), (4, 19, 20, 21160), (6, 13, 14, 6034), (6, 15, 16, 1691), (13, 14, 15, 22510), (13, 14, 16, 22511), (13, 14, 20, 34473)

X(38021) = CENTROID OF TRIANGLE {X(1), X(2), X(4)}

Barycentrics    a^4-3*(b+c)*a^3-(5*b^2-6*b*c+5*c^2)*a^2+3*(b^2-c^2)*(b-c)*a+4*(b^2-c^2)^2 : :
X(38021) = X(1)+2*X(381) = X(1)+8*X(9955) = 5*X(1)+4*X(18480) = 4*X(1)+5*X(18492) = X(1)-10*X(18493) = 7*X(1)+2*X(18525) = 11*X(1)-2*X(18526) = X(1)+5*X(30308) = 17*X(1)-8*X(32900) = X(381)-4*X(9955) = 5*X(381)-2*X(18480) = 8*X(381)-5*X(18492) = X(381)+5*X(18493) = 7*X(381)-X(18525) = 11*X(381)+X(18526) = 2*X(381)-5*X(30308) = 17*X(381)+4*X(32900) = 10*X(9955)-X(18480) = 4*X(9955)+5*X(18493) = 8*X(9955)-5*X(30308) = 17*X(9955)+X(32900)

X(38021) lies on these lines: {1,381}, {2,40}, {3,28202}, {4,551}, {5,3656}, {9,11813}, {10,5071}, {11,11529}, {30,1699}, {36,28444}, {57,1727}, {84,10266}, {140,9589}, {165,5054}, {226,37704}, {355,5066}, {376,1125}, {382,30389}, {497,18406}, {515,3839}, {516,3524}, {517,4731}, {518,38072}, {519,3545}, {524,38035}, {527,38036}, {528,38038}, {529,38039}, {542,16475}, {547,1698}, {549,3624}, {553,3086}, {944,12571}, {952,38071}, {971,38024}, {1385,3830}, {1420,37735}, {1482,4677}, {1503,38023}, {1519,6173}, {1656,7991}, {1697,3584}, {1702,13846}, {1703,13847}, {1836,5298}, {2043,36462}, {2044,36444}, {2077,16417}, {2100,13626}, {2101,13627}, {2807,16226}, {2829,38026}, {3058,9614}, {3090,3828}, {3091,3241}, {3149,4428}, {3333,4654}, {3338,7701}, {3340,7741}, {3428,16857}, {3525,5493}, {3534,7987}, {3543,3616}, {3544,34641}, {3579,15694}, {3583,13384}, {3586,15950}, {3601,5443}, {3622,31673}, {3628,9588}, {3632,11737}, {3633,18357}, {3655,3845}, {3751,5476}, {3829,34647}, {3832,5882}, {3843,15178}, {3850,37727}, {3851,24680}, {3929,12704}, {4297,15682}, {4312,15325}, {4342,8164}, {4423,7688}, {4668,11278}, {4669,5818}, {4745,12245}, {4995,12701}, {5056,11362}, {5068,5734}, {5072,16189}, {5079,30315}, {5087,9623}, {5219,10056}, {5259,28466}, {5274,15933}, {5290,11373}, {5309,9575}, {5434,9612}, {5438,34629}, {5550,15692}, {5563,37234}, {5655,12261}, {5657,10171}, {5690,10109}, {5693,13374}, {5715,11113}, {5790,11224}, {5842,38027}, {5844,14892}, {6054,12258}, {6174,14217}, {6175,19861}, {6246,10031}, {6264,10711}, {6326,10707}, {6361,15702}, {6841,11518}, {6866,37723}, {6990,11523}, {7681,17530}, {7951,7962}, {8226,31146}, {9619,11648}, {9778,15708}, {9812,10165}, {9819,31479}, {9856,15016}, {9880,11724}, {9956,11531}, {10164,15709}, {10199,12705}, {10246,14269}, {10247,37712}, {10248,15640}, {10283,23046}, {10385,13411}, {10389,37701}, {10516,28538}, {10527,17781}, {10531,34701}, {10532,34716}, {10595,19925}, {10706,33535}, {10864,12608}, {11012,16418}, {11014,17556}, {11049,12696}, {11218,37713}, {11374,15170}, {11496,16371}, {11525,17757}, {11539,28174}, {11545,16236}, {11645,38029}, {11723,12407}, {12243,21636}, {12512,15698}, {12611,18540}, {12702,15703}, {13624,15681}, {14893,34773}, {15022,31399}, {15673,16113}, {15678,16125}, {15687,18481}, {15688,28146}, {15689,17502}, {15693,16192}, {15699,26446}, {15701,31663}, {15829,25639}, {16370,22753}, {17532,22835}, {17800,31666}, {18398,31937}, {18519,37602}, {18991,35823}, {18992,35822}, {19541,34486}, {19708,34638}, {21164,24644}, {24541,31156}, {25542,35239}, {26286,28453}, {28160,30392}, {28628,37428}, {31140,37569}, {31165,37625}, {31755,31961}, {33179,34748}, {33697,35403}

X(38021) = midpoint of X(i) and X(j) for these {i,j}: {1699, 25055}, {3545, 5603}, {9812, 10304}, {10246, 14269}, {10283, 23046}
X(38021) = reflection of X(i) in X(j) for these (i,j): (165, 5054), (3524, 19883), (3545, 3817), (3576, 25055), (3653, 38022), (5054, 11230), (5587, 3545), (10304, 10165), (15689, 17502), (19875, 5055), (25055, 5886), (26446, 15699), (38066, 38083), (38074, 38076)
X(38021) = anticomplement of X(38068)
X(38021) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 30308, 381), (2, 946, 31162), (2, 31162, 40), (2, 34632, 6684), (5, 3656, 3679), (5, 11522, 7982), (381, 9955, 30308), (946, 8227, 40), (962, 31423, 40), (1699, 5886, 3576), (3653, 5886, 38022), (3653, 38022, 25055), (3656, 3679, 7982), (3679, 11522, 3656), (3845, 5901, 3655), (4654, 10072, 3333), (4870, 11238, 1), (5818, 34631, 4669), (5886, 38034, 1699), (7988, 19875, 5055), (8227, 31162, 2), (9955, 18493, 1), (18393, 23708, 57)

X(38022) = CENTROID OF TRIANGLE {X(1), X(2), X(5)}

Barycentrics    8*a^4-6*(b+c)*a^3-(13*b^2-12*b*c+13*c^2)*a^2+6*(b^2-c^2)*(b-c)*a+5*(b^2-c^2)^2 : :
X(38022) = X(1)+2*X(547) = 5*X(2)+X(1482) = 4*X(2)-X(5690) = X(2)+2*X(5901) = 7*X(2)+5*X(10595) = 13*X(2)-X(12245) = 11*X(2)+X(34631) = 7*X(2)-X(34718) = 4*X(1482)+5*X(5690) = X(1482)-10*X(5901) = 13*X(1482)+5*X(12245) = 11*X(1482)-5*X(34631) = 7*X(1482)+5*X(34718) = 3*X(1482)+5*X(38066) = X(5690)+8*X(5901) = 7*X(5690)+20*X(10595) = 13*X(5690)-4*X(12245) = 11*X(5690)+4*X(34631) = 7*X(5690)-4*X(34718) = 3*X(5690)-4*X(38066)

X(38022) lies on these lines: {1,547}, {2,1482}, {5,551}, {8,15703}, {30,1699}, {40,11812}, {140,3656}, {355,10109}, {376,18493}, {381,3616}, {515,38071}, {516,38080}, {517,11539}, {518,38079}, {519,10172}, {524,38040}, {527,38041}, {528,38044}, {529,38045}, {549,1125}, {632,13464}, {944,19709}, {946,8703}, {952,5055}, {962,15693}, {1385,3845}, {1387,10056}, {1656,3241}, {3058,37735}, {3338,19919}, {3524,28174}, {3530,11522}, {3545,10246}, {3564,38023}, {3582,5425}, {3622,5071}, {3624,3654}, {3628,3679}, {3636,37705}, {3655,5066}, {3817,23046}, {3828,24680}, {3839,28186}, {3853,30389}, {3860,5691}, {4297,33699}, {4301,14869}, {4428,6924}, {4669,33179}, {4870,6147}, {5054,5603}, {5067,31145}, {5434,5443}, {5550,15694}, {5731,14269}, {5762,38025}, {5818,34748}, {5843,38024}, {5844,19875}, {5881,12812}, {6361,15700}, {7982,16239}, {7987,15690}, {7988,14892}, {8148,15723}, {9778,15706}, {9779,28190}, {9812,15689}, {9955,15687}, {10072,37737}, {10165,17504}, {10171,38138}, {10680,17542}, {11278,19878}, {11376,15170}, {11849,36006}, {12100,16192}, {12101,34628}, {12699,34200}, {12702,15702}, {13624,15686}, {14893,18481}, {15688,28178}, {15701,34632}, {15714,31730}, {16417,33814}, {16858,22765}, {18483,35404}, {19710,22793}, {19711,31663}, {26286,28463}, {28461,37535}, {34627,37624}, {35018,37727}

X(38022) = midpoint of X(i) and X(j) for these {i,j}: {3545, 10246}, {3653, 38021}, {5054, 5603}, {5731, 14269}, {5886, 25055}, {9812, 15689}, {10283, 15699}
X(38022) = reflection of X(i) in X(j) for these (i,j): (11539, 19883), (15699, 11230), (17504, 10165), (23046, 3817), (38028, 25055), (38042, 15699), (38081, 38083)
X(38022) = complement of X(38066)
X(38022) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 10595, 34718), (3624, 3654, 10124), (3653, 5886, 38021), (3655, 8227, 5066), (5886, 38028, 38034), (10283, 11230, 38042), (15699, 38081, 38083), (18481, 30308, 14893), (25055, 38021, 3653), (38081, 38083, 38042), (38084, 38085, 5055)

X(38023) = CENTROID OF TRIANGLE {X(1), X(2), X(6)}

Barycentrics    7*a^3+4*(b+c)*a^2+4*(b^2+c^2)*a+(b+c)*(b^2+c^2) : :
X(38023) = X(1)+2*X(597) = X(2)+2*X(1386) = 4*X(2)-X(3416) = 7*X(2)-4*X(3844) = X(6)+2*X(551) = 2*X(182)+X(3656) = 8*X(1386)+X(3416) = 7*X(1386)+2*X(3844) = 7*X(3416)-16*X(3844) = 2*X(38029)+X(38035) = X(38029)+2*X(38040) = X(38035)-4*X(38040) = X(38047)-4*X(38049) = 3*X(38047)-2*X(38087) = 3*X(38047)-4*X(38089) = 7*X(38047)-4*X(38191) = 6*X(38049)-X(38087) = 3*X(38049)-X(38089) = 7*X(38049)-X(38191) = 7*X(38087)-6*X(38191)

X(38023) lies on these lines: {1,597}, {2,1386}, {6,551}, {30,38029}, {182,3656}, {511,3653}, {515,38072}, {516,38086}, {517,38064}, {518,38025}, {519,38047}, {524,16475}, {527,38046}, {528,38050}, {529,38051}, {542,5886}, {599,1125}, {952,38079}, {1385,20423}, {1503,38021}, {1992,3616}, {2796,17301}, {3241,3618}, {3246,26626}, {3564,38022}, {3589,3679}, {3622,4663}, {3624,20582}, {3654,10168}, {3655,5476}, {4301,10541}, {4428,36741}, {4702,17014}, {5085,28194}, {5845,38024}, {5846,19875}, {5847,19883}, {5848,38026}, {5849,38027}, {8550,9624}, {10246,14848}, {11725,18800}, {11735,15303}, {14561,28204}, {25555,37727}, {26230,31179}, {29648,31143}

X(38023) = midpoint of X(i) and X(j) for these {i,j}: {10246, 14848}, {16475, 25055}
X(38023) = reflection of X(i) in X(j) for these (i,j): (21358, 19883), (38087, 38089)
X(38023) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38029, 38040, 38035), (38087, 38089, 38047), (38090, 38091, 38079)

X(38024) = CENTROID OF TRIANGLE {X(1), X(2), X(7)}

Barycentrics    a^3-10*(b+c)*a^2-(-5*b^2+18*b*c-5*c^2)*a+4*(b^2-c^2)*(b-c) : :
X(38024) = 5*X(1)+4*X(5880) = X(1)+2*X(6173) = X(1)+8*X(25557) = 4*X(2)-X(5223) = X(2)+2*X(5542) = X(7)+2*X(551) = 4*X(142)-X(3679) = X(5223)+8*X(5542) = 3*X(5223)-8*X(38101) = 3*X(5542)+X(38101) = 2*X(5880)-5*X(6173) = X(5880)-10*X(25557) = X(6173)-4*X(25557) = 2*X(11038)+X(38052) = X(11038)+2*X(38054) = 3*X(11038)+X(38092) = 3*X(11038)+2*X(38094) = 7*X(11038)+2*X(38201) = 3*X(19875)-2*X(38097) = 2*X(38030)+X(38036) = X(38030)+2*X(38041) = X(38036)-4*X(38041) = 3*X(38093)-X(38097)

X(38024) lies on these lines: {1,528}, {2,5223}, {7,551}, {30,38030}, {142,3679}, {515,38073}, {516,30392}, {517,38065}, {518,3921}, {519,11038}, {524,38046}, {527,17561}, {529,38056}, {535,4321}, {537,27475}, {952,38080}, {971,38021}, {1001,37587}, {1125,6172}, {2801,7988}, {2951,20330}, {3243,4677}, {3582,10398}, {3622,30424}, {3653,5762}, {3656,31657}, {4355,11111}, {5045,5696}, {5220,34595}, {5249,30350}, {5735,30389}, {5843,38022}, {5845,38023}, {5850,19883}, {5851,38026}, {5852,38027}, {10072,30330}, {10582,31164}, {11240,30343}, {11274,20119}, {13159,15678}, {14151,38207}, {15709,38130}, {18421,30379}, {19876,20195}, {21151,28194}, {28204,38107}

X(38024) = reflection of X(i) in X(j) for these (i,j): (19875, 38093), (25055, 38053), (38092, 38094)
X(38024) = anticomplement of X(38101)
X(38024) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11038, 38054, 38052), (38030, 38041, 38036), (38092, 38094, 38052), (38095, 38096, 38080)

X(38025) = CENTROID OF TRIANGLE {X(1), X(2), X(9)}

Barycentrics    7*a^3-7*(b+c)*a^2-(b^2+18*b*c+c^2)*a+(b^2-c^2)*(b-c) : :
X(38025) = 5*X(2)+X(390) = X(2)+2*X(1001) = 4*X(2)-X(2550) = 7*X(2)-4*X(3826) = X(9)+2*X(551) = X(390)-10*X(1001) = 4*X(390)+5*X(2550) = 7*X(390)+20*X(3826) = 3*X(390)+5*X(38092) = 8*X(1001)+X(2550) = 7*X(1001)+2*X(3826) = 6*X(1001)+X(38092) = 7*X(2550)-16*X(3826) = 3*X(2550)-4*X(38092) = 12*X(3826)-7*X(38092) = 3*X(19883)-X(38094) = 2*X(38031)+X(38037) = X(38031)+2*X(38043) = X(38037)-4*X(38043) = 3*X(38093)-2*X(38094)

X(38025) lies on these lines: {2,11}, {7,4870}, {9,551}, {30,38031}, {142,18393}, {405,34610}, {443,4330}, {515,38075}, {516,3524}, {517,38067}, {518,38023}, {519,38057}, {524,38048}, {527,17561}, {529,38061}, {537,24497}, {952,38082}, {971,3653}, {1125,5698}, {2094,3683}, {3241,3759}, {3246,5308}, {3616,3758}, {3622,5220}, {3656,31658}, {3679,6666}, {3828,30331}, {3945,8692}, {4208,34706}, {4299,5259}, {4648,15485}, {5129,11236}, {5434,8232}, {5550,5880}, {5762,38022}, {5766,11376}, {5853,19875}, {5856,38026}, {5857,38027}, {7288,8543}, {7676,36006}, {7677,16858}, {11106,34620}, {11108,34619}, {11239,17547}, {11495,15692}, {12848,15950}, {15672,17768}, {15675,16133}, {15702,35514}, {16020,17301}, {21153,28194}, {25072,35227}, {28204,38108}

X(38025) = reflection of X(i) in X(j) for these (i,j): (38053, 25055), (38093, 19883), (38097, 38101)
X(38025) = complement of X(38092)
X(38025) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38031, 38043, 38037), (38097, 38101, 38057), (38102, 38103, 38082)

X(38026) = CENTROID OF TRIANGLE {X(1), X(2), X(11)}

Barycentrics    8*a^4-6*(b+c)*a^3-13*(b-c)^2*a^2+2*(b+c)*(3*b^2-5*b*c+3*c^2)*a+5*(b^2-c^2)^2 : :
X(38026) = 4*X(2)-X(1145) = 5*X(2)+X(1320) = X(2)+2*X(1387) = X(11)+2*X(551) = 7*X(11)+2*X(33337) = 7*X(551)-X(33337) = 5*X(1145)+4*X(1320) = X(1145)+8*X(1387) = X(1320)-10*X(1387) = 4*X(32557)-X(34122) = 6*X(32557)-X(38099) = 3*X(32557)-X(38104) = 7*X(32557)-X(38213) = 3*X(34122)-2*X(38099) = 3*X(34122)-4*X(38104) = 7*X(34122)-4*X(38213) = 2*X(38032)+X(38038) = X(38032)+2*X(38044) = X(38038)-4*X(38044) = 7*X(38099)-6*X(38213)

X(38026) lies on these lines: {2,1000}, {11,551}, {30,38032}, {515,38077}, {516,38095}, {517,38069}, {518,38090}, {519,32557}, {524,38050}, {527,38055}, {528,15015}, {529,38063}, {952,5055}, {1086,24871}, {1125,6174}, {1317,33709}, {2802,19883}, {2829,38021}, {3241,31272}, {3616,10609}, {3622,10031}, {3636,11274}, {3653,5840}, {3656,6713}, {3679,6667}, {4428,10090}, {5440,38202}, {5848,38023}, {5851,38024}, {5854,19875}, {5856,38025}, {8164,12735}, {9624,20418}, {11112,11376}, {12690,25525}, {13747,34640}, {13996,19862}, {14150,24541}, {17525,33593}, {18240,24473}, {21154,28194}, {23513,28204}

X(38026) = midpoint of X(16173) and X(25055)
X(38026) = reflection of X(i) in X(j) for these (i,j): (34123, 25055), (38099, 38104)
X(38026) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38032, 38044, 38038), (38099, 38104, 34122)

X(38027) = CENTROID OF TRIANGLE {X(1), X(2), X(12)}

Barycentrics    8*a^4-6*(b+c)*a^3-(13*b^2+2*b*c+13*c^2)*a^2+2*(b+c)*(3*b^2-7*b*c+3*c^2)*a+5*(b^2-c^2)^2 : :
X(38027) = X(2)+2*X(37737) = X(12)+2*X(551) = 4*X(1125)-X(31157) = X(3656)+2*X(31659) = X(3679)-4*X(6668) = 2*X(4870)+X(37298) = 2*X(38033)+X(38039) = X(38033)+2*X(38045) = X(38039)-4*X(38045) = X(38056)+2*X(38061) = X(38058)-4*X(38062) = 3*X(38058)-2*X(38100) = 3*X(38058)-4*X(38105) = 7*X(38058)-4*X(38214) = 6*X(38062)-X(38100) = 3*X(38062)-X(38105) = 7*X(38062)-X(38214) = 7*X(38100)-6*X(38214) = 7*X(38105)-3*X(38214)

X(38027) lies on these lines: {2,4930}, {12,551}, {30,38033}, {515,38078}, {516,38096}, {517,38070}, {518,38091}, {519,38058}, {524,38051}, {527,38056}, {528,38063}, {529,25055}, {758,19883}, {952,5055}, {1125,31157}, {2975,16857}, {3656,31659}, {3679,6668}, {4870,37298}, {5842,38021}, {5849,38023}, {5852,38024}, {5855,19875}, {5857,38025}, {7483,34647}, {10197,15950}, {10609,17532}, {11113,11375}, {21155,28194}, {28204,38109}

X(38027) = midpoint of X(25055) and X(37701)
X(38027) = reflection of X(38100) in X(38105)
X(38027) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38033, 38045, 38039), (38100, 38105, 38058)

X(38028) = CENTROID OF TRIANGLE {X(1), X(3), X(5)}

Barycentrics    4*a^4-2*(b+c)*a^3-(5*b^2-4*b*c+5*c^2)*a^2+2*(b^2-c^2)*(b-c)*a+(b^2-c^2)^2 : :
X(38028) = X(1)+2*X(140) = 2*X(1)+X(5690) = 5*X(1)+7*X(31423) = 3*X(2)+X(7967) = 5*X(3)+X(962) = X(3)+5*X(3616) = X(3)+2*X(5901) = 7*X(3)-X(6361) = 3*X(3)-X(9778) = 2*X(3)+X(22791) = 4*X(140)-X(5690) = 10*X(140)-7*X(31423) = X(962)-5*X(5603) = X(962)-10*X(5901) = 7*X(962)+5*X(6361) = 3*X(962)+5*X(9778) = 2*X(962)-5*X(22791) = 5*X(5690)-14*X(31423) = X(5790)+3*X(10246) = 2*X(5790)-3*X(38042) = X(7967)-3*X(10246) = 2*X(7967)+3*X(38042) = 2*X(10246)+X(38042) = 5*X(26446)-7*X(31423)

X(38028) lies on these lines: {1,140}, {2,952}, {3,962}, {4,28190}, {5,515}, {8,3526}, {10,632}, {11,37525}, {12,21842}, {20,18493}, {21,33668}, {30,1699}, {36,7508}, {40,3530}, {55,1387}, {56,6147}, {84,16009}, {104,5284}, {145,3525}, {165,3656}, {214,1484}, {226,5126}, {355,3624}, {376,28178}, {381,5731}, {392,10202}, {404,37621}, {405,16203}, {474,16202}, {495,1319}, {496,2646}, {497,37606}, {498,1388}, {499,34471}, {511,38040}, {516,8703}, {517,549}, {518,38110}, {519,11231}, {546,8227}, {547,3655}, {548,7987}, {550,946}, {590,35762}, {615,35763}, {631,1482}, {758,31650}, {942,31838}, {944,1656}, {960,13373}, {971,38043}, {978,37698}, {999,3475}, {1001,6914}, {1006,22765}, {1159,5435}, {1317,5326}, {1368,24301}, {1420,11374}, {1511,11735}, {1537,6950}, {1595,11363}, {1698,16239}, {1737,37728}, {2320,6980}, {2948,13392}, {3058,16173}, {3086,12433}, {3090,18525}, {3147,11396}, {3241,15694}, {3311,13959}, {3312,13902}, {3333,26921}, {3338,16137}, {3434,9945}, {3476,31479}, {3485,24470}, {3523,10595}, {3533,3617}, {3534,9812}, {3564,38029}, {3579,13464}, {3601,11373}, {3612,11376}, {3627,4297}, {3634,13607}, {3636,6684}, {3649,12104}, {3654,11812}, {3679,10124}, {3753,17564}, {3817,3845}, {3820,32213}, {3826,22935}, {3828,38176}, {3830,9779}, {3850,5691}, {3858,31673}, {3877,37298}, {3878,5885}, {3884,35004}, {3890,25413}, {3897,4187}, {3925,37726}, {4301,31663}, {4304,7743}, {4305,9669}, {4423,22758}, {4666,37533}, {4881,11112}, {4999,30144}, {5045,31837}, {5054,5657}, {5066,7988}, {5070,5818}, {5248,32612}, {5250,37612}, {5265,5708}, {5298,5902}, {5305,9619}, {5330,37291}, {5428,11281}, {5434,37701}, {5443,7354}, {5450,31649}, {5453,21214}, {5499,26287}, {5658,6913}, {5692,31157}, {5694,12005}, {5703,7373}, {5722,13384}, {5762,28466}, {5763,11249}, {5771,15934}, {5780,17552}, {5840,38044}, {5843,38030}, {5881,34595}, {5882,9956}, {6261,16617}, {6265,11219}, {6284,37616}, {6583,31806}, {6675,19861}, {6690,6713}, {6691,30147}, {6857,24558}, {6861,10785}, {6889,10586}, {6920,26321}, {6923,22938}, {6924,10267}, {6929,22799}, {6940,11849}, {6946,18524}, {6951,10738}, {6965,10742}, {6967,10587}, {6989,14986}, {6996,29592}, {7288,34753}, {7294,18395}, {7510,17923}, {7555,9625}, {7968,8981}, {7969,13966}, {7982,12108}, {7989,12812}, {8148,15720}, {8236,38121}, {8572,24159}, {8583,37700}, {8727,13151}, {8728,10943}, {8983,19117}, {9780,12645}, {10021,33858}, {10035,28364}, {10172,28236}, {10175,15699}, {10186,28915}, {10198,26492}, {10200,26487}, {10264,11720}, {10299,20070}, {10303,12245}, {10386,12053}, {10543,37720}, {10572,10593}, {10582,37611}, {10609,11680}, {10806,37462}, {10942,17527}, {10950,24926}, {11019,15935}, {11362,33179}, {11375,18990}, {11522,33923}, {11545,37740}, {11698,11715}, {11723,12041}, {11724,12042}, {11725,33813}, {12047,37605}, {12101,30308}, {12135,37119}, {12261,34153}, {12263,32448}, {12266,21230}, {12571,33697}, {12647,12735}, {12811,18492}, {12898,15059}, {13411,24928}, {13465,15676}, {13971,19116}, {14150,34862}, {14893,34628}, {15015,34612}, {15251,36477}, {15326,18393}, {15686,28150}, {15687,28164}, {15700,34632}, {15702,34718}, {15703,34627}, {15704,22793}, {15709,38066}, {15713,28234}, {15714,28232}, {15721,34631}, {15723,34748}, {16475,34380}, {16831,19512}, {17043,25523}, {17044,20328}, {17504,28194}, {17605,21578}, {17768,28463}, {19710,28154}, {19919,24467}, {22837,32426}, {23410,34634}, {24953,37733}, {25405,31397}, {25917,31835}, {25935,31186}, {26201,31803}, {26476,30538}, {26686,30140}, {28172,33699}, {28208,38071}, {28216,31162}, {29648,37360}, {30384,37600}, {31419,32214}, {32789,35788}, {32790,35789}, {33152,37617}, {33591,34643}, {35255,35775}, {35256,35774}, {37571,37722}, {37602,37703}, {38170,38204}

X(38028) = midpoint of X(i) and X(j) for these {i,j}: {1, 26446}, {2, 10246}, {3, 5603}, {165, 3656}, {381, 5731}, {392, 10202}, {549, 10283}, {551, 10165}, {1385, 11230}, {3534, 9812}, {3576, 5886}, {3653, 25055}, {3654, 16200}, {3655, 5587}, {5657, 10247}, {5790, 7967}, {5882, 38155}, {6265, 11219}, {8236, 38121}, {34123, 38032}, {38031, 38053}
X(38028) = reflection of X(i) in X(j) for these (i,j): (5, 11230), (165, 12100), (549, 10165), (3845, 3817), (5587, 547), (5603, 5901), (5690, 26446), (8703, 17502), (10283, 551), (11230, 1125), (15699, 19883), (22791, 5603), (26446, 140), (37705, 38155), (38022, 25055), (38034, 5886), (38041, 38053), (38042, 2), (38112, 11231), (38138, 10175), (38140, 10171), (38155, 9956), (38170, 38204), (38176, 3828)
X(38028) = complement of X(5790)
X(38028) = X(22791)-Gibert-Moses centroid
X(38028) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 140, 5690), (1, 5444, 5432), (2, 7967, 5790), (3, 3616, 5901), (3, 5901, 22791), (5, 1385, 34773), (10, 15178, 1483), (56, 37737, 6147), (104, 5284, 7489), (632, 1483, 10), (946, 13624, 550), (1125, 1385, 5), (3526, 37624, 8), (3576, 25055, 5886), (3653, 5886, 3576), (4297, 9955, 3627), (5790, 10246, 7967), (7987, 12699, 548), (8227, 18481, 546), (10171, 38140, 5), (10267, 25524, 6924), (11230, 38140, 10171), (15699, 38138, 10175), (17614, 24541, 8728), (34126, 38114, 2), (37616, 37735, 6284), (38022, 38034, 5886), (38032, 38033, 10246)

X(38029) = CENTROID OF TRIANGLE {X(1), X(3), X(6)}

Barycentrics    a*(3*a^5-(b+c)*a^4-2*(b^2-b*c+c^2)*a^3-2*b*c*(b+c)*a^2-(b^4+c^4-2*b*c*(-3*b*c+c^2+b^2))*a+(b^4-c^4)*(b-c)) : :
X(38029) = X(1)+2*X(182) = X(3)+2*X(1386) = X(6)+2*X(1385) = X(40)-4*X(5092) = 4*X(140)-X(3416) = X(355)-4*X(3589) = 2*X(551)+X(11179) = 4*X(575)-X(3751) = 2*X(576)+7*X(30389) = 2*X(597)+X(3655) = X(944)+5*X(3618) = 4*X(1125)-X(1352) = 3*X(14561)-2*X(38146) = 3*X(38023)-X(38035) = 3*X(38023)-2*X(38040) = 3*X(38047)-2*X(38165) = 3*X(38049)-X(38146) = 3*X(38064)-X(38116) = 3*X(38064)-2*X(38118) = 3*X(38110)-X(38165)

X(38029) lies on these lines: {1,182}, {3,1386}, {6,1385}, {30,38023}, {40,5092}, {140,3416}, {165,17508}, {355,3589}, {511,3576}, {515,14561}, {516,38115}, {517,5085}, {518,5050}, {519,38064}, {524,3653}, {542,25055}, {551,11179}, {572,16972}, {575,3751}, {576,30389}, {597,3655}, {611,1319}, {613,2646}, {614,37527}, {944,3618}, {952,38047}, {971,38048}, {997,17976}, {1125,1352}, {1350,13624}, {1469,37618}, {1482,12017}, {1503,5886}, {1571,5116}, {1572,1691}, {1699,29012}, {2836,32609}, {3056,3612}, {3098,7987}, {3242,15178}, {3526,3844}, {3564,38028}, {3616,6776}, {3624,24206}, {3679,10168}, {3745,16434}, {3818,8227}, {3827,10202}, {4265,32612}, {4297,31670}, {5054,28538}, {5096,32613}, {5102,31662}, {5138,18443}, {5480,18481}, {5603,25406}, {5691,19130}, {5731,14853}, {5762,38046}, {5840,38050}, {5845,38030}, {5846,26446}, {5847,10165}, {5848,38032}, {5849,38033}, {7290,31394}, {7968,19145}, {7969,19146}, {7982,20190}, {9778,33750}, {9955,36990}, {9970,11709}, {10267,36741}, {10269,36740}, {10516,11230}, {10541,24680}, {11579,11720}, {11645,38021}, {11699,16010}, {11710,12177}, {12261,32233}, {12329,16202}, {12407,20301}, {13373,24476}, {16203,22769}, {17502,31884}, {18583,34773}, {19140,33535}, {28186,38136}, {28204,38144}, {28208,38072}

X(38029) = midpoint of X(i) and X(j) for these {i,j}: {3576, 16475}, {5050, 10246}, {5603, 25406}, {5731, 14853}
X(38029) = reflection of X(i) in X(j) for these (i,j): (165, 17508), (10516, 11230), (14561, 38049), (31884, 17502), (38035, 38040), (38047, 38110), (38116, 38118), (38144, 38167)
X(38029) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38023, 38035, 38040), (38064, 38116, 38118), (38119, 38120, 38110)

X(38030) = CENTROID OF TRIANGLE {X(1), X(3), X(7)}

Barycentrics    a^6-5*(b+c)*a^5+5*(b^2+c^2)*a^4+4*(b^3+c^3)*a^3-(7*b^2+4*b*c+7*c^2)*(b-c)^2*a^2+(b^2-c^2)*(b-c)^3*a+(b^2-c^2)^2*(b-c)^2 : :
X(38030) = X(1)+2*X(31657) = X(3)+2*X(5542) = X(7)+2*X(1385) = 4*X(140)-X(5223) = 4*X(142)-X(355) = X(390)-4*X(15178) = 4*X(1125)-X(5779) = 3*X(3653)-2*X(38031) = 3*X(5886)-2*X(38037) = 3*X(26446)-2*X(38126) = 3*X(38024)-X(38036) = 3*X(38024)-2*X(38041) = X(38037)-3*X(38053) = 3*X(38052)-2*X(38170) = 3*X(38054)-X(38151) = 3*X(38065)-X(38121) = 3*X(38065)-2*X(38123) = 3*X(38107)-2*X(38151) = 3*X(38111)-X(38170) = 3*X(38122)-X(38126)

X(38030) lies on these lines: {1,31657}, {2,38179}, {3,5542}, {7,1385}, {30,38024}, {140,5223}, {142,355}, {390,15178}, {511,38046}, {515,38054}, {516,3534}, {517,11038}, {518,10202}, {519,38065}, {527,3653}, {952,38052}, {954,10269}, {971,5886}, {1001,16203}, {1125,5779}, {1387,10384}, {2346,26285}, {2550,37727}, {2801,38108}, {2951,22791}, {3062,9624}, {3475,11227}, {3576,5762}, {3616,36996}, {3655,6173}, {4297,31671}, {4321,18443}, {5054,38130}, {5055,38158}, {5587,38171}, {5686,11231}, {5728,13373}, {5732,12699}, {5759,13624}, {5790,38204}, {5791,12005}, {5805,18481}, {5817,11230}, {5840,7675}, {5843,38028}, {5845,38029}, {5850,10165}, {5851,38032}, {5852,38033}, {5885,7672}, {5901,11372}, {6067,37438}, {7988,38139}, {8581,11374}, {9955,36991}, {10156,25568}, {10398,15325}, {10427,12737}, {11373,14100}, {11495,16202}, {12669,26201}, {14986,15008}, {15298,21154}, {19875,38175}, {24680,35514}, {25055,38043}, {28186,38137}, {28204,38149}, {28208,38073}, {30331,37624}, {38093,38154}

X(38030) = midpoint of X(11038) and X(21151)
X(38030) = reflection of X(i) in X(j) for these (i,j): (5587, 38171), (5686, 11231), (5790, 38204), (5817, 11230), (5886, 38053), (26446, 38122), (38036, 38041), (38052, 38111), (38107, 38054), (38121, 38123), (38149, 38172)
X(38030) = anticomplement of X(38179)
X(38030) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5732, 20330, 12699), (38024, 38036, 38041), (38065, 38121, 38123), (38124, 38125, 38111)

X(38031) = CENTROID OF TRIANGLE {X(1), X(3), X(9)}

Barycentrics    a*(3*a^5-6*(b+c)*a^4+2*b*c*a^3+2*(b+c)*(3*b^2-2*b*c+3*c^2)*a^2-3*(b^2+c^2)*(b-c)^2*a-4*(b^2-c^2)*(b-c)*b*c) : :
X(38031) = X(1)+2*X(31658) = X(3)+2*X(1001) = 5*X(3)-2*X(11495) = X(9)+2*X(1385) = 4*X(140)-X(2550) = X(355)-4*X(6666) = X(390)+5*X(631) = X(944)+5*X(18230) = 5*X(1001)+X(11495) = 4*X(1125)-X(5805) = 3*X(3653)-X(38030) = 3*X(5054)-X(38121) = 3*X(10165)-X(38123) = 3*X(38025)-X(38037) = 3*X(38025)-2*X(38043) = 3*X(38059)-X(38158) = 3*X(38067)-X(38126) = 3*X(38067)-2*X(38130) = 3*X(38108)-2*X(38158) = 3*X(38122)-2*X(38123)

X(38031) lies on these lines: {1,15837}, {2,38149}, {3,142}, {7,37737}, {9,1385}, {21,10307}, {30,38025}, {55,31231}, {140,2550}, {165,10156}, {355,6666}, {390,496}, {405,5731}, {511,38048}, {515,16857}, {517,21153}, {518,5050}, {519,38067}, {527,3653}, {528,5054}, {938,3295}, {944,18230}, {952,6883}, {954,999}, {971,3576}, {1319,15298}, {1376,38201}, {1617,17718}, {1621,6244}, {1890,3517}, {2646,15299}, {3243,15178}, {3358,31435}, {3523,35514}, {3526,3826}, {3612,14100}, {3616,5759}, {4297,31672}, {4312,5204}, {4321,5126}, {4421,24386}, {4423,7988}, {4428,10164}, {4512,11227}, {4881,10861}, {5220,37733}, {5251,30283}, {5259,8273}, {5284,7580}, {5587,11108}, {5686,7967}, {5698,31657}, {5705,9709}, {5729,30284}, {5732,13624}, {5762,28466}, {5779,15254}, {5840,38060}, {5850,11194}, {5852,16203}, {5853,26446}, {5856,38032}, {5857,38033}, {6594,12737}, {6600,16202}, {6684,30331}, {6767,28234}, {6796,16863}, {6875,8543}, {6889,9669}, {6913,28160}, {6936,9655}, {6937,7678}, {6940,7676}, {6947,31479}, {6963,7679}, {7675,37606}, {7742,37701}, {7987,11372}, {8171,13405}, {8227,18482}, {8232,18990}, {8581,37618}, {10172,11500}, {10283,22770}, {10384,30282}, {10398,13384}, {10902,16408}, {11230,38150}, {11374,12573}, {11539,38170}, {12114,16866}, {12560,37582}, {15171,37407}, {15185,31837}, {15626,16373}, {15709,38092}, {15726,28444}, {16370,21151}, {16417,38052}, {17768,28443}, {18412,34471}, {19883,38151}, {20116,31806}, {24393,37727}, {25055,38036}, {28186,38139}, {28204,38154}, {28208,38075}, {28534,38065}, {38093,38172}

X(38031) = midpoint of X(i) and X(j) for these {i,j}: {5657, 8236}, {5686, 7967}, {5731, 5817}, {11038, 21168}
X(38031) = reflection of X(i) in X(j) for these (i,j): (38037, 38043), (38053, 38028), (38057, 38113), (38108, 38059), (38122, 10165), (38126, 38130), (38150, 11230), (38154, 38179), (38200, 11231)
X(38031) = complement of X(38149)
X(38031) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (954, 7677, 999), (3616, 5759, 20330), (4423, 15931, 19541), (38025, 38037, 38043), (38067, 38126, 38130), (38131, 38132, 38113)

X(38032) = CENTROID OF TRIANGLE {X(1), X(3), X(11)}

Barycentrics    4*a^7-6*(b+c)*a^6-(7*b^2-22*b*c+7*c^2)*a^5+(b+c)*(13*b^2-24*b*c+13*c^2)*a^4+2*(b^4+c^4-2*b*c*(5*b^2-8*b*c+5*c^2))*a^3-2*(b^2-c^2)*(b-c)*(4*b^2-5*b*c+4*c^2)*a^2+(b^2-c^2)^2*(b-c)^2*a+(b^2-c^2)^3*(b-c) : :
X(38032) = X(1)+2*X(6713) = X(3)+2*X(1387) = X(11)+2*X(1385) = X(104)+5*X(3616) = 5*X(104)+X(9809) = X(104)+2*X(11729) = X(119)-4*X(1125) = X(119)+2*X(11715) = 4*X(140)-X(1145) = 2*X(214)+X(37726) = X(355)-4*X(6667) = 2*X(1125)+X(11715) = 5*X(3616)-2*X(11729) = X(9809)-10*X(11729) = 3*X(23513)-2*X(38161) = 3*X(32557)-X(38161) = 3*X(34122)-2*X(38177) = 3*X(34126)-X(38177) = 3*X(38026)-X(38038) = 3*X(38026)-2*X(38044)

X(38032) lies on these lines: {1,6713}, {2,952}, {3,1387}, {11,1385}, {30,38026}, {100,16202}, {104,3560}, {119,1125}, {140,1145}, {149,6897}, {153,6898}, {214,24299}, {355,6667}, {511,38050}, {515,23513}, {516,38124}, {517,5298}, {518,38119}, {519,38069}, {528,3653}, {551,2800}, {631,1320}, {944,31272}, {971,38060}, {1317,10039}, {1319,6882}, {1388,26492}, {1482,6961}, {1484,10609}, {1537,5901}, {1621,18861}, {2802,10165}, {2829,5886}, {3035,12737}, {3036,37727}, {3576,5840}, {3622,6977}, {3624,12751}, {3636,25485}, {4297,16174}, {4308,11929}, {5533,37525}, {5690,25416}, {5731,32558}, {5762,7677}, {5848,38029}, {5851,38030}, {5854,26446}, {5856,38031}, {5882,6702}, {5887,15528}, {6246,33709}, {6265,20418}, {6699,31523}, {6850,10738}, {6892,13226}, {6893,10742}, {6920,13257}, {6937,12690}, {6940,33814}, {6941,34773}, {6981,18525}, {7491,37618}, {7968,13913}, {7969,13977}, {7987,14217}, {8068,21842}, {9624,34789}, {9840,10035}, {10058,10269}, {10074,22759}, {10090,10267}, {10993,21630}, {11570,13373}, {11713,29008}, {12119,30389}, {12515,37612}, {12735,19914}, {12758,34339}, {13462,37826}, {13607,15863}, {13624,24466}, {13729,22799}, {13902,19081}, {13959,19082}, {15558,37562}, {15808,21635}, {18240,24474}, {18857,30384}, {22938,37437}, {24928,32554}, {26287,37722}, {28186,38141}, {28204,38156}, {28208,38077}, {28444,38053}, {30392,37718}

X(38032) = midpoint of X(3576) and X(16173)
X(38032) = reflection of X(i) in X(j) for these (i,j): (23513, 32557), (34122, 34126), (34123, 38028), (38038, 38044), (38128, 38133), (38156, 38182)
X(38032) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (104, 3616, 11729), (1125, 11715, 119), (10246, 38028, 38033), (12619, 15178, 1317), (19914, 37624, 12735), (38026, 38038, 38044), (38069, 38128, 38133)

X(38033) = CENTROID OF TRIANGLE {X(1), X(3), X(12)}

Barycentrics    4*a^7-6*(b+c)*a^6-(7*b^2-10*b*c+7*c^2)*a^5+(b+c)*(13*b^2-16*b*c+13*c^2)*a^4-2*(-b^4-c^4+4*b*c*(b-c)^2)*a^3-2*(b^2-c^2)*(b-c)*(4*b^2-b*c+4*c^2)*a^2+(b^2-c^2)^2*(b-c)^2*a+(b^2-c^2)^3*(b-c) : :
X(38033) = X(1)+2*X(31659) = X(3)+2*X(37737) = X(12)+2*X(1385) = X(355)-4*X(6668) = 4*X(1125)-X(26470) = 2*X(2646)+X(6842) = 5*X(3616)+X(11491) = 2*X(4999)+X(37733) = 4*X(13624)-X(30264) = 4*X(15178)-X(37734) = 3*X(38027)-X(38039) = 3*X(38027)-2*X(38045) = 3*X(38058)-2*X(38178) = 3*X(38062)-X(38162) = 3*X(38070)-X(38129) = 3*X(38070)-2*X(38134) = 3*X(38109)-2*X(38162) = 3*X(38114)-X(38178)

X(38033) lies on these lines: {1,31659}, {2,952}, {3,3474}, {12,1385}, {30,38027}, {140,4511}, {355,6668}, {511,38051}, {515,38062}, {516,38125}, {517,4995}, {518,38120}, {519,38070}, {529,3653}, {758,10165}, {971,38061}, {997,4999}, {1006,37535}, {1125,6881}, {1482,6954}, {1621,11729}, {1737,15178}, {2646,6842}, {2975,6883}, {3576,28459}, {3616,6911}, {3622,6880}, {3897,10942}, {3925,22935}, {4313,11928}, {5326,12619}, {5444,6713}, {5703,10680}, {5719,22765}, {5761,35252}, {5762,38056}, {5795,24927}, {5840,38063}, {5842,5886}, {5849,38029}, {5852,38030}, {5855,26446}, {5857,38031}, {5901,6905}, {6265,6690}, {6830,34773}, {6841,37837}, {6859,18525}, {6923,37606}, {6947,20060}, {7491,11375}, {8068,37525}, {10164,10273}, {13624,30264}, {14988,37298}, {15950,32613}, {18391,37624}, {26487,34471}, {28186,38142}, {28204,38157}, {28208,38078}

X(38033) = midpoint of X(3576) and X(37701)
X(38033) = reflection of X(i) in X(j) for these (i,j): (38039, 38045), (38058, 38114), (38109, 38062), (38129, 38134), (38157, 38183)
X(38033) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10246, 38028, 38032), (38027, 38039, 38045), (38070, 38129, 38134)

X(38034) = CENTROID OF TRIANGLE {X(1), X(4), X(5)}

Barycentrics    2*(b+c)*a^3+(3*b^2-4*b*c+3*c^2)*a^2-2*(b^2-c^2)*(b-c)*a-3*(b^2-c^2)^2 : :
X(38034) = X(1)+2*X(546) = 5*X(3)-11*X(5550) = 5*X(4)+7*X(3622) = X(4)+2*X(5901) = X(4)+5*X(18493) = 2*X(4)+X(34773) = 5*X(5)-2*X(10) = X(5)+2*X(946) = 7*X(5)+2*X(4301) = 4*X(5)-X(5690) = X(5)-4*X(9955) = 7*X(5)-4*X(9956) = 3*X(5)-2*X(10175) = 11*X(5)-2*X(11362) = 2*X(5)+X(22791) = 19*X(5)-10*X(31399) = 3*X(5)-X(38112) = 7*X(5)-2*X(38127) = 7*X(3622)-10*X(5901) = 7*X(3622)-5*X(10246) = 14*X(3622)-5*X(34773) = 11*X(5550)+5*X(9812) = 2*X(5901)-5*X(18493) = 4*X(5901)-X(34773) = X(10246)-5*X(18493) = 10*X(18493)-X(34773)

X(38034) lies on these lines: {1,546}, {2,28174}, {3,5284}, {4,3622}, {5,10}, {8,3851}, {11,5902}, {30,1699}, {40,3628}, {65,10593}, {140,165}, {145,3855}, {226,5049}, {354,496}, {355,3633}, {376,28182}, {381,952}, {382,3616}, {495,5919}, {497,5719}, {515,3845}, {516,549}, {518,38136}, {519,38071}, {547,7988}, {550,1125}, {551,15687}, {632,3579}, {758,3829}, {944,3843}, {962,1656}, {971,38041}, {1385,3627}, {1387,1478}, {1479,37737}, {1482,3091}, {1483,3858}, {1484,2801}, {1503,38040}, {1519,8727}, {1537,6830}, {1539,11735}, {1698,35018}, {1702,13925}, {1703,13993}, {1836,15325}, {2099,12019}, {2807,5946}, {2829,38044}, {3057,10592}, {3058,37701}, {3086,24470}, {3090,12702}, {3338,11544}, {3485,9669}, {3526,6361}, {3530,3624}, {3544,3617}, {3545,5790}, {3564,38035}, {3582,11246}, {3583,15950}, {3614,5697}, {3649,37720}, {3654,10109}, {3655,14893}, {3656,4677}, {3679,11737}, {3681,24390}, {3754,3847}, {3816,3833}, {3830,5731}, {3832,10595}, {3839,7967}, {3848,12609}, {3853,9624}, {3856,18492}, {3857,19925}, {3859,5881}, {3861,5691}, {3877,17530}, {3947,31792}, {4295,34753}, {4297,31662}, {4536,20117}, {5054,9778}, {5055,5657}, {5067,20070}, {5068,12245}, {5072,5818}, {5073,10248}, {5074,20328}, {5079,9780}, {5131,5433}, {5226,6767}, {5274,15934}, {5434,16173}, {5443,6284}, {5714,7373}, {5734,12645}, {5762,38037}, {5842,38045}, {5843,38036}, {5903,7173}, {5927,10943}, {6583,31803}, {6839,9963}, {6841,7704}, {6859,8166}, {6911,33814}, {6912,22765}, {6914,22753}, {6915,11849}, {6924,11496}, {6946,35000}, {7294,37572}, {7354,37735}, {7377,29572}, {7516,9911}, {7526,11365}, {7547,12135}, {7982,12811}, {7987,12103}, {7991,12812}, {8703,10165}, {9589,16239}, {9612,11373}, {9614,10389}, {10113,11723}, {10164,11539}, {10171,11231}, {10172,28228}, {10273,12672}, {10386,13411}, {10589,36279}, {10739,15735}, {10896,37730}, {11218,34746}, {11220,37447}, {11375,15171}, {11376,18990}, {11545,25415}, {11724,22515}, {11725,22505}, {11729,22938}, {12005,31828}, {12108,35242}, {12701,37692}, {12785,20584}, {13374,24475}, {13624,15704}, {14869,19862}, {15170,17718}, {15178,31673}, {15251,36663}, {15686,28154}, {15703,34632}, {15712,31730}, {15714,34638}, {15726,27869}, {15935,18527}, {16160,33592}, {16191,37714}, {16408,26129}, {17504,19883}, {17579,34123}, {18538,35775}, {18762,35774}, {19710,28158}, {21669,37535}, {23046,28204}, {24220,29349}, {26286,31649}, {28168,33699}, {28172,35404}, {30305,31479}, {30311,38055}, {30957,37365}, {32153,37234}, {34036,37729}, {37610,37691}

X(38034) = midpoint of X(i) and X(j) for these {i,j}: {3, 9812}, {4, 10246}, {165, 12699}, {355, 16200}, {381, 5603}, {946, 3817}, {1699, 5886}, {3656, 5587}, {3830, 5731}, {3845, 10283}, {4301, 38127}, {10273, 12672}, {10739, 15735}, {17502, 22793}, {22791, 38042}, {26446, 31162}
X(38034) = reflection of X(i) in X(j) for these (i,j): (5, 3817), (165, 140), (549, 11230), (550, 17502), (3817, 9955), (4297, 31662), (5587, 5066), (5690, 38042), (8703, 10165), (10246, 5901), (11231, 10171), (17502, 1125), (17504, 19883), (26446, 547), (34773, 10246), (38028, 5886), (38042, 5), (38112, 10175), (38127, 9956), (38138, 38140)
X(38034) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 5901, 34773), (4, 18493, 5901), (5, 946, 22791), (5, 22791, 5690), (5, 38112, 10175), (496, 12047, 6147), (946, 9955, 5), (946, 22835, 7956), (1125, 22793, 550), (1699, 38021, 5886), (5603, 9779, 381), (5886, 38028, 38022), (7988, 26446, 547), (7988, 31162, 26446), (8227, 12699, 140), (10175, 38112, 38042), (12611, 16174, 1484), (13464, 18480, 1483), (17605, 30384, 495), (19862, 31663, 14869), (38038, 38039, 5603), (38071, 38138, 38140), (38141, 38142, 381)

X(38035) = CENTROID OF TRIANGLE {X(1), X(4), X(6)}

Barycentrics    a^6-(b+c)*a^5-(3*b^2-2*b*c+3*c^2)*a^4-2*b*c*(b+c)*a^3+(b^2+4*b*c+c^2)*(b-c)^2*a^2+(b^4-c^4)*(b-c)*a+(b^4-c^4)*(b^2-c^2) : :
X(38035) = X(1)+2*X(5480) = X(4)+2*X(1386) = 4*X(5)-X(3416) = X(6)+2*X(946) = X(40)-4*X(3589) = 2*X(141)-5*X(8227) = 2*X(182)+X(12699) = X(355)-4*X(19130) = 2*X(597)+X(31162) = X(962)+5*X(3618) = 4*X(1125)-X(1350) = X(1351)+5*X(18493) = 3*X(14561)-X(38116) = 3*X(14561)-2*X(38167) = 3*X(38023)-2*X(38029) = 3*X(38023)-4*X(38040) = 3*X(38047)-2*X(38116) = 3*X(38047)-4*X(38167) = 3*X(38072)-X(38144) = 3*X(38072)-2*X(38146)

X(38035) lies on these lines: {1,5480}, {4,1386}, {5,3416}, {6,946}, {30,38023}, {40,3589}, {141,8227}, {182,12699}, {355,19130}, {511,5886}, {516,5085}, {517,14561}, {518,5603}, {519,38072}, {524,38021}, {597,31162}, {611,30384}, {613,12047}, {952,38136}, {962,3618}, {971,38046}, {1125,1350}, {1351,18493}, {1352,9955}, {1385,31670}, {1428,1836}, {1469,11376}, {1503,1699}, {1702,13910}, {1703,13972}, {2330,12701}, {2829,38050}, {3056,11375}, {3086,24471}, {3090,3844}, {3242,13464}, {3545,28538}, {3564,38034}, {3576,29181}, {3653,19924}, {3656,5476}, {3751,11522}, {3817,5847}, {4657,6210}, {5102,34379}, {5138,5805}, {5587,5846}, {5820,22835}, {5842,38051}, {5845,38036}, {5848,38038}, {5849,38039}, {5901,21850}, {8229,17723}, {9053,16200}, {9812,25406}, {9970,12261}, {10165,31884}, {10387,13411}, {10752,32238}, {11496,36741}, {12588,17605}, {13374,24476}, {17301,29057}, {17792,25681}, {18483,36990}, {18583,22791}, {22753,36740}, {28174,38110}, {28194,38118}, {28198,38064}

X(38035) = midpoint of X(i) and X(j) for these {i,j}: {1699, 16475}, {5603, 14853}, {9812, 25406}
X(38035) = reflection of X(i) in X(j) for these (i,j): (5085, 38049), (10516, 3817), (31884, 10165), (38029, 38040), (38047, 14561), (38116, 38167), (38144, 38146)
X(38035) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14561, 38116, 38167), (38029, 38040, 38023), (38072, 38144, 38146), (38116, 38167, 38047), (38147, 38148, 38136)

X(38036) = CENTROID OF TRIANGLE {X(1), X(4), X(7)}

Barycentrics    a^6+(b+c)*a^5-4*(b^2+b*c+c^2)*a^4-2*(b^2-c^2)*(b-c)*a^3+(5*b^2+2*b*c+5*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)^2 : :
X(38036) = X(1)+2*X(5805) = X(1)-4*X(20330) = X(4)+2*X(5542) = 4*X(5)-X(5223) = X(7)+2*X(946) = 2*X(7)+X(11372) = 2*X(9)-5*X(8227) = X(40)-4*X(142) = X(390)-4*X(13464) = 4*X(946)-X(11372) = X(4312)+5*X(11522) = 3*X(5587)-2*X(38154) = X(5805)+2*X(20330) = 2*X(6173)+X(31162) = 3*X(38024)-2*X(38030) = 3*X(38024)-4*X(38041) = 3*X(38052)-2*X(38121) = 3*X(38052)-4*X(38172) = 3*X(38107)-X(38121) = 3*X(38107)-2*X(38172)

X(38036) lies on these lines: {1,5805}, {2,38130}, {4,5542}, {5,5223}, {7,84}, {9,6832}, {11,10398}, {30,38024}, {40,142}, {56,4312}, {165,38122}, {354,971}, {376,516}, {390,13464}, {480,6918}, {496,30330}, {515,11038}, {517,38052}, {518,5587}, {519,38073}, {527,38021}, {952,38137}, {954,22753}, {960,5833}, {990,2191}, {1001,5735}, {1125,5759}, {1156,16174}, {1385,31671}, {1503,38046}, {2346,6796}, {2550,7982}, {2829,4321}, {2886,5785}, {2951,12699}, {3062,5557}, {3243,5881}, {3254,6326}, {3338,3358}, {3545,38158}, {3624,31658}, {3646,5758}, {3817,5817}, {4301,35514}, {5055,38179}, {5219,15298}, {5290,5806}, {5657,38204}, {5686,10175}, {5691,18482}, {5715,5728}, {5732,25557}, {5762,5886}, {5779,9955}, {5804,18492}, {5842,38056}, {5843,38034}, {5845,38035}, {5851,38038}, {5852,38039}, {5853,16200}, {7672,31870}, {7965,30304}, {7988,31142}, {8273,9589}, {8581,9612}, {8727,10980}, {9581,18412}, {9614,14100}, {9779,31164}, {9845,11037}, {10384,30384}, {10392,10591}, {10427,14217}, {10595,30331}, {11544,18222}, {12005,12669}, {13159,21669}, {15251,16469}, {15841,21628}, {16173,24644}, {18483,30340}, {19875,38126}, {20119,25485}, {20195,24468}, {21168,38059}, {25055,38031}, {26446,38171}, {27475,29054}, {28174,38111}, {28194,38123}, {28198,38065}, {30275,31393}

X(38036) = reflection of X(i) in X(j) for these (i,j): (165, 38122), (3576, 38053), (5587, 38150), (5657, 38204), (5686, 10175), (5817, 3817), (21151, 38054), (21168, 38059), (26446, 38171), (38030, 38041), (38052, 38107), (38121, 38172), (38149, 38151)
X(38036) = anticomplement of X(38130)
X(38036) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 946, 11372), (5805, 20330, 1), (12699, 31657, 2951), (38030, 38041, 38024), (38073, 38149, 38151), (38107, 38121, 38172), (38121, 38172, 38052), (38152, 38153, 38137)

X(38037) = CENTROID OF TRIANGLE {X(1), X(4), X(9)}

Barycentrics    a^6-2*(b+c)*a^5-(b^2-6*b*c+c^2)*a^4+4*(b^3+c^3)*a^3-(b-c)^4*a^2-2*(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)*a+(b^2-c^2)^2*(b-c)^2 : :
X(38037) = X(4)+2*X(1001) = 4*X(5)-X(2550) = 3*X(5)-X(38170) = X(9)+2*X(946) = X(40)-4*X(6666) = 2*X(142)-5*X(8227) = 2*X(142)+X(11372) = X(390)+5*X(3091) = 3*X(2550)-4*X(38170) = 3*X(3817)-X(38151) = 3*X(7988)+X(24644) = 3*X(7988)-X(38052) = 5*X(8227)+X(11372) = 3*X(38025)-2*X(38031) = 3*X(38025)-4*X(38043) = 3*X(38057)-2*X(38126) = 3*X(38057)-4*X(38179) = 3*X(38108)-X(38126) = 3*X(38108)-2*X(38179) = 3*X(38150)-2*X(38151)

X(38037) lies on these lines: {1,5809}, {2,165}, {4,1001}, {5,2550}, {7,90}, {9,946}, {20,5259}, {30,38025}, {40,6666}, {142,6847}, {144,10527}, {238,3332}, {390,1479}, {443,7958}, {480,5082}, {497,954}, {499,4312}, {517,38057}, {518,5603}, {519,38075}, {527,38021}, {528,3545}, {631,11495}, {673,36662}, {938,12617}, {952,38139}, {962,6886}, {971,5886}, {990,16020}, {1125,5732}, {1210,12560}, {1385,31672}, {1445,4295}, {1503,38048}, {1709,9776}, {1890,3089}, {2801,11038}, {2829,38060}, {2951,3624}, {3090,3826}, {3243,13464}, {3254,16174}, {3475,5927}, {3485,5728}, {3487,5572}, {3616,36991}, {3855,10786}, {4026,36682}, {4293,6912}, {4294,6835}, {4326,13411}, {4423,7965}, {5055,38121}, {5068,5552}, {5219,10384}, {5223,11522}, {5263,36660}, {5284,10431}, {5542,14986}, {5584,17552}, {5587,5853}, {5686,5692}, {5698,5805}, {5726,19925}, {5735,26363}, {5759,6832}, {5762,38034}, {5777,15185}, {5779,18493}, {5832,22835}, {5840,6826}, {5842,38061}, {5856,38038}, {5857,38039}, {5880,6833}, {5942,10004}, {6594,14217}, {6825,18482}, {6828,7678}, {6843,26333}, {6855,7681}, {6861,31671}, {6864,11496}, {6873,10598}, {6887,12699}, {6888,37524}, {6908,18483}, {6915,7676}, {6926,20195}, {6935,21154}, {6939,7680}, {6945,7679}, {6953,30332}, {6957,10590}, {6989,22793}, {6990,10531}, {7080,7989}, {7402,16593}, {7982,24393}, {8581,11376}, {8727,26105}, {9612,12573}, {9856,28629}, {10157,25568}, {10175,38200}, {10198,12571}, {10442,19863}, {10443,19858}, {10785,16112}, {11025,12528}, {11036,20116}, {11230,38122}, {11375,14100}, {12701,15837}, {12756,34790}, {12848,18393}, {15298,30384}, {15587,25681}, {15726,21151}, {28174,38113}, {28194,38130}, {28198,38067}, {28534,38073}, {31394,36670}, {38093,38123}, {38107,38205}

X(38037) = midpoint of X(i) and X(j) for these {i,j}: {5603, 5817}, {24644, 38052}
X(38037) = reflection of X(i) in X(j) for these (i,j): (21153, 38059), (38031, 38043), (38053, 5886), (38057, 38108), (38122, 11230), (38126, 38179), (38150, 3817), (38154, 38158), (38200, 10175), (38204, 10171)
X(38037) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (499, 4312, 8732), (946, 6846, 19843), (962, 6886, 19855), (3090, 35514, 3826), (5779, 18493, 20330), (7988, 24644, 38052), (8227, 11372, 142), (12047, 15299, 7), (16112, 25557, 36996), (38031, 38043, 38025), (38075, 38154, 38158), (38108, 38126, 38179), (38126, 38179, 38057), (38159, 38160, 38139)

X(38038) = CENTROID OF TRIANGLE {X(1), X(4), X(11)}

Barycentrics    2*(b+c)*a^6+(b^2-10*b*c+c^2)*a^5-(b+c)*(7*b^2-16*b*c+7*c^2)*a^4-2*(b^2-5*b*c+c^2)*(b-c)^2*a^3+2*(b^2-c^2)*(b-c)*(4*b^2-3*b*c+4*c^2)*a^2+(b^2-c^2)^2*(-4*b*c+c^2+b^2)*a-3*(b^2-c^2)^3*(b-c) : :
X(38038) = X(4)+2*X(1387) = 4*X(5)-X(1145) = X(11)+2*X(946) = 2*X(11)+X(1537) = 5*X(11)-2*X(10265) = X(11)-4*X(16174) = X(40)-4*X(6667) = X(80)+5*X(11522) = X(119)-4*X(9955) = 2*X(355)+X(25416) = 4*X(946)-X(1537) = 5*X(946)+X(10265) = X(946)+2*X(16174) = 5*X(1537)+4*X(10265) = X(1537)+8*X(16174) = X(10265)-10*X(16174) = X(12672)+2*X(12736) = 3*X(23513)-X(38128) = 3*X(23513)-2*X(38182) = 3*X(38026)-2*X(38032) = 3*X(38026)-4*X(38044)

X(38038) lies on these lines: {4,1387}, {5,1145}, {11,65}, {30,38026}, {40,6667}, {80,11522}, {100,6918}, {104,5556}, {119,9955}, {149,6835}, {355,25416}, {381,952}, {516,21154}, {517,23513}, {518,38147}, {519,38077}, {528,38021}, {962,31272}, {971,1519}, {1071,18240}, {1125,24466}, {1317,6246}, {1320,3091}, {1482,10598}, {1503,38050}, {1532,22835}, {1699,2829}, {2802,3817}, {3035,8227}, {3036,7982}, {3086,24465}, {3616,10724}, {4301,6702}, {5533,18393}, {5541,20400}, {5587,5854}, {5734,12531}, {5804,12247}, {5840,5886}, {5842,38063}, {5848,38035}, {5851,38036}, {5856,38037}, {5901,22938}, {6265,12690}, {6713,12699}, {6917,10531}, {6971,22791}, {7681,8068}, {7686,12758}, {7687,31523}, {9624,12119}, {9812,32558}, {10058,22753}, {10090,11496}, {10532,10742}, {10595,12735}, {10698,12019}, {10956,17605}, {11570,13374}, {11715,18483}, {11813,38211}, {12611,13257}, {12688,15528}, {12764,26332}, {13273,26333}, {16417,34474}, {18238,33593}, {20330,30311}, {20418,34789}, {21630,37725}, {26726,37714}, {28174,34126}, {28194,38133}, {28198,38069}

X(38038) = midpoint of X(1699) and X(16173)
X(38038) = reflection of X(i) in X(j) for these (i,j): (21154, 32557), (34122, 23513), (34123, 5886), (38032, 38044), (38128, 38182), (38156, 38161)
X(38038) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11, 946, 1537), (946, 16174, 11), (5603, 38034, 38039), (6246, 13464, 1317), (8227, 14217, 3035), (10738, 11729, 10609), (10738, 18493, 11729), (12611, 37726, 13257), (22835, 30384, 1532), (23513, 38128, 38182), (38032, 38044, 38026), (38077, 38156, 38161), (38128, 38182, 34122)

X(38039) = CENTROID OF TRIANGLE {X(1), X(4), X(12)}

Barycentrics    2*(b+c)*a^6+(b^2-6*b*c+c^2)*a^5-(b+c)*(7*b^2-8*b*c+7*c^2)*a^4-2*(b^2-b*c+c^2)*(b-c)^2*a^3+2*(b^2-c^2)*(b-c)*(4*b^2+b*c+4*c^2)*a^2+(b^4-c^4)*(b^2-c^2)*a-3*(b^2-c^2)^3*(b-c) : :
X(38039) = X(4)+2*X(37737) = X(12)+2*X(946) = X(40)-4*X(6668) = 4*X(1125)-X(30264) = X(1537)+2*X(8068) = 2*X(4999)-5*X(8227) = X(6831)+2*X(12047) = 4*X(9955)-X(26470) = 5*X(11522)+X(37710) = X(12699)+2*X(31659) = 4*X(13464)-X(37734) = 3*X(38027)-2*X(38033) = 3*X(38027)-4*X(38045) = 3*X(38058)-2*X(38129) = 3*X(38058)-4*X(38183) = 3*X(38078)-X(38157) = 3*X(38078)-2*X(38162) = 3*X(38109)-X(38129) = 3*X(38109)-2*X(38183)

X(38039) lies on these lines: {4,37737}, {5,3869}, {11,18389}, {12,946}, {30,38027}, {40,6668}, {381,952}, {515,4870}, {516,21155}, {517,17530}, {518,38148}, {519,38078}, {529,38021}, {758,3817}, {971,38056}, {1125,30264}, {1389,18357}, {1482,10599}, {1503,38051}, {1537,7680}, {1699,5842}, {2829,38063}, {2975,6913}, {3090,18231}, {3574,5777}, {3683,31260}, {4999,8227}, {5587,5855}, {5730,6867}, {5849,38035}, {5852,38036}, {5857,38037}, {5886,11113}, {6001,6831}, {6879,36279}, {6907,10129}, {6929,10532}, {6957,20060}, {6980,22791}, {7173,31870}, {7678,20330}, {10175,31165}, {11230,15670}, {11375,37468}, {11491,19541}, {11522,37710}, {11544,26877}, {12608,12671}, {12699,31659}, {13464,37734}, {28174,38114}, {28194,38134}, {28198,38070}

X(38039) = midpoint of X(1699) and X(37701)
X(38039) = reflection of X(i) in X(j) for these (i,j): (21155, 38062), (38033, 38045), (38058, 38109), (38129, 38183), (38157, 38162)
X(38039) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (946, 17605, 1532), (5603, 38034, 38038), (7680, 18393, 1537), (38033, 38045, 38027), (38078, 38157, 38162), (38109, 38129, 38183), (38129, 38183, 38058)

X(38040) = CENTROID OF TRIANGLE {X(1), X(5), X(6)}

Barycentrics    4*a^6-2*(b+c)*a^5-(5*b^2-4*b*c+5*c^2)*a^4-4*b*c*(b+c)*a^3+4*b*c*(-3*b*c+c^2+b^2)*a^2+2*(b^4-c^4)*(b-c)*a+(b^4-c^4)*(b^2-c^2) : :
X(38040) = X(1)+2*X(18583) = X(5)+2*X(1386) = X(6)+2*X(5901) = 2*X(182)+X(22791) = X(962)+5*X(12017) = X(1351)+5*X(3616) = 2*X(1385)+X(21850) = X(1482)+5*X(3618) = X(3416)-4*X(3628) = 4*X(3589)-X(5690) = 2*X(5480)+X(34773) = X(6776)+5*X(18493) = 5*X(8227)-2*X(18358) = 3*X(14561)-X(38144) = 3*X(38023)-X(38029) = 3*X(38023)+X(38035) = 3*X(38049)-X(38118) = 3*X(38079)-X(38165) = 3*X(38079)-2*X(38167) = 3*X(38110)-2*X(38118)

X(38040) lies on these lines: {1,18583}, {5,1386}, {6,5901}, {30,38023}, {182,22791}, {511,38028}, {515,38136}, {516,38164}, {517,38049}, {518,10283}, {519,38079}, {524,38022}, {611,1387}, {613,37737}, {952,14561}, {962,12017}, {1351,3616}, {1385,21850}, {1482,3618}, {1503,38034}, {3416,3628}, {3564,5886}, {3589,5690}, {5050,5603}, {5085,28174}, {5480,34773}, {5762,38048}, {5843,38046}, {5844,38047}, {5845,38041}, {5846,38042}, {5847,11230}, {5848,38044}, {5849,38045}, {6776,18493}, {8227,18358}, {10246,14853}, {15699,28538}, {28204,38146}

X(38040) = midpoint of X(i) and X(j) for these {i,j}: {5050, 5603}, {5886, 16475}, {10246, 14853}, {38029, 38035}
X(38040) = reflection of X(i) in X(j) for these (i,j): (38110, 38049), (38165, 38167)
X(38040) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38023, 38035, 38029), (38079, 38165, 38167), (38168, 38169, 14561)

X(38041) = CENTROID OF TRIANGLE {X(1), X(5), X(7)}

Barycentrics    6*(b+c)*a^5-(9*b^2+4*b*c+9*c^2)*a^4-2*(b+c)*(3*b^2-4*b*c+3*c^2)*a^3+6*(2*b^2+b*c+2*c^2)*(b-c)^2*a^2+8*(b^2-c^2)*(b-c)*b*c*a-3*(b^2-c^2)^2*(b-c)^2 : :
X(38041) = X(5)+2*X(5542) = 3*X(5)-2*X(38158) = X(7)+2*X(5901) = 4*X(142)-X(5690) = 3*X(5542)+X(38158) = 4*X(20330)-X(22791) = X(20330)+2*X(25557) = 2*X(20330)+X(31657) = X(22791)+8*X(25557) = X(22791)+2*X(31657) = 4*X(25557)-X(31657) = 3*X(38022)-2*X(38043) = 3*X(38024)-X(38030) = 3*X(38024)+X(38036) = 3*X(38042)-2*X(38175) = 3*X(38054)-X(38123) = 3*X(38080)-X(38170) = 3*X(38080)-2*X(38172) = 3*X(38111)-2*X(38123) = 3*X(38171)-X(38175)

X(38041) lies on these lines: {5,5542}, {7,5901}, {30,38024}, {142,5690}, {515,38137}, {516,550}, {517,38054}, {518,38042}, {519,38080}, {527,38022}, {952,1056}, {971,38034}, {2801,38139}, {3333,5843}, {3564,38046}, {3628,5223}, {5709,38122}, {5762,28466}, {5763,10165}, {5779,30340}, {5805,34773}, {5844,38052}, {5845,38040}, {5850,11230}, {5851,38044}, {5852,38045}, {8257,15296}, {11539,38130}, {15699,38179}, {18412,38109}, {18493,36996}, {21151,28174}, {28204,38151}, {38093,38126}, {38112,38204}

X(38041) = midpoint of X(i) and X(j) for these {i,j}: {11038, 38107}, {38030, 38036}
X(38041) = reflection of X(i) in X(j) for these (i,j): (38028, 38053), (38042, 38171), (38111, 38054), (38112, 38204), (38170, 38172)
X(38041) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (20330, 25557, 31657), (20330, 31657, 22791), (38024, 38036, 38030), (38055, 38056, 11038), (38080, 38170, 38172), (38173, 38174, 38107)

X(38042) = CENTROID OF TRIANGLE {X(1), X(5), X(8)}

Barycentrics    2*(b+c)*a^3-(3*b^2+4*b*c+3*c^2)*a^2-2*(b^2-c^2)*(b-c)*a+3*(b^2-c^2)^2 : :
X(38042) = X(1)-4*X(3628) = 5*X(2)-X(7967) = X(3)+5*X(5818) = X(3)-7*X(9780) = X(3)+2*X(18357) = X(5)+2*X(10) = 5*X(5)-2*X(946) = 3*X(5)-2*X(3817) = 11*X(5)-2*X(4301) = 2*X(5)+X(5690) = 7*X(5)-4*X(9955) = X(5)-4*X(9956) = 7*X(5)+2*X(11362) = 4*X(5)-X(22791) = X(5)-10*X(31399) = 3*X(5)+2*X(38127) = 5*X(5790)+X(7967) = 3*X(5790)+X(10246) = 2*X(5790)+X(38028) = 5*X(5818)+7*X(9780) = 5*X(5818)-2*X(18357) = 3*X(7967)-5*X(10246) = 2*X(7967)-5*X(38028) = 7*X(9780)+2*X(18357) = 2*X(10246)-3*X(38028)

X(38042) lies on these lines: {1,3628}, {2,952}, {3,5260}, {4,28178}, {5,10}, {8,1656}, {12,5902}, {30,165}, {40,546}, {55,12019}, {65,10592}, {80,5432}, {100,7489}, {104,9342}, {116,20328}, {119,3925}, {140,355}, {145,5067}, {354,495}, {376,28190}, {381,5657}, {388,34753}, {405,32141}, {442,25005}, {474,32153}, {496,5919}, {498,37730}, {515,549}, {516,3845}, {518,38041}, {519,10172}, {528,38082}, {547,3679}, {548,5691}, {550,6684}, {590,35789}, {615,35788}, {631,18525}, {632,1385}, {758,38183}, {944,3526}, {958,6924}, {962,3851}, {1006,18524}, {1125,1483}, {1145,11680}, {1159,5226}, {1210,5049}, {1216,23841}, {1376,6914}, {1387,12647}, {1482,3090}, {1484,3816}, {1512,8727}, {1573,34460}, {1621,12331}, {1699,3654}, {1772,24431}, {1788,9654}, {2550,6929}, {2551,6917}, {2771,15064}, {2800,3968}, {2801,3826}, {2802,3829}, {2807,15060}, {3036,19907}, {3057,10593}, {3058,37718}, {3085,12433}, {3091,12702}, {3241,15703}, {3416,18583}, {3530,18481}, {3545,38066}, {3560,9709}, {3564,38047}, {3579,3627}, {3614,5903}, {3616,5070}, {3624,37727}, {3625,33179}, {3626,24680}, {3655,10124}, {3656,7988}, {3681,6881}, {3697,24474}, {3698,5887}, {3753,10273}, {3754,5694}, {3812,24475}, {3830,9778}, {3833,25466}, {3843,6361}, {3847,3884}, {3848,32213}, {3850,7989}, {3853,9588}, {3855,20070}, {3857,18483}, {3858,22793}, {3859,9589}, {3918,20117}, {3940,6858}, {3947,31794}, {4002,37562}, {4015,31870}, {4026,7611}, {4187,7705}, {4297,15712}, {4413,22758}, {4668,9624}, {4678,7486}, {4680,20575}, {4691,13464}, {4721,13624}, {4745,10171}, {4769,20576}, {5047,37621}, {5054,5731}, {5055,5603}, {5056,12245}, {5071,34718}, {5079,8148}, {5090,21841}, {5131,5445}, {5251,7508}, {5252,15325}, {5261,5708}, {5326,37525}, {5428,6796}, {5433,37710}, {5444,9897}, {5453,37699}, {5499,18242}, {5550,37624}, {5552,6861}, {5686,38107}, {5692,38109}, {5697,7173}, {5704,7373}, {5719,18391}, {5722,10389}, {5762,38057}, {5771,6826}, {5780,6856}, {5791,37281}, {5817,38121}, {5843,38052}, {5846,38040}, {5847,38167}, {5850,38172}, {5853,38043}, {5854,38044}, {5855,38045}, {5881,16239}, {5883,21357}, {5927,6907}, {6668,30147}, {6867,8165}, {6887,7080}, {6911,9708}, {6912,35000}, {6920,11849}, {6923,22799}, {6940,26321}, {6946,22765}, {6951,10742}, {6959,19843}, {6965,10738}, {6980,33108}, {7294,21842}, {7505,12135}, {7516,9798}, {7525,8185}, {7583,13973}, {7584,13911}, {7713,16198}, {7982,12812}, {7987,12108}, {7991,12811}, {8164,15934}, {8193,13861}, {8227,11224}, {8254,12785}, {8703,10164}, {8728,10202}, {8976,19065}, {9581,15172}, {9779,19709}, {9947,13369}, {10165,11539}, {10272,13211}, {10516,38116}, {10573,37737}, {10590,36279}, {10826,15171}, {10827,18990}, {10943,17527}, {11737,31162}, {11801,12778}, {12034,17369}, {12103,35242}, {12106,15177}, {12115,13226}, {12512,31447}, {12610,28633}, {12780,20253}, {12781,20252}, {13145,31803}, {13405,15935}, {13607,19878}, {13883,19116}, {13925,18991}, {13936,19117}, {13951,19066}, {13993,18992}, {14891,34628}, {15048,31398}, {15079,37722}, {15174,37721}, {15178,19862}, {15686,28168}, {15687,28146}, {15694,34627}, {15704,31663}, {16202,16842}, {16203,16862}, {17308,19512}, {17504,28208}, {17531,37535}, {17591,37716}, {17592,37715}, {17714,37557}, {18538,35774}, {18762,35775}, {19710,28172}, {23046,28198}, {26037,37365}, {26285,31649}, {28154,33699}, {28158,35404}, {28194,38071}, {28538,38079}, {28915,36551}, {29679,37360}, {31650,38134}, {32789,35763}, {32790,35762}, {37290,37828}, {38108,38200}, {38111,38204}, {38122,38154}, {38126,38150}

X(38042) = midpoint of X(i) and X(j) for these {i,j}: {2, 5790}, {5, 38112}, {8, 10247}, {10, 10175}, {355, 3576}, {381, 5657}, {549, 38138}, {1699, 3654}, {3545, 38066}, {3655, 37712}, {3679, 5886}, {3817, 38127}, {3830, 9778}, {4745, 10171}, {5054, 38074}, {5587, 26446}, {5686, 38107}, {5690, 38034}, {5817, 38121}, {10165, 38155}, {10202, 18908}, {10516, 38116}, {11230, 38176}, {15699, 38081}, {38108, 38200}, {38122, 38154}, {38126, 38150}, {38171, 38175}
X(38042) = reflection of X(i) in X(j) for these (i,j): (5, 10175), (549, 11231), (1699, 5066), (3576, 140), (3845, 38140), (5690, 38112), (5886, 547), (8703, 10164), (10175, 9956), (10247, 5901), (10283, 11230), (11230, 10172), (11231, 3828), (15699, 38083), (17504, 38068), (22791, 38034), (23046, 38076), (34773, 3576), (38022, 15699), (38028, 2), (38034, 5), (38041, 38171), (38111, 38204), (38112, 10)
X(38042) = complement of X(10246)
X(38042) = X(8)-Beth conjugate of-X(38112)
X(38042) = X(5)-of-cross-triangle-of-Fuhrmann-and-K798i-triangles
X(38042) = center of the Vu pedal-centroidal circle of X(8)
X(38042) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5818, 18357), (5, 10, 5690), (5, 5690, 22791), (8, 1656, 5901), (10, 3817, 38127), (10, 5123, 3820), (10, 9956, 5), (10, 25639, 8256), (10, 31399, 9956), (140, 355, 34773), (355, 1698, 140), (1385, 3634, 632), (3090, 3617, 1482), (5070, 12645, 3616), (5587, 19875, 26446), (5818, 9780, 3), (6684, 18480, 550), (7989, 12699, 3850), (10039, 17606, 496), (10172, 38176, 10283), (10175, 38112, 38034), (10175, 38127, 3817), (11230, 38083, 10172), (18391, 31479, 5719), (18481, 31423, 3530), (31447, 33697, 12512), (34122, 38058, 2), (38177, 38178, 5790)

X(38043) = CENTROID OF TRIANGLE {X(1), X(5), X(9)}

Barycentrics    4*a^6-8*(b+c)*a^5-(b^2-8*b*c+c^2)*a^4+2*(b+c)*(5*b^2-4*b*c+5*c^2)*a^3-2*(2*b^2-b*c+2*c^2)*(b-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 : :
X(38043) = X(5)+2*X(1001) = X(9)+2*X(5901) = X(390)+5*X(1656) = 4*X(1125)-X(31657) = X(1482)+5*X(18230) = X(2550)-4*X(3628) = 7*X(3526)-X(35514) = 5*X(3616)+X(5779) = 3*X(5055)-X(38149) = 3*X(11230)-X(38172) = 3*X(15699)-X(38170) = 3*X(38022)-X(38041) = 3*X(38025)-X(38031) = 3*X(38025)+X(38037) = 3*X(38059)-X(38130) = 3*X(38082)-X(38175) = 3*X(38082)-2*X(38179) = 3*X(38108)-X(38154) = 3*X(38113)-2*X(38130) = 3*X(38171)-2*X(38172)

X(38043) lies on these lines: {2,38121}, {5,1001}, {9,5901}, {30,38025}, {390,1656}, {515,38139}, {516,549}, {517,38059}, {518,10283}, {519,38082}, {527,38022}, {528,15699}, {952,38060}, {971,38028}, {1125,31657}, {1387,15298}, {1482,18230}, {2550,3628}, {3526,35514}, {3564,38048}, {3616,5779}, {4423,37364}, {5055,38149}, {5284,8727}, {5686,10247}, {5690,6666}, {5694,20116}, {5759,18493}, {5762,5886}, {5790,8236}, {5817,10246}, {5843,38053}, {5844,38057}, {5853,38042}, {5856,38044}, {5857,38045}, {6887,15172}, {7489,7677}, {8158,17554}, {8227,15911}, {9956,30331}, {10861,34123}, {11495,15712}, {12560,34753}, {15185,31835}, {15254,20330}, {15299,37737}, {19883,38123}, {21153,28174}, {22791,31658}, {25055,38030}, {28204,38158}, {28534,38080}

X(38043) = midpoint of X(i) and X(j) for these {i,j}: {5686, 10247}, {5790, 8236}, {5817, 10246}, {38031, 38037}
X(38043) = reflection of X(i) in X(j) for these (i,j): (38113, 38059), (38171, 11230), (38175, 38179)
X(38043) = complement of X(38121)
X(38043) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38025, 38037, 38031), (38082, 38175, 38179), (38180, 38181, 38108)

X(38044) = CENTROID OF TRIANGLE {X(1), X(5), X(11)}

Barycentrics    2*a^7-4*(b+c)*a^6-4*(b^2-4*b*c+c^2)*a^5+10*(b^2-c^2)*(b-c)*a^4+(2*b^4+2*c^4-b*c*(17*b^2-28*b*c+17*c^2))*a^3-8*(b^3+c^3)*(b-c)^2*a^2+(b^2-c^2)^2*b*c*a+2*(b^2-c^2)^3*(b-c) : :
X(38044) = X(5)+2*X(1387) = X(11)+2*X(5901) = 2*X(11)+X(19907) = X(104)+5*X(18493) = 4*X(1125)-X(33814) = X(1145)-4*X(3628) = X(1320)+5*X(1656) = X(1484)+2*X(11729) = X(5660)-7*X(5886) = X(5660)+7*X(16173) = 4*X(5901)-X(19907) = X(6265)-7*X(9624) = 5*X(8227)+X(12737) = 3*X(23513)-X(38156) = 3*X(32557)-X(38133) = 3*X(34126)-2*X(38133) = 3*X(38026)-X(38032) = 3*X(38026)+X(38038) = 3*X(38084)-X(38177) = 3*X(38084)-2*X(38182)

This triangle has collinear vertices.

X(38044) lies on these lines: {1,5}, {30,38026}, {104,18493}, {515,38141}, {516,23961}, {517,32557}, {518,38168}, {519,38084}, {528,38022}, {1125,33814}, {1145,3628}, {1320,1656}, {1385,16174}, {1482,31272}, {1621,34474}, {2802,11230}, {2829,38034}, {3564,38050}, {3616,10738}, {3884,11231}, {3898,38114}, {5123,38213}, {5603,32558}, {5690,6667}, {5762,38060}, {5840,38028}, {5843,38055}, {5844,34122}, {5848,38040}, {5851,38041}, {5854,38042}, {5856,38043}, {6246,15178}, {6702,24680}, {6713,22791}, {6959,18220}, {9955,11715}, {10165,26086}, {10595,19914}, {11522,12515}, {12619,13464}, {15863,33179}, {18240,24475}, {20304,31523}, {21154,28174}, {22835,24042}, {23960,28234}, {28204,38161}, {31649,33593}, {33895,38176}

X(38044) = midpoint of X(i) and X(j) for these {i,j}: {5886, 16173}, {38032, 38038}
X(38044) = reflection of X(i) in X(j) for these (i,j): (34126, 32557), (38177, 38182)
X(38044) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11, 5901, 19907), (1385, 16174, 22938), (5886, 10283, 38045), (5886, 11373, 37713), (9955, 11715, 22799), (13464, 33709, 12619), (38026, 38038, 38032), (38063, 38184, 38045), (38084, 38177, 38182)

X(38045) = CENTROID OF TRIANGLE {X(1), X(5), X(12)}

Barycentrics    2*a^7-4*(b+c)*a^6-4*(b-c)^2*a^5+2*(b+c)*(5*b^2-6*b*c+5*c^2)*a^4+(2*b^4+2*c^4-b*c*(7*b^2-12*b*c+7*c^2))*a^3-8*(b^4-c^4)*(b-c)*a^2-(b^2-c^2)^2*b*c*a+2*(b^2-c^2)^3*(b-c) : :
X(38045) = X(5)+2*X(37737) = X(12)+2*X(5901) = X(5690)-4*X(6668) = 2*X(8068)+X(19907) = 5*X(8227)+X(37733) = X(11491)+5*X(18493) = X(22791)+2*X(31659) = 3*X(38027)-X(38033) = 3*X(38027)+X(38039) = 3*X(38062)-X(38134) = 3*X(38085)-X(38178) = 3*X(38085)-2*X(38183) = 3*X(38109)-X(38157) = 3*X(38114)-2*X(38134)

This triangle has collinear vertices.

X(38045) lies on these lines: {1,5}, {30,38027}, {515,38142}, {516,33862}, {517,38062}, {518,38169}, {519,38085}, {529,38022}, {632,28628}, {758,11230}, {3564,38051}, {3754,11231}, {4870,14988}, {5428,10165}, {5690,6668}, {5762,38061}, {5842,38034}, {5843,38056}, {5844,38058}, {5849,38040}, {5852,38041}, {5855,38042}, {5857,38043}, {5883,34126}, {8543,38209}, {11491,18493}, {21155,28174}, {22791,31659}, {26725,31650}, {28204,38162}

X(38045) = midpoint of X(i) and X(j) for these {i,j}: {5886, 37701}, {38033, 38039}
X(38045) = reflection of X(i) in X(j) for these (i,j): (38114, 38062), (38178, 38183)
X(38045) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5886, 10283, 38044), (38027, 38039, 38033), (38063, 38184, 38044), (38085, 38178, 38183)

X(38046) = CENTROID OF TRIANGLE {X(1), X(6), X(7)}

Barycentrics    a^5-4*(b+c)*a^4+(b^2-10*b*c+c^2)*a^3-(b+c)*(b^2+4*b*c+c^2)*a^2+2*(b^2+c^2)*(b-c)^2*a+(b^4-c^4)*(b-c) : :
X(38046) = X(6)+2*X(5542) = X(7)+2*X(1386) = 4*X(142)-X(3416) = 4*X(3589)-X(5223) = 3*X(38023)-2*X(38048) = 3*X(38047)-2*X(38190) = 3*X(38086)-X(38185) = 3*X(38086)-2*X(38187) = 3*X(38186)-X(38190)

X(38046) lies on these lines: {2,210}, {6,4989}, {7,1386}, {142,3416}, {223,4321}, {511,38030}, {515,38143}, {516,17301}, {517,38115}, {519,38086}, {524,38024}, {527,38023}, {952,38164}, {971,38035}, {1503,38036}, {2550,4402}, {2801,38145}, {3243,4966}, {3564,38041}, {3589,5223}, {5762,38029}, {5843,38040}, {5845,16475}, {5846,38052}, {5847,38054}, {5848,38055}, {5849,38056}, {5850,38049}, {5851,38050}, {5852,38051}

X(38046) = reflection of X(i) in X(j) for these (i,j): (38047, 38186), (38185, 38187)
X(38046) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38086, 38185, 38187), (38188, 38189, 38164)

X(38047) = CENTROID OF TRIANGLE {X(1), X(6), X(8)}

Barycentrics    a^3+2*(b+c)*a^2+(b+c)*(b^2+c^2) : :
X(38047) = X(1)-4*X(3589) = X(6)+2*X(10) = 2*X(6)+X(3416) = X(8)+2*X(1386) = X(8)+5*X(3618) = 4*X(10)-X(3416) = X(40)+2*X(5480) = X(69)-4*X(3844) = X(69)+2*X(4663) = X(69)-7*X(9780) = 2*X(125)+X(32278) = 2*X(141)-5*X(1698) = 2*X(141)+X(3751) = 2*X(182)+X(355) = 2*X(1386)-5*X(3618) = 5*X(1698)+X(3751) = 2*X(3844)+X(4663) = 4*X(3844)-7*X(9780) = 2*X(4663)+7*X(9780) = X(38046)+2*X(38190)

X(38047) lies on these lines: {1,3589}, {2,210}, {6,10}, {8,1386}, {9,4026}, {38,29663}, {40,5480}, {42,26061}, {43,32780}, {55,5294}, {65,28739}, {69,3844}, {72,19784}, {81,29679}, {125,32278}, {141,1698}, {145,17358}, {165,29181}, {182,355}, {238,29659}, {344,15569}, {387,3714}, {405,12329}, {474,22769}, {495,16799}, {511,26446}, {515,5085}, {516,36721}, {517,14561}, {519,38023}, {524,19875}, {528,38088}, {583,37148}, {584,33745}, {594,16972}, {597,3679}, {599,3828}, {607,1861}, {611,1737}, {613,10039}, {698,3097}, {726,17301}, {730,13331}, {740,17281}, {748,29685}, {756,29647}, {758,38198}, {894,4429}, {952,38029}, {958,36741}, {964,19133}, {984,4657}, {993,5096}, {1001,17353}, {1009,15624}, {1125,3242}, {1150,26251}, {1215,3772}, {1220,5135}, {1279,36479}, {1329,27384}, {1350,6684}, {1352,9956}, {1376,36740}, {1428,5252}, {1469,17077}, {1503,5587}, {1691,10791}, {1738,4363}, {1757,4643}, {1788,24471}, {1836,4972}, {1837,2330}, {1974,5090}, {2264,2550}, {2308,33074}, {2325,4356}, {2345,3696}, {2551,5800}, {2802,38197}, {2836,25316}, {2948,25328}, {3006,17723}, {3240,32779}, {3333,25914}, {3555,19836}, {3564,38042}, {3579,31670}, {3616,17263}, {3619,19877}, {3634,3763}, {3654,5476}, {3655,10168}, {3666,33163}, {3685,17354}, {3701,18147}, {3703,5256}, {3717,17023}, {3729,28556}, {3745,10327}, {3753,3827}, {3755,5695}, {3758,4645}, {3773,17299}, {3779,26115}, {3790,4360}, {3811,17698}, {3812,24476}, {3821,17276}, {3823,4670}, {3826,10436}, {3836,4675}, {3867,7713}, {3920,30615}, {3923,4085}, {3966,29667}, {4078,16777}, {4202,10404}, {4259,26066}, {4260,5791}, {4265,25440}, {4357,5220}, {4358,29829}, {4437,16831}, {4438,6685}, {4640,26065}, {4641,26034}, {4649,4851}, {4660,4672}, {4684,29596}, {4722,33080}, {4795,31151}, {4850,33170}, {4863,24552}, {4966,17284}, {5026,13178}, {5050,5790}, {5092,18481}, {5123,5820}, {5222,5772}, {5223,17306}, {5263,17368}, {5268,6703}, {5302,13725}, {5657,14853}, {5690,18583}, {5718,29857}, {5722,16792}, {5818,6776}, {5844,38040}, {5845,38052}, {5848,34122}, {5849,38058}, {5850,38187}, {5852,17274}, {5853,38048}, {5854,38050}, {5855,38051}, {5883,34378}, {5902,9021}, {6593,13211}, {6679,29670}, {7174,29598}, {7222,7613}, {7672,28780}, {8818,37159}, {9041,25055}, {9709,37492}, {9791,17336}, {9902,32449}, {10164,31884}, {10175,10516}, {10477,22277}, {11269,30818}, {11375,28741}, {11574,23841}, {12017,18525}, {12586,17619}, {12589,17606}, {12699,19130}, {12782,24256}, {13280,28343}, {13910,18991}, {13972,18992}, {14848,38066}, {15059,32238}, {15069,31399}, {15254,26685}, {15481,17257}, {15988,25005}, {16468,33076}, {16587,23543}, {16706,24349}, {16791,36568}, {16793,37717}, {16796,17721}, {16823,17352}, {16830,17381}, {16973,17398}, {17011,32862}, {17012,33089}, {17017,33162}, {17018,33157}, {17165,32774}, {17278,24325}, {17282,25557}, {17290,24231}, {17321,27549}, {17350,24723}, {17351,24248}, {17359,28581}, {17367,32922}, {17369,36404}, {17382,28582}, {17383,31302}, {17397,32029}, {17526,37080}, {17592,33164}, {17717,29861}, {17719,29856}, {17720,29631}, {17724,29855}, {17725,29859}, {17776,37593}, {17792,26042}, {18183,24046}, {18227,27539}, {18743,29837}, {19561,24809}, {19786,32937}, {19868,24393}, {19876,20582}, {19925,36990}, {24295,32941}, {24331,31289}, {24703,27064}, {24789,29850}, {24892,31264}, {24988,26627}, {25144,26029}, {25329,32261}, {25441,25688}, {25496,29673}, {26037,37676}, {28174,38136}, {28194,38072}, {28204,38064}, {28369,31340}, {28595,32946}, {28606,33166}, {29636,32927}, {29654,32920}, {29821,33169}, {29828,35466}, {29852,32923}, {29867,33127}, {29868,33133}, {30768,30811}, {31017,31098}, {31079,33070}, {31161,33143}, {32772,33117}, {32776,32938}, {32781,32912}, {32913,33174}, {32940,33125}, {33078,37685}, {35026,36231}

X(38047) = midpoint of X(i) and X(j) for these {i,j}: {3679, 16475}, {5050, 5790}, {5085, 38144}, {5657, 14853}, {14561, 38116}, {14848, 38066}, {38049, 38191}, {38110, 38165}, {38186, 38190}
X(38047) = reflection of X(i) in X(j) for these (i,j): (5085, 38118), (10516, 10175), (14561, 38167), (16475, 597), (31884, 10164), (38029, 38110), (38035, 14561), (38046, 38186)
X(38047) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 33159, 17279), (6, 10, 3416), (8, 3618, 1386), (8, 26083, 17289), (42, 26061, 32777), (69, 9780, 3844), (894, 4429, 5880), (984, 29633, 4657), (1215, 25453, 3772), (1698, 3751, 141), (1757, 32784, 4643), (1757, 36478, 32784), (3755, 17355, 5695), (3821, 32935, 17276), (3844, 4663, 69), (4972, 26223, 1836), (29631, 32931, 17720), (29667, 32911, 3966), (38087, 38089, 38023), (38089, 38191, 38049), (38116, 38167, 38035), (38192, 38193, 38165)

X(38048) = CENTROID OF TRIANGLE {X(1), X(6), X(9)}

Barycentrics    a*(3*a^4-3*(b+c)*a^3+(b^2-10*b*c+c^2)*a^2-(b+c)^3*a-4*b*c*(b^2+c^2)) : :
X(38048) = X(6)+2*X(1001) = X(9)+2*X(1386) = X(390)+5*X(3618) = X(2550)-4*X(3589) = X(3416)-4*X(6666) = 3*X(38023)-X(38046) = 3*X(38049)-X(38187) = 3*X(38088)-X(38190) = 3*X(38088)-2*X(38194) = 3*X(38186)-2*X(38187)

This triangle has collinear vertices.

X(38048) lies on these lines: {1,6}, {390,3618}, {511,38031}, {515,38145}, {516,5085}, {517,38117}, {519,38088}, {524,38025}, {527,38023}, {528,38090}, {952,38166}, {971,38029}, {1503,38037}, {2550,3589}, {3416,6666}, {3564,38043}, {5762,38040}, {5845,38053}, {5846,38057}, {5847,29600}, {5848,38060}, {5849,38061}, {5853,38047}, {5856,38050}, {5857,38051}, {20155,27475}, {28534,38086}, {31191,38204}, {32941,38191}, {33682,38054}

X(38048) = reflection of X(i) in X(j) for these (i,j): (38186, 38049), (38190, 38194)
X(38048) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38088, 38190, 38194), (38195, 38196, 38166)

X(38049) = CENTROID OF TRIANGLE {X(1), X(6), X(10)}

Barycentrics    4*a^3+3*(b+c)*a^2+2*(b^2+c^2)*a+(b+c)*(b^2+c^2) : :
X(38049) = X(1)+5*X(3618) = X(6)+2*X(1125) = X(10)+2*X(1386) = X(10)-4*X(3589) = X(69)-7*X(3624) = 2*X(141)-5*X(19862) = 2*X(182)+X(946) = X(193)+11*X(5550) = X(551)+2*X(597) = X(1386)+2*X(3589) = 3*X(38023)+X(38047) = 5*X(38023)+X(38087) = 2*X(38023)+X(38089) = 6*X(38023)+X(38191) = 2*X(38029)+X(38146) = 2*X(38040)+X(38118) = 5*X(38047)-3*X(38087) = 2*X(38047)-3*X(38089) = 2*X(38048)+X(38187) = 2*X(38087)-5*X(38089) = 6*X(38087)-5*X(38191)

X(38049) lies on these lines: {1,344}, {2,5847}, {6,1125}, {10,1386}, {69,3624}, {141,19862}, {142,5138}, {182,946}, {193,5550}, {226,1428}, {238,17023}, {354,34378}, {511,10165}, {515,14561}, {516,5085}, {517,38040}, {518,551}, {519,38023}, {524,19883}, {613,13411}, {726,13331}, {758,38051}, {908,29636}, {952,38167}, {1385,18583}, {1503,3817}, {1699,25406}, {1738,17367}, {2308,29684}, {2321,24295}, {2330,12053}, {2800,38119}, {2801,38195}, {2802,38050}, {3242,3636}, {3313,31757}, {3416,3634}, {3452,29645}, {3564,11230}, {3576,14853}, {3616,3751}, {3619,34595}, {3653,14848}, {3663,4672}, {3742,34381}, {3758,24231}, {3763,19878}, {3821,28508}, {3827,5883}, {3836,4349}, {3879,29637}, {3883,29633}, {3923,3946}, {4104,32911}, {4133,4852}, {4297,5480}, {4353,32935}, {4356,4432}, {4357,16468}, {4416,16477}, {4645,29630}, {4663,6329}, {4676,17380}, {4684,29660}, {4697,24177}, {4966,16666}, {4989,5750}, {5009,17200}, {5026,11599}, {5050,5886}, {5092,31730}, {5135,12609}, {5248,36741}, {5249,29852}, {5294,17017}, {5745,29650}, {5845,38054}, {5848,32557}, {5849,38062}, {5850,38046}, {5882,25555}, {6593,13605}, {6776,8227}, {6789,16503}, {10171,10516}, {10186,21153}, {11720,15118}, {11735,32300}, {12017,12699}, {12571,36990}, {13624,21850}, {15254,17045}, {16469,29598}, {16472,20806}, {17301,28526}, {17304,24695}, {17355,32921}, {17364,26150}, {17382,17768}, {17594,35261}, {17772,29594}, {19130,31673}, {20985,28256}, {23659,28288}, {24597,29826}, {25501,37676}, {25524,37492}, {26723,32772}, {28160,38136}, {28194,38064}, {28204,38079}, {28234,38116}, {28236,38144}, {29571,31289}, {29596,32846}, {29666,37685}, {30768,33070}

X(38049) = midpoint of X(i) and X(j) for these {i,j}: {2, 16475}, {1699, 25406}, {3576, 14853}, {3653, 14848}, {5050, 5886}, {5085, 38035}, {14561, 38029}, {38040, 38110}, {38048, 38186}
X(38049) = reflection of X(i) in X(j) for these (i,j): (10516, 10171), (38118, 38110), (38146, 14561), (38187, 38186), (38191, 38047)
X(38049) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 17353, 4078), (1386, 3589, 10), (4349, 31191, 3836), (4989, 5750, 16825), (16468, 29646, 4357), (38089, 38191, 38047), (38197, 38198, 38167)

X(38050) = CENTROID OF TRIANGLE {X(1), X(6), X(11)}

Barycentrics    4*a^6-2*(b+c)*a^5-5*(b-c)^2*a^4-2*b*c*(b+c)*a^3+4*b*c*(2*b^2-3*b*c+2*c^2)*a^2+2*(b^4-c^4)*(b-c)*a+(b^4-c^4)*(b^2-c^2) : :
X(38050) = X(6)+2*X(1387) = X(11)+2*X(1386) = X(1145)-4*X(3589) = X(1320)+5*X(3618) = X(3416)-4*X(6667) = 5*X(3616)+X(10755) = 2*X(15118)+X(31523) = 4*X(18240)-X(24476) = 3*X(38090)-X(38192) = 3*X(38090)-2*X(38197)

X(38050) lies on these lines: {6,1387}, {11,1386}, {511,38032}, {515,38147}, {516,38188}, {517,38119}, {518,38060}, {519,38090}, {524,38026}, {528,38023}, {952,14561}, {1145,3589}, {1320,3618}, {1503,38038}, {2802,38049}, {2829,38035}, {3416,6667}, {3564,38044}, {3616,10755}, {5840,38029}, {5845,38055}, {5846,34122}, {5847,32557}, {5848,16173}, {5849,38063}, {5851,38046}, {5854,38047}, {5856,38048}, {9024,34123}, {15118,31523}, {18240,24476}

X(38050) = midpoint of X(16173) and X(16475)
X(38050) = reflection of X(38192) in X(38197)
X(38050) = {X(38090), X(38192)}-harmonic conjugate of X(38197)

X(38051) = CENTROID OF TRIANGLE {X(1), X(6), X(12)}

Barycentrics    4*a^6-2*(b+c)*a^5-(5*b^2+2*b*c+5*c^2)*a^4-6*b*c*(b+c)*a^3-12*b^2*c^2*a^2+2*(b^4-c^4)*(b-c)*a+(b^4-c^4)*(b^2-c^2) : :
X(38051) = X(6)+2*X(37737) = X(12)+2*X(1386) = X(3416)-4*X(6668) = 3*X(38091)-X(38193) = 3*X(38091)-2*X(38198)

X(38051) lies on these lines: {6,37737}, {12,1386}, {511,38033}, {515,38148}, {516,38189}, {517,38120}, {518,38061}, {519,38091}, {524,38027}, {529,38023}, {758,38049}, {952,14561}, {1503,38039}, {3416,6668}, {3564,38045}, {5842,38035}, {5845,38056}, {5846,38058}, {5847,38062}, {5848,38063}, {5849,16475}, {5852,38046}, {5855,38047}, {5857,38048}

X(38051) = midpoint of X(16475) and X(37701)
X(38051) = reflection of X(38193) in X(38198)
X(38051) = {X(38091), X(38193)}-harmonic conjugate of X(38198)

X(38052) = CENTROID OF TRIANGLE {X(1), X(7), X(8)}

Barycentrics    a^3+(b^2+6*b*c+c^2)*a-2*(b^2-c^2)*(b-c) : :
X(38052) = X(1)-4*X(142) = X(1)+2*X(2550) = 2*X(4)+X(2951) = 4*X(5)-X(11372) = X(7)+2*X(10) = 2*X(7)+X(5223) = X(8)+2*X(5542) = 2*X(9)-5*X(1698) = X(9)-4*X(3826) = 2*X(9)+X(4312) = X(9)+2*X(5880) = 4*X(10)-X(5223) = 2*X(142)+X(2550) = 5*X(1698)-8*X(3826) = 5*X(1698)+X(4312) = 5*X(1698)+4*X(5880) = X(5290)+2*X(5833) = 3*X(7988)-X(24644) = 3*X(7988)-2*X(38037) = X(38059)-3*X(38204)

X(38052) lies on these lines: {1,142}, {2,165}, {4,2951}, {5,11372}, {7,10}, {8,5542}, {9,46}, {11,10384}, {35,474}, {40,5805}, {55,37271}, {57,3925}, {78,26060}, {80,10427}, {144,9780}, {200,5249}, {210,4654}, {214,20119}, {226,8580}, {238,31183}, {329,30393}, {354,10855}, {355,31657}, {377,5691}, {390,1125}, {405,11495}, {475,1890}, {480,3824}, {515,21151}, {517,38036}, {518,599}, {519,11038}, {527,19875}, {528,15015}, {551,8236}, {612,23681}, {673,25351}, {740,27475}, {758,38208}, {936,12560}, {942,3059}, {946,17582}, {952,38030}, {954,1376}, {971,5587}, {984,4862}, {1056,4915}, {1086,7174}, {1156,6702}, {1159,36922}, {1210,30330}, {1445,3841}, {1706,25466}, {1709,25973}, {1736,24341}, {1737,10398}, {1836,7308}, {2346,8715}, {2801,10861}, {2802,38207}, {2886,5437}, {3008,4307}, {3062,5177}, {3243,3632}, {3254,5541}, {3305,20292}, {3306,5231}, {3333,6067}, {3359,6881}, {3361,8732}, {3419,8255}, {3434,10582}, {3576,38122}, {3579,31671}, {3616,30331}, {3617,17288}, {3626,30340}, {3633,4716}, {3634,18230}, {3635,12630}, {3646,12699}, {3663,7613}, {3696,17296}, {3698,8581}, {3731,24248}, {3740,28609}, {3742,24392}, {3751,4888}, {3754,7672}, {3782,7322}, {3812,5696}, {3819,10439}, {3820,36973}, {3822,7679}, {3823,4363}, {3825,7678}, {3828,6172}, {3833,7671}, {3836,17284}, {3838,30827}, {3848,11235}, {3870,27186}, {3874,34784}, {3884,7673}, {3886,17234}, {3914,17022}, {3929,11246}, {3932,4659}, {3973,24695}, {4197,5735}, {4223,24309}, {4292,5234}, {4321,9623}, {4349,5222}, {4356,5308}, {4384,4645}, {4413,5219}, {4423,9580}, {4429,10436}, {4666,33110}, {4731,11237}, {4847,9776}, {4854,25430}, {4882,21620}, {5226,20103}, {5248,7676}, {5259,16410}, {5263,17282}, {5268,17889}, {5269,24789}, {5272,33109}, {5287,33131}, {5438,28628}, {5439,5572}, {5698,6666}, {5715,5759}, {5750,5819}, {5762,26446}, {5779,9956}, {5784,13750}, {5817,10175}, {5818,36996}, {5832,5856}, {5843,38042}, {5844,38041}, {5845,38047}, {5846,38046}, {5847,16833}, {5851,34122}, {5852,38058}, {5854,38055}, {5855,38056}, {5886,38171}, {6675,35242}, {6701,16133}, {6743,11036}, {6762,9710}, {6824,10270}, {6826,30503}, {6835,12565}, {6857,16192}, {6871,25011}, {6904,7987}, {6989,10268}, {7290,17278}, {7982,20330}, {7991,37436}, {8056,24239}, {8226,10860}, {8227,10310}, {8583,11522}, {9342,30852}, {9441,37075}, {9581,14100}, {9589,17529}, {9814,10590}, {9843,31418}, {9955,16863}, {10004,31994}, {10389,34612}, {10434,16056}, {10442,37153}, {10826,17668}, {10895,31391}, {10940,24982}, {11037,15841}, {11108,18482}, {11526,16236}, {11530,32049}, {11680,31249}, {11684,13159}, {12512,17558}, {12563,20007}, {15185,18398}, {15254,19872}, {15346,17057}, {15601,17337}, {15726,17532}, {15803,19854}, {15931,37270}, {16112,17619}, {16417,38031}, {16475,38186}, {16593,24715}, {16825,28512}, {16830,17304}, {16831,20533}, {16845,31730}, {16853,22793}, {16857,28146}, {16862,25522}, {17064,17122}, {17252,20059}, {17294,27478}, {17313,28581}, {17502,19706}, {17559,18483}, {17567,34595}, {17605,20196}, {18141,35613}, {18421,30275}, {18492,31672}, {18788,37097}, {19862,30332}, {19925,36991}, {20116,30628}, {21168,38130}, {21949,37674}, {23511,26098}, {24280,25101}, {25509,32942}, {28174,38137}, {28194,38073}, {28204,38065}, {28538,38086}, {30147,30284}, {30350,36845}, {31231,31245}, {31423,31658}, {37701,38206}

X(38052) = midpoint of X(i) and X(j) for these {i,j}: {7, 5686}, {2550, 38053}, {6173, 38200}, {21151, 38149}, {38054, 38201}, {38107, 38121}, {38111, 38170}
X(38052) = reflection of X(i) in X(j) for these (i,j): (1, 38053), (2, 38204), (1699, 38150), (3576, 38122), (3679, 38200), (5223, 5686), (5686, 10), (5817, 10175), (5886, 38171), (8236, 551), (11038, 38054), (16173, 38205), (16475, 38186), (21151, 38123), (21168, 38130), (24644, 38037), (25055, 38093), (37701, 38206), (38030, 38111), (38036, 38107), (38053, 142), (38107, 38172)
X(38052) = anticomplement of X(38059)
X(38052) = X(8)-Beth conjugate of-X(5686)
X(38052) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 10, 5223), (9, 3826, 1698), (9, 5880, 4312), (142, 2550, 1), (226, 26040, 8580), (1001, 20195, 3624), (1698, 4312, 9), (3008, 4307, 16469), (3306, 33108, 5231), (3755, 4648, 1), (3826, 5880, 9), (4208, 11024, 10), (5177, 8582, 7989), (7988, 24644, 38037), (8732, 12573, 3361), (11038, 38054, 38024), (24154, 24155, 3946), (38092, 38094, 38024), (38094, 38201, 38054), (38121, 38172, 38036), (38202, 38203, 38170)

X(38053) = CENTROID OF TRIANGLE {X(1), X(7), X(9)}

Barycentrics    a^3-3*(b+c)*a^2+(b^2-6*b*c+c^2)*a+(b^2-c^2)*(b-c) : :
X(38053) = X(1)+2*X(142) = 2*X(1)+X(2550) = X(3)+2*X(20330) = X(7)+2*X(1001) = X(7)+5*X(3616) = 2*X(7)+X(5698) = X(7)-4*X(25557) = X(8)-4*X(3826) = X(9)-4*X(1125) = X(9)+2*X(5542) = 4*X(142)-X(2550) = 2*X(1001)-5*X(3616) = 4*X(1001)-X(5698) = X(1001)+2*X(25557) = X(3189)+2*X(6601) = 10*X(3616)-X(5698) = 5*X(3616)+4*X(25557) = X(5686)+3*X(11038) = 2*X(5686)-3*X(38057) = X(5698)+8*X(25557) = 2*X(11038)+X(38057)

X(38053) lies on these lines: {1,142}, {2,210}, {3,20330}, {6,16020}, {7,21}, {8,3826}, {9,1125}, {10,3243}, {11,5809}, {37,4310}, {48,5819}, {55,9776}, {65,8732}, {69,16823}, {105,36740}, {144,15254}, {145,15570}, {214,3254}, {226,4321}, {238,4644}, {329,4423}, {344,24349}, {376,516}, {388,21617}, {390,2646}, {404,2346}, {452,10404}, {474,6600}, {480,27383}, {497,4666}, {499,18412}, {515,38150}, {517,38122}, {519,38093}, {527,17561}, {528,8236}, {553,4512}, {614,5712}, {631,12704}, {673,20131}, {938,25466}, {944,6900}, {946,5732}, {948,1458}, {952,38171}, {954,5856}, {958,11037}, {960,11036}, {962,8273}, {971,5886}, {1058,12609}, {1156,25558}, {1191,28014}, {1279,4307}, {1319,30275}, {1376,10578}, {1385,5805}, {1386,3945}, {1387,10427}, {1420,12573}, {1445,7288}, {1474,31900}, {1621,3474}, {1698,24393}, {1788,7672}, {1997,26103}, {2194,8025}, {2303,22127}, {2345,24325}, {2360,28619}, {2551,21620}, {2801,5817}, {2886,10580}, {2951,11522}, {3059,17609}, {3085,5439}, {3086,5728}, {3242,17245}, {3247,4353}, {3306,5218}, {3338,6857}, {3416,4869}, {3434,27186}, {3486,30284}, {3555,19855}, {3624,5223}, {3636,30331}, {3664,7290}, {3672,15569}, {3683,9965}, {3720,33143}, {3748,17784}, {3751,37650}, {3757,18141}, {3811,17582}, {3816,5226}, {3817,5658}, {3824,31418}, {3828,38210}, {3836,36479}, {3838,5274}, {3870,26040}, {3883,17298}, {3886,24199}, {3889,34784}, {3913,11024}, {3923,7222}, {3925,36845}, {3932,29627}, {4223,22769}, {4298,5436}, {4326,12053}, {4384,4684}, {4413,37703}, {4419,24231}, {4428,9778}, {4441,18157}, {4640,21454}, {4663,37681}, {4667,16469}, {4860,5744}, {4883,24789}, {4989,16667}, {5045,15185}, {5049,34625}, {5084,13407}, {5220,5550}, {5239,30344}, {5240,30345}, {5253,26357}, {5284,5905}, {5302,17554}, {5308,16593}, {5435,6690}, {5437,13405}, {5528,21630}, {5572,5784}, {5657,5883}, {5695,31995}, {5703,25524}, {5716,28082}, {5745,10980}, {5759,11012}, {5761,31658}, {5762,28466}, {5839,16825}, {5843,38043}, {5845,38048}, {5846,17313}, {5851,38060}, {5852,6172}, {5857,38056}, {5901,31657}, {5902,34744}, {6349,23207}, {6764,9710}, {6824,13373}, {6837,12669}, {6846,12675}, {6904,37080}, {6908,13374}, {7174,29571}, {7284,34919}, {7321,24280}, {7671,10861}, {7674,17580}, {7679,10956}, {7965,10430}, {7967,38149}, {8167,18228}, {8232,8581}, {8257,15298}, {8758,26635}, {9624,11372}, {9780,17283}, {9858,20790}, {9955,31672}, {10072,26725}, {10165,21153}, {10171,38158}, {10175,38154}, {10198,30329}, {10202,14647}, {10246,28452}, {10247,38121}, {10283,38111}, {10389,34607}, {10390,11523}, {10394,10586}, {10527,11025}, {10529,30628}, {10588,30318}, {10589,31266}, {10595,35514}, {11019,25525}, {11230,38108}, {11231,38126}, {11269,17450}, {11373,17668}, {11376,14100}, {11520,24564}, {11730,24203}, {12563,15829}, {12630,20057}, {15296,37787}, {15297,29007}, {15485,24695}, {15668,19288}, {16828,22312}, {17140,17776}, {17263,27549}, {17316,32922}, {17559,21077}, {17740,29830}, {18240,33993}, {18450,31019}, {18481,18482}, {19785,29814}, {19822,33173}, {20116,26363}, {20533,29570}, {24177,37553}, {26098,29820}, {26102,33144}, {26228,37633}, {28444,38032}, {28808,30947}, {29851,33163}, {30341,30557}, {30342,30556}, {31479,38211}, {34379,37654}

X(38053) = midpoint of X(i) and X(j) for these {i,j}: {1, 38052}, {2, 11038}, {551, 38054}, {3576, 38036}, {5542, 38059}, {5603, 21151}, {5886, 38030}, {7671, 10861}, {7967, 38149}, {10246, 38107}, {10247, 38121}, {10283, 38111}, {25055, 38024}, {34123, 38055}, {38028, 38041}
X(38053) = reflection of X(i) in X(j) for these (i,j): (9, 38059), (2550, 38052), (6173, 38054), (21153, 10165), (38025, 25055), (38031, 38028), (38037, 5886), (38052, 142), (38057, 2), (38059, 1125), (38108, 11230), (38126, 11231), (38154, 10175), (38158, 10171), (38200, 38204), (38210, 3828)
X(38053) = complement of X(5686)
X(38053) = intersection, other than A,B,C, of conics {{A, B, C, X(8), X(17201)}} and {{A, B, C, X(86), X(6601)}}
X(38053) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 142, 2550), (1, 443, 3189), (1, 4859, 3755), (2, 354, 24477), (2, 3475, 25568), (7, 1001, 5698), (7, 3616, 1001), (226, 10582, 26105), (443, 6601, 2550), (1001, 25557, 7), (1125, 3333, 30478), (1125, 5542, 9), (1279, 4675, 4307), (3243, 20195, 10), (3616, 25557, 5698), (4666, 5249, 497), (8581, 11375, 8232), (24154, 24155, 4648), (27186, 29817, 3434), (38093, 38200, 38204), (38205, 38206, 38171)

X(38054) = CENTROID OF TRIANGLE {X(1), X(7), X(10)}

Barycentrics    5*(b+c)*a^2-2*(b^2-6*b*c+c^2)*a-3*(b^2-c^2)*(b-c) : :
X(38054) = X(7)+2*X(1125) = 2*X(9)-5*X(19862) = X(10)-4*X(142) = 5*X(10)-8*X(3826) = X(10)+2*X(5542) = 7*X(10)-4*X(24393) = X(10)+8*X(25557) = 3*X(10)-2*X(38210) = 5*X(142)-2*X(3826) = 2*X(142)+X(5542) = 7*X(142)-X(24393) = X(142)+2*X(25557) = 6*X(142)-X(38210) = 4*X(3826)+5*X(5542) = 14*X(3826)-5*X(24393) = X(3826)+5*X(25557) = 4*X(3826)-5*X(38204) = 12*X(3826)-5*X(38210) = 7*X(5542)+2*X(24393) = X(5542)-4*X(25557) = 3*X(5542)+X(38210)

X(38054) lies on these lines: {1,7613}, {2,5850}, {7,1125}, {9,5551}, {10,141}, {144,3624}, {238,4896}, {376,516}, {390,3636}, {392,3671}, {515,38030}, {517,38041}, {519,11038}, {527,19883}, {726,27475}, {758,38056}, {946,31657}, {952,38172}, {971,3742}, {1001,5267}, {1086,4356}, {1156,33709}, {2550,3244}, {2800,38124}, {2801,38158}, {2802,38055}, {3059,3881}, {3243,3625}, {3306,15298}, {3616,4312}, {3634,5223}, {3664,16475}, {3683,4114}, {3828,5686}, {3841,6067}, {3947,5439}, {3950,28516}, {3982,4423}, {4297,5805}, {4301,10179}, {4321,30275}, {4349,4675}, {4353,4648}, {4669,38200}, {4847,27186}, {4888,16020}, {5045,15587}, {5249,10861}, {5550,20059}, {5696,11025}, {5762,10165}, {5784,20116}, {5785,10980}, {5817,10171}, {5833,11036}, {5843,11230}, {5845,38049}, {5847,38046}, {5851,32557}, {5852,38062}, {5880,30331}, {7672,33815}, {8227,36996}, {8232,10200}, {8732,10198}, {9776,13405}, {9843,21617}, {10164,38122}, {10175,38171}, {10177,11263}, {10427,21630}, {12571,36991}, {12577,28629}, {12609,21625}, {15673,17768}, {18230,19878}, {20119,33812}, {24231,29571}, {24929,33558}, {28160,38137}, {28194,38065}, {28204,38080}, {28234,38121}, {28236,38149}, {33682,38048}, {38057,38093}

X(38054) = midpoint of X(i) and X(j) for these {i,j}: {5542, 38204}, {6173, 38053}, {11038, 38052}, {21151, 38036}, {38030, 38107}, {38041, 38111}
X(38054) = reflection of X(i) in X(j) for these (i,j): (10, 38204), (551, 38053), (4669, 38200), (5686, 3828), (5817, 10171), (10164, 38122), (10175, 38171), (38123, 38111), (38151, 38107), (38201, 38052), (38204, 142)
X(38054) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (142, 5542, 10), (142, 25557, 5542), (21255, 24325, 10), (38024, 38052, 11038), (38094, 38201, 38052), (38207, 38208, 38172)

X(38055) = CENTROID OF TRIANGLE {X(1), X(7), X(11)}

Barycentrics    2*(b+c)*a^5-(3*b^2+2*b*c+3*c^2)*a^4-2*(b^2-3*b*c+c^2)*(b+c)*a^3+4*(b^2+c^2)*(b-c)^2*a^2+2*(b^2-c^2)*(b-c)*b*c*a-(b^2-c^2)^2*(b-c)^2 : :
X(38055) = X(7)+2*X(1387) = X(11)+2*X(5542) = 4*X(142)-X(1145) = 4*X(1125)-X(6068) = X(1156)+5*X(30340) = X(1537)-4*X(20330) = 2*X(2550)+X(25416) = 2*X(3254)+X(10609) = X(5223)-4*X(6667) = X(5728)-4*X(18240) = X(10427)-4*X(25557) = 3*X(11038)-X(14151) = 2*X(12735)+X(20119) = 3*X(34122)-2*X(38211) = 3*X(38026)-2*X(38060) = 3*X(38095)-X(38202) = 3*X(38095)-2*X(38207) = 3*X(38205)-X(38211)

X(38055) lies on these lines: {1,528}, {3,36976}, {7,104}, {11,118}, {30,18450}, {56,36971}, {80,10390}, {100,9776}, {119,21617}, {142,1145}, {355,30318}, {390,6948}, {474,34894}, {496,7704}, {499,5220}, {515,38152}, {516,1319}, {517,30379}, {518,1737}, {519,38095}, {527,38026}, {651,15251}, {673,36942}, {938,9654}, {952,1056}, {954,5856}, {971,1519}, {1001,22767}, {1020,3333}, {1125,6068}, {1156,3296}, {1420,5735}, {1445,6713}, {1484,11025}, {1617,3474}, {2095,8732}, {2346,33814}, {2550,25416}, {2802,38054}, {2829,4321}, {3035,5437}, {3036,30286}, {3086,5729}, {3582,38102}, {3649,4934}, {3660,37374}, {3813,5696}, {4326,14217}, {5045,37726}, {5049,18801}, {5223,6667}, {5249,12915}, {5265,5763}, {5572,25558}, {5687,11023}, {5690,30312}, {5732,9580}, {5748,31272}, {5762,7677}, {5784,24390}, {5840,7675}, {5843,38044}, {5845,38050}, {5848,38046}, {5850,32557}, {5851,16173}, {5852,38063}, {5854,38052}, {5886,8545}, {5901,8543}, {6174,13405}, {6594,13411}, {7091,11522}, {7679,38171}, {8068,18412}, {8544,12699}, {10580,10707}, {10738,18530}, {10980,11219}, {11508,18223}, {11715,12573}, {12735,20119}, {12832,30329}, {15252,33148}, {15325,37787}, {15726,30384}, {21620,37725}, {21630,24009}, {28174,30295}, {30311,38034}, {30353,31162}, {31434,38093}

X(38055) = reflection of X(i) in X(j) for these (i,j): (34122, 38205), (34123, 38053), (37787, 15325), (38202, 38207)
X(38055) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11038, 30275, 1056), (11038, 38041, 38056), (38095, 38202, 38207)

X(38056) = CENTROID OF TRIANGLE {X(1), X(7), X(12)}

Barycentrics    6*(b+c)*a^5-(9*b^2+2*b*c+9*c^2)*a^4-2*(b+c)*(3*b^2+b*c+3*c^2)*a^3+12*(b^3-c^3)*(b-c)*a^2+10*(b^2-c^2)*(b-c)*b*c*a-3*(b^2-c^2)^2*(b-c)^2 : :
X(38056) = X(7)+2*X(37737) = X(12)+2*X(5542) = X(5223)-4*X(6668) = 3*X(38027)-2*X(38061) = 3*X(38058)-2*X(38212) = 3*X(38096)-X(38203) = 3*X(38096)-2*X(38208) = 3*X(38206)-X(38212)

X(38056) lies on these lines: {7,37737}, {9,583}, {12,5542}, {515,38153}, {516,2646}, {517,38125}, {518,38058}, {519,38096}, {527,38027}, {529,38024}, {758,38054}, {952,1056}, {971,38039}, {2801,38160}, {5223,6668}, {5265,6147}, {5762,38033}, {5842,38036}, {5843,38045}, {5845,38051}, {5849,38046}, {5850,38062}, {5851,38063}, {5855,38052}, {5857,38053}, {7679,38175}, {8236,20330}, {21617,38109}

X(38056) = reflection of X(i) in X(j) for these (i,j): (38058, 38206), (38203, 38208)
X(38056) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11038, 30275, 38149), (11038, 38041, 38055), (38096, 38203, 38208)

X(38057) = CENTROID OF TRIANGLE {X(1), X(8), X(9)}

Barycentrics    a^3+(b+c)*a^2-3*(b+c)^2*a+(b^2-c^2)*(b-c) : :
X(38057) = X(1)-4*X(6666) = X(1)+2*X(24393) = X(7)-4*X(3826) = X(7)+2*X(5220) = X(7)-7*X(9780) = X(9)+2*X(10) = 2*X(9)+X(2550) = 4*X(9)-X(5698) = 3*X(9)+2*X(38201) = 4*X(10)-X(2550) = 8*X(10)+X(5698) = 3*X(10)-X(38201) = 2*X(2550)+X(5698) = 3*X(2550)-4*X(38201) = 3*X(5686)+X(11038) = 2*X(5686)+X(38053) = X(5698)+4*X(38200) = 3*X(5698)+8*X(38201) = X(5759)+5*X(5818) = 2*X(6666)+X(24393) = 2*X(11038)-3*X(38053) = 3*X(38200)-2*X(38201)

X(38057) lies on these lines: {1,4878}, {2,210}, {4,9}, {7,12}, {8,344}, {20,5302}, {55,5809}, {63,26040}, {65,8232}, {72,19855}, {75,27549}, {80,6594}, {142,1698}, {144,5880}, {165,5325}, {329,3715}, {346,3696}, {355,31658}, {374,38145}, {388,1445}, {390,1837}, {391,3416}, {405,3189}, {497,3305}, {498,18412}, {515,21153}, {517,38037}, {519,38025}, {527,19875}, {528,38099}, {631,17857}, {756,33128}, {758,38217}, {936,10165}, {952,6883}, {954,18391}, {958,5731}, {960,6886}, {962,9710}, {971,14647}, {984,4000}, {1125,3243}, {1212,11200}, {1265,16824}, {1329,6991}, {1376,5273}, {1386,37681}, {1656,20330}, {1737,15298}, {1738,4419}, {1757,4644}, {1898,12706}, {2095,3820}, {2801,21151}, {2802,38216}, {2886,9779}, {2951,9588}, {3008,7174}, {3059,3983}, {3085,3697}, {3161,5695}, {3214,4343}, {3219,3474}, {3242,16020}, {3254,6702}, {3358,12667}, {3421,8257}, {3434,27065}, {3452,7988}, {3476,7677}, {3485,3876}, {3486,5260}, {3487,3678}, {3579,31672}, {3616,17352}, {3618,16830}, {3622,15570}, {3626,30331}, {3634,5542}, {3679,5853}, {3683,17784}, {3717,4384}, {3731,3755}, {3748,20015}, {3751,4648}, {3753,34744}, {3811,16845}, {3823,4643}, {3828,5850}, {3841,5714}, {3842,3949}, {3844,5232}, {3886,25101}, {3911,4321}, {3921,15733}, {3945,4663}, {3974,5271}, {4015,10198}, {4026,5296}, {4042,34255}, {4078,17314}, {4223,12329}, {4310,17278}, {4326,10392}, {4349,16670}, {4356,16676}, {4361,28472}, {4383,17726}, {4413,5744}, {4423,36845}, {4429,17257}, {4437,20156}, {4512,34607}, {4651,17776}, {4662,5572}, {4682,37666}, {4748,32784}, {4847,7308}, {4848,12560}, {4966,29627}, {5044,5761}, {5084,6601}, {5123,5832}, {5129,7674}, {5218,7675}, {5231,5316}, {5233,30741}, {5235,7474}, {5263,26685}, {5268,37642}, {5278,10327}, {5297,24597}, {5436,6743}, {5603,10176}, {5646,37660}, {5658,15064}, {5705,10172}, {5732,6684}, {5735,31399}, {5745,8580}, {5748,31245}, {5762,38042}, {5770,5791}, {5779,37401}, {5784,12669}, {5785,38123}, {5790,28459}, {5795,37712}, {5805,9956}, {5815,25466}, {5839,17772}, {5843,37438}, {5844,38043}, {5845,17251}, {5846,38048}, {5847,37654}, {5854,38060}, {5855,38061}, {5856,34122}, {5857,38058}, {5901,31494}, {6172,6175}, {7222,32935}, {7613,17276}, {8167,10580}, {8581,8732}, {9330,33139}, {9578,12573}, {9623,28234}, {10039,15299}, {10175,38150}, {10177,34619}, {10398,31434}, {10404,37436}, {10527,28748}, {10528,30628}, {10588,21617}, {10916,17559}, {11201,15853}, {11495,18253}, {14555,29641}, {14872,37407}, {15185,34790}, {15587,18231}, {16593,29611}, {17307,19877}, {17316,20154}, {17335,32850}, {17336,24280}, {19822,33166}, {19854,22375}, {20533,29593}, {21060,25525}, {21677,23904}, {22793,31420}, {24389,31435}, {24953,27383}, {25917,28778}, {26037,33163}, {28174,38139}, {28194,38075}, {28204,38067}, {28534,38092}, {28538,38088}, {30416,30417}, {31018,33108}, {38054,38093}

X(38057) = midpoint of X(i) and X(j) for these {i,j}: {2, 5686}, {8, 8236}, {9, 38200}, {5657, 5817}, {21153, 38154}, {21168, 38149}, {38059, 38210}, {38108, 38126}, {38113, 38175}
X(38057) = reflection of X(i) in X(j) for these (i,j): (2550, 38200), (6173, 38204), (8236, 1001), (21153, 38130), (38031, 38113), (38037, 38108), (38053, 2), (38108, 38179), (38122, 11231), (38150, 10175), (38200, 10), (38204, 3828)
X(38057) = complement of X(11038)
X(38057) = barycentric product X(10)*X(16053)
X(38057) = trilinear product X(37)*X(16053)
X(38057) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(16053)}} and {{A, B, C, X(7), X(1839)}}
X(38057) = X(8)-Beth conjugate of-X(38200)
X(38057) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 210, 25568), (2, 3681, 3475), (7, 9780, 3826), (8, 18230, 1001), (9, 10, 2550), (9, 2550, 5698), (10, 18249, 1706), (72, 19855, 28629), (1698, 5223, 142), (1837, 15837, 390), (2346, 5047, 1001), (3305, 25006, 497), (3715, 3925, 329), (3826, 5220, 7), (5880, 15481, 144), (6666, 24393, 1), (38097, 38101, 38025), (38101, 38210, 38059), (38126, 38179, 38037), (38211, 38212, 38175)

X(38058) = CENTROID OF TRIANGLE {X(1), X(8), X(12)}

Barycentrics    (b+c)*(2*a^3-3*(b+c)*a^2-2*(b^2-b*c+c^2)*a+3*(b^2-c^2)*(b-c)) : :
X(38058) = X(1)-4*X(6668) = X(8)+2*X(37737) = 2*X(10)+X(12) = X(355)+2*X(31659) = X(2975)-7*X(9780) = 5*X(5818)+X(11491) = 3*X(38027)-4*X(38062) = X(38027)+2*X(38100) = X(38027)-4*X(38105) = 3*X(38027)+4*X(38214) = X(38033)+2*X(38178) = X(38039)+2*X(38129) = X(38039)-4*X(38183) = X(38056)+2*X(38212) = 2*X(38062)+3*X(38100) = X(38062)-3*X(38105) = X(38100)+2*X(38105) = 3*X(38100)-2*X(38214) = X(38129)+2*X(38183) = 2*X(38134)+X(38157) = X(38203)+2*X(38217)

X(38058) lies on these lines: {1,6668}, {2,952}, {5,3877}, {8,31479}, {10,12}, {11,3898}, {21,18357}, {55,12690}, {80,5426}, {140,4881}, {214,5326}, {355,7483}, {392,10175}, {405,5818}, {443,20060}, {474,2975}, {495,3873}, {515,21155}, {516,38160}, {517,17530}, {518,38056}, {519,38027}, {528,38103}, {529,19875}, {860,7140}, {1125,37734}, {1145,2886}, {1482,6933}, {1512,8226}, {1537,6980}, {1621,12019}, {1698,4999}, {1737,3742}, {1772,24434}, {1962,37715}, {2476,5690}, {2802,38219}, {3158,3419}, {3214,31880}, {3434,12732}, {3614,3878}, {3617,3940}, {3634,17614}, {3679,5855}, {3826,38211}, {3828,31157}, {3869,10592}, {3880,10039}, {3884,7173}, {3890,10593}, {3892,15888}, {3894,37719}, {3897,37705}, {3899,7951}, {3943,21943}, {4187,9956}, {4512,5587}, {5141,22791}, {5260,37308}, {5432,10609}, {5657,17532}, {5724,17734}, {5730,10588}, {5844,38045}, {5846,38051}, {5847,38198}, {5849,38047}, {5850,38208}, {5852,38052}, {5853,38061}, {5854,38063}, {5857,38057}, {5901,7504}, {6684,30264}, {6871,12702}, {6910,18525}, {7705,17527}, {8728,25005}, {10827,26066}, {10959,17606}, {11112,21165}, {12647,25416}, {13373,17529}, {17549,28186}, {17575,17619}, {17577,28174}, {17665,34501}, {18395,25466}, {19860,37733}, {21674,21682}, {24386,31397}, {25522,30315}, {28190,37299}, {28194,38078}, {28204,38070}, {28538,38091}

X(38058) = midpoint of X(i) and X(j) for these {i,j}: {3679, 37701}, {21155, 38157}, {38062, 38214}, {38109, 38129}, {38114, 38178}, {38206, 38212}
X(38058) = reflection of X(i) in X(j) for these (i,j): (21155, 38134), (38033, 38114), (38039, 38109), (38056, 38206), (38109, 38183)
X(38058) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 38042, 34122), (1698, 37710, 4999), (9956, 24987, 4187), (38100, 38105, 38027), (38105, 38214, 38062), (38129, 38183, 38039)

X(38059) = CENTROID OF TRIANGLE {X(1), X(9), X(10)}

Barycentrics    4*a^3-3*(b+c)*a^2-2*(b^2+6*b*c+c^2)*a+(b^2-c^2)*(b-c) : :
X(38059) = X(1)+5*X(18230) = X(7)-7*X(3624) = X(9)+2*X(1125) = 2*X(9)+X(5542) = X(10)+2*X(1001) = X(10)-4*X(6666) = 2*X(10)+X(30331) = X(72)+2*X(20116) = X(142)+2*X(15254) = 2*X(142)-5*X(19862) = 4*X(142)-X(30424) = X(1001)+2*X(6666) = 4*X(1001)-X(30331) = 4*X(1125)-X(5542) = X(5686)-5*X(18230) = 8*X(6666)+X(30331) = 4*X(15254)+5*X(19862) = 8*X(15254)+X(30424) = 10*X(19862)-X(30424) = 2*X(38052)-3*X(38204)

X(38059) lies on these lines: {1,4924}, {2,165}, {7,3624}, {9,1125}, {10,1001}, {72,20116}, {142,3647}, {144,5550}, {238,4349}, {390,1698}, {405,4297}, {451,1890}, {515,16857}, {517,38043}, {518,551}, {519,38025}, {527,19883}, {528,38104}, {631,11372}, {758,38061}, {846,24175}, {946,31658}, {952,38179}, {954,4423}, {960,30329}, {971,10165}, {1210,25542}, {1445,3671}, {2550,3634}, {2646,10392}, {2800,38131}, {2801,34123}, {2802,38060}, {2951,3523}, {3008,4356}, {3216,4343}, {3243,3636}, {3244,4974}, {3254,33709}, {3305,21060}, {3576,5817}, {3616,5223}, {3678,15185}, {3679,8236}, {3707,4966}, {3731,4353}, {3742,5325}, {3755,17337}, {3826,3847}, {3828,38200}, {3883,17263}, {3947,12573}, {3950,16825}, {4078,17769}, {4098,32921}, {4298,8232}, {4301,17552}, {4312,34595}, {4314,5259}, {4315,5251}, {4648,15601}, {4656,33143}, {4668,12630}, {4684,17335}, {4847,5284}, {4989,16777}, {5044,5572}, {5129,19925}, {5218,10384}, {5220,15808}, {5234,12577}, {5308,16469}, {5436,12447}, {5698,19878}, {5703,30330}, {5728,25917}, {5732,17558}, {5745,8167}, {5759,8227}, {5762,11230}, {5847,29600}, {5856,32557}, {5857,38062}, {5904,11025}, {6594,21630}, {6684,16853}, {6688,10440}, {7308,13405}, {7987,36991}, {8582,17536}, {10443,16844}, {10578,30393}, {11038,25055}, {11357,38145}, {11495,16408}, {12053,15837}, {12512,17582}, {12575,19855}, {13411,15299}, {15828,32935}, {16823,25101}, {16849,21629}, {16859,24564}, {16866,31672}, {17194,25889}, {17542,38155}, {17570,24987}, {17588,25881}, {19227,31423}, {19872,30332}, {21168,38036}, {28160,38139}, {28194,38067}, {28204,38082}, {28234,38126}, {28236,38154}, {28534,38094}, {31191,31289}

X(38059) = midpoint of X(i) and X(j) for these {i,j}: {1, 5686}, {9, 38053}, {3576, 5817}, {3679, 8236}, {21153, 38037}, {21168, 38036}, {38031, 38108}, {38043, 38113}
X(38059) = reflection of X(i) in X(j) for these (i,j): (5542, 38053), (38053, 1125), (38130, 38113), (38150, 10171), (38158, 38108), (38200, 3828), (38204, 2), (38210, 38057)
X(38059) = complement of X(38052)
X(38059) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9, 1125, 5542), (10, 1001, 30331), (238, 29571, 4349), (1001, 6666, 10), (3731, 16020, 4353), (15254, 19862, 30424), (38101, 38210, 38057), (38216, 38217, 38179)

X(38060) = CENTROID OF TRIANGLE {X(1), X(9), X(11)}

Barycentrics    4*a^6-8*(b+c)*a^5-(b^2-14*b*c+c^2)*a^4+2*(b+c)*(5*b^2-8*b*c+5*c^2)*a^3-4*(b^4+c^4-3*b*c*(b-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 : :
X(38060) = X(9)+2*X(1387) = X(11)+2*X(1001) = X(390)+5*X(31272) = 4*X(1125)-X(10427) = X(1145)-4*X(6666) = X(1156)+5*X(3616) = X(1320)+5*X(18230) = X(2550)-4*X(6667) = X(6068)-4*X(15254) = 2*X(6702)+X(30331) = 5*X(11025)+X(12532) = X(15185)+2*X(18254) = 2*X(24393)+X(25416) = 3*X(32557)-X(38207) = 3*X(38026)-X(38055) = 3*X(38102)-X(38211) = 3*X(38102)-2*X(38216) = 3*X(38205)-2*X(38207)

X(38060) lies on these lines: {2,11}, {9,1387}, {515,38159}, {516,21154}, {517,38131}, {518,38050}, {519,38102}, {527,38026}, {551,2801}, {952,38043}, {971,38032}, {1125,10427}, {1145,6666}, {1156,3616}, {1320,18230}, {1532,38077}, {2802,38059}, {2829,38037}, {5298,28534}, {5762,38044}, {5840,38031}, {5848,38048}, {5851,38053}, {5853,34122}, {5854,38057}, {5856,16173}, {5857,38063}, {5886,6173}, {6068,15254}, {6702,30331}, {6735,38099}, {8545,15950}, {11025,12532}, {15185,18254}, {24393,25416}

X(38060) = reflection of X(i) in X(j) for these (i,j): (38205, 32557), (38211, 38216)
X(38060) = {X(38102), X(38211)}-harmonic conjugate of X(38216)

X(38061) = CENTROID OF TRIANGLE {X(1), X(9), X(12)}

Barycentrics    4*a^6-8*(b+c)*a^5-(b-c)^2*a^4+10*(b+c)*(b^2+c^2)*a^3-4*(b^4+c^4-2*b*c*(b^2+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 : :
X(38061) = X(9)+2*X(37737) = X(12)+2*X(1001) = X(2550)-4*X(6668) = 3*X(38027)-X(38056) = 3*X(38062)-X(38208) = 3*X(38103)-X(38212) = 3*X(38103)-2*X(38217) = 3*X(38206)-2*X(38208)

X(38061) lies on these lines: {9,37737}, {12,1001}, {515,38160}, {516,21155}, {517,38132}, {518,38051}, {519,38103}, {527,38027}, {528,38106}, {529,38025}, {758,38059}, {952,38043}, {971,38033}, {2550,6668}, {5762,38045}, {5842,38037}, {5849,38048}, {5852,6172}, {5853,38058}, {5855,38057}, {5856,38063}, {5857,37701}, {15296,15950}, {28534,38096}

X(38061) = reflection of X(i) in X(j) for these (i,j): (38206, 38062), (38212, 38217)
X(38061) = {X(38103), X(38212)}-harmonic conjugate of X(38217)

X(38062) = CENTROID OF TRIANGLE {X(1), X(10), X(12)}

Barycentrics    2*a^4-(b+c)*a^3-2*(2*b^2+b*c+2*c^2)*a^2+(b+c)*(b^2-3*b*c+c^2)*a+2*(b^2-c^2)^2 : :
X(38062) = X(10)-4*X(6668) = X(10)+2*X(37737) = X(12)+2*X(1125) = X(214)+2*X(8068) = X(946)+2*X(31659) = 5*X(1656)+X(37733) = X(2975)-7*X(3624) = 2*X(6668)+X(37737) = 3*X(38027)+X(38058) = 5*X(38027)+X(38100) = 2*X(38027)+X(38105) = 6*X(38027)+X(38214) = 2*X(38033)+X(38162) = 2*X(38045)+X(38134) = 5*X(38058)-3*X(38100) = 2*X(38058)-3*X(38105) = 2*X(38061)+X(38208) = 2*X(38100)-5*X(38105) = 6*X(38100)-5*X(38214) = 3*X(38105)-X(38214)

X(38062) lies on these lines: {1,7504}, {2,758}, {5,35016}, {10,5855}, {12,1125}, {65,20104}, {79,37291}, {140,11263}, {214,3822}, {226,4973}, {404,6701}, {515,38033}, {516,21155}, {517,38045}, {518,38198}, {519,38027}, {529,19883}, {547,551}, {946,31659}, {993,5219}, {1656,30143}, {2800,38135}, {2801,38218}, {2802,3584}, {2975,3624}, {3216,31880}, {3305,6763}, {3452,4999}, {3616,37710}, {3628,11281}, {3636,37734}, {3647,7483}, {3817,5842}, {3841,27385}, {3874,11374}, {3878,11375}, {3881,37731}, {3884,5443}, {3892,17718}, {3898,5886}, {3918,27529}, {3919,11231}, {5010,10129}, {5141,37571}, {5248,37692}, {5249,6681}, {5426,37375}, {5439,20107}, {5550,20060}, {5847,38051}, {5849,38049}, {5850,38056}, {5852,38054}, {5857,38059}, {6175,15015}, {6690,11813}, {6862,31803}, {6888,31871}, {8227,11491}, {13411,25639}, {15064,37713}, {19878,31260}, {24160,33147}, {24950,25645}, {25568,26363}, {28160,38142}, {28194,38070}, {28204,38085}, {28234,38129}, {28236,38157}, {31262,34772}

X(38062) = midpoint of X(i) and X(j) for these {i,j}: {2, 37701}, {21155, 38039}, {38033, 38109}, {38045, 38114}, {38061, 38206}
X(38062) = reflection of X(i) in X(j) for these (i,j): (38134, 38114), (38162, 38109), (38208, 38206), (38214, 38058)
X(38062) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 26725, 3833), (551, 11230, 32557), (5886, 10197, 3898), (6668, 37737, 10), (38105, 38214, 38058)

X(38063) = CENTROID OF TRIANGLE {X(1), X(11), X(12)}

Barycentrics    a^7-2*(b+c)*a^6-2*(b^2-3*b*c+c^2)*a^5+(b+c)*(5*b^2-8*b*c+5*c^2)*a^4+(b^2-5*b*c+c^2)*(b^2-b*c+c^2)*a^3-2*(b^2-c^2)*(b-c)*(2*b^2-b*c+2*c^2)*a^2+(b^2-c^2)^3*(b-c) : :
X(38063) = X(1)+2*X(8068) = X(11)+2*X(37737) = X(12)+2*X(1387) = 4*X(1125)-X(4996) = X(1145)-4*X(6668) = 3*X(38106)-X(38215) = 3*X(38106)-2*X(38219)

This triangle has collinear vertices.

X(38063) lies on these lines: {1,5}, {214,6701}, {515,38163}, {516,38209}, {517,38135}, {518,38199}, {519,38106}, {528,38027}, {529,38026}, {758,3582}, {1125,4996}, {1145,6668}, {2771,4870}, {2802,3584}, {2829,38039}, {3336,6713}, {3485,11571}, {4857,16174}, {5131,21154}, {5270,11715}, {5425,20118}, {5441,22938}, {5840,38033}, {5842,38038}, {5848,38051}, {5849,38050}, {5851,38056}, {5852,38055}, {5854,38058}, {5855,34122}, {5856,38061}, {5857,38060}, {10058,18393}, {10072,32558}, {10165,14792}, {10738,37571}, {13273,37525}, {18976,24926}, {26725,34123}, {31659,37563}

X(38063) = midpoint of X(16173) and X(37701)
X(38063) = reflection of X(38215) in X(38219)
X(38063) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38044, 38045, 38184), (38106, 38215, 38219)

X(38064) = CENTROID OF TRIANGLE {X(2), X(3), X(6)}

Barycentrics    7*a^6-7*(b^2+c^2)*a^4-(b^4+18*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2) : :
X(38064) = X(2)+2*X(182) = 4*X(2)-X(1352) = 13*X(2)-X(5921) = 5*X(2)+X(6776) = X(2)-4*X(10168) = 5*X(2)-2*X(11178) = 2*X(2)+X(11179) = 7*X(2)-X(11180) = 7*X(2)-4*X(24206) = 8*X(182)+X(1352) = 10*X(182)-X(6776) = X(182)+2*X(10168) = 5*X(182)+X(11178) = 4*X(182)-X(11179) = 14*X(182)+X(11180) = 7*X(182)+2*X(24206) = 13*X(1352)-4*X(5921) = 5*X(1352)+4*X(6776) = X(1352)-16*X(10168) = 5*X(1352)-8*X(11178) = X(1352)+2*X(11179) = 7*X(1352)-4*X(11180) = 7*X(1352)-16*X(24206)

The locus of the centroid in a Lemoine porism (triangles sharing circumcircle and Lemoine inellipse with ABC) is a circle with center X(38064). (Randy Hutson, May 19, 2020)

X(38064) lies on these lines: {2,98}, {3,597}, {4,20190}, {5,10541}, {6,549}, {20,25555}, {30,5085}, {69,15702}, {140,599}, {141,15694}, {146,25566}, {193,15721}, {373,26255}, {376,3618}, {381,3589}, {511,3524}, {515,38089}, {517,38023}, {518,3653}, {519,38029}, {524,5050}, {527,38115}, {528,38119}, {529,38120}, {575,631}, {576,3523}, {611,5298}, {613,4995}, {632,15069}, {952,38087}, {971,38088}, {1350,12100}, {1351,15693}, {1353,15533}, {1386,3654}, {1428,10056}, {1503,5055}, {2030,7736}, {2330,10072}, {2549,6034}, {2930,11694}, {3066,37904}, {3090,25561}, {3091,25565}, {3098,15053}, {3525,34507}, {3526,8550}, {3530,11477}, {3534,5480}, {3543,19130}, {3545,11645}, {3564,11539}, {3763,10124}, {3818,5071}, {3839,29012}, {5026,11632}, {5032,10519}, {5033,7753}, {5066,36990}, {5067,18553}, {5093,15707}, {5097,15719}, {5486,32154}, {5544,15448}, {5569,22677}, {5652,9175}, {5762,38086}, {5840,38090}, {5845,38065}, {5846,38066}, {5847,38068}, {5848,38069}, {5849,38070}, {5965,33748}, {5969,7618}, {6329,15700}, {6593,20126}, {6699,15303}, {6795,34094}, {7606,7615}, {7622,14645}, {8541,35486}, {8584,15701}, {8703,18583}, {9041,10246}, {9044,32232}, {9971,13363}, {10182,10250}, {10303,11160}, {10304,14853}, {10510,15361}, {10516,15699}, {11163,37450}, {11183,21732}, {11284,35266}, {11812,15534}, {12007,22165}, {13334,32985}, {13335,33215}, {13339,19127}, {14216,14787}, {14810,15698}, {14912,15709}, {15688,29181}, {15697,33751}, {15703,18440}, {15720,20583}, {15723,34573}, {15805,34351}, {15812,19129}, {16962,36758}, {16963,36757}, {17504,31884}, {19145,32788}, {19146,32787}, {21850,34200}, {26446,28538}, {28194,38049}, {28198,38035}, {28204,38047}, {28208,38167}, {28466,36741}, {29317,33750}, {31152,37649}, {33273,35424}

X(38064) = midpoint of X(i) and X(j) for these {i,j}: {3, 14848}, {3545, 25406}, {5032, 10519}, {5050, 5054}, {10304, 14853}, {14912, 21356}
X(38064) = reflection of X(i) in X(j) for these (i,j): (10304, 17508), (10516, 15699), (14848, 597), (20423, 14848), (21358, 11539), (31884, 17504), (38072, 38079)
X(38064) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 182, 11179), (2, 6776, 11178), (2, 11179, 1352), (2, 11180, 24206), (3, 597, 20423), (182, 10168, 2), (182, 22112, 11579), (376, 3618, 5476), (376, 5476, 31670), (3618, 5092, 31670), (5032, 15708, 10519), (5085, 38110, 14561), (5092, 5476, 376), (5642, 22112, 2), (6036, 18800, 19905), (14912, 15709, 21356), (38029, 38118, 38116), (38072, 38079, 14561)

X(38065) = CENTROID OF TRIANGLE {X(2), X(3), X(7)}

Barycentrics    a^6-8*(b+c)*a^5+2*(4*b^2-7*b*c+4*c^2)*a^4+10*(b+c)*(b^2+c^2)*a^3-(13*b^2+4*b*c+13*c^2)*(b-c)^2*a^2-2*(b^2-c^2)^2*(b+c)*a+4*(b^2-c^2)^2*(b-c)^2 : :
X(38065) = 4*X(2)-X(5779) = X(2)+2*X(31657) = 5*X(2)+X(36996) = 7*X(3)+2*X(5735) = X(3)+2*X(6173) = X(7)+2*X(549) = 2*X(9)-5*X(15694) = X(5735)-7*X(6173) = X(5779)+8*X(31657) = 5*X(5779)+4*X(36996) = 3*X(5779)-8*X(38082) = 3*X(21151)+X(38073) = 3*X(21151)+2*X(38080) = 2*X(21151)+X(38107) = X(21151)+2*X(38111) = 7*X(21151)+2*X(38137) = 10*X(31657)-X(36996) = 3*X(31657)+X(38082) = 3*X(36996)+10*X(38082) = 2*X(38073)-3*X(38107) = X(38073)-6*X(38111) = 7*X(38073)-6*X(38137)

X(38065) lies on these lines: {2,5779}, {3,5735}, {7,549}, {9,15694}, {30,21151}, {140,6172}, {142,381}, {144,15702}, {376,31671}, {511,38086}, {515,38094}, {516,3653}, {517,38024}, {518,38066}, {519,38030}, {524,38115}, {527,5054}, {528,10246}, {529,38125}, {952,38092}, {971,5055}, {3524,5762}, {3534,5805}, {3545,38171}, {3654,5542}, {3830,5732}, {5066,36991}, {5759,12100}, {5817,15699}, {5840,38095}, {5843,11539}, {5845,38064}, {5850,38068}, {5851,38069}, {5852,38070}, {6666,15723}, {10124,18230}, {12702,25557}, {14269,38150}, {14848,38186}, {15684,18482}, {15701,31658}, {15703,20195}, {15707,21153}, {15708,21168}, {15709,38113}, {15721,20059}, {15934,30379}, {18541,30275}, {19924,38143}, {28194,38054}, {28198,38036}, {28204,38052}, {28208,38172}, {28534,38031}

X(38065) = reflection of X(i) in X(j) for these (i,j): (3545, 38171), (5054, 38122), (5055, 38093), (5817, 15699), (14269, 38150), (14848, 38186), (38073, 38080)
X(38065) = anticomplement of X(38082)
X(38065) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (21151, 38111, 38107), (38030, 38123, 38121), (38073, 38080, 38107)

X(38066) = CENTROID OF TRIANGLE {X(2), X(3), X(8)}

Barycentrics    a^4+6*(b+c)*a^3-(5*b^2+12*b*c+5*c^2)*a^2-6*(b^2-c^2)*(b-c)*a+4*(b^2-c^2)^2 : :
X(38066) = 2*X(1)-5*X(15694) = 4*X(2)-X(1482) = X(2)+2*X(5690) = 7*X(2)-4*X(5901) = 11*X(2)-5*X(10595) = 5*X(2)+X(12245) = 7*X(2)-X(34631) = 2*X(2)+X(34718) = X(1482)+8*X(5690) = 7*X(1482)-16*X(5901) = 11*X(1482)-20*X(10595) = 5*X(1482)+4*X(12245) = 7*X(1482)-4*X(34631) = X(1482)+2*X(34718) = 3*X(1482)-8*X(38022) = 7*X(5690)+2*X(5901) = 10*X(5690)-X(12245) = 14*X(5690)+X(34631) = 4*X(5690)-X(34718) = 3*X(5690)+X(38022)

X(38066) lies on these lines: {1,15694}, {2,1482}, {3,3679}, {8,549}, {9,35460}, {10,381}, {30,5657}, {40,3830}, {140,3241}, {145,15702}, {165,15689}, {355,3534}, {376,3617}, {511,38087}, {515,15688}, {517,4731}, {518,38065}, {519,3653}, {524,38116}, {527,38121}, {528,38128}, {529,38129}, {547,9780}, {551,3526}, {631,31145}, {944,12100}, {952,3524}, {962,5066}, {971,38097}, {1125,15723}, {1159,31434}, {1385,4677}, {1483,11812}, {1656,3656}, {1698,8148}, {3543,18357}, {3545,38042}, {3576,15707}, {3579,15681}, {3616,10124}, {3621,15721}, {3626,15700}, {3655,4669}, {3839,28174}, {3845,5818}, {3851,7991}, {4301,5079}, {4421,28443}, {4668,13624}, {4678,15692}, {4691,14093}, {4816,32900}, {4995,10573}, {5050,28538}, {5070,7982}, {5071,22791}, {5072,31399}, {5073,37714}, {5076,5493}, {5298,12647}, {5434,37545}, {5554,15670}, {5587,14269}, {5603,15699}, {5687,28466}, {5691,15685}, {5708,10039}, {5731,17504}, {5762,38092}, {5840,38099}, {5844,11539}, {5846,38064}, {5853,38067}, {5854,38069}, {5855,38070}, {6174,19914}, {6361,15687}, {7967,15708}, {8668,37621}, {8703,34627}, {9623,35459}, {9708,28444}, {9779,14892}, {9812,23046}, {9956,19709}, {10056,15934}, {10127,34656}, {10164,15706}, {10247,11231}, {10303,20049}, {10679,16857}, {11237,36279}, {11238,18395}, {11248,28453}, {11849,16418}, {14070,34713}, {14848,38047}, {15177,37922}, {15178,34747}, {15684,18480}, {15695,31663}, {15709,38028}, {15720,34641}, {16417,22765}, {19883,28234}, {19924,38144}, {21161,32141}, {22697,32447}, {25413,31165}, {28212,38071}, {31423,37624}, {33697,35400}, {34006,37557}, {34200,37705}

X(38066) = reflection of X(i) in X(j) for these (i,j): (3545, 38042), (3653, 38068), (5054, 26446), (5055, 19875), (5603, 15699), (5731, 17504), (9812, 23046), (10246, 5054), (10247, 25055), (14269, 5587), (14848, 38047), (15689, 165), (25055, 11231), (38021, 38083), (38074, 38081)
X(38066) = anticomplement of X(38022)
X(38066) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5690, 34718), (2, 34631, 5901), (2, 34718, 1482), (10, 3654, 381), (381, 3654, 12702), (1385, 4677, 34748), (3653, 26446, 38068), (3653, 38068, 5054), (3655, 4669, 12645), (3655, 6684, 15693), (3656, 3828, 1656), (3828, 11362, 3656), (4669, 6684, 3655), (5657, 38112, 5790), (5818, 34632, 3845), (12645, 15693, 3655), (15701, 34748, 1385), (19875, 38021, 38083), (31663, 34628, 15695), (38021, 38083, 5055), (38074, 38081, 5790)

X(38067) = CENTROID OF TRIANGLE {X(2), X(3), X(9)}

Barycentrics    7*a^6-11*(b+c)*a^5-(7*b^2+8*b*c+7*c^2)*a^4+16*(b+c)*(b^2+c^2)*a^3-(b^2-8*b*c+c^2)*(b-c)^2*a^2-5*(b^2-c^2)^2*(b+c)*a+(b^2-c^2)^2*(b-c)^2 : :
X(38067) = 5*X(2)+X(5759) = 4*X(2)-X(5805) = X(2)+2*X(31658) = X(7)-7*X(15702) = X(9)+2*X(549) = 4*X(5759)+5*X(5805) = X(5759)-10*X(31658) = 3*X(5759)+5*X(38073) = X(5805)+8*X(31658) = 3*X(5805)-4*X(38073) = 3*X(21153)+X(38075) = 3*X(21153)+2*X(38082) = 2*X(21153)+X(38108) = X(21153)+2*X(38113) = 7*X(21153)+2*X(38139) = 6*X(31658)+X(38073) = 2*X(38075)-3*X(38108) = X(38075)-6*X(38113) = 7*X(38075)-6*X(38139) = 4*X(38082)-3*X(38108) = X(38082)-3*X(38113) = 7*X(38082)-3*X(38139)

X(38067) lies on these lines: {2,5759}, {7,15702}, {9,549}, {30,21153}, {140,6173}, {142,15694}, {144,15721}, {376,18230}, {381,6666}, {511,38088}, {515,38101}, {516,5055}, {517,38025}, {518,3653}, {519,38031}, {524,38117}, {527,5054}, {528,26446}, {529,38132}, {631,6172}, {632,5735}, {952,38097}, {971,3524}, {1001,3654}, {4312,5326}, {4995,15299}, {5071,18482}, {5298,15298}, {5732,12100}, {5762,11539}, {5779,15693}, {5817,10304}, {5840,38102}, {5853,38066}, {5856,38069}, {5857,38070}, {5880,20104}, {10072,15837}, {10124,20195}, {11812,31657}, {15699,38150}, {15703,31671}, {15708,21151}, {15709,21168}, {19708,36991}, {19924,38145}, {28194,38059}, {28198,38037}, {28204,38057}, {28208,38179}

X(38067) = midpoint of X(5817) and X(10304)
X(38067) = reflection of X(i) in X(j) for these (i,j): (38075, 38082), (38093, 11539), (38122, 5054), (38150, 15699)
X(38067) = complement of X(38073)
X(38067) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (21153, 38113, 38108), (38031, 38130, 38126), (38075, 38082, 38108)

X(38068) = CENTROID OF TRIANGLE {X(2), X(3), X(10)}

Barycentrics    8*a^4+3*(b+c)*a^3-(13*b^2+6*b*c+13*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+5*(b^2-c^2)^2 : :
X(38068) = X(1)-7*X(15702) = 5*X(2)+X(40) = 4*X(2)-X(946) = 13*X(2)-X(962) = X(2)+2*X(6684) = 11*X(2)-5*X(8227) = 7*X(2)-X(31162) = X(2)-7*X(31423) = 11*X(2)+X(34632) = 4*X(40)+5*X(946) = 13*X(40)+5*X(962) = X(40)-10*X(6684) = 7*X(40)+5*X(31162) = 11*X(40)-5*X(34632) = 3*X(40)+5*X(38021) = 13*X(946)-4*X(962) = X(946)+8*X(6684) = 11*X(946)-20*X(8227) = 7*X(946)-4*X(31162) = 11*X(946)+4*X(34632) = 3*X(946)-4*X(38021)

X(38068) lies on these lines: {1,15702}, {2,40}, {3,3828}, {4,19876}, {5,28202}, {8,15721}, {10,549}, {30,10164}, {140,551}, {165,3545}, {355,15693}, {376,1698}, {381,3634}, {498,553}, {511,38089}, {515,3524}, {516,5055}, {517,11539}, {519,3653}, {524,38118}, {527,38123}, {528,38133}, {529,38134}, {547,3579}, {581,36634}, {631,3679}, {632,4301}, {758,38070}, {952,38098}, {971,38101}, {1125,3654}, {1210,4995}, {1385,4669}, {2077,16858}, {2802,38069}, {3091,31425}, {3241,10303}, {3337,3584}, {3525,9588}, {3526,3656}, {3529,30315}, {3533,7991}, {3534,19925}, {3543,19877}, {3576,15708}, {3628,5493}, {3655,4745}, {3817,15699}, {3830,12512}, {3839,28150}, {3845,31663}, {3911,10056}, {4297,12100}, {4677,13607}, {5071,18483}, {5298,31397}, {5306,31396}, {5587,10304}, {5657,11224}, {5690,15713}, {5691,19708}, {5762,38094}, {5790,15707}, {5818,15698}, {5840,38104}, {5844,14890}, {5847,38064}, {5850,38065}, {5901,11540}, {6174,10265}, {6260,13089}, {6361,19872}, {7688,9342}, {7987,15719}, {7988,28232}, {7989,15682}, {8582,15670}, {8703,9956}, {9780,15692}, {10109,22793}, {10124,19862}, {10299,37714}, {10310,17542}, {11001,16192}, {11012,36006}, {12699,15703}, {12702,15723}, {13912,32788}, {13975,32787}, {14869,34641}, {14891,18357}, {14892,28178}, {15683,18492}, {15686,22266}, {15688,28164}, {15691,33697}, {15700,18481}, {15718,18525}, {17502,38155}, {17504,28208}, {17525,17619}, {17781,27529}, {18480,34200}, {19924,38146}, {25440,28466}, {26086,28463}, {28146,38071}, {31188,31393}

X(38068) = midpoint of X(i) and X(j) for these {i,j}: {165, 3545}, {3524, 19875}, {3653, 38066}, {5054, 26446}, {5587, 10304}, {5657, 25055}, {17504, 38042}
X(38068) = reflection of X(i) in X(j) for these (i,j): (3545, 10172), (3817, 15699), (10165, 5054), (19883, 11539), (38076, 38083)
X(38068) = complement of X(38021)
X(38068) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 34632, 8227), (3525, 9588, 13464), (3628, 31447, 5493), (3653, 26446, 38066), (3654, 15694, 1125), (3845, 31663, 34638), (5054, 38066, 3653), (5657, 15709, 25055), (5818, 15698, 34628), (8703, 9956, 34648), (10164, 11231, 10175), (10165, 26446, 38127), (15719, 34627, 7987), (38076, 38083, 10175)

X(38069) = CENTROID OF TRIANGLE {X(2), X(3), X(11)}

Barycentrics    8*a^7-8*(b+c)*a^6+3*(-7*b^2+10*b*c-7*c^2)*a^5+(b+c)*(21*b^2-22*b*c+21*c^2)*a^4+2*(9*b^4+9*c^4-7*b*c*(3*b^2-2*b*c+3*c^2))*a^3-2*(b^2-c^2)*(b-c)*(9*b^2+2*b*c+9*c^2)*a^2-(b^2-c^2)^2*(5*b^2-12*b*c+5*c^2)*a+5*(b^2-c^2)^3*(b-c) : :
X(38069) = 5*X(2)+X(104) = 4*X(2)-X(119) = 13*X(2)-X(153) = X(2)+2*X(6713) = 7*X(2)-X(10711) = X(11)+2*X(549) = 4*X(104)+5*X(119) = 13*X(104)+5*X(153) = X(104)-10*X(6713) = 7*X(104)+5*X(10711) = 13*X(119)-4*X(153) = X(119)+8*X(6713) = 7*X(119)-4*X(10711) = 7*X(153)-13*X(10711) = 14*X(6713)+X(10711) = 2*X(21154)+X(23513) = X(21154)+2*X(34126) = 3*X(21154)+X(38077) = 3*X(21154)+2*X(38084) = 7*X(21154)+2*X(38141)

X(38069) lies on these lines: {2,104}, {11,549}, {30,21154}, {100,15702}, {140,6174}, {149,15721}, {376,31272}, {381,6667}, {511,38090}, {515,38104}, {517,38026}, {519,38032}, {524,38119}, {527,38124}, {528,5054}, {529,38135}, {631,10707}, {632,37725}, {952,3653}, {971,38102}, {1317,5444}, {1387,3654}, {1484,15713}, {2783,9167}, {2800,19883}, {2802,38068}, {2829,5055}, {3035,15694}, {3524,5840}, {3526,20418}, {3828,11715}, {5437,12515}, {5762,38095}, {5848,38064}, {5851,38065}, {5854,38066}, {5856,38067}, {10090,28466}, {10109,22799}, {10124,31235}, {10724,19708}, {10738,15693}, {10742,15703}, {11812,33814}, {12100,24466}, {12751,19876}, {12773,15723}, {13913,32788}, {13977,32787}, {15708,34474}, {16858,18861}, {17564,26470}, {19924,38147}, {22938,34200}, {24644,38137}, {28194,32557}, {28198,38038}, {28204,34122}, {28208,38182}

X(38069) = reflection of X(38077) in X(38084)
X(38069) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3653, 11539, 38070), (21154, 34126, 23513), (38032, 38133, 38128), (38077, 38084, 23513)

X(38070) = CENTROID OF TRIANGLE {X(2), X(3), X(12)}

Barycentrics    8*a^7-8*(b+c)*a^6-(21*b^2-2*b*c+21*c^2)*a^5+(b+c)*(21*b^2-10*b*c+21*c^2)*a^4+2*(9*b^4+9*c^4-b*c*(5*b^2-2*b*c+5*c^2))*a^3-2*(b^2-c^2)*(b-c)*(9*b^2+8*b*c+9*c^2)*a^2-(b^2-c^2)^2*(5*b^2-8*b*c+5*c^2)*a+5*(b^2-c^2)^3*(b-c) : :
X(38070) = 5*X(2)+X(11491) = 4*X(2)-X(26470) = X(2)+2*X(31659) = X(12)+2*X(549) = 4*X(11491)+5*X(26470) = X(11491)-10*X(31659) = 3*X(21155)+X(38078) = 3*X(21155)+2*X(38085) = 2*X(21155)+X(38109) = X(21155)+2*X(38114) = 7*X(21155)+2*X(38142) = X(26470)+8*X(31659) = 2*X(38078)-3*X(38109) = X(38078)-6*X(38114) = 7*X(38078)-6*X(38142) = 4*X(38085)-3*X(38109) = X(38085)-3*X(38114) = 7*X(38085)-3*X(38142) = X(38109)-4*X(38114) = 7*X(38109)-4*X(38142)

X(38070) lies on these lines: {2,10267}, {12,549}, {30,21155}, {140,5258}, {381,6668}, {511,38091}, {515,38105}, {517,38027}, {519,38033}, {524,38120}, {527,38125}, {528,38135}, {529,5054}, {547,5259}, {758,38068}, {952,3653}, {971,38103}, {2975,15702}, {3654,37737}, {4999,15694}, {5055,5842}, {5762,38096}, {5840,38106}, {5849,38064}, {5852,38065}, {5855,38066}, {5857,38067}, {10124,31260}, {12100,30264}, {15721,20060}, {19924,38148}, {28194,38062}, {28198,38039}, {28204,38058}, {28208,38183}

X(38070) = reflection of X(38078) in X(38085)
X(38070) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3653, 11539, 38069), (21155, 38114, 38109), (38033, 38134, 38129), (38078, 38085, 38109)

X(38071) = CENTROID OF TRIANGLE {X(2), X(4), X(5)}

Barycentrics    4*a^4+7*(b^2+c^2)*a^2-11*(b^2-c^2)^2 : :
X(38071) = 11*X(2)-5*X(3) = 7*X(2)+5*X(4) = 2*X(2)-5*X(5) = 13*X(2)-10*X(140) = 17*X(2)-5*X(376) = X(2)+5*X(381) = 5*X(2)+X(382) = X(2)+2*X(546) = 7*X(2)-10*X(547) = 8*X(2)-5*X(549) = 4*X(2)-X(550) = 9*X(2)-5*X(3524) = 19*X(2)-7*X(3528) = 13*X(2)-X(3529) = 7*X(2)-4*X(3530) = 19*X(2)+5*X(3543) = 5*X(2)-17*X(3544) = X(2)-5*X(3545) = 16*X(2)+5*X(3627) = 17*X(2)-20*X(3628) = 13*X(2)+5*X(3830) = 3*X(2)+5*X(3839) = 4*X(2)+5*X(3845) = X(2)+20*X(3850) = X(2)-7*X(3851) = X(2)+11*X(3855) = 7*X(2)+20*X(3860) = 19*X(2)+20*X(3861) = 7*X(2)-5*X(5054) = 3*X(2)-5*X(5055) = X(2)-10*X(5066) = 7*X(2)-13*X(5079) = 14*X(2)-5*X(8703) = 11*X(2)-20*X(10109) = 13*X(2)-5*X(10304) = 6*X(2)-5*X(11539) = X(2)-4*X(11737) = 19*X(2)-10*X(12100) = 17*X(2)+10*X(12101) = 10*X(2)-7*X(14869) = 3*X(2)-10*X(14892) = 11*X(2)+10*X(14893) = 7*X(2)-X(15681) = 2*X(2)+X(15687) = 19*X(2)-5*X(15689) = 4*X(2)-5*X(15699) = 13*X(2)-7*X(15700) = 5*X(2)-3*X(15707) = 19*X(2)-15*X(15709) = 7*X(2)-3*X(15710) = 17*X(2)-11*X(15720) = 2*X(2)+5*X(23046) = 5*X(2)-2*X(34200) = 5*X(2)-8*X(35018)

This triangle has collinear vertices.

X(38071) lies on these lines: {2,3}, {52,11017}, {355,30308}, {515,38022}, {516,38082}, {517,38076}, {519,38034}, {524,38136}, {527,38137}, {528,38141}, {529,38142}, {538,20112}, {754,16509}, {946,34641}, {952,38021}, {971,38080}, {1353,5476}, {1483,19925}, {1503,38079}, {1699,38112}, {2829,38084}, {3058,10592}, {3244,9955}, {3564,38072}, {3584,10386}, {3626,22791}, {3629,19130}, {3631,11178}, {3632,3656}, {3636,18480}, {3653,7988}, {3654,7989}, {3655,18492}, {3817,10283}, {3818,6329}, {3828,22793}, {5318,16963}, {5321,16962}, {5434,10593}, {5461,22505}, {5475,14075}, {5480,25561}, {5655,11801}, {5663,14845}, {5690,12571}, {5762,38075}, {5842,38085}, {5843,38073}, {5844,9779}, {6055,15092}, {6435,6565}, {6436,6564}, {6498,31412}, {7745,18362}, {7753,34571}, {7776,32868}, {8584,18553}, {9220,14836}, {9781,31834}, {10095,14831}, {10170,13570}, {10175,28198}, {10896,15170}, {11381,32205}, {11459,13451}, {11591,21969}, {11645,38110}, {11648,31406}, {11693,34153}, {12162,18874}, {13363,16194}, {13364,15030}, {13630,27355}, {14449,15056}, {15038,15052}, {15048,18424}, {15058,16881}, {15619,34598}, {15808,34648}, {16267,16809}, {16268,16808}, {18358,20423}, {18493,20057}, {19116,35822}, {19117,35823}, {19875,28174}, {19883,28160}, {22515,35022}, {22796,35019}, {22797,35020}, {22938,35023}, {28146,38068}, {28194,38042}, {28208,38028}, {28212,38066}

X(38071) = midpoint of X(i) and X(j) for these {i,j}: {2, 14269}, {4, 5054}, {5, 23046}, {381, 3545}, {3543, 15689}, {3830, 10304}, {3839, 5055}, {3845, 15699}, {15687, 17504}, {18403, 37907}
X(38071) = reflection of X(i) in X(j) for these (i,j): (5, 3545), (549, 15699), (550, 17504), (3545, 5066), (3845, 23046), (5054, 547), (5055, 14892), (8703, 5054), (10304, 140), (11539, 5055), (14269, 546), (15686, 10304), (15687, 14269), (15689, 12100), (15699, 5), (15704, 15689), (17504, 2), (23046, 381), (34153, 11693)
X(38071) = complement of X(15688)
X(38071) = orthocentroidal circle-inverse of-X(15681)
X(38071) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11737, 5), (4, 12811, 5), (5, 15687, 2), (140, 5072, 5), (381, 3851, 2), (381, 5066, 5), (381, 19709, 4), (546, 3530, 4), (546, 11737, 2), (547, 3530, 2), (547, 3860, 4), (3090, 15715, 2), (3529, 5071, 2), (3845, 35404, 4), (3859, 12811, 4), (5055, 15688, 2), (5079, 15681, 2), (10109, 14893, 3), (15690, 35417, 4), (34200, 35018, 2)

X(38072) = CENTROID OF TRIANGLE {X(2), X(4), X(6)}

Barycentrics    a^6-10*(b^2+c^2)*a^4+(5*b^4-18*b^2*c^2+5*c^4)*a^2+4*(b^4-c^4)*(b^2-c^2) : :
X(38072) = 4*X(2)-X(1350) = X(2)+2*X(5480) = X(4)+2*X(597) = 4*X(5)-X(599) = 8*X(5)+X(11477) = 2*X(5)+X(20423) = X(6)+2*X(381) = 5*X(6)+4*X(3818) = X(6)-4*X(5476) = 7*X(6)+2*X(18440) = X(6)+8*X(19130) = 5*X(381)-2*X(3818) = X(381)+2*X(5476) = 7*X(381)-X(18440) = X(381)-4*X(19130) = X(399)-4*X(25566) = 2*X(599)+X(11477) = X(599)+2*X(20423) = X(1350)+8*X(5480) = X(11477)-4*X(20423)

X(38072) lies on these lines: {2,1350}, {4,597}, {5,599}, {6,13}, {30,5085}, {69,32893}, {141,5071}, {154,23049}, {182,3830}, {262,5503}, {373,32216}, {376,3589}, {382,10541}, {511,5055}, {515,38023}, {516,38088}, {517,38087}, {518,38021}, {519,38035}, {524,3545}, {527,38143}, {528,38147}, {529,38148}, {547,3763}, {549,31670}, {575,3843}, {576,3851}, {971,38086}, {1351,11178}, {1352,5066}, {1503,3839}, {1656,25565}, {1992,3091}, {2453,16279}, {2781,5640}, {2829,38090}, {2882,13240}, {3053,37345}, {3090,20582}, {3098,15694}, {3124,9759}, {3524,29181}, {3534,10168}, {3543,3618}, {3564,38071}, {3584,10387}, {3751,30308}, {3832,8550}, {3845,11179}, {3855,20583}, {5050,11645}, {5052,18362}, {5054,19924}, {5068,11160}, {5072,34507}, {5073,20190}, {5092,15681}, {5096,28444}, {5182,11317}, {5587,28538}, {5603,9041}, {5621,31861}, {5842,38091}, {5845,38073}, {5846,38074}, {5847,38076}, {5848,38077}, {5849,38078}, {6144,18358}, {6811,13783}, {6813,13663}, {7394,17809}, {7426,31860}, {7540,37476}, {7565,34117}, {7610,9753}, {8176,14645}, {8584,11180}, {9752,15597}, {9755,10033}, {9830,14639}, {10249,37077}, {10601,31133}, {10706,16010}, {11163,13862}, {11284,13857}, {11482,18553}, {11801,25336}, {12017,15684}, {13626,15162}, {13627,15163}, {13632,37499}, {14787,17834}, {14810,15701}, {14892,34380}, {15688,29317}, {15689,17508}, {15703,33878}, {15709,21167}, {16226,34146}, {17825,31152}, {18382,19132}, {18405,19153}, {20126,32271}, {22236,37332}, {22238,37333}, {28194,38047}, {28198,38167}, {28208,38029}

X(38072) = midpoint of X(i) and X(j) for these {i,j}: {381, 14848}, {3545, 14853}, {5050, 14269}
X(38072) = reflection of X(i) in X(j) for these (i,j): (6, 14848), (10516, 3545), (14848, 5476), (15689, 17508), (21358, 5055), (31884, 5054), (38064, 38079)
X(38072) = Kiepert-hyperbola-inverse of X(15484)
X(38072) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 20423, 599), (13, 14, 15484), (381, 5476, 6), (382, 25555, 10541), (599, 20423, 11477), (1351, 11178, 15533), (1351, 19709, 11178), (3845, 11179, 36990), (3845, 18583, 11179), (5476, 19130, 381), (10516, 14853, 5102), (14561, 38064, 38079), (38035, 38146, 38144)

X(38073) = CENTROID OF TRIANGLE {X(2), X(4), X(7)}

Barycentrics    5*a^6-4*(b+c)*a^5-(5*b^2-2*b*c+5*c^2)*a^4-4*(b+c)*(b^2+c^2)*a^3+(7*b^2-2*b*c+7*c^2)*(b-c)^2*a^2+8*(b^2-c^2)^2*(b+c)*a-7*(b^2-c^2)^2*(b-c)^2 : :
X(38073) = 4*X(2)-X(5759) = X(2)+2*X(5805) = 7*X(2)-4*X(31658) = X(4)+2*X(6173) = 4*X(5)-X(6172) = X(7)+2*X(381) = 2*X(9)-5*X(5071) = X(5759)+8*X(5805) = 7*X(5759)-16*X(31658) = 3*X(5759)-8*X(38067) = 7*X(5805)+2*X(31658) = 3*X(5805)+X(38067) = 3*X(21151)-4*X(38065) = 3*X(21151)-8*X(38080) = X(21151)-4*X(38107) = 5*X(21151)-8*X(38111) = X(21151)+8*X(38137) = 6*X(31658)-7*X(38067) = X(38065)-3*X(38107) = 5*X(38065)-6*X(38111) = X(38065)+6*X(38137)

X(38073) lies on these lines: {2,5759}, {4,6173}, {5,6172}, {7,381}, {9,5071}, {30,21151}, {142,376}, {515,38024}, {516,3524}, {517,38092}, {518,38074}, {519,38036}, {524,38143}, {527,3545}, {528,5603}, {529,38153}, {547,18230}, {549,31671}, {971,3839}, {1503,38086}, {2094,8226}, {2829,38095}, {3090,5735}, {3241,20330}, {3488,30275}, {3543,18482}, {3830,31657}, {3845,36991}, {4312,10589}, {5054,38171}, {5055,5762}, {5066,5779}, {5732,15682}, {5842,38096}, {5843,38071}, {5845,38072}, {5850,38076}, {5851,38077}, {5852,38078}, {10304,38122}, {11038,28204}, {11645,38115}, {15702,20195}, {15709,21153}, {17254,36660}, {18480,30340}, {18493,30332}, {28194,38052}, {28198,38172}, {28208,38030}, {28534,38037}, {31162,35514}

X(38073) = reflection of X(i) in X(j) for these (i,j): (3524, 38093), (3545, 38150), (5054, 38171), (5817, 3545), (10304, 38122), (38065, 38080)
X(38073) = anticomplement of X(38067)
X(38073) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38036, 38151, 38149), (38065, 38107, 38080)

X(38074) = CENTROID OF TRIANGLE {X(2), X(4), X(8)}

Barycentrics    5*a^4-6*(b+c)*a^3+2*(b^2+6*b*c+c^2)*a^2+6*(b^2-c^2)*(b-c)*a-7*(b^2-c^2)^2 : :
X(38074) = 2*X(1)-5*X(5071) = X(2)+2*X(355) = 4*X(2)-X(944) = 7*X(2)-4*X(1385) = 5*X(2)-2*X(3655) = 2*X(2)-5*X(5818) = 5*X(2)-8*X(9956) = 2*X(2)+X(34627) = 3*X(2)-4*X(38083) = 8*X(355)+X(944) = 7*X(355)+2*X(1385) = 3*X(355)+X(3653) = 5*X(355)+X(3655) = 4*X(355)+5*X(5818) = 5*X(355)+4*X(9956) = 4*X(355)-X(34627) = 3*X(355)+2*X(38083) = 7*X(944)-16*X(1385) = 3*X(944)-8*X(3653) = 5*X(944)-8*X(3655) = X(944)-10*X(5818) = X(944)+2*X(34627) = 3*X(944)-16*X(38083)

X(38074) lies on these lines: {1,5071}, {2,355}, {4,3679}, {5,3241}, {7,11545}, {8,381}, {10,376}, {30,5657}, {40,4745}, {80,3488}, {100,28444}, {104,16417}, {495,15933}, {515,3524}, {516,38097}, {517,3839}, {518,38073}, {519,3545}, {524,38144}, {527,38149}, {528,5817}, {529,38157}, {547,3616}, {549,9780}, {551,3090}, {631,3828}, {946,4677}, {952,5055}, {962,3845}, {971,38092}, {1389,4930}, {1482,5066}, {1483,10109}, {1503,38087}, {1698,15702}, {2829,38099}, {3091,3656}, {3476,3582}, {3486,3584}, {3487,10827}, {3525,19876}, {3543,3617}, {3544,13464}, {3576,15709}, {3579,15683}, {3621,9955}, {3626,18492}, {3632,30308}, {3829,34717}, {3830,5690}, {3851,5734}, {3855,7982}, {4297,15698}, {4421,28461}, {4428,6920}, {4668,18483}, {4669,12245}, {4678,12699}, {5054,5731}, {5056,37727}, {5067,5882}, {5068,20049}, {5175,34629}, {5260,18518}, {5550,15703}, {5554,6175}, {5691,11001}, {5714,10573}, {5727,8164}, {5842,38100}, {5844,9779}, {5846,38072}, {5853,38075}, {5854,38077}, {5855,38078}, {5901,34748}, {6684,19708}, {6829,37725}, {6845,21031}, {6990,12607}, {7319,15171}, {7486,15178}, {7967,10175}, {7989,10595}, {9041,10516}, {9588,17538}, {9812,14269}, {9864,12243}, {10039,10385}, {10072,37710}, {10164,15710}, {10246,15699}, {10267,16861}, {10304,26446}, {10588,37711}, {10589,37708}, {10590,11041}, {10711,12247}, {11231,15708}, {11237,18391}, {11278,20052}, {11491,16418}, {11499,17549}, {11500,21161}, {11539,28224}, {11645,38116}, {11737,18493}, {12645,19709}, {12702,15687}, {13587,22758}, {13624,15721}, {14831,23841}, {14853,28538}, {15177,37939}, {15688,28186}, {15692,18481}, {15694,19877}, {15697,31663}, {15719,31423}, {17310,36662}, {18516,33110}, {19065,35822}, {19066,35823}, {19883,28236}, {28158,35409}, {28198,38176}, {28453,32141}, {34697,37430}

X(38074) = midpoint of X(25055) and X(37712)
X(38074) = reflection of X(i) in X(j) for these (i,j): (3524, 19875), (3545, 5587), (3653, 38083), (5054, 38042), (5603, 3545), (5731, 5054), (7967, 25055), (9812, 14269), (10246, 15699), (10304, 26446), (25055, 10175), (38021, 38076), (38066, 38081)
X(38074) = anticomplement of X(3653)
X(38074) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 355, 34627), (2, 34627, 944), (40, 34648, 15682), (355, 5818, 944), (946, 4677, 34631), (3091, 31145, 3656), (3543, 3617, 3654), (3543, 3654, 6361), (3617, 18480, 6361), (3653, 38083, 2), (3654, 18480, 3543), (3655, 9956, 2), (3830, 5690, 34632), (3845, 34718, 962), (4745, 34648, 40), (5587, 38021, 38076), (5790, 38066, 38081), (5818, 34627, 2), (9778, 38112, 5657), (10175, 37712, 7967), (38021, 38076, 3545)

X(38075) = CENTROID OF TRIANGLE {X(2), X(4), X(9)}

Barycentrics    a^6+(b+c)*a^5-2*(5*b^2-2*b*c+5*c^2)*a^4+10*(b+c)*(b^2+c^2)*a^3+(5*b^2+14*b*c+5*c^2)*(b-c)^2*a^2-11*(b^2-c^2)^2*(b+c)*a+4*(b^2-c^2)^2*(b-c)^2 : :
X(38075) = 4*X(2)-X(5732) = 5*X(2)+X(36991) = 4*X(5)-X(6173) = 3*X(5)-X(38080) = X(9)+2*X(381) = 5*X(9)+4*X(18482) = 7*X(9)+2*X(31671) = 5*X(381)-2*X(18482) = 7*X(381)-X(31671) = 5*X(5732)+4*X(36991) = 3*X(6173)-4*X(38080) = 14*X(18482)-5*X(31671) = 3*X(21153)-4*X(38067) = 3*X(21153)-8*X(38082) = X(21153)-4*X(38108) = 5*X(21153)-8*X(38113) = X(21153)+8*X(38139) = X(38067)-3*X(38108) = 5*X(38067)-6*X(38113) = X(38067)+6*X(38139)

X(38075) lies on these lines: {2,1750}, {5,6173}, {9,381}, {30,21153}, {142,5071}, {376,6666}, {515,38025}, {516,3839}, {517,38097}, {518,38021}, {519,38037}, {524,38145}, {527,3545}, {528,5587}, {529,38160}, {547,20195}, {549,31672}, {971,5055}, {1503,38088}, {2093,30311}, {2801,7988}, {2829,38102}, {2951,19876}, {3091,5735}, {3543,18230}, {3582,4321}, *{3584,4326}, {3830,31658}, {5066,5805}, {5223,11680}, {5762,38071}, {5779,19709}, {5842,38103}, {5843,14892}, {5853,38074}, {5856,38077}, {5857,38078}, {7705,7989}, {8226,31142}, {10109,31657}, {11645,38117}, {15254,18492}, {15699,38122}, {16861,34628}, {28194,38057}, {28198,38179}, {28208,38031}, {30326,31164}

X(38075) = midpoint of X(3545) and X(5817)
X(38075) = reflection of X(i) in X(j) for these (i,j): (38067, 38082), (38093, 5055), (38122, 15699), (38150, 3545)
X(38075) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38037, 38158, 38154), (38067, 38108, 38082)

X(38076) = CENTROID OF TRIANGLE {X(2), X(4), X(10)}

Barycentrics    4*a^4-3*(b+c)*a^3+(7*b^2+6*b*c+7*c^2)*a^2+3*(b^2-c^2)*(b-c)*a-11*(b^2-c^2)^2 : :
X(38076) = 4*X(2)-X(4297) = 5*X(2)+X(5691) = 11*X(2)-5*X(7987) = X(2)-7*X(7989) = X(2)+2*X(19925) = 7*X(2)-X(34628) = 2*X(2)+X(34648) = 5*X(4297)+4*X(5691) = 11*X(4297)-20*X(7987) = X(4297)+8*X(19925) = 7*X(4297)-4*X(34628) = X(4297)+2*X(34648) = X(5691)-10*X(19925) = 7*X(5691)+5*X(34628) = 2*X(5691)-5*X(34648) = 10*X(7987)+11*X(34648) = 7*X(7989)+2*X(19925) = 14*X(7989)+X(34648) = 14*X(19925)+X(34628) = 4*X(19925)-X(34648)

X(38076) lies on these lines: {2,4297}, {4,3828}, {5,551}, {8,30308}, {10,381}, {20,19876}, {30,10164}, {355,19709}, {376,3634}, {515,3653}, {516,3839}, {517,38071}, {519,3545}, {524,38146}, {527,38151}, {528,38161}, {529,38162}, {546,5493}, {547,18480}, {549,31673}, {553,10895}, {758,38078}, {946,4669}, {952,14892}, {971,38094}, {1125,5071}, {1385,10109}, {1503,38089}, {1698,3543}, {1737,30424}, {2784,9166}, {2796,14639}, {2801,38095}, {2802,38077}, {2829,38104}, {3091,3679}, {3146,30315}, {3241,5068}, {3244,11737}, {3524,28164}, {3544,5881}, {3579,14893}, {3582,4315}, {3584,4314}, {3625,9955}, {3656,3851}, {3830,6684}, {3845,9956}, {3850,11362}, {3854,7991}, {3860,22793}, {3947,10826}, {4745,5818}, {5054,10172}, {5072,13464}, {5542,10590}, {5842,38105}, {5847,38072}, {5850,38073}, {6175,8582}, {7988,28236}, {8227,34627}, {10056,30331}, {10165,15699}, {10171,25055}, {11001,31423}, {11019,11237}, {11522,31145}, {11539,28160}, {11645,38118}, {11648,31396}, {12102,31447}, {12512,15682}, {12617,17577}, {14269,26446}, {15683,19877}, {15684,22266}, {15687,31730}, {15688,28172}, {15702,31253}, {15703,18481}, {15721,19872}, {15808,18525}, {15931,17547}, {21969,31752}, {23046,28198}, {31663,33699}, {33697,34200}

X(38076) = midpoint of X(i) and X(j) for these {i,j}: {3545, 5587}, {3839, 19875}, {14269, 26446}, {23046, 38042}, {38021, 38074}
X(38076) = reflection of X(i) in X(j) for these (i,j): (3817, 3545), (5054, 10172), (10165, 15699), (19883, 5055), (25055, 10171), (38068, 38083)
X(38076) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 19925, 34648), (2, 34648, 4297), (381, 3654, 18483), (546, 31399, 5493), (3545, 38074, 38021), (3817, 5587, 38155), (3830, 6684, 34638), (4745, 12571, 31162), (5587, 38021, 38074), (5818, 31162, 4745), (10175, 38068, 38083)

X(38077) = CENTROID OF TRIANGLE {X(2), X(4), X(11)}

Barycentrics    4*a^7-4*(b+c)*a^6+3*(b+c)^2*a^5-(b+c)*(3*b^2+2*b*c+3*c^2)*a^4-2*(9*b^4+9*c^4-2*b*c*(6*b^2-b*c+6*c^2))*a^3+2*(b^2-c^2)*(b-c)*(9*b^2+8*b*c+9*c^2)*a^2+(b^2-c^2)^2*(11*b^2-30*b*c+11*c^2)*a-11*(b^2-c^2)^3*(b-c) : :
X(38077) = 5*X(2)+X(10724) = 4*X(2)-X(24466) = 4*X(5)-X(6174) = 10*X(5)-X(10993) = X(11)+2*X(381) = 7*X(11)+2*X(10742) = 11*X(11)-2*X(12773) = 7*X(381)-X(10742) = 11*X(381)+X(12773) = 5*X(6174)-2*X(10993) = 4*X(10724)+5*X(24466) = 11*X(10742)+7*X(12773) = X(21154)-4*X(23513) = 5*X(21154)-8*X(34126) = 3*X(21154)-4*X(38069) = 3*X(21154)-8*X(38084) = X(21154)+8*X(38141) = 5*X(23513)-2*X(34126) = 3*X(23513)-X(38069) = 3*X(23513)-2*X(38084) = X(23513)+2*X(38141)

X(38077) lies on these lines: {2,10724}, {5,6174}, {11,381}, {30,21154}, {80,30308}, {119,5066}, {376,6667}, {515,38026}, {516,38102}, {517,38099}, {519,38038}, {524,38147}, {527,38152}, {528,3545}, {529,38163}, {547,22938}, {952,38021}, {971,38095}, {1503,38090}, {1532,38060}, {2802,38076}, {2829,3839}, {3035,5071}, {3091,10707}, {3543,31272}, {3830,6713}, {3832,20418}, {3847,37430}, {3850,37726}, {3860,22799}, {5055,5840}, {5068,20400}, {5842,38106}, {5848,38072}, {5851,38073}, {5854,38074}, {5856,38075}, {6154,11737}, {6973,31140}, {10109,33814}, {10598,11236}, {10738,19709}, {10896,34746}, {11645,38119}, {11928,34720}, {15908,17556}, {16174,17618}, {28194,34122}, {28198,38182}, {28208,38032}

X(38077) = reflection of X(38069) in X(38084)
X(38077) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (23513, 38069, 38084), (38021, 38071, 38078), (38038, 38161, 38156)

X(38078) = CENTROID OF TRIANGLE {X(2), X(4), X(12)}

Barycentrics    4*a^7-4*(b+c)*a^6+(b+3*c)*(3*b+c)*a^5-(b+c)*(3*b^2+14*b*c+3*c^2)*a^4+2*(-9*b^4-9*c^4+2*b*c*(b^2+5*b*c+c^2))*a^3+2*(b^2-c^2)*(b-c)*(9*b^2+14*b*c+9*c^2)*a^2+(b^2-c^2)^2*(11*b^2-14*b*c+11*c^2)*a-11*(b^2-c^2)^3*(b-c) : :
X(38078) = 4*X(2)-X(30264) = 4*X(5)-X(31157) = X(12)+2*X(381) = X(376)-4*X(6668) = 3*X(21155)-4*X(38070) = 3*X(21155)-8*X(38085) = X(21155)-4*X(38109) = 5*X(21155)-8*X(38114) = X(21155)+8*X(38142) = 2*X(38039)+X(38157) = X(38039)+2*X(38162) = X(38070)-3*X(38109) = 5*X(38070)-6*X(38114) = X(38070)+6*X(38142) = 2*X(38085)-3*X(38109) = 5*X(38085)-3*X(38114) = X(38085)+3*X(38142) = 5*X(38109)-2*X(38114) = X(38109)+2*X(38142) = X(38114)+5*X(38142)

X(38078) lies on these lines: {2,30264}, {5,31157}, {12,381}, {30,21155}, {376,6668}, {515,38027}, {516,38103}, {517,38100}, {519,38039}, {524,38148}, {527,38153}, {528,38163}, {529,3545}, {547,31260}, {758,38076}, {952,38021}, {971,38096}, {1503,38091}, {2829,38106}, {3614,28452}, {3830,31659}, {3839,5842}, {4999,5071}, {5066,26470}, {5849,38072}, {5852,38073}, {5855,38074}, {5857,38075}, {6867,31141}, {10599,11235}, {10895,34697}, {11645,38120}, {11929,34689}, {28194,38058}, {28198,38183}, {28208,38033}, {30308,37710}

X(38078) = reflection of X(38070) in X(38085)
X(38078) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38021, 38071, 38077), (38039, 38162, 38157), (38070, 38109, 38085)

X(38079) = CENTROID OF TRIANGLE {X(2), X(5), X(6)}

Barycentrics    8*a^6-17*(b^2+c^2)*a^4+4*(b^4-9*b^2*c^2+c^4)*a^2+5*(b^4-c^4)*(b^2-c^2) : :
X(38079) = 5*X(2)+X(1351) = X(2)+2*X(18583) = 5*X(5)+4*X(575) = X(5)+2*X(597) = 7*X(5)+2*X(8550) = 13*X(5)-4*X(18553) = X(5)+8*X(25555) = 7*X(5)-4*X(25561) = 5*X(5)-8*X(25565) = 19*X(5)+8*X(33749) = 2*X(575)-5*X(597) = 14*X(575)-5*X(8550) = 13*X(575)+5*X(18553) = X(575)-10*X(25555) = 7*X(575)+5*X(25561) = X(575)+2*X(25565) = 19*X(575)-10*X(33749) = 7*X(597)-X(8550) = 13*X(597)+2*X(18553) = X(597)-4*X(25555) = 7*X(597)+2*X(25561) = 5*X(597)+4*X(25565) = 19*X(597)-4*X(33749) = X(1351)-5*X(14848) = X(1351)-10*X(18583)

X(38079) lies on these lines: {2,1351}, {5,542}, {6,547}, {30,5085}, {69,15703}, {140,20423}, {182,3845}, {373,14984}, {381,3618}, {511,11539}, {517,38089}, {518,38022}, {519,38040}, {524,15520}, {527,38164}, {528,38168}, {529,38169}, {549,3098}, {576,20582}, {599,3628}, {952,38023}, {1350,11812}, {1352,10109}, {1353,11178}, {1503,38071}, {1656,1992}, {3363,5182}, {3543,12017}, {3545,5050}, {3564,5055}, {3853,10541}, {3860,36990}, {5054,14853}, {5066,11179}, {5067,11160}, {5071,18358}, {5092,15686}, {5093,21356}, {5097,22165}, {5480,8703}, {5762,38088}, {5843,38086}, {5844,38087}, {5845,38080}, {5846,38081}, {5847,38083}, {5848,38084}, {5849,38085}, {5969,12040}, {6034,15048}, {6776,19709}, {8584,24206}, {9041,10283}, {9771,14645}, {11477,16239}, {11645,23046}, {12812,15069}, {14269,25406}, {14389,32227}, {14810,19711}, {15303,20304}, {15687,19130}, {15702,33878}, {17504,19924}, {20583,34507}, {21358,34380}, {28198,38118}, {28204,38049}, {28208,38146}, {28538,38042}, {31670,34200}

X(38079) = midpoint of X(i) and X(j) for these {i,j}: {2, 14848}, {3545, 5050}, {5054, 14853}, {5093, 21356}, {14269, 25406}, {38064, 38072}
X(38079) = reflection of X(14848) in X(18583)
X(38079) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (549, 5476, 21850), (3589, 5476, 549), (5480, 10168, 8703), (14561, 38064, 38072), (14561, 38110, 38136), (38040, 38167, 38165), (38090, 38091, 38023)

X(38080) = CENTROID OF TRIANGLE {X(2), X(5), X(7)}

Barycentrics    4*a^6+4*(b+c)*a^5-(13*b^2-16*b*c+13*c^2)*a^4-14*(b+c)*(b^2+c^2)*a^3+2*(10*b^2+b*c+10*c^2)*(b-c)^2*a^2+10*(b^2-c^2)^2*(b+c)*a-11*(b^2-c^2)^2*(b-c)^2 : :
X(38080) = X(5)+2*X(6173) = 3*X(5)-2*X(38075) = X(7)+2*X(547) = 4*X(142)-X(549) = X(144)-7*X(15703) = 3*X(6173)+X(38075) = 3*X(21151)-5*X(38065) = 3*X(21151)+5*X(38073) = X(21151)+5*X(38107) = 2*X(21151)-5*X(38111) = 4*X(21151)+5*X(38137) = X(38065)+3*X(38107) = 2*X(38065)-3*X(38111) = 4*X(38065)+3*X(38137) = X(38073)-3*X(38107) = 2*X(38073)+3*X(38111) = 4*X(38073)-3*X(38137) = 2*X(38107)+X(38111) = 4*X(38107)-X(38137) = 2*X(38111)+X(38137)

X(38080) lies on these lines: {5,6173}, {7,547}, {30,21151}, {142,549}, {144,15703}, {516,38022}, {517,38094}, {518,38081}, {519,38041}, {524,38164}, {527,15699}, {528,10283}, {529,38174}, {952,38024}, {971,38071}, {3564,38086}, {3628,6172}, {3845,31657}, {3860,36991}, {5055,5843}, {5732,33699}, {5735,14869}, {5759,11812}, {5762,11539}, {5779,10109}, {5805,8703}, {5844,38092}, {5845,38079}, {5850,38083}, {5851,38084}, {5852,38085}, {17504,38122}, {18482,35404}, {19709,36996}, {23046,38150}, {25557,37705}, {28198,38123}, {28204,38054}, {28208,38151}, {28534,38043}, {31671,34200}

X(38080) = midpoint of X(38065) and X(38073)
X(38080) = reflection of X(i) in X(j) for these (i,j): (11539, 38093), (15699, 38171), (17504, 38122), (23046, 38150)
X(38080) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38041, 38172, 38170), (38065, 38107, 38073), (38095, 38096, 38024), (38107, 38111, 38137)

X(38081) = CENTROID OF TRIANGLE {X(2), X(5), X(8)}

Barycentrics    4*a^4-12*(b+c)*a^3+(7*b^2+24*b*c+7*c^2)*a^2+12*(b^2-c^2)*(b-c)*a-11*(b^2-c^2)^2 : :
X(38081) = 4*X(2)-X(1483) = 5*X(2)+X(12645) = 7*X(2)-X(34748) = 11*X(2)-5*X(37624) = 5*X(5)-2*X(3656) = X(5)+2*X(3679) = 11*X(5)-2*X(7982) = 19*X(5)-10*X(11522) = 3*X(5)-2*X(38021) = 5*X(1483)+4*X(12645) = 7*X(1483)-4*X(34748) = 11*X(1483)-20*X(37624) = X(3656)+5*X(3679) = 11*X(3656)-5*X(7982) = 3*X(3656)-5*X(38021) = 11*X(3679)+X(7982) = 19*X(3679)+5*X(11522) = 3*X(3679)+X(38021) = 3*X(7982)-11*X(38021) = 7*X(12645)+5*X(34748)

X(38081) lies on these lines: {2,1483}, {5,3656}, {8,547}, {10,549}, {30,5657}, {40,33699}, {145,15703}, {355,8703}, {381,3617}, {517,38071}, {518,38080}, {519,10172}, {524,38165}, {527,38170}, {528,38177}, {529,38178}, {632,3828}, {944,11812}, {952,3653}, {962,3860}, {1482,10109}, {1656,31145}, {3241,3628}, {3524,28224}, {3564,38087}, {3627,5493}, {3654,15687}, {3655,15713}, {3839,28212}, {3845,4745}, {3858,11362}, {4669,9956}, {4677,5901}, {4678,5071}, {4691,22791}, {5055,5844}, {5066,5818}, {5067,20049}, {5587,23046}, {5762,38097}, {5843,38092}, {5846,38079}, {5853,38082}, {5854,38084}, {5855,38085}, {5881,14869}, {6684,15711}, {9780,10124}, {10056,11545}, {12100,34627}, {12101,34632}, {12245,19709}, {12331,16858}, {12702,14893}, {15702,18526}, {15714,18481}, {17504,26446}, {18480,35404}, {18525,34200}, {19876,37727}, {28198,38127}, {28208,38155}, {28463,32141}, {31399,34641}

X(38081) = midpoint of X(38066) and X(38074)
X(38081) = reflection of X(i) in X(j) for these (i,j): (10283, 15699), (11539, 19875), (15699, 38042), (17504, 26446), (23046, 5587), (38022, 38083)
X(38081) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3654, 18357, 15687), (5790, 38066, 38074), (5790, 38112, 38138), (5818, 34718, 5066), (38022, 38042, 38083), (38022, 38083, 15699), (38099, 38100, 19875)

X(38082) = CENTROID OF TRIANGLE {X(2), X(5), X(9)}

Barycentrics    8*a^6-10*(b+c)*a^5-(17*b^2+4*b*c+17*c^2)*a^4+26*(b+c)*(b^2+c^2)*a^3+2*(2*b^2+11*b*c+2*c^2)*(b-c)^2*a^2-16*(b^2-c^2)^2*(b+c)*a+5*(b^2-c^2)^2*(b-c)^2 : :
X(38082) = 5*X(2)+X(5779) = 4*X(2)-X(31657) = 13*X(2)-X(36996) = X(7)-7*X(15703) = X(9)+2*X(547) = 4*X(5779)+5*X(31657) = 13*X(5779)+5*X(36996) = 3*X(5779)+5*X(38065) = 3*X(21153)-5*X(38067) = 3*X(21153)+5*X(38075) = X(21153)+5*X(38108) = 2*X(21153)-5*X(38113) = 4*X(21153)+5*X(38139) = 13*X(31657)-4*X(36996) = 3*X(31657)-4*X(38065) = 3*X(36996)-13*X(38065) = X(38067)+3*X(38108) = 2*X(38067)-3*X(38113) = 4*X(38067)+3*X(38139) = X(38075)-3*X(38108) = 2*X(38075)+3*X(38113) = 4*X(38075)-3*X(38139)

X(38082) lies on these lines: {2,5779}, {7,15703}, {9,547}, {30,21153}, {381,18230}, {516,38071}, {517,38101}, {518,38022}, {519,38043}, {524,38166}, {527,15699}, {528,38042}, {529,38181}, {549,6666}, {952,38025}, {971,11539}, {1656,6172}, {3564,38088}, {3628,6173}, {3845,31658}, {5054,5817}, {5055,5762}, {5732,11812}, {5735,12812}, {5759,19709}, {5805,10109}, {5843,38093}, {5844,38097}, {5853,38081}, {5856,38084}, {5857,38085}, {15693,36991}, {28198,38130}, {28204,38059}, {28208,38158}, {31672,34200}

X(38082) = midpoint of X(i) and X(j) for these {i,j}: {5054, 5817}, {38067, 38075}
X(38082) = reflection of X(38171) in X(15699)
X(38082) = complement of X(38065)
X(38082) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38043, 38179, 38175), (38067, 38108, 38075), (38102, 38103, 38025), (38108, 38113, 38139)

X(38083) = CENTROID OF TRIANGLE {X(2), X(5), X(10)}

Barycentrics    2*a^4+3*(b+c)*a^3-2*(5*b^2+3*b*c+5*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+8*(b^2-c^2)^2 : :
X(38083) = X(1)-7*X(15703) = 5*X(2)+X(355) = 13*X(2)-X(944) = 4*X(2)-X(1385) = 7*X(2)-X(3655) = 7*X(2)+5*X(5818) = X(2)+2*X(9956) = 11*X(2)+X(34627) = 3*X(2)+X(38074) = 13*X(355)+5*X(944) = 4*X(355)+5*X(1385) = 3*X(355)+5*X(3653) = 7*X(355)+5*X(3655) = X(355)-10*X(9956) = 11*X(355)-5*X(34627) = 3*X(355)-5*X(38074) = 4*X(944)-13*X(1385) = 3*X(944)-13*X(3653) = 7*X(944)-13*X(3655) = 11*X(944)+13*X(34627) = 3*X(944)+13*X(38074)

X(38083) lies on these lines: {1,15703}, {2,355}, {3,19876}, {5,3828}, {10,547}, {30,10164}, {40,19709}, {165,14269}, {376,19877}, {381,1698}, {515,11539}, {516,38071}, {517,4731}, {519,10172}, {524,38167}, {527,38172}, {528,38182}, {529,38183}, {546,31447}, {549,3634}, {551,3628}, {553,10592}, {758,38085}, {946,10109}, {952,19883}, {1656,3679}, {2802,38084}, {3090,3656}, {3241,5067}, {3524,28160}, {3526,31666}, {3534,31423}, {3545,9812}, {3564,38089}, {3584,17606}, {3654,5071}, {3830,7989}, {3839,28146}, {3844,5476}, {3845,6684}, {4297,11812}, {4301,12812}, {4413,28444}, {4669,5901}, {4677,33179}, {4870,18395}, {5054,5587}, {5066,22793}, {5070,15178}, {5072,9588}, {5076,31425}, {5493,12811}, {5550,32900}, {5691,15693}, {5762,38101}, {5790,25055}, {5843,38094}, {5844,38098}, {5847,38079}, {5850,38080}, {6175,22936}, {8227,34718}, {8703,19925}, {10124,18357}, {10171,38112}, {10225,17532}, {11362,35018}, {11499,17542}, {11648,31430}, {12100,34648}, {12101,34638}, {12512,33699}, {12702,30308}, {13624,15694}, {14892,28174}, {14893,22266}, {15681,18492}, {15684,35242}, {15685,16192}, {15688,28168}, {15701,34628}, {15702,18481}, {15723,18525}, {16417,23961}, {16418,33862}, {16857,32613}, {19878,37705}, {26086,28453}, {31253,34773}, {31673,34200}

X(38083) = midpoint of X(i) and X(j) for these {i,j}: {165, 14269}, {3545, 26446}, {3653, 38074}, {5054, 5587}, {5055, 19875}, {5790, 25055}, {15699, 38042}, {38021, 38066}, {38022, 38081}, {38068, 38076}
X(38083) = reflection of X(i) in X(j) for these (i,j): (11230, 15699), (15699, 10172), (17502, 5054)
X(38083) = complement of X(3653)
X(38083) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5818, 3655), (2, 38074, 3653), (3654, 5071, 9955), (5055, 38066, 38021), (5071, 9780, 3654), (10172, 38042, 11230), (10175, 11231, 38140), (10175, 38068, 38076), (11230, 38042, 38176), (15699, 38081, 38022), (19875, 38021, 38066), (38022, 38042, 38081), (38104, 38105, 19883)

X(38084) = CENTROID OF TRIANGLE {X(2), X(5), X(11)}

Barycentrics    2*a^7-2*(b+c)*a^6-12*(b^2-b*c+c^2)*a^5+2*(b+c)*(6*b^2-5*b*c+6*c^2)*a^4+(18*b^4+18*c^4-(33*b^2-16*b*c+33*c^2)*b*c)*a^3-2*(b^2-c^2)*(b-c)*(9*b^2+5*b*c+9*c^2)*a^2-(b^2-c^2)^2*(8*b^2-21*b*c+8*c^2)*a+8*(b^2-c^2)^3*(b-c) : :
X(38084) = 5*X(2)+X(10738) = 13*X(2)-X(13199) = 4*X(2)-X(33814) = 7*X(5)+2*X(20418) = X(11)+2*X(547) = 13*X(10738)+5*X(13199) = 4*X(10738)+5*X(33814) = 4*X(13199)-13*X(33814) = X(21154)+5*X(23513) = 2*X(21154)-5*X(34126) = 3*X(21154)-5*X(38069) = 3*X(21154)+5*X(38077) = 4*X(21154)+5*X(38141) = 2*X(23513)+X(34126) = 3*X(23513)+X(38069) = 3*X(23513)-X(38077) = 4*X(23513)-X(38141) = 3*X(34126)-2*X(38069) = 3*X(34126)+2*X(38077) = 2*X(34126)+X(38141)

X(38084) lies on these lines: {2,10738}, {5,10199}, {11,547}, {30,21154}, {100,15703}, {104,19709}, {119,10109}, {381,10728}, {517,38104}, {519,38044}, {524,38168}, {527,38173}, {528,15699}, {529,38184}, {549,6667}, {952,5055}, {1656,10707}, {2802,38083}, {2829,38071}, {3564,38090}, {3628,6174}, {3845,6713}, {5066,22799}, {5690,26129}, {5762,38102}, {5840,11539}, {5843,38095}, {5844,38099}, {5848,38079}, {5851,38080}, {5854,38081}, {5856,38082}, {10724,15693}, {11812,24466}, {12515,30308}, {12812,37725}, {28198,38133}, {28204,32557}, {28208,38161}, {35018,37726}

X(38084) = midpoint of X(38069) and X(38077)
X(38084) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5055, 38022, 38085), (23513, 34126, 38141), (23513, 38069, 38077), (38044, 38182, 38177)

X(38085) = CENTROID OF TRIANGLE {X(2), X(5), X(12)}

Barycentrics    2*a^7-2*(b+c)*a^6-4*(3*b^2+b*c+3*c^2)*a^5+2*(b+c)*(6*b^2+b*c+6*c^2)*a^4+(18*b^4+18*c^4-(7*b^2+8*b*c+7*c^2)*b*c)*a^3-2*(b^2-c^2)*(b-c)*(9*b^2+11*b*c+9*c^2)*a^2-(b^2-c^2)^2*(8*b^2-11*b*c+8*c^2)*a+8*(b^2-c^2)^3*(b-c) : :
X(38085) = X(12)+2*X(547) = X(549)-4*X(6668) = 3*X(21155)-5*X(38070) = 3*X(21155)+5*X(38078) = X(21155)+5*X(38109) = 2*X(21155)-5*X(38114) = 4*X(21155)+5*X(38142) = 2*X(38045)+X(38178) = X(38045)+2*X(38183) = X(38070)+3*X(38109) = 2*X(38070)-3*X(38114) = 4*X(38070)+3*X(38142) = X(38078)-3*X(38109) = 2*X(38078)+3*X(38114) = 4*X(38078)-3*X(38142) = 2*X(38109)+X(38114) = 4*X(38109)-X(38142) = 2*X(38114)+X(38142) = X(38174)+2*X(38181) = X(38178)-4*X(38183)

X(38085) lies on these lines: {5,10197}, {12,547}, {30,21155}, {517,38105}, {519,38045}, {524,38169}, {527,38174}, {528,38184}, {529,15699}, {549,6668}, {758,38083}, {952,5055}, {2975,15703}, {3564,38091}, {3628,31157}, {3845,31659}, {5762,38103}, {5842,38071}, {5843,38096}, {5844,38100}, {5849,38079}, {5852,38080}, {5855,38081}, {5857,38082}, {10109,26470}, {11491,19709}, {11812,30264}, {17532,33814}, {28198,38134}, {28204,38062}, {28208,38162}

X(38085) = midpoint of X(38070) and X(38078)
X(38085) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5055, 38022, 38084), (38045, 38183, 38178), (38070, 38109, 38078), (38109, 38114, 38142)

X(38086) = CENTROID OF TRIANGLE {X(2), X(6), X(7)}

Barycentrics    a^4-8*(b+c)*a^3+(5*b^2-14*b*c+5*c^2)*a^2-2*(b+c)*(b^2+c^2)*a+4*(b^2+c^2)*(b-c)^2 : :
X(38086) = X(6)+2*X(6173) = X(7)+2*X(597) = 4*X(142)-X(599) = 4*X(3589)-X(6172) = 2*X(5735)+7*X(10541) = X(20423)+2*X(31657) = 2*X(38046)+X(38185) = X(38046)+2*X(38187) = X(38088)-3*X(38186) = 2*X(38115)+X(38143) = X(38115)+2*X(38164) = X(38143)-4*X(38164) = X(38185)-4*X(38187)

X(38086) lies on these lines: {6,4859}, {7,597}, {30,38115}, {142,599}, {511,38065}, {516,38023}, {518,3921}, {519,38046}, {527,38088}, {528,38188}, {529,38189}, {542,38107}, {971,38072}, {1503,38073}, {3564,38080}, {3589,6172}, {5735,10541}, {5762,38064}, {5843,38079}, {5846,38092}, {5847,38094}, {5848,38095}, {5849,38096}, {5850,38089}, {5851,38090}, {5852,38091}, {9041,11038}, {20423,31657}, {28534,38048}, {28538,38052}

X(38086) = reflection of X(21358) in X(38093)
X(38086) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38046, 38187, 38185), (38115, 38164, 38143)

X(38087) = CENTROID OF TRIANGLE {X(2), X(6), X(8)}

Barycentrics    a^3+7*(b+c)*a^2-2*(b^2+c^2)*a+4*(b+c)*(b^2+c^2) : :
X(38087) = 4*X(2)-X(3242) = X(6)+2*X(3679) = X(8)+2*X(597) = 4*X(10)-X(599) = 2*X(1386)+X(4677) = X(1992)+5*X(3617) = X(3241)-4*X(3589) = X(38023)-3*X(38047) = 5*X(38023)-6*X(38049) = X(38023)+6*X(38191) = 5*X(38047)-2*X(38049) = 3*X(38047)-2*X(38089) = X(38047)+2*X(38191) = 3*X(38049)-5*X(38089) = X(38049)+5*X(38191) = X(38089)+3*X(38191) = 2*X(38116)+X(38144) = X(38116)+2*X(38165) = X(38144)-4*X(38165) = X(38185)+2*X(38190)

X(38087) lies on these lines: {2,1280}, {6,3679}, {8,597}, {10,599}, {30,38116}, {45,29659}, {511,38066}, {517,38072}, {518,3921}, {519,38023}, {527,38185}, {528,38192}, {529,38193}, {542,5790}, {952,38064}, {1386,4677}, {1503,38074}, {1992,3617}, {3241,3589}, {3416,4745}, {3564,38081}, {3618,31145}, {3751,15533}, {3763,3828}, {4731,9004}, {5085,28204}, {5476,34718}, {5690,20423}, {5844,38079}, {5845,38092}, {5847,38098}, {5848,38099}, {5849,38100}, {5853,38088}, {5854,38090}, {5855,38091}, {5881,10541}, {9780,20582}, {16777,33165}, {18493,25565}, {27777,29861}, {31079,31179}

X(38087) = reflection of X(i) in X(j) for these (i,j): (21358, 19875), (38023, 38089)
X(38087) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38023, 38047, 38089), (38116, 38165, 38144)

X(38088) = CENTROID OF TRIANGLE {X(2), X(6), X(9)}

Barycentrics    7*a^4-11*(b+c)*a^3+8*(b^2-b*c+c^2)*a^2-5*(b+c)*(b^2+c^2)*a+(b^2+c^2)*(b-c)^2 : :
X(38088) = X(9)+2*X(597) = X(599)-4*X(6666) = X(1992)+5*X(18230) = 4*X(3589)-X(6173) = 5*X(3618)+X(6172) = X(20423)+2*X(31658) = 2*X(38048)+X(38190) = X(38048)+2*X(38194) = 2*X(38086)-3*X(38186) = 2*X(38117)+X(38145) = X(38117)+2*X(38166) = X(38145)-4*X(38166) = X(38190)-4*X(38194)

X(38088) lies on these lines: {6,4909}, {9,597}, {30,38117}, {511,38067}, {516,38072}, {518,38023}, {519,38048}, {527,38086}, {528,38047}, {529,38196}, {542,38108}, {599,6666}, {971,38064}, {1503,38075}, {1992,17317}, {3564,38082}, {3589,6173}, {3618,6172}, {5762,38079}, {5845,38093}, {5846,38097}, {5847,38101}, {5848,38102}, {5849,38103}, {5853,38087}, {5856,38090}, {5857,38091}, {20423,31658}, {28538,38057}

X(38088) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38048, 38194, 38190), (38117, 38166, 38145)

X(38089) = CENTROID OF TRIANGLE {X(2), X(6), X(10)}

Barycentrics    8*a^3+11*(b+c)*a^2+2*(b^2+c^2)*a+5*(b+c)*(b^2+c^2) : :
X(38089) = 5*X(2)+X(3751) = X(6)+2*X(3828) = X(10)+2*X(597) = X(69)-7*X(19876) = X(551)-4*X(3589) = 4*X(575)+5*X(31399) = X(599)-4*X(3634) = X(38023)+3*X(38047) = 2*X(38023)-3*X(38049) = 4*X(38023)+3*X(38191) = 2*X(38047)+X(38049) = 3*X(38047)-X(38087) = 4*X(38047)-X(38191) = 3*X(38049)+2*X(38087) = 2*X(38049)+X(38191) = 4*X(38087)-3*X(38191) = 2*X(38118)+X(38146) = X(38118)+2*X(38167) = X(38146)-4*X(38167) = X(38187)+2*X(38194)

X(38089) lies on these lines: {2,3751}, {6,3828}, {10,597}, {30,38118}, {69,19876}, {511,38068}, {515,38064}, {516,38072}, {517,38079}, {518,19883}, {519,38023}, {527,38187}, {528,38197}, {529,38198}, {542,10175}, {551,3589}, {575,31399}, {599,3634}, {758,38091}, {1386,4669}, {1503,38076}, {1698,1992}, {2796,6034}, {2802,38090}, {3241,17268}, {3564,38083}, {3618,3679}, {3844,8584}, {4663,20582}, {5845,38094}, {5846,38098}, {5847,19875}, {5848,38104}, {5849,38105}, {5850,38086}, {6684,20423}, {11160,19877}, {11362,25555}, {14561,28194}, {14848,26446}, {21358,34379}, {26083,29617}, {28202,38136}, {28204,38110}, {29574,33159}, {30768,31179}

X(38089) = midpoint of X(i) and X(j) for these {i,j}: {14848, 26446}, {38023, 38087}
X(38089) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38023, 38047, 38087), (38047, 38049, 38191), (38118, 38167, 38146)

X(38090) = CENTROID OF TRIANGLE {X(2), X(6), X(11)}

Barycentrics    8*a^5-8*(b+c)*a^4-3*(3*b-c)*(b-3*c)*a^3+(b+c)*(9*b^2-22*b*c+9*c^2)*a^2-(b^2+c^2)*(5*b^2-12*b*c+5*c^2)*a+(b^4-c^4)*(5*b-5*c) : :
X(38090) = 5*X(2)+X(10755) = X(11)+2*X(597) = X(599)-4*X(6667) = X(1992)+5*X(31272) = 4*X(3589)-X(6174) = 5*X(3618)+X(10707) = 2*X(6713)+X(20423) = 8*X(25555)+X(37726) = 2*X(38050)+X(38192) = X(38050)+2*X(38197) = 2*X(38119)+X(38147) = X(38119)+2*X(38168) = X(38147)-4*X(38168) = X(38188)+2*X(38195) = X(38192)-4*X(38197)

X(38090) lies on these lines: {2,10755}, {11,597}, {30,38119}, {511,38069}, {518,38026}, {519,38050}, {527,38188}, {528,38048}, {529,38199}, {542,23513}, {599,6667}, {952,38023}, {1503,38077}, {1992,31272}, {2802,38089}, {2829,38072}, {3564,38084}, {3589,6174}, {3618,10707}, {5840,38064}, {5845,38095}, {5846,38099}, {5847,38104}, {5849,38106}, {5851,38086}, {5854,38087}, {5856,38088}, {6713,20423}, {25555,37726}, {28538,34122}

X(38090) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38023, 38079, 38091), (38050, 38197, 38192), (38119, 38168, 38147)

X(38091) = CENTROID OF TRIANGLE {X(2), X(6), X(12)}

Barycentrics    8*a^6-(17*b^2+14*b*c+17*c^2)*a^4-8*b*c*(b+c)*a^3+4*(b^4+c^4-b*c*(2*b^2+9*b*c+2*c^2))*a^2-2*b*c*(b+c)*(b^2+c^2)*a+5*(b^4-c^4)*(b^2-c^2) : :
X(38091) = X(12)+2*X(597) = X(599)-4*X(6668) = 4*X(3589)-X(31157) = X(20423)+2*X(31659) = 2*X(38051)+X(38193) = X(38051)+2*X(38198) = 2*X(38120)+X(38148) = X(38120)+2*X(38169) = X(38148)-4*X(38169) = X(38189)+2*X(38196) = X(38193)-4*X(38198)

X(38091) lies on these lines: {12,597}, {30,38120}, {511,38070}, {518,38027}, {519,38051}, {527,38189}, {528,38199}, {542,38109}, {599,6668}, {758,38089}, {952,38023}, {1503,38078}, {3564,38085}, {3589,31157}, {5842,38072}, {5845,38096}, {5846,38100}, {5847,38105}, {5848,38106}, {5852,38086}, {5855,38087}, {5857,38088}, {20423,31659}, {28538,38058}

X(38091) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38023, 38079, 38090), (38051, 38198, 38193), (38120, 38169, 38148)

X(38092) = CENTROID OF TRIANGLE {X(2), X(7), X(8)}

Barycentrics    5*a^3-5*(b+c)*a^2+(7*b^2+18*b*c+7*c^2)*a-7*(b^2-c^2)*(b-c) : :
X(38092) = 4*X(2)-X(390) = 7*X(2)-4*X(1001) = X(2)+2*X(2550) = 5*X(2)-8*X(3826) = X(7)+2*X(3679) = X(8)+2*X(6173) = 4*X(8)+5*X(30340) = 4*X(10)-X(6172) = 7*X(390)-16*X(1001) = X(390)+8*X(2550) = 3*X(390)-8*X(38025) = 2*X(1001)+7*X(2550) = 5*X(1001)-14*X(3826) = 6*X(1001)-7*X(38025) = 5*X(2550)+4*X(3826) = 3*X(2550)+X(38025) = 12*X(3826)-5*X(38025) = 8*X(6173)-5*X(30340) = 2*X(6174)+X(20119) = 2*X(38121)+X(38149) = X(38121)+2*X(38170)

X(38092) lies on these lines: {2,11}, {7,3679}, {8,6173}, {10,6172}, {30,38121}, {142,3241}, {381,35514}, {516,3839}, {517,38073}, {519,11038}, {524,38185}, {527,5686}, {529,38203}, {952,38065}, {971,38074}, {1698,30332}, {2094,25006}, {2951,34648}, {3600,5288}, {3617,4741}, {3621,25557}, {3698,10394}, {3820,30311}, {3828,18230}, {3918,5696}, {4208,34619}, {4677,5542}, {4731,15726}, {4745,5223}, {5261,17528}, {5265,30312}, {5762,38066}, {5843,38081}, {5844,38080}, {5845,38087}, {5846,38086}, {5850,38098}, {5851,38099}, {5852,38100}, {5853,38093}, {5854,38095}, {5855,38096}, {7676,16418}, {7677,16417}, {8165,17577}, {8236,25055}, {8543,9709}, {9623,18450}, {9708,30295}, {9710,34610}, {11236,37161}, {11240,26060}, {11495,15683}, {12632,37436}, {15709,38031}, {16845,34707}, {17373,31145}, {20330,34631}, {21151,28204}, {24473,34784}, {28534,38057}, {34746,37108}

X(38092) = reflection of X(i) in X(j) for these (i,j): (8236, 25055), (25055, 38204), (38024, 38094)
X(38092) = anticomplement of X(38025)
X(38092) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38024, 38052, 38094), (38121, 38170, 38149)

X(38093) = CENTROID OF TRIANGLE {X(2), X(7), X(9)}

Barycentrics    a^2-5*(b+c)*a+4*(b-c)^2 : :
X(38093) = 5*X(2)+X(7) = 4*X(2)-X(9) = X(2)+2*X(142) = 13*X(2)-X(144) = 7*X(2)-X(6172) = 2*X(2)+X(6173) = 7*X(2)-4*X(6666) = 11*X(2)-5*X(18230) = 2*X(2)-5*X(20195) = 4*X(7)+5*X(9) = X(7)-10*X(142) = 13*X(7)+5*X(144) = 7*X(7)+5*X(6172) = 2*X(7)-5*X(6173) = 7*X(7)+20*X(6666) = X(9)+8*X(142) = 13*X(9)-4*X(144) = 7*X(9)-4*X(6172) = X(9)+2*X(6173) = 7*X(9)-16*X(6666) = 11*X(9)-20*X(18230) = X(9)-10*X(20195)

This triangle has collinear vertices.

X(38093) lies on these lines: {2,7}, {30,38122}, {140,5735}, {381,5732}, {443,34701}, {516,3524}, {518,3921}, {519,38053}, {524,38186}, {528,15015}, {529,38206}, {547,31657}, {549,5805}, {551,2550}, {971,5055}, {1001,5010}, {1086,16676}, {1449,17278}, {1698,25557}, {2801,38104}, {2951,30308}, {3243,3679}, {3247,4859}, {3254,6174}, {3340,30312}, {3534,18482}, {3545,21151}, {3624,5880}, {3654,20330}, {3828,5542}, {3834,16832}, {3848,10177}, {3875,29575}, {3925,31146}, {4034,4869}, {4321,11237}, {4326,11238}, {4384,17297}, {4395,29602}, {4648,4909}, {4659,17264}, {4675,16670}, {4725,16833}, {4870,12560}, {4873,29627}, {4888,17337}, {4902,16814}, {4971,29573}, {5054,21153}, {5066,31672}, {5131,28534}, {5223,19876}, {5308,17067}, {5436,11112}, {5528,10707}, {5564,17234}, {5698,19862}, {5759,15702}, {5762,11539}, {5779,15703}, {5843,38082}, {5845,38088}, {5850,38101}, {5851,38102}, {5852,38103}, {5853,38092}, {5856,38095}, {5857,38096}, {7988,15726}, {10012,31169}, {10582,31140}, {10956,38099}, {11523,17529}, {12625,37436}, {15254,34595}, {15570,34747}, {15693,31671}, {15694,31658}, {15699,38108}, {16831,17399}, {17132,36911}, {17237,31244}, {17265,17359}, {17284,34824}, {17298,17346}, {17307,28650}, {17384,31312}, {17504,38137}, {17605,30353}, {18065,30044}, {19877,30340}, {19878,30424}, {28313,29600}, {29598,36834}, {31434,38055}, {38030,38154}, {38031,38172}, {38037,38123}, {38041,38126}, {38054,38057}

X(38093) = midpoint of X(i) and X(j) for these {i,j}: {3524, 38073}, {3545, 21151}, {5054, 38107}, {5055, 38065}, {11539, 38080}, {15699, 38111}, {17504, 38137}, {19875, 38024}, {19883, 38094}, {21358, 38086}, {25055, 38052}
X(38093) = reflection of X(i) in X(j) for these (i,j): (21153, 5054), (38025, 19883), (38067, 11539), (38075, 5055), (38097, 19875), (38108, 15699)
X(38093) = barycentric product X(8)*X(20121)
X(38093) = trilinear product X(9)*X(20121)
X(38093) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 142, 6173), (2, 5249, 31142), (2, 6172, 6666), (2, 6173, 9), (2, 27186, 31164), (2, 31164, 7308), (142, 20195, 9), (4675, 31183, 16670), (4859, 17245, 3247), (6173, 20195, 2), (38053, 38204, 38200), (38122, 38171, 38150)

X(38094) = CENTROID OF TRIANGLE {X(2), X(7), X(10)}

Barycentrics    4*a^3+5*(b+c)*a^2+2*(b^2+18*b*c+c^2)*a-11*(b^2-c^2)*(b-c) : :
X(38094) = 5*X(2)+X(4312) = X(7)+2*X(3828) = X(10)+2*X(6173) = 3*X(10)-2*X(38097) = 4*X(142)-X(551) = 10*X(142)-X(30331) = X(144)-7*X(19876) = 5*X(551)-2*X(30331) = 3*X(6173)+X(38097) = 3*X(11038)-5*X(38024) = X(11038)+5*X(38052) = 2*X(11038)-5*X(38054) = 3*X(11038)+5*X(38092) = 4*X(11038)+5*X(38201) = 3*X(19883)-2*X(38025) = X(38024)+3*X(38052) = 2*X(38024)-3*X(38054) = 4*X(38024)+3*X(38201) = X(38025)-3*X(38093) = 2*X(38123)+X(38151) = X(38123)+2*X(38172) = X(38151)-4*X(38172)

X(38094) lies on these lines: {2,4312}, {7,3828}, {10,6173}, {30,38123}, {142,214}, {144,19876}, {515,38065}, {516,3524}, {517,38080}, {518,38098}, {519,11038}, {524,38187}, {527,38101}, {529,38208}, {758,38096}, {971,38076}, {1125,30332}, {2802,38095}, {3625,25557}, {3634,6172}, {4669,5542}, {4691,30340}, {5493,37407}, {5762,38068}, {5843,38083}, {5845,38089}, {5847,38086}, {5850,19875}, {5851,38104}, {5852,38105}, {5880,19862}, {28194,38107}, {28202,38137}, {28204,38111}, {28534,38059}

X(38094) = midpoint of X(38024) and X(38092)
X(38094) = reflection of X(19883) in X(38093)
X(38094) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38024, 38052, 38092), (38052, 38054, 38201), (38123, 38172, 38151)

X(38095) = CENTROID OF TRIANGLE {X(2), X(7), X(11)}

Barycentrics    4*a^5-(13*b^2-14*b*c+13*c^2)*a^3-(b+c)*(b^2-6*b*c+c^2)*a^2+(21*b^2-10*b*c+21*c^2)*(b-c)^2*a-11*(b^2-c^2)*(b-c)^3 : :
X(38095) = 4*X(2)-X(6068) = X(11)+2*X(6173) = 4*X(142)-X(6174) = X(6172)-4*X(6667) = 2*X(38055)+X(38202) = X(38055)+2*X(38207) = X(38102)-3*X(38205) = 2*X(38124)+X(38152) = X(38124)+2*X(38173) = X(38152)-4*X(38173) = X(38202)-4*X(38207)

X(38095) lies on these lines: {2,6068}, {11,6173}, {30,38124}, {142,6174}, {516,38026}, {518,38099}, {519,38055}, {524,38188}, {527,38102}, {528,8236}, {529,38209}, {952,38024}, {971,38077}, {2801,38076}, {2802,38094}, {2829,38073}, {5298,28534}, {5762,38069}, {5840,38065}, {5843,38084}, {5845,38090}, {5848,38086}, {5850,38104}, {5852,38106}, {5854,38092}, {5856,38093}, {6172,6667}, {11237,30275}

X(38095) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38024, 38080, 38096), (38055, 38207, 38202), (38124, 38173, 38152)

X(38096) = CENTROID OF TRIANGLE {X(2), X(7), X(12)}

Barycentrics    4*a^6+4*(b+c)*a^5-(13*b^2-18*b*c+13*c^2)*a^4-2*(b+c)*(7*b^2+10*b*c+7*c^2)*a^3+4*(5*b^4+5*c^4-7*b*c*(b^2+c^2))*a^2+2*(b^2-c^2)*(b-c)*(5*b^2+14*b*c+5*c^2)*a-11*(b^2-c^2)^2*(b-c)^2 : :
X(38096) = X(12)+2*X(6173) = 4*X(142)-X(31157) = X(6172)-4*X(6668) = 2*X(38056)+X(38203) = X(38056)+2*X(38208) = X(38103)-3*X(38206) = 2*X(38125)+X(38153) = X(38125)+2*X(38174) = X(38153)-4*X(38174) = X(38203)-4*X(38208)

X(38096) lies on these lines: {12,6173}, {30,38125}, {142,31157}, {516,38027}, {518,38100}, {519,38056}, {524,38189}, {527,38103}, {528,38209}, {758,38094}, {952,38024}, {971,38078}, {5762,38070}, {5842,38073}, {5843,38085}, {5845,38091}, {5849,38086}, {5850,38105}, {5851,38106}, {5855,38092}, {5857,38093}, {6172,6668}, {28534,38061}

X(38096) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38024, 38080, 38095), (38056, 38208, 38203), (38125, 38174, 38153)

X(38097) = CENTROID OF TRIANGLE {X(2), X(8), X(9)}

Barycentrics    a^3+8*(b+c)*a^2-(13*b^2+18*b*c+13*c^2)*a+4*(b^2-c^2)*(b-c) : :
X(38097) = 4*X(2)-X(3243) = X(2)+2*X(24393) = X(9)+2*X(3679) = 4*X(10)-X(6173) = 3*X(10)-X(38094) = X(3243)+8*X(24393) = 3*X(6173)-4*X(38094) = 3*X(19875)-X(38024) = 2*X(38024)-3*X(38093) = X(38025)-3*X(38057) = 5*X(38025)-6*X(38059) = X(38025)+6*X(38210) = 5*X(38057)-2*X(38059) = 3*X(38057)-2*X(38101) = X(38057)+2*X(38210) = 3*X(38059)-5*X(38101) = X(38059)+5*X(38210) = 2*X(38126)+X(38154) = X(38126)+2*X(38175) = X(38154)-4*X(38175)

X(38097) lies on these lines: {2,3243}, {9,80}, {10,6173}, {30,38126}, {516,38074}, {517,38075}, {518,3921}, {519,38025}, {524,38190}, {527,5686}, {529,38212}, {952,38067}, {971,38066}, {1001,4677}, {2550,4745}, {3241,6666}, {3617,6172}, {3828,20195}, {4668,15254}, {4691,5698}, {4711,10177}, {4866,11236}, {5762,38081}, {5844,38082}, {5846,38088}, {5854,38102}, {5855,38103}, {5856,38099}, {5857,38100}, {18230,31145}, {21153,28204}, {25006,31142}

X(38097) = reflection of X(i) in X(j) for these (i,j): (38025, 38101), (38093, 19875)
X(38097) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38025, 38057, 38101), (38126, 38175, 38154)

X(38098) = CENTROID OF TRIANGLE {X(2), X(8), X(10)}

Barycentrics    4*a-11*b-11*c : :
Trilinears    15 r - 22 R sin B sin C : :
X(38098) = 5*X(1)-11*X(2) = 7*X(1)+11*X(8) = 2*X(1)-11*X(10) = 8*X(1)-11*X(551) = 17*X(1)-11*X(3241) = 20*X(1)-11*X(3244) = 16*X(1)+11*X(3625) = X(1)+11*X(3679) = 4*X(1)+11*X(4669) = 13*X(1)+11*X(4677) = 3*X(1)-11*X(19875) = 6*X(1)-11*X(19883) = 7*X(1)-11*X(25055) = 19*X(1)+11*X(31145) = 7*X(2)+5*X(8) = 2*X(2)-5*X(10) = 8*X(2)-5*X(551) = 13*X(2)-10*X(1125) = 17*X(2)-5*X(3241) = 4*X(2)-X(3244) = 16*X(2)+5*X(3625) = X(2)+2*X(3626) = 5*X(2)+X(3632) = 17*X(2)-20*X(3634) = 7*X(2)-4*X(3636) = X(2)+5*X(3679) = 7*X(2)-10*X(3828) = 4*X(2)+5*X(4669) = 13*X(2)+5*X(4677) = X(2)+20*X(4691) = X(2)-10*X(4745) = 19*X(2)+20*X(4746) = 10*X(2)-7*X(15808) = 3*X(2)-5*X(19875) = 6*X(2)-5*X(19883) = 13*X(2)-X(20050) = 19*X(2)-7*X(20057) = 7*X(2)-5*X(25055) = 19*X(2)+5*X(31145) = 2*X(8)+7*X(10)

This triangle has collinear vertices.

X(38098) lies on these lines: {1,2}, {30,38127}, {355,15681}, {382,3654}, {515,15688}, {516,38074}, {517,38071}, {518,38094}, {524,38191}, {527,38201}, {528,38213}, {529,38214}, {546,11362}, {550,28208}, {594,16590}, {758,38100}, {946,11737}, {952,38068}, {966,36911}, {2802,3921}, {3524,28236}, {3528,34627}, {3529,34638}, {3655,15720}, {3681,4744}, {3839,28228}, {3851,4301}, {3855,31162}, {3913,16860}, {3918,24473}, {3982,11237}, {3992,4793}, {4015,31165}, {4084,4662}, {4297,34200}, {4421,17571}, {4540,10914}, {4681,4709}, {4686,4732}, {4711,5883}, {4733,28558}, {4796,10022}, {5055,28234}, {5232,36588}, {5288,36006}, {5690,15687}, {5790,14269}, {5844,38083}, {5846,38089}, {5847,38087}, {5850,38092}, {5853,38101}, {5854,38104}, {5855,38105}, {5881,10299}, {5882,14869}, {6684,15700}, {8715,19526}, {10109,11278}, {10164,17504}, {10304,37712}, {11194,17573}, {11545,30331}, {13607,15694}, {15707,26446}, {15863,35023}, {16857,25439}, {17251,28301}, {17320,32101}, {28202,38138}, {31399,35018}, {34632,37714}

X(38098) = midpoint of X(i) and X(j) for these {i,j}: {8, 25055}, {10304, 37712}
X(38098) = reflection of X(i) in X(j) for these (i,j): (19883, 19875), (25055, 3828)
X(38098) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3244, 551), (2, 3679, 3626), (2, 31145, 20057), (8, 10, 19862), (10, 3626, 3244), (10, 3679, 4669), (10, 4669, 551), (10, 19883, 19875), (551, 4669, 3625), (3244, 19862, 3636), (3617, 3679, 4745), (3617, 4691, 10), (3626, 3636, 8), (3632, 15808, 3244), (3636, 3828, 2), (3654, 34648, 5493), (3679, 4677, 4678), (3679, 4745, 10), (4691, 4745, 3679), (38127, 38176, 38155)

X(38099) = CENTROID OF TRIANGLE {X(2), X(8), X(11)}

Barycentrics    4*a^4-12*(b+c)*a^3+(7*b^2+22*b*c+7*c^2)*a^2+4*(b+c)*(3*b^2-8*b*c+3*c^2)*a-11*(b^2-c^2)^2 : :
X(38099) = 4*X(2)-X(1317) = X(2)+2*X(3036) = 5*X(2)+X(12531) = 4*X(10)-X(6174) = 10*X(10)-X(10609) = X(11)+2*X(3679) = 11*X(11)-2*X(12653) = X(1317)+8*X(3036) = 5*X(1317)+4*X(12531) = 10*X(3036)-X(12531) = 11*X(3679)+X(12653) = 5*X(6174)-2*X(10609) = 2*X(32557)-5*X(34122) = 6*X(32557)-5*X(38026) = 3*X(32557)-5*X(38104) = X(32557)+5*X(38213) = 3*X(34122)-X(38026) = 3*X(34122)-2*X(38104) = X(34122)+2*X(38213) = 2*X(38128)+X(38156) = X(38128)+2*X(38177) = X(38156)-4*X(38177)

X(38099) lies on these lines: {2,1317}, {10,6174}, {11,3679}, {30,38128}, {80,31508}, {517,38077}, {518,38095}, {519,32557}, {524,38192}, {527,38202}, {528,38057}, {529,38215}, {952,3653}, {1145,4745}, {1387,4677}, {2801,4731}, {2802,3921}, {2829,38074}, {3241,6667}, {3617,10707}, {3634,11274}, {3828,15863}, {4669,6702}, {5840,38066}, {5844,38084}, {5846,38090}, {5848,38087}, {5851,38092}, {5853,38102}, {5855,38106}, {5856,38097}, {6735,38060}, {6931,34710}, {7972,19876}, {9780,10031}, {10087,16857}, {10956,38093}, {21154,28204}, {25416,34641}, {31145,31272}, {37438,37725}

X(38099) = reflection of X(38026) in X(38104)
X(38099) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (19875, 38081, 38100), (34122, 38026, 38104), (38128, 38177, 38156)

X(38100) = CENTROID OF TRIANGLE {X(2), X(8), X(12)}

Barycentrics    4*a^4-12*(b+c)*a^3+(7*b^2+26*b*c+7*c^2)*a^2+4*(b+c)*(3*b^2-4*b*c+3*c^2)*a-11*(b^2-c^2)^2 : :
X(38100) = 4*X(2)-X(37734) = 4*X(10)-X(31157) = X(12)+2*X(3679) = X(3241)-4*X(6668) = 8*X(3828)-5*X(31260) = X(4677)+2*X(37737) = X(38027)-3*X(38058) = 5*X(38027)-6*X(38062) = X(38027)+6*X(38214) = 5*X(38058)-2*X(38062) = 3*X(38058)-2*X(38105) = X(38058)+2*X(38214) = 3*X(38062)-5*X(38105) = X(38062)+5*X(38214) = X(38105)+3*X(38214) = 2*X(38129)+X(38157) = X(38129)+2*X(38178) = X(38157)-4*X(38178) = X(38203)+2*X(38212)

X(38100) lies on these lines: {2,37734}, {10,31157}, {12,3340}, {30,38129}, {517,38078}, {518,38096}, {519,38027}, {524,38193}, {527,38203}, {528,38215}, {758,38098}, {952,3653}, {3241,6668}, {3828,31260}, {4677,37737}, {5842,38074}, {5844,38085}, {5846,38091}, {5849,38087}, {5852,38092}, {5853,38103}, {5854,38106}, {5857,38097}, {6933,34743}, {21155,28204}

X(38100) = reflection of X(38027) in X(38105)
X(38100) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (19875, 38081, 38099), (38027, 38058, 38105), (38129, 38178, 38157)

X(38101) = CENTROID OF TRIANGLE {X(2), X(9), X(10)}

Barycentrics    8*a^3+(b+c)*a^2-2*(7*b^2+18*b*c+7*c^2)*a+5*(b^2-c^2)*(b-c) : :
X(38101) = 5*X(2)+X(5223) = 4*X(2)-X(5542) = X(7)-7*X(19876) = X(9)+2*X(3828) = 5*X(10)+4*X(15254) = 4*X(5223)+5*X(5542) = 3*X(5223)+5*X(38024) = 3*X(5542)-4*X(38024) = 3*X(19875)-X(38092) = X(38025)+3*X(38057) = 2*X(38025)-3*X(38059) = 4*X(38025)+3*X(38210) = 2*X(38057)+X(38059) = 3*X(38057)-X(38097) = 4*X(38057)-X(38210) = 3*X(38059)+2*X(38097) = 2*X(38059)+X(38210) = 2*X(38130)+X(38158) = X(38130)+2*X(38179) = X(38158)-4*X(38179)

X(38101) lies on these lines: {2,5223}, {7,19876}, {9,3828}, {10,528}, {30,38130}, {480,17542}, {515,38067}, {516,3839}, {517,38082}, {518,19883}, {519,38025}, {524,38194}, {527,38094}, {529,38217}, {551,3940}, {758,38103}, {971,38068}, {1001,4669}, {1698,6172}, {2802,38102}, {3634,6173}, {3679,18230}, {3956,10177}, {4301,6886}, {4995,10392}, {5686,25055}, {5762,38083}, {5847,38088}, {5850,38093}, {5853,38098}, {5856,38104}, {5857,38105}, {28194,38108}, {28202,38139}, {28204,38113}

X(38101) = midpoint of X(i) and X(j) for these {i,j}: {5686, 25055}, {38025, 38097}
X(38101) = complement of X(38024)
X(38101) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38025, 38057, 38097), (38057, 38059, 38210), (38130, 38179, 38158)

X(38102) = CENTROID OF TRIANGLE {X(2), X(9), X(11)}

Barycentrics    8*a^5-18*(b+c)*a^4+(b^2+46*b*c+c^2)*a^3+5*(b+c)*(5*b^2-12*b*c+5*c^2)*a^2-(21*b^2+20*b*c+21*c^2)*(b-c)^2*a+5*(b^2-c^2)*(b-c)^3 : :
X(38102) = 5*X(2)+X(1156) = 4*X(2)-X(10427) = 4*X(1156)+5*X(10427) = X(6172)+5*X(31272) = X(6173)-4*X(6667) = X(6174)-4*X(6666) = X(10707)+5*X(18230) = 2*X(38060)+X(38211) = X(38060)+2*X(38216) = 2*X(38095)-3*X(38205) = 2*X(38131)+X(38159) = X(38131)+2*X(38180) = X(38159)-4*X(38180) = X(38211)-4*X(38216)

X(38102) lies on these lines: {2,1156}, {30,38131}, {516,38077}, {518,38026}, {519,38060}, {524,38195}, {527,38095}, {528,19875}, {529,38218}, {952,38025}, {971,38069}, {2801,19883}, {2802,38101}, {2829,38075}, {3582,38055}, {5762,38084}, {5840,38067}, {5848,38088}, {5851,38093}, {5853,38099}, {5854,38097}, {5857,38106}, {6172,31272}, {6173,6667}, {6174,6666}, {10707,18230}

X(38102) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38025, 38082, 38103), (38060, 38216, 38211), (38131, 38180, 38159)

X(38103) = CENTROID OF TRIANGLE {X(2), X(9), X(12)}

Barycentrics    8*a^6-10*(b+c)*a^5-(17*b^2+18*b*c+17*c^2)*a^4+2*(b+c)*(13*b^2+7*b*c+13*c^2)*a^3+4*(b^4+c^4+4*b*c*(b^2+c^2))*a^2-2*(b^2-c^2)*(b-c)*(8*b^2+17*b*c+8*c^2)*a+5*(b^2-c^2)^2*(b-c)^2 : :
X(38103) = X(6173)-4*X(6668) = 4*X(6666)-X(31157) = 2*X(38061)+X(38212) = X(38061)+2*X(38217) = 2*X(38096)-3*X(38206) = 2*X(38132)+X(38160) = X(38132)+2*X(38181) = X(38160)-4*X(38181) = X(38212)-4*X(38217)

X(38103) lies on these lines: {30,38132}, {516,38078}, {518,38027}, {519,38061}, {524,38196}, {527,38096}, {528,38058}, {758,38101}, {952,38025}, {971,38070}, {5762,38085}, {5842,38075}, {5849,38088}, {5852,38093}, {5853,38100}, {5855,38097}, {5856,38106}, {6173,6668}, {6666,31157}

X(38103) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38025, 38082, 38102), (38061, 38217, 38212), (38132, 38181, 38160)

X(38104) = CENTROID OF TRIANGLE {X(2), X(10), X(11)}

Barycentrics    2*a^4+3*(b+c)*a^3-2*(5*b^2-b*c+5*c^2)*a^2-(b+c)*(3*b^2-11*b*c+3*c^2)*a+8*(b^2-c^2)^2 : :
X(38104) = 5*X(2)+X(80) = 4*X(2)-X(214) = 13*X(2)-X(6224) = X(2)+2*X(6702) = X(11)+2*X(3828) = 4*X(80)+5*X(214) = 13*X(80)+5*X(6224) = X(80)-10*X(6702) = X(100)-7*X(19876) = 13*X(214)-4*X(6224) = X(214)+8*X(6702) = X(32557)+2*X(34122) = 3*X(32557)-2*X(38026) = 3*X(32557)+2*X(38099) = 2*X(32557)+X(38213) = 3*X(34122)+X(38026) = 3*X(34122)-X(38099) = 4*X(34122)-X(38213) = 2*X(38133)+X(38161) = X(38133)+2*X(38182) = X(38161)-4*X(38182)

X(38104) lies on these lines: {2,80}, {11,3828}, {30,38133}, {100,19876}, {515,38069}, {516,38077}, {517,38084}, {519,32557}, {524,38197}, {527,3814}, {528,38059}, {529,38219}, {547,12619}, {549,6246}, {551,6667}, {758,38106}, {952,19883}, {1125,11274}, {1387,4669}, {1698,10707}, {2800,5055}, {2801,38093}, {2802,19875}, {2829,38076}, {3624,10031}, {3634,6174}, {3654,16174}, {3679,31272}, {4745,33709}, {4973,31160}, {5840,38068}, {5847,38090}, {5848,38089}, {5850,38095}, {5851,38094}, {5854,38098}, {5856,38101}, {6265,15703}, {8988,32788}, {10109,12611}, {10609,31253}, {12119,15702}, {12515,19709}, {12747,15723}, {13976,32787}, {23513,28194}, {28202,38141}, {28204,34126}

X(38104) = midpoint of X(38026) and X(38099)
X(38104) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (19883, 38083, 38105), (32557, 34122, 38213), (34122, 38026, 38099), (38133, 38182, 38161)

X(38105) = CENTROID OF TRIANGLE {X(2), X(10), X(12)}

Barycentrics    2*a^4+3*(b+c)*a^3-2*(5*b^2+7*b*c+5*c^2)*a^2-(b+c)*(3*b^2-b*c+3*c^2)*a+8*(b^2-c^2)^2 : :
X(38105) = 5*X(2)+X(37710) = X(12)+2*X(3828) = X(551)-4*X(6668) = X(2975)-7*X(19876) = 4*X(3634)-X(31157) = X(4669)+2*X(37737) = X(38027)+3*X(38058) = 2*X(38027)-3*X(38062) = 4*X(38027)+3*X(38214) = 2*X(38058)+X(38062) = 3*X(38058)-X(38100) = 4*X(38058)-X(38214) = 3*X(38062)+2*X(38100) = 2*X(38062)+X(38214) = 4*X(38100)-3*X(38214) = 2*X(38134)+X(38162) = X(38134)+2*X(38183) = X(38162)-4*X(38183) = X(38208)+2*X(38217)

X(38105) lies on these lines: {2,21842}, {12,553}, {30,38134}, {515,38070}, {516,38078}, {517,38085}, {519,38027}, {524,38198}, {527,38208}, {528,38219}, {551,6668}, {758,19875}, {952,19883}, {2802,38106}, {2975,19876}, {3634,31157}, {4669,37737}, {5722,10197}, {5842,38076}, {5847,38091}, {5849,38089}, {5850,38096}, {5852,38094}, {5855,38098}, {5857,38101}, {28194,38109}, {28202,38142}, {28204,38114}

X(38105) = midpoint of X(38027) and X(38100)
X(38105) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (19883, 38083, 38104), (38027, 38058, 38100), (38058, 38062, 38214), (38134, 38183, 38162)

X(38106) = CENTROID OF TRIANGLE {X(2), X(11), X(12)}

Barycentrics    a^7-(b+c)*a^6-2*(3*b^2-b*c+3*c^2)*a^5+2*(b+c)*(3*b^2-b*c+3*c^2)*a^4+(9*b^4+9*c^4-5*b*c*(2*b^2+b*c+2*c^2))*a^3-(b+c)*(9*b^4+9*c^4-b*c*(10*b^2-3*b*c+10*c^2))*a^2-4*(b^2-c^2)^2*(b-c)^2*a+4*(b^2-c^2)^3*(b-c) : :
X(38106) = 4*X(2)-X(4996) = X(2)+2*X(8068) = X(4996)+8*X(8068) = X(6174)-4*X(6668) = 4*X(6667)-X(31157) = 2*X(38063)+X(38215) = X(38063)+2*X(38219) = 2*X(38135)+X(38163) = X(38135)+2*X(38184) = X(38163)-4*X(38184) = X(38209)+2*X(38218) = X(38215)-4*X(38219)

X(38106) lies on these lines: {2,4996}, {30,38135}, {519,38063}, {524,38199}, {527,38209}, {528,38061}, {758,38104}, {952,5055}, {999,31272}, {2802,38105}, {2829,38078}, {3295,10707}, {5172,37375}, {5425,6702}, {5840,38070}, {5842,38077}, {5848,38091}, {5849,38090}, {5851,38096}, {5852,38095}, {5854,38100}, {5855,38099}, {5856,38103}, {5857,38102}, {6174,6668}, {6667,31157}, {17100,17532}

X(38106) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38063, 38219, 38215), (38135, 38184, 38163)

X(38107) = CENTROID OF TRIANGLE {X(3), X(4), X(7)}

Barycentrics    a^6-2*(b^2-b*c+c^2)*a^4-2*(b+c)*(b^2+c^2)*a^3+3*(b^2+c^2)*(b-c)^2*a^2+2*(b^2-c^2)^2*(b+c)*a-2*(b^2-c^2)^2*(b-c)^2 : :
X(38107) = X(3)-4*X(142) = X(3)+2*X(5805) = 2*X(3)+X(31671) = X(4)+2*X(31657) = 2*X(5)+X(7) = 4*X(5)-X(5779) = 2*X(7)+X(5779) = 2*X(9)-5*X(1656) = 2*X(142)+X(5805) = 8*X(142)+X(31671) = 4*X(5805)-X(31671) = 4*X(5880)+5*X(18493) = 2*X(21151)-3*X(38065) = X(21151)+3*X(38073) = X(21151)-6*X(38080) = X(21151)+2*X(38137) = X(21168)-6*X(38171) = X(31671)+4*X(38122) = X(38065)+2*X(38073) = X(38065)-4*X(38080) = 3*X(38065)-4*X(38111) = 3*X(38065)+4*X(38137) = X(38113)-3*X(38171)

X(38107) lies on these lines: {2,5762}, {3,142}, {4,31657}, {5,7}, {9,1656}, {30,21151}, {57,7082}, {140,5759}, {144,3090}, {355,5542}, {381,971}, {382,5732}, {390,5901}, {511,38143}, {515,38030}, {517,38036}, {518,5790}, {527,5055}, {542,38086}, {546,36991}, {547,6172}, {942,5290}, {952,1056}, {954,6911}, {1482,2550}, {1503,38115}, {1699,5918}, {2095,3820}, {2346,32141}, {2800,38207}, {2829,38124}, {2951,22793}, {3091,36996}, {3243,12645}, {3254,12331}, {3358,37612}, {3526,5735}, {3545,38139}, {3564,38164}, {3628,18230}, {3652,13159}, {3843,31672}, {3940,6854}, {4307,15251}, {4312,8227}, {4654,10157}, {5044,5833}, {5050,38186}, {5054,21153}, {5056,20059}, {5070,6666}, {5223,9956}, {5249,19541}, {5603,35272}, {5657,8728}, {5686,38042}, {5709,11231}, {5731,20420}, {5733,17366}, {5763,17582}, {5791,10172}, {5812,16853}, {5840,38152}, {5842,38125}, {5844,38170}, {5845,14561}, {5850,10175}, {5851,23513}, {5852,38109}, {5853,10247}, {6147,6864}, {6824,8732}, {6846,24470}, {6855,34753}, {6859,12848}, {6913,18541}, {6918,21617}, {6944,8232}, {7580,27186}, {7680,33558}, {7717,21841}, {7743,10384}, {8236,10283}, {8255,18530}, {8727,9776}, {9669,14100}, {9955,11372}, {10246,28452}, {10427,10738}, {10861,17532}, {11502,17718}, {11518,37712}, {11928,17668}, {12650,37615}, {13374,15587}, {15298,31479}, {16593,24833}, {17313,29016}, {18357,30340}, {18443,28160}, {18525,25557}, {19907,20119}, {21164,24644}, {22791,35514}, {24474,38200}, {26446,38204}, {26806,36652}, {27475,29010}, {28194,38094}, {28204,38024}, {28234,38201}, {30274,37718}, {38037,38205}

X(38107) = midpoint of X(i) and X(j) for these {i,j}: {7, 5817}, {5805, 38122}, {6173, 38150}, {11038, 38149}, {38036, 38052}, {38054, 38151}, {38111, 38137}
X(38107) = reflection of X(i) in X(j) for these (i,j): (2, 38171), (3, 38122), (381, 38150), (5050, 38186), (5054, 38093), (5686, 38042), (5779, 5817), (5817, 5), (8236, 10283), (10246, 38053), (11038, 38041), (21151, 38111), (21168, 38113), (26446, 38204), (38030, 38054), (38052, 38172), (38121, 38052), (38122, 142)
X(38107) = anticomplement of X(38113)
X(38107) = complement of X(21168)
X(38107) = X(31671)-Gibert-Moses centroid
X(38107) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 21168, 38113), (3, 5805, 31671), (5, 7, 5779), (5, 5708, 5789), (142, 5805, 3), (2550, 20330, 1482), (5732, 18482, 382), (5735, 20195, 31658), (20195, 31658, 3526), (21151, 38111, 38065), (38036, 38172, 38121), (38073, 38080, 38065), (38080, 38137, 38111), (38173, 38174, 38041)

X(38108) = CENTROID OF TRIANGLE {X(3), X(4), X(9)}

Barycentrics    a^6-(b+c)*a^5-3*(b^2+c^2)*a^4+4*(b+c)*(b^2+c^2)*a^3+(b^2+4*b*c+c^2)*(b-c)^2*a^2-3*(b^2-c^2)^2*(b+c)*a+(b^2-c^2)^2*(b-c)^2 : :
X(38108) = X(3)-4*X(6666) = 2*X(3)+X(31672) = X(4)+5*X(18230) = X(4)+2*X(31658) = 2*X(5)+X(9) = 10*X(5)-X(5735) = 4*X(5)-X(5805) = 3*X(5)-X(38137) = 5*X(9)+X(5735) = 2*X(9)+X(5805) = 3*X(9)+2*X(38137) = 2*X(5735)-5*X(5805) = 3*X(5735)-10*X(38137) = X(5735)-5*X(38150) = 3*X(5805)-4*X(38137) = 3*X(5817)+X(21151) = 2*X(5817)+X(38122) = 8*X(6666)+X(31672) = 5*X(18230)-2*X(31658) = 2*X(21151)-3*X(38122)

X(38108) lies on these lines: {2,971}, {3,6666}, {4,18230}, {5,9}, {7,3090}, {11,15298}, {12,15299}, {30,21153}, {72,6886}, {140,5732}, {142,1656}, {144,5056}, {355,1001}, {381,516}, {390,5818}, {498,14100}, {499,8581}, {511,38145}, {515,16857}, {517,38037}, {518,5886}, {527,5055}, {542,38088}, {547,5843}, {631,36991}, {942,8232}, {952,38043}, {954,5722}, {990,17337}, {1479,15837}, {1482,24393}, {1503,38117}, {1698,11372}, {2550,6893}, {2800,38216}, {2801,38030}, {2829,38131}, {2951,31423}, {3091,5759}, {3243,5901}, {3305,8226}, {3358,6259}, {3545,21168}, {3564,38166}, {3628,20195}, {3634,9842}, {3824,5811}, {3826,6842}, {3851,31671}, {4208,22792}, {4312,24914}, {4321,15325}, {5044,6846}, {5067,36996}, {5071,6172}, {5219,10398}, {5223,8227}, {5296,36682}, {5587,34746}, {5603,5686}, {5659,30308}, {5698,6867}, {5728,6832}, {5729,21617}, {5733,16669}, {5777,6887}, {5784,6862}, {5785,30827}, {5787,11108}, {5789,9843}, {5790,5853}, {5809,24929}, {5840,38159}, {5842,38132}, {5844,38175}, {5850,10171}, {5856,23513}, {5857,38109}, {6245,16853}, {6594,10738}, {6705,16863}, {6829,37787}, {6852,10394}, {6864,31445}, {6881,8257}, {6928,15254}, {6975,29007}, {7174,15251}, {7308,8727}, {7988,31142}, {9654,12573}, {9780,35514}, {9856,19855}, {10172,38204}, {10384,31434}, {10392,13411}, {10739,28345}, {10883,35595}, {10884,17590}, {11230,38053}, {11375,18412}, {12618,17259}, {13727,17338}, {15587,26364}, {15699,38093}, {17260,36652}, {17582,34862}, {19875,24644}, {20117,30329}, {25722,27529}, {26685,36660}, {28194,38101}, {28204,38025}, {28234,38210}, {28629,31821}, {29335,36661}, {38042,38200}

X(38108) = midpoint of X(i) and X(j) for these {i,j}: {2, 5817}, {9, 38150}, {5603, 5686}, {38037, 38057}, {38059, 38158}, {38113, 38139}
X(38108) = reflection of X(i) in X(j) for these (i,j): (5805, 38150), (6173, 38171), (21153, 38113), (38031, 38059), (38053, 11230), (38057, 38179), (38093, 15699), (38122, 2), (38126, 38057), (38150, 5), (38171, 547), (38200, 38042), (38204, 10172)
X(38108) = complement of X(21151)
X(38108) = X(31672)-Gibert-Moses centroid
X(38108) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 18230, 31658), (5, 9, 5805), (1656, 5779, 142), (3091, 5759, 18482), (3628, 31657, 20195), (5223, 8227, 20330), (21153, 38113, 38067), (38037, 38179, 38126), (38075, 38082, 38067), (38082, 38139, 38113), (38180, 38181, 38043)

X(38109) = CENTROID OF TRIANGLE {X(3), X(4), X(12)}

Barycentrics    (3*b^2+2*b*c+3*c^2)*a^5-(b+c)*(3*b^2+2*b*c+3*c^2)*a^4-2*(3*b^4+3*c^4-b*c*(b+c)^2)*a^3+2*(b^2-c^2)*(b-c)*(3*b^2+4*b*c+3*c^2)*a^2+(b^2-c^2)^2*(3*b^2-4*b*c+3*c^2)*a-3*(b^2-c^2)^3*(b-c) : :
X(38109) = X(3)-4*X(6668) = X(4)+2*X(31659) = 2*X(5)+X(12) = 4*X(5)-X(26470) = 2*X(12)+X(26470) = X(119)+2*X(8068) = X(355)+2*X(37737) = 4*X(5901)-X(37734) = 5*X(8227)+X(37710) = 2*X(21155)-3*X(38070) = X(21155)+3*X(38078) = X(21155)-6*X(38085) = X(21155)+2*X(38142) = 2*X(38045)+X(38157) = X(38070)+2*X(38078) = X(38070)-4*X(38085) = 3*X(38070)-4*X(38114) = 3*X(38070)+4*X(38142) = X(38078)+2*X(38085) = 3*X(38078)+2*X(38114) = 3*X(38078)-2*X(38142)

X(38109) lies on these lines: {1,5}, {3,6668}, {4,31659}, {30,21155}, {140,30264}, {329,6829}, {381,5842}, {442,11231}, {511,38148}, {515,38033}, {516,6842}, {517,17530}, {529,5055}, {542,38091}, {547,31157}, {758,10175}, {908,9956}, {971,38125}, {1503,38120}, {1512,9955}, {1656,4999}, {2475,34474}, {2476,5657}, {2800,38219}, {2829,38135}, {2975,3090}, {3091,11491}, {3564,38169}, {3628,5251}, {3814,5745}, {3822,6882}, {3829,10247}, {3850,18406}, {4930,5790}, {4996,6946}, {5056,20060}, {5690,17057}, {5692,38042}, {5731,6830}, {5762,38153}, {5840,17577}, {5843,38174}, {5844,38178}, {5849,14561}, {5852,38107}, {5857,38108}, {5885,12691}, {6831,28160}, {6859,10590}, {6860,18761}, {6862,10895}, {6863,10894}, {6867,10588}, {6871,11248}, {6874,11681}, {6879,10269}, {6933,10599}, {6941,9779}, {6971,25466}, {6980,7680}, {10267,10585}, {11929,26363}, {15908,28212}, {17757,38176}, {18412,38041}, {21617,38056}, {25639,28234}, {28194,38105}, {28204,38027}, {31479,37820}

X(38109) = midpoint of X(i) and X(j) for these {i,j}: {5587, 37701}, {38039, 38058}, {38062, 38162}, {38114, 38142}, {38160, 38206}
X(38109) = reflection of X(i) in X(j) for these (i,j): (21155, 38114), (38033, 38062), (38058, 38183), (38125, 38206), (38129, 38058)
X(38109) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 12, 26470), (5, 5886, 23513), (5, 5901, 7173), (5, 7951, 119), (5, 10592, 355), (12, 7173, 37734), (5219, 7951, 10592), (6867, 10588, 11499), (6933, 10599, 11249), (10592, 37737, 12), (21155, 38114, 38070), (38039, 38183, 38129), (38078, 38085, 38070), (38085, 38142, 38114)

X(38110) = CENTROID OF TRIANGLE {X(3), X(5), X(6)}

Barycentrics    4*a^6-5*(b^2+c^2)*a^4-12*b^2*c^2*a^2+(b^4-c^4)*(b^2-c^2) : :
X(38110) = 3*X(2)+X(14912) = 5*X(2)+3*X(33748) = X(3)+5*X(3618) = X(3)+2*X(18583) = 2*X(3)+X(21850) = X(4)+5*X(12017) = X(5)+2*X(182) = X(5)-4*X(3589) = 5*X(5)-2*X(3818) = X(6)+2*X(140) = X(182)+2*X(3589) = 5*X(182)+X(3818) = 10*X(3589)-X(3818) = 5*X(3618)-X(14853) = 5*X(3618)-2*X(18583) = 10*X(3618)-X(21850) = 3*X(5050)-X(14912) = 5*X(5050)-3*X(33748) = 5*X(14912)-9*X(33748) = 4*X(18583)-X(21850)

X(38110) lies on these lines: {2,3167}, {3,3618}, {4,12017}, {5,182}, {6,140}, {30,5085}, {69,3526}, {141,575}, {143,3313}, {147,16987}, {154,10128}, {184,35283}, {193,3525}, {230,5034}, {371,32494}, {372,32497}, {373,13394}, {381,25406}, {395,36757}, {396,36758}, {427,19128}, {495,1428}, {496,2330}, {511,549}, {515,38167}, {517,38040}, {518,38028}, {524,11539}, {542,15699}, {546,10541}, {547,10516}, {548,31670}, {550,5092}, {576,6329}, {599,10124}, {611,15325}, {631,1351}, {952,38029}, {971,38166}, {973,9967}, {1176,7403}, {1350,3530}, {1352,3628}, {1368,19131}, {1386,5690}, {1511,15118}, {1513,7875}, {1595,1974}, {1596,19124}, {1656,6776}, {1691,18907}, {1692,3815}, {1899,11548}, {1992,15694}, {2456,7792}, {2930,13392}, {3066,37897}, {3090,18440}, {3098,15712}, {3147,12167}, {3329,37450}, {3398,8362}, {3523,33878}, {3524,14848}, {3533,3620}, {3534,33750}, {3541,19118}, {3619,11898}, {3627,19130}, {3629,15516}, {3763,16239}, {3843,14927}, {3845,29012}, {3850,36990}, {3867,7715}, {5012,37439}, {5020,35260}, {5032,15709}, {5033,7745}, {5054,5093}, {5067,5921}, {5096,7508}, {5102,11812}, {5135,37438}, {5157,19154}, {5306,15819}, {5422,7499}, {5462,11574}, {5476,8703}, {5622,14643}, {5656,11479}, {5762,38117}, {5840,38168}, {5843,38115}, {5844,38116}, {5845,38111}, {5846,38112}, {5847,11231}, {5848,34126}, {5849,38114}, {5943,10154}, {6036,15491}, {6403,15028}, {6593,10264}, {6656,10359}, {6661,7709}, {6676,10601}, {6677,17825}, {6688,10192}, {6696,34779}, {6699,32300}, {6823,16657}, {7383,13142}, {7399,12022}, {7405,13353}, {7495,15018}, {7509,31802}, {7516,31521}, {7583,19146}, {7584,19145}, {7606,16509}, {8259,16773}, {8260,16772}, {8369,11171}, {8549,31267}, {8550,24206}, {9729,23328}, {9825,37476}, {9969,15026}, {10127,23041}, {10182,11695}, {10272,11579}, {10282,15583}, {10301,15080}, {10303,11482}, {10984,16654}, {11064,22112}, {11174,35429}, {11180,15703}, {11272,13354}, {11427,16419}, {11477,12108}, {11540,15534}, {11585,19129}, {11645,38071}, {11649,16532}, {11801,32233}, {12006,19161}, {12007,34507}, {12100,20423}, {12220,15024}, {12584,22251}, {13331,32515}, {13355,20576}, {13363,34351}, {13747,15988}, {13910,19117}, {13972,19116}, {14216,19132}, {14389,30739}, {14677,32271}, {14786,18914}, {15462,32423}, {15520,15713}, {16475,26446}, {18911,37454}, {19119,26944}, {19139,36752}, {20299,34774}, {20806,36753}, {22234,32455}, {23049,31833}, {24256,32448}, {25320,32609}, {26341,37343}, {26348,37342}, {28160,38146}, {28174,38035}, {28204,38089}, {28224,38144}, {35458,37455}, {37458,37513}

X(38110) = midpoint of X(i) and X(j) for these {i,j}: {2, 5050}, {3, 14853}, {381, 25406}, {3524, 14848}, {5085, 14561}, {5093, 10519}, {5476, 17508}, {5622, 14643}, {10516, 11179}, {16475, 26446}, {20423, 31884}, {23041, 23327}, {25320, 32609}, {38029, 38047}, {38049, 38118}, {38117, 38186}
X(38110) = reflection of X(i) in X(j) for these (i,j): (8703, 17508), (10516, 547), (14853, 18583), (21850, 14853), (31884, 12100), (38040, 38049), (38136, 14561), (38164, 38186), (38165, 38047)
X(38110) = complement of the complement of X(14912)
X(38110) = X(21850)-Gibert-Moses centroid
X(38110) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 3618, 18583), (3, 18583, 21850), (141, 575, 1353), (182, 3589, 5), (597, 10168, 549), (632, 1353, 141), (1656, 6776, 18358), (5054, 5093, 10519), (5092, 5480, 550), (5092, 25555, 5480), (7405, 13353, 31804), (8981, 13966, 31406), (12007, 34573, 34507), (14561, 38064, 5085), (31521, 37488, 7516), (38079, 38136, 14561), (38119, 38120, 38029)

X(38111) = CENTROID OF TRIANGLE {X(3), X(5), X(7)}

Barycentrics    4*(b+c)*a^5-(5*b^2-8*b*c+5*c^2)*a^4-6*(b+c)*(b^2+c^2)*a^3+2*(4*b^2+b*c+4*c^2)*(b-c)^2*a^2+2*(b^2-c^2)^2*(b+c)*a-3*(b^2-c^2)^2*(b-c)^2 : :
X(38111) = X(5)-4*X(142) = X(5)+2*X(31657) = 3*X(5)-2*X(38139) = X(7)+2*X(140) = 2*X(9)-5*X(632) = 2*X(142)+X(31657) = 6*X(142)-X(38139) = X(21151)-3*X(38065) = 5*X(21151)+3*X(38073) = 2*X(21151)+3*X(38080) = 2*X(21151)+X(38137) = 3*X(31657)+X(38139) = 5*X(38065)+X(38073) = 2*X(38065)+X(38080) = 3*X(38065)+X(38107) = 6*X(38065)+X(38137) = 2*X(38073)-5*X(38080) = 3*X(38073)-5*X(38107) = 6*X(38073)-5*X(38137) = X(38139)-3*X(38171)

X(38111) lies on these lines: {2,5843}, {5,142}, {7,140}, {9,632}, {30,21151}, {144,3526}, {511,38164}, {515,38172}, {516,8703}, {517,38041}, {518,38112}, {527,11539}, {547,5817}, {548,31671}, {549,5762}, {550,5805}, {952,38030}, {1483,2550}, {1484,10427}, {1656,36996}, {3525,20059}, {3530,5759}, {3564,38115}, {3627,5732}, {3628,5779}, {3845,38150}, {3850,36991}, {3858,31672}, {5054,21168}, {5249,37364}, {5542,5690}, {5763,33575}, {5840,38173}, {5844,11038}, {5845,38110}, {5850,11231}, {5851,34126}, {5852,38114}, {5886,24644}, {6172,10124}, {6883,30275}, {8727,27186}, {10283,38053}, {13373,15587}, {14869,31658}, {15699,38093}, {16239,18230}, {17768,31650}, {25557,34352}, {28160,38151}, {28174,38036}, {28204,38094}, {28224,38149}, {32613,33558}, {38042,38204}

X(38111) = midpoint of X(i) and X(j) for these {i,j}: {6173, 38122}, {11038, 38121}, {21151, 38107}, {31657, 38171}, {38030, 38052}, {38054, 38123}
X(38111) = reflection of X(i) in X(j) for these (i,j): (5, 38171), (549, 38122), (3845, 38150), (5817, 547), (10283, 38053), (15699, 38093), (38041, 38054), (38042, 38204), (38137, 38107), (38170, 38052), (38171, 142)
X(38111) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (142, 31657, 5), (38065, 38107, 21151), (38080, 38137, 38107), (38124, 38125, 38030)

X(38112) = CENTROID OF TRIANGLE {X(3), X(5), X(8)}

Barycentrics    4*(b+c)*a^3-(3*b^2+8*b*c+3*c^2)*a^2-4*(b^2-c^2)*(b-c)*a+3*(b^2-c^2)^2 : :
X(38112) = 2*X(1)-5*X(632) = X(3)+5*X(3617) = 2*X(3)+X(37705) = X(5)-4*X(10) = 7*X(5)-4*X(946) = 5*X(5)-4*X(3817) = 13*X(5)-4*X(4301) = X(5)+2*X(5690) = 11*X(5)-8*X(9955) = 5*X(5)-8*X(9956) = 3*X(5)-4*X(10175) = 5*X(5)+4*X(11362) = 5*X(5)-2*X(22791) = 11*X(5)-20*X(31399) = 3*X(5)-2*X(38034) = X(5)+4*X(38127) = 7*X(10)-X(946) = 5*X(10)-X(3817) = 13*X(10)-X(4301) = 2*X(10)+X(5690) = 11*X(10)-2*X(9955) = 5*X(10)-2*X(9956) = 3*X(10)-X(10175) = 5*X(10)+X(11362) = 10*X(10)-X(22791) = 11*X(10)-5*X(31399) = 6*X(10)-X(38034) = 10*X(3617)-X(37705) = 2*X(10247)-3*X(10283)

X(38112) lies on these lines: {1,632}, {2,5844}, {3,3617}, {4,28216}, {5,10}, {8,140}, {30,5657}, {40,3627}, {55,11545}, {72,10273}, {100,7508}, {145,3526}, {165,355}, {210,14988}, {354,10039}, {381,28212}, {495,5902}, {496,18395}, {511,38165}, {515,4745}, {516,15687}, {518,38111}, {519,11231}, {546,5818}, {547,5603}, {548,18525}, {549,952}, {631,4678}, {944,3530}, {962,3850}, {971,38175}, {993,3036}, {997,19907}, {1006,12331}, {1145,1484}, {1159,8164}, {1353,3416}, {1385,3626}, {1482,3628}, {1656,12245}, {1698,5901}, {1699,38071}, {1706,26921}, {1737,5919}, {1837,10386}, {2800,3956}, {3090,8148}, {3241,10124}, {3523,18526}, {3525,3621}, {3533,3623}, {3564,38116}, {3579,15704}, {3616,16239}, {3625,15178}, {3634,24680}, {3654,3845}, {3678,35004}, {3681,37438}, {3697,31835}, {3828,11230}, {3843,20070}, {3853,6361}, {3857,7991}, {3858,12699}, {3876,25413}, {3877,34122}, {3932,7611}, {3983,5887}, {4002,24474}, {4015,5694}, {4421,28463}, {4540,20117}, {4662,34339}, {4668,30392}, {4669,10165}, {4691,6684}, {4746,13607}, {4848,6147}, {5049,31397}, {5054,7967}, {5070,10595}, {5090,7715}, {5119,12019}, {5131,37710}, {5260,11849}, {5396,31855}, {5428,32141}, {5432,37728}, {5445,10944}, {5499,10942}, {5554,6675}, {5686,5843}, {5697,10593}, {5719,31434}, {5731,12100}, {5762,38126}, {5840,38177}, {5846,38110}, {5853,38113}, {5854,34126}, {5855,38114}, {5882,31662}, {5886,11224}, {5903,10592}, {6734,32214}, {6735,10202}, {6907,11698}, {6914,9708}, {6924,9709}, {7575,15177}, {8193,37440}, {9578,24470}, {9588,18481}, {9624,16191}, {9779,11737}, {9952,18446}, {10164,17504}, {10171,38083}, {10263,23841}, {10389,12433}, {10627,16980}, {11248,31649}, {12000,16842}, {12001,16862}, {12034,17330}, {12647,15325}, {12785,36966}, {13911,19117}, {13973,19116}, {14893,34632}, {15174,31452}, {15686,28160}, {15694,31145}, {15702,34748}, {15703,34631}, {15935,18391}, {17527,25005}, {18480,28150}, {18493,35018}, {18908,37424}, {19710,28190}, {19862,33179}, {19925,28232}, {23046,28194}, {23410,34656}, {24393,31657}, {24475,34790}, {25006,37364}, {28146,33699}, {28154,34648}, {28182,35404}, {28473,28602}, {32612,33559}, {32789,35810}, {32790,35811}, {33591,34713}, {34200,34627}, {38041,38204}

X(38112) = midpoint of X(i) and X(j) for these {i,j}: {8, 10246}, {10, 38127}, {72, 10273}, {165, 355}, {3654, 5587}, {3679, 26446}, {3817, 11362}, {4669, 10165}, {5603, 34718}, {5657, 5790}, {5686, 38121}, {5690, 38042}, {9812, 12702}, {38126, 38200}
X(38112) = reflection of X(i) in X(j) for these (i,j): (5, 38042), (549, 26446), (550, 165), (1483, 10246), (3817, 9956), (3845, 5587), (5603, 547), (5690, 38127), (5731, 12100), (5882, 31662), (9812, 546), (10246, 140), (10283, 2), (11230, 3828), (15699, 19875), (16200, 5901), (17502, 6684), (22791, 3817), (34773, 17502), (38028, 11231), (38034, 10175), (38041, 38204), (38042, 10), (38138, 5790), (38170, 38200), (38176, 4745)
X(38112) = complement of X(10247)
X(38112) = X(37705)-Gibert-Moses centroid
X(38112) = X(8)-Beth conjugate of-X(38042)
X(38112) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 140, 1483), (10, 5690, 5), (10, 8256, 31419), (10, 11362, 9956), (40, 18357, 3627), (631, 4678, 12645), (1482, 9780, 3628), (3525, 3621, 37624), (3697, 37562, 31835), (5657, 38074, 9778), (5690, 22791, 11362), (5790, 38066, 5657), (5818, 12702, 546), (9956, 11362, 22791), (9956, 22791, 5), (10175, 38034, 5), (11231, 38028, 11539), (38034, 38042, 10175), (38081, 38138, 5790), (38128, 38129, 26446)

X(38113) = CENTROID OF TRIANGLE {X(3), X(5), X(9)}

Barycentrics    4*a^6-6*(b+c)*a^5-(5*b^2+4*b*c+5*c^2)*a^4+10*(b+c)*(b^2+c^2)*a^3+6*b*c*(b-c)^2*a^2-4*(b^2-c^2)^2*(b+c)*a+(b^2-c^2)^2*(b-c)^2 : :
X(38113) = 3*X(2)+X(21168) = X(3)+5*X(18230) = 5*X(3)+X(36991) = X(5)-4*X(6666) = 5*X(5)-2*X(18482) = X(5)+2*X(31658) = X(7)-7*X(3526) = X(9)+2*X(140) = 2*X(9)+X(31657) = 4*X(140)-X(31657) = X(5817)-5*X(18230) = 5*X(5817)-X(36991) = 10*X(6666)-X(18482) = 2*X(6666)+X(31658) = X(18482)+5*X(31658) = X(21153)-3*X(38067) = 5*X(21153)+3*X(38075) = 2*X(21153)+3*X(38082) = 2*X(21153)+X(38139) = 2*X(21168)+3*X(38171) = 2*X(38107)-3*X(38171)

X(38113) lies on these lines: {2,5762}, {3,5817}, {5,516}, {7,3526}, {9,140}, {30,21153}, {142,632}, {144,3525}, {496,15837}, {511,38166}, {515,38179}, {517,38043}, {518,38028}, {527,11539}, {547,38150}, {548,31672}, {549,971}, {631,5779}, {952,6883}, {1001,5690}, {1445,6147}, {1483,24393}, {1484,6594}, {1656,5759}, {1709,7308}, {3090,31671}, {3452,38123}, {3530,5732}, {3564,38117}, {3628,5805}, {3646,5763}, {5044,10165}, {5054,21151}, {5325,10156}, {5657,5804}, {5686,10246}, {5719,11038}, {5840,38180}, {5844,38126}, {5853,38112}, {5856,34126}, {5857,38114}, {6172,15694}, {6173,10124}, {6827,38149}, {8232,24470}, {8236,12433}, {8257,15296}, {9342,19541}, {10283,31837}, {10303,36996}, {15298,15325}, {15699,38137}, {15709,38065}, {16239,20195}, {28160,38158}, {28174,38037}, {28204,38101}, {28224,38154}

X(38113) = midpoint of X(i) and X(j) for these {i,j}: {3, 5817}, {9, 38122}, {5686, 10246}, {21153, 38108}, {21168, 38107}, {38031, 38057}, {38059, 38130}
X(38113) = reflection of X(i) in X(j) for these (i,j): (31657, 38122), (38043, 38059), (38122, 140), (38139, 38108), (38150, 547), (38171, 2), (38175, 38057)
X(38113) = complement of X(38107)
X(38113) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 21168, 38107), (9, 140, 31657), (6666, 31658, 5), (38067, 38108, 21153), (38082, 38139, 38108), (38131, 38132, 38031)

X(38114) = CENTROID OF TRIANGLE {X(3), X(5), X(12)}

Barycentrics    2*a^7-2*(b+c)*a^6-6*(b^2+c^2)*a^5+2*(b+c)*(3*b^2-b*c+3*c^2)*a^4+3*(2*b^4+2*c^4-b*c*(b^2+c^2))*a^3-6*(b^3-c^3)*(b^2-c^2)*a^2-(b^2-c^2)^2*(2*b^2-3*b*c+2*c^2)*a+2*(b^2-c^2)^3*(b-c) : :
X(38114) = X(5)-4*X(6668) = X(5)+2*X(31659) = X(12)+2*X(140) = X(5690)+2*X(37737) = 2*X(6668)+X(31659) = X(21155)-3*X(38070) = 5*X(21155)+3*X(38078) = 2*X(21155)+3*X(38085) = 2*X(21155)+X(38142) = 2*X(38033)+X(38178) = X(38045)+2*X(38134) = 5*X(38070)+X(38078) = 2*X(38070)+X(38085) = 3*X(38070)+X(38109) = 6*X(38070)+X(38142) = 2*X(38078)-5*X(38085) = 3*X(38078)-5*X(38109) = 6*X(38078)-5*X(38142) = 3*X(38085)-2*X(38109) = 3*X(38085)-X(38142)

X(38114) lies on these lines: {2,952}, {5,5248}, {12,36}, {30,21155}, {498,2099}, {511,38169}, {515,38183}, {517,38045}, {529,11539}, {632,3820}, {758,11231}, {971,38181}, {1385,20104}, {1389,5901}, {1656,5284}, {1698,37733}, {2975,3526}, {3525,20060}, {3530,30264}, {3564,38120}, {3584,5844}, {3628,26470}, {3898,38044}, {5432,8068}, {5762,38132}, {5840,38184}, {5843,38125}, {5849,38110}, {5852,38111}, {5855,38112}, {5857,38113}, {6863,22791}, {6914,22799}, {6980,22938}, {7504,37621}, {7508,7951}, {10124,31157}, {10197,10283}, {11545,37734}, {16239,31260}, {26446,37701}, {26487,34773}, {28160,38162}, {28174,38039}, {28204,38105}, {28224,38157}, {31650,33961}

X(38114) = midpoint of X(i) and X(j) for these {i,j}: {21155, 38109}, {26446, 37701}, {38033, 38058}, {38062, 38134}, {38132, 38206}
X(38114) = reflection of X(i) in X(j) for these (i,j): (38045, 38062), (38142, 38109), (38174, 38206), (38178, 38058)
X(38114) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 38028, 34126), (6668, 31659, 5), (38070, 38109, 21155), (38085, 38142, 38109)

X(38115) = CENTROID OF TRIANGLE {X(3), X(6), X(7)}

Barycentrics    a^8-4*(b+c)*a^7+2*(b^2-3*b*c+c^2)*a^6+4*(b+c)*(b^2+c^2)*a^5-2*(b^2+c^2)*(b^2-3*b*c+c^2)*a^4+8*b^2*c^2*(b+c)*a^3-2*(b^4+c^4+b*c*(b^2+6*b*c+c^2))*(b-c)^2*a^2+(b^4-c^4)*(b^2-c^2)*(b-c)^2 : :
X(38115) = X(6)+2*X(31657) = X(7)+2*X(182) = 4*X(142)-X(1352) = 4*X(3589)-X(5779) = 5*X(3618)+X(36996) = 4*X(5092)-X(5759) = 2*X(5732)+X(31670) = X(6172)-4*X(10168) = 2*X(6173)+X(11179) = 3*X(14561)-2*X(38145) = 4*X(19130)-X(36991) = 3*X(38064)-2*X(38117) = 3*X(38086)-X(38143) = 3*X(38086)-2*X(38164) = X(38145)-3*X(38186)

X(38115) lies on these lines: {6,31657}, {7,182}, {30,38086}, {142,1352}, {511,21151}, {515,38187}, {516,38029}, {517,38046}, {518,10202}, {524,38065}, {527,38064}, {952,38185}, {971,14561}, {1503,38107}, {3564,38111}, {3589,5779}, {3618,36996}, {5050,5845}, {5085,5762}, {5092,5759}, {5732,31670}, {5840,38188}, {5843,38110}, {5846,38121}, {5847,38123}, {5848,38124}, {5849,38125}, {5850,38118}, {5851,38119}, {5852,38120}, {6172,10168}, {6173,11179}, {10516,38171}, {11645,38073}, {19130,36991}

X(38115) = reflection of X(i) in X(j) for these (i,j): (10516, 38171), (14561, 38186), (38143, 38164)
X(38115) = {X(38086), X(38143)}-harmonic conjugate of X(38164)

X(38116) = CENTROID OF TRIANGLE {X(3), X(6), X(8)}

Barycentrics    a^6+2*(b+c)*a^5-(3*b^2+4*b*c+3*c^2)*a^4+4*b*c*(b+c)*a^3+(b^4+c^4-2*b*c*(2*b^2+3*b*c+2*c^2))*a^2-2*(b^4-c^4)*(b-c)*a+(b^4-c^4)*(b^2-c^2) : :
X(38116) = X(6)+2*X(5690) = X(8)+2*X(182) = 4*X(10)-X(1352) = 2*X(40)+X(31670) = 4*X(140)-X(3242) = 2*X(597)+X(34718) = X(944)-4*X(5092) = X(962)-4*X(19130) = X(1482)-4*X(3589) = X(3241)-4*X(10168) = 5*X(3617)+X(6776) = 5*X(3618)+X(12245) = 3*X(14561)-2*X(38035) = 3*X(14561)-4*X(38167) = 2*X(38029)-3*X(38064) = X(38035)-3*X(38047) = 3*X(38047)-2*X(38167) = 3*X(38064)-4*X(38118) = 3*X(38087)-X(38144) = 3*X(38087)-2*X(38165)

X(38116) lies on these lines: {6,5690}, {8,182}, {10,1352}, {30,38087}, {40,31670}, {140,3242}, {511,5657}, {515,38191}, {517,14561}, {518,10202}, {519,38029}, {524,38066}, {597,34718}, {944,5092}, {952,5085}, {962,19130}, {971,38190}, {1428,12647}, {1482,3589}, {1503,5790}, {2330,10573}, {3241,10168}, {3564,38112}, {3617,6776}, {3618,12245}, {3654,20423}, {3679,11179}, {3818,5818}, {5050,5846}, {5054,9041}, {5480,12702}, {5731,17508}, {5762,38185}, {5840,38192}, {5844,38110}, {5845,38121}, {5847,38127}, {5848,38128}, {5849,38129}, {5853,38117}, {5854,38119}, {5855,38120}, {6211,29659}, {9053,10246}, {9778,29317}, {9780,24206}, {10327,37527}, {10516,38042}, {11645,38074}, {12017,12645}, {12589,18395}, {18357,36990}, {28194,38146}, {28212,38136}, {28234,38049}, {33163,37619}

X(38116) = reflection of X(i) in X(j) for these (i,j): (5731, 17508), (10516, 38042), (14561, 38047), (38029, 38118), (38035, 38167), (38144, 38165)
X(38116) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38029, 38118, 38064), (38035, 38047, 38167), (38035, 38167, 14561), (38087, 38144, 38165)

X(38117) = CENTROID OF TRIANGLE {X(3), X(6), X(9)}

Barycentrics    a*(3*a^7-5*(b+c)*a^6-(b^2+4*b*c+c^2)*a^5+5*(b+c)*(b^2+c^2)*a^4-(3*b^4+3*c^4-2*b*c*(2*b^2-7*b*c+2*c^2))*a^3+(b+c)*(b^4+14*b^2*c^2+c^4)*a^2+(b^4+c^4+2*b*c*(b^2-b*c+c^2))*(b-c)^2*a-(b^4-c^4)*(b^2-c^2)*(b+c)) : :
X(38117) = X(6)+2*X(31658) = X(9)+2*X(182) = X(1352)-4*X(6666) = 4*X(3589)-X(5805) = 5*X(3618)+X(5759) = 4*X(5092)-X(5732) = X(5779)+5*X(12017) = X(6173)-4*X(10168) = X(6776)+5*X(18230) = 3*X(38064)-X(38115) = 3*X(38088)-X(38145) = 3*X(38088)-2*X(38166) = 3*X(38110)-X(38164) = 2*X(38164)-3*X(38186)

X(38117) lies on these lines: {6,31658}, {9,182}, {30,38088}, {511,21153}, {515,38194}, {516,14561}, {517,38048}, {518,5050}, {524,38067}, {527,38064}, {613,15837}, {952,38190}, {971,5085}, {1352,6666}, {1428,15298}, {1503,38108}, {2330,15299}, {3564,38113}, {3589,5805}, {3618,5759}, {5092,5732}, {5762,38110}, {5779,12017}, {5817,25406}, {5840,38195}, {5845,38122}, {5846,38126}, {5847,38130}, {5848,38131}, {5849,38132}, {5853,38116}, {5856,38119}, {5857,38120}, {6173,10168}, {6776,18230}, {11645,38075}, {13329,36404}

X(38117) = midpoint of X(5817) and X(25406)
X(38117) = reflection of X(i) in X(j) for these (i,j): (38145, 38166), (38186, 38110)
X(38117) = {X(38088), X(38145)}-harmonic conjugate of X(38166)

X(38118) = CENTROID OF TRIANGLE {X(3), X(6), X(10)}

Barycentrics    4*a^6+(b+c)*a^5-(5*b^2+2*b*c+5*c^2)*a^4+2*b*c*(b+c)*a^3-2*b*c*(b^2+6*b*c+c^2)*a^2-(b^4-c^4)*(b-c)*a+(b^4-c^4)*(b^2-c^2) : :
X(38118) = X(6)+2*X(6684) = X(10)+2*X(182) = X(40)+5*X(3618) = X(69)-7*X(31423) = X(355)+5*X(12017) = X(551)-4*X(10168) = 5*X(631)+X(3751) = X(946)-4*X(3589) = X(1352)-4*X(3634) = 2*X(1386)+X(11362) = 5*X(1698)+X(6776) = X(3313)+2*X(31760) = 3*X(5085)+X(38144) = X(38029)-3*X(38064) = 2*X(38040)-3*X(38049) = X(38040)-3*X(38110) = 3*X(38047)-X(38144) = 3*X(38064)+X(38116) = 3*X(38089)-X(38146) = 3*X(38089)-2*X(38167)

X(38118) lies on these lines: {6,6684}, {10,182}, {30,38089}, {40,3618}, {69,31423}, {165,14853}, {355,12017}, {511,10164}, {515,5085}, {516,14561}, {517,38040}, {518,10165}, {519,38029}, {524,38068}, {551,10168}, {611,3911}, {631,3751}, {758,38120}, {946,3589}, {952,38191}, {971,38194}, {1210,2330}, {1352,3634}, {1386,11362}, {1428,31397}, {1503,10175}, {1692,31398}, {1698,6776}, {2802,38119}, {3313,31760}, {3564,11231}, {3579,18583}, {3828,11179}, {3844,8550}, {4297,5092}, {5050,5847}, {5480,31730}, {5493,25555}, {5587,25406}, {5657,16475}, {5762,38187}, {5840,38197}, {5845,38123}, {5846,38127}, {5848,38133}, {5849,38134}, {5850,38115}, {5921,19877}, {6211,17023}, {10172,10516}, {10202,34378}, {11180,19876}, {11645,38076}, {12512,31670}, {13883,19146}, {13936,19145}, {14927,18492}, {17355,24257}, {21850,31663}, {28146,38136}, {28194,38035}, {28198,38079}, {28204,38165}

X(38118) = midpoint of X(i) and X(j) for these {i,j}: {165, 14853}, {5050, 26446}, {5085, 38047}, {5587, 25406}, {5657, 16475}, {38029, 38116}
X(38118) = reflection of X(i) in X(j) for these (i,j): (10516, 10172), (38049, 38110), (38146, 38167)
X(38118) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13912, 13975, 31396), (38064, 38116, 38029), (38089, 38146, 38167)

X(38119) = CENTROID OF TRIANGLE {X(3), X(6), X(11)}

Barycentrics    4*a^9-4*(b+c)*a^8-(9*b^2-14*b*c+9*c^2)*a^7+(b+c)*(9*b^2-10*b*c+9*c^2)*a^6+(5*b^4+5*c^4-2*b*c*(7*b^2-b*c+7*c^2))*a^5-(b+c)*(5*b^4+5*c^4-2*b*c*(5*b^2-b*c+5*c^2))*a^4+(b^6+c^6-(2*b^4+2*c^4-3*b*c*(5*b^2-12*b*c+5*c^2))*b*c)*a^3-(b^2-c^2)*(b-c)*(b^4+14*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2)*(b-c)^2*a+(b^4-c^4)*(b^2-c^2)^2*(b-c) : :
X(38119) = X(6)+2*X(6713) = X(11)+2*X(182) = X(104)+5*X(3618) = X(119)-4*X(3589) = 5*X(631)+X(10755) = X(1352)-4*X(6667) = 4*X(5092)-X(24466) = X(6174)-4*X(10168) = X(6776)+5*X(31272) = X(10738)+5*X(12017) = 3*X(38090)-X(38147) = 3*X(38090)-2*X(38168)

X(38119) lies on these lines: {6,6713}, {11,182}, {30,38090}, {104,3618}, {119,3589}, {511,21154}, {515,38197}, {517,38050}, {518,38032}, {524,38069}, {528,38064}, {631,10755}, {952,38029}, {971,38195}, {1352,6667}, {1503,23513}, {2800,38049}, {2802,38118}, {2829,14561}, {3564,34126}, {5050,5848}, {5085,5840}, {5092,24466}, {5762,38188}, {5845,38124}, {5846,38128}, {5847,38133}, {5849,38135}, {5851,38115}, {5854,38116}, {5856,38117}, {6174,10168}, {6776,31272}, {10738,12017}, {11645,38077}

X(38119) = reflection of X(38147) in X(38168)
X(38119) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38029, 38110, 38120), (38090, 38147, 38168)

X(38120) = CENTROID OF TRIANGLE {X(3), X(6), X(12)}

Barycentrics    4*a^9-4*(b+c)*a^8-(9*b^2-2*b*c+9*c^2)*a^7+3*(b+c)*(3*b^2-2*b*c+3*c^2)*a^6+(5*b^4+5*c^4-6*(b^2+b*c+c^2)*b*c)*a^5-(b+c)*(5*b^4+5*c^4-2*(5*b^2+3*b*c+5*c^2)*b*c)*a^4+(b^6+c^6+(2*b^4+2*c^4+(7*b^2-12*b*c+7*c^2)*b*c)*b*c)*a^3-(b^2-c^2)*(b-c)*(b^4+c^4+2*b*c*(2*b^2+7*b*c+2*c^2))*a^2-(b^4-c^4)*(b^2-c^2)*(b-c)^2*a+(b^4-c^4)*(b^2-c^2)^2*(b-c) : :
X(38120) = X(6)+2*X(31659) = X(12)+2*X(182) = X(1352)-4*X(6668) = 4*X(3589)-X(26470) = 5*X(3618)+X(11491) = 4*X(5092)-X(30264) = 4*X(10168)-X(31157) = 3*X(38091)-X(38148) = 3*X(38091)-2*X(38169)

X(38120) lies on these lines: {6,31659}, {12,182}, {30,38091}, {511,21155}, {515,38198}, {517,38051}, {518,38033}, {524,38070}, {529,38064}, {758,38118}, {952,38029}, {971,38196}, {1352,6668}, {1503,38109}, {3564,38114}, {3589,26470}, {3618,11491}, {5050,5849}, {5092,30264}, {5762,38189}, {5840,38199}, {5842,14561}, {5845,38125}, {5846,38129}, {5847,38134}, {5848,38135}, {5852,38115}, {5855,38116}, {5857,38117}, {10168,31157}, {11645,38078}

X(38120) = reflection of X(38148) in X(38169)
X(38120) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38029, 38110, 38119), (38091, 38148, 38169)

X(38121) = CENTROID OF TRIANGLE {X(3), X(7), X(8)}

Barycentrics    a^6-2*(b+c)*a^5+2*(b^2+7*b*c+c^2)*a^4-2*(b+c)*(b^2+4*b*c+c^2)*a^3-(b^2+4*b*c+c^2)*(b-c)^2*a^2+4*(b^4-c^4)*(b-c)*a-2*(b^2-c^2)^2*(b-c)^2 : :
X(38121) = X(3)+2*X(2550) = 2*X(5)+X(35514) = X(7)+2*X(5690) = X(8)+2*X(31657) = 4*X(10)-X(5779) = 2*X(40)+X(31671) = 4*X(140)-X(390) = 4*X(142)-X(1482) = 3*X(5054)-2*X(38031) = 3*X(5790)-2*X(38154) = 3*X(26446)-2*X(38130) = 2*X(38030)-3*X(38065) = X(38036)-3*X(38052) = 2*X(38036)-3*X(38107) = 3*X(38052)-2*X(38172) = 3*X(38065)-4*X(38123) = 3*X(38066)-2*X(38126) = 3*X(38092)-X(38149) = 3*X(38092)-2*X(38170) = 3*X(38107)-4*X(38172)

X(38121) lies on these lines: {2,38043}, {3,1602}, {5,35514}, {7,5690}, {8,31657}, {10,5779}, {30,38092}, {40,31671}, {140,390}, {142,1482}, {381,516}, {443,8158}, {511,38185}, {515,38201}, {517,38036}, {519,38030}, {527,38066}, {528,5054}, {952,21151}, {971,5790}, {1001,3526}, {1656,3826}, {1657,11495}, {2829,9708}, {2951,18480}, {3059,34339}, {3174,37615}, {3617,36996}, {3925,6244}, {4312,9654}, {5055,38037}, {5450,31494}, {5603,38171}, {5657,5762}, {5686,5843}, {5732,18525}, {5789,37560}, {5805,12702}, {5817,38042}, {5840,38202}, {5844,11038}, {5845,38116}, {5846,38115}, {5850,38127}, {5851,38128}, {5852,38129}, {5853,10246}, {5854,38124}, {5855,38125}, {5886,38204}, {6173,34718}, {8148,20330}, {8236,38028}, {9669,31423}, {9956,11372}, {10156,24392}, {10247,38053}, {10427,19914}, {10531,16855}, {10827,31391}, {12573,37545}, {17532,21168}, {18357,36991}, {19855,31777}, {19875,38179}, {20119,33814}, {24475,34784}, {28194,38151}, {28212,38137}, {28234,38054}

X(38121) = reflection of X(i) in X(j) for these (i,j): (5603, 38171), (5686, 38112), (5790, 38200), (5817, 38042), (5886, 38204), (8236, 38028), (10246, 38122), (10247, 38053), (11038, 38111), (38030, 38123), (38036, 38172), (38107, 38052), (38149, 38170)
X(38121) = anticomplement of X(38043)
X(38121) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38030, 38123, 38065), (38036, 38052, 38172), (38036, 38172, 38107), (38092, 38149, 38170)

X(38122) = CENTROID OF TRIANGLE {X(3), X(7), X(9)}

Barycentrics    a^6-3*(b+c)*a^5+(b^2-4*b*c+c^2)*a^4+4*(b+c)*(b^2+c^2)*a^3-3*(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(38122) = X(3)+2*X(142) = 2*X(3)+X(5805) = 5*X(3)+X(31671) = 2*X(5)+X(5732) = 2*X(5)-5*X(20195) = 4*X(5)-X(31672) = X(7)+5*X(631) = X(7)+2*X(31658) = 4*X(142)-X(5805) = 10*X(142)-X(31671) = X(5732)+5*X(20195) = 2*X(5732)+X(31672) = 5*X(5805)-2*X(31671) = X(5817)+3*X(21151) = 2*X(5817)-3*X(38108) = 2*X(11495)+X(12699) = 10*X(20195)-X(31672) = 2*X(21151)+X(38108) = X(31671)-5*X(38107) = X(38031)+2*X(38123)

X(38122) lies on these lines: {2,971}, {3,142}, {5,5732}, {7,631}, {9,140}, {20,18482}, {30,38093}, {40,20330}, {57,5432}, {144,10303}, {165,38036}, {277,11200}, {355,3826}, {390,11373}, {443,5731}, {495,4321}, {496,4326}, {498,8581}, {499,14100}, {511,38186}, {515,38204}, {517,38053}, {518,10202}, {527,5054}, {528,3653}, {547,38139}, {549,5762}, {942,5657}, {952,9623}, {954,30379}, {990,17245}, {991,17278}, {1385,2550}, {1484,5528}, {2801,38133}, {2951,8227}, {3062,34595}, {3090,36991}, {3243,5690}, {3254,33814}, {3358,6675}, {3523,5759}, {3525,18230}, {3526,5779}, {3530,5735}, {3576,38052}, {3587,28212}, {3616,35514}, {3624,11372}, {3634,5789}, {3824,6865}, {4312,11375}, {4675,13329}, {5122,30275}, {5129,22792}, {5223,31423}, {5433,15299}, {5438,5833}, {5439,37112}, {5542,5708}, {5587,5787}, {5660,11407}, {5709,38041}, {5722,7675}, {5728,6889}, {5770,5791}, {5806,37108}, {5840,38205}, {5845,38117}, {5850,38130}, {5851,38131}, {5852,37612}, {5853,10246}, {5856,38124}, {5857,38125}, {6172,15702}, {6245,10172}, {6259,11108}, {6260,16853}, {6713,10427}, {6826,28160}, {6846,31805}, {6853,10394}, {6885,13624}, {7679,18450}, {7988,8727}, {8127,8389}, {8128,8388}, {8236,24929}, {8703,38137}, {9352,9776}, {9858,19843}, {10164,38054}, {10283,37531}, {10304,38073}, {10383,17726}, {10398,31231}, {10884,17529}, {11230,38037}, {12246,17554}, {12560,37737}, {12618,17265}, {12679,25542}, {13373,15185}, {13727,27147}, {15587,26363}, {15699,38075}, {15803,37701}, {15934,28234}, {16173,30282}, {16845,34862}, {17502,38172}, {17504,38080}, {17668,26492}, {17768,28465}, {18412,24914}, {21154,21164}, {25525,37364}, {25557,37532}, {30284,30312}, {31884,38143}, {38042,38154}

X(38122) = midpoint of X(i) and X(j) for these {i,j}: {2, 21151}, {3, 38107}, {7, 21168}, {165, 38036}, {549, 38111}, {3576, 38052}, {5054, 38065}, {5657, 11038}, {5731, 38149}, {6173, 21153}, {8703, 38137}, {10164, 38054}, {10165, 38123}, {10246, 38121}, {10304, 38073}, {17502, 38172}, {17504, 38080}, {26446, 38030}, {31657, 38113}, {31884, 38143}
X(38122) = reflection of X(i) in X(j) for these (i,j): (9, 38113), (5805, 38107), (6173, 38111), (21153, 549), (21168, 31658), (38031, 10165), (38037, 11230), (38057, 11231), (38067, 5054), (38075, 15699), (38107, 142), (38108, 2), (38113, 140), (38126, 26446), (38139, 547), (38150, 38171), (38154, 38042), (38158, 10172)
X(38122) = complement of X(5817)
X(38122) = X(5805)-Gibert-Moses centroid
X(38122) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 142, 5805), (5, 5732, 31672), (7, 631, 31658), (140, 31657, 9), (3525, 36996, 18230), (3526, 5779, 6666), (5732, 20195, 5), (5770, 6989, 11231), (5770, 11231, 5791), (6989, 9940, 5791), (8726, 8728, 5787), (9940, 11231, 5770), (38093, 38150, 38171)

X(38123) = CENTROID OF TRIANGLE {X(3), X(7), X(10)}

Barycentrics    3*(b+c)*a^5-(3*b^2-14*b*c+3*c^2)*a^4-2*(b+c)*(3*b^2+2*b*c+3*c^2)*a^3+6*(b^2+c^2)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a-3*(b^2-c^2)^2*(b-c)^2 : :
X(38123) = X(7)+2*X(6684) = X(10)+2*X(31657) = 3*X(10)-2*X(38175) = 4*X(142)-X(946) = X(144)-7*X(31423) = 5*X(631)+X(4312) = 2*X(5805)+X(31730) = 3*X(10165)-2*X(38031) = 2*X(12512)+X(31671) = 3*X(21151)+X(38149) = 3*X(31657)+X(38175) = X(38030)-3*X(38065) = X(38031)-3*X(38122) = 2*X(38041)-3*X(38054) = X(38041)-3*X(38111) = 3*X(38052)-X(38149) = 3*X(38065)+X(38121) = 3*X(38068)-2*X(38130) = 3*X(38094)-X(38151) = 3*X(38094)-2*X(38172)

X(38123) lies on these lines: {3,142}, {7,6684}, {10,31657}, {30,38094}, {144,31423}, {226,21168}, {511,38187}, {515,21151}, {517,38041}, {518,38127}, {519,38030}, {527,38068}, {631,4312}, {758,38125}, {952,38201}, {971,10175}, {1698,36996}, {2550,5882}, {2802,38124}, {2951,18483}, {3059,12005}, {3062,3090}, {3339,5657}, {3452,38113}, {3634,5779}, {3817,10156}, {3826,31399}, {4208,5587}, {5542,11362}, {5732,31673}, {5762,10164}, {5785,38057}, {5817,6260}, {5840,38207}, {5843,11231}, {5845,38118}, {5847,38115}, {5850,26446}, {5851,38133}, {5852,38134}, {6937,10392}, {8728,9948}, {9940,15587}, {10265,10427}, {11038,11526}, {11200,24181}, {13464,35514}, {19883,38043}, {28146,38137}, {28194,38036}, {28198,38080}, {28204,38170}, {30424,31658}, {38037,38093}

X(38123) = midpoint of X(i) and X(j) for these {i,j}: {21151, 38052}, {38030, 38121}
X(38123) = reflection of X(i) in X(j) for these (i,j): (3817, 38171), (5817, 10172), (10165, 38122), (10175, 38204), (38054, 38111), (38151, 38172)
X(38123) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38065, 38121, 38030), (38094, 38151, 38172)

X(38124) = CENTROID OF TRIANGLE {X(3), X(7), X(11)}

Barycentrics    4*(b+c)*a^8-(9*b^2+2*b*c+9*c^2)*a^7-5*(b^2-c^2)*(b-c)*a^6+(23*b^4+23*c^4-6*b*c*(5*b^2-7*b*c+5*c^2))*a^5-(b+c)*(5*b^4+5*c^4-2*b*c*(6*b^2-11*b*c+6*c^2))*a^4-(19*b^4+19*c^4-2*b*c*(4*b^2-11*b*c+4*c^2))*(b-c)^2*a^3+(b^2-c^2)*(b-c)^3*(9*b^2+2*b*c+9*c^2)*a^2+(b^2-c^2)^2*(b-c)^2*(5*b^2-4*b*c+5*c^2)*a-3*(b^2-c^2)^3*(b-c)^3 : :
X(38124) = X(7)+2*X(6713) = X(11)+2*X(31657) = X(119)-4*X(142) = 4*X(140)-X(6068) = 2*X(3254)+X(10993) = X(5779)-4*X(6667) = 2*X(10427)+X(37726) = 3*X(23513)-2*X(38159) = 5*X(31272)+X(36996) = 3*X(38069)-2*X(38131) = 3*X(38095)-X(38152) = 3*X(38095)-2*X(38173) = X(38159)-3*X(38205)

X(38124) lies on these lines: {7,6713}, {11,31657}, {30,38095}, {116,119}, {140,6068}, {511,38188}, {515,38207}, {516,38032}, {517,30379}, {518,38128}, {527,38069}, {528,10246}, {952,38030}, {971,23513}, {1387,3576}, {2800,38054}, {2802,38123}, {2829,38107}, {3254,10993}, {5762,21154}, {5779,6667}, {5840,21151}, {5843,34126}, {5845,38119}, {5848,38115}, {5850,38133}, {5852,38135}, {5854,38121}, {5856,38122}, {5886,6173}, {10427,37726}, {11219,38180}, {15726,22835}, {31272,36996}

X(38124) = reflection of X(i) in X(j) for these (i,j): (23513, 38205), (38152, 38173)
X(38124) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38030, 38111, 38125), (38095, 38152, 38173)

X(38125) = CENTROID OF TRIANGLE {X(3), X(7), X(12)}

Barycentrics    4*(b+c)*a^8-(9*b^2-2*b*c+9*c^2)*a^7-5*(b+c)^3*a^6+(23*b^4+23*c^4-2*b*c*(5*b^2-21*b*c+5*c^2))*a^5-(b+c)*(5*b^4+5*c^4-2*b*c*(14*b^2-19*b*c+14*c^2))*a^4-(b^2+c^2)*(19*b^2+20*b*c+19*c^2)*(b-c)^2*a^3+3*(b^2-c^2)*(b-c)^3*(3*b^2+2*b*c+3*c^2)*a^2+5*(b^4-c^4)*(b^2-c^2)*(b-c)^2*a-3*(b^2-c^2)^3*(b-c)^3 : :
X(38125) = X(7)+2*X(31659) = X(12)+2*X(31657) = 4*X(142)-X(26470) = X(5779)-4*X(6668) = 3*X(38070)-2*X(38132) = 3*X(38096)-X(38153) = 3*X(38096)-2*X(38174) = 3*X(38109)-2*X(38160) = X(38160)-3*X(38206)

X(38125) lies on these lines: {7,31659}, {12,31657}, {30,38096}, {142,26470}, {511,38189}, {515,38208}, {516,38033}, {517,38056}, {518,38129}, {527,38070}, {529,38065}, {758,38123}, {952,38030}, {971,38109}, {5762,21155}, {5779,6668}, {5840,38209}, {5842,38107}, {5843,38114}, {5845,38120}, {5849,38115}, {5850,38134}, {5851,38135}, {5855,38121}, {5857,38122}

X(38125) = reflection of X(i) in X(j) for these (i,j): (38109, 38206), (38153, 38174)
X(38125) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38030, 38111, 38124), (38096, 38153, 38174)

X(38126) = CENTROID OF TRIANGLE {X(3), X(8), X(9)}

Barycentrics    a^6+(b+c)*a^5-(7*b^2+12*b*c+7*c^2)*a^4+4*(b+c)^3*a^3+(5*b^2+8*b*c+5*c^2)*(b-c)^2*a^2-(b^2-c^2)*(b-c)*(5*b^2+2*b*c+5*c^2)*a+(b^2-c^2)^2*(b-c)^2 : :
X(38126) = X(3)+2*X(24393) = X(8)+2*X(31658) = X(9)+2*X(5690) = 4*X(10)-X(5805) = 3*X(10)-X(38151) = 2*X(40)+X(31672) = 4*X(140)-X(3243) = X(1482)-4*X(6666) = 3*X(5805)-4*X(38151) = 3*X(26446)-X(38030) = 2*X(38030)-3*X(38122) = 2*X(38031)-3*X(38067) = X(38037)-3*X(38057) = 2*X(38037)-3*X(38108) = 3*X(38057)-2*X(38179) = 3*X(38066)-X(38121) = 3*X(38067)-4*X(38130) = 3*X(38097)-X(38154) = 3*X(38097)-2*X(38175) = 3*X(38108)-4*X(38179)

X(38126) lies on these lines: {3,24393}, {8,31658}, {9,5690}, {10,5805}, {30,38097}, {40,31672}, {140,3243}, {511,38190}, {515,38210}, {516,3654}, {517,38037}, {518,10202}, {519,38031}, {527,38066}, {952,21153}, {971,5657}, {1001,12000}, {1482,6666}, {1698,20330}, {2550,10526}, {3617,5759}, {5762,38112}, {5818,18482}, {5840,38211}, {5844,38113}, {5846,38117}, {5854,38131}, {5855,38132}, {5856,38128}, {5857,38129}, {6594,19914}, {7672,11374}, {7956,30393}, {10573,15837}, {11231,38053}, {11495,18518}, {12245,18230}, {19875,38036}, {28194,38158}, {28212,38139}, {28234,38059}, {38041,38093}, {38042,38150}

X(38126) = midpoint of X(5657) and X(5686)
X(38126) = reflection of X(i) in X(j) for these (i,j): (38031, 38130), (38037, 38179), (38053, 11231), (38108, 38057), (38122, 26446), (38150, 38042), (38154, 38175), (38200, 38112)
X(38126) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38031, 38130, 38067), (38037, 38057, 38179), (38037, 38179, 38108), (38097, 38154, 38175)

X(38127) = CENTROID OF TRIANGLE {X(3), X(8), X(10)}

Barycentrics    5*(b+c)*a^3-(b+3*c)*(3*b+c)*a^2-5*(b^2-c^2)*(b-c)*a+3*(b^2-c^2)^2 : :
X(38127) = 5*X(1)-11*X(3525) = X(3)+2*X(3626) = 2*X(5)-5*X(10) = 8*X(5)-5*X(946) = 6*X(5)-5*X(3817) = 14*X(5)-5*X(4301) = X(5)+5*X(5690) = 13*X(5)-10*X(9955) = 7*X(5)-10*X(9956) = 4*X(5)-5*X(10175) = 4*X(5)+5*X(11362) = 11*X(5)-5*X(22791) = 7*X(5)-5*X(38034) = 3*X(5)-5*X(38042) = X(5)-5*X(38112) = 4*X(10)-X(946) = 3*X(10)-X(3817) = 7*X(10)-X(4301) = X(10)+2*X(5690) = 13*X(10)-4*X(9955) = 7*X(10)-4*X(9956) = 2*X(10)+X(11362) = 11*X(10)-2*X(22791) = 8*X(10)-5*X(31399) = 7*X(10)-2*X(38034) = 3*X(10)-2*X(38042)

X(38127) lies on these lines: {1,3525}, {2,16200}, {3,3626}, {4,28232}, {5,10}, {8,3523}, {30,38098}, {40,3146}, {140,3244}, {145,31423}, {165,376}, {210,2800}, {354,31397}, {355,1657}, {381,28228}, {511,38191}, {516,3654}, {518,38123}, {519,3653}, {549,31662}, {551,5844}, {631,3632}, {632,15808}, {758,10273}, {944,4668}, {952,4669}, {962,3854}, {971,38210}, {1064,31855}, {1125,10247}, {1145,4847}, {1210,5919}, {1385,3625}, {1482,3634}, {1698,11224}, {2801,24393}, {2802,24386}, {3036,4640}, {3090,11531}, {3305,12703}, {3488,30286}, {3526,3636}, {3579,12103}, {3628,11278}, {3655,15718}, {3656,10171}, {3678,37562}, {3681,6735}, {3697,20117}, {3698,31870}, {3707,12034}, {3828,5886}, {3839,5587}, {3860,28212}, {3911,12647}, {3918,24474}, {3983,12672}, {4015,5887}, {4067,35004}, {4078,7611}, {4134,14988}, {4297,28224}, {4662,31788}, {4677,7967}, {4678,5881}, {4701,37727}, {4746,12645}, {4816,30389}, {4848,5902}, {5245,11752}, {5246,11789}, {5288,6940}, {5295,22004}, {5432,36920}, {5493,18480}, {5603,10172}, {5688,6280}, {5689,6279}, {5731,15705}, {5762,38201}, {5818,7991}, {5840,38213}, {5846,38118}, {5847,38116}, {5850,38121}, {5853,38130}, {5854,38133}, {5855,38134}, {5884,34790}, {6361,37714}, {6883,25439}, {7294,33176}, {7982,9780}, {8164,18421}, {9519,12618}, {9624,19877}, {9778,28172}, {10196,28292}, {10202,10915}, {10283,19883}, {10303,20050}, {10389,18391}, {10595,16191}, {10916,12640}, {11010,26878}, {12053,18395}, {12102,18357}, {12511,18518}, {12512,18525}, {12702,19925}, {13405,14563}, {14893,28174}, {15702,34747}, {16189,19872}, {18239,18908}, {18492,20070}, {19710,28160}, {19862,24680}, {19876,34631}, {28146,34648}, {28168,34638}, {28198,38081}, {31663,37705}

X(38127) = midpoint of X(i) and X(j) for these {i,j}: {8, 3576}, {376, 37712}, {3654, 5790}, {3679, 5657}, {4669, 10164}, {4677, 7967}, {5690, 38112}, {5886, 34718}, {10175, 11362}, {11224, 12245}
X(38127) = reflection of X(i) in X(j) for these (i,j): (10, 38112), (551, 11231), (946, 10175), (3576, 6684), (3656, 10171), (3817, 38042), (4301, 38034), (5603, 10172), (5790, 4745), (5882, 3576), (5886, 3828), (10165, 26446), (10175, 10), (10247, 1125), (11224, 13464), (34648, 38138), (38034, 9956), (38155, 38176)
X(38127) = complement of X(16200)
X(38127) = X(8)-Beth conjugate of-X(10175)
X(38127) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 6684, 5882), (10, 946, 31399), (10, 3817, 38042), (10, 4301, 9956), (10, 5690, 11362), (10, 11362, 946), (631, 3632, 13607), (632, 33179, 15808), (1698, 12245, 13464), (3817, 38042, 10175), (4668, 9588, 944), (4848, 10039, 21620), (5603, 19875, 10172), (5818, 7991, 18483), (10165, 26446, 38068), (38098, 38155, 38176)

X(38128) = CENTROID OF TRIANGLE {X(3), X(8), X(11)}

Barycentrics    4*(b+c)*a^6-7*(b+c)^2*a^5-(5*b-c)*(b-5*c)*(b+c)*a^4+2*(7*b^4+7*c^4-b*c*(b^2+18*b*c+c^2))*a^3-2*(b^2-c^2)*(b-c)*(b^2+12*b*c+c^2)*a^2-(b^2-c^2)^2*(7*b^2-16*b*c+7*c^2)*a+3*(b^2-c^2)^3*(b-c) : :
X(38128) = X(3)+2*X(3036) = X(8)+2*X(6713) = 4*X(10)-X(119) = 7*X(10)-X(21635) = X(11)+2*X(5690) = 2*X(80)+X(10993) = X(104)+5*X(3617) = 7*X(119)-4*X(21635) = 4*X(140)-X(1317) = 5*X(631)+X(12531) = X(1145)+2*X(12619) = 2*X(1145)+X(37726) = 2*X(18254)+X(37562) = 3*X(23513)-2*X(38038) = 3*X(23513)-4*X(38182) = 3*X(34122)-X(38038) = 3*X(34122)-2*X(38182) = 2*X(38032)-3*X(38069) = 3*X(38069)-4*X(38133) = 3*X(38099)-X(38156) = 3*X(38099)-2*X(38177)

X(38128) lies on these lines: {3,3036}, {8,6713}, {10,119}, {11,5690}, {30,38099}, {80,10993}, {100,6875}, {104,3617}, {140,1317}, {511,38192}, {515,38213}, {517,23513}, {518,38124}, {519,38032}, {528,38066}, {549,952}, {631,12531}, {936,6265}, {971,38211}, {1145,6734}, {1482,6667}, {1484,13996}, {1537,9956}, {1698,11729}, {2802,24386}, {2829,5790}, {3035,19914}, {3582,5844}, {3626,11715}, {3634,25485}, {3697,17654}, {4853,12737}, {5657,5840}, {5762,38202}, {5846,38119}, {5848,38116}, {5851,38121}, {5853,38131}, {5855,38135}, {5856,38126}, {6684,15863}, {6702,11362}, {6883,10087}, {7330,12515}, {7972,31423}, {8256,26470}, {9588,12119}, {9780,10698}, {9952,12738}, {10039,12832}, {10057,31515}, {11231,34123}, {12245,31272}, {12751,16209}, {18802,24390}, {19907,31235}, {28194,38161}, {28212,38141}, {28234,32557}

X(38128) = reflection of X(i) in X(j) for these (i,j): (23513, 34122), (34123, 11231), (38032, 38133), (38038, 38182), (38156, 38177)
X(38128) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1145, 12619, 37726), (26446, 38112, 38129), (34122, 38038, 38182), (38032, 38133, 38069), (38038, 38182, 23513), (38099, 38156, 38177)

X(38129) = CENTROID OF TRIANGLE {X(3), X(8), X(12)}

Barycentrics    4*(b+c)*a^6-(7*b^2+18*b*c+7*c^2)*a^5-(b+c)*(5*b^2-22*b*c+5*c^2)*a^4+2*(7*b^4+7*c^4+b*c*(3*b^2-14*b*c+3*c^2))*a^3-2*(b^2-c^2)*(b-c)*(b^2+10*b*c+c^2)*a^2-(b^2-c^2)^2*(7*b^2-12*b*c+7*c^2)*a+3*(b^2-c^2)^3*(b-c) : :
X(38129) = X(8)+2*X(31659) = 4*X(10)-X(26470) = X(12)+2*X(5690) = 4*X(140)-X(37734) = X(1482)-4*X(6668) = 5*X(3617)+X(11491) = 2*X(38033)-3*X(38070) = X(38039)-3*X(38058) = 2*X(38039)-3*X(38109) = 3*X(38058)-2*X(38183) = 3*X(38070)-4*X(38134) = 3*X(38100)-X(38157) = 3*X(38100)-2*X(38178) = 3*X(38109)-4*X(38183)

X(38129) lies on these lines: {8,31659}, {10,6882}, {12,5690}, {30,38100}, {140,24926}, {200,37733}, {355,31424}, {511,38193}, {515,38214}, {517,17530}, {518,38125}, {519,38033}, {529,38066}, {549,952}, {758,10273}, {971,38212}, {1482,6668}, {3584,5844}, {3617,11491}, {3877,23513}, {5657,17579}, {5762,38203}, {5790,5842}, {5840,38215}, {5846,38120}, {5849,38116}, {5852,38121}, {5853,38132}, {5854,38135}, {5857,38126}, {6735,34339}, {10959,18395}, {28194,38162}, {28212,38142}, {28234,38062}, {31434,37737}, {34352,37722}

X(38129) = reflection of X(i) in X(j) for these (i,j): (38033, 38134), (38039, 38183), (38109, 38058), (38157, 38178)
X(38129) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (26446, 38112, 38128), (38033, 38134, 38070), (38039, 38058, 38183), (38039, 38183, 38109), (38100, 38157, 38178)

X(38130) = CENTROID OF TRIANGLE {X(3), X(9), X(10)}

Barycentrics    4*a^6-5*(b+c)*a^5-(7*b^2+10*b*c+7*c^2)*a^4+2*(b+c)*(5*b^2+2*b*c+5*c^2)*a^3+2*(b^2+4*b*c+c^2)*(b-c)^2*a^2-(b^2-c^2)*(b-c)*(5*b^2+6*b*c+5*c^2)*a+(b^2-c^2)^2*(b-c)^2 : :
X(38130) = X(7)-7*X(31423) = X(9)+2*X(6684) = X(10)+2*X(31658) = X(40)+5*X(18230) = 4*X(140)-X(5542) = 5*X(631)+X(5223) = X(946)-4*X(6666) = 2*X(1001)+X(11362) = 5*X(1698)+X(5759) = 4*X(3634)-X(5805) = 3*X(21153)+X(38154) = 3*X(26446)-X(38121) = X(38031)-3*X(38067) = 2*X(38043)-3*X(38059) = X(38043)-3*X(38113) = 3*X(38057)-X(38154) = 3*X(38067)+X(38126) = 3*X(38068)-X(38123) = 3*X(38101)-X(38158) = 3*X(38101)-2*X(38179)

X(38130) lies on these lines: {2,38036}, {7,31423}, {9,1158}, {10,31658}, {30,38101}, {35,10392}, {40,18230}, {140,5542}, {142,15296}, {165,5817}, {381,516}, {511,38194}, {515,21153}, {517,38043}, {518,10165}, {519,38031}, {527,38068}, {631,5223}, {758,38132}, {946,6666}, {952,38210}, {971,3740}, {1001,11362}, {1210,15837}, {1445,21620}, {1698,5759}, {2802,38131}, {3576,5686}, {3634,5805}, {3911,15298}, {4312,10588}, {5054,38030}, {5218,10398}, {5690,30331}, {5762,11231}, {5840,38216}, {5847,38117}, {5850,38122}, {5853,38127}, {5856,38133}, {5857,38134}, {5882,24393}, {6594,10265}, {9588,35514}, {10172,38150}, {10863,35595}, {11539,38041}, {12512,31672}, {15709,38024}, {15841,34753}, {19862,20330}, {19875,38149}, {21168,38052}, {28146,38139}, {28194,38037}, {28198,38082}, {28204,38175}, {30329,31837}, {35242,36991}

X(38130) = midpoint of X(i) and X(j) for these {i,j}: {165, 5817}, {3576, 5686}, {21153, 38057}, {21168, 38052}, {38031, 38126}
X(38130) = reflection of X(i) in X(j) for these (i,j): (38059, 38113), (38150, 10172), (38158, 38179), (38204, 11231)
X(38130) = complement of X(38036)
X(38130) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38067, 38126, 38031), (38101, 38158, 38179)

X(38131) = CENTROID OF TRIANGLE {X(3), X(9), X(11)}

Barycentrics    4*a^9-10*(b+c)*a^8-(3*b^2-22*b*c+3*c^2)*a^7+(b+c)*(25*b^2-28*b*c+25*c^2)*a^6-(11*b^4+11*c^4+2*b*c*(10*b^2+7*b*c+10*c^2))*a^5-(b+c)*(19*b^4+19*c^4-2*b*c*(16*b^2-5*b*c+16*c^2))*a^4+3*(5*b^4+5*c^4+2*b*c*(4*b^2+9*b*c+4*c^2))*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(3*b^4+3*c^4+2*b*c*(3*b^2-13*b*c+3*c^2))*a^2-(b^2-c^2)^2*(b-c)^2*(5*b^2+6*b*c+5*c^2)*a+(b^2-c^2)^3*(b-c)^3 : :
X(38131) = X(9)+2*X(6713) = X(11)+2*X(31658) = X(104)+5*X(18230) = X(119)-4*X(6666) = 4*X(140)-X(10427) = 5*X(631)+X(1156) = X(5759)+5*X(31272) = X(5805)-4*X(6667) = 2*X(6594)+X(37726) = 3*X(34126)-X(38173) = 3*X(38069)-X(38124) = 3*X(38102)-X(38159) = 3*X(38102)-2*X(38180) = 2*X(38173)-3*X(38205)

X(38131) lies on these lines: {2,14646}, {9,6713}, {11,31658}, {30,38102}, {104,18230}, {119,6666}, {140,10427}, {511,38195}, {515,38216}, {516,6882}, {517,38060}, {518,38032}, {527,38069}, {528,26446}, {631,1156}, {952,6883}, {971,21154}, {2800,38059}, {2801,10165}, {2802,38130}, {2829,38108}, {5660,7308}, {5759,31272}, {5762,34126}, {5805,6667}, {5817,37249}, {5840,21153}, {5848,38117}, {5851,38122}, {5853,38128}, {5854,38126}, {5857,38135}, {5886,8257}, {6594,37726}, {6963,30312}, {15325,37787}

X(38131) = reflection of X(i) in X(j) for these (i,j): (38159, 38180), (38205, 34126)
X(38131) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38031, 38113, 38132), (38102, 38159, 38180)

X(38132) = CENTROID OF TRIANGLE {X(3), X(9), X(12)}

Barycentrics    4*a^9-10*(b+c)*a^8-(3*b-c)*(b-3*c)*a^7+(b+c)*(25*b^2-4*b*c+25*c^2)*a^6-(b^2+c^2)*(11*b^2+20*b*c+11*c^2)*a^5-(b+c)*(19*b^4+19*c^4-2*b*c*(4*b^2+3*b*c+4*c^2))*a^4+3*(5*b^4+5*c^4+6*b*c*(2*b^2+3*b*c+2*c^2))*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(3*b^4+3*c^4+2*b*c*(3*b^2-5*b*c+3*c^2))*a^2-(b^2-c^2)^2*(b-c)^2*(5*b^2+6*b*c+5*c^2)*a+(b^2-c^2)^3*(b-c)^3 : :
X(38132) = X(9)+2*X(31659) = X(12)+2*X(31658) = X(5805)-4*X(6668) = 4*X(6666)-X(26470) = X(11491)+5*X(18230) = 3*X(38070)-X(38125) = 3*X(38103)-X(38160) = 3*X(38103)-2*X(38181) = 3*X(38114)-X(38174) = 2*X(38174)-3*X(38206)

X(38132) lies on these lines: {9,31659}, {12,31658}, {30,38103}, {511,38196}, {515,38217}, {516,6842}, {517,38061}, {518,38033}, {527,38070}, {529,38067}, {758,38130}, {952,6883}, {971,21155}, {2478,38149}, {3678,10165}, {5762,38114}, {5805,6668}, {5817,37284}, {5840,38218}, {5842,38108}, {5849,38117}, {5852,37612}, {5853,38129}, {5855,38126}, {5856,38135}, {6666,26470}, {6889,8232}, {11491,18230}

X(38132) = reflection of X(i) in X(j) for these (i,j): (38160, 38181), (38206, 38114)
X(38132) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38031, 38113, 38131), (38103, 38160, 38181)

X(38133) = CENTROID OF TRIANGLE {X(3), X(10), X(11)}

Barycentrics    2*a^7-(b+c)*a^6-(7*b^2-4*b*c+7*c^2)*a^5+(b+c)*(4*b^2+b*c+4*c^2)*a^4+(8*b^4+8*c^4-b*c*(11*b^2+2*b*c+11*c^2))*a^3-(b^2-c^2)*(b-c)*(5*b^2+7*b*c+5*c^2)*a^2-(b^2-c^2)^2*(3*b^2-7*b*c+3*c^2)*a+2*(b^2-c^2)^3*(b-c) : :
X(38133) = 2*X(3)+X(6246) = X(3)+2*X(6702) = X(10)+2*X(6713) = 2*X(10)+X(11715) = X(11)+2*X(6684) = X(40)+2*X(16174) = X(40)+5*X(31272) = X(80)+5*X(631) = X(100)-7*X(31423) = 4*X(140)-X(214) = 2*X(140)+X(12619) = X(214)+2*X(12619) = 2*X(1385)+X(15863) = 2*X(3035)+X(10265) = 2*X(3036)+X(5882) = X(6246)-4*X(6702) = 4*X(6713)-X(11715) = 2*X(16174)-5*X(31272) = 3*X(38104)-X(38161) = 3*X(38104)-2*X(38182)

X(38133) lies on these lines: {2,2800}, {3,6246}, {10,140}, {11,6684}, {30,38104}, {40,16174}, {80,631}, {100,31423}, {104,1698}, {119,3634}, {153,19877}, {371,13976}, {372,8988}, {404,5450}, {442,10172}, {498,5083}, {499,5445}, {511,38197}, {515,21154}, {516,6882}, {517,32557}, {519,38032}, {528,38068}, {758,38135}, {946,6667}, {971,38216}, {1006,3586}, {1125,25485}, {1387,11362}, {1656,12515}, {2771,34128}, {2801,38122}, {2802,26446}, {2829,10175}, {3090,34789}, {3523,12119}, {3525,12247}, {3526,6265}, {3624,10698}, {3628,12611}, {3754,5886}, {4881,38215}, {5048,28234}, {5251,18861}, {5432,20118}, {5444,12647}, {5660,12691}, {5762,38207}, {5770,15528}, {5840,10164}, {5847,38119}, {5848,38118}, {5850,38124}, {5851,38123}, {5854,38127}, {5856,38130}, {5884,18254}, {6224,10303}, {7989,10728}, {9540,19077}, {9780,12751}, {10320,15556}, {10711,19876}, {10724,35242}, {11108,12332}, {11729,19862}, {12005,27529}, {12532,15016}, {12736,24914}, {12747,15720}, {12832,13411}, {13226,20400}, {13253,34595}, {13883,13977}, {13893,19081}, {13913,13936}, {13935,19078}, {13947,19082}, {15017,19872}, {16408,22775}, {17654,25917}, {20107,37562}, {22938,31663}, {28146,38141}, {28194,38038}, {28198,38084}, {28204,38177}

X(38133) = midpoint of X(i) and X(j) for these {i,j}: {5657, 16173}, {21154, 34122}, {34474, 37718}, {38032, 38128}
X(38133) = reflection of X(i) in X(j) for these (i,j): (32557, 34126), (38161, 38182)
X(38133) = X(6246)-Gibert-Moses centroid
X(38133) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6702, 6246), (10, 6713, 11715), (40, 31272, 16174), (140, 12619, 214), (10165, 11231, 38134), (38069, 38128, 38032), (38104, 38161, 38182)

X(38134) = CENTROID OF TRIANGLE {X(3), X(10), X(12)}

Barycentrics    2*a^7-(b+c)*a^6-(7*b^2+4*b*c+7*c^2)*a^5+(b+c)*(4*b^2+3*b*c+4*c^2)*a^4+(8*b^4+8*c^4-b*c*(b^2+6*b*c+c^2))*a^3-(b^2-c^2)*(b-c)*(5*b^2+9*b*c+5*c^2)*a^2-(b^2-c^2)^2*(3*b^2-5*b*c+3*c^2)*a+2*(b^2-c^2)^3*(b-c) : :
X(38134) = X(10)+2*X(31659) = X(12)+2*X(6684) = 5*X(631)+X(37710) = X(946)-4*X(6668) = 5*X(1698)+X(11491) = X(2975)-7*X(31423) = 4*X(3634)-X(26470) = X(11362)+2*X(37737) = 3*X(21155)+X(38157) = X(38033)-3*X(38070) = 2*X(38045)-3*X(38062) = X(38045)-3*X(38114) = 3*X(38058)-X(38157) = 3*X(38070)+X(38129) = 3*X(38105)-X(38162) = 3*X(38105)-2*X(38183)

X(38134) lies on these lines: {10,140}, {12,4292}, {21,5587}, {30,38105}, {40,10129}, {498,3485}, {511,38198}, {515,21155}, {516,6842}, {517,38045}, {519,38033}, {529,38068}, {631,37710}, {758,26446}, {946,6668}, {971,38217}, {1698,11491}, {2802,38135}, {2975,31423}, {3634,26470}, {3884,5886}, {4187,10172}, {5552,15016}, {5660,20117}, {5731,37291}, {5762,38208}, {5840,38219}, {5842,10175}, {5847,38120}, {5849,38118}, {5850,38125}, {5852,38123}, {5855,38127}, {5857,38130}, {5884,26487}, {6940,7280}, {10283,34352}, {11011,13411}, {11362,37737}, {15865,17718}, {28146,38142}, {28194,38039}, {28198,38085}, {28204,38178}, {31650,38042}

X(38134) = midpoint of X(i) and X(j) for these {i,j}: {5657, 37701}, {21155, 38058}, {38033, 38129}
X(38134) = reflection of X(i) in X(j) for these (i,j): (38062, 38114), (38162, 38183)
X(38134) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10165, 11231, 38133), (38070, 38129, 38033), (38105, 38162, 38183)

X(38135) = CENTROID OF TRIANGLE {X(3), X(11), X(12)}

Barycentrics    a^10-2*(b+c)*a^9-3*(b-c)^2*a^8+8*(b^3+c^3)*a^7+(2*b^4+2*c^4-b*c*(16*b^2-9*b*c+16*c^2))*a^6-2*(b+c)*(6*b^4+6*c^4-b*c*(12*b^2-11*b*c+12*c^2))*a^5+(2*b^6+2*c^6+(12*b^4+12*c^4-b*c*(15*b^2-4*b*c+15*c^2))*b*c)*a^4+2*(b^2-c^2)*(b-c)*(4*b^4+4*c^4-b*c*(4*b^2+b*c+4*c^2))*a^3-3*(b^2-c^2)^4*a^2-2*(b^2-c^2)^3*(b-c)^3*a+(b^2-c^2)^4*(b-c)^2 : :
X(38135) = X(3)+2*X(8068) = X(11)+2*X(31659) = X(12)+2*X(6713) = X(119)-4*X(6668) = 4*X(140)-X(4996) = 4*X(6667)-X(26470) = X(11491)+5*X(31272) = 3*X(38106)-X(38163) = 3*X(38106)-2*X(38184)

X(38135) lies on these lines: {2,952}, {3,8068}, {11,31659}, {12,6713}, {30,38106}, {119,6668}, {140,4996}, {498,19914}, {511,38199}, {515,38219}, {517,38063}, {528,38070}, {529,38069}, {758,38133}, {971,38218}, {1482,10320}, {2800,38062}, {2802,38134}, {2829,38109}, {4293,6958}, {4294,6863}, {5762,38209}, {5840,21155}, {5842,23513}, {5848,38120}, {5849,38119}, {5851,38125}, {5852,38124}, {5854,38129}, {5855,38128}, {5856,38132}, {5857,38131}, {6667,26470}, {6853,33814}, {6862,10742}, {6888,22799}, {6960,22938}, {6980,10058}, {10265,20104}, {11491,31272}, {12331,31493}

X(38135) = reflection of X(38163) in X(38184)
X(38135) = {X(38106), X(38163)}-harmonic conjugate of X(38184)

X(38136) = CENTROID OF TRIANGLE {X(4), X(5), X(6)}

Barycentrics    7*(b^2+c^2)*a^4-4*(b^4-3*b^2*c^2+c^4)*a^2-3*(b^4-c^4)*(b^2-c^2) : :
X(38136) = X(4)+2*X(18583) = 5*X(5)-2*X(141) = X(5)+2*X(5480) = X(5)-4*X(19130) = 2*X(5)+X(21850) = 7*X(5)-4*X(24206) = X(6)+2*X(546) = X(141)+5*X(5480) = X(141)-10*X(19130) = 4*X(141)+5*X(21850) = 7*X(141)-10*X(24206) = X(5085)-3*X(14561) = 7*X(5085)-9*X(38064) = X(5085)-9*X(38072) = 4*X(5085)-9*X(38079) = 2*X(5085)-3*X(38110) = X(5480)+2*X(19130) = 4*X(5480)-X(21850) = 7*X(5480)+2*X(24206) = 8*X(19130)+X(21850) = 7*X(19130)-X(24206) = 7*X(21850)+8*X(24206)

X(38136) lies on these lines: {4,5050}, {5,141}, {6,546}, {30,5085}, {69,3851}, {140,31670}, {182,3627}, {193,3855}, {373,1368}, {381,1992}, {382,3618}, {427,5640}, {428,6800}, {515,38040}, {516,38166}, {517,38146}, {518,38034}, {524,38071}, {542,23046}, {549,29181}, {550,3589}, {568,7403}, {576,3857}, {597,15687}, {599,11737}, {632,3098}, {952,38035}, {971,38164}, {1350,3628}, {1351,3091}, {1352,3850}, {1353,3818}, {1469,10593}, {1503,3845}, {1539,15118}, {1595,9730}, {1907,15072}, {2829,38168}, {3056,10592}, {3066,5159}, {3090,33878}, {3146,12017}, {3544,3620}, {3619,5079}, {3629,18553}, {3763,35018}, {3830,25406}, {3832,18440}, {3839,14848}, {3843,6776}, {3859,15069}, {3861,36990}, {5055,10519}, {5066,10516}, {5076,14927}, {5092,15704}, {5133,11002}, {5254,6249}, {5318,36758}, {5321,36757}, {5762,38145}, {5842,38169}, {5843,38143}, {5844,38144}, {5845,38137}, {5846,38138}, {5847,38140}, {5848,38141}, {5849,38142}, {5921,11482}, {5946,34146}, {6090,6997}, {6756,37506}, {7405,13340}, {7998,37439}, {8703,29317}, {8705,11563}, {9825,37497}, {9970,11801}, {9993,37451}, {10095,19161}, {10168,15686}, {10264,32271}, {10301,14389}, {10982,19139}, {11179,14893}, {11459,31802}, {11477,12811}, {11539,19924}, {11548,33586}, {13331,15048}, {13363,23335}, {14810,14869}, {14984,36518}, {15681,33750}, {16981,37353}, {18383,34774}, {20300,33332}, {28146,38118}, {28160,38049}, {28174,38047}, {28186,38029}, {28202,38089}, {28212,38116}, {29323,33699}, {33540,34817}, {33884,37990}, {35268,37649}

X(38136) = midpoint of X(i) and X(j) for these {i,j}: {4, 5050}, {381, 14853}, {1352, 5102}, {3818, 15520}, {3830, 25406}, {3839, 14848}, {10516, 20423}, {31670, 31884}
X(38136) = reflection of X(i) in X(j) for these (i,j): (550, 17508), (1353, 15520), (5050, 18583), (10516, 5066), (17508, 3589), (31884, 140), (38110, 14561)
X(38136) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 5480, 21850), (1351, 3091, 18358), (1353, 3858, 3818), (5480, 19130, 5), (14561, 38110, 38079), (38147, 38148, 38035)

X(38137) = CENTROID OF TRIANGLE {X(4), X(5), X(7)}

Barycentrics    4*a^6-4*(b+c)*a^5-3*(b^2+c^2)*a^4-2*(b+c)*(b^2+c^2)*a^3+2*(2*b^2-b*c+2*c^2)*(b-c)^2*a^2+6*(b^2-c^2)^2*(b+c)*a-5*(b^2-c^2)^2*(b-c)^2 : :
X(38137) = 5*X(5)-2*X(9) = 7*X(5)+2*X(5735) = X(5)+2*X(5805) = 3*X(5)-2*X(38108) = X(7)+2*X(546) = 7*X(9)+5*X(5735) = X(9)+5*X(5805) = 3*X(9)-5*X(38108) = X(9)-5*X(38150) = X(5735)-7*X(5805) = 3*X(5735)+7*X(38108) = X(5735)+7*X(38150) = 3*X(5805)+X(38108) = 7*X(21151)-9*X(38065) = X(21151)-9*X(38073) = 4*X(21151)-9*X(38080) = X(21151)-3*X(38107) = 2*X(21151)-3*X(38111) = X(38065)-7*X(38073) = 4*X(38065)-7*X(38080) = 3*X(38065)-7*X(38107) = 6*X(38065)-7*X(38111) = X(38108)-3*X(38150)

X(38137) lies on these lines: {5,9}, {7,546}, {30,21151}, {140,31671}, {142,550}, {144,3851}, {381,5843}, {515,38041}, {516,549}, {517,38151}, {518,38138}, {527,38071}, {952,38036}, {971,3845}, {1483,15570}, {1503,38164}, {2829,38173}, {3564,38143}, {3627,18482}, {3628,5759}, {3843,36996}, {3850,5779}, {3855,20059}, {3861,36991}, {4312,34753}, {5055,21168}, {5066,5817}, {5842,38174}, {5844,38149}, {5845,38136}, {5850,38140}, {5851,38141}, {5852,38142}, {6172,11737}, {6173,15687}, {8703,38122}, {8727,27003}, {11038,28224}, {11539,21153}, {14869,20195}, {15699,38113}, {17504,38093}, {18230,35018}, {24644,38069}, {28146,38123}, {28160,38054}, {28174,38052}, {28186,38030}, {28202,38094}, {28212,38121}

X(38137) = midpoint of X(5805) and X(38150)
X(38137) = reflection of X(i) in X(j) for these (i,j): (5, 38150), (549, 38171), (5817, 5066), (8703, 38122), (17504, 38093), (38111, 38107)
X(38137) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18482, 31657, 3627), (38107, 38111, 38080), (38152, 38153, 38036)

X(38138) = CENTROID OF TRIANGLE {X(4), X(5), X(8)}

Barycentrics    4*a^4-4*(b+c)*a^3+(b^2+8*b*c+c^2)*a^2+4*(b^2-c^2)*(b-c)*a-5*(b^2-c^2)^2 : :
X(38138) = 2*X(1)-5*X(5) = X(1)+5*X(355) = 8*X(1)-5*X(1483) = X(1)-5*X(5587) = 7*X(1)+5*X(5881) = 3*X(1)-5*X(5886) = 7*X(1)-10*X(5901) = 7*X(1)-15*X(7988) = 4*X(1)-5*X(10283) = X(1)-10*X(18357) = 4*X(1)+5*X(37705) = 3*X(1)+5*X(37712) = 11*X(1)-5*X(37727) = X(5)+2*X(355) = 4*X(5)-X(1483) = 7*X(5)+2*X(5881) = 3*X(5)-2*X(5886) = 7*X(5)-4*X(5901) = 7*X(5)-6*X(7988) = 11*X(5)-14*X(7989) = 13*X(5)-10*X(8227) = X(5)-4*X(18357) = 2*X(5)+X(37705) = 3*X(5)+2*X(37712) = X(5)-10*X(37714) = 11*X(5)-2*X(37727)

X(38138) lies on these lines: {1,5}, {2,28224}, {4,4678}, {8,546}, {10,550}, {30,5657}, {40,28182}, {140,5731}, {145,3851}, {165,15686}, {381,5844}, {382,3617}, {515,549}, {516,3627}, {517,3845}, {518,38137}, {519,38034}, {547,10246}, {632,9956}, {944,3628}, {946,3857}, {962,3861}, {971,38170}, {1385,10172}, {1478,11545}, {1482,3850}, {1503,38165}, {1698,14869}, {1699,23046}, {2829,38177}, {3036,22799}, {3090,18526}, {3091,12645}, {3241,11737}, {3530,9780}, {3544,3623}, {3545,10247}, {3564,38144}, {3576,11539}, {3579,28172}, {3616,35018}, {3621,3855}, {3622,5079}, {3626,22793}, {3654,28178}, {3679,15687}, {3830,28216}, {3832,8148}, {3843,12245}, {3853,12702}, {3858,4701}, {4745,28150}, {5055,7967}, {5056,37624}, {5066,5603}, {5072,10595}, {5076,20070}, {5428,11500}, {5432,37006}, {5691,15704}, {5762,38154}, {5766,5817}, {5770,37281}, {5816,16675}, {5842,38178}, {5843,6917}, {5846,38136}, {5853,38139}, {5854,38141}, {5855,38142}, {6102,23841}, {6912,12331}, {6946,12773}, {7508,18524}, {7979,20584}, {8256,38213}, {8703,26446}, {9613,34753}, {9656,11544}, {9812,14893}, {10039,10386}, {10164,28208}, {10171,38022}, {10175,15699}, {11230,28236}, {11278,12571}, {11591,16980}, {12811,18493}, {13624,22266}, {15177,37936}, {15712,18481}, {15714,34628}, {17504,19875}, {17563,25005}, {17564,34122}, {19710,28164}, {24475,30329}, {28146,34648}, {28154,31673}, {28202,38098}, {31649,32141}

X(38138) = midpoint of X(i) and X(j) for these {i,j}: {355, 5587}, {5731, 18525}, {5886, 37712}, {9812, 34718}, {10246, 34627}, {10283, 37705}, {18480, 38176}, {34648, 38127}
X(38138) = reflection of X(i) in X(j) for these (i,j): (5, 5587), (549, 38042), (1385, 10172), (1483, 10283), (5587, 18357), (5603, 5066), (5690, 38176), (5731, 140), (8703, 26446), (9812, 14893), (10165, 9956), (10246, 547), (10283, 5), (15686, 165), (17502, 3828), (17504, 19875), (34773, 10165), (38028, 10175), (38034, 38140), (38112, 5790)
X(38138) = X(549)-of-Fuhrmann-triangle
X(38138) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 355, 37705), (5, 37705, 1483), (355, 5886, 37712), (355, 18357, 5), (355, 37714, 18357), (5587, 5881, 7988), (5587, 37712, 5886), (5690, 18480, 3627), (5790, 38112, 38081), (5818, 18525, 140), (9956, 34773, 632), (10175, 38028, 15699), (19925, 22791, 3858), (38034, 38140, 38071), (38156, 38157, 5587)

X(38139) = CENTROID OF TRIANGLE {X(4), X(5), X(9)}

Barycentrics    2*(b+c)*a^5-(7*b^2-4*b*c+7*c^2)*a^4+6*(b+c)*(b^2+c^2)*a^3+2*(b+2*c)*(2*b+c)*(b-c)^2*a^2-8*(b^2-c^2)^2*(b+c)*a+3*(b^2-c^2)^2*(b-c)^2 : :
X(38139) = 5*X(5)-2*X(142) = 4*X(5)-X(31657) = 3*X(5)-X(38111) = X(7)-7*X(3851) = X(9)+2*X(546) = 8*X(142)-5*X(31657) = 6*X(142)-5*X(38111) = 4*X(142)-5*X(38171) = 7*X(21153)-9*X(38067) = X(21153)-9*X(38075) = 4*X(21153)-9*X(38082) = X(21153)-3*X(38108) = 2*X(21153)-3*X(38113) = 3*X(31657)-4*X(38111) = X(38067)-7*X(38075) = 4*X(38067)-7*X(38082) = 3*X(38067)-7*X(38108) = 6*X(38067)-7*X(38113) = 4*X(38075)-X(38082) = 3*X(38075)-X(38108) = 6*X(38075)-X(38113) = 2*X(38111)-3*X(38171)

X(38139) lies on these lines: {5,142}, {7,3851}, {9,546}, {30,21153}, {140,31672}, {144,3855}, {381,5762}, {382,18230}, {515,38043}, {516,3845}, {517,38158}, {518,38034}, {527,38071}, {547,38122}, {550,6666}, {952,38037}, {1503,38166}, {1656,36991}, {2801,38041}, {2829,38180}, {3091,5779}, {3545,38107}, {3564,38145}, {3627,31658}, {3628,5732}, {3832,31671}, {3839,21168}, {3843,5759}, {3850,5805}, {3858,18482}, {3859,5735}, {3947,15008}, {3988,20117}, {5055,21151}, {5066,5843}, {5068,36996}, {5587,24644}, {5690,16616}, {5719,5809}, {5842,38181}, {5844,38154}, {5853,38138}, {5856,38141}, {5857,38142}, {6173,11737}, {7988,38030}, {8226,31053}, {8232,12433}, {8581,10593}, {10592,14100}, {12618,28633}, {20195,35018}, {28146,38130}, {28160,38059}, {28174,38057}, {28186,38031}, {28202,38101}, {28212,38126}

X(38139) = midpoint of X(381) and X(5817)
X(38139) = reflection of X(i) in X(j) for these (i,j): (31657, 38171), (38113, 38108), (38122, 547), (38150, 5066), (38171, 5)
X(38139) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38108, 38113, 38082), (38159, 38160, 38037)

X(38140) = CENTROID OF TRIANGLE {X(4), X(5), X(10)}

Barycentrics    2*a^4-(b+c)*a^3+2*(b^2+b*c+c^2)*a^2+(b^2-c^2)*(b-c)*a-4*(b^2-c^2)^2 : :
X(38140) = X(1)-7*X(3851) = X(3)-7*X(7989) = X(3)+5*X(18492) = 2*X(3)+X(33697) = 2*X(4)+X(3579) = 3*X(4)+X(9778) = 5*X(4)+7*X(9780) = X(4)+2*X(9956) = 5*X(5)-2*X(1125) = 4*X(5)-X(1385) = 3*X(5)-2*X(10171) = 2*X(5)+X(18480) = X(5)+2*X(19925) = 7*X(5)-X(34773) = 3*X(5)-X(38028) = 3*X(3579)-2*X(9778) = 5*X(3579)-14*X(9780) = X(3579)-4*X(9956) = 7*X(7989)+5*X(18492) = 14*X(7989)+X(33697) = X(9778)-6*X(9956) = X(9778)-3*X(26446) = 7*X(9780)-10*X(9956) = 7*X(9780)-5*X(26446) = 10*X(18492)-X(33697)

X(38140) lies on these lines: {1,3851}, {2,17502}, {3,7989}, {4,2355}, {5,515}, {8,3855}, {10,546}, {30,10164}, {40,3843}, {80,17605}, {119,8226}, {140,28190}, {145,355}, {165,3830}, {210,381}, {226,12019}, {354,37718}, {382,1698}, {392,37375}, {495,18527}, {516,3845}, {519,38034}, {547,10165}, {549,10172}, {550,3634}, {551,11737}, {758,38142}, {942,10826}, {944,5068}, {946,3625}, {950,10592}, {952,3817}, {971,38172}, {1478,17728}, {1482,37714}, {1503,38167}, {1538,6968}, {1656,5691}, {1657,31423}, {1737,11246}, {1829,7547}, {1871,7559}, {1902,35488}, {2475,17619}, {2771,5927}, {2801,38173}, {2802,38141}, {2829,38182}, {3085,31795}, {3090,18481}, {3338,9656}, {3475,5722}, {3529,19877}, {3544,3616}, {3545,5886}, {3560,33862}, {3564,38146}, {3576,5055}, {3585,17606}, {3586,31479}, {3614,10572}, {3624,5079}, {3627,6684}, {3628,4297}, {3654,9812}, {3656,9779}, {3683,17057}, {3753,17577}, {3828,15687}, {3832,5818}, {3839,5657}, {3853,31447}, {3856,11362}, {3857,4746}, {3858,5690}, {3859,4301}, {3860,28212}, {3861,31399}, {3947,12433}, {4663,18553}, {5045,9581}, {5049,11237}, {5070,7987}, {5071,5731}, {5072,8227}, {5073,30315}, {5122,12943}, {5154,17614}, {5229,31776}, {5252,7743}, {5439,26201}, {5560,37571}, {5658,6843}, {5694,7686}, {5726,6767}, {5762,38158}, {5806,10894}, {5842,38183}, {5843,38151}, {5847,38136}, {5850,38137}, {5876,31760}, {5881,18493}, {5887,7548}, {5901,12811}, {6001,23325}, {6246,22935}, {6583,14872}, {6702,22799}, {6826,18516}, {6835,37821}, {6846,26487}, {6849,10526}, {6893,18517}, {6911,23961}, {6913,18491}, {6918,18761}, {6929,18407}, {6957,37820}, {6964,26492}, {7377,29628}, {7384,29587}, {7545,9625}, {7741,24928}, {7748,31430}, {7951,24929}, {7988,10246}, {8703,28172}, {9578,9669}, {9612,31794}, {9624,18526}, {9626,34864}, {9668,31434}, {9957,10827}, {10095,31732}, {10106,10593}, {10222,10609}, {10247,37712}, {10254,24301}, {10263,31752}, {10283,28236}, {10742,11219}, {10914,26200}, {11363,16868}, {11522,12645}, {11709,15088}, {11710,15092}, {11928,13600}, {12101,28182}, {12262,32767}, {12616,22792}, {12664,31828}, {12688,13145}, {12747,15017}, {13373,18542}, {13464,37705}, {13743,26086}, {14128,31738}, {14269,19875}, {14869,31253}, {14893,28178}, {15079,32636}, {15681,19876}, {15703,34628}, {15720,19872}, {16128,20292}, {16192,17800}, {16200,30308}, {16616,31837}, {17359,29032}, {18443,18529}, {18514,37568}, {18586,34557}, {18587,34556}, {19862,35018}, {22936,37230}, {23046,28194}, {23261,31439}, {23708,25405}, {26285,37234}, {31758,31824}, {31937,35004}, {33152,37717}

X(38140) = midpoint of X(i) and X(j) for these {i,j}: {4, 26446}, {165, 3830}, {355, 5603}, {381, 5587}, {946, 38155}, {1699, 5790}, {3654, 9812}, {3845, 38042}, {10165, 34648}, {10247, 37712}, {10742, 11219}, {11230, 18480}, {14269, 19875}, {38034, 38138}
X(38140) = reflection of X(i) in X(j) for these (i,j): (549, 10172), (1385, 11230), (3579, 26446), (3817, 5066), (5603, 9955), (10165, 547), (11230, 5), (11231, 10175), (17502, 2), (24680, 5603), (26446, 9956), (38028, 10171), (38155, 18357)
X(38140) = X(33697)-Gibert-Moses centroid
X(38140) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 9956, 3579), (5, 18480, 1385), (5, 19925, 18480), (5, 38028, 10171), (10, 546, 22793), (381, 5790, 1699), (382, 1698, 31663), (1656, 5691, 13624), (1699, 5587, 5790), (3585, 17606, 37582), (3832, 5818, 12699), (3850, 18357, 946), (3857, 22791, 12571), (3858, 5690, 18483), (7989, 18492, 3), (9581, 9654, 5045), (10171, 38028, 11230), (10175, 11231, 38083), (10246, 19709, 7988), (10826, 10895, 942), (10827, 10896, 9957), (37712, 38021, 10247), (38071, 38138, 38034), (38161, 38162, 3817)

X(38141) = CENTROID OF TRIANGLE {X(4), X(5), X(11)}

Barycentrics    2*a^7-2*(b+c)*a^6+4*b*c*a^5-2*b*c*(b+c)*a^4-(6*b^4+6*c^4-7*b*c*(b^2+c^2))*a^3+6*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-c^2)^2*(4*b^2-11*b*c+4*c^2)*a-4*(b^2-c^2)^3*(b-c) : :
X(38141) = 5*X(5)-2*X(3035) = 2*X(5)+X(22938) = 4*X(5)-X(33814) = X(11)+2*X(546) = 2*X(11)+X(22799) = 4*X(546)-X(22799) = 4*X(3035)+5*X(22938) = 8*X(3035)-5*X(33814) = X(21154)-3*X(23513) = 2*X(21154)-3*X(34126) = 7*X(21154)-9*X(38069) = X(21154)-9*X(38077) = 4*X(21154)-9*X(38084) = 2*X(22938)+X(33814) = 7*X(23513)-3*X(38069) = X(23513)-3*X(38077) = 4*X(23513)-3*X(38084) = 7*X(34126)-6*X(38069) = X(34126)-6*X(38077) = 2*X(34126)-3*X(38084)

X(38141) lies on these lines: {5,3035}, {11,546}, {30,21154}, {100,3851}, {104,3843}, {119,3850}, {149,3855}, {355,26726}, {381,952}, {382,31272}, {515,38044}, {516,38180}, {517,38161}, {528,38071}, {550,6667}, {971,38173}, {1387,10896}, {1484,3858}, {1503,38168}, {1656,10724}, {1862,7547}, {2802,38140}, {2829,3845}, {3091,10738}, {3564,38147}, {3627,6713}, {3628,24466}, {3832,10742}, {3856,37726}, {3857,11698}, {3859,37725}, {5055,34474}, {5068,13199}, {5762,38159}, {5842,38184}, {5843,38152}, {5844,38156}, {5848,38136}, {5851,38137}, {5854,38138}, {5856,38139}, {6174,11737}, {6246,9955}, {6702,22793}, {10893,22791}, {10895,12735}, {12138,35488}, {12571,12611}, {12619,18483}, {12737,18492}, {16174,18480}, {17577,34123}, {28146,38133}, {28160,32557}, {28174,34122}, {28186,38032}, {28202,38104}, {28212,38128}, {31235,35018}

X(38141) = reflection of X(34126) in X(23513)
X(38141) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 22938, 33814), (11, 546, 22799), (381, 38034, 38142), (6246, 9955, 19907), (23513, 34126, 38084)

X(38142) = CENTROID OF TRIANGLE {X(4), X(5), X(12)}

Barycentrics    2*a^7-2*(b+c)*a^6+4*b*c*a^5-6*b*c*(b+c)*a^4-(6*b^4+6*c^4-b*c*(b^2+8*b*c+c^2))*a^3+2*(b^2-c^2)*(b-c)*(3*b^2+5*b*c+3*c^2)*a^2+(b^2-c^2)^2*(4*b^2-5*b*c+4*c^2)*a-4*(b^2-c^2)^3*(b-c) : :
X(38142) = 5*X(5)-2*X(4999) = X(12)+2*X(546) = X(550)-4*X(6668) = 7*X(21155)-9*X(38070) = X(21155)-9*X(38078) = 4*X(21155)-9*X(38085) = X(21155)-3*X(38109) = 2*X(21155)-3*X(38114) = X(38070)-7*X(38078) = 4*X(38070)-7*X(38085) = 3*X(38070)-7*X(38109) = 6*X(38070)-7*X(38114) = 4*X(38078)-X(38085) = 3*X(38078)-X(38109) = 6*X(38078)-X(38114) = 3*X(38085)-4*X(38109) = 3*X(38085)-2*X(38114) = 4*X(38162)-X(38178) = 5*X(38162)-X(38214) = 5*X(38178)-4*X(38214)

X(38142) lies on these lines: {5,993}, {12,546}, {30,21155}, {381,952}, {515,38045}, {516,38181}, {517,38162}, {529,38071}, {550,6668}, {758,38140}, {971,38174}, {1503,38169}, {2829,38184}, {2975,3851}, {3564,38148}, {3627,31659}, {3628,30264}, {3843,11491}, {3845,5842}, {3850,26470}, {3855,20060}, {5762,38160}, {5843,38153}, {5844,38157}, {5849,38136}, {5852,38137}, {5855,38138}, {5857,38139}, {7548,18357}, {7680,22938}, {8068,22799}, {10894,22791}, {10895,37737}, {11737,31157}, {17577,28174}, {18492,37733}, {28146,38134}, {28160,38062}, {28186,38033}, {28202,38105}, {28212,38129}, {31260,35018}

X(38142) = reflection of X(38114) in X(38109)
X(38142) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 38034, 38141), (38109, 38114, 38085)

X(38143) = CENTROID OF TRIANGLE {X(4), X(6), X(7)}

Barycentrics    a^8+(b+c)^2*a^6-6*(b+c)*(b^2+c^2)*a^5+(b^4+c^4-2*b*c*(4*b^2-3*b*c+4*c^2))*a^4+4*(b^2-c^2)^2*(b+c)*a^3-(b^4-10*b^2*c^2+c^4)*(b-c)^2*a^2+2*(b^4-c^4)*(b^2-c^2)*(b+c)*a+(b^2-c^2)*(b-c)^2*(-2*b^4+2*c^4) : :
X(38143) = X(6)+2*X(5805) = X(7)+2*X(5480) = 4*X(142)-X(1350) = 2*X(182)+X(31671) = X(3242)-4*X(20330) = 4*X(3589)-X(5759) = X(5779)-4*X(19130) = 3*X(14561)-2*X(38166) = 4*X(18482)-X(36990) = 2*X(31657)+X(31670) = 3*X(38072)-2*X(38145) = 3*X(38086)-2*X(38115) = 3*X(38086)-4*X(38164)

X(38143) lies on these lines: {6,5805}, {7,5480}, {30,38086}, {142,1350}, {182,31671}, {511,38107}, {515,38046}, {516,5085}, {517,38185}, {518,5587}, {524,38073}, {527,38072}, {2829,38188}, {3242,20330}, {3564,38137}, {3589,5759}, {5762,14561}, {5779,19130}, {5842,38189}, {5843,38136}, {5845,14853}, {5846,38149}, {5847,38151}, {5848,38152}, {5849,38153}, {5850,38146}, {5851,38147}, {5852,38148}, {18482,36990}, {19924,38065}, {21151,29181}, {31657,31670}, {31884,38122}

X(38143) = reflection of X(i) in X(j) for these (i,j): (5085, 38186), (10516, 38150), (31884, 38122), (38115, 38164)
X(38143) = {X(38115), X(38164)}-harmonic conjugate of X(38086)

X(38144) = CENTROID OF TRIANGLE {X(4), X(6), X(8)}

Barycentrics    a^6-2*(b+c)*a^5+4*(b^2+b*c+c^2)*a^4-4*b*c*(b+c)*a^3-(3*b^4+3*c^4-2*b*c*(2*b^2+3*b*c+2*c^2))*a^2+2*(b^4-c^4)*(b-c)*a-2*(b^4-c^4)*(b^2-c^2) : :
X(38144) = 4*X(5)-X(3242) = X(6)+2*X(355) = X(8)+2*X(5480) = 4*X(10)-X(1350) = 2*X(141)-5*X(5818) = 2*X(182)+X(18525) = 2*X(597)+X(34627) = X(944)-4*X(3589) = X(1352)-4*X(18357) = 2*X(1386)+X(5881) = X(1482)-4*X(19130) = 2*X(3416)+X(11477) = 2*X(3751)+X(15069) = X(3751)+5*X(37714) = 3*X(5085)-4*X(38118) = 3*X(14561)-2*X(38040) = 2*X(38035)-3*X(38072) = 3*X(38047)-2*X(38118) = 3*X(38072)-4*X(38146) = 3*X(38087)-2*X(38116) = 3*X(38087)-4*X(38165)

X(38144) lies on these lines: {5,3242}, {6,355}, {8,5480}, {10,1350}, {30,38087}, {80,611}, {141,5818}, {182,18525}, {511,5790}, {515,5085}, {516,38190}, {518,5587}, {519,38035}, {524,38074}, {597,34627}, {613,37710}, {944,3589}, {952,14561}, {971,38185}, {1352,18357}, {1386,5881}, {1482,19130}, {2771,25330}, {2829,38192}, {3416,11477}, {3545,9041}, {3564,38138}, {3751,15069}, {3763,9956}, {4265,11499}, {4437,36662}, {5096,22758}, {5102,5847}, {5603,9053}, {5657,29181}, {5690,31670}, {5842,38193}, {5844,38136}, {5845,38149}, {5846,14853}, {5848,38156}, {5849,38157}, {5853,38145}, {5854,38147}, {5855,38148}, {10039,10387}, {12368,16010}, {16475,37712}, {18480,36990}, {18583,37705}, {19161,23841}, {19924,38066}, {26446,31884}, {28204,38029}, {28224,38110}, {28236,38049}

X(38144) = midpoint of X(16475) and X(37712)
X(38144) = reflection of X(i) in X(j) for these (i,j): (5085, 38047), (10516, 5587), (31884, 26446), (38029, 38167), (38035, 38146), (38116, 38165)
X(38144) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38035, 38146, 38072), (38116, 38165, 38087)

X(38145) = CENTROID OF TRIANGLE {X(4), X(6), X(9)}

Barycentrics    a^8-(b+c)*a^7-4*(b^2+c^2)*a^6+7*(b+c)*(b^2+c^2)*a^5-2*(b^4+c^4-b*c*(3*b^2-8*b*c+3*c^2))*a^4-(b+c)*(3*b^4-14*b^2*c^2+3*c^4)*a^3+4*(b^2-c^2)*(b-c)*(b^3+c^3)*a^2-3*(b^4-c^4)*(b^2-c^2)*(b+c)*a+(b^4-c^4)*(b^2-c^2)*(b-c)^2 : :
X(38145) = X(9)+2*X(5480) = 2*X(182)+X(31672) = X(1350)-4*X(6666) = 4*X(3589)-X(5732) = 5*X(3618)+X(36991) = X(5805)-4*X(19130) = 3*X(14561)-X(38115) = 2*X(31658)+X(31670) = 3*X(38072)-X(38143) = 3*X(38088)-2*X(38117) = 3*X(38088)-4*X(38166) = 2*X(38115)-3*X(38186)

X(38145) lies on these lines: {9,5480}, {30,38088}, {182,31672}, {374,38057}, {511,38108}, {515,38048}, {516,36721}, {517,38190}, {518,5603}, {524,38075}, {527,38072}, {971,14561}, {1350,6666}, {2801,38046}, {2829,38195}, {3564,38139}, {3589,5732}, {3618,36991}, {5762,38136}, {5805,19130}, {5842,38196}, {5845,38150}, {5846,38154}, {5847,38158}, {5848,38159}, {5849,38160}, {5853,38144}, {5856,38147}, {5857,38148}, {11357,38059}, {19924,38067}, {21153,29181}, {31658,31670}

X(38145) = midpoint of X(5817) and X(14853)
X(38145) = reflection of X(i) in X(j) for these (i,j): (38117, 38166), (38186, 14561)
X(38145) = {X(38117), X(38166)}-harmonic conjugate of X(38088)

X(38146) = CENTROID OF TRIANGLE {X(4), X(6), X(10)}

Barycentrics    (b+c)*a^5-(7*b^2+2*b*c+7*c^2)*a^4+2*b*c*(b+c)*a^3+2*(2*b-c)*(b-2*c)*(b+c)^2*a^2-(b^4-c^4)*(b-c)*a+3*(b^4-c^4)*(b^2-c^2) : :
X(38146) = X(6)+2*X(19925) = X(10)+2*X(5480) = X(69)-7*X(7989) = 2*X(182)+X(31673) = 2*X(597)+X(34648) = X(946)-4*X(19130) = X(1350)-4*X(3634) = 5*X(3091)+X(3751) = 4*X(3589)-X(4297) = 5*X(3618)+X(5691) = 2*X(6684)+X(31670) = 3*X(14561)-X(38029) = 2*X(38029)-3*X(38049) = X(38035)-3*X(38072) = 3*X(38072)+X(38144) = 3*X(38089)-2*X(38118) = 3*X(38089)-4*X(38167) = 3*X(38136)+X(38165) = 4*X(38136)+X(38191) = 4*X(38165)-3*X(38191)

X(38146) lies on these lines: {6,19925}, {10,5480}, {30,38089}, {69,7989}, {182,31673}, {511,10175}, {515,14561}, {516,36721}, {517,38136}, {518,3817}, {519,38035}, {524,38076}, {597,34648}, {758,38148}, {946,19130}, {971,38187}, {1350,3634}, {2784,6034}, {2801,38188}, {2802,38147}, {2829,38197}, {3091,3751}, {3564,38140}, {3589,4297}, {3618,5691}, {5085,28164}, {5587,5847}, {5842,38198}, {5845,38151}, {5846,38155}, {5848,38161}, {5849,38162}, {5850,38143}, {6684,31670}, {6776,18492}, {9956,21850}, {10164,29181}, {10516,34379}, {18480,18583}, {19924,38068}, {28160,38110}, {28194,38116}, {28204,38040}, {28208,38079}

X(38146) = midpoint of X(i) and X(j) for these {i,j}: {5587, 14853}, {38035, 38144}
X(38146) = reflection of X(i) in X(j) for these (i,j): (38049, 14561), (38118, 38167)
X(38146) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38072, 38144, 38035), (38118, 38167, 38089)

X(38147) = CENTROID OF TRIANGLE {X(4), X(6), X(11)}

Barycentrics    (7*b^2-2*b*c+7*c^2)*a^7-(b+c)*(7*b^2-2*b*c+7*c^2)*a^6-(11*b^4+11*c^4-2*b*c*(10*b^2-3*b*c+10*c^2))*a^5+(b^2-c^2)*(b-c)*(11*b^2+8*b*c+11*c^2)*a^4+(b^4+c^4-2*b*c*(4*b^2+13*b*c+4*c^2))*(b-c)^2*a^3-(b^2-c^2)*(b-c)*(b^4+c^4-2*b*c*(2*b^2+9*b*c+2*c^2))*a^2+(b^4-c^4)*(b^2-c^2)*(3*b^2-8*b*c+3*c^2)*a+(b^4-c^4)*(b^2-c^2)^2*(-3*b+3*c) : :
X(38147) = X(11)+2*X(5480) = X(119)-4*X(19130) = X(1350)-4*X(6667) = 5*X(3091)+X(10755) = 4*X(3589)-X(24466) = 5*X(3618)+X(10724) = 2*X(6713)+X(31670) = 2*X(18583)+X(22938) = 3*X(38090)-2*X(38119) = 3*X(38090)-4*X(38168)

X(38147) lies on these lines: {11,1469}, {30,38090}, {119,19130}, {511,23513}, {515,38050}, {516,38195}, {517,38192}, {518,38038}, {524,38077}, {528,38072}, {952,38035}, {971,38188}, {1350,6667}, {2802,38146}, {3091,10755}, {3564,38141}, {3589,24466}, {3618,10724}, {5840,14561}, {5842,38199}, {5845,38152}, {5846,38156}, {5847,38161}, {5848,14853}, {5849,38163}, {5851,38143}, {5854,38144}, {5856,38145}, {6713,31670}, {18583,22938}, {19924,38069}, {21154,29181}

X(38147) = reflection of X(38119) in X(38168)
X(38147) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38035, 38136, 38148), (38119, 38168, 38090)

X(38148) = CENTROID OF TRIANGLE {X(4), X(6), X(12)}

Barycentrics    (7*b^2+2*b*c+7*c^2)*a^7-(b+c)*(7*b^2+2*b*c+7*c^2)*a^6-(11*b^4+11*c^4-2*b*c*(4*b^2+b*c+4*c^2))*a^5+(b+c)*(11*b^4+11*c^4-2*b*c*(7*b^2+b*c+7*c^2))*a^4+(b^2-c^2)^2*(b^2-6*b*c+c^2)*a^3-(b^2-c^2)^2*(b+c)*(b^2-10*b*c+c^2)*a^2+(b^4-c^4)*(b^2-c^2)*(3*b^2-4*b*c+3*c^2)*a+(b^4-c^4)*(b^2-c^2)^2*(-3*b+3*c) : :
X(38148) = X(12)+2*X(5480) = X(1350)-4*X(6668) = 4*X(3589)-X(30264) = 4*X(19130)-X(26470) = 2*X(31659)+X(31670) = 3*X(38091)-2*X(38120) = 3*X(38091)-4*X(38169)

X(38148) lies on these lines: {12,3056}, {30,38091}, {511,38109}, {515,38051}, {516,38196}, {517,38193}, {518,38039}, {524,38078}, {529,38072}, {758,38146}, {952,38035}, {971,38189}, {1350,6668}, {2829,38199}, {3564,38142}, {3589,30264}, {5845,38153}, {5846,38157}, {5847,38162}, {5848,38163}, {5849,14853}, {5852,38143}, {5855,38144}, {5857,38145}, {19130,26470}, {19924,38070}, {21155,29181}, {31659,31670}

X(38148) = reflection of X(38120) in X(38169)
X(38148) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38035, 38136, 38147), (38120, 38169, 38091)

X(38149) = CENTROID OF TRIANGLE {X(4), X(7), X(8)}

Barycentrics    3*a^6-6*(b+c)*a^5+(b+3*c)*(3*b+c)*a^4-8*b*c*(b+c)*a^3-3*(b^2-c^2)^2*a^2+2*(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a-3*(b^2-c^2)^2*(b-c)^2 : :
X(38149) = X(4)+2*X(2550) = 2*X(4)+X(35514) = 4*X(5)-X(390) = X(7)+2*X(355) = X(8)+2*X(5805) = 2*X(9)-5*X(5818) = 4*X(10)-X(5759) = 2*X(119)+X(20119) = 4*X(142)-X(944) = X(145)-4*X(20330) = 4*X(2550)-X(35514) = 3*X(5587)-2*X(38158) = 3*X(5817)-4*X(38158) = X(11372)-4*X(19925) = 3*X(21151)-4*X(38123) = 2*X(38036)-3*X(38073) = 3*X(38052)-2*X(38123) = 3*X(38073)-4*X(38151) = 3*X(38092)-2*X(38121) = 3*X(38092)-4*X(38170)

X(38149) lies on these lines: {2,38031}, {4,9}, {5,390}, {7,355}, {8,5805}, {30,38092}, {119,20119}, {142,944}, {145,20330}, {388,37712}, {443,5731}, {497,7988}, {515,21151}, {519,38036}, {527,38074}, {528,3545}, {631,3826}, {673,7402}, {952,1056}, {954,6843}, {962,18482}, {971,3753}, {1001,3090}, {1058,5703}, {1389,6601}, {1478,30286}, {1503,38185}, {2478,38132}, {2829,38202}, {2951,31673}, {3059,7686}, {3091,5687}, {3158,3817}, {3421,6839}, {3434,5748}, {3529,11495}, {3560,7676}, {3576,38204}, {4312,4848}, {5055,38043}, {5082,5730}, {5225,7989}, {5263,36682}, {5542,5881}, {5603,5853}, {5686,5762}, {5690,31671}, {5714,12560}, {5735,24393}, {5763,5780}, {5779,18357}, {5787,11024}, {5795,5833}, {5842,38203}, {5843,6917}, {5844,38137}, {5845,38144}, {5846,38143}, {5850,38155}, {5851,38156}, {5852,38157}, {5854,38152}, {5855,38153}, {5880,36996}, {6173,34627}, {6253,19855}, {6827,38113}, {6855,11499}, {6858,18524}, {6865,11231}, {6867,8543}, {6885,30312}, {6896,7704}, {6911,7677}, {6916,28160}, {6935,34474}, {6939,37820}, {7967,38053}, {8226,17784}, {8227,30331}, {8732,37281}, {9780,31658}, {9812,10157}, {9956,18230}, {10165,17582}, {10172,17559}, {10246,38171}, {11729,12730}, {12630,24680}, {12669,34339}, {18480,36991}, {18525,31657}, {19875,38130}, {20533,36662}, {24474,34784}, {28204,38030}, {28224,38111}, {28236,38054}, {30340,37705}

X(38149) = reflection of X(i) in X(j) for these (i,j): (3576, 38204), (5603, 38150), (5657, 38200), (5686, 5790), (5731, 38122), (5817, 5587), (7967, 38053), (8236, 5886), (10246, 38171), (11038, 38107), (21151, 38052), (21168, 38057), (38030, 38172), (38036, 38151), (38121, 38170)
X(38149) = anticomplement of X(38031)
X(38149) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 2550, 35514), (11038, 30275, 38056), (38036, 38151, 38073), (38121, 38170, 38092)

X(38150) = CENTROID OF TRIANGLE {X(4), X(7), X(9)}

Barycentrics    a^6-(b+c)*a^5-2*(b+c)*(b^2+c^2)*a^3+(b-c)^4*a^2+3*(b^2-c^2)^2*(b+c)*a-2*(b^2-c^2)^2*(b-c)^2 : :
X(38150) = X(3)+2*X(18482) = 2*X(3)-5*X(20195) = X(4)+2*X(142) = 2*X(4)+X(5732) = 4*X(5)-X(9) = 8*X(5)+X(5735) = 2*X(5)+X(5805) = X(7)+5*X(3091) = 2*X(9)+X(5735) = X(9)+2*X(5805) = X(9)+4*X(38137) = 4*X(142)-X(5732) = 2*X(5715)+X(5833) = X(5735)-4*X(5805) = X(5735)+4*X(38108) = X(5735)-8*X(38137) = 4*X(18482)+5*X(20195) = X(24644)-5*X(30308) = X(38037)+2*X(38151) = X(38108)+2*X(38137)

X(38150) lies on these lines: {1,6835}, {2,165}, {3,18482}, {4,142}, {5,9}, {7,1210}, {30,38093}, {40,3826}, {57,8226}, {78,11522}, {119,3254}, {144,5068}, {355,3243}, {381,971}, {390,9614}, {411,3624}, {515,38053}, {517,38200}, {518,5587}, {527,3545}, {528,38021}, {546,31657}, {547,38113}, {920,1445}, {936,946}, {938,5290}, {954,5219}, {990,4859}, {1001,3149}, {1478,4321}, {1479,4326}, {1490,6849}, {1503,38186}, {1656,31658}, {1698,6991}, {1750,5249}, {2801,37718}, {2829,38205}, {2951,6836}, {3008,3332}, {3062,6870}, {3086,12573}, {3090,5759}, {3306,10883}, {3339,12617}, {3358,6841}, {3586,6839}, {3832,36991}, {3845,38111}, {3847,5880}, {3851,5779}, {3855,36996}, {4292,8732}, {4357,36660}, {4654,5927}, {5056,18230}, {5066,5843}, {5071,21168}, {5223,6734}, {5436,20420}, {5437,8727}, {5528,10738}, {5603,5853}, {5691,6894}, {5698,6855}, {5703,30331}, {5704,30424}, {5714,9842}, {5728,9581}, {5733,16667}, {5750,36682}, {5785,6843}, {5809,30275}, {5818,24393}, {5842,38206}, {5845,38145}, {5850,38158}, {5851,38159}, {5852,38160}, {5856,38152}, {5857,38153}, {5905,30326}, {6282,6854}, {6601,6765}, {6622,7717}, {6766,9710}, {6830,8257}, {6837,15803}, {6846,31424}, {6865,18483}, {6918,9955}, {6945,8545}, {6957,30379}, {7385,10444}, {7548,10394}, {7679,31434}, {7741,15299}, {7951,15298}, {7958,31435}, {7965,10860}, {8068,15518}, {8544,13729}, {8581,10895}, {9624,33597}, {9843,12571}, {10157,28609}, {10172,38130}, {10175,38057}, {10431,10857}, {10436,36652}, {10826,18412}, {10861,17577}, {10893,17668}, {10896,14100}, {11230,38031}, {12047,12560}, {12436,37434}, {12618,25590}, {12649,37714}, {13329,31183}, {13374,15185}, {13727,17282}, {14269,38065}, {15699,38067}, {15933,28236}, {16189,20013}, {17529,37551}, {18492,25557}, {19541,25525}, {23046,38080}, {29016,29573}, {37447,37526}, {38042,38126}

X(38150) = midpoint of X(i) and X(j) for these {i,j}: {4, 21151}, {5, 38137}, {381, 38107}, {1699, 38052}, {3545, 38073}, {3817, 38151}, {3845, 38111}, {5587, 38036}, {5603, 38149}, {5805, 38108}, {10516, 38143}, {14269, 38065}, {23046, 38080}
X(38150) = reflection of X(i) in X(j) for these (i,j): (9, 38108), (5732, 21151), (5805, 38137), (6173, 38107), (21151, 142), (21153, 2), (38031, 11230), (38037, 3817), (38057, 10175), (38059, 10171), (38067, 15699), (38075, 3545), (38108, 5), (38113, 547), (38122, 38171), (38126, 38042), (38130, 10172), (38139, 5066), (38154, 5587)
X(38150) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 142, 5732), (5, 5805, 9), (9, 5805, 5735), (355, 20330, 3243), (546, 31657, 31672), (946, 6864, 936), (1656, 31671, 31658), (3090, 5759, 6666), (38122, 38171, 38093)

X(38151) = CENTROID OF TRIANGLE {X(4), X(7), X(10)}

Barycentrics    4*a^6-5*(b+c)*a^5-(b^2-6*b*c+c^2)*a^4-2*(b+c)^3*a^3+2*(b-c)^4*a^2+(b^2-c^2)*(b-c)*(7*b^2+10*b*c+7*c^2)*a-5*(b^2-c^2)^2*(b-c)^2 : :
X(38151) = X(7)+2*X(19925) = X(10)+2*X(5805) = 3*X(10)-2*X(38126) = 4*X(142)-X(4297) = X(144)-7*X(7989) = 2*X(2550)+X(4301) = 3*X(3817)-2*X(38037) = 3*X(5805)+X(38126) = 3*X(9779)-X(24644) = 2*X(38030)-3*X(38054) = X(38030)-3*X(38107) = X(38036)-3*X(38073) = X(38037)-3*X(38150) = 3*X(38073)+X(38149) = 3*X(38076)-2*X(38158) = 3*X(38094)-2*X(38123) = 3*X(38094)-4*X(38172) = 3*X(38137)+X(38170) = 4*X(38137)+X(38201) = 4*X(38170)-3*X(38201)

X(38151) lies on these lines: {2,165}, {7,19925}, {10,5805}, {30,38094}, {142,4297}, {144,7989}, {388,5542}, {515,38030}, {517,38137}, {518,38155}, {519,38036}, {527,38076}, {758,38153}, {971,5883}, {1503,38187}, {2550,4301}, {2802,38152}, {2829,38207}, {3062,3832}, {3091,4312}, {3244,20330}, {3634,5759}, {3671,6835}, {5221,5729}, {5587,5850}, {5735,8165}, {5762,10175}, {5806,15587}, {5833,18250}, {5842,38208}, {5843,38140}, {5845,38146}, {5847,38143}, {5851,38161}, {5852,38162}, {5853,34640}, {6173,34648}, {6684,31671}, {10165,38171}, {11038,28236}, {11372,12571}, {12053,15950}, {18492,36996}, {19883,38031}, {21151,28164}, {28160,38111}, {28194,38121}, {28204,38041}, {28208,38080}, {31657,31673}

X(38151) = midpoint of X(38036) and X(38149)
X(38151) = reflection of X(i) in X(j) for these (i,j): (3817, 38150), (10164, 38204), (10165, 38171), (38054, 38107), (38123, 38172)
X(38151) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38073, 38149, 38036), (38123, 38172, 38094)

X(38152) = CENTROID OF TRIANGLE {X(4), X(7), X(11)}

Barycentrics    4*a^9-8*(b+c)*a^8-3*(b^2-6*b*c+c^2)*a^7+(b+c)*(9*b^2-14*b*c+9*c^2)*a^6+(5*b^4+5*c^4-2*b*c*(10*b^2-11*b*c+10*c^2))*a^5+(b^2-c^2)*(b-c)*(b^2+10*b*c+c^2)*a^4-(17*b^4+17*c^4+2*b*c*(4*b^2+19*b*c+4*c^2))*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(3*b^4+3*c^4-2*b*c*(4*b^2-13*b*c+4*c^2))*a^2+(b^2-c^2)^2*(b-c)^2*(11*b^2-2*b*c+11*c^2)*a-5*(b^2-c^2)^3*(b-c)^3 : :
X(38152) = 4*X(5)-X(6068) = X(11)+2*X(5805) = 4*X(142)-X(24466) = X(1317)-4*X(20330) = 2*X(3254)+X(37725) = X(5759)-4*X(6667) = 2*X(6713)+X(31671) = 3*X(23513)-2*X(38180) = 3*X(38077)-2*X(38159) = 3*X(38095)-2*X(38124) = 3*X(38095)-4*X(38173)

X(38152) lies on these lines: {5,6068}, {11,57}, {30,38095}, {142,24466}, {515,38055}, {516,21154}, {517,38202}, {518,38156}, {527,38077}, {528,5603}, {952,38036}, {1317,20330}, {1503,38188}, {2802,38151}, {3254,37725}, {5759,6667}, {5762,23513}, {5840,38107}, {5842,38209}, {5843,38141}, {5845,38147}, {5848,38143}, {5850,38161}, {5852,38163}, {5854,38149}, {5856,38150}, {6713,31671}

X(38152) = reflection of X(i) in X(j) for these (i,j): (21154, 38205), (38124, 38173)
X(38152) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38036, 38137, 38153), (38124, 38173, 38095)

X(38153) = CENTROID OF TRIANGLE {X(4), X(7), X(12)}

Barycentrics    4*a^9-8*(b+c)*a^8-(3*b^2-14*b*c+3*c^2)*a^7+9*(b^2-c^2)*(b-c)*a^6+(5*b^4+5*c^4-2*b*c*(2*b^2-19*b*c+2*c^2))*a^5+(b^2-c^2)*(b-c)*(b^2+18*b*c+c^2)*a^4-(17*b^4+17*c^4+2*b*c*(14*b^2+23*b*c+14*c^2))*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(3*b^4+3*c^4-2*b*c*(6*b^2-b*c+6*c^2))*a^2+(b^2-c^2)^2*(b-c)^2*(11*b^2+6*b*c+11*c^2)*a-5*(b^2-c^2)^3*(b-c)^3 : :
X(38153) = X(12)+2*X(5805) = 4*X(142)-X(30264) = X(5759)-4*X(6668) = 4*X(20330)-X(37734) = 2*X(31659)+X(31671) = 3*X(38078)-2*X(38160) = 3*X(38096)-2*X(38125) = 3*X(38096)-4*X(38174) = 3*X(38109)-2*X(38181)

X(38153) lies on these lines: {12,5805}, {30,38096}, {142,30264}, {515,38056}, {516,21155}, {517,38203}, {518,38157}, {527,38078}, {529,38073}, {758,38151}, {952,38036}, {1503,38189}, {2829,38209}, {5759,6668}, {5762,38109}, {5843,38142}, {5845,38148}, {5849,38143}, {5850,38162}, {5851,38163}, {5855,38149}, {5857,38150}, {20330,37734}, {31659,31671}

X(38153) = reflection of X(i) in X(j) for these (i,j): (21155, 38206), (38125, 38174)
X(38153) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38036, 38137, 38152), (38125, 38174, 38096)

X(38154) = CENTROID OF TRIANGLE {X(4), X(8), X(9)}

Barycentrics    a^6-5*(b+c)*a^5+8*(b^2+b*c+c^2)*a^4-2*(b+c)*(b^2+4*b*c+c^2)*a^3-(7*b^2+10*b*c+7*c^2)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(7*b^2+6*b*c+7*c^2)*a-2*(b^2-c^2)^2*(b-c)^2 : :
X(38154) = X(4)+2*X(24393) = 4*X(5)-X(3243) = X(9)+2*X(355) = 4*X(10)-X(5732) = 2*X(142)-5*X(5818) = X(944)-4*X(6666) = 2*X(1001)+X(5881) = 2*X(5223)+X(5735) = X(5223)+5*X(37714) = 3*X(5587)-X(38036) = 3*X(5790)-X(38121) = 3*X(21153)-4*X(38130) = 2*X(38036)-3*X(38150) = 2*X(38037)-3*X(38075) = 2*X(38043)-3*X(38108) = 3*X(38057)-2*X(38130) = 3*X(38074)-X(38149) = 3*X(38075)-4*X(38158) = 3*X(38097)-2*X(38126) = 3*X(38097)-4*X(38175) = 2*X(38121)-3*X(38200)

X(38154) lies on these lines: {1,6886}, {4,24393}, {5,3243}, {9,355}, {10,5732}, {30,38097}, {80,15298}, {142,5818}, {515,21153}, {516,3543}, {518,5587}, {519,38037}, {527,38074}, {944,6666}, {952,38043}, {954,5727}, {971,5790}, {1001,5881}, {1445,9613}, {1503,38190}, {1699,3681}, {1737,4321}, {1750,25006}, {2550,6256}, {2801,10861}, {2829,38211}, {3434,30326}, {3436,5223}, {3617,36991}, {4326,10039}, {4882,12617}, {5261,5542}, {5290,30329}, {5690,31672}, {5728,9578}, {5762,38138}, {5805,18357}, {5809,31397}, {5815,19925}, {5817,5853}, {5842,38212}, {5844,38139}, {5846,38145}, {5854,38159}, {5855,38160}, {5856,38156}, {5857,38157}, {6173,38172}, {7672,9612}, {7675,31434}, {7988,31146}, {9624,15570}, {9956,20195}, {10157,24392}, {10175,38053}, {10573,12560}, {10827,18412}, {11372,33559}, {11500,31446}, {12629,24389}, {15299,37710}, {18525,31658}, {28204,38031}, {28224,38113}, {28236,38059}, {38030,38093}, {38042,38122}

X(38154) = reflection of X(i) in X(j) for these (i,j): (21153, 38057), (38031, 38179), (38037, 38158), (38053, 10175), (38122, 38042), (38126, 38175), (38150, 5587), (38200, 5790)
X(38154) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38037, 38158, 38075), (38126, 38175, 38097)

X(38155) = CENTROID OF TRIANGLE {X(4), X(8), X(10)}

Barycentrics    4*a^4-5*(b+c)*a^3+(b^2+10*b*c+c^2)*a^2+5*(b^2-c^2)*(b-c)*a-5*(b^2-c^2)^2 : :
X(38155) = 5*X(1)-11*X(5056) = 5*X(2)-3*X(30392) = 2*X(3)-5*X(10) = X(3)+5*X(355) = 8*X(3)-5*X(4297) = X(3)-5*X(5790) = 7*X(3)-10*X(6684) = 4*X(3)-5*X(10164) = 11*X(3)-5*X(18481) = 7*X(3)+5*X(18525) = 3*X(3)-5*X(26446) = X(10)+2*X(355) = 4*X(10)-X(4297) = 7*X(10)-4*X(6684) = 11*X(10)-2*X(18481) = 7*X(10)+2*X(18525) = 3*X(10)-2*X(26446) = 8*X(355)+X(4297) = 7*X(355)+2*X(6684) = 4*X(355)+X(10164) = 11*X(355)+X(18481) = 7*X(355)-X(18525) = 3*X(355)+X(26446) = 11*X(5056)-10*X(10171) = 3*X(30392)+5*X(37712)

X(38155) lies on these lines: {1,5056}, {2,28236}, {3,10}, {4,3626}, {5,3244}, {7,30286}, {8,1699}, {30,38098}, {40,4691}, {80,2346}, {145,7989}, {381,28234}, {516,3543}, {517,3845}, {518,38151}, {519,3545}, {547,551}, {758,18908}, {912,3919}, {944,3533}, {946,3625}, {962,4668}, {971,38201}, {1071,3918}, {1125,5067}, {1210,37710}, {1385,16239}, {1478,30424}, {1503,38191}, {1656,13607}, {1737,4315}, {2801,3753}, {2802,38156}, {2829,38213}, {3090,3636}, {3091,3632}, {3241,7988}, {3475,6738}, {3576,3828}, {3579,28190}, {3586,5766}, {3617,5059}, {3621,11522}, {3635,8227}, {3654,28150}, {3655,15723}, {3671,10573}, {3754,14872}, {3822,37725}, {3829,33956}, {3830,28232}, {3853,11362}, {3880,10157}, {3947,10827}, {3950,5816}, {4002,12680}, {4015,14110}, {4311,18395}, {4314,10039}, {4342,12647}, {4678,7991}, {4701,7982}, {4731,10167}, {4745,5657}, {4746,12245}, {4847,5176}, {4848,11246}, {5068,20050}, {5086,6736}, {5102,5847}, {5252,11019}, {5261,12563}, {5288,6915}, {5493,5690}, {5534,30143}, {5542,18391}, {5550,30315}, {5727,13405}, {5731,15708}, {5836,9947}, {5842,38214}, {5846,38146}, {5850,38149}, {5853,38158}, {5854,38161}, {5855,38162}, {5882,9956}, {6347,21569}, {6348,21564}, {6431,13883}, {6432,13936}, {6913,25439}, {8582,17535}, {9590,13620}, {9779,11224}, {10106,17728}, {10165,11539}, {10172,10246}, {10443,17275}, {11219,36006}, {11231,11812}, {12645,13464}, {13411,37711}, {15022,20057}, {15686,28160}, {15690,28186}, {15863,21635}, {16189,20053}, {16859,24987}, {16980,31752}, {17502,38068}, {17542,38059}, {17857,30147}, {19877,30389}, {21627,32426}, {23841,31732}, {28168,31730}, {28172,34638}, {28208,38081}, {28463,38178}

X(38155) = midpoint of X(i) and X(j) for these {i,j}: {2, 37712}, {8, 1699}, {355, 5790}, {3576, 34627}, {5691, 9778}, {5881, 7967}, {11224, 31145}, {37705, 38028}
X(38155) = reflection of X(i) in X(j) for these (i,j): (1, 10171), (10, 5790), (551, 10175), (946, 38140), (1699, 19925), (3576, 3828), (3817, 5587), (4297, 10164), (4301, 1699), (5657, 4745), (5882, 38028), (7967, 1125), (10164, 10), (10165, 38042), (10246, 10172), (38028, 9956), (38127, 38176), (38140, 18357)
X(38155) = X(8)-Beth conjugate of-X(10164)
X(38155) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 3832, 11531), (8, 19925, 4301), (8, 37714, 19925), (1656, 13607, 15808), (3817, 5587, 38076), (3850, 11278, 946), (4701, 12571, 7982), (5587, 16200, 3545), (5690, 31673, 5493), (5818, 5881, 1125), (5836, 9947, 31803), (5882, 9956, 19862), (9779, 31145, 11224), (9956, 37705, 5882), (10172, 10246, 19883), (11539, 31662, 10165), (38127, 38176, 38098)

X(38156) = CENTROID OF TRIANGLE {X(4), X(8), X(11)}

Barycentrics    4*a^7-8*(b+c)*a^6+(b^2+26*b*c+c^2)*a^5+(b+c)*(11*b^2-34*b*c+11*c^2)*a^4-2*(7*b^4+7*c^4+2*b*c*(b^2-11*b*c+c^2))*a^3+2*(b^2-c^2)*(b-c)*(b^2+14*b*c+c^2)*a^2+(b^2-c^2)^2*(9*b^2-22*b*c+9*c^2)*a-5*(b^2-c^2)^3*(b-c) : :
X(38156) = X(4)+2*X(3036) = 4*X(5)-X(1317) = 4*X(10)-X(24466) = X(11)+2*X(355) = 5*X(11)-2*X(12737) = 5*X(80)+X(5531) = X(80)+5*X(37714) = 2*X(80)+X(37725) = X(119)-4*X(18357) = 5*X(355)+X(12737) = 2*X(1387)+X(5881) = 2*X(5531)-5*X(37725) = X(7972)-7*X(7989) = 5*X(8227)-2*X(12735) = 2*X(12019)+X(12751) = 3*X(21154)-4*X(38133) = 3*X(23513)-2*X(38044) = 10*X(37714)-X(37725) = 3*X(38099)-2*X(38128) = 3*X(38099)-4*X(38177)

X(38156) lies on these lines: {1,5}, {4,3036}, {10,24466}, {30,38099}, {515,21154}, {516,38211}, {518,38152}, {519,38038}, {528,5817}, {944,6667}, {971,38202}, {1145,6246}, {1376,6950}, {1503,38192}, {1537,15863}, {2800,5927}, {2802,38155}, {2829,14647}, {3035,5818}, {3091,12531}, {3617,10724}, {4188,12114}, {5603,34717}, {5657,11826}, {5731,17566}, {5790,5840}, {5842,38215}, {5843,11545}, {5844,38141}, {5846,38147}, {5848,38144}, {5851,38149}, {5853,38159}, {5854,11235}, {5855,38163}, {5856,38154}, {6224,20400}, {6713,18525}, {6797,9947}, {6826,12763}, {6893,13274}, {6913,10087}, {6918,10074}, {9956,31235}, {10165,17619}, {10172,17614}, {10175,34123}, {10738,13996}, {10914,14740}, {12736,14872}, {16174,25416}, {17613,28172}, {28204,38032}, {28224,34126}, {28236,32557}

X(38156) = midpoint of X(16173) and X(37712)
X(38156) = reflection of X(i) in X(j) for these (i,j): (21154, 34122), (34123, 10175), (38032, 38182), (38038, 38161), (38128, 38177)
X(38156) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5587, 38138, 38157), (6797, 9947, 12665), (15863, 19925, 1537), (38038, 38161, 38077), (38128, 38177, 38099)

X(38157) = CENTROID OF TRIANGLE {X(4), X(8), X(12)}

Barycentrics    4*a^7-8*(b+c)*a^6+(b^2+22*b*c+c^2)*a^5+(b+c)*(11*b^2-30*b*c+11*c^2)*a^4-2*(7*b^4+7*c^4+2*b*c*(2*b^2-9*b*c+2*c^2))*a^3+2*(b^2-c^2)*(b-c)*(b^2+12*b*c+c^2)*a^2+(b^2-c^2)^2*(9*b^2-14*b*c+9*c^2)*a-5*(b^2-c^2)^3*(b-c) : :
X(38157) = 4*X(5)-X(37734) = 4*X(10)-X(30264) = X(12)+2*X(355) = 5*X(12)-2*X(37733) = 5*X(355)+X(37733) = X(944)-4*X(6668) = 2*X(4999)-5*X(5818) = X(5881)+2*X(37737) = 8*X(9956)-5*X(31260) = 4*X(18357)-X(26470) = X(18525)+2*X(31659) = 3*X(21155)-4*X(38134) = X(37710)+5*X(37714) = 2*X(38039)-3*X(38078) = 2*X(38045)-3*X(38109) = 3*X(38058)-2*X(38134) = 3*X(38078)-4*X(38162) = 3*X(38100)-2*X(38129) = 3*X(38100)-4*X(38178)

X(38157) lies on these lines: {1,5}, {10,30264}, {30,38100}, {515,21155}, {516,38212}, {518,38153}, {519,38039}, {529,38074}, {758,18908}, {944,6668}, {958,6942}, {971,38203}, {1503,38193}, {2475,12762}, {2829,38215}, {4189,11500}, {4999,5818}, {5603,34700}, {5657,11827}, {5790,34606}, {5842,11114}, {5844,38142}, {5846,38148}, {5849,38144}, {5852,38149}, {5853,38160}, {5854,38163}, {5855,11236}, {5857,38154}, {9956,31260}, {18525,31659}, {21677,38176}, {28204,38033}, {28224,38114}, {28236,38062}

X(38157) = midpoint of X(37701) and X(37712)
X(38157) = reflection of X(i) in X(j) for these (i,j): (21155, 38058), (38033, 38183), (38039, 38162), (38129, 38178)
X(38157) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5587, 38138, 38156), (38039, 38162, 38078), (38129, 38178, 38100)

X(38158) = CENTROID OF TRIANGLE {X(4), X(9), X(10)}

Barycentrics    3*(b+c)*a^5-(9*b^2+2*b*c+9*c^2)*a^4+2*(b+c)*(3*b^2+2*b*c+3*c^2)*a^3+6*(b^2-c^2)^2*a^2-(b^2-c^2)*(b-c)*(9*b^2+14*b*c+9*c^2)*a+3*(b^2-c^2)^2*(b-c)^2 : :
X(38158) = 4*X(5)-X(5542) = 3*X(5)-X(38041) = X(7)-7*X(7989) = X(9)+2*X(19925) = 2*X(355)+X(30331) = X(390)+5*X(37714) = 3*X(5542)-4*X(38041) = 3*X(5587)-X(38149) = X(5759)+5*X(18492) = 3*X(5817)+X(38149) = 5*X(5818)+X(11372) = 2*X(38031)-3*X(38059) = X(38031)-3*X(38108) = X(38037)-3*X(38075) = 3*X(38075)+X(38154) = 3*X(38101)-2*X(38130) = 3*X(38101)-4*X(38179) = 3*X(38139)+X(38175) = 4*X(38139)+X(38210) = 4*X(38175)-3*X(38210)

This triangle has collinear vertices.

X(38158) lies on these lines: {4,9}, {5,5542}, {7,7989}, {12,10392}, {30,38101}, {355,30331}, {390,7319}, {515,16857}, {517,38139}, {518,3817}, {519,38037}, {527,38076}, {758,38160}, {971,10175}, {1210,15841}, {1503,38194}, {1698,36991}, {1699,5686}, {2801,38054}, {2802,38159}, {2829,38216}, {2951,9780}, {3091,5223}, {3545,38036}, {3634,5732}, {3947,5728}, {4297,6666}, {4301,24393}, {4847,9779}, {5055,38030}, {5226,7988}, {5261,30330}, {5572,9947}, {5691,18230}, {5731,17554}, {5762,38140}, {5777,30329}, {5779,30424}, {5809,13405}, {5842,38217}, {5847,38145}, {5850,38150}, {5853,38155}, {5856,38161}, {5857,38162}, {5886,21625}, {6245,10172}, {6684,31672}, {6738,8232}, {8226,21060}, {8236,37712}, {10171,38053}, {10398,10590}, {12635,24389}, {14872,20116}, {21153,28164}, {28160,38113}, {28194,38126}, {28204,38043}, {28208,38082}, {31658,31673}

X(38158) = midpoint of X(i) and X(j) for these {i,j}: {1699, 5686}, {5587, 5817}, {8236, 37712}, {38037, 38154}
X(38158) = reflection of X(i) in X(j) for these (i,j): (38053, 10171), (38059, 38108), (38122, 10172), (38130, 38179), (38204, 10175)
X(38158) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38075, 38154, 38037), (38130, 38179, 38101)

X(38159) = CENTROID OF TRIANGLE {X(4), X(9), X(11)}

Barycentrics    2*(b+c)*a^8-(9*b^2-2*b*c+9*c^2)*a^7+(b+c)*(11*b^2-4*b*c+11*c^2)*a^6+(7*b^4+7*c^4-6*b*c*(5*b^2-b*c+5*c^2))*a^5-(b+c)*(25*b^4+25*c^4-2*b*c*(18*b^2-7*b*c+18*c^2))*a^4+(13*b^4+13*c^4+10*b*c*(4*b^2+7*b*c+4*c^2))*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(9*b^4+9*c^4-2*b*c*(b^2+19*b*c+c^2))*a^2-(b^2-c^2)^2*(b-c)^2*(11*b^2+8*b*c+11*c^2)*a+3*(b^2-c^2)^3*(b-c)^3 : :
X(38159) = 4*X(5)-X(10427) = X(1156)+5*X(3091) = X(5528)-4*X(20400) = X(5732)-4*X(6667) = 4*X(6666)-X(24466) = 2*X(6713)+X(31672) = X(10724)+5*X(18230) = 3*X(23513)-X(38124) = 5*X(31272)+X(36991) = 3*X(38077)-X(38152) = 3*X(38102)-2*X(38131) = 3*X(38102)-4*X(38180) = 2*X(38124)-3*X(38205)

X(38159) lies on these lines: {5,10427}, {11,118}, {30,38102}, {381,14647}, {515,38060}, {516,34122}, {517,38211}, {518,38038}, {527,38077}, {528,5587}, {952,38037}, {971,23513}, {1156,3091}, {1503,38195}, {1532,15726}, {2550,6929}, {2802,38158}, {5528,20400}, {5732,6667}, {5762,38141}, {5817,5856}, {5840,38108}, {5842,38218}, {5848,38145}, {5851,38150}, {5853,38156}, {5854,38154}, {5857,38163}, {6666,24466}, {6713,31672}, {10724,18230}, {31272,36991}

X(38159) = reflection of X(i) in X(j) for these (i,j): (38131, 38180), (38205, 23513)
X(38159) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38037, 38139, 38160), (38131, 38180, 38102)

X(38160) = CENTROID OF TRIANGLE {X(4), X(9), X(12)}

Barycentrics    2*(b+c)*a^8-(9*b^2+2*b*c+9*c^2)*a^7+(b+c)*(11*b^2+4*b*c+11*c^2)*a^6+(7*b^4+7*c^4-2*b*c*(13*b^2+9*b*c+13*c^2))*a^5-(b+c)*(25*b^4+25*c^4-2*b*c*(10*b^2+b*c+10*c^2))*a^4+(b^2-c^2)^2*(13*b^2+18*b*c+13*c^2)*a^3+3*(b^2-c^2)^2*(b+c)*(3*b^2-4*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b-c)^2*(11*b^2+12*b*c+11*c^2)*a+3*(b^2-c^2)^3*(b-c)^3 : :
X(38160) = X(5732)-4*X(6668) = 4*X(6666)-X(30264) = 2*X(31659)+X(31672) = 3*X(38078)-X(38153) = 3*X(38103)-2*X(38132) = 3*X(38103)-4*X(38181) = 3*X(38109)-X(38125) = 2*X(38125)-3*X(38206)

X(38160) lies on these lines: {12,14100}, {30,38103}, {515,38061}, {516,38058}, {517,38212}, {518,38039}, {527,38078}, {529,38075}, {758,38158}, {952,38037}, {971,38109}, {1503,38196}, {2801,38056}, {2829,38218}, {5732,6668}, {5762,38142}, {5817,5857}, {5849,38145}, {5852,38150}, {5853,38157}, {5855,38154}, {5856,38163}, {6666,30264}, {31659,31672}

X(38160) = reflection of X(i) in X(j) for these (i,j): (38132, 38181), (38206, 38109)
X(38160) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38037, 38139, 38159), (38132, 38181, 38103)

X(38161) = CENTROID OF TRIANGLE {X(4), X(10), X(11)}

Barycentrics    2*a^7-3*(b+c)*a^6+(b^2+8*b*c+c^2)*a^5+(b+c)*(2*b^2-9*b*c+2*c^2)*a^4-(8*b^4+8*c^4-5*b*c*(b+c)^2)*a^3+(b^2-c^2)*(b-c)*(5*b^2+11*b*c+5*c^2)*a^2+(b^2-c^2)^2*(5*b^2-13*b*c+5*c^2)*a-4*(b^2-c^2)^3*(b-c) : :
X(38161) = X(4)+2*X(6702) = 4*X(5)-X(214) = 2*X(5)+X(6246) = X(11)+2*X(19925) = X(80)+5*X(3091) = X(100)-7*X(7989) = X(104)+5*X(18492) = X(214)+2*X(6246) = X(355)+2*X(16174) = 2*X(546)+X(12619) = 2*X(946)+X(15863) = 3*X(23513)-X(38032) = 3*X(32557)-2*X(38032) = X(38038)-3*X(38077) = 3*X(38077)+X(38156) = 3*X(38104)-2*X(38133) = 3*X(38104)-4*X(38182) = 3*X(38141)+X(38177) = 4*X(38141)+X(38213) = 4*X(38177)-3*X(38213)

X(38161) lies on these lines: {4,6702}, {5,214}, {11,10106}, {30,38104}, {80,3091}, {100,7989}, {104,18492}, {355,16174}, {381,2800}, {515,23513}, {516,34122}, {517,38141}, {519,38038}, {528,38076}, {546,12619}, {758,38163}, {946,15863}, {952,3817}, {971,38207}, {1320,37714}, {1503,38197}, {1537,12571}, {1698,10724}, {2801,37718}, {2802,5587}, {3036,4301}, {3070,13976}, {3071,8988}, {3090,12119}, {3614,12743}, {3634,24466}, {3832,34789}, {3843,12515}, {3850,12611}, {3851,6265}, {3855,12247}, {4297,6667}, {5068,6224}, {5072,12747}, {5083,10895}, {5691,31272}, {5818,14217}, {5840,10175}, {5842,38219}, {5847,38147}, {5848,38146}, {5850,38152}, {5851,38151}, {5854,38155}, {5856,38158}, {6713,31673}, {7173,18976}, {9581,18240}, {9842,12019}, {9955,25485}, {9956,22938}, {10171,34123}, {10826,12736}, {10896,15558}, {11522,12531}, {11715,18480}, {19077,31412}, {21154,28164}, {28160,34126}, {28194,38128}, {28204,38044}, {28208,38084}

X(38161) = midpoint of X(38038) and X(38156)
X(38161) = reflection of X(i) in X(j) for these (i,j): (32557, 23513), (34123, 10171), (38133, 38182)
X(38161) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 6246, 214), (3817, 38140, 38162), (38077, 38156, 38038), (38133, 38182, 38104)

X(38162) = CENTROID OF TRIANGLE {X(4), X(10), X(12)}

Barycentrics    2*a^7-3*(b+c)*a^6+(b^2+8*b*c+c^2)*a^5+(b+c)*(2*b^2-11*b*c+2*c^2)*a^4-(8*b^2-15*b*c+8*c^2)*(b+c)^2*a^3+(b^2-c^2)*(b-c)*(5*b^2+13*b*c+5*c^2)*a^2+(b^2-c^2)^2*(5*b^2-7*b*c+5*c^2)*a-4*(b^2-c^2)^3*(b-c) : :
X(38162) = X(12)+2*X(19925) = X(2975)-7*X(7989) = 5*X(3091)+X(37710) = 4*X(3634)-X(30264) = X(4297)-4*X(6668) = X(11491)+5*X(18492) = 2*X(31659)+X(31673) = 2*X(38033)-3*X(38062) = X(38033)-3*X(38109) = X(38039)-3*X(38078) = 3*X(38078)+X(38157) = 3*X(38105)-2*X(38134) = 3*X(38105)-4*X(38183) = 3*X(38142)+X(38178) = 4*X(38142)+X(38214) = 4*X(38178)-3*X(38214)

X(38162) lies on these lines: {12,950}, {30,38105}, {165,15679}, {515,38033}, {516,38058}, {517,38142}, {519,38039}, {529,38076}, {758,5587}, {952,3817}, {971,38208}, {1503,38198}, {2801,38209}, {2802,38163}, {2829,38219}, {2975,7989}, {3091,37710}, {3634,30264}, {4297,6668}, {5847,38148}, {5849,38146}, {5850,38153}, {5852,38151}, {5855,38155}, {5857,38158}, {10895,31803}, {11491,18492}, {21155,28164}, {28160,38114}, {28194,38129}, {28204,38045}, {28208,38085}, {31659,31673}

X(38162) = midpoint of X(38039) and X(38157)
X(38162) = reflection of X(i) in X(j) for these (i,j): (38062, 38109), (38134, 38183)
X(38162) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3817, 38140, 38161), (38078, 38157, 38039), (38134, 38183, 38105)

X(38163) = CENTROID OF TRIANGLE {X(4), X(11), X(12)}

Barycentrics    a^10-2*(b+c)*a^9+6*b*c*a^8+2*(b+c)*(b^2-4*b*c+c^2)*a^7-(4*b^4+4*c^4-b*c*(2*b^2+13*b*c+2*c^2))*a^6+2*(b+c)*(3*b^4-7*b^2*c^2+3*c^4)*a^5+2*(b^4+c^4-b*c*(7*b^2+13*b*c+7*c^2))*(b-c)^2*a^4-2*(b^2-c^2)*(b-c)*(5*b^4+5*c^4-b*c*(2*b^2+7*b*c+2*c^2))*a^3+(b^2-c^2)^2*(3*b^4+3*c^4+b*c*(6*b^2-17*b*c+6*c^2))*a^2+4*(b^2-c^2)^3*(b-c)^3*a-2*(b^2-c^2)^4*(b-c)^2 : :
X(38163) = X(4)+2*X(8068) = 4*X(5)-X(4996) = 4*X(6667)-X(30264) = 4*X(6668)-X(24466) = 3*X(38106)-2*X(38135) = 3*X(38106)-4*X(38184)

X(38163) lies on these lines: {4,8068}, {5,4996}, {30,38106}, {104,9655}, {119,7548}, {149,10599}, {381,952}, {515,38063}, {516,38218}, {517,38215}, {528,38078}, {529,38077}, {758,38161}, {971,38209}, {1503,38199}, {2802,38162}, {5840,17577}, {5848,38148}, {5849,38147}, {5851,38153}, {5852,38152}, {5854,38157}, {5855,38156}, {5856,38160}, {5857,38159}, {6667,30264}, {6668,24466}, {6830,13273}, {6845,12761}, {9668,11491}, {10598,20060}, {10698,10894}, {11929,12531}, {17532,34474}, {23513,37375}

X(38163) = reflection of X(38135) in X(38184)
X(38163) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6830, 13273, 18861), (38135, 38184, 38106)

X(38164) = CENTROID OF TRIANGLE {X(5), X(6), X(7)}

Barycentrics    4*(b+c)*a^7-(b^2-8*b*c+c^2)*a^6-10*(b+c)*(b^2+c^2)*a^5+(3*b^4+3*c^4-2*b*c*(7*b^2-5*b*c+7*c^2))*a^4+4*(b+c)*(b^4-4*b^2*c^2+c^4)*a^3+(b^4+c^4+2*b*c*(b^2+11*b*c+c^2))*(b-c)^2*a^2+2*(b^4-c^4)*(b^2-c^2)*(b+c)*a+(b^2-c^2)*(b-c)^2*(-3*b^4+3*c^4) : :
X(38164) = X(7)+2*X(18583) = X(21850)+2*X(31657) = 3*X(38079)-2*X(38166) = 3*X(38086)-X(38115) = 3*X(38086)+X(38143) = 3*X(38110)-2*X(38117) = X(38117)-3*X(38186)

X(38164) lies on these lines: {7,18583}, {30,38086}, {511,38111}, {516,38040}, {517,38187}, {518,38041}, {524,38080}, {527,38079}, {952,38046}, {971,38136}, {1503,38137}, {3564,38107}, {5762,38110}, {5843,14561}, {5844,38185}, {5846,38170}, {5847,38172}, {5848,38173}, {5849,38174}, {5850,38167}, {5851,38168}, {5852,38169}, {21850,31657}

X(38164) = midpoint of X(38115) and X(38143)
X(38164) = reflection of X(38110) in X(38186)
X(38164) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38086, 38143, 38115), (38188, 38189, 38046)

X(38165) = CENTROID OF TRIANGLE {X(5), X(6), X(8)}

Barycentrics    4*(b+c)*a^5-(7*b^2+8*b*c+7*c^2)*a^4+8*b*c*(b+c)*a^3+4*(b^4+c^4-b*c*(2*b^2+3*b*c+2*c^2))*a^2-4*(b^4-c^4)*(b-c)*a+3*(b^4-c^4)*(b^2-c^2) : :
X(38165) = X(8)+2*X(18583) = 2*X(182)+X(37705) = X(1351)+5*X(3617) = X(1483)-4*X(3589) = X(3242)-4*X(3628) = 5*X(3618)+X(12645) = 2*X(5690)+X(21850) = 5*X(5818)-2*X(18358) = X(38029)-3*X(38047) = 2*X(38029)-3*X(38110) = 2*X(38040)-3*X(38079) = 3*X(38079)-4*X(38167) = 3*X(38087)-X(38116) = 3*X(38087)+X(38144) = 3*X(38136)-4*X(38146) = X(38136)+4*X(38191) = X(38146)+3*X(38191)

X(38165) lies on these lines: {8,18583}, {30,38087}, {182,37705}, {511,38112}, {517,38136}, {518,38041}, {519,38040}, {524,38081}, {611,11545}, {952,38029}, {1351,3617}, {1483,3589}, {1503,38138}, {3242,3628}, {3564,5790}, {3618,12645}, {5085,28224}, {5690,21850}, {5762,38190}, {5818,18358}, {5843,38185}, {5844,14561}, {5845,38170}, {5847,38176}, {5848,38177}, {5849,38178}, {5853,38166}, {5854,38168}, {5855,38169}, {9041,15699}, {9053,10283}, {28204,38118}

X(38165) = midpoint of X(38116) and X(38144)
X(38165) = reflection of X(i) in X(j) for these (i,j): (38040, 38167), (38110, 38047)
X(38165) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38040, 38167, 38079), (38087, 38144, 38116), (38192, 38193, 38047)

X(38166) = CENTROID OF TRIANGLE {X(5), X(6), X(9)}

Barycentrics    4*a^8-6*(b+c)*a^7-(5*b^2+4*b*c+5*c^2)*a^6+12*(b+c)*(b^2+c^2)*a^5-5*(b^4+c^4-2*b*c*(b^2-3*b*c+c^2))*a^4-2*(b+c)*(b^2-4*b*c+c^2)*(b^2+4*b*c+c^2)*a^3+(5*b^4+5*c^4+2*b*c*(3*b^2-b*c+3*c^2))*(b-c)^2*a^2-4*(b^4-c^4)*(b^2-c^2)*(b+c)*a+(b^4-c^4)*(b^2-c^2)*(b-c)^2 : :
X(38166) = X(9)+2*X(18583) = X(1351)+5*X(18230) = 4*X(3589)-X(31657) = 5*X(3618)+X(5779) = 5*X(12017)+X(36991) = 3*X(14561)-X(38143) = X(21850)+2*X(31658) = 3*X(38079)-X(38164) = 3*X(38088)-X(38117) = 3*X(38088)+X(38145)

X(38166) lies on these lines: {9,18583}, {30,38088}, {511,38113}, {516,38136}, {517,38194}, {518,10283}, {524,38082}, {527,38079}, {952,38048}, {971,38110}, {1351,18230}, {1503,38139}, {3564,38108}, {3589,31657}, {3618,5779}, {5050,5817}, {5762,14561}, {5843,38186}, {5844,38190}, {5845,38171}, {5846,38175}, {5847,38179}, {5848,38180}, {5849,38181}, {5853,38165}, {5856,38168}, {5857,38169}, {12017,36991}, {21850,31658}

X(38166) = midpoint of X(i) and X(j) for these {i,j}: {5050, 5817}, {38117, 38145}
X(38166) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38088, 38145, 38117), (38195, 38196, 38048)

X(38167) = CENTROID OF TRIANGLE {X(5), X(6), X(10)}

Barycentrics    2*a^6+(b+c)*a^5-2*(3*b^2+b*c+3*c^2)*a^4+2*b*c*(b+c)*a^3+2*(b^4+c^4-b*c*(b^2+6*b*c+c^2))*a^2-(b^4-c^4)*(b-c)*a+2*(b^4-c^4)*(b^2-c^2) : :
X(38167) = X(6)+2*X(9956) = X(10)+2*X(18583) = 2*X(182)+X(18480) = X(355)+5*X(3618) = X(576)+2*X(3844) = X(1351)+5*X(1698) = X(1385)-4*X(3589) = X(1386)-4*X(25555) = 5*X(1656)+X(3751) = X(3579)+2*X(5480) = X(4663)+2*X(24206) = X(5691)+5*X(12017) = 3*X(14561)-X(38035) = 3*X(14561)+X(38116) = X(38035)+3*X(38047) = X(38040)-3*X(38079) = 3*X(38047)-X(38116) = 3*X(38079)+X(38165) = 3*X(38089)-X(38118) = 3*X(38089)+X(38146)

X(38167) lies on these lines: {6,9956}, {10,18583}, {30,38089}, {182,18480}, {355,3618}, {511,11231}, {515,38110}, {516,38136}, {517,14561}, {518,11230}, {519,38040}, {524,38083}, {576,3844}, {758,38169}, {952,38049}, {1351,1698}, {1385,3589}, {1386,25555}, {1503,38140}, {1656,3751}, {2802,38168}, {3564,10175}, {3579,5480}, {4663,24206}, {5050,5587}, {5085,28160}, {5691,12017}, {5762,38194}, {5790,16475}, {5843,38187}, {5844,38191}, {5845,38172}, {5846,38176}, {5847,38042}, {5848,38182}, {5849,38183}, {5850,38164}, {6684,21850}, {7989,18440}, {10172,34379}, {14848,19875}, {14853,26446}, {15988,17619}, {19130,22793}, {28198,38072}, {28204,38029}, {28208,38064}, {31423,33878}, {31663,31670}

X(38167) = midpoint of X(i) and X(j) for these {i,j}: {5050, 5587}, {5790, 16475}, {14561, 38047}, {14848, 19875}, {14853, 26446}, {38029, 38144}, {38035, 38116}, {38040, 38165}, {38118, 38146}
X(38167) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14561, 38116, 38035), (38035, 38047, 38116), (38079, 38165, 38040), (38089, 38146, 38118), (38197, 38198, 38049)

X(38168) = CENTROID OF TRIANGLE {X(5), X(6), X(11)}

Barycentrics    2*a^9-2*(b+c)*a^8-8*(b^2-b*c+c^2)*a^7+2*(b+c)*(4*b^2-3*b*c+4*c^2)*a^6+(8*b^4+8*c^4-b*c*(17*b^2-4*b*c+17*c^2))*a^5-4*(b+c)*(2*b^4+2*c^4-b*c*(3*b^2-b*c+3*c^2))*a^4+4*(b^4+c^4+3*b*c*(b^2-3*b*c+c^2))*b*c*a^3-2*(b^2-c^2)*(b-c)*b*c*(b^2+8*b*c+c^2)*a^2-(b^4-c^4)*(b^2-c^2)*(b-2*c)*(2*b-c)*a+(b^4-c^4)*(b^2-c^2)^2*(2*b-2*c) : :
X(38168) = X(11)+2*X(18583) = 2*X(182)+X(22938) = X(1351)+5*X(31272) = 5*X(1656)+X(10755) = 4*X(3589)-X(33814) = 5*X(3618)+X(10738) = 2*X(6713)+X(21850) = X(10724)+5*X(12017) = 4*X(19130)-X(22799) = 3*X(38090)-X(38119) = 3*X(38090)+X(38147)

X(38168) lies on these lines: {11,18583}, {30,38090}, {182,22938}, {511,34126}, {517,38197}, {518,38044}, {524,38084}, {528,38079}, {952,14561}, {1351,31272}, {1503,38141}, {1656,10755}, {2802,38167}, {2829,38136}, {3564,23513}, {3589,33814}, {3618,10738}, {5762,38195}, {5840,38110}, {5843,38188}, {5844,38192}, {5845,38173}, {5846,38177}, {5847,38182}, {5849,38184}, {5851,38164}, {5854,38165}, {5856,38166}, {6713,21850}, {10724,12017}, {19130,22799}

X(38168) = midpoint of X(38119) and X(38147)
X(38168) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14561, 38040, 38169), (38090, 38147, 38119)

X(38169) = CENTROID OF TRIANGLE {X(5), X(6), X(12)}

Barycentrics    2*a^9-2*(b+c)*a^8-8*(b^2+c^2)*a^7+2*(b+c)*(4*b^2-b*c+4*c^2)*a^6+(8*b^4+8*c^4-b*c*(7*b^2+4*b*c+7*c^2))*a^5-4*(2*b-c)*(b-2*c)*(b+c)*(b^2+b*c+c^2)*a^4+4*(b^4+c^4+b*c*(b^2-3*b*c+c^2))*b*c*a^3-2*(b^2-c^2)*(b-c)*b*c*(3*b^2+8*b*c+3*c^2)*a^2-(b^4-c^4)*(b^2-c^2)*(2*b^2-3*b*c+2*c^2)*a+(b^4-c^4)*(b^2-c^2)^2*(2*b-2*c) : :
X(38169) = X(12)+2*X(18583) = X(21850)+2*X(31659) = 3*X(38091)-X(38120) = 3*X(38091)+X(38148)

X(38169) lies on these lines: {12,18583}, {30,38091}, {511,38114}, {517,38198}, {518,38045}, {524,38085}, {529,38079}, {758,38167}, {952,14561}, {1503,38142}, {3564,38109}, {5762,38196}, {5842,38136}, {5843,38189}, {5844,38193}, {5845,38174}, {5846,38178}, {5847,38183}, {5848,38184}, {5852,38164}, {5855,38165}, {5857,38166}, {21850,31659}

X(38169) = midpoint of X(38120) and X(38148)
X(38169) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14561, 38040, 38168), (38091, 38148, 38120)

X(38170) = CENTROID OF TRIANGLE {X(5), X(7), X(8)}

Barycentrics    4*a^6-8*(b+c)*a^5+(5*b^2+24*b*c+5*c^2)*a^4-2*(b+c)*(b^2+8*b*c+c^2)*a^3-2*(b+2*c)*(2*b+c)*(b-c)^2*a^2+2*(b^2-c^2)*(b-c)*(5*b^2+2*b*c+5*c^2)*a-5*(b^2-c^2)^2*(b-c)^2 : :
X(38170) = X(5)+2*X(2550) = 3*X(5)-2*X(38037) = 4*X(142)-X(1483) = X(390)-4*X(3628) = 2*X(546)+X(35514) = 5*X(632)-8*X(3826) = 3*X(2550)+X(38037) = 3*X(15699)-2*X(38043) = X(38030)-3*X(38052) = 2*X(38030)-3*X(38111) = 2*X(38041)-3*X(38080) = 3*X(38042)-2*X(38179) = 3*X(38080)-4*X(38172) = 3*X(38081)-2*X(38175) = 3*X(38092)-X(38121) = 3*X(38092)+X(38149) = 3*X(38112)-2*X(38126) = 3*X(38137)-4*X(38151) = X(38137)+4*X(38201) = X(38151)+3*X(38201)

X(38170) lies on these lines: {5,2550}, {30,38092}, {142,1483}, {390,3628}, {516,3845}, {517,38137}, {519,38041}, {527,38081}, {528,15699}, {546,35514}, {632,3826}, {952,38030}, {971,38138}, {3564,38185}, {5762,38112}, {5790,5843}, {5844,38107}, {5845,38165}, {5846,38164}, {5850,38176}, {5851,38177}, {5852,38178}, {5853,10283}, {5854,38173}, {5855,38174}, {11539,38031}, {21151,28224}, {28204,38123}, {31657,37705}, {38028,38204}

X(38170) = midpoint of X(38121) and X(38149)
X(38170) = reflection of X(i) in X(j) for these (i,j): (10283, 38171), (38028, 38204), (38041, 38172), (38111, 38052), (38112, 38200)
X(38170) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38041, 38172, 38080), (38092, 38149, 38121), (38202, 38203, 38052)

X(38171) = CENTROID OF TRIANGLE {X(5), X(7), X(9)}

Barycentrics    2*(b+c)*a^5-(b^2-4*b*c+c^2)*a^4-6*(b+c)*(b^2+c^2)*a^3+2*(2*b^2-b*c+2*c^2)*(b-c)^2*a^2+4*(b^2-c^2)^2*(b+c)*a-3*(b^2-c^2)^2*(b-c)^2 : :
X(38171) = 5*X(2)-X(21168) = X(5)+2*X(142) = 2*X(5)+X(31657) = X(7)+5*X(1656) = X(9)-4*X(3628) = 2*X(140)+X(5805) = 2*X(140)-5*X(20195) = 4*X(142)-X(31657) = 4*X(142)+X(38139) = X(144)-13*X(5067) = X(5805)+5*X(20195) = 5*X(20195)-X(21153) = X(21168)+5*X(38107) = 2*X(21168)-5*X(38113) = 2*X(38041)+X(38175) = X(38043)+2*X(38172) = 3*X(38093)-X(38122) = 3*X(38093)+X(38150) = 2*X(38107)+X(38113) = 2*X(38111)+X(38139)

X(38171) lies on these lines: {2,5762}, {5,142}, {7,1656}, {9,3628}, {30,38093}, {140,5805}, {144,5067}, {381,21151}, {516,549}, {517,38204}, {518,38041}, {527,15699}, {528,38022}, {546,5732}, {547,5843}, {550,18482}, {631,27355}, {632,31658}, {946,33575}, {952,38053}, {1001,6924}, {2550,5901}, {2801,38182}, {3090,5779}, {3526,5759}, {3545,38065}, {3564,38186}, {3817,10156}, {3826,3918}, {3850,31672}, {3851,36991}, {5054,38073}, {5055,5817}, {5056,36996}, {5070,18230}, {5542,9956}, {5587,38030}, {5603,38121}, {5735,16239}, {5790,11038}, {5844,38200}, {5845,38166}, {5850,10172}, {5851,38180}, {5852,38181}, {5853,10283}, {5856,38173}, {5857,38174}, {5886,38052}, {6147,21617}, {6172,15703}, {6858,30275}, {6887,8732}, {7679,38055}, {8227,24644}, {8581,10592}, {10165,38151}, {10175,38054}, {10246,38149}, {10516,38115}, {10593,14100}, {10861,17530}, {16417,34474}, {18493,35514}, {26446,38036}

X(38171) = midpoint of X(i) and X(j) for these {i,j}: {2, 38107}, {5, 38111}, {381, 21151}, {549, 38137}, {3545, 38065}, {3817, 38123}, {5054, 38073}, {5587, 38030}, {5603, 38121}, {5790, 11038}, {5805, 21153}, {5886, 38052}, {6173, 38108}, {10165, 38151}, {10175, 38054}, {10246, 38149}, {10283, 38170}, {10516, 38115}, {11230, 38172}, {15699, 38080}, {26446, 38036}, {31657, 38139}, {38041, 38042}, {38122, 38150}
X(38171) = reflection of X(i) in X(j) for these (i,j): (21153, 140), (31657, 38111), (38043, 11230), (38082, 15699), (38108, 547), (38111, 142), (38113, 2), (38139, 5), (38175, 38042), (38179, 10172)
X(38171) = center of Vu pedal-centroidal circle of X(7)
X(38171) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 142, 31657), (3826, 20330, 5690), (5805, 20195, 140), (38093, 38150, 38122), (38205, 38206, 38053)

X(38172) = CENTROID OF TRIANGLE {X(5), X(7), X(10)}

Barycentrics    2*a^6-(b+c)*a^5-2*(b^2-5*b*c+c^2)*a^4-4*(b+c)*(b^2+b*c+c^2)*a^3+2*(2*b^2-b*c+2*c^2)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(5*b^2+6*b*c+5*c^2)*a-4*(b^2-c^2)^2*(b-c)^2 : :
X(38172) = X(7)+2*X(9956) = 4*X(142)-X(1385) = 5*X(1656)+X(4312) = 2*X(2550)+X(24680) = X(3059)+2*X(6583) = X(3062)-7*X(3851) = X(3579)+2*X(5805) = 2*X(5732)+X(33697) = 3*X(11230)-2*X(38043) = 3*X(11231)-2*X(38130) = X(38036)+3*X(38052) = X(38036)-3*X(38107) = X(38041)-3*X(38080) = X(38043)-3*X(38171) = 3*X(38052)-X(38121) = 3*X(38080)+X(38170) = 3*X(38083)-2*X(38179) = 3*X(38094)-X(38123) = 3*X(38094)+X(38151) = 3*X(38107)+X(38121)

X(38172) lies on these lines: {7,9956}, {30,38094}, {142,1385}, {515,38111}, {516,549}, {517,38036}, {518,38176}, {519,38041}, {527,38083}, {758,38174}, {952,38054}, {971,38140}, {1001,33862}, {1656,4312}, {2550,24680}, {2802,38173}, {3059,6583}, {3062,3851}, {3564,38187}, {3579,5805}, {5732,33697}, {5762,11231}, {5843,10175}, {5844,38201}, {5845,38167}, {5847,38164}, {5850,38042}, {5851,38182}, {5852,38183}, {5880,6862}, {6173,38154}, {6926,9955}, {11278,20330}, {17502,38122}, {18480,31657}, {21151,28160}, {28198,38073}, {28204,38030}, {28208,38065}, {31663,31671}, {38031,38093}

X(38172) = midpoint of X(i) and X(j) for these {i,j}: {38030, 38149}, {38036, 38121}, {38041, 38170}, {38052, 38107}, {38123, 38151}
X(38172) = reflection of X(i) in X(j) for these (i,j): (11230, 38171), (11231, 38204), (17502, 38122)
X(38172) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38036, 38052, 38121), (38080, 38170, 38041), (38094, 38151, 38123), (38107, 38121, 38036), (38207, 38208, 38054)

X(38173) = CENTROID OF TRIANGLE {X(5), X(7), X(11)}

Barycentrics    2*a^9-2*(b+c)*a^8-2*(3*b^2-4*b*c+3*c^2)*a^7+2*(b^3+c^3)*a^6+(14*b^4+14*c^4-b*c*(25*b^2-32*b*c+25*c^2))*a^5-2*(b+c)*(b^4+c^4-5*b*c*(b-c)^2)*a^4-6*(3*b^4+5*b^2*c^2+3*c^4)*(b-c)^2*a^3+2*(b^2-c^2)*(b-c)*(3*b^4+3*c^4-2*b*c*(3*b^2-5*b*c+3*c^2))*a^2+(b^2-c^2)^2*(b-c)^2*(8*b^2-3*b*c+8*c^2)*a-4*(b^2-c^2)^3*(b-c)^3 : :
X(38173) = 4*X(142)-X(33814) = 4*X(3628)-X(6068) = X(22938)+2*X(31657) = 3*X(34126)-2*X(38131) = 3*X(38084)-2*X(38180) = 3*X(38095)-X(38124) = 3*X(38095)+X(38152) = X(38131)-3*X(38205)

X(38173) lies on these lines: {30,38095}, {142,33814}, {516,23961}, {517,38207}, {518,38177}, {527,38084}, {528,10283}, {952,1056}, {971,38141}, {2801,38140}, {2802,38172}, {2829,38137}, {3564,38188}, {3628,6068}, {5762,34126}, {5840,38111}, {5843,23513}, {5844,38202}, {5845,38168}, {5848,38164}, {5850,38182}, {5852,38184}, {5854,38170}, {5856,38171}, {22938,31657}

X(38173) = midpoint of X(38124) and X(38152)
X(38173) = reflection of X(34126) in X(38205)
X(38173) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38041, 38107, 38174), (38095, 38152, 38124)

X(38174) = CENTROID OF TRIANGLE {X(5), X(7), X(12)}

Barycentrics    2*a^9-2*(b+c)*a^8-2*(3*b^2-4*b*c+3*c^2)*a^7+2*(b+c)*(b^2-7*b*c+c^2)*a^6+(14*b^4+14*c^4-b*c*(7*b^2-40*b*c+7*c^2))*a^5-2*(b+c)*(b^4+c^4-b*c*(11*b^2-18*b*c+11*c^2))*a^4-6*(b^3-c^3)*(b-c)*(3*b^2+b*c+3*c^2)*a^3+2*(b^2-c^2)*(b-c)*(3*b^4+3*c^4-2*b*c*(3*b^2-b*c+3*c^2))*a^2+(b^2-c^2)^2*(b-c)^2*(8*b^2+3*b*c+8*c^2)*a-4*(b^2-c^2)^3*(b-c)^3 : :
X(38174) = 3*X(38085)-2*X(38181) = 3*X(38096)-X(38125) = 3*X(38096)+X(38153) = 3*X(38114)-2*X(38132) = X(38132)-3*X(38206)

X(38174) lies on these lines: {30,38096}, {516,33862}, {517,38208}, {518,38178}, {527,38085}, {529,38080}, {758,38172}, {952,1056}, {971,38142}, {3564,38189}, {5762,38114}, {5842,38137}, {5843,38109}, {5844,38203}, {5845,38169}, {5849,38164}, {5850,38183}, {5851,38184}, {5855,38170}, {5857,38171}

X(38174) = midpoint of X(38125) and X(38153)
X(38174) = reflection of X(38114) in X(38206)
X(38174) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38041, 38107, 38173), (38096, 38153, 38125)

X(38175) = CENTROID OF TRIANGLE {X(5), X(8), X(9)}

Barycentrics    6*(b+c)*a^5-5*(3*b^2+4*b*c+3*c^2)*a^4+2*(b+c)*(3*b^2+8*b*c+3*c^2)*a^3+6*(2*b^2+3*b*c+2*c^2)*(b-c)^2*a^2-4*(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a+3*(b^2-c^2)^2*(b-c)^2 : :
X(38175) = X(5)+2*X(24393) = 4*X(10)-X(31657) = 3*X(10)-X(38123) = X(1483)-4*X(6666) = X(3243)-4*X(3628) = 5*X(3617)+X(5779) = 3*X(5686)+X(38149) = 3*X(31657)-4*X(38123) = X(38031)-3*X(38057) = 2*X(38031)-3*X(38113) = X(38041)-3*X(38042) = 2*X(38041)-3*X(38171) = 2*X(38043)-3*X(38082) = 3*X(38081)-X(38170) = 3*X(38082)-4*X(38179) = 3*X(38097)-X(38126) = 3*X(38097)+X(38154) = 3*X(38139)-4*X(38158) = X(38139)+4*X(38210) = X(38158)+3*X(38210)

X(38175) lies on these lines: {5,24393}, {10,31657}, {30,38097}, {516,3627}, {517,38139}, {518,38041}, {519,38043}, {527,38081}, {952,6883}, {971,38112}, {1483,6666}, {3243,3628}, {3564,38190}, {3617,5779}, {4866,5587}, {5686,5762}, {5843,38200}, {5844,38108}, {5846,38166}, {5854,38180}, {5855,38181}, {5856,38177}, {5857,38178}, {7679,38056}, {8164,11038}, {11545,15298}, {12645,18230}, {19875,38030}, {21153,28224}, {28204,38130}, {31658,37705}

X(38175) = midpoint of X(i) and X(j) for these {i,j}: {5686, 5790}, {38126, 38154}
X(38175) = reflection of X(i) in X(j) for these (i,j): (38043, 38179), (38113, 38057), (38171, 38042)
X(38175) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38043, 38179, 38082), (38097, 38154, 38126), (38211, 38212, 38057)

X(38176) = CENTROID OF TRIANGLE {X(5), X(8), X(10)}

Barycentrics    2*a^4-5*(b+c)*a^3+2*(b^2+5*b*c+c^2)*a^2+5*(b^2-c^2)*(b-c)*a-4*(b^2-c^2)^2 : :
X(38176) = 5*X(1)-11*X(5070) = X(5)+2*X(3626) = 4*X(5)-X(11278) = 5*X(8)+7*X(3090) = X(8)+2*X(9956) = 7*X(8)+5*X(10595) = 2*X(8)+X(24680) = 5*X(10)-2*X(140) = 4*X(10)-X(1385) = 7*X(10)-X(5882) = 3*X(10)-X(10165) = 8*X(140)-5*X(1385) = 14*X(140)-5*X(5882) = 6*X(140)-5*X(10165) = 4*X(140)-5*X(11231) = 7*X(3090)-5*X(5886) = 7*X(3090)-10*X(9956) = 14*X(3090)-5*X(24680) = 8*X(3626)+X(11278) = 7*X(5886)-5*X(10595) = 14*X(9956)-5*X(10595) = 4*X(9956)-X(24680) = 10*X(10595)-7*X(24680)

X(38176) lies on these lines: {1,5070}, {3,37712}, {5,3626}, {8,3090}, {10,140}, {20,355}, {30,38098}, {40,5073}, {165,15689}, {210,381}, {480,37622}, {515,4745}, {516,3627}, {518,38172}, {519,10172}, {547,34641}, {549,28236}, {632,13607}, {758,38178}, {944,31666}, {946,12811}, {958,33862}, {1159,5726}, {1376,23961}, {1482,3711}, {1483,3634}, {1656,3632}, {1698,12645}, {2771,18908}, {2802,38177}, {3057,37718}, {3244,3628}, {3419,6976}, {3524,5731}, {3564,38191}, {3576,15701}, {3625,5901}, {3654,15682}, {3655,15721}, {3698,5885}, {3817,14892}, {3826,32213}, {3828,38028}, {3845,28228}, {3851,11531}, {3861,11362}, {3918,24475}, {4297,31447}, {4669,5844}, {4677,10247}, {4678,5068}, {4731,10202}, {4746,13464}, {4816,9624}, {5054,31662}, {5055,16200}, {5067,20050}, {5126,37708}, {5251,12331}, {5660,19914}, {5694,5836}, {5762,38210}, {5784,5833}, {5843,38201}, {5846,38167}, {5847,38165}, {5850,38170}, {5853,38179}, {5854,38182}, {5855,38183}, {5881,13624}, {6684,37705}, {7951,36920}, {7989,8148}, {9578,31794}, {9708,32613}, {9709,32612}, {9710,10942}, {9711,10943}, {9779,12245}, {9780,37727}, {9897,37600}, {9947,31828}, {10164,14891}, {10246,19875}, {10573,17718}, {11545,31397}, {12101,28174}, {12702,37714}, {13145,14872}, {15685,28168}, {15687,28232}, {15691,28186}, {15694,30392}, {15703,34747}, {15934,30286}, {16173,17606}, {17590,24987}, {17757,38109}, {18251,35004}, {18395,24928}, {18481,21735}, {18525,31663}, {18526,31423}, {19876,34748}, {21677,38157}, {22936,37829}, {23513,24390}, {28178,34648}, {28182,31673}, {28198,38074}, {33895,38044}, {37582,37710}

X(38176) = midpoint of X(i) and X(j) for these {i,j}: {3, 37712}, {8, 5886}, {355, 5657}, {1699, 34718}, {3679, 5790}, {4669, 10175}, {4677, 10247}, {5660, 19914}, {5690, 38138}, {38127, 38155}
X(38176) = reflection of X(i) in X(j) for these (i,j): (1385, 11231), (3579, 5657), (5886, 9956), (10283, 10172), (11230, 38042), (11231, 10), (17502, 26446), (18480, 38138), (24680, 5886), (38028, 3828), (38112, 4745)
X(38176) = X(8)-Beth conjugate of-X(11231)
X(38176) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 9956, 24680), (1656, 3632, 33179), (1698, 12645, 15178), (3625, 31399, 5901), (10172, 10283, 11230), (10283, 38042, 10172), (11230, 38042, 38083), (11362, 18357, 22793), (38098, 38155, 38127), (38213, 38214, 10)

X(38177) = CENTROID OF TRIANGLE {X(5), X(8), X(11)}

Barycentrics    2*a^7-6*(b+c)*a^6+4*(b^2+5*b*c+c^2)*a^5+2*(b+c)*(4*b^2-15*b*c+4*c^2)*a^4-(14*b^4+14*c^4+b*c*(b^2-40*b*c+c^2))*a^3+2*(b^2-c^2)*(b-c)*(b^2+13*b*c+c^2)*a^2+(b^2-c^2)^2*(8*b^2-19*b*c+8*c^2)*a-4*(b^2-c^2)^3*(b-c) : :
X(38177) = X(5)+2*X(3036) = 4*X(10)-X(33814) = X(1317)-4*X(3628) = X(1483)-4*X(6667) = 5*X(1656)+X(12531) = 5*X(3617)+X(10738) = 2*X(5690)+X(22938) = 5*X(5818)+X(19914) = 2*X(6713)+X(37705) = 2*X(9956)+X(15863) = 4*X(9956)-X(19907) = 3*X(34122)-X(38032) = 3*X(34126)-2*X(38032) = 2*X(38044)-3*X(38084) = 3*X(38084)-4*X(38182) = 3*X(38099)-X(38128) = 3*X(38099)+X(38156) = 3*X(38141)-4*X(38161) = X(38141)+4*X(38213) = X(38161)+3*X(38213)

X(38177) lies on these lines: {2,952}, {5,3036}, {10,26086}, {30,38099}, {517,38141}, {518,38173}, {519,38044}, {528,38081}, {1317,3628}, {1483,6667}, {1656,12531}, {2802,38176}, {2829,38138}, {3564,38192}, {3617,10738}, {5690,10525}, {5762,38211}, {5818,19914}, {5840,38112}, {5843,38202}, {5844,23513}, {5846,38168}, {5848,38165}, {5851,38170}, {5853,38180}, {5855,38184}, {5856,38175}, {6713,37705}, {9956,15863}, {12019,13274}, {12515,37714}, {12645,31272}, {18357,22799}, {21154,28224}, {28204,38133}

X(38177) = midpoint of X(38128) and X(38156)
X(38177) = reflection of X(i) in X(j) for these (i,j): (34126, 34122), (38044, 38182)
X(38177) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5790, 38042, 38178), (9956, 15863, 19907), (38044, 38182, 38084), (38099, 38156, 38128)

X(38178) = CENTROID OF TRIANGLE {X(5), X(8), X(12)}

Barycentrics    2*a^7-6*(b+c)*a^6+4*(b^2+5*b*c+c^2)*a^5+2*(b+c)*(4*b^2-13*b*c+4*c^2)*a^4-(14*b^4+14*c^4+b*c*(7*b^2-32*b*c+7*c^2))*a^3+2*(b^2-c^2)*(b-c)*(b^2+11*b*c+c^2)*a^2+(b^2-c^2)^2*(8*b^2-13*b*c+8*c^2)*a-4*(b^2-c^2)^3*(b-c) : :
X(38178) = X(1483)-4*X(6668) = 4*X(3628)-X(37734) = 2*X(31659)+X(37705) = X(38033)-3*X(38058) = 2*X(38033)-3*X(38114) = 2*X(38045)-3*X(38085) = 3*X(38085)-4*X(38183) = 3*X(38100)-X(38129) = 3*X(38100)+X(38157) = 3*X(38142)-4*X(38162) = X(38142)+4*X(38214) = X(38162)+3*X(38214)

X(38178) lies on these lines: {2,952}, {30,38100}, {355,7508}, {517,38142}, {518,38174}, {519,38045}, {529,38081}, {758,38176}, {1483,6668}, {3564,38193}, {3628,37734}, {5690,10526}, {5762,38212}, {5842,38138}, {5843,38203}, {5844,38109}, {5846,38169}, {5849,38165}, {5852,38170}, {5853,38181}, {5854,38184}, {5857,38175}, {21155,28224}, {24914,37710}, {28204,38134}, {28463,38155}, {31659,37705}

X(38178) = midpoint of X(38129) and X(38157)
X(38178) = reflection of X(i) in X(j) for these (i,j): (38045, 38183), (38114, 38058)
X(38178) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5790, 38042, 38177), (38045, 38183, 38085), (38100, 38157, 38129)

X(38179) = CENTROID OF TRIANGLE {X(5), X(9), X(10)}

Barycentrics    2*a^6-(b+c)*a^5-2*(4*b^2+3*b*c+4*c^2)*a^4+4*(b+c)*(2*b^2+b*c+2*c^2)*a^3+2*(b+2*c)*(2*b+c)*(b-c)^2*a^2-(b^2-c^2)*(b-c)*(7*b^2+10*b*c+7*c^2)*a+2*(b^2-c^2)^2*(b-c)^2 : :
X(38179) = X(9)+2*X(9956) = X(355)+5*X(18230) = X(1385)-4*X(6666) = 5*X(1656)+X(5223) = 5*X(1698)+X(5779) = 4*X(3628)-X(5542) = 4*X(3634)-X(31657) = 3*X(5055)-X(38036) = 7*X(7989)-X(31671) = 3*X(10175)-X(38151) = X(38037)+3*X(38057) = X(38037)-3*X(38108) = 3*X(38042)-X(38170) = X(38043)-3*X(38082) = 3*X(38057)-X(38126) = 3*X(38082)+X(38175) = 3*X(38083)-X(38172) = 3*X(38101)-X(38130) = 3*X(38101)+X(38158) = 3*X(38108)+X(38126)

X(38179) lies on these lines: {2,38030}, {9,9956}, {30,38101}, {210,11218}, {355,18230}, {515,38113}, {516,3845}, {517,38037}, {518,11230}, {519,38043}, {527,38083}, {758,38181}, {952,38059}, {971,11231}, {1385,6666}, {1656,5223}, {1698,5779}, {2802,38180}, {3085,15008}, {3564,38194}, {3628,5542}, {3634,31657}, {5055,38036}, {5445,31391}, {5686,5886}, {5762,10175}, {5817,26446}, {5843,38204}, {5844,38210}, {5847,38166}, {5850,10172}, {5853,38176}, {5856,38182}, {5857,38183}, {7989,31671}, {8232,31794}, {10398,31479}, {11495,31447}, {15699,38041}, {18480,31658}, {19855,31821}, {19875,38121}, {19877,36996}, {21153,28160}, {24393,24680}, {28198,38075}, {28204,38031}, {28208,38067}, {30329,31835}, {31663,31672}

X(38179) = midpoint of X(i) and X(j) for these {i,j}: {5686, 5886}, {5817, 26446}, {38031, 38154}, {38037, 38126}, {38043, 38175}, {38057, 38108}, {38130, 38158}
X(38179) = reflection of X(38171) in X(10172)
X(38179) = complement of X(38030)
X(38179) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38037, 38057, 38126), (38082, 38175, 38043), (38101, 38158, 38130), (38108, 38126, 38037), (38216, 38217, 38059)

X(38180) = CENTROID OF TRIANGLE {X(5), X(9), X(11)}

Barycentrics    2*a^9-4*(b+c)*a^8-6*(b-c)^2*a^7+2*(b+c)*(9*b^2-8*b*c+9*c^2)*a^6-(b^2+c^2)*(2*b^2+25*b*c+2*c^2)*a^5-2*(b+c)*(11*b^4+11*c^4-b*c*(17*b^2-6*b*c+17*c^2))*a^4+2*(7*b^4+7*c^4+b*c*(16*b^2+31*b*c+16*c^2))*(b-c)^2*a^3+2*(b^2-c^2)*(b-c)*(3*b^4+3*c^4+b*c*(b^2-16*b*c+c^2))*a^2-(b^2-c^2)^2*(b-c)^2*(8*b^2+7*b*c+8*c^2)*a+2*(b^2-c^2)^3*(b-c)^3 : :
X(38180) = X(1156)+5*X(1656) = 4*X(3628)-X(10427) = X(5779)+5*X(31272) = 4*X(6666)-X(33814) = 4*X(6667)-X(31657) = X(10738)+5*X(18230) = X(22938)+2*X(31658) = 3*X(23513)-X(38152) = 3*X(38084)-X(38173) = 3*X(38102)-X(38131) = 3*X(38102)+X(38159)

X(38180) lies on these lines: {5,1158}, {30,38102}, {516,38141}, {517,38216}, {518,38044}, {527,38084}, {528,38042}, {952,38043}, {971,34126}, {1156,1656}, {2801,11230}, {2802,38179}, {2829,38139}, {3564,38195}, {3628,10427}, {5762,23513}, {5779,31272}, {5840,38113}, {5843,38205}, {5844,38211}, {5848,38166}, {5851,38171}, {5853,38177}, {5854,38175}, {5857,38184}, {6666,33814}, {6667,31657}, {10738,18230}, {11219,38124}, {22938,31658}, {26446,38202}

X(38180) = midpoint of X(38131) and X(38159)
X(38180) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38043, 38108, 38181), (38102, 38159, 38131)

X(38181) = CENTROID OF TRIANGLE {X(5), X(9), X(12)}

Barycentrics    2*a^9-4*(b+c)*a^8-2*(3*b^2-2*b*c+3*c^2)*a^7+18*(b+c)*(b^2+c^2)*a^6-(2*b^4+2*c^4+b*c*(23*b^2+20*b*c+23*c^2))*a^5-2*(b+c)*(11*b^4+11*c^4-b*c*(7*b^2+2*b*c+7*c^2))*a^4+2*(7*b^4+7*c^4+b*c*(20*b^2+29*b*c+20*c^2))*(b-c)^2*a^3+2*(b^2-c^2)*(b-c)*(3*b^4+3*c^4+b*c*(3*b^2-4*b*c+3*c^2))*a^2-(b^2-c^2)^2*(b-c)^2*(8*b^2+9*b*c+8*c^2)*a+2*(b^2-c^2)^3*(b-c)^3 : :
X(38181) = 4*X(6668)-X(31657) = 3*X(38085)-X(38174) = 3*X(38103)-X(38132) = 3*X(38103)+X(38160) = 3*X(38109)-X(38153)

X(38181) lies on these lines: {5,15296}, {30,38103}, {516,38142}, {517,38217}, {518,38045}, {527,38085}, {529,38082}, {758,38179}, {952,38043}, {971,38114}, {3564,38196}, {5762,38109}, {5842,38139}, {5843,38206}, {5844,38212}, {5849,38166}, {5852,38171}, {5853,38178}, {5855,38175}, {5856,38184}, {6668,31657}

X(38181) = midpoint of X(38132) and X(38160)
X(38181) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38043, 38108, 38180), (38103, 38160, 38132)

X(38182) = CENTROID OF TRIANGLE {X(5), X(10), X(11)}

Barycentrics    (b+c)*a^6-2*(2*b^2+b*c+2*c^2)*a^5+(b+c)*(b^2+5*b*c+c^2)*a^4+2*(4*b^4+4*c^4-b*c*(4*b^2+3*b*c+4*c^2))*a^3-(b^2-c^2)*(b-c)*(5*b^2+9*b*c+5*c^2)*a^2-2*(b^2-c^2)^2*(b-2*c)*(2*b-c)*a+3*(b^2-c^2)^3*(b-c) : :
X(38182) = X(5)+2*X(6702) = 4*X(5)-X(12611) = 2*X(5)+X(12619) = X(11)+2*X(9956) = X(80)+5*X(1656) = 2*X(140)+X(6246) = X(214)-4*X(3628) = X(355)+5*X(31272) = X(1385)-4*X(6667) = 8*X(6702)+X(12611) = 4*X(6702)-X(12619) = X(12611)+2*X(12619) = 3*X(23513)-X(38038) = 3*X(23513)+X(38128) = 3*X(34122)+X(38038) = 3*X(34122)-X(38128) = X(38044)-3*X(38084) = 3*X(38084)+X(38177) = 3*X(38104)-X(38133) = 3*X(38104)+X(38161)

X(38182) lies on these lines: {5,2800}, {11,9956}, {30,38104}, {80,1656}, {119,9947}, {140,6246}, {214,3628}, {355,31272}, {515,34126}, {516,38141}, {517,23513}, {519,38044}, {528,38083}, {547,551}, {758,38184}, {971,18856}, {1385,6667}, {1698,10738}, {2771,10157}, {2801,38171}, {2802,3829}, {2829,38140}, {3036,24680}, {3090,6265}, {3091,12515}, {3526,12119}, {3564,38197}, {3614,11570}, {3634,33814}, {3851,34789}, {5056,12247}, {5067,6224}, {5070,12747}, {5083,10592}, {5690,16174}, {5694,12736}, {5762,38216}, {5790,16173}, {5818,12737}, {5840,11231}, {5843,38207}, {5844,38213}, {5847,38168}, {5848,38167}, {5850,38173}, {5851,38172}, {5854,38176}, {5856,38179}, {5901,15863}, {6684,22938}, {6713,18480}, {7173,12758}, {7505,12137}, {7583,13976}, {7584,8988}, {7951,20118}, {7989,10742}, {8068,17606}, {8227,19914}, {8976,19077}, {10593,15558}, {11715,18357}, {12019,22935}, {13199,19877}, {13951,19078}, {16858,34474}, {21154,28160}, {21630,31399}, {28198,38077}, {28204,38032}, {28208,38069}

X(38182) = midpoint of X(i) and X(j) for these {i,j}: {5790, 16173}, {23513, 34122}, {38032, 38156}, {38038, 38128}, {38044, 38177}, {38133, 38161}
X(38182) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 6702, 12619), (5, 12619, 12611), (10175, 11230, 38183), (23513, 38128, 38038), (34122, 38038, 38128), (38084, 38177, 38044), (38104, 38161, 38133)

X(38183) = CENTROID OF TRIANGLE {X(5), X(10), X(12)}

Barycentrics    (b+c)*a^6-2*(2*b^2+3*b*c+2*c^2)*a^5+(b+c)*(b^2+7*b*c+c^2)*a^4+2*(4*b^4-5*b^2*c^2+4*c^4)*a^3-(b^2-c^2)*(b-c)*(5*b^2+11*b*c+5*c^2)*a^2-2*(b^2-c^2)^2*(2*b^2-3*b*c+2*c^2)*a+3*(b^2-c^2)^3*(b-c) : :
X(38183) = X(12)+2*X(9956) = X(1385)-4*X(6668) = 5*X(1656)+X(37710) = 5*X(5818)+X(37733) = X(18480)+2*X(31659) = X(38039)+3*X(38058) = X(38039)-3*X(38109) = X(38045)-3*X(38085) = 3*X(38058)-X(38129) = 3*X(38085)+X(38178) = 3*X(38105)-X(38134) = 3*X(38105)+X(38162) = 3*X(38109)+X(38129)

X(38183) lies on these lines: {5,3884}, {12,942}, {30,38105}, {515,38114}, {516,38142}, {517,17530}, {519,38045}, {529,38083}, {547,551}, {758,38042}, {1385,6668}, {1388,1656}, {2802,38184}, {3564,38198}, {3822,12619}, {5426,5587}, {5694,10592}, {5762,38217}, {5790,37701}, {5818,37733}, {5842,38140}, {5843,38208}, {5844,38214}, {5847,38169}, {5849,38167}, {5850,38174}, {5852,38172}, {5855,38176}, {5857,38179}, {6175,26446}, {8068,17636}, {18242,22798}, {18480,31659}, {21155,28160}, {28198,38078}, {28204,38033}, {28208,38070}

X(38183) = midpoint of X(i) and X(j) for these {i,j}: {5790, 37701}, {38033, 38157}, {38039, 38129}, {38045, 38178}, {38058, 38109}, {38134, 38162}
X(38183) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10175, 11230, 38182), (38039, 38058, 38129), (38085, 38178, 38045), (38105, 38162, 38134), (38109, 38129, 38039)

X(38184) = CENTROID OF TRIANGLE {X(5), X(11), X(12)}

Barycentrics    3*(b^2+c^2)*a^8-6*(b+c)*(b^2+c^2)*a^7-2*(3*b^4+3*c^4-b*c*(9*b^2+2*b*c+9*c^2))*a^6+2*(b+c)*(9*b^4+9*c^4-4*b*c*(3*b^2-b*c+3*c^2))*a^5-(30*b^4+30*c^4-b*c*(19*b^2+20*b*c+19*c^2))*b*c*a^4-2*(b^2-c^2)*(b-c)*(9*b^4+9*c^4-2*b*c*(3*b^2+4*b*c+3*c^2))*a^3+(b^2-c^2)^2*(6*b^4+6*c^4+b*c*(6*b^2-23*b*c+6*c^2))*a^2+6*(b^2-c^2)^3*(b-c)^3*a-3*(b^2-c^2)^4*(b-c)^2 : :
X(38184) = X(5)+2*X(8068) = 4*X(3628)-X(4996) = 4*X(6668)-X(33814) = X(22938)+2*X(31659) = 3*X(38106)-X(38135) = 3*X(38106)+X(38163)

This triangle has collinear vertices.

X(38184) lies on these lines: {1,5}, {30,38106}, {517,38219}, {528,38085}, {529,38084}, {758,38182}, {2802,38183}, {2829,38142}, {3564,38199}, {3628,4996}, {5762,38218}, {5840,38114}, {5842,38141}, {5843,38209}, {5844,38215}, {5848,38169}, {5849,38168}, {5851,38174}, {5852,38173}, {5854,38178}, {5855,38177}, {5856,38181}, {5857,38180}, {6668,33814}, {22938,31659}

X(38184) = midpoint of X(38135) and X(38163)
X(38184) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38044, 38045, 38063), (38106, 38163, 38135)

X(38185) = CENTROID OF TRIANGLE {X(6), X(7), X(8)}

Barycentrics    a^5-(b+c)*a^4+5*(b+c)^2*a^3-(b+c)*(5*b^2-6*b*c+5*c^2)*a^2+2*(b^2+c^2)*(b+c)^2*a+(b^4-c^4)*(-2*b+2*c) : :
X(38185) = X(6)+2*X(2550) = 4*X(142)-X(3242) = X(390)-4*X(3589) = 5*X(3763)-8*X(3826) = 2*X(5480)+X(35514) = 2*X(38046)-3*X(38086) = 3*X(38047)-2*X(38194) = 3*X(38086)-4*X(38187) = 3*X(38087)-2*X(38190)

X(38185) lies on these lines: {6,2550}, {142,3242}, {390,3589}, {511,38121}, {516,36721}, {517,38143}, {518,599}, {519,38046}, {524,38092}, {527,38087}, {528,38048}, {952,38115}, {971,38144}, {1001,29633}, {1503,38149}, {3564,38170}, {3763,3826}, {5480,35514}, {5686,17251}, {5762,38116}, {5843,38165}, {5844,38164}, {5845,35578}, {5847,38201}, {5848,38202}, {5849,38203}, {5850,38191}, {5851,38192}, {5852,38193}, {5853,38186}, {5854,38188}, {5855,38189}, {9053,11038}

X(38185) = reflection of X(38046) in X(38187)
X(38185) = {X(38046), X(38187)}-harmonic conjugate of X(38086)

X(38186) = CENTROID OF TRIANGLE {X(6), X(7), X(9)}

Barycentrics    a^4-3*(b+c)*a^3+2*(b-c)^2*a^2-(b+c)*(b^2+c^2)*a+(b^2+c^2)*(b-c)^2 : :
X(38186) = X(6)+2*X(142) = X(7)+5*X(3618) = X(9)-4*X(3589) = 2*X(141)-5*X(20195) = 2*X(182)+X(5805) = 2*X(597)+X(6173) = 2*X(1386)+X(2550) = X(3416)-4*X(3826) = 2*X(5480)+X(5732) = 5*X(12017)+X(31671) = 2*X(18583)+X(31657) = 4*X(19130)-X(31672) = 2*X(38046)+X(38190) = X(38048)+2*X(38187) = 2*X(38086)+X(38088) = 2*X(38115)+X(38145) = X(38117)+2*X(38164)

X(38186) lies on these lines: {1,16593}, {2,210}, {6,142}, {7,3618}, {9,3589}, {141,16832}, {144,17383}, {182,5805}, {511,38122}, {516,5085}, {524,38093}, {527,38086}, {528,38023}, {597,4795}, {673,1492}, {971,14561}, {1001,17023}, {1086,36404}, {1386,2550}, {1503,38150}, {2801,38197}, {2886,24600}, {2999,10383}, {3242,29571}, {3243,17284}, {3306,26007}, {3416,3826}, {3564,38171}, {3751,31183}, {5050,38107}, {5480,5732}, {5542,31191}, {5762,38110}, {5843,38166}, {5846,16833}, {5847,38204}, {5848,38205}, {5849,38206}, {5850,38194}, {5851,38195}, {5852,38196}, {5853,38185}, {5856,38188}, {5857,38189}, {6600,21526}, {8732,24471}, {9053,29573}, {11019,30825}, {11025,28757}, {11376,26964}, {12017,31671}, {14848,38065}, {14853,21151}, {15570,17316}, {16475,38052}, {16972,17366}, {16973,17245}, {17244,32029}, {17322,18230}, {17609,28740}, {18583,31657}, {19130,31672}, {19512,20330}, {22769,37272}, {24393,29604}

X(38186) = midpoint of X(i) and X(j) for these {i,j}: {5050, 38107}, {5085, 38143}, {14561, 38115}, {14848, 38065}, {14853, 21151}, {16475, 38052}, {38046, 38047}, {38049, 38187}, {38110, 38164}
X(38186) = reflection of X(i) in X(j) for these (i,j): (38048, 38049), (38117, 38110), (38145, 14561), (38190, 38047)

X(38187) = CENTROID OF TRIANGLE {X(6), X(7), X(10)}

Barycentrics    3*(b+c)*a^4+4*(b^2+5*b*c+c^2)*a^3-2*(2*b-c)*(b-2*c)*(b+c)*a^2+8*b*c*(b^2+c^2)*a+(b^4-c^4)*(-3*b+3*c) : :
X(38187) = 5*X(3618)+X(4312) = X(38046)-3*X(38086) = 2*X(38048)-3*X(38049) = X(38048)-3*X(38186) = 3*X(38086)+X(38185) = 3*X(38089)-2*X(38194)

X(38187) lies on these lines: {10,141}, {511,38123}, {515,38115}, {516,5085}, {517,38164}, {519,38046}, {524,38094}, {527,38089}, {758,38189}, {971,38146}, {1503,38151}, {1738,4307}, {2261,5819}, {2802,38188}, {3564,38172}, {3618,4312}, {5762,38118}, {5843,38167}, {5846,38201}, {5847,16833}, {5848,38207}, {5849,38208}, {5850,38047}, {5851,38197}, {5852,38198}

X(38187) = midpoint of X(38046) and X(38185)
X(38187) = reflection of X(38049) in X(38186)
X(38187) = {X(38086), X(38185)}-harmonic conjugate of X(38046)

X(38188) = CENTROID OF TRIANGLE {X(6), X(7), X(11)}

Barycentrics    4*(b+c)*a^6-(5*b^2+2*b*c+5*c^2)*a^5-5*(b^2-c^2)*(b-c)*a^4+4*(2*b^4+2*c^4-b*c*(4*b^2-5*b*c+4*c^2))*a^3-2*(b^2-c^2)*(b-c)*(b-2*c)*(2*b-c)*a^2+(b^2+c^2)*(5*b^2-4*b*c+5*c^2)*(b-c)^2*a-3*(b^4-c^4)*(b-c)^3 : :
X(38188) = 4*X(3589)-X(6068) = 3*X(38090)-2*X(38195)

X(38188) lies on these lines: {511,38124}, {516,38050}, {518,1737}, {524,38095}, {527,38090}, {528,38086}, {952,38046}, {971,38147}, {1503,38152}, {2801,38146}, {2802,38187}, {2829,38143}, {3564,38173}, {3589,6068}, {5762,38119}, {5840,38115}, {5843,38168}, {5846,38202}, {5847,38207}, {5849,38209}, {5850,38197}, {5852,38199}, {5854,38185}, {5856,38186}

X(38188) = {X(38046), X(38164)}-harmonic conjugate of X(38189)

X(38189) = CENTROID OF TRIANGLE {X(6), X(7), X(12)}

Barycentrics    4*(b+c)*a^7-(b^2-10*b*c+c^2)*a^6-2*(b+c)*(5*b^2+4*b*c+5*c^2)*a^5+(3*b^4+3*c^4-2*b*c*(6*b^2+5*b*c+6*c^2))*a^4+2*(b+c)*(2*b^4+2*c^4-b*c*(b^2+12*b*c+c^2))*a^3+(b^4+c^4+2*b*c*(3*b^2+11*b*c+3*c^2))*(b-c)^2*a^2+2*(b^4-c^4)*(b-c)*(b^2+3*b*c+c^2)*a+(b^2-c^2)*(b-c)^2*(-3*b^4+3*c^4) : :
X(38189) = 3*X(38091)-2*X(38196)

X(38189) lies on these lines: {511,38125}, {516,38051}, {518,38056}, {524,38096}, {527,38091}, {529,38086}, {758,38187}, {952,38046}, {971,38148}, {1503,38153}, {3564,38174}, {5762,38120}, {5842,38143}, {5843,38169}, {5846,38203}, {5847,38208}, {5848,38209}, {5850,38198}, {5851,38199}, {5855,38185}, {5857,38186}

X(38189) = {X(38046), X(38164)}-harmonic conjugate of X(38188)

X(38190) = CENTROID OF TRIANGLE {X(6), X(8), X(9)}

Barycentrics    a^5+2*(b+c)*a^4-5*(b+c)^2*a^3+(b+c)*(5*b^2-4*b*c+5*c^2)*a^2-4*(b^2+c^2)*(b^2+b*c+c^2)*a+(b^4-c^4)*(b-c) : :
X(38190) = X(6)+2*X(24393) = X(3242)-4*X(6666) = X(3243)-4*X(3589) = X(38046)-3*X(38047) = 2*X(38046)-3*X(38186) = 2*X(38048)-3*X(38088) = 3*X(38087)-X(38185) = 3*X(38088)-4*X(38194)

X(38190) lies on these lines: {2,210}, {6,24393}, {511,38126}, {516,38144}, {517,38145}, {519,38048}, {524,38097}, {527,38087}, {952,38117}, {971,38116}, {1001,3717}, {1503,38154}, {2550,5772}, {3242,6666}, {3243,3589}, {3564,38175}, {5223,32784}, {5762,38165}, {5844,38166}, {5845,38200}, {5847,38210}, {5848,38211}, {5849,38212}, {5853,17281}, {5854,38195}, {5855,38196}, {5856,38192}, {5857,38193}, {8236,17264}

X(38190) = reflection of X(i) in X(j) for these (i,j): (38048, 38194), (38186, 38047)
X(38190) = {X(38048), X(38194)}-harmonic conjugate of X(38088)

X(38191) = CENTROID OF TRIANGLE {X(6), X(8), X(10)}

Barycentrics    5*(b+c)*a^2-2*(b^2+c^2)*a+3*(b+c)*(b^2+c^2) : :
X(38191) = X(6)+2*X(3626) = 5*X(10)-2*X(141) = 7*X(10)-4*X(3844) = 7*X(141)-10*X(3844) = 2*X(597)+X(34641) = 2*X(1386)+X(3625) = X(1992)+5*X(3679) = X(3242)-4*X(3634) = 3*X(38023)-7*X(38047) = 6*X(38023)-7*X(38049) = X(38023)-7*X(38087) = 4*X(38023)-7*X(38089) = X(38047)-3*X(38087) = 4*X(38047)-3*X(38089) = X(38049)-6*X(38087) = 2*X(38049)-3*X(38089) = 4*X(38087)-X(38089) = 4*X(38136)-5*X(38146) = X(38136)-5*X(38165) = X(38146)-4*X(38165)

X(38191) lies on these lines: {6,3626}, {8,16475}, {10,141}, {511,38127}, {515,38116}, {516,38144}, {517,38136}, {519,38023}, {524,38098}, {551,9053}, {597,34641}, {758,38193}, {952,38118}, {1386,3625}, {1503,38155}, {1992,3679}, {2802,38192}, {3242,3634}, {3244,3589}, {3416,4691}, {3564,38176}, {3617,3751}, {3618,3632}, {3717,29659}, {3753,34378}, {3755,28522}, {3918,24476}, {3919,9021}, {4060,16972}, {4078,33165}, {4085,28516}, {4104,29667}, {4356,4439}, {4669,5846}, {4745,15533}, {4899,32784}, {5085,28236}, {5844,38167}, {5845,38201}, {5848,38213}, {5849,38214}, {5850,38185}, {5853,38194}, {5854,38197}, {5855,38198}, {14561,28234}, {25406,37712}, {32941,38048}, {33163,35258}

X(38191) = midpoint of X(i) and X(j) for these {i,j}: {8, 16475}, {25406, 37712}
X(38191) = reflection of X(38049) in X(38047)
X(38191) = {X(38047), X(38049)}-harmonic conjugate of X(38089)

X(38192) = CENTROID OF TRIANGLE {X(6), X(8), X(11)}

Barycentrics    4*(b+c)*a^5-(7*b^2+6*b*c+7*c^2)*a^4+12*b*c*(b+c)*a^3+4*(b^4+c^4-b*c*(2*b^2+3*b*c+2*c^2))*a^2-2*(2*b-c)*(b-2*c)*(b+c)*(b^2+c^2)*a+3*(b^4-c^4)*(b^2-c^2) : :
X(38192) = X(6)+2*X(3036) = X(1317)-4*X(3589) = X(3242)-4*X(6667) = 5*X(3617)+X(10755) = 5*X(3618)+X(12531) = 2*X(38050)-3*X(38090) = 3*X(38090)-4*X(38197)

X(38192) lies on these lines: {6,3036}, {11,32931}, {511,38128}, {517,38147}, {518,1737}, {519,38050}, {524,38099}, {528,38087}, {952,38029}, {1317,3589}, {1503,38156}, {2802,38191}, {2829,38144}, {3242,6667}, {3564,38177}, {3617,10755}, {3618,12531}, {5840,38116}, {5844,38168}, {5845,38202}, {5847,38213}, {5849,38215}, {5851,38185}, {5853,38195}, {5855,38199}, {5856,38190}

X(38192) = reflection of X(38050) in X(38197)
X(38192) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38047, 38165, 38193), (38050, 38197, 38090)

X(38193) = CENTROID OF TRIANGLE {X(6), X(8), X(12)}

Barycentrics    4*(b+c)*a^5-(7*b^2+10*b*c+7*c^2)*a^4+4*b*c*(b+c)*a^3+4*(b^4+c^4-b*c*(2*b^2+3*b*c+2*c^2))*a^2-2*(b+c)*(2*b^2-3*b*c+2*c^2)*(b^2+c^2)*a+3*(b^4-c^4)*(b^2-c^2) : :
X(38193) = X(3242)-4*X(6668) = 4*X(3589)-X(37734) = 2*X(38051)-3*X(38091) = 3*X(38091)-4*X(38198)

X(38193) lies on these lines: {12,28109}, {511,38129}, {517,38148}, {518,38056}, {519,38051}, {524,38100}, {529,38087}, {758,38191}, {952,38029}, {1503,38157}, {3242,6668}, {3564,38178}, {3589,37734}, {5842,38144}, {5844,38169}, {5845,38203}, {5847,38214}, {5848,38215}, {5852,38185}, {5853,38196}, {5854,38199}, {5857,38190}

X(38193) = reflection of X(38051) in X(38198)
X(38193) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38047, 38165, 38192), (38051, 38198, 38091)

X(38194) = CENTROID OF TRIANGLE {X(6), X(9), X(10)}

Barycentrics    4*a^5-(b+c)*a^4-4*(b^2+5*b*c+c^2)*a^3+2*(b+c)*(2*b^2-3*b*c+2*c^2)*a^2-4*(b^2+c^2)*(b+c)^2*a+(b^4-c^4)*(b-c) : :
X(38194) = 4*X(3589)-X(5542) = 5*X(3618)+X(5223) = X(3751)+5*X(18230) = 3*X(38047)-X(38185) = X(38048)-3*X(38088) = 3*X(38088)+X(38190) = 3*X(38089)-X(38187)

X(38194) lies on these lines: {511,38130}, {515,38117}, {516,36721}, {517,38166}, {518,551}, {519,38048}, {524,38101}, {527,38089}, {758,38196}, {971,38118}, {1503,38158}, {2330,10392}, {2802,38195}, {3564,38179}, {3589,5542}, {3618,5223}, {3751,18230}, {5686,16475}, {5762,38167}, {5845,38204}, {5846,38210}, {5847,37654}, {5848,38216}, {5849,38217}, {5850,38186}, {5853,38191}, {5856,38197}, {5857,38198}

X(38194) = midpoint of X(i) and X(j) for these {i,j}: {5686, 16475}, {38048, 38190}
X(38194) = {X(38088), X(38190)}-harmonic conjugate of X(38048)

X(38195) = CENTROID OF TRIANGLE {X(6), X(9), X(11)}

Barycentrics    4*a^7-10*(b+c)*a^6+(5*b^2+22*b*c+5*c^2)*a^5+7*(b+c)*(b^2-4*b*c+c^2)*a^4-2*(3*b^2-8*b*c+3*c^2)*(2*b^2+b*c+2*c^2)*a^3+2*(b+c)*(5*b^4+5*c^4-2*b*c*(7*b^2-8*b*c+7*c^2))*a^2-(b^2+c^2)*(5*b^2+6*b*c+5*c^2)*(b-c)^2*a+(b^4-c^4)*(b-c)^3 : :
X(38195) = X(1156)+5*X(3618) = 4*X(3589)-X(10427) = X(10755)+5*X(18230) = 3*X(38090)-X(38188)

X(38195) lies on these lines: {511,38131}, {516,38147}, {518,38050}, {524,38102}, {527,38090}, {528,38047}, {952,38048}, {971,38119}, {1156,3618}, {1503,38159}, {2801,38049}, {2802,38194}, {2829,38145}, {3564,38180}, {3589,10427}, {5762,38168}, {5840,38117}, {5845,38205}, {5846,38211}, {5847,38216}, {5849,38218}, {5851,38186}, {5853,38192}, {5854,38190}, {5857,38199}, {10755,18230}

X(38195) = {X(38048), X(38166)}-harmonic conjugate of X(38196)

X(38196) = CENTROID OF TRIANGLE {X(6), X(9), X(12)}

Barycentrics    4*a^8-6*(b+c)*a^7-5*(b+c)^2*a^6+6*(b+c)*(2*b^2+b*c+2*c^2)*a^5-(b^2+c^2)*(5*b^2-8*b*c+5*c^2)*a^4-2*(b+c)*(b^4+c^4-b*c*(b^2+16*b*c+c^2))*a^3+(5*b^4+5*c^4-2*b*c*(7*b^2-11*b*c+7*c^2))*(b+c)^2*a^2-4*(b^4-c^4)*(b^2-c^2)*(b+c)*a+(b^4-c^4)*(b^2-c^2)*(b-c)^2 : :
X(38196) = 3*X(38091)-X(38189)

X(38196) lies on these lines: {511,38132}, {516,38148}, {518,38051}, {524,38103}, {527,38091}, {529,38088}, {758,38194}, {952,38048}, {971,38120}, {1503,38160}, {3564,38181}, {5762,38169}, {5842,38145}, {5845,38206}, {5846,38212}, {5847,38217}, {5848,38218}, {5852,38186}, {5853,38193}, {5855,38190}, {5856,38199}

X(38196) = {X(38048), X(38166)}-harmonic conjugate of X(38195)

X(38197) = CENTROID OF TRIANGLE {X(6), X(10), X(11)}

Barycentrics    2*a^6+(b+c)*a^5-2*(3*b^2-b*c+3*c^2)*a^4+5*b*c*(b+c)*a^3+2*((b^2-c^2)^2-4*b^2*c^2)*a^2-(b+c)*(b^2+c^2)*(b^2-3*b*c+c^2)*a+2*(b^4-c^4)*(b^2-c^2) : :
X(38197) = X(6)+2*X(6702) = X(80)+5*X(3618) = 2*X(182)+X(6246) = X(214)-4*X(3589) = 2*X(1386)+X(15863) = 5*X(1698)+X(10755) = X(3751)+5*X(31272) = X(12619)+2*X(18583) = X(38050)-3*X(38090) = 3*X(38090)+X(38192)

X(38197) lies on these lines: {6,6702}, {80,3618}, {182,6246}, {214,3589}, {511,38133}, {515,38119}, {516,38147}, {517,38168}, {518,32557}, {519,38050}, {524,38104}, {528,38089}, {758,38199}, {952,38049}, {1386,15863}, {1503,38161}, {1698,10755}, {2800,14561}, {2801,38186}, {2802,38047}, {2829,38146}, {3564,38182}, {3751,31272}, {5840,38118}, {5845,38207}, {5846,38213}, {5847,34122}, {5849,38219}, {5850,38188}, {5851,38187}, {5854,38191}, {5856,38194}, {12619,18583}

X(38197) = midpoint of X(38050) and X(38192)
X(38197) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38049, 38167, 38198), (38090, 38192, 38050)

X(38198) = CENTROID OF TRIANGLE {X(6), X(10), X(12)}

Barycentrics    2*a^6+(b+c)*a^5-6*(b^2+b*c+c^2)*a^4-b*c*(b+c)*a^3+2*(b^2-4*b*c+c^2)*(b+c)^2*a^2-(b^3+c^3)*(b^2+c^2)*a+2*(b^4-c^4)*(b^2-c^2) : :
X(38198) = 5*X(3618)+X(37710) = X(38051)-3*X(38091) = 3*X(38091)+X(38193)

X(38198) lies on these lines: {511,38134}, {515,38120}, {516,38148}, {517,38169}, {518,38062}, {519,38051}, {524,38105}, {529,38089}, {758,38047}, {952,38049}, {1503,38162}, {2802,38199}, {3564,38183}, {3618,37710}, {5842,38146}, {5845,38208}, {5846,38214}, {5847,38058}, {5848,38219}, {5850,38189}, {5852,38187}, {5855,38191}, {5857,38194}

X(38198) = midpoint of X(38051) and X(38193)
X(38198) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38049, 38167, 38197), (38091, 38193, 38051)

X(38199) = CENTROID OF TRIANGLE {X(6), X(11), X(12)}

Barycentrics    a^9-(b+c)*a^8-2*(2*b^2-b*c+2*c^2)*a^7+2*(b+c)*(2*b^2-b*c+2*c^2)*a^6+(4*b^4+4*c^4-3*(2*b^2+b*c+2*c^2)*b*c)*a^5-(b+c)*(4*b^4+4*c^4-(6*b^2-b*c+6*c^2)*b*c)*a^4+2*(b^4+c^4+(b^2-6*b*c+c^2)*b*c)*b*c*a^3-2*(b^2-c^2)*(b-c)*b*c*(b^2+4*b*c+c^2)*a^2-(b^4-c^4)*(b^2-c^2)*(b-c)^2*a+(b^4-c^4)*(b^2-c^2)^2*(b-c) : :
X(38199) = X(6)+2*X(8068) = 4*X(3589)-X(4996)

X(38199) lies on these lines: {6,8068}, {511,38135}, {518,38063}, {524,38106}, {528,38091}, {529,38090}, {758,38197}, {952,14561}, {1503,38163}, {2802,38198}, {2829,38148}, {3564,38184}, {3589,4996}, {5840,38120}, {5842,38147}, {5845,38209}, {5846,38215}, {5847,38219}, {5851,38189}, {5852,38188}, {5854,38193}, {5855,38192}, {5856,38196}, {5857,38195}

X(38200) = CENTROID OF TRIANGLE {X(7), X(8), X(9)}

Barycentrics    a^3-2*(b+c)*a^2+3*(b+c)^2*a-2*(b^2-c^2)*(b-c) : :
X(38200) = X(1)-4*X(3826) = 2*X(1)-5*X(20195) = X(7)+5*X(3617) = X(7)+2*X(24393) = X(8)+2*X(142) = 2*X(8)+X(3243) = X(9)-4*X(10) = X(9)+2*X(2550) = 5*X(9)-2*X(5698) = X(9)+4*X(38201) = 2*X(10)+X(2550) = 10*X(10)-X(5698) = 4*X(142)-X(3243) = 5*X(2550)+X(5698) = 5*X(3617)-2*X(24393) = 8*X(3826)-5*X(20195) = X(5698)-5*X(38057) = X(5698)+10*X(38201) = 5*X(5818)+X(35514) = X(38057)+2*X(38201)

X(38200) lies on these lines: {1,3826}, {2,3158}, {4,9}, {7,3617}, {8,142}, {12,12560}, {57,25006}, {75,4901}, {80,5528}, {200,3925}, {210,28609}, {355,5732}, {390,6666}, {443,6762}, {517,38150}, {518,599}, {519,38053}, {527,5686}, {528,19875}, {612,33128}, {673,17308}, {936,5886}, {952,9623}, {954,31434}, {971,5790}, {1001,1698}, {1145,3254}, {1279,31183}, {1376,15931}, {1699,3740}, {1738,7174}, {1788,12573}, {1837,4326}, {2136,6601}, {2801,38213}, {2886,7988}, {2951,37714}, {2999,17726}, {3036,10427}, {3059,3698}, {3174,12625}, {3242,4859}, {3247,3755}, {3305,9580}, {3340,21617}, {3416,4034}, {3434,7308}, {3452,9779}, {3579,31446}, {3621,17312}, {3622,12630}, {3626,5542}, {3633,15570}, {3634,30331}, {3681,4654}, {3696,4007}, {3697,9612}, {3717,4659}, {3812,15185}, {3823,17284}, {3828,38059}, {3914,7322}, {3918,30329}, {3921,17532}, {3932,4873}, {4002,5728}, {4307,16670}, {4312,5220}, {4321,5252}, {4384,32850}, {4413,5231}, {4429,17306}, {4512,34612}, {4662,5290}, {4668,25557}, {4669,38054}, {4678,17287}, {4731,15733}, {4745,5850}, {4847,5437}, {4853,6067}, {4863,10582}, {4882,25466}, {4923,29616}, {5178,37723}, {5219,7679}, {5223,5852}, {5260,7676}, {5268,32865}, {5325,9778}, {5436,19855}, {5438,10165}, {5690,5805}, {5691,11495}, {5705,9709}, {5727,7675}, {5762,38112}, {5784,5833}, {5785,5832}, {5794,37712}, {5843,38175}, {5844,38171}, {5845,38190}, {5846,16833}, {5851,38211}, {5854,38205}, {5855,38206}, {5856,38202}, {5857,38203}, {6594,20119}, {6743,28629}, {6765,8728}, {7677,31231}, {7989,9711}, {7994,8226}, {8255,30286}, {8582,24389}, {8730,19520}, {8732,10106}, {9708,28160}, {10005,31995}, {10175,38037}, {10176,31162}, {10826,15254}, {11024,24391}, {11221,21020}, {11491,31423}, {11680,20196}, {12702,18482}, {13624,31494}, {15299,18395}, {16208,37828}, {16593,16832}, {17151,28472}, {17294,27475}, {18357,31672}, {18483,31420}, {20533,29576}, {21153,26446}, {24474,38107}, {24564,37556}, {24703,30393}, {28580,36911}, {28581,29573}, {30312,37709}, {38042,38108}

X(38200) = midpoint of X(i) and X(j) for these {i,j}: {8, 11038}, {10, 38201}, {2550, 38057}, {3679, 38052}, {4669, 38054}, {5657, 38149}, {5790, 38121}, {38112, 38170}
X(38200) = reflection of X(i) in X(j) for these (i,j): (9, 38057), (2550, 38201), (3243, 11038), (6173, 38052), (11038, 142), (21153, 26446), (38031, 11231), (38037, 10175), (38053, 38204), (38057, 10), (38059, 3828), (38108, 38042), (38126, 38112), (38154, 5790), (38210, 4745)
X(38200) = complement of X(8236)
X(38200) = X(8)-Beth conjugate of-X(38057)
X(38200) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3826, 20195), (7, 3617, 24393), (8, 142, 3243), (10, 2550, 9), (200, 3925, 25525), (2886, 8580, 30827), (3305, 33110, 9580), (4413, 5231, 31190), (4847, 26040, 5437), (38053, 38204, 38093)

X(38201) = CENTROID OF TRIANGLE {X(7), X(8), X(10)}

Barycentrics    4*a^3-5*(b+c)*a^2+6*(b+c)^2*a-5*(b^2-c^2)*(b-c) : :
X(38201) = X(7)+2*X(3626) = 2*X(9)-5*X(10) = X(9)+5*X(2550) = 11*X(9)-5*X(5698) = 3*X(9)-5*X(38057) = X(9)-5*X(38200) = X(10)+2*X(2550) = 11*X(10)-2*X(5698) = 3*X(10)-2*X(38057) = 4*X(142)-X(3244) = 5*X(142)-2*X(15570) = X(390)-4*X(3634) = 11*X(2550)+X(5698) = 3*X(2550)+X(38057) = 5*X(3244)-8*X(15570) = 8*X(3826)-5*X(19862) = 4*X(3826)-X(30331) = 3*X(5698)-11*X(38057) = X(5698)-11*X(38200) = 2*X(19925)+X(35514) = X(38057)-3*X(38200)

X(38201) lies on these lines: {4,9}, {7,3626}, {142,3244}, {390,3634}, {496,3826}, {515,38121}, {517,38137}, {518,3919}, {519,11038}, {527,38098}, {528,38059}, {551,5853}, {758,38203}, {952,38123}, {971,38155}, {1001,16855}, {1125,8236}, {1376,38031}, {2802,38202}, {3059,3754}, {3174,30143}, {3416,4545}, {3617,4312}, {3625,5542}, {3679,5850}, {3723,3755}, {3918,5728}, {4002,14100}, {4297,9710}, {4349,16666}, {4356,16672}, {4691,5223}, {4745,5686}, {4847,27003}, {5762,38127}, {5833,37712}, {5843,38176}, {5844,38172}, {5845,38191}, {5846,38187}, {5847,38185}, {5851,38213}, {5852,24393}, {5854,38207}, {5855,38208}, {5884,15587}, {6173,34641}, {7988,20103}, {8580,9779}, {9708,28172}, {9709,10172}, {10165,31419}, {12571,31420}, {15808,20195}, {21151,28236}, {28234,38107}

X(38201) = midpoint of X(2550) and X(38200)
X(38201) = reflection of X(i) in X(j) for these (i,j): (10, 38200), (551, 38204), (5686, 4745), (8236, 1125), (38054, 38052)
X(38201) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3826, 30331, 19862), (38052, 38054, 38094)

X(38202) = CENTROID OF TRIANGLE {X(7), X(8), X(11)}

Barycentrics    4*a^6-8*(b+c)*a^5+(b+5*c)*(5*b+c)*a^4-2*(b+c)*(b^2+8*b*c+c^2)*a^3-4*((b^2-c^2)^2-4*b^2*c^2)*a^2+10*(b^4-c^4)*(b-c)*a-5*(b^2-c^2)^2*(b-c)^2 : :
X(38202) = X(7)+2*X(3036) = 4*X(10)-X(6068) = X(11)+2*X(2550) = 4*X(142)-X(1317) = X(390)-4*X(6667) = 2*X(3035)+X(20119) = X(3059)+2*X(12736) = 2*X(3254)+X(13996) = 8*X(3826)-5*X(31235) = 3*X(34122)-2*X(38216) = 2*X(38055)-3*X(38095) = 3*X(38095)-4*X(38207) = 3*X(38099)-2*X(38211)

X(38202) lies on these lines: {2,11}, {7,3036}, {10,6068}, {142,1317}, {516,34122}, {517,38152}, {519,38055}, {527,38099}, {952,38030}, {971,38156}, {2801,3753}, {2802,38201}, {2829,38149}, {3059,12736}, {3254,13996}, {3845,11372}, {5252,6173}, {5440,38026}, {5762,38128}, {5840,38121}, {5843,38177}, {5844,38173}, {5845,38192}, {5846,38188}, {5848,38185}, {5850,38213}, {5852,38215}, {5853,38205}, {5855,38209}, {5856,38200}, {17528,37725}, {26446,38180}, {34123,38204}

X(38202) = reflection of X(i) in X(j) for these (i,j): (34123, 38204), (38055, 38207)
X(38202) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38052, 38170, 38203), (38055, 38207, 38095)

X(38203) = CENTROID OF TRIANGLE {X(7), X(8), X(12)}

Barycentrics    4*a^6-8*(b+c)*a^5+(5*b^2+22*b*c+5*c^2)*a^4-2*(b+c)*(b^2+8*b*c+c^2)*a^3-4*(b^3+c^3)*(b+c)*a^2+2*(b^2-c^2)*(b-c)*(5*b^2+4*b*c+5*c^2)*a-5*(b^2-c^2)^2*(b-c)^2 : :
X(38203) = X(12)+2*X(2550) = 4*X(142)-X(37734) = X(390)-4*X(6668) = 8*X(3826)-5*X(31260) = 2*X(38056)-3*X(38096) = 3*X(38058)-2*X(38217) = 3*X(38096)-4*X(38208) = 3*X(38100)-2*X(38212)

X(38203) lies on these lines: {12,480}, {142,37734}, {390,6668}, {516,38058}, {517,38153}, {519,38056}, {527,38100}, {528,38061}, {529,38092}, {758,38201}, {952,38030}, {971,38157}, {3826,17566}, {5762,38129}, {5842,38149}, {5843,38178}, {5844,38174}, {5845,38193}, {5846,38189}, {5849,38185}, {5850,38214}, {5851,38215}, {5853,38206}, {5854,38209}, {5857,38200}

X(38203) = reflection of X(38056) in X(38208)
X(38203) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38052, 38170, 38202), (38056, 38208, 38096)

X(38204) = CENTROID OF TRIANGLE {X(7), X(9), X(10)}

Barycentrics    (b+c)*a^2+2*(b^2+6*b*c+c^2)*a-3*(b^2-c^2)*(b-c) : :
X(38204) = 7*X(1)-X(12630) = X(7)+5*X(1698) = X(9)-4*X(3634) = 2*X(9)+X(30424) = X(10)+2*X(142) = X(10)-4*X(3826) = 2*X(10)+X(5542) = 5*X(10)-2*X(24393) = 5*X(10)+4*X(25557) = X(142)+2*X(3826) = 4*X(142)-X(5542) = 5*X(142)+X(24393) = 5*X(142)-2*X(25557) = 4*X(142)+X(38210) = 8*X(3634)+X(30424) = 4*X(3812)-X(30329) = 8*X(3826)+X(5542) = 10*X(3826)-X(24393) = 5*X(3826)+X(25557) = 4*X(3826)+X(38054) = 8*X(3826)-X(38210) = 2*X(38052)+X(38059)

X(38204) lies on these lines: {1,12630}, {2,165}, {7,1698}, {9,3634}, {10,141}, {144,19877}, {390,3624}, {392,4301}, {443,4297}, {515,38122}, {517,38171}, {519,38053}, {527,38094}, {528,19883}, {551,5853}, {740,29600}, {758,38206}, {954,4413}, {971,10175}, {1001,16408}, {1058,1125}, {1738,4356}, {1757,4896}, {2801,34122}, {2802,38205}, {2951,3091}, {3008,4349}, {3059,5439}, {3090,11372}, {3243,3626}, {3576,38149}, {3614,31391}, {3671,5692}, {3679,11038}, {3715,3982}, {3731,7613}, {3755,17245}, {3821,3986}, {3828,5850}, {3841,9843}, {3848,24386}, {3925,11019}, {3950,28522}, {4061,18139}, {4078,28516}, {4197,7705}, {4208,5732}, {4298,8732}, {4307,31183}, {4312,18230}, {4353,4859}, {5223,9780}, {5249,21060}, {5259,7676}, {5587,21151}, {5657,38036}, {5686,19875}, {5698,31253}, {5704,30330}, {5759,31423}, {5762,11231}, {5790,38030}, {5805,6684}, {5843,38179}, {5845,38194}, {5847,38186}, {5851,38216}, {5852,38217}, {5856,38207}, {5857,38208}, {5880,6666}, {5886,38121}, {6172,19876}, {6702,10427}, {7679,30379}, {7989,36991}, {8227,35514}, {8236,25055}, {9956,31657}, {10172,38108}, {10384,10589}, {10392,17606}, {11108,11495}, {11362,20330}, {12447,28629}, {12512,16845}, {12571,17559}, {12573,19854}, {13159,18253}, {13405,26040}, {15481,22266}, {16593,25351}, {16853,18483}, {16857,28150}, {17282,19868}, {17612,25973}, {18398,34784}, {18482,31730}, {20103,25525}, {20790,21625}, {24248,25072}, {26446,38107}, {27475,29594}, {31191,38048}, {34123,38202}, {35262,37462}, {38028,38170}, {38041,38112}, {38042,38111}

X(38204) = midpoint of X(i) and X(j) for these {i,j}: {2, 38052}, {10, 38054}, {551, 38201}, {3576, 38149}, {3679, 11038}, {5542, 38210}, {5587, 21151}, {5657, 38036}, {5790, 38030}, {5886, 38121}, {6173, 38057}, {10164, 38151}, {10175, 38123}, {11231, 38172}, {25055, 38092}, {26446, 38107}, {34123, 38202}, {38028, 38170}, {38041, 38112}, {38042, 38111}, {38053, 38200}
X(38204) = reflection of X(i) in X(j) for these (i,j): (5542, 38054), (38037, 10171), (38054, 142), (38057, 3828), (38059, 2), (38108, 10172), (38130, 11231), (38158, 10175), (38210, 10)
X(38204) = X(8)-Beth conjugate of-X(38210)
X(38204) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 142, 5542), (142, 3826, 10), (142, 24393, 25557), (1125, 2550, 30331), (1738, 29571, 4356), (3059, 5439, 20116), (15841, 21620, 5542), (38093, 38200, 38053)

X(38205) = CENTROID OF TRIANGLE {X(7), X(9), X(11)}

Barycentrics    2*(b+c)*a^4-(3*b^2+2*b*c+3*c^2)*a^3-(b+c)*(3*b^2-8*b*c+3*c^2)*a^2+7*(b^2+c^2)*(b-c)^2*a-3*(b^2-c^2)*(b-c)^3 : :
X(38205) = X(7)+5*X(31272) = X(9)-4*X(6667) = X(11)+2*X(142) = 2*X(11)+X(10427) = 4*X(142)-X(10427) = X(1145)-4*X(3826) = 2*X(1387)+X(2550) = 2*X(3035)+X(3254) = 2*X(3035)-5*X(20195) = 2*X(3036)+X(3243) = X(3254)+5*X(20195) = 5*X(3616)+X(20119) = 7*X(3622)-X(12730) = X(5542)+2*X(6702) = X(5805)+2*X(6713) = 2*X(38055)+X(38211) = X(38060)+2*X(38207) = 2*X(38095)+X(38102) = 2*X(38124)+X(38159) = X(38131)+2*X(38173)

X(38205) lies on these lines: {2,5856}, {7,31272}, {9,6667}, {11,142}, {516,21154}, {518,1737}, {527,38095}, {528,15015}, {952,38053}, {971,23513}, {1001,10090}, {1145,3826}, {1387,2550}, {2801,38054}, {2802,38204}, {2829,38150}, {3035,3254}, {3036,3243}, {3616,20119}, {3622,12730}, {5542,6702}, {5762,34126}, {5805,6713}, {5840,38122}, {5843,38180}, {5845,38195}, {5848,38186}, {5850,38216}, {5851,6173}, {5852,38218}, {5853,38202}, {5854,38200}, {5857,38209}, {5880,23708}, {6068,6666}, {6594,31235}, {6692,25606}, {7951,25557}, {8732,24465}, {15185,18240}, {17100,32558}, {38037,38107}

X(38205) = midpoint of X(i) and X(j) for these {i,j}: {16173, 38052}, {21154, 38152}, {23513, 38124}, {32557, 38207}, {34122, 38055}, {34126, 38173}
X(38205) = reflection of X(i) in X(j) for these (i,j): (38060, 32557), (38131, 34126), (38159, 23513), (38211, 34122)
X(38205) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11, 142, 10427), (3254, 20195, 3035), (38053, 38171, 38206)

X(38206) = CENTROID OF TRIANGLE {X(7), X(9), X(12)}

Barycentrics    2*(b+c)*a^5-(b^2-6*b*c+c^2)*a^4-6*(b+c)*(b^2+b*c+c^2)*a^3+4*(b^4+c^4-2*b*c*(b^2+c^2))*a^2+2*(b^2-c^2)*(b-c)*(2*b+c)*(b+2*c)*a-3*(b^2-c^2)^2*(b-c)^2 : :
X(38206) = X(9)-4*X(6668) = X(12)+2*X(142) = X(2550)+2*X(37737) = 2*X(4999)-5*X(20195) = X(5805)+2*X(31659) = 2*X(8068)+X(10427) = 2*X(38056)+X(38212) = X(38061)+2*X(38208) = 2*X(38096)+X(38103) = 2*X(38125)+X(38160) = X(38132)+2*X(38174)

X(38206) lies on these lines: {2,5857}, {9,6668}, {12,142}, {516,21155}, {518,38056}, {527,38096}, {528,38027}, {529,38093}, {758,38204}, {952,38053}, {971,38109}, {2550,37737}, {2801,38219}, {4999,20195}, {5762,38114}, {5805,31659}, {5842,38150}, {5843,38181}, {5845,38196}, {5849,38186}, {5850,38217}, {5851,38218}, {5852,6173}, {5853,38203}, {5855,38200}, {5856,38209}, {7679,38211}, {8068,10427}, {37701,38052}

X(38206) = midpoint of X(i) and X(j) for these {i,j}: {21155, 38153}, {37701, 38052}, {38056, 38058}, {38062, 38208}, {38109, 38125}, {38114, 38174}
X(38206) = reflection of X(i) in X(j) for these (i,j): (38061, 38062), (38132, 38114), (38160, 38109), (38212, 38058)
X(38206) = {X(38053), X(38171)}-harmonic conjugate of X(38205)

X(38207) = CENTROID OF TRIANGLE {X(7), X(10), X(11)}

Barycentrics    2*a^6-(b+c)*a^5-2*(b^2-5*b*c+c^2)*a^4-(b+c)*(4*b^2-b*c+4*c^2)*a^3+4*(b^4+c^4-3*b*c*(b-c)^2)*a^2+(b^2-c^2)*(b-c)*(5*b^2+3*b*c+5*c^2)*a-4*(b^2-c^2)^2*(b-c)^2 : :
X(38207) = X(7)+2*X(6702) = 4*X(142)-X(214) = 4*X(3634)-X(6068) = X(4312)+5*X(31272) = 2*X(5542)+X(15863) = X(6246)+2*X(31657) = X(14151)-3*X(38024) = 3*X(32557)-2*X(38060) = X(38055)-3*X(38095) = X(38060)-3*X(38205) = 3*X(38095)+X(38202) = 3*X(38104)-2*X(38216)

X(38207) lies on these lines: {7,6702}, {142,214}, {515,38124}, {516,21154}, {517,38173}, {518,38213}, {519,38055}, {527,3814}, {535,30379}, {758,38209}, {952,38054}, {971,38161}, {2800,38107}, {2801,5587}, {2802,38052}, {2829,38151}, {3634,6068}, {3679,12736}, {4312,31272}, {5542,15863}, {5762,38133}, {5840,38123}, {5843,38182}, {5845,38197}, {5847,38188}, {5848,38187}, {5850,34122}, {5852,38219}, {5854,38201}, {5856,38204}, {5880,10199}, {6246,31657}, {14151,38024}, {18240,31146}

X(38207) = midpoint of X(38055) and X(38202)
X(38207) = reflection of X(32557) in X(38205)
X(38207) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38054, 38172, 38208), (38095, 38202, 38055)

X(38208) = CENTROID OF TRIANGLE {X(7), X(10), X(12)}

Barycentrics    2*a^6-(b+c)*a^5-2*(b^2-5*b*c+c^2)*a^4-(b+c)*(4*b^2+9*b*c+4*c^2)*a^3+4*(b^4+c^4-2*b*c*(b^2+c^2))*a^2+(b^2-c^2)*(b-c)*(5*b^2+9*b*c+5*c^2)*a-4*(b^2-c^2)^2*(b-c)^2 : :
X(38208) = X(38056)-3*X(38096) = 2*X(38061)-3*X(38062) = X(38061)-3*X(38206) = 3*X(38096)+X(38203) = 3*X(38105)-2*X(38217)

X(38208) lies on these lines: {515,38125}, {516,21155}, {517,38174}, {518,38214}, {519,38056}, {527,38105}, {529,38094}, {758,38052}, {952,38054}, {971,38162}, {2802,38209}, {5762,38134}, {5842,38151}, {5843,38183}, {5845,38198}, {5847,38189}, {5849,38187}, {5850,38058}, {5851,38219}, {5855,38201}, {5857,38204}

X(38208) = midpoint of X(38056) and X(38203)
X(38208) = reflection of X(38062) in X(38206)
X(38208) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38054, 38172, 38207), (38096, 38203, 38056)

X(38209) = CENTROID OF TRIANGLE {X(7), X(11), X(12)}

Barycentrics    a^9-(b+c)*a^8-(3*b^2-4*b*c+3*c^2)*a^7+(b+c)*(b^2-4*b*c+c^2)*a^6+(7*b^4+7*c^4-b*c*(8*b^2-19*b*c+8*c^2))*a^5-(b+c)*(b^4+c^4-b*c*(8*b^2-19*b*c+8*c^2))*a^4-3*(b^3-c^3)*(b-c)*(3*b^2-b*c+3*c^2)*a^3+(b^2-c^2)*(b-c)*(3*b^4+3*c^4-b*c*(6*b^2-7*b*c+6*c^2))*a^2+4*(b^4-c^4)*(b^2-c^2)*(b-c)^2*a-2*(b^2-c^2)^3*(b-c)^3 : :
X(38209) = X(7)+2*X(8068) = 4*X(142)-X(4996) = X(6068)-4*X(6668) = 3*X(38106)-2*X(38218)

X(38209) lies on these lines: {7,8068}, {142,4996}, {516,38063}, {518,38215}, {527,38106}, {528,38096}, {529,38095}, {758,38207}, {952,1056}, {971,38163}, {2801,38162}, {2802,38208}, {2829,38153}, {5762,38135}, {5840,38125}, {5842,38152}, {5843,38184}, {5845,38199}, {5848,38189}, {5849,38188}, {5850,38219}, {5854,38203}, {5855,38202}, {5856,38206}, {5857,38205}, {6068,6668}, {8543,38045}

X(38210) = CENTROID OF TRIANGLE {X(8), X(9), X(10)}

Barycentrics    7*(b+c)*a^2-2*(5*b^2+6*b*c+5*c^2)*a+3*(b^2-c^2)*(b-c) : :
X(38210) = 2*X(8)+X(30331) = X(9)+2*X(3626) = 5*X(10)-2*X(142) = 7*X(10)-4*X(3826) = 4*X(10)-X(5542) = X(10)+2*X(24393) = 13*X(10)-4*X(25557) = 3*X(10)-X(38054) = 7*X(142)-10*X(3826) = 8*X(142)-5*X(5542) = X(142)+5*X(24393) = 13*X(142)-10*X(25557) = 6*X(142)-5*X(38054) = 4*X(142)-5*X(38204) = 16*X(3826)-7*X(5542) = 2*X(3826)+7*X(24393) = 13*X(3826)-7*X(25557) = 12*X(3826)-7*X(38054) = 8*X(3826)-7*X(38204) = X(5542)+8*X(24393) = 13*X(5542)-16*X(25557) = 3*X(5542)-4*X(38054)

X(38210) lies on these lines: {8,25101}, {9,3626}, {10,141}, {210,3817}, {390,4668}, {515,38126}, {516,3543}, {517,38139}, {519,38025}, {527,38098}, {758,38212}, {952,38130}, {971,38127}, {1001,3625}, {2550,4691}, {2802,38211}, {3243,3634}, {3244,6666}, {3617,5223}, {3632,18230}, {3828,38053}, {3947,7672}, {4301,5692}, {4669,5853}, {4677,8236}, {4745,5850}, {4816,12630}, {4866,19925}, {5762,38176}, {5844,38179}, {5846,38194}, {5847,38190}, {5854,38216}, {5855,38217}, {5856,38213}, {5857,38214}, {11038,19875}, {15570,15808}, {20330,31399}, {21060,25006}, {21153,28236}, {28234,38108}

X(38210) = midpoint of X(i) and X(j) for these {i,j}: {3679, 5686}, {4677, 8236}
X(38210) = reflection of X(i) in X(j) for these (i,j): (5542, 38204), (38053, 3828), (38059, 38057), (38200, 4745), (38204, 10)
X(38210) = X(8)-Beth conjugate of-X(38204)
X(38210) = {X(38057), X(38059)}-harmonic conjugate of X(38101)

X(38211) = CENTROID OF TRIANGLE {X(8), X(9), X(11)}

Barycentrics    (-a+b+c)*(2*(b+c)*a^4-(3*b^2+2*b*c+3*c^2)*a^3-(b+c)*(b^2-4*b*c+c^2)*a^2+(3*b^2+4*b*c+3*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)^3) : :
X(38211) = X(9)+2*X(3036) = 4*X(10)-X(10427) = X(11)+2*X(24393) = X(1156)+5*X(3617) = X(1317)-4*X(6666) = X(3243)-4*X(6667) = X(12531)+5*X(18230) = 3*X(34122)-X(38055) = 2*X(38055)-3*X(38205) = 2*X(38060)-3*X(38102) = 3*X(38099)-X(38202) = 3*X(38102)-4*X(38216)

X(38211) lies on these lines: {2,14151}, {8,4578}, {9,80}, {10,1071}, {11,210}, {100,5273}, {142,12832}, {516,38156}, {517,38159}, {518,1737}, {519,38060}, {527,38099}, {936,20418}, {952,6883}, {958,10609}, {971,38128}, {1001,12647}, {1156,3617}, {1317,6666}, {1329,15079}, {1387,3940}, {1537,18254}, {2550,5779}, {2551,12019}, {2802,38210}, {2829,38154}, {3035,5531}, {3243,6667}, {3254,4866}, {3826,38058}, {3925,12831}, {4092,6068}, {4915,5854}, {5044,37726}, {5176,37787}, {5252,8257}, {5686,5856}, {5690,5698}, {5705,20400}, {5726,6173}, {5745,6174}, {5762,38177}, {5780,19843}, {5795,6594}, {5837,13996}, {5840,38126}, {5844,38180}, {5846,38195}, {5848,38190}, {5851,38200}, {5855,38218}, {5857,38215}, {5880,10827}, {6735,15733}, {7672,31053}, {7679,38206}, {8256,37714}, {9965,24465}, {10177,31397}, {10707,18228}, {10993,31445}, {11813,38038}, {12531,18230}, {15558,24389}, {25557,37719}, {30326,34789}, {31479,38053}

X(38211) = midpoint of X(5176) and X(37787)
X(38211) = reflection of X(i) in X(j) for these (i,j): (38060, 38216), (38205, 34122)
X(38211) = complement of X(14151)
X(38211) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8580, 11219, 3035), (38057, 38175, 38212), (38060, 38216, 38102)

X(38212) = CENTROID OF TRIANGLE {X(8), X(9), X(12)}

Barycentrics    6*(b+c)*a^5-(15*b^2+22*b*c+15*c^2)*a^4+2*(b+c)*(3*b^2+7*b*c+3*c^2)*a^3+12*(b^4+c^4)*a^2-2*(b^2-c^2)*(b-c)*(6*b^2+5*b*c+6*c^2)*a+3*(b^2-c^2)^2*(b-c)^2 : :
X(38212) = X(12)+2*X(24393) = X(3243)-4*X(6668) = 4*X(6666)-X(37734) = X(38056)-3*X(38058) = 2*X(38056)-3*X(38206) = 2*X(38061)-3*X(38103) = 3*X(38100)-X(38203) = 3*X(38103)-4*X(38217)

X(38212) lies on these lines: {12,24393}, {516,38157}, {517,38160}, {518,38056}, {519,38061}, {527,38100}, {529,38097}, {758,38210}, {952,6883}, {971,38129}, {3243,6668}, {5223,5852}, {5686,5857}, {5762,38178}, {5842,38154}, {5844,38181}, {5846,38196}, {5849,38190}, {5854,38218}, {5856,38215}, {6666,37734}

X(38212) = reflection of X(i) in X(j) for these (i,j): (38061, 38217), (38206, 38058)
X(38212) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38057, 38175, 38211), (38061, 38217, 38103)

X(38213) = CENTROID OF TRIANGLE {X(8), X(10), X(11)}

Barycentrics    2*a^4-5*(b+c)*a^3+2*(b^2+5*b*c+c^2)*a^2+(b+c)*(5*b^2-13*b*c+5*c^2)*a-4*(b^2-c^2)^2 : :
X(38213) = X(8)+2*X(6702) = 5*X(8)+X(26726) = 4*X(10)-X(214) = 5*X(10)-2*X(3035) = X(10)+2*X(3036) = 2*X(10)+X(15863) = 7*X(10)-X(33337) = X(11)+2*X(3626) = 5*X(214)-8*X(3035) = X(214)+8*X(3036) = X(214)+2*X(15863) = 7*X(214)-4*X(33337) = X(3035)+5*X(3036) = 4*X(3035)+5*X(15863) = 14*X(3035)-5*X(33337) = 4*X(3036)-X(15863) = 14*X(3036)+X(33337) = 10*X(6702)-X(26726) = 7*X(15863)+2*X(33337) = 5*X(16173)-X(26726)

X(38213) lies on these lines: {8,6702}, {10,140}, {11,3626}, {80,3617}, {100,17574}, {515,38128}, {516,38156}, {517,38141}, {518,38207}, {519,32557}, {528,38098}, {758,38215}, {993,34474}, {1145,4691}, {1317,3634}, {1320,4668}, {1387,3625}, {1484,9711}, {1698,12531}, {2800,5790}, {2801,38200}, {2802,3679}, {2829,38155}, {3244,6667}, {3632,31272}, {3697,17636}, {3814,23513}, {3828,11274}, {3878,11928}, {3918,11570}, {4002,17660}, {4662,6797}, {4669,5854}, {4701,25416}, {4745,17525}, {4973,5176}, {5123,38044}, {5587,18254}, {5690,6246}, {5840,38127}, {5844,38182}, {5846,38197}, {5847,38192}, {5848,38191}, {5850,38202}, {5851,38201}, {5853,38216}, {5855,38219}, {5856,38210}, {7972,9780}, {8256,38138}, {9710,11698}, {9956,25485}, {9963,31446}, {11729,31399}, {12735,19862}, {21154,28236}, {25440,37712}, {31145,32558}, {31235,33812}

X(38213) = midpoint of X(8) and X(16173)
X(38213) = reflection of X(i) in X(j) for these (i,j): (11274, 34123), (16173, 6702), (32557, 34122), (34123, 3828)
X(38213) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 3036, 15863), (10, 15863, 214), (10, 38176, 38214), (4701, 33709, 25416), (32557, 34122, 38104)

X(38214) = CENTROID OF TRIANGLE {X(8), X(10), X(12)}

Barycentrics    2*a^4-5*(b+c)*a^3+2*(b^2+5*b*c+c^2)*a^2+(b+c)*(5*b^2-7*b*c+5*c^2)*a-4*(b^2-c^2)^2 : :
X(38214) = 5*X(10)-2*X(4999) = X(12)+2*X(3626) = X(3244)-4*X(6668) = 5*X(3617)+X(37710) = X(3625)+2*X(37737) = 4*X(3634)-X(37734) = 3*X(38027)-7*X(38058) = 6*X(38027)-7*X(38062) = X(38027)-7*X(38100) = 4*X(38027)-7*X(38105) = X(38058)-3*X(38100) = 4*X(38058)-3*X(38105) = X(38062)-6*X(38100) = 2*X(38062)-3*X(38105) = 4*X(38100)-X(38105) = 4*X(38142)-5*X(38162) = X(38142)-5*X(38178) = X(38162)-4*X(38178)

X(38214) lies on these lines: {5,15862}, {8,17057}, {10,140}, {12,3626}, {355,3647}, {515,38129}, {516,38157}, {517,38142}, {518,38208}, {519,38027}, {529,38098}, {758,3679}, {993,37712}, {2802,38215}, {3244,6668}, {3617,4293}, {3625,37737}, {3634,37734}, {3878,5587}, {3884,37718}, {4669,5855}, {5790,10176}, {5842,38155}, {5844,38183}, {5846,38198}, {5847,38193}, {5849,38191}, {5850,38203}, {5852,24393}, {5853,38217}, {5854,38219}, {5857,38210}, {21155,28236}, {25639,28234}

X(38214) = midpoint of X(8) and X(37701)
X(38214) = reflection of X(38062) in X(38058)
X(38214) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 38176, 38213), (38058, 38062, 38105)

X(38215) = CENTROID OF TRIANGLE {X(8), X(11), X(12)}

Barycentrics    a^7-3*(b+c)*a^6+2*(b^2+5*b*c+c^2)*a^5+2*(b+c)*(2*b^2-7*b*c+2*c^2)*a^4-(7*b^4+7*c^4+b*c*(2*b^2-19*b*c+2*c^2))*a^3+(b+c)*(b^4+c^4+b*c*(10*b^2-21*b*c+10*c^2))*a^2+4*(b^2-c^2)^2*(b-c)^2*a-2*(b^2-c^2)^3*(b-c) : :
X(38215) = X(8)+2*X(8068) = 4*X(10)-X(4996) = X(12)+2*X(3036) = X(1317)-4*X(6668) = 4*X(6667)-X(37734) = 2*X(38063)-3*X(38106) = 3*X(38106)-4*X(38219)

X(38215) lies on these lines: {2,952}, {8,8068}, {10,4996}, {12,3036}, {80,5248}, {517,38163}, {518,38209}, {519,38063}, {528,38100}, {529,38099}, {758,38213}, {1317,6668}, {2476,19914}, {2802,38214}, {2829,38157}, {3617,10522}, {4881,38133}, {5141,10698}, {5840,38129}, {5842,38156}, {5844,38184}, {5846,38199}, {5848,38193}, {5849,38192}, {5851,38203}, {5852,38202}, {5853,38218}, {5856,38212}, {5857,38211}, {6667,37734}, {7504,19907}, {10827,12532}, {25005,37710}

X(38215) = reflection of X(38063) in X(38219)
X(38215) = {X(38063), X(38219)}-harmonic conjugate of X(38106)

X(38216) = CENTROID OF TRIANGLE {X(9), X(10), X(11)}

Barycentrics    2*a^6-(b+c)*a^5-2*(4*b^2+b*c+4*c^2)*a^4+(b+c)*(8*b^2+b*c+8*c^2)*a^3+4*((b^2-c^2)^2-4*b^2*c^2)*a^2-(b^2-c^2)*(b-c)*(7*b^2+9*b*c+7*c^2)*a+2*(b^2-c^2)^2*(b-c)^2 : :
X(38216) = X(9)+2*X(6702) = X(80)+5*X(18230) = X(214)-4*X(6666) = 2*X(1001)+X(15863) = X(1156)+5*X(1698) = 2*X(3036)+X(30331) = 4*X(3634)-X(10427) = X(5223)+5*X(31272) = X(5542)-4*X(6667) = X(6246)+2*X(31658) = X(14151)-3*X(25055) = 2*X(18254)+X(30329) = 3*X(34122)-X(38202) = X(38060)-3*X(38102) = 3*X(38102)+X(38211) = 3*X(38104)-X(38207)

X(38216) lies on these lines: {2,2801}, {9,6702}, {10,528}, {80,18230}, {214,6666}, {515,38131}, {516,34122}, {517,38180}, {518,32557}, {519,38060}, {527,3814}, {758,38218}, {952,38059}, {971,38133}, {1001,15863}, {1156,1698}, {2800,38108}, {2802,38057}, {2829,38158}, {3036,30331}, {3305,10707}, {3634,10427}, {5223,31272}, {5542,6667}, {5686,16173}, {5762,38182}, {5840,38130}, {5847,38195}, {5848,38194}, {5850,38205}, {5851,38204}, {5853,38213}, {5854,38210}, {5857,38219}, {6246,31658}, {7705,7989}, {10199,11374}, {14151,25055}, {18254,30329}, {31160,37787}

X(38216) = midpoint of X(i) and X(j) for these {i,j}: {5686, 16173}, {31160, 37787}, {38060, 38211}
X(38216) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38059, 38179, 38217), (38102, 38211, 38060)

X(38217) = CENTROID OF TRIANGLE {X(9), X(10), X(12)}

Barycentrics    2*a^6-(b+c)*a^5-2*(4*b^2+5*b*c+4*c^2)*a^4+(b+c)*(8*b^2+7*b*c+8*c^2)*a^3+4*(b^3+c^3)*(b+c)*a^2-(b^2-c^2)*(b-c)*(7*b^2+11*b*c+7*c^2)*a+2*(b^2-c^2)^2*(b-c)^2 : :
X(38217) = X(5542)-4*X(6668) = 5*X(18230)+X(37710) = 3*X(38058)-X(38203) = X(38061)-3*X(38103) = 3*X(38103)+X(38212) = 3*X(38105)-X(38208)

X(38217) lies on these lines: {515,38132}, {516,38058}, {517,38181}, {518,38062}, {519,38061}, {527,38105}, {529,38101}, {758,38057}, {952,38059}, {971,38134}, {2802,38218}, {5542,6668}, {5686,37701}, {5762,38183}, {5842,38158}, {5847,38196}, {5849,38194}, {5850,38206}, {5852,38204}, {5853,38214}, {5855,38210}, {5856,38219}, {18230,37710}

X(38217) = midpoint of X(i) and X(j) for these {i,j}: {5686, 37701}, {38061, 38212}
X(38217) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (38059, 38179, 38216), (38103, 38212, 38061)

X(38218) = CENTROID OF TRIANGLE {X(9), X(11), X(12)}

Barycentrics    a^9-2*(b+c)*a^8-(3*b^2-4*b*c+3*c^2)*a^7+(b+c)*(9*b^2-4*b*c+9*c^2)*a^6-(b^4+c^4+3*(4*b^2+3*b*c+4*c^2)*b*c)*a^5-(b+c)*(11*b^4+11*c^4-2*(6*b^2+b*c+6*c^2)*b*c)*a^4+(7*b^6+7*c^6+2*(2*b^4-9*b^2*c^2+2*c^4)*b*c)*a^3+(b^2-c^2)*(b-c)^3*(3*b^2+8*b*c+3*c^2)*a^2-4*(b^2-c^2)^2*(b-c)*(b^3-c^3)*a+(b^2-c^2)^3*(b-c)^3 : :
X(38218) = X(9)+2*X(8068) = X(4996)-4*X(6666) = 4*X(6668)-X(10427) = 3*X(38106)-X(38209)

X(38218) lies on these lines: {9,6506}, {516,38163}, {518,38063}, {527,38106}, {528,38058}, {529,38102}, {758,38216}, {952,38043}, {971,38135}, {2801,38062}, {2802,38217}, {2829,38160}, {4996,6666}, {5762,38184}, {5840,38132}, {5842,38159}, {5848,38196}, {5849,38195}, {5851,38206}, {5852,38205}, {5853,38215}, {5854,38212}, {5855,38211}, {6668,10427}

X(38219) = CENTROID OF TRIANGLE {X(10), X(11), X(12)}

Barycentrics    (b+c)*a^6-4*(b^2+b*c+c^2)*a^5+(b+c)*(b^2+6*b*c+c^2)*a^4+4*(2*b^4+2*c^4-b*c*(b^2+3*b*c+c^2))*a^3-(b+c)*(5*b^4-9*b^2*c^2+5*c^4)*a^2-4*(b^2-c^2)^2*(b-c)^2*a+3*(b^2-c^2)^3*(b-c) : :
X(38219) = X(10)+2*X(8068) = X(12)+2*X(6702) = X(214)-4*X(6668) = 4*X(3634)-X(4996) = X(6246)+2*X(31659) = X(15863)+2*X(37737) = 5*X(31272)+X(37710) = X(38063)-3*X(38106) = 3*X(38106)+X(38215)

X(38219) lies on these lines: {10,8068}, {12,5083}, {214,6668}, {515,38135}, {516,38163}, {517,38184}, {519,38063}, {528,38105}, {529,38104}, {547,551}, {758,34122}, {2800,38109}, {2801,38206}, {2802,38058}, {2829,38162}, {3634,4996}, {5840,38134}, {5842,38161}, {5847,38199}, {5848,38198}, {5849,38197}, {5850,38209}, {5851,38208}, {5852,38207}, {5854,38214}, {5855,38213}, {5856,38217}, {5857,38216}, {6246,31659}, {10197,37718}, {11263,12619}, {15863,37737}, {31272,37710}

X(38219) = midpoint of X(38063) and X(38215)
X(38219) = {X(38106), X(38215)}-harmonic conjugate of X(38063)

X(38220) = CENTROID OF TRIANGLE {X(1), X(13), X(14)}

Barycentrics    a^5-(b^2+c^2)*a^3+(2*b^4-3*b^2*c^2+2*c^4)*a+(b^2-c^2)^2*(b+c) : :
X(38220) = X(1)+2*X(115) = X(1)-4*X(11725) = 2*X(1)+X(13178) = 4*X(2)-X(9881) = X(2)+2*X(12258) = X(4)+2*X(11710) = 4*X(5)-X(9864) = 2*X(10)+X(7983) = 2*X(10)-5*X(14061) = X(13)+2*X(11706) = X(14)+2*X(11705) = X(40)-4*X(6036) = X(98)+2*X(946) = X(99)-4*X(1125) = X(99)+2*X(11599) = X(115)+2*X(11725) = 4*X(115)-X(13178) = X(7983)+5*X(14061) = X(9881)+8*X(12258) = 8*X(11725)+X(13178)

X(38220) lies on these lines: {1,115}, {2,9881}, {4,11710}, {5,9864}, {10,7983}, {13,11706}, {14,11705}, {30,38221}, {40,6036}, {98,946}, {99,1125}, {114,8227}, {148,3616}, {214,10769}, {230,5184}, {238,5127}, {405,22514}, {474,13173}, {515,14639}, {516,34473}, {517,38224}, {518,6034}, {519,9166}, {542,16475}, {543,25055}, {551,671}, {620,3624}, {952,38229}, {985,18393}, {1319,13182}, {1385,6321}, {1386,11646}, {1698,6722}, {1699,2794}, {1702,8980}, {1703,13967}, {1916,12263}, {2023,12782}, {2643,24957}, {2646,13183}, {2782,5886}, {2784,3817}, {2787,16173}, {2795,26725}, {2796,19883}, {3023,11376}, {3027,11375}, {3029,19858}, {3086,24472}, {3120,24617}, {3576,23698}, {3679,5461}, {4297,10723}, {4368,19935}, {5182,38049}, {5587,23514}, {5603,14651}, {5988,24161}, {6033,9955}, {6055,31162}, {6669,12780}, {6670,12781}, {7970,13464}, {7982,20398}, {7988,36519}, {8983,19109}, {9478,12783}, {9624,11724}, {9830,38023}, {9860,11522}, {9875,36523}, {10053,30384}, {10069,12047}, {10165,21166}, {10722,18483}, {10768,16174}, {11230,15561}, {11602,11739}, {11603,11740}, {11606,12264}, {11707,23004}, {11708,23005}, {11735,16278}, {12042,12699}, {12184,17605}, {12188,18493}, {12261,18332}, {13180,17614}, {13211,15359}, {13605,15342}, {13971,19108}, {14844,33147}, {14971,19875}, {18481,22515}, {23944,25687}, {26446,34127}, {31274,34595}, {38028,38222}

X(38220) = midpoint of X(5603) and X(14651)
X(38220) = reflection of X(i) in X(j) for these (i,j): (5182, 38049), (5587, 23514), (15561, 11230), (19875, 14971), (21166, 10165), (26446, 34127)
X(38220) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 115, 13178), (115, 11725, 1), (148, 3616, 11711), (1125, 11599, 99), (3624, 13174, 620), (7983, 14061, 10)

X(38221) = CENTROID OF TRIANGLE {X(1), X(15), X(16)}

Barycentrics    a*(3*a^4+2*(b+c)*a^3-2*(b^2+c^2)*a^2-(b+c)*(b^2+c^2)*a+b^4-b^2*c^2+c^4) : :
X(38221) = X(1)+2*X(187) = 2*X(1)+X(5184) = X(15)+2*X(11708) = X(16)+2*X(11707) = 4*X(187)-X(5184) = 4*X(230)-X(13178) = X(316)-4*X(1125) = X(355)-4*X(14693) = X(385)+2*X(11711) = 4*X(625)-7*X(3624) = 2*X(1385)+X(2080) = 2*X(1386)+X(5104) = 4*X(2021)-X(12782) = 4*X(2030)-X(3751) = 5*X(3616)+X(14712) = X(6781)+2*X(11725) = 5*X(7987)-2*X(18860) = 5*X(8227)-2*X(13449) = X(9855)+2*X(12258) = X(9864)-4*X(37459)

X(38221) lies on these lines: {1,187}, {15,11708}, {16,11707}, {30,38220}, {214,238}, {230,13178}, {316,1125}, {355,14693}, {385,11711}, {511,3576}, {512,25569}, {515,38227}, {517,38225}, {518,1691}, {519,26613}, {625,3624}, {952,38230}, {985,37525}, {1326,4653}, {1385,2080}, {1386,5104}, {1420,5194}, {1570,9592}, {2021,12782}, {2030,3751}, {2031,9575}, {2076,11368}, {2459,35774}, {2460,35775}, {3601,5148}, {3616,14712}, {3849,25055}, {5006,5692}, {5215,19875}, {5426,30571}, {6781,11725}, {7987,18860}, {8227,13449}, {9855,12258}, {9864,37459}, {9881,27088}, {10631,10789}, {11676,11710}, {13174,32456}, {13624,35002}, {15177,32762}, {31275,34595}

X(38221) = reflection of X(19875) in X(5215)
X(38221) = {X(1), X(187)}-harmonic conjugate of X(5184)

X(38222) = CENTROID OF TRIANGLE {X(1), X(17), X(18)}

Barycentrics    11*a^5+4*(b+c)*a^4-17*(b^2+c^2)*a^3-6*(b+c)*(b^2+c^2)*a^2+3*(4*b^4-7*b^2*c^2+4*c^4)*a+5*(b^2-c^2)^2*(b+c) : :
X(38222) = X(1)+2*X(12815) = X(17)+2*X(11740) = X(18)+2*X(11739) = 4*X(6673)-X(22851) = 4*X(6674)-X(22896)

X(38222) lies on these lines: {1,12815}, {17,11740}, {18,11739}, {515,38228}, {517,38226}, {518,38232}, {519,38223}, {952,38231}, {5965,16475}, {6673,22851}, {6674,22896}, {38028,38220}

X(38223) = CENTROID OF TRIANGLE {X(2), X(17), X(18)}

Barycentrics    19*a^4-29*(b^2+c^2)*a^2+22*b^4-41*b^2*c^2+22*c^4 : :
X(38223) = X(2)+2*X(12815) = 2*X(38226)+X(38228) = X(38226)+2*X(38231) = X(38228)-4*X(38231)

X(38223) lies on these lines: {2,7765}, {30,38226}, {519,38222}, {524,38232}, {598,5055}, {7809,17004}, {9166,11539}, {14892,26613}

X(38223) = {X(38226), X(38231)}-harmonic conjugate of X(38228)

X(38224) = CENTROID OF TRIANGLE {X(3), X(13), X(14)}

Barycentrics    a^8-2*(b^2+c^2)*a^6+(3*b^4-b^2*c^2+3*c^4)*a^4-(b^2+c^2)*(3*b^4-5*b^2*c^2+3*c^4)*a^2+(b^2-c^2)^4 : :
X(38224) = 4*X(2)-X(8724) = 2*X(2)+X(11632) = 5*X(2)+X(12243) = X(3)+2*X(115) = X(3)-4*X(6036) = 2*X(3)+X(6321) = X(3)+8*X(20398) = X(115)+2*X(6036) = 4*X(115)-X(6321) = X(115)-4*X(20398) = X(8724)+2*X(11632) = 5*X(8724)+4*X(12243) = X(8724)+4*X(14651) = X(8724)-8*X(34127) = 5*X(11632)-2*X(12243) = X(11632)+4*X(34127) = X(12243)-5*X(14651) = 2*X(12243)+5*X(15561) = X(12243)+10*X(34127) = 2*X(14651)+X(15561) = X(14651)+2*X(34127) = X(15561)-4*X(34127)

Let Q be the cyclic quadrilateral ABCX(98). Taking the vertices 3 at a time yields four triangles whose nine-point centers are the vertices of a cyclic quadrilateral homothetic to Q at X(38224). (Randy Hutson, May 19, 2020)

X(38224) lies on these lines: {2,2782}, {3,115}, {4,12042}, {5,83}, {11,10053}, {12,10069}, {13,6774}, {14,6771}, {20,22515}, {30,9166}, {35,13183}, {36,13182}, {99,140}, {110,15535}, {114,1656}, {125,15545}, {147,3090}, {148,631}, {182,11646}, {230,2080}, {265,15359}, {355,11710}, {381,2794}, {498,3027}, {499,3023}, {511,6034}, {517,38220}, {542,5050}, {543,5054}, {546,10722}, {547,6054}, {549,671}, {550,10723}, {575,7603}, {597,19905}, {620,3526}, {632,23235}, {690,15061}, {1385,13178}, {1482,11725}, {1569,31455}, {1576,34989}, {2023,3095}, {2072,13557}, {2482,15694}, {2548,12829}, {2784,10175}, {2796,38068}, {3091,9862}, {3311,8980}, {3312,13967}, {3523,13172}, {3524,26614}, {3534,9880}, {3541,5186}, {3542,12131}, {3589,12177}, {3628,7859}, {3654,12258}, {3788,13108}, {3851,10991}, {3933,8781}, {4027,16921}, {4045,35464}, {4193,5985}, {4995,10070}, {5025,10104}, {5056,5984}, {5070,6721}, {5071,11177}, {5182,38110}, {5298,10054}, {5309,32447}, {5432,10086}, {5433,10089}, {5465,20126}, {5469,21156}, {5470,21157}, {5475,11842}, {5613,6669}, {5617,6670}, {5690,7983}, {5901,7970}, {5976,32832}, {5986,37990}, {6230,6289}, {6231,6290}, {6248,7886}, {6684,11599}, {6699,16278}, {6777,20416}, {6778,20415}, {7529,9861}, {7583,19055}, {7584,19056}, {7607,7771}, {7612,16041}, {7741,12185}, {7769,32448}, {7785,36864}, {7797,11272}, {7806,10796}, {7951,12184}, {8227,9860}, {8591,15702}, {8596,15721}, {8981,19109}, {9722,37893}, {9734,11648}, {9830,38064}, {9864,9956}, {10242,14041}, {10264,15342}, {10303,20094}, {10352,32992}, {10359,33002}, {10576,35825}, {10577,35824}, {10753,18583}, {10769,33814}, {10992,15720}, {11005,20304}, {11676,14693}, {12100,12117}, {12355,15693}, {13174,31423}, {13349,23005}, {13350,23004}, {13966,19108}, {14160,14537}, {14568,32515}, {14880,37446}, {15059,22265}, {15597,16508}, {15699,23234}, {15723,22247}, {18502,20576}, {26316,37348}

X(38224) = midpoint of X(i) and X(j) for these {i,j}: {2, 14651}, {671, 21166}, {5469, 21156}, {5470, 21157}, {6055, 23514}, {11632, 15561}, {14041, 21445}, {14639, 34473}, {14643, 14849}
X(38224) = reflection of X(i) in X(j) for these (i,j): (2, 34127), (381, 23514), (3524, 26614), (5055, 14971), (5182, 38110), (8724, 15561), (10242, 14041), (11632, 14651), (14639, 38229), (15561, 2), (21166, 549), (23234, 15699), (23514, 5461)
X(38224) = X(6321)-Gibert-Moses centroid
X(38224) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11632, 8724), (3, 115, 6321), (5, 98, 6033), (98, 14061, 5), (114, 6722, 1656), (114, 11623, 12188), (115, 6036, 3), (125, 18332, 15545), (125, 33511, 18332), (148, 631, 33813), (230, 15980, 2080), (381, 6055, 14830), (1656, 12188, 114), (3526, 13188, 620), (5461, 6055, 381), (6036, 20398, 115), (6722, 11623, 114), (9166, 14639, 38229), (9166, 34473, 14639), (14651, 34127, 15561), (15092, 22505, 3091), (20415, 25560, 6778), (20416, 25559, 6777), (22510, 22511, 6034)

X(38225) = CENTROID OF TRIANGLE {X(3), X(15), X(16)}

Barycentrics    a^2*(3*a^6-7*(b^2+c^2)*a^4+(6*b^4+b^2*c^2+6*c^4)*a^2-(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)) : :
X(38225) = X(3)+2*X(187) = 2*X(3)+X(2080) = 5*X(3)+X(9301) = 5*X(3)-2*X(18860) = 4*X(3)-X(35002) = X(15)+2*X(13349) = X(16)+2*X(13350) = 2*X(182)+X(5104) = 4*X(187)-X(2080) = 10*X(187)-X(9301) = 5*X(187)+X(18860) = 8*X(187)+X(35002) = 4*X(575)-X(8586) = X(576)-4*X(8590) = X(1351)-4*X(2030) = X(1691)+2*X(35375) = 5*X(1691)-2*X(35377) = 2*X(1691)+X(35383) = 4*X(2021)-X(3095) = 2*X(2076)+X(2456)

This triangle has collinear vertices.

X(38225) lies on these lines: {3,6}, {4,14693}, {30,9166}, {140,316}, {186,14565}, {230,6321}, {385,33813}, {517,38221}, {547,38226}, {625,3526}, {631,14712}, {691,7575}, {842,18571}, {843,8600}, {1003,7697}, {1385,5184}, {1503,14830}, {1656,13449}, {2070,15563}, {2482,5965}, {2782,13586}, {3517,5140}, {3552,10104}, {3564,8593}, {3849,5054}, {5055,5215}, {5191,35265}, {5640,37457}, {6033,37459}, {6036,6781}, {6090,35302}, {6671,20429}, {6672,20428}, {7607,8181}, {8598,11632}, {9155,35296}, {9751,12100}, {10242,33228}, {10359,33022}, {11676,12042}, {13188,32456}, {13196,15483}, {14041,34127}, {14561,37809}, {14651,33265}, {15561,35297}, {15694,31173}, {18502,37334}, {21166,32515}, {21292,34417}, {21843,37348}

X(38225) = midpoint of X(i) and X(j) for these {i,j}: {2076, 5085}, {13586, 21445}, {14651, 33265}
X(38225) = reflection of X(i) in X(j) for these (i,j): (2456, 5085), (5055, 5215), (5093, 1692), (10242, 33228), (14041, 34127), (15561, 35297), (38227, 38230)
X(38225) = isogonal conjugate of the antigonal conjugate of X(7607)
X(38225) = isogonal conjugate of the antitomic conjugate of X(7607)
X(38225) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(14565)}} and {{A, B, C, X(74), X(8586)}}
X(38225) = circumcircle-inverse of-X(576)
X(38225) = Schoute circle-inverse of-X(8586)
X(38225) = crossdifference of every pair of points on line {X(523), X(3054)}
X(38225) = X(i)-Hirst inverse of-X(j) for these {i,j}: {6, 576}, {576, 6}
X(38225) = X(512)-vertex conjugate of-X(576)
X(38225) = centroid of X(3)PU(2)
X(38225) = radical trace of circumcircle and 5th Lozada circle
X(38225) = circle-{{X(1687),X(1688),PU(1),PU(2)}}-inverse of X(10631)
X(38225) = X(2080)-Gibert-Moses centroid
X(38225) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 187, 2080), (3, 2080, 35002), (3, 3053, 3095), (3, 9301, 18860), (3, 11842, 574), (3, 13335, 12054), (3, 32447, 9734), (15, 16, 8586), (32, 9734, 32447), (182, 8588, 3), (187, 2021, 3053), (187, 8588, 5104), (187, 15513, 5162), (574, 10631, 5111), (575, 13349, 16), (575, 13350, 15), (1379, 1380, 576), (1687, 1688, 10631), (2459, 2460, 6), (13335, 15513, 3)

X(38226) = CENTROID OF TRIANGLE {X(3), X(17), X(18)}

Barycentrics    11*a^8-32*(b^2+c^2)*a^6+(39*b^4+17*b^2*c^2+39*c^4)*a^4-(b^2+c^2)*(23*b^4-43*b^2*c^2+23*c^4)*a^2+5*(b^2-c^2)^4 : :
X(38226) = X(3)+2*X(12815) = 4*X(6673)-X(16627) = 4*X(6674)-X(16626) = 3*X(38223)-X(38228) = 3*X(38223)-2*X(38231)

X(38226) lies on these lines: {2,34510}, {3,12815}, {30,38223}, {140,7799}, {381,37810}, {511,38232}, {517,38222}, {547,38225}, {549,671}, {599,5050}, {3398,11539}, {5054,8716}, {5092,11646}, {6673,16627}, {6674,16626}, {10124,31168}, {11171,15702}

X(38226) = reflection of X(38228) in X(38231)
X(38226) = {X(38223), X(38228)}-harmonic conjugate of X(38231)

X(38227) = CENTROID OF TRIANGLE {X(4), X(15), X(16)}

Barycentrics    a^8-4*(b^2+c^2)*a^6+(2*b^2-3*b*c+2*c^2)*(2*b^2+3*b*c+2*c^2)*a^4-2*(b^4-c^4)*(b^2-c^2)*a^2+(b^4-b^2*c^2+c^4)*(b^2-c^2)^2 : :
X(38227) = X(3)-4*X(14693) = X(4)+2*X(187) = 4*X(5)-X(316) = 2*X(5)+X(2080) = X(15)+2*X(7685) = X(16)+2*X(7684) = X(23)+2*X(16188) = X(98)-4*X(230) = X(98)+2*X(1513) = X(99)-4*X(37459) = 2*X(114)+X(385) = 2*X(115)+X(11676) = 4*X(140)-X(35002) = 2*X(230)+X(1513) = 2*X(237)+X(34175) = X(316)+2*X(2080) = X(325)-4*X(10011) = 4*X(468)-X(842) = 3*X(26613)-2*X(38225) = 3*X(26613)-4*X(38230)

X(38227) lies on these lines: {2,51}, {3,7790}, {4,187}, {5,316}, {15,7685}, {16,7684}, {22,32762}, {23,137}, {30,9166}, {32,37446}, {76,37466}, {83,20576}, {98,230}, {99,37459}, {107,468}, {114,385}, {115,11676}, {140,7859}, {182,7806}, {183,37071}, {237,34175}, {325,10011}, {381,8860}, {383,5479}, {419,35282}, {515,38221}, {524,23234}, {542,8859}, {546,38228}, {547,31168}, {576,7777}, {625,3090}, {631,7834}, {671,37461}, {691,11799}, {754,36519}, {946,5184}, {1080,5478}, {1348,1380}, {1349,1379}, {1352,17008}, {1570,7736}, {1656,3096}, {1692,7735}, {2021,3767}, {2030,6776}, {2076,9993}, {2456,7792}, {2459,6560}, {2460,6561}, {2782,14568}, {2794,21445}, {3054,5104}, {3055,7608}, {3085,5148}, {3086,5194}, {3089,5140}, {3091,13449}, {3095,7769}, {3523,7932}, {3524,5215}, {3545,3849}, {3564,6054}, {3628,7944}, {3788,12251}, {3815,5111}, {3972,37348}, {5025,5171}, {5031,15271}, {5067,31275}, {5071,31173}, {5093,11163}, {5102,11184}, {5162,7749}, {5188,7886}, {5207,34229}, {5305,32467}, {5309,7709}, {5475,10631}, {5999,6036}, {6055,29012}, {6671,14539}, {6672,14538}, {6721,7925}, {7610,10516}, {7612,14458}, {7753,22521}, {7771,37242}, {7797,13334}, {7799,15561}, {7813,20399}, {7817,21163}, {7827,11171}, {7844,8722}, {7907,9737}, {8724,11054}, {9734,33274}, {9755,35006}, {9774,25406}, {9855,9880}, {10358,16921}, {10753,15993}, {11668,14484}, {11669,14494}, {11674,31850}, {11675,18322}, {11811,37353}, {12117,27088}, {13172,32456}, {13586,23698}, {13862,17004}, {14041,23514}, {14061,15980}, {14236,26331}, {14240,26330}, {15514,31489}, {16984,37455}, {17006,19130}, {21166,35297}, {21167,37450}

X(38227) = reflection of X(i) in X(j) for these (i,j): (2456, 38110), (3524, 5215), (7799, 15561), (14041, 23514), (14912, 1692), (21166, 35297), (38225, 38230)
X(38227) = orthoptic circle of Steiner inellipse-inverse of-X(51)
X(38227) = crosspoint of X(98) and X(7608)
X(38227) = crosssum of X(511) and X(575)
X(38227) = centroid of X(4)PU(2)
X(38227) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9752, 9753), (2, 9753, 262), (5, 2080, 316), (230, 1513, 98), (3091, 14712, 13449), (9752, 9754, 262), (9753, 9754, 2), (38225, 38230, 26613)

X(38228) = CENTROID OF TRIANGLE {X(4), X(17), X(18)}

Barycentrics    a^8+(b^2+c^2)*a^6-(7*b^4+b^2*c^2+7*c^4)*a^4+9*(b^4-c^4)*(b^2-c^2)*a^2-(4*b^4-7*b^2*c^2+4*c^4)*(b^2-c^2)^2 : :
X(38228) = X(4)+2*X(12815) = X(17)+2*X(22831) = X(18)+2*X(22832) = 4*X(6673)-X(22843) = 4*X(6674)-X(22890) = 3*X(38223)-2*X(38226) = 3*X(38223)-4*X(38231)

X(38228) lies on these lines: {4,5206}, {5,99}, {17,22831}, {18,22832}, {30,38223}, {115,32467}, {193,576}, {262,3851}, {381,10104}, {515,38222}, {546,38227}, {1503,38232}, {3398,9166}, {3545,18546}, {3850,6287}, {3855,9753}, {5066,11054}, {5182,33013}, {6673,22843}, {6674,22890}, {8550,11646}, {9737,33011}, {12122,15980}, {14045,15819}, {16044,23514}, {18424,37446}, {21166,32967}

X(38228) = reflection of X(38226) in X(38231)
X(38228) = {X(38226), X(38231)}-harmonic conjugate of X(38223)

X(38229) = CENTROID OF TRIANGLE {X(5), X(13), X(14)}

Barycentrics    (b^2+c^2)*a^6-4*(b^4-b^2*c^2+c^4)*a^4+(b^2+c^2)*(6*b^4-11*b^2*c^2+6*c^4)*a^2-(3*b^4-5*b^2*c^2+3*c^4)*(b^2-c^2)^2 : :
X(38229) = 5*X(2)+X(12355) = 5*X(5)-2*X(114) = X(5)+2*X(115) = 11*X(5)-2*X(14981) = 5*X(5)-8*X(15092) = 3*X(5)-2*X(36519) = X(114)+5*X(115) = 11*X(114)-5*X(14981) = X(114)-4*X(15092) = X(114)-5*X(23514) = 3*X(114)-5*X(36519) = 11*X(115)+X(14981) = 5*X(115)+4*X(15092) = 3*X(115)+X(36519) = 4*X(2023)-X(32448) = X(14981)-11*X(23514) = 3*X(14981)-11*X(36519) = 4*X(15092)-5*X(23514) = 12*X(15092)-5*X(36519) = 3*X(23514)-X(36519)

X(38229) lies on these lines: {2,12355}, {5,39}, {13,20253}, {14,20252}, {30,9166}, {98,546}, {99,3628}, {140,6321}, {147,3851}, {148,1656}, {381,9755}, {542,38071}, {543,15699}, {547,671}, {548,10723}, {549,5461}, {550,6036}, {632,6722}, {952,38220}, {1483,11725}, {2784,38140}, {2794,3845}, {2796,38083}, {3023,10593}, {3027,10592}, {3090,13188}, {3091,12188}, {3526,13172}, {3564,6034}, {3627,12042}, {3843,9862}, {3850,6033}, {3855,5984}, {3858,11623}, {3861,10722}, {5066,11632}, {5067,20094}, {5469,16267}, {5470,16268}, {5901,13178}, {6054,11737}, {6055,15687}, {7516,13175}, {8591,15703}, {8703,9880}, {8724,10109}, {9478,22712}, {9830,38079}, {9956,11599}, {10113,33511}, {10264,15359}, {10280,13187}, {10796,18424}, {11539,14971}, {11646,18583}, {11801,18332}, {11812,12117}, {12243,19709}, {12812,23235}, {13182,15325}, {13925,19109}, {13993,19108}, {14830,14893}, {16278,20304}, {32515,33228}

X(38229) = midpoint of X(i) and X(j) for these {i,j}: {115, 23514}, {381, 14651}, {671, 15561}, {6321, 21166}, {14639, 38224}
X(38229) = reflection of X(i) in X(j) for these (i,j): (5, 23514), (549, 34127), (11539, 14971), (15561, 547), (21166, 140), (34127, 5461)
X(38229) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (114, 15092, 5), (6036, 22515, 550), (6321, 14061, 140), (6722, 33813, 632), (9166, 14639, 38224)

X(38230) = CENTROID OF TRIANGLE {X(5), X(15), X(16)}

Barycentrics    4*a^8-11*(b^2+c^2)*a^6+10*(b^4+c^4)*a^4-(b^2+c^2)*(4*b^4-7*b^2*c^2+4*c^4)*a^2+(b^4-b^2*c^2+c^4)*(b^2-c^2)^2 : :
X(38230) = X(5)+2*X(187) = 5*X(5)-2*X(13449) = X(5)-4*X(14693) = 2*X(140)+X(2080) = 5*X(187)+X(13449) = X(187)+2*X(14693) = X(316)-4*X(3628) = 5*X(631)+X(9301) = X(691)+2*X(25338) = X(842)-4*X(22249) = X(1353)-4*X(2030) = 5*X(1656)+X(14712) = 4*X(2021)-X(32448) = 4*X(3530)-X(35002) = X(5104)+2*X(18583) = X(5162)+2*X(20576) = X(5184)+2*X(5901) = X(13449)-10*X(14693) = 3*X(26613)-X(38225) = 3*X(26613)+X(38227)

X(38230) lies on these lines: {3,7797}, {5,187}, {30,9166}, {83,140}, {316,3628}, {511,549}, {631,9301}, {691,25338}, {842,22249}, {952,38221}, {1353,2030}, {1656,14712}, {1691,3564}, {2021,32448}, {2076,14561}, {3530,35002}, {3815,10631}, {3849,15699}, {5066,38231}, {5104,18583}, {5162,20576}, {5184,5901}, {5215,11539}, {5961,7575}, {7502,32762}, {7622,15520}, {7807,10333}, {8550,8590}, {8587,37461}, {10616,36759}, {10617,36760}, {12042,29012}, {15712,18860}, {29181,35375}, {32151,37466}, {32447,33274}, {32515,35297}

X(38230) = midpoint of X(i) and X(j) for these {i,j}: {2076, 14561}, {38225, 38227}
X(38230) = reflection of X(11539) in X(5215)
X(38230) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (187, 14693, 5), (26613, 38227, 38225)

X(38231) = CENTROID OF TRIANGLE {X(5), X(17), X(18)}

Barycentrics    8*a^8-35*(b^2+c^2)*a^6+20*(3*b^4+b^2*c^2+3*c^4)*a^4-(b^2+c^2)*(50*b^4-97*b^2*c^2+50*c^4)*a^2+(17*b^4-31*b^2*c^2+17*c^4)*(b^2-c^2)^2 : :
X(38231) = X(5)+2*X(12815) = 3*X(38223)-X(38226) = 3*X(38223)+X(38228)

X(38231) lies on these lines: {2,12355}, {5,12815}, {30,38223}, {547,14568}, {952,38222}, {3564,38232}, {5066,38230}, {5965,8584}, {10109,12156}

X(38231) = midpoint of X(38226) and X(38228)
X(38231) = {X(38223), X(38228)}-harmonic conjugate of X(38226)

X(38232) = CENTROID OF TRIANGLE {X(6), X(17), X(18)}

Barycentrics    11*a^6-13*(b^2+c^2)*a^4+3*(2*b^4-11*b^2*c^2+2*c^4)*a^2+5*(b^4-c^4)*(b^2-c^2) : :
X(38232) = X(6)+2*X(12815)

This triangle has collinear vertices.

X(38232) lies on these lines: {6,17}, {511,38226}, {518,38222}, {524,38223}, {1503,38228}, {3564,38231}, {6034,38110}

X(38233) = X(112)X(23616)∩X(127)X(38240)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 + c^2)*(-(a^2*(a^2 - b^2)^2*(a^2 + b^2 - c^2)^2*(-a^2 + c^2)^2*(a^2 - b^2 + c^2)^2) + b^2*(a^2 - b^2)^2*(b^2 - c^2)^2*(a^2 + b^2 - c^2)^2*(-a^2 + b^2 + c^2)^2 - (a^2 - b^2)*(a^2 - b^2 - c^2)*(b^2 - c^2)^2*(a^2 + b^2 - c^2)*(-a^2 + c^2)*(a^2 - b^2 + c^2)*(-a^2 + b^2 + c^2)^2 + c^2*(b^2 - c^2)^2*(-a^2 + c^2)^2*(a^2 - b^2 + c^2)^2*(-a^2 + b^2 + c^2)^2) : :
Barycentrics    (a^18-2*(b^2+c^2)*a^16-2*(b^4-5*b^2*c^2+c^4)*a^14+(b^2+c^2)*(7*b^4-16*b^2*c^2+7*c^4)*a^12-(4*b^8+4*c^8+9*(b^4-3*b^2*c^2+c^4)*b^2*c^2)*a^10-(b^4-c^4)*(b^2-c^2)*(b^4-12*b^2*c^2+c^4)*a^8+2*(b^2-c^2)^2*(b^8-5*b^4*c^4+c^8)*a^6-(b^4-c^4)*(b^2-c^2)^3*(3*b^4+4*b^2*c^2+3*c^4)*a^4+(b^2-c^2)^4*(3*b^8+3*c^8+b^2*c^2*(7*b^4+9*b^2*c^2+7*c^4))*a^2-(b^4-c^4)^3*(b^2-c^2)*(b^4+c^4))*(a^2-b^2-c^2)*(b^2-c^2) : :
X(38233) = X(112)+3*X(23616), X(127)-3*X(38240), X(13219)-9*X(34767)

X(38233) is the focus of the circumparabola given by

(-a^2 + b^2 + c^2)^2*(b^2 - c^2)^2*y*z + (-b^2 + c^2 + a^2)^2*(c^2 - a^2)^2*z*x + (-c^2 + a^2 + b^2)^2*(a^2 - b^2)^2*x*y = 0.

This circumparabola is the isogonal conjugate of line X(112)X(1576) (the tangent to the circumcircle at X(112)), and the isotomic conjugate of line X(107)X(110) (the tangent to the Steiner circumellipse at X(648)). This circumparabola passes through the points X(i) for these i: 525, 850, 2867, 3265, 16077, 17932, 34767. (Randy Hutson, May 19, 2020)

X(38233) lies on the curve Q077, the nine-point circle of the cevian triangle of X(525), and these lines: {112, 23616}, {127, 38240}, {525, 6720}, {13219, 34767}

X(38233) = perspector of ABC and orthic triangle of cevian triangle of X(525)

X(38234) = PERSPECTOR OF THESE TRIANGLES: ABC AND VIJAY-PAASCHE REFLECTION TRIANGLE

Barycentrics    a / (a S + 2 b SB +2 c SC - a b c ) : b / (b S + 2 a SA +2 c SC - a b c ) : c / (c S + 2 a SA +2 b SB - a b c )

Let Ab = 0 : 2R : c, Ac = 0 : b : 2R, Bc = a : 0 : 2R, Ba = 2R : 0 : c, Ca = 2R : b : 0, Cb = a : 2R : 0; these are points on the Paasche ellipse, as at X(37861) and X(37881). Let B'a and C'a be the reflections of Ba and Ca in the line BC, and define B'c, A'c, A'b, C'b cyclically. Let

Va = B'cA'c∩A'bC'b,     Vb = C'aB'a∩B'cA'c,     Vc = C'aB'a∩A'bC'b.

The triangle VaVbVc, here named the Vijay-Paasch reflection triangle, is perspective to ABC, and the perspector is X(38234).

See X(38234). (Dasari Naga Vijay Krishna)

X(38234) lies on these lines: {}

X(38234) = isogonal conjugate of X(38235)
X(38234) = isotomic conjugate of X(38236)
X(38234) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(1132)}} and {{A, B, C, X(2), X(15446)}}

X(38235) = ISOGONAL CONJUGATE OF X(38234)

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

X(38235) lies on these lines: {1, 1152}, {6, 5903}, {46, 7969}, {55, 35611}, {56, 35641}, {65, 1335}, {80, 23251}, {371, 37567}, {372, 2099}, {484, 1151}, {517, 1124}, {1377, 3869}, {1388, 35810}, {1482, 6502}, {1703, 3340}, {1788, 9661}, {2066, 12702}, {2067, 36279}, {2093, 18991}, {2098, 35769}, {3070, 10573}, {3297, 5697}, {3298, 5902}, {3474, 9647}, {3753, 30557}, {3878, 31473}, {4295, 19065}, {5128, 9583}, {5172, 35772}, {5204, 35763}, {5221, 35768}, {5420, 15950}, {5443, 8252}, {5445, 8253}, {5657, 9646}, {5690, 31472}, {6361, 9660}, {6396, 34471}, {6409, 37572}, {6410, 37525}, {6412, 37616}, {6560, 10950}, {7968, 25415}, {10895, 35789}, {12047, 13973}, {12245, 31408}, {18995, 35642}, {19038, 35610}

X(38235) = reflection of X(1124) in X(2362)
X(38235) = isogonal conjugate of X(38234) X(38235) = {X(65), X(35774)}-harmonic conjugate of X(1335)

X(38236) = ISOTOMIC CONJUGATE OF X(38234)

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

X(38236) lies on these lines: {2, 20920}, {75, 1271}, {1267, 3262}, {1270, 17791}, {1441, 5391}, {20895, 32797}

X(38236) = isotomic conjugate of X(38234)

X(38237) = MIDPOINT OF X(2) AND X(23610)

Barycentrics    a^2*(b^2 - c^2)*(a^4*b^4 - 3*a^4*b^2*c^2 + a^2*b^4*c^2 + a^4*c^4 + a^2*b^2*c^4 - b^4*c^4) : :
X(38237) = X[115] + 2 X[38017]

X(38237) lies on these lines: {2, 512}, {51, 3221}, {115, 38017}, {647, 881}, {669, 20965}, {688, 10191}, {1915, 9426}, {1962, 9402}, {9429, 11176}

X(38237) is the centroid of the tangential triangle of hyperbola {{A,B,C,X(2),X(6)}}. (Randy Hutson, May 19, 2020)

X(38237) = midpoint of X(2) and X(23610)
X(38237) = X(i)-Ceva conjugate of X(j) for these (i,j): {1084, 512}, {9428, 25054}
X(38237) = X(25054)-cross conjugate of X(512)
X(38237) = crosspoint of X(9428) and X(25054)
X(38237) = crosssum of X(2) and X(14824)
X(38237) = barycentric product X(i)*X(j) for these {i,j}: {512, 25054}, {523, 9431}, {1084, 9428}, {2501, 23180}
X(38237) = barycentric quotient X(i)/X(j) for these {i,j}: {9431, 99}, {23180, 4563}, {25054, 670}

X(38238) = MIDPOINT OF X(2) AND X(8027)

Barycentrics    a*(b - c)*(a^2*b^2 - 3*a^2*b*c + a*b^2*c + a^2*c^2 + a*b*c^2 - b^2*c^2) : :
X(38238) = 5 X[2] - 3 X[14434], X[2] - 3 X[14474], X[11] + 2 X[38018], 4 X[1015] - X[9267], 5 X[8027] + 3 X[14434], X[8027] + 3 X[14474], X[14434] - 5 X[14474]

X(38238) lies on these lines: {2, 513}, {11, 38018}, {171, 667}, {354, 4083}, {512, 10180}, {649, 2666}, {650, 3572}, {1015, 9267}, {1638, 3808}, {4164, 21786}, {4763, 6373}, {4782, 37520}, {4871, 23825}, {4932, 27854}, {6164, 18191}, {6686, 31286}, {14433, 29198}

X(38238) = midpoint of X(2) and X(8027)
X(38238) = X(i)-Ceva conjugate of X(j) for these (i,j): {1015, 513}, {9266, 9359}, {9296, 9263}, {9362, 21893}
X(38238) = X(9263)-cross conjugate of X(513)
X(38238) = X(i)-isoconjugate of X(j) for these (i,j): {100, 9361}, {101, 9295}, {190, 9265}, {765, 9267}, {7035, 9299}
X(38238) = crosspoint of X(i) and X(j) for these (i,j): {9263, 9296}, {9266, 9359}
X(38238) = crosssum of X(i) and X(j) for these (i,j): {9265, 9299}, {9267, 9361}
X(38238) = crossdifference of every pair of points on line {2664, 3230}
X(38238) = barycentric product X(i)*X(j) for these {i,j}: {1, 21211}, {244, 9362}, {513, 9263}, {514, 9359}, {649, 18149}, {693, 1979}, {1015, 9296}, {1019, 21100}, {1086, 9266}, {7192, 21893}, {17924, 22158}
X(38238) = barycentric quotient X(i)/X(j) for these {i,j}: {513, 9295}, {649, 9361}, {667, 9265}, {1015, 9267}, {1977, 9299}, {1979, 100}, {9263, 668}, {9266, 1016}, {9296, 31625}, {9359, 190}, {9362, 7035}, {18149, 1978}, {21100, 4033}, {21211, 75}, {21893, 3952}, {22158, 1332}
X(38238) = {X(8027),X(14474)}-harmonic conjugate of X(2)

X(38239) = MIDPOINT OF X(2) AND X(8030)

Barycentrics    (2*a^2 - b^2 - c^2)*(5*a^4 - 5*a^2*b^2 - b^4 - 5*a^2*c^2 + 7*b^2*c^2 - c^4) : :
X(38239) = X[2] - 3 X[1641], 5 X[2] - 3 X[1648], X[2] + 3 X[5468], 2 X[2] - 3 X[11053], X[126] + 2 X[38020], 5 X[1641] - X[1648], 3 X[1641] + X[8030], X[1648] + 5 X[5468], 3 X[1648] + 5 X[8030], 2 X[1648] - 5 X[11053], 3 X[5468] - X[8030], 2 X[5468] + X[11053], 2 X[8030] + 3 X[11053]

X(38239) lies on these lines: {2, 6}, {126, 8787}, {690, 10190}, {5026, 12036}, {8352, 22254}, {8598, 17941}, {9225, 11054}, {10278, 33921}, {10488, 10553}

X(38239) = midpoint of X(i) and X(j) for these {i,j}: {2, 8030}, {1641, 5468}
X(38239) = reflection of X(11053) in X(1641)
X(38239) = X(2482)-Ceva conjugate of X(524)
X(38239) = X(8591)-cross conjugate of X(524)
X(38239) = crosssum of X(9178) and X(21906)
X(38239) = barycentric product X(524)*X(8591)
X(38239) = barycentric quotient X(8591)/X(671)
X(38239) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5468, 8030}, {1641, 8030, 2}

X(38240) = MIDPOINT OF X(2) AND X(23616)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 + c^2)*(-a^8 + a^6*b^2 + 2*a^4*b^4 - 3*a^2*b^6 + b^8 + a^6*c^2 - 5*a^4*b^2*c^2 + 3*a^2*b^4*c^2 + b^6*c^2 + 2*a^4*c^4 + 3*a^2*b^2*c^4 - 4*b^4*c^4 - 3*a^2*c^6 + b^2*c^6 + c^8) : :
X(38240) = 5 X[2] - 3 X[14401], X[2] + 3 X[34767], X[127] + 2 X[38233], 3 X[14401] + 5 X[23616], X[14401] + 5 X[34767], X[23616] - 3 X[34767]

X(38240) lies on these lines: {2, 525}, {127, 38233}, {343, 3265}, {520, 3819}, {523, 23332}, {1499, 35450}, {6368, 10184}, {9007, 22165}

X(38240) = midpoint of X(2) and X(23616)
X(38240) = X(15526)-Ceva conjugate of X(525)
X(38240) = X(i)-isoconjugate of X(j) for these (i,j): {112, 9390}, {9392, 23964}, {15351, 32676}
X(38240) = barycentric product X(i)*X(j) for these {i,j}: {402, 34767}, {2629, 14208}, {2633, 17879}
X(38240) = barycentric quotient X(i)/X(j) for these {i,j}: {402, 4240}, {525, 15351}, {656, 9390}, {2629, 162}, {2632, 9392}, {2633, 24000}, {19208, 933}
X(38240) = {X(2),X(34767)}-harmonic conjugate of X(23616)

X(38241) = X(511)-CROSS CONJUGATE OF X(512)

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

X(38241) = X[15631] + 3 X[23610]

X(38241) lies on the curve Q079 and these lines: {512, 620}, {669, 20976}, {15631, 23610}

x(38241) = X(511)-cross conjugate of X(512)
x(38241) = cevapoint of X(2491) and X(23610)
X(38241) = vertex of parabola {{A,B,C,X(512),X(669)}}

X(38242) = X(517)-CROSS CONJUGATE OF X(513)

Barycentrics    a*(b - c)*(3*a^4*b - 3*a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - a^4*c - 4*a^3*b*c + 12*a^2*b^2*c - 4*a*b^3*c - b^4*c + a^3*c^2 - 4*a^2*b*c^2 - 4*a*b^2*c^2 + b^3*c^2 + a^2*c^3 + 4*a*b*c^3 + b^2*c^3 - a*c^4 - b*c^4)*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4 - 3*a^4*c + 4*a^3*b*c + 4*a^2*b^2*c - 4*a*b^3*c + b^4*c + 3*a^3*c^2 - 12*a^2*b*c^2 + 4*a*b^2*c^2 - b^3*c^2 + 3*a^2*c^3 + 4*a*b*c^3 - b^2*c^3 - 3*a*c^4 + b*c^4) : :
X(38242) = 3 X[8027] + X[15632]

X(38242) lies on the curve Q079 and these lines: {513, 3035}, {649, 17439}, {8027, 15632}

X(38242) = X(517)-cross conjugate of X(513)
X(38242) = cevapoint of X(3310) and X(8027)

X(38243) = X(513)-CROSS CONJUGATE OF X(517)

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

X(38243) lies on the curve Q079 and these lines: {517, 1387}, {901, 37136}

X(38243) = X(513)-cross conjugate of X(517)
X(38243) = X(33646)-isoconjugate of X(36037)
X(38243) = barycentric quotient X(3310)/X(33646)

X(38244) = X(3667)-CROSS CONJUGATE OF X(519)

Barycentrics    (a - b)*(a - c)*(2*a - b - c)*(4*a^3 - 9*a^2*b - 9*a*b^2 + 4*b^3 - 7*a^2*c + 37*a*b*c - 7*b^2*c - 8*a*c^2 - 8*b*c^2 + 3*c^3)*(4*a^3 - 7*a^2*b - 8*a*b^2 + 3*b^3 - 9*a^2*c + 37*a*b*c - 8*b^2*c - 9*a*c^2 - 7*b*c^2 + 4*c^3)

X(38244) = X[8028] + X[15637]

X(38244) lies on the curve Q079 and these lines: {519, 6715}, {8028, 15637}

X(38244) = midpoint of X(8028) and X(15637)
X(39244) = X(3667)-cross conjugate of X(519)
X(39244) = cevapoint of X(8028) and X(14425)

X(38245) = X(1499)-CROSS CONJUGATE OF X(524)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(2*a^2 - b^2 - c^2)*(4*a^6 - 15*a^4*b^2 - 15*a^2*b^4 + 4*b^6 - 9*a^4*c^2 + 63*a^2*b^2*c^2 - 9*b^4*c^2 - 12*a^2*c^4 - 12*b^2*c^4 + c^6)*(4*a^6 - 9*a^4*b^2 - 12*a^2*b^4 + b^6 - 15*a^4*c^2 + 63*a^2*b^2*c^2 - 12*b^4*c^2 - 15*a^2*c^4 - 9*b^2*c^4 + 4*c^6) : :
X(38245) = 3 X[8030] + X[15638]

X(38245) lies on the curve Q079 and these lines: {524, 6719}, {8030, 15638}

X(38245) = X(1499)-cross conjugate of X(524)
X(38245) = cevapoint of X(8030) and X(9125)

X(38246) = X(30)X(6699)∩X(74)X(3081)

Barycentrics    (S^2-3*SB*SC)*((39*R^2+SA-9*SW)*S^2-216*R^4*(6*R^2-5*SW+2*SA)+27*(2*SA^2+6*SA*SW-11*SW^2)*R^2+3*(4*SA+9*SW)*(SB+SC)*SW) : :
X(38246) = X(74)+3*X(3081), X(113)-3*X(34582), X(146)-9*X(4240), 5*X(146)-9*X(12369), 3*X(1511)-X(20123), 5*X(4240)-X(12369), 3*X(14847)-X(34297), 3*X(15774)-X(16163)

X(38246) is the focus of the circumparabola that is the isogonal conjugate of line X(74)X(526) (the tangent to the circumcircle at X(74)), and the isotomic conjugate of line X(1494)X(3268) (the tangent to the Steiner circumellipse at X(1494)). This circumparabola passes through the points X(i) for these i: 30, 476, 4240, 9141, 16077. (Randy Hutson, May 19, 2020)

X(38246) lies on the curve Q077, the nine-point circle of the cevian triangle of X(30), and these lines: {30, 6699}, {74, 3081}, {113, 34582}, {146, 4240}, {265, 16080}, {1511, 3163}, {14847, 15774}

X(38246) = perspector of ABC and orthic triangle of cevian triangle of X(30)

leftri

Vu-Lozada QA-points: X(38247)-X(38304)

rightri

This preamble and centers X(38247)-X(38304) were contributed by César Eliud Lozada, May 1, 2020.

The following theorem is enunciated in the preamble just before X(36598):

Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P and ℭ a conic through A', B', C'. If A", B", C" are the points, others than A', B', C', at which ℭ cuts BC, CA, AB, respectively, then AA", BB", CC" are concurrent.

It is also written in that preamble that if P = x : y : z and ℭ has perspector X' = x' : y' : z' with respect to ABC (barycentrics), then the triangles ABC and A"B"C" have perspector:

Q(P, X') = x (x' y z + x (z' y - 3 y' z)) (x' y z + x (y' z - 3 z' y)) : :

Vu Thanh Tung observed that if the previous construction is applied to a quadrangle P1P2P3P4 and every Qi is calculated as above in the triangle PjPkPℓ, then the four lines PnQn are also concurrent (Quadri-and-Poly-Geometry #243). César Lozada found that this point of concurrence T(P, X') has barycentrics coordinates:

T(P, X') = x' (x' y z + x (y' z - 3 z' y)) (x' y z + x (z' y - 3 y' z)) : :

The point T(P, X') is nere named the Vu-Lozada QA-point of (P, X').

Notes:

The appearance of (i, j, n) in the following partial list means that the Vu-Lozada QA-point of (X(i), X(j)) is X(n):
(1, 2, 38247), (1, 3, 38248), (1, 4, 38249), (1, 6, 3445), (1, 7, 38250), (1, 8, 38251), (1, 31, 38252), (2, 1, 8056), (2, 3, 1073), (2, 4, 38253), (2, 6, 8770), (2, 7, 38254), (2, 8, 38255), (2, 111, 38280), (3, 1, 36600), (3, 2, 38256), (3, 4, 38257), (3, 6, 3532), (3, 31, 38258), (4, 1, 36599), (4, 2, 38259), (4, 3, 38260), (4, 6, 64), (4, 7, 38261), (6, 1, 36598), (6, 2, 38262), (6, 3, 38263), (6, 4, 38264), (6, 8, 38265), (6, 31, 38266), (7, 1, 3062), (7, 2, 36606), (7, 3, 38267), (7, 4, 38268), (7, 6, 38269), (7, 8, 38270), (8, 1, 38271), (8, 2, 36605), (8, 4, 38272), (8, 6, 38273), (8, 7, 38274), (31, 1, 38275), (31, 2, 38276), (31, 6, 36614), (111, 1, 38277), (111, 2, 38278), (111, 3, 38279), (111, 6, 111)

X(38247) = VU-LOZADA QA-POINT OF (X(1), X(2))

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

X(38247) lies on the circumconic with center X(1015) and on these lines: {1,4704}, {2,17448}, {57,4393}, {81,36614}, {88,34063}, {105,7766}, {145,291}, {194,3227}, {239,8056}, {274,4772}, {330,1278}, {1002,3623}, {1022,17753}, {1432,20090}, {2176,23560}, {3304,16995}, {3622,30571}, {4788,32005}, {9263,18135}, {16834,36603}, {16975,32009}, {25430,29570}, {32104,36871}

X(38247) = isogonal conjugate of X(16969)
X(38247) = isotomic conjugate of X(1278)
X(38247) = cyclocevian conjugate of the isogonal conjugate of X(23857)
X(38247) = antitomic conjugate of the isogonal conjugate of X(20669)
X(38247) = antitomic conjugate of the isotomic conjugate of X(20530)
X(38247) = barycentric product X(i)*X(j) for these {i, j}: {75, 36598}, {76, 36614}, {85, 36630}, {693, 29227}
X(38247) = barycentric quotient X(i)/X(j) for these (i, j): (1, 16569), (3, 22149), (7, 17090), (8, 4903), (9, 4050), (10, 4135)
X(38247) = trilinear product X(i)*X(j) for these {i, j}: {2, 36598}, {7, 36630}, {75, 36614}, {514, 29227}
X(38247) = trilinear quotient X(i)/X(j) for these (i, j): (2, 16569), (8, 4050), (10, 21868), (63, 22149), (76, 20943), (85, 17090)
X(38247) = trilinear pole of the line {513, 6687}
X(38247) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(2)}} and {{A, B, C, X(4), X(6630)}}
X(38247) = Cevapoint of X(i) and X(j) for these {i,j}: {2, 4788}, {1015, 4083}
X(38247) = X(192)-cross conjugate of-X(2)
X(38247) = X(i)-isoconjugate-of-X(j) for these {i,j}: {6, 16569}, {19, 22149}, {32, 20943}, {41, 17090}
X(38247) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 16569), (3, 22149), (7, 17090), (8, 4903)
X(38247) = {X(17448), X(31999)}-harmonic conjugate of X(2)

X(38248) = VU-LOZADA QA-POINT OF (X(1), X(3))

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

X(38248) lies on these lines: {1,1898}, {29,36610}, {77,13369}, {102,10680}, {945,12001}, {947,16202}, {1936,36600}, {3244,10570}, {3345,24474}, {6585,20419}

X(38248) = isogonal conjugate of X(38295)
X(38248) = barycentric product X(i)*X(j) for these {i, j}: {63, 36599}, {394, 36610}
X(38248) = barycentric quotient X(3)/X(20078)
X(38248) = trilinear product X(i)*X(j) for these {i, j}: {3, 36599}, {255, 36610}
X(38248) = trilinear quotient X(63)/X(20078)
X(38248) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(3)}} and {{A, B, C, X(4), X(1898)}}
X(38248) = X(19)-isoconjugate-of-X(20078)
X(38248) = X(3)-reciprocal conjugate of-X(20078)

X(38249) = VU-LOZADA QA-POINT OF (X(1), X(4))

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

X(38249) lies on the Feuerbach hyperbola and these lines: {1,8762}, {4,11189}, {21,36607}, {243,36599}

X(38249) = isogonal conjugate of X(38284)
X(38249) = polar conjugate of the anticomplement of X(6360)
X(38249) = barycentric product X(i)*X(j) for these {i, j}: {92, 36600}, {2052, 36607}
X(38249) = trilinear product X(i)*X(j) for these {i, j}: {4, 36600}, {158, 36607}
X(38249) = X(1148)-cross conjugate of-X(4)

X(38250) = VU-LOZADA QA-POINT OF (X(1), X(7))

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

X(38250) lies on the Feuerbach hyperbola and these lines: {1,36601}, {9,25716}, {885,17090}, {3062,14189}, {3160,9442}, {31527,38261}

X(38250) = isogonal conjugate of X(38285)
X(38250) = barycentric product X(i)*X(j) for these {i, j}: {85, 36601}, {1088, 36628}
X(38250) = barycentric quotient X(7)/X(20089)
X(38250) = trilinear product X(i)*X(j) for these {i, j}: {7, 36601}, {279, 36628}
X(38250) = trilinear quotient X(85)/X(20089)
X(38250) = X(41)-isoconjugate-of-X(20089)
X(38250) = X(7)-reciprocal conjugate of-X(20089)

X(38251) = VU-LOZADA QA-POINT OF (X(1), X(8))

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

X(38251) lies on the Feuerbach hyperbola and these lines: {1,36602}, {79,30947}, {256,27680}, {2298,36619}, {3551,26093}, {5205,38271}, {9365,27383}

X(38251) = isogonal conjugate of X(38286)
X(38251) = barycentric product X(312)*X(36602)
X(38251) = trilinear product X(i)*X(j) for these {i, j}: {8, 36602}, {312, 36619}
X(38251) = trilinear quotient X(8)/X(6048)
X(38251) = intersection, other than A,B,C, of Feuerbach hyperbola and conic {{A, B, C, X(280), X(38265)}}
X(38251) = X(56)-isoconjugate-of-X(6048)

X(38252) = VU-LOZADA QA-POINT OF (X(1), X(31))

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

X(38252) lies on these lines: {1,1958}, {42,8770}, {255,36051}, {304,17876}, {741,3565}, {923,2172}, {1096,8772}, {1245,6391}, {1580,38275}, {1895,36120}, {2268,16524}, {18826,35136}, {34065,37132}

X(38252) = isogonal conjugate of X(18156)
X(38252) = anticomplement of the complementary conjugate of X(16605)
X(38252) = barycentric product X(i)*X(j) for these {i, j}: {1, 8770}, {6, 8769}, {19, 6391}, {31, 2996}, {48, 34208}, {63, 14248}
X(38252) = barycentric quotient X(i)/X(j) for these (i, j): (31, 193), (32, 1707), (48, 6337), (213, 4028), (560, 3053), (604, 17081)
X(38252) = trilinear product X(i)*X(j) for these {i, j}: {3, 14248}, {6, 8770}, {25, 6391}, {31, 8769}, {32, 2996}, {184, 34208}
X(38252) = trilinear quotient X(i)/X(j) for these (i, j): (3, 6337), (6, 193), (25, 6353), (31, 1707), (32, 3053), (42, 4028)
X(38252) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(31)}} and {{A, B, C, X(25), X(961)}}
X(38252) = crosspoint of X(1) and X(2129)
X(38252) = crosssum of X(1) and X(2128)
X(38252) = X(48)-cross conjugate of-X(31)
X(38252) = X(i)-isoconjugate-of-X(j) for these {i,j}: {2, 193}, {4, 6337}, {8, 17081}, {69, 6353}
X(38252) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (31, 193), (32, 1707), (48, 6337), (213, 4028)

X(38253) = VU-LOZADA QA-POINT OF ( X(2), X(4))

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

X(38253) lies on the Kiepert hyperbola, cubic K709 and these lines: {2,15851}, {4,1192}, {125,3079}, {226,25993}, {275,37643}, {297,38259}, {376,37877}, {458,18845}, {459,33630}, {1131,3535}, {1132,3536}, {2996,14952}, {3090,31363}, {3424,6353}, {5067,13599}, {6820,13579}, {8889,14484}, {11606,37187}, {13585,37192}, {14361,16080}

X(38253) = isogonal conjugate of X(38292)
X(38253) = polar conjugate of X(3146)
X(38253) = barycentric product X(i)*X(j) for these {i, j}: {4, 35510}, {253, 33893}, {264, 3532}, {2052, 36609}
X(38253) = barycentric quotient X(i)/X(j) for these (i, j): (4, 3146), (19, 18594), (20, 27082), (125, 13611), (278, 18624), (393, 33630)
X(38253) = trilinear product X(i)*X(j) for these {i, j}: {19, 35510}, {92, 3532}, {158, 36609}, {2184, 33893}
X(38253) = trilinear quotient X(i)/X(j) for these (i, j): (4, 18594), (92, 3146), (158, 33630), (273, 18624)
X(38253) = trilinear pole of the line {523, 13473}
X(38253) = intersection, other than A,B,C, of Kiepert hyperbola and conic {{A, B, C, X(3), X(1192)}}
X(38253) = Cevapoint of X(i) and X(j) for these {i,j}: {6, 15750}, {125, 6587}, {1249, 33893}
X(38253) = X(1249)-cross conjugate of-X(4)
X(38253) = X(i)-isoconjugate-of-X(j) for these {i,j}: {3, 18594}, {48, 3146}, {212, 18624}, {255, 33630}
X(38253) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (4, 3146), (19, 18594), (20, 27082), (125, 13611)

X(38254) = VU-LOZADA QA-POINT OF ( X(2), X(7))

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

X(38254) lies on the circumhyperbola dual of Yff parabola and these lines: {2,25716}, {7,3817}, {75,36625}, {9436,36606}

X(38254) = isogonal conjugate of X(38293)
X(38254) = barycentric product X(i)*X(j) for these {i, j}: {7, 36605}, {279, 36625}, {1088, 36627}
X(38254) = barycentric quotient X(i)/X(j) for these (i, j): (7, 20059), (269, 33633)
X(38254) = trilinear product X(i)*X(j) for these {i, j}: {57, 36605}, {269, 36625}, {279, 36627}
X(38254) = trilinear quotient X(i)/X(j) for these (i, j): (85, 20059), (279, 33633)
X(38254) = X(i)-isoconjugate-of-X(j) for these {i,j}: {41, 20059}, {220, 33633}
X(38254) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (7, 20059), (269, 33633)

X(38255) = VU-LOZADA QA-POINT OF ( X(2), X(8))

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

X(38255) lies on the circumconic with center X(1146) and on these lines: {2,4488}, {8,18220}, {11,15519}, {85,36621}, {189,30852}, {333,5328}, {1121,1997}, {1220,5550}, {1311,8699}, {3912,36605}, {4102,28808}, {4997,8055}, {5748,34234}, {10405,29627}, {18228,30608}, {19877,31359}, {20057,37662}

X(38255) = isogonal conjugate of X(38296)
X(38255) = barycentric product X(i)*X(j) for these {i, j}: {8, 36606}, {312, 36603}, {346, 36621}
X(38255) = barycentric quotient X(i)/X(j) for these (i, j): (8, 3621), (9, 3973), (55, 21000), (219, 22147), (312, 20942), (522, 4962)
X(38255) = trilinear product X(i)*X(j) for these {i, j}: {8, 36603}, {9, 36606}, {200, 36621}
X(38255) = trilinear quotient X(i)/X(j) for these (i, j): (8, 3973), (9, 21000), (78, 22147), (312, 3621), (522, 2516)
X(38255) = lies on the circumconic with center X(1146))
X(38255) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(8)}} and {{A, B, C, X(7), X(4862)}}
X(38255) = Cevapoint of X(11) and X(4521)
X(38255) = X(i)-isoconjugate-of-X(j) for these {i,j}: {34, 22147}, {56, 3973}, {57, 21000}, {109, 2516}
X(38255) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (8, 3621), (9, 3973), (55, 21000), (219, 22147)

X(38256) = VU-LOZADA QA-POINT OF ( X(3), X(2))

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

X(38256) lies on the circumconic with center X(35071)) and on these lines: {3,36608}, {20,14941}, {394,36617}, {401,1073}, {35941,36609}

X(38256) = isogonal conjugate of X(38297)
X(38256) = isotomic conjugate of the anticomplement of X(3164)
X(38256) = isotomic conjugate of the isogonal conjugate of X(36617)
X(38256) = isotomic conjugate of the polar conjugate of X(38264)
X(38256) = polar conjugate of the isogonal conjugate of X(36608)
X(38256) = X(19)-isoconjugate of X(38283)
X(38256) = barycentric product X(i)*X(j) for these {i, j}: {69, 38264}, {264, 36608}
X(38256) = trilinear product X(i)*X(j) for these {i, j}: {63, 38264}, {75, 36617}, {92, 36608}
X(38256) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(3)}} and {{A, B, C, X(20), X(401)}}

X(38257) = VU-LOZADA QA-POINT OF ( X(3), X(4))

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

X(38257) lies on the Jerabek hyperbola and these lines: {4,30263}, {73,8762}, {450,38260}, {1942,3147}

X(38257) = isogonal conjugate of X(38281)
X(38257) = intersection, other than A,B,C, of Jerabek hyperbola and conic {{A, B, C, X(450), X(3147)}}

X(38258) = VU-LOZADA QA-POINT OF ( X(3), X(31))

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

X(38258) lies on the line {3207,7085}

X(38258) = isogonal conjugate of X(38298)
X(38258) = barycentric quotient X(31)/X(20061)
X(38258) = trilinear quotient X(6)/X(20061)
X(38258) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(31)}} and {{A, B, C, X(73), X(7118)}}
X(38258) = X(2)-isoconjugate-of-X(20061)
X(38258) = X(31)-reciprocal conjugate of-X(20061)

X(38259) = VU-LOZADA QA-POINT OF ( X(4), X(2))

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

X(38259) lies on the Kiepert hyperbola and these lines: {2,15815}, {4,1353}, {5,10155}, {6,18845}, {10,24280}, {20,7612}, {76,32982}, {83,32979}, {98,3146}, {115,439}, {148,8781}, {226,25716}, {262,3832}, {297,38253}, {321,25278}, {459,37174}, {598,5286}, {625,32876}, {671,6392}, {1916,20105}, {2996,20080}, {3091,14494}, {3407,14068}, {3424,17578}, {3522,7607}, {5068,7608}, {5254,5395}, {6655,32872}, {7388,34091}, {7389,34089}, {7620,10302}, {7748,32838}, {7762,32532}, {7812,33698}, {10159,11185}, {11172,33192}, {11669,15022}, {14063,32840}, {18840,32974}, {18841,32971}

X(38259) = isogonal conjugate of X(5023)
X(38259) = isotomic conjugate of X(20080)
X(38259) = cyclocevian conjugate of the isogonal conjugate of X(19588)
X(38259) = cyclocevian conjugate of the isotomic conjugate of X(19583)
X(38259) = cyclocevian conjugate of the polar conjugate of X(6392)
X(38259) = antigonal conjugate of the isogonal conjugate of X(1570)
X(38259) = polar conjugate of the isogonal conjugate of X(38263)
X(38259) = antitomic conjugate of the isogonal conjugate of X(1570)
X(38259) = barycentric product X(i)*X(j) for these {i, j}: {69, 36611}, {76, 36616}, {264, 38263}
X(38259) = barycentric quotient X(1)/X(16570)
X(38259) = polar conjugate of X(38282)
X(38259) = trilinear product X(i)*X(j) for these {i, j}: {63, 36611}, {75, 36616}, {92, 38263}
X(38259) = trilinear quotient X(2)/X(16570)
X(38259) = trilinear pole of the line {523, 14341} (the radical axis of polar circle and complement of polar circle)
X(38259) = intersection, other than A,B,C, of Kiepert hyperbola and conic {{A, B, C, X(3), X(5093)}}
X(38259) = Cevapoint of X(i) and X(j) for these {i,j}: {6, 20850}, {115, 3566}
X(38259) = X(193)-cross conjugate of-X(2)
X(38259) = X(6)-isoconjugate-of-X(16570)
X(38259) = X(1)-reciprocal conjugate of-X(16570)

X(38260) = VU-LOZADA QA-POINT OF ( X(4), X(3))

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

X(38260) lies on the Jerabek hyperbola and these lines: {3,12235}, {4,13292}, {6,7529}, {52,64}, {54,5422}, {65,8757}, {66,1351}, {69,11585}, {74,12085}, {265,12164}, {381,14457}, {450,38257}, {454,20975}, {569,14528}, {1177,7517}, {1209,13622}, {1656,5486}, {1995,13472}, {3167,15317}, {3426,34783}, {3431,17928}, {3527,18445}, {3532,17834}, {3549,10602}, {4846,6146}, {5073,10293}, {5504,9937}, {6391,9926}, {6515,23335}, {8549,34207}, {9967,34817}, {10297,15077}, {11270,11413}, {11432,14542}, {11744,12295}, {11800,32321}, {18474,22466}, {19456,35603}, {34436,37488}, {34438,34751}

X(38260) = reflection of X(3) in X(19360)
X(38260) = isogonal conjugate of X(3147)
X(38260) = barycentric product X(394)*X(36612)
X(38260) = trilinear product X(255)*X(36612)
X(38260) = intersection, other than A,B,C, of conic {{A, B, C, X(2), X(7529)}} and Jerabek hyperbola
X(38260) = X(155)-cross conjugate of-X(3)

X(38261) = VU-LOZADA QA-POINT OF ( X(4), X(7))

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

X(38261) lies on the Feuerbach hyperbola and these lines: {9,25719}, {31527,38250}

X(38261) = isogonal conjugate of X(38287)

X(38262) = VU-LOZADA QA-POINT OF ( X(6), X(2))

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

X(38262) lies on the circumconic with center X(1084) and on these lines: {2,32746}, {6,3552}, {25,7766}, {37,24524}, {42,17350}, {193,694}, {385,8770}, {2998,20081}, {3228,8264}, {14614,36616}, {20105,36648}, {25054,32747}

X(38262) = isogonal conjugate of X(21001)
X(38262) = isotomic conjugate of X(20081)
X(38262) = cyclocevian conjugate of the isotomic conjugate of X(32548)
X(38262) = anticomplement of X(32746)
X(38262) = barycentric product X(i)*X(j) for these {i, j}: {75, 38275}, {76, 36615}
X(38262) = barycentric quotient X(i)/X(j) for these (i, j): (1, 16571), (3, 22152), (7, 17091), (10, 21095), (75, 20945), (194, 32746)
X(38262) = trilinear product X(i)*X(j) for these {i, j}: {2, 38275}, {75, 36615}
X(38262) = trilinear quotient X(i)/X(j) for these (i, j): (2, 16571), (63, 22152), (76, 20945), (85, 17091), (321, 21095), (693, 21206)
X(38262) = trilinear pole of the line {512, 31286}
X(38262) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(6)}} and {{A, B, C, X(4), X(3225)}}
X(38262) = Cevapoint of X(i) and X(j) for these {i,j}: {2, 20105}, {1084, 3221}
X(38262) = X(194)-cross conjugate of-X(2)
X(38262) = X(i)-isoconjugate-of-X(j) for these {i,j}: {6, 16571}, {19, 22152}, {32, 20945}, {41, 17091}
X(38262) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 16571), (3, 22152), (7, 17091), (10, 21095)

X(38263) = VU-LOZADA QA-POINT OF ( X(6), X(3))

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

X(38263) lies on the Jerabek hyperbola and these lines: {3,14914}, {4,1353}, {6,8780}, {54,11443}, {64,1351}, {66,3629}, {67,6144}, {69,30771}, {193,16774}, {511,3532}, {576,22334}, {895,19588}, {1176,10602}, {1177,34777}, {2519,10097}, {3167,21971}, {3531,18445}, {3564,15077}, {3589,5486}, {3763,13622}, {3818,22466}, {5050,14528}, {5644,6677}, {6391,21639}, {6776,31371}, {13418,31282}, {14498,22146}, {22660,32533}, {34435,37492}, {34436,37491}

X(38263) = isogonal conjugate of X(38282)
X(38263) = midpoint of X(193) and X(16774)
X(38263) = isogonal conjugate of the polar conjugate of X(38259)
X(38263) = isotomic conjugate of the polar conjugate of X(36616)
X(38263) = barycentric product X(i)*X(j) for these {i, j}: {3, 38259}, {69, 36616}, {394, 36611}
X(38263) = barycentric quotient X(i)/X(j) for these (i, j): (3, 20080), (48, 16570), (184, 5023)
X(38263) = trilinear product X(i)*X(j) for these {i, j}: {48, 38259}, {63, 36616}, {255, 36611}
X(38263) = trilinear quotient X(i)/X(j) for these (i, j): (3, 16570), (48, 5023), (63, 20080)
X(38263) = intersection, other than A,B,C, of Jerabek hyperbola and conic {{A, B, C, X(25), X(30771)}}
X(38263) = X(i)-isoconjugate-of-X(j) for these {i,j}: {4, 16570}, {19, 20080}, {92, 5023}
X(38263) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (3, 20080), (48, 16570), (184, 5023)

X(38264) = VU-LOZADA QA-POINT OF ( X(6), X(4))

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

X(38264) lies on the Jerabek hyperbola and these lines: {3,36608}, {6,36617}, {1249,1987}, {22334,33971}

X(38264) = isogonal conjugate of X(38283)
X(38264) = polar conjugate of the anticomplement of X(3164)
X(38264) = polar conjugate of the isogonal conjugate of X(36617)
X(38264) = polar conjugate of the isotomic conjugate of X(38256)
X(38264) = barycentric product X(i)*X(j) for these {i, j}: {4, 38256}, {264, 36617}, {2052, 36608}
X(38264) = trilinear product X(i)*X(j) for these {i, j}: {19, 38256}, {92, 36617}, {158, 36608}
X(38264) = X(63)-isoconjugate of X(38297)

X(38265) = VU-LOZADA QA-POINT OF ( X(6), X(8))

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

X(38265) lies on these lines: {6,36618}, {1193,17350}

X(38265) = isogonal conjugate of X(38299)
X(38265) = trilinear product X(312)*X(36618)
X(38265) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(17350)}} and {{A, B, C, X(6), X(8)}}

X(38266) = VU-LOZADA QA-POINT OF ( X(6), X(31))

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

X(38266) lies on these lines: {1,23617}, {6,1201}, {31,5042}, {41,34543}, {42,7050}, {48,9456}, {81,2999}, {572,739}, {604,16945}, {608,1404}, {1253,3248}, {1419,1462}, {1449,2298}, {1914,36614}, {2162,2280}, {2214,16666}, {2347,32577}, {2991,25930}, {3451,9316}, {4052,19738}, {4373,14621}, {5381,5382}, {16779,20332}

X(38266) = isogonal conjugate of X(18743)
X(38266) = anticomplement of the complementary conjugate of X(16602)
X(38266) = complement of the anticomplementary conjugate of X(17490)
X(38266) = barycentric product X(i)*X(j) for these {i, j}: {1, 3445}, {8, 16945}, {31, 4373}, {41, 27818}, {55, 19604}, {56, 3680}
X(38266) = barycentric quotient X(i)/X(j) for these (i, j): (31, 145), (32, 1743), (41, 3161), (184, 4855), (213, 3950), (512, 4404)
X(38266) = trilinear product X(i)*X(j) for these {i, j}: {6, 3445}, {9, 16945}, {32, 4373}, {41, 19604}, {513, 34080}, {604, 3680}
X(38266) = trilinear quotient X(i)/X(j) for these (i, j): (6, 145), (31, 1743), (32, 3052), (39, 4884), (41, 3158), (42, 3950)
X(38266) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(1106)}} and {{A, B, C, X(6), X(31)}}
X(38266) = Cevapoint of X(6) and X(21785)
X(38266) = crossdifference of every pair of points on line {X(3667), X(4404)}
X(38266) = crosspoint of X(1293) and X(5382)
X(38266) = crosssum of X(i) and X(j) for these {i,j}: {2, 8055}, {145, 3161}, {1743, 4855}
X(38266) = X(284)-Beth conjugate of-X(3217)
X(38266) = X(41)-cross conjugate of-X(31)
X(38266) = X(i)-isoconjugate-of-X(j) for these {i,j}: {2, 145}, {7, 3161}, {8, 5435}, {43, 27496}
X(38266) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (31, 145), (32, 1743), (41, 3161), (184, 4855)

X(38267) = VU-LOZADA QA-POINT OF ( X(7), X(3))

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

X(38267) lies on the line {942,6180}

X(38267) = isogonal conjugate of X(38300)
X(38267) = barycentric quotient X(3)/X(20110)
X(38267) = trilinear quotient X(63)/X(20110)
X(38267) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(7)}} and {{A, B, C, X(63), X(13404)}}
X(38267) = X(19)-isoconjugate-of-X(20110)
X(38267) = X(3)-reciprocal conjugate of-X(20110)

X(38268) = VU-LOZADA QA-POINT OF ( X(7), X(4))

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

X(38268) lies on the Feuerbach hyperbola and these lines: {7,36622}, {278,3062}, {1148,38272}, {7952,38271}

X(38268) = isogonal conjugate of X(38288)
X(38268) = polar conjugate of the anticomplement of X(347)
X(38268) = barycentric product X(281)*X(36622)
X(38268) = barycentric quotient X(19)/X(1750)
X(38268) = trilinear product X(33)*X(36622)
X(38268) = trilinear quotient X(4)/X(1750)
X(38268) = intersection, other than A,B,C, of Feuerbach hyperbola and conic {{A, B, C, X(278), X(38253)}}
X(38268) = X(196)-cross conjugate of-X(4)
X(38268) = X(3)-isoconjugate-of-X(1750)
X(38268) = X(19)-reciprocal conjugate of-X(1750)

X(38269) = VU-LOZADA QA-POINT OF ( X(7), X(6))

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

X(38269) lies on these lines: {46,518}, {672,2178}, {840,1602}, {1406,1458}, {10679,34230}

X(38269) = isogonal conjugate of X(20075)
X(38269) = anticomplement of the complementary conjugate of X(3434)
X(38269) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(11510)}} and {{A, B, C, X(3), X(20615)}}
X(38269) = X(1486)-cross conjugate of-X(6)

X(38270) = VU-LOZADA QA-POINT OF ( X(7), X(8))

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

X(38270) lies on the Feuerbach hyperbola and the line {1,4488}

X(38270) = isogonal conjugate of X(38289)

X(38271) = VU-LOZADA QA-POINT OF ( X(8), X(1))

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

X(38271) lies on the Feuerbach hyperbola, cubic K807 and these lines: {1,1864}, {4,2093}, {7,1210}, {8,3586}, {9,3697}, {21,936}, {35,30393}, {46,3062}, {72,3680}, {79,3339}, {80,7991}, {84,1728}, {90,165}, {104,1490}, {226,3296}, {329,9614}, {442,3255}, {484,36599}, {950,1000}, {1103,2310}, {1172,1743}, {1320,11682}, {1392,36846}, {1479,5223}, {1698,34919}, {1702,7133}, {1708,10308}, {1736,8809}, {1737,7992}, {1768,34256}, {1785,7149}, {1898,30503}, {2320,19861}, {2335,3731}, {2956,9355}, {3361,7284}, {3419,34918}, {3427,5691}, {3487,18490}, {3577,12672}, {3601,5780}, {4866,5119}, {4882,10092}, {4900,5697}, {5205,38251}, {5551,5714}, {5553,10395}, {5557,7741}, {5558,14986}, {5559,9819}, {5715,10598}, {5728,10390}, {5825,37421}, {5927,7091}, {6048,9365}, {6260,6969}, {7082,10268}, {7160,18908}, {7285,7580}, {7952,38268}, {7987,15446}, {9581,37822}, {9851,37618}, {11224,21398}, {11501,30223}, {12625,12641}, {13407,30330}, {13606,30337}, {15556,16615}, {17098,18421}, {18529,37550}, {30343,37735}

X(38271) = isogonal conjugate of X(15803)
X(38271) = barycentric product X(i)*X(j) for these {i, j}: {7, 36629}, {57, 36624}
X(38271) = barycentric quotient X(i)/X(j) for these (i, j): (1, 9965), (9, 27383), (31, 37519), (42, 21866), (48, 23072)
X(38271) = trilinear product X(i)*X(j) for these {i, j}: {56, 36624}, {57, 36629}
X(38271) = trilinear quotient X(i)/X(j) for these (i, j): (2, 9965), (3, 23072), (6, 37519), (8, 27383), (37, 21866)
X(38271) = intersection, other than A,B,C, of Feuerbach hyperbola and conic {{A, B, C, X(3), X(2093)}}
X(38271) = X(40)-cross conjugate of-X(1)
X(38271) = X(i)-isoconjugate-of-X(j) for these {i,j}: {2, 37519}, {4, 23072}, {6, 9965}, {56, 27383}
X(38271) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 9965), (9, 27383), (31, 37519), (42, 21866)
X(38271) = {X(1728), X(1750)}-harmonic conjugate of X(15803)

X(38272) = VU-LOZADA QA-POINT OF ( X(8), X(4))

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

X(38272) lies on the Feuerbach hyperbola and the line {1148,38268}

X(38272) = isogonal conjugate of X(38290)

X(38273) = VU-LOZADA QA-POINT OF ( X(8), X(6))

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

X(38273) lies on these lines: {960,5119}, {1193,11509}

X(38273) = isogonal conjugate of X(20076)
X(38273) = anticomplement of the complementary conjugate of X(3436)
X(38273) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(2164)}} and {{A, B, C, X(6), X(8)}}
X(38273) = X(197)-cross conjugate of-X(6)
X(38273) = X(56)-vertex conjugate of-X(34447)

X(38274) = VU-LOZADA QA-POINT OF ( X(8), X(7))

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

X(38274) lies on the Feuerbach hyperbola and the line {9,25718}

X(38274) = isogonal conjugate of X(38291)

X(38275) = VU-LOZADA QA-POINT OF ( X(31), X(1))

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

X(38275) lies on the circumconic with center X(23505) and on these lines: {42,17350}, {213,3550}, {1580,38252}, {1707,1967}, {3223,16571}, {33782,37132}

X(38275) = isogonal conjugate of X(16571)
X(38275) = isotomic conjugate of X(20945)
X(38275) = barycentric product X(i)*X(j) for these {i, j}: {1, 38262}, {75, 36615}
X(38275) = barycentric quotient X(i)/X(j) for these (i, j): (1, 20081), (31, 21001), (37, 21095), (48, 22152), (57, 17091), (513, 21206)
X(38275) = trilinear product X(i)*X(j) for these {i, j}: {2, 36615}, {6, 38262}
X(38275) = trilinear quotient X(i)/X(j) for these (i, j): (2, 20081), (3, 22152), (6, 21001), (7, 17091), (10, 21095), (194, 32746)
X(38275) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(31)}} and {{A, B, C, X(43), X(292)}}
X(38275) = Cevapoint of X(513) and X(23505)
X(38275) = X(i)-isoconjugate-of-X(j) for these {i,j}: {2, 21001}, {4, 22152}, {6, 20081}, {55, 17091}
X(38275) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 20081), (31, 21001), (37, 21095), (48, 22152)

X(38276) = VU-LOZADA QA-POINT OF ( X(31), X(2))

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

X(38276) lies on these lines: {2276,25286}, {14945,20064}

X(38276) = isogonal conjugate of X(38301)
X(38276) = isotomic conjugate of the anticomplement of X(17486)
X(38276) = trilinear pole of the line {788, 31288}

X(38277) = VU-LOZADA QA-POINT OF ( X(111), X(1))

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

X(38277) lies on the circumconic with center X(3647) and on these lines: {4427,5297}, {16785,35342}

X(38277) = isogonal conjugate of X(38302)
X(38277) = trilinear pole of the line {1100, 4879}
X(38277) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(111)}} and {{A, B, C, X(100), X(1929)}}

X(38278) = VU-LOZADA QA-POINT OF ( X(111), X(2))

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

X(38278) = lies on the circumconics with center X(1084), and on the circuimconic with center X(6292), and on these lines: {6,10330}, {83,31128}, {99,3108}, {9178,22105}, {14948,20099}, {16055,21448}

X(38278) = isogonal conjugate of X(38303)
X(38278) = barycentric product X(83)*X(25322)
X(38278) = trilinear product X(82)*X(25322)
X(38278) = trilinear quotient X(308)/X(18075)
X(38278) = trilinear pole of the line {512, 3589}
X(38278) = lies on the circumconics with center X(i) for i in {1084, 6292}
X(38278) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(6)}} and {{A, B, C, X(99), X(10330)}}
X(38278) = X(524)-cross conjugate of-X(83)

X(38279) = VU-LOZADA QA-POINT OF ( X(111), X(3))

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

X(38279) lies on the Johnson circumconic and these lines: {1384,1613}, {2080,10836}, {3186,4232}, {9126,19909}, {20794,23180}

X(38279) = isogonal conjugate of X(38294)
X(38279) = reflection of X(19909) in X(9126)
X(38279) = isogonal conjugate of the polar conjugate of X(9227)
X(38279) = barycentric product X(3)*X(9227)
X(38279) = barycentric quotient X(184)/X(9225)
X(38279) = trilinear product X(48)*X(9227)
X(38279) = trilinear quotient X(48)/X(9225)
X(38279) = trilinear pole of the line {216, 2524}
X(38279) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(111)}} and {{A, B, C, X(69), X(694)}}
X(38279) = X(92)-isoconjugate-of-X(9225)
X(38279) = X(184)-reciprocal conjugate of-X(9225)

X(38280) = VU-LOZADA QA-POINT OF ( X(2), X(111))

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

X(38280) lies on the circumconic with center X(1084) and on this line: {8791,15269}

X(38280) = isogonal conjugate of X(38304)
X(38280) = barycentric quotient X(111)/X(20099)
X(38280) = trilinear quotient X(897)/X(20099)
X(38280) = X(896)-isoconjugate-of-X(20099)
X(38280) = X(111)-reciprocal conjugate of-X(20099)

X(38281) = ISOGONAL CONJUGATE OF X(38257)

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

X(38281) lies on these lines: {2,3}, {511,14059}, {1935,20764}, {1942,38260}, {2055,9306}, {3157,8763}, {6509,10110}, {6760,13346}, {9781,13409}, {14673,32321}

X(38281) = isogonal conjugate of X(38257)
X(38281) = pole of the trilinear polar of X(13855) with respect to MacBeath circumconic
X(38281) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 6638, 3), (24, 15781, 3), (1598, 6617, 3)

X(38282) = ISOGONAL CONJUGATE OF X(38263)

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

As a point on the Euler line, X(38282) has Shinagawa coefficients (-4*F, E+F)

X(38282) lies on these lines: {2,3}, {98,38253}, {99,6340}, {125,11206}, {154,23291}, {184,18950}, {230,1249}, {232,33630}, {242,17917}, {275,10155}, {317,34803}, {393,37637}, {459,7612}, {511,15010}, {1007,32001}, {1395,17123}, {1495,32064}, {1611,8743}, {1620,5893}, {1660,5622}, {1698,7718}, {1799,10603}, {1829,5550}, {1843,6688}, {1853,15448}, {1870,5272}, {1899,35260}, {1974,3619}, {2211,21001}, {2212,17122}, {2356,16569}, {3087,31489}, {3168,21910}, {3618,8541}, {3620,19118}, {5090,19877}, {5268,6198}, {5284,11383}, {5306,5702}, {5410,13941}, {5411,8972}, {5412,32786}, {5413,32785}, {5921,8780}, {5943,6403}, {5972,37669}, {6524,14165}, {6723,14927}, {6776,10192}, {7581,34516}, {7582,34515}, {7585,13937}, {7586,13884}, {7713,19862}, {7717,20195}, {8739,11488}, {8740,11489}, {8854,10881}, {8855,10880}, {9306,19128}, {9780,11363}, {10984,22750}, {11427,15004}, {11433,13366}, {13567,14912}, {14826,37638}, {15011,19161}, {15471,15533}, {16252,18913}, {17821,18945}, {19583,37803}, {19596,36851}, {32000,34229}, {34208,36611}, {34966,37784}

X(38282) = isogonal conjugate of X(38263)
X(38282) = polar conjugate of X(38259)
X(38282) = barycentric product X(i)*X(j) for these {i, j}: {4, 20080}, {92, 16570}, {264, 5023}
X(38282) = barycentric quotient X(i)/X(j) for these (i, j): (4, 38259), (25, 36616), (393, 36611)
X(38282) = trilinear product X(i)*X(j) for these {i, j}: {4, 16570}, {19, 20080}, {92, 5023}
X(38282) = trilinear quotient X(i)/X(j) for these (i, j): (19, 36616), (92, 38259), (158, 36611)
X(38282) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(17568)}} and {{A, B, C, X(2), X(20080)}}
X(38282) = orthoptic circle of Steiner inellipse-inverse of-X(13473)
X(38282) = polar circle-inverse of-X(37911)
X(38282) = X(i)-isoconjugate-of-X(j) for these {i,j}: {48, 38259}, {63, 36616}, {255, 36611}
X(38282) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (4, 38259), (25, 36616), (393, 36611)
X(38282) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 6353, 4), (3, 6622, 4), (24, 3090, 4), (25, 8889, 4), (376, 403, 4), (427, 7714, 4), (451, 7521, 4), (461, 4212, 4), (468, 37453, 2), (631, 3542, 4), (3144, 7498, 4), (3147, 7505, 4), (3545, 18533, 4), (3855, 6240, 4), (4213, 7490, 4), (4232, 7408, 25), (10154, 30771, 20), (10192, 26958, 6776), (16051, 37777, 4), (30771, 37911, 2), (33703, 35488, 4)

X(38283) = ISOGONAL CONJUGATE OF X(38264)

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

X(38283) lies on these lines: {2,3}, {52,14059}, {216,6688}, {1073,1351}, {1624,1853}, {2972,3060}, {3289,20233}, {5640,13409}, {5644,15851}, {5943,6509}, {6760,13352}, {15466,32428}, {22143,34966}, {23071,38284}, {34147,34986}

X(38283) = isogonal conjugate of X(38264)
X(38283) = isotomic conjugate of polar conjugate of X(38297)
X(38283) = barycentric quotient X(i)/X(j) for these (i, j): (3, 38256), (184, 36617), (577, 36608)
X(38283) = trilinear quotient X(i)/X(j) for these (i, j): (48, 36617), (63, 38256), (255, 36608)
X(38283) = intersection, other than A,B,C, of conics {{A, B, C, X(20), X(14941)}} and {{A, B, C, X(401), X(1073)}}
X(38283) = X(i)-isoconjugate-of-X(j) for these {i,j}: {19, 38256}, {92, 36617}, {158, 36608}
X(38283) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (3, 38256), (184, 36617), (577, 36608)
X(38283) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 6638, 3), (441, 11328, 3), (1589, 23266, 3), (1590, 23272, 3), (5020, 6617, 3), (6644, 15781, 3)

X(38284) = ISOGONAL CONJUGATE OF X(38249)

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

X(38284) lies on these lines: {1,3}, {145,856}, {296,38248}, {342,37411}, {653,7049}, {1069,17975}, {3157,8763}, {20793,26921}, {22147,38292}, {23071,38283}

X(38284) = isogonal conjugate of X(38249)
X(38284) = barycentric quotient X(i)/X(j) for these (i, j): (48, 36600), (577, 36607)
X(38284) = trilinear quotient X(i)/X(j) for these (i, j): (3, 36600), (255, 36607)
X(38284) = X(i)-isoconjugate-of-X(j) for these {i,j}: {4, 36600}, {158, 36607}
X(38284) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (48, 36600), (577, 36607)
X(38284) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 20764, 3), (3295, 7011, 3), (38288, 38290, 3)

X(38285) = ISOGONAL CONJUGATE OF X(38250)

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

X(38285) lies on these lines: {1,3}, {3022,4253}, {4336,20992}, {5432,27253}, {12513,28053}

X(38285) = isogonal conjugate of X(38250)
X(38285) = barycentric product X(55)*X(20089)
X(38285) = barycentric quotient X(i)/X(j) for these (i, j): (41, 36601), (1253, 36628)
X(38285) = trilinear product X(41)*X(20089)
X(38285) = trilinear quotient X(i)/X(j) for these (i, j): (55, 36601), (220, 36628)
X(38285) = X(i)-isoconjugate-of-X(j) for these {i,j}: {7, 36601}, {279, 36628}
X(38285) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (41, 36601), (1253, 36628)
X(38285) = {X(55), X(38291)}-harmonic conjugate of X(38287)

X(38286) = ISOGONAL CONJUGATE OF X(38251)

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

X(38286) lies on these lines: {1,3}, {12,4429}, {100,36508}, {344,1284}, {3210,3913}, {3813,28036}, {4972,36513}, {5433,26093}, {5687,32920}, {7080,21320}, {11237,17678}, {17054,23844}, {20760,24440}, {24443,28109}

X(38286) = isogonal conjugate of X(38251)
X(38286) = barycentric quotient X(i)/X(j) for these (i, j): (604, 36602), (1397, 36619)
X(38286) = trilinear product X(56)*X(6048)
X(38286) = trilinear quotient X(i)/X(j) for these (i, j): (56, 36602), (604, 36619)
X(38286) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(6048)}} and {{A, B, C, X(6), X(37603)}}
X(38286) = X(i)-isoconjugate-of-X(j) for these {i,j}: {8, 36602}, {312, 36619}
X(38286) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (604, 36602), (1397, 36619)
X(38286) = {X(65), X(1403)}-harmonic conjugate of X(56)

X(38287) = ISOGONAL CONJUGATE OF X(38261)

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

X(38287) lies on these lines: {1,3}, {2975,28053}

X(38287) = isogonal conjugate of X(38261)
X(38287) = {X(55), X(38291)}-harmonic conjugate of X(38285)

X(38288) = ISOGONAL CONJUGATE OF X(38268)

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

X(38288) lies on these lines: {1,3}, {109,2192}, {219,22117}, {278,15252}, {347,7580}, {1073,3190}, {1433,23072}, {2256,15905}, {2318,7078}, {2968,36845}, {5812,20264}, {6349,10578}, {6350,10580}, {6913,34231}, {7515,14986}, {7952,37411}, {13405,17073}, {16596,25568}, {23122,35350}

X(38288) = isogonal conjugate of X(38268)
X(38288) = barycentric product X(63)*X(1750)
X(38288) = barycentric quotient X(i)/X(j) for these (i, j): (222, 36622), (1750, 92)
X(38288) = trilinear product X(3)*X(1750)
X(38288) = trilinear quotient X(i)/X(j) for these (i, j): (77, 36622), (1750, 4)
X(38288) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(1750)}} and {{A, B, C, X(77), X(10857)}}
X(38288) = pole of the trilinear polar of X(268) with respect to MacBeath circumconic
X(38288) = X(268)-Ceva conjugate of-X(3)
X(38288) = X(33)-isoconjugate-of-X(36622)
X(38288) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (222, 36622), (1750, 92)
X(38288) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 38284, 38290), (55, 7011, 3), (3295, 20764, 3)

X(38289) = ISOGONAL CONJUGATE OF X(38270)

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

X(38289) lies on these lines: {1,3}, {1376,31995}

X(38289) = isogonal conjugate of X(38270)

X(38290) = ISOGONAL CONJUGATE OF X(38272)

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

X(38290) lies on these lines: {1,3}, {347,9122}, {515,37818}, {934,9799}, {1056,18641}, {1071,7053}, {1490,6611}, {1870,13737}, {3157,20818}, {3868,6617}, {4254,8555}, {6349,11037}, {9538,35987}, {11036,21482}, {17073,21620}, {23072,23089}

X(38290) = isogonal conjugate of X(38272)
X(38290) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7011, 3), (3, 38284, 38288), (999, 20764, 3)

X(38291) = ISOGONAL CONJUGATE OF X(38274)

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

X(38291) lies on the line {1,3}

X(38291) = isogonal conjugate of X(38274)
X(38291) = {X(38285), X(38287)}-harmonic conjugate of X(55)

X(38292) = ISOGONAL CONJUGATE OF X(38253)

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

X(38292) lies on these lines: {3,6}, {4,36413}, {30,1249}, {53,3830}, {112,5896}, {157,11216}, {193,441}, {219,22117}, {232,20850}, {233,15703}, {248,38263}, {268,2323}, {381,3087}, {382,393}, {440,37666}, {524,20208}, {550,5702}, {1033,12085}, {1073,15400}, {1368,5304}, {1576,33582}, {1990,5073}, {1993,6617}, {2003,7011}, {3068,19039}, {3069,19040}, {3146,33630}, {3157,20818}, {3163,15684}, {3167,8779}, {3211,23072}, {3289,20233}, {3527,14152}, {3553,18447}, {3554,18455}, {3629,6389}, {3843,6748}, {3851,6749}, {3964,22151}, {4667,17073}, {5054,36427}, {5159,37689}, {5523,34725}, {6144,15526}, {6415,10132}, {6416,10133}, {6676,37665}, {7494,14930}, {7735,30771}, {8584,34828}, {8745,18534}, {9722,10255}, {9777,23606}, {9909,10313}, {10602,14575}, {11405,23635}, {12164,14642}, {15013,22253}, {15291,28783}, {16318,34609}, {16667,17102}, {19136,33580}, {19588,22143}, {21482,37685}, {22147,38284}

X(38292) = isogonal conjugate of X(38253)
X(38292) = complement of the isotomic conjugate of X(15749)
X(38292) = barycentric product X(i)*X(j) for these {i, j}: {3, 3146}, {63, 18594}, {64, 27082}, {219, 18624}, {250, 13611}, {394, 33630}
X(38292) = barycentric quotient X(i)/X(j) for these (i, j): (3, 35510), (154, 33893), (184, 3532), (577, 36609)
X(38292) = trilinear product X(i)*X(j) for these {i, j}: {3, 18594}, {48, 3146}, {212, 18624}, {255, 33630}
X(38292) = trilinear quotient X(i)/X(j) for these (i, j): (48, 3532), (63, 35510), (255, 36609), (610, 33893)
X(38292) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(15851)}} and {{A, B, C, X(3), X(3146)}}
X(38292) = Brocard circle-inverse of-X(15851)
X(38292) = pole of the trilinear polar of X(1073) with respect to MacBeath circumconic
X(38292) = crossdifference of every pair of points on line {X(523), X(13473)}
X(38292) = crosspoint of X(2) and X(15749)
X(38292) = crosssum of X(i) and X(j) for these {i,j}: {6, 15750}, {125, 6587}, {1249, 33893}
X(38292) = X(i)-Ceva conjugate of-X(j) for these (i,j): (2, 15748), (1073, 3)
X(38292) = X(31)-complementary conjugate of-X(15748)
X(38292) = circle {{X(371),X(372),PU(1),PU(39)}}-inverse of X(1192)
X(38292) = X(i)-isoconjugate-of-X(j) for these {i,j}: {19, 35510}, {92, 3532}, {158, 36609}
X(38292) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (3, 35510), (154, 33893), (184, 3532), (577, 36609)
X(38292) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6, 15851), (3, 33636, 15905), (6, 3284, 15905), (6, 15905, 3), (6, 18365, 8553), (6, 36748, 5158), (371, 372, 1192), (1384, 14961, 3), (1576, 34777, 33582), (2055, 11432, 3), (3284, 15905, 33636), (3311, 3312, 389), (11485, 11486, 11438), (22143, 23163, 19588), (23115, 30435, 3)

X(38293) = ISOGONAL CONJUGATE OF X(38254)

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

X(38293) lies on these lines: {6,31}, {11,37681}, {480,2323}, {1155,1419}, {1191,8163}, {1449,15837}, {1456,37567}, {1743,4907}, {2098,7290}, {3057,16469}, {3332,10895}, {3945,5432}, {4319,16669}, {4336,16885}, {4413,37659}, {5204,13329}, {14100,16670}, {36971,37800}

X(38293) = isogonal conjugate of X(38254)
X(38293) = barycentric product X(i)*X(j) for these {i, j}: {55, 20059}, {200, 33633}
X(38293) = barycentric quotient X(i)/X(j) for these (i, j): (55, 36605), (220, 36625), (1253, 36627)
X(38293) = trilinear product X(i)*X(j) for these {i, j}: {41, 20059}, {220, 33633}
X(38293) = trilinear quotient X(i)/X(j) for these (i, j): (9, 36605), (200, 36625), (220, 36627)
X(38293) = X(i)-isoconjugate-of-X(j) for these {i,j}: {57, 36605}, {269, 36625}, {279, 36627}
X(38293) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (55, 36605), (220, 36625), (1253, 36627)
X(38293) = {X(19037), X(19038)}-harmonic conjugate of X(672)

X(38294) = ISOGONAL CONJUGATE OF X(38279)

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

X(38294) is the perspector of the circumconic passing through the polar conjugates of PU(7). (Randy Hutson, May 19, 2020)

X(38294) lies on these lines: {4,524}, {25,385}, {186,523}, {230,1249}, {264,8541}, {325,8889}, {340,8754}, {378,36207}, {393,15993}, {419,648}, {420,1990}, {427,7840}, {458,11405}, {468,8859}, {892,8753}, {1597,32515}, {6353,22329}, {7378,7779}, {16066,37792}, {32001,34208}, {35360,37962}

X(38294) = isogonal conjugate of X(38279)
X(38294) = polar conjugate of X(9227)
X(38294) = barycentric product X(264)*X(9225)
X(38294) = barycentric quotient X(4)/X(9227)
X(38294) = trilinear product X(92)*X(9225)
X(38294) = trilinear quotient X(92)/X(9227)
X(38294) = crossdifference of every pair of points on line {X(216), X(2524)}
X(38294) = X(48)-isoconjugate-of-X(9227)
X(38294) = X(4)-reciprocal conjugate of-X(9227)

X(38295) = ISOGONAL CONJUGATE OF X(38248)

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

X(38295) lies on these lines: {1,4}, {28,11401}, {108,11510}, {145,860}, {186,196}, {347,3651}, {377,18447}, {406,10587}, {475,1897}, {631,37565}, {651,1069}, {653,35486}, {999,37117}, {1060,6897}, {1062,6899}, {1148,3147}, {1249,8609}, {1319,14257}, {1426,9957}, {1482,37414}, {1824,5045}, {1825,18398}, {1835,5697}, {1871,16216}, {1937,38249}, {3011,38282}, {3176,7505}, {3295,7414}, {3518,33925}, {3520,26357}, {3622,5136}, {4185,7373}, {6353,26228}, {6767,37194}, {6834,15252}, {6835,37729}, {6836,18455}, {6851,9538}, {6896,37696}, {6898,37697}, {6925,32047}, {6977,17102}, {7040,36610}, {7046,10527}, {7412,16202}, {7577,26481}, {8144,10431}, {10267,37441}, {10680,37305}, {10916,17917}, {11809,13619}, {12704,22465}, {14018,15934}, {14794,23040}, {17562,26377}, {21844,36152}, {24299,37028}, {24474,37417}, {34028,36996}

X(38295) = isogonal conjugate of X(38248)
X(38295) = polar conjugate of the isotomic conjugate of X(20078)
X(38295) = barycentric product X(4)*X(20078)
X(38295) = barycentric quotient X(i)/X(j) for these (i, j): (19, 36599), (393, 36610)
X(38295) = trilinear product X(19)*X(20078)
X(38295) = trilinear quotient X(i)/X(j) for these (i, j): (4, 36599), (158, 36610)
X(38295) = intersection, other than A,B,C, of conics {{A, B, C, X(226), X(20078)}} and {{A, B, C, X(243), X(38249)}}
X(38295) = X(i)-isoconjugate-of-X(j) for these {i,j}: {3, 36599}, {255, 36610}
X(38295) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (19, 36599), (393, 36610)
X(38295) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1068, 4), (1, 23710, 1068), (278, 6198, 4), (1870, 7952, 4), (9538, 37798, 6851), (37696, 37800, 6896)

X(38296) = ISOGONAL CONJUGATE OF X(38255)

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

X(38296) lies on these lines: {6,41}, {9,1388}, {65,16667}, {69,31230}, {226,19739}, {346,1317}, {391,5433}, {572,5217}, {1319,1743}, {1419,5575}, {1449,2099}, {2223,38293}, {2285,16666}, {2323,11510}, {3451,4251}, {3973,22147}, {4559,21785}, {5120,5172}, {5232,31221}, {5749,10944}, {17355,37738}, {17439,34524}

X(38296) = isogonal conjugate of X(38255)
X(38296) = barycentric product X(i)*X(j) for these {i, j}: {7, 21000}, {56, 3621}, {57, 3973}, {109, 4962}, {278, 22147}, {604, 20942}
X(38296) = barycentric quotient X(i)/X(j) for these (i, j): (56, 36606), (604, 36603), (1407, 36621)
X(38296) = trilinear product X(i)*X(j) for these {i, j}: {34, 22147}, {56, 3973}, {57, 21000}, {109, 2516}, {604, 3621}, {1397, 20942}
X(38296) = trilinear quotient X(i)/X(j) for these (i, j): (56, 36603), (57, 36606), (269, 36621), (1415, 8699)
X(38296) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(8572)}} and {{A, B, C, X(6), X(3973)}}
X(38296) = crosssum of X(11) and X(4521)
X(38296) = X(i)-isoconjugate-of-X(j) for these {i,j}: {8, 36603}, {9, 36606}, {200, 36621}
X(38296) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (56, 36606), (604, 36603)
X(38296) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2067, 6502, 8572), (18995, 18996, 1193)

X(38297) = ISOGONAL CONJUGATE OF X(38256)

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

X(38297) lies on these lines: {4,6}, {39,13474}, {64,1987}, {232,11381}, {237,5023}, {382,1625}, {389,33842}, {1968,1971}, {1970,6759}, {1988,36617}, {2076,5167}, {2422,36615}, {3146,3289}, {3199,6000}, {3269,12290}, {3763,37186}, {5013,32444}, {6241,33885}, {7747,13419}, {8571,11563}, {11439,22240}, {11672,31952}, {12279,15355}, {14157,14585}, {15305,22416}, {22332,22334}, {33537,36751}

X(38297) = isogonal conjugate of X(38256)
X(38297) = polar conjugate of the isotomic conjugate of X(38283)
X(38297) = barycentric product X(4)*X(38283)
X(38297) = barycentric quotient X(i)/X(j) for these (i, j): (25, 38264), (32, 36617), (184, 36608)
X(38297) = trilinear product X(19)*X(38283)
X(38297) = trilinear quotient X(i)/X(j) for these (i, j): (19, 38264), (31, 36617), (48, 36608)
X(38297) = pole of the trilinear polar of X(1988) with respect to circumcircle
X(38297) = X(i)-isoconjugate-of-X(j) for these {i,j}: {63, 38264}, {75, 36617}, {92, 36608}
X(38297) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (25, 38264), (32, 36617), (184, 36608)
X(38297) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 3331, 32445), (4, 32445, 6), (1968, 26883, 1971)

X(38298) = ISOGONAL CONJUGATE OF X(38258)

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

X(38298) lies on these lines: {4,75}, {85,2478}, {169,16568}, {312,10405}, {349,5342}, {469,18738}, {857,18743}, {4687,27250}, {5179,6376}, {6554,17289}, {18147,33780}, {20921,37185}, {30854,37445}

X(38298) = isogonal conjugate of X(38258)
X(38298) = barycentric product X(75)*X(20061)
X(38298) = trilinear product X(2)*X(20061)
X(38298) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 20914, 75), (30807, 31042, 312)

X(38299) = ISOGONAL CONJUGATE OF X(38265)

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

X(38299) lies on these lines: {2,12}, {221,38286}, {1402,15485}

X(38299) = isogonal conjugate of X(38265)
X(38299) = barycentric quotient X(1397)/X(36618)
X(38299) = trilinear quotient X(604)/X(36618)
X(38299) = X(312)-isoconjugate-of-X(36618)
X(38299) = X(1397)-reciprocal conjugate of-X(36618)

X(38300) = ISOGONAL CONJUGATE OF X(38267)

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

X(38300) lies on these lines: {1,7521}, {4,12}, {19,13405}, {165,5236}, {278,5218}, {390,37372}, {475,5687}, {495,37395}, {497,37799}, {631,1214}, {954,11406}, {1056,7501}, {1058,7537}, {1148,3147}, {1249,8608}, {1375,38288}, {1435,10164}, {1838,31452}, {1861,3158}, {1870,5657}, {3011,17903}, {3487,6197}, {4219,5281}, {5089,6353}, {7046,26227}, {10587,37253}, {20075,37371}, {23171,24580}

X(38300) = isogonal conjugate of X(38267)
X(38300) = polar conjugate of the isotomic conjugate of X(20110)
X(38300) = barycentric product X(4)*X(20110)
X(38300) = trilinear product X(19)*X(20110)

X(38301) = ISOGONAL CONJUGATE OF X(38276)

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

X(38301) lies on these lines: {6,75}, {8620,21001}, {21790,28365}

X(38301) = isogonal conjugate of X(38276)
X(38301) = crossdifference of every pair of points on line {X(788), X(31288)}
X(38301) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (75, 8621, 21776), (75, 21776, 6), (4361, 18278, 6), (4363, 34251, 6)

X(38302) = ISOGONAL CONJUGATE OF X(38277)

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

X(38302) lies on these lines: {1,524}, {42,4416}, {484,1734}, {1743,2238}, {5223,6007}, {17207,22174}, {17272,26102}, {18206,20984}

X(38302) = isogonal conjugate of X(38277)
X(38302) = crossdifference of every pair of points on line {X(1100), X(4879)}
X(38302) = X(524)-Zayin conjugate of-X(1)

X(38303) = ISOGONAL CONJUGATE OF X(38278)

Barycentrics    a^2*(b^2+c^2)*(a^4+(b^2+c^2)*a^2-3*b^2*c^2) : :
X(38303) = X(6)-4*X(3231)

X(38303) lies on these lines: {2,6}, {512,2076}, {1634,8623}, {3229,5201}, {3787,29959}, {5023,23208}, {9019,36827}, {9225,18374}, {35325,36824}

X(38303) = isogonal conjugate of X(38278)
X(38303) = barycentric product X(1964)*X(18075)
X(38303) = barycentric quotient X(39)/X(25322)
X(38303) = trilinear quotient X(38)/X(25322)
X(38303) = intersection, other than A,B,C, of conics {{A, B, C, X(39), X(597)}} and {{A, B, C, X(69), X(36824)}}
X(38303) = crossdifference of every pair of points on line {X(512), X(3589)}
X(38303) = X(111)-Ceva conjugate of-X(39)
X(38303) = X(82)-isoconjugate-of-X(25322)
X(38303) = X(i)-line conjugate of-X(j) for these (i,j): (2, 3589), (6, 3589), (69, 3589), (81, 3589)
X(38303) = X(39)-reciprocal conjugate of-X(25322)
X(38303) = {X(599), X(1613)}-harmonic conjugate of X(6)

X(38304) = ISOGONAL CONJUGATE OF X(38280)

Barycentrics    (3*a^6-(b^2+c^2)*a^4-5*(b^4-b^2*c^2+c^4)*a^2-(b^2+c^2)*(b^4-4*b^2*c^2+c^4))*(2*a^2-b^2-c^2) : :
X(38304) = X(6)+3*X(1641) = X(141)-3*X(11053) = 5*X(3618)+3*X(5468)

X(38304) lies on these lines: {2,6}, {126,6593}, {9177,32459}

X(38304) = isogonal conjugate of X(38280)
X(38304) = barycentric product X(524)*X(20099)
X(38304) = trilinear product X(896)*X(20099)

leftri

Vu antipedal translations: X(38305)-X(38309)

rightri

This preamble is based on notes from Vu Thanh Tung, May 1, 2020.

Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC, but not on the circumcircle. Let

A'B'C' = antipedal triangle of P
V = 2 * vector OP'
A1B1C1 = V(A'B'C')

Then A1B1C1 is perspective to ABC, and the perspector, here named the Vu antipedal translation of P, is the point

V(P) = 1 / ( c^4 q (2 p - r) + b^4 (2 p - q) r - 3 a^4 q r - 2 a^2 (c^2 q (p - 2 r) + b^2 (p - 2 q) r) + 2 b^2 c^2 (q r + p (q + r)) ) : :

See Vu Antipedal Translation.

The appearance of (i,j) in the following list means that V(X(i)) = X(j):
(1,3577), (2,14484), (4,4), (5,38305), (6,3531), (7,38306), (8,38307), (9,38308), (10,38309)

See the preamble just before X(38005) for Vu pedal translation.

X(38305) = VU ANTIPEDAL TRANSLATION OF X(5)

Barycentrics    (2 a^8 + a^6 b^2 - 6 a^4 b^4 + a^2 b^6 + 2 b^8 - 3 a^6 c^2 - 3 a^4 b^2 c^2 - 3 a^2 b^4 c^2 - 3 b^6 c^2 - 3 a^4 c^4 - 5 a^2 b^2 c^4 - 3 b^4 c^4 + 7 a^2 c^6 + 7 b^2 c^6 - 3 c^8) (2 a^8 - 3 a^6 b^2 - 3 a^4 b^4 + 7 a^2 b^6 - 3 b^8 + a^6 c^2 - 3 a^4 b^2 c^2 - 5 a^2 b^4 c^2 + 7 b^6 c^2 - 6 a^4 c^4 - 3 a^2 b^2 c^4 - 3 b^4 c^4 + a^2 c^6 - 3 b^2 c^6 + 2 c^8) : :
X(38305) = 2*X(4)+X(33992)

X(38305) lies on the circumconic with center X(137) and on these lines: {4,33992}, {5,32223}, {53,14836}, {546,25043}, {1263,3845}, {3153,17500}, {3574,15619}, {3627,22335}, {3861,32535}, {14141,23046}

X(38305) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(5)}} and {{A, B, C, X(6), X(18401)}}
X(38305) = X(253)-vertex conjugate of-X(7488)

X(38306) = VU ANTIPEDAL TRANSLATION OF X(7)

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

X(38306) lies on the Feuerbach hyperbola and these lines: {9,6843}, {21,22753}, {79,5768}, {943,33993}, {1836,3427}, {2320,9812}, {3255,26333}, {5805,34919}, {6598,16125}, {6601,37820}, {7285,18483}, {7319,7686}, {10248,10266}, {12331,34894}

X(38306) = intersection, other than A,B,C, of Feuerbach hyperbola and conic {{A, B, C, X(27), X(6843)}}

X(38307) = VU ANTIPEDAL TRANSLATION OF X(8)

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

X(38307) lies on the Feuerbach hyperbola and these lines: {9,38127}, {104,33994}, {1158,7285}, {1476,22753}, {1837,10309}, {3254,6246}, {5556,7686}, {5811,6598}, {12641,26333}

X(38307) = intersection, other than A,B,C, of Feuerbach hyperbola and conic {{A, B, C, X(6), X(2745)}}

X(38308) = VU ANTIPEDAL TRANSLATION OF X(9)

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

X(38308) lies on the Feuerbach hyperbola and these lines: {7,7682}, {84,33995}, {1156,2950}, {1476,1490}, {1750,7284}, {3296,8166}, {4900,30326}, {6601,26333}, {7091,22753}

X(38308) = isogonal conjugate of X(21164)

X(38309) = VU ANTIPEDAL TRANSLATION OF X(10)

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

X(38309) lies on the Kiepert hyperbola and these lines: {226,5724}, {1834,3429}, {13478,33996}

X(38309) = intersection, other than A,B,C, of Kiepert hyperbola and conic {{A, B, C, X(80), X(5724)}}

leftri

Points Associated with generalized Paasche conics: X(38310)-X(38313)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, May 4, 2020.

As stated at X(37861), if P = p : q : r and U = u : v : w (barycentrics) are triangle centers having the same degree of homogeneity in a,b,c, then the (p,u)-generalized Paasche conic, GPC(p,u), is the conic that passes through these six points: 0 : r : w, 0 : v : q, u : 0 : p, r : 0 : w, q : v : 0, u : p : 0 and is given by the equation p v w x^2 + q w u y^2 + r u v z^2 - (q r u + u v w) y z - (r p v + u v w) z x - (p q w + u v w) x y = 0.

The perspector of GPC(p,u) is the barycentric quotient P/U = p v w : q w u : r u v, and the center of GPC(p,u) is 2 p^2 v^2 w^2 - u v w (2 p q r + 2 q r u + 2 p r v + 2 p q w + r u v + q u w) : :

Note, for example, that GPC(p,2u) is not the same conic as GPC(p,u); i.e., the definition of generalized Paasche conic depends on the representations of centers P and U. Nevertheless, we write GPC(P,U) instead of GPC(p,u) in cases where p and u are the first barycentrics as shown in ETC; in particular, this is the case when p and u are polynomials in a,b,c with relatively prime coefficients that agree in parity. Examples of this kind include the following:

Degree 2 of homogeneity:

GPC(X(9),X(192)) passes through X(10030).
GPC(X(75),X(9)) passes through X(3307) and X(3308), and has center X(650).
GPC(141),X(6)) passes through X(1662) and X(1663), a hyperbola with center X(182).
GPC(X(192),X(9)) passes through X(3685).
GPC(X(239),X(320)) passes through X(320).
GPC(X(320),X(239)) passes through X(239) and its antipode, X(38311).

Degree 3 of homogeneity:

GPC(X(11),X(55)) passes through X(101) and X(3939), and has center X(15260).
GPC(X(11),X(100)) passes through X(765), X(4564), and has center X(38310).
GPC(X(38), X(31)) = GPC(X(141),X(6)).
GPC(X(42),X(171)) passes through X(4128).
GPC(X(55),X(11))) passes through X(514) and X(522), a hyperbola with center X(15280)
GPC(X(55),X(43)) passes through X(10030).
GPC(X(55),X(312)) passes through X(1921).
GPC(X(100),X(11))) passes through X(514) and X(522), a hyperbola with center X(11)
GPC(X(100),X(244)) passes through X(514).
GPC(X(171),X(42)) passes through X(2643).
GPC(X(244),X(100)) passes through X(1016).
GPC(X(321),X(210)) = GPC(X(75),X(9)).
GPC(X(354),X(210)) passes through X(3932).

GPC(p,u} passes through u : v : w if q r + r p + p q + v w + w u + u v = p u + q v + r w.

GPC(p,u} passes through p : q : r if q^2 r^2 u + p^2 r^2 v + p^2 q^2 w + (q r + r p + p q) u v w = p^3 v w + q^3 w u + r^3 u v.

As noted at X(37861), César Lozada observed that the Paasche inner conic (i.e., the Paasche ellipse), passes through the points indicated by the notation GCP(1,sin A). To generalize, the locus of the center of GCP(t, sinA) as t goes through the real numbers is the quartic curve given by

a*b*(b - c)^2*c*x^4 + 2*(a - b)*b*(a - 2*c)*(b - c)*c*x^3*y + a*b*c*(a^2 - 4*a*b + b^2 + 2*a*c + 2*b*c - 2*c^2)*x^2*y^2 - 2*a*(a - b)*(b - 2*c)*(a - c)*c*x*y^3 + a*b*(a - c)^2*c*y^4 - 2*(a - 2*b)*b*(a - c)*(b - c)*c*x^3*z - 2*a*b*c*(a^2 - a*b + b^2 - a*c - b*c + c^2)*x^2*y*z - 2*a*b*c*(a^2 - a*b + b^2 - a*c - b*c + c^2)*x*y^2*z + 2*a*(2*a - b)*(a - c)*(b - c)*c*y^3*z + a*b*c*(a^2 + 2*a*b - 2*b^2 - 4*a*c + 2*b*c + c^2)*x^2*z^2 - 2*a*b*c*(a^2 - a*b + b^2 - a*c - b*c + c^2)*x*y*z^2 - a*b*c*(2*a^2 - 2*a*b - b^2 - 2*a*c + 4*b*c - c^2)*y^2*z^2 + 2*a*(a - b)*b*(a - c)*(2*b - c)*x*z^3 - 2*a*(a - b)*b*(2*a - c)*(b - c)*y*z^3 + a*(a - b)^2*b*c*z^4 = 0.

This curve passes through the points X(i) for i = 2, 9, 3218, 8183, 37861, 37862, 38312, 38313.

The following table identifies the conics GPC(t, sin A) for several choices of t:

t GPC(t, sin A) center
0 Steiner circumellipse X(2)
1 Paasche elllipse X(37861)
-1 Paasche outer conic X(37862)
infinity circumellipse centered at X(9) X(9)
-2/(3R) hyperbola X(3218)
-W^(1/2) (see below) (pending) X(38312)
W^(1/2) (see below) (pending) X(38313)

In the table, W = 3S/(4Rs) = 3(-a+b+c)(a-b+c)(a+b-c)/(4 a b c).

The locus of the center of GCP(sin A, t) as t goes through the real numbers is the quartic curve given by

a^2*(b - c)^2*x^4 - 2*a*(a - b)*(2*a - c)*(b - c)*x^3*y + (-2*a^2*b^2 + 2*a^2*b*c + 2*a*b^2*c + a^2*c^2 - 4*a*b*c^2 + b^2*c^2)*x^2*y^2 + 2*(a - b)*b*(a - c)*(2*b - c)*x*y^3 + b^2*(a - c)^2*y^4 + 2*a*(2*a - b)*(a - c)*(b - c)*x^3*z - 2*(a^2*b^2 - a^2*b*c - a*b^2*c + a^2*c^2 - a*b*c^2 + b^2*c^2)*x^2*y*z - 2*(a^2*b^2 - a^2*b*c - a*b^2*c + a^2*c^2 - a*b*c^2 + b^2*c^2)*x*y^2*z - 2*(a - 2*b)*b*(a - c)*(b - c)*y^3*z + (a^2*b^2 + 2*a^2*b*c - 4*a*b^2*c - 2*a^2*c^2 + 2*a*b*c^2 + b^2*c^2)*x^2*z^2 - 2*(a^2*b^2 - a^2*b*c - a*b^2*c + a^2*c^2 - a*b*c^2 + b^2*c^2)*x*y*z^2 + (a^2*b^2 - 4*a^2*b*c + 2*a*b^2*c + a^2*c^2 + 2*a*b*c^2 - 2*b^2*c^2)*y^2*z^2 - 2*(a - b)*(b - 2*c)*(a - c)*c*x*z^3 + 2*(a - b)*(a - 2*c)*(b - c)*c*y*z^3 + (a - b)^2*c^2*z^4 = 0.

X(38310) = CENTER OF GPC(X(11),X(100))

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

X(38310) lies on these lines: {2,5377}, {59,3911}, {100,919}, {765,3218}, {3935,4564}, {4998,8047}, {6065,6745}, {6551,9081}, {14513,36167}, {31633,33110}

X(38310) = barycentric product X(1016)*X(38530)
X(38310) = trilinear product X(765)*X(38530)

X(38311) = GPC(X(320),X(239))-ANTIPODE OF X(239)

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

X(38311) lies on these lines: {}

X(38312) = CENTER OF GPC(- sqrt(3S/(4Rs)), sin A))

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

X(38312) lies on this line: {2,37}

X(38312) = reflection of X(38313) in X(1575)
X(38312) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 37, 38313}, {2276, 4688, 38313}, {3752, 17281, 38313}, {4908, 16610, 38313}, {28244, 35652, 38313}

X(38313) = CENTER OF GPC(sqrt(3S/(4Rs)), sin A))

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

X(38313) lies on this line: {2,37}

X(38313) = reflection of X(38312) in X(1575)
X(38313) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 37, 38312}, {2276, 4688, 38312}, {3752, 17281, 38312}, {4908, 16610, 38312}, {28244, 35652, 38312}

X(38314) =  X(1)X(2)∩X(3)X(5734)

Barycentrics    7*a+b+c : :
Trilinears    3 r + R sin B sin C : :

X(38314) = 2*X(1)+X(2),8*X(1)+X(8),7*X(1)+2*X(10),10*X(1)-X(145),X(1)+2*X(551),5*X(1)+4*X(1125),4*X(1)-X(3241),4*X(1)+5*X(3616),2*X(1)+7*X(3622),X(1)+8*X(3636),5*X(1)+X(3679),3*X(1)+X(19875),3*X(1)+2*X(19883),4*X(2)-X(8),7*X(2)-4*X(10),5*X(2)+X(145),X(2)-4*X(551),5*X(2)-8*X(1125),2*X(2)+X(3241),2*X(2)-5*X(3616),X(2)-7*X(3622),7*X(2)+5*X(3623),5*X(2)-2*X(3679),10*X(2)-7*X(9780),3*X(2)-4*X(19883),8*X(2)+7*X(20057),7*X(2)-X(31145),4*X(3)+5*X(5734),4*X(3)-X(34632),X(4)+2*X(3655),X(4)+8*X(15178),4*X(5)-X(34627),5*X(8)+4*X(145),X(8)+2*X(3241),X(8)-10*X(3616),5*X(8)-8*X(3679),3*X(8)-8*X(19875) (and many others)

See Kadir Altintas and Ercole Suppa, Euclid 856 .

X(38314) lies on these lines: {1,2}, {3,5734}, {4,3655}, {5,34627}, {7,1319}, {11,10031}, {20,13464}, {21,3304}, {30,5603}, {40,15692}, {55,4345}, {56,4323}, {75,4742}, {80,11274}, {81,16483}, {86,16711}, {89,24857}, {100,6767}, {104,28444}, {115,9884}, {140,34718}, {148,12258}, {165,15705}, {192,28554}, {193,16491}, {214,9802}, {279,25723}, {348,5543}, {354,3877}, {355,5071}, {376,962}, {381,944}, {388,4870}, {390,6173}, {392,3873}, {404,3303}, {405,19738}, {452,28609}, {495,17533}, {496,17530}, {515,3839}, {516,30392}, {517,3524}, {527,11038}, {528,8236}, {529,3475}, {537,24508}, {545,35578}, {547,1483}, {549,1482}, {553,1420}, {597,3242}, {631,3654}, {664,31721}, {671,11725}, {758,15672}, {940,16486}, {942,3890}, {946,3543}, {950,18220}, {952,5055}, {956,5284}, {958,16861}, {960,3889}, {993,37602}, {999,1621}, {1001,6172}, {1010,19819}, {1056,5080}, {1100,5296}, {1120,12035}, {1279,4715}, {1318,4618}, {1320,6174}, {1376,8162}, {1386,1992}, {1387,3488}, {1388,3485}, {1655,31999}, {1656,34748}, {2098,4995}, {2099,5298}, {2476,3829}, {2482,7983}, {2646,9785}, {2975,7373}, {3058,4313}, {3090,37727}, {3091,5882}, {3146,11522}, {3160,17079}, {3161,3247}, {3227,17794}, {3295,5253}, {3306,31393}, {3340,5265}, {3436,26127}, {3445,19336}, {3476,5226}, {3486,11238}, {3487,11113}, {3522,4301}, {3523,7982}, {3534,22791}, {3545,5886}, {3576,9778}, {3579,15698}, {3600,4654}, {3648,16137}, {3649,15678}, {3685,28301}, {3723,17281}, {3742,5919}, {3746,4188}, {3812,3885}, {3813,4197}, {3826,12630}, {3830,34773}, {3832,30308}, {3845,18493}, {3868,17609}, {3869,5045}, {3871,25524}, {3876,34791}, {3884,18398}, {3892,4430}, {3895,5437}, {3897,20323}, {3898,5902}, {3902,19804}, {3913,17531}, {3928,5250}, {3945,17274}, {3984,17554}, {4189,5563}, {4193,15888}, {4216,18613}, {4295,21842}, {4297,15683}, {4305,24926}, {4309,37256}, {4317,15680}, {4344,17392}, {4353,24280}, {4361,28641}, {4370,16672}, {4371,6707}, {4392,4694}, {4402,15668}, {4419,4795}, {4423,17547}, {4432,17487}, {4448,9269}, {4460,28626}, {4479,34284}, {4644,24441}, {4645,17399}, {4648,17382}, {4653,8025}, {4658,17588}, {4661,10176}, {4664,15569}, {4671,4975}, {4720,25507}, {4723,30829}, {4740,24325}, {4862,30712}, {4916,17327}, {4921,11110}, {4930,15670}, {5032,16475}, {5047,12513}, {5048,5218}, {5054,5657}, {5056,5881}, {5064,7718}, {5066,18525}, {5082,26060}, {5119,27003}, {5141,37720}, {5154,37719}, {5258,16859}, {5260,17542}, {5273,5289}, {5281,7962}, {5315,37685}, {5325,15829}, {5328,37703}, {5330,34744}, {5439,14923}, {5459,7975}, {5460,7974}, {5493,21734}, {5556,7354}, {5625,9791}, {5642,7984}, {5690,15694}, {5719,5748}, {5744,15934}, {5749,16777}, {5790,15699}, {5844,11539}, {5846,21358}, {6054,11724}, {6055,7970}, {6175,11235}, {6361,8703}, {6646,31313}, {6684,15721}, {6829,37726}, {6921,7320}, {6933,37724}, {6940,37622}, {7229,17319}, {7288,11011}, {7714,11363}, {7734,34656}, {7743,10129}, {7810,34738}, {7968,19054}, {7969,19053}, {7976,9466}, {7987,20070}, {7988,28236}, {7991,15717}, {8148,15693}, {8164,10584}, {8227,13607}, {8591,11711}, {8596,11599}, {8666,16865}, {8715,17572}, {9140,11735}, {9143,11720}, {9172,10704}, {9327,16783}, {9336,25092}, {9460,27922}, {9589,34638}, {9708,19536}, {9776,24929}, {9782,37571}, {9803,19907}, {9809,11715}, {9939,34645}, {10022,17318}, {10109,37705}, {10154,34730}, {10164,11224}, {10165,15708}, {10171,37712}, {10181,32065}, {10186,28854}, {10303,11362}, {10389,35262}, {10430,18444}, {10525,33657}, {10543,15679}, {10585,37739}, {10588,37738}, {10589,37740}, {10680,28466}, {10706,11723}, {10708,11726}, {10709,11727}, {10710,11728}, {10711,11729}, {10712,11730}, {10713,11731}, {10714,11732}, {10715,11733}, {10716,11734}, {11001,12699}, {11024,17614}, {11037,11111}, {11041,15325}, {11049,12626}, {11112,15170}, {11115,28619}, {11177,11710}, {11236,26129}, {11249,21161}, {11278,15719}, {11346,19722}, {11352,19719}, {11354,19684}, {11520,17558}, {11684,15673}, {11717,17777}, {12100,12702}, {12245,15702}, {12437,37436}, 12526,30343}, {12645,15703}, {12735,31272}, {13462,21454}, {13624,19708}, {13667,33456}, {13787,33457}, {13846,19066}, {13847,19065}, {13902,32787}, {13959,32788}, {14450,15677}, {14942,35110}, {14996,16489}, {14997,16490}, {15022,37714}, {15246,37546}, {15485,37677}, {15621,19292}, {15671,24477}, {15676,16126}, {15682,18481}, {15688,28174}, {15709,26446}, {15710,17502}, {15715,31663}, {16484,17379}, {16498,17778}, {16676,31722}, {16712,17169}, {16801,20145}, {16884,17330}, {16971,37657}, {17045,17313}, {17051,31188}, {17095,32003}, {17180,18600}, {17257,17488}, {17273,17321}, {17322,32099}, {17393,32087}, {17395,31139}, {17556,37737}, {17589,28620}, {17592,35269}, {18135,25303}, {18526,19709}, {19325,37580}, {20085,33812}, {21356,28538}, {21735,31666}, {21806,24620}, {24654,26978}, {25557,30332}, {26062,34711}, {28309,32922}, {28453,32153}, {28458,34629}, {28459,34617}, {28629,34612}, {30305,36004}, {30384,31019}, {31141,36977}, {31227,36593}, {31266,37704}, {32577,37573}, {34603,34634}, {34604,34636}, {34607,34640}, {34608,34643}, {37290,37518}

X(38314) = midpoint of X(i) and X(j) for these {i,j}: {1,25055}, {3545,7967}, {5054,10247}
X(38314) = reflection of X(i) in X(j) for these (i,j): (2,25055), (3524,3653), (3545,5886), (5032,16475), (5657,5054), (5790,15699),(9778,10304), (10304,3576), (19875,19883), (25055,551)
X(38314) = anticomplement of X(19875)
X(38314) = X(643)-beth conjugate of X(6767)
X(38314) = X(649)-he conjugate of X(9359)
X(38314) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,2,3241), (1,8,20057), (1,10,3623), (1,551,2), (1,614,17015), (1,995,17018), (1,997,3957), (1,1125,145), (1,1201,19767), (1,1698,3635), (1,3616,8), (1,3622,3616), (1,3624,3244), (1,3636,3622), (1,10582,3872), (1,15808,3621), (1,16826,36534), (1,24331,4393), (1,28011,5262), (2,145,3679), (2,551,3616), (2,3241,8), (2,3617,3828), (2,3622,551), (2,3623,31145), (2,3679,9780), (2,17310,29611), (2,20049,3617), (2,29585,17310), (2,31145,10), (8,3616,5550), (8,5550,19877), (10,3623,20050), (10,20050,8), (56,4428,17549), (145,1125,9780), (145,4746,20053), (145,9780,8), (348,5543,32098), (354,10179,3877), (376,3656,962), (376,10595,3656), (392,5049,3873), (551,4669,15808), (631,34631,3654), (1125,3679,2), (1385,3656,376), (1385,10595,962), (1388,3485,4308), (1698,3635,3621), (3241,3616,2), (3244,3617,20053), (3244,3624,3617), (3244,3828,4677), (3244,4677,20049), (3476,15950,5226), (3582,10197,2), (3584,10199,2), (3616,9780,1125), (3616,20053,3624), (3616,20057,19877), (3617,20049,4677), (3617,20053,8), (3624,3828,2), (3624,4677,3828), (3632,19876,4745), (3633,3634,4678), (3635,15808,1698), (3654,10222,34631), (3679,4677,4746), (3828,4677,3617), (3892,5692,4430), (4301,30389,3522), (4677,20049,20053), (4745,19862,19876), (4861,27383,8), (5222,36534,8), (5439,31792,14923), (5550,20057,8), (5603,5731,9812), (5603,10246,5731), (5882,9624,3091), (5901,37624,944), (10246,10283,5603), (11112,15170,34611), (17023,29624,29627), (17310,17397,2), (17397,29585,29611), (19862,19876,2), (19875,19883,2), (19875,25055,19883), (26626,29570,5308), (30308,34648,3832), (36444,36462,3828)

X(38315) =  X(1)X(6)∩X(2)X(5846)

Barycentrics    a*(3*a^2+a*b+2*b^2+a*c+2*c^2) : :
X(38315) = 2*X(1)+X(6),X(1)+2*X(1386),4*X(1)-X(3242),5*X(1)+X(3751),7*X(1)+2*X(4663),X(1)+5*X(16491),7*X(1)-X(16496),X(6)-4*X(1386),2*X(6)+X(3242),5*X(6)-2*X(3751),7*X(6)-4*X(4663),X(6)-10*X(16491),7*X(6)+2*X(16496),X(8)-4*X(3589),X(67)-4*X(11735),X(69)-7*X(3622),2*X(125)+X(32298),2*X(141)-5*X(3616),X(145)+5*X(3618),2*X(182)+X(1482) (and many others)

See Kadir Altintas and Ercole Suppa, Euclid 856 .

X(38315) lies on these lines: {1,6}, {2,5846}, {8,3589}, {10,4989}, {31,17599}, {55,5096}, {56,1631}, {67,11735}, {69,3622}, {81,17024}, {100,17025}, {105,28895}, {125,32298}, {141,3616}, {145,3618}, {154,354}, {182,1482}, {511,10246}, {516,17301}, {517,5085}, {551,599}, {597,3241}, {612,37679}, {614,3745}, {748,29816}, {940,7191}, {944,5480}, {946,36990}, {952,14561}, {995,35272}, {999,2097}, {1058,5800}, {1086,4307}, {1125,3416}, {1150,29823}, {1350,1385}, {1351,37624}, {1352,5901}, {1376,17716}, {1387,5820}, {1388,1469}, {1420,24471}, {1428,2099}, {1456,4327}, {1483,18583}, {1503,5603}, {1621,20182}, {1691,10800}, {1961,8167}, {1974,11396}, {2076,11368}, {2098,2330}, {2352,35289}, {2550,17366}, {2646,10387}, {2930,11720}, {3011,17723}, {3052,3666}, {3056,34471}, {3295,36741}, {3303,12329}, {3304,22769}, {3315,14996}, {3564,10283}, {3576,31884}, {3624,3844}, {3685,17318}, {3744,5256}, {3752,5269}, {3791,29652}, {3818,18493}, {3846,29842}, {3867,7718}, {3873,9021}, {3883,4657}, {3886,4852}, {3892,34378}, {3920,4383}, {3923,28516}, {3936,29831}, {3941,37575}, {3976,18183}, {4000,4344}, {4252,37592}, {4255,5266}, {4310,17365}, {4318,5228}, {4321,6610}, {4349,4675}, {4353,17276}, {4361,5263}, {4363,32922}, {4387,32928}, {4417,29838}, {4423,5311}, {4428,17592}, {4437,29585}, {4645,17290}, {4666,37595}, {4676,17262}, {4682,5272}, {4719,37552}, {4850,37540}, {4865,29654}, {4884,26065}, {4974,36480}, {4981,19723}, {5026,7983}, {5045,24476}, {5049,34381}, {5050,10247}, {5092,12702}, {5132,37590}, {5138,15934}, {5262,5710}, {5429,11194}, {5550,34573}, {5695,28522}, {5698,17246}, {5711,5883}, {5718,26228}, {5731,29181}, {5733,20330}, {5845,11038}, {5886,10516}, {6329,20057}, {6593,7984}, {6776,10595}, {7050,17612}, {7221,14100}, {7292,9347}, {7373,37492}, {7716,11363}, {7967,14853}, {7976,24256}, {7982,10541}, {8148,12017}, {8616,17600}, {8679,28382}, {10168,34718}, {10704,28662}, {10705,28343}, {11235,33135}, {11365,20987}, {11477,15178}, {11646,11725}, {11723,14982}, {12264,24273}, {12588,15950}, {13331,14839}, {13605,25335}, {13910,19066}, {13972,19065}, {15534,34379}, {15668,16823}, {16020,17245}, {16830,17259}, {16834,27474}, {17014,20533}, {17016,37542}, {17061,26098}, {17126,17595}, {17150,24552}, {17274,28570}, {17275,19868}, {17323,24723}, {17367,32850}, {17382,28566}, {17594,21000}, {17722,29658}, {18525,19130}, {19145,35642}, {19146,35641}, {20470,21010}, {21358,25055}, {21747,36263}, {25760,29834}, {26230,30811}, {27949,32029}, {29580,31319}, {29634,33071}, {29636,32844}, {29639,31187}, {29646,33076}, {29648,33075}, {29660,32846}, {29666,33078}, {29684,33074}, {29686,32852}, {29814,37676}, {29815,32911}, {29852,33072}, {31140,33128}, {31670,34773}, {33844,37606}, {35262,37539}

X(38315) = midpoint of X(i) and X(j) for these {i,j}: {1,16475}, {5050,10247}, {7967,14853}
X(38315) = reflection of X(i) in X(j) for these (i,j): (6,16475), (10516,5886), (16475,1386), (21358,25055), (31884,3576)
X(38315) = X(513)-he conjugate of X(1054)
X(38315) = crosssum of X(1) and X(7174)
X(38315) = barycentric product X(i)*X(j) for these (i,j): (1,29598), (72,31918)
X(38315) = trilinear product X(i)*X(j) for these (i,j): (6,29598), (71,31918)
X(38315) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,6,3242), (1,1001,16777), (1,1386,6), (1,3246,16672), (1,7290,37), (1,11370,7968), (1,11371,7969), (1,16469,7174), (1,16478,958), (1,16491,1386), (1,16667,3243), (1,18991,5605), (1,18992,5604), (31,29819,17599), (81,17024,17597), (614,3745,37674), (1100,16972,6), (1125,3416,3763), (4682,5272,37682), (7968,7969,9575), (16666,36404,6), (17017,17469,55), (17716,29821,1376), (26230,33070,30811)

X(38316) =  X(1)X(6)∩X(2)X(3158)

Barycentrics    a*(3*a^2-4*a*b+b^2-4*a*c-6*b*c+c^2) : :
X(38316) = 2*X(1)+X(9),X(1)+2*X(1001),4*X(1)-X(3243),7*X(1)+2*X(5220),5*X(1)+X(5223),5*X(1)+4*X(15254),7*X(1)-4*X(15570),X(7)-7*X(3622),X(8)-4*X(6666),X(9)-4*X(1001),2*X(9)+X(3243),7*X(9)-4*X(5220),5*X(9)-2*X(5223),5*X(9)-8*X(15254),7*X(9)+8*X(15570),2*X(142)+X(390),2*X(142)-5*X(3616),X(145)+5*X(18230),X(145)+2*X(24393),4*X(214)-X(5528),X(390)+5*X(3616),4*X(551)-X(6173),2*X(960)+X(15185),8*X(1001)+X(3243),7*X(1001)-X(5220),10*X(1001)-X(5223),5*X(1001)-2*X(15254),7*X(1001)+2*X(15570) (and many others)

See Kadir Altintas and Ercole Suppa, Euclid 856 .

X(38316) lies on these lines: {1,6}, {2,3158}, {3,23397}, {7,1420}, {8,6666}, {21,10390}, {40,5883}, {55,5437}, {56,12560}, {57,1621}, {63,29817}, {105,28879}, {142,390}, {145,18230}, {165,3742}, {200,3748}, {214,5528}, {344,4901}, {354,3928}, {376,516}, {480,4853}, {497,25525}, {517,21153}, {527,11038}, {528,15015}, {614,37553}, {673,16831}, {968,3677}, {971,10246}, {1058,1125}, {1201,4343}, {1319,4321}, {1320,6594}, {1385,5732}, {1387,3254}, {1388,8581}, {1445,3340}, {1482,31658}, {1706,3295}, {2098,15837}, {2136,3303}, {2320,18450}, {2346,3680}, {2646,4326}, {2802,31393}, {2951,30389}, {3174,8583}, {3241,5686}, {3305,3957}, {3306,35445}, {3333,5248}, {3358,7971}, {3452,10578}, {3475,28609}, {3485,12573}, {3612,5880}, {3617,12630}, {3623,17121}, {3624,3826}, {3636,5542}, {3646,3811}, {3685,4659}, {3720,5269}, {3744,17022}, {3749,26102}, {3750,5272}, {3755,16020}, {3848,4421}, {3869,11025}, {3870,5284}, {3873,3929}, {3878,20116}, {3883,17296}, {3884,30329}, {3886,16823}, {3890,7672}, {3898,16200}, {3920,25430}, {3921,19536}, {3938,7322}, {4312,25557}, {4313,15006}, {4335,15839}, {4640,10980}, {4679,37703}, {4779,31995}, {5045,31424}, {5049,16418}, {5218,31190}, {5249,9580}, {5250,11518}, {5253,7676}, {5268,17715}, {5281,6692}, {5426,17525}, {5432,31249}, {5573,17594}, {5735,20330}, {5745,10580}, {5759,10595}, {5762,10283}, {5779,37624}, {5805,5901}, {5817,7967}, {5846,29573}, {6764,17554}, {6765,11108}, {6767,9623}, {6896,8227}, {7674,21627}, {7675,10384}, {7962,8257}, {7982,30143}, {7987,11495}, {8167,8580}, {8232,10106}, {8273,12651}, {9451,14439}, {9578,10587}, {9841,11496}, {10268,13374}, {10383,17612}, {10434,18613}, {10695,28345}, {11281,11522}, {11372,16132}, {11407,17613}, {11526,37787}, {12575,28629}, {12625,24389}, {12730,31272}, {13405,26105}, {14100,34471}, {15726,24644}, {16593,29598}, {16833,28581}, {17313,28566}, {17397,20533}, {18482,18493}, {21625,30478}, {24929,35272}, {24987,37723}, {27475,29597}, {27484,29584}, {28071,35293}, {29007,30318}, {29812,37554}, {30282,35271}, {31434,34122}, {31672,34773}

X(38316) = midpoint of X(i) and X(j) for these {i,j}: {2,8236}, {3241,5686}, {5817,7967}
X(38316) = X(513)-he conjugate of X(1054)
X(38316) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,9,3243), (1,405,6762), (1,1001,9), (1,3731,3242), (1,5234,34791), (1,7290,1449), (1,15485,3751), (1,16487,1386), (1,16496,15600), (1,31435,11523), (2,10389,3158), (55,10582,5437), (145,18230,24393), (354,4512,3928), (390,3616,142), (1125,2550,20195), (1125,30331,2550), (1621,4666,57), (3742,4428,165), (3748,4423,200), (3751,15485,15601), (3870,5284,7308), (5223,15254,9), (10384,13384,7675), (13405,26105,30827), (16484,35227,3247), (17594,29820,5573), (19860,37556,3680)

leftri

Centers of Vu pedal-centroidal circles: X(38317)-X(38319)

rightri

This preamble is based on notes from Vu Thanh Tung, May 4, 2020.

Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC. Let

G = X(2) = centroid of ABC
A0B0C0 = medial triangle of ABC
A1B1C1 = pedal triangle of P
A2 = centroid of A1B0C0
B2 = centroid of B1A0C0
C2 = centroid of C1A0B0

The points G, A2, B2, C2 lie on a circle, here named the Vu pedal-centroidal circle of P. The center of this circle is the point

V(P) = a^4 (2 p + q + r) + (b^2 - c^2)^2 (3 p + 2 (q + r)) - a^2 (b^2 + c^2) (5 p + 3 (q + r)) : :

See PedalCentroidCircle

The appearance of (i,j) in the following list means that V(X(i)) = X(j):
(1,11230), (2,15699), (3,2), (4,5), (5,547), (6,38317), (7,38171), (8,38042), (9,38318), (10,10172), (11,38319), (20,549), (74,34128), (98,34127), (140, 3628), (376, 11539), (381, 5055), (382, 381)

If P* is the circumcircle-inverse of P, then the Vu pedal-centroidal circles of P and P* are tangent at X(2). (Randy Hutson, May 5, 2020)

If P lies on the Euler line, then V(P) also lies on the Euler line. If P lies on the line at infinity, then V(P) = P. (Randy Hutson, May 5, 2020)

V maps the circumcircle onto the circle of the points V(X(104)) = X(34126), V(X(98)) = X(34127), and V(X(74)) = X(32128); the center of this circle is X(2). (Randy Hutson, May 5, 2020)

X(38317) = CENTER OF THE VU PEDAL-CENTROIDAL CIRCLE OF X(6)

Barycentrics    a^6 - 6 a^2 b^2 c^2 - 2 a^4 (b^2 + c^2) + (b^2 - c^2)^2 (b^2 + c^2) : :
X(38317) = 2*X(2)+X(5476) = 5*X(2)-X(10519) = 3*X(2)+X(14853) = 5*X(2)+X(20423) = X(3)+2*X(19130) = X(4)+2*X(5092) = 2*X(5)+X(182) = X(5)+2*X(3589) = 4*X(5)-X(3818) = X(182)-4*X(3589) = 2*X(182)+X(3818) = X(262)+2*X(32149) = 5*X(5476)+2*X(10519) = 3*X(5476)-2*X(14853) = 5*X(5476)-2*X(20423) = X(10519)+5*X(14561) = 3*X(10519)+5*X(14853) = 3*X(14561)-X(14853) = 5*X(14561)-X(20423) = 5*X(14853)-3*X(20423)

X(38317) lies on these lines: {2,51}, {3,7889}, {4,5092}, {5,182}, {6,17}, {30,17508}, {66,32767}, {67,25556}, {69,5067}, {83,3406}, {98,7875}, {110,20301}, {113,32305}, {114,11174}, {125,19140}, {140,3098}, {141,576}, {161,5020}, {184,37990}, {193,22330}, {230,5039}, {381,5085}, {389,14786}, {403,19124}, {518,11230}, {524,15520}, {542,5050}, {547,597}, {549,29181}, {575,1352}, {578,7405}, {599,5093}, {626,35431}, {631,14810}, {632,21850}, {858,22112}, {1177,32743}, {1350,3526}, {1351,3763}, {1353,6329}, {1386,9956}, {1428,7951}, {1511,32273}, {1594,1974}, {1595,13347}, {1657,33751}, {1691,5475}, {1692,7603}, {1843,7505}, {2030,31415}, {2072,19131}, {2330,7741}, {2777,9818}, {2781,34128}, {2916,18378}, {3066,32223}, {3088,17704}, {3091,20190}, {3094,31455}, {3095,7822}, {3543,33750}, {3545,11645}, {3549,11574}, {3788,11272}, {3817,38118}, {3851,12017}, {3855,14927}, {3867,21841}, {4045,35930}, {4260,6861}, {5017,7749}, {5031,35377}, {5054,19924}, {5056,6776}, {5071,11179}, {5072,10541}, {5079,18440}, {5096,7489}, {5102,14848}, {5103,7761}, {5116,7748}, {5138,6881}, {5157,5576}, {5171,8362}, {5309,7697}, {5422,7571}, {5449,19139}, {5544,6723}, {5569,38230}, {5587,38029}, {5603,38116}, {5621,38789}, {5651,14389}, {5790,38315}, {5817,38115}, {5845,38166}, {5846,38040}, {5847,10172}, {5848,38319}, {5886,38047}, {5892,23329}, {5972,11284}, {5999,16987}, {6034,15561}, {6248,7803}, {6393,37647}, {6403,14940}, {6593,20304}, {6639,9967}, {6640,37511}, {6642,10182}, {6656,10358}, {6680,35424}, {6683,37466}, {6689,15577}, {6697,34117}, {6699,32271}, {6721,31489}, {6771,9749}, {6774,9750}, {7380,17352}, {7383,13598}, {7387,31521}, {7392,35260}, {7394,22352}, {7399,16657}, {7403,37515}, {7404,9729}, {7486,15516}, {7495,34417}, {7503,32600}, {7533,15080}, {7539,10601}, {7570,15018}, {7577,19128}, {7583,13972}, {7584,13910}, {7606,8176}, {7759,8177}, {7769,18906}, {7801,32447}, {7804,37242}, {7811,22521}, {7815,20576}, {7819,9737}, {7831,10788}, {7846,37334}, {7852,13354}, {7867,35389}, {7876,12110}, {7884,14651}, {7913,15980}, {8369,9734}, {8550,18358}, {9024,38168}, {9041,38022}, {9053,10283}, {9306,35283}, {9730,14787}, {9735,37341}, {9736,37340}, {9970,15059}, {9976,12900}, {9993,37455}, {10011,15491}, {10127,11202}, {10128,10192}, {10165,38146}, {10175,38049}, {10224,19154}, {10246,38144}, {10255,19129}, {10272,25328}, {10282,31267}, {10984,16658}, {11064,16187}, {11180,33748}, {11286,23698}, {11548,13567}, {11550,37353}, {11585,19126}, {11646,32135}, {11669,35005}, {12022,14788}, {12177,14061}, {12294,37119}, {13329,36530}, {14036,21166}, {14160,16041}, {14162,32984}, {14356,34130}, {14644,15462}, {14994,32832}, {15004,37636}, {15026,32191}, {15045,16223}, {15088,32274}, {15482,37459}, {15805,19149}, {18382,34785}, {18383,36989}, {18390,37347}, {18400,23041}, {18418,18537}, {19138,33547}, {19141,20302}, {20126,25566}, {22510,22690}, {22511,22688}, {25563,34778}, {26446,38035}, {28538,38083}, {29633,31394}, {29663,37619}, {31239,35439}, {31395,33159}, {33220,38748}, {34218,35282}, {34573,37517}, {36757,37835}, {36758,37832}, {37454,37648}, {38108,38186}, {38117,38150}, {38122,38145}, {38147,38760}

X(38317) = midpoint of X(i) and X(j) for these {i,j}: {2, 14561}, {5, 38110}, {381, 5085}, {549, 38136}, {599, 5093}, {1352, 14912}, {3545, 38064}, {3817, 38118}, {5050, 10516}, {5054, 38072}, {5480, 21167}, {5587, 38029}, {5603, 38116}, {5621, 38789}, {5790, 38315}, {5817, 38115}, {5886, 38047}, {6034, 15561}, {7697, 13331}, {10165, 38146}, {10175, 38049}, {10246, 38144}, {10283, 38165}, {10519, 20423}, {11230, 38167}, {14644, 15462}, {14848, 21358}, {15699, 38079}, {23042, 23325}, {26446, 38035}, {38040, 38042}, {38108, 38186}, {38117, 38150}, {38122, 38145}, {38147, 38760}, {38166, 38171}
X(38317) = reflection of X(i) in X(j) for these (i,j): (182, 38110), (3098, 21167), (5085, 10168), (5476, 14561), (14912, 575), (21167, 140), (38110, 3589)
X(38317) = complement of the isogonal conjugate of X(14495)
X(38317) = complement of the complement of X(14853)
X(38317) = intersection, other than A,B,C, of conics {{A, B, C, X(182), X(2979)}} and {{A, B, C, X(262), X(2963)}}
X(38317) = crossdifference of every pair of points on line {X(1510), X(3288)}
X(38317) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7605, 5640), (2, 9753, 15819), (5, 182, 3818), (5, 3589, 182), (6, 1656, 24206), (6, 24206, 34507), (140, 5480, 3098), (141, 18583, 576), (206, 20300, 18381), (547, 597, 11178), (631, 31670, 14810), (1351, 5070, 3763), (1352, 3618, 575), (1656, 25555, 34507), (3090, 3618, 1352), (3628, 18583, 141), (5050, 5055, 10516), (10168, 25565, 381), (24206, 25555, 6), (33478, 33479, 262)

X(38318) = CENTER OF THE VU PEDAL-CENTROIDAL CIRCLE OF X(9)

Barycentrics    2 a^6 + 6 a^2 b (b - c)^2 c - 3 a^5 (b + c) + 2 (b - c)^4 (b + c)^2 - 5 a (b - c)^2 (b + c)^3 - 2 a^4 (2 b^2 + b c + 2 c^2) + 8 a^3 (b^3 + b^2 c + b c^2 + c^3) : :
X(38318) = 3*X(2)+X(5817) = 5*X(2)-X(21151) = X(5)+2*X(6666) = 4*X(5)-X(18482) = 2*X(5)+X(31658) = X(7)-13*X(5067) = X(9)+5*X(1656) = X(142)-4*X(3628) = 5*X(1656)-X(38107) = 5*X(5817)+3*X(21151) = X(5817)-3*X(38108) = 8*X(6666)+X(18482) = 4*X(6666)-X(31658) = 5*X(15699)-X(38080) = 3*X(15699)-X(38171) = X(18482)+2*X(31658) = X(18482)+4*X(38113) = X(21151)+5*X(38108) = 3*X(21151)-5*X(38122) = 3*X(38108)+X(38122)

X(38318) lies on these lines: {2,971}, {5,516}, {7,5067}, {9,1656}, {142,3628}, {381,21153}, {518,11230}, {527,15699}, {528,38083}, {547,5762}, {549,38139}, {631,31672}, {1001,9956}, {2801,34126}, {3090,5805}, {3525,36991}, {3526,5732}, {3545,38067}, {3817,38130}, {4860,5219}, {5044,5761}, {5054,38075}, {5055,38150}, {5056,5759}, {5070,5779}, {5079,31671}, {5084,38149}, {5587,11108}, {5603,38126}, {5657,5806}, {5722,8236}, {5731,17552}, {5790,38316}, {5853,38042}, {5856,38319}, {5901,24393}, {6173,15703}, {6883,28160}, {6886,31793}, {7308,7988}, {7741,15837}, {8728,22792}, {10165,38158}, {10175,38059}, {10200,13373}, {10246,38154}, {10283,38175}, {10516,38117}, {10592,12573}, {10679,38200}, {10915,38176}, {11038,11374}, {17768,38172}, {19876,24644}, {19877,35514}, {26446,38037}, {30331,31399}, {38133,38204}, {38159,38760}

X(38318) = midpoint of X(i) and X(j) for these {i,j}: {2, 38108}, {5, 38113}, {9, 38107}, {381, 21153}, {549, 38139}, {3545, 38067}, {3817, 38130}, {5054, 38075}, {5587, 38031}, {5603, 38126}, {5790, 38316}, {5805, 21168}, {5817, 38122}, {5886, 38057}, {10165, 38158}, {10175, 38059}, {10246, 38154}, {10283, 38175}, {10516, 38117}, {11230, 38179}, {15699, 38082}, {26446, 38037}, {38042, 38043}, {38159, 38760}
X(38318) = reflection of X(i) in X(j) for these (i,j): (31658, 38113), (38113, 6666)
X(38318) = complement of X(38122)
X(38318) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5817, 38122), (5, 6666, 31658), (5, 31658, 18482), (3090, 18230, 5805), (5070, 5779, 20195), (38108, 38122, 5817)

X(38319) = CENTER OF THE VU PEDAL-CENTROIDAL CIRCLE OF X(11)

Barycentrics    2 a^7 - 2 a^6 (b + c) + 5 (b - c)^4 (b + c)^3 + a^5 (-9 b^2 + 8 b c - 9 c^2) - a (b^2 - c^2)^2 (5 b^2 - 14 b c + 5 c^2) + 3 a^4 (3 b^3 + b^2 c + b c^2 + 3 c^3) - 4 a^2 (b - c)^2 (3 b^3 + 5 b^2 c + 5 b c^2 + 3 c^3) + 2 a^3 (6 b^4 - 11 b^3 c + 4 b^2 c^2 - 11 b c^3 + 6 c^4) : :
X(38319) = 5*X(2)-X(34474) = X(5)+2*X(6667) = 2*X(5)+X(6713) = 7*X(5)-X(22799) = 5*X(5)+X(38602) = X(11)+5*X(1656) = 5*X(11)+X(12331) = 5*X(1656)-X(38752) = 4*X(6667)-X(6713) = 14*X(6667)+X(22799) = 10*X(6667)-X(38602) = 7*X(6713)+2*X(22799) = 5*X(6713)-2*X(38602) = X(12331)-5*X(38752) = X(22799)+7*X(34126) = 5*X(22799)+7*X(38602) = 5*X(23513)+X(34474) = 3*X(23513)+X(38760) = 5*X(34126)-X(38602) = 3*X(34474)-5*X(38760)

X(38319) lies on these lines: {2,5840}, {5,2829}, {11,498}, {100,5067}, {104,5056}, {119,3090}, {149,38763}, {381,21154}, {528,15699}, {546,38759}, {547,551}, {549,38141}, {632,22938}, {1387,9956}, {1484,20400}, {2800,10171}, {2802,10172}, {3035,3628}, {3036,5901}, {3091,38761}, {3525,10724}, {3526,24466}, {3545,38069}, {3634,16174}, {3817,38133}, {3825,31659}, {5054,38077}, {5055,38755}, {5068,10728}, {5070,10738}, {5072,38753}, {5079,10742}, {5587,38032}, {5603,38128}, {5817,38124}, {5848,38317}, {5851,38171}, {5854,38042}, {5856,38318}, {5886,34122}, {6174,15703}, {6246,19862}, {6702,11729}, {7486,10587}, {10165,38161}, {10246,38156}, {10283,38177}, {10516,38119}, {12119,34595}, {12812,38757}, {12832,37692}, {20107,37290}, {20418,35018}, {26446,38038}, {38108,38205}, {38122,38159}, {38131,38150}

X(38319) = midpoint of X(i) and X(j) for these {i,j}: {2, 23513}, {5, 34126}, {11, 38752}, {381, 21154}, {549, 38141}, {3545, 38069}, {3817, 38133}, {5054, 38077}, {5587, 38032}, {5603, 38128}, {5817, 38124}, {5886, 34122}, {10165, 38161}, {10175, 32557}, {10246, 38156}, {10283, 38177}, {10516, 38119}, {11230, 38182}, {15699, 38084}, {26446, 38038}, {38042, 38044}, {38108, 38205}, {38122, 38159}, {38131, 38150}, {38171, 38180}
X(38319) = reflection of X(i) in X(j) for these (i,j): (6713, 34126), (34126, 6667)
X(38319) = complement of X(38760)
X(38319) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 6667, 6713), (3090, 31272, 119), (5070, 10738, 31235)

X(38320) =  EULER LINE INTERCEPT OF X(11745)X(12278)

Barycentrics    6 a^10+2 a^2 b^2 c^2 (b^2-c^2)^2-9 a^8 (b^2+c^2)-3 (b^2-c^2)^4 (b^2+c^2)-2 a^6 (3 b^4-7 b^2 c^2+3 c^4)+4 a^4 (3 b^6-4 b^4 c^2-4 b^2 c^4+3 c^6) : :
Barycentrics    S^2 (14 R^2-3 SW)-3 SB SC (8 R^2-3 SW) : :
X(38320) = X(2)+2*X(3575),5*X(2)-8*X(9825),4*X(2)-X(12225),7*X(2)-4*X(12362),X(376)-4*X(31833),2*X(381)+X(6240),7*X(381)-X(18562),X(381)-4*X(31830),X(3534)+2*X(11819),X(3543)-4*X(6756),5*X(3575)+4*X(9825),8*X(3575)+X(12225),7*X(3575)+2*X(12362),5*X(3839)-3*X(37077),4*X(3845)-X(18560),4*X(5066)-X(18563),5*X(5071)-2*X(12605),7*X(6240)+2*X(18562),X(6240)+8*X(31830),2*X(7540)-5*X(7576),8*X(7540)-5*X(34603),2*X(7553)+X(11001),4*X(7576)-X(34603),7*X(7576)-X(34613),2*X(7667)-3*X(15705),8*X(11745)+X(12278),X(14516)+2*X(14831),X(15683)-4*X(31829),X(18564)-4*X(23410),7*X(34603)-4*X(34613)

As a point on the Euler line, X(38320) has Shinagawa coefficients (E-6*F,6*(E+3*F)).

See Kadir Altintas and Ercole Suppa, Euclid 859 .

X(38320) lies on these lines: {2,3}, {11745,12278}, {14516,14831}, {16226,18400}

X(38320) = midpoint of X(3545) and X(18559)
X(38320) = reflection of X(14269) in X(13490)
X(38320) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381,3515,2), (3515,3575,6240)

X(38321) =  EULER LINE INTERCEPT OF X(49)X(12233)

Barycentrics    2 a^10-3 a^8 (b^2+c^2)-(b^2-c^2)^4 (b^2+c^2)-2 a^6 (b^4-3 b^2 c^2+c^4)-a^4 (-4 b^6+6 b^4 c^2+6 b^2 c^4-4 c^6) : :
Barycentrics    S^2 (5 R^2-SW)-3 SB SC (3 R^2-SW) : :
X(38321) = X(3)+2*X(3575),3*X(3)-2*X(7667),X(3)-4*X(31833),X(4)-4*X(31830),2*X(5)+X(6240),4*X(5)-X(18563),X(20)+2*X(11819),4*X(140)-X(12225),X(382)-4*X(6756),4*X(546)-X(18560),5*X(1656)-8*X(9825),5*X(1656)-2*X(12605),X(1657)+2*X(7553),X(1657)-4*X(31829),2*X(1885)-5*X(3843),2*X(1885)+X(18565),5*X(3091)+X(34797),7*X(3526)-4*X(12362),3*X(3545)-4*X(23410),5*X(3567)+X(12278),5*X(3567)-2*X(12370),3*X(3575)+X(7667),X(3575)+2*X(31833),5*X(3843)+X(18565),7*X(3851)-X(18562),5*X(3858)-8*X(13163),9*X(5054)-8*X(7734),3*X(5055)-4*X(10127),3*X(5055)-X(18564),3*X(5055)-2*X(34664),4*X(5462)-X(21659),2*X(6102)+X(14516),2*X(6146)-5*X(37481),2*X(6240)+X(18563),3*X(7540)-2*X(34603),5*X(7540)-2*X(34613),X(7553)+2*X(31829),3*X(7576)-X(34603),5*X(7576)-X(34613),X(7667)-6*X(31833),4*X(9729)-X(11750),4*X(9825)-X(12605),3*X(9825)-2*X(13361),4*X(10127)-X(18564),X(10575)+2*X(13419),4*X(10691)-5*X(15693),X(12134)+2*X(13568),2*X(12134)+X(34783),X(12278)+2*X(12370),X(12289)-7*X(15043),3*X(12605)-8*X(13361),2*X(13491)+X(16659),4*X(13568)-X(34783),4*X(13630)-X(34224),X(18561)-5*X(19709),5*X(34603)-3*X(34613)

As a point on the Euler line, X(38321) has Shinagawa coefficients (E-4*F,3*(E+4*F)).

See Kadir Altintas and Ercole Suppa, Euclid 859 .

X(38321) lies on these lines: {2,3}, {49,12233}, {51,16222}, {68,37490}, {184,7706}, {265,13567}, {343,3581}, {539,14831}, {541,21650}, {542,11562}, {569,34785}, {1236,37671}, {1994,12383}, {2777,16194}, {2883,3521}, {3313,19924}, {3567,12278}, {3574,12038}, {4846,31383}, {5050,10169}, {5462,21659}, {5480,12121}, {5504,11597}, {5946,12022}, {6102,14516}, {6146,37481}, {6288,12359}, {6723,15432}, {6781,36412}, {9729,11750}, {9730,18400}, {9786,25738}, {10575,13419}, {11179,36989}, {11438,18474}, {11561,12236}, {12118,36749}, {12134,13568}, {12289,15043}, {12918,18876}, {13491,16659}, {13561,22804}, {13630,34224}, {14542,15317}, {14805,37649}, {14855,29012}, {14910,18373}, {15053,25739}, {15061,23332}, {15367,23320}, {15466,16263}, {17845,36752}, {18350,22660}, {19467,36753}, {21243,32110}, {21850,22151}, {28198,34657}, {30714,34986}

X(38321) = midpoint of X(2) and X(18559)
X(38321) = reflection of X(i) in X(j) for these (i,j): (4,13490), (3830,428), (7540,7576), (12022,5946), (13490,31830), (18564,34664), (34664,10127)
X(38321) = (2,186,34477), (2,34608,15818), (3,3830,34609), (3,18420,37347), (3,18494,31723), (4,6644,2072), (4,22467,13371), (5,6240,18563), (5,16532,34330), (5,34477,2), (20,37349,13596), (550,7403,14130), (2043,2044,7526), (3518,34007,15761), (3567,12278,12370), (3575,31833,3), (3830,34609,31723), (3843,18565,1885), (5055,18564,34664), (5066,18579,2), (5133,10295,18570), (6642,12173,18404), (7394,35481,31861), (7544,35471,7526), (7545,31726,1596), (7553,31829,1657), (9825,12605,1656), (10127,34664,5055), (12134,13568,34783), (15760,37458,2070), (15765,18585,1594), (16238,23047,10255), (16532,34330,10018), (17928,18569,37452), (18420,18533,3), (18494,34609,3830), (18586,18587,18404), (31236,35472,18580), (36437,36455,31181)

X(38322) =  EULER LINE INTERCEPT OF X(51)X(30522)

Barycentrics    2 a^10-2 a^6 (b^2-c^2)^2+a^2 b^2 c^2 (b^2-c^2)^2-3 a^8 (b^2+c^2)-(b^2-c^2)^4 (b^2+c^2)-a^4 (-4 b^6+5 b^4 c^2+5 b^2 c^4-4 c^6) : :
Barycentrics    S^2 (9 R^2-2 SW)-3 SB SC (5 R^2-2 SW) : :
X(38322) = 5*X(4)+X(18565),X(5)+2*X(3575),5*X(5)-2*X(12605),3*X(5)-4*X(23410),X(5)-4*X(31830),3*X(5)-2*X(34664),5*X(381)-X(18561),2*X(546)+X(6240),5*X(549)-4*X(10691),X(550)+2*X(11819),X(550)-4*X(31833),5*X(632)-8*X(9825),5*X(3091)-8*X(13163),3*X(3545)-X(18564),5*X(3575)+X(12605),3*X(3575)+2*X(23410),X(3575)+2*X(31830),3*X(3575)+X(34664),X(3627)-4*X(6756),4*X(3628)-X(12225),7*X(3832)-X(18562),5*X(3843)+X(34797),5*X(3843)-3*X(37077),4*X(3850)-X(18563),4*X(3861)-X(18560),5*X(5946)-4*X(32068),X(7540)-3*X(7576),5*X(7540)-3*X(34603),3*X(7540)-X(34613),2*X(7553)+X(15704),5*X(7576)-X(34603),9*X(7576)-X(34613),2*X(7667)-3*X(17504),4*X(10095)-X(21659),5*X(10127)-4*X(13361),4*X(10127)-3*X(15699),4*X(11745)-X(12370),X(11750)-4*X(12006),X(11819)+2*X(31833),3*X(12605)-10*X(23410),X(12605)-10*X(31830),3*X(12605)-5*X(34664),2*X(13419)+X(13491),3*X(15686)-2*X(34614),5*X(18559)+X(18561),X(23410)-3*X(31830),6*X(31830)-X(34664),4*X(32165)-X(34799),9*X(34603)-5*X(34613),X(34797)+3*X(37077)

As a point on the Euler line, X(38322) has Shinagawa coefficients (E-8*F,9*E+24*F).

See Kadir Altintas and Ercole Suppa, Euclid 859 .

X(38322) lies on these lines: {2,3}, {51,30522}, {143,25711}, {539,13368}, {542,6102}, {568,7730}, {1147,20424}, {2493,7747}, {3410,32608}, {3574,32171}, {5449,22804}, {5476,34785}, {5946,18400}, {7816,34827}, {8262,18553}, {9545,22051}, {10095,21659}, {10264,11438}, {11745,12370}, {11750,12006}, {11801,18430}, {12118,20423}, {13352,34153}, {13419,13491}, {15361,34826}, {15738,32137}, {16776,29012}, {18121,32134}, {18356,37490}, {18488,32210}, {28160,34633}, {28174,34657}, {28224,34668}, {32165,34799}

X(38322) = midpoint of X(381) and X(18559)
X(38322) = reflection of X(i) in X(j) for these (i,j): (3845,13490),(15687,428),(34664,23410)
X(38322) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,37922,15330), (3,382,5189), (3,14789,140), (4,12106,5), (5,3627,18572), (5,37458,7575), (24,13621,12106), (24,16868,468), (376,7544,14787), (381,37922,2), (468,546,5), (468,3575,6240), (3575,31830,5), (7506,18377,5), (11438,34514,10264), (11818,18533,18570), (11819,31833,550), (18281,37814,549), (18586,18587,3153) ,(23410,34664,5)

X(38323) =  EULER LINE INTERCEPT OF X(64)X(34118)

Barycentrics    2 a^10-2 a^2 b^2 c^2 (b^2-c^2)^2-3 a^8 (b^2+c^2)-(b^2-c^2)^4 (b^2+c^2)-2 a^6 (b^4-5 b^2 c^2+c^4)+4 a^4 (b^6-2 b^4 c^2-2 b^2 c^4+c^6) : :
Barycentrics    S^2 (6 R^2-SW)-3 SB SC (4 R^2-SW) : :
X(38323) = 2*X(3)+X(6240),4*X(3)-X(12225),X(4)-4*X(31833),4*X(5)-X(18560),X(20)+2*X(3575),X(20)-4*X(31829),3*X(20)-2*X(34614),4*X(140)-X(18563),2*X(185)+X(14516),3*X(381)-4*X(23410),X(382)-4*X(31830),5*X(631)-2*X(12605),5*X(631)+X(34797),5*X(1656)+X(18565),X(1657)+2*X(11819),2*X(1885)-5*X(3091),X(1885)-4*X(9825),2*X(1885)-3*X(37077),5*X(3091)-8*X(9825),5*X(3091)-3*X(37077),X(3146)-4*X(6756),7*X(3523)-4*X(12362),7*X(3526)-X(18562),X(3529)+2*X(7553),3*X(3545)-4*X(10127),X(3575)+2*X(31829),3*X(3575)+X(34614),7*X(3832)-4*X(13488),3*X(5054)-X(18564),3*X(5640)-2*X(16657),3*X(5731)-2*X(34634),X(5889)-4*X(13568),2*X(6146)-5*X(10574),2*X(6146)+X(12278),2*X(6240)+X(12225),X(6241)+2*X(12134),2*X(7540)-3*X(7576),4*X(7540)-3*X(34603),3*X(7576)-X(34613),2*X(7667)-3*X(10304),8*X(7734)-9*X(15708),4*X(9729)-X(21659),8*X(9825)-3*X(37077),5*X(10574)+X(12278),2*X(10575)+X(16659),4*X(10691)-5*X(15692),2*X(11591)+X(34798),4*X(12241)-7*X(15043),X(12279)+2*X(16655),2*X(12370)-5*X(37481),X(12528)-4*X(31832),2*X(12605)+X(34797),5*X(15694)-X(18561),4*X(18914)-X(34799),6*X(31829)-X(34614),3*X(34603)-2*X(34613)

As a point on the Euler line, X(38323) has Shinagawa coefficients (E-2*F,6*F).

See Kadir Altintas and Ercole Suppa, Euclid 859 .

X(38323) lies on these lines: {2,3}, {64,34118}, {67,15054}, {146,15052}, {185,542}, {515,34668}, {516,34657}, {524,5889}, {541,12162}, {566,26216}, {567,12121}, {597,13434}, {895,8550}, {1154,11660}, {1503,7729}, {1514,10546}, {1975,5877}, {2777,15030}, {2883,15139}, {3260,7750}, {3521,5655}, {3580,11438}, {3818,26156}, {4846,11456}, {5085,23049}, {5434,9630}, {5476,11424}, {5640,16657}, {5731,34634}, {5894,15062}, {6146,10574}, {6241,12134}, {6247,15138}, {6288,20126}, {7592,12118}, {7706,13352}, {8549,17845}, {9729,21659}, {9730,12022}, {9927,26879}, {10575,16659}, {10605,11442}, {10733,37648}, {10984,34785}, {10990,18553}, {11179,19467}, {11430,14389}, {11591,34798}, {12233,34148}, {12241,15043}, {12279,16655}, {12293,18912}, {12370,37481}, {12383,15032}, {12528,31832}, {13567,15053}, {14915,16658}, {14983,18876}, {15055,23328}, {15131,35904}, {15136,22660}, {15305,15311}, {15873,20192}, {16252,35266}, {18392,26913}, {18396,18911}, {18914,34799}, {21243,21663}, {26206,31670}, {28164,34633}, {29959,36201}, {37487,37638}

X(38323) = midpoint of X(376) and X(18559)
X(38323) = reflection of X(i) in X(j) for these (i,j): (3543,428), (3830,13490), (12022,9730), (34603,7576), (34613,7540)
X(38323) = anticomplement of X(34664)
X(38323) = X(6240)-Gibert-Moses centroid
X(38323) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,37941,549), (3,4,858), (3,5,37118), (3,381,18281), (3,382,14791), (3,1656,18580), (3,6240,12225), (3,12173,37444), (3,21284,7488), (3,34725,31152), (3,37196,20), (4,37777,235), (5,549,34331), (20,6815,7503), (20,7503,34005), (20,7544,1593), (186,7552,34351), (378,18420,5133), (381,18281,1594), (381,34622,1593), (550,7399,14118), (631,34797,12605), (1885,9825,3091), (2043,2044,378), (3543,37444,34725), (3575,31829,20), (3575,37196,6240), (5133,16386,378), (6240,37118,18560), (6676,37931,10298), (7540,34613,34603), (7558,35503,3), (7576,34613,7540), (10024,37814,10018), (10574,12278,6146), (12173,31152,34725), (12173,34725,3543), (14118,37978,3), (14709,14710,7503), (14788,35491,7526), (15760,34351,7552), (15765,18585,2072), (18396,37475,18911), (18494,21312,7391), (22467,34007,5), (31152,34725,37444), (36437,36455,31180)

X(38324) =  X(3)X(2820)∩X(40)X(1635)

Barycentrics    a*(b - c)*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5 - a^4*c + 3*a^3*b*c - 3*a^2*b^2*c + a*b^3*c - 2*a^3*c^2 - 3*a^2*b*c^2 + b^3*c^2 + 2*a^2*c^3 + a*b*c^3 + b^2*c^3 + a*c^4 - c^5) : :
X(38324) = X[40] - 3 X[1635], 3 X[4728] - 5 X[8227], 3 X[4763] - 2 X[6684]

X(38324) lies on these lines: {3, 2820}, {40, 1635}, {650, 28292}, {659, 2814}, {676, 2826}, {812, 946}, {2821, 9508}, {2827, 19916}, {3309, 4794}, {3743, 11615}, {4728, 8227}, {4763, 6684}, {8645, 10902}

X(38324) = reflection of X(3743) in X(11615)
X(38324) = center of circle {{X(11), X(101), X(105), X(5513), X(5580), X(6326)}}
X(38324) = {X(40),X(1635)}-harmonic conjugate of X(38327)

X(38325) =  X(11)X(244)∩X(100)X(101)

Barycentrics    a*(b - c)*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4 + a^3*c - 3*a^2*c^2 + 3*a*c^3 - c^4) : :
X(38325) = 4 X[11] - 3 X[4728], 2 X[100] - 3 X[1635]

X(38325) lies on these lines: {11, 244}, {100, 101}, {149, 812}, {659, 8674}, {661, 37998}, {1768, 2820}, {2826, 21115}, {2827, 13243}, {3315, 3960}, {3716, 33115}, {3722, 4895}, {3738, 13256}, {9897, 13259}

X(38325) = reflection of X(2254) in X(13277)
X(38325) = crossdifference of every pair of points on line {101, 244}

X(38326) =  X(88)X(655)∩X(527)X(4370)

Barycentrics    6*a^5 - 9*a^4*b + 2*a^3*b^2 + 8*a^2*b^3 - 8*a*b^4 + b^5 - 9*a^4*c + 14*a^3*b*c - 11*a^2*b^2*c + 9*a*b^3*c + b^4*c + 2*a^3*c^2 - 11*a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 + 8*a^2*c^3 + 9*a*b*c^3 - 2*b^2*c^3 - 8*a*c^4 + b*c^4 + c^5 : :
X(38326) = 3 X[1155] + X[4542]

X(38326) lies on these lines: {88, 655}, {527, 4370}, {1155, 4542}, {2325, 3218}, {5541, 5853}

X(38326) = midpoint of X(2325) and X(3218)
X(38326) = reflection of X(17067) in X(3911)

X(38327) =  X(40)X(1635)∩X(812)X(6684)

Barycentrics    a*(b - c)*(3*a^5 - a^4*b - 6*a^3*b^2 + 2*a^2*b^3 + 3*a*b^4 - b^5 - a^4*c - 3*a^3*b*c + 7*a^2*b^2*c - a*b^3*c - 2*b^4*c - 6*a^3*c^2 + 7*a^2*b*c^2 - 8*a*b^2*c^2 + 3*b^3*c^2 + 2*a^2*c^3 - a*b*c^3 + 3*b^2*c^3 + 3*a*c^4 - 2*b*c^4 - c^5) : :
x(38327) = X[40] + 3 X[1635], X[946] - 3 X[4763], 3 X[4728] - 7 X[31423]

X(38327) lies on these lines: {40, 1635}, {812, 6684}, {946, 4763}, {2814, 9508}, {2820, 3579}, {4394, 28292}, {4728, 31423}

X(38327) = {X(40),X(1635)}-harmonic conjugate of X(38324)

X(38328) =  X(30)X(511)∩X(40)X(1635)

Barycentrics    a*(b - c)*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5 + a^4*c - 9*a^3*b*c + 13*a^2*b^2*c - 3*a*b^3*c - 2*b^4*c - 2*a^3*c^2 + 13*a^2*b*c^2 - 8*a*b^2*c^2 + b^3*c^2 - 2*a^2*c^3 - 3*a*b*c^3 + b^2*c^3 + a*c^4 - 2*b*c^4 + c^5) : :

X(38328) lies on these lines: {30, 511}, {40, 1635}, {946, 4928}, {962, 21297}, {4728, 31162}, {7688, 8645}, {35280, 35281}

X(38329) =  X(1)X(2820)∩X(3)X(8645)

Barycentrics    a*(b - c)*(a^4*b - 2*a^2*b^3 + b^5 + a^4*c - 6*a^3*b*c + 8*a^2*b^2*c - 2*a*b^3*c - b^4*c + 8*a^2*b*c^2 - 4*a*b^2*c^2 - 2*a^2*c^3 - 2*a*b*c^3 - b*c^4 + c^5) : :
X(38329) = 2 X[40] - 3 X[1635], 4 X[946] - 3 X[4728]

X(38329) lies on these lines: {1, 2820}, {3, 8645}, {40, 1635}, {661, 28292}, {812, 962}, {946, 4728}, {1938, 6608}, {2254, 2821}, {2826, 23764}

X(38330) =  EULER LINE INTERCEPT OF X(519)X(27871)

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

As a point on the Euler line, X(38330) has Shinagawa coefficients (20 r^4 + 32 r^3 R + S^2, -12 r^4 + 9 S^2).

See Kadir Altintas and Ercole Suppa, Euclid 865 .

X(38330) lies on these lines: {2,3}, {519,27871}, {524,17733}, {542,946}, {2796,15349}, {3178,18480}, {18357,20653}, {22791,27368}, {32431,34528}

leftri

Vu circumcevian-orthocenter perspectors: X(38331)-X(38334)

rightri

This preamble is based on notes from Vu Thanh Tung, May 9, 2020.

In the plane of a triangle ABC, let P = p : q : r (barycentrics), and let
A1B1C1 = circumcevian triangle of P
Hbc = orthocenter of PB1C, and define Hca and Hab cyclically
Hcb = orthocenter of PC1B, and define Hac and Hba cyclically
A' = HcaHac∩HabHba, and define B' and C' cyclically.
Then A'B'C' is perspective to ABC, and the perspector is given by

V(P) = (2 b^2 c^2 p^2 + a^2 c^2 p q + b^2 c^2 p q - c^4 p q + a^2 b^2 p r - b^4 p r + b^2 c^2 p r + a^4 q r - a^2 b^2 q r - a^2 c^2 q r) * (-a^2 b^2 c^2 p q + b^4 c^2 p q - a^2 c^4 p q - 2 b^2 c^4 p q + c^6 p q - a^4 c^2 q^2 + a^2 b^2 c^2 q^2 - a^2 c^4 q^2 + a^4 b^2 p r - 2 a^2 b^4 p r + b^6 p r - 2 b^4 c^2 p r + b^2 c^4 p r + a^6 q r - 2 a^4 b^2 q r + a^2 b^4 q r - a^4 c^2 q r - a^2 b^2 c^2 q r) * (a^4 c^2 p q + b^4 c^2 p q - 2 a^2 c^4 p q - 2 b^2 c^4 p q + c^6 p q - a^2 b^4 p r + b^6 p r - a^2 b^2 c^2 p r - 2 b^4 c^2 p r + b^2 c^4 p r + a^6 q r - a^4 b^2 q r - 2 a^4 c^2 q r - a^2 b^2 c^2 q r + a^2 c^4 q r - a^4 b^2 r^2 - a^2 b^4 r^2 + a^2 b^2 c^2 r^2) : :

See Vu Circumcevian-Orthocenter Perspector.

The appearance of (i,j) in the following list means that V(X(i)) = X(j): (1,79), (2,38331), (3,3), (4,14111), (5,38332), (6,22100) (7,38333), (8,38334)

X(38331) =  VU CIRCUMCEVIAN-ORTHOCENTER PERSPECTOR OF X(2)

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

X(38331) lies on the circumconic with center X(5512) and on these lines: {2,38337}, {4,9019}, {5,111}

X(38331) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(34165)}} and {{A, B, C, X(4), X(111)}}

X(38332) =  VU CIRCUMCEVIAN-ORTHOCENTER PERSPECTOR OF X(5)

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

X(38332) lies on this line: {4,32338}

X(38333) =  VU CIRCUMCEVIAN-ORTHOCENTER PERSPECTOR OF X(7)

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

X(38333) lies on these lines: {}

X(38334) =  VU CIRCUMCEVIAN-ORTHOCENTER PERSPECTOR OF X(8)

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

X(38334) lies on these lines: {}

X(38335) =  EULER LINE INTERCEPT OF X(156)X(13482)

Barycentrics    11 a^4-10 (b^2-c^2)^2-a^2 (b^2+c^2) : :
Barycentrics    S^2-21 SB SC : :
Trilinears    21 cos A - 20 sin B sin C : :
X(38335) = 10*X(2)-7*X(3),X(2)-7*X(4),4*X(2)-7*X(381),8*X(2)+7*X(382),7*X(2)-4*X(548),4*X(2)-X(1657),9*X(2)-7*X(3524),5*X(2)+7*X(3543),5*X(2)-7*X(3545),X(2)+2*X(3627),2*X(2)+7*X(3830),3*X(2)-7*X(3839),2*X(2)-5*X(3843),5*X(2)-8*X(3850),8*X(2)-7*X(5054),6*X(2)-7*X(5055),8*X(2)-5*X(14093),2*X(2)-7*X(14269),9*X(2)-8*X(14890),3*X(2)-4*X(14892),X(2)-4*X(14893),2*X(2)+X(15684),5*X(2)-2*X(15686),4*X(2)-3*X(15706),5*X(2)+X(33703),X(3)-10*X(4),2*X(3)-5*X(381),4*X(3)+5*X(382),5*X(3)-8*X(547),9*X(3)-10*X(3524),8*X(3)-5*X(3534),X(3)+2*X(3543),X(3)+5*X(3830),3*X(3)-10*X(3839),X(3)-4*X(3845),X(3)+8*X(3853),4*X(3)-5*X(5054),3*X(3)-5*X(5055),5*X(3)-2*X(11001),3*X(3)-4*X(11539),X(3)-5*X(14269),7*X(3)+5*X(15684),7*X(3)-4*X(15686),6*X(3)-5*X(15688),7*X(3)-5*X(15689),5*X(3)-6*X(15708),7*X(3)+2*X(33703),5*X(3)+X(35400),4*X(4)-X(381),8*X(4)+X(382),9*X(4)-X(3524),5*X(4)+X(3543),5*X(4)-X(3545),7*X(4)+2*X(3627),2*X(4)+X(3830),3*X(4)-X(3839),5*X(4)-2*X(3845),5*X(4)+4*X(3853),8*X(4)-X(5054),6*X(4)-X(5055),4*X(4)+5*X(5076),X(4)-4*X(12101),X(4)+8*X(12102),7*X(4)-4*X(14893),X(4)+2*X(15687),7*X(4)-2*X(23046),2*X(4)-5*X(35403),10*X(5)-X(5059) (and many others)

As a point on the Euler line, X(38335) has Shinagawa coefficients (1,-21).

See Kadir Altintas and Ercole Suppa, Euclid 871 .

X(38335) lies on these lines: {2,3}, {156,13482}, {265,14490}, {538,22728}, {542,5102}, {568,32062}, {1327,3311}, {1328,3312}, {1699,28208}, {3625,12699}, {3630,31670}, {3633,11278}, {3635,3656}, {3653,28164}, {3655,18483}, {4114,5722}, {4668,18480}, {4691,12702}, {5041,11648}, {5097,36990}, {5229,15170}, {5587,28202}, {5655,12295}, {5691,33179}, {5790,28198}, {6000,13321}, {6033,12355}, {6144,18440}, {6417,23253}, {6418,23263}, {6429,8976}, {6430,13951}, {6431,35822}, {6432,35823}, {6437,6564}, {6438,6565}, {6500,23269}, {6501,23275}, {8148,20053}, {9655,11238}, {9668,11237}, {9779,28190}, {9880,12188}, {9955,34628}, {10242,19924}, {10540,11935}, {10706,12902}, {10723,22566}, {11472,32608}, {11531,12645}, {11645,14848}, {11693,12121}, {12816,16964}, {12817,16965}, {12943,37587}, {13202,20126}, {13391,16261}, {13570,14855}, {13846,35786}, {13847,35787}, {14483,18550}, {14831,18439}, {14915,16226}, {14929,32874}, {14961,33880}, {15602,31489}, {16200,28204}, {16267,34754}, {16268,34755}, {16962,19107}, {16963,19106}, {18357,34632}, {18493,33697}, {19875,28146}, {19883,28172}, {20423,32455}, {21358,29317}, {21849,34783}, {22615,32787}, {22644,32788}, {22791,34748}, {23251,35771}, {23261,35770}, {23324,35450}, {25055,31662}, {28160,30392}, {32006,32878}, {33541,37490}

X(38335) = midpoint of X(i) and X(j) for these {i,j}: {382,5054}, {3543,3545}, {3627,23046}, {3830,14269}, {10304,15682}, {15684,15689}, {15699,33699}, {17504,35404}
X(38335) = reflection of X(i) in X(j) for these (i,j): (2,23046), (3,3545), (20,17504), (376,15699), (381,14269), (1657,15689), (3534,5054), (3545,3845), (5054,381), (5055,3839), (10304,5), (12121,11693), (14269,4), (15681,10304), (15688,5055), (15689,2), (15699,546), (17504,5066), (23046,14893)
X(38335) = trisector nearest X(381) of segment X(381)X(382)
X(38335) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,14893), (2,376,15712), (2,381,5072), (2,548,15718), (2,1657,14093), (2,3524,14890), (2,3543,33703), (2,3627,15684), (2,3843,381), (2,12108,15694), (2,14892,5055), (2,14893,3843), (2,15684,1657), (2,15686,3), (2,15689,15706), (2,15706,5054), (2,17538,14891), (2,21735,549), (2,33703,15686), (3,3830,3543), (3,3843,3850), (3,3845,381), (3,3851,5067), (3,5055,11539), (3,5056,3526), (3,15681,15690), (3,15694,15719), (3,15702,15693), (3,16239,15720), (3,33703,1657), (3,35400,11001), (3,35401,3830), (4,3146,3861), (4,3543,3845), (4,3627,3843), (4,3830,381), (4,3853,3), (4,5076,382), (4,12101,35403), (4,13473,1597), (4,15687,3830), (4,17578,546), (5,5059,3), (5,14891,2), (5,15681,15693), (5,15682,15681), (5,15690,15702), (20,5066,15694), (140,15682,35411), (140,15683,15695), (376,546,19709), (376,3526,15716), (376,5056,11812), (376,11812,3), (376,15699,15707), (376,17578,33699), (376,19709,3526), (376,33699,5073), (381,382,3534), (381,1657,2), (381,3526,19709), (381,3534,1656), (381,3830,382), (381,5073,15716), (381,5076,3830), (381,5079,5066), (381,15684,14093), (381,15688,5055), (381,15693,5), (381,35402,3543), (382,3830,35434), (382,3845,15723), (382,5072,1657), (546,5073,3526), (546,17578,5073), (546,19709,381), (546,33699,376), (547,3543,35400), (547,3845,3832), (547,11001,3), (547,19711,3533), (549,3146,15685), (549,12812,2), (549,15685,15696), (550,3860,5071), (550,5071,15701), (1656,3534,15700), (1657,3627,382), (1657,3843,5072), (1657,14093,3534), (1657,15706,15689), (2043,2044,632), (3091,8703,15703), (3091,17800,15720), (3146,3851,15696), (3146,3861,3851), (3526,15707,5054), (3526,17578,382), (3529,3858,5070), (3534,15723,3), (3534,35382,15720), (3534,35434,382), (3543,3832,11001), (3543,3845,3), (3543,3853,3830), (3543,5059,15682), (3543,11812,5073), (3543,15686,15684), (3543,15687,35401), (3543,35401,35402), (3545,11001,15708), (3545,11539,5055), (3545,15708,547), (3627,3843,1657), (3627,3845,15686), (3627,3850,33703), (3627,3861,21735), (3627,12812,3146), (3627,14269,15706), (3627,14891,15682), (3627,14893,2), (3830,3843,15684), (3830,3853,35402), (3830,14893,1657), (3830,15684,3627), (3830,15687,5076), (3830,19709,33699), (3830,35401,3853), (3830,35403,4), (3832,11001,547), (3832,15708,3545), (3839,5055,381), (3843,15684,2), (3845,3853,3543), (3845,15686,3850), (3845,15687,3853), (3845,33699,11812), (3845,35402,382), (3850,3853,3627), (3850,14893,3845), (3850,15686,2), (3850,15712,5056), (3850,33703,3), (3851,15685,549), (3851,35419,20), (3854,11541,3530), (3855,15692,10109), (3860,15640,15701), (3861,15685,381), (5054,14093,15706), (5054,15716,15707), (5055,14269,3839), (5055,15688,5054), (5056,15716,15723), (5059,15690,15681), (5059,15702,15690), (5066,15694,5079), (5066,35404,20), (5071,15640,550), (5072,14093,2), (5072,15684,3534), (5073,15712,1657), (5073,19709,376), (8703,15703,15720), (10109,15704,15692), (11159,14041,33240), (11737,19710,631), (12101,12102,15687), (12101,15687,4), (13620,18570,3), (14269,15689,23046), (14269,15707,546), (14893,15684,381), (15681,35411,15683), (15682,15702,5059), (15684,23046,15706), (15686,15690,17538), (15687,35403,381), (15689,15706,14093), (15690,15702,3), (15693,17538,14093), (15694,35418,15706), (15699,15707,3526), (15703,17800,8703), (15706,23046,5072), (15707,19709,15699), (15709,15720,5054), (15765,18585,3855), (17538,33703,5059), (18586,18587,5073)

leftri

2nd Vu circumcevian-orthocenter perspectors and trilinear poles: X(38336)-X(38343)

rightri

This preamble is based on notes from Vu Thanh Tung, May 10, 2020.

As in the preamble just before X(38331), let P = p : q : r (barycentrics), and let
A1B1C1 = circumcevian triangle of P
Hbc = orthocenter of PB1C, and define Hca and Hab cyclically
Hcb = orthocenter of PC1B, and define Hac and Hba cyclically
A' = HcaHac∩HabHba, and define B' and C' cyclically.
Then A'B'C' is perspective to A1B1C1, and the perspector, here named the 2nd Vu circumcevian-orthocenter perspector of P, denoted by V22(P). The points P, V(P), V2(P) are collinear in a line d(P), here named the Vu circumcevian-orthocenter perspectrix. The trilinear pole of d(P) is denoted by T(P). Barycentrics for V(P) are given in the preamble just before X(38331), and barycentrics for V2(P) and T(P) are given here.

See Second Vu Circumcevian-Orthocenter Perspector.

. The appearance of (i,j) in the following list means that V2(X(i)) = X(j): (1,38336), (2,38337), (3,3), (4,6242), (6,38339), (25,38338).

The appearance of (i,j) in the following list means that T(X(i)) = X(j): (1,38340), (2,38341), (4,38342), (6,38343)

If TX=pedal-of-X and TU=pedal-of-U, then E(TX,TU) is the orthopole of line XU. (Randy Hutson, May 19, 2020)

X(38336) =  2ND VU CIRCUMCEVIAN-ORTHOCENTER PERSPECTOR OF X(1)

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

X(38336) lies on these lines: {1,30}, {8,27548}, {11,33178}, {33,429}, {46,37729}, {55,2915}, {58,16141}, {65,74}, {78,9640}, {81,17637}, {226,9627}, {229,2646}, {515,9628}, {546,1718}, {946,9630}, {950,9629}, {1062,11375}, {1770,18447}, {1858,2906}, {2099,8283}, {2594,3465}, {2607,24307}, {2658,2667}, {2960,3601}, {3340,9577}, {3485,9538}, {3486,9539}, {3652,6149}, {4354,24929}, {5492,16140}, {7072,16471}, {9639,19860}, {9644,37724}, {12047,18455}, {16142,33100}, {24914,37696}, {28450,37606}, {34977,35979}

X(38336) = midpoint of X(1) and X(1717)
X(38336) = intersection, other than A,B,C, of conics {{A, B, C, X(30), X(6198)}} and {{A, B, C, X(74), X(7100)}}
X(38336) = crosssum of X(3) and X(8614)
X(38336) = X(21)-Beth conjugate of-X(7100)
X(38336) = {X(3649), X(10149)}-harmonic conjugate of X(1)

X(38337) =  2ND VU CIRCUMCEVIAN-ORTHOCENTER PERSPECTOR OF X(2)

Barycentrics    a^14*b^2 - 6*a^12*b^4 + 5*a^10*b^6 + 10*a^8*b^8 - 13*a^6*b^10 - 2*a^4*b^12 + 7*a^2*b^14 - 2*b^16 + a^14*c^2 + a^12*b^2*c^2 - 7*a^10*b^4*c^2 + 2*a^8*b^6*c^2 + 25*a^6*b^8*c^2 - 13*a^4*b^10*c^2 - 19*a^2*b^12*c^2 + 10*b^14*c^2 - 6*a^12*c^4 - 7*a^10*b^2*c^4 + 25*a^8*b^4*c^4 + 16*a^6*b^6*c^4 - a^4*b^8*c^4 + a^2*b^10*c^4 - 12*b^12*c^4 + 5*a^10*c^6 + 2*a^8*b^2*c^6 + 16*a^6*b^4*c^6 + 48*a^4*b^6*c^6 + 11*a^2*b^8*c^6 - 10*b^10*c^6 + 10*a^8*c^8 + 25*a^6*b^2*c^8 - a^4*b^4*c^8 + 11*a^2*b^6*c^8 + 28*b^8*c^8 - 13*a^6*c^10 - 13*a^4*b^2*c^10 + a^2*b^4*c^10 - 10*b^6*c^10 - 2*a^4*c^12 - 19*a^2*b^2*c^12 - 12*b^4*c^12 + 7*a^2*c^14 + 10*b^2*c^14 - 2*c^16 : :

X(38337) lies on these lines: {2, 38331}, {4, 6031}, {23, 31606}, {3849, 7565}, {5169, 12505}, {6236, 12506}, {7488, 10163}, {7527, 14682}

X(38338) =  2ND VU CIRCUMCEVIAN-ORTHOCENTER PERSPECTOR OF X(25)

Barycentrics    a^2*(4*a^14 - 20*a^12*b^2 + 6*a^10*b^4 + 42*a^8*b^6 - 24*a^6*b^8 - 24*a^4*b^10 + 14*a^2*b^12 + 2*b^14 - 20*a^12*c^2 + 119*a^10*b^2*c^2 - 117*a^8*b^4*c^2 - 74*a^6*b^6*c^2 + 170*a^4*b^8*c^2 - 45*a^2*b^10*c^2 - 33*b^12*c^2 + 6*a^10*c^4 - 117*a^8*b^2*c^4 + 227*a^6*b^4*c^4 - 126*a^4*b^6*c^4 - 99*a^2*b^8*c^4 + 125*b^10*c^4 + 42*a^8*c^6 - 74*a^6*b^2*c^6 - 126*a^4*b^4*c^6 + 244*a^2*b^6*c^6 - 94*b^8*c^6 - 24*a^6*c^8 + 170*a^4*b^2*c^8 - 99*a^2*b^4*c^8 - 94*b^6*c^8 - 24*a^4*c^10 - 45*a^2*b^2*c^10 + 125*b^4*c^10 + 14*a^2*c^12 - 33*b^2*c^12 + 2*c^14) : :
X(38338) = 7 X[3523] - 6 X[11628]

X(38338) lies on these lines: {23, 14262}, {3523, 11628}, {8705, 34795}, {9716, 31962}, {10102, 14002}

X(38339) =  2ND VU CIRCUMCEVIAN-ORTHOCENTER PERSPECTOR OF X(6)

Barycentrics    a^2 (4 a^14 - 22 a^12 (b^2 + c^2) + a^10 (42 b^4 + 73 b^2 c^2 + 42 c^4) - a^8 (24 b^6 + 59 b^4 c^2 + 59 b^2 c^4 + 24 c^6) - 2 a^6 (12 b^8 - 5 b^6 c^2 + 18 b^4 c^4 - 5 b^2 c^6 + 12 c^8) - a^2 (b^2 - c^2)^2 (22 b^8 + b^6 c^2 - 33 b^4 c^4 + b^2 c^6 + 22 c^8) + (b^2 - c^2)^2 (4 b^10 - 8 b^8 c^2 + b^6 c^4 + b^4 c^6 - 8 b^2 c^8 + 4 c^10) + a^4 (42 b^10 - 29 b^8 c^2 + 3 b^6 c^4 + 3 b^4 c^6 - 29 b^2 c^8 + 42 c^10)) : :

X(38339) lies on these lines: {6,22100}, {576,8705}, {10166,15037}, {11004,31962}, {15032,31731}, {31739,37513}

X(38340) =  TRILINEAR POLE OF COLLINEAR POINTS X(1), V(X(1)), AND V2(X(1))

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

X(38340) lies on the cirumconics with centers X(9), X(553), X(3163), and X(8818), and on these lines: {2,2349}, {7,8287}, {57,24624}, {79,1156}, {88,26723}, {100,4458}, {142,7110}, {162,4240}, {190,15455}, {655,1020}, {662,2407}, {664,37211}, {673,2160}, {897,9214}, {1086,1989}, {1099,3464}, {1302,36064}, {1443,11076}, {3615,37142}, {4384,37202}, {4552,37212}, {4565,7178}, {4606,22003}, {6740,9405}, {7100,23707}, {17092,37128}, {27003,30690}, {27833,27834}

X(38340) = isogonal conjugate of X(9404)
X(38340) = barycentric product X(i)*X(j) for these {i, j}: {7, 6742}, {57, 15455}, {75, 26700}, {79, 664}, {109, 20565}, {651, 30690}
X(38340) = barycentric quotient X(i)/X(j) for these (i, j): (1, 35057), (7, 4467), (56, 2605), (57, 14838), (85, 18160), (100, 4420)
X(38340) = trilinear product X(i)*X(j) for these {i, j}: {2, 26700}, {56, 15455}, {57, 6742}, {79, 651}, {109, 30690}, {226, 13486}
X(38340) = trilinear quotient X(i)/X(j) for these (i, j): (7, 14838), (57, 2605), (79, 650), (109, 2174), (110, 35192), (190, 4420)
X(38340) = trilinear pole of the line {1, 30}
X(38340) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(648)}} and {{A, B, C, X(7), X(4616)}}
X(38340) = Cevapoint of X(i) and X(j) for these {i,j}: {1, 9404}, {57, 7178}, {514, 553}, {523, 8818}
X(38340) = X(333)-Beth conjugate of-X(2349)
X(38340) = X(i)-cross conjugate of-X(j) for these (i,j): (57, 35049), (523, 7), (583, 59)
X(38340) = X(i)-isoconjugate-of-X(j) for these {i,j}: {6, 35057}, {9, 2605}, {35, 650}, {55, 14838}
X(38340) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 35057), (7, 4467), (56, 2605), (57, 14838)
X(38340) = X(1717)-Zayin conjugate of-X(652)

X(38341) =  TRILINEAR POLE OF COLLINEAR POINTS X(2), V(X(2)), AND V2(X(2))

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

X(38341) lies on the Steiner circumellipse and these lines: {}

X(38341) = trilinear pole of the line {2, 38331}

X(38342) =  TRILINEAR POLE OF COLLINEAR POINTS X(4), V(X(4)), AND V2(X(4))

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

X(38342) lies on the circumconic with center X(1249) and on these lines: {93,17983}, {107,930}, {685,32737}, {2963,16081}, {11140,16080}, {14590,16813}, {18314,18315}, {32036,36309}, {32037,36306}

X(38342) = polar conjugate of X(1510)
X(38342) = isotomic conjugate of the complement of X(18314)
X(38342) = barycentric product X(i)*X(j) for these {i, j}: {93, 99}, {110, 20572}, {264, 930}, {562, 35139}, {648, 11140}, {811, 2962}
X(38342) = barycentric quotient X(i)/X(j) for these (i, j): (3, 37084), (4, 1510), (93, 523), (107, 3518), (110, 49), (112, 2965)
X(38342) = trilinear product X(i)*X(j) for these {i, j}: {92, 930}, {93, 662}, {162, 11140}, {163, 20572}, {264, 36148}, {562, 32680}
X(38342) = trilinear quotient X(i)/X(j) for these (i, j): (63, 37084), (92, 1510), (93, 661), (162, 2965), (562, 2624), (648, 2964)
X(38342) = trilinear pole of the line {4, 93}
X(38342) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(14570)}} and {{A, B, C, X(107), X(648)}}
X(38342) = Cevapoint of X(i) and X(j) for these {i,j}: {2, 18314}, {1594, 2501}
X(38342) = X(i)-isoconjugate-of-X(j) for these {i,j}: {19, 37084}, {48, 1510}, {49, 661}, {647, 2964}
X(38342) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (3, 37084), (4, 1510), (93, 523), (107, 3518)

X(38343) =  TRILINEAR POLE OF COLLINEAR POINTS X(6), V(X(6)), AND V2(X(6))

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

X(38343) lies on the circumcircle and these lines: {}

X(38343) = trilinear pole of the line {6, 22100}

leftri

Equicenters: X(38344)-X(38429)

rightri

This preamble and centers X(38344)-X(38429) were contributed by César Eliud Lozada, May 10, 2020.

Let A'B'C' and A"B"C" be triangles inscribed in ABC. The affine transformation sending A' to A", B' to B", C' to C" has a fixed point E named the equicenter of triangles A'B'C' and A"B"C". (Reference: The Triangle Web by Quim Castellsaguer).

The fixed point E is unique, and if T' and T" are homothetic, then E is their homothetic center.

The equicenter of the affine transformation sending (A', B', C') to (A", B", C") coincides with the equicenter of the inverse affine transformation sending (A", B", C") to (A', B', C'). Therefore, the equicenter of T' and T" may be referred simply as the equicenter of triangles T' and T", regardless of the order in which the triangles are listed.

The equicenter of T' and T" is also the similarity image of T' and T"..

Some particular results:

For an almost complete list of equicenters related to ABC see here. Also, definitions of triangles mentioned can be found in the index of triangles.

Open problem: give a geometric construction of the equicenter of two arbitrary triangles.

Indeed, on October 29, 2005, François Rideau provided and proved a method, based strictly on intersections of lines, for constructing the fixed point of an affine transformation (see Francois Rideau - Les points fixes d'une application affine.pdf (in French)). This is his construction:

Given two non-homothetic triangles A'B'C' and A"B"C" and the affine transformation ƒ({ A', B', C' }) → { A", B", C" }, we complete the parallelograms A'B'C'D' and A"B"C"D"; i.e., D' is the reflection of B' in the midpoint of segment A'C' and D" is the reflection of B" in the midpoint of segment A"C". Let A* = A'B' ∩ A"B", B* = B'C' ∩ B"C", C* = C'D' ∩ C"D", D* = D'A' ∩ D"A". Then the fixed point M of the affine transformation ƒ is M = A*C* ∩ B*D*.

Additionally, Rideau provided a very simple method for finding ƒ(X) of a given point X: let U = A*B* ∩ parallelLine(X, A'B') and V = B*D* ∩ parallelLine(X, A'D'). Then ƒ(X) = parallelLine(U, A"B") ∩ parallelLine(V, A"D").

Many thanks to Francisco Javier García Capitán for his notes with simplifications of Rideua conclusions and to Angel Montesdeoca for sending me these notes.

César Lozada, January 26, 2021.

If TX=pedal-of-X and TU=pedal-of-U, then E(TX,TU) is the orthopole of line XU. (Randy Hutson, May 19, 2020)

X(38344) = EQUICENTER OF THESE TRIANGLES: INCENTRAL AND CEVIAN-OF-X(3)

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

X(38344) lies on these lines: {71,22052}, {1364,7117}, {1795,32660}, {2269,17454}, {3270,22096}, {3708,7004}, {20750,22066}, {22065,22361}

X(38344) = isogonal conjugate of the polar conjugate of X(34589)
X(38344) = barycentric product X(i)*X(j) for these {i, j}: {3, 34589}, {63, 11998}, {219, 24237}, {521, 21173}, {522, 23187}, {572, 26932}
X(38344) = trilinear product X(i)*X(j) for these {i, j}: {3, 11998}, {11, 22118}, {48, 34589}, {212, 24237}, {572, 7004}, {650, 23187}
X(38344) = trilinear quotient X(572)/X(7012)
X(38344) = crossdifference of every pair of points on line {X(17906), X(23706)}
X(38344) = crosssum of X(573) and X(4551)
X(38344) = X(48)-Ceva conjugate of-X(652)
X(38344) = X(2051)-isoconjugate-of-X(7012)

X(38345) = EQUICENTER OF THESE TRIANGLES: INCENTRAL AND ORTHIC

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

X(38345) lies on these lines: {1,1415}, {11,3125}, {19,32677}, {55,3735}, {101,10703}, {172,33177}, {244,17419}, {573,34242}, {649,38344}, {654,2170}, {1146,8735}, {1361,34457}, {1573,24431}, {1772,34460}, {1831,1854}, {1837,3959}, {2269,4016}, {2294,14749}, {2310,2643}, {2646,3727}, {2800,4559}, {3022,31892}, {3057,3721}, {3270,11918}, {3726,5048}, {3910,17880}, {4534,17435}, {9259,12740}, {11376,20271}, {11700,36075}, {17452,17465}, {17466,21334}, {17606,21951}, {18061,28798}, {20594,20599}, {21044,35015}, {21429,23876}, {21859,24028}

X(38345) = polar conjugate of the isotomic conjugate of X(34588)
X(38345) = barycentric product X(i)*X(j) for these {i, j}: {1, 124}, {4, 34588}, {11, 3869}, {522, 21189}, {573, 4858}, {1146, 17080}
X(38345) = barycentric quotient X(i)/X(j) for these (i, j): (11, 2995), (31, 15386), (124, 75), (573, 4564), (663, 36050), (2170, 13478)
X(38345) = trilinear product X(i)*X(j) for these {i, j}: {6, 124}, {11, 573}, {19, 34588}, {522, 6589}, {650, 21189}, {1146, 10571}
X(38345) = trilinear quotient X(i)/X(j) for these (i, j): (6, 15386), (124, 2), (573, 59), (650, 36050), (663, 32653), (1146, 10570)
X(38345) = intersection, other than A,B,C, of conics {{A, B, C, X(65), X(17880)}} and {{A, B, C, X(124), X(18191)}}
X(38345) = pole of the trilinear polar of X(19) with respect to Feuerbach hyperbola
X(38345) = crossdifference of every pair of points on line {X(4551), X(36050)}
X(38345) = crosspoint of X(i) and X(j) for these {i,j}: {1, 4391}, {4, 4560}, {522, 2051}
X(38345) = crosssum of X(i) and X(j) for these {i,j}: {1, 1415}, {3, 4559}, {109, 572}
X(38345) = X(19)-Ceva conjugate of-X(650)
X(38345) = X(i)-isoconjugate-of-X(j) for these {i,j}: {2, 15386}, {59, 13478}, {651, 36050}, {664, 32653}
X(38345) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (11, 2995), (31, 15386), (124, 75), (573, 4564)
X(38345) = {X(2170), X(7004)}-harmonic conjugate of X(11998)

X(38346) = EQUICENTER OF THESE TRIANGLES: INCENTRAL AND SYMMEDIAL

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

X(38346) lies on these lines: {6,23404}, {31,2350}, {42,20457}, {88,25577}, {244,649}, {902,20459}, {1015,1977}, {1149,23531}, {1193,23530}, {1201,23443}, {1438,32739}, {1475,21747}, {1962,14751}, {2170,2611}, {2319,30957}, {3121,8054}, {3248,4117}, {3271,8645}, {3924,23535}, {3952,24491}, {4253,30653}, {4598,31002}, {14752,17475}, {17761,26846}, {20045,20372}, {23417,23632}, {23647,38364}

X(38346) = isogonal conjugate of the isotomic conjugate of X(17761)
X(38346) = barycentric product X(i)*X(j) for these {i, j}: {6, 17761}, {7, 38365}, {57, 38347}, {58, 2486}, {244, 1621}, {513, 4040}
X(38346) = barycentric quotient X(i)/X(j) for these (i, j): (1015, 17758), (1621, 7035), (1977, 2350), (2486, 313)
X(38346) = trilinear product X(i)*X(j) for these {i, j}: {31, 17761}, {56, 38347}, {57, 38365}, {244, 4251}, {513, 21007}, {649, 4040}
X(38346) = trilinear quotient X(i)/X(j) for these (i, j): (244, 17758), (1015, 13476), (1621, 1016), (2486, 321)
X(38346) = crossdifference of every pair of points on line {X(1026), X(3952)}
X(38346) = crosspoint of X(i) and X(j) for these {i,j}: {1, 10566}, {6, 1019}
X(38346) = crosssum of X(i) and X(j) for these {i,j}: {2, 1018}, {8, 35341}
X(38346) = X(31)-Ceva conjugate of-X(649)
X(38346) = X(i)-isoconjugate-of-X(j) for these {i,j}: {765, 17758}, {1016, 13476}
X(38346) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1015, 17758), (1621, 7035), (1977, 2350)
X(38346) = {X(1015), X(23470)}-harmonic conjugate of X(1977)

X(38347) = EQUICENTER OF THESE TRIANGLES: INCENTRAL AND CEVIAN-OF-X(9)

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

X(38347) lies on these lines: {1,35326}, {11,1566}, {37,2246}, {100,5701}, {528,23988}, {654,2170}, {661,38390}, {693,27009}, {1015,35076}, {1146,6741}, {2264,14749}, {2310,24012}, {3058,16588}, {3271,7063}, {3748,14746}, {4251,20616}, {4762,23989}, {6154,6184}, {11246,23653}, {17494,26846}, {18785,23404}, {24486,26073}, {24562,26565}, {25577,34583}

X(38347) = reflection of X(35310) in X(23988)
X(38347) = barycentric product X(i)*X(j) for these {i, j}: {9, 17761}, {11, 1621}, {21, 2486}, {75, 38365}, {244, 3996}, {312, 38346}
X(38347) = barycentric quotient X(i)/X(j) for these (i, j): (1621, 4998), (2170, 17758), (2486, 1441)
X(38347) = trilinear product X(i)*X(j) for these {i, j}: {2, 38365}, {8, 38346}, {11, 4251}, {55, 17761}, {284, 2486}, {522, 21007}
X(38347) = trilinear quotient X(i)/X(j) for these (i, j): (11, 17758), (1621, 4564), (2170, 13476), (2486, 226)
X(38347) = intersection, other than A,B,C, of conics {{A, B, C, X(11), X(1621)}} and {{A, B, C, X(55), X(26846)}}
X(38347) = pole of the trilinear polar of X(55) with respect to Feuerbach hyperbola
X(38347) = crossdifference of every pair of points on line {X(2283), X(4551)}
X(38347) = crosspoint of X(i) and X(j) for these {i,j}: {9, 4560}, {1621, 17494}
X(38347) = crosssum of X(i) and X(j) for these {i,j}: {1, 35326}, {57, 4559}
X(38347) = X(55)-Ceva conjugate of-X(650)
X(38347) = X(59)-isoconjugate-of-X(17758)
X(38347) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1621, 4998), (2170, 17758)
X(38347) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11, 14936, 650), (2310, 38375, 38358)

X(38348) = EQUICENTER OF THESE TRIANGLES: INCENTRAL AND STEINER

Barycentrics    a*(b-c)*(a^2-b*c)*(a^2+(b+c)*a-b^2-b*c-c^2) : :
X(38348) = 4*X(21261)-5*X(30795)

X(38348) lies on these lines: {1,512}, {214,3126}, {523,4360}, {663,1193}, {830,4983}, {885,6654}, {900,4366}, {1027,25426}, {1045,3737}, {1960,3802}, {1964,2605}, {2646,4162}, {3733,4068}, {3837,21303}, {4010,4107}, {4132,17457}, {4435,8632}, {5029,9508}, {8034,17011}, {21261,30795}

X(38348) = reflection of X(i) in X(j) for these (i,j): (659, 8632), (21303, 3837)
X(38348) = barycentric product X(i)*X(j) for these {i, j}: {1, 27929}, {238, 2786}, {350, 5029}, {513, 6651}, {514, 8298}, {659, 6542}
X(38348) = barycentric quotient X(i)/X(j) for these (i, j): (238, 35148), (649, 9505), (659, 6650), (667, 9506), (812, 18032), (1326, 4584)
X(38348) = trilinear product X(i)*X(j) for these {i, j}: {6, 27929}, {239, 5029}, {513, 8298}, {649, 6651}, {659, 1757}, {812, 17735}
X(38348) = trilinear quotient X(i)/X(j) for these (i, j): (238, 37135), (239, 35148), (513, 9505), (649, 9506), (659, 1929), (812, 6650)
X(38348) = intersection, other than A,B,C, of conics {{A, B, C, X(239), X(1931)}} and {{A, B, C, X(256), X(8843)}}
X(38348) = crossdifference of every pair of points on line {X(291), X(1757)}
X(38348) = crosspoint of X(i) and X(j) for these {i,j}: {1, 3573}, {99, 239}
X(38348) = crosssum of X(i) and X(j) for these {i,j}: {1, 876}, {100, 37135}, {292, 512}, {523, 20531}
X(38348) = X(2109)-anticomplementary conjugate of-X(21221)
X(38348) = X(i)-Ceva conjugate of-X(j) for these (i,j): (99, 1931), (513, 659), (662, 2238)
X(38348) = X(659)-Hirst inverse of-X(21832)
X(38348) = X(i)-isoconjugate-of-X(j) for these {i,j}: {100, 9505}, {190, 9506}, {291, 37135}, {292, 35148}
X(38348) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (238, 35148), (649, 9505), (659, 6650), (667, 9506)
X(38348) = {X(1), X(24286)}-harmonic conjugate of X(4367)

X(38349) = EQUICENTER OF THESE TRIANGLES: INCENTRAL AND YFF CONTACT

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

X(38349) lies on these lines: {1,649}, {55,4057}, {522,3158}, {890,891}, {1635,8299}, {1962,4132}, {3720,14474}, {3795,4893}, {4375,17780}, {14437,23343}, {17494,25264}

X(38349) = crosspoint of X(1) and X(23343)
X(38349) = X(i)-Ceva conjugate of-X(j) for these (i,j): (1, 16507), (100, 899), (1019, 3768)

X(38350) = EQUICENTER OF THESE TRIANGLES: INCENTRAL AND MACBEATH

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

X(38350) lies on these lines: {2632,21340}, {18210,34589}

X(38351) = EQUICENTER OF THESE TRIANGLES: MEDIAL AND 2nd HATZIPOLAKIS

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

X(38351) lies on the line {3270,7358}

X(38351) = center of the circumconic {{ A, B, C, X(1119), X(1265), X(17054) }}
X(38351) = crosssum of X(1260) and X(29163)

X(38352) = EQUICENTER OF THESE TRIANGLES: CEVIAN-OF-X(3) AND SYMMEDIAL

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

X(38352) lies on these lines: {39,5191}, {110,5661}, {115,137}, {125,647}, {248,13198}, {542,23584}, {1084,15527}, {1194,12829}, {11672,24981}, {13366,14773}, {20975,23216}, {23878,23962}, {31296,36901}

X(38352) = reflection of X(35319) in X(23584)
X(38352) = isogonal conjugate of the polar conjugate of X(7668)
X(38352) = barycentric product X(i)*X(j) for these {i, j}: {3, 7668}, {125, 5012}, {184, 36901}, {525, 3050}, {647, 31296}, {1078, 20975}
X(38352) = barycentric quotient X(i)/X(j) for these (i, j): (184, 27867), (647, 11794)
X(38352) = trilinear product X(i)*X(j) for these {i, j}: {48, 7668}, {656, 3050}, {810, 31296}, {1629, 37754}
X(38352) = trilinear quotient X(i)/X(j) for these (i, j): (48, 27867), (656, 11794)
X(38352) = pole of the trilinear polar of X(184) with respect to Jerabek hyperbola
X(38352) = crossdifference of every pair of points on line {X(4230), X(11794)}
X(38352) = crosspoint of X(i) and X(j) for these {i,j}: {3, 4580}, {6, 15412}
X(38352) = crosssum of X(i) and X(j) for these {i,j}: {2, 1625}, {4, 35325}
X(38352) = X(184)-Ceva conjugate of-X(647)
X(38352) = X(i)-isoconjugate-of-X(j) for these {i,j}: {92, 27867}, {162, 11794}
X(38352) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (184, 27867), (647, 11794)

X(38353) = EQUICENTER OF THESE TRIANGLES: CEVIAN-OF-X(3) AND EXTOUCH

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

X(38353) lies on these lines: {36,2323}, {7117,36054}, {8611,34591}

X(38353) = barycentric product X(1845)*X(24031)
X(38353) = barycentric quotient X(1845)/X(24032)
X(38353) = trilinear product X(i)*X(j) for these {i, j}: {1845, 35072}, {2323, 35014}
X(38353) = trilinear quotient X(i)/X(j) for these (i, j): (654, 36110), (1845, 23984)
X(38353) = crossdifference of every pair of points on line {X(23987), X(36110)}
X(38353) = X(655)-isoconjugate-of-X(36110)
X(38353) = X(1845)-reciprocal conjugate of-X(24032)

X(38354) = EQUICENTER OF THESE TRIANGLES: CEVIAN-OF-X(3) AND STEINER

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

X(38354) lies on these lines: {3,525}, {520,20775}, {523,8266}, {577,3049}, {4580,23286}, {8723,34952}, {22078,37084}

X(38355) = EQUICENTER OF THESE TRIANGLES: CEVIAN-OF-X(3) AND YFF CONTACT

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

X(38355) lies on the line {228,4064}

X(38356) = EQUICENTER OF THESE TRIANGLES: ORTHIC AND SYMMEDIAL

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

X(38356) lies on these lines: {6,1112}, {22,11610}, {23,34137}, {110,10766}, {115,125}, {127,18187}, {184,5028}, {248,13558}, {525,36793}, {647,2972}, {1899,3981}, {2781,35325}, {3448,35902}, {9700,14585}, {9909,22135}, {12310,22146}, {14580,34146}, {15341,32269}, {17434,20625}, {20975,23216}, {21637,23642}

X(38356) = isogonal conjugate of the isotomic conjugate of X(127)
X(38356) = barycentric product X(i)*X(j) for these {i, j}: {6, 127}, {22, 125}, {37, 18187}, {115, 20806}, {206, 339}, {315, 20975}
X(38356) = barycentric quotient X(i)/X(j) for these (i, j): (22, 18020), (32, 15388), (125, 18018), (127, 76), (206, 250), (512, 1289)
X(38356) = trilinear product X(i)*X(j) for these {i, j}: {22, 3708}, {31, 127}, {42, 18187}, {125, 2172}, {206, 20902}, {339, 17453}
X(38356) = trilinear quotient X(i)/X(j) for these (i, j): (31, 15388), (127, 75), (661, 1289), (1760, 18020), (2172, 250)
X(38356) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(36793)}} and {{A, B, C, X(22), X(868)}}
X(38356) = pole of the trilinear polar of X(25) with respect to Jerabek hyperbola
X(38356) = crossdifference of every pair of points on line {X(110), X(1289)}
X(38356) = crosspoint of X(i) and X(j) for these {i,j}: {4, 4580}, {6, 525}, {22, 33294}
X(38356) = crosssum of X(i) and X(j) for these {i,j}: {2, 112}, {3, 35325}, {110, 10316}, {441, 15639}
X(38356) = X(i)-Ceva conjugate of-X(j) for these (i,j): (25, 647), (339, 20975)
X(38356) = X(i)-isoconjugate-of-X(j) for these {i,j}: {75, 15388}, {662, 1289}
X(38356) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (22, 18020), (32, 15388), (125, 18018), (127, 76)
X(38356) = {X(3124), X(3269)}-harmonic conjugate of X(125)

X(38357) = EQUICENTER OF THESE TRIANGLES: ORTHIC AND INTOUCH

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

X(38357) lies on these lines: {1,1537}, {4,1854}, {7,4617}, {11,244}, {33,1836}, {34,12679}, {55,4415}, {64,1118}, {65,1830}, {100,26611}, {109,15252}, {124,522}, {125,20620}, {158,6247}, {190,27542}, {221,7952}, {225,12688}, {227,6260}, {243,1503}, {278,2192}, {318,20306}, {329,7074}, {390,33151}, {497,3782}, {516,16870}, {523,2632}, {651,9809}, {908,9371}, {952,10703}, {982,15845}, {1040,24703}, {1146,8735}, {1364,3326}, {1365,3022}, {1413,10309}, {1456,23710}, {1532,1735}, {1699,15430}, {1745,18243}, {1776,35466}, {1785,6001}, {1807,5840}, {1829,11921}, {1834,1858}, {1837,5151}, {1845,2778}, {1853,1857}, {1864,3914}, {1897,33650}, {1936,17768}, {1940,6696}, {2654,3649}, {2818,21664}, {2822,15902}, {2886,24430}, {2969,3270}, {3100,5057}, {3139,22094}, {3318,6087}, {3465,5842}, {3700,23970}, {3772,30223}, {3925,7069}, {4336,24725}, {4551,13257}, {4854,14547}, {5274,33146}, {5514,13612}, {5603,15306}, {5660,15737}, {5880,9817}, {6259,21147}, {10374,17832}, {10391,24210}, {10394,33134}, {12608,17102}, {13567,21924}, {21635,24025}, {24026,26932}, {24028,37725}, {24410,25968}

X(38357) = midpoint of X(i) and X(j) for these {i,j}: {1897, 33650}, {10703, 18340}
X(38357) = reflection of X(i) in X(j) for these (i,j): (109, 15252), (2968, 124)
X(38357) = polar conjugate of the isotomic conjugate of X(16596)
X(38357) = barycentric product X(i)*X(j) for these {i, j}: {4, 16596}, {7, 5514}, {11, 329}, {40, 4858}, {189, 3318}, {196, 2968}
X(38357) = barycentric quotient X(i)/X(j) for these (i, j): (40, 4564), (198, 59), (208, 7128), (221, 1262), (223, 7045), (244, 1422)
X(38357) = trilinear product X(i)*X(j) for these {i, j}: {11, 40}, {19, 16596}, {34, 7358}, {57, 5514}, {78, 38362}, {84, 3318}
X(38357) = trilinear quotient X(i)/X(j) for these (i, j): (40, 59), (196, 7128), (198, 2149), (221, 24027), (223, 1262), (244, 1413)
X(38357) = orthojoin of X(652)
X(38357) = Zosma transform of X(108)
X(38357) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(2968)}} and {{A, B, C, X(11), X(5514)}}
X(38357) = center of the circumconic {{ A, B, C, X(158), X(1440), X(3086), X(7080), X(10309) }}
X(38357) = pole of the trilinear polar of X(158) with respect to Kiepert hyperbola
X(38357) = pole of the trilinear polar of X(278) with respect to Feuerbach hyperbola
X(38357) = pole of the trilinear polar of X(1440) with respect to circumhyperbola dual of Yff parabola
X(38357) = orthopole of line X(1)X(4)
X(38357) = crossdifference of every pair of points on line {X(101), X(2425)}
X(38357) = crosspoint of X(i) and X(j) for these {i,j}: {4, 522}, {7, 4391}, {329, 17896}, {347, 14837}
X(38357) = crosssum of X(i) and X(j) for these {i,j}: {3, 109}, {55, 1415}, {101, 7074}, {577, 692}
X(38357) = X(i)-Ceva conjugate of-X(j) for these (i,j): (158, 523), (269, 21120), (278, 650), (329, 14298)
X(38357) = X(i)-isoconjugate-of-X(j) for these {i,j}: {101, 37141}, {109, 13138}, {189, 2149}, {268, 7128}
X(38357) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (40, 4564), (198, 59), (208, 7128), (221, 1262)
X(38357) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2310, 3120, 11), (7004, 35015, 11)

X(38358) = EQUICENTER OF THESE TRIANGLES: ORTHIC AND CEVIAN-OF-X(9)

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

X(38358) lies on these lines: {11,17435}, {661,21339}, {1146,8735}, {2310,24012}, {2801,35326}, {3119,7004}, {3239,34589}, {3700,24026}, {4413,24274}, {5083,25069}, {11998,34591}, {16588,24431}, {17463,20974}, {20901,25259}, {23988,24433}

X(38358) = barycentric product X(i)*X(j) for these {i, j}: {8, 17463}, {9, 116}, {11, 3681}, {21, 21045}, {55, 20901}, {210, 17198}
X(38358) = barycentric quotient X(i)/X(j) for these (i, j): (41, 15378), (116, 85), (1734, 664), (2170, 14377)
X(38358) = trilinear product X(i)*X(j) for these {i, j}: {8, 20974}, {9, 17463}, {41, 20901}, {55, 116}, {210, 18184}, {281, 22084}
X(38358) = trilinear quotient X(i)/X(j) for these (i, j): (11, 14377), (55, 15378), (116, 7), (1734, 651)
X(38358) = pole of the trilinear polar of X(33) with respect to Feuerbach hyperbola
X(38358) = crosspoint of X(i) and X(j) for these {i,j}: {9, 4391}, {522, 17758}
X(38358) = crosssum of X(i) and X(j) for these {i,j}: {3, 35326}, {57, 1415}, {109, 4251}
X(38358) = X(i)-Ceva conjugate of-X(j) for these (i,j): (33, 650), (116, 17463)
X(38358) = X(i)-isoconjugate-of-X(j) for these {i,j}: {7, 15378}, {59, 14377}
X(38358) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (41, 15378), (116, 85), (1734, 664)
X(38358) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2310, 38375, 38347), (3119, 7004, 650), (17435, 36197, 11)

X(38359) = EQUICENTER OF THESE TRIANGLES: ORTHIC AND STEINER

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

X(38359) lies on the circumconic with center X(6132) and on these lines: {4,3566}, {52,512}, {113,114}, {155,525}, {193,523}, {826,13431}, {924,1843}, {20580,32605}

X(38359) = barycentric product X(99)*X(36472)
X(38359) = trilinear product X(i)*X(j) for these {i, j}: {662, 36472}, {1733, 6132}
X(38359) = crossdifference of every pair of points on line {X(1570), X(34382)}
X(38359) = crosspoint of X(4) and X(4226)
X(38359) = crosssum of X(i) and X(j) for these {i,j}: {3, 35364}, {512, 32654}
X(38359) = X(i)-Ceva conjugate of-X(j) for these (i,j): (99, 35296), (648, 230)
X(38359) = orthocenter of X(4)X(6)X(155)
X(38359) = orthocenter of X(52)X(185)X(193)

X(38360) = EQUICENTER OF THESE TRIANGLES: ORTHIC AND YFF CONTACT

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

X(38360) lies on these lines: {118,4120}, {522,2900}, {1824,15313}, {2901,4024}

X(38360) = crosssum of X(3) and X(35365)

X(38361) = EQUICENTER OF THESE TRIANGLES: ORTHIC AND LEMOINE

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

X(38361) lies on these lines: {125,136}, {3014,11007}, {9145,36194}

X(38361) = barycentric product X(i)*X(j) for these {i, j}: {115, 7850}, {338, 7492}
X(38361) = trilinear product X(1109)*X(7492)
X(38361) = crosspoint of X(598) and X(850)
X(38361) = crosssum of X(i) and X(j) for these {i,j}: {3, 35357}, {574, 1576}

X(38362) = EQUICENTER OF THESE TRIANGLES: ORTHIC AND 2nd HATZIPOLAKIS

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

X(38362) lies on these lines: {11,7649}, {196,3195}, {244,2969}, {1109,2501}, {3125,8735}, {6591,14936}, {17924,21666}, {18344,38389}

X(38362) = barycentric product X(i)*X(j) for these {i, j}: {11, 196}, {198, 2973}, {208, 4858}, {278, 38357}, {281, 38374}, {329, 2969}
X(38362) = barycentric quotient X(i)/X(j) for these (i, j): (196, 4998), (208, 4564), (1015, 1433), (1358, 34400)
X(38362) = trilinear product X(i)*X(j) for these {i, j}: {11, 208}, {33, 38374}, {34, 38357}, {40, 2969}, {196, 2170}, {223, 8735}
X(38362) = trilinear quotient X(i)/X(j) for these (i, j): (11, 271), (196, 4564), (208, 59), (244, 1433), (342, 4998)
X(38362) = intersection, other than A,B,C, of conics {{A, B, C, X(11), X(1118)}} and {{A, B, C, X(244), X(8735)}}
X(38362) = crosspoint of X(i) and X(j) for these {i,j}: {1086, 38374}, {1118, 7649}
X(38362) = crosssum of X(i) and X(j) for these {i,j}: {3, 35350}, {906, 1260}, {1259, 1331}
X(38362) = X(1086)-Ceva conjugate of-X(8735)
X(38362) = X(i)-isoconjugate-of-X(j) for these {i,j}: {59, 271}, {268, 4564}, {765, 1433}, {1331, 13138}
X(38362) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (196, 4998), (208, 4564), (1015, 1433), (1358, 34400)

X(38363) = EQUICENTER OF THESE TRIANGLES: SYMMEDIAL AND INTOUCH

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

X(38363) lies on these lines: {6,32666}, {663,1015}, {667,7117}, {1356,15615}, {1362,1457}, {1565,4905}, {2223,2356}, {2254,35094}, {2310,4017}, {3271,20980}, {20455,20972}, {20456,20662}, {20982,38365}, {24488,34067}

X(38363) = crosspoint of X(i) and X(j) for these {i,j}: {6, 2254}, {513, 5089}
X(38363) = crosssum of X(i) and X(j) for these {i,j}: {2, 36086}, {100, 1814}
X(38363) = X(56)-Ceva conjugate of-X(665)

X(38364) = EQUICENTER OF THESE TRIANGLES: SYMMEDIAL AND EXTOUCH

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

X(38364) lies on these lines: {14936,35506}, {23647,38346}

X(38364) = barycentric product X(830)*X(17420)
X(38364) = trilinear quotient X(830)/X(6648)
X(38364) = crosssum of X(2) and X(36147)
X(38364) = X(831)-isoconjugate-of-X(6648)

X(38365) = EQUICENTER OF THESE TRIANGLES: SYMMEDIAL AND CEVIAN-OF-X(9)

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

X(38365) lies on these lines: {663,2170}, {1111,4724}, {3271,7117}, {4040,17761}, {20982,38363}

X(38365) = barycentric product X(i)*X(j) for these {i, j}: {1, 38347}, {8, 38346}, {11, 4251}, {55, 17761}, {284, 2486}, {522, 21007}
X(38365) = trilinear product X(i)*X(j) for these {i, j}: {6, 38347}, {9, 38346}, {41, 17761}, {650, 21007}, {663, 4040}, {1621, 3271}
X(38365) = trilinear quotient X(i)/X(j) for these (i, j): (1621, 4998), (2170, 17758)
X(38365) = crossdifference of every pair of points on line {X(1025), X(4552)}
X(38365) = crosspoint of X(55) and X(3737)
X(38365) = crosssum of X(i) and X(j) for these {i,j}: {2, 35338}, {7, 4551}
X(38365) = X(41)-Ceva conjugate of-X(663)

X(38366) = EQUICENTER OF THESE TRIANGLES: SYMMEDIAL AND STEINER

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

X(38366) lies on these lines: {6,669}, {110,14606}, {351,36213}, {512,3051}, {525,3167}, {3288,6195}, {8711,11205}

X(38366) = crosspoint of X(6) and X(5118)
X(38366) = X(110)-Ceva conjugate of-X(3231)

X(38367) = EQUICENTER OF THESE TRIANGLES: SYMMEDIAL AND YFF CONTACT

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

X(38367) lies on these lines: {6,1919}, {101,5378}, {213,667}, {1017,20671}, {1924,20970}, {2275,23572}, {4040,6196}, {4064,29078}, {4435,8632}, {20467,20662}

X(38367) = barycentric product X(i)*X(j) for these {i, j}: {238, 6373}, {513, 20663}, {649, 17475}, {659, 3009}, {663, 8850}, {667, 17793}
X(38367) = trilinear product X(i)*X(j) for these {i, j}: {649, 20663}, {659, 21760}, {667, 17475}, {1914, 6373}, {1919, 17793}
X(38367) = trilinear quotient X(i)/X(j) for these (i, j): (659, 32020), (1575, 4583), (1914, 8709)
X(38367) = crossdifference of every pair of points on line {X(291), X(350)}
X(38367) = crosspoint of X(i) and X(j) for these {i,j}: {101, 2210}, {667, 8632}
X(38367) = crosssum of X(i) and X(j) for these {i,j}: {334, 514}, {668, 4562}
X(38367) = X(101)-Ceva conjugate of-X(3009)
X(38367) = X(i)-isoconjugate-of-X(j) for these {i,j}: {335, 8709}, {660, 32020}

X(38368) = EQUICENTER OF THESE TRIANGLES: SYMMEDIAL AND MACBEATH

Barycentrics    a^2*(b^2-c^2)^2*((b^2+c^2)*a^2-b^4-c^4)*(a^8-(b^2+c^2)*a^6-(b^4+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(38368) lies on these lines: {30,2967}, {115,15451}, {232,237}, {338,523}, {512,3269}, {684,35088}, {11988,11998}, {20410,21177}

X(38368) = reflection of X(237) in X(232)
X(38368) = barycentric product X(i)*X(j) for these {i, j}: {232, 3150}, {868, 10313}
X(38368) = intersection, other than A,B,C, of conics {{A, B, C, X(232), X(10313)}} and {{A, B, C, X(237), X(3150)}}
X(38368) = crossdifference of every pair of points on line {X(14966), X(34211)}
X(38368) = crosspoint of X(232) and X(523)
X(38368) = crosssum of X(110) and X(287)
X(38368) = X(4)-Ceva conjugate of-X(3569)

X(38369) = EQUICENTER OF THESE TRIANGLES: SYMMEDIAL AND LEMOINE

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

X(38369) lies on these lines: {512,3124}, {1499,8288}, {6593,8627}

X(38369) = crosspoint of X(598) and X(22105)
X(38369) = crosssum of X(574) and X(36827)
X(38369) = X(1383)-Ceva conjugate of-X(351)

X(38370) = EQUICENTER OF THESE TRIANGLES: INTOUCH AND STEINER

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

X(38370) lies on these lines: {7,4897}, {1366,34194}

X(38371) = EQUICENTER OF THESE TRIANGLES: INTOUCH AND YFF CONTACT

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

X(38371) lies on these lines: {1,30719}, {7,3667}, {65,3309}, {145,514}, {224,6332}, {522,3174}, {900,3126}, {1317,2826}, {3649,4170}, {6161,6362}, {14100,30198}, {16236,28292}

X(38371) = X(664)-Ceva conjugate of-X(3008)

X(38372) = EQUICENTER OF THESE TRIANGLES: INTOUCH AND MACBEATH

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

X(38372) lies on these lines: {693,1565}, {1367,3326}, {2973,35012}

X(38372) = crosspoint of X(264) and X(36038)
X(38372) = X(331)-Ceva conjugate of-X(10015)

X(38373) = EQUICENTER OF THESE TRIANGLES: INTOUCH AND LEMOINE

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

X(38373) lies on the line {1365,4170}

X(38374) = EQUICENTER OF THESE TRIANGLES: INTOUCH AND 2nd HATZIPOLAKIS

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

X(38374) lies on these lines: {221,14256}, {244,1358}, {1086,8735}, {1146,7178}, {1565,3676}, {3669,7117}, {24002,34387}

X(38374) = barycentric product X(i)*X(j) for these {i, j}: {11, 14256}, {196, 1565}, {221, 23989}, {223, 1111}, {227, 16727}, {279, 38357}
X(38374) = barycentric quotient X(i)/X(j) for these (i, j): (196, 15742), (198, 6065), (221, 1252), (223, 765), (244, 282), (329, 4076)
X(38374) = trilinear product X(i)*X(j) for these {i, j}: {40, 1358}, {77, 38362}, {196, 3942}, {208, 1565}, {221, 1111}, {223, 1086}
X(38374) = trilinear quotient X(i)/X(j) for these (i, j): (40, 6065), (221, 1110), (223, 1252), (244, 2192), (322, 4076), (342, 15742)
X(38374) = intersection, other than A,B,C, of conics {{A, B, C, X(34), X(1146)}} and {{A, B, C, X(244), X(8735)}}
X(38374) = crosssum of X(1260) and X(3939)
X(38374) = X(514)-Beth conjugate of-X(1146)
X(38374) = X(1111)-Ceva conjugate of-X(1358)
X(38374) = X(i)-isoconjugate-of-X(j) for these {i,j}: {84, 6065}, {280, 1110}, {282, 1252}, {309, 6066}
X(38374) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (196, 15742), (198, 6065), (221, 1252), (223, 765)

X(38375) = EQUICENTER OF THESE TRIANGLES: EXTOUCH AND CEVIAN-OF-X(9)

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

X(38375) lies on these lines: {9,14740}, {11,1146}, {101,37736}, {657,38346}, {1768,2291}, {2310,24012}, {3239,4939}, {3271,14935}, {3676,24775}, {3711,33299}, {3938,5452}, {3942,20974}, {5531,5540}, {7004,14936}, {15657,34784}, {17451,17718}, {21808,37703}

X(38375) = barycentric product X(i)*X(j) for these {i, j}: {9, 4904}, {11, 3870}, {21, 21945}, {218, 4858}, {344, 2170}, {522, 3309}
X(38375) = barycentric quotient X(i)/X(j) for these (i, j): (218, 4564), (650, 37206), (663, 1292), (884, 36041), (1015, 17107), (1445, 1275)
X(38375) = trilinear product X(i)*X(j) for these {i, j}: {11, 218}, {55, 4904}, {284, 21945}, {344, 3271}, {650, 3309}, {657, 31605}
X(38375) = trilinear quotient X(i)/X(j) for these (i, j): (11, 277), (218, 59), (244, 17107), (344, 4998), (522, 37206), (650, 1292)
X(38375) = intersection, other than A,B,C, of conics {{A, B, C, X(9), X(23760)}} and {{A, B, C, X(11), X(34894)}}
X(38375) = pole of the trilinear polar of X(200) with respect to Feuerbach hyperbola
X(38375) = crossdifference of every pair of points on line {X(109), X(1292)}
X(38375) = crosspoint of X(i) and X(j) for these {i,j}: {9, 514}, {1445, 3309}
X(38375) = crosssum of X(i) and X(j) for these {i,j}: {56, 35326}, {57, 101}, {651, 17092}
X(38375) = X(1021)-Beth conjugate of-X(244)
X(38375) = X(i)-Ceva conjugate of-X(j) for these (i,j): (200, 650), (514, 23760), (1111, 2310), (1445, 3309)
X(38375) = X(i)-isoconjugate-of-X(j) for these {i,j}: {59, 277}, {109, 37206}, {651, 1292}, {765, 17107}
X(38375) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (218, 4564), (650, 37206), (663, 1292), (884, 36041)
X(38375) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2170, 3119, 11), (14936, 17435, 7004), (38347, 38358, 2310)

X(38376) = EQUICENTER OF THESE TRIANGLES: EXTOUCH AND YFF CONTACT

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

X(38376) lies on these lines: {8,3239}, {200,522}, {210,521}, {3667,6546}, {3887,30565}, {4046,4086}, {4152,6068}, {4468,30625}, {4847,14476}, {25604,26364}

X(38376) = trilinear product X(522)*X(6594)
X(38376) = crosspoint of X(190) and X(30806)
X(38376) = crosssum of X(649) and X(34068)
X(38376) = X(1461)-isoconjugate-of-X(15734)

X(38377) = EQUICENTER OF THESE TRIANGLES: EXTOUCH AND MACBEATH

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

X(38377) lies on the line {7068,24026}

X(38378) = EQUICENTER OF THESE TRIANGLES: EXTOUCH AND LEMOINE

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

X(38378) lies on the line {4092,4965}

X(38379) = EQUICENTER OF THESE TRIANGLES: CEVIAN-OF-X(9) AND YFF CONTACT

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

X(38379) lies on these lines: {9,522}, {220,663}, {514,16552}, {672,3126}, {1334,3900}

X(38379) = X(644)-Ceva conjugate of-X(2340)

X(38380) = EQUICENTER OF THESE TRIANGLES: STEINER AND MACBEATH

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

X(38380) lies on these lines: {264,6563}, {3265,34291}, {14264,34767}, {14417,34834}, {34333,34336}

X(38381) = EQUICENTER OF THESE TRIANGLES: STEINER AND LEMOINE

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

X(38381) lies on these lines: {523,1992}, {598,1499}

X(38381) = crosspoint of X(598) and X(34245)

X(38382) = EQUICENTER OF THE CEVIAN TRIANGLES OF BROCARD POINTS

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

X(38382) lies on the cubic K739 and these lines: {2,3511}, {3,10342}, {6,19585}, {39,83}, {194,3499}, {239,19580}, {287,6467}, {385,3978}, {1003,32524}, {1207,9427}, {2309,4366}, {7787,9431}, {7824,21444}, {19571,23642}

X(38382) = intersection, other than A,B,C, of conics {{A, B, C, X(83), X(8870)}} and {{A, B, C, X(385), X(733)}}
X(38382) = X(6)-Ceva conjugate of-X(385)
X(38382) = X(i)-Hirst inverse of-X(j) for these {i,j}: {2, 3511}, {385, 8623}
X(38382) = {X(39), X(10341)}-harmonic conjugate of X(384)

X(38383) = EQUICENTER OF THE PEDAL TRIANGLES OF BROCARD POINTS

Barycentrics    ((b^2+c^2)*a^2-b^4-c^4)*(a^8+2*(b^2+c^2)*a^6-(b^4+3*b^2*c^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) : :
X(38383) = X(98)-3*X(262) = 3*X(262)-2*X(2023) = 2*X(3934)-3*X(36519) = 4*X(6721)-3*X(15819) = 5*X(7786)-3*X(34473) = 3*X(8724)-X(19910) = X(8782)-3*X(9772) = X(9821)-3*X(15561) = X(13188)+3*X(22728) = 3*X(23234)-X(33706)

X(38383) lies on the cubic K1113 and these lines: {3,10349}, {4,147}, {6,98}, {39,2794}, {114,325}, {620,5188}, {1351,36849}, {1503,12830}, {2025,5477}, {2456,4027}, {3094,9744}, {3104,9749}, {3105,9750}, {3314,37446}, {3329,3398}, {3543,11152}, {3934,36519}, {5149,30270}, {5969,6054}, {5989,12177}, {6036,7792}, {6721,15819}, {7762,35436}, {7778,22712}, {7786,34473}, {7807,35430}, {7838,35437}, {7840,22566}, {8290,11676}, {8724,19910}, {8782,9742}, {9475,9862}, {9753,13330}, {9821,15561}, {10722,11257}, {12184,12836}, {12185,12837}, {12251,37668}, {13111,35464}, {13188,22728}, {13354,37450}, {22678,26316}, {23234,33706}, {31670,35705}

X(38383) = midpoint of X(i) and X(j) for these {i,j}: {147, 1916}, {3095, 6033}, {3543, 11152}, {10722, 11257}
X(38383) = reflection of X(i) in X(j) for these (i,j): (98, 2023), (5188, 620), (5976, 114), (12042, 11272)
X(38383) = orthojoin of X(385)
X(38383) = orthopole of PU(1) (line X(39)X(512))
X(38383) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(5976)}} and {{A, B, C, X(262), X(32458)}}
X(38383) = {X(98), X(262)}-harmonic conjugate of X(2023)

X(38384) = EQUICENTER OF THESE TRIANGLES: INTOUCH AND PEDAL-OF-X(2)

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

X(38384) lies on these lines: {3,25919}, {4,145}, {515,37743}, {1357,16185}, {1699,33103}, {3120,38386}, {3667,3756}, {3699,34548}, {9812,15519}, {17777,28915}

X(38384) = midpoint of X(3699) and X(34548)
X(38384) = reflection of X(3756) in X(5510)
X(38384) = trilinear product X(1339)*X(15637)
X(38384) = orthojoin of X(649)
X(38384) = orthopole of Nagel line
X(38384) = crossdifference of every pair of points on line {X(2429), X(22086)}
X(38384) = crosspoint of X(7) and X(4462)
X(38384) = crosssum of X(55) and X(34080)
X(38384) = X(1440)-Ceva conjugate of-X(30719)

X(38385) = EQUICENTER OF THESE TRIANGLES: INTOUCH AND PEDAL-OF-X(5)

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

X(38385) lies on these lines: {3,6}, {940,21909}, {1124,37607}, {1335,37573}, {1702,10476}, {3299,37608}, {3301,37574}, {4383,21992}, {5706,36715}, {7969,37529}, {11292,37676}, {21991,37633}, {35631,35775}

X(38386) = EQUICENTER OF THESE TRIANGLES: INTOUCH AND PEDAL-OF-X(6)

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

X(38386) lies on these lines: {4,218}, {644,34547}, {1358,16184}, {1537,2801}, {1565,2310}, {2170,38392}, {3120,38384}, {3309,4904}, {4534,28292}, {6831,34848}, {14109,30231}, {14661,15521}, {21044,38387}

X(38386) = midpoint of X(i) and X(j) for these {i,j}: {644, 34547}, {14661, 15521}
X(38386) = reflection of X(4904) in X(5511)
X(38386) = orthojoin of X(513)
X(38386) = orthopole of PU(28) (line X(1)X(6))
X(38386) = crosspoint of X(7) and X(4468)

X(38387) = EQUICENTER OF THESE TRIANGLES: PEDAL-OF-X(2) AND PEDAL-OF-X(7)

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

X(38387) lies on these lines: {4,653}, {1512,1536}, {21044,38386}

X(38387) = orthojoin of X(663)
X(38387) = orthopole of line X(2)X(7)

X(38388) = EQUICENTER OF THESE TRIANGLES: ORTHIC AND PEDAL-OF-X(7)

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

X(38388) lies on these lines: {11,15607}, {1146,3270}, {1863,5185}, {2310,3125}, {3022,21044}, {3900,5514}, {7215,8760}

X(38388) = barycentric product X(1146)*X(7580)
X(38388) = Zosma transform of X(36118)
X(38388) = crosssum of X(i) and X(j) for these {i,j}: {3, 934}, {165, 1020}
X(38388) = orthopole of line X(4)X(7)

X(38389) = EQUICENTER OF THESE TRIANGLES: ORTHIC AND PEDAL-OF-X(8)

Barycentrics    a*(b-c)^2*(a^3-(b+c)^2*a+2*(b+c)*b*c) : :
X(38389) = X(3937)-4*X(38390) = 4*X(6667)-3*X(34583)

X(38389) lies on these lines: {4,151}, {11,3025}, {51,1836}, {80,2841}, {100,29349}, {124,867}, {125,5521}, {149,2810}, {221,17516}, {223,15503}, {244,4014}, {373,5880}, {511,5057}, {512,1109}, {661,14936}, {667,34467}, {908,15310}, {1331,36280}, {1357,1647}, {1364,35015}, {1479,23154}, {1562,8735}, {1699,26892}, {1777,28077}, {1828,12688}, {1829,2778}, {1878,6001}, {2310,18210}, {2611,8672}, {2617,37019}, {2771,15906}, {2807,34789}, {2808,9809}, {2815,24026}, {2835,22321}, {2969,3270}, {3038,9458}, {3042,24410}, {3120,3271}, {3248,21963}, {3888,17777}, {3917,24703}, {4124,17888}, {4679,5650}, {4813,20974}, {5146,6000}, {5840,31847}, {5943,20292}, {6007,32843}, {6667,34583}, {8050,36791}, {9052,17484}, {9519,14740}, {11381,12679}, {11918,11988}, {12109,14450}, {12699,16980}, {13744,34586}, {17477,19945}, {18344,38362}, {20295,23989}, {21746,24725}, {22376,27627}, {23638,33094}

X(38389) = reflection of X(i) in X(j) for these (i,j): (11, 38390), (3937, 11)
X(38389) = barycentric product X(i)*X(j) for these {i, j}: {11, 34048}, {649, 17894}, {905, 16228}, {1086, 5687}
X(38389) = trilinear product X(i)*X(j) for these {i, j}: {244, 5687}, {667, 17894}, {1459, 16228}, {2170, 34048}
X(38389) = orthojoin of X(22383)
X(38389) = orthopole of line X(4)X(8)
X(38389) = Zosma transform of X(1897)
X(38389) = crosspoint of X(4) and X(513)
X(38389) = crosssum of X(3) and X(100)
X(38389) = X(i)-Ceva conjugate of-X(j) for these (i,j): (4, 16228), (459, 6591)

X(38390) = EQUICENTER OF THESE TRIANGLES: PEDAL-OF-X(5) AND PEDAL-OF-X(10)

Barycentrics    a*(b-c)^2*(a^3-(b^2+3*b*c+c^2)*a+3*(b+c)*b*c) : :
X(38390) = 3*X(11)-X(3937) = X(3937)+3*X(38389) = 5*X(31272)-3*X(34583)

X(38390) lies on these lines: {4,34434}, {11,3025}, {517,6246}, {661,38347}, {1109,4132}, {2779,18483}, {2841,12019}, {3035,29349}, {3756,4014}, {3781,24703}, {4106,23989}, {4388,14973}, {4553,17777}, {4776,27009}, {5057,20718}, {5087,15310}, {5400,23845}, {10738,31847}, {13476,24725}, {16228,21666}, {22300,22793}, {31272,34583}, {34462,34789}

X(38390) = midpoint of X(i) and X(j) for these {i,j}: {11, 38389}, {10738, 31847}, {34462, 34789}
X(38390) = crosssum of X(100) and X(5303)
X(38390) = orthopole of line X(5)X(10)

X(38391) = EQUICENTER OF THESE TRIANGLES: PEDAL-OF-X(7) AND PEDAL-OF-X(8)

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

X(38391) lies on these lines: {3,25920}, {4,150}, {1358,30199}, {3309,4904}

X(38391) = orthojoin of X(3063)
X(38391) = orthopole of line X(7)X(8)

X(38392) = EQUICENTER OF THESE TRIANGLES: PEDAL-OF-X(8) AND PEDAL-OF-X(9)

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

X(38392) lies on these lines: {4,10743}, {2170,38386}

X(38392) = orthopole of line X(8)X(9)

X(38393) = EQUICENTER OF THESE TRIANGLES: CEVIAN-OF-X(13) AND CEVIAN-OF-X(14)

Barycentrics    (b^2-c^2)^2*(a^4+(b^2+c^2)*a^2-2*b^4+b^2*c^2-2*c^4) : :
X(38393) = X(2407)-3*X(14995)

X(38393) lies on these lines: {4,23347}, {6,15358}, {30,14356}, {50,35727}, {53,232}, {115,804}, {381,6795}, {395,36298}, {396,36299}, {511,35346}, {523,868}, {597,3845}, {1576,10722}, {1648,10278}, {2088,15543}, {2407,14995}, {3134,3258}, {6593,22505}, {9220,34845}, {10412,12079}, {13595,16984}, {15109,35896}, {17500,37349}, {23283,30452}, {23284,30453}

X(38393) = reflection of X(35345) in X(24975)
X(38393) = barycentric product X(i)*X(j) for these {i, j}: {115, 7809}, {338, 15107}, {1109, 18722}
X(38393) = trilinear product X(i)*X(j) for these {i, j}: {115, 18722}, {1109, 15107}
X(38393) = pole of the trilinear polar of X(1989) with respect to Kiepert hyperbola
X(38393) = crosssum of X(i) and X(j) for these {i,j}: {15, 35330}, {16, 35329}
X(38393) = X(1989)-Ceva conjugate of-X(523)
X(38393) = {X(7668), X(34981)}-harmonic conjugate of X(38394)

X(38394) = EQUICENTER OF THESE TRIANGLES: CEVIAN-OF-X(17) AND CEVIAN-OF-X(18)

Barycentrics    (b^2-c^2)^2*(a^4-3*(b^2+c^2)*a^2+2*b^4-3*b^2*c^2+2*c^4) : :
X(38394) = 4*X(231)-X(35727)

X(38394) lies on these lines: {54,32737}, {115,804}, {137,8901}, {231,35727}, {1576,14639}, {1879,3613}, {1989,15358}, {4577,15031}, {18353,34845}, {34989,38229}

X(38394) = pole of the trilinear polar of X(2963) with respect to Kiepert hyperbola
X(38394) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115, 34981, 7668), (7668, 34981, 38393)

X(38395) = EQUICENTER OF THESE TRIANGLES: PEDAL-OF-X(13) AND PEDAL-OF-X(14)

Barycentrics    (b^2-c^2)^2*(2*a^2-b^2-c^2)*(3*a^4-2*(b^2+c^2)*a^2-b^4+3*b^2*c^2-c^4) : :
X(38395) = 4*X(15359)-3*X(35605)

X(38395) lies on these lines: {115,1499}, {512,16278}, {690,2682}, {1503,1570}, {1561,32111}, {3566,14120}, {7728,14559}, {11615,18334}, {11645,15303}, {15342,36174}, {15359,32478}

X(38395) = midpoint of X(15342) and X(36174)
X(38395) = reflection of X(i) in X(j) for these (i,j): (5099, 2682), (15357, 14120)
X(38395) = orthojoin of X(526)
X(38395) = orthopole of Fermat axis

X(38396) = EQUICENTER OF THESE TRIANGLES: ABC AND ANTI-ASCELLA

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

X(38396) lies on these lines: {3,37645}, {25,5480}, {54,7393}, {110,9818}, {125,32621}, {141,26869}, {182,394}, {1147,6090}, {1385,37246}, {1593,6759}, {3515,12242}, {3516,15105}, {5094,19459}, {5544,12099}, {5650,31521}, {5890,21766}, {5972,11284}, {6642,8907}, {6800,8718}, {7592,7998}, {8717,18475}, {11245,11898}, {11403,34782}, {11410,13171}, {13198,15106}, {28419,30739}

X(38396) = circumnormal isogonal conjugate of X(35243)

X(38397) = EQUICENTER OF THESE TRIANGLES: ABC AND 3rd ANTI-EULER

Barycentrics    (b^2+c^2)*a^4-(3*b^4-b^2*c^2+3*c^4)*a^2+2*(b^4-c^4)*(b^2-c^2) : :
X(38397) = 4*X(5)-3*X(7699) = 8*X(5)-3*X(36852) = X(20)+3*X(18387) = 5*X(631)-3*X(3431) = 7*X(3526)-3*X(11935) = 3*X(7703)-2*X(31857) = 2*X(12062)+X(37484)

X(38397) lies on these lines: {2,575}, {3,9140}, {4,15360}, {5,568}, {20,11454}, {54,3526}, {68,631}, {69,11443}, {76,850}, {110,15069}, {125,7998}, {141,15531}, {182,18882}, {183,30789}, {343,858}, {382,15062}, {511,7703}, {548,12289}, {599,895}, {1154,7579}, {1209,14789}, {1352,10546}, {1656,5643}, {2888,11449}, {3060,5169}, {3410,37760}, {3448,15080}, {3740,32782}, {3832,15741}, {3843,33539}, {4197,5885}, {5067,11487}, {5070,12161}, {5094,8537}, {5181,32248}, {5449,11444}, {5486,21356}, {5876,12824}, {5925,11440}, {6032,13330}, {6792,7746}, {7486,11431}, {7493,11206}, {7539,12834}, {7999,21230}, {8262,11188}, {8548,15059}, {9159,12079}, {10296,18392}, {10516,10545}, {10574,12359}, {11178,16042}, {11412,23330}, {11450,32547}, {11464,23236}, {11550,20063}, {11591,11704}, {11799,15305}, {12062,37484}, {12111,12827}, {12505,32228}, {12828,16868}, {14002,18553}, {14076,32338}, {15067,20396}, {15072,16003}, {15141,32244}, {15717,19467}, {16239,32358}, {18134,37762}, {19127,25335}, {21358,38402}, {22712,31127}, {26913,30739}

X(38397) = reflection of X(36852) in X(7699)
X(38397) = complement of X(9716)
X(38397) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (343, 23293, 2979), (18553, 32225, 14002)

X(38398) = EQUICENTER OF THESE TRIANGLES: ABC AND AOA

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

X(38398) lies on these lines: {1593,22962}, {1899,5159}, {3548,18396}, {5094,5943}, {5972,26864}, {13416,37638}, {15128,32257}

X(38399) = EQUICENTER OF THESE TRIANGLES: ABC AND ASCELLA

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

X(38399) lies on these lines: {1,5744}, {2,1750}, {9,17603}, {21,3306}, {57,1001}, {100,9623}, {142,30353}, {200,5218}, {224,936}, {405,37526}, {411,18219}, {443,31673}, {452,12436}, {1466,5436}, {1490,7483}, {1768,11407}, {2999,3736}, {3358,11227}, {3601,3913}, {3653,6265}, {3897,12629}, {4312,9776}, {4326,5231}, {5437,13615}, {6261,6705}, {6847,12565}, {6935,30503}, {7171,11108}, {9841,37224}, {9940,31435}, {10179,11518}, {14022,20195}, {16112,17612}, {21164,37306}

X(38399) = {X(6857), X(8726)}-harmonic conjugate of X(8583)

X(38400) = EQUICENTER OF THESE TRIANGLES: ABC AND BANKOFF

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

X(38400) lies on these lines: {2,33358}, {624,33368}, {640,36770}, {641,3366}, {5859,35731}, {16627,35738}, {22113,35730}

X(38401) = EQUICENTER OF THESE TRIANGLES: ABC AND EHRMANN-CROSS

Barycentrics    SA*(SB-SC)*((6*R^2+SA-2*SW)*S^2-3*(3*R^2-SW)*SB*SC) : :
X(38401) = 3*X(376)+X(18808)

X(38401) lies on these lines: {3,523}, {69,3265}, {141,30511}, {376,18808}, {441,18310}, {520,1216}, {525,35254}, {526,6334}, {924,5907}, {5489,11821}, {6699,8552}, {8057,12359}, {8675,11574}, {9003,32257}, {22104,24975}

X(38401) = barycentric product X(525)*X(12383)
X(38401) = barycentric quotient X(647)/X(35372)
X(38401) = trilinear product X(656)*X(12383)
X(38401) = trilinear quotient X(656)/X(35372)
X(38401) = crossdifference of every pair of points on line {X(3003), X(14581)}
X(38401) = X(162)-isoconjugate-of-X(35372)
X(38401) = X(647)-reciprocal conjugate of-X(35372)

X(38402) = EQUICENTER OF THESE TRIANGLES: ABC AND 1st EHRMANN

Barycentrics    a^2*(a^6+(b^2+c^2)*a^4-(b^4+17*b^2*c^2+c^4)*a^2-(b^2+c^2)*(b^4+8*b^2*c^2+c^4)) : :
X(38402) = X(6)+2*X(5888) = 2*X(182)+X(14926) = 4*X(3589)-X(7693) = 5*X(12017)+X(18551)

X(38402) lies on the Stammler hyperbola, cubic K282 and these lines: {2,2930}, {3,5476}, {6,5888}, {23,2916}, {159,11284}, {182,399}, {195,575}, {576,15047}, {597,7496}, {599,16511}, {1498,10249}, {2918,12106}, {2935,15578}, {3618,37827}, {3763,19588}, {5085,31861}, {5092,35001}, {5621,38064}, {5643,9019}, {5898,12584}, {8542,22112}, {10627,11477}, {12017,18551}, {13474,18374}, {15534,16419}, {16042,19596}, {21358,38397}, {24206,25330}, {33532,38072}

X(38402) = isogonal conjugate of the cyclocevian conjugate of X(598)
X(38402) = barycentric product X(598)*X(8561)
X(38402) = pole of the trilinear polar of X(597) with respect to circumcircle
X(38402) = pole of the trilinear polar of X(7496) with respect to MacBeath circumconic
X(38402) = crossdifference of every pair of points on line {X(12073), X(24976)}
X(38402) = crosssum of X(523) and X(20389)
X(38402) = X(597)-Ceva conjugate of-X(6)

X(38403) = EQUICENTER OF THESE TRIANGLES: ABC AND 1st FERMAT-DAO

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

X(38403) lies on the cubic K341b and these lines: {2,11080}, {3,3440}, {6,2981}, {14,99}, {15,10411}, {16,16459}, {98,25213}, {323,8603}, {471,8741}, {1511,3439}, {2378,10409}, {2379,25211}, {3005,5888}, {8739,14590}, {9203,25209}, {11086,11146}

X(38403) = isogonal conjugate of X(8014)
X(38403) = anticomplement of X(16536)
X(38403) = barycentric product X(i)*X(j) for these {i, j}: {298, 2981}, {323, 11117}
X(38403) = barycentric quotient X(i)/X(j) for these (i, j): (15, 396), (186, 23714), (323, 532), (526, 14446)
X(38403) = trilinear product X(1094)*X(11119)
X(38403) = trilinear pole of the line {323, 6137}
X(38403) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(11131)}} and {{A, B, C, X(6), X(14)}}
X(38403) = Cevapoint of X(15) and X(11131)
X(38403) = X(15)-cross conjugate of-X(2981)
X(38403) = X(396)-isoconjugate-of-X(2153)
X(38403) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (15, 396), (186, 23714), (323, 532), (526, 14446)

X(38404) = EQUICENTER OF THESE TRIANGLES: ABC AND 2nd FERMAT-DAO

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

X(38404) lies on the cubic K341a and these lines: {2,11085}, {3,3441}, {6,6151}, {13,99}, {15,16460}, {16,10411}, {98,25216}, {323,8604}, {470,8742}, {1511,3438}, {2378,25212}, {2379,10410}, {3005,5888}, {8740,14590}, {9202,25210}, {11081,11145}

X(38404) = isogonal conjugate of X(8015)
X(38404) = anticomplement of X(16537)
X(38404) = barycentric product X(i)*X(j) for these {i, j}: {299, 6151}, {323, 11118}
X(38404) = barycentric quotient X(i)/X(j) for these (i, j): (16, 395), (186, 23715), (323, 533), (526, 14447)
X(38404) = trilinear product X(1095)*X(11120)
X(38404) = trilinear pole of the line {323, 6138}
X(38404) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(11130)}} and {{A, B, C, X(6), X(13)}}
X(38404) = Cevapoint of X(16) and X(11130)
X(38404) = X(16)-cross conjugate of-X(6151)
X(38404) = X(395)-isoconjugate-of-X(2154)
X(38404) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (16, 395), (186, 23715), (323, 533), (526, 14447)

X(38405) = EQUICENTER OF THESE TRIANGLES: ABC AND 1st HYACINTH

Barycentrics    3*S^4+(R^2*(50*R^2-19*SA-7*SW)+3*SA^2-3*SB*SC)*S^2+(10*R^2*(R^2-SW)+SW^2)*SB*SC : :

X(38405) lies on these lines: {1657,10605}, {5878,6102}, {10095,14982}

X(38406) = EQUICENTER OF THESE TRIANGLES: ABC AND JENKINS-CONTACT

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

X(38406) lies on these lines: {2,18654}, {4,9}, {6,21019}, {8,21030}, {37,5123}, {75,21244}, {322,26012}, {346,21044}, {579,21066}, {594,1329}, {604,5176}, {958,5124}, {1400,25005}, {1737,21074}, {2092,23903}, {2171,11681}, {2324,6048}, {3036,17362}, {3169,6735}, {3247,5530}, {3596,4858}, {3661,21246}, {3687,4007}, {4193,17452}, {4268,17303}, {4361,30826}, {5783,5790}, {9564,15877}, {17229,30812}, {17299,25681}, {17314,17748}, {17754,24996}, {18395,21061}, {21273,31032}, {25007,27626}, {26594,29965}

X(38406) = barycentric product X(10)*X(14011)
X(38406) = trilinear product X(37)*X(14011)
X(38406) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(14011)}} and {{A, B, C, X(242), X(3596)}}

X(38407) = EQUICENTER OF THESE TRIANGLES: ABC AND JENKINS-TANGENTIAL

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

X(38407) lies on these lines: {321,21033}, {3210,21857}, {3662,4359}, {27269,28606}

X(38407) = crosspoint of X(75) and X(30022)

X(38408) = EQUICENTER OF THESE TRIANGLES: ABC AND 2nd JENKINS

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

X(38408) lies on these lines: {6,8258}, {8,21014}, {10,37}, {71,4109}, {261,284}, {346,21044}, {573,1761}, {3125,34895}, {3169,3705}, {3564,5771}, {3707,18253}, {4165,21798}, {5530,5750}, {9560,15349}, {17275,18755}, {18697,27691}, {36974,37508}

X(38408) = barycentric product X(i)*X(j) for these {i, j}: {333, 34528}, {2321, 26840}
X(38408) = barycentric quotient X(i)/X(j) for these (i, j): (8, 18812), (2321, 34527)
X(38408) = trilinear product X(i)*X(j) for these {i, j}: {21, 34528}, {210, 26840}, {314, 9560}
X(38408) = trilinear quotient X(312)/X(18812)
X(38408) = intersection, other than A,B,C, of conics {{A, B, C, X(10), X(261)}} and {{A, B, C, X(37), X(2185)}}
X(38408) = X(i)-isoconjugate-of-X(j) for these {i,j}: {604, 18812}, {1408, 34527}
X(38408) = X(8)-reciprocal conjugate of-X(18812)
X(38408) = {X(3694), X(4136)}-harmonic conjugate of X(2321)

X(38409) = EQUICENTER OF THESE TRIANGLES: ABC AND 3rd JENKINS

Barycentrics    (b+c)*(-a+b+c)*(4*(b+c)*a^5+3*(3*b^2+4*b*c+3*c^2)*a^4+2*(b+c)*(3*b^2+7*b*c+3*c^2)*a^3+(b^2+b*c+c^2)*(b^2+15*b*c+c^2)*a^2+4*(b+c)*(b^2+b*c+c^2)*b*c*a+(b+c)^2*b^2*c^2) : :

X(38409) lies on the line {2321,37868}

X(38410) = EQUICENTER OF THESE TRIANGLES: ABC AND K798E

Barycentrics    3*a^4-(b+c)*a^3-(6*b^2+b*c+6*c^2)*a^2+(b+c)*(b^2-4*b*c+c^2)*a+3*(b^2-c^2)^2 : :
X(38410) = 10*X(19862)-X(32633)

X(38410) lies on these lines: {1,6668}, {2,3754}, {5,3576}, {21,19862}, {80,1125}, {191,31260}, {499,18412}, {1698,15950}, {2476,5444}, {3616,15079}, {3628,6265}, {3646,5128}, {3753,38411}, {3868,37701}, {3881,37731}, {3890,37735}, {4423,37251}, {5259,6905}, {5426,7173}, {5439,8261}, {5880,7483}, {5901,38129}, {6933,37525}, {7548,10171}, {7988,37468}, {10039,26726}, {10950,25055}, {11813,19878}, {15228,37291}, {17530,37616}, {31424,37692}

X(38410) = {X(1125), X(7504)}-harmonic conjugate of X(80)

X(38411) = EQUICENTER OF THESE TRIANGLES: ABC AND K798I

Barycentrics    3*a^4+(b+c)*a^3-(6*b^2+b*c+6*c^2)*a^2-(b+c)*(b^2-4*b*c+c^2)*a+3*(b^2-c^2)^2 : :
X(38411) = 16*X(3634)-X(17501)

X(38411) lies on these lines: {2,3884}, {10,7972}, {21,3634}, {140,355}, {498,38053}, {1125,5559}, {3624,8275}, {3646,19872}, {3753,38410}, {3812,26725}, {3826,4187}, {3869,5445}, {3874,27529}, {3901,24914}, {4999,34918}, {5251,6940}, {5441,6702}, {5442,11681}, {5444,25005}, {6667,37563}, {6684,31263}, {6842,31423}, {11231,25917}, {16173,20107}, {16239,34352}, {17057,19877}, {17566,37710}, {19854,38134}, {19876,37298}, {19907,31235}, {34122,37616}

X(38412) = EQUICENTER OF THESE TRIANGLES: ABC AND LARGEST-CIRCUMSCRIBED-EQUILATERAL

Barycentrics    -2*sqrt(3)*(a^2+4*b^2+4*c^2)*S+7*a^4-13*(b^2+c^2)*a^2-8*b^2*c^2+2*(b^2-c^2)^2 : :

X(38412) lies on these lines: {2,9112}, {141,36770}, {298,36764}, {302,7757}, {376,618}, {530,31683}, {620,5464}, {623,5473}, {625,6779}, {5463,31693}, {6054,32553}, {6773,21156}, {21358,31274}, {22712,37463}

X(38413) = EQUICENTER OF THESE TRIANGLES: ABC AND INNER-LE VIET AN

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

X(38413) lies on the MacBeath circumconic and these lines: {14,2986}, {110,5994}, {476,16806}, {647,32661}, {648,23896}, {895,36297}, {2987,3458}

X(38413) = isogonal conjugate of the polar conjugate of X(23896)
X(38413) = isotomic conjugate of the polar conjugate of X(5994)
X(38413) = barycentric product X(i)*X(j) for these {i, j}: {3, 23896}, {14, 4558}, {69, 5994}, {99, 36297}, {265, 17402}, {298, 32662}
X(38413) = barycentric quotient X(i)/X(j) for these (i, j): (3, 23871), (14, 14618), (110, 471), (184, 6138), (647, 30468), (1576, 8740)
X(38413) = trilinear product X(i)*X(j) for these {i, j}: {14, 4575}, {15, 36061}, {48, 23896}, {63, 5994}, {255, 36309}, {662, 36297}
X(38413) = trilinear quotient X(i)/X(j) for these (i, j): (14, 24006), (48, 6138), (63, 23871), (163, 8740), (656, 30468), (662, 471)
X(38413) = trilinear pole of the line {3, 36297}
X(38413) = intersection, other than A,B,C, of conic {{A, B, C, X(3), X(9203)}} and MacBeath circumconic
X(38413) = crossdifference of every pair of points on line {X(30468), X(35235)}
X(38413) = X(i)-isoconjugate-of-X(j) for these {i,j}: {16, 24006}, {19, 23871}, {92, 6138}, {162, 30468}
X(38413) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (3, 23871), (14, 14618), (110, 471), (184, 6138)

X(38414) = EQUICENTER OF THESE TRIANGLES: ABC AND OUTER-LE VIET AN

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

X(38414) lies on the MacBeath circumconic and these lines: {13,2986}, {110,5995}, {476,16807}, {647,32661}, {648,23895}, {895,36296}, {2987,3457}

X(38414) = isogonal conjugate of the polar conjugate of X(23895)
X(38414) = isotomic conjugate of the polar conjugate of X(5995)
X(38414) = barycentric product X(i)*X(j) for these {i, j}: {3, 23895}, {13, 4558}, {69, 5995}, {99, 36296}, {265, 17403}, {299, 32662}
X(38414) = barycentric quotient X(i)/X(j) for these (i, j): (3, 23870), (13, 14618), (110, 470), (184, 6137), (647, 30465), (1576, 8739)
X(38414) = trilinear product X(i)*X(j) for these {i, j}: {13, 4575}, {16, 36061}, {48, 23895}, {63, 5995}, {255, 36306}, {662, 36296}
X(38414) = trilinear quotient X(i)/X(j) for these (i, j): (13, 24006), (48, 6137), (63, 23870), (163, 8739), (656, 30465), (662, 470)
X(38414) = trilinear pole of the line {3, 36296}
X(38414) = intersection, other than A,B,C, of conic {{A, B, C, X(3), X(9202)}} and MacBeath circumconic
X(38414) = crossdifference of every pair of points on line {X(30465), X(35235)}
X(38414) = X(i)-isoconjugate-of-X(j) for these {i,j}: {15, 24006}, {19, 23870}, {92, 6137}, {162, 30465}
X(38414) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (3, 23870), (13, 14618), (110, 470), (184, 6137)

X(38415) = EQUICENTER OF THESE TRIANGLES: ABC AND 1st MORLEY-MIDPOINT

Barycentrics    (b*u*x^2-a*v*y^2)*(c*u-a*w)*v*z-(c*u*x^2-a*w*z^2)*(b*u-a*v)*w*y : : , where x:y:z=cos(A/3) : : and u:v:w=sin(A/3) : :
Barycentrics    Sec[A/3 + Pi/6]*Sin[A/3] : :    (Peter Moses, May 11, 2020)

X(38415) lies on the circumconic with center X(16271) and these lines: {2,3603}, {357,5456}, {5390,13593}

X(38415) = intersection, other than A,B,C, of conics {{A, B, C, X(13), X(5457)}} and {{A, B, C, X(14), X(357)}}

X(38416) = EQUICENTER OF THESE TRIANGLES: ABC AND 2nd MORLEY-MIDPOINT

Barycentrics    (b*u*x^2-a*v*y^2)*(c*u-a*w)*v*z-(c*u*x^2-a*w*z^2)*(b*u-a*v)*w*y : : , where x:y:z=cos((A-2*Pi)/3) : : and u:v:w=sin((A-2*Pi)/3) : :
Barycentrics    Cos[A/3 - Pi/6]*Csc[A/3] : :    (Peter Moses, May 11, 2020)

X(38416) lies on these lines: {2,3604}, {357,8065}, {1136,16871}, {10258,13593}

X(38417) = EQUICENTER OF THESE TRIANGLES: ABC AND 3rd MORLEY-MIDPOINT

Barycentrics    (b*u*x^2-a*v*y^2)*(c*u-a*w)*v*z-(c*u*x^2-a*w*z^2)*(b*u-a*v)*w*y : : , where x:y:z=cos((A-4*Pi)/3) : : and u:v:w=sin((A-4*Pi)/3) : :
Barycentrics    Cos[A/3 + Pi/6]*Sec[A/3 - Pi/6] : :    (Peter Moses, May 11, 2020)

X(38417) lies on these lines: {2,3602}, {1134,7309}, {10259,13593}

X(38418) = EQUICENTER OF THESE TRIANGLES: ABC AND MOSES-STEINER REFLECTION

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

X(38418) lies on these lines: {2,8786}, {3763,11188}

X(38419) = EQUICENTER OF THESE TRIANGLES: ABC AND 1st PRZYBYŁOWSKI-BOLLIN

Barycentrics    a*(SW+sqrt(SW+sqrt(3)*S)*(b+c)+sqrt(3)*S)*((a+2*sqrt(SW+sqrt(3)*S))*S+sqrt(3)*SA*a) : :

X(38419) lies on the line {993,5239}

X(38420) = EQUICENTER OF THESE TRIANGLES: ABC AND 2nd PRZYBYŁOWSKI-BOLLIN

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

X(38420) lies on the line {993,5239}

X(38421) = EQUICENTER OF THESE TRIANGLES: ABC AND 3rd PRZYBYŁOWSKI-BOLLIN

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

X(38421) lies on the line {993,5240}

X(38422) = EQUICENTER OF THESE TRIANGLES: ABC AND 4th PRZYBYŁOWSKI-BOLLIN

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

X(38422) lies on the line {993,5240}

X(38423) = EQUICENTER OF THESE TRIANGLES: ABC AND 1st TRI-SQUARES-CENTRAL

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

X(38423) lies on these lines: {371,12297}, {487,31411}, {1587,3102}, {3068,35949}, {3543,6561}, {11288,32806}, {12221,19103}, {12601,19117}, {13644,32815}, {13665,13674}, {15484,37350}, {18512,21850}, {19054,33457}, {32787,33456}

X(38424) = EQUICENTER OF THESE TRIANGLES: ABC AND 2nd TRI-SQUARES-CENTRAL

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

X(38424) lies on these lines: {372,12296}, {1588,3103}, {3069,35948}, {3543,6560}, {11288,32805}, {12222,19104}, {12602,19116}, {13763,32815}, {13785,13794}, {15484,37350}, {18510,21850}, {19053,33456}, {32788,33457}

X(38425) = EQUICENTER OF THESE TRIANGLES: ABC AND 1st TRI-SQUARES

Barycentrics    (2*(5*a^2+2*b^2+2*c^2))*S+(9*(b^2+c^2))*a^2-(b^2-c^2)^2 : :

X(38425) lies on these lines: {6,13771}, {371,5870}, {376,3068}, {485,15883}, {491,31168}, {590,13711}, {1384,13712}, {2549,13835}, {3103,5418}, {3311,6280}, {6054,13653}, {6221,36734}, {6396,8975}, {7583,12124}, {7585,26288}, {8981,12306}, {13669,13720}

X(38426) = EQUICENTER OF THESE TRIANGLES: ABC AND 2nd TRI-SQUARES

Barycentrics    -2*(5*a^2+2*b^2+2*c^2)*S+9*(b^2+c^2)*a^2-(b^2-c^2)^2 : :

X(38426) lies on these lines: {6,13650}, {372,5871}, {376,3069}, {486,15884}, {492,31168}, {615,13834}, {1384,13835}, {2549,13712}, {3102,5420}, {3312,6279}, {6054,13773}, {6200,13949}, {6398,36718}, {7584,12123}, {7586,26289}, {12305,13966}, {13789,13843}

X(38427) = EQUICENTER OF THESE TRIANGLES: ABC AND VU-DAO-X(15)-ISODYNAMIC

Barycentrics    (sqrt(3)*b^2-2*S)*(sqrt(3)*c^2-2*S)*SB*SC : :

X(38427) lies on the circumconic with center X(136) and on these lines: {4,617}, {393,6151}, {427,36898}, {463,32085}, {470,8742}, {471,8738}, {472,648}, {1300,10410}, {2381,6116}, {6110,16460}

X(38427) = polar conjugate of X(395)
X(38427) = isotomic conjugate of the anticomplement of X(11543)
X(38427) = barycentric product X(i)*X(j) for these {i, j}: {264, 6151}, {470, 11118}, {471, 11120}
X(38427) = barycentric quotient X(i)/X(j) for these (i, j): (4, 395), (112, 35330), (186, 19295), (340, 14921), (393, 462), (468, 9117)
X(38427) = trilinear product X(92)*X(6151)
X(38427) = trilinear quotient X(i)/X(j) for these (i, j): (92, 395), (158, 462), (162, 35330), (811, 35315)
X(38427) = trilinear pole of the line {470, 2501}
X(38427) = lies on the circumconic with center X(136))
X(38427) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(93)}} and {{A, B, C, X(14), X(617)}}
X(38427) = Cevapoint of X(4) and X(471)
X(38427) = X(340)-cross conjugate of-X(38428)
X(38427) = X(i)-isoconjugate-of-X(j) for these {i,j}: {48, 395}, {255, 462}, {656, 35330}, {810, 35315}
X(38427) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (4, 395), (112, 35330), (186, 19295), (340, 14921)

X(38428) = EQUICENTER OF THESE TRIANGLES: ABC AND VU-DAO-X(16)-ISODYNAMIC

Barycentrics    (sqrt(3)*b^2+2*S)*(sqrt(3)*c^2+2*S)*SB*SC : :

X(38428) lies on the circumconic with center X(136) and on these lines: {4,616}, {393,2981}, {427,36898}, {462,32085}, {470,8737}, {471,8741}, {473,648}, {1300,10409}, {2380,6117}, {6111,16459}

X(38428) = polar conjugate of X(396)
X(38428) = isotomic conjugate of the anticomplement of X(11542)
X(38428) = barycentric product X(i)*X(j) for these {i, j}: {264, 2981}, {470, 11119}, {471, 11117}
X(38428) = barycentric quotient X(i)/X(j) for these (i, j): (4, 396), (112, 35329), (186, 19294), (340, 14922), (393, 463), (468, 9115)
X(38428) = trilinear product X(92)*X(2981)
X(38428) = trilinear quotient X(i)/X(j) for these (i, j): (92, 396), (158, 463), (162, 35329), (811, 35314)
X(38428) = trilinear pole of the line {471, 2501}
X(38428) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(93)}} and {{A, B, C, X(13), X(616)}}
X(38428) = Cevapoint of X(4) and X(470)
X(38428) = X(340)-cross conjugate of-X(38427)
X(38428) = X(i)-isoconjugate-of-X(j) for these {i,j}: {48, 396}, {255, 463}, {656, 35329}, {810, 35314}
X(38428) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (4, 396), (112, 35329), (186, 19294), (340, 14922)

X(38429) = EQUICENTER OF THESE TRIANGLES: ABC AND YIU TANGENTS

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

X(38429) lies on these lines: {128,549}, {233,3054}, {373,18875}, {427,14918}, {550,1209}, {5965,33992}, {14073,21357}, {15699,16336}, {32348,35728}

X(38430) =  EULER LINE INTERCEPT OF X(519)X(15349)

Barycentrics    2 a^7+6 a^6 b+5 a^5 b^2-4 a^4 b^3-7 a^3 b^4-a^2 b^5-b^7+6 a^6 c+14 a^5 b c-13 a^3 b^3 c-6 a^2 b^4 c-a b^5 c+5 a^5 c^2-10 a^3 b^2 c^2-7 a^2 b^3 c^2+2 b^5 c^2-4 a^4 c^3-13 a^3 b c^3-7 a^2 b^2 c^3+2 a b^3 c^3-b^4 c^3-7 a^3 c^4-6 a^2 b c^4-b^3 c^4-a^2 c^5-a b c^5+2 b^2 c^5-c^7 : :

As a point on the Euler line, X(38430) has Shinagawa coefficients {7$a$(E+F)+5$aSA$+13a b c,-9$a$(E+F)-3$aSA$-15a b c).

See Kadir Altintas and Ercole Suppa, Euclid 875 .

X(38430) lies on these lines: {2,3}, {519,15349}, {540,17748}, {986,37631}, {1326,3017}, {3178,3579}, {6002,28602}, {27368,34773}, {34528,37508}

X(38431) =  EULER LINE INTERCEPT OF X(15)X(110)

Barycentrics    Sqrt[3] a^2 (a-b-c) (a+b-c) (a-b+c) (a+b+c) (a^6-2 a^4 b^2+2 a^2 b^4-b^6-2 a^4 c^2+4 a^2 b^2 c^2+3 b^4 c^2+2 a^2 c^4+3 b^2 c^4-c^6)-2 a^2 (a^8-a^6 b^2+a^2 b^6-b^8-a^6 c^2-6 a^4 b^2 c^2+15 a^2 b^4 c^2-4 b^6 c^2+15 a^2 b^2 c^4+12 b^4 c^4+a^2 c^6-4 b^2 c^6-c^8) S : :
Barycentrics    (SB+SC) (9 S^4 +6 Sqrt[3] S^3 (SA+SW)+ 3 S^2 (3 SA^2+2 SA SW+SW^2)+2 Sqrt[3] SA S(3 SA-SW)SW + SA (3 SA-2 SW) SW^2) : :

See Kadir Altintas and Ercole Suppa, Euclid 886 .

X(38431) lies on these lines: {2,3}, {15,110}, {16,5640}, {61,11422}, {62,15019}, {182,14169}, {187,37776}, {323,5611}, {373,13349}, {511,11131}, {574,37775}, {576,11126}, {1495,13350}, {2981,36757}, {3066,11481}, {3106,3458}, {3292,11127}, {3457,16461}, {5201,35315}, {5463,15360}, {5615,11002}, {5617,11092}, {5643,34425}, {5650,36756}, {5651,9735}, {5943,21402}, {6105,13859}, {6772,13233}, {7998,14539}, {8546,14179}, {9194,14270}, {9736,34417}, {9885,34013}, {10545,10646}, {10546,10645}, {11141,21467}, {11480,35259}, {14174,14184}, {14187,22687}, {14538,15107}, {15801,15961}

X(38431) = crossdifference of every pair of points on line {647, 9200}
X(38431) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,23,34008), (3,3129,23), (3,1995,38432), (3,31861,35470), (23,11146,3), (23,38432,38431), (1495,13350,14170), (3129,11146,34008), (5651,9735,11130)

X(38432) =  EULER LINE INTERCEPT OF X(16)X(110)

Barycentrics    Sqrt[3] a^2 (a-b-c) (a+b-c) (a-b+c) (a+b+c) (a^6-2 a^4 b^2+2 a^2 b^4-b^6-2 a^4 c^2+4 a^2 b^2 c^2+3 b^4 c^2+2 a^2 c^4+3 b^2 c^4-c^6)+2 a^2 (a^8-a^6 b^2+a^2 b^6-b^8-a^6 c^2-6 a^4 b^2 c^2+15 a^2 b^4 c^2-4 b^6 c^2+15 a^2 b^2 c^4+12 b^4 c^4+a^2 c^6-4 b^2 c^6-c^8) S : :
Barycentrics    (SB+SC) (9 S^4-6 Sqrt[3] S^3 (SA+SW)+3 S^2 (3 SA^2+2 SA SW+SW^2)-2 Sqrt[3] S SA (3 SA-SW) SW+SA (3 SA-2 SW) SW^2) : :

See Kadir Altintas and Ercole Suppa, Euclid 886 .

X(38432) lies on these lines: {2,3}, {15,5640}, {16,110}, {61,15019}, {62,11422}, {182,14170}, {187,37775}, {323,5615}, {373,13350}, {511,11130}, {574,37776}, {576,11127}, {1495,13349}, {3066,11480}, {3107,3457}, {3292,11126}, {3458,16462}, {5201,35314}, {5464,15360}, {5611,11002}, {5613,11078}, {5643,34424}, {5650,36755}, {5651,9736}, {5943,21401}, {6104,13858}, {6151,36758}, {6775,13233}, {7998,14538}, {8546,14173}, {9195,14270}, {9735,34417}, {9886,34013}, {10545,10645}, {10546,10646}, {11142,21466}, {11481,35259}, {14180,14183}, {14185,22689}, {14539,15107}, {15801,15962}

X(38432) = crossdifference of every pair of points on line {647, 9201}
X(38432) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,23,34009), (3,1995,38431), (3,3130,23), (3,31861,35469), (23,11145,3), (23,35298,38431), (1495,13349,14169), (3130,11145,34009), (5651,9736,11131)

leftri

Points associated with Vu (k)-conics: X(38433)-X(38449)

rightri

This preamble is based on notes from Vu Thanh Tung, May 13-14, 2020 and Peter Moses, May 13, 2020.

Suppose that 0 < k < π/2. Let A' and A" be the points on line BC such that |AA'| = |AA"| and angle A'AA" has measure 2k. Likewise, let B' and B" on CA and C' and C" on AB be the points such that the triangles B'BB" and C'CC" are similar to A'AA". Then the points A', A", B', B", C', C" lie on a conic, here named the Vu (k)-conic.

Peter Moses found that these conics all have center X(6) and that the perspector, V(k), of the Vu (k)-conic is given by

V(k) = 1/((a^2 - b^2 - c^2)^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*Cos[k]^2 - 4*(a^4 - b^4 - c^4)*S^2*Sin[k]^2) : : (barycentrics)

Moses also found that he ratio of minor axis to major axis is Sqrt[(J-1) (J+3) / ((J-3) (J+1))], where J = |OH|/R, as at X(1113).

Accordingly, the isotomic conjugate, T(k), and the isogonal conjugate, U(k), are given by

T(k) = (a^2 - b^2 - c^2)^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*Cos[k]^2 - 4*(a^4 - b^4 - c^4)*S^2*Sin[k]^2 : :

U(k) = a^2 ((a^2 - b^2 - c^2)^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*Cos[k]^2 - 4*(a^4 - b^4 - c^4)*S^2*Sin[k]^2) : :

V(k) lies on the Jerabek rectangular hyperbola {{A,B,C,X(3),X(4)}}.
T(k) lies on the line X(69)X(264).
U(k) lies on the Euler line.

X(6145) = perspector of the Vu (π/4) conic, and X(18434) = perspector of the Vu (π/6) conic.

For every real number x, the perspector of the Vu (acrctan(x))-conic is

1/((a^2 - b^2 - c^2)^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 4*(a^4 - b^4 - c^4)*S^2*x^2) : : , with isogonal conjugate

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

on the Euler line. (Peter Moses, May 15, 2020)

X(38433) = PERSPECTOR OF VU (π/3)-CONIC

Barycentrics    1/(2*a^8 - 4*a^6*b^2 + 4*a^2*b^6 - 2*b^8 - 4*a^6*c^2 - a^4*b^2*c^2 + 2*a^2*b^4*c^2 + 3*b^6*c^2 + 2*a^2*b^2*c^4 - 2*b^4*c^4 + 4*a^2*c^6 + 3*b^2*c^6 - 2*c^8) : :

X(38433) lies on the Jerabek right circumhyperbola and these lines: {3, 14864}, {54, 1853}, {1176, 10516}, {2435, 7950}, {3431, 34224}, {3519, 18381}, {6247, 11270}, {11738, 15105}, {13452, 18559}

X(38434) = ISOTOMIC CONJUGATE OF X(38433)

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

X(38434) lies on these lines: {4, 69}, {95, 7814}, {3964, 7917}, {7871, 9723}, {11548, 37688}

X(38435) = ISOGONAL CONJUGATE OF X(38433)

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

X(38435) lies on these lines: {2, 3}, {8, 9626}, {52, 11003}, {61, 11421}, {62, 11420}, {69, 15582}, {146, 22109}, {184, 15801}, {323, 9707}, {389, 15080}, {511, 9545}, {569, 1173}, {575, 12220}, {576, 19121}, {578, 15107}, {962, 9591}, {1131, 35776}, {1132, 35777}, {1216, 26882}, {1297, 7954}, {1493, 6243}, {1495, 11444}, {1614, 7712}, {2888, 9833}, {2916, 25406}, {2917, 11206}, {2979, 10282}, {3060, 37505}, {3431, 37495}, {3616, 9625}, {3746, 9538}, {5007, 22240}, {5092, 15028}, {5286, 9700}, {5562, 26881}, {5609, 12219}, {5921, 15581}, {5944, 37484}, {6030, 10984}, {6102, 23060}, {6194, 21458}, {6419, 11418}, {6420, 11417}, {6759, 7691}, {6800, 17834}, {7689, 8718}, {7772, 10313}, {9544, 11412}, {9699, 31400}, {9781, 37513}, {9925, 20080}, {10625, 11464}, {11416, 22234}, {11449, 15644}, {11468, 14641}, {13219, 15562}, {13340, 32171}, {13452, 32138}, {13472, 32046}, {14683, 25714}, {14927, 15579}, {15043, 22352}, {15054, 17856}, {16981, 36749}, {34513, 37472}

X(38436) = PERSPECTOR OF VU (ARCTAN(1/2))-CONIC

Barycentrics    1/(5 a^8 + 14 a^4 b^2 c^2 - 10 a^6 (b^2 + c^2) - (b^2 - c^2)^2 (5 b^4 + 8 b^2 c^2 + 5 c^4) + 2 a^2 (5 b^6 - 3 b^4 c^2 - 3 b^2 c^4 + 5 c^6)) : :

X(38436) lies on the Jerabek right circumhyperbola and these lines: {3, 23324}, {68, 18376}, {3519, 32369}, {4846, 18383}, {13851, 14542}, {14216, 18550}, {18951, 32533}

X(38437) = ISOTOMIC CONJUGATE OF X(38436)

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

X(38437) lies on these lines: {4, 69}, {253, 7802}, {10513, 33651}

X(38438) = ISOGONAL CONJUGATE OF X(38436)

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

X(38438) lies on these lines: {2, 3}, {64, 26881}, {154, 11440}, {1192, 5012}, {1204, 6800}, {1498, 11454}, {1620, 3796}, {5023, 22240}, {6409, 11418}, {6410, 11417}, {7592, 32110}, {7689, 9707}, {7691, 35602}, {8567, 12279}, {10313, 15815}, {10541, 11416}, {10546, 33537}, {10574, 37487}, {11202, 11441}, {11422, 14528}, {11464, 12163}, {12111, 17821}, {12219, 15040}, {12278, 37638}, {15053, 37476}, {17845, 23293}, {19121, 31884}

X(38439) = PERSPECTOR OF VU (ARCTAN(SQRT(1/12))-CONIC

Barycentrics    1/(13 a^8 + 46 a^4 b^2 c^2 - 26 a^6 (b^2 + c^2) - (b^2 - c^2)^2 (13 b^4 + 24 b^2 c^2 + 13 c^4) + a^2 (26 b^6 - 22 b^4 c^2 - 22 b^2 c^4 + 26 c^6))) : :

X(38439) lies on the Jerabek right circumhyperbola and these lines: (none)

X(38440) = ISOTOMIC CONJUGATE OF X(38439)

Barycentrics    13 a^8 + 46 a^4 b^2 c^2 - 26 a^6 (b^2 + c^2) - (b^2 - c^2)^2 (13 b^4 + 24 b^2 c^2 + 13 c^4) + a^2 (26 b^6 - 22 b^4 c^2 - 22 b^2 c^4 + 26 c^6) : :

X(38440) lies on this line: {4, 69}

X(38441) = ISOGONAL CONJUGATE OF X(38439)

Barycentrics    a^2 (13 a^8 + 46 a^4 b^2 c^2 - 26 a^6 (b^2 + c^2) - (b^2 - c^2)^2 (13 b^4 + 24 b^2 c^2 + 13 c^4) + a^2 (26 b^6 - 22 b^4 c^2 - 22 b^2 c^4 + 26 c^6)) : :

X(38441) lies on these lines: {2, 3}, {5585, 22240}, {12219, 15042}

X(38442) = PERSPECTOR OF VU (ARCTAN(SQRT(2))-CONIC

Barycentrics    1/((a^2 - b^2 - c^2)^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 8*(a^4 - b^4 - c^4)*S^2) : :

X(38442) lies on the Jerabek right circumhyperbola and these lines: {3, 32064}, {54, 8889}, {69, 15606}, {1176, 6759}, {1853, 14528}, {3519, 32346}, {3521, 6225}, {3532, 6247}, {4846, 12324}, {5446, 32533}, {6000, 31371}, {6403, 16774}, {14216, 15740}, {14457, 18918}, {14542, 18909}, {15749, 18383}, {18296, 18376}, {34787, 34817}

X(38442) = isogonal conjugate of X(9715)
X(38442) = barycentric quotient X(6)/X(9715)

X(38443) = PERSPECTOR OF VU (ARCTAN(SQRT(1/2))-CONIC

Barycentrics    1/((a^2 - b^2 - c^2)^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 2*(a^4 - b^4 - c^4)*S^2) : :

X(38443) lies on the Jerabek right circumhyperbola and these lines: {2, 34472}, {3, 16254}, {4, 32392}, {6, 23047}, {54, 18945}, {66, 11572}, {68, 18383}, {1853, 3532}, {3521, 14216}, {4846, 18381}, {5878, 18550}, {5889, 15077}, {13851, 14457}, {14528, 19467}, {16625, 18376}, {22967, 31371}

X(38443) = isogonal conjugate of X(38444)
X(38443) = anticomplement of X(34472)

X(38444) = ISOGONAL CONJUGATE OF X(38443)

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

X(38444) lies on these lines: {2, 3}, {35, 9643}, {54, 37489}, {64, 11454}, {110, 17821}, {154, 12111}, {155, 9705}, {185, 6800}, {394, 7691}, {973, 15043}, {1092, 15606}, {1151, 11418}, {1152, 11417}, {1192, 3796}, {1273, 9723}, {1350, 19121}, {1498, 11440}, {1614, 12163}, {1993, 13367}, {2917, 38397}, {2979, 35602}, {3053, 22240}, {3060, 11425}, {3100, 5217}, {3567, 37506}, {3580, 19467}, {3581, 12161}, {4296, 5204}, {5012, 9786}, {5013, 10313}, {5085, 12220}, {5562, 11202}, {5889, 9706}, {5907, 35264}, {5944, 18445}, {7592, 18475}, {7689, 11456}, {7731, 15035}, {8567, 13445}, {8718, 11468}, {8907, 23358}, {9544, 12164}, {9545, 12160}, {9590, 37714}, {9659, 15888}, {9672, 37722}, {9682, 35812}, {9704, 32608}, {9707, 13754}, {9936, 19908}, {10037, 31410}, {10282, 11441}, {10606, 12279}, {10610, 36753}, {11062, 36748}, {11362, 15177}, {11420, 11481}, {11421, 11480}, {11442, 34782}, {11935, 12316}, {12038, 37478}, {12219, 32609}, {12289, 14852}, {12893, 16003}, {13289, 15063}, {13394, 13568}, {13630, 34513}, {15036, 33543}, {15053, 37514}, {15055, 15738}, {15056, 35259}, {15069, 15577}, {15080, 37487}, {17834, 34148}, {17845, 37638}, {18451, 26882}, {32046, 37490}, {33556, 34116}

X(38444) = isogonal conjugate of X(38443)

X(38445) = PERSPECTOR OF VU (ARCTAN(SQRT(1/6))-CONIC

Barycentrics    1/(3*(a^2 - b^2 - c^2)^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 2*(a^4 - b^4 - c^4)*S^2) : :

X(38445) lies on the Jerabek right circumhyperbola and these lines: {69,18392}, {4846,18376}

X(38445) = isogonal conjugate of X(38446)

X(38446) = ISOGONAL CONJUGATE OF X(38445)

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

X(38446) lies on these lines: {2, 3}, {154, 11454}, {1620, 10574}, {5012, 37487}, {5085, 11416}, {5210, 22240}, {6411, 11418}, {6412, 11417}, {6800, 21663}, {10606, 26881}, {11440, 17821}

X(38446) = isogonal conjugate of X(38445)

X(38447) = PERSPECTOR OF VU (ARCTAN(SQRT(1/5))-CONIC

Barycentrics    1/(5*(a^2 - b^2 - c^2)^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 4*(a^4 - b^4 - c^4)*S^2) : :

X(38447) lies on the Jerabek right circumhyperbola and these lines: {3, 18376}, {68, 18379}, {3519, 32365}, {3521, 18383}, {6102, 32533}, {11572, 11744}, {14528, 21659}, {18381, 18550}, {32369, 33565}

X(38447) = isogonal conjugate of X(38448)

X(38448) = ISOGONAL CONJUGATE OF X(38447)

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

X(38448) lies on these lines: {2, 3}, {54, 32110}, {146, 16252}, {323, 12038}, {1216, 11562}, {1495, 15062}, {3357, 26881}, {3431, 12161}, {5010, 9643}, {5206, 22240}, {5217, 9538}, {5972, 35240}, {6759, 11454}, {7689, 11464}, {7691, 15606}, {9544, 12163}, {9705, 13754}, {9706, 13367}, {9729, 12226}, {10282, 11440}, {10313, 37512}, {11202, 12111}, {11204, 12279}, {11416, 20190}, {12162, 35265}, {12220, 17508}, {12606, 14708}, {13470, 15061}, {14531, 34148}, {14810, 19121}, {15055, 21650}, {15069, 35228}, {20191, 25739}, {23293, 34785}, {31834, 32609}

X(38448) = isogonal conjugate of X(38447)

X(38449) = PERSPECTOR OF VU (ω)-CONIC

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

X(38449) lies on the Jerabek right circumhyperbola and these lines: {1987, 36990}, {2435, 23878}

X(38450) =  EULER LINE INTERCEPT OF X(1154)X(19908)

Barycentrics    a^2*(a^14-3*a^12*(b^2+c^2)-(b^2-c^2)^4*(b^2+c^2)^3+a^10*(b^4+10*b^2*c^2+c^4)+a^8*(5*b^6-9*b^4*c^2-9*b^2*c^4+5*c^6)+a^2*(b^2-c^2)^2*(3*b^8-2*b^4*c^4+3*c^8)-a^6*(5*b^8+4*b^6*c^2-10*b^4*c^4+4*b^2*c^6+5*c^8)-a^4*(b^10-11*b^8*c^2+6*b^6*c^4+6*b^4*c^6-11*b^2*c^8+c^10)) : :
Barycentrics    (SB+SC) (R^2 S^2+SA (26 R^4+R^2 SA-16 R^2 SW+2 SW^2)) : :

As a point on the Euler line, X(38450) has Shinagawa coefficients (e^2-16 f^2,-3 e^2+16 f^2).

See Kadir Altintas and Ercole Suppa, Euclid 892 .

X(38450) lies on these lines: {2,3}, {1154,19908}, {5504,16266}, {5663,32321}, {9937,32423}, {12164,12412}, {12228,35603}, {14852,32345}, {19155,34117}, {30522,32048}

X(38450) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,9714,37954), (26,7526,15761), (26,12084,4), (1658,11250,140), (1658,13371,6644), (6644,12084,13371), (7387,12085,5073), (7556,11541,23), (11250,15761,7526)

leftri

Points associated with Vu P-cirumcircle points: X(38451)-X(38456)

rightri

This preamble is based on notes from Vu Thanh Tung, May 16, 2020.

Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC, but not X(1) and not on the circumcircle. Let

P' = isogonal conjugate of P
Ta = line tangent to the circle APP' at A, and define Tb and Tc cyclically.

The lines Ta, Tb, Tc concur in a point V(P) on the circumcircle:

V(P) = V(P') = a^2 / (c^2 p^2 q + b^2 p^2 r - a^2 q^2 r - a^2 q r^2) : :

The point V(P) is here named the Vu P-circumcircle point.

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

(2,111), (3,74), (4,74), (5,14979), (6,111), (9,2291), (7,38451), (8,38452), (10,38453)

See Vu Circumcircle Point.

V(P) is the trilinear pole of the line through X(6) and the PK-transform of P. (Randy Hutson, May 19, 2020)

X(38451) = VU X(7)-CIRCUMCIRCLE POINT

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

X(38451) lies on the circumcircle and these lines: {3, 20219}, {100, 3059}, {101, 8012}, {108, 1827}, {109, 2293}, {354, 934}, {1308, 5527}, {1385, 14074}, {11012, 28291}, {14110, 30237}

X(38451) = isogonal conjugate of X(38454)
X(38451) = trilinear pole of line X(6)X(10581)

X(38452) = VU X(8)-CIRCUMCIRCLE POINT

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

The trilinear polar of X(38452) passes through X(6) and the PK-transform of X(8). (Randy Hutson, May 19, 2020)

X(38452) lies on the circumcircle and these lines: {8, 8706}, {99, 17183}, {100, 3057}, {101, 2347}, {108, 1828}, {109, 1201}, {934, 1122}, {2716, 32486}, {7191, 9058}

X(38452) = isogonal conjugate of X(38455)

X(38453) = VU X(10)-CIRCUMCIRCLE POINT

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

The trilinear polar of X(38453) passes through X(6) and the PK-transform of X(10). (Randy Hutson, May 19, 2020)

X(38453) lies on the circumcircle and these lines: {10, 8707}, {99, 4357}, {100, 2292}, {101, 2092}, {110, 1193}, {112, 2354}, {386, 5975}, {831, 30115}, {1400, 8687}, {3920, 9070}

X(38453) = isogonal conjugate of X(38456)

X(38454) = ISOGONAL CONJUGATE OF X(38451)

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

X(38454) lies on these lines: {1, 34917}, {3, 25557}, {4, 5220}, {7, 55}, {9, 1699}, {11, 37787}, {30, 511}, {36, 38055}, {40, 5735}, {71, 5829}, {142, 6690}, {144, 3434}, {165, 6173}, {388, 20070}, {390, 2099}, {497, 12848}, {553, 10178}, {946, 15254}, {958, 962}, {1001, 1006}, {1086, 9441}, {1155, 30379}, {1389, 34919}, {1445, 17728}, {1479, 5729}, {1721, 17276}, {1742, 17365}, {1754, 17061}, {1836, 8545}, {2550, 6839}, {3058, 7671}, {3219, 7965}, {3254, 5536}, {3419, 5223}, {3485, 5766}, {3486, 30332}, {3522, 30340}, {3668, 30621}, {3816, 8257}, {3826, 5805}, {4312, 5119}, {5057, 6068}, {5173, 5572}, {5218, 30275}, {5249, 7964}, {5493, 21620}, {5537, 10427}, {5542, 24929}, {5559, 9613}, {5696, 11661}, {5732, 37569}, {5779, 37820}, {5784, 7957}, {5790, 31671}, {5794, 7991}, {6147, 12511}, {6172, 9812}, {6284, 10394}, {6666, 10171}, {6831, 24468}, {6951, 35514}, {7992, 28646}, {8158, 13463}, {11230, 25379}, {12512, 24470}, {12702, 15346}, {14942, 17950}, {15326, 18450}, {15837, 21617}, {17351, 21629}, {18230, 31245}, {18482, 38140}, {20059, 20075}, {20330, 38028}, {21153, 38036}, {21168, 38037}, {24393, 38155}, {24466, 25558}, {31657, 32613}, {36991, 36999}, {36996, 37000}

X(38454) = isogonal conjugate of X(38451)
X(38454) = crossdifference of every pair of points on line X(6)X(10581)

X(38455) = ISOGONAL CONJUGATE OF X(38452)

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

X(38455) lies on these lines: {1, 1329}, {3, 32157}, {4, 10912}, {5, 22837}, {8, 56}, {10, 6691}, {11, 5176}, {12, 4861}, {30, 511}, {36, 1145}, {46, 3632}, {55, 12648}, {78, 37738}, {80, 11256}, {104, 32198}, {145, 497}, {165, 34716}, {191, 5559}, {355, 3813}, {908, 5048}, {944, 3913}, {946, 33895}, {956, 8069}, {962, 37001}, {1056, 25557}, {1317, 4511}, {1319, 3035}, {1320, 5080}, {1385, 10915}, {1387, 3814}, {1388, 5552}, {1420, 37828}, {1470, 15813}, {1482, 26333}, {1483, 22836}, {1532, 12751}, {1699, 34640}, {1737, 3036}, {1750, 11519}, {1828, 12135}, {1837, 36846}, {1846, 5081}, {2136, 10860}, {2886, 3872}, {3218, 18802}, {3241, 25568}, {3244, 21616}, {3304, 5554}, {3419, 37708}, {3421, 5289}, {3582, 34122}, {3586, 3633}, {3616, 31246}, {3621, 17784}, {3625, 24391}, {3635, 12433}, {3680, 5691}, {3681, 34689}, {3811, 37727}, {3826, 9623}, {3829, 5587}, {3847, 11373}, {3870, 37740}, {3873, 34749}, {3877, 34606}, {3885, 6284}, {3943, 16561}, {4297, 12640}, {4298, 10107}, {4421, 5731}, {4853, 5794}, {4881, 6174}, {4919, 17747}, {4999, 10039}, {5123, 6667}, {5258, 18253}, {5541, 36975}, {5603, 11236}, {5657, 11194}, {5690, 8666}, {5795, 18227}, {5836, 10106}, {5881, 12629}, {6690, 31397}, {6738, 16215}, {6882, 12737}, {6905, 22560}, {6909, 13205}, {7354, 14923}, {7962, 24703}, {7967, 34619}, {8668, 12114}, {8715, 34773}, {9711, 19861}, {9778, 34620}, {9812, 34739}, {10222, 21077}, {10284, 37290}, {10430, 12536}, {10528, 34471}, {10680, 12645}, {11224, 28609}, {11246, 34605}, {11281, 15888}, {11415, 20050}, {11545, 15863}, {12245, 37002}, {12690, 37006}, {13996, 15326}, {16173, 17533}, {16200, 34647}, {17751, 27657}, {18357, 24387}, {18526, 35448}, {20323, 24982}, {24386, 38155}, {24390, 37710}, {24392, 37712}, {25438, 35000}, {34625, 34717}, {34772, 37734}

X(38455) = isogonal conjugate of X(38452)

X(38456) = ISOGONAL CONJUGATE OF X(38453)

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

X(38456) lies on these lines: {1, 1330}, {2, 5429}, {3, 17748}, {4, 17733}, {8, 1046}, {10, 58}, {21, 3178}, {30, 511}, {69, 24291}, {145, 33100}, {172, 4109}, {551, 33124}, {581, 22836}, {946, 37823}, {950, 10381}, {956, 4865}, {993, 29671}, {1104, 1125}, {1333, 21076}, {1430, 5081}, {1468, 5016}, {1478, 4362}, {1757, 16086}, {2305, 38408}, {2475, 27368}, {3125, 4987}, {3419, 32853}, {3430, 4297}, {3634, 6693}, {3704, 24850}, {3743, 12579}, {3769, 37716}, {3771, 37817}, {4028, 4304}, {4071, 5291}, {4234, 33160}, {4298, 35650}, {4434, 17757}, {4511, 32843}, {4680, 29673}, {5080, 17763}, {5251, 29653}, {5725, 32916}, {7683, 19925}, {8669, 21077}, {10180, 13745}, {10974, 17647}, {11114, 32915}, {11115, 20653}, {12746, 32844}, {13735, 33158}, {14829, 37717}, {15971, 35099}, {16821, 33109}, {17015, 32947}, {17539, 27558}, {17579, 32860}, {17588, 27577}, {17592, 37038}, {17677, 33135}, {17678, 33132}, {17799, 32847}, {20067, 32842}, {33071, 37617}

X(38456) = isogonal conjugate of X(38453)

leftri

Points associated with Vu (P,U)-circles points: X(38457)-X(38468)

rightri

This preamble is based on notes from Vu Thanh Tung, May 18-19, 2020.

Let P = p : q : r and U = u : v : w be distinct points in the plane of a triangle ABC, not both on the circumcircle. Let A' be the point, other than A, that lies on both circles (ABC) and (APU), and define B' and C' cyclically. The four lines AA', BB", CC", PU concur in a point, V(P,U), here named the Vu {P,U}-circles point point, given by

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

The appearance of (i,j,k) in the following list means that V(X(i),X(j)) = V(X(j),X(i)) = X(k): (1,2,7292), (1,3,36), (1,6,16784), (1,4,1870 ), (1,5,38458), (1,6,16784), (1,7,38459), (1,8,38460), (2,3,23), (2,4,468), (2,5,37760), (2,6 11580), (2 7,37761), (2,8,37762), (3,4,186), (3,5,2070), (3,6,187), (3,7,32624), (3,8,17100), (4,5,37943), (4,6,8744), (4,75,38457), (13,14,1989), (15,16,6), (61,62,35006), (371,372,1692)

See Vu PU Circles Point.

Let V2(P,U) denote the point introduced in the preamble just before X(37756), given by

V2( (P, U) = q r (a^2 (q r u (u+v+w) - p v w (p+q+r)) - b^2 p u (w (p+q) - r (u+v)) - c^2 p u (v (p+r) - q (u+w))) : :

Then V(P,U) = barycentric product P* V2(P,U)
V(P,U) = barycentric product U* V2(U,P)
V2(P,U) = barycentric product U*V2(U,P)
V(X(2),U) = V(U,X(2)) = V2(X(2),U)
V(X(3),U) = V(U,X(3)) = circumcircle-inverse of U

The Vu {P,U}-circles point is related to the Vu circlecevian point as follows: Let P' and U' be the isogonal conjugates of P and U, resp. Then the Vu {P,U}-circles point is the isogonal conjugate of the Vu circlecevian point V(P',U'). (Randy Hutson, May 20, 2020)

X(38457) = VU {X(4),X(75)}-CIRCLES POINT

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

X(38457) lies on these lines: {4,75}, {350,37168}, {693,905}, {1842,20888}, {3263,38462}, {3520,34884}, {6381,8756}

X(38457) = barycentric product X(1969)*X(5161)
X(38457) = barycentric quotient X(92)/X(37842)
X(38457) = trilinear product X(264)*X(5161)
X(38457) = trilinear quotient X(264)/X(37842)
X(38457) = X(184)-isoconjugate-of-X(37842)
X(38457) = X(92)-reciprocal conjugate of-X(37842)

X(38458) = VU {X(1),X(5)}-CIRCLES POINT

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

X(38458) lies on these lines: {1,5}, {34,3518}, {35,34864}, {36,2070}, {56,13621}, {58,229}, {106,26711}, {109,1393}, {143,7356}, {484,1772}, {614,13595}, {942,6126}, {990,5561}, {999,21308}, {1870,3582}, {3065,7004}, {3086,21451}, {3100,3153}, {3216,24916}, {3737,4960}, {4351,7292}, {5010,35921}, {5259,34977}, {5433,34577}, {5563,18369}, {6286,11591}, {6583,14627}, {7280,7488}, {9642,10896}, {10095,18984}, {10096,15325}, {13079,14128}, {13163,18990}, {16784,38463}, {18282,32047}, {18398,36750}, {18426,33178}, {18514,31724}, {24028,37563}, {38459,38464}, {38460,38465}

X(38458) = barycentric product X(1)*X(24145)
X(38458) = trilinear product X(6)*X(24145)
X(38458) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1393, 2964, 3336), (1421, 1718, 1)

X(38459) = VU {X(1),X(7)}-CIRCLES POINT

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

X(38459) lies on these lines: {1,7}, {2,34525}, {35,34865}, {36,32624}, {57,1055}, {104,24016}, {144,34526}, {220,26669}, {241,1252}, {663,3676}, {664,38460}, {840,934}, {948,31019}, {1088,29817}, {1108,34028}, {1407,28606}, {1420,7177}, {1427,17011}, {1447,4566}, {1736,38666}, {1870,36118}, {2099,23839}, {3008,37797}, {3660,10426}, {3811,32003}, {3935,37757}, {3957,17093}, {4511,9436}, {4850,5228}, {4861,9312}, {4881,6516}, {5222,8776}, {5526,15730}, {5723,33129}, {5744,25930}, {6180,24554}, {6604,34772}, {6610,34056}, {7292,37761}, {7365,17019}, {9364,35293}, {14256,34489}, {16784,38466}, {22129,24635}, {26224,34055}, {38458,38464}

X(38459) = barycentric product X(i)*X(j) for these {i, j}: {1, 37757}, {7, 37787}, {85, 2078}, {269, 17264}, {279, 3935}, {658, 3887}
X(38459) = barycentric quotient X(i)/X(j) for these (i, j): (57, 3254), (269, 34578), (658, 35171), (934, 37143), (1461, 1308), (2078, 9)
X(38459) = trilinear product X(i)*X(j) for these {i, j}: {6, 37757}, {7, 2078}, {57, 37787}, {269, 3935}, {279, 5526}, {658, 22108}
X(38459) = trilinear quotient X(i)/X(j) for these (i, j): (7, 3254), (279, 34578), (658, 37143), (934, 1308), (2078, 55)
X(38459) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(840)}} and {{A, B, C, X(7), X(4564)}}
X(38459) = crosspoint of X(1170) and X(34056)
X(38459) = crosssum of X(i) and X(j) for these {i,j}: {1, 5527}, {1212, 6603}
X(38459) = X(99)-Beth conjugate of-X(38468)
X(38459) = X(i)-isoconjugate-of-X(j) for these {i,j}: {55, 3254}, {220, 34578}, {657, 37143}, {1308, 3900}
X(38459) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (57, 3254), (269, 34578), (658, 35171), (934, 37143)
X(38459) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 4350, 279), (1319, 34855, 934)

X(38460) = VU {X(1),X(8)}-CIRCLES POINT

Barycentrics    a (a^3 - a^2 b - a b^2 + b^3 - a^2 c + 5 a b c - 2 b^2 c - a c^2 - 2 b c^2 + c^3) : :
X(38460) = 4*X(1)-X(3935) = 2*X(10)-3*X(3582) = X(145)+2*X(26015) = 7*X(3622)-4*X(6745)

X(38460) lies on these lines: {1,2}, {3,3885}, {4,36977}, {11,5176}, {21,9957}, {35,34758}, {36,2802}, {56,8668}, {63,7962}, {65,33895}, {72,5330}, {77,4452}, {90,1392}, {100,1319}, {104,517}, {106,1739}, {149,515}, {153,1519}, {224,12536}, {269,4373}, {329,4345}, {346,3554}, {355,6945}, {392,27065}, {404,10914}, {484,12653}, {516,20067}, {518,5048}, {522,4318}, {528,18450}, {529,5057}, {664,38459}, {758,2611}, {765,1279}, {902,13541}, {912,10698}, {944,6925}, {946,20060}, {952,1532}, {956,3219}, {958,3890}, {962,20076}, {999,27003}, {1012,1482}, {1056,31019}, {1120,1411}, {1145,15325}, {1158,7982}, {1222,4696}, {1266,1443}, {1280,34056}, {1331,10700}, {1385,3871}, {1387,17757}, {1388,3913}, {1420,3680}, {1422,6553}, {1442,3875}, {1447,21272}, {1457,37759}, {1464,4442}, {1467,37267}, {1483,6907}, {1621,5919}, {1697,4189}, {1706,17572}, {1727,30323}, {1776,2098}, {1836,34605}, {1870,1897}, {2077,11715}, {2078,27086}, {2093,23958}, {2099,3873}, {2136,4855}, {2475,10106}, {2975,3057}, {3035,32426}, {3146,12650}, {3210,24806}, {3243,8545}, {3245,4973}, {3295,3897}, {3421,27131}, {3434,3476}, {3555,5887}, {3576,3895}, {3583,21630}, {3681,5289}, {3684,17439}, {3699,4487}, {3813,5086}, {3814,16173}, {3874,11009}, {3878,5288}, {3879,7269}, {3881,20612}, {3884,5258}, {3892,5425}, {3898,5251}, {4002,17535}, {4018,11278}, {4051,9310}, {4084,11280}, {4190,4308}, {4193,11373}, {4296,17480}, {4297,20066}, {4311,37256}, {4328,32093}, {4430,16200}, {4513,26690}, {4720,18465}, {4867,18254}, {4996,32760}, {5046,12053}, {5080,30384}, {5081,15500}, {5123,31272}, {5126,13587}, {5141,9578}, {5172,22560}, {5193,37789}, {5252,11680}, {5253,5836}, {5267,37563}, {5284,10179}, {5298,13996}, {5303,37568}, {5434,20292}, {5440,25405}, {5603,6957}, {5697,8666}, {5731,20075}, {5770,6935}, {5795,37162}, {5844,25416}, {5853,30379}, {5882,11014}, {5883,37602}, {6001,38669}, {6049,12541}, {6762,11682}, {6872,9785}, {6905,13279}, {6906,23340}, {6913,10247}, {6916,7967}, {6932,21740}, {6939,10595}, {6940,24927}, {6969,37700}, {7176,20244}, {7354,13463}, {7743,37375}, {7993,20085}, {8715,21842}, {9369,25253}, {9580,34716}, {9708,35595}, {9802,21578}, {9819,35258}, {10129,11237}, {10524,10591}, {10624,15680}, {10950,15845}, {11011,34195}, {11015,34773}, {11220,30283}, {11256,12531}, {11376,11681}, {12665,25485}, {12690,28224}, {12699,34617}, {13528,38693}, {14151,15733}, {15617,20842}, {16784,38467}, {17296,18261}, {17606,32537}, {18525,18549}, {18857,34474}, {24203,30806}, {24387,37710}, {24440,32577}, {25413,26877}, {25439,37525}, {26087,37733}, {31263,32557}, {32900,33858}, {35460,38602}, {38458,38465}

X(38460) = midpoint of X(484) and X(12653)
X(38460) = reflection of X(i) in X(j) for these (i,j): (8, 1737), (100, 1319), (153, 1519), (765, 1279), (1145, 15325), (2077, 11715), (3245, 4973), (3583, 21630), (3935, 4511), (4511, 1), (5080, 30384), (5176, 11), (5440, 25405), (17757, 1387), (35460, 38602)
X(38460) = anticomplement of X(6735)
X(38460) = barycentric product X(i)*X(j) for these {i, j}: {1, 37758}, {8, 37789}, {190, 2827}, {312, 5193}
X(38460) = barycentric quotient X(i)/X(j) for these (i, j): (9, 12641), (101, 2743)
X(38460) = trilinear product X(i)*X(j) for these {i, j}: {6, 37758}, {8, 5193}, {9, 37789}, {100, 2827}
X(38460) = trilinear quotient X(i)/X(j) for these (i, j): (8, 12641), (100, 2743)
X(38460) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(5193)}} and {{A, B, C, X(2), X(5382)}}
X(38460) = crosspoint of X(664) and X(5376)
X(38460) = crosssum of X(663) and X(2087)
X(38460) = X(i)-anticomplementary conjugate of-X(j) for these (i,j): (56, 153), (104, 3436), (909, 329)
X(38460) = X(643)-Beth conjugate of-X(1319)
X(38460) = X(i)-isoconjugate-of-X(j) for these {i,j}: {56, 12641}, {513, 2743}
X(38460) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (9, 12641), (101, 2743)
X(38460) = circumconic-centered-at-X(1)-inverse of X(145)
X(38460) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 145, 34772), (1, 3241, 3957), (1, 3632, 30144), (1, 3633, 22836), (1, 3872, 2), (1, 4853, 19861), (1, 12629, 78), (1, 15955, 5262), (1, 17015, 17011), (1, 19860, 3622), (1, 22837, 4861), (1, 36846, 145), (2, 145, 12648), (8, 3086, 25005), (56, 10912, 14923), (78, 12629, 3621), (145, 10529, 8), (3057, 11260, 2975), (3241, 36845, 145), (3632, 30144, 4420), (4853, 19861, 3617), (5836, 20323, 5253), (10914, 24928, 404)

X(38461) = VU {X(4),X(7)}-CIRCLES POINT

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

X(38461) lies on these lines: {4,7}, {186,32624}, {225,10481}, {279,1068}, {347,37427}, {468,37761}, {514,3064}, {527,37805}, {860,9436}, {912,4566}, {1323,23710}, {1434,14016}, {1441,17528}, {1870,36118}, {3160,38295}, {3218,26003}, {3520,34865}, {7195,14257}, {8744,38466}, {18026,18821}, {31019,37448}, {37943,38464}

X(38461) = polar conjugate of the isogonal conjugate of X(6610)
X(38461) = polar conjugate of the isotomic conjugate of X(37780)
X(38461) = barycentric product X(i)*X(j) for these {i, j}: {4, 37780}, {7, 37805}, {85, 23710}, {92, 1323}, {264, 6610}, {273, 527}
X(38461) = barycentric quotient X(i)/X(j) for these (i, j): (19, 4845), (25, 18889), (34, 2291), (273, 1121), (278, 1156), (527, 78)
X(38461) = trilinear product X(i)*X(j) for these {i, j}: {4, 1323}, {7, 23710}, {19, 37780}, {34, 30806}, {57, 37805}, {92, 6610}
X(38461) = trilinear quotient X(i)/X(j) for these (i, j): (4, 4845), (19, 18889), (273, 1156), (278, 2291), (331, 1121), (527, 219)
X(38461) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(10394)}} and {{A, B, C, X(4), X(3064)}}
X(38461) = X(i)-isoconjugate-of-X(j) for these {i,j}: {3, 4845}, {63, 18889}, {212, 1156}, {219, 2291}
X(38461) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (19, 4845), (25, 18889), (34, 2291), (273, 1121)

X(38462) = VU {X(4),X(8)}-CIRCLES POINT

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

The trilinear polar of X(38462) passes through X(1639), the isotomic conjugate of X(4622), and the polar conjugates of PU(50). (Randy Hutson, May 20, 2020)

X(38462) lies on these lines: {4,8}, {5,23661}, {10,7069}, {19,21372}, {25,26262}, {28,32017}, {33,997}, {80,23580}, {104,1309}, {108,2757}, {186,17100}, {225,596}, {240,522}, {242,4076}, {273,36588}, {278,34625}, {280,6848}, {281,17281}, {297,26594}, {406,9371}, {468,37762}, {475,3086}, {515,24026}, {519,1877}, {758,1830}, {956,37391}, {1068,4200}, {1089,1842}, {1145,4723}, {1158,1726}, {1319,36944}, {1441,17532}, {1532,2968}, {1751,7008}, {1837,17869}, {1862,1884}, {1870,1897}, {1883,4968}, {1895,5704}, {2074,36797}, {2325,3992}, {3263,38457}, {3520,34758}, {3701,4186}, {3702,37226}, {4358,5440}, {5587,17860}, {5722,17862}, {6198,11109}, {6734,20879}, {6905,10538}, {7020,37417}, {8668,11398}, {8744,38467}, {9581,20320}, {14018,19792}, {15500,36123}, {17555,25005}, {17916,17920}, {18026,18821}, {24537,37696}, {36058,36112}, {37943,38465}

X(38462) = isogonal conjugate of X(36058)
X(38462) = polar conjugate of X(88)
X(38462) = barycentric product X(i)*X(j) for these {i, j}: {4, 4358}, {8, 37790}, {19, 3264}, {27, 3992}, {44, 264}, {75, 8756}
X(38462) = barycentric quotient X(i)/X(j) for these (i, j): (1, 1797), (4, 88), (19, 106), (25, 9456), (31, 32659), (33, 2316)
X(38462) = trilinear product X(i)*X(j) for these {i, j}: {2, 8756}, {4, 519}, {8, 1877}, {9, 37790}, {10, 37168}, {19, 4358}
X(38462) = trilinear quotient X(i)/X(j) for these (i, j): (2, 1797), (4, 106), (6, 32659), (19, 9456), (34, 1417), (44, 48)
X(38462) = Mimosa transform of X(2370)
X(38462) = trilinear pole of the line {1639, 17465}
X(38462) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(14923)}} and {{A, B, C, X(4), X(1877)}}
X(38462) = crossdifference of every pair of points on line {X(48), X(22383)}
X(38462) = crosspoint of X(1309) and X(15742)
X(38462) = crosssum of X(i) and X(j) for these {i,j}: {3, 23169}, {48, 23202}
X(38462) = X(29)-Beth conjugate of-X(1878)
X(38462) = X(44)-cross conjugate of-X(4358)
X(38462) = X(i)-isoconjugate-of-X(j) for these {i,j}: {2, 32659}, {3, 106}, {6, 1797}, {48, 88}, {63, 9456
X(38462) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 1797), (4, 88), (19, 106), (25, 9456)
X(38462) = {X(1785), X(1861)}-harmonic conjugate of X(860)

X(38463) = VU {X(5),X(6)}-CIRCLES POINT

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

X(38463) lies on these lines: {5,6}, {32,18369}, {111,11635}, {112,23096}, {115,22121}, {187,2070}, {230,8744}, {574,34864}, {827,14659}, {1383,8770}, {1384,13621}, {1611,16317}, {3053,3518}, {3288,6587}, {5023,12107}, {5210,7488}, {11580,37760}, {16784,38458}, {21308,21309}, {21451,37689}, {38464,38466}, {38465,38467}

X(38463) = crossdifference of every pair of points on line {X(924), X(3819)}

X(38464) = VU {X(5),X(7)}-CIRCLES POINT

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

X(38464) lies on these lines: {5,7}, {2070,32624}, {34864,34865}, {37760,37761}, {37943,38461}, {38458,38459}, {38463,38466}, {38465,38468}

X(38465) = VU {X(5),X(8)}-CIRCLES POINT

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

X(38465) lies on these lines: {5,8}, {2070,17100}, {34758,34864}, {37760,37762}, {37943,38462}, {38458,38460}, {38463,38467}, {38464,38468}

X(38466) = VU {X(6),X(7)}-CIRCLES POINT

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

X(38466) lies on these lines: {6,7}, {187,32624}, {574,34865}, {8744,38461}, {11580,37761}, {16784,38459}, {38463,38464}, {38467,38468}

X(38467) = VU {X(6),X(8)}-CIRCLES POINT

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

X(38467) lies on these lines: {6,8}, {187,17100}, {323,26594}, {574,34758}, {5526,18254}, {8744,38462}, {11580,37762}, {16784,38460}, {31460,37675}, {38463,38465}, {38466,38468}

X(38468) = VU {X(7),X(8)}-CIRCLES POINT

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

X(38468) lies on these lines: {7,8}, {57,28930}, {104,927}, {279,10529}, {348,3086}, {664,38459}, {693,6362}, {1111,1737}, {1447,6516}, {3218,30807}, {5744,30854}, {8732,20927}, {10481,10916}, {14942,18461}, {17079,34625}, {17100,32624}, {17134,33524}, {18026,18821}, {25005,26563}, {26591,31019}, {30379,37788}, {34758,34865}, {35160,35174}, {37758,37797}, {37761,37762}, {38464,38465}, {38466,38467}

X(38468) = isotomic conjugate of X(34894)
X(38468) = barycentric product X(i)*X(j) for these {i, j}: {7, 37788}, {75, 30379}, {76, 3660}, {85, 26015}
X(38468) = barycentric quotient X(i)/X(j) for these (i, j): (279, 15728), (651, 2742)
X(38468) = trilinear product X(i)*X(j) for these {i, j}: {2, 30379}, {7, 26015}, {57, 37788}, {75, 3660}, {664, 2826}, {1088, 15733}
X(38468) = trilinear quotient X(i)/X(j) for these (i, j): (664, 2742), (1088, 15728)
X(38468) = intersection, other than A,B,C, of conics {{A, B, C, X(7), X(30379)}} and {{A, B, C, X(8), X(693)}}
X(38468) = crossdifference of every pair of points on line {X(3063), X(14827)}
X(38468) = X(99)-Beth conjugate of-X(38459)
X(38468) = X(i)-isoconjugate-of-X(j) for these {i,j}: {663, 2742}, {1253, 15728}
X(38468) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (279, 15728), (651, 2742)

X(38469) = X(8)X(4581)∩X(30)X(511)

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

X(38469) lies on these lines: {8, 4581}, {30, 511}, {650, 1919}, {656, 4367}, {693, 21304}, {810, 2605}, {1459, 1491}, {2533, 21300}, {3733, 7234}, {3737, 4705}, {4147, 6133}, {4378, 23800}, {4874, 20316}, {4885, 21262}, {8062, 21051}, {20293, 25301}

X(38469) = isogonal conjugate of X(38470)
X(38469) = isogonal conjugate of the anticomplement of X(5993)
X(38469) = crosssum of X(i) and X(j) for these (i,j): {513, 37607}, {667, 16470}, {4455, 19557}
X(38469) = crosspoint of X(4589) and X(24479)
X(38469) = crossdifference of every pair of points on line {6, 986}

X(38470) = ANTICOMPLEMENT OF X(5993)

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

X(38470) lies on the circumcircle and these lines: {1, 38453}, {2, 5993}, {98, 3757}, {100, 8052}, {101, 21383}, {104, 9840}, {409, 759}, {1961, 28482}, {3573, 29119}, {7191, 28479}, {9077, 26230}

X(38470) = isogonal conjugate of X(38469)
X(38470) = anticomplement of X(5993)
X(38470) = Collings transform of X(37607)
X(38470) = cevapoint of X(i) and X(j) for these (i,j): {513, 37607}, {667, 16470}, {4455, 19557}
X(38470) = trilinear pole of line {6, 986}
X(38470) = X(i)-isoconjugate of X(j) for these (i,j): {6, 8045}, {513, 5293}
X(38470) = barycentric product X(i)*X(j) for these {i,j}: {1, 8052}, {75, 34076}, {662, 34920}
X(38470) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 8045}, {101, 5293}, {8052, 75}, {34076, 1}, {34920, 1577}

X(38471) = X(1)X(2)∩X(517)X(3030)

Barycentrics    a^3*b + 3*a^2*b^2 + a*b^3 - b^4 + a^3*c + 4*a^2*b*c - 7*a*b^2*c + 3*a^2*c^2 - 7*a*b*c^2 + 2*b^2*c^2 + a*c^3 - c^4 : :
X(38471) = 2 X[10] + X[5212], 7 X[5121] - 4 X[23869]

X(38471) lies on these lines: {1, 2}, {517, 3030}, {908, 4695}, {1329, 21896}, {1734, 3667}, {1738, 17757}, {1739, 24231}, {3820, 24210}, {4646, 9711}, {4731, 5718}, {10175, 32865}, {12607, 24178}, {13161, 21031}, {21075, 24440}

X(38471) = complement of X(38475)
X(38471) = incircle-inverse of X(21625)
X(38471) = Spieker-radical-circle-inverse of X(1)
X(38471) = orthoptic-circle-of-Steiner-inellipse-inverse of X(5268)
X(38471) = Conway-circle-inverse of X(39584)

X(38472) = X(5)X(10)∩X(44)X(513)

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

X(38472) lies on these lines: {5, 10}, {6, 5061}, {36, 3216}, {43, 3185}, {44, 513}, {51, 5432}, {181, 37662}, {197, 4383}, {210, 33162}, {375, 5745}, {511, 3035}, {516, 38390}, {674, 6745}, {692, 33849}, {908, 20718}, {1193, 1319}, {1724, 2933}, {2392, 8258}, {3006, 15632}, {3293, 23846}, {3687, 14973}, {3911, 8679}, {4023, 22271}, {4551, 20470}, {4553, 5205}, {4999, 23841}, {5048, 10459}, {5433, 16980}, {5462, 31659}, {5741, 22275}, {5752, 26364}, {5943, 6690}, {6681, 20108}, {18191, 20962}, {18839, 29639}, {20986, 32911}, {21865, 32931}, {22313, 33136}, {23638, 37646}, {25048, 37764}, {26028, 26030}

X(38472) = Spieker-radical-circle-inverse of X(2051)
X(38472) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {970, 1329, 22299}, {3452, 10440, 22276}

leftri

Points associated with the Conway circle: X(38473)-X(38485)

rightri

This preamble is contributed by Peter Moses, May 20, 2020.

Let P = p : q : r be a point on the circumcircle of a triangle ABC. Then the complement of the Conway-circle-inverse of P, denoted by CC(P) lies on the Apollonius circle. The appearance of (i,j) in the following list means that CC(X(i)) = X(j):

(101,3032), (106,3034), (109, 38485), (111,6044), (729,5213), (739,3030),(2291,34458), (8693,3033), (26715,34456), (28841,3029), (32722,34459), (32726,34455),

X(38473) = CONWAY-CIRCLE-INVERSE OF X(2)

Barycentrics    a^3 + 3*a^2*b - 2*a*b^2 + 3*a^2*c + 3*a*b*c - 3*b^2*c - 2*a*c^2 - 3*b*c^2 : :
X(38473) = 3 X[5205] - 2 X[5524]

X(38473) lies on these: {1, 2}, {314, 16741}, {333, 4891}, {740, 18201}, {896, 3685}, {3667, 5214}, {3886, 37684}, {3999, 17160}, {4663, 27064}, {4684, 37759}, {4689, 14829}, {4956, 17491}, {4966, 17070}, {5208, 17616}, {15601, 37652}, {17268, 33114}, {17288, 33134}, {17312, 33108}, {17377, 17721}, {17777, 34379}, {25722, 35892}, {32915, 36263}, {36277, 37683}

X(38473) = anticomplement of X(5212)
X(38473) = Conway-circle-inverse of X(2)
X(38473) = orthoptic-circle-of-Steiner-inellipe-inverse of X(39580)
X(38473) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(39581)
X(38473) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7292, 17162, 239}, {17162, 29824, 7292}

X(38474) = CONWAY-CIRCLE-INVERSE OF X(3)

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

X(38474) lies on these lines: {1, 3}, {4, 31738}, {79, 11573}, {386, 28268}, {511, 3583}, {513, 4960}, {515, 35649}, {519, 35636}, {758, 32919}, {912, 13244}, {1203, 18178}, {1737, 29311}, {2392, 5057}, {3585, 15488}, {3741, 11813}, {3814, 10479}, {3874, 32928}, {4880, 20718}, {5080, 10449}, {5123, 35628}, {5176, 35614}, {5180, 10453}, {5258, 22299}, {5259, 18180}, {5752, 7741}, {5891, 18406}, {6001, 12551}, {6902, 31760}, {6903, 31728}, {9037, 10477}, {14988, 35638}, {16473, 37415}, {18407, 23039}, {28534, 35892}, {35633, 35637}

X(38474) = circumcircle-inverse of X(39578)
X(38474) = Conway-circle-inverse of X(3)

X(38475) = CONWAY-CIRCLE-INVERSE OF X(8)

Barycentrics    a^4 - a^2*b^2 + 9*a^2*b*c - 2*a*b^2*c - b^3*c - a^2*c^2 - 2*a*b*c^2 - 2*b^2*c^2 - b*c^3 : :
X(38475) = 5 X[3616] - 2 X[5212]

X(38475) lies on these lines: {1, 2}, {333, 10179}, {999, 32932}, {1043, 20323}, {3304, 4673}, {3685, 4742}, {5919, 14829}, {6762, 19582}

X(38475) = anticomplement of X(38471)
X(38475) = Conway-circle-inverse of X(8)
X(38475) = incircle-of-anticomplementary-triangle-inverse of X(4882)

X(38476) = CONWAY-CIRCLE-INVERSE OF X(10)

Barycentrics    a^4 + 2*a^3*b - a*b^3 + 2*a^3*c + 9*a^2*b*c - 2*a*b^2*c - 2*b^3*c - 2*a*b*c^2 - 4*b^2*c^2 - a*c^3 - 2*b*c^3 : :
X(38476) = 7 X[3624] - 4 X[5212]

X(38476) lies on these lines: {1, 2}, {1757, 4975}, {3702, 32940}, {4742, 32919}, {5429, 32943}, {12688, 35631}

X(38476) = Conway-circle-inverse of X(10)
X(38476) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 35613, 3679}, {1, 35629, 3633}

X(38477) = CONWAY-CIRCLE-INVERSE OF X(99)

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

X(38477) lies on these lines: {1, 99}, {2, 5213}, {80, 313}, {86, 4424}, {517, 5209}, {645, 5540}, {811, 1845}, {1018, 36800}, {1356, 10473}, {1764, 6010}, {2802, 7257}, {3037, 35628}, {3741, 30992}, {4674, 10455}, {5195, 30941}

X(38477) = anticomplement of X(5213)
X(38477) = Conway-circle-inverse of X(99)

X(38478) = CONWAY-CIRCLE-INVERSE OF X(100)

Barycentrics    a*(a^3*b^2 - a*b^4 + a^3*b*c - 2*a^2*b^2*c + 2*a*b^3*c - b^4*c + a^3*c^2 - 2*a^2*b*c^2 - 3*a*b^2*c^2 + 2*b^3*c^2 + 2*a*b*c^3 + 2*b^2*c^3 - a*c^4 - b*c^4) : :
X(38478) = 3 X[1054] - 4 X[12014]

X(38478) lies on these lines: {1, 88}, {2, 3030}, {121, 10479}, {149, 3888}, {314, 4576}, {329, 2810}, {1293, 1764}, {1357, 10473}, {1999, 32029}, {2796, 5208}, {2827, 35649}, {3038, 35628}, {3057, 24627}, {3681, 35613}, {3741, 11814}, {3757, 35626}, {3786, 31136}, {3794, 32943}, {3873, 32118}, {3909, 10707}, {4919, 23622}, {5211, 35104}, {5510, 10478}, {6018, 10480}, {7998, 21283}, {9519, 10439}, {10434, 14664}, {10446, 34548}, {10449, 21290}, {14942, 21334}, {20059, 35892}, {24392, 25308}

X(38478) = anticomplement of X(3030)
X(38478) = Conway-circle-inverse of X(100)
X(38478) = {X(10453),X(35645)}-harmonic conjugate of X(35614)

X(38479) = CONWAY-CIRCLE-INVERSE OF X(105)

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

X(38479) lies on these lines: {1, 41}, {2, 3033}, {27, 295}, {103, 1764}, {116, 10479}, {118, 10478}, {150, 10449}, {152, 10446}, {329, 2810}, {1362, 10473}, {2801, 10439}, {2808, 9799}, {2823, 12555}, {2876, 20539}, {3022, 10480}, {3041, 35628}, {3786, 31027}, {3887, 35636}, {10025, 34381}, {10697, 11521}, {10882, 11714}, {16560, 20778}, {32118, 32913}

X(38479) = anticomplement of X(3033)
X(38479) = Conway-circle-inverse of X(105)

X(38480) = CONWAY-CIRCLE-INVERSE OF X(110)

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

X(38480) lies on these lines: {1, 60}, {2, 6044}, {27, 295}, {1283, 35623}, {1365, 10473}, {1764, 6011}, {3741, 30995}, {5546, 21382}, {10479, 31845}, {10480, 34194}, {35636, 35637}

X(38480) = anticomplement of X(6044)
X(38480) = Conway-circle-inverse of X(110)

X(38481) = CONWAY-CIRCLE-INVERSE OF X(741)

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

X(38481) lies on these lines: {1, 99}, {2, 3029}, {98, 1764}, {114, 10478}, {115, 10479}, {147, 10446}, {148, 10449}, {314, 32035}, {1281, 35623}, {2108, 36800}, {2311, 20603}, {2782, 10441}, {2783, 35649}, {2784, 12545}, {2787, 35636}, {2795, 35637}, {2796, 5208}, {3023, 10480}, {3027, 10473}, {3741, 11599}, {5969, 10477}, {7970, 11521}, {10454, 23698}, {10470, 21166}, {10882, 11710}, {18417, 35103}, {19863, 38220}, {35614, 36862}

X(38481) = anticomplement of X(3029)
X(38481) = Conway-circle-inverse of X(741)

X(38482) = CONWAY-CIRCLE-INVERSE OF X(759)

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

X(38482) lies on these lines: {1, 60}, {2, 3031}, {74, 1764}, {113, 10478}, {125, 10479}, {146, 10446}, {314, 32032}, {2771, 35631}, {2836, 18417}, {2854, 10477}, {3024, 10480}, {3028, 10473}, {3448, 10449}, {3741, 13605}, {5663, 10441}, {7978, 11521}, {8674, 35636}, {10454, 17702}, {10470, 15035}, {10476, 33535}, {10882, 11709}

X(38482) = anticomplement of X(3031)
X(38482) = Conway-circle-inverse of X(759)

X(38483) = CONWAY-CIRCLE-INVERSE OF X(942)

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

X(38483) lies on these lines: {1, 3}, {30, 13244}, {79, 31774}, {758, 12551}, {1999, 20067}, {2941, 37469}, {2999, 28268}, {3814, 18229}, {5080, 11679}, {5441, 31782}, {5844, 12550}, {16143, 37482}, {28174, 35638}

X(38483) = Conway-circle-inverse of X(942)

X(38484) = CONWAY-CIRCLE-INVERSE OF X(1054)

Barycentrics    a*(a^3*b^2 - a*b^4 - a^2*b^2*c + a*b^3*c - b^4*c + a^3*c^2 - a^2*b*c^2 - 2*a*b^2*c^2 + 2*b^3*c^2 + a*b*c^3 + 2*b^2*c^3 - a*c^4 - b*c^4) : :
X(38484) = 3 X[10439] + X[13244], 3 X[10439] - X[35649]

X(38484) lies on these lines: {1, 88}, {11, 1211}, {80, 10449}, {104, 10476}, {149, 6327}, {758, 32919}, {952, 35631}, {1125, 3032}, {1150, 3878}, {1768, 35621}, {2796, 3937}, {2800, 10441}, {2801, 10439}, {2810, 21093}, {2829, 12545}, {3631, 9024}, {3738, 4010}, {3884, 32917}, {4362, 35645}, {4432, 18191}, {5083, 10473}, {5903, 37684}, {6702, 10479}, {10446, 34789}, {10480, 15558}, {11679, 14740}, {11715, 37620}, {13205, 23853}, {19863, 32557}, {29311, 34458}

X(38484) = midpoint of X(i) and X(j) for these {i,j}: {1, 35636}, {10441, 35638}, {13244, 35649}
X(38484) = reflection of X(3032) in X(1125)
X(38484) = Conway-circle-inverse of X(1054)
X(38484) = barycentric product X(1)*X(30019)
X(38484) = barycentric quotient X(30019)/X(75)
X(38484) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10439, 13244, 35649}, {21334, 35626, 3741}

X(38485) = CONWAY-CIRCLE-INVERSE OF X(1083)

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

X(38485) lies on these lines: {1, 6}, {35, 16574}, {79, 314}, {209, 33158}, {386, 28256}, {513, 4960}, {674, 32846}, {758, 3685}, {894, 3874}, {942, 24342}, {1045, 3670}, {1125, 3786}, {1463, 4654}, {2895, 20961}, {3678, 17260}, {3681, 29651}, {3736, 4022}, {3741, 5208}, {3750, 22275}, {3779, 29674}, {3792, 4966}, {3811, 21371}, {3834, 10472}, {3836, 10479}, {3868, 3923}, {3873, 29652}, {3886, 5903}, {3932, 9054}, {4259, 33087}, {4260, 29637}, {4645, 10449}, {4693, 20718}, {5905, 10453}, {6007, 32857}, {9047, 17374}, {9052, 32847}, {10436, 18398}, {10441, 15310}, {10916, 29967}, {12717, 15071}, {17770, 35633}, {17772, 25048}, {17781, 35614}, {21746, 33082}, {22277, 33159}

X(38485) = reflection of X(3792) in X(4966)
X(38485) = Conway-circle-inverse of X(1083)
X(38485) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10477, 35892, 1}

X(38486) = COMPLEMENT OF CONWAY-CIRCLE-INVERSE OF X(2291)

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

Let A'B'C' be as at X(10440). Then X(38486) = X(111)-of-A'B'C'. (Randy Hutson, May 31, 2020)

X(38486) lies on the Apollonius circle and these lines: {10, 31844}, {181, 3321}, {386, 14074}, {573, 15731}, {5851, 34458}, {10440, 34457}

X(38486) = Spieker-radical-circle-inverse of X(31844)

leftri

Points associated with Vijay-Paasche-Hutson triangles: X(38487)-X(38494

rightri

This preamble was contributed by Dasari Naga Vijay Krishna, May 21, 2020.

In the preamble just before X(37994), six points are defined as follows:

Ab = 0 : 2R : c, Ac = 0 : b : 2R, Bc = a : 0 : 2R, Ba = 2R : 0 : c, Ca = 2R : b : 0, Cb = a : 2R : 0 (These points lie on the Paasche ellipse; see X(37861) and X(3788)

Here we define six more points:

A'b = 0 : 4R + c : c and A'c = 0 : b : 4R + b
B'c = a : 0 : 4R + a and B'a = 4R + c : 0 : c
C'a = 4R+b : b : 0 and C'b = a : 4R+a : 0
Qa = midpoint of A'bA'c, Qb = midpoint of B'cB'a, Qc = midpoint of C'aC'b

Geometrically, A'b = midpoint of B and Ab, A'c = midpoint of C and Ac, etc.. Using the notation in the preamble just before X(38310), these midpoints are given by taking (p,q,r) = (4R + a, 4R + b, 4R + c) and (u,v,w) = (a,b,c).

Define 21 points as follows:

L'a = A'cC'a∩A'bB'a, L'b = A'bB'a∩B'cC'b, L'c = B'cC'b∩A'cC'a
P'a = C'aB'c∩C'bB'a, P'b = C'bA'c∩C'aA'b, P'c = B'aA'c∩B'cA'b
K'a = C'bA'c∩A'bB'c, K'b = A'cB'a∩B'cC'a, K'c = B'aC'b∩C'aA'b
Ma = C'bAc∩B'cAb, Mb = C'aBc∩A'cBa, Mc = A'bCa∩B'aCb
Na = B'aCa∩C'aBa, Nb = C'bAb∩A'bCb, Nc = A'cBc∩B'cAc
Va = A'cB'c∩A'bC'b, Vb = B'aC'a∩B'cA'c, Vc = A'bC'b∩B'aC'a
Ra= B'aCb∩C'aBc, Rb = A'bCa∩C'bAc, Rc = B'cAb∩A'cBa

First barycentrics representing the 21 points follow:

L'a = 2R (4R + b + c) : b (2R + c) : c (2R + b)
P'a = ((4R + a)(16R^2 + 4R (a + b + c) + (a b + b c + c a)) + a b c : b (4R + a)(4R + a + c) : c (4R + a)( 4R + a + b)
K'a = 2R a (4R + b + c) : (4R + a)(2R + b)(4R + c): (4R + a)(2R + c)(4R + b)
Ma = a (4R^2 - b c) : 2R(4R + a)(2R + b) : 2R(4R + a)(2R + c)
Na = 12R^2 + 4R b + 4R c + b c : b (2R + c) : c (2R + b)
Va = -2R a (4R + b + c) : b (4R + a)(2R + c) : c (4R + a)(2R + b)
Ra = a^2 b c - 4R^2 (4R + b)(4R + c) : 2R a b c - 4R^2 b (4R + c) : 2R a b c - 4R^2 c (4R + b)

Also, from the preamble just before X(37944),

La = 4R^2 - b c : b (2R + c) : c (2R + b) )
Pa = 16R^4 - a^2 b c : 2R b (4 R^2 - a c) : 2R c ( 4R^2 - a b)
Ka = a (4R^2 - b c) : 4R^2 (2R + b) : 4R^2 (2R + c)
Ha = -a (4 R^2 - b c) : 2 R b (2R + c) : 2 R c (2R + b) )

Related triangles are here given names as follows:

L'aL'bL'c = 4th Vijay-Paasche-Hutson triangle
P'aP'bP'c = 5th Vijay-Paasche-Hutson triangle
K'aK'bK'c = 6th Vijay-Paasche-Hutson triangle
MaMbMc = 7th Vijay-Paasche-Hutson triangle
NaNbNc = 8th Vijay-Paasche-Hutson triangle
VaVbVc = 9th Vijay-Paasche-Hutson triangle
RaRbRc = 10th Vijay-Paasche-Hutson triangle
QaQbQc = 11th Vijay-Paasche-Hutson triangle

Collinearities:

L'a, Hb, Hc
A, Na, Va, La, L'a, Ha
A, Ma, Ka are collinear
P'a, L'a, K'a are collinear
Va, K'a, Qa
Ra, Na, Pa, are collinear
(Each list of collinearities represents a family of collinearities; e.g., the list L'a, Hb, Hc also represents L'b, Hc, Ha and L'c, Ha, Hb.)

Perspectors of triangles: X(1123) = ANaVaL'aLaHa ∩ BNbVbL'bLbHa ∩ CNcVcL'cLcHc (the Paasche point)
X(3083) = AMaKa ∩ BMbKb ∩ CMcKc = X(1)X(2)∩X(37)X(494)
X(3086) = APa ∩ BPb ∩ CPc = X(1)X(2)∩X(4)X(11)
X(37884) = ATa ∩ BTb ∩ CTc
X(37861) = HaTaKa ∩ HbTbKb ∩ HcTcKc = center of Paasche conic
X(37994) = PaLaKa ∩ PbLbKb ∩ PcLcKc
X(37995) = TaLa ∩ TbLb ∩ TcLc
X(37996) = HaPa ∩ HbPb ∩ HcPc
X(37997) = TaPa ∩ TbPb ∩ TcPc
X(38487) = AP'a ∩ BP'b ∩ CP'c
X(38488) = AK'a ∩ BK'b ∩ CK'c
X(38489) = P'aL'aK'a ∩ P'bL'bK'b ∩ P'cL'cK'c
X(38490) = VaK'aQa ∩ VbK'bQb ∩ VcK'cQc
X(38491) = K'aLa ∩ K'bLb ∩ K'cLc
X(38492) = MaHa ∩ MbHb ∩ McHc
X(38493)= P'aVa ∩ P'bVb ∩ P'cVc
X(38494) = RaP'a ∩ RbP'b ∩ RcP'c
X(38495) = NaK'a∩NbK'b∩NcK'c

X(38487) = PERSPECTOR OF THESE TRIANGLES: ABC AND 5th VIJAY-PAASCHE-HUTSON

Barycentrics    a(4R+b+c) : :
X(38487) = (a + b + c) (a + b + c + 4*R)*X[1] - (a^2 + b^2 + c^2)*X[6]

For a construction, X(38487). (Vijay Krishna, May 22, 2020)

X(38487) lies on these lines: {1, 6}, {2, 38488}, {56, 7133}, {176, 241}, {482, 1418}, {517, 8953}, {1123, 3086}, {1427, 1659}, {2067, 6457}, {2362, 13456}, {3070, 31533}, {3083, 38494}, {3300, 3582}, {3666, 33365}, {3752, 5393}, {5045, 32589}, {6351, 14986}, {6459, 30333}, {8957, 13911}, {17092, 31601}

X(38487) = (X(i)-complementary conjugate of X(j) for these (i,j): {15892, 21244}, {30335, 1329}
X(38487) = (crosspoint of X(1) and X(1123)
X(38487) = (crosssum of X(1) and X(1124)
X(38487) = ({X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 9, 3297}, {1, 1335, 1100}, {1, 8965, 37}

X(38488) = PERSPECTOR OF THESE TRIANGLES: ABC AND 6th VIJAY-PAASCHE-HUTSON

Barycentrics    (2 R + a)(4 R + b)(4 R + c) : :

For a construction, X(38488). (Vijay Krishna, May 22, 2020)

X(38488) lies on these lines: {2,38487}, {1123,38489}

X(38488) = barycentric quotient X(i)/X(j) for these (i, j): (1124, 3297), (1267, 32793)
X(38488) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(1267)}} and {{A, B, C, X(333), X(13425)}}
X(38488) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1124, 3297), (1267, 32793)
X(38488) = {X(38491), X(38495)}-harmonic conjugate of X(38489)

X(38489) = PERSPECTOR OF EACH PAIR OF THESE TRIANGLES: 4TH-, 5th-, AND 6th- VIJAY-PAASCHE-HUTSON

Barycentrics    128 R^4 + 64 R^3(b + c + 2a) + 8 R^2(b^2 + c^2 + 3 a^2 + 8 a b+ 8 a c + 3 b c)+2 R(5 a^2(b + c) + 4 a (b^2 + c^2) +b c(b + c + 12a)) + 2 a b c (a + b + c) + a^2(b^2 + c^2) : :

For a construction, X(38489). (Vijay Krishna, May 22, 2020)

X(38489) lies on these lines: {1123,38488}

X(38489) = {X(38491), X(38495)}-harmonic conjugate of X(38488)

X(38490) = PERSPECTOR OF EACH PAIR OF THESE TRIANGLES: 6th-, 9th-, AND 11TH- VIJAY-PAASCHE-HUTSON

Barycentrics    a ((16 R^4( b + c - a) + 4 R^3(b^2 + c^2 + 8 b c - a(b + c)) + 8 R^2 b c(a + b + c) + b c R (3 a(b + c) + 2 b c)) + a b^2 c^2 ) : :

For a construction, X(38490). (Vijay Krishna, May 22, 2020)

X(38490) lies on the line {1123,3086}

X(38491) = PERSPECTOR OF THESE TRIANGLES: 1st VIJAY-PAASCHE-HUTSON AND 6th VIJAY-PAASCHE-HUTSON

Barycentrics    256R^6 + 64 R^5 (b + c + 4a) + 16R^4 (2a (a + 2b + 2c) - 3b c) - 8 R^3 ((a^2 + 2b c)(b + c) + 6a b c )-4 R^2 (a^2 b^2 + b^2 c^2 + c^2 a^2 + a b c (7a + 4b + 4c)) - 2a b c R (3a b + 3 a c + 2b c) - a^2 b^2 c^2 : :

For a construction, X(38491). (Vijay Krishna, May 22, 2020)

X(38491) lies on the line {1123,38488}

X(38491) = {X(38488), X(38489)}-harmonic conjugate of X(38495)

X(38492) = PERSPECTOR OF THESE TRIANGLES: PAASCHE-HUTSON AND 7th VIJAY-PAASCHE-HUTSON

Barycentrics    a(16 R^3 (b + c - a) + 24 b c R^2 + 2 b c R (b + c + 4a) + a b c (b + c)) : :

For a construction, X(38492). (Vijay Krishna, May 22, 2020)

X(38492) lies on these lines: {9,13389}, {1123,14986}

X(38493) = PERSPECTOR OF THESE TRIANGLES: 5TH VIJAY-PAASCHE-HUTSON AND 9TH VIJAY-PAASCHE-HUTSON

Barycentrics    a(4R+b)(4R+c)(4R+b+c)(32R^3+16(a+b+c)R^2+2R(a^2+4ab+4ac+3a b c) +a(ab+ac+2a b c)) : :

For a construction, X(38493). (Vijay Krishna, May 22, 2020)

X(38493) lies on the line {1123,38488}

X(38494) = PERSPECTOR OF THESE TRIANGLES: 5TH VIJAY-PAASCHE-HUTSON AND 10TH VIJAY-PAASCHE-HUTSON

Barycentrics    256R^5+64R^4(2a+b+c)+16R^3a b c-8R^2(a^2(b+c)+a(b^2+c^2+a b c))-2R(a^2(b^2+c^2)+a b c(b+c-2a))+a^2a b c(b+c) : :

For a construction, X(38494). (Vijay Krishna, May 22, 2020)

X(38494) lies on this line: {3083,38487}

X(38495) = PERSPECTOR OF THESE TRIANGLES: 6TH VIJAY-PAASCHE-HUTSON AND 8TH VIJAY-PAASCHE-HUTSON

Barycentrics    768 R^6 + 64 R^5(12 a + 7 b + 7 c) + 16 R^4(10 a^2 + 4 b^2 + 4 c^2 + 28 a b + 28 a c + 15 b c) + 8 R^3(11 a^2 b + 11 a^2 c + 8 a b^2 + 8 a c^2 + 4 b^2 c + 4 b c^2 + 30 a b c) + 4 R^2(3 a^2 b^2 + 3 a^2 c^2 + 11 a^2 b c + 8 a b^2 c + 8 a b c^2 + b^2 c^2) + 2 R a b c(3 a b + 3 a c + 2 b c) + a^2 b^2 c^2 : :

For a construction, X(38495). (Vijay Krishna, May 22, 2020)

X(38495) lies on this line: {1123,38488}

X(38495) = {X(38488), X(38489)}-harmonic conjugate of X(38491)

X(38496) =  X(1)X(474)∩X(8)X(2885)

Barycentrics    a c (a^3-4 a^2 b-3 a b^2+2 b^3-4 a^2 c+22 a b c-6 b^2 c-3 a c^2-6 b c^2+2 c^3) : :
X(38496) = 3*X(1)-X(11512), X(2899)+3*X(3241), 3*X(3445)-2*X(11512)

See Kadir Altintas and Ercole Suppa, Euclid 895 .

X(38496) lies on these lines: {1,474}, {8,2885}, {56,8683}, {145,3699}, {1120,19582}, {1482,10700}, {1616,36846}, {2137,3340}, {2899,3241}, {3242,9026}, {3895,8572}, {4327,11011}, {4344,15590}, {4849,12127}, {7262,12513}, {12629,37679}

X(38496) = reflection of X(i) in X(j) for these (i,j): (8,2885), (3445,1)

leftri

Dilations of points on the circumcircle to other circles: X(38497)-X(38529)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, May 23, 2020.

Suppose that P and U are distinct points in the plane of a triangle ABC. Let Γ(P,U)) denote the circle with center P and pass-through point U. Suppose that V is a point distinct from U, and let D(P,U,V) denote the dilation from P that maps Γ(P,U)) onto Γ(P,V)). Centers X(38497)-X(38519) are dilations from the circumcircle to Γ(X(3),X(8)), and centers X(38520)-X(38529) are dilations from the circumcircle to Γ(X(3),X(76))

The appearance of {{i1, i2, . . . ik}} in the following list means that the k points all lie on the circle Γ(X(3),X(i1)):

{{1,40,13534,22939}}
{{2,376,11006,14916}}
{{4,20,18337,18339,32616,32617}}
{{6,1350,2453}}
{{8,944,18340}}
{{11,3025,5520,24466}}
{{69,6776,35902}}
{{76,11257,13325,13326}}
{{113,16111,25641}}
{{122,3184,16177}}
{{125,3258,16163}}
{{146,12244,34193}}
{{265,12121,20957}}
{{399,8008,8009,10620,11258,12188,12331,12773,13115,13188,13310,13512,14663,15154,15155,32595}}, the Stammler circle
{{684,9409,31953}}
{{1511,12041,12042,14650,33813,33814,35231,35232}}, has radius R/2
{{1670,1671,2554,2555,2556,2557,32481,32482}}, the 2nd Brocard circle
{{1768,6326,34464}}
{{2421,7418,11634}}
{2979,5890,15111}}
{{3109,13868,36158}}
{{3448,12383,14731}}

X(38497) = DILATION FROM X(3) OF X(74) TO THE CIRCLE Γ(X(3),X(8))

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

X(38497) lies on these lines: {3, 74}, {8, 2771}, {113, 4193}, {125, 6941}, {146, 5046}, {541, 11114}, {542, 37430}, {944, 8674}, {1388, 3024}, {1479, 10767}, {2098, 3028}, {2778, 11670}, {2779, 9904}, {3448, 37437}, {5697, 19470}, {5903, 18340}, {6699, 17566}, {6932, 16003}, {6963, 15063}, {7727, 11709}, {10525, 10778}, {10706, 17556}, {11014, 33535}, {12133, 17516}, {12371, 12904}, {12381, 26358}

X(38497) = reflection of X(38508) in X(3)

X(38498) = DILATION FROM X(3) OF X(98) TO THE CIRCLE Γ(X(3),X(8))

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

X(38498) lies on these lines: {3, 76}, {8, 2783}, {114, 4193}, {115, 6941}, {147, 5046}, {148, 37437}, {542, 11114}, {543, 37430}, {944, 2787}, {1388, 3023}, {1479, 10768}, {2098, 3027}, {5984, 15680}, {6036, 17566}, {6054, 17556}, {6949, 14651}, {6963, 14981}, {9860, 11010}, {10525, 10769}, {11177, 37299}, {11710, 21842}, {12131, 17516}, {12182, 13183}, {12189, 26358}, {18340, 37598}

X(38498) = reflection of X(38499) in X(3)

X(38499) = DILATION FROM X(3) OF X(99) TO THE CIRCLE Γ(X(3),X(8))

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

X(38499) lies on these lines: {3, 76}, {8, 2787}, {114, 6941}, {115, 4193}, {147, 37437}, {148, 5046}, {542, 37430}, {543, 11114}, {620, 17566}, {671, 17556}, {944, 2783}, {1388, 3027}, {1479, 10769}, {2098, 3023}, {2795, 15680}, {5186, 17516}, {6932, 14981}, {8591, 37299}, {10525, 10768}, {11010, 13174}, {11711, 21842}, {12185, 13180}, {13189, 26358}

X(38499) = reflection of X(38498) in X(3)

X(38500) = DILATION FROM X(3) OF X(101) TO THE CIRCLE Γ(X(3),X(8))

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

X(38500) lies on these lines: {3, 101}, {8, 3762}, {116, 4193}, {118, 6941}, {150, 5046}, {152, 37437}, {544, 11114}, {944, 2801}, {1282, 11010}, {1362, 1388}, {1479, 10770}, {2098, 3022}, {2809, 5697}, {5185, 17516}, {6161, 9320}, {6710, 17566}, {10525, 10772}, {10708, 17556}, {11712, 21842}, {15680, 20096}

X(38500) = reflection of X(38502) in X(3)

X(38501) = DILATION FROM X(3) OF X(101) TO THE CIRCLE Γ(X(3),X(8))

Barycentrics    a^2*(a^9*b^2 - a^8*b^3 - 4*a^7*b^4 + 4*a^6*b^5 + 6*a^5*b^6 - 6*a^4*b^7 - 4*a^3*b^8 + 4*a^2*b^9 + a*b^10 - b^11 + a^9*b*c - 4*a^8*b^2*c + 5*a^7*b^3*c + 9*a^6*b^4*c - 21*a^5*b^5*c - 3*a^4*b^6*c + 23*a^3*b^7*c - 5*a^2*b^8*c - 8*a*b^9*c + 3*b^10*c + a^9*c^2 - 4*a^8*b*c^2 + 9*a^7*b^2*c^2 - 15*a^6*b^3*c^2 - 3*a^5*b^4*c^2 + 39*a^4*b^5*c^2 - 25*a^3*b^6*c^2 - 17*a^2*b^7*c^2 + 18*a*b^8*c^2 - 3*b^9*c^2 - a^8*c^3 + 5*a^7*b*c^3 - 15*a^6*b^2*c^3 + 36*a^5*b^3*c^3 - 30*a^4*b^4*c^3 - 27*a^3*b^5*c^3 + 49*a^2*b^6*c^3 - 14*a*b^7*c^3 - 3*b^8*c^3 - 4*a^7*c^4 + 9*a^6*b*c^4 - 3*a^5*b^2*c^4 - 30*a^4*b^3*c^4 + 66*a^3*b^4*c^4 - 31*a^2*b^5*c^4 - 19*a*b^6*c^4 + 12*b^7*c^4 + 4*a^6*c^5 - 21*a^5*b*c^5 + 39*a^4*b^2*c^5 - 27*a^3*b^3*c^5 - 31*a^2*b^4*c^5 + 44*a*b^5*c^5 - 8*b^6*c^5 + 6*a^5*c^6 - 3*a^4*b*c^6 - 25*a^3*b^2*c^6 + 49*a^2*b^3*c^6 - 19*a*b^4*c^6 - 8*b^5*c^6 - 6*a^4*c^7 + 23*a^3*b*c^7 - 17*a^2*b^2*c^7 - 14*a*b^3*c^7 + 12*b^4*c^7 - 4*a^3*c^8 - 5*a^2*b*c^8 + 18*a*b^2*c^8 - 3*b^3*c^8 + 4*a^2*c^9 - 8*a*b*c^9 - 3*b^2*c^9 + a*c^10 + 3*b*c^10 - c^11) : :

X(38501) lies on these lines: {3, 102}, {8, 153}, {117, 4193}, {124, 6941}, {151, 5046}, {517, 18340}, {944, 3738}, {1361, 2098}, {1364, 1388}, {1479, 10771}, {1845, 14257}, {2779, 11014}, {2817, 5697}, {6001, 18339}, {6711, 17566}, {10525, 10777}, {10709, 17556}, {11713, 21842}

X(38501) = reflection of X(38507) in X(3)

X(38502) = DILATION FROM X(3) OF X(103) TO THE CIRCLE Γ(X(3),X(8))

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

X(38502) lies on these lines: {3, 101}, {8, 2801}, {65, 18340}, {116, 6941}, {118, 4193}, {150, 37437}, {152, 5046}, {544, 37430}, {944, 3887}, {971, 18328}, {1362, 2098}, {1388, 3022}, {1479, 10772}, {6712, 17566}, {10525, 10770}, {10710, 17556}, {11714, 21842}

X(38502) = reflection of X(38500) in X(3)

X(38503) = DILATION FROM X(3) OF X(105) TO THE CIRCLE Γ(X(3),X(8))

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

X(38503) lies on these lines: {3, 105}, {8, 190}, {120, 4193}, {516, 18343}, {944, 2826}, {1358, 1388}, {1479, 10773}, {1697, 18340}, {2098, 3021}, {2795, 15680}, {2809, 5697}, {3322, 14733}, {3730, 5540}, {5046, 20344}, {5433, 33970}, {5511, 6941}, {6714, 17566}, {10712, 17556}, {11716, 21842}, {13589, 26228}, {34547, 37437}

X(38502) = reflection of X(38500) in X(3)

X(38504) = DILATION FROM X(3) OF X(106) TO THE CIRCLE Γ(X(3),X(8))

Barycentrics    a^2*(a^3*b^2 + a^2*b^3 - a*b^4 - b^5 + a^3*b*c - 6*a^2*b^2*c - 2*a*b^3*c + 5*b^4*c + a^3*c^2 - 6*a^2*b*c^2 + 17*a*b^2*c^2 - 6*b^3*c^2 + a^2*c^3 - 2*a*b*c^3 - 6*b^2*c^3 - a*c^4 + 5*b*c^4 - c^5) : :

X(38504) lies on these lines: {3, 106}, {8, 80}, {121, 4193}, {517, 6788}, {944, 2827}, {1054, 11010}, {1357, 1388}, {1364, 2098}, {1697, 15737}, {2222, 13756}, {2842, 13541}, {3057, 18340}, {5510, 6941}, {6715, 17566}, {10713, 17556}, {11717, 21842}, {15680, 20098}, {34548, 37437}

X(38504) = reflection of X(38515) in X(3)

X(38505) = DILATION FROM X(3) OF X(107) TO THE CIRCLE Γ(X(3),X(8))

Barycentrics    a^14*b^2 - 5*a^12*b^4 + 10*a^10*b^6 - 10*a^8*b^8 + 5*a^6*b^10 - a^4*b^12 - a^14*b*c - a^13*b^2*c + a^12*b^3*c + a^11*b^4*c + 2*a^10*b^5*c + 2*a^9*b^6*c - 2*a^8*b^7*c - 2*a^7*b^8*c - a^6*b^9*c - a^5*b^10*c + a^4*b^11*c + a^3*b^12*c + a^14*c^2 - a^13*b*c^2 + 6*a^12*b^2*c^2 + a^11*b^3*c^2 - 9*a^10*b^4*c^2 + 2*a^9*b^5*c^2 - 13*a^8*b^6*c^2 - 2*a^7*b^7*c^2 + 19*a^6*b^8*c^2 - a^5*b^9*c^2 + a^3*b^11*c^2 - 3*a^2*b^12*c^2 - b^14*c^2 + a^12*b*c^3 + a^11*b^2*c^3 - 5*a^10*b^3*c^3 - 5*a^9*b^4*c^3 + 2*a^8*b^5*c^3 + 2*a^7*b^6*c^3 + 6*a^6*b^7*c^3 + 6*a^5*b^8*c^3 - 3*a^4*b^9*c^3 - 3*a^3*b^10*c^3 - a^2*b^11*c^3 - a*b^12*c^3 - 5*a^12*c^4 + a^11*b*c^4 - 9*a^10*b^2*c^4 - 5*a^9*b^3*c^4 + 46*a^8*b^4*c^4 + 2*a^7*b^5*c^4 - 24*a^6*b^6*c^4 + 6*a^5*b^7*c^4 - 23*a^4*b^8*c^4 - 3*a^3*b^9*c^4 + 9*a^2*b^10*c^4 - a*b^11*c^4 + 6*b^12*c^4 + 2*a^10*b*c^5 + 2*a^9*b^2*c^5 + 2*a^8*b^3*c^5 + 2*a^7*b^4*c^5 - 10*a^6*b^5*c^5 - 10*a^5*b^6*c^5 + 2*a^4*b^7*c^5 + 2*a^3*b^8*c^5 + 4*a^2*b^9*c^5 + 4*a*b^10*c^5 + 10*a^10*c^6 + 2*a^9*b*c^6 - 13*a^8*b^2*c^6 + 2*a^7*b^3*c^6 - 24*a^6*b^4*c^6 - 10*a^5*b^5*c^6 + 48*a^4*b^6*c^6 + 2*a^3*b^7*c^6 - 6*a^2*b^8*c^6 + 4*a*b^9*c^6 - 15*b^10*c^6 - 2*a^8*b*c^7 - 2*a^7*b^2*c^7 + 6*a^6*b^3*c^7 + 6*a^5*b^4*c^7 + 2*a^4*b^5*c^7 + 2*a^3*b^6*c^7 - 6*a^2*b^7*c^7 - 6*a*b^8*c^7 - 10*a^8*c^8 - 2*a^7*b*c^8 + 19*a^6*b^2*c^8 + 6*a^5*b^3*c^8 - 23*a^4*b^4*c^8 + 2*a^3*b^5*c^8 - 6*a^2*b^6*c^8 - 6*a*b^7*c^8 + 20*b^8*c^8 - a^6*b*c^9 - a^5*b^2*c^9 - 3*a^4*b^3*c^9 - 3*a^3*b^4*c^9 + 4*a^2*b^5*c^9 + 4*a*b^6*c^9 + 5*a^6*c^10 - a^5*b*c^10 - 3*a^3*b^3*c^10 + 9*a^2*b^4*c^10 + 4*a*b^5*c^10 - 15*b^6*c^10 + a^4*b*c^11 + a^3*b^2*c^11 - a^2*b^3*c^11 - a*b^4*c^11 - a^4*c^12 + a^3*b*c^12 - 3*a^2*b^2*c^12 - a*b^3*c^12 + 6*b^4*c^12 - b^2*c^14 : :

X(38505) lies on these lines: {3, 107}, {8, 2803}, {122, 4193}, {133, 6941}, {944, 2828}, {1388, 3324}, {1479, 10775}, {2098, 7158}, {5046, 34186}, {6716, 17566}, {9528, 15680}, {9530, 11114}, {10714, 17556}, {11718, 21842}, {34549, 37437}

X(38505) = reflection of X(38516) in X(3)

X(38506) = DILATION FROM X(3) OF X(108) TO THE CIRCLE Γ(X(3),X(8))

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

X(38506) lies on these lines: {3, 108}, {4, 35014}, {8, 2804}, {123, 4193}, {499, 11798}, {944, 1317}, {953, 3326}, {1068, 14127}, {1359, 1388}, {1479, 10776}, {2778, 11670}, {2817, 5697}, {5046, 34188}, {6717, 17566}, {6941, 25640}, {9539, 36171}, {10715, 17556}, {11719, 21842}, {18340, 23757}, {34550, 37437}

X(38506) = reflection of X(38517) in X(3)

X(38507) = DILATION FROM X(3) OF X(109) TO THE CIRCLE Γ(X(3),X(8))

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

X(38507) lies on these lines: {3, 102}, {8, 3738}, {117, 6941}, {124, 4193}, {151, 37437}, {513, 18340}, {944, 2800}, {1361, 1388}, {1364, 2098}, {1479, 10777}, {1777, 14127}, {2718, 3025}, {2779, 9904}, {5046, 33650}, {6718, 17566}, {10525, 10771}, {10716, 17556}, {11700, 21842}

X(38507) = reflection of X(38501) in X(3)

X(38508) = DILATION FROM X(3) OF X(110) TO THE CIRCLE Γ(X(3),X(8))

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

X(38508) lies on these lines: {1, 23341}, {3, 74}, {8, 8674}, {113, 6941}, {125, 4193}, {146, 37437}, {517, 36171}, {541, 37430}, {542, 11114}, {944, 2771}, {1112, 17516}, {1388, 3028}, {1479, 10778}, {1482, 13744}, {2098, 3024}, {2779, 11014}, {2842, 13541}, {2948, 11010}, {3448, 5046}, {5697, 7727}, {5972, 17566}, {6003, 36154}, {6902, 12317}, {6932, 15063}, {6963, 16003}, {9140, 17556}, {9143, 37299}, {10525, 10767}, {11720, 19470}, {12374, 13213}, {12826, 28098}, {13217, 26358}

X(38508) = reflection of X(38497) in X(3)
X(38508) = reflection of X(38514) in line X(1)X(3)

X(38509) = DILATION FROM X(3) OF X(111) TO THE CIRCLE Γ(X(3),X(8))

Barycentrics    a^2*(a^6*b^2 + a^4*b^4 - a^2*b^6 - b^8 + a^6*b*c + a^5*b^2*c - a^4*b^3*c - a^3*b^4*c - 2*a^2*b^5*c - 2*a*b^6*c + a^6*c^2 + a^5*b*c^2 - 14*a^4*b^2*c^2 - a^3*b^3*c^2 + 10*a^2*b^4*c^2 - 2*a*b^5*c^2 + 7*b^6*c^2 - a^4*b*c^3 - a^3*b^2*c^3 + 5*a^2*b^3*c^3 + 5*a*b^4*c^3 + a^4*c^4 - a^3*b*c^4 + 10*a^2*b^2*c^4 + 5*a*b^3*c^4 - 20*b^4*c^4 - 2*a^2*b*c^5 - 2*a*b^2*c^5 - a^2*c^6 - 2*a*b*c^6 + 7*b^2*c^6 - c^8) : :

X(38509) lies on these lines: {3, 111}, {8, 2805}, {126, 4193}, {543, 11114}, {944, 2830}, {1388, 3325}, {1479, 10779}, {2098, 6019}, {5046, 14360}, {5512, 6941}, {6719, 17566}, {10717, 17556}, {11721, 21842}, {15680, 20099}

X(38509) = reflection of X(38518) in X(3)

X(38510) = DILATION FROM X(3) OF X(112) TO THE CIRCLE Γ(X(3),X(8))

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

X(38510) lies on these lines: {3, 112}, {8, 2806}, {127, 4193}, {132, 6941}, {944, 2831}, {1388, 3320}, {1479, 10780}, {2098, 6020}, {5046, 13219}, {6720, 17566}, {9530, 37430}, {10718, 17556}, {11010, 13221}, {11722, 21842}, {12384, 37437}, {12955, 13294}, {13166, 17516}, {13313, 26358}

X(38510) = reflection of X(38519) in X(3)

X(38511) = DILATION FROM X(3) OF X(759) TO THE CIRCLE Γ(X(3),X(8))

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

X(38511) lies on these lines: {1, 36171}, {3, 759}, {8, 191}, {55, 13744}, {1283, 14795}, {1365, 1388}, {2098, 34194}, {4193, 31845}, {5046, 25645}, {5445, 34311}, {5697, 7727}, {6284, 10777}, {31524, 34921}

X(38512) = DILATION FROM X(3) OF X(901) TO THE CIRCLE Γ(X(3),X(8))

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

X(38512) lies on these lines: {3, 901}, {8, 513}, {20, 145}, {36, 3915}, {59, 221}, {100, 2841}, {484, 3216}, {962, 31849}, {1318, 3445}, {1320, 3937}, {1357, 8686}, {1388, 13756}, {1479, 31512}, {1482, 26914}, {2098, 3025}, {2842, 5541}, {3259, 4193}, {3616, 34583}, {3869, 6790}, {4345, 33647}, {5657, 31847}, {5687, 14513}, {5902, 13752}, {6941, 31841}, {13753, 15016}, {17566, 22102}, {21842, 23153}

X(38512) = reflection of X(38513) in X(3)
X(38512) = reflection of X(8) in line X(1)X(3)

X(38513) = DILATION FROM X(3) OF X(953) TO THE CIRCLE Γ(X(3),X(8))

Barycentrics    a^2*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*b*c - 5*a^5*b^2*c + 2*a^4*b^3*c + 10*a^3*b^4*c - 7*a^2*b^5*c - 5*a*b^6*c + 4*b^7*c + a^6*c^2 - 5*a^5*b*c^2 + 13*a^4*b^2*c^2 - 12*a^3*b^3*c^2 - 9*a^2*b^4*c^2 + 17*a*b^5*c^2 - 5*b^6*c^2 + 2*a^4*b*c^3 - 12*a^3*b^2*c^3 + 26*a^2*b^3*c^3 - 12*a*b^4*c^3 - 4*b^5*c^3 - 3*a^4*c^4 + 10*a^3*b*c^4 - 9*a^2*b^2*c^4 - 12*a*b^3*c^4 + 12*b^4*c^4 - 7*a^2*b*c^5 + 17*a*b^2*c^5 - 4*b^3*c^5 + 3*a^2*c^6 - 5*a*b*c^6 - 5*b^2*c^6 + 4*b*c^7 - c^8) : :

X(38513) lies on these lines: {3, 901}, {4, 8}, {104, 2841}, {108, 1361}, {513, 944}, {1388, 3025}, {1482, 13744}, {2098, 13756}, {2779, 13253}, {2800, 16110}, {2818, 10698}, {2842, 6264}, {3259, 6941}, {4193, 31841}, {5603, 31849}, {5697, 18340}, {5902, 13753}, {7428, 22765}, {8148, 10263}, {10246, 26910}, {11010, 34464}, {11280, 31825}, {13752, 15016}

X(38513) = reflection of X(3851) in X(3)
X(38513) = reflection of X(944) in line X(1)X(3)

X(38514) = DILATION FROM X(3) OF X(1290) TO THE CIRCLE Γ(X(3),X(8))

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

X(38514) lies on these lines: {1, 36171}, {3, 1290}, {4, 18115}, {8, 523}, {23, 11809}, {30, 944}, {56, 1325}, {691, 35915}, {858, 30741}, {964, 2453}, {1388, 31524}, {1479, 36175}, {2098, 31522}, {2690, 36007}, {2752, 16048}, {3017, 3336}, {3109, 3616}, {3258, 27686}, {4193, 5520}, {5189, 29832}, {7952, 37964}, {9778, 36158}, {9780, 36155}, {11248, 36001}, {11604, 18210}, {11681, 30447}, {17479, 20060}

X(38514) = reflection of X(8) in the Euler line
X(38514) = reflection of X(38508) in line X(1)X(3)

X(38515) = DILATION FROM X(3) OF X(1293) TO THE CIRCLE Γ(X(3),X(8))

Barycentrics    a^2*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 - a^6*b*c - 4*a^5*b^2*c + 10*a^4*b^3*c + 10*a^3*b^4*c - 15*a^2*b^5*c - 6*a*b^6*c + 6*b^7*c + a^6*c^2 - 4*a^5*b*c^2 + 11*a^4*b^2*c^2 - 29*a^3*b^3*c^2 - 2*a^2*b^4*c^2 + 33*a*b^5*c^2 - 10*b^6*c^2 + 10*a^4*b*c^3 - 29*a^3*b^2*c^3 + 56*a^2*b^3*c^3 - 31*a*b^4*c^3 - 6*b^5*c^3 - 3*a^4*c^4 + 10*a^3*b*c^4 - 2*a^2*b^2*c^4 - 31*a*b^3*c^4 + 22*b^4*c^4 - 15*a^2*b*c^5 + 33*a*b^2*c^5 - 6*b^3*c^5 + 3*a^2*c^6 - 6*a*b*c^6 - 10*b^2*c^6 + 6*b*c^7 - c^8) : :

X(38515) lies on these lines: {3, 106}, {8, 2827}, {121, 6941}, {944, 2802}, {1054, 2943}, {1357, 2098}, {1388, 6018}, {4193, 5510}, {5046, 34548}, {10525, 10774}, {21290, 37437}

X(38515) = reflection of X(38504) in X(3)

X(38516) = DILATION FROM X(3) OF X(1294) TO THE CIRCLE Γ(X(3),X(8))

Barycentrics    a^17*b^2 - a^16*b^3 - 6*a^15*b^4 + 6*a^14*b^5 + 15*a^13*b^6 - 15*a^12*b^7 - 20*a^11*b^8 + 20*a^10*b^9 + 15*a^9*b^10 - 15*a^8*b^11 - 6*a^7*b^12 + 6*a^6*b^13 + a^5*b^14 - a^4*b^15 + a^17*b*c - a^16*b^2*c + a^15*b^3*c + 4*a^14*b^4*c - 19*a^13*b^5*c - 5*a^12*b^6*c + 45*a^11*b^7*c - 45*a^9*b^9*c + 5*a^8*b^10*c + 19*a^7*b^11*c - 4*a^6*b^12*c - a^5*b^13*c + a^4*b^14*c - a^3*b^15*c + a^17*c^2 - a^16*b*c^2 + 4*a^15*b^2*c^2 - 6*a^14*b^3*c^2 - 10*a^13*b^4*c^2 + 20*a^12*b^5*c^2 - 14*a^11*b^6*c^2 - 6*a^10*b^7*c^2 + 42*a^9*b^8*c^2 - 22*a^8*b^9*c^2 - 24*a^7*b^10*c^2 + 14*a^6*b^11*c^2 - 2*a^5*b^12*c^2 + 4*a^4*b^13*c^2 + 2*a^3*b^14*c^2 - 2*a^2*b^15*c^2 + a*b^16*c^2 - b^17*c^2 - a^16*c^3 + a^15*b*c^3 - 6*a^14*b^2*c^3 + 31*a^13*b^3*c^3 - 2*a^12*b^4*c^3 - 43*a^11*b^5*c^3 + 32*a^10*b^6*c^3 - 49*a^9*b^7*c^3 - 16*a^8*b^8*c^3 + 83*a^7*b^9*c^3 - 14*a^6*b^10*c^3 - 11*a^5*b^11*c^3 + 2*a^4*b^12*c^3 - 9*a^3*b^13*c^3 + 4*a^2*b^14*c^3 - 3*a*b^15*c^3 + b^16*c^3 - 6*a^15*c^4 + 4*a^14*b*c^4 - 10*a^13*b^2*c^4 - 2*a^12*b^3*c^4 + 64*a^11*b^4*c^4 - 46*a^10*b^5*c^4 - 57*a^9*b^6*c^4 + 83*a^8*b^7*c^4 - 18*a^7*b^8*c^4 - 20*a^6*b^9*c^4 + 32*a^5*b^10*c^4 - 32*a^4*b^11*c^4 + 6*a^2*b^13*c^4 - 5*a*b^14*c^4 + 7*b^15*c^4 + 6*a^14*c^5 - 19*a^13*b*c^5 + 20*a^12*b^2*c^5 - 43*a^11*b^3*c^5 - 46*a^10*b^4*c^5 + 188*a^9*b^5*c^5 - 35*a^8*b^6*c^5 - 102*a^7*b^7*c^5 + 66*a^6*b^8*c^5 - 75*a^5*b^9*c^5 + 14*a^4*b^10*c^5 + 33*a^3*b^11*c^5 - 18*a^2*b^12*c^5 + 18*a*b^13*c^5 - 7*b^14*c^5 + 15*a^13*c^6 - 5*a^12*b*c^6 - 14*a^11*b^2*c^6 + 32*a^10*b^3*c^6 - 57*a^9*b^4*c^6 - 35*a^8*b^5*c^6 + 96*a^7*b^6*c^6 - 48*a^6*b^7*c^6 - 31*a^5*b^8*c^6 + 77*a^4*b^9*c^6 - 18*a^3*b^10*c^6 + 9*a*b^12*c^6 - 21*b^13*c^6 - 15*a^12*c^7 + 45*a^11*b*c^7 - 6*a^10*b^2*c^7 - 49*a^9*b^3*c^7 + 83*a^8*b^4*c^7 - 102*a^7*b^5*c^7 - 48*a^6*b^6*c^7 + 174*a^5*b^7*c^7 - 65*a^4*b^8*c^7 - 23*a^3*b^9*c^7 + 30*a^2*b^10*c^7 - 45*a*b^11*c^7 + 21*b^12*c^7 - 20*a^11*c^8 + 42*a^9*b^2*c^8 - 16*a^8*b^3*c^8 - 18*a^7*b^4*c^8 + 66*a^6*b^5*c^8 - 31*a^5*b^6*c^8 - 65*a^4*b^7*c^8 + 32*a^3*b^8*c^8 - 20*a^2*b^9*c^8 - 5*a*b^10*c^8 + 35*b^11*c^8 + 20*a^10*c^9 - 45*a^9*b*c^9 - 22*a^8*b^2*c^9 + 83*a^7*b^3*c^9 - 20*a^6*b^4*c^9 - 75*a^5*b^5*c^9 + 77*a^4*b^6*c^9 - 23*a^3*b^7*c^9 - 20*a^2*b^8*c^9 + 60*a*b^9*c^9 - 35*b^10*c^9 + 15*a^9*c^10 + 5*a^8*b*c^10 - 24*a^7*b^2*c^10 - 14*a^6*b^3*c^10 + 32*a^5*b^4*c^10 + 14*a^4*b^5*c^10 - 18*a^3*b^6*c^10 + 30*a^2*b^7*c^10 - 5*a*b^8*c^10 - 35*b^9*c^10 - 15*a^8*c^11 + 19*a^7*b*c^11 + 14*a^6*b^2*c^11 - 11*a^5*b^3*c^11 - 32*a^4*b^4*c^11 + 33*a^3*b^5*c^11 - 45*a*b^7*c^11 + 35*b^8*c^11 - 6*a^7*c^12 - 4*a^6*b*c^12 - 2*a^5*b^2*c^12 + 2*a^4*b^3*c^12 - 18*a^2*b^5*c^12 + 9*a*b^6*c^12 + 21*b^7*c^12 + 6*a^6*c^13 - a^5*b*c^13 + 4*a^4*b^2*c^13 - 9*a^3*b^3*c^13 + 6*a^2*b^4*c^13 + 18*a*b^5*c^13 - 21*b^6*c^13 + a^5*c^14 + a^4*b*c^14 + 2*a^3*b^2*c^14 + 4*a^2*b^3*c^14 - 5*a*b^4*c^14 - 7*b^5*c^14 - a^4*c^15 - a^3*b*c^15 - 2*a^2*b^2*c^15 - 3*a*b^3*c^15 + 7*b^4*c^15 + a*b^2*c^16 + b^3*c^16 - b^2*c^17 : :

X(38516) lies on these lines: {3, 107}, {8, 2828}, {122, 6941}, {133, 4193}, {944, 2803}, {1388, 7158}, {2098, 3324}, {5046, 34549}, {9530, 37430}, {10525, 10775}, {17566, 34842}, {34186, 37437}

X(38516) = reflection of X(38505) in X(3)

X(38517) = DILATION FROM X(3) OF X(1295) TO THE CIRCLE Γ(X(3),X(8))

Barycentrics    3*a^13 - 5*a^12*b - 6*a^11*b^2 + 16*a^10*b^3 - 5*a^9*b^4 - 15*a^8*b^5 + 20*a^7*b^6 - 15*a^5*b^8 + 5*a^4*b^9 + 2*a^3*b^10 + a*b^12 - b^13 - 5*a^12*c + 23*a^11*b*c - 18*a^10*b^2*c - 37*a^9*b^3*c + 69*a^8*b^4*c - 18*a^7*b^5*c - 52*a^6*b^6*c + 46*a^5*b^7*c - 3*a^4*b^8*c - 5*a^3*b^9*c + 6*a^2*b^10*c - 9*a*b^11*c + 3*b^12*c - 6*a^11*c^2 - 18*a^10*b*c^2 + 84*a^9*b^2*c^2 - 54*a^8*b^3*c^2 - 92*a^7*b^4*c^2 + 132*a^6*b^5*c^2 - 32*a^5*b^6*c^2 - 28*a^4*b^7*c^2 + 34*a^3*b^8*c^2 - 34*a^2*b^9*c^2 + 12*a*b^10*c^2 + 2*b^11*c^2 + 16*a^10*c^3 - 37*a^9*b*c^3 - 54*a^8*b^2*c^3 + 180*a^7*b^3*c^3 - 80*a^6*b^4*c^3 - 94*a^5*b^5*c^3 + 116*a^4*b^6*c^3 - 76*a^3*b^7*c^3 + 16*a^2*b^8*c^3 + 27*a*b^9*c^3 - 14*b^10*c^3 - 5*a^9*c^4 + 69*a^8*b*c^4 - 92*a^7*b^2*c^4 - 80*a^6*b^3*c^4 + 190*a^5*b^4*c^4 - 90*a^4*b^5*c^4 - 36*a^3*b^6*c^4 + 96*a^2*b^7*c^4 - 57*a*b^8*c^4 + 5*b^9*c^4 - 15*a^8*c^5 - 18*a^7*b*c^5 + 132*a^6*b^2*c^5 - 94*a^5*b^3*c^5 - 90*a^4*b^4*c^5 + 162*a^3*b^5*c^5 - 84*a^2*b^6*c^5 - 18*a*b^7*c^5 + 25*b^8*c^5 + 20*a^7*c^6 - 52*a^6*b*c^6 - 32*a^5*b^2*c^6 + 116*a^4*b^3*c^6 - 36*a^3*b^4*c^6 - 84*a^2*b^5*c^6 + 88*a*b^6*c^6 - 20*b^7*c^6 + 46*a^5*b*c^7 - 28*a^4*b^2*c^7 - 76*a^3*b^3*c^7 + 96*a^2*b^4*c^7 - 18*a*b^5*c^7 - 20*b^6*c^7 - 15*a^5*c^8 - 3*a^4*b*c^8 + 34*a^3*b^2*c^8 + 16*a^2*b^3*c^8 - 57*a*b^4*c^8 + 25*b^5*c^8 + 5*a^4*c^9 - 5*a^3*b*c^9 - 34*a^2*b^2*c^9 + 27*a*b^3*c^9 + 5*b^4*c^9 + 2*a^3*c^10 + 6*a^2*b*c^10 + 12*a*b^2*c^10 - 14*b^3*c^10 - 9*a*b*c^11 + 2*b^2*c^11 + a*c^12 + 3*b*c^12 - c^13 : :

X(38517) lies on these lines: {3, 108}, {8, 2829}, {40, 18340}, {123, 6941}, {944, 2804}, {1359, 2098}, {1388, 3318}, {4193, 25640}, {5046, 34550}, {9528, 15680}, {10525, 10776}, {34188, 37437}

X(38517) = reflection of X(38506) in X(3)

X(38518) = DILATION FROM X(3) OF X(1296) TO THE CIRCLE Γ(X(3),X(8))

Barycentrics    a^2*(a^9*b^2 - a^8*b^3 - 2*a^5*b^6 + 2*a^4*b^7 + a*b^10 - b^11 - a^9*b*c - a^8*b^2*c + 7*a^7*b^3*c - 2*a^6*b^4*c + 3*a^5*b^5*c - 7*a^3*b^7*c + 2*a^2*b^8*c - 2*a*b^9*c + b^10*c + a^9*c^2 - a^8*b*c^2 - 16*a^7*b^2*c^2 + 14*a^6*b^3*c^2 + 23*a^5*b^4*c^2 - 25*a^4*b^5*c^2 - 2*a^3*b^6*c^2 + 4*a^2*b^7*c^2 - 6*a*b^8*c^2 + 8*b^9*c^2 - a^8*c^3 + 7*a^7*b*c^3 + 14*a^6*b^2*c^3 - 63*a^5*b^3*c^3 + 5*a^4*b^4*c^3 + 45*a^3*b^5*c^3 - 18*a^2*b^6*c^3 + 19*a*b^7*c^3 - 8*b^8*c^3 - 2*a^6*b*c^4 + 23*a^5*b^2*c^4 + 5*a^4*b^3*c^4 - 40*a^3*b^4*c^4 + 20*a^2*b^5*c^4 + 13*a*b^6*c^4 - 27*b^7*c^4 + 3*a^5*b*c^5 - 25*a^4*b^2*c^5 + 45*a^3*b^3*c^5 + 20*a^2*b^4*c^5 - 66*a*b^5*c^5 + 27*b^6*c^5 - 2*a^5*c^6 - 2*a^3*b^2*c^6 - 18*a^2*b^3*c^6 + 13*a*b^4*c^6 + 27*b^5*c^6 + 2*a^4*c^7 - 7*a^3*b*c^7 + 4*a^2*b^2*c^7 + 19*a*b^3*c^7 - 27*b^4*c^7 + 2*a^2*b*c^8 - 6*a*b^2*c^8 - 8*b^3*c^8 - 2*a*b*c^9 + 8*b^2*c^9 + a*c^10 + b*c^10 - c^11) : :

X(38518) lies on these lines: {3, 111}, {8, 2830}, {126, 6941}, {543, 37430}, {944, 2805}, {1388, 6019}, {2098, 3325}, {4193, 5512}, {10525, 10779}, {14360, 37437}, {37299, 37749}

X(38518) = reflection of X(38509) in X(3)

X(38519) = DILATION FROM X(3) OF X(1297) TO THE CIRCLE Γ(X(3),X(8))

Barycentrics    a^2*(a^13*b^2 - a^12*b^3 - 2*a^11*b^4 + 2*a^10*b^5 - a^9*b^6 + a^8*b^7 + 4*a^7*b^8 - 4*a^6*b^9 - a^5*b^10 + a^4*b^11 - 2*a^3*b^12 + 2*a^2*b^13 + a*b^14 - b^15 + a^13*b*c - a^12*b^2*c + a^11*b^3*c - 2*a^9*b^5*c + 3*a^8*b^6*c - 6*a^7*b^7*c + 5*a^5*b^9*c - 3*a^4*b^10*c + 5*a^3*b^11*c - 4*a*b^13*c + b^14*c + a^13*c^2 - a^12*b*c^2 - 4*a^11*b^2*c^2 + 2*a^10*b^3*c^2 + 6*a^9*b^4*c^2 - 4*a^8*b^5*c^2 - 4*a^7*b^6*c^2 + 8*a^6*b^7*c^2 + a^5*b^8*c^2 - 5*a^4*b^9*c^2 - 2*a^2*b^11*c^2 + 2*b^13*c^2 - a^12*c^3 + a^11*b*c^3 + 2*a^10*b^2*c^3 - 3*a^9*b^3*c^3 - 2*a^8*b^4*c^3 + 8*a^7*b^5*c^3 - 2*a^6*b^6*c^3 - 8*a^5*b^7*c^3 + 5*a^4*b^8*c^3 - a^3*b^9*c^3 + 3*a*b^11*c^3 - 2*b^12*c^3 - 2*a^11*c^4 + 6*a^9*b^2*c^4 - 2*a^8*b^3*c^4 - 4*a^7*b^4*c^4 - 2*a^6*b^5*c^4 + 6*a^4*b^7*c^4 - 2*a^3*b^8*c^4 + 2*a^2*b^9*c^4 + 2*a*b^10*c^4 - 4*b^11*c^4 + 2*a^10*c^5 - 2*a^9*b*c^5 - 4*a^8*b^2*c^5 + 8*a^7*b^3*c^5 - 2*a^6*b^4*c^5 + 6*a^5*b^5*c^5 - 4*a^4*b^6*c^5 - 4*a^3*b^7*c^5 + 4*a^2*b^8*c^5 - 8*a*b^9*c^5 + 4*b^10*c^5 - a^9*c^6 + 3*a^8*b*c^6 - 4*a^7*b^2*c^6 - 2*a^6*b^3*c^6 - 4*a^4*b^5*c^6 + 8*a^3*b^6*c^6 - 6*a^2*b^7*c^6 - 3*a*b^8*c^6 + 9*b^9*c^6 + a^8*c^7 - 6*a^7*b*c^7 + 8*a^6*b^2*c^7 - 8*a^5*b^3*c^7 + 6*a^4*b^4*c^7 - 4*a^3*b^5*c^7 - 6*a^2*b^6*c^7 + 18*a*b^7*c^7 - 9*b^8*c^7 + 4*a^7*c^8 + a^5*b^2*c^8 + 5*a^4*b^3*c^8 - 2*a^3*b^4*c^8 + 4*a^2*b^5*c^8 - 3*a*b^6*c^8 - 9*b^7*c^8 - 4*a^6*c^9 + 5*a^5*b*c^9 - 5*a^4*b^2*c^9 - a^3*b^3*c^9 + 2*a^2*b^4*c^9 - 8*a*b^5*c^9 + 9*b^6*c^9 - a^5*c^10 - 3*a^4*b*c^10 + 2*a*b^4*c^10 + 4*b^5*c^10 + a^4*c^11 + 5*a^3*b*c^11 - 2*a^2*b^2*c^11 + 3*a*b^3*c^11 - 4*b^4*c^11 - 2*a^3*c^12 - 2*b^3*c^12 + 2*a^2*c^13 - 4*a*b*c^13 + 2*b^2*c^13 + a*c^14 + b*c^14 - c^15) : :

X(38519) lies on these lines: {3, 112}, {8, 2831}, {127, 6941}, {132, 4193}, {944, 2806}, {1388, 6020}, {2098, 3320}, {5046, 12384}, {9530, 11114}, {10525, 10780}, {11010, 12408}, {12145, 17516}, {12265, 21842}, {12925, 13297}, {13118, 26358}, {13219, 37437}, {17566, 34841}

X(38519) = reflection of X(38510) in X(3)

X(38520) = DILATION FROM X(3) OF X(74) TO THE CIRCLE Γ(X(3),X(76))

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

X(38520) lies on these lines: {3, 74}, {39, 15920}, {76, 542}, {113, 5025}, {125, 37446}, {146, 6655}, {376, 2396}, {541, 7833}, {690, 11257}, {1503, 18304}, {2781, 32445}, {5890, 7418}, {6000, 10568}, {6699, 7907}, {7841, 10706}, {7891, 33512}, {12192, 14901}, {12203, 15342}, {12383, 37889}, {13754, 15915}, {14157, 37930}, {14880, 18332}, {14915, 36182}, {32228, 34792}

X(38520) = reflection of X(38523) in X(3)
X(38520) = 2nd-Brocard-circle-inverse of X(74)

X(38521) = DILATION FROM X(3) OF X(100) TO THE CIRCLE Γ(X(3),X(76))

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

X(38521) lies on these lines: {3, 8}, {11, 5025}, {76, 2787}, {119, 37446}, {149, 6655}, {528, 7833}, {2783, 11257}, {3035, 7907}, {6154, 33275}, {6174, 33274}, {7748, 10769}, {7841, 10707}, {7887, 31272}, {20095, 33260}, {32452, 32454}

X(38521) = 2nd-Brocard-circle-inverse of X(100)

X(38522) = DILATION FROM X(3) OF X(101) TO THE CIRCLE Γ(X(3),X(76))

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

X(38522) lies on these lines: {3, 101}, {76, 2786}, {116, 5025}, {118, 37446}, {150, 6655}, {544, 7833}, {2784, 11257}, {6710, 7907}, {7841, 10708}, {7887, 31273}, {20096, 33260}

X(38522) = 2nd-Brocard-circle-inverse of X(101)

X(38523) = DILATION FROM X(3) OF X(110) TO THE CIRCLE Γ(X(3),X(76))

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

X(38523) lies on these lines: {3, 74}, {76, 690}, {113, 37446}, {125, 5025}, {511, 10568}, {542, 7833}, {575, 15918}, {626, 15357}, {1078, 15342}, {1499, 36165}, {2088, 6787}, {2979, 11634}, {3448, 6655}, {5012, 35936}, {5642, 33274}, {5972, 7907}, {6000, 15915}, {7418, 15305}, {7801, 11006}, {7841, 9140}, {7887, 15059}, {9517, 32547}, {10104, 18332}, {11442, 35923}, {12505, 32228}, {13334, 15920}, {13754, 37991}, {14509, 34359}, {14683, 33260}, {15107, 37915}, {24981, 33275}

X(38523) = reflection of X(38520) in X(3)
X(38523) = 2nd-Brocard-circle-inverse of X(110)

X(38524) = DILATION FROM X(3) OF X(111) TO THE CIRCLE Γ(X(3),X(76))

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

X(38524) lies on these lines: {3, 111}, {39, 36182}, {76, 543}, {126, 5025}, {2793, 11257}, {5512, 37446}, {6655, 14360}, {6719, 7907}, {7841, 10717}, {9172, 33274}, {9465, 11634}, {13334, 15921}, {20099, 33260}

X(38524) = 2nd-Brocard-circle-inverse of X(111)

X(38525) = DILATION FROM X(3) OF X(112) TO THE CIRCLE Γ(X(3),X(76))

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

X(38525) lies on these lines: {3, 112}, {39, 37991}, {76, 2799}, {127, 5025}, {132, 37446}, {1569, 2794}, {2781, 32445}, {6655, 13219}, {6720, 7907}, {7841, 10718}, {9517, 32547}, {10312, 37930}, {10766, 15073}

X(38525) = reflection of X(38529) in X(3)
X(38525) = 2nd-Brocard-circle-inverse of X(112)

X(38526) = DILATION FROM X(3) OF X(691) TO THE CIRCLE Γ(X(3),X(76))

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

X(38526) lies on these lines: {2, 19663}, {3, 691}, {4, 16237}, {6, 37915}, {23, 32}, {30, 3095}, {39, 36182}, {76, 523}, {83, 1316}, {250, 8743}, {315, 2396}, {316, 3001}, {468, 7857}, {576, 15032}, {671, 20975}, {729, 3124}, {827, 11641}, {858, 7752}, {1078, 9832}, {1634, 16175}, {2207, 36176}, {2452, 7760}, {2453, 7770}, {2770, 16055}, {2971, 14061}, {3096, 11007}, {3972, 36156}, {5025, 5099}, {5188, 15915}, {5189, 7785}, {6795, 12203}, {7464, 9737}, {7472, 7782}, {7748, 36174}, {7775, 10989}, {7786, 36157}, {7807, 16320}, {7835, 16316}, {7862, 30745}, {7883, 36194}, {8705, 13330}, {8753, 14060}, {13334, 37991}, {14700, 21906}, {16188, 37446}, {16308, 37927}, {20063, 20088}, {34604, 37901}

X(38526) = reflection of X(38528) in X(3)
X(38526) = reflection of X(76) in the Euler line
X(38526) = 2nd-Brocard-circle-inverse of X(691)

X(38527) = DILATION FROM X(3) OF X(805) TO THE CIRCLE Γ(X(3),X(76))

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

X(38527) lies on these lines: {3, 805}, {20, 185}, {76, 512}, {249, 3202}, {699, 9427}, {1975, 14509}, {2396, 2979}, {2679, 5025}, {2896, 34214}, {3098, 15923}, {3111, 7786}, {3934, 6787}, {7748, 31513}, {7907, 22103}, {22712, 31848}, {33330, 37446}

X(38527) = reflection of X(76) in the Brocard axis
X(38527) = reflection of X(194) in line PU(1)
X(38527) = 2nd-Brocard-circle-inverse of X(805)
X(38527) = X(76)-of-dual-of-1st-Brocard-triangle

X(38528) = DILATION FROM X(3) OF X(842) TO THE CIRCLE Γ(X(3),X(76))

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

X(38528) lies on these lines: {3, 691}, {20, 15112}, {23, 5171}, {30, 76}, {39, 37991}, {186, 1968}, {262, 36157}, {511, 10568}, {523, 11257}, {1316, 12110}, {2452, 32467}, {3053, 37930}, {5025, 16188}, {5099, 37446}, {7464, 30270}, {7907, 16760}, {9409, 14223}, {22712, 36165}

X(38528) = reflection of X(38526) in X(3)
X(38528) = 2nd-Brocard-circle-inverse of X(842)

X(38529) = DILATION FROM X(3) OF X(1297) TO THE CIRCLE Γ(X(3),X(76))

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

X(38529) lies on these lines: {3, 112}, {20, 877}, {76, 2794}, {127, 37446}, {132, 5025}, {2799, 11257}, {5188, 15915}, {6655, 12384}, {7833, 9530}, {7907, 34841}, {8722, 21395}

X(38529) = reflection of X(38525) in X(3)
X(38529) = 2nd-Brocard-circle-inverse of X(1297)

X(38530) = DILATION FROM X(3) OF X(36735) TO THE CIRCLE Γ(X(3),X(6))

Barycentrics    a*(a^4 - a^3*b + a^2*b^2 - a*b^3 - a^3*c - a^2*b*c + a*b^2*c + 2*b^3*c + a^2*c^2 + a*b*c^2 - 4*b^2*c^2 - a*c^3 + 2*b*c^3) : :
X(38530) = 5 X[3763] - 4 X[24250]

X(38530) lies on these lines: {3, 36735}, {6, 513}, {7, 3446}, {31, 19945}, {36, 1001}, {45, 1155}, {46, 24433}, {55, 24405}, {59, 6180}, {100, 545}, {105, 1086}, {171, 24338}, {183, 5990}, {190, 31073}, {484, 984}, {517, 990}, {692, 4014}, {901, 9081}, {999, 1308}, {1290, 2721}, {1319, 2263}, {1376, 23343}, {2222, 37541}, {2687, 2722}, {3000, 7113}, {3550, 24406}, {3763, 24250}, {3821, 24288}, {4265, 24248}, {4389, 5078}, {4429, 5080}, {4436, 13174}, {5048, 7221}, {5057, 17290}, {5096, 17768}, {5255, 24397}, {5264, 24399}, {6075, 26866}, {7083, 7336}, {17276, 24309}, {20872, 32857}, {20994, 33868}, {24418, 37588}, {24429, 37522}, {31847, 36745}, {34583, 37674}

X(38530) = reflection of X(6) in X(5091)
X(38530) = reflection of X(38531) in X(3)
X(38530) = reflection of X(6) in the OI line
X(38530) = barycentric product X(1086)*X(38310)
X(38530) = barycentric quotient X(38310)/X(1016)
X(38530) = {X(1086),X(1633)}-harmonic conjugate of X(16686)

X(38531) = DILATION FROM X(3) OF X(36736) TO THE CIRCLE Γ(X(3),X(6))

Barycentrics    a*(a^8 - a^7*b - a^6*b^2 + a^5*b^3 - a^4*b^4 + a^3*b^5 + a^2*b^6 - a*b^7 - a^7*c + 7*a^6*b*c - 5*a^5*b^2*c - 4*a^4*b^3*c + 5*a^3*b^4*c - a^2*b^5*c + a*b^6*c - 2*b^7*c - a^6*c^2 - 5*a^5*b*c^2 + 6*a^4*b^2*c^2 + 2*a^3*b^3*c^2 - a^2*b^4*c^2 - 5*a*b^5*c^2 + 4*b^6*c^2 + a^5*c^3 - 4*a^4*b*c^3 + 2*a^3*b^2*c^3 - 6*a^2*b^3*c^3 + 5*a*b^4*c^3 + 2*b^5*c^3 - a^4*c^4 + 5*a^3*b*c^4 - a^2*b^2*c^4 + 5*a*b^3*c^4 - 8*b^4*c^4 + a^3*c^5 - a^2*b*c^5 - 5*a*b^2*c^5 + 2*b^3*c^5 + a^2*c^6 + a*b*c^6 + 4*b^2*c^6 - a*c^7 - 2*b*c^7) :
X(38531) = 3 X[5085] - 2 X[5091], 3 X[10516] - 4 X[24250]

X(38531) lies on these lines: {3, 36735}, {6, 517}, {36, 4862}, {104, 545}, {513, 1350}, {1290, 2747}, {2077, 11495}, {2687, 2746}, {2717, 2743}, {5085, 5091}, {9058, 26611}, {10516, 24250}, {22765, 28464}

X(38531) = reflection of X(38530) in X(3)
X(38531) = reflection of X(1350) in the OI line

X(38532) =  X(22)X(1296)∩X(23)X(14262)

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

See Kadir Altintas and Ercole Suppa, Euclid 896 .

X(38532) lies on the cubic K108 and these lines: {22,1296}, {23,14262}, {25,187}, {154,1177}, {394,2434}, {468,32133}, {3455,10355}, {5210,10354}, {7493,34165}

X(38532) = isogonal conjugate of the isotomic conjugate of X(34165)
X(38532) = X(23)-Ceva conjugate of X(10355)
X(38532) = barycentric product X(i)*X(j) for these (i,j): (6,34165), (5485,19153)
X(38532) = trilinear product X(31)*X(34165)

X(38533) =  X(23)X(10355)∩X(187)X(2930)

Barycentrics    a^2 (5 a^2-b^2-c^2) (a^6+b^6-9 b^4 c^2-3 b^2 c^4+7 c^6+3 a^4 (b^2-3 c^2)+3 a^2 (b^4+3 b^2 c^2-c^4)) (a^6+7 b^6-3 b^4 c^2-9 b^2 c^4+c^6+a^4 (-9 b^2+3 c^2)+a^2 (-3 b^4+9 b^2 c^2+3 c^4)) : :
Barycentrics    (SB+SC) (3 SA-2 SW) (27 R^2 S^2+SW (9 SC^2-6 SC SW-SW^2)) (9 S^2 (3 R^2-SW)+SW (9 SA SC-3 SA SW-3 SC SW+2 SW^2)) : :

See Kadir Altintas and Ercole Suppa, Euclid 896 .

X(38533) lies on the cubic K108 and these lines: {23,10355}, {187,2930}

X(38533) = isogonal conjugate of X(34164)
X(38533) = anticomplement of the complementary conjugate of X(34581)
X(38533) = X(75)-isoconjugate of X(10355)
X(38533) = X(i)-reciprocal conjugate of X(j) for these {i,j}: {6,34164}, {32,10355}
X(38533) = barycentric quotient X(32)/X(10355)
X(38533) = trilinear quotient X(31)/X(10355)

X(38534) =  ISOGONAL CONJUGATE OF X(2072)

Barycentrics    a^2*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^8-a^6*(b^2+4*c^2)+(b^2+c^2)*(-(b^2*c)+c^3)^2+a^4*(-b^4+b^2*c^2+6*c^4)+a^2*(b^6+b^2*c^4-4*c^6))*(a^8-a^6*(4*b^2+c^2)+(b^2+c^2)*(b^3-b*c^2)^2+a^4*(6*b^4+b^2*c^2-c^4)+a^2*(-4*b^6+b^4*c^2+c^6)) : :
Trilinears    1/((J^2 + 1) cos A + 2(J^2 - 1) cos B cos C), J is as at X(1113) : :
X(38534) = 3*X(15061)-2*X(34115)

See Thanh Tung Vu and Ercole Suppa, Euclid 897 .

X(38534) lies on the Jerabek circumhyperbola and these lines: {3,1986}, {4,9934}, {24,12236}, {64,17854}, {66,5622}, {68,110}, {69,3043}, {70,125}, {186,5504}, {265,403}, {542,18124}, {567,3521}, {568,16867}, {895,19128}, {1112,32046}, {2781,34436}, {2931,8907}, {3047,3542}, {3519,10018}, {3532,17835}, {4846,5012}, {6145,20303}, {6391,19138}, {11579,18125}, {12022,22466}, {12825,34801}, {14644,16000}, {14861,35491}, {15044,32533}, {15061,34115}, {21400,37197}, {34438,35603}

X(38534) = isogonal conjugate of X(2072)
X(38534) = antigonal conjugate of X(70)
X(38534) = complement of the anticomplementary conjugate of X(186)
X(38534) = reflection of X(i) in X(j) for these (i,j): (70,125), (110,34116)
X(38534) = X(5961)-cross conjugate of X(1299)
X(38534) = X(6)-reciprocal conjugate of X(2072)
X(38534) = X(i)-vertex conjugate of X(j) for these (i,j): (4,5504),(1177,33565),(5504,4),(6344,14910)
X(38534) = cevapoint of X(i)and X(j) for these {i,j}: {6,34397}, {184,3003}
X(38534) = Jerabek hyperbola antipode of X(70)
X(38534) = trilinear pole of line X(571)X(647)
X(38534) = {X(1986),X(12228)}-harmonic conjugate of X(15463)

X(38535) =  ISOGONAL CONJUGATE OF X(2073)

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

See Thanh Tung Vu and Ercole Suppa, Euclid 897 .

The trilinear polar of X(38535) passes through X(647) and the crosspoint of X(2) and X(71). (Randy Hutson, May 31, 2020)

X(38535) lies on the Jerabek circumhyperbola and these lines: {3,4466}, {4,2772}, {6,3120}, {54,11263}, {67,674}, {71,125}, {72,20902}, {74,516}, {110,34830}, {265,916}, {895,9028}, {1175,12047}, {1243,2771}, {3448,8044}, {5057,37142}, {10693,20718}, {16686,34435}

X(38535) = isogonal conjugate of X(2073)
X(38535) = antigonal conjugate of X(71)
X(38535) = midpoint of X(3448) and X(17220)
X(38535) = reflection of X(i) in X(j) for these (i,j): (71,125), (110,34830)
X(38535) = X(i)-isoconjugate of X(j) for these (i,j): (162,2774), (2774,162)
X(38535) = Jerabek-hyperbola-antipode of X(71)
X(38535) = X(i)-reciprocal conjugate of X(j) for these {i,j}: {6,2073}, {647,2774}, {2690,648}
X(38535) = barycentric product X(525)*X(2690)
X(38535) = barycentric quotient X(i)/X(j) for these {i,j}: {647,2774}, {2690,648}
X(38535) = trilinear product X(656)*X(2690)
X(38535) = trilinear quotient X(i)/X(j) for these (i,j): (656,2774), (2690,162)

leftri

Vu (P,U)-circles perspectors: X(38536)-X(38550)

rightri

This preamble is based on notes contributed by Thanh Tung, May 23, 2020.

Let P = p : q : r (barycentrics) and U = u : v : w be points in the plane of a triangle ABC such that P, U, P1, U1, where P1 and U1 are the respective isogonal conjugates of P and U, are distinct finite points. Let A' be the point, other than A, in which the circles (APU) and (AP1U1) intersect, and define B' and C' cyclically. The triangles ABC and A'B'C' are perspective, and their perspector is given by

V(P,U) = V(U,P) = V(P1,U1) = V(U1,P1) = (c^2 p q u^2 + b^2 p r u^2 + a^2 q r u^2 - c^2 p^2 u v + b^2 p r u v - c^2 p r u v + a^2 q r u v - b^2 p^2 u w - b^2 p q u w + c^2 p q u w + a^2 q r u w - a^2 p^2 v w - a^2 p q v w - a^2 p r v w)*(-a^2 c^2 p q u v + c^4 p q u v - a^2 c^2 q^2 u v + b^2 c^2 p r u v - a^2 c^2 p q v^2 - a^2 c^2 q^2 v^2 - a^2 c^2 q r v^2 + b^2 c^2 p q u w + b^4 p r u w + a^2 b^2 q r u w - a^2 c^2 q^2 v w + a^2 b^2 p r v w + a^4 q r v w - a^2 c^2 q r v w)*(c^4 p q u v + b^2 c^2 p r u v + a^2 c^2 q r u v + b^2 c^2 p q u w - a^2 b^2 p r u w + b^4 p r u w - a^2 b^2 r^2 u w + a^2 c^2 p q v w + a^4 q r v w - a^2 b^2 q r v w - a^2 b^2 r^2 v w - a^2 b^2 p r w^2 - a^2 b^2 q r w^2 - a^2 b^2 r^2 w^2) : :

The point V(P,U)) is here named the Vu (P,U)-circles perspector.

See Vu Circles Perspector

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

(2,3,14246) and (2,4,14246), (4,6,14246) and (3,6,14246), (2,5,38536), (2,7,38537), (2,8,38538), (3,5,38539), (3,7,38540), (3,8,38541), (4,5,38542), (4,7,38543), (4,8,38544), (5,6,38545), (5,7,38546), (5,8,38547), (6,7,38548), (6,8,38549), (7,8,38550)

X(38536) =  VU (X(2),X(5))-CIRCLES PERSPECTOR

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

X(38536) lies on these lines: {24,9925}, {68,23096}

X(38536) = isogonal conjugate of X(38545)
X(38536) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(32240)}} and {{A, B, C, X(24), X(23096)}}

X(38537) =  VU (X(2),X(7))-CIRCLES PERSPECTOR

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

X(38537) lies on these lines: {}

X(38537) = isogonal conjugate of X(38548)

X(38538) =  VU (X(2),X(8))-CIRCLES PERSPECTOR

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

X(38538) lies on these lines: {}

X(38538) = isogonal conjugate of X(38549)

X(38539) =  VU (X(3),X(5))-CIRCLES PERSPECTOR

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

X(38539) lies on the cubic K028, on the circumconic with center X(24977), and on these lines: {3,1291}, {4,195}, {6,11071}, {381,15392}, {2937,3447}, {3432,14367}, {13621,25044}, {14254,14859}, {15087,34302}

X(38539) = isogonal conjugate of X(38542)
X(38539) = barycentric product X(i)*X(j) for these {i, j}: {1291, 24978}, {2070, 13582}
X(38539) = barycentric quotient X(2070)/X(37779)
X(38539) = trilinear quotient X(2070)/X(1749)
X(38539) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(32423)}} and {{A, B, C, X(4), X(2070)}}
X(38539) = X(1749)-isoconjugate-of-X(33565)
X(38539) = X(2070)-reciprocal conjugate of-X(37779)

X(38540) =  VU (X(3),X(7))-CIRCLES PERSPECTOR

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

X(38540) lies on these lines: {942,14733}, {943,1156}

X(38540) = isogonal conjugate of X(38543)

X(38541) =  VU (X(3),X(8))-CIRCLES PERSPECTOR

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

X(38541) lies on these lines: {104,517}, {1318,1417}, {4674,27247}, {5563,16944}, {14190,32636}, {14260,37535}

X(38541) = isogonal conjugate of X(38544)
X(38541) = barycentric product X(88)*X(17100)
X(38541) = trilinear product X(106)*X(17100)
X(38541) = trilinear quotient X(106)/X(17101)
X(38541) = X(i)-isoconjugate-of-X(j) for these {i,j}: {519, 17101}, {1877, 34901}
X(38541) = {X(901), X(10428)}-harmonic conjugate of X(1320)

X(38542) =  VU (X(4),X(5))-CIRCLES PERSPECTOR

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

X(38542) lies on the cubic K009 (Lemoine cubic) and these lines: {2,15392}, {3,2888}, {4,14979}, {3459,14940}

X(38542) = isogonal conjugate of X(38539)
X(38542) = trilinear product X(1749)*X(33565)
X(38542) = trilinear quotient X(1749)/X(2070)
X(38542) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(1138)}} and {{A, B, C, X(4), X(32423)}}
X(38542) = {X(34418), X(34900)}-harmonic conjugate of X(33565)

X(38543) =  VU (X(4),X(7))-CIRCLES PERSPECTOR

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

X(38543) lies on these lines: {}

X(38543) = isogonal conjugate of X(38540)
X(38543) = trilinear quotient X(1323)/X(32624)

X(38544) =  VU (X(4),X(8))-CIRCLES PERSPECTOR

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

X(38544) lies on these lines: {517,10742}, {1769,24443}

X(38544) = isogonal conjugate of X(38541)
X(38544) = barycentric quotient X(44)/X(17100)
X(38544) = trilinear product X(i)*X(j) for these {i, j}: {519, 17101}, {1877, 34901}
X(38544) = trilinear quotient X(519)/X(17100)
X(38544) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(10742)}} and {{A, B, C, X(100), X(24443)}}
X(38544) = X(106)-isoconjugate-of-X(17100)
X(38544) = X(44)-reciprocal conjugate of-X(17100)

X(38545) =  VU (X(5),X(6))-CIRCLES PERSPECTOR

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

X(38545) lies on the line {24,23096}

X(38545) = isogonal conjugate of X(38536)

X(38546) =  VU (X(5),X(7))-CIRCLES PERSPECTOR

Barycentrics    (a^8 - 2a^7(b + c) + 2a(b - c)^4(b + c)^3 - 2a^6(b^2 + a b c + c^2) - (b - c)^4(b + c)^2(b^2 + 4a b c + c^2) - a^4a b c(2b^2 + 7a b c + 2c^2) + 6a^5(b^3 + b^2c + a b c^2 + c^3) + a^2(b - c)^2(2b^4 + 10b^3c + 13b^2c^2 + 10a b c^3 + 2c^4) - 2a^3(3b^5 + b^4c - 3b^3c^2 - 3b^2c^3 + a b c^4 + 3c^5)) (a^10 - 2a^9c - 2a(b - c)^3c^2(b + c)^2(b^2 + c^2) + (b - c)^5(b + c)^3(b^2 + c^2) - a^8(5b^2 + c^2) + a^7(4b^3 + 6b^2c - 2a b c^2 + 4c^3) + a^6(4b^4 + 2b^3c + b^2c^2 + 2a b c^3 - 2c^4) - 2a^5b(4b^4 + 3b^3c + b^2c^2 + 2a b c^3 - c^4) - a^2(b - c)^2(5b^6 + 4b^5c + 2b^4c^2 + 4b^3c^3 + 2b^2c^4 - c^6) + 2a^4(2b^6 - 3b^5c + 4b^4c^2 - 2b^3c^3 + 2b^2c^4 - a b c^5 + c^6) + 2a^3(2b^7 + b^6c - b^5c^2 - 2b^4c^3 + b^3c^4 + a b c^6 - 2c^7)) (a^10 - 2a^9b + 2ab^2(b - c)^3(b + c)^2(b^2 + c^2) - (b - c)^5(b + c)^3(b^2 + c^2) - a^8(b^2 + 5c^2) + a^7(4b^3 - 2b^2c + 6a b c^2 + 4c^3) + a^6(-2b^4 + 2b^3c + b^2c^2 + 2a b c^3 + 4c^4) - 2a^5c(-b^4 + 2b^3c + b^2c^2 + 3a b c^3 + 4c^4) + a^2(b - c)^2(b^6 - 2b^4c^2 - 4b^3c^3 - 2b^2c^4 - 4a b c^5 - 5c^6) + 2a^4(b^6 - b^5c + 2b^4c^2 - 2b^3c^3 + 4b^2c^4 - 3a b c^5 + 2c^6) + 2a^3(-2b^7 + b^6c + b^4c^3 - 2b^3c^4 - b^2c^5 + a b c^6 + 2c^7)) : :

X(38546) lies on these lines: {}

X(38547) =  VU (X(5),X(8))-CIRCLES PERSPECTOR

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

X(38547) lies on these lines: {}

X(38548) =  VU (X(6),X(7))-CIRCLES PERSPECTOR

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

X(38548) lies on these lines: {}

X(38548) = isogonal conjugate of X(38537)

X(38549) =  VU (X(6),X(8))-CIRCLES PERSPECTOR

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

X(38549) lies on these lines: {}

X(38549) = isogonal conjugate of X(38538)

X(38550) =  VU (X(7),X(8))-CIRCLES PERSPECTOR

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

X(38550) lies on these lines: {}

X(38550) = intersection, other than A,B,C, of conics {{A, B, C, X(21), X(2826)}} and {{A, B, C, X(65), X(3660)}}

X(38551) = DILATION FROM X(3) OF X(110) TO THE CIRCLE Γ(X(3),X(2967))

Barycentrics    a^2*(a^14*b^4 - 5*a^12*b^6 + 9*a^10*b^8 - 5*a^8*b^10 - 5*a^6*b^12 + 9*a^4*b^14 - 5*a^2*b^16 + b^18 - 6*a^14*b^2*c^2 + 13*a^12*b^4*c^2 - 5*a^10*b^6*c^2 - 9*a^8*b^8*c^2 + 18*a^6*b^10*c^2 - 19*a^4*b^12*c^2 + 9*a^2*b^14*c^2 - b^16*c^2 + a^14*c^4 + 13*a^12*b^2*c^4 - 32*a^10*b^4*c^4 + 22*a^8*b^6*c^4 - 13*a^6*b^8*c^4 + 13*a^4*b^10*c^4 - 4*b^14*c^4 - 5*a^12*c^6 - 5*a^10*b^2*c^6 + 22*a^8*b^4*c^6 - 4*a^6*b^6*c^6 - 3*a^4*b^8*c^6 - 13*a^2*b^10*c^6 + 8*b^12*c^6 + 9*a^10*c^8 - 9*a^8*b^2*c^8 - 13*a^6*b^4*c^8 - 3*a^4*b^6*c^8 + 18*a^2*b^8*c^8 - 4*b^10*c^8 - 5*a^8*c^10 + 18*a^6*b^2*c^10 + 13*a^4*b^4*c^10 - 13*a^2*b^6*c^10 - 4*b^8*c^10 - 5*a^6*c^12 - 19*a^4*b^2*c^12 + 8*b^6*c^12 + 9*a^4*c^14 + 9*a^2*b^2*c^14 - 4*b^4*c^14 - 5*a^2*c^16 - b^2*c^16 + c^18) : :

X(38551) lies on these lines: {3, 74}, {113, 1560}, {542, 3269}, {1976, 19457}, {2967, 9517}, {3124, 7687}, {9408, 25556}, {9979, 34334}, {12227, 20976}, {16163, 36790}

X(38552) = DILATION FROM X(3) OF X(935) TO THE CIRCLE Γ(X(3),X(2967))

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

X(38552) lies on the MacBeath inconic and these lines: {3, 935}, {23, 32428}, {25, 476}, {30, 339}, {98, 30716}, {186, 12042}, {264, 842}, {403, 2971}, {468, 2970}, {523, 2967}, {850, 34336}, {858, 2972}, {2782, 7482}, {2868, 10098}, {2974, 36170}, {5094, 33927}, {5133, 24977}, {5191, 7473}, {14618, 34334}, {16188, 18312}, {18403, 19163}, {23635, 36183}, {34338, 37981}

X(38552) = reflection of X(2967) in the Euler line
X(38552) = polar conjugate of isogonal conjugate of X(23967)
X(38552) = pole wrt polar circle of line X(842)X(7418)

X(38553) = DILATION FROM X(3) OF X(1295) TO THE CIRCLE Γ(X(3),X(2967))

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

X(38553) lies on these lines: {3, 112}, {20, 2782}, {22, 5191}, {30, 339}, {74, 3565}, {376, 3164}, {378, 23635}, {511, 3269}, {2071, 35002}, {2799, 9409}, {2971, 7418}, {2972, 4230}, {5092, 9408}, {6033, 13219}, {9155, 37918}, {9301, 10313}, {10722, 38528}, {21851, 31390}, {34186, 35922}

X(38553) = reflection of X(2967) in X(3)

X(38554) = DILATION FROM X(3) OF X(100) TO THE CIRCLE Γ(X(3),X(2968))

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

X(38554) lies on these lines: {3, 8}, {4, 15252}, {20, 1897}, {30, 21664}, {77, 1060}, {117, 515}, {123, 37725}, {125, 18641}, {153, 10746}, {934, 5932}, {1012, 7071}, {1071, 3937}, {1317, 35014}, {1359, 24034}, {1398, 3149}, {2804, 24466}, {2811, 3184}, {5882, 17102}, {6326, 16596}, {7718, 37252}, {10786, 34120}, {18447, 21740}, {21312, 26706}, {31866, 38357}

X(38554) = reflection of X(2968) in X(3)
X(38554) = isotomic conjugate of polar conjugate of X(23986)

X(38555) = DILATION FROM X(3) OF X(74) TO THE CIRCLE Γ(X(3),X(2975))

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

X(38555) lies on these lines: {1, 14127}, {3, 74}, {113, 5046}, {125, 6949}, {146, 15680}, {541, 37299}, {2771, 2975}, {3448, 6960}, {5697, 7978}, {6284, 10767}, {6941, 14644}, {7727, 14795}, {8674, 11491}, {10091, 21842}, {10706, 11114}, {17702, 37437}, {22265, 38498}

X(38555) = reflection of X(38566) in X(3)

X(38556) = DILATION FROM X(3) OF X(98) TO THE CIRCLE Γ(X(3),X(2975))

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

X(38556) lies on these lines: {3, 76}, {114, 5046}, {115, 6949}, {147, 15680}, {148, 6960}, {542, 37299}, {2783, 2975}, {2787, 11491}, {5697, 7970}, {6054, 11114}, {6284, 10768}, {6941, 14639}, {10089, 21842}, {12117, 37430}, {17556, 23234}, {23698, 37437}

X(38556) = reflection of X(38557) in X(3)

X(38557) = DILATION FROM X(3) OF X(99) TO THE CIRCLE Γ(X(3),X(2975))

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

X(38557) lies on these lines: {3, 76}, {114, 6949}, {115, 5046}, {147, 6960}, {148, 15680}, {543, 37299}, {671, 11114}, {1281, 26270}, {2783, 11491}, {2787, 2975}, {2794, 37437}, {4193, 14061}, {4612, 25051}, {5697, 7983}, {6284, 10769}, {6902, 14651}, {9166, 17556}, {10069, 21842}, {13589, 26227}, {15342, 38508}

X(38557) = reflection of X(38556) in X(3)

X(38558) = DILATION FROM X(3) OF X(101) TO THE CIRCLE Γ(X(3),X(2975))

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

X(38588) lies on these lines: {3, 101}, {116, 5046}, {118, 6949}, {150, 15680}, {152, 6960}, {544, 37299}, {2801, 11491}, {2809, 11010}, {2975, 3887}, {4193, 31273}, {5697, 10695}, {6284, 10770}, {10708, 11114}

X(38558) = reflection of X(38560) in X(3)

X(38559) = DILATION FROM X(3) OF X(102) TO THE CIRCLE Γ(X(3),X(2975))

Barycentrics    a^2*(a^11 - 2*a^10*b - 3*a^9*b^2 + 8*a^8*b^3 + 2*a^7*b^4 - 12*a^6*b^5 + 2*a^5*b^6 + 8*a^4*b^7 - 3*a^3*b^8 - 2*a^2*b^9 + a*b^10 - 2*a^10*c + 6*a^9*b*c - 19*a^7*b^3*c + 13*a^6*b^4*c + 21*a^5*b^5*c - 19*a^4*b^6*c - 9*a^3*b^7*c + 9*a^2*b^8*c + a*b^9*c - b^10*c - 3*a^9*c^2 + 13*a^7*b^2*c^2 + 3*a^6*b^3*c^2 - 27*a^5*b^4*c^2 - 3*a^4*b^5*c^2 + 27*a^3*b^6*c^2 - 3*a^2*b^7*c^2 - 10*a*b^8*c^2 + 3*b^9*c^2 + 8*a^8*c^3 - 19*a^7*b*c^3 + 3*a^6*b^2*c^3 + 8*a^5*b^3*c^3 + 14*a^4*b^4*c^3 - 3*a^3*b^5*c^3 - 25*a^2*b^6*c^3 + 14*a*b^7*c^3 + 2*a^7*c^4 + 13*a^6*b*c^4 - 27*a^5*b^2*c^4 + 14*a^4*b^3*c^4 - 24*a^3*b^4*c^4 + 21*a^2*b^5*c^4 + 9*a*b^6*c^4 - 8*b^7*c^4 - 12*a^6*c^5 + 21*a^5*b*c^5 - 3*a^4*b^2*c^5 - 3*a^3*b^3*c^5 + 21*a^2*b^4*c^5 - 30*a*b^5*c^5 + 6*b^6*c^5 + 2*a^5*c^6 - 19*a^4*b*c^6 + 27*a^3*b^2*c^6 - 25*a^2*b^3*c^6 + 9*a*b^4*c^6 + 6*b^5*c^6 + 8*a^4*c^7 - 9*a^3*b*c^7 - 3*a^2*b^2*c^7 + 14*a*b^3*c^7 - 8*b^4*c^7 - 3*a^3*c^8 + 9*a^2*b*c^8 - 10*a*b^2*c^8 - 2*a^2*c^9 + a*b*c^9 + 3*b^2*c^9 + a*c^10 - b*c^10) : :

X(38559) lies on these lines: {3, 102}, {117, 5046}, {124, 6949}, {151, 15680}, {1795, 21842}, {2800, 2975}, {2817, 11010}, {3738, 11491}, {5697, 10696}, {6284, 10771}, {6960, 33650}, {10709, 11114}, {31866, 36037}

X(38559) = reflection of X(38565) in X(3)

X(38560) = DILATION FROM X(3) OF X(103) TO THE CIRCLE Γ(X(3),X(2975))

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

X(38560) lies on these lines: {3, 101}, {116, 6949}, {118, 5046}, {150, 6960}, {152, 15680}, {2801, 2975}, {2809, 11014}, {3887, 11491}, {5697, 10697}, {6284, 10772}, {10710, 11114}

X(38560) = reflection of X(38558) in X(3)

X(38561) = DILATION FROM X(3) OF X(105) TO THE CIRCLE Γ(X(3),X(2975))

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

X(38561) lies on these lines: {3, 105}, {120, 5046}, {528, 2975}, {2809, 11010}, {2826, 11491}, {5511, 6949}, {5540, 24047}, {5697, 10699}, {6284, 10773}, {6960, 34547}, {10712, 11114}, {15680, 20344}

X(38562) = DILATION FROM X(3) OF X(106) TO THE CIRCLE Γ(X(3),X(2975))

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

X(106) lies on these lines: {3, 106}, {121, 5046}, {595, 23832}, {2802, 2975}, {2827, 11491}, {5510, 6949}, {5697, 10700}, {6284, 10774}, {6960, 34548}, {8715, 23831}, {10713, 11114}, {15680, 21290}

X(38563) = DILATION FROM X(3) OF X(107) TO THE CIRCLE Γ(X(3),X(2975))

Barycentrics    a^16 - a^14*b^2 - 9*a^12*b^4 + 25*a^10*b^6 - 25*a^8*b^8 + 9*a^6*b^10 + a^4*b^12 - a^2*b^14 - a^14*b*c - a^13*b^2*c + a^12*b^3*c + a^11*b^4*c + 2*a^10*b^5*c + 2*a^9*b^6*c - 2*a^8*b^7*c - 2*a^7*b^8*c - a^6*b^9*c - a^5*b^10*c + a^4*b^11*c + a^3*b^12*c - a^14*c^2 - a^13*b*c^2 + 19*a^12*b^2*c^2 + a^11*b^3*c^2 - 25*a^10*b^4*c^2 + 2*a^9*b^5*c^2 - 23*a^8*b^6*c^2 - 2*a^7*b^7*c^2 + 45*a^6*b^8*c^2 - a^5*b^9*c^2 - 11*a^4*b^10*c^2 + a^3*b^11*c^2 - 3*a^2*b^12*c^2 - b^14*c^2 + a^12*b*c^3 + a^11*b^2*c^3 - 5*a^10*b^3*c^3 - 5*a^9*b^4*c^3 + 2*a^8*b^5*c^3 + 2*a^7*b^6*c^3 + 6*a^6*b^7*c^3 + 6*a^5*b^8*c^3 - 3*a^4*b^9*c^3 - 3*a^3*b^10*c^3 - a^2*b^11*c^3 - a*b^12*c^3 - 9*a^12*c^4 + a^11*b*c^4 - 25*a^10*b^2*c^4 - 5*a^9*b^3*c^4 + 96*a^8*b^4*c^4 + 2*a^7*b^5*c^4 - 54*a^6*b^6*c^4 + 6*a^5*b^7*c^4 - 29*a^4*b^8*c^4 - 3*a^3*b^9*c^4 + 15*a^2*b^10*c^4 - a*b^11*c^4 + 6*b^12*c^4 + 2*a^10*b*c^5 + 2*a^9*b^2*c^5 + 2*a^8*b^3*c^5 + 2*a^7*b^4*c^5 - 10*a^6*b^5*c^5 - 10*a^5*b^6*c^5 + 2*a^4*b^7*c^5 + 2*a^3*b^8*c^5 + 4*a^2*b^9*c^5 + 4*a*b^10*c^5 + 25*a^10*c^6 + 2*a^9*b*c^6 - 23*a^8*b^2*c^6 + 2*a^7*b^3*c^6 - 54*a^6*b^4*c^6 - 10*a^5*b^5*c^6 + 78*a^4*b^6*c^6 + 2*a^3*b^7*c^6 - 11*a^2*b^8*c^6 + 4*a*b^9*c^6 - 15*b^10*c^6 - 2*a^8*b*c^7 - 2*a^7*b^2*c^7 + 6*a^6*b^3*c^7 + 6*a^5*b^4*c^7 + 2*a^4*b^5*c^7 + 2*a^3*b^6*c^7 - 6*a^2*b^7*c^7 - 6*a*b^8*c^7 - 25*a^8*c^8 - 2*a^7*b*c^8 + 45*a^6*b^2*c^8 + 6*a^5*b^3*c^8 - 29*a^4*b^4*c^8 + 2*a^3*b^5*c^8 - 11*a^2*b^6*c^8 - 6*a*b^7*c^8 + 20*b^8*c^8 - a^6*b*c^9 - a^5*b^2*c^9 - 3*a^4*b^3*c^9 - 3*a^3*b^4*c^9 + 4*a^2*b^5*c^9 + 4*a*b^6*c^9 + 9*a^6*c^10 - a^5*b*c^10 - 11*a^4*b^2*c^10 - 3*a^3*b^3*c^10 + 15*a^2*b^4*c^10 + 4*a*b^5*c^10 - 15*b^6*c^10 + a^4*b*c^11 + a^3*b^2*c^11 - a^2*b^3*c^11 - a*b^4*c^11 + a^4*c^12 + a^3*b*c^12 - 3*a^2*b^2*c^12 - a*b^3*c^12 + 6*b^4*c^12 - a^2*c^14 - b^2*c^14 : :

X(38563) lies on these lines: {3, 107}, {122, 5046}, {133, 6949}, {2803, 2975}, {2828, 11491}, {5697, 10701}, {6284, 10775}, {6960, 34549}, {9530, 37299}, {10714, 11114}, {15680, 34186}

X(38564) = DILATION FROM X(3) OF X(108) TO THE CIRCLE Γ(X(3),X(2975))

Barycentrics    a*(a^12 - a^11*b - 3*a^10*b^2 + 3*a^9*b^3 + 2*a^8*b^4 - 2*a^7*b^5 + 2*a^6*b^6 - 2*a^5*b^7 - 3*a^4*b^8 + 3*a^3*b^9 + a^2*b^10 - a*b^11 - a^11*c + 6*a^10*b*c - 3*a^9*b^2*c - 11*a^8*b^3*c + 14*a^7*b^4*c - 2*a^6*b^5*c - 14*a^5*b^6*c + 12*a^4*b^7*c + 3*a^3*b^8*c - 4*a^2*b^9*c + a*b^10*c - b^11*c - 3*a^10*c^2 - 3*a^9*b*c^2 + 21*a^8*b^2*c^2 - 13*a^7*b^3*c^2 - 25*a^6*b^4*c^2 + 31*a^5*b^5*c^2 + a^4*b^6*c^2 - 11*a^3*b^7*c^2 + 4*a^2*b^8*c^2 - 4*a*b^9*c^2 + 2*b^10*c^2 + 3*a^9*c^3 - 11*a^8*b*c^3 - 13*a^7*b^2*c^3 + 50*a^6*b^3*c^3 - 15*a^5*b^4*c^3 - 28*a^4*b^5*c^3 + 21*a^3*b^6*c^3 - 14*a^2*b^7*c^3 + 4*a*b^8*c^3 + 3*b^9*c^3 + 2*a^8*c^4 + 14*a^7*b*c^4 - 25*a^6*b^2*c^4 - 15*a^5*b^3*c^4 + 36*a^4*b^4*c^4 - 16*a^3*b^5*c^4 - 5*a^2*b^6*c^4 + 17*a*b^7*c^4 - 8*b^8*c^4 - 2*a^7*c^5 - 2*a^6*b*c^5 + 31*a^5*b^2*c^5 - 28*a^4*b^3*c^5 - 16*a^3*b^4*c^5 + 36*a^2*b^5*c^5 - 17*a*b^6*c^5 - 2*b^7*c^5 + 2*a^6*c^6 - 14*a^5*b*c^6 + a^4*b^2*c^6 + 21*a^3*b^3*c^6 - 5*a^2*b^4*c^6 - 17*a*b^5*c^6 + 12*b^6*c^6 - 2*a^5*c^7 + 12*a^4*b*c^7 - 11*a^3*b^2*c^7 - 14*a^2*b^3*c^7 + 17*a*b^4*c^7 - 2*b^5*c^7 - 3*a^4*c^8 + 3*a^3*b*c^8 + 4*a^2*b^2*c^8 + 4*a*b^3*c^8 - 8*b^4*c^8 + 3*a^3*c^9 - 4*a^2*b*c^9 - 4*a*b^2*c^9 + 3*b^3*c^9 + a^2*c^10 + a*b*c^10 + 2*b^2*c^10 - a*c^11 - b*c^11) : :

X(38564) lies on these lines: {3, 108}, {123, 5046}, {2804, 2975}, {2817, 11010}, {2829, 11491}, {5697, 10702}, {6284, 10776}, {6949, 25640}, {6960, 34550}, {10715, 11114}, {15680, 34188}

X(38565) = DILATION FROM X(3) OF X(109) TO THE CIRCLE Γ(X(3),X(2975))

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

X(38565) lies on these lines: {3, 102}, {117, 6949}, {124, 5046}, {151, 6960}, {2800, 11010}, {2817, 11014}, {2975, 3738}, {5697, 10703}, {6284, 10777}, {10716, 11114}, {15680, 33650}

X(38565) = reflection of X(38559) in X(3)

X(38566) = DILATION FROM X(3) OF X(110) TO THE CIRCLE Γ(X(3),X(2975))

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

X(38566) lies on these lines: {3, 74}, {40, 13589}, {113, 6949}, {125, 5046}, {146, 6960}, {542, 37299}, {2771, 11491}, {2777, 37437}, {2975, 8674}, {3448, 15680}, {4193, 15059}, {5330, 31525}, {5697, 7984}, {6284, 10778}, {6963, 15057}, {9140, 11114}, {10081, 21842}, {14795, 19470}

X(38566) = reflection of X(38555) in X(3)

X(38567) = DILATION FROM X(3) OF X(112) TO THE CIRCLE Γ(X(3),X(2975))

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

X(38567) lies on these lines: {3, 112}, {127, 5046}, {132, 6949}, {2806, 2975}, {2831, 11491}, {5697, 10705}, {6284, 10780}, {6960, 12384}, {10718, 11114}, {13117, 21842}, {13219, 15680}

X(38567) = reflection of X(38571) in X(3)

X(38568) = DILATION FROM X(3) OF X(901) TO THE CIRCLE Γ(X(3),X(2975))

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

X(38568) lies on these lines: {1, 13589}, {3, 901}, {8, 14513}, {36, 595}, {59, 10571}, {411, 517}, {513, 2975}, {944, 14511}, {1319, 4296}, {2841, 4996}, {3259, 5046}, {3336, 13752}, {5253, 34583}, {6284, 31512}, {6949, 31841}, {14795, 23153}, {22767, 26910}

X(38568) = reflection of X(38569) in X(3)

X(38569) = DILATION FROM X(3) OF X(953) TO THE CIRCLE Γ(X(3),X(2975))

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

X(38569) lies on these lines: {3, 901}, {40, 14127}, {100, 31847}, {513, 11491}, {517, 2975}, {1614, 1618}, {3259, 6949}, {3336, 13753}, {5046, 31841}, {11508, 26914}, {14511, 32153}, {14513, 32141}, {28219, 34139}

X(38569) = reflection of X(38568) in X(3)

X(38570) = DILATION FROM X(3) OF X(1290) TO THE CIRCLE Γ(X(3),X(2975))

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

X(38570) lies on these lines: {1, 229}, {3, 1290}, {30, 11491}, {55, 36171}, {100, 36154}, {523, 2975}, {1621, 3109}, {2752, 17522}, {2766, 17555}, {5046, 5520}, {6284, 36175}, {26285, 36001}, {26711, 35221}, {27529, 30447}

X(38571) = DILATION FROM X(3) OF X(1297) TO THE CIRCLE Γ(X(3),X(2975))

Barycentrics    a^2*(a^15 - a^14*b - 4*a^13*b^2 + 4*a^12*b^3 + 5*a^11*b^4 - 5*a^10*b^5 - 5*a^7*b^8 + 5*a^6*b^9 + 4*a^5*b^10 - 4*a^4*b^11 - a^3*b^12 + a^2*b^13 - a^14*c + a^13*b*c + 2*a^12*b^2*c - 5*a^11*b^3*c + a^10*b^4*c + 4*a^9*b^5*c - 4*a^8*b^6*c + 6*a^7*b^7*c + a^6*b^8*c - 7*a^5*b^9*c + 2*a^4*b^10*c - a^3*b^11*c - a^2*b^12*c + 2*a*b^13*c - 4*a^13*c^2 + 2*a^12*b*c^2 + 11*a^11*b^2*c^2 - 5*a^10*b^3*c^2 - 11*a^9*b^4*c^2 + 7*a^8*b^5*c^2 + 6*a^7*b^6*c^2 - 10*a^6*b^7*c^2 - 2*a^5*b^8*c^2 + 8*a^4*b^9*c^2 - a^3*b^10*c^2 - a^2*b^11*c^2 + a*b^12*c^2 - b^13*c^2 + 4*a^12*c^3 - 5*a^11*b*c^3 - 5*a^10*b^2*c^3 + 9*a^9*b^3*c^3 + a^8*b^4*c^3 - 10*a^7*b^5*c^3 + 2*a^6*b^6*c^3 + 10*a^5*b^7*c^3 - 6*a^4*b^8*c^3 - a^3*b^9*c^3 + 3*a^2*b^10*c^3 - 3*a*b^11*c^3 + b^12*c^3 + 5*a^11*c^4 + a^10*b*c^4 - 11*a^9*b^2*c^4 + a^8*b^3*c^4 + 6*a^7*b^4*c^4 + 2*a^6*b^5*c^4 - 2*a^5*b^6*c^4 - 6*a^4*b^7*c^4 + 5*a^3*b^8*c^4 - 3*a^2*b^9*c^4 - 3*a*b^10*c^4 + 5*b^11*c^4 - 5*a^10*c^5 + 4*a^9*b*c^5 + 7*a^8*b^2*c^5 - 10*a^7*b^3*c^5 + 2*a^6*b^4*c^5 - 6*a^5*b^5*c^5 + 6*a^4*b^6*c^5 + 2*a^3*b^7*c^5 - 5*a^2*b^8*c^5 + 10*a*b^9*c^5 - 5*b^10*c^5 - 4*a^8*b*c^6 + 6*a^7*b^2*c^6 + 2*a^6*b^3*c^6 - 2*a^5*b^4*c^6 + 6*a^4*b^5*c^6 - 6*a^3*b^6*c^6 + 6*a^2*b^7*c^6 + 2*a*b^8*c^6 - 10*b^9*c^6 + 6*a^7*b*c^7 - 10*a^6*b^2*c^7 + 10*a^5*b^3*c^7 - 6*a^4*b^4*c^7 + 2*a^3*b^5*c^7 + 6*a^2*b^6*c^7 - 18*a*b^7*c^7 + 10*b^8*c^7 - 5*a^7*c^8 + a^6*b*c^8 - 2*a^5*b^2*c^8 - 6*a^4*b^3*c^8 + 5*a^3*b^4*c^8 - 5*a^2*b^5*c^8 + 2*a*b^6*c^8 + 10*b^7*c^8 + 5*a^6*c^9 - 7*a^5*b*c^9 + 8*a^4*b^2*c^9 - a^3*b^3*c^9 - 3*a^2*b^4*c^9 + 10*a*b^5*c^9 - 10*b^6*c^9 + 4*a^5*c^10 + 2*a^4*b*c^10 - a^3*b^2*c^10 + 3*a^2*b^3*c^10 - 3*a*b^4*c^10 - 5*b^5*c^10 - 4*a^4*c^11 - a^3*b*c^11 - a^2*b^2*c^11 - 3*a*b^3*c^11 + 5*b^4*c^11 - a^3*c^12 - a^2*b*c^12 + a*b^2*c^12 + b^3*c^12 + a^2*c^13 + 2*a*b*c^13 - b^2*c^13) : :

X(38571) lies on these lines: {3, 112}, {127, 6949}, {132, 5046}, {2794, 37437}, {2806, 11491}, {2831, 2975}, {5697, 13099}, {6960, 13219}, {9530, 37299}, {12384, 15680}, {13312, 21842}

X(38571) = reflection of X(38567) in X(3)

X(38572) = DILATION FROM X(3) OF X(101) TO THE CIRCLE Γ(X(3),X(399))

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

X(38572) lies on these lines: {3, 101}, {4, 20096}, {5, 150}, {30, 152}, {116, 1656}, {118, 381}, {140, 32008}, {355, 2784}, {382, 10741}, {399, 2774}, {517, 1282}, {952, 14942}, {971, 17976}, {999, 1362}, {1001, 2801}, {1351, 2810}, {1482, 2809}, {1598, 5185}, {1657, 33520}, {2427, 20761}, {2772, 10620}, {2786, 13188}, {2813, 11258}, {2825, 13115}, {3022, 3295}, {3033, 9567}, {3041, 9708}, {3046, 9704}, {3526, 6710}, {3732, 18329}, {3830, 10710}, {3887, 12331}, {4845, 6767}, {5054, 6712}, {5055, 10708}, {5070, 31273}, {5073, 10727}, {5079, 20401}, {5093, 10756}, {8148, 10697}, {9518, 13310}, {9566, 34457}, {10247, 10695}, {11028, 15730}, {11917, 34112}, {14663, 20430}, {15720, 35024}, {26446, 28346}

X(38572) = reflection of X(38574) in X(3)

X(38573) = DILATION FROM X(3) OF X(102) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a^2*(a^8 - a^7*b + 3*a^5*b^3 - 6*a^4*b^4 - 3*a^3*b^5 + 8*a^2*b^6 + a*b^7 - 3*b^8 - a^7*c + 3*a^6*b*c - 5*a^5*b^2*c - a^4*b^3*c + 13*a^3*b^4*c - 7*a^2*b^5*c - 7*a*b^6*c + 5*b^7*c - 5*a^5*b*c^2 + 16*a^4*b^2*c^2 - 10*a^3*b^3*c^2 - 12*a^2*b^4*c^2 + 15*a*b^5*c^2 - 4*b^6*c^2 + 3*a^5*c^3 - a^4*b*c^3 - 10*a^3*b^2*c^3 + 22*a^2*b^3*c^3 - 9*a*b^4*c^3 - 5*b^5*c^3 - 6*a^4*c^4 + 13*a^3*b*c^4 - 12*a^2*b^2*c^4 - 9*a*b^3*c^4 + 14*b^4*c^4 - 3*a^3*c^5 - 7*a^2*b*c^5 + 15*a*b^2*c^5 - 5*b^3*c^5 + 8*a^2*c^6 - 7*a*b*c^6 - 4*b^2*c^6 + a*c^7 + 5*b*c^7 - 3*c^8) : :

X(38573) lies on these lines: {3, 102}, {5, 151}, {30, 33650}, {117, 1656}, {124, 381}, {382, 10747}, {399, 2779}, {517, 3465}, {999, 1364}, {1361, 3295}, {1482, 2817}, {2773, 10620}, {2785, 12188}, {2792, 13188}, {2800, 11500}, {2816, 12699}, {2819, 11258}, {2841, 35448}, {2853, 13115}, {3040, 9709}, {3042, 9708}, {3526, 6711}, {3738, 12773}, {3830, 10716}, {5054, 6718}, {5055, 10709}, {5073, 10732}, {5093, 10757}, {5708, 12016}, {8148, 10703}, {9532, 13310}, {9566, 34459}, {9567, 34455}, {10246, 11713}, {10247, 10696}, {13532, 18525}

X(38573) = reflection of X(38579) in X(3)

X(38574) = DILATION FROM X(3) OF X(103) TO THE CIRCLE Γ(X(3),X(399))

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

X(38574) lies on these lines: {3, 101}, {5, 152}, {30, 150}, {116, 381}, {118, 1656}, {376, 20096}, {382, 10739}, {399, 2772}, {517, 5527}, {544, 3534}, {999, 3022}, {1282, 3579}, {1362, 3295}, {1597, 5185}, {1657, 33521}, {2095, 2823}, {2774, 10620}, {2784, 13188}, {2786, 12188}, {2801, 11495}, {2807, 10679}, {2809, 12702}, {2810, 33878}, {2824, 11258}, {2825, 13310}, {2826, 19921}, {3033, 9566}, {3041, 9709}, {3046, 9703}, {3526, 6712}, {3830, 10708}, {3887, 12773}, {5054, 6710}, {5055, 10710}, {5073, 10725}, {5093, 10758}, {5708, 11028}, {8148, 10695}, {9518, 13115}, {9567, 34457}, {10246, 11714}, {10247, 10697}, {15700, 35024}, {18413, 36279}, {22765, 35454}

X(38574) = reflection of X(38572) in X(3)

X(38575) = DILATION FROM X(3) OF X(105) TO THE CIRCLE Γ(X(3),X(399))

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

X(38575) lies on these lines: {3, 105}, {4, 20097}, {5, 20344}, {30, 34547}, {119, 381}, {120, 1656}, {382, 15521}, {399, 2836}, {517, 2348}, {999, 1358}, {1482, 2809}, {2095, 2835}, {2775, 10620}, {2788, 12188}, {2795, 13188}, {2826, 12773}, {2834, 18534}, {2837, 11258}, {2838, 13310}, {3021, 3295}, {3034, 9567}, {3039, 9708}, {3526, 6714}, {3830, 10729}, {5055, 10712}, {5093, 10760}, {5620, 31394}, {8692, 9519}, {9523, 13115}, {10246, 11716}, {10247, 10699}, {10269, 34578}, {14661, 14839}, {16408, 34124}, {18535, 21664}

X(38575) = reflection of X(38589) in X(3)

X(38576) = DILATION FROM X(3) OF X(106) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a^2*(a^5 - 2*a^4*b - a^3*b^2 + 5*a^2*b^3 - 3*b^5 - 2*a^4*c + 9*a^3*b*c - 10*a^2*b^2*c - 9*a*b^3*c + 12*b^4*c - a^3*c^2 - 10*a^2*b*c^2 + 28*a*b^2*c^2 - 11*b^3*c^2 + 5*a^2*c^3 - 9*a*b*c^3 - 11*b^2*c^3 + 12*b*c^4 - 3*c^5) : :

X(38576) lies on these lines: {3, 106}, {4, 20098}, {5, 21290}, {30, 34548}, {121, 1656}, {381, 5510}, {382, 15522}, {399, 2842}, {517, 1054}, {952, 1120}, {999, 1357}, {1351, 2810}, {1482, 2802}, {2776, 10620}, {2789, 12188}, {2796, 3656}, {2827, 12773}, {2841, 10680}, {2843, 11258}, {2844, 13310}, {3030, 9567}, {3038, 9708}, {3295, 6018}, {3526, 6715}, {3830, 10730}, {5055, 10713}, {5093, 10761}, {5603, 17777}, {5886, 11814}, {8688, 12702}, {9519, 10246}, {9527, 13115}, {10222, 13541}, {10247, 10700}, {14260, 22148}, {22765, 34139}

X(38576) = reflection of X(38590) in X(3)

X(38577) = DILATION FROM X(3) OF X(107) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a^16 - 4*a^14*b^2 + 6*a^12*b^4 - 5*a^10*b^6 + 5*a^8*b^8 - 6*a^6*b^10 + 4*a^4*b^12 - a^2*b^14 - 4*a^14*c^2 + a^12*b^2*c^2 + 2*a^10*b^4*c^2 + 16*a^8*b^6*c^2 - 12*a^6*b^8*c^2 - 11*a^4*b^10*c^2 + 6*a^2*b^12*c^2 + 2*b^14*c^2 + 6*a^12*c^4 + 2*a^10*b^2*c^4 - 42*a^8*b^4*c^4 + 18*a^6*b^6*c^4 + 40*a^4*b^8*c^4 - 12*a^2*b^10*c^4 - 12*b^12*c^4 - 5*a^10*c^6 + 16*a^8*b^2*c^6 + 18*a^6*b^4*c^6 - 66*a^4*b^6*c^6 + 7*a^2*b^8*c^6 + 30*b^10*c^6 + 5*a^8*c^8 - 12*a^6*b^2*c^8 + 40*a^4*b^4*c^8 + 7*a^2*b^6*c^8 - 40*b^8*c^8 - 6*a^6*c^10 - 11*a^4*b^2*c^10 - 12*a^2*b^4*c^10 + 30*b^6*c^10 + 4*a^4*c^12 + 6*a^2*b^2*c^12 - 12*b^4*c^12 - a^2*c^14 + 2*b^2*c^14 : :

X(38577) lies on these lines: {3, 107}, {4, 19774}, {5, 34186}, {30, 5667}, {64, 265}, {122, 1656}, {127, 133}, {339, 18535}, {399, 9033}, {999, 3324}, {1657, 23240}, {2070, 14703}, {2790, 12188}, {2797, 13188}, {2803, 12331}, {2816, 12699}, {2828, 12773}, {2847, 11258}, {2848, 13310}, {3184, 3534}, {3295, 7158}, {3526, 6716}, {3830, 10152}, {5054, 34842}, {5055, 10714}, {5079, 36520}, {5093, 10762}, {6528, 20477}, {6699, 14847}, {7517, 14673}, {9528, 13743}, {10246, 11718}, {10247, 10701}, {15061, 24930}

X(38577) = reflection of X(38591) in X(3)

X(38578) = DILATION FROM X(3) OF X(108) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a*(a^12 - a^11*b - 3*a^10*b^2 + 3*a^9*b^3 + 2*a^8*b^4 - 2*a^7*b^5 + 2*a^6*b^6 - 2*a^5*b^7 - 3*a^4*b^8 + 3*a^3*b^9 + a^2*b^10 - a*b^11 - a^11*c + a^10*b*c + 3*a^9*b^2*c - 4*a^8*b^3*c - 4*a^7*b^4*c + 4*a^6*b^5*c + 4*a^5*b^6*c + 2*a^4*b^7*c - 3*a^3*b^8*c - 5*a^2*b^9*c + a*b^10*c + 2*b^11*c - 3*a^10*c^2 + 3*a^9*b*c^2 - 2*a^8*b^2*c^2 + 6*a^7*b^3*c^2 + 4*a^6*b^4*c^2 - 12*a^5*b^5*c^2 + 6*a^4*b^6*c^2 - 6*a^3*b^7*c^2 - a^2*b^8*c^2 + 9*a*b^9*c^2 - 4*b^10*c^2 + 3*a^9*c^3 - 4*a^8*b*c^3 + 6*a^7*b^2*c^3 - 20*a^6*b^3*c^3 + 10*a^5*b^4*c^3 + 6*a^4*b^5*c^3 - 10*a^3*b^6*c^3 + 24*a^2*b^7*c^3 - 9*a*b^8*c^3 - 6*b^9*c^3 + 2*a^8*c^4 - 4*a^7*b*c^4 + 4*a^6*b^2*c^4 + 10*a^5*b^3*c^4 - 22*a^4*b^4*c^4 + 16*a^3*b^5*c^4 - 22*a*b^7*c^4 + 16*b^8*c^4 - 2*a^7*c^5 + 4*a^6*b*c^5 - 12*a^5*b^2*c^5 + 6*a^4*b^3*c^5 + 16*a^3*b^4*c^5 - 38*a^2*b^5*c^5 + 22*a*b^6*c^5 + 4*b^7*c^5 + 2*a^6*c^6 + 4*a^5*b*c^6 + 6*a^4*b^2*c^6 - 10*a^3*b^3*c^6 + 22*a*b^5*c^6 - 24*b^6*c^6 - 2*a^5*c^7 + 2*a^4*b*c^7 - 6*a^3*b^2*c^7 + 24*a^2*b^3*c^7 - 22*a*b^4*c^7 + 4*b^5*c^7 - 3*a^4*c^8 - 3*a^3*b*c^8 - a^2*b^2*c^8 - 9*a*b^3*c^8 + 16*b^4*c^8 + 3*a^3*c^9 - 5*a^2*b*c^9 + 9*a*b^2*c^9 - 6*b^3*c^9 + a^2*c^10 + a*b*c^10 - 4*b^2*c^10 - a*c^11 + 2*b*c^11) : :

X(38578) lies on these lines: {3, 108}, {5, 34188}, {30, 34550}, {123, 1656}, {381, 10746}, {382, 2829}, {399, 2850}, {999, 1359}, {1482, 2817}, {2095, 2823}, {2778, 10620}, {2791, 12188}, {2798, 13188}, {2804, 12331}, {2834, 18534}, {2851, 11258}, {3295, 3318}, {3526, 6717}, {3830, 10731}, {5055, 10715}, {5093, 10763}, {9528, 16117}, {9567, 34456}, {10246, 11719}, {10247, 10702}, {10271, 16202}

X(38578) = reflection of X(38592) in X(3)

X(38579) = DILATION FROM X(3) OF X(109) TO THE CIRCLE Γ(X(3),X(399))

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

X(38579) lies on these lines: {3, 102}, {5, 33650}, {30, 151}, {117, 381}, {119, 36280}, {124, 1656}, {382, 10740}, {399, 2773}, {952, 22148}, {999, 1361}, {1364, 3295}, {1482, 2800}, {1845, 36279}, {2095, 2835}, {2779, 10620}, {2785, 13188}, {2792, 12188}, {2807, 10679}, {2817, 12702}, {2841, 10680}, {2852, 11258}, {2853, 13310}, {3040, 9708}, {3042, 9709}, {3526, 6718}, {3738, 12331}, {3830, 10709}, {5054, 6711}, {5055, 10716}, {5073, 10726}, {5093, 10764}, {5790, 13532}, {8148, 10696}, {9532, 13115}, {9566, 34455}, {9567, 34459}, {10246, 11700}, {10247, 10703}, {12016, 15934}

X(38579) = reflection of X(38573) in X(3)

X(38580) = DILATION FROM X(3) OF X(476) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a^16 - 5*a^14*b^2 + 11*a^12*b^4 - 15*a^10*b^6 + 15*a^8*b^8 - 11*a^6*b^10 + 5*a^4*b^12 - a^2*b^14 - 5*a^14*c^2 + 8*a^12*b^2*c^2 - 5*a^10*b^4*c^2 + 8*a^8*b^6*c^2 - a^6*b^8*c^2 - 10*a^4*b^10*c^2 + 3*a^2*b^12*c^2 + 2*b^14*c^2 + 11*a^12*c^4 - 5*a^10*b^2*c^4 - 21*a^8*b^4*c^4 + 9*a^6*b^6*c^4 + 21*a^4*b^8*c^4 - 3*a^2*b^10*c^4 - 12*b^12*c^4 - 15*a^10*c^6 + 8*a^8*b^2*c^6 + 9*a^6*b^4*c^6 - 32*a^4*b^6*c^6 + a^2*b^8*c^6 + 30*b^10*c^6 + 15*a^8*c^8 - a^6*b^2*c^8 + 21*a^4*b^4*c^8 + a^2*b^6*c^8 - 40*b^8*c^8 - 11*a^6*c^10 - 10*a^4*b^2*c^10 - 3*a^2*b^4*c^10 + 30*b^6*c^10 + 5*a^4*c^12 + 3*a^2*b^2*c^12 - 12*b^4*c^12 - a^2*c^14 + 2*b^2*c^14 : :

X(38580) lies on these lines: {3, 476}, {4, 18319}, {5, 14731}, {30, 3448}, {125, 14993}, {140, 11749}, {381, 2453}, {399, 523}, {999, 33964}, {1656, 3258}, {2070, 13558}, {2452, 15087}, {3233, 15039}, {3295, 33965}, {3526, 22104}, {5054, 31379}, {5055, 34312}, {5073, 14989}, {5201, 12188}, {6699, 14851}, {7471, 32609}, {7574, 13115}, {14934, 15040}, {15112, 18378}, {17511, 34209}, {30716, 34334}, {31874, 31876}, {36188, 37496}

X(38580) = reflection of X(38581) in X(3)

X(38581) = DILATION FROM X(3) OF X(477) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a^16 - a^14*b^2 - 9*a^12*b^4 + 25*a^10*b^6 - 25*a^8*b^8 + 9*a^6*b^10 + a^4*b^12 - a^2*b^14 - a^14*c^2 + 16*a^12*b^2*c^2 - 21*a^10*b^4*c^2 - 16*a^8*b^6*c^2 + 31*a^6*b^8*c^2 - 6*a^4*b^10*c^2 - a^2*b^12*c^2 - 2*b^14*c^2 - 9*a^12*c^4 - 21*a^10*b^2*c^4 + 75*a^8*b^4*c^4 - 39*a^6*b^6*c^4 - 27*a^4*b^8*c^4 + 9*a^2*b^10*c^4 + 12*b^12*c^4 + 25*a^10*c^6 - 16*a^8*b^2*c^6 - 39*a^6*b^4*c^6 + 64*a^4*b^6*c^6 - 7*a^2*b^8*c^6 - 30*b^10*c^6 - 25*a^8*c^8 + 31*a^6*b^2*c^8 - 27*a^4*b^4*c^8 - 7*a^2*b^6*c^8 + 40*b^8*c^8 + 9*a^6*c^10 - 6*a^4*b^2*c^10 + 9*a^2*b^4*c^10 - 30*b^6*c^10 + a^4*c^12 - a^2*b^2*c^12 + 12*b^4*c^12 - a^2*c^14 - 2*b^2*c^14 : :

X(38581) lies on these lines: {2, 18319}, {3, 476}, {5, 34193}, {30, 146}, {125, 14851}, {381, 3258}, {382, 20957}, {523, 10620}, {999, 33965}, {1656, 25641}, {2070, 14703}, {3233, 14934}, {3295, 33964}, {3526, 31379}, {3534, 15919}, {3830, 9717}, {5054, 22104}, {6699, 14993}, {7471, 15040}, {10272, 33855}, {12121, 14559}, {12308, 14480}, {12902, 17511}, {13188, 35001}, {13512, 35452}, {15041, 36164}, {15046, 36169}

X(38581) = reflection of X(38580) in X(3)

X(38582) = DILATION FROM X(3) OF X(691) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a^2*(a^12 - 5*a^10*b^2 + 5*a^8*b^4 + 4*a^6*b^6 - 7*a^4*b^8 + a^2*b^10 + b^12 - 5*a^10*c^2 + 20*a^8*b^2*c^2 - 24*a^6*b^4*c^2 + 18*a^4*b^6*c^2 - 4*a^2*b^8*c^2 - 5*b^10*c^2 + 5*a^8*c^4 - 24*a^6*b^2*c^4 + 3*a^4*b^4*c^4 + 17*b^8*c^4 + 4*a^6*c^6 + 18*a^4*b^2*c^6 - 26*b^6*c^6 - 7*a^4*c^8 - 4*a^2*b^2*c^8 + 17*b^4*c^8 + a^2*c^10 - 5*b^2*c^10 + c^12) : :

X(38582) lies on these lines: {3, 691}, {30, 148}, {187, 2070}, {381, 16188}, {399, 512}, {511, 10620}, {523, 13188}, {999, 6023}, {1384, 5941}, {1511, 9218}, {1656, 5099}, {2080, 11258}, {3295, 6027}, {3534, 15919}, {5054, 16760}, {9181, 32609}, {13115, 18859}, {37760, 38230}, {37958, 38225}

X(38582) = reflection of X(38583) in X(3)

X(38583) = DILATION FROM X(3) OF X(842) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a^2*(a^12 - a^10*b^2 + a^8*b^4 - 4*a^6*b^6 + a^4*b^8 + 5*a^2*b^10 - 3*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 - 4*a^2*b^8*c^2 + 7*b^10*c^2 + a^8*c^4 + 8*a^6*b^2*c^4 + 3*a^4*b^4*c^4 - 15*b^8*c^4 - 4*a^6*c^6 - 6*a^4*b^2*c^6 + 22*b^6*c^6 + a^4*c^8 - 4*a^2*b^2*c^8 - 15*b^4*c^8 + 5*a^2*c^10 + 7*b^2*c^10 - 3*c^12) : :

X(38583) lies on these lines: {3, 691}, {23, 9301}, {30, 147}, {187, 37958}, {381, 2453}, {399, 511}, {512, 10620}, {523, 12188}, {868, 14731}, {999, 6027}, {1656, 16188}, {2070, 2080}, {3295, 6023}, {3526, 16760}, {9181, 15040}, {11842, 37930}, {18859, 18860}, {35001, 35002}

X(38583) = reflection of X(38582) in X(3)

X(38584) = DILATION FROM X(3) OF X(901) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a^2*(a^8 - 2*a^7*b - 4*a^6*b^2 + 6*a^5*b^3 + 6*a^4*b^4 - 6*a^3*b^5 - 4*a^2*b^6 + 2*a*b^7 + b^8 - 2*a^7*c + 8*a^6*b*c + 4*a^5*b^2*c - 22*a^4*b^3*c - 2*a^3*b^4*c + 20*a^2*b^5*c - 6*b^7*c - 4*a^6*c^2 + 4*a^5*b*c^2 - 3*a^4*b^2*c^2 + 20*a^3*b^3*c^2 - 3*a^2*b^4*c^2 - 24*a*b^5*c^2 + 10*b^6*c^2 + 6*a^5*c^3 - 22*a^4*b*c^3 + 20*a^3*b^2*c^3 - 32*a^2*b^3*c^3 + 22*a*b^4*c^3 + 6*b^5*c^3 + 6*a^4*c^4 - 2*a^3*b*c^4 - 3*a^2*b^2*c^4 + 22*a*b^3*c^4 - 22*b^4*c^4 - 6*a^3*c^5 + 20*a^2*b*c^5 - 24*a*b^2*c^5 + 6*b^3*c^5 - 4*a^2*c^6 + 10*b^2*c^6 + 2*a*c^7 - 6*b*c^7 + c^8) : :

X(38584) lies on these lines: {3, 901}, {46, 23152}, {56, 23153}, {381, 31841}, {513, 12331}, {517, 1768}, {999, 13756}, {1656, 3259}, {2937, 10016}, {3025, 3295}, {3526, 22102}, {3579, 34464}, {5708, 24201}, {15934, 33645}, {22765, 34139}

X(38584) = reflection of X(38586) in X(3)

X(38585) = DILATION FROM X(3) OF X(933) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a^2*(a^20 - 7*a^18*b^2 + 19*a^16*b^4 - 22*a^14*b^6 + 28*a^10*b^10 - 28*a^8*b^12 + 6*a^6*b^14 + 7*a^4*b^16 - 5*a^2*b^18 + b^20 - 7*a^18*c^2 + 32*a^16*b^2*c^2 - 56*a^14*b^4*c^2 + 48*a^12*b^6*c^2 - 27*a^10*b^8*c^2 + 17*a^8*b^10*c^2 - 2*a^6*b^12*c^2 - 14*a^4*b^14*c^2 + 12*a^2*b^16*c^2 - 3*b^18*c^2 + 19*a^16*c^4 - 56*a^14*b^2*c^4 + 57*a^12*b^4*c^4 - 28*a^10*b^6*c^4 + 16*a^8*b^8*c^4 - 12*a^6*b^10*c^4 + 9*a^4*b^12*c^4 - 8*a^2*b^14*c^4 + 3*b^16*c^4 - 22*a^14*c^6 + 48*a^12*b^2*c^6 - 28*a^10*b^4*c^6 - 10*a^8*b^6*c^6 + 8*a^6*b^8*c^6 + 10*a^4*b^10*c^6 + 2*a^2*b^12*c^6 - 8*b^14*c^6 - 27*a^10*b^2*c^8 + 16*a^8*b^4*c^8 + 8*a^6*b^6*c^8 - 24*a^4*b^8*c^8 - a^2*b^10*c^8 + 28*b^12*c^8 + 28*a^10*c^10 + 17*a^8*b^2*c^10 - 12*a^6*b^4*c^10 + 10*a^4*b^6*c^10 - a^2*b^8*c^10 - 42*b^10*c^10 - 28*a^8*c^12 - 2*a^6*b^2*c^12 + 9*a^4*b^4*c^12 + 2*a^2*b^6*c^12 + 28*b^8*c^12 + 6*a^6*c^14 - 14*a^4*b^2*c^14 - 8*a^2*b^4*c^14 - 8*b^6*c^14 + 7*a^4*c^16 + 12*a^2*b^2*c^16 + 3*b^4*c^16 - 5*a^2*c^18 - 3*b^2*c^18 + c^20) : :

X(38585) lies on these lines: {3, 933}, {20, 35311}, {195, 2935}, {381, 18402}, {1656, 20625}, {2070, 8157}, {7545, 10214}, {11587, 18570}, {12188, 31723}, {18445, 22552}, {19553, 31726}

X(38586) = DILATION FROM X(3) OF X(953) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a^2*(a^8 - 2*a^7*b + 6*a^5*b^3 - 6*a^4*b^4 - 6*a^3*b^5 + 8*a^2*b^6 + 2*a*b^7 - 3*b^8 - 2*a^7*c + 8*a^6*b*c - 12*a^5*b^2*c - 6*a^4*b^3*c + 30*a^3*b^4*c - 12*a^2*b^5*c - 16*a*b^6*c + 10*b^7*c - 12*a^5*b*c^2 + 37*a^4*b^2*c^2 - 28*a^3*b^3*c^2 - 27*a^2*b^4*c^2 + 40*a*b^5*c^2 - 10*b^6*c^2 + 6*a^5*c^3 - 6*a^4*b*c^3 - 28*a^3*b^2*c^3 + 64*a^2*b^3*c^3 - 26*a*b^4*c^3 - 10*b^5*c^3 - 6*a^4*c^4 + 30*a^3*b*c^4 - 27*a^2*b^2*c^4 - 26*a*b^3*c^4 + 26*b^4*c^4 - 6*a^3*c^5 - 12*a^2*b*c^5 + 40*a*b^2*c^5 - 10*b^3*c^5 + 8*a^2*c^6 - 16*a*b*c^6 - 10*b^2*c^6 + 2*a*c^7 + 10*b*c^7 - 3*c^8) : :

X(38586) lies on these lines: {1, 23152}, {3, 901}, {55, 23153}, {381, 3259}, {513, 12773}, {517, 3689}, {999, 3025}, {1656, 31841}, {2070, 10016}, {2841, 35451}, {3295, 13756}, {5054, 22102}, {5180, 13512}, {5708, 33645}, {15934, 24201}

X(38586) = reflection of X(38584) in X(3)

X(38587) = DILATION FROM X(3) OF X(1141) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a^16 - 3*a^14*b^2 + a^12*b^4 + 5*a^10*b^6 - 5*a^8*b^8 - a^6*b^10 + 3*a^4*b^12 - a^2*b^14 - 3*a^14*c^2 + 4*a^12*b^2*c^2 + 5*a^10*b^4*c^2 - 12*a^8*b^6*c^2 + 13*a^6*b^8*c^2 - 14*a^4*b^10*c^2 + 9*a^2*b^12*c^2 - 2*b^14*c^2 + a^12*c^4 + 5*a^10*b^2*c^4 - 5*a^8*b^4*c^4 - 3*a^6*b^6*c^4 + 11*a^4*b^8*c^4 - 21*a^2*b^10*c^4 + 12*b^12*c^4 + 5*a^10*c^6 - 12*a^8*b^2*c^6 - 3*a^6*b^4*c^6 + 13*a^2*b^8*c^6 - 30*b^10*c^6 - 5*a^8*c^8 + 13*a^6*b^2*c^8 + 11*a^4*b^4*c^8 + 13*a^2*b^6*c^8 + 40*b^8*c^8 - a^6*c^10 - 14*a^4*b^2*c^10 - 21*a^2*b^4*c^10 - 30*b^6*c^10 + 3*a^4*c^12 + 9*a^2*b^2*c^12 + 12*b^4*c^12 - a^2*c^14 - 2*b^2*c^14 : :

X(38587) lies on these lines: {2, 12026}, {3, 252}, {4, 195}, {30, 11671}, {128, 1656}, {137, 381}, {140, 14073}, {539, 19552}, {631, 6592}, {999, 3327}, {1154, 13504}, {1478, 14101}, {1658, 34418}, {2070, 15959}, {2937, 13558}, {3090, 23237}, {3091, 25147}, {3295, 7159}, {3526, 34837}, {5054, 13372}, {5071, 25339}, {5072, 23516}, {6102, 13505}, {7502, 14652}, {7517, 15960}, {10285, 36966}, {10619, 18370}, {13188, 14674}, {16766, 27868}, {22335, 30531}, {25042, 32551}, {25148, 32535}

X(38587) = reflection of X(13512) in X(3)
X(38587) = anticomplement of X(14072)

X(38588) = DILATION FROM X(3) OF X(1290) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a*(a^12 - a^11*b - 3*a^10*b^2 + 3*a^9*b^3 + 2*a^8*b^4 - 2*a^7*b^5 + 2*a^6*b^6 - 2*a^5*b^7 - 3*a^4*b^8 + 3*a^3*b^9 + a^2*b^10 - a*b^11 - a^11*c - a^10*b*c + 5*a^9*b^2*c - 10*a^7*b^4*c + 4*a^6*b^5*c + 10*a^5*b^6*c - 2*a^4*b^7*c - 5*a^3*b^8*c - 3*a^2*b^9*c + a*b^10*c + 2*b^11*c - 3*a^10*c^2 + 5*a^9*b*c^2 + a^8*b^2*c^2 - a^7*b^3*c^2 - 8*a^5*b^5*c^2 + 5*a^4*b^6*c^2 - a^3*b^7*c^2 + a^2*b^8*c^2 + 5*a*b^9*c^2 - 4*b^10*c^2 + 3*a^9*c^3 - a^7*b^2*c^3 - 5*a^6*b^3*c^3 + 6*a^5*b^4*c^3 + a^4*b^5*c^3 - 3*a^3*b^6*c^3 + 10*a^2*b^7*c^3 - 5*a*b^8*c^3 - 6*b^9*c^3 + 2*a^8*c^4 - 10*a^7*b*c^4 + 6*a^5*b^3*c^4 - 8*a^4*b^4*c^4 + 6*a^3*b^5*c^4 - 2*a^2*b^6*c^4 - 10*a*b^7*c^4 + 16*b^8*c^4 - 2*a^7*c^5 + 4*a^6*b*c^5 - 8*a^5*b^2*c^5 + a^4*b^3*c^5 + 6*a^3*b^4*c^5 - 14*a^2*b^5*c^5 + 10*a*b^6*c^5 + 4*b^7*c^5 + 2*a^6*c^6 + 10*a^5*b*c^6 + 5*a^4*b^2*c^6 - 3*a^3*b^3*c^6 - 2*a^2*b^4*c^6 + 10*a*b^5*c^6 - 24*b^6*c^6 - 2*a^5*c^7 - 2*a^4*b*c^7 - a^3*b^2*c^7 + 10*a^2*b^3*c^7 - 10*a*b^4*c^7 + 4*b^5*c^7 - 3*a^4*c^8 - 5*a^3*b*c^8 + a^2*b^2*c^8 - 5*a*b^3*c^8 + 16*b^4*c^8 + 3*a^3*c^9 - 3*a^2*b*c^9 + 5*a*b^2*c^9 - 6*b^3*c^9 + a^2*c^10 + a*b*c^10 - 4*b^2*c^10 - a*c^11 + 2*b*c^11) : :

X(38588) lies on these lines: {1, 23152}, {3, 1290}, {30, 149}, {381, 14686}, {399, 513}, {517, 9904}, {523, 12331}, {999, 31524}, {1656, 5520}, {2070, 14667}, {3295, 31522}, {5899, 20999}, {14663, 22765}

X(38589) = DILATION FROM X(3) OF X(1292) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a*(a^7 - 2*a^6*b + a^5*b^2 - a^3*b^4 + 2*a^2*b^5 - a*b^6 - 2*a^6*c - 3*a^5*b*c + 11*a^4*b^2*c - 13*a^3*b^3*c + 5*a^2*b^4*c + 2*b^6*c + a^5*c^2 + 11*a^4*b*c^2 - 2*a^3*b^2*c^2 - a^2*b^3*c^2 - 3*a*b^4*c^2 - 6*b^5*c^2 - 13*a^3*b*c^3 - a^2*b^2*c^3 + 8*a*b^3*c^3 + 4*b^4*c^3 - a^3*c^4 + 5*a^2*b*c^4 - 3*a*b^2*c^4 + 4*b^3*c^4 + 2*a^2*c^5 - 6*b^2*c^5 - a*c^6 + 2*b*c^6) : :

X(38589) lies on these lines: {3, 105}, {5, 34547}, {30, 20344}, {120, 381}, {376, 20097}, {382, 10743}, {399, 2775}, {528, 3534}, {999, 3021}, {1358, 3295}, {1656, 5511}, {2788, 13188}, {2795, 12188}, {2809, 12702}, {2826, 12331}, {2836, 10620}, {2838, 13115}, {3034, 9566}, {3039, 9709}, {3579, 5540}, {3830, 10712}, {5054, 6714}, {5073, 10729}, {8148, 10699}, {9519, 10246}, {9522, 11258}, {9523, 13310}, {11108, 34124}

X(38589) = reflection of X(38575) in X(3)

X(38590) = DILATION FROM X(3) OF X(1293) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a^2*(a^5 - 2*a^4*b - 5*a^3*b^2 + a^2*b^3 + 4*a*b^4 + b^5 - 2*a^4*c + 9*a^3*b*c + 10*a^2*b^2*c - 9*a*b^3*c - 8*b^4*c - 5*a^3*c^2 + 10*a^2*b*c^2 - 20*a*b^2*c^2 + 13*b^3*c^2 + a^2*c^3 - 9*a*b*c^3 + 13*b^2*c^3 + 4*a*c^4 - 8*b*c^4 + c^5) : :

X(38590) lies on these lines: {3, 106}, {5, 34548}, {30, 21290}, {40, 14663}, {121, 381}, {376, 20098}, {382, 10744}, {399, 2776}, {517, 13541}, {999, 6018}, {1054, 3579}, {1357, 3295}, {1656, 5510}, {2789, 13188}, {2796, 12188}, {2802, 11256}, {2810, 33878}, {2827, 12331}, {2841, 35448}, {2842, 10620}, {2844, 13115}, {3030, 9566}, {3038, 9709}, {3699, 18326}, {3830, 10713}, {5054, 6715}, {5073, 10730}, {6361, 17777}, {8148, 10700}, {8692, 9519}, {9526, 11258}, {9527, 13310}, {11814, 12699}

X(38590) = reflection of X(38576) in X(3)

X(38591) = DILATION FROM X(3) OF X(1294) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a^16 - 14*a^12*b^4 + 35*a^10*b^6 - 35*a^8*b^8 + 14*a^6*b^10 - a^2*b^14 + 25*a^12*b^2*c^2 - 34*a^10*b^4*c^2 - 36*a^8*b^6*c^2 + 64*a^6*b^8*c^2 - 11*a^4*b^10*c^2 - 6*a^2*b^12*c^2 - 2*b^14*c^2 - 14*a^12*c^4 - 34*a^10*b^2*c^4 + 142*a^8*b^4*c^4 - 78*a^6*b^6*c^4 - 52*a^4*b^8*c^4 + 24*a^2*b^10*c^4 + 12*b^12*c^4 + 35*a^10*c^6 - 36*a^8*b^2*c^6 - 78*a^6*b^4*c^6 + 126*a^4*b^6*c^6 - 17*a^2*b^8*c^6 - 30*b^10*c^6 - 35*a^8*c^8 + 64*a^6*b^2*c^8 - 52*a^4*b^4*c^8 - 17*a^2*b^6*c^8 + 40*b^8*c^8 + 14*a^6*c^10 - 11*a^4*b^2*c^10 + 24*a^2*b^4*c^10 - 30*b^6*c^10 - 6*a^2*b^2*c^12 + 12*b^4*c^12 - a^2*c^14 - 2*b^2*c^14 : :

X(38591) lies on these lines: {3, 107}, {5, 34549}, {20, 35311}, {30, 34186}, {122, 381}, {133, 1656}, {382, 10745}, {399, 1498}, {550, 5667}, {999, 7158}, {2790, 13188}, {2797, 12188}, {2803, 12773}, {2828, 12331}, {2848, 13115}, {2937, 14703}, {3184, 15696}, {3295, 3324}, {3526, 34842}, {3534, 9530}, {3830, 10714}, {5054, 6716}, {5072, 36520}, {5073, 10152}, {8148, 10701}, {9033, 10620}, {9528, 16117}, {9529, 11258}, {12083, 14673}

X(38591) = reflection of X(38577) in X(3)

X(38592) = DILATION FROM X(3) OF X(1295) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a*(a^12 - a^11*b - 3*a^10*b^2 + 3*a^9*b^3 + 2*a^8*b^4 - 2*a^7*b^5 + 2*a^6*b^6 - 2*a^5*b^7 - 3*a^4*b^8 + 3*a^3*b^9 + a^2*b^10 - a*b^11 - a^11*c + 9*a^10*b*c - 5*a^9*b^2*c - 16*a^8*b^3*c + 20*a^7*b^4*c - 4*a^6*b^5*c - 20*a^5*b^6*c + 18*a^4*b^7*c + 5*a^3*b^8*c - 5*a^2*b^9*c + a*b^10*c - 2*b^11*c - 3*a^10*c^2 - 5*a^9*b*c^2 + 30*a^8*b^2*c^2 - 18*a^7*b^3*c^2 - 36*a^6*b^4*c^2 + 44*a^5*b^5*c^2 - 2*a^4*b^6*c^2 - 14*a^3*b^7*c^2 + 7*a^2*b^8*c^2 - 7*a*b^9*c^2 + 4*b^10*c^2 + 3*a^9*c^3 - 16*a^8*b*c^3 - 18*a^7*b^2*c^3 + 76*a^6*b^3*c^3 - 22*a^5*b^4*c^3 - 42*a^4*b^5*c^3 + 30*a^3*b^6*c^3 - 24*a^2*b^7*c^3 + 7*a*b^8*c^3 + 6*b^9*c^3 + 2*a^8*c^4 + 20*a^7*b*c^4 - 36*a^6*b^2*c^4 - 22*a^5*b^3*c^4 + 58*a^4*b^4*c^4 - 24*a^3*b^5*c^4 - 8*a^2*b^6*c^4 + 26*a*b^7*c^4 - 16*b^8*c^4 - 2*a^7*c^5 - 4*a^6*b*c^5 + 44*a^5*b^2*c^5 - 42*a^4*b^3*c^5 - 24*a^3*b^4*c^5 + 58*a^2*b^5*c^5 - 26*a*b^6*c^5 - 4*b^7*c^5 + 2*a^6*c^6 - 20*a^5*b*c^6 - 2*a^4*b^2*c^6 + 30*a^3*b^3*c^6 - 8*a^2*b^4*c^6 - 26*a*b^5*c^6 + 24*b^6*c^6 - 2*a^5*c^7 + 18*a^4*b*c^7 - 14*a^3*b^2*c^7 - 24*a^2*b^3*c^7 + 26*a*b^4*c^7 - 4*b^5*c^7 - 3*a^4*c^8 + 5*a^3*b*c^8 + 7*a^2*b^2*c^8 + 7*a*b^3*c^8 - 16*b^4*c^8 + 3*a^3*c^9 - 5*a^2*b*c^9 - 7*a*b^2*c^9 + 6*b^3*c^9 + a^2*c^10 + a*b*c^10 + 4*b^2*c^10 - a*c^11 - 2*b*c^11) : :

X(38592) lies on these lines: {3, 108}, {5, 34550}, {30, 34188}, {123, 381}, {382, 10746}, {399, 2778}, {999, 3318}, {1359, 3295}, {1656, 25640}, {1657, 2829}, {2791, 13188}, {2798, 12188}, {2804, 12773}, {2817, 12702}, {2850, 10620}, {3830, 10715}, {5054, 6717}, {5073, 10731}, {8148, 10702}, {9528, 13743}, {9531, 11258}, {9566, 34456}, {33903, 35457}

X(38592) = reflection of X(38578) in X(3)

X(38593) = DILATION FROM X(3) OF X(1296) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a^2*(a^8 - 6*a^6*b^2 - 2*a^4*b^4 + 6*a^2*b^6 + b^8 - 6*a^6*c^2 + 49*a^4*b^2*c^2 - 35*a^2*b^4*c^2 - 12*b^6*c^2 - 2*a^4*c^4 - 35*a^2*b^2*c^4 + 46*b^4*c^4 + 6*a^2*c^6 - 12*b^2*c^6 + c^8) : :

X(38593) lies on these lines: {3, 111}, {30, 5971}, {126, 381}, {376, 20099}, {382, 10748}, {399, 2780}, {511, 37751}, {543, 3534}, {550, 14654}, {999, 6019}, {1656, 5512}, {1657, 23699}, {2793, 13188}, {2805, 12773}, {2830, 12331}, {2854, 10620}, {2937, 14657}, {3048, 9703}, {3295, 3325}, {3830, 10717}, {5050, 14688}, {5054, 6719}, {5073, 10734}, {6221, 11835}, {6398, 11836}, {7711, 36182}, {8148, 10704}, {9129, 15040}, {9146, 18346}, {9172, 15693}, {9690, 11833}, {12017, 28662}, {12149, 15066}, {14666, 15688}, {15681, 32424}, {18440, 36883}, {35001, 35002}

X(38593) = reflection of X(11258) in X(3)

X(38594) = DILATION FROM X(3) OF X(1298) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a^2*(2*a^18*b^6 - 14*a^16*b^8 + 42*a^14*b^10 - 70*a^12*b^12 + 70*a^10*b^14 - 42*a^8*b^16 + 14*a^6*b^18 - 2*a^4*b^20 + a^20*b^2*c^2 - 5*a^18*b^4*c^2 + 7*a^16*b^6*c^2 + a^14*b^8*c^2 - 4*a^12*b^10*c^2 - 16*a^10*b^12*c^2 + 35*a^8*b^14*c^2 - 27*a^6*b^16*c^2 + 9*a^4*b^18*c^2 - a^2*b^20*c^2 - 5*a^18*b^2*c^4 + 20*a^16*b^4*c^4 - 33*a^14*b^6*c^4 + 34*a^12*b^8*c^4 - 28*a^10*b^10*c^4 + 12*a^8*b^12*c^4 + 7*a^6*b^14*c^4 - 10*a^4*b^16*c^4 + 3*a^2*b^18*c^4 + 2*a^18*c^6 + 7*a^16*b^2*c^6 - 33*a^14*b^4*c^6 + 41*a^12*b^6*c^6 - 17*a^10*b^8*c^6 - 7*a^8*b^10*c^6 + 7*a^6*b^12*c^6 + a^4*b^14*c^6 + a^2*b^16*c^6 - 2*b^18*c^6 - 14*a^16*c^8 + a^14*b^2*c^8 + 34*a^12*b^4*c^8 - 17*a^10*b^6*c^8 + 4*a^8*b^8*c^8 - a^6*b^10*c^8 - 8*a^4*b^12*c^8 - 11*a^2*b^14*c^8 + 12*b^16*c^8 + 42*a^14*c^10 - 4*a^12*b^2*c^10 - 28*a^10*b^4*c^10 - 7*a^8*b^6*c^10 - a^6*b^8*c^10 + 20*a^4*b^10*c^10 + 8*a^2*b^12*c^10 - 30*b^14*c^10 - 70*a^12*c^12 - 16*a^10*b^2*c^12 + 12*a^8*b^4*c^12 + 7*a^6*b^6*c^12 - 8*a^4*b^8*c^12 + 8*a^2*b^10*c^12 + 40*b^12*c^12 + 70*a^10*c^14 + 35*a^8*b^2*c^14 + 7*a^6*b^4*c^14 + a^4*b^6*c^14 - 11*a^2*b^8*c^14 - 30*b^10*c^14 - 42*a^8*c^16 - 27*a^6*b^2*c^16 - 10*a^4*b^4*c^16 + a^2*b^6*c^16 + 12*b^8*c^16 + 14*a^6*c^18 + 9*a^4*b^2*c^18 + 3*a^2*b^4*c^18 - 2*b^6*c^18 - 2*a^4*c^20 - a^2*b^2*c^20) : :

X(38594) lies on these lines: {3, 1298}, {129, 1656}, {130, 381}, {195, 13310}, {399, 32438}, {568, 21661}, {3526, 34838}, {5054, 34839}, {7517, 22551}, {13188, 18436}, {13512, 20477}, {18445, 22552}

X(38595) = DILATION FROM X(3) OF X(1304) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a^2*(a^20 - 5*a^18*b^2 + 7*a^16*b^4 + 6*a^14*b^6 - 28*a^12*b^8 + 28*a^10*b^10 - 22*a^6*b^14 + 19*a^4*b^16 - 7*a^2*b^18 + b^20 - 5*a^18*c^2 + 16*a^16*b^2*c^2 - 26*a^14*b^4*c^2 + 42*a^12*b^6*c^2 - 33*a^10*b^8*c^2 - 45*a^8*b^10*c^2 + 104*a^6*b^12*c^2 - 64*a^4*b^14*c^2 + 8*a^2*b^16*c^2 + 3*b^18*c^2 + 7*a^16*c^4 - 26*a^14*b^2*c^4 - 3*a^12*b^4*c^4 + 2*a^10*b^6*c^4 + 164*a^8*b^8*c^4 - 210*a^6*b^10*c^4 + 21*a^4*b^12*c^4 + 66*a^2*b^14*c^4 - 21*b^16*c^4 + 6*a^14*c^6 + 42*a^12*b^2*c^6 + 2*a^10*b^4*c^6 - 238*a^8*b^6*c^6 + 128*a^6*b^8*c^6 + 220*a^4*b^10*c^6 - 176*a^2*b^12*c^6 + 16*b^14*c^6 - 28*a^12*c^8 - 33*a^10*b^2*c^8 + 164*a^8*b^4*c^8 + 128*a^6*b^6*c^8 - 392*a^4*b^8*c^8 + 109*a^2*b^10*c^8 + 52*b^12*c^8 + 28*a^10*c^10 - 45*a^8*b^2*c^10 - 210*a^6*b^4*c^10 + 220*a^4*b^6*c^10 + 109*a^2*b^8*c^10 - 102*b^10*c^10 + 104*a^6*b^2*c^12 + 21*a^4*b^4*c^12 - 176*a^2*b^6*c^12 + 52*b^8*c^12 - 22*a^6*c^14 - 64*a^4*b^2*c^14 + 66*a^2*b^4*c^14 + 16*b^6*c^14 + 19*a^4*c^16 + 8*a^2*b^2*c^16 - 21*b^4*c^16 - 7*a^2*c^18 + 3*b^2*c^18 + c^20) : :

X(38595) lies on these lines: {3, 1304}, {30, 34186}, {64, 13997}, {381, 18809}, {382, 20957}, {399, 520}, {1656, 16177}, {2070, 6000}, {6760, 18859}, {6761, 11563}, {11799, 12188}, {13115, 37924}, {34147, 35001}

X(38596) = DILATION FROM X(3) OF X(1379) TO THE CIRCLE Γ(X(3),X(399))

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

X(38596) lies on these lines: {3, 6}, {30, 6190}, {381, 2040}, {385, 6040}, {1656, 2039}, {1993, 21032}, {3413, 13188}, {3414, 12188}, {6033, 14501}, {14502, 15561}, {19659, 21850}

X(38596) = reflection of X(38597) in X(3)
X(38596) = {X(6),X(9301)}-harmonic conjugate of X(38597)

X(38597) = DILATION FROM X(3) OF X(1380) TO THE CIRCLE Γ(X(3),X(399))

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

X(38597) lies on these lines: {3, 6}, {30, 6189}, {381, 2039}, {385, 6039}, {1656, 2040}, {1993, 21036}, {3413, 12188}, {3414, 13188}, {6033, 14502}, {14501, 15561}, {19660, 21850}

X(38597) = reflection of X(38596) in X(3)
X(38597) = {X(6),X(9301)}-harmonic conjugate of X(38596)

X(38598) = DILATION FROM X(3) OF X(2696) TO THE CIRCLE Γ(X(3),X(399))

Barycentrics    a^16 - 7*a^14*b^2 + 5*a^12*b^4 + 13*a^10*b^6 - 13*a^8*b^8 - 5*a^6*b^10 + 7*a^4*b^12 - a^2*b^14 - 7*a^14*c^2 + 60*a^12*b^2*c^2 - 93*a^10*b^4*c^2 - 44*a^8*b^6*c^2 + 111*a^6*b^8*c^2 - 18*a^4*b^10*c^2 - 11*a^2*b^12*c^2 + 2*b^14*c^2 + 5*a^12*c^4 - 93*a^10*b^2*c^4 + 299*a^8*b^4*c^4 - 159*a^6*b^6*c^4 - 99*a^4*b^8*c^4 + 47*a^2*b^10*c^4 - 4*b^12*c^4 + 13*a^10*c^6 - 44*a^8*b^2*c^6 - 159*a^6*b^4*c^6 + 244*a^4*b^6*c^6 - 35*a^2*b^8*c^6 - 2*b^10*c^6 - 13*a^8*c^8 + 111*a^6*b^2*c^8 - 99*a^4*b^4*c^8 - 35*a^2*b^6*c^8 + 8*b^8*c^8 - 5*a^6*c^10 - 18*a^4*b^2*c^10 + 47*a^2*b^4*c^10 - 2*b^6*c^10 + 7*a^4*c^12 - 11*a^2*b^2*c^12 - 4*b^4*c^12 - a^2*c^14 + 2*b^2*c^14 : :

X(38598) lies on these lines: {3, 2696}, {30, 11258}, {381, 31655}, {399, 1499}, {524, 10620}, {691, 14614}, {3830, 34320}, {5913, 18325}, {5971, 37950}, {6090, 18346}, {12188, 35001}, {33998, 37924}

X(38599) = DILATION FROM X(3) OF X(101) TO THE CIRCLE Γ(X(3),X(1511))

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

X(38599) lies on these lines: lies on these lines: {2, 10739}, {3, 101}, {5, 6710}, {20, 10741}, {24, 5185}, {30, 118}, {35, 3022}, {36, 1362}, {41, 14520}, {116, 140}, {150, 631}, {152, 376}, {182, 2810}, {381, 10725}, {515, 28346}, {517, 5144}, {544, 549}, {550, 35024}, {971, 28345}, {993, 3041}, {1282, 3576}, {1385, 2809}, {1511, 2774}, {1657, 10727}, {2646, 18413}, {2772, 12041}, {2784, 6684}, {2786, 33813}, {2801, 15481}, {2813, 14650}, {3523, 20096}, {3526, 31273}, {3534, 10710}, {3627, 20401}, {3887, 33814}, {5050, 10756}, {5054, 10708}, {5122, 15730}, {10246, 10695}, {10697, 12702}, {10758, 33878}, {11028, 24929}, {11714, 13624}, {11726, 38028}, {11728, 22791}, {23585, 32656}, {26348, 34112}, {33521, 33923}, {34457, 35203}

X(38599) = reflection of X(38601) in X(3)
X(38599) = complement of X(10739)

X(38600) = DILATION FROM X(3) OF X(102) TO THE CIRCLE Γ(X(3),X(1511))

Barycentrics    a^2*(2*a^8 - 2*a^7*b - 3*a^6*b^2 + 6*a^5*b^3 - 3*a^4*b^4 - 6*a^3*b^5 + 7*a^2*b^6 + 2*a*b^7 - 3*b^8 - 2*a^7*c + 6*a^6*b*c - 4*a^5*b^2*c - 8*a^4*b^3*c + 14*a^3*b^4*c - 2*a^2*b^5*c - 8*a*b^6*c + 4*b^7*c - 3*a^6*c^2 - 4*a^5*b*c^2 + 20*a^4*b^2*c^2 - 8*a^3*b^3*c^2 - 15*a^2*b^4*c^2 + 12*a*b^5*c^2 - 2*b^6*c^2 + 6*a^5*c^3 - 8*a^4*b*c^3 - 8*a^3*b^2*c^3 + 20*a^2*b^3*c^3 - 6*a*b^4*c^3 - 4*b^5*c^3 - 3*a^4*c^4 + 14*a^3*b*c^4 - 15*a^2*b^2*c^4 - 6*a*b^3*c^4 + 10*b^4*c^4 - 6*a^3*c^5 - 2*a^2*b*c^5 + 12*a*b^2*c^5 - 4*b^3*c^5 + 7*a^2*c^6 - 8*a*b*c^6 - 2*b^2*c^6 + 2*a*c^7 + 4*b*c^7 - 3*c^8) : :

X(38600) lies on these lines: lies on these lines: {2, 10740}, {3, 102}, {5, 6711}, {20, 10747}, {30, 124}, {35, 1361}, {36, 1364}, {56, 11438}, {117, 140}, {151, 631}, {185, 3417}, {376, 33650}, {381, 10726}, {517, 11713}, {549, 6718}, {993, 3042}, {1125, 2816}, {1385, 2817}, {1511, 2779}, {1657, 10732}, {1795, 5204}, {1845, 2646}, {2773, 12041}, {2785, 12042}, {2792, 33813}, {2800, 3579}, {2819, 14650}, {3040, 25440}, {3534, 10716}, {5050, 10757}, {5054, 10709}, {10246, 10696}, {10703, 12702}, {10764, 33878}, {11700, 13624}, {11727, 38028}, {11734, 22791}, {12016, 37582}, {13532, 18481}, {14127, 38389}, {14690, 31663}, {34459, 35203}

X(38600) = reflection of X(38607) in X(3)
X(38600) = complement of X(10740)

X(38601) = DILATION FROM X(3) OF X(103) TO THE CIRCLE Γ(X(3),X(1511))

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

X(38601) lies on these lines: {2, 10741}, {3, 101}, {5, 6712}, {20, 10739}, {30, 116}, {35, 1362}, {36, 3022}, {118, 140}, {150, 376}, {152, 631}, {378, 5185}, {381, 10727}, {517, 11714}, {544, 8703}, {549, 6710}, {1155, 18413}, {1282, 35242}, {1511, 2772}, {1657, 10725}, {2280, 14520}, {2774, 12041}, {2784, 33813}, {2786, 12042}, {2801, 33814}, {2807, 32613}, {2809, 3579}, {2810, 3098}, {2824, 14650}, {3033, 35203}, {3041, 25440}, {3046, 22115}, {3534, 10708}, {5050, 10758}, {5054, 10710}, {10246, 10697}, {10304, 20096}, {10695, 12702}, {10756, 33878}, {11028, 37582}, {11712, 13624}, {11726, 22791}, {11728, 38028}, {14869, 20401}, {17504, 35024}, {22440, 37741}, {33520, 33923}, {34112, 35247}

X(38601) = reflection of X(38599) in X(3)
X(38601) = complement of X(10741)

X(38602) = DILATION FROM X(3) OF X(104) TO THE CIRCLE Γ(X(3),X(1511))

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

X(38602) lies on these lines: {1, 12515}, {2, 10742}, {3, 8}, {4, 38141}, {5, 2829}, {11, 30}, {12, 14800}, {20, 10738}, {21, 33860}, {24, 12138}, {35, 1317}, {40, 12653}, {55, 10074}, {56, 1387}, {79, 38063}, {80, 5442}, {119, 140}, {149, 376}, {153, 631}, {165, 6264}, {214, 960}, {355, 38177}, {378, 1862}, {381, 10728}, {404, 18357}, {496, 34880}, {498, 12763}, {499, 12764}, {515, 12619}, {517, 4973}, {528, 8703}, {546, 23513}, {547, 38069}, {548, 11012}, {549, 993}, {550, 1484}, {946, 38044}, {1001, 6914}, {1006, 13257}, {1012, 38034}, {1125, 12611}, {1319, 12758}, {1320, 12702}, {1385, 2800}, {1483, 25439}, {1537, 5901}, {1657, 10724}, {1768, 3576}, {2077, 5844}, {2080, 12199}, {2646, 11570}, {2783, 33813}, {2787, 12042}, {2801, 15481}, {2802, 3579}, {2830, 14650}, {3032, 35203}, {3036, 25440}, {3045, 22115}, {3098, 9024}, {3218, 35459}, {3311, 19081}, {3312, 19082}, {3522, 13199}, {3530, 37725}, {3534, 10707}, {3612, 12739}, {3651, 12690}, {3655, 5010}, {3656, 37587}, {4188, 18525}, {4297, 10265}, {4299, 13273}, {4302, 13274}, {4413, 22758}, {4999, 5499}, {5050, 10759}, {5054, 10711}, {5083, 24929}, {5119, 20586}, {5122, 6797}, {5126, 17010}, {5204, 10090}, {5217, 10087}, {5249, 33594}, {5253, 13743}, {5284, 28453}, {5433, 15446}, {5480, 38168}, {5533, 6284}, {5541, 35242}, {5805, 38173}, {5854, 8666}, {5882, 26086}, {5884, 26287}, {5886, 34789}, {6001, 18857}, {6147, 22766}, {6154, 7688}, {6174, 12100}, {6200, 35857}, {6221, 19113}, {6246, 28160}, {6326, 7987}, {6396, 35856}, {6398, 19112}, {6642, 9913}, {6702, 18480}, {6830, 38142}, {6905, 28186}, {6909, 22765}, {6924, 12114}, {6950, 10246}, {6952, 38135}, {6958, 37002}, {6959, 33898}, {7354, 8068}, {7583, 13913}, {7584, 13977}, {7993, 16192}, {8674, 12041}, {9821, 32454}, {9955, 32557}, {9956, 38133}, {9963, 37105}, {10165, 21635}, {10267, 12332}, {10281, 31750}, {10304, 20095}, {10572, 20118}, {10755, 33878}, {10767, 20127}, {10775, 23240}, {10778, 12121}, {10882, 35638}, {10956, 14803}, {10993, 33923}, {11194, 13205}, {11219, 12119}, {11571, 37525}, {12054, 13194}, {12499, 26316}, {12513, 25438}, {12665, 31835}, {12699, 16173}, {12736, 37582}, {12738, 15015}, {12740, 37618}, {12751, 26446}, {12752, 26451}, {12753, 26341}, {12754, 26348}, {12761, 26492}, {12762, 26487}, {12767, 30389}, {12775, 16203}, {12776, 16202}, {12832, 37730}, {13222, 35243}, {13228, 35244}, {13230, 35245}, {13235, 35248}, {13243, 37106}, {13268, 35241}, {13269, 35246}, {13270, 35247}, {13271, 35249}, {13272, 35250}, {13278, 35251}, {13279, 35252}, {13587, 18524}, {13922, 35255}, {13991, 35256}, {14869, 20400}, {14893, 38077}, {15178, 25485}, {15528, 24475}, {15558, 24928}, {15863, 28204}, {16174, 22793}, {16371, 18519}, {16617, 31936}, {17504, 35023}, {17638, 37605}, {17660, 37600}, {18483, 33709}, {18518, 19537}, {18583, 38119}, {18976, 21578}, {19925, 38182}, {21630, 31730}, {22560, 35239}, {25416, 35000}, {28444, 38022}, {30282, 37736}, {33668, 37737}, {33858, 37616}, {35460, 38460}

X(38602) = reflection of X(33814) in X(3)
X(38602) = complement of X(10742)

X(38603) = DILATION FROM X(3) OF X(105) TO THE CIRCLE Γ(X(3),X(1511))

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

X(38603) lies on these lines: {2, 10743}, {3, 105}, {5, 6714}, {20, 15521}, {30, 5511}, {35, 3021}, {36, 1358}, {120, 140}, {376, 34547}, {381, 10729}, {404, 34124}, {517, 11716}, {528, 549}, {631, 20344}, {993, 3039}, {1385, 2809}, {1511, 2836}, {2775, 12041}, {2788, 12042}, {2795, 5428}, {2834, 6644}, {2837, 14650}, {3523, 20097}, {3576, 5540}, {5050, 10760}, {5054, 10712}, {5840, 33970}, {10246, 10699}, {11730, 38028}

X(38603) = reflection of X(38619) in X(3)
X(38603) = complement of X(10743)

X(38604) = DILATION FROM X(3) OF X(106) TO THE CIRCLE Γ(X(3),X(1511))

Barycentrics    a^2*(2*a^5 - 4*a^4*b - 5*a^3*b^2 + 7*a^2*b^3 + 3*a*b^4 - 3*b^5 - 4*a^4*c + 18*a^3*b*c - 5*a^2*b^2*c - 18*a*b^3*c + 9*b^4*c - 5*a^3*c^2 - 5*a^2*b*c^2 + 20*a*b^2*c^2 - 4*b^3*c^2 + 7*a^2*c^3 - 18*a*b*c^3 - 4*b^2*c^3 + 3*a*c^4 + 9*b*c^4 - 3*c^5) : :

X(38604) lies on these lines: {2, 10744}, {3, 106}, {5, 6715}, {20, 15522}, {30, 5510}, {35, 6018}, {36, 1357}, {121, 140}, {182, 2810}, {376, 34548}, {381, 10730}, {517, 11717}, {631, 21290}, {993, 3038}, {1054, 3576}, {1385, 2802}, {1511, 2842}, {2776, 12041}, {2789, 12042}, {2796, 33813}, {2841, 32612}, {2843, 14650}, {3523, 20098}, {5050, 10761}, {5054, 10713}, {9519, 17502}, {10165, 11814}, {10246, 10700}, {11731, 38028}, {13352, 37999}

X(38604) = reflection of X(38620) in X(3)
X(38604) = complement of X(10744)

X(38605) = DILATION FROM X(3) OF X(107) TO THE CIRCLE Γ(X(3),X(1511))

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

X(38605) lies on these lines: {2, 5667}, {3, 107}, {4, 23240}, {5, 1539}, {20, 22337}, {30, 133}, {35, 7158}, {36, 3324}, {113, 402}, {122, 140}, {157, 1605}, {376, 34549}, {381, 10152}, {517, 11718}, {549, 6720}, {631, 34186}, {1125, 2816}, {1511, 9033}, {2797, 33813}, {2803, 33814}, {2847, 14650}, {3628, 36520}, {3843, 23241}, {5050, 10762}, {5054, 10714}, {5428, 9528}, {5961, 14254}, {6642, 14673}, {8703, 20207}, {10246, 10701}, {11732, 38028}, {14847, 15774}, {14920, 34601}, {17702, 24930}, {33892, 34286}

X(38605) = reflection of X(38621) in X(3)
X(38605) = complement of X(10745)

X(38606) = DILATION FROM X(3) OF X(108) TO THE CIRCLE Γ(X(3),X(1511))

Barycentrics    a*(2*a^12 - 2*a^11*b - 6*a^10*b^2 + 6*a^9*b^3 + 4*a^8*b^4 - 4*a^7*b^5 + 4*a^6*b^6 - 4*a^5*b^7 - 6*a^4*b^8 + 6*a^3*b^9 + 2*a^2*b^10 - 2*a*b^11 - 2*a^11*c + 8*a^10*b*c - 17*a^8*b^3*c + 10*a^7*b^4*c + 2*a^6*b^5*c - 10*a^5*b^6*c + 16*a^4*b^7*c - 10*a^2*b^9*c + 2*a*b^10*c + b^11*c - 6*a^10*c^2 + 20*a^8*b^2*c^2 - 6*a^7*b^3*c^2 - 22*a^6*b^4*c^2 + 18*a^5*b^5*c^2 + 6*a^4*b^6*c^2 - 18*a^3*b^7*c^2 + 4*a^2*b^8*c^2 + 6*a*b^9*c^2 - 2*b^10*c^2 + 6*a^9*c^3 - 17*a^8*b*c^3 - 6*a^7*b^2*c^3 + 32*a^6*b^3*c^3 - 4*a^5*b^4*c^3 - 24*a^4*b^5*c^3 + 10*a^3*b^6*c^3 + 12*a^2*b^7*c^3 - 6*a*b^8*c^3 - 3*b^9*c^3 + 4*a^8*c^4 + 10*a^7*b*c^4 - 22*a^6*b^2*c^4 - 4*a^5*b^3*c^4 + 16*a^4*b^4*c^4 + 2*a^3*b^5*c^4 - 6*a^2*b^6*c^4 - 8*a*b^7*c^4 + 8*b^8*c^4 - 4*a^7*c^5 + 2*a^6*b*c^5 + 18*a^5*b^2*c^5 - 24*a^4*b^3*c^5 + 2*a^3*b^4*c^5 - 4*a^2*b^5*c^5 + 8*a*b^6*c^5 + 2*b^7*c^5 + 4*a^6*c^6 - 10*a^5*b*c^6 + 6*a^4*b^2*c^6 + 10*a^3*b^3*c^6 - 6*a^2*b^4*c^6 + 8*a*b^5*c^6 - 12*b^6*c^6 - 4*a^5*c^7 + 16*a^4*b*c^7 - 18*a^3*b^2*c^7 + 12*a^2*b^3*c^7 - 8*a*b^4*c^7 + 2*b^5*c^7 - 6*a^4*c^8 + 4*a^2*b^2*c^8 - 6*a*b^3*c^8 + 8*b^4*c^8 + 6*a^3*c^9 - 10*a^2*b*c^9 + 6*a*b^2*c^9 - 3*b^3*c^9 + 2*a^2*c^10 + 2*a*b*c^10 - 2*b^2*c^10 - 2*a*c^11 + b*c^11) : :

X(38606) lies on these lines: {2, 10746}, {3, 108}, {5, 2829}, {20, 33566}, {30, 25640}, {35, 3318}, {36, 1359}, {123, 140}, {376, 34550}, {381, 10731}, {517, 11719}, {631, 34188}, {1385, 2817}, {1511, 2850}, {2778, 12041}, {2791, 12042}, {2798, 33813}, {2804, 33814}, {2834, 6644}, {2851, 14650}, {4242, 38554}, {5050, 10763}, {5054, 10715}, {10246, 10702}, {10271, 26285}, {11733, 38028}

X(38606) = reflection of X(38622) in X(3)
X(38606) = complement of X(10746)

X(38607) = DILATION FROM X(3) OF X(109) TO THE CIRCLE Γ(X(3),X(1511))

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

X(38607) lies on these lines: {2, 10747}, {3, 102}, {5, 6718}, {20, 10740}, {30, 117}, {35, 1364}, {36, 1361}, {55, 1795}, {124, 140}, {151, 376}, {381, 10732}, {389, 1399}, {517, 11700}, {549, 6711}, {578, 11509}, {631, 33650}, {692, 12332}, {993, 3040}, {1155, 1845}, {1385, 2800}, {1511, 2773}, {1657, 10726}, {2779, 12041}, {2785, 33813}, {2792, 12042}, {2807, 32613}, {2816, 12512}, {2817, 3579}, {2841, 32612}, {2852, 14650}, {3042, 25440}, {3534, 10709}, {3738, 33814}, {5050, 10764}, {5054, 10716}, {10246, 10703}, {10310, 37480}, {10696, 12702}, {10757, 33878}, {11713, 13624}, {11727, 22791}, {11734, 38028}, {12016, 24929}, {13532, 26446}, {34455, 35203}

X(38607) = reflection of X(38600) in X(3)
X(38607) = complement of X(10747)

X(38608) = DILATION FROM X(3) OF X(112) TO THE CIRCLE Γ(X(3),X(1511))

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

X(38608) lies on these lines: {2, 10749}, {3, 112}, {5, 2794}, {20, 12918}, {24, 13166}, {30, 132}, {35, 6020}, {36, 3320}, {52, 16225}, {55, 13312}, {56, 13311}, {127, 140}, {143, 16224}, {182, 2781}, {376, 12384}, {378, 12145}, {381, 10735}, {498, 13296}, {499, 13297}, {511, 28343}, {517, 11722}, {549, 34841}, {631, 13219}, {1511, 9517}, {2070, 20410}, {2080, 13195}, {2799, 33813}, {2806, 33814}, {2881, 25644}, {3311, 19114}, {3312, 19115}, {3522, 12253}, {3576, 13221}, {4299, 12945}, {4302, 12955}, {5020, 9157}, {5050, 10766}, {5054, 10718}, {5204, 13117}, {5217, 13116}, {5663, 17974}, {6200, 35881}, {6221, 19094}, {6396, 35880}, {6398, 19093}, {6642, 11641}, {6644, 14649}, {7514, 18876}, {7583, 13923}, {7584, 13992}, {8703, 9530}, {9818, 11637}, {10246, 10705}, {10267, 13206}, {10269, 19162}, {12054, 12207}, {12131, 36156}, {12265, 13624}, {12340, 35238}, {12408, 35242}, {12413, 35243}, {12478, 35244}, {12479, 35245}, {12503, 35248}, {12702, 13099}, {12784, 18481}, {12796, 35241}, {12805, 35246}, {12806, 35247}, {12925, 35249}, {12935, 35250}, {13118, 35251}, {13119, 35252}, {13236, 26316}, {13280, 26446}, {13281, 26451}, {13282, 26341}, {13283, 26348}, {13294, 26492}, {13295, 26487}, {13313, 16203}, {13314, 16202}, {13918, 35255}, {13985, 35256}, {19159, 35239}, {34217, 37814}

X(38608) = reflection of X(38624) in X(3)
X(38608) = complement of X(10749)

X(38609) = DILATION FROM X(3) OF X(476) TO THE CIRCLE Γ(X(3),X(1511))

Barycentrics    2*a^16 - 7*a^14*b^2 + 7*a^12*b^4 - 7*a^6*b^10 + 7*a^4*b^12 - 2*a^2*b^14 - 7*a^14*c^2 + 22*a^12*b^2*c^2 - 22*a^10*b^4*c^2 - 2*a^8*b^6*c^2 + 22*a^6*b^8*c^2 - 17*a^4*b^10*c^2 + 3*a^2*b^12*c^2 + b^14*c^2 + 7*a^12*c^4 - 22*a^10*b^2*c^4 + 30*a^8*b^4*c^4 - 18*a^6*b^6*c^4 + 6*a^4*b^8*c^4 + 3*a^2*b^10*c^4 - 6*b^12*c^4 - 2*a^8*b^2*c^6 - 18*a^6*b^4*c^6 + 8*a^4*b^6*c^6 - 4*a^2*b^8*c^6 + 15*b^10*c^6 + 22*a^6*b^2*c^8 + 6*a^4*b^4*c^8 - 4*a^2*b^6*c^8 - 20*b^8*c^8 - 7*a^6*c^10 - 17*a^4*b^2*c^10 + 3*a^2*b^4*c^10 + 15*b^6*c^10 + 7*a^4*c^12 + 3*a^2*b^2*c^12 - 6*b^4*c^12 - 2*a^2*c^14 + b^2*c^14 : :

X(38609) lies on these lines: {2, 20957}, {3, 476}, {5, 22104}, {30, 125}, {35, 33965}, {36, 33964}, {74, 36193}, {140, 3258}, {143, 16978}, {186, 2970}, {376, 34193}, {523, 1511}, {549, 31379}, {550, 18319}, {631, 14731}, {1539, 36169}, {1657, 14989}, {2453, 6644}, {3111, 15536}, {3154, 34128}, {3233, 5609}, {3581, 36188}, {5054, 34312}, {5627, 12902}, {5642, 33505}, {5663, 7471}, {6070, 32423}, {6699, 16340}, {6795, 34513}, {7575, 12042}, {7687, 21315}, {9179, 33962}, {11749, 15712}, {12006, 36161}, {12052, 15026}, {12121, 14993}, {13630, 36159}, {14480, 32609}, {14508, 15041}, {14650, 18579}, {14809, 16171}, {15061, 17511}, {17702, 34209}, {20127, 36172}, {20304, 36184}

X(38609) = reflection of X(38610) in X(3)
X(38609) = complement of X(20957)

X(38610) = DILATION FROM X(3) OF X(477) TO THE CIRCLE Γ(X(3),X(1511))

Barycentrics    (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(a^12 - 2*a^10*b^2 - 2*a^8*b^4 + 8*a^6*b^6 - 7*a^4*b^8 + 2*a^2*b^10 - 2*a^10*c^2 + 10*a^8*b^2*c^2 - 10*a^6*b^4*c^2 - a^4*b^6*c^2 + 2*a^2*b^8*c^2 + b^10*c^2 - 2*a^8*c^4 - 10*a^6*b^2*c^4 + 17*a^4*b^4*c^4 - 4*a^2*b^6*c^4 - 4*b^8*c^4 + 8*a^6*c^6 - a^4*b^2*c^6 - 4*a^2*b^4*c^6 + 6*b^6*c^6 - 7*a^4*c^8 + 2*a^2*b^2*c^8 - 4*b^4*c^8 + 2*a^2*c^10 + b^2*c^10) : :

X(38610) lies on these lines: {3, 476}, {5, 31379}, {20, 20957}, {30, 113}, {35, 33964}, {36, 33965}, {140, 25641}, {186, 34334}, {376, 14731}, {381, 14989}, {399, 14508}, {523, 12041}, {549, 18319}, {631, 34193}, {3154, 10113}, {3534, 34312}, {5663, 14611}, {6699, 34209}, {6723, 21315}, {7687, 21269}, {7740, 18577}, {8703, 11749}, {10620, 14480}, {12028, 34178}, {12121, 14851}, {13391, 16978}, {14643, 36172}, {14677, 32417}, {15035, 36193}, {15063, 33505}, {16340, 17702}, {20304, 34150}, {21316, 23515}, {33813, 37950}

X(38610) = reflection of X(38609) in X(3)

X(38611) = DILATION FROM X(3) OF X(691) TO THE CIRCLE Γ(X(3),X(1511))

Barycentrics    a^2*(2*a^12 - 7*a^10*b^2 + 7*a^8*b^4 + 2*a^6*b^6 - 8*a^4*b^8 + 5*a^2*b^10 - b^12 - 7*a^10*c^2 + 22*a^8*b^2*c^2 - 24*a^6*b^4*c^2 + 18*a^4*b^6*c^2 - 8*a^2*b^8*c^2 - b^10*c^2 + 7*a^8*c^4 - 24*a^6*b^2*c^4 + 6*a^4*b^4*c^4 + 10*b^8*c^4 + 2*a^6*c^6 + 18*a^4*b^2*c^6 - 16*b^6*c^6 - 8*a^4*c^8 - 8*a^2*b^2*c^8 + 10*b^4*c^8 + 5*a^2*c^10 - b^2*c^10 - c^12) : :

X(38611) lies on these lines: {3, 691}, {23, 38225}, {30, 115}, {35, 6027}, {36, 6023}, {74, 9218}, {140, 5099}, {186, 2971}, {246, 323}, {249, 399}, {511, 11806}, {512, 1511}, {523, 33813}, {549, 16760}, {2071, 35002}, {2080, 7464}, {2696, 5970}, {2782, 7472}, {4235, 38552}, {5191, 7468}, {5663, 9181}, {7575, 14650}, {8703, 11749}, {9139, 22143}, {9301, 18859}, {11171, 15918}, {11799, 14693}, {14120, 34127}, {18325, 38227}, {22505, 36170}, {26316, 32531}, {36174, 38224}

X(38611) = reflection of X(38613) in X(3)
X(38611) = Schoute-circle-inverse of X(115)

X(38612) = DILATION FROM X(3) OF X(759) TO THE CIRCLE Γ(X(3),X(1511))

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

X(38612) lies on these lines: {3, 759}, {21, 125}, {35, 34194}, {36, 1365}, {140, 31845}, {1283, 2077}, {1385, 1511}, {3576, 21381}, {5428, 6684}, {12119, 34311}, {14664, 31663}, {19642, 37106}, {34196, 35242}

X(38613) = DILATION FROM X(3) OF X(842) TO THE CIRCLE Γ(X(3),X(1511))

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

X(38613) lies on these lines: {3, 691}, {5, 16760}, {23, 7711}, {30, 114}, {35, 6023}, {36, 6027}, {140, 16188}, {186, 2080}, {187, 16308}, {249, 15040}, {511, 1511}, {512, 12041}, {523, 12042}, {549, 31379}, {625, 18572}, {1350, 14729}, {2782, 36166}, {6644, 14687}, {9218, 15036}, {9301, 37958}, {11454, 18321}, {12308, 33803}, {14120, 22515}, {14881, 36156}, {15561, 36173}, {16324, 18579}, {37952, 38225}

X(38613) = reflection of X(38611) in X(3)
X(38613) = complement of X(38953)

X(38614) = DILATION FROM X(3) OF X(901) TO THE CIRCLE Γ(X(3),X(1511))

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

X(38614) lies on these lines: {3, 901}, {5, 22102}, {30, 31841}, {35, 3025}, {36, 13756}, {140, 3259}, {513, 33814}, {517, 4973}, {2077, 12041}, {7280, 23153}, {15626, 35000}, {24201, 37582}, {24929, 33645}, {34464, 35242}

X(38614) = reflection of X(38617) in X(3)
X(38614) = complement of X(40100)

X(38615) = DILATION FROM X(3) OF X(930) TO THE CIRCLE Γ(X(3),X(1511))

Barycentrics    2*a^16 - 11*a^14*b^2 + 27*a^12*b^4 - 40*a^10*b^6 + 40*a^8*b^8 - 27*a^6*b^10 + 11*a^4*b^12 - 2*a^2*b^14 - 11*a^14*c^2 + 38*a^12*b^2*c^2 - 50*a^10*b^4*c^2 + 26*a^8*b^6*c^2 + 6*a^6*b^8*c^2 - 13*a^4*b^10*c^2 + 3*a^2*b^12*c^2 + b^14*c^2 + 27*a^12*c^4 - 50*a^10*b^2*c^4 + 30*a^8*b^4*c^4 - 6*a^6*b^6*c^4 + 2*a^4*b^8*c^4 + 3*a^2*b^10*c^4 - 6*b^12*c^4 - 40*a^10*c^6 + 26*a^8*b^2*c^6 - 6*a^6*b^4*c^6 - 4*a^2*b^8*c^6 + 15*b^10*c^6 + 40*a^8*c^8 + 6*a^6*b^2*c^8 + 2*a^4*b^4*c^8 - 4*a^2*b^6*c^8 - 20*b^8*c^8 - 27*a^6*c^10 - 13*a^4*b^2*c^10 + 3*a^2*b^4*c^10 + 15*b^6*c^10 + 11*a^4*c^12 + 3*a^2*b^2*c^12 - 6*b^4*c^12 - 2*a^2*c^14 + b^2*c^14 : :

X(38615) lies on these lines: {3, 252}, {5, 13372}, {20, 31656}, {30, 128}, {35, 3327}, {36, 7159}, {137, 140}, {548, 12041}, {549, 1263}, {550, 14072}, {631, 11671}, {632, 25147}, {1154, 14071}, {2979, 13505}, {3530, 12026}, {3627, 23237}, {3628, 23516}, {6150, 24147}, {6636, 34418}, {7502, 23320}, {8703, 14073}, {10285, 31376}, {15960, 35243}

X(38615) = reflection of X(38618) in X(3)

X(38616) = DILATION FROM X(3) OF X(933) TO THE CIRCLE Γ(X(3),X(1511))

Barycentrics    a^2*(2*a^20 - 11*a^18*b^2 + 23*a^16*b^4 - 20*a^14*b^6 + 14*a^10*b^10 - 14*a^8*b^12 + 12*a^6*b^14 - 10*a^4*b^16 + 5*a^2*b^18 - b^20 - 11*a^18*c^2 + 46*a^16*b^2*c^2 - 70*a^14*b^4*c^2 + 42*a^12*b^6*c^2 - 3*a^10*b^8*c^2 + a^8*b^10*c^2 - 16*a^6*b^12*c^2 + 20*a^4*b^14*c^2 - 12*a^2*b^16*c^2 + 3*b^18*c^2 + 23*a^16*c^4 - 70*a^14*b^2*c^4 + 78*a^12*b^4*c^4 - 38*a^10*b^6*c^4 - a^8*b^8*c^4 + 18*a^6*b^10*c^4 - 12*a^4*b^12*c^4 + 2*a^2*b^14*c^4 - 20*a^14*c^6 + 42*a^12*b^2*c^6 - 38*a^10*b^4*c^6 + 28*a^8*b^6*c^6 - 14*a^6*b^8*c^6 - 4*a^4*b^10*c^6 + 16*a^2*b^12*c^6 - 10*b^14*c^6 - 3*a^10*b^2*c^8 - a^8*b^4*c^8 - 14*a^6*b^6*c^8 + 12*a^4*b^8*c^8 - 11*a^2*b^10*c^8 + 17*b^12*c^8 + 14*a^10*c^10 + a^8*b^2*c^10 + 18*a^6*b^4*c^10 - 4*a^4*b^6*c^10 - 11*a^2*b^8*c^10 - 18*b^10*c^10 - 14*a^8*c^12 - 16*a^6*b^2*c^12 - 12*a^4*b^4*c^12 + 16*a^2*b^6*c^12 + 17*b^8*c^12 + 12*a^6*c^14 + 20*a^4*b^2*c^14 + 2*a^2*b^4*c^14 - 10*b^6*c^14 - 10*a^4*c^16 - 12*a^2*b^2*c^16 + 5*a^2*c^18 + 3*b^2*c^18 - c^20) : :

X(38616) lies on these lines: {3, 933}, {5, 11701}, {30, 18402}, {140, 20625}, {186, 24977}, {548, 3184}, {2071, 11587}, {6150, 15646}, {7575, 10214}, {8157, 18570}, {10610, 10628}

X(38617) = DILATION FROM X(3) OF X(953) TO THE CIRCLE Γ(X(3),X(1511))

Barycentrics    a^2*(a^2 - b^2 + b*c - c^2)*(2*a^6 - 4*a^5*b - a^4*b^2 + 8*a^3*b^3 - 4*a^2*b^4 - 4*a*b^5 + 3*b^6 - 4*a^5*c + 14*a^4*b*c - 12*a^3*b^2*c - 9*a^2*b^3*c + 16*a*b^4*c - 5*b^5*c - a^4*c^2 - 12*a^3*b*c^2 + 28*a^2*b^2*c^2 - 12*a*b^3*c^2 - 3*b^4*c^2 + 8*a^3*c^3 - 9*a^2*b*c^3 - 12*a*b^2*c^3 + 10*b^3*c^3 - 4*a^2*c^4 + 16*a*b*c^4 - 3*b^2*c^4 - 4*a*c^5 - 5*b*c^5 + 3*c^6) : :

X(38617) lies on these lines: {3, 901}, {30, 3259}, {35, 13756}, {36, 1464}, {101, 35128}, {140, 31841}, {214, 517}, {549, 22102}, {3576, 34464}, {5010, 23153}, {6644, 10016}, {12006, 37535}, {12331, 14511}, {12773, 14513}, {23152, 37600}, {24201, 24929}, {33645, 37582}

X(38617) = reflection of X(38614) in X(3)
X(38617) = complement of X(38954)

X(38618) = DILATION FROM X(3) OF X(1141) TO THE CIRCLE Γ(X(3),X(1511))

Barycentrics    2*a^16 - 9*a^14*b^2 + 17*a^12*b^4 - 20*a^10*b^6 + 20*a^8*b^8 - 17*a^6*b^10 + 9*a^4*b^12 - 2*a^2*b^14 - 9*a^14*c^2 + 26*a^12*b^2*c^2 - 26*a^10*b^4*c^2 + 6*a^8*b^6*c^2 + 14*a^6*b^8*c^2 - 19*a^4*b^10*c^2 + 9*a^2*b^12*c^2 - b^14*c^2 + 17*a^12*c^4 - 26*a^10*b^2*c^4 + 14*a^8*b^4*c^4 - 6*a^6*b^6*c^4 + 10*a^4*b^8*c^4 - 15*a^2*b^10*c^4 + 6*b^12*c^4 - 20*a^10*c^6 + 6*a^8*b^2*c^6 - 6*a^6*b^4*c^6 + 8*a^2*b^8*c^6 - 15*b^10*c^6 + 20*a^8*c^8 + 14*a^6*b^2*c^8 + 10*a^4*b^4*c^8 + 8*a^2*b^6*c^8 + 20*b^8*c^8 - 17*a^6*c^10 - 19*a^4*b^2*c^10 - 15*a^2*b^4*c^10 - 15*b^6*c^10 + 9*a^4*c^12 + 9*a^2*b^2*c^12 + 6*b^4*c^12 - 2*a^2*c^14 - b^2*c^14 : :

X(38618) lies on these lines: {2, 31656}, {3, 252}, {5, 11701}, {30, 137}, {35, 7159}, {36, 3327}, {54, 13856}, {125, 128}, {186, 2970}, {376, 11671}, {546, 23516}, {549, 13372}, {550, 1263}, {632, 23237}, {1154, 24147}, {1594, 15367}, {3530, 6592}, {3627, 25147}, {5066, 25339}, {5890, 13504}, {6642, 15960}, {6644, 15959}, {6689, 32904}, {7604, 34599}, {10574, 13505}, {10615, 18400}, {12902, 34308}, {14073, 15712}, {14101, 15326}, {14769, 37347}, {18016, 18807}, {18284, 23320}, {22467, 34418}, {23280, 36837}

X(38618) = reflection of X(38615) in X(3)
X(38618) = complement of X(31656)

X(38619) = DILATION FROM X(3) OF X(1292) TO THE CIRCLE Γ(X(3),X(1511))

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

X(38619) lies on these lines: {2, 15521}, {3, 105}, {20, 10743}, {21, 34124}, {30, 120}, {35, 1358}, {36, 3021}, {140, 5511}, {376, 20344}, {528, 8703}, {549, 6714}, {631, 34547}, {1511, 2775}, {1657, 10729}, {2788, 33813}, {2795, 12042}, {2809, 3579}, {2826, 33814}, {2836, 12041}, {3034, 35203}, {3039, 25440}, {3534, 10712}, {5540, 35242}, {9519, 17502}, {9522, 14650}, {10304, 20097}, {10699, 12702}, {10760, 33878}, {11716, 13624}, {11730, 22791}

X(38619) = reflection of X(38603) in X(3)
X(38619) = complement of X(15521)

X(38620) = DILATION FROM X(3) OF X(1293) TO THE CIRCLE Γ(X(3),X(1511))

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

X(38620) lies on these lines: {2, 15522}, {3, 106}, {20, 10744}, {30, 121}, {35, 1357}, {36, 6018}, {40, 13541}, {140, 5510}, {376, 21290}, {549, 6715}, {631, 34548}, {901, 22082}, {1054, 35242}, {1511, 2776}, {1657, 10730}, {2789, 33813}, {2796, 12042}, {2802, 3579}, {2810, 3098}, {2827, 33814}, {2842, 12041}, {3030, 35203}, {3038, 25440}, {3534, 10713}, {9526, 14650}, {10304, 20098}, {10700, 12702}, {10761, 33878}, {11717, 13624}, {11731, 22791}, {11814, 31730}, {14664, 31663}, {34139, 35239}, {37470, 37999}

X(38620) = reflection of X(38604) in X(3)
X(38620) = complement of X(15522)

X(38621) = DILATION FROM X(3) OF X(1294) TO THE CIRCLE Γ(X(3),X(1511))

Barycentrics    2*a^16 - 3*a^14*b^2 - 13*a^12*b^4 + 40*a^10*b^6 - 40*a^8*b^8 + 13*a^6*b^10 + 3*a^4*b^12 - 2*a^2*b^14 - 3*a^14*c^2 + 32*a^12*b^2*c^2 - 41*a^10*b^4*c^2 - 33*a^8*b^6*c^2 + 71*a^6*b^8*c^2 - 22*a^4*b^10*c^2 - 3*a^2*b^12*c^2 - b^14*c^2 - 13*a^12*c^4 - 41*a^10*b^2*c^4 + 146*a^8*b^4*c^4 - 84*a^6*b^6*c^4 - 35*a^4*b^8*c^4 + 21*a^2*b^10*c^4 + 6*b^12*c^4 + 40*a^10*c^6 - 33*a^8*b^2*c^6 - 84*a^6*b^4*c^6 + 108*a^4*b^6*c^6 - 16*a^2*b^8*c^6 - 15*b^10*c^6 - 40*a^8*c^8 + 71*a^6*b^2*c^8 - 35*a^4*b^4*c^8 - 16*a^2*b^6*c^8 + 20*b^8*c^8 + 13*a^6*c^10 - 22*a^4*b^2*c^10 + 21*a^2*b^4*c^10 - 15*b^6*c^10 + 3*a^4*c^12 - 3*a^2*b^2*c^12 + 6*b^4*c^12 - 2*a^2*c^14 - b^2*c^14 : :

X(38621) lies on these lines: {2, 22337}, {3, 107}, {5, 34842}, {20, 10745}, {30, 122}, {35, 3324}, {36, 7158}, {133, 140}, {376, 23240}, {546, 36520}, {548, 3184}, {549, 6716}, {550, 1511}, {631, 34549}, {1657, 10152}, {2790, 33813}, {2797, 12042}, {2816, 12512}, {2828, 33814}, {3522, 5667}, {3534, 10714}, {8703, 9530}, {9033, 12041}, {9529, 14650}, {10701, 12702}, {10762, 33878}, {11718, 13624}, {11732, 22791}, {14673, 35243}

X(38621) = reflection of X(38605) in X(3)
X(38621) = complement of X(22337)

X(38622) = DILATION FROM X(3) OF X(1295) TO THE CIRCLE Γ(X(3),X(1511))

Barycentrics    a*(2*a^12 - 2*a^11*b - 6*a^10*b^2 + 6*a^9*b^3 + 4*a^8*b^4 - 4*a^7*b^5 + 4*a^6*b^6 - 4*a^5*b^7 - 6*a^4*b^8 + 6*a^3*b^9 + 2*a^2*b^10 - 2*a*b^11 - 2*a^11*c + 12*a^10*b*c - 4*a^9*b^2*c - 23*a^8*b^3*c + 22*a^7*b^4*c - 2*a^6*b^5*c - 22*a^5*b^6*c + 24*a^4*b^7*c + 4*a^3*b^8*c - 10*a^2*b^9*c + 2*a*b^10*c - b^11*c - 6*a^10*c^2 - 4*a^9*b*c^2 + 36*a^8*b^2*c^2 - 18*a^7*b^3*c^2 - 42*a^6*b^4*c^2 + 46*a^5*b^5*c^2 + 2*a^4*b^6*c^2 - 22*a^3*b^7*c^2 + 8*a^2*b^8*c^2 - 2*a*b^9*c^2 + 2*b^10*c^2 + 6*a^9*c^3 - 23*a^8*b*c^3 - 18*a^7*b^2*c^3 + 80*a^6*b^3*c^3 - 20*a^5*b^4*c^3 - 48*a^4*b^5*c^3 + 30*a^3*b^6*c^3 - 12*a^2*b^7*c^3 + 2*a*b^8*c^3 + 3*b^9*c^3 + 4*a^8*c^4 + 22*a^7*b*c^4 - 42*a^6*b^2*c^4 - 20*a^5*b^3*c^4 + 56*a^4*b^4*c^4 - 18*a^3*b^5*c^4 - 10*a^2*b^6*c^4 + 16*a*b^7*c^4 - 8*b^8*c^4 - 4*a^7*c^5 - 2*a^6*b*c^5 + 46*a^5*b^2*c^5 - 48*a^4*b^3*c^5 - 18*a^3*b^4*c^5 + 44*a^2*b^5*c^5 - 16*a*b^6*c^5 - 2*b^7*c^5 + 4*a^6*c^6 - 22*a^5*b*c^6 + 2*a^4*b^2*c^6 + 30*a^3*b^3*c^6 - 10*a^2*b^4*c^6 - 16*a*b^5*c^6 + 12*b^6*c^6 - 4*a^5*c^7 + 24*a^4*b*c^7 - 22*a^3*b^2*c^7 - 12*a^2*b^3*c^7 + 16*a*b^4*c^7 - 2*b^5*c^7 - 6*a^4*c^8 + 4*a^3*b*c^8 + 8*a^2*b^2*c^8 + 2*a*b^3*c^8 - 8*b^4*c^8 + 6*a^3*c^9 - 10*a^2*b*c^9 - 2*a*b^2*c^9 + 3*b^3*c^9 + 2*a^2*c^10 + 2*a*b*c^10 + 2*b^2*c^10 - 2*a*c^11 - b*c^11) : :

X(38622) lies on these lines: {2, 33566}, {3, 108}, {20, 10746}, {30, 123}, {35, 1359}, {36, 3318}, {140, 25640}, {376, 34188}, {549, 6717}, {550, 2829}, {631, 34550}, {1511, 2778}, {1657, 10731}, {2791, 33813}, {2798, 12042}, {2817, 3579}, {2850, 12041}, {3534, 10715}, {5428, 9528}, {9531, 14650}, {10702, 12702}, {10763, 33878}, {11719, 13624}, {11733, 22791}, {34456, 35203}

X(38622) = reflection of X(38606) in X(3)
X(38622) = complement of X(33566)

X(38623) = DILATION FROM X(3) OF X(1296) TO THE CIRCLE Γ(X(3),X(1511))

Barycentrics    a^2*(2*a^8 - 9*a^6*b^2 - a^4*b^4 + 9*a^2*b^6 - b^8 - 9*a^6*c^2 + 56*a^4*b^2*c^2 - 40*a^2*b^4*c^2 - 3*b^6*c^2 - a^4*c^4 - 40*a^2*b^2*c^4 + 32*b^4*c^4 + 9*a^2*c^6 - 3*b^2*c^6 - c^8) : :

X(38623) lies on these lines: {2, 22338}, {3, 111}, {6, 11835}, {20, 10748}, {30, 126}, {35, 3325}, {36, 6019}, {110, 35447}, {140, 5512}, {376, 6031}, {511, 14688}, {543, 8703}, {549, 6719}, {550, 23699}, {1511, 2780}, {1657, 10734}, {2793, 33813}, {2830, 33814}, {2854, 3098}, {3048, 22115}, {3522, 14654}, {3524, 37749}, {3534, 10717}, {5085, 37751}, {5092, 28662}, {6445, 11833}, {6446, 11834}, {9172, 12100}, {10304, 14666}, {10704, 12702}, {10765, 33878}, {11721, 13624}, {12017, 36696}, {12149, 15080}, {14691, 26316}, {32456, 34227}

X(38623) = reflection of X(14650) in X(3)
X(38623) = complement of X(22338)
X(38623) = {X(11835),X(11836)}-harmonic conjugate of X(6)

X(38624) = DILATION FROM X(3) OF X(1297) TO THE CIRCLE Γ(X(3),X(1511))

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

X(38624) lies on these lines: {2, 12253}, {3, 112}, {5, 19160}, {20, 10749}, {24, 12145}, {30, 127}, {35, 3320}, {36, 6020}, {55, 13117}, {56, 13116}, {132, 140}, {206, 1511}, {376, 13219}, {378, 13166}, {498, 12945}, {499, 12955}, {517, 12265}, {548, 14689}, {549, 6720}, {550, 2794}, {631, 12384}, {1350, 32661}, {1657, 10735}, {2080, 12207}, {2799, 12042}, {2831, 33814}, {2881, 8552}, {3311, 19093}, {3312, 19094}, {3522, 13200}, {3534, 10718}, {3576, 12408}, {4299, 13296}, {4302, 13297}, {5092, 28343}, {5204, 13312}, {5217, 13311}, {6200, 35829}, {6221, 19115}, {6396, 35828}, {6398, 19114}, {6642, 12413}, {7583, 13918}, {7584, 13985}, {9517, 12041}, {10246, 13099}, {10267, 12340}, {10269, 19159}, {10705, 12702}, {10766, 33878}, {11641, 35243}, {11722, 13624}, {12006, 16224}, {12054, 13195}, {12503, 26316}, {12784, 26446}, {12796, 26451}, {12805, 26341}, {12806, 26348}, {12925, 26492}, {12935, 26487}, {13118, 16203}, {13119, 16202}, {13206, 35238}, {13221, 35242}, {13229, 35244}, {13231, 35245}, {13236, 35248}, {13280, 18481}, {13281, 35241}, {13282, 35246}, {13283, 35247}, {13294, 35249}, {13295, 35250}, {13313, 35251}, {13314, 35252}, {13923, 35255}, {13992, 35256}, {14791, 30794}, {14900, 33923}, {18324, 18876}, {19162, 35239}

X(38624) = reflection of X(38608) in X(3)
X(38624) = complement of X(12918)

X(38625) = DILATION FROM X(3) OF X(1304) TO THE CIRCLE Γ(X(3),X(1511))

Barycentrics    a^2*(2*a^20 - 7*a^18*b^2 - a^16*b^4 + 36*a^14*b^6 - 56*a^12*b^8 + 14*a^10*b^10 + 42*a^8*b^12 - 44*a^6*b^14 + 14*a^4*b^16 + a^2*b^18 - b^20 - 7*a^18*c^2 + 38*a^16*b^2*c^2 - 58*a^14*b^4*c^2 - 30*a^12*b^6*c^2 + 177*a^10*b^8*c^2 - 183*a^8*b^10*c^2 + 52*a^6*b^12*c^2 + 28*a^4*b^14*c^2 - 20*a^2*b^16*c^2 + 3*b^18*c^2 - a^16*c^4 - 58*a^14*b^2*c^4 + 198*a^12*b^4*c^4 - 194*a^10*b^6*c^4 - 41*a^8*b^8*c^4 + 198*a^6*b^10*c^4 - 132*a^4*b^12*c^4 + 30*a^2*b^14*c^4 + 36*a^14*c^6 - 30*a^12*b^2*c^6 - 194*a^10*b^4*c^6 + 364*a^8*b^6*c^6 - 206*a^6*b^8*c^6 + 20*a^4*b^10*c^6 + 20*a^2*b^12*c^6 - 10*b^14*c^6 - 56*a^12*c^8 + 177*a^10*b^2*c^8 - 41*a^8*b^4*c^8 - 206*a^6*b^6*c^8 + 140*a^4*b^8*c^8 - 31*a^2*b^10*c^8 + 17*b^12*c^8 + 14*a^10*c^10 - 183*a^8*b^2*c^10 + 198*a^6*b^4*c^10 + 20*a^4*b^6*c^10 - 31*a^2*b^8*c^10 - 18*b^10*c^10 + 42*a^8*c^12 + 52*a^6*b^2*c^12 - 132*a^4*b^4*c^12 + 20*a^2*b^6*c^12 + 17*b^8*c^12 - 44*a^6*c^14 + 28*a^4*b^2*c^14 + 30*a^2*b^4*c^14 - 10*b^6*c^14 + 14*a^4*c^16 - 20*a^2*b^2*c^16 + a^2*c^18 + 3*b^2*c^18 - c^20) : :

X(38625) lies on these lines: {3, 1304}, {5, 31379}, {30, 122}, {140, 16177}, {186, 2972}, {402, 25641}, {520, 1511}, {6000, 12041}, {6644, 14687}, {6759, 13997}, {11589, 37968}, {14670, 37814}, {18571, 34147}

X(38626) = DILATION FROM X(3) OF X(74) TO THE CIRCLE Γ(X(3),X(5609))

Barycentrics    a^2*(4*a^8 + a^6*b^2 - 27*a^4*b^4 + 35*a^2*b^6 - 13*b^8 + a^6*c^2 + 38*a^4*b^2*c^2 - 30*a^2*b^4*c^2 - 9*b^6*c^2 - 27*a^4*c^4 - 30*a^2*b^2*c^4 + 44*b^4*c^4 + 35*a^2*c^6 - 9*b^2*c^6 - 13*c^8) : :

X(38626) lies on these lines: {3, 74}, {125, 3857}, {146, 15088}, {541, 546}, {542, 12103}, {632, 15063}, {1539, 15027}, {3091, 20126}, {3448, 11541}, {3627, 16003}, {3628, 20417}, {5076, 9140}, {5079, 10706}, {5655, 10303}, {6000, 12105}, {6488, 10819}, {6489, 10820}, {10990, 15704}, {12102, 36253}, {12108, 16534}, {12133, 26863}, {12811, 20304}, {12812, 20397}, {13148, 14865}, {22234, 32305}, {22334, 34802}

X(38626) = reflection of X(38632) in X(3)

-

X(38627) = DILATION FROM X(3) OF X(98) TO THE CIRCLE Γ(X(3),X(5609))

Barycentrics    4*a^8 + a^6*b^2 - a^4*b^4 - 4*a^2*b^6 + a^6*c^2 - 14*a^4*b^2*c^2 + 9*a^2*b^4*c^2 - 9*b^6*c^2 - a^4*c^4 + 9*a^2*b^2*c^4 + 18*b^4*c^4 - 4*a^2*c^6 - 9*b^2*c^6 : :

X(38627) lies on these lines: {3, 76}, {115, 3857}, {147, 15092}, {148, 11541}, {542, 546}, {543, 12103}, {632, 14981}, {671, 5076}, {2023, 33694}, {3091, 11632}, {3146, 12243}, {3529, 11177}, {3628, 11623}, {3861, 36523}, {5072, 22566}, {5079, 6054}, {5984, 22515}, {5985, 17543}, {6055, 14869}, {8724, 10303}, {10991, 15704}, {12131, 26863}, {12812, 20398}, {14651, 15022}, {14830, 17538}

X(38627) = reflection of X(38628) in X(3)

X(38628) = DILATION FROM X(3) OF X(99) TO THE CIRCLE Γ(X(3),X(5609))

Barycentrics    4*a^8 - 17*a^6*b^2 + 17*a^4*b^4 - 4*a^2*b^6 - 17*a^6*c^2 + 22*a^4*b^2*c^2 - 9*a^2*b^4*c^2 + 9*b^6*c^2 + 17*a^4*c^4 - 9*a^2*b^2*c^4 - 18*b^4*c^4 - 4*a^2*c^6 + 9*b^2*c^6 : :

X(38628) lies on these lines: {3, 76}, {114, 3857}, {147, 11541}, {542, 12103}, {543, 546}, {550, 15300}, {632, 9167}, {671, 5079}, {2482, 14869}, {3091, 8596}, {3529, 7946}, {3530, 36521}, {3627, 14981}, {3628, 5461}, {5076, 6054}, {5186, 26863}, {10303, 11632}, {10992, 15704}, {11623, 12108}, {12812, 20399}, {15022, 15092}, {20094, 22505}, {35018, 36523}

X(38628) = reflection of X(38627) in X(3)

X(38629) = DILATION FROM X(3) OF X(100) TO THE CIRCLE Γ(X(3),X(5609))

Barycentrics    a*(4*a^6 - 4*a^5*b - 8*a^4*b^2 + 8*a^3*b^3 + 4*a^2*b^4 - 4*a*b^5 - 4*a^5*c - 6*a^4*b*c + 18*a^3*b^2*c - 3*a^2*b^3*c - 14*a*b^4*c + 9*b^5*c - 8*a^4*c^2 + 18*a^3*b*c^2 - 28*a^2*b^2*c^2 + 18*a*b^3*c^2 + 8*a^3*c^3 - 3*a^2*b*c^3 + 18*a*b^2*c^3 - 18*b^3*c^3 + 4*a^2*c^4 - 14*a*b*c^4 - 4*a*c^5 + 9*b*c^5) : :

X(38629) lies on these lines: {3, 8}, {5, 34719}, {119, 3857}, {153, 11541}, {528, 546}, {632, 37726}, {1862, 26863}, {3627, 37725}, {3746, 12019}, {3918, 15178}, {5076, 10711}, {5079, 10707}, {5537, 28186}, {6174, 14869}, {6265, 16189}, {7991, 12738}, {10993, 15704}, {12108, 20418}, {12812, 20400}, {19647, 27756}, {20095, 22799}

X(38629) = reflection of X(38631) in X(3)

X(38630) = DILATION FROM X(3) OF X(101) TO THE CIRCLE Γ(X(3),X(5609))

Barycentrics    a^2*(4*a^6 - 4*a^5*b - 13*a^4*b^2 + 18*a^3*b^3 + 4*a^2*b^4 - 14*a*b^5 + 5*b^6 - 4*a^5*c + 4*a^4*b*c + 8*a^3*b^2*c - 8*a^2*b^3*c - 4*a*b^4*c + 4*b^5*c - 13*a^4*c^2 + 8*a^3*b*c^2 - 18*a^2*b^2*c^2 + 18*a*b^3*c^2 + 5*b^4*c^2 + 18*a^3*c^3 - 8*a^2*b*c^3 + 18*a*b^2*c^3 - 28*b^3*c^3 + 4*a^2*c^4 - 4*a*b*c^4 + 5*b^2*c^4 - 14*a*c^5 + 4*b*c^5 + 5*c^6) : :

X(38630) lies on these lines: {3, 101}, {118, 3857}, {152, 11541}, {544, 546}, {2801, 15178}, {3304, 34931}, {5076, 10710}, {5079, 10708}, {5185, 26863}, {12812, 20401}, {15704, 33520}

X(38631) = DILATION FROM X(3) OF X(104) TO THE CIRCLE Γ(X(3),X(5609))

Barycentrics    a*(4*a^6 - 4*a^5*b - 8*a^4*b^2 + 8*a^3*b^3 + 4*a^2*b^4 - 4*a*b^5 - 4*a^5*c + 30*a^4*b*c - 18*a^3*b^2*c - 21*a^2*b^3*c + 22*a*b^4*c - 9*b^5*c - 8*a^4*c^2 - 18*a^3*b*c^2 + 44*a^2*b^2*c^2 - 18*a*b^3*c^2 + 8*a^3*c^3 - 21*a^2*b*c^3 - 18*a*b^2*c^3 + 18*b^3*c^3 + 4*a^2*c^4 + 22*a*b*c^4 - 4*a*c^5 - 9*b*c^5) : :

X(38631) lies on these lines: {3, 8}, {11, 3857}, {149, 11541}, {528, 12103}, {632, 37725}, {2801, 15178}, {3627, 37726}, {3628, 20418}, {5076, 10707}, {5079, 10711}, {5563, 12019}, {12138, 26863}, {12737, 16189}, {12738, 30389}

X(38631) = reflection of X(38629) in X(3)

X(38632) = DILATION FROM X(3) OF X(110) TO THE CIRCLE Γ(X(3),X(5609))

Barycentrics    a^2*(4*a^8 - 17*a^6*b^2 + 27*a^4*b^4 - 19*a^2*b^6 + 5*b^8 - 17*a^6*c^2 + 2*a^4*b^2*c^2 + 6*a^2*b^4*c^2 + 9*b^6*c^2 + 27*a^4*c^4 + 6*a^2*b^2*c^4 - 28*b^4*c^4 - 19*a^2*c^6 + 9*b^2*c^6 + 5*c^8) : :

X(38632) lies on these lines: {3, 74}, {113, 3857}, {146, 11541}, {541, 12103}, {542, 546}, {632, 16003}, {1112, 26863}, {1154, 37967}, {1539, 14683}, {3090, 20396}, {3091, 5655}, {3146, 23236}, {3518, 13148}, {3529, 9143}, {3627, 15063}, {3628, 13393}, {5076, 10706}, {5079, 9140}, {5642, 14869}, {6053, 12811}, {6102, 14002}, {10272, 20397}, {10303, 20126}, {10540, 37953}, {12006, 16042}, {12102, 32423}, {12105, 13754}, {12108, 20417}, {12812, 20304}, {13364, 18451}, {13391, 37946}, {14984, 15083}, {15022, 15027}, {15704, 30714}, {19140, 22234}, {20125, 34128}

X(38632) = reflection of X(38626) in X(3)

X(38633) = DILATION FROM X(3) OF X(74) TO THE CIRCLE Γ(X(3),X(15041))

Barycentrics    a^2*(9*a^8 - 14*a^6*b^2 - 12*a^4*b^4 + 30*a^2*b^6 - 13*b^8 - 14*a^6*c^2 + 53*a^4*b^2*c^2 - 35*a^2*b^4*c^2 - 4*b^6*c^2 - 12*a^4*c^4 - 35*a^2*b^2*c^4 + 34*b^4*c^4 + 30*a^2*c^6 - 4*b^2*c^6 - 13*c^8) : :

X(38633) lies on these lines: {3, 74}, {125, 17800}, {146, 15720}, {541, 15707}, {1656, 14677}, {2777, 5055}, {3526, 12244}, {3532, 17855}, {3830, 15061}, {3843, 20127}, {3851, 6699}, {5070, 36518}, {5073, 16111}, {6455, 35827}, {6456, 35826}, {6496, 12375}, {6497, 12376}, {9691, 19111}, {10264, 15696}, {12316, 37497}, {12317, 33923}, {12902, 37853}, {14269, 34584}, {14643, 15701}, {14644, 15684}, {14915, 37922}, {15688, 32423}, {15689, 17702}, {15695, 20126}, {19709, 34128}, {21663, 35452}, {25335, 33751}

X(38633) = reflection of X(38638) in X(3)

X(38634) = DILATION FROM X(3) OF X(98) TO THE CIRCLE Γ(X(3),X(15041))

Barycentrics    9*a^8 - 14*a^6*b^2 + 14*a^4*b^4 - 9*a^2*b^6 - 14*a^6*c^2 + a^4*b^2*c^2 + 4*a^2*b^4*c^2 - 4*b^6*c^2 + 14*a^4*c^4 + 4*a^2*b^2*c^4 + 8*b^4*c^4 - 9*a^2*c^6 - 4*b^2*c^6 : :

X(38634) lies on these lines: {3, 76}, {115, 17800}, {147, 15720}, {382, 38229}, {542, 15707}, {2794, 5055}, {3526, 9862}, {3530, 5984}, {3534, 14651}, {3830, 38224}, {3843, 23514}, {3851, 6036}, {5013, 33694}, {5070, 36519}, {5985, 17573}, {6055, 15681}, {6455, 35825}, {6456, 35824}, {6496, 35878}, {6497, 35879}, {9691, 19109}, {11177, 15700}, {11632, 15695}, {14639, 15684}, {14830, 15694}, {15561, 15701}, {15689, 23698}, {19709, 34127}

X(38634) = reflection of X(38635) in X(3)

X(38635) = DILATION FROM X(3) OF X(99) TO THE CIRCLE Γ(X(3),X(15041))

Barycentrics    9*a^8 - 22*a^6*b^2 + 22*a^4*b^4 - 9*a^2*b^6 - 22*a^6*c^2 + 17*a^4*b^2*c^2 - 4*a^2*b^4*c^2 + 4*b^6*c^2 + 22*a^4*c^4 - 4*a^2*b^2*c^4 - 8*b^4*c^4 - 9*a^2*c^6 + 4*b^2*c^6 : :

X(38635) lies on these lines: {3, 76}, {114, 17800}, {148, 15720}, {543, 15707}, {549, 8596}, {550, 7947}, {620, 3851}, {2482, 15681}, {2794, 15689}, {3526, 13172}, {3530, 20094}, {3830, 15561}, {3843, 36519}, {5055, 9167}, {5070, 23514}, {5461, 15694}, {6455, 35879}, {6456, 35878}, {6496, 35824}, {6497, 35825}, {7373, 15452}, {8591, 15700}, {8724, 15695}, {9691, 19056}, {12117, 19709}, {12243, 15716}, {12355, 34127}, {14639, 15703}, {14651, 15693}, {15701, 38224}

X(38635) = reflection of X(38634) in X(3)

X(38636) = DILATION FROM X(3) OF X(100) TO THE CIRCLE Γ(X(3),X(15041))

Barycentrics    a*(9*a^6 - 9*a^5*b - 18*a^4*b^2 + 18*a^3*b^3 + 9*a^2*b^4 - 9*a*b^5 - 9*a^5*c + 19*a^4*b*c + 8*a^3*b^2*c - 23*a^2*b^3*c + a*b^4*c + 4*b^5*c - 18*a^4*c^2 + 8*a^3*b*c^2 + 2*a^2*b^2*c^2 + 8*a*b^3*c^2 + 18*a^3*c^3 - 23*a^2*b*c^3 + 8*a*b^2*c^3 - 8*b^3*c^3 + 9*a^2*c^4 + a*b*c^4 - 9*a*c^5 + 4*b*c^5) : :

X(38636) lies on these lines: {3, 8}, {119, 17800}, {149, 15720}, {528, 15707}, {2077, 28154}, {2829, 15689}, {3035, 3851}, {3526, 13199}, {3530, 20095}, {5055, 5840}, {5070, 23513}, {5073, 24466}, {5217, 37718}, {6174, 15681}, {6455, 35883}, {6456, 35882}, {6496, 35856}, {6497, 35857}, {7988, 26285}, {9691, 19082}, {11698, 15696}, {15718, 21154}

X(38636) = reflection of X(38637) in X(3)

X(38637) = DILATION FROM X(3) OF X(104) TO THE CIRCLE Γ(X(3),X(15041))

Barycentrics    a*(9*a^6 - 9*a^5*b - 18*a^4*b^2 + 18*a^3*b^3 + 9*a^2*b^4 - 9*a*b^5 - 9*a^5*c + 35*a^4*b*c - 8*a^3*b^2*c - 31*a^2*b^3*c + 17*a*b^4*c - 4*b^5*c - 18*a^4*c^2 - 8*a^3*b*c^2 + 34*a^2*b^2*c^2 - 8*a*b^3*c^2 + 18*a^3*c^3 - 31*a^2*b*c^3 - 8*a*b^2*c^3 + 8*b^3*c^3 + 9*a^2*c^4 + 17*a*b*c^4 - 9*a*c^5 - 4*b*c^5) : :

X(38637) lies on these lines: {3, 8}, {11, 17800}, {36, 28154}, {153, 15720}, {1484, 15696}, {2829, 5055}, {3526, 12248}, {3843, 23513}, {3851, 6713}, {5204, 37718}, {5840, 15689}, {6455, 35857}, {6456, 35856}, {6496, 35882}, {6497, 35883}, {7988, 32612}, {9691, 19113}, {12515, 37624}, {15694, 21154}, {19709, 34126}, {35403, 38141}

X(38637) = reflection of X(38636) in X(3)

X(38638) = DILATION FROM X(3) OF X(110) TO THE CIRCLE Γ(X(3),X(15041))

Barycentrics    a^2*(9*a^8 - 22*a^6*b^2 + 12*a^4*b^4 + 6*a^2*b^6 - 5*b^8 - 22*a^6*c^2 + 37*a^4*b^2*c^2 - 19*a^2*b^4*c^2 + 4*b^6*c^2 + 12*a^4*c^4 - 19*a^2*b^2*c^4 + 2*b^4*c^4 + 6*a^2*c^6 + 4*b^2*c^6 - 5*c^8) : :

X(38638) lies on these lines: {3, 74}, {4, 22251}, {20, 13392}, {113, 17800}, {373, 12038}, {511, 37922}, {542, 15707}, {548, 20125}, {1656, 34153}, {1657, 10272}, {2777, 11693}, {3043, 15750}, {3448, 15720}, {3523, 13393}, {3526, 12383}, {3530, 14683}, {3534, 11694}, {3830, 14643}, {3843, 12121}, {3851, 5972}, {5054, 32423}, {5055, 17702}, {5070, 12902}, {5073, 16163}, {5092, 32254}, {5093, 15462}, {5642, 15681}, {5655, 15695}, {6102, 7666}, {6417, 10820}, {6418, 10819}, {6455, 12376}, {6456, 12375}, {6496, 35826}, {6497, 35827}, {9143, 15700}, {9691, 19060}, {9919, 17821}, {11202, 34006}, {11597, 15748}, {12017, 12584}, {12165, 21844}, {12315, 25564}, {12317, 15712}, {12778, 37624}, {13293, 14530}, {14269, 15046}, {14644, 15703}, {15061, 15701}, {18378, 37497}, {18436, 33556}, {18859, 35265}, {22250, 33923}, {37477, 37923}

X(38638) = reflection of X(38633) in X(3)

X(38639) = DILATION FROM X(3) OF X(112) TO THE CIRCLE Γ(X(3),X(15041))

Barycentrics    a^2*(9*a^12 - 22*a^10*b^2 + 13*a^8*b^4 + 8*a^6*b^6 - 17*a^4*b^8 + 14*a^2*b^10 - 5*b^12 - 22*a^10*c^2 + 35*a^8*b^2*c^2 - 21*a^6*b^4*c^2 + 21*a^4*b^6*c^2 - 9*a^2*b^8*c^2 - 4*b^10*c^2 + 13*a^8*c^4 - 21*a^6*b^2*c^4 - 8*a^4*b^4*c^4 - 5*a^2*b^6*c^4 + 21*b^8*c^4 + 8*a^6*c^6 + 21*a^4*b^2*c^6 - 5*a^2*b^4*c^6 - 24*b^6*c^6 - 17*a^4*c^8 - 9*a^2*b^2*c^8 + 21*b^4*c^8 + 14*a^2*c^10 - 4*b^2*c^10 - 5*c^12) : :

X(38639) lies on these lines: {3, 112}, {132, 17800}, {2794, 5055}, {3526, 13200}, {3851, 6720}, {5073, 14689}, {6455, 35881}, {6456, 35880}, {6496, 35828}, {6497, 35829}, {9691, 19094}, {13219, 15720}

X(38640) = DILATION FROM X(3) OF X(930) TO THE CIRCLE Γ(X(3),X(15041))

Barycentrics    9*a^16 - 49*a^14*b^2 + 119*a^12*b^4 - 175*a^10*b^6 + 175*a^8*b^8 - 119*a^6*b^10 + 49*a^4*b^12 - 9*a^2*b^14 - 49*a^14*c^2 + 168*a^12*b^2*c^2 - 219*a^10*b^4*c^2 + 112*a^8*b^6*c^2 + 29*a^6*b^8*c^2 - 60*a^4*b^10*c^2 + 15*a^2*b^12*c^2 + 4*b^14*c^2 + 119*a^12*c^4 - 219*a^10*b^2*c^4 + 131*a^8*b^4*c^4 - 27*a^6*b^6*c^4 + 11*a^4*b^8*c^4 + 9*a^2*b^10*c^4 - 24*b^12*c^4 - 175*a^10*c^6 + 112*a^8*b^2*c^6 - 27*a^6*b^4*c^6 - 15*a^2*b^8*c^6 + 60*b^10*c^6 + 175*a^8*c^8 + 29*a^6*b^2*c^8 + 11*a^4*b^4*c^8 - 15*a^2*b^6*c^8 - 80*b^8*c^8 - 119*a^6*c^10 - 60*a^4*b^2*c^10 + 9*a^2*b^4*c^10 + 60*b^6*c^10 + 49*a^4*c^12 + 15*a^2*b^2*c^12 - 24*b^4*c^12 - 9*a^2*c^14 + 4*b^2*c^14 : :

X(38640) lies on these lines: {3, 252}, {128, 17800}, {382, 23237}, {1657, 6592}, {3526, 25147}, {3528, 14073}, {3851, 13372}, {5070, 23516}, {11671, 15720}, {14072, 15696}, {15688, 32423}

X(38641) = DILATION FROM X(3) OF X(74) TO THE CIRCLE Γ(X(3),X(5976))

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

X(38641) lies on these lines: {3, 74}, {113, 6656}, {125, 1513}, {146, 7791}, {376, 4576}, {541, 8356}, {542, 5976}, {1539, 37243}, {1691, 12192}, {1986, 35476}, {2781, 3094}, {3124, 7418}, {3448, 37182}, {5621, 11653}, {5972, 37450}, {6699, 7807}, {7728, 37242}, {8363, 12900}, {10706, 11287}, {11325, 12133}, {14915, 37927}, {15059, 37071}, {15061, 37466}

X(38641) = reflection of X(38650) in X(3)

X(38642) = DILATION FROM X(3) OF X(98) TO THE CIRCLE Γ(X(3),X(5976))

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

X(38642) lies on these lines: {3, 76}, {4, 2023}, {20, 1916}, {39, 2794}, {114, 6656}, {115, 1513}, {147, 7791}, {148, 37182}, {262, 10722}, {376, 5969}, {542, 8356}, {620, 21163}, {736, 18860}, {1569, 10991}, {1691, 11676}, {3094, 3269}, {3095, 36998}, {3522, 8782}, {5149, 37479}, {5984, 32965}, {6033, 9744}, {6036, 6248}, {6054, 11287}, {6055, 35297}, {6683, 36519}, {6721, 8363}, {7849, 14981}, {11152, 11177}, {11272, 22505}, {11325, 12131}, {12829, 34870}, {14061, 37071}, {15092, 22681}, {24256, 35925}, {37466, 38224}

X(38642) = reflection of X(5976) in X(3)

X(38643) = DILATION FROM X(3) OF X(100) TO THE CIRCLE Γ(X(3),X(5976))

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

X(38643) lies on these lines: {3, 8}, {11, 6656}, {119, 1513}, {149, 7791}, {153, 37182}, {528, 8356}, {1691, 13194}, {1862, 11325}, {2787, 5976}, {3035, 7807}, {3094, 9024}, {6174, 35297}, {6667, 8363}, {6713, 37450}, {7866, 31272}, {10707, 11287}, {10738, 37242}, {20095, 32965}, {22938, 37243}

X(38643) = reflection of X(38646) in X(3)

X(38644) = DILATION FROM X(3) OF X(101) TO THE CIRCLE Γ(X(3),X(5976))

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

X(38644) lies on these lines: {3, 101}, {116, 6656}, {118, 1513}, {150, 7791}, {152, 37182}, {295, 2276}, {544, 8356}, {2786, 5976}, {2810, 3094}, {5185, 11325}, {6710, 7807}, {6712, 37450}, {7866, 31273}, {10708, 11287}, {10739, 37242}, {20096, 32965}

X(38644) = reflection of X(38645) in X(3)

X(38645) = DILATION FROM X(3) OF X(103) TO THE CIRCLE Γ(X(3),X(5976))

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

X(38645) lies on these lines: {3, 101}, {116, 1513}, {118, 6656}, {150, 37182}, {152, 7791}, {2784, 5976}, {6710, 37450}, {6712, 7807}, {10710, 11287}, {10741, 37242}, {31273, 37071}

X(38645) = reflection of X(38644) in X(3)

X(38646) = DILATION FROM X(3) OF X(104) TO THE CIRCLE Γ(X(3),X(5976))

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

X(38646) lies on these lines: {3, 8}, {11, 1513}, {119, 6656}, {149, 37182}, {153, 7791}, {1691, 12199}, {2783, 5976}, {3035, 37450}, {3094, 12499}, {6713, 7807}, {10711, 11287}, {10742, 37242}, {11325, 12138}, {22799, 37243}, {31272, 37071}

X(38646) = reflection of X(38643) in X(3)

X(38647) = DILATION FROM X(3) OF X(105) TO THE CIRCLE Γ(X(3),X(5976))

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

X(38647) lies on these lines: {3, 105}, {120, 6656}, {528, 8356}, {1513, 5511}, {2795, 5976}, {6714, 7807}, {7791, 20344}, {10712, 11287}, {10743, 37242}, {20097, 32965}, {34547, 37182}

X(38648) = DILATION FROM X(3) OF X(106) TO THE CIRCLE Γ(X(3),X(5976))

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

X(38648) lies on these lines: {3, 106}, {121, 6656}, {1513, 5510}, {2796, 5976}, {2810, 3094}, {6715, 7807}, {7791, 21290}, {10713, 11287}, {10744, 37242}, {20098, 32965}, {34548, 37182}

X(38649) = DILATION FROM X(3) OF X(107) TO THE CIRCLE Γ(X(3),X(5976))

Barycentrics    a^14*b^2 + 3*a^12*b^4 - 14*a^10*b^6 + 14*a^8*b^8 - 3*a^6*b^10 - a^4*b^12 + a^14*c^2 - 8*a^12*b^2*c^2 + 13*a^10*b^4*c^2 + 9*a^8*b^6*c^2 - 25*a^6*b^8*c^2 + 6*a^4*b^10*c^2 + 3*a^2*b^12*c^2 + b^14*c^2 + 3*a^12*c^4 + 13*a^10*b^2*c^4 - 44*a^8*b^4*c^4 + 28*a^6*b^6*c^4 + 13*a^4*b^8*c^4 - 9*a^2*b^10*c^4 - 4*b^12*c^4 - 14*a^10*c^6 + 9*a^8*b^2*c^6 + 28*a^6*b^4*c^6 - 36*a^4*b^6*c^6 + 6*a^2*b^8*c^6 + 7*b^10*c^6 + 14*a^8*c^8 - 25*a^6*b^2*c^8 + 13*a^4*b^4*c^8 + 6*a^2*b^6*c^8 - 8*b^8*c^8 - 3*a^6*c^10 + 6*a^4*b^2*c^10 - 9*a^2*b^4*c^10 + 7*b^6*c^10 - a^4*c^12 + 3*a^2*b^2*c^12 - 4*b^4*c^12 + b^2*c^14 : :

X(38649) lies on these lines: {3, 107}, {122, 6656}, {133, 1513}, {2797, 5976}, {6716, 7807}, {7791, 34186}, {8356, 9530}, {10714, 11287}, {10745, 37242}, {34549, 37182}, {34842, 37450}

X(38650) = DILATION FROM X(3) OF X(110) TO THE CIRCLE Γ(X(3),X(5976))

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

X(38650) lies on these lines: {2, 14605}, {3, 74}, {113, 1513}, {125, 6656}, {146, 37182}, {265, 37242}, {511, 37927}, {542, 8356}, {690, 5976}, {1112, 11325}, {1691, 6593}, {1976, 35936}, {2001, 18570}, {2493, 6787}, {2781, 35924}, {2854, 3094}, {3448, 7791}, {5181, 6393}, {5642, 35297}, {5972, 7807}, {6699, 37450}, {6723, 8363}, {7866, 15059}, {9140, 11287}, {9143, 33008}, {10113, 37243}, {10272, 37459}, {11634, 36790}, {12292, 35476}, {12824, 21177}, {14643, 37466}, {14683, 32965}

X(38650) = reflection of X(38641) in X(3)

X(38651) = DILATION FROM X(3) OF X(111) TO THE CIRCLE Γ(X(3),X(5976))

Barycentrics    a^2*(3*a^6*b^2 - a^4*b^4 - 5*a^2*b^6 - b^8 + 3*a^6*c^2 - 16*a^4*b^2*c^2 + 14*a^2*b^4*c^2 + 3*b^6*c^2 - a^4*c^4 + 14*a^2*b^2*c^4 - 10*b^4*c^4 - 5*a^2*c^6 + 3*b^2*c^6 - c^8) : :

X(38651) lies on these lines: {3, 111}, {126, 6656}, {543, 5976}, {574, 12093}, {1513, 5512}, {1691, 28662}, {2021, 37927}, {2854, 3094}, {3124, 11634}, {6719, 7807}, {7791, 14360}, {9172, 35297}, {10717, 11287}, {10748, 37242}, {20099, 32965}

X(38652) = DILATION FROM X(3) OF X(112) TO THE CIRCLE Γ(X(3),X(5976))

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

X(38652) lies on these lines: {3, 112}, {6, 3425}, {32, 34217}, {39, 2794}, {114, 34349}, {127, 6656}, {132, 232}, {216, 34841}, {1180, 9157}, {1691, 13195}, {1916, 10684}, {2023, 36183}, {2491, 2799}, {2781, 3094}, {5133, 14768}, {6720, 7807}, {7772, 15562}, {7791, 13219}, {9605, 11641}, {10547, 14885}, {10718, 11287}, {10749, 37242}, {10766, 19164}, {11325, 13166}, {12384, 37182}, {14689, 14961}, {15355, 37071}, {17907, 34129}, {19163, 37243}, {20410, 21177}

X(38653) = DILATION FROM X(3) OF X(74) TO THE CIRCLE Γ(X(3),X(5989))

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

X(38653) lies on these lines: {3, 74}, {6, 12192}, {113, 7770}, {125, 13860}, {146, 384}, {376, 10330}, {378, 35325}, {541, 1003}, {542, 5989}, {2076, 9984}, {2781, 5017}, {2935, 3499}, {3448, 5999}, {5116, 5621}, {6593, 35423}, {6699, 11285}, {7418, 20998}, {7470, 12383}, {7728, 35930}, {10117, 33877}, {10706, 11286}, {11676, 12244}, {14915, 37903}

X(38653) = reflection of X(38661) in X(3)

X(38654) = DILATION FROM X(3) OF X(98) TO THE CIRCLE Γ(X(3),X(5989))

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

X(38654) lies on these lines: {3, 76}, {6, 12176}, {114, 7770}, {115, 13860}, {147, 384}, {148, 5999}, {542, 1003}, {1503, 2076}, {3148, 22735}, {3552, 5984}, {4048, 6776}, {5026, 35423}, {5085, 19120}, {5116, 7709}, {5149, 14981}, {5254, 14651}, {6033, 35930}, {6036, 11285}, {6054, 11286}, {6770, 35917}, {6773, 35918}, {7470, 13172}, {7783, 10998}, {7816, 35385}, {7851, 38224}, {8178, 18860}, {11177, 13586}, {33430, 35939}, {33431, 35938}

X(38655) = DILATION FROM X(3) OF X(100) TO THE CIRCLE Γ(X(3),X(5989))

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

X(38655) lies on these lines: {3, 8}, {6, 13194}, {11, 7770}, {119, 13860}, {149, 384}, {153, 5999}, {528, 1003}, {2076, 13235}, {2787, 5989}, {3035, 11285}, {3552, 20095}, {5017, 9024}, {6154, 33235}, {7470, 12248}, {10707, 11286}, {10738, 35930}, {11676, 13199}, {18047, 23402}

X(38655) = reflection of X(38657) in X(3)

X(38656) = DILATION FROM X(3) OF X(101) TO THE CIRCLE Γ(X(3),X(5989))

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

X(38656) lies on these lines: {3, 101}, {116, 7770}, {118, 13860}, {150, 384}, {152, 5999}, {197, 25577}, {544, 1003}, {813, 20871}, {2786, 5989}, {2810, 5017}, {3552, 20096}, {6710, 11285}, {10708, 11286}, {10739, 35930}

X(38657) = DILATION FROM X(3) OF X(104) TO THE CIRCLE Γ(X(3),X(5989))

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

X(38657) lies on these lines: {3, 8}, {6, 12199}, {11, 13860}, {119, 7770}, {149, 5999}, {153, 384}, {2076, 12499}, {2783, 5989}, {6713, 11285}, {7470, 13199}, {10711, 11286}, {10742, 35930}, {11676, 12248}

X(38657) = reflection of X(38655) in X(3)

X(38658) = DILATION FROM X(3) OF X(105) TO THE CIRCLE Γ(X(3),X(5989))

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

X(38658) lies on these lines: {3, 105}, {120, 7770}, {384, 20344}, {528, 1003}, {2795, 5989}, {3552, 20097}, {5511, 13860}, {5999, 34547}, {6714, 11285}, {10712, 11286}, {10743, 35930}

X(38659) = DILATION FROM X(3) OF X(106) TO THE CIRCLE Γ(X(3),X(5989))

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

X(38659) lies on these lines: {3, 106}, {121, 7770}, {384, 21290}, {2796, 5989}, {2810, 5017}, {3552, 20098}, {5510, 13860}, {5999, 34548}, {6715, 11285}, {10713, 11286}, {10744, 35930}

X(38660) = DILATION FROM X(3) OF X(107) TO THE CIRCLE Γ(X(3),X(5989))

Barycentrics    a^16 - 6*a^12*b^4 + 11*a^10*b^6 - 11*a^8*b^8 + 6*a^6*b^10 - a^2*b^14 + 11*a^12*b^2*c^2 - 12*a^10*b^4*c^2 - 14*a^8*b^6*c^2 + 20*a^6*b^8*c^2 - 5*a^4*b^10*c^2 - 6*a^12*c^4 - 12*a^10*b^2*c^4 + 52*a^8*b^4*c^4 - 26*a^6*b^6*c^4 - 16*a^4*b^8*c^4 + 6*a^2*b^10*c^4 + 2*b^12*c^4 + 11*a^10*c^6 - 14*a^8*b^2*c^6 - 26*a^6*b^4*c^6 + 42*a^4*b^6*c^6 - 5*a^2*b^8*c^6 - 8*b^10*c^6 - 11*a^8*c^8 + 20*a^6*b^2*c^8 - 16*a^4*b^4*c^8 - 5*a^2*b^6*c^8 + 12*b^8*c^8 + 6*a^6*c^10 - 5*a^4*b^2*c^10 + 6*a^2*b^4*c^10 - 8*b^6*c^10 + 2*b^4*c^12 - a^2*c^14 : :

X(38660) lies on these lines: {3, 107}, {122, 7770}, {133, 13860}, {384, 34186}, {1003, 9530}, {2797, 5989}, {5667, 11676}, {5999, 34549}, {6716, 11285}, {10714, 11286}, {10745, 35930}, {14703, 37123}

X(38661) = DILATION FROM X(3) OF X(110) TO THE CIRCLE Γ(X(3),X(5989))

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

X(38661) lies on these lines: {3, 74}, {6, 13193}, {113, 13860}, {125, 7770}, {146, 5999}, {265, 35930}, {384, 3448}, {511, 37903}, {542, 1003}, {690, 5989}, {2076, 2930}, {2421, 2936}, {2854, 5017}, {2931, 37123}, {2935, 8925}, {3552, 14683}, {4235, 25046}, {5149, 15357}, {5152, 15342}, {5972, 11285}, {7470, 12244}, {9140, 11286}, {9143, 13586}, {11061, 12215}, {11676, 12383}, {12317, 35925}, {24981, 33235}, {35936, 36213}

X(38661) = reflection of X(38653) in X(3)

X(38662) = DILATION FROM X(3) OF X(111) TO THE CIRCLE Γ(X(3),X(5989))

Barycentrics    a^2*(a^8 - 2*a^6*b^2 - 2*a^4*b^4 - b^8 - 2*a^6*c^2 + 19*a^4*b^2*c^2 - 11*a^2*b^4*c^2 - 2*b^6*c^2 - 2*a^4*c^4 - 11*a^2*b^2*c^4 + 16*b^4*c^4 - 2*b^2*c^6 - c^8) : :

X(38662) lies on these lines: {3, 111}, {126, 7770}, {384, 14360}, {543, 1003}, {2854, 5017}, {3552, 20099}, {5116, 38402}, {5162, 37903}, {5512, 13860}, {6719, 11285}, {10717, 11286}, {10748, 35930}, {11634, 20998}, {11676, 14654}, {14657, 37123}, {14688, 35423}, {32526, 35001}

X(38663) = DILATION FROM X(3) OF X(112) TO THE CIRCLE Γ(X(3),X(5989))

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

X(38663) lies on these lines: {3, 112}, {6, 13195}, {22, 35325}, {127, 7770}, {132, 13860}, {187, 14676}, {384, 13219}, {2076, 13236}, {2781, 5017}, {2799, 5989}, {5999, 12384}, {6720, 11285}, {7470, 12253}, {10718, 11286}, {10749, 35930}, {11676, 13200}, {15013, 34163}

X(38664) = DILATION FROM X(3) OF X(98) TO THE CIRCLE Γ(X(3),X(14094))

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

X(38664) lies on these lines: {2, 11623}, {3, 76}, {4, 542}, {5, 6054}, {13, 36252}, {14, 36251}, {20, 543}, {23, 5986}, {30, 11054}, {83, 575}, {114, 3090}, {115, 147}, {140, 8724}, {148, 2794}, {194, 22664}, {287, 18338}, {316, 3564}, {338, 2930}, {376, 10992}, {512, 14510}, {538, 5999}, {546, 6033}, {550, 12117}, {620, 10303}, {631, 6055}, {632, 15561}, {690, 15054}, {1352, 7790}, {1513, 14568}, {1656, 23234}, {2142, 32483}, {2482, 3523}, {2784, 13178}, {2795, 33557}, {2796, 5493}, {2936, 17928}, {3023, 3304}, {3027, 3303}, {3455, 7488}, {3515, 9876}, {3522, 8591}, {3525, 6036}, {3529, 9862}, {3592, 19109}, {3594, 19108}, {3627, 6321}, {3628, 7859}, {3746, 10053}, {3839, 36523}, {3851, 22566}, {3934, 9772}, {3972, 9755}, {4857, 10070}, {5026, 10541}, {5028, 5921}, {5056, 5461}, {5059, 8596}, {5073, 12355}, {5076, 22515}, {5182, 7770}, {5186, 11403}, {5188, 17129}, {5198, 12131}, {5254, 11646}, {5270, 10054}, {5309, 13862}, {5563, 10069}, {5609, 18332}, {5613, 20415}, {5617, 20416}, {5882, 9884}, {5985, 16865}, {5987, 14002}, {6194, 17131}, {6419, 19056}, {6420, 19055}, {6453, 35878}, {6454, 35879}, {6776, 11185}, {7486, 14971}, {7527, 13233}, {7748, 9863}, {7754, 10754}, {7757, 13860}, {7765, 37336}, {7781, 9888}, {7809, 15980}, {7815, 15483}, {7824, 11152}, {7841, 11161}, {7850, 11898}, {7883, 19905}, {7894, 11482}, {7970, 10222}, {7982, 7983}, {7991, 9860}, {8289, 17128}, {8370, 8550}, {9154, 14999}, {9302, 10159}, {9466, 37455}, {9512, 36841}, {9756, 31859}, {9830, 34505}, {10304, 15300}, {10358, 22234}, {10359, 32135}, {11005, 36253}, {11006, 20417}, {11289, 36776}, {11522, 12258}, {11656, 16534}, {11711, 30389}, {12122, 32521}, {12150, 35930}, {12189, 37622}, {12811, 38229}, {13172, 17538}, {13335, 35950}, {14928, 25406}, {14931, 20081}, {14957, 23061}, {15022, 36519}, {15025, 15359}, {15027, 15535}, {15692, 36521}, {19662, 33230}, {19911, 34506}, {21636, 38220}, {22735, 37338}, {25330, 38361}, {31276, 37479}, {32448, 32469}, {32907, 36362}, {32909, 36363}, {34624, 37182}

X(38664) = reflection of X(23235) in X(3)
X(38664) = anticomplement of X(14981)

X(38665) = DILATION FROM X(3) OF X(100) TO THE CIRCLE Γ(X(3),X(14094))

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

X(38665) lies on these lines: {1, 6946}, {2, 37726}, {3, 8}, {4, 528}, {5, 10707}, {10, 34486}, {11, 1058}, {20, 10993}, {40, 2801}, {80, 943}, {105, 24808}, {119, 149}, {145, 11499}, {153, 3146}, {214, 5438}, {355, 3871}, {376, 34689}, {404, 10031}, {495, 20119}, {515, 5537}, {517, 3935}, {519, 6905}, {546, 10738}, {576, 10755}, {631, 6174}, {942, 14151}, {946, 5660}, {962, 18518}, {1006, 3679}, {1012, 34627}, {1156, 18908}, {1293, 16528}, {1317, 3304}, {1320, 1389}, {1376, 7967}, {1387, 5703}, {1483, 5253}, {1484, 3628}, {1490, 2800}, {1512, 5853}, {1621, 5790}, {1737, 12750}, {1862, 5198}, {2077, 28236}, {2136, 2802}, {2771, 15054}, {2787, 23235}, {2829, 3529}, {3035, 3525}, {3058, 6965}, {3241, 6911}, {3295, 5818}, {3421, 37000}, {3428, 8168}, {3434, 6982}, {3592, 19082}, {3594, 19081}, {3617, 10267}, {3621, 11249}, {3625, 11012}, {3626, 10902}, {3627, 10724}, {3632, 6796}, {3651, 11362}, {3654, 7411}, {3753, 35985}, {3813, 6949}, {3893, 37837}, {3895, 5720}, {4295, 12831}, {4421, 6950}, {4571, 21290}, {4662, 26878}, {4669, 21161}, {5076, 22799}, {5082, 10786}, {5220, 31980}, {5260, 37621}, {5284, 38042}, {5552, 6978}, {5563, 7972}, {5587, 25439}, {5722, 12730}, {5758, 12732}, {5761, 9802}, {5842, 12762}, {5844, 18524}, {5881, 6906}, {5882, 6940}, {6223, 6361}, {6419, 19113}, {6420, 19112}, {6453, 35856}, {6454, 35857}, {6684, 11219}, {6713, 10303}, {6826, 11239}, {6829, 10056}, {6848, 12632}, {6880, 34625}, {6901, 15888}, {6902, 21031}, {6909, 28204}, {6913, 38074}, {6929, 34611}, {6938, 34607}, {6942, 12513}, {6951, 34612}, {6970, 11240}, {6976, 10385}, {7080, 12116}, {7993, 11715}, {8674, 14094}, {9024, 10759}, {9342, 38028}, {9780, 16202}, {9897, 10058}, {10591, 10965}, {10680, 20050}, {10778, 36253}, {10914, 21740}, {10915, 12751}, {10943, 27529}, {11403, 12138}, {11500, 12245}, {11518, 12736}, {11849, 37705}, {12115, 17784}, {12248, 17538}, {12515, 13243}, {12536, 37302}, {12653, 16189}, {12691, 14740}, {12737, 15178}, {12739, 17636}, {14217, 21635}, {14923, 37700}, {15017, 16174}, {15022, 23513}, {15931, 38127}, {16842, 34122}, {16862, 34123}, {18391, 33925}, {20117, 37563}, {21154, 35023}, {28224, 35000}, {31649, 31660}

X(38665) = reflection of X(38669) in X(3)
X(38665) = anticomplement of X(37726)

X(38666) = DILATION FROM X(3) OF X(101) TO THE CIRCLE Γ(X(3),X(14094))

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

X(38666) lies on these lines: {1, 651}, {3, 101}, {4, 544}, {5, 10708}, {20, 33520}, {116, 3090}, {118, 150}, {152, 3146}, {376, 33521}, {546, 10739}, {576, 10756}, {952, 18328}, {971, 6603}, {1170, 6915}, {1282, 7991}, {1362, 3304}, {1736, 38459}, {1768, 35293}, {2340, 5537}, {2772, 15054}, {2774, 14094}, {2784, 13178}, {2786, 23235}, {2809, 7982}, {2810, 10758}, {3022, 3303}, {3525, 6710}, {3627, 10725}, {3628, 31273}, {5185, 5198}, {5526, 13329}, {5705, 34933}, {5709, 34925}, {5714, 34929}, {5723, 13257}, {5779, 34522}, {6712, 10303}, {6734, 34932}, {9327, 14520}, {10222, 10695}, {10902, 34927}, {11012, 34928}, {11028, 11518}, {11714, 30389}

X(38666) = reflection of X(38668) in X(3)

X(38667) = DILATION FROM X(3) OF X(102) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a^2*(a^8 - a^7*b + a^6*b^2 + 3*a^5*b^3 - 9*a^4*b^4 - 3*a^3*b^5 + 11*a^2*b^6 + a*b^7 - 4*b^8 - a^7*c + 3*a^6*b*c - 7*a^5*b^2*c + a^4*b^3*c + 17*a^3*b^4*c - 11*a^2*b^5*c - 9*a*b^6*c + 7*b^7*c + a^6*c^2 - 7*a^5*b*c^2 + 20*a^4*b^2*c^2 - 14*a^3*b^3*c^2 - 15*a^2*b^4*c^2 + 21*a*b^5*c^2 - 6*b^6*c^2 + 3*a^5*c^3 + a^4*b*c^3 - 14*a^3*b^2*c^3 + 30*a^2*b^3*c^3 - 13*a*b^4*c^3 - 7*b^5*c^3 - 9*a^4*c^4 + 17*a^3*b*c^4 - 15*a^2*b^2*c^4 - 13*a*b^3*c^4 + 20*b^4*c^4 - 3*a^3*c^5 - 11*a^2*b*c^5 + 21*a*b^2*c^5 - 7*b^3*c^5 + 11*a^2*c^6 - 9*a*b*c^6 - 6*b^2*c^6 + a*c^7 + 7*b*c^7 - 4*c^8) : :

X(38667) lies on these lines: {3, 102}, {4, 10716}, {5, 10709}, {117, 3090}, {124, 151}, {546, 10740}, {576, 10757}, {1361, 3303}, {1364, 3304}, {1490, 2800}, {2773, 15054}, {2779, 14094}, {2792, 23235}, {2817, 7982}, {3146, 10732}, {3525, 6711}, {3627, 10726}, {5691, 36921}, {6718, 10303}, {10222, 10696}, {10764, 11477}, {11700, 30389}

X(38667) = reflection of X(38674) in X(3)

X(38668) = DILATION FROM X(3) OF X(103) TO THE CIRCLE Γ(X(3),X(14094))

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

X(38668) lies on these lines: {1, 23056}, {2, 20401}, {3, 101}, {4, 10708}, {5, 10710}, {20, 544}, {35, 34927}, {40, 2801}, {56, 4845}, {116, 152}, {118, 3090}, {150, 3146}, {376, 33520}, {514, 14512}, {546, 10741}, {576, 10758}, {971, 5011}, {1362, 3303}, {1385, 15735}, {2772, 14094}, {2774, 15054}, {2784, 23235}, {2809, 7991}, {3022, 3304}, {3340, 34930}, {3525, 6712}, {3601, 15730}, {3627, 10727}, {5185, 11403}, {6710, 10303}, {7982, 10695}, {10222, 10697}, {10756, 11477}, {11712, 30389}, {20367, 36002}

X(38668) = reflection of X(38666) in X(3)

X(38669) = DILATION FROM X(3) OF X(104) TO THE CIRCLE Γ(X(3),X(14094))

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

X(38669) lies on these lines: {1, 651}, {2, 20400}, {3, 8}, {4, 10707}, {5, 10711}, {10, 11219}, {11, 153}, {20, 528}, {21, 5882}, {36, 28236}, {80, 1210}, {84, 1320}, {106, 5400}, {119, 3090}, {145, 12114}, {149, 2829}, {150, 934}, {214, 936}, {355, 5253}, {376, 10993}, {404, 5881}, {411, 8666}, {513, 14511}, {515, 13279}, {519, 5537}, {529, 6840}, {546, 1484}, {576, 10759}, {962, 12246}, {991, 16499}, {999, 12019}, {1006, 3655}, {1012, 3241}, {1071, 4861}, {1125, 5660}, {1158, 3885}, {1317, 3303}, {1385, 5260}, {1387, 3487}, {1389, 24475}, {1483, 26321}, {1537, 9809}, {1621, 7967}, {1768, 2802}, {1862, 11403}, {2771, 7984}, {2783, 23235}, {2826, 13252}, {2827, 38329}, {3035, 10303}, {3428, 35986}, {3476, 33925}, {3485, 12831}, {3523, 6174}, {3525, 6713}, {3529, 5840}, {3533, 38069}, {3592, 19113}, {3594, 19112}, {3621, 10310}, {3623, 11496}, {3627, 10728}, {3628, 11698}, {3646, 3897}, {3681, 37611}, {3746, 7972}, {3813, 37437}, {3817, 37602}, {3854, 38077}, {3871, 5450}, {3890, 7330}, {3918, 35010}, {4293, 20119}, {4297, 5288}, {4304, 12730}, {4511, 17615}, {4578, 6790}, {5076, 22938}, {5083, 11518}, {5198, 12138}, {5229, 18967}, {5258, 6986}, {5270, 7548}, {5284, 10246}, {5330, 5693}, {5434, 6839}, {5603, 18519}, {5777, 6265}, {5784, 18444}, {5790, 9342}, {5818, 16203}, {5842, 20067}, {5919, 16140}, {6001, 38460}, {6419, 19082}, {6420, 19081}, {6453, 35882}, {6454, 35883}, {6888, 15888}, {6905, 28204}, {6906, 37727}, {6911, 34627}, {6913, 38314}, {6925, 34625}, {6935, 11239}, {6938, 34611}, {6945, 10072}, {6966, 34619}, {6972, 12607}, {6978, 10785}, {6982, 11680}, {7966, 35258}, {7995, 12559}, {8674, 15054}, {9623, 11407}, {9845, 9859}, {9897, 10090}, {9963, 12119}, {9964, 17660}, {10265, 12751}, {10306, 20050}, {10529, 12667}, {10572, 12750}, {10595, 18761}, {10755, 11477}, {11037, 38055}, {11260, 12680}, {12648, 14647}, {12653, 12767}, {12762, 20060}, {12775, 37622}, {13199, 17538}, {13407, 16173}, {13729, 37722}, {15017, 32557}, {15071, 22837}, {16842, 34123}, {16862, 34122}, {17638, 20586}, {20085, 22775}, {20095, 24466}, {21630, 34789}, {22765, 28224}, {24297, 36279}, {30315, 38104}, {32633, 33862}, {37535, 37705}

X(38669) = reflection of X(38665) in X(3)
X(38669) = anticomplement of X(37725)

X(38670) = DILATION FROM X(3) OF X(105) TO THE CIRCLE Γ(X(3),X(14094))

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

X(38670) lies on these lines: {3, 105}, {4, 528}, {5, 10712}, {40, 9519}, {120, 3090}, {546, 10743}, {576, 10760}, {1072, 34486}, {1358, 3304}, {2775, 15054}, {2795, 21669}, {2809, 7982}, {2826, 13252}, {2836, 14094}, {3021, 3303}, {3091, 5511}, {3146, 20097}, {3525, 6714}, {3627, 10729}, {5540, 7991}, {10222, 10699}, {16862, 34124}

X(38670) = reflection of X(38684) in X(3)

X(38671) = DILATION FROM X(3) OF X(106) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a^2*(a^5 - 2*a^4*b + 6*a^2*b^3 - a*b^4 - 4*b^5 - 2*a^4*c + 9*a^3*b*c - 15*a^2*b^2*c - 9*a*b^3*c + 17*b^4*c - 15*a^2*b*c^2 + 40*a*b^2*c^2 - 17*b^3*c^2 + 6*a^2*c^3 - 9*a*b*c^3 - 17*b^2*c^3 - a*c^4 + 17*b*c^4 - 4*c^5) : :

X(38671) lies on these lines: {1, 9519}, {3, 106}, {5, 10713}, {121, 3090}, {546, 10744}, {576, 10761}, {1054, 7991}, {1357, 3304}, {2136, 2802}, {2776, 15054}, {2796, 4301}, {2810, 10758}, {2827, 38329}, {2842, 14094}, {3091, 5510}, {3146, 20098}, {3303, 6018}, {3525, 6715}, {3627, 10730}, {5497, 10222}, {13541, 16189}

X(38671) = reflection of X(38685) in X(3)

X(38672) = DILATION FROM X(3) OF X(107) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a^16 - 5*a^14*b^2 + 11*a^12*b^4 - 15*a^10*b^6 + 15*a^8*b^8 - 11*a^6*b^10 + 5*a^4*b^12 - a^2*b^14 - 5*a^14*c^2 - 5*a^12*b^2*c^2 + 11*a^10*b^4*c^2 + 29*a^8*b^6*c^2 - 31*a^6*b^8*c^2 - 11*a^4*b^10*c^2 + 9*a^2*b^12*c^2 + 3*b^14*c^2 + 11*a^12*c^4 + 11*a^10*b^2*c^4 - 88*a^8*b^4*c^4 + 42*a^6*b^6*c^4 + 63*a^4*b^8*c^4 - 21*a^2*b^10*c^4 - 18*b^12*c^4 - 15*a^10*c^6 + 29*a^8*b^2*c^6 + 42*a^6*b^4*c^6 - 114*a^4*b^6*c^6 + 13*a^2*b^8*c^6 + 45*b^10*c^6 + 15*a^8*c^8 - 31*a^6*b^2*c^8 + 63*a^4*b^4*c^8 + 13*a^2*b^6*c^8 - 60*b^8*c^8 - 11*a^6*c^10 - 11*a^4*b^2*c^10 - 21*a^2*b^4*c^10 + 45*b^6*c^10 + 5*a^4*c^12 + 9*a^2*b^2*c^12 - 18*b^4*c^12 - a^2*c^14 + 3*b^2*c^14 : :

X(38672) lies on these lines: {3, 107}, {4, 9530}, {5, 10714}, {30, 23241}, {122, 3090}, {133, 3091}, {546, 10745}, {576, 10762}, {1515, 15312}, {2592, 15157}, {2593, 15156}, {2777, 3146}, {2797, 23235}, {3183, 3529}, {3184, 17538}, {3303, 7158}, {3304, 3324}, {3525, 6716}, {3627, 10152}, {9033, 14094}, {9528, 21669}, {10222, 10701}, {10303, 34842}, {15022, 36520}, {15704, 23240}

X(38672) = reflection of X(38686) in X(3)

X(38673) = DILATION FROM X(3) OF X(108) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a*(a^12 - a^11*b - 3*a^10*b^2 + 3*a^9*b^3 + 2*a^8*b^4 - 2*a^7*b^5 + 2*a^6*b^6 - 2*a^5*b^7 - 3*a^4*b^8 + 3*a^3*b^9 + a^2*b^10 - a*b^11 - a^11*c - a^10*b*c + 5*a^9*b^2*c - a^8*b^3*c - 10*a^7*b^4*c + 6*a^6*b^5*c + 10*a^5*b^6*c - 2*a^4*b^7*c - 5*a^3*b^8*c - 5*a^2*b^9*c + a*b^10*c + 3*b^11*c - 3*a^10*c^2 + 5*a^9*b*c^2 - 10*a^8*b^2*c^2 + 12*a^7*b^3*c^2 + 14*a^6*b^4*c^2 - 26*a^5*b^5*c^2 + 8*a^4*b^6*c^2 - 4*a^3*b^7*c^2 - 3*a^2*b^8*c^2 + 13*a*b^9*c^2 - 6*b^10*c^2 + 3*a^9*c^3 - a^8*b*c^3 + 12*a^7*b^2*c^3 - 44*a^6*b^3*c^3 + 18*a^5*b^4*c^3 + 18*a^4*b^5*c^3 - 20*a^3*b^6*c^3 + 36*a^2*b^7*c^3 - 13*a*b^8*c^3 - 9*b^9*c^3 + 2*a^8*c^4 - 10*a^7*b*c^4 + 14*a^6*b^2*c^4 + 18*a^5*b^3*c^4 - 42*a^4*b^4*c^4 + 26*a^3*b^5*c^4 + 2*a^2*b^6*c^4 - 34*a*b^7*c^4 + 24*b^8*c^4 - 2*a^7*c^5 + 6*a^6*b*c^5 - 26*a^5*b^2*c^5 + 18*a^4*b^3*c^5 + 26*a^3*b^4*c^5 - 62*a^2*b^5*c^5 + 34*a*b^6*c^5 + 6*b^7*c^5 + 2*a^6*c^6 + 10*a^5*b*c^6 + 8*a^4*b^2*c^6 - 20*a^3*b^3*c^6 + 2*a^2*b^4*c^6 + 34*a*b^5*c^6 - 36*b^6*c^6 - 2*a^5*c^7 - 2*a^4*b*c^7 - 4*a^3*b^2*c^7 + 36*a^2*b^3*c^7 - 34*a*b^4*c^7 + 6*b^5*c^7 - 3*a^4*c^8 - 5*a^3*b*c^8 - 3*a^2*b^2*c^8 - 13*a*b^3*c^8 + 24*b^4*c^8 + 3*a^3*c^9 - 5*a^2*b*c^9 + 13*a*b^2*c^9 - 9*b^3*c^9 + a^2*c^10 + a*b*c^10 - 6*b^2*c^10 - a*c^11 + 3*b*c^11) : :

X(38673) lies on these lines: {3, 108}, {5, 10715}, {123, 3090}, {149, 2829}, {546, 10746}, {576, 10763}, {1359, 3304}, {2778, 15054}, {2798, 23235}, {2817, 7982}, {2850, 14094}, {3091, 25640}, {3303, 3318}, {3525, 6717}, {3627, 10731}, {9528, 33557}, {10222, 10702}

X(38673) = reflection of X(38687) in X(3)

X(38674) = DILATION FROM X(3) OF X(109) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a^2*(a^8 - a^7*b - 5*a^6*b^2 + 3*a^5*b^3 + 9*a^4*b^4 - 3*a^3*b^5 - 7*a^2*b^6 + a*b^7 + 2*b^8 - a^7*c + 3*a^6*b*c + 5*a^5*b^2*c - 11*a^4*b^3*c - 7*a^3*b^4*c + 13*a^2*b^5*c + 3*a*b^6*c - 5*b^7*c - 5*a^6*c^2 + 5*a^5*b*c^2 - 4*a^4*b^2*c^2 + 10*a^3*b^3*c^2 + 3*a^2*b^4*c^2 - 15*a*b^5*c^2 + 6*b^6*c^2 + 3*a^5*c^3 - 11*a^4*b*c^3 + 10*a^3*b^2*c^3 - 18*a^2*b^3*c^3 + 11*a*b^4*c^3 + 5*b^5*c^3 + 9*a^4*c^4 - 7*a^3*b*c^4 + 3*a^2*b^2*c^4 + 11*a*b^3*c^4 - 16*b^4*c^4 - 3*a^3*c^5 + 13*a^2*b*c^5 - 15*a*b^2*c^5 + 5*b^3*c^5 - 7*a^2*c^6 + 3*a*b*c^6 + 6*b^2*c^6 + a*c^7 - 5*b*c^7 + 2*c^8) : :

X(38674) lies on these lines: {3, 102}, {4, 10709}, {5, 10716}, {40, 651}, {84, 1320}, {117, 3091}, {124, 3090}, {151, 3146}, {546, 10747}, {576, 10764}, {946, 34234}, {947, 34043}, {1361, 3304}, {1364, 3303}, {1795, 5563}, {2182, 5011}, {2773, 14094}, {2779, 13217}, {2785, 23235}, {2817, 2956}, {3525, 6718}, {3627, 10732}, {6711, 10303}, {10222, 10703}, {10757, 11477}, {11518, 12016}, {11713, 30389}, {15178, 20324}

X(38674) = reflection of X(38667) in X(3)

X(38675) = DILATION FROM X(3) OF X(111) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a^2*(a^8 - a^6*b^2 + 3*a^4*b^4 + a^2*b^6 - 4*b^8 - a^6*c^2 - 21*a^4*b^2*c^2 + 15*a^2*b^4*c^2 + 23*b^6*c^2 + 3*a^4*c^4 + 15*a^2*b^2*c^4 - 54*b^4*c^4 + a^2*c^6 + 23*b^2*c^6 - 4*c^8) : :

X(38675) lies on these lines: {3, 111}, {4, 543}, {5, 10717}, {126, 3090}, {382, 32424}, {511, 9871}, {538, 13168}, {546, 10748}, {550, 14666}, {575, 36696}, {576, 10765}, {631, 9172}, {1499, 14515}, {2780, 15054}, {2854, 10752}, {3091, 5512}, {3146, 20099}, {3303, 6019}, {3304, 3325}, {3522, 37749}, {3525, 6719}, {3529, 14654}, {3627, 10734}, {5969, 15098}, {6453, 11833}, {6454, 11834}, {7550, 15560}, {7841, 11162}, {9129, 15034}, {9156, 11615}, {10222, 10704}, {10541, 14688}, {12082, 33900}, {12505, 34505}, {14671, 34010}

X(38675) = reflection of X(38688) in X(3)

X(38676) = DILATION FROM X(3) OF X(112) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a^2*(a^12 - 5*a^10*b^2 + 4*a^8*b^4 + 6*a^6*b^6 - 7*a^4*b^8 - a^2*b^10 + 2*b^12 - 5*a^10*c^2 + 9*a^8*b^2*c^2 - 10*a^6*b^4*c^2 + 10*a^4*b^6*c^2 - a^2*b^8*c^2 - 3*b^10*c^2 + 4*a^8*c^4 - 10*a^6*b^2*c^4 - 6*a^4*b^4*c^4 + 2*a^2*b^6*c^4 + 10*b^8*c^4 + 6*a^6*c^6 + 10*a^4*b^2*c^6 + 2*a^2*b^4*c^6 - 18*b^6*c^6 - 7*a^4*c^8 - a^2*b^2*c^8 + 10*b^4*c^8 - a^2*c^10 - 3*b^2*c^10 + 2*c^12) : :

X(38676) lies on these lines: {3, 112}, {5, 10718}, {20, 648}, {23, 9157}, {64, 895}, {127, 3090}, {132, 3091}, {148, 2794}, {546, 10749}, {576, 10766}, {2386, 19158}, {2799, 23235}, {3303, 6020}, {3304, 3320}, {3525, 6720}, {3529, 13200}, {3592, 19094}, {3594, 19093}, {3627, 10735}, {3746, 13311}, {5076, 19160}, {5198, 13166}, {5563, 13312}, {6419, 19115}, {6420, 19114}, {6453, 35828}, {6454, 35829}, {7982, 13099}, {7991, 13221}, {9149, 19164}, {9517, 14094}, {10222, 10705}, {10303, 34841}, {11403, 12145}, {12253, 14689}, {12265, 30389}, {13313, 37622}, {15012, 16225}

X(38676) = reflection of X(38689) in X(3)

X(38677) = DILATION FROM X(3) OF X(476) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a^16 - 6*a^14*b^2 + 16*a^12*b^4 - 25*a^10*b^6 + 25*a^8*b^8 - 16*a^6*b^10 + 6*a^4*b^12 - a^2*b^14 - 6*a^14*c^2 + 6*a^12*b^2*c^2 - a^10*b^4*c^2 + 14*a^8*b^6*c^2 - 9*a^6*b^8*c^2 - 11*a^4*b^10*c^2 + 4*a^2*b^12*c^2 + 3*b^14*c^2 + 16*a^12*c^4 - a^10*b^2*c^4 - 45*a^8*b^4*c^4 + 21*a^6*b^6*c^4 + 33*a^4*b^8*c^4 - 6*a^2*b^10*c^4 - 18*b^12*c^4 - 25*a^10*c^6 + 14*a^8*b^2*c^6 + 21*a^6*b^4*c^6 - 56*a^4*b^6*c^6 + 3*a^2*b^8*c^6 + 45*b^10*c^6 + 25*a^8*c^8 - 9*a^6*b^2*c^8 + 33*a^4*b^4*c^8 + 3*a^2*b^6*c^8 - 60*b^8*c^8 - 16*a^6*c^10 - 11*a^4*b^2*c^10 - 6*a^2*b^4*c^10 + 45*b^6*c^10 + 6*a^4*c^12 + 4*a^2*b^2*c^12 - 18*b^4*c^12 - a^2*c^14 + 3*b^2*c^14 : :

X(38677) lies on these lines: {3, 476}, {5, 34312}, {30, 15054}, {523, 14094}, {546, 18319}, {632, 11749}, {1995, 15111}, {3090, 3258}, {3091, 14731}, {3146, 14989}, {3303, 33965}, {3304, 33964}, {3525, 22104}, {5609, 14480}, {5627, 17511}, {7471, 15034}, {7530, 15112}, {9158, 16619}, {10303, 31379}, {14536, 16534}, {14934, 15020}, {14993, 16340}, {15027, 34209}, {15044, 36184}

X(38677) = reflection of X(38678) in X(3)

X(38678) = DILATION FROM X(3) OF X(477) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a^16 - 14*a^12*b^4 + 35*a^10*b^6 - 35*a^8*b^8 + 14*a^6*b^10 - a^2*b^14 + 18*a^12*b^2*c^2 - 25*a^10*b^4*c^2 - 22*a^8*b^6*c^2 + 39*a^6*b^8*c^2 - 5*a^4*b^10*c^2 - 2*a^2*b^12*c^2 - 3*b^14*c^2 - 14*a^12*c^4 - 25*a^10*b^2*c^4 + 99*a^8*b^4*c^4 - 51*a^6*b^6*c^4 - 39*a^4*b^8*c^4 + 12*a^2*b^10*c^4 + 18*b^12*c^4 + 35*a^10*c^6 - 22*a^8*b^2*c^6 - 51*a^6*b^4*c^6 + 88*a^4*b^6*c^6 - 9*a^2*b^8*c^6 - 45*b^10*c^6 - 35*a^8*c^8 + 39*a^6*b^2*c^8 - 39*a^4*b^4*c^8 - 9*a^2*b^6*c^8 + 60*b^8*c^8 + 14*a^6*c^10 - 5*a^4*b^2*c^10 + 12*a^2*b^4*c^10 - 45*b^6*c^10 - 2*a^2*b^2*c^12 + 18*b^4*c^12 - a^2*c^14 - 3*b^2*c^14 : :

X(38678) lies on these lines: {3, 476}, {4, 34312}, {30, 14094}, {523, 14508}, {3090, 25641}, {3091, 3258}, {3146, 14731}, {3154, 15025}, {3303, 33964}, {3304, 33965}, {3470, 3627}, {3525, 31379}, {3628, 18319}, {5627, 15027}, {7471, 15020}, {10303, 22104}, {14851, 34209}, {14934, 15034}, {15021, 36164}, {15029, 36169}, {15044, 34150}, {15111, 31861}

X(38678) = reflection of X(38677) in X(3)

X(38679) = DILATION FROM X(3) OF X(691) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a^2*(a^12 - 6*a^10*b^2 + 6*a^8*b^4 + 6*a^6*b^6 - 9*a^4*b^8 + 2*b^12 - 6*a^10*c^2 + 26*a^8*b^2*c^2 - 32*a^6*b^4*c^2 + 24*a^4*b^6*c^2 - 4*a^2*b^8*c^2 - 8*b^10*c^2 + 6*a^8*c^4 - 32*a^6*b^2*c^4 + 3*a^4*b^4*c^4 + 25*b^8*c^4 + 6*a^6*c^6 + 24*a^4*b^2*c^6 - 38*b^6*c^6 - 9*a^4*c^8 - 4*a^2*b^2*c^8 + 25*b^4*c^8 - 8*b^2*c^10 + 2*c^12) : :

X(38679) lies on these lines: {3, 691}, {23, 14669}, {30, 11054}, {187, 37953}, {511, 15054}, {512, 14094}, {523, 23235}, {576, 36182}, {2080, 37967}, {3090, 5099}, {3091, 16188}, {3303, 6027}, {3304, 6023}, {5171, 13225}, {9181, 15034}, {9218, 15020}, {10303, 16760}, {11477, 33987}

X(38679) = reflection of X(38680) in X(3)

X(38680) = DILATION FROM X(3) OF X(842) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a^2*(a^12 - 6*a^6*b^6 + 3*a^4*b^8 + 6*a^2*b^10 - 4*b^12 - 10*a^8*b^2*c^2 + 16*a^6*b^4*c^2 - 12*a^4*b^6*c^2 - 4*a^2*b^8*c^2 + 10*b^10*c^2 + 16*a^6*b^2*c^4 + 3*a^4*b^4*c^4 - 23*b^8*c^4 - 6*a^6*c^6 - 12*a^4*b^2*c^6 + 34*b^6*c^6 + 3*a^4*c^8 - 4*a^2*b^2*c^8 - 23*b^4*c^8 + 6*a^2*c^10 + 10*b^2*c^10 - 4*c^12) : :

X(38680) lies on these lines: {3, 691}, {23, 9157}, {30, 23235}, {187, 37957}, {249, 15034}, {511, 12112}, {512, 15054}, {2080, 12105}, {3090, 16188}, {3091, 5099}, {3303, 6023}, {3304, 6027}, {3525, 16760}, {5609, 33803}, {9181, 15020}, {11477, 37915}, {11638, 18114}, {15156, 23109}, {15157, 23110}

X(38680) = reflection of X(38679) in X(3)

X(38681) = DILATION FROM X(3) OF X(930) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a^16 - 8*a^14*b^2 + 26*a^12*b^4 - 45*a^10*b^6 + 45*a^8*b^8 - 26*a^6*b^10 + 8*a^4*b^12 - a^2*b^14 - 8*a^14*c^2 + 34*a^12*b^2*c^2 - 55*a^10*b^4*c^2 + 38*a^8*b^6*c^2 - 7*a^6*b^8*c^2 + a^4*b^10*c^2 - 6*a^2*b^12*c^2 + 3*b^14*c^2 + 26*a^12*c^4 - 55*a^10*b^2*c^4 + 35*a^8*b^4*c^4 - 3*a^6*b^6*c^4 - 9*a^4*b^8*c^4 + 24*a^2*b^10*c^4 - 18*b^12*c^4 - 45*a^10*c^6 + 38*a^8*b^2*c^6 - 3*a^6*b^4*c^6 - 17*a^2*b^8*c^6 + 45*b^10*c^6 + 45*a^8*c^8 - 7*a^6*b^2*c^8 - 9*a^4*b^4*c^8 - 17*a^2*b^6*c^8 - 60*b^8*c^8 - 26*a^6*c^10 + a^4*b^2*c^10 + 24*a^2*b^4*c^10 + 45*b^6*c^10 + 8*a^4*c^12 - 6*a^2*b^2*c^12 - 18*b^4*c^12 - a^2*c^14 + 3*b^2*c^14 : :

X(38681) lies on these lines: {3, 252}, {30, 23238}, {128, 3091}, {137, 3090}, {546, 14072}, {632, 6592}, {1263, 3628}, {3303, 3327}, {3304, 7159}, {3525, 13372}, {3627, 14073}, {7604, 15345}, {10303, 34837}, {12026, 14869}, {12811, 23237}, {12812, 25147}, {15022, 23516}, {15027, 34308}, {15054, 15704}

X(38681) = reflection of X(38683) in X(3)

X(38682) = DILATION FROM X(3) OF X(953) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a^2*(a^8 - 2*a^7*b + a^6*b^2 + 6*a^5*b^3 - 9*a^4*b^4 - 6*a^3*b^5 + 11*a^2*b^6 + 2*a*b^7 - 4*b^8 - 2*a^7*c + 8*a^6*b*c - 16*a^5*b^2*c - 2*a^4*b^3*c + 38*a^3*b^4*c - 20*a^2*b^5*c - 20*a*b^6*c + 14*b^7*c + a^6*c^2 - 16*a^5*b*c^2 + 47*a^4*b^2*c^2 - 40*a^3*b^3*c^2 - 33*a^2*b^4*c^2 + 56*a*b^5*c^2 - 15*b^6*c^2 + 6*a^5*c^3 - 2*a^4*b*c^3 - 40*a^3*b^2*c^3 + 88*a^2*b^3*c^3 - 38*a*b^4*c^3 - 14*b^5*c^3 - 9*a^4*c^4 + 38*a^3*b*c^4 - 33*a^2*b^2*c^4 - 38*a*b^3*c^4 + 38*b^4*c^4 - 6*a^3*c^5 - 20*a^2*b*c^5 + 56*a*b^2*c^5 - 14*b^3*c^5 + 11*a^2*c^6 - 20*a*b*c^6 - 15*b^2*c^6 + 2*a*c^7 + 14*b*c^7 - 4*c^8) : :

X(38682) lies on these lines: {3, 901}, {513, 14511}, {517, 3935}, {3025, 3304}, {3090, 31841}, {3091, 3259}, {3303, 13756}, {3746, 23153}, {7991, 34464}, {10303, 22102}, {11518, 24201}

X(38683) = DILATION FROM X(3) OF X(1141) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a^16 - 2*a^14*b^2 - 4*a^12*b^4 + 15*a^10*b^6 - 15*a^8*b^8 + 4*a^6*b^10 + 2*a^4*b^12 - a^2*b^14 - 2*a^14*c^2 - 2*a^12*b^2*c^2 + 17*a^10*b^4*c^2 - 22*a^8*b^6*c^2 + 17*a^6*b^8*c^2 - 17*a^4*b^10*c^2 + 12*a^2*b^12*c^2 - 3*b^14*c^2 - 4*a^12*c^4 + 17*a^10*b^2*c^4 - 13*a^8*b^4*c^4 - 3*a^6*b^6*c^4 + 15*a^4*b^8*c^4 - 30*a^2*b^10*c^4 + 18*b^12*c^4 + 15*a^10*c^6 - 22*a^8*b^2*c^6 - 3*a^6*b^4*c^6 + 19*a^2*b^8*c^6 - 45*b^10*c^6 - 15*a^8*c^8 + 17*a^6*b^2*c^8 + 15*a^4*b^4*c^8 + 19*a^2*b^6*c^8 + 60*b^8*c^8 + 4*a^6*c^10 - 17*a^4*b^2*c^10 - 30*a^2*b^4*c^10 - 45*b^6*c^10 + 2*a^4*c^12 + 12*a^2*b^2*c^12 + 18*b^4*c^12 - a^2*c^14 - 3*b^2*c^14 : :

X(38683) lies on these lines: {3, 252}, {128, 3090}, {137, 3091}, {140, 23238}, {546, 1263}, {632, 12026}, {3146, 11671}, {3303, 7159}, {3304, 3327}, {3525, 34837}, {3627, 7728}, {3628, 14072}, {6592, 14869}, {7527, 23235}, {10303, 13372}, {12811, 25147}, {12812, 23237}, {34308, 36253}

X(38683) = reflection of X(38681) in X(3)

X(38684) = DILATION FROM X(3) OF X(1292) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a*(a^7 - 2*a^6*b + a^5*b^2 - a^3*b^4 + 2*a^2*b^5 - a*b^6 - 2*a^6*c - 5*a^5*b*c + 15*a^4*b^2*c - 18*a^3*b^3*c + 8*a^2*b^4*c - a*b^5*c + 3*b^6*c + a^5*c^2 + 15*a^4*b*c^2 - 2*a^3*b^2*c^2 - 2*a^2*b^3*c^2 - 3*a*b^4*c^2 - 9*b^5*c^2 - 18*a^3*b*c^3 - 2*a^2*b^2*c^3 + 10*a*b^3*c^3 + 6*b^4*c^3 - a^3*c^4 + 8*a^2*b*c^4 - 3*a*b^2*c^4 + 6*b^3*c^4 + 2*a^2*c^5 - a*b*c^5 - 9*b^2*c^5 - a*c^6 + 3*b*c^6) : :

X(38684) lies on these lines: {1, 9519}, {3, 105}, {4, 10712}, {20, 528}, {120, 3091}, {546, 15521}, {1358, 3303}, {2775, 14094}, {2788, 23235}, {2795, 33557}, {2809, 7991}, {2836, 7957}, {3021, 3304}, {3090, 5511}, {3146, 10729}, {3627, 10743}, {5205, 36002}, {6714, 10303}, {7982, 10699}, {10760, 11477}, {11716, 30389}, {16842, 34124}, {26245, 35986}, {34578, 37328}

X(38684) = reflection of X(38670) in X(3)

X(38685) = DILATION FROM X(3) OF X(1293) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a^2*(a^5 - 2*a^4*b - 6*a^3*b^2 + 5*a*b^4 + 2*b^5 - 2*a^4*c + 9*a^3*b*c + 15*a^2*b^2*c - 9*a*b^3*c - 13*b^4*c - 6*a^3*c^2 + 15*a^2*b*c^2 - 32*a*b^2*c^2 + 19*b^3*c^2 - 9*a*b*c^3 + 19*b^2*c^3 + 5*a*c^4 - 13*b*c^4 + 2*c^5) : :

X(38685) lies on these lines: {3, 106}, {4, 10713}, {40, 9519}, {121, 3091}, {546, 15522}, {1357, 3303}, {1768, 2802}, {2776, 14094}, {2789, 23235}, {2796, 5493}, {2842, 15054}, {3090, 5510}, {3146, 10730}, {3304, 6018}, {3627, 10744}, {3699, 31853}, {6715, 10303}, {7982, 10700}, {10761, 11477}, {11717, 30389}

X(38685) = reflection of X(38671) in X(3)

X(38686) = DILATION FROM X(3) OF X(1294) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a^16 + a^14*b^2 - 19*a^12*b^4 + 45*a^10*b^6 - 45*a^8*b^8 + 19*a^6*b^10 - a^4*b^12 - a^2*b^14 + a^14*c^2 + 31*a^12*b^2*c^2 - 43*a^10*b^4*c^2 - 49*a^8*b^6*c^2 + 83*a^6*b^8*c^2 - 11*a^4*b^10*c^2 - 9*a^2*b^12*c^2 - 3*b^14*c^2 - 19*a^12*c^4 - 43*a^10*b^2*c^4 + 188*a^8*b^4*c^4 - 102*a^6*b^6*c^4 - 75*a^4*b^8*c^4 + 33*a^2*b^10*c^4 + 18*b^12*c^4 + 45*a^10*c^6 - 49*a^8*b^2*c^6 - 102*a^6*b^4*c^6 + 174*a^4*b^6*c^6 - 23*a^2*b^8*c^6 - 45*b^10*c^6 - 45*a^8*c^8 + 83*a^6*b^2*c^8 - 75*a^4*b^4*c^8 - 23*a^2*b^6*c^8 + 60*b^8*c^8 + 19*a^6*c^10 - 11*a^4*b^2*c^10 + 33*a^2*b^4*c^10 - 45*b^6*c^10 - a^4*c^12 - 9*a^2*b^2*c^12 + 18*b^4*c^12 - a^2*c^14 - 3*b^2*c^14 : :

X(38686) lies on these lines: {3, 107}, {4, 10714}, {20, 648}, {122, 3091}, {133, 3090}, {546, 22337}, {1657, 23241}, {2777, 3529}, {2790, 23235}, {3146, 3346}, {3303, 3324}, {3304, 7158}, {3525, 34842}, {3627, 10745}, {5667, 17538}, {6716, 10303}, {7982, 10701}, {9033, 15054}, {9528, 33557}, {10762, 11477}, {11718, 30389}, {12103, 23240}

X(38686) = reflection of X(38672) in X(3)

X(38687) = DILATION FROM X(3) OF X(1295) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a*(a^12 - a^11*b - 3*a^10*b^2 + 3*a^9*b^3 + 2*a^8*b^4 - 2*a^7*b^5 + 2*a^6*b^6 - 2*a^5*b^7 - 3*a^4*b^8 + 3*a^3*b^9 + a^2*b^10 - a*b^11 - a^11*c + 11*a^10*b*c - 7*a^9*b^2*c - 19*a^8*b^3*c + 26*a^7*b^4*c - 6*a^6*b^5*c - 26*a^5*b^6*c + 22*a^4*b^7*c + 7*a^3*b^8*c - 5*a^2*b^9*c + a*b^10*c - 3*b^11*c - 3*a^10*c^2 - 7*a^9*b*c^2 + 38*a^8*b^2*c^2 - 24*a^7*b^3*c^2 - 46*a^6*b^4*c^2 + 58*a^5*b^5*c^2 - 4*a^4*b^6*c^2 - 16*a^3*b^7*c^2 + 9*a^2*b^8*c^2 - 11*a*b^9*c^2 + 6*b^10*c^2 + 3*a^9*c^3 - 19*a^8*b*c^3 - 24*a^7*b^2*c^3 + 100*a^6*b^3*c^3 - 30*a^5*b^4*c^3 - 54*a^4*b^5*c^3 + 40*a^3*b^6*c^3 - 36*a^2*b^7*c^3 + 11*a*b^8*c^3 + 9*b^9*c^3 + 2*a^8*c^4 + 26*a^7*b*c^4 - 46*a^6*b^2*c^4 - 30*a^5*b^3*c^4 + 78*a^4*b^4*c^4 - 34*a^3*b^5*c^4 - 10*a^2*b^6*c^4 + 38*a*b^7*c^4 - 24*b^8*c^4 - 2*a^7*c^5 - 6*a^6*b*c^5 + 58*a^5*b^2*c^5 - 54*a^4*b^3*c^5 - 34*a^3*b^4*c^5 + 82*a^2*b^5*c^5 - 38*a*b^6*c^5 - 6*b^7*c^5 + 2*a^6*c^6 - 26*a^5*b*c^6 - 4*a^4*b^2*c^6 + 40*a^3*b^3*c^6 - 10*a^2*b^4*c^6 - 38*a*b^5*c^6 + 36*b^6*c^6 - 2*a^5*c^7 + 22*a^4*b*c^7 - 16*a^3*b^2*c^7 - 36*a^2*b^3*c^7 + 38*a*b^4*c^7 - 6*b^5*c^7 - 3*a^4*c^8 + 7*a^3*b*c^8 + 9*a^2*b^2*c^8 + 11*a*b^3*c^8 - 24*b^4*c^8 + 3*a^3*c^9 - 5*a^2*b*c^9 - 11*a*b^2*c^9 + 9*b^3*c^9 + a^2*c^10 + a*b*c^10 + 6*b^2*c^10 - a*c^11 - 3*b*c^11) : :

X(38687) lies on these lines: {3, 108}, {4, 10715}, {123, 3091}, {546, 33566}, {1359, 3303}, {2778, 14094}, {2791, 23235}, {2817, 2956}, {2829, 3529}, {2850, 15054}, {3090, 25640}, {3146, 10731}, {3304, 3318}, {3627, 10746}, {6717, 10303}, {7982, 10702}, {9528, 21669}, {10763, 11477}, {11719, 30389}

X(38687) = reflection of X(38673) in X(3)

X(38688) = DILATION FROM X(3) OF X(1296) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a^2*(a^8 - 7*a^6*b^2 - 3*a^4*b^4 + 7*a^2*b^6 + 2*b^8 - 7*a^6*c^2 + 63*a^4*b^2*c^2 - 45*a^2*b^4*c^2 - 19*b^6*c^2 - 3*a^4*c^4 - 45*a^2*b^2*c^4 + 66*b^4*c^4 + 7*a^2*c^6 - 19*b^2*c^6 + 2*c^8) : :

X(38688) lies on these lines: {3, 111}, {4, 10717}, {20, 543}, {126, 3091}, {524, 14514}, {546, 22338}, {548, 14666}, {576, 35687}, {1657, 32424}, {2780, 14094}, {2793, 23235}, {2854, 15054}, {3090, 5512}, {3146, 10734}, {3303, 3325}, {3304, 6019}, {3523, 9172}, {3529, 23699}, {3627, 10748}, {6453, 11835}, {6454, 11836}, {6519, 11833}, {6522, 11834}, {6719, 10303}, {7833, 11162}, {7982, 10704}, {9129, 15020}, {9146, 15098}, {10541, 28662}, {10765, 11477}, {11615, 19901}, {11721, 30389}, {14654, 17538}, {14688, 36696}

X(38688) = reflection of X(38675) in X(3)

X(38689) = DILATION FROM X(3) OF X(1297) TO THE CIRCLE Γ(X(3),X(14094))

Barycentrics    a^2*(a^12 + a^10*b^2 - 2*a^8*b^4 - 6*a^6*b^6 + 5*a^4*b^8 + 5*a^2*b^10 - 4*b^12 + a^10*c^2 - 3*a^8*b^2*c^2 + 8*a^6*b^4*c^2 - 8*a^4*b^6*c^2 - a^2*b^8*c^2 + 3*b^10*c^2 - 2*a^8*c^4 + 8*a^6*b^2*c^4 + 6*a^4*b^4*c^4 - 4*a^2*b^6*c^4 - 8*b^8*c^4 - 6*a^6*c^6 - 8*a^4*b^2*c^6 - 4*a^2*b^4*c^6 + 18*b^6*c^6 + 5*a^4*c^8 - a^2*b^2*c^8 - 8*b^4*c^8 + 5*a^2*c^10 + 3*b^2*c^10 - 4*c^12) : :

X(38689) lies on these lines: {3, 112}, {4, 9530}, {127, 3091}, {132, 3090}, {185, 13247}, {376, 14900}, {511, 13509}, {546, 12918}, {1498, 2781}, {2794, 3529}, {3146, 10735}, {3303, 3320}, {3304, 6020}, {3525, 34841}, {3592, 19115}, {3594, 19114}, {3627, 10749}, {3746, 13116}, {5076, 19163}, {5198, 12145}, {5523, 15312}, {5563, 13117}, {6419, 19094}, {6420, 19093}, {6453, 35880}, {6454, 35881}, {6720, 10303}, {7982, 10705}, {7991, 12408}, {9517, 15054}, {10222, 13099}, {10541, 28343}, {10766, 11477}, {11403, 13166}, {11722, 30389}, {12088, 15562}, {13118, 37622}, {13200, 17538}

X(38689) = reflection of X(38676) in X(3)

X(38690) = DILATION FROM X(3) OF X(101) TO THE CIRCLE Γ(X(3),X(15035))

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

X(38690) lies on these lines: {3, 101}, {4, 6710}, {5, 10725}, {20, 118}, {36, 59}, {40, 10697}, {116, 631}, {140, 10739}, {150, 3523}, {152, 3522}, {165, 15735}, {182, 10756}, {376, 10710}, {544, 3524}, {549, 10708}, {550, 10741}, {664, 31852}, {962, 11728}, {1282, 7987}, {1350, 10758}, {1362, 5204}, {1385, 10695}, {2772, 15055}, {2774, 15035}, {2784, 10164}, {2786, 21166}, {2801, 15015}, {2809, 3576}, {2810, 5085}, {2811, 23239}, {3022, 5217}, {3515, 5185}, {3528, 35024}, {3529, 20401}, {3601, 11028}, {3612, 18413}, {3887, 34474}, {4297, 28346}, {4845, 5010}, {5732, 28345}, {6713, 10770}, {10470, 38479}, {10772, 24466}, {15717, 20096}, {15856, 16192}, {21735, 33521}

X(38690) = reflection of X(38692) in X(3)

X(38691) = DILATION FROM X(3) OF X(102) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    ^2*(3*a^8 - 3*a^7*b - 5*a^6*b^2 + 9*a^5*b^3 - 3*a^4*b^4 - 9*a^3*b^5 + 9*a^2*b^6 + 3*a*b^7 - 4*b^8 - 3*a^7*c + 9*a^6*b*c - 5*a^5*b^2*c - 13*a^4*b^3*c + 19*a^3*b^4*c - a^2*b^5*c - 11*a*b^6*c + 5*b^7*c - 5*a^6*c^2 - 5*a^5*b*c^2 + 28*a^4*b^2*c^2 - 10*a^3*b^3*c^2 - 21*a^2*b^4*c^2 + 15*a*b^5*c^2 - 2*b^6*c^2 + 9*a^5*c^3 - 13*a^4*b*c^3 - 10*a^3*b^2*c^3 + 26*a^2*b^3*c^3 - 7*a*b^4*c^3 - 5*b^5*c^3 - 3*a^4*c^4 + 19*a^3*b*c^4 - 21*a^2*b^2*c^4 - 7*a*b^3*c^4 + 12*b^4*c^4 - 9*a^3*c^5 - a^2*b*c^5 + 15*a*b^2*c^5 - 5*b^3*c^5 + 9*a^2*c^6 - 11*a*b*c^6 - 2*b^2*c^6 + 3*a*c^7 + 5*b*c^7 - 4*c^8) : :

X(38691) lies on these lines: {3, 102}, {4, 6711}, {5, 10726}, {20, 124}, {40, 10703}, {117, 631}, {140, 10740}, {151, 3523}, {165, 2800}, {182, 10757}, {376, 10716}, {549, 10709}, {550, 10747}, {962, 11734}, {1350, 10764}, {1361, 5217}, {1364, 5204}, {1385, 10696}, {1795, 7280}, {1845, 3612}, {2773, 15055}, {2779, 15035}, {2785, 34473}, {2792, 21166}, {2816, 10165}, {2817, 3576}, {2849, 14414}, {3522, 33650}, {4297, 13532}, {6713, 10771}, {7987, 11700}, {10777, 24466}, {12016, 15803}, {14110, 34242}, {14690, 35242}

X(38691) = reflection of X(38697) in X(3)

X(38692) = DILATION FROM X(3) OF X(103) TO THE CIRCLE Γ(X(3),X(15035))

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

X(38692) lies on these lines: {3, 101}, {4, 6712}, {5, 10727}, {20, 116}, {40, 10695}, {118, 631}, {140, 10741}, {150, 3522}, {152, 3523}, {165, 2809}, {182, 10758}, {376, 10708}, {544, 10304}, {549, 10710}, {550, 10739}, {962, 11726}, {1092, 3046}, {1282, 16192}, {1350, 10756}, {1362, 5217}, {1385, 10697}, {2772, 15035}, {2774, 15055}, {2784, 21166}, {2786, 34473}, {2801, 21165}, {2810, 31884}, {2822, 21162}, {2823, 21164}, {3022, 5204}, {3361, 14760}, {3516, 5185}, {6713, 10772}, {7987, 11712}, {10770, 24466}, {11028, 15803}, {15735, 17502}, {20096, 21734}, {21735, 33520}

X(38692) = reflection of X(38690) in X(3)

X(38693) = DILATION FROM X(3) OF X(104) TO THE CIRCLE Γ(X(3),X(15035))

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

X(38693) lies on these lines: {2, 2829}, {3, 8}, {4, 6713}, {5, 10728}, {11, 20}, {21, 10165}, {35, 10074}, {36, 516}, {40, 1320}, {80, 4297}, {119, 631}, {140, 10742}, {149, 3522}, {153, 2551}, {165, 2802}, {182, 10759}, {214, 1768}, {371, 19081}, {372, 19082}, {376, 5840}, {381, 34126}, {404, 5450}, {411, 7280}, {515, 13587}, {528, 10304}, {548, 1484}, {549, 10711}, {550, 10738}, {962, 1387}, {963, 1811}, {971, 35271}, {1006, 37822}, {1012, 9779}, {1071, 12532}, {1092, 3045}, {1125, 34789}, {1151, 19113}, {1152, 19112}, {1156, 5732}, {1317, 5217}, {1350, 10755}, {1385, 10698}, {1420, 15558}, {1537, 3616}, {1587, 13913}, {1588, 13977}, {1621, 6950}, {1656, 22799}, {1657, 22938}, {1699, 32557}, {1862, 3516}, {2077, 13278}, {2771, 10167}, {2783, 21166}, {2787, 34473}, {2800, 3576}, {2801, 15015}, {2828, 23239}, {2950, 5250}, {3091, 6667}, {3184, 10775}, {3361, 18240}, {3486, 12832}, {3515, 12138}, {3528, 13199}, {3530, 11698}, {3545, 38069}, {3579, 12737}, {3601, 5083}, {3612, 11570}, {3830, 38141}, {3868, 15528}, {3876, 12665}, {4188, 12114}, {4189, 22775}, {4293, 6966}, {4299, 6943}, {4302, 5533}, {4881, 6001}, {4973, 5538}, {4999, 37163}, {5010, 10087}, {5044, 17661}, {5046, 12761}, {5126, 17613}, {5171, 12199}, {5188, 32454}, {5218, 10956}, {5253, 5886}, {5260, 6940}, {5267, 5660}, {5284, 6914}, {5432, 12763}, {5433, 12764}, {5541, 16192}, {5584, 22560}, {5603, 38032}, {5691, 6702}, {5817, 37249}, {5842, 36004}, {5848, 25406}, {5851, 37106}, {5854, 11194}, {6174, 15692}, {6256, 17566}, {6264, 35242}, {6265, 13624}, {6326, 10884}, {6684, 12751}, {6691, 13729}, {6840, 13273}, {6905, 23961}, {6912, 7988}, {6948, 11680}, {6952, 38109}, {6955, 33108}, {6961, 11681}, {6972, 7354}, {8666, 25438}, {8674, 15055}, {9024, 31884}, {9342, 18515}, {9812, 32558}, {9943, 17638}, {10057, 21578}, {10172, 17531}, {10175, 36006}, {10265, 12119}, {10267, 12776}, {10283, 37535}, {10303, 31235}, {10470, 35649}, {10767, 16111}, {10778, 16163}, {10780, 14689}, {10993, 21735}, {11012, 13279}, {11230, 28461}, {11500, 37307}, {11571, 37616}, {12332, 17548}, {12512, 21630}, {12528, 18254}, {12619, 18481}, {12700, 26286}, {12736, 15803}, {12739, 18444}, {12740, 37605}, {12758, 37618}, {13194, 37479}, {13222, 37198}, {13253, 30389}, {13274, 15338}, {13528, 38460}, {13747, 33898}, {14151, 30282}, {14217, 31730}, {14269, 38084}, {14800, 37701}, {14853, 38119}, {15717, 37725}, {16370, 21151}, {16371, 34122}, {17652, 31798}, {17654, 31786}, {20095, 21734}, {20586, 37568}, {22765, 28212}, {22935, 26878}, {24813, 36237}, {25440, 37712}, {26321, 38138}, {26877, 34195}, {28172, 36002}, {33970, 34547}, {35010, 35016}, {35202, 35204}, {37306, 38122}

X(38693) = reflection of X(34474) in X(3)

X(38694) = DILATION FROM X(3) OF X(105) TO THE CIRCLE Γ(X(3),X(15035))

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

X(38694) lies on these lines: {3, 105}, {4, 6714}, {5, 10729}, {20, 5511}, {40, 11716}, {120, 631}, {140, 10743}, {182, 10760}, {528, 3524}, {549, 10712}, {550, 15521}, {1358, 5204}, {1385, 10699}, {2077, 37815}, {2775, 15055}, {2788, 34473}, {2795, 21161}, {2809, 3576}, {2833, 23239}, {2835, 21164}, {2836, 15035}, {3021, 5217}, {3522, 34547}, {3523, 20344}, {5540, 7987}, {6713, 10773}, {15717, 20097}, {16371, 34124}, {24466, 33970}

X(38694) = reflection of X(38712) in X(3)

X(38695) = DILATION FROM X(3) OF X(106) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    a^2*(3*a^5 - 6*a^4*b - 8*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 - 4*b^5 - 6*a^4*c + 27*a^3*b*c - 5*a^2*b^2*c - 27*a*b^3*c + 11*b^4*c - 8*a^3*c^2 - 5*a^2*b*c^2 + 24*a*b^2*c^2 - 3*b^3*c^2 + 10*a^2*c^3 - 27*a*b*c^3 - 3*b^2*c^3 + 5*a*c^4 + 11*b*c^4 - 4*c^5) : :

X(38695) lies on these lines: {1, 14664}, {3, 106}, {4, 6715}, {5, 10730}, {20, 5510}, {40, 11717}, {121, 631}, {140, 10744}, {182, 10761}, {549, 10713}, {550, 15522}, {1054, 7987}, {1357, 5204}, {1385, 10700}, {2776, 15055}, {2789, 34473}, {2796, 21166}, {2802, 3576}, {2810, 5085}, {2839, 23239}, {2842, 15035}, {3522, 34548}, {3523, 21290}, {5217, 6018}, {6713, 10774}, {10470, 38478}, {13541, 30389}, {15717, 20098}, {37497, 37999}

X(38695) = reflection of X(38713) in X(3)

X(38696) = DILATION FROM X(3) OF X(108) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    a*(3*a^12 - 3*a^11*b - 9*a^10*b^2 + 9*a^9*b^3 + 6*a^8*b^4 - 6*a^7*b^5 + 6*a^6*b^6 - 6*a^5*b^7 - 9*a^4*b^8 + 9*a^3*b^9 + 3*a^2*b^10 - 3*a*b^11 - 3*a^11*c + 13*a^10*b*c - a^9*b^2*c - 27*a^8*b^3*c + 18*a^7*b^4*c + 2*a^6*b^5*c - 18*a^5*b^6*c + 26*a^4*b^7*c + a^3*b^8*c - 15*a^2*b^9*c + 3*a*b^10*c + b^11*c - 9*a^10*c^2 - a^9*b*c^2 + 34*a^8*b^2*c^2 - 12*a^7*b^3*c^2 - 38*a^6*b^4*c^2 + 34*a^5*b^5*c^2 + 8*a^4*b^6*c^2 - 28*a^3*b^7*c^2 + 7*a^2*b^8*c^2 + 7*a*b^9*c^2 - 2*b^10*c^2 + 9*a^9*c^3 - 27*a^8*b*c^3 - 12*a^7*b^2*c^3 + 60*a^6*b^3*c^3 - 10*a^5*b^4*c^3 - 42*a^4*b^5*c^3 + 20*a^3*b^6*c^3 + 12*a^2*b^7*c^3 - 7*a*b^8*c^3 - 3*b^9*c^3 + 6*a^8*c^4 + 18*a^7*b*c^4 - 38*a^6*b^2*c^4 - 10*a^5*b^3*c^4 + 34*a^4*b^4*c^4 - 2*a^3*b^5*c^4 - 10*a^2*b^6*c^4 - 6*a*b^7*c^4 + 8*b^8*c^4 - 6*a^7*c^5 + 2*a^6*b*c^5 + 34*a^5*b^2*c^5 - 42*a^4*b^3*c^5 - 2*a^3*b^4*c^5 + 6*a^2*b^5*c^5 + 6*a*b^6*c^5 + 2*b^7*c^5 + 6*a^6*c^6 - 18*a^5*b*c^6 + 8*a^4*b^2*c^6 + 20*a^3*b^3*c^6 - 10*a^2*b^4*c^6 + 6*a*b^5*c^6 - 12*b^6*c^6 - 6*a^5*c^7 + 26*a^4*b*c^7 - 28*a^3*b^2*c^7 + 12*a^2*b^3*c^7 - 6*a*b^4*c^7 + 2*b^5*c^7 - 9*a^4*c^8 + a^3*b*c^8 + 7*a^2*b^2*c^8 - 7*a*b^3*c^8 + 8*b^4*c^8 + 9*a^3*c^9 - 15*a^2*b*c^9 + 7*a*b^2*c^9 - 3*b^3*c^9 + 3*a^2*c^10 + 3*a*b*c^10 - 2*b^2*c^10 - 3*a*c^11 + b*c^11) : :

X(38696) lies on these lines: {2, 2829}, {3, 108}, {4, 6717}, {5, 10731}, {20, 25640}, {40, 11719}, {123, 631}, {140, 10746}, {182, 10763}, {549, 10715}, {550, 33566}, {1359, 5204}, {1385, 10702}, {2778, 15055}, {2791, 34473}, {2798, 21166}, {2804, 34474}, {2817, 3576}, {2823, 21164}, {2845, 23239}, {2850, 15035}, {3318, 5217}, {3522, 34550}, {3523, 34188}, {6713, 10776}

X(38696) = reflection of X(38715) in X(3)

X(38697) = DILATION FROM X(3) OF X(109) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    a^2*(3*a^8 - 3*a^7*b - 7*a^6*b^2 + 9*a^5*b^3 + 3*a^4*b^4 - 9*a^3*b^5 + 3*a^2*b^6 + 3*a*b^7 - 2*b^8 - 3*a^7*c + 9*a^6*b*c - a^5*b^2*c - 17*a^4*b^3*c + 11*a^3*b^4*c + 7*a^2*b^5*c - 7*a*b^6*c + b^7*c - 7*a^6*c^2 - a^5*b*c^2 + 20*a^4*b^2*c^2 - 2*a^3*b^3*c^2 - 15*a^2*b^4*c^2 + 3*a*b^5*c^2 + 2*b^6*c^2 + 9*a^5*c^3 - 17*a^4*b*c^3 - 2*a^3*b^2*c^3 + 10*a^2*b^3*c^3 + a*b^4*c^3 - b^5*c^3 + 3*a^4*c^4 + 11*a^3*b*c^4 - 15*a^2*b^2*c^4 + a*b^3*c^4 - 9*a^3*c^5 + 7*a^2*b*c^5 + 3*a*b^2*c^5 - b^3*c^5 + 3*a^2*c^6 - 7*a*b*c^6 + 2*b^2*c^6 + 3*a*c^7 + b*c^7 - 2*c^8) : :

X(38697) lies on these lines: {1, 14690}, {3, 102}, {4, 6718}, {5, 10732}, {20, 117}, {35, 947}, {40, 10696}, {59, 2077}, {124, 631}, {140, 10747}, {151, 3522}, {165, 2817}, {182, 10764}, {376, 10709}, {549, 10716}, {550, 10740}, {962, 11727}, {1350, 10757}, {1361, 5204}, {1364, 5217}, {1385, 10703}, {2773, 15035}, {2779, 15055}, {2785, 21166}, {2792, 34473}, {2800, 3576}, {2835, 21164}, {2846, 23239}, {3523, 6711}, {3601, 12016}, {3738, 34474}, {6684, 13532}, {6713, 10777}, {7987, 11713}, {10771, 24466}, {14217, 29008}, {18339, 38554}, {34242, 34339}

X(38697) = reflection of X(38691) in X(3)

X(38698) = DILATION FROM X(3) OF X(111) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    a^2*(3*a^8 - 11*a^6*b^2 + a^4*b^4 + 11*a^2*b^6 - 4*b^8 - 11*a^6*c^2 + 49*a^4*b^2*c^2 - 35*a^2*b^4*c^2 + 13*b^6*c^2 + a^4*c^4 - 35*a^2*b^2*c^4 - 2*b^4*c^4 + 11*a^2*c^6 + 13*b^2*c^6 - 4*c^8) : :

X(38698) lies on these lines: {2, 23699}, {3, 111}, {4, 6719}, {5, 10734}, {20, 5512}, {40, 11721}, {74, 9129}, {126, 631}, {140, 10748}, {182, 10765}, {186, 15560}, {371, 11833}, {372, 11834}, {376, 9172}, {477, 9179}, {511, 5166}, {543, 3524}, {549, 10717}, {550, 22338}, {1350, 28662}, {1385, 10704}, {2696, 36168}, {2780, 15055}, {2793, 34473}, {2805, 34474}, {2847, 23239}, {2854, 5085}, {3325, 5204}, {3523, 14360}, {5054, 32424}, {5217, 6019}, {6713, 10779}, {8722, 34241}, {9126, 9156}, {14657, 17928}, {15717, 20099}, {21163, 34010}

X(38698) = reflection of X(38716) in X(3)
X(38698) = X(111)-Gibert-Moses centroid
X(38698) = centroid of X(74)X(110)X(111)
X(38698) = centroid of X(98)X(99)X(111)

X(38699) = DILATION FROM X(3) OF X(112) TO THE CIRCLE Γ(X(3),X(15035))

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

X(38699) lies on these lines: {2, 2794}, {3, 112}, {4, 6720}, {5, 10735}, {20, 132}, {35, 13312}, {36, 13311}, {40, 11722}, {127, 631}, {140, 10749}, {182, 10766}, {250, 2071}, {371, 19114}, {372, 19115}, {511, 16225}, {549, 10718}, {550, 12918}, {1151, 19094}, {1152, 19093}, {1350, 28343}, {1385, 10705}, {1587, 13923}, {1588, 13992}, {1656, 19163}, {1657, 19160}, {2077, 13118}, {2781, 5085}, {2799, 21166}, {2806, 34474}, {2848, 23239}, {3060, 16224}, {3320, 5204}, {3515, 13166}, {3516, 12145}, {3522, 12384}, {3523, 13219}, {3528, 12253}, {4297, 12784}, {5010, 13116}, {5171, 13195}, {5217, 6020}, {5432, 13296}, {5433, 13297}, {5481, 18876}, {5584, 19159}, {5968, 6091}, {6684, 13280}, {6713, 10780}, {7280, 13117}, {7503, 19164}, {7987, 12265}, {9517, 15035}, {9530, 10304}, {10267, 13314}, {10269, 13313}, {11012, 13119}, {11610, 13335}, {12207, 14676}, {12408, 16192}, {12413, 37198}, {12945, 15326}, {12955, 15338}, {14070, 20410}, {17928, 19165}, {22467, 34217}

X(38699) = reflection of X(38717) in X(3)
X(38699) = X(112)-Gibert-Moses centroid
X(38699) = centroid of X(74)X(110)X(112)
X(38699) = centroid of X(98)X(99)X(112)

X(38700) = DILATION FROM X(3) OF X(476) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    3*a^16 - 10*a^14*b^2 + 8*a^12*b^4 + 5*a^10*b^6 - 5*a^8*b^8 - 8*a^6*b^10 + 10*a^4*b^12 - 3*a^2*b^14 - 10*a^14*c^2 + 34*a^12*b^2*c^2 - 35*a^10*b^4*c^2 - 6*a^8*b^6*c^2 + 37*a^6*b^8*c^2 - 25*a^4*b^10*c^2 + 4*a^2*b^12*c^2 + b^14*c^2 + 8*a^12*c^4 - 35*a^10*b^2*c^4 + 57*a^8*b^4*c^4 - 33*a^6*b^6*c^4 + 3*a^4*b^8*c^4 + 6*a^2*b^10*c^4 - 6*b^12*c^4 + 5*a^10*c^6 - 6*a^8*b^2*c^6 - 33*a^6*b^4*c^6 + 24*a^4*b^6*c^6 - 7*a^2*b^8*c^6 + 15*b^10*c^6 - 5*a^8*c^8 + 37*a^6*b^2*c^8 + 3*a^4*b^4*c^8 - 7*a^2*b^6*c^8 - 20*b^8*c^8 - 8*a^6*c^10 - 25*a^4*b^2*c^10 + 6*a^2*b^4*c^10 + 15*b^6*c^10 + 10*a^4*c^12 + 4*a^2*b^2*c^12 - 6*b^4*c^12 - 3*a^2*c^14 + b^2*c^14 : :

X(38700) lies on these lines: {3, 476}, {4, 22104}, {20, 14989}, {30, 14644}, {74, 7471}, {140, 20957}, {186, 30716}, {523, 15035}, {548, 18319}, {549, 34312}, {631, 3258}, {1296, 9179}, {1511, 14480}, {1553, 12244}, {3233, 14094}, {3522, 34193}, {3523, 14731}, {3567, 16978}, {5204, 33964}, {5217, 33965}, {5627, 17702}, {6070, 12383}, {6699, 17511}, {9158, 18579}, {10721, 36169}, {12041, 14508}, {12052, 15024}, {12121, 34209}, {14611, 15034}, {14934, 15051}, {15059, 36184}, {15078, 15111}, {16111, 36172}, {32110, 36188}

X(38700) = reflection of X(38701) in X(3)
X(38700) = X(476)-Gibert-Moses centroid
X(38700) = centroid of X(74)X(110)X(476)
X(38700) = centroid of X(98)X(99)X(476)

X(38701) = DILATION FROM X(3) OF X(477) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    3*a^16 - 8*a^14*b^2 - 2*a^12*b^4 + 25*a^10*b^6 - 25*a^8*b^8 + 2*a^6*b^10 + 8*a^4*b^12 - 3*a^2*b^14 - 8*a^14*c^2 + 38*a^12*b^2*c^2 - 43*a^10*b^4*c^2 - 18*a^8*b^6*c^2 + 53*a^6*b^8*c^2 - 23*a^4*b^10*c^2 + 2*a^2*b^12*c^2 - b^14*c^2 - 2*a^12*c^4 - 43*a^10*b^2*c^4 + 105*a^8*b^4*c^4 - 57*a^6*b^6*c^4 - 21*a^4*b^8*c^4 + 12*a^2*b^10*c^4 + 6*b^12*c^4 + 25*a^10*c^6 - 18*a^8*b^2*c^6 - 57*a^6*b^4*c^6 + 72*a^4*b^6*c^6 - 11*a^2*b^8*c^6 - 15*b^10*c^6 - 25*a^8*c^8 + 53*a^6*b^2*c^8 - 21*a^4*b^4*c^8 - 11*a^2*b^6*c^8 + 20*b^8*c^8 + 2*a^6*c^10 - 23*a^4*b^2*c^10 + 12*a^2*b^4*c^10 - 15*b^6*c^10 + 8*a^4*c^12 + 2*a^2*b^2*c^12 + 6*b^4*c^12 - 3*a^2*c^14 - b^2*c^14 : :

X(38701) lies on these lines: {3, 476}, {4, 31379}, {5, 14989}, {20, 3258}, {30, 14643}, {74, 14480}, {110, 14508}, {376, 34312}, {523, 15055}, {550, 20957}, {631, 25641}, {2071, 21166}, {3154, 10733}, {3233, 15020}, {3522, 14731}, {3523, 22104}, {3530, 18319}, {5204, 33965}, {5217, 33964}, {5627, 15061}, {5972, 36172}, {7471, 15051}, {12079, 15057}, {12121, 16340}, {14611, 15054}, {15059, 34150}, {16163, 17511}, {23239, 37941}

X(38701) = reflection of X(38700) in X(3)
X(38701) = X(477)-Gibert-Moses centroid
X(38701) = centroid of X(74)X(110)X(477)
X(38701) = centroid of X(98)X(99)X(477)

X(38702) = DILATION FROM X(3) OF X(691) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    a^2*(3*a^12 - 10*a^10*b^2 + 10*a^8*b^4 + 2*a^6*b^6 - 11*a^4*b^8 + 8*a^2*b^10 - 2*b^12 - 10*a^10*c^2 + 30*a^8*b^2*c^2 - 32*a^6*b^4*c^2 + 24*a^4*b^6*c^2 - 12*a^2*b^8*c^2 + 10*a^8*c^4 - 32*a^6*b^2*c^4 + 9*a^4*b^4*c^4 + 11*b^8*c^4 + 2*a^6*c^6 + 24*a^4*b^2*c^6 - 18*b^6*c^6 - 11*a^4*c^8 - 12*a^2*b^2*c^8 + 11*b^4*c^8 + 8*a^2*c^10 - 2*c^12) : :

X(38702) lies on these lines: {3, 691}, {20, 16188}, {30, 9166}, {74, 9181}, {98, 7472}, {186, 15560}, {187, 7464}, {249, 5663}, {316, 15122}, {511, 2071}, {512, 15035}, {523, 21166}, {631, 5099}, {2080, 37950}, {2693, 2715}, {3523, 16760}, {5204, 6023}, {5217, 6027}, {6036, 36174}, {10420, 23700}, {10722, 36170}, {12117, 16092}, {13449, 30745}, {14693, 18325}, {32609, 33803}, {35268, 37918}

X(38702) = reflection of X(38704) in X(3)

X(38703) = DILATION FROM X(3) OF X(805) TO THE CIRCLE Γ(X(3),X(15035))

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

X(38703) lies on these lines: {3, 805}, {4, 22103}, {20, 33330}, {511, 21445}, {512, 21166}, {631, 2679}, {3567, 16979}, {6036, 31513}, {6071, 13172}, {6072, 9862}, {12042, 14510}, {14509, 33813}

X(38704) = DILATION FROM X(3) OF X(842) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    a^2*(3*a^12 - 8*a^10*b^2 + 8*a^8*b^4 - 2*a^6*b^6 - 7*a^4*b^8 + 10*a^2*b^10 - 4*b^12 - 8*a^10*c^2 + 18*a^8*b^2*c^2 - 16*a^6*b^4*c^2 + 12*a^4*b^6*c^2 - 12*a^2*b^8*c^2 + 6*b^10*c^2 + 8*a^8*c^4 - 16*a^6*b^2*c^4 + 9*a^4*b^4*c^4 - 5*b^8*c^4 - 2*a^6*c^6 + 12*a^4*b^2*c^6 + 6*b^6*c^6 - 7*a^4*c^8 - 12*a^2*b^2*c^8 - 5*b^4*c^8 + 10*a^2*c^10 + 6*b^2*c^10 - 4*c^12) : :

X(38704) lies on these lines: {3, 691}, {4, 16760}, {20, 5099}, {23, 18860}, {30, 10242}, {99, 36166}, {186, 249}, {187, 37952}, {316, 10295}, {512, 15055}, {523, 34473}, {620, 36173}, {625, 10296}, {631, 16188}, {1304, 2710}, {2080, 18571}, {3098, 12157}, {5204, 6027}, {5217, 6023}, {5663, 33803}, {7575, 35002}, {9181, 15051}, {9734, 37991}, {10425, 32710}, {10723, 14120}, {15646, 38225}

X(38704) = reflection of X(38702) in X(3)

X(38705) = DILATION FROM X(3) OF X(901) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    a^2*(3*a^8 - 6*a^7*b - 7*a^6*b^2 + 18*a^5*b^3 + 3*a^4*b^4 - 18*a^3*b^5 + 3*a^2*b^6 + 6*a*b^7 - 2*b^8 - 6*a^7*c + 24*a^6*b*c - 8*a^5*b^2*c - 46*a^4*b^3*c + 34*a^3*b^4*c + 20*a^2*b^5*c - 20*a*b^6*c + 2*b^7*c - 7*a^6*c^2 - 8*a^5*b*c^2 + 41*a^4*b^2*c^2 - 39*a^2*b^4*c^2 + 8*a*b^5*c^2 + 5*b^6*c^2 + 18*a^5*c^3 - 46*a^4*b*c^3 + 24*a^2*b^3*c^3 + 6*a*b^4*c^3 - 2*b^5*c^3 + 3*a^4*c^4 + 34*a^3*b*c^4 - 39*a^2*b^2*c^4 + 6*a*b^3*c^4 - 6*b^4*c^4 - 18*a^3*c^5 + 20*a^2*b*c^5 + 8*a*b^2*c^5 - 2*b^3*c^5 + 3*a^2*c^6 - 20*a*b*c^6 + 5*b^2*c^6 + 6*a*c^7 + 2*b*c^7 - 2*c^8) : :

X(38705) lies on these lines: {3, 901}, {4, 22102}, {20, 31841}, {513, 34474}, {631, 3259}, {3025, 5217}, {3601, 33645}, {5204, 13756}, {6073, 12248}, {6075, 13199}, {6713, 31512}, {10016, 10323}, {14513, 33814}, {15803, 24201}, {16192, 34464}

X(38705) = reflection of X(38707) in X(3)

X(38706) = DILATION FROM X(3) OF X(930) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    3*a^16 - 16*a^14*b^2 + 38*a^12*b^4 - 55*a^10*b^6 + 55*a^8*b^8 - 38*a^6*b^10 + 16*a^4*b^12 - 3*a^2*b^14 - 16*a^14*c^2 + 54*a^12*b^2*c^2 - 69*a^10*b^4*c^2 + 34*a^8*b^6*c^2 + 11*a^6*b^8*c^2 - 21*a^4*b^10*c^2 + 6*a^2*b^12*c^2 + b^14*c^2 + 38*a^12*c^4 - 69*a^10*b^2*c^4 + 41*a^8*b^4*c^4 - 9*a^6*b^6*c^4 + 5*a^4*b^8*c^4 - 6*b^12*c^4 - 55*a^10*c^6 + 34*a^8*b^2*c^6 - 9*a^6*b^4*c^6 - 3*a^2*b^8*c^6 + 15*b^10*c^6 + 55*a^8*c^8 + 11*a^6*b^2*c^8 + 5*a^4*b^4*c^8 - 3*a^2*b^6*c^8 - 20*b^8*c^8 - 38*a^6*c^10 - 21*a^4*b^2*c^10 + 15*b^6*c^10 + 16*a^4*c^12 + 6*a^2*b^2*c^12 - 6*b^4*c^12 - 3*a^2*c^14 + b^2*c^14 : :

X(38706) lies on these lines: {2, 23516}, {3, 252}, {4, 13372}, {20, 128}, {30, 23237}, {137, 631}, {140, 25147}, {548, 14072}, {550, 6592}, {1263, 3530}, {3327, 5217}, {3523, 11671}, {5204, 7159}, {7512, 23320}, {8703, 15055}, {10126, 11016}, {10323, 15959}, {12026, 15712}, {12307, 14071}, {14769, 23335}, {15061, 34308}, {15960, 37198}

X(38706) = reflection of X(38710) in X(3)

X(38707) = DILATION FROM X(3) OF X(953) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    a^2*(3*a^8 - 6*a^7*b - 5*a^6*b^2 + 18*a^5*b^3 - 3*a^4*b^4 - 18*a^3*b^5 + 9*a^2*b^6 + 6*a*b^7 - 4*b^8 - 6*a^7*c + 24*a^6*b*c - 16*a^5*b^2*c - 38*a^4*b^3*c + 50*a^3*b^4*c + 4*a^2*b^5*c - 28*a*b^6*c + 10*b^7*c - 5*a^6*c^2 - 16*a^5*b*c^2 + 61*a^4*b^2*c^2 - 24*a^3*b^3*c^2 - 51*a^2*b^4*c^2 + 40*a*b^5*c^2 - 5*b^6*c^2 + 18*a^5*c^3 - 38*a^4*b*c^3 - 24*a^3*b^2*c^3 + 72*a^2*b^3*c^3 - 18*a*b^4*c^3 - 10*b^5*c^3 - 3*a^4*c^4 + 50*a^3*b*c^4 - 51*a^2*b^2*c^4 - 18*a*b^3*c^4 + 18*b^4*c^4 - 18*a^3*c^5 + 4*a^2*b*c^5 + 40*a*b^2*c^5 - 10*b^3*c^5 + 9*a^2*c^6 - 28*a*b*c^6 - 5*b^2*c^6 + 6*a*c^7 + 10*b*c^7 - 4*c^8) : :

X(38707) lies on these lines: {3, 901}, {20, 3259}, {36, 59}, {100, 14511}, {104, 14513}, {517, 4881}, {631, 31841}, {3025, 5204}, {3523, 22102}, {3601, 24201}, {5217, 13756}, {7987, 34464}, {10016, 17928}, {15803, 33645}, {24466, 31512}

X(38707) = reflection of X(38705) in X(3)

X(38708) = DILATION FROM X(3) OF X(1113) TO THE CIRCLE Γ(X(3),X(15035))

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

X(38708) lies on these lines: {2, 3}, {40, 2103}, {182, 2104}, {1350, 2105}, {1385, 2102}, {2100, 7987}, {2101, 16192}, {2574, 15035}, {2575, 15055}, {6713, 10781}, {10782, 24466}, {13414, 21663}, {14500, 37853}

X(38708) = reflection of X(38709) in X(3)
X(38708) = trisector nearest X(1113) of segment X(1113)X(1114)
X(38708) = {X(2),X(37941)}-harmonic conjugate of X(38709)

X(38709) = DILATION FROM X(3) OF X(1114) TO THE CIRCLE Γ(X(3),X(15035))

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

X(38709) lies on these lines: {2, 3}, {40, 2102}, {182, 2105}, {1350, 2104}, {1385, 2103}, {2100, 16192}, {2101, 7987}, {2574, 15055}, {2575, 15035}, {6713, 10782}, {10781, 24466}, {13415, 21663}, {14499, 37853}

X(38709) = reflection of X(38708) in X(3)
X(38709) = trisector nearest X(1114) of segment X(1113)X(1114)
X(38709) = {X(2),X(37941)}-harmonic conjugate of X(38708)

X(38710) = DILATION FROM X(3) OF X(1141) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    3*a^16 - 14*a^14*b^2 + 28*a^12*b^4 - 35*a^10*b^6 + 35*a^8*b^8 - 28*a^6*b^10 + 14*a^4*b^12 - 3*a^2*b^14 - 14*a^14*c^2 + 42*a^12*b^2*c^2 - 45*a^10*b^4*c^2 + 14*a^8*b^6*c^2 + 19*a^6*b^8*c^2 - 27*a^4*b^10*c^2 + 12*a^2*b^12*c^2 - b^14*c^2 + 28*a^12*c^4 - 45*a^10*b^2*c^4 + 25*a^8*b^4*c^4 - 9*a^6*b^6*c^4 + 13*a^4*b^8*c^4 - 18*a^2*b^10*c^4 + 6*b^12*c^4 - 35*a^10*c^6 + 14*a^8*b^2*c^6 - 9*a^6*b^4*c^6 + 9*a^2*b^8*c^6 - 15*b^10*c^6 + 35*a^8*c^8 + 19*a^6*b^2*c^8 + 13*a^4*b^4*c^8 + 9*a^2*b^6*c^8 + 20*b^8*c^8 - 28*a^6*c^10 - 27*a^4*b^2*c^10 - 18*a^2*b^4*c^10 - 15*b^6*c^10 + 14*a^4*c^12 + 12*a^2*b^2*c^12 + 6*b^4*c^12 - 3*a^2*c^14 - b^2*c^14 : :

X(38710) lies on these lines: {3, 252}, {4, 23516}, {20, 137}, {30, 25147}, {128, 631}, {140, 23237}, {548, 1263}, {549, 9140}, {550, 12026}, {3327, 5204}, {3522, 11671}, {3523, 13372}, {3530, 14072}, {3858, 25339}, {5217, 7159}, {6592, 15712}, {7399, 14769}, {7691, 27196}, {10574, 13504}, {13160, 23319}, {13371, 15367}, {14652, 22467}, {14674, 15331}, {15959, 17928}, {17702, 34308}, {24147, 35449}

X(38710) = reflection of X(38706) in X(3)

X(38711) = DILATION FROM X(3) OF X(1290) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    a*(3*a^12 - 3*a^11*b - 9*a^10*b^2 + 9*a^9*b^3 + 6*a^8*b^4 - 6*a^7*b^5 + 6*a^6*b^6 - 6*a^5*b^7 - 9*a^4*b^8 + 9*a^3*b^9 + 3*a^2*b^10 - 3*a*b^11 - 3*a^11*c + 7*a^10*b*c + 5*a^9*b^2*c - 15*a^8*b^3*c + 2*a^6*b^5*c + 14*a^4*b^7*c - 5*a^3*b^8*c - 9*a^2*b^9*c + 3*a*b^10*c + b^11*c - 9*a^10*c^2 + 5*a^9*b*c^2 + 23*a^8*b^2*c^2 - 13*a^7*b^3*c^2 - 20*a^6*b^4*c^2 + 16*a^5*b^5*c^2 + 5*a^4*b^6*c^2 - 13*a^3*b^7*c^2 + 3*a^2*b^8*c^2 + 5*a*b^9*c^2 - 2*b^10*c^2 + 9*a^9*c^3 - 15*a^8*b*c^3 - 13*a^7*b^2*c^3 + 35*a^6*b^3*c^3 - 2*a^5*b^4*c^3 - 22*a^4*b^5*c^3 + 11*a^3*b^6*c^3 + 5*a^2*b^7*c^3 - 5*a*b^8*c^3 - 3*b^9*c^3 + 6*a^8*c^4 - 20*a^6*b^2*c^4 - 2*a^5*b^3*c^4 + 16*a^4*b^4*c^4 - 2*a^3*b^5*c^4 - 6*a^2*b^6*c^4 + 8*b^8*c^4 - 6*a^7*c^5 + 2*a^6*b*c^5 + 16*a^5*b^2*c^5 - 22*a^4*b^3*c^5 - 2*a^3*b^4*c^5 + 8*a^2*b^5*c^5 + 2*b^7*c^5 + 6*a^6*c^6 + 5*a^4*b^2*c^6 + 11*a^3*b^3*c^6 - 6*a^2*b^4*c^6 - 12*b^6*c^6 - 6*a^5*c^7 + 14*a^4*b*c^7 - 13*a^3*b^2*c^7 + 5*a^2*b^3*c^7 + 2*b^5*c^7 - 9*a^4*c^8 - 5*a^3*b*c^8 + 3*a^2*b^2*c^8 - 5*a*b^3*c^8 + 8*b^4*c^8 + 9*a^3*c^9 - 9*a^2*b*c^9 + 5*a*b^2*c^9 - 3*b^3*c^9 + 3*a^2*c^10 + 3*a*b*c^10 - 2*b^2*c^10 - 3*a*c^11 + b*c^11) : :

X(38711) lies on these lines: {3, 1290}, {36, 36001}, {104, 36167}, {513, 15035}, {517, 15055}, {523, 34474}, {631, 5520}, {1325, 23961}, {5204, 31524}, {5217, 31522}, {6713, 36175}

X(38712) = DILATION FROM X(3) OF X(1292) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    a*(3*a^7 - 6*a^6*b + 3*a^5*b^2 - 3*a^3*b^4 + 6*a^2*b^5 - 3*a*b^6 - 6*a^6*c + a^5*b*c + 13*a^4*b^2*c - 14*a^3*b^3*c + 5*a*b^5*c + b^6*c + 3*a^5*c^2 + 13*a^4*b*c^2 - 6*a^3*b^2*c^2 + 2*a^2*b^3*c^2 - 9*a*b^4*c^2 - 3*b^5*c^2 - 14*a^3*b*c^3 + 2*a^2*b^2*c^3 + 14*a*b^3*c^3 + 2*b^4*c^3 - 3*a^3*c^4 - 9*a*b^2*c^4 + 2*b^3*c^4 + 6*a^2*c^5 + 5*a*b*c^5 - 3*b^2*c^5 - 3*a*c^6 + b*c^6) : :

X(38712) lies on these lines: {3, 105}, {20, 120}, {40, 10699}, {140, 15521}, {165, 2809}, {376, 10712}, {528, 10304}, {550, 10743}, {631, 5511}, {962, 11730}, {1350, 10760}, {1358, 5217}, {2775, 15035}, {2788, 21166}, {2795, 34473}, {2826, 34474}, {2836, 15055}, {3021, 5204}, {3522, 20344}, {3523, 6714}, {5540, 16192}, {7987, 11716}, {9520, 23239}, {10773, 24466}, {16370, 34124}, {20097, 21734}

X(38712) = reflection of X(38694) in X(3)

X(38713) = DILATION FROM X(3) OF X(1293) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    a^2*(3*a^5 - 6*a^4*b - 10*a^3*b^2 + 8*a^2*b^3 + 7*a*b^4 - 2*b^5 - 6*a^4*c + 27*a^3*b*c + 5*a^2*b^2*c - 27*a*b^3*c + b^4*c - 10*a^3*c^2 + 5*a^2*b*c^2 + 9*b^3*c^2 + 8*a^2*c^3 - 27*a*b*c^3 + 9*b^2*c^3 + 7*a*c^4 + b*c^4 - 2*c^5) : :

X(38713) lies on these lines: {3, 106}, {20, 121}, {40, 10700}, {140, 15522}, {165, 2802}, {376, 10713}, {550, 10744}, {631, 5510}, {962, 11731}, {1054, 16192}, {1350, 10761}, {1357, 5217}, {2776, 15035}, {2789, 21166}, {2796, 34473}, {2810, 31884}, {2827, 34474}, {2842, 15055}, {3522, 21290}, {3523, 6715}, {5204, 6018}, {5584, 34139}, {7987, 11717}, {9524, 23239}, {10774, 24466}, {11814, 12512}, {14664, 35242}, {20098, 21734}

X(38713) = reflection of X(38695) in X(3)

X(38714) = DILATION FROM X(3) OF X(1294) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    (3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 2*b^2*c^2 - c^4)*(a^12 - a^10*b^2 - 6*a^8*b^4 + 14*a^6*b^6 - 11*a^4*b^8 + 3*a^2*b^10 - a^10*c^2 + 13*a^8*b^2*c^2 - 14*a^6*b^4*c^2 - 6*a^4*b^6*c^2 + 7*a^2*b^8*c^2 + b^10*c^2 - 6*a^8*c^4 - 14*a^6*b^2*c^4 + 34*a^4*b^4*c^4 - 10*a^2*b^6*c^4 - 4*b^8*c^4 + 14*a^6*c^6 - 6*a^4*b^2*c^6 - 10*a^2*b^4*c^6 + 6*b^6*c^6 - 11*a^4*c^8 + 7*a^2*b^2*c^8 - 4*b^4*c^8 + 3*a^2*c^10 + b^2*c^10) : :

X(38714) lies on these lines: {3, 107}, {4, 34842}, {20, 122}, {40, 10701}, {133, 631}, {140, 22337}, {376, 2777}, {548, 23240}, {550, 10745}, {962, 11732}, {1350, 10762}, {2071, 30716}, {2790, 21166}, {2797, 34473}, {2828, 34474}, {3184, 3522}, {3324, 5217}, {3523, 6716}, {3528, 5667}, {5204, 7158}, {7987, 11718}, {9033, 15055}, {9530, 10304}, {10323, 14703}, {10775, 24466}, {14673, 37198}, {15696, 23241}

X(38714) = reflection of X(23239) in X(3)

X(38715) = DILATION FROM X(3) OF X(1295) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    a*(3*a^12 - 3*a^11*b - 9*a^10*b^2 + 9*a^9*b^3 + 6*a^8*b^4 - 6*a^7*b^5 + 6*a^6*b^6 - 6*a^5*b^7 - 9*a^4*b^8 + 9*a^3*b^9 + 3*a^2*b^10 - 3*a*b^11 - 3*a^11*c + 17*a^10*b*c - 5*a^9*b^2*c - 33*a^8*b^3*c + 30*a^7*b^4*c - 2*a^6*b^5*c - 30*a^5*b^6*c + 34*a^4*b^7*c + 5*a^3*b^8*c - 15*a^2*b^9*c + 3*a*b^10*c - b^11*c - 9*a^10*c^2 - 5*a^9*b*c^2 + 50*a^8*b^2*c^2 - 24*a^7*b^3*c^2 - 58*a^6*b^4*c^2 + 62*a^5*b^5*c^2 + 4*a^4*b^6*c^2 - 32*a^3*b^7*c^2 + 11*a^2*b^8*c^2 - a*b^9*c^2 + 2*b^10*c^2 + 9*a^9*c^3 - 33*a^8*b*c^3 - 24*a^7*b^2*c^3 + 108*a^6*b^3*c^3 - 26*a^5*b^4*c^3 - 66*a^4*b^5*c^3 + 40*a^3*b^6*c^3 - 12*a^2*b^7*c^3 + a*b^8*c^3 + 3*b^9*c^3 + 6*a^8*c^4 + 30*a^7*b*c^4 - 58*a^6*b^2*c^4 - 26*a^5*b^3*c^4 + 74*a^4*b^4*c^4 - 22*a^3*b^5*c^4 - 14*a^2*b^6*c^4 + 18*a*b^7*c^4 - 8*b^8*c^4 - 6*a^7*c^5 - 2*a^6*b*c^5 + 62*a^5*b^2*c^5 - 66*a^4*b^3*c^5 - 22*a^3*b^4*c^5 + 54*a^2*b^5*c^5 - 18*a*b^6*c^5 - 2*b^7*c^5 + 6*a^6*c^6 - 30*a^5*b*c^6 + 4*a^4*b^2*c^6 + 40*a^3*b^3*c^6 - 14*a^2*b^4*c^6 - 18*a*b^5*c^6 + 12*b^6*c^6 - 6*a^5*c^7 + 34*a^4*b*c^7 - 32*a^3*b^2*c^7 - 12*a^2*b^3*c^7 + 18*a*b^4*c^7 - 2*b^5*c^7 - 9*a^4*c^8 + 5*a^3*b*c^8 + 11*a^2*b^2*c^8 + a*b^3*c^8 - 8*b^4*c^8 + 9*a^3*c^9 - 15*a^2*b*c^9 - a*b^2*c^9 + 3*b^3*c^9 + 3*a^2*c^10 + 3*a*b*c^10 + 2*b^2*c^10 - 3*a*c^11 - b*c^11) : :

X(38715) lies on these lines: {3, 108}, {20, 123}, {40, 10702}, {140, 33566}, {165, 2817}, {376, 2829}, {550, 10746}, {631, 25640}, {962, 11733}, {1350, 10763}, {1359, 5217}, {2778, 15035}, {2791, 21166}, {2798, 34473}, {2849, 14414}, {2850, 15055}, {3318, 5204}, {3522, 34188}, {3523, 6717}, {7987, 11719}, {9528, 21161}, {10776, 24466}

X(38715) = reflection of X(38696) in X(3)

X(38716) = DILATION FROM X(3) OF X(1296) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    a^2*(3*a^8 - 13*a^6*b^2 - a^4*b^4 + 13*a^2*b^6 - 2*b^8 - 13*a^6*c^2 + 77*a^4*b^2*c^2 - 55*a^2*b^4*c^2 - b^6*c^2 - a^4*c^4 - 55*a^2*b^2*c^4 + 38*b^4*c^4 + 13*a^2*c^6 - b^2*c^6 - 2*c^8) : :

X(38716) lies on these lines: {3, 111}, {20, 126}, {40, 10704}, {140, 22338}, {371, 11836}, {372, 11835}, {376, 10717}, {543, 10304}, {550, 10748}, {631, 5512}, {1092, 3048}, {1350, 10765}, {1511, 35447}, {2780, 15035}, {2793, 21166}, {2830, 34474}, {2854, 5621}, {3325, 5217}, {3522, 14360}, {3523, 6719}, {3528, 14654}, {5085, 36696}, {5204, 6019}, {6449, 11833}, {6450, 11834}, {7987, 11721}, {9129, 15051}, {9172, 15692}, {9529, 23239}, {9734, 15921}, {10323, 14657}, {10779, 24466}, {14666, 34200}, {15688, 32424}, {20099, 21734}, {28662, 37751}

X(38716) = reflection of X(38698) in X(3)

X(38717) = DILATION FROM X(3) OF X(1297) TO THE CIRCLE Γ(X(3),X(15035))

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

X(38717) lies on these lines: {3, 112}, {4, 34841}, {20, 127}, {35, 13117}, {36, 13116}, {40, 10705}, {132, 631}, {140, 12918}, {186, 12096}, {371, 19093}, {372, 19094}, {376, 2482}, {550, 10749}, {1151, 19115}, {1152, 19114}, {1350, 10766}, {1385, 13099}, {1587, 13918}, {1588, 13985}, {1656, 19160}, {1657, 19163}, {2077, 13313}, {2781, 15035}, {2799, 34473}, {2831, 34474}, {3320, 5217}, {3515, 12145}, {3516, 13166}, {3522, 13219}, {3523, 6720}, {3524, 9530}, {3528, 13200}, {4297, 13280}, {5010, 13311}, {5171, 12207}, {5204, 6020}, {5432, 12945}, {5433, 12955}, {5584, 19162}, {6676, 14983}, {6684, 12784}, {7280, 13312}, {7512, 19164}, {7987, 11722}, {9155, 9157}, {9517, 15055}, {10267, 13119}, {10269, 13118}, {10298, 18876}, {10323, 19165}, {10780, 24466}, {11012, 13314}, {11610, 30270}, {11641, 37198}, {13195, 37479}, {13221, 16192}, {13296, 15326}, {13297, 15338}, {14900, 21735}, {15045, 16224}, {16225, 16836}

X(38717) = reflection of X(38699) in X(3)

X(38718) = DILATION FROM X(3) OF X(1300) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    3*a^16 - 11*a^14*b^2 + 13*a^12*b^4 - 5*a^10*b^6 + 5*a^8*b^8 - 13*a^6*b^10 + 11*a^4*b^12 - 3*a^2*b^14 - 11*a^14*c^2 + 37*a^12*b^2*c^2 - 39*a^10*b^4*c^2 + a^8*b^6*c^2 + 27*a^6*b^8*c^2 - 21*a^4*b^10*c^2 + 7*a^2*b^12*c^2 - b^14*c^2 + 13*a^12*c^4 - 39*a^10*b^2*c^4 + 48*a^8*b^4*c^4 - 22*a^6*b^6*c^4 - 3*a^4*b^8*c^4 - 3*a^2*b^10*c^4 + 6*b^12*c^4 - 5*a^10*c^6 + a^8*b^2*c^6 - 22*a^6*b^4*c^6 + 26*a^4*b^6*c^6 - a^2*b^8*c^6 - 15*b^10*c^6 + 5*a^8*c^8 + 27*a^6*b^2*c^8 - 3*a^4*b^4*c^8 - a^2*b^6*c^8 + 20*b^8*c^8 - 13*a^6*c^10 - 21*a^4*b^2*c^10 - 3*a^2*b^4*c^10 - 15*b^6*c^10 + 11*a^4*c^12 + 7*a^2*b^2*c^12 + 6*b^4*c^12 - 3*a^2*c^14 - b^2*c^14 : :

X(38718) lies on these lines: {2, 14644}, {3, 847}, {4, 34840}, {20, 136}, {131, 631}, {550, 13556}, {3523, 34844}, {3528, 21667}, {5961, 22467}, {13496, 14118}, {13558, 17928}, {15078, 23239}, {22823, 34007}

X(38719) = DILATION FROM X(3) OF X(1304) TO THE CIRCLE Γ(X(3),X(15035))

Barycentrics    a^2*(3*a^20 - 10*a^18*b^2 - 4*a^16*b^4 + 58*a^14*b^6 - 84*a^12*b^8 + 14*a^10*b^10 + 70*a^8*b^12 - 66*a^6*b^14 + 17*a^4*b^16 + 4*a^2*b^18 - 2*b^20 - 10*a^18*c^2 + 58*a^16*b^2*c^2 - 88*a^14*b^4*c^2 - 64*a^12*b^6*c^2 + 306*a^10*b^8*c^2 - 290*a^8*b^10*c^2 + 52*a^6*b^12*c^2 + 68*a^4*b^14*c^2 - 36*a^2*b^16*c^2 + 4*b^18*c^2 - 4*a^16*c^4 - 88*a^14*b^2*c^4 + 331*a^12*b^4*c^4 - 324*a^10*b^6*c^4 - 123*a^8*b^8*c^4 + 400*a^6*b^10*c^4 - 227*a^4*b^12*c^4 + 28*a^2*b^14*c^4 + 7*b^16*c^4 + 58*a^14*c^6 - 64*a^12*b^2*c^6 - 324*a^10*b^4*c^6 + 686*a^8*b^6*c^6 - 386*a^6*b^8*c^6 - 40*a^4*b^10*c^6 + 92*a^2*b^12*c^6 - 22*b^14*c^6 - 84*a^12*c^8 + 306*a^10*b^2*c^8 - 123*a^8*b^4*c^8 - 386*a^6*b^6*c^8 + 364*a^4*b^8*c^8 - 88*a^2*b^10*c^8 + 11*b^12*c^8 + 14*a^10*c^10 - 290*a^8*b^2*c^10 + 400*a^6*b^4*c^10 - 40*a^4*b^6*c^10 - 88*a^2*b^8*c^10 + 4*b^10*c^10 + 70*a^8*c^12 + 52*a^6*b^2*c^12 - 227*a^4*b^4*c^12 + 92*a^2*b^6*c^12 + 11*b^8*c^12 - 66*a^6*c^14 + 68*a^4*b^2*c^14 + 28*a^2*b^4*c^14 - 22*b^6*c^14 + 17*a^4*c^16 - 36*a^2*b^2*c^16 + 7*b^4*c^16 + 4*a^2*c^18 + 4*b^2*c^18 - 2*c^20) : :

X(38719) lies on these lines: {3, 1304}, {20, 18809}, {186, 12096}, {250, 15404}, {520, 15035}, {523, 23239}, {631, 16177}, {1294, 31510}, {6000, 15055}, {6760, 15646}, {10282, 13997}, {15078, 33927}

X(38720) = CIRCUMCIRCLE-INVERSE OF X(1670)

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

X(38720) lies on these lines: {3, 6}, {316, 11000}, {384, 5403}, {5404, 37334}, {11673, 15245}, {15244, 33873}

X(38720) = reflection of X(38721) in X(3)
X(38720) = circumcircle-inverse of X(1670)
X(38720) = inner-Montesdeoca-Lemoine-circle-inverse of X(1342)
X(38720) = {X(187),X(2076)}-harmonic conjugate of X(38721)
X(38720) = {X(1379),X(1380)}-harmonic conjugate of X(1670)

X(38721) = CIRCUMCIRCLE-INVERSE OF X(1671)

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

X(38721) lies on these lines: {3, 6}, {316, 10999}, {384, 5404}, {5403, 37334}, {11673, 15244}, {15245, 33873}

X(38721) = reflection of X(38720) in X(3)
X(38721) = circumcircle-inverse of X(1671)
X(38721) = outer-Montesdeoca-Lemoine-circle-inverse of X(1343)
X(38721) = {X(187),X(2076)}-harmonic conjugate of X(38720)
X(38721) = {X(1379),X(1380)}-harmonic conjugate of X(1671)

X(38722) = CIRCUMCIRCLE-INVERSE OF X(19914)

Barycentrics    a^2*(a^8 - 2*a^7*b - 2*a^6*b^2 + 6*a^5*b^3 - 6*a^3*b^5 + 2*a^2*b^6 + 2*a*b^7 - b^8 - 2*a^7*c + 7*a^6*b*c - 3*a^5*b^2*c - 12*a^4*b^3*c + 12*a^3*b^4*c + 3*a^2*b^5*c - 7*a*b^6*c + 2*b^7*c - 2*a^6*c^2 - 3*a^5*b*c^2 + 12*a^4*b^2*c^2 - 2*a^3*b^3*c^2 - 11*a^2*b^4*c^2 + 5*a*b^5*c^2 + b^6*c^2 + 6*a^5*c^3 - 12*a^4*b*c^3 - 2*a^3*b^2*c^3 + 10*a^2*b^3*c^3 - 2*b^5*c^3 + 12*a^3*b*c^4 - 11*a^2*b^2*c^4 - 6*a^3*c^5 + 3*a^2*b*c^5 + 5*a*b^2*c^5 - 2*b^3*c^5 + 2*a^2*c^6 - 7*a*b*c^6 + b^2*c^6 + 2*a*c^7 + 2*b*c^7 - c^8) : :
X(38722) = X[16128] - 3 X[37713]

X(38722) lies on these lines: {3, 8}, {5, 10058}, {11, 6924}, {30, 12761}, {35, 6265}, {36, 12737}, {55, 19907}, {119, 5432}, {149, 6942}, {153, 6950}, {214, 32613}, {474, 34126}, {960, 22935}, {1012, 22799}, {1317, 14793}, {1320, 22765}, {1387, 8069}, {1483, 10074}, {1484, 5172}, {1768, 37700}, {2077, 4867}, {2476, 38135}, {2771, 26086}, {2800, 26285}, {2802, 26286}, {2886, 6713}, {3149, 22938}, {3820, 7508}, {5010, 6326}, {5844, 25438}, {6264, 7280}, {6905, 10738}, {6906, 10742}, {7972, 14792}, {8071, 12735}, {10087, 37564}, {10164, 17009}, {10698, 11849}, {10942, 12762}, {11248, 22775}, {11249, 13205}, {11571, 37733}, {11715, 32612}, {12619, 25440}, {12740, 32760}, {16128, 37713}, {19525, 38042}, {36152, 37726}, {37293, 37621}

X(38722) = midpoint of X(i) and X(j) for these {i,j}: {1768, 37700}, {11248, 22775}, {11249, 13205}
X(38722) = circumcircle-inverse of X(19914)
X(38722) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 2932, 33814}, {3, 12331, 4996}, {3, 12773, 18861}, {100, 104, 19914}

leftri

Circum-Euler-points: X(38723)-X(38807)

rightri

This preamble and centers X(38723)-X(38807) were contributed by César Eliud Lozada, May 28, 2020.

The centroids of the four triangles determined by four concyclic points are concyclic.. (Reference: Halsted, George Bruce, Elementary Synthetic Geometry, 1892, problem 24, pp. 84.)

Indeed, the above result is true for other points distinct from the centroids, although the exact locus is still unknown.

Part of this locus is as follows: let Q be a point on the circumcircle of ABC and P a point on the Euler line of ABC such that |OP|/|OH| = λ, λ being a real number not depending on ABC. Denote Pa = P-of-QBC, Pb = P-of-QCA and Pc = P-of-QAB. Then the points P, Pa, Pb, Pc are concyclic on a circle with radius r' = |λ| R and whose center O'(Q, P) is here named the Q-circum-Euler-point of P. In this case:

   O'(Q, P) = λ (Q + X(4)) + (1 - 2 λ) X(3)

and it lies on the line {X(3), midpoint(X(4), Q) }. Moreover, when P is fixed and Q varies, the locus of O'(Q, P) is another circle with the same radius r' and center P.

There are other P such that P, Pa, Pb, Pc are concyclic. A numerical calculus shows that a partial list of such points contains P=X(n) for n ∈ {1, 13, 14, 15, 16, 23, 26, 36, 40, 80, 125, 155, 165, 186, 265, 368, 369, 370, 399, 1144, 1147, 1385, 1482, 1511, 1658, 2070, 2071, 2072, 2077, 3167, 3232, 3576, 3579, 5159, 5373, 5394, 5473, 5474, 5537, 5609, 5611, 5615, 5626, 5899, 5961, 5962, 5963, 5964, 6104, 6105, 6699, 6771, 6774, 7387, 7464, 7575, 7689, 7982, 7987, 7991, 8008, 8009, 8148, 8697, 9909} (n<10000). When P=X(1), P, Pa, Pb, Pc are vertices of a rectangle.

Curiously, when P=X(15) and Q varies, the locus of O'(Q,P) are the sides of a central triangle having A-vertex with barycentric coordinates A' = (sqrt(3)*S+SA)*(SB+SC): S^2-SA*SC : S^2-SA*SB. This triangle is perspective to the following triangles with perspector X(3): (ABC, ABC-X3 reflections, 2nd anti-extouch, 2nd Hyacinth, Lucas antipodal(±1), Lucas central(±1), X3-ABC reflections). It is also perspective to the following triangles with the given perspectors: (circumsymmedial, 11485), (outer-Le Viet An, 3129), (symmedial, 61), (tangential, 22236). A similar locus does not occur for P=X(16).

The following table contains the Q-circum-Euler-point of P for selected P and Q:

P (λ) \ Q X(74) X(98) X(99) X(100) X(101) X(102) X(110) X(111)
X(2) (λ = 1/3) X(15061) X(38224) X(15561) X(38752) X(38764) X(38776) X(14643) X(38796)
X(4) (λ = 1) X(265) X(6321) X(6033) X(10742) X(10741) X(10747) X(7728) X(22338)
X(5) (λ = 1/2) X(125) X(115) X(114) X(119) X(118) X(124) X(113) X(5512)
X(20) (λ = -1) X(12121) X(38730) X(38741) X(38753) X(38765) X(38777) X(20127) X(38797)
X(140) (λ = 1/4) X(6699) X(6036) X(620) X(3035) X(6710) X(6711) X(5972) X(6719)
X(376) (λ = -1/3) X(38723) X(38731) X(38742) X(38754) X(38766) X(38778) X(38788) X(38798)
X(381) (λ = 2/3) X(38724) X(38732) X(38743) X(38755) X(38767) X(38779) X(38789) X(38799)
X(382) (λ = 2) X(12902) X(38733) X(38744) X(38756) X(38768) X(38780) X(38790) X(38800)
X(546) (λ = 3/4) X(36253) X(38734) X(38745) X(38757) X(38769) X(38781) X(38791) X(38801)
X(547) (λ = 5/12) X(38725) X(38735) X(38746) X(38758) X(38770) X(38782) X(38792) X(38802)
X(548) (λ = -1/4) X(38726) X(38736) X(38747) X(38759) X(38771) X(38783) X(37853) X(38803)
X(549) (λ = 1/6) X(38727) X(38737) X(38748) X(38760) X(38772) X(38784) X(38793) X(38804)
X(550) (λ = -1/2) X(16163) X(38738) X(38749) X(38761) X(38773) X(38785) X(16111) X(38805)
X(631) (λ = 1/5) X(38728) X(38739) X(38750) X(38762) X(38774) X(38786) X(38794) X(38806)
X(632) (λ = 3/10) X(38729) X(38740) X(38751) X(38763) X(38775) X(38787) X(38795) X(38807)

X(38723) = X(74)-CIRCUM-EULER-POINT OF X(376)

Barycentrics    (-a^2+b^2+c^2)*(5*a^8-5*(b^2+c^2)*a^6-(4*b^4-13*b^2*c^2+4*c^4)*a^4+3*(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4) : :
X(38723) = 5*X(3)-2*X(125) = 4*X(3)-X(265) = 7*X(3)-4*X(6699) = 2*X(3)+X(12121) = 7*X(3)-X(12902) = 14*X(3)-5*X(15027) = X(3)+2*X(16163) = 17*X(3)-8*X(20397) = 13*X(3)-4*X(36253) = 3*X(3)-X(38724) = 9*X(3)-4*X(38725) = X(3)-4*X(38726) = 3*X(3)-2*X(38727) = 8*X(3)-5*X(38728) = 19*X(3)-10*X(38729) = 8*X(125)-5*X(265) = 7*X(125)-10*X(6699) = 4*X(125)+5*X(12121) = 14*X(125)-5*X(12902) = 4*X(125)-5*X(15061) = X(125)+5*X(16163) = 17*X(125)-20*X(20397) = 13*X(125)-10*X(36253) = 6*X(125)-5*X(38724) = 9*X(125)-10*X(38725) = X(125)-10*X(38726) = 3*X(125)-5*X(38727)

X(38723) lies on these lines: {3,125}, {4,38794}, {5,15051}, {20,1511}, {30,14643}, {40,12898}, {67,14810}, {74,548}, {110,550}, {113,1657}, {140,10733}, {146,17538}, {154,2777}, {376,5663}, {381,38793}, {382,5972}, {399,15696}, {541,15689}, {542,15041}, {549,14644}, {631,10113}, {632,15023}, {690,38731}, {1151,19052}, {1152,19051}, {1216,22584}, {1350,16176}, {1539,3529}, {1656,12295}, {1986,35503}, {2077,12905}, {2771,38754}, {2772,38766}, {2773,38778}, {2780,38798}, {3098,32233}, {3448,3528}, {3516,12140}, {3521,12038}, {3522,12041}, {3523,20304}, {3524,34128}, {3525,15088}, {3526,7687}, {3530,15059}, {3830,36518}, {3843,12900}, {4297,12778}, {4316,12373}, {4324,12374}, {4549,35257}, {5010,12903}, {5054,23515}, {5204,12896}, {5217,18968}, {5447,21650}, {5504,14861}, {5584,19478}, {5609,12244}, {5642,15681}, {6053,15039}, {6101,7722}, {6243,14708}, {6288,10226}, {6455,8994}, {6456,13969}, {6723,15720}, {7280,12904}, {7987,12261}, {8703,15055}, {8717,10293}, {9129,38797}, {9140,34200}, {9833,11598}, {10088,15326}, {10091,15338}, {10264,33923}, {10272,10721}, {10282,11744}, {10295,15463}, {10540,16386}, {10620,11850}, {10627,12219}, {10706,15686}, {10990,12308}, {11012,12906}, {11559,26861}, {11562,15644}, {11694,19710}, {11801,15712}, {12103,15034}, {12108,15044}, {12201,37479}, {12228,37495}, {12407,16192}, {12412,37198}, {13202,17800}, {13211,31663}, {13289,13564}, {13391,35489}, {14094,14677}, {14849,34473}, {14850,21166}, {14984,25406}, {14989,21317}, {15081,15717}, {15462,29181}, {16063,32227}, {16534,38790}, {18332,38738}, {18474,35495}, {18859,19596}, {20379,21734}, {20773,34350}, {25320,33750}, {25564,34785}, {27082,34783}, {30522,37948}, {32305,33751}, {33511,38733}, {33512,38744}, {35447,38803}

X(38723) = midpoint of X(i) and X(j) for these {i,j}: {3534, 32609}, {12121, 15061}, {15681, 38789}
X(38723) = reflection of X(i) in X(j) for these (i,j): (265, 15061), (381, 38793), (3830, 36518), (5655, 32609), (14643, 15035), (14644, 549), (14849, 34473), (14850, 21166), (15055, 8703), (15061, 3), (20126, 15055), (38724, 38727), (38788, 376), (38789, 5642)
X(38723) = X(12121)-Gibert-Moses centroid
X(38723) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 265, 38728), (3, 12121, 265), (3, 12902, 6699), (3, 16163, 12121), (3, 38724, 38727), (20, 1511, 7728), (74, 34153, 23236), (110, 550, 20127), (399, 15696, 16111), (548, 34153, 74), (1657, 15040, 113), (3522, 12383, 12041), (6699, 12902, 15027), (10272, 15704, 10721), (10721, 15020, 10272), (10733, 15036, 140), (12902, 15027, 265), (16163, 38726, 3), (30714, 37853, 10620), (38724, 38727, 15061)

X(38724) = X(74)-CIRCUM-EULER-POINT OF X(381)

Barycentrics    (-a^2+b^2+c^2)*(a^8-(b^2+c^2)*a^6+(b^4-b^2*c^2+c^4)*a^4-3*(b^4-c^4)*(b^2-c^2)*a^2+2*(b^2-c^2)^4) : :
X(38724) = 7*X(2)-4*X(11694) = X(3)-4*X(125) = X(3)+2*X(265) = 5*X(3)-8*X(6699) = 5*X(3)-2*X(12121) = 2*X(3)+X(12902) = X(3)-10*X(15027) = 7*X(3)-4*X(16163) = 7*X(3)-16*X(20397) = X(3)+8*X(36253) = 3*X(3)-2*X(38723) = 3*X(3)-8*X(38725) = 11*X(3)-8*X(38726) = 3*X(3)-4*X(38727) = 7*X(3)-10*X(38728) = 11*X(3)-20*X(38729) = 2*X(125)+X(265) = 5*X(125)-2*X(6699) = 10*X(125)-X(12121) = 8*X(125)+X(12902) = 2*X(125)-5*X(15027) = 7*X(125)-X(16163) = 7*X(125)-4*X(20397) = X(125)+2*X(36253) = 6*X(125)-X(38723) = 3*X(125)-2*X(38725) = 11*X(125)-2*X(38726) = 3*X(125)-X(38727) = 14*X(125)-5*X(38728) = 11*X(125)-5*X(38729) = 8*X(11694)-7*X(32609)

X(38724) lies on these lines: {2,11694}, {3,125}, {4,10264}, {5,399}, {6,7579}, {30,15041}, {67,1351}, {68,23306}, {74,382}, {110,1656}, {113,3851}, {115,15538}, {140,12383}, {143,7731}, {146,546}, {155,33547}, {156,11704}, {195,10224}, {355,13605}, {381,5640}, {541,14269}, {542,5050}, {547,9143}, {568,10628}, {569,11597}, {631,34153}, {690,38732}, {858,37496}, {895,11898}, {974,26944}, {999,12903}, {1147,15089}, {1151,35835}, {1152,35834}, {1209,5898}, {1350,32273}, {1352,25328}, {1385,12407}, {1482,12261}, {1511,3526}, {1539,15054}, {1594,14627}, {1657,10733}, {1853,2777}, {1899,10254}, {1986,7507}, {2070,25739}, {2072,3564}, {2771,5587}, {2772,38767}, {2773,38779}, {2780,38799}, {2930,24206}, {2935,20299}, {2948,9956}, {3024,9669}, {3028,9654}, {3090,10272}, {3091,12317}, {3146,14677}, {3153,32608}, {3295,12904}, {3517,12140}, {3530,15042}, {3534,15055}, {3567,15100}, {3580,7574}, {3627,12244}, {3628,15039}, {3763,12584}, {3818,16010}, {3843,7687}, {3858,13393}, {5054,15035}, {5056,20125}, {5070,5972}, {5072,14094}, {5073,12295}, {5076,10721}, {5079,5609}, {5094,15463}, {5097,16176}, {5504,12429}, {5621,5899}, {5627,16168}, {5642,15703}, {5644,5655}, {5654,10255}, {5876,12284}, {5889,13358}, {6070,20957}, {6102,12281}, {6243,11800}, {6321,15357}, {6417,19051}, {6418,19052}, {6723,30714}, {7506,12412}, {7517,13171}, {7545,34514}, {7547,7722}, {7577,15087}, {7727,10896}, {7984,12645}, {8227,11699}, {9655,10081}, {9668,10065}, {9904,22793}, {9934,34780}, {9976,15069}, {10088,31479}, {10117,18378}, {10125,12254}, {10263,13201}, {10280,14695}, {10778,12331}, {10895,19470}, {11005,12188}, {11006,12355}, {11061,18583}, {11438,18430}, {11561,15043}, {11562,37481}, {11564,11935}, {11579,18440}, {11591,12273}, {11645,15362}, {11735,12898}, {11744,13093}, {11806,21650}, {11821,12319}, {12017,32233}, {12079,36184}, {12163,19479}, {12270,13630}, {12292,37197}, {12307,36853}, {12334,37621}, {12900,24981}, {13413,34545}, {13561,14130}, {13881,14901}, {13915,19111}, {13979,19110}, {15057,15696}, {15141,20300}, {15359,18332}, {15681,38788}, {15694,38793}, {15738,22584}, {16111,17800}, {16239,22251}, {17511,34209}, {17835,19506}, {17847,32743}, {18400,37922}, {18404,18933}, {18436,21649}, {18480,33535}, {18550,34802}, {18565,26937}, {18566,34796}, {18912,32341}, {19140,25335}, {19478,37535}, {21316,36172}, {21850,32247}, {23251,35827}, {23261,35826}, {25150,34308}, {25331,34155}, {25336,25556}, {30522,37955}, {32223,37923}, {32305,36990}, {32767,37472}, {33533,38397}, {34826,34864}, {35447,38800}

X(38724) = midpoint of X(i) and X(j) for these {i,j}: {265, 15061}, {9140, 14644}, {10516, 25330}
X(38724) = reflection of X(i) in X(j) for these (i,j): (3, 15061), (381, 14644), (3534, 15055), (5655, 36518), (14643, 23515), (15035, 34128), (15061, 125), (15681, 38788), (16222, 12099), (25331, 34155), (32609, 2), (38723, 38727), (38727, 38725), (38789, 381)
X(38724) = orthocentroidal circle-inverse of X(5946)
X(38724) = X(12902)-Gibert-Moses centroid
X(38724) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 265, 12902), (3, 31676, 13558), (4, 10264, 10620), (4, 10620, 38790), (5, 3448, 399), (125, 265, 3), (125, 16163, 20397), (125, 36253, 265), (125, 38727, 38725), (265, 15027, 125), (3448, 15081, 5), (3851, 12308, 113), (5609, 15025, 5079), (6699, 12121, 3), (7687, 7728, 3843), (10113, 20379, 74), (10264, 11801, 4), (10733, 12041, 1657), (11804, 33565, 195), (12261, 13211, 1482), (12295, 20127, 5073), (15027, 36253, 3), (15035, 34128, 5054), (15043, 15102, 11561), (15061, 38723, 38727), (16163, 20397, 38728), (16163, 38728, 3), (38723, 38727, 3), (38725, 38727, 15061)

X(38725) = X(74)-CIRCUM-EULER-POINT OF X(547)

Barycentrics    (-a^2+b^2+c^2)*(2*a^8-2*(b^2+c^2)*a^6-(7*b^4-16*b^2*c^2+7*c^4)*a^4+12*(b^4-c^4)*(b^2-c^2)*a^2-5*(b^2-c^2)^4) : :
X(38725) = X(3)+5*X(125) = 7*X(3)+5*X(265) = 2*X(3)-5*X(6699) = 17*X(3)-5*X(12121) = 19*X(3)+5*X(12902) = X(3)-5*X(15061) = 11*X(3)-5*X(16163) = X(3)-10*X(20397) = 4*X(3)+5*X(36253) = 9*X(3)-5*X(38723) = 3*X(3)+5*X(38724) = 8*X(3)-5*X(38726) = 3*X(3)-5*X(38727) = 7*X(125)-X(265) = 2*X(125)+X(6699) = 17*X(125)+X(12121) = 19*X(125)-X(12902) = 11*X(125)-5*X(15027) = 11*X(125)+X(16163) = X(125)+2*X(20397) = 4*X(125)-X(36253) = 9*X(125)+X(38723) = 3*X(125)-X(38724) = 8*X(125)+X(38726) = 3*X(125)+X(38727) = 13*X(125)+5*X(38728) = 7*X(125)+5*X(38729)

X(38725) lies on these lines: {3,125}, {74,3832}, {110,3533}, {113,5056}, {155,19348}, {541,3545}, {542,11539}, {547,5663}, {690,38735}, {2771,38758}, {2772,38770}, {2773,38782}, {2777,3845}, {2780,38802}, {3526,24981}, {3543,14644}, {3564,14156}, {3628,6053}, {3850,20304}, {3853,7687}, {5059,12295}, {5067,12900}, {5097,15118}, {5642,15723}, {5972,16239}, {6723,10264}, {9140,15702}, {10628,15113}, {11001,15055}, {11735,33179}, {11801,37853}, {11812,32423}, {12244,15025}, {12317,38795}, {15035,15708}, {15041,38335}, {15046,15063}, {15057,15081}, {15088,38791}, {20126,36518}, {29012,37936}

X(38725) = midpoint of X(i) and X(j) for these {i,j}: {125, 15061}, {9140, 38793}, {20126, 36518}, {38724, 38727}
X(38725) = reflection of X(i) in X(j) for these (i,j): (6699, 15061), (15061, 20397), (38792, 547)
X(38725) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (125, 6699, 36253), (125, 16163, 15027), (125, 20397, 6699), (125, 38727, 38724), (125, 38729, 265), (6699, 36253, 38726), (6723, 10264, 16534), (15057, 15081, 16111), (15059, 16003, 12900), (15061, 38724, 38727)

X(38726) = X(74)-CIRCUM-EULER-POINT OF X(548)

Barycentrics    (-a^2+b^2+c^2)*(6*a^8-6*(b^2+c^2)*a^6-(5*b^4-16*b^2*c^2+5*c^4)*a^4+4*(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4) : :
X(38726) = 3*X(2)-7*X(15036) = 3*X(3)-X(125) = 5*X(3)-X(265) = 3*X(3)+X(12121) = 9*X(3)-X(12902) = 17*X(3)-5*X(15027) = 7*X(3)-3*X(15061) = 5*X(3)-2*X(20397) = 4*X(3)-X(36253) = X(3)+3*X(38723) = 11*X(3)-3*X(38724) = 8*X(3)-3*X(38725) = 5*X(3)-3*X(38727) = 9*X(3)-5*X(38728) = 11*X(3)-5*X(38729) = 5*X(125)-3*X(265) = 2*X(125)-3*X(6699) = 3*X(125)-X(12902) = 17*X(125)-15*X(15027) = 7*X(125)-9*X(15061) = X(125)+3*X(16163) = 5*X(125)-6*X(20397) = 4*X(125)-3*X(36253) = X(125)+9*X(38723) = 11*X(125)-9*X(38724) = 8*X(125)-9*X(38725) = 5*X(125)-9*X(38727) = 3*X(125)-5*X(38728) = 11*X(125)-15*X(38729) = X(12295)-7*X(15036)

X(38726) lies on these lines: {2,12295}, {3,125}, {4,12900}, {20,113}, {30,5972}, {74,3522}, {110,376}, {140,7687}, {146,15034}, {382,36518}, {399,10990}, {511,14708}, {516,11723}, {539,21663}, {542,8703}, {548,1216}, {549,6723}, {550,1511}, {631,10733}, {690,38736}, {974,11577}, {1038,12888}, {1040,19469}, {1112,37931}, {1539,15704}, {1656,15042}, {1657,13202}, {1986,10625}, {2771,31805}, {2772,38771}, {2773,38783}, {2780,38803}, {2794,33512}, {2979,7722}, {3090,15023}, {3448,10304}, {3520,12140}, {3523,14644}, {3524,15059}, {3528,12383}, {3529,38795}, {3530,20304}, {3534,5642}, {3917,7723}, {4550,35485}, {5010,18968}, {5092,15118}, {5095,33878}, {5446,9826}, {5447,12358}, {5504,15740}, {5609,14677}, {5655,15689}, {5891,12292}, {5892,11746}, {6409,8994}, {6410,13969}, {6449,19052}, {6450,19051}, {6560,8998}, {6561,13990}, {6716,32162}, {6823,23306}, {7280,12896}, {7978,9778}, {9129,38805}, {9140,19708}, {9541,19110}, {9729,12236}, {10111,18128}, {10117,35243}, {10272,12103}, {10295,10564}, {10299,15081}, {10519,32275}, {10575,12825}, {10620,15688}, {10628,13348}, {10706,11693}, {10996,12319}, {11495,22583}, {11541,15029}, {11598,34782}, {11656,12117}, {11694,15691}, {11720,31730}, {11735,13624}, {11744,17821}, {11800,16836}, {11801,12100}, {11806,14984}, {12118,18931}, {12227,16266}, {12228,13346}, {12261,17502}, {12307,14049}, {12308,15695}, {12317,15021}, {12897,37814}, {13211,35242}, {13293,15577}, {13393,15605}, {13417,36987}, {14093,20126}, {14791,19506}, {14855,17854}, {15020,17538}, {15041,23236}, {15063,15696}, {15113,15122}, {15115,31829}, {15463,35503}, {15686,22251}, {15712,34128}, {15760,33547}, {16278,38730}, {16386,32111}, {18332,38731}, {18400,34152}, {18570,24206}, {18571,32223}, {19504,37483}, {19924,32217}, {21167,32274}, {21850,32300}, {22978,23358}, {23698,33511}, {27866,35489}, {29012,37950}, {31884,32233}

X(38726) = midpoint of X(i) and X(j) for these {i,j}: {3, 16163}, {20, 113}, {74, 30714}, {110, 16111}, {125, 12121}, {399, 10990}, {550, 1511}, {1539, 15704}, {1657, 13202}, {1986, 10625}, {3534, 5642}, {5095, 33878}, {5609, 14677}, {9129, 38805}, {10272, 12103}, {10295, 10564}, {10575, 12825}, {10620, 24981}, {11598, 34782}, {11656, 12117}, {11694, 15691}, {11720, 31730}, {12041, 34153}, {12307, 14049}, {12383, 16003}, {15063, 20127}, {16278, 38730}
X(38726) = reflection of X(i) in X(j) for these (i,j): (4, 12900), (265, 20397), (5446, 9826), (6699, 3), (7687, 140), (10111, 18128), (10113, 6723), (11735, 13624), (12236, 9729), (12358, 5447), (15118, 5092), (16534, 1511), (20304, 3530), (21850, 32300), (32223, 18571), (36253, 6699), (37853, 548), (38791, 10272)
X(38726) = complement of X(12295)
X(38726) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 265, 38727), (3, 12121, 125), (3, 12902, 38728), (3, 38723, 16163), (4, 15051, 38793), (4, 38793, 12900), (20, 15035, 113), (110, 376, 16111), (125, 16163, 12121), (265, 38727, 20397), (382, 38794, 36518), (399, 38788, 10990), (549, 10113, 6723), (631, 10733, 23515), (1657, 14643, 13202), (6699, 36253, 38725), (8703, 34153, 12041), (12121, 38728, 12902), (12902, 38728, 125), (20127, 32609, 15063), (20397, 38727, 6699)

X(38727) = X(74)-CIRCUM-EULER-POINT OF X(549)

Barycentrics    (-a^2+b^2+c^2)*(4*a^8-4*(b^2+c^2)*a^6-(5*b^4-14*b^2*c^2+5*c^4)*a^4+6*(b^4-c^4)*(b^2-c^2)*a^2-(b^2-c^2)^4) : :
X(38727) = 2*X(3)+X(125) = 5*X(3)+X(265) = X(3)+2*X(6699) = 7*X(3)-X(12121) = 11*X(3)+X(12902) = 13*X(3)+5*X(15027) = 4*X(3)-X(16163) = 5*X(3)+4*X(20397) = 7*X(3)+2*X(36253) = 3*X(3)-X(38723) = 3*X(3)+X(38724) = 3*X(3)+2*X(38725) = 5*X(3)-2*X(38726) = X(3)+5*X(38728) = 4*X(3)+5*X(38729) = 5*X(125)-2*X(265) = X(125)-4*X(6699) = 7*X(125)+2*X(12121) = 11*X(125)-2*X(12902) = 13*X(125)-10*X(15027) = 2*X(125)+X(16163) = 5*X(125)-8*X(20397) = 7*X(125)-4*X(36253) = 3*X(125)+2*X(38723) = 3*X(125)-2*X(38724) = 3*X(125)-4*X(38725) = 5*X(125)+4*X(38726) = X(125)-10*X(38728) = 2*X(125)-5*X(38729) = 2*X(15055)+X(36518)

X(38727) lies on the Walsmith rectangular hyperbola and these lines: {2,2777}, {3,125}, {4,6723}, {5,13202}, {20,7687}, {30,23515}, {40,11735}, {74,631}, {110,3523}, {113,140}, {146,10303}, {182,5095}, {185,12358}, {186,29012}, {371,13969}, {372,8994}, {376,14644}, {381,38788}, {468,1533}, {541,5054}, {542,3524}, {548,10113}, {549,5642}, {550,12295}, {632,14677}, {690,38737}, {974,5562}, {1092,13198}, {1204,5654}, {1294,24930}, {1350,15118}, {1511,3530}, {1514,37911}, {1531,5159}, {1539,3628}, {1553,12068}, {1568,10257}, {1656,20127}, {1986,9729}, {2771,38760}, {2772,38772}, {2773,38784}, {2780,38804}, {2781,16223}, {2854,21167}, {2935,7395}, {3090,10721}, {3448,15051}, {3522,10733}, {3525,12244}, {3526,7728}, {3528,15081}, {3541,15473}, {3564,16976}, {3574,23336}, {3627,15088}, {3796,15693}, {5447,11806}, {5622,10519}, {5655,15701}, {5892,16222}, {5907,17854}, {6036,16278}, {6053,15054}, {6449,19051}, {6450,19052}, {6684,11709}, {6696,15647}, {6776,32257}, {6803,13203}, {7399,23315}, {7464,32223}, {7503,13293}, {7987,13211}, {8567,11744}, {8674,21154}, {9140,15692}, {9540,19059}, {9826,13417}, {10182,15072}, {10212,10610}, {10264,15712}, {10272,12108}, {10299,12383}, {10516,36201}, {10574,12219}, {10620,15720}, {10625,12236}, {10628,16836}, {10706,15702}, {10752,32300}, {10996,18933}, {11579,32114}, {11695,11807}, {11793,12825}, {11800,13348}, {11801,33923}, {12099,36987}, {12100,32423}, {12140,32534}, {12162,17856}, {12228,13336}, {12261,31663}, {12317,15034}, {12368,31423}, {13160,32743}, {13289,17928}, {13403,35497}, {13935,19060}, {14448,14708}, {14683,15020}, {14855,34477}, {14869,38795}, {15030,23328}, {15078,23329}, {15122,32110}, {15359,38738}, {15694,38789}, {16270,21649}, {17701,22352}, {18400,37941}, {18580,37470}, {19504,37514}, {20379,34153}, {20725,37984}, {22104,36164}, {22467,25563}, {26913,35493}, {31762,32311}, {33511,38739}, {33512,38750}, {33547,35240}, {35447,38806}

X(38727) = midpoint of X(i) and X(j) for these {i,j}: {2, 15055}, {3, 15061}, {376, 14644}, {381, 38788}, {5622, 10519}, {14643, 15041}, {20126, 32609}, {38723, 38724}
X(38727) = reflection of X(i) in X(j) for these (i,j): (125, 15061), (5642, 38793), (10706, 38792), (15061, 6699), (16222, 5892), (23515, 34128), (36518, 2), (38724, 38725), (38793, 549)
X(38727) = X(125)-Gibert-Moses centroid
X(38727) = centroid of X(74)X(110)X(125)
X(38727) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 125, 16163), (3, 265, 38726), (3, 6699, 125), (3, 38724, 38723), (3, 38728, 6699), (5, 16111, 13202), (20, 15059, 7687), (74, 631, 5972), (74, 5972, 15063), (113, 12041, 10990), (125, 6699, 38729), (140, 12041, 113), (265, 20397, 125), (6699, 38726, 20397), (6723, 37853, 4), (10257, 21663, 1568), (15051, 15057, 3448), (15061, 38723, 38724), (15061, 38724, 38725), (16163, 38729, 125), (20397, 38726, 265), (38724, 38725, 125)

X(38728) = X(74)-CIRCUM-EULER-POINT OF X(631)

Barycentrics    (-a^2+b^2+c^2)*(3*a^8-3*(b^2+c^2)*a^6-(4*b^4-11*b^2*c^2+4*c^4)*a^4+5*(b^4-c^4)*(b^2-c^2)*a^2-(b^2-c^2)^4) : :
X(38728) = 6*X(2)-X(7728) = 3*X(2)+2*X(12041) = 9*X(2)+X(12244) = 3*X(3)+2*X(125) = 4*X(3)+X(265) = X(3)+4*X(6699) = 6*X(3)-X(12121) = 9*X(3)+X(12902) = 2*X(3)+X(15027) = 2*X(3)+3*X(15061) = 7*X(3)-2*X(16163) = 7*X(3)+8*X(20397) = 11*X(3)+4*X(36253) = 8*X(3)-3*X(38723) = 7*X(3)+3*X(38724) = 13*X(3)+12*X(38725) = 9*X(3)-4*X(38726) = X(3)-6*X(38727) = X(3)+2*X(38729) = X(7728)+4*X(12041) = 3*X(7728)+2*X(12244) = 6*X(12041)-X(12244)

X(38728) lies on these lines: {2,7728}, {3,125}, {4,15088}, {5,10721}, {20,20304}, {30,15059}, {67,5092}, {74,140}, {98,14850}, {99,14849}, {110,549}, {113,3526}, {146,3525}, {165,12261}, {376,10113}, {381,6723}, {382,23515}, {399,15720}, {476,14851}, {477,14993}, {541,15694}, {542,15040}, {548,10733}, {550,14644}, {631,5663}, {632,15021}, {690,38739}, {974,18436}, {1151,19051}, {1152,19052}, {1511,3523}, {1539,3090}, {1656,2777}, {1657,7687}, {2771,38762}, {2772,38774}, {2773,38786}, {2780,38806}, {3091,34584}, {3147,12133}, {3311,13969}, {3312,8994}, {3448,3524}, {3522,15081}, {3528,20396}, {3530,10264}, {3534,12295}, {3576,12898}, {3581,15122}, {3628,14677}, {3851,13202}, {3917,11806}, {4550,10293}, {5010,12904}, {5054,5646}, {5070,36518}, {5432,10081}, {5433,10065}, {5447,21649}, {5504,34483}, {5622,15089}, {5642,12308}, {5891,17855}, {5892,13417}, {6459,13979}, {6460,13915}, {6696,9934}, {6719,35447}, {7280,12903}, {7978,38028}, {7998,12284}, {8273,12334}, {8703,11801}, {8722,12201}, {8981,19059}, {9140,12100}, {9143,15719}, {9144,26614}, {10164,12778}, {10193,13293}, {10272,14869}, {10706,11539}, {10752,38110}, {10902,12906}, {10990,12900}, {11231,12368}, {11562,16836}, {11585,18442}, {11709,26446}, {11735,12702}, {12103,15025}, {12108,14094}, {12140,15750}, {12219,13630}, {12228,37471}, {12236,37484}, {12281,20791}, {12358,18931}, {12383,15717}, {12905,37561}, {13198,22115}, {13201,15045}, {13211,13624}, {13289,25563}, {13353,15463}, {13416,23039}, {13966,19060}, {14708,18580}, {14989,21315}, {15046,38791}, {15051,15712}, {15118,33878}, {15131,37470}, {15359,38730}, {15535,21166}, {15707,24981}, {16003,32609}, {16220,36739}, {17508,32233}, {17854,18435}, {18332,38737}, {18390,35495}, {19110,35256}, {19111,35255}, {23315,37347}, {25739,37968}, {29012,37958}, {32223,35001}

X(38728) = midpoint of X(3522) and X(15081)
X(38728) = reflection of X(i) in X(j) for these (i,j): (110, 22251), (265, 15027), (15051, 15712), (38729, 6699), (38794, 631)
X(38728) = X(15027)-Gibert-Moses centroid
X(38728) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12041, 7728), (3, 125, 12121), (3, 265, 38723), (3, 6699, 15061), (3, 12902, 38726), (3, 15061, 265), (3, 38724, 16163), (3, 38729, 15027), (5, 15055, 20127), (74, 140, 14643), (125, 12121, 265), (125, 38726, 12902), (3526, 15041, 113), (6699, 38727, 3), (6723, 16111, 381), (12121, 15061, 125), (12902, 38726, 12121), (16163, 20397, 38724), (20417, 38793, 399), (23515, 37853, 382)

X(38729) = X(74)-CIRCUM-EULER-POINT OF X(632)

Barycentrics    (-a^2+b^2+c^2)*(4*a^8-4*(b^2+c^2)*a^6-(7*b^4-18*b^2*c^2+7*c^4)*a^4+10*(b^4-c^4)*(b^2-c^2)*a^2-3*(b^2-c^2)^4) : :
X(38729) = 9*X(2)+X(15054) = 3*X(2)+7*X(15057) = 6*X(2)-X(15063) = 3*X(2)+2*X(20417) = 2*X(3)+3*X(125) = 7*X(3)+3*X(265) = X(3)-6*X(6699) = 13*X(3)-3*X(12121) = 17*X(3)+3*X(12902) = X(3)+9*X(15061) = 8*X(3)-3*X(16163) = X(3)+4*X(20397) = 3*X(3)+2*X(36253) = 19*X(3)-9*X(38723) = 11*X(3)+9*X(38724) = 7*X(3)+18*X(38725) = 11*X(3)-6*X(38726) = 4*X(3)-9*X(38727) = X(3)-3*X(38728) = 2*X(15054)+3*X(15063) = X(15054)-6*X(20417) = 14*X(15057)+X(15063) = 7*X(15057)-2*X(20417) = X(15063)+4*X(20417)

X(38729) lies on these lines: {2,15054}, {3,125}, {5,10990}, {20,15044}, {67,10541}, {74,3090}, {110,10303}, {113,3628}, {140,5609}, {146,15029}, {541,1656}, {542,631}, {546,12041}, {549,20379}, {550,20396}, {575,5095}, {632,5663}, {690,38740}, {1092,5622}, {1511,12108}, {1539,12811}, {1568,5159}, {1986,15012}, {1995,23329}, {2771,38763}, {2772,38775}, {2773,38787}, {2777,3091}, {2780,38807}, {3146,7687}, {3292,10257}, {3448,15020}, {3523,9140}, {3525,5972}, {3526,16534}, {3529,14644}, {3544,12244}, {3549,34802}, {3627,16111}, {3857,14677}, {5054,15039}, {5067,10706}, {5072,15041}, {5079,7728}, {5562,16270}, {6036,31854}, {6419,8994}, {6420,13969}, {6447,19051}, {6448,19052}, {7527,25563}, {7982,11735}, {9730,14448}, {10113,12103}, {10193,26913}, {10264,14869}, {10297,21663}, {10620,12900}, {11477,15118}, {11693,11812}, {11694,13393}, {11695,12824}, {12140,35479}, {12162,17853}, {12295,15704}, {13211,30389}, {13336,15132}, {13416,21649}, {15081,17538}, {15106,37514}, {15720,23236}, {16278,20398}, {18400,37952}, {19357,32272}, {25714,32348}, {29012,37953}, {32223,37946}, {32247,32300}, {32250,35486}

X(38729) = midpoint of X(i) and X(j) for these {i,j}: {3, 15027}, {3091, 15021}, {10264, 22251}
X(38729) = reflection of X(i) in X(j) for these (i,j): (20125, 5972), (38728, 6699), (38795, 632)
X(38729) = barycentric product X(525)*X(30221)
X(38729) = trilinear product X(656)*X(30221)
X(38729) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 15057, 20417), (2, 20417, 15063), (3, 15061, 20397), (3, 20397, 125), (74, 3090, 38791), (74, 6723, 36518), (125, 6699, 38727), (125, 38727, 16163), (140, 16003, 5642), (265, 38725, 125), (549, 20379, 30714), (3090, 38791, 36518), (3146, 15025, 7687), (6699, 15061, 125), (6699, 20397, 3), (6723, 38791, 3090), (12041, 23515, 13202), (15021, 15059, 3091), (15025, 15055, 3146), (15027, 38728, 3)

X(38730) = X(98)-CIRCUM-EULER-POINT OF X(20)

Barycentrics    3*a^8-6*(b^2+c^2)*a^6+5*(b^4+b^2*c^2+c^4)*a^4-(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^2-(b^2-c^2)^4 : :
X(38730) = 9*X(2)-8*X(15092) = 3*X(3)-2*X(115) = 5*X(3)-4*X(6036) = 11*X(3)-8*X(20398) = 4*X(3)-3*X(38224) = 2*X(3)-3*X(38731) = 5*X(3)-3*X(38732) = 3*X(3)-X(38733) = 7*X(3)-4*X(38734) = 17*X(3)-12*X(38735) = 3*X(3)-4*X(38736) = 7*X(3)-6*X(38737) = 6*X(3)-5*X(38739) = 13*X(3)-10*X(38740) = 5*X(115)-6*X(6036) = 4*X(115)-3*X(6321) = 11*X(115)-12*X(20398) = 8*X(115)-9*X(38224) = 4*X(115)-9*X(38731) = 10*X(115)-9*X(38732) = 7*X(115)-6*X(38734) = 17*X(115)-18*X(38735) = 7*X(115)-9*X(38737) = X(115)-3*X(38738) = 4*X(115)-5*X(38739) = 13*X(115)-15*X(38740) = 4*X(15092)-3*X(22515)

Let LA, LB, LC be the lines through A, B, C, resp. parallel to the Brocard axis. Let MA, MB, MC be the reflections of lines 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 Brocard axis. The triangle A"B"C" is homothetic to ABC, with center of homothety X(115) and circumcenter X(38730). (See Hyacinthos #16741/16782, Sep 2008.) (Randy Hutson, May 31, 2020)

X(38730) lies on these lines: {2,15092}, {3,115}, {4,15561}, {5,10723}, {20,2782}, {30,99}, {35,13182}, {36,13183}, {98,550}, {114,382}, {140,14639}, {147,3529}, {148,376}, {381,620}, {542,15681}, {543,3534}, {548,34473}, {549,14061}, {671,8703}, {690,12121}, {1478,15452}, {1656,38748}, {1657,2794}, {2482,3830}, {2549,26316}, {2783,38753}, {2784,38765}, {2785,38777}, {2793,38797}, {2797,23240}, {3023,4302}, {3027,4299}, {3054,38226}, {3095,7737}, {3098,11646}, {3146,22505}, {3398,15048}, {3522,14651}, {3523,34127}, {3526,23514}, {3530,38229}, {3579,13178}, {3843,36519}, {3851,6721}, {4027,33257}, {5026,31670}, {5054,6722}, {5055,31274}, {5073,38743}, {5076,20399}, {5092,6034}, {5182,21850}, {5186,18533}, {5461,15693}, {5473,6777}, {5474,6778}, {5611,9112}, {5615,9113}, {6055,12355}, {6284,10089}, {6449,8980}, {6450,13967}, {6658,14881}, {6781,9301}, {7354,10086}, {7816,37243}, {7970,28174}, {7983,34773}, {8591,11001}, {8596,15697}, {8997,13665}, {9155,36181}, {9166,12100}, {9167,19709}, {9864,28160}, {9881,28208}, {10053,15338}, {10069,15326}, {10352,18502}, {10483,12184}, {11711,12699}, {12177,29181}, {12243,35369}, {12902,15357}, {13202,33512}, {13624,38220}, {13785,13989}, {14692,15704}, {14971,15701}, {14981,17800}, {15055,15535}, {15300,15685}, {15342,34153}, {15359,38728}, {15545,17702}, {15682,22566}, {15687,23234}, {15696,38747}, {15698,26614}, {15993,35383}, {16163,18332}, {16278,38726}, {21636,28150}, {23004,36755}, {23005,36756}, {29012,35456}, {29317,35458}

X(38730) = midpoint of X(i) and X(j) for these {i,j}: {20, 13172}, {147, 3529}, {1657, 13188}, {8591, 11001}, {9862, 20094}, {17800, 38744}
X(38730) = reflection of X(i) in X(j) for these (i,j): (3, 38738), (4, 33813), (98, 550), (115, 38736), (148, 12042), (382, 114), (671, 8703), (3146, 22505), (3830, 2482), (6033, 99), (6321, 3), (7983, 34773), (9301, 6781), (10723, 5), (11632, 376), (11646, 3098), (12188, 38749), (12355, 6055), (12699, 11711), (12902, 15357), (13178, 3579), (13188, 10992), (13202, 33512), (14692, 23235), (14830, 3534), (15342, 34153), (15682, 22566), (16278, 38726), (18332, 16163), (23004, 36755), (23005, 36756), (31670, 5026), (38224, 38731), (38733, 115), (38741, 20), (38744, 14981)
X(38730) = anticomplement of X(22515)
X(38730) = Stammler circle-inverse of-X(35453)
X(38730) = crossdifference of every pair of points on line {X(6041), X(6132)}
X(38730) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 115, 38739), (3, 6321, 38224), (3, 38732, 6036), (3, 38733, 115), (3, 38738, 38731), (4, 33813, 15561), (5, 21166, 38750), (20, 20094, 9862), (98, 550, 38742), (99, 6033, 8724), (115, 38733, 6321), (115, 38736, 3), (115, 38738, 38736), (115, 38739, 38224), (148, 12042, 11632), (6321, 38731, 3), (6321, 38739, 115), (9862, 13172, 20094), (10723, 21166, 5), (12355, 15688, 6055), (38733, 38736, 38739)

X(38731) = X(98)-CIRCUM-EULER-POINT OF X(376)

Barycentrics    5*a^8-10*(b^2+c^2)*a^6+(9*b^4+7*b^2*c^2+9*c^4)*a^4-(b^2+c^2)*(3*b^4-b^2*c^2+3*c^4)*a^2-(b^2-c^2)^4 : :
X(38731) = 5*X(3)-2*X(115) = 7*X(3)-4*X(6036) = 4*X(3)-X(6321) = 17*X(3)-8*X(20398) = 2*X(3)+X(38730) = 3*X(3)-X(38732) = 7*X(3)-X(38733) = 13*X(3)-4*X(38734) = 9*X(3)-4*X(38735) = X(3)-4*X(38736) = 3*X(3)-2*X(38737) = X(3)+2*X(38738) = 8*X(3)-5*X(38739) = 19*X(3)-10*X(38740) = 7*X(115)-10*X(6036) = 8*X(115)-5*X(6321) = 17*X(115)-20*X(20398) = 4*X(115)-5*X(38224) = 4*X(115)+5*X(38730) = 6*X(115)-5*X(38732) = 14*X(115)-5*X(38733) = 13*X(115)-10*X(38734) = 9*X(115)-10*X(38735) = X(115)-10*X(38736) = 3*X(115)-5*X(38737) = X(115)+5*X(38738)

X(38731) lies on these lines: {3,115}, {4,38750}, {20,6033}, {30,10242}, {98,548}, {99,550}, {114,1657}, {140,10723}, {147,17538}, {148,3528}, {376,2782}, {381,22247}, {382,620}, {542,15689}, {543,15688}, {549,14639}, {631,22515}, {671,34200}, {690,38723}, {2482,15681}, {2783,38754}, {2784,38766}, {2785,38778}, {2793,38798}, {2794,3534}, {3398,9607}, {3522,12042}, {3524,34127}, {3525,15092}, {3529,22505}, {3530,14061}, {3830,36519}, {3843,6721}, {3851,31274}, {4027,33268}, {4299,15452}, {4316,12184}, {4324,12185}, {5010,13182}, {5054,23514}, {5461,15700}, {6034,17508}, {6054,15686}, {6055,14093}, {6287,7816}, {6455,8980}, {6456,13967}, {6722,15720}, {7280,13183}, {8703,11632}, {9166,17504}, {9167,14269}, {9862,14692}, {9880,15693}, {10086,15326}, {10089,15338}, {10304,14651}, {10352,33250}, {10722,15704}, {10992,12188}, {11646,14810}, {12100,38229}, {12121,15545}, {13178,31663}, {13188,15696}, {14971,15707}, {15683,22566}, {15705,26614}, {17502,38220}, {18332,38726}

X(38731) = midpoint of X(i) and X(j) for these {i,j}: {12117, 34473}, {15681, 38743}, {38224, 38730}
X(38731) = reflection of X(i) in X(j) for these (i,j): (381, 38748), (3830, 36519), (6034, 17508), (6321, 38224), (9166, 17504), (11632, 34473), (14269, 9167), (14639, 549), (14830, 38742), (15561, 21166), (34473, 8703), (38220, 17502), (38224, 3), (38229, 12100), (38732, 38737), (38742, 376), (38743, 2482)
X(38731) = X(38730)-Gibert-Moses centroid
X(38731) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6321, 38739), (3, 38730, 6321), (3, 38732, 38737), (3, 38733, 6036), (3, 38738, 38730), (20, 33813, 6033), (99, 550, 38741), (3522, 13172, 12042), (8703, 12117, 11632), (10992, 38747, 12188), (13188, 15696, 38749), (38732, 38737, 38224), (38736, 38738, 3)

X(38732) = X(98)-CIRCUM-EULER-POINT OF X(381)

Barycentrics    a^8-2*(b^2+c^2)*a^6+5*b^2*c^2*a^4+(b^2+c^2)*(3*b^4-7*b^2*c^2+3*c^4)*a^2-2*(b^2-c^2)^4 : :
X(38732) = 2*X(2)+X(12355) = X(3)-4*X(115) = 5*X(3)-8*X(6036) = X(3)+2*X(6321) = 7*X(3)-16*X(20398) = 5*X(3)-2*X(38730) = 3*X(3)-2*X(38731) = 2*X(3)+X(38733) = X(3)+8*X(38734) = 3*X(3)-8*X(38735) = 11*X(3)-8*X(38736) = 3*X(3)-4*X(38737) = 7*X(3)-4*X(38738) = 7*X(3)-10*X(38739) = 11*X(3)-20*X(38740) = 5*X(115)-2*X(6036) = 2*X(115)+X(6321) = 7*X(115)-4*X(20398) = 10*X(115)-X(38730) = 6*X(115)-X(38731) = 8*X(115)+X(38733) = X(115)+2*X(38734) = 3*X(115)-2*X(38735) = 11*X(115)-2*X(38736) = 3*X(115)-X(38737) = 7*X(115)-X(38738) = 14*X(115)-5*X(38739) = 11*X(115)-5*X(38740) = X(12355)+4*X(38229)

X(38732) lies on these lines: {2,12355}, {3,115}, {4,5984}, {5,148}, {13,13102}, {14,13103}, {30,8859}, {98,382}, {99,1656}, {114,3851}, {140,13172}, {147,546}, {262,381}, {265,16278}, {355,11599}, {542,5093}, {543,5055}, {547,8591}, {616,20253}, {617,20252}, {620,5070}, {690,38724}, {754,10242}, {999,13182}, {1351,11646}, {1482,13178}, {1657,10723}, {1916,13108}, {2482,15703}, {2777,14849}, {2783,38755}, {2784,38767}, {2785,38779}, {2793,38799}, {2794,3830}, {2936,21308}, {3023,9669}, {3027,9654}, {3090,20094}, {3295,13183}, {3526,14061}, {3534,34473}, {3627,9862}, {3843,6033}, {3845,12243}, {3934,19910}, {5050,6034}, {5054,9166}, {5056,35369}, {5071,8596}, {5072,23235}, {5073,11623}, {5076,10722}, {5079,15092}, {5461,15694}, {5469,16963}, {5470,16962}, {5611,23004}, {5615,23005}, {6055,15681}, {6722,10992}, {6777,16001}, {6778,16002}, {7697,18546}, {7773,32520}, {7983,12645}, {8724,19709}, {9655,10069}, {9668,10053}, {9734,18362}, {9735,22846}, {9736,22891}, {9755,34682}, {9830,14848}, {9860,22793}, {9956,13174}, {9993,33997}, {10086,31479}, {10113,22265}, {10246,38220}, {10620,15535}, {10754,11898}, {10769,12331}, {11171,11648}, {11177,15687}, {11272,15031}, {11602,16629}, {11603,16628}, {11608,37826}, {11725,37624}, {11801,18331}, {12117,15693}, {12121,33511}, {12902,18332}, {13449,32457}, {13846,35699}, {13847,35698}, {13860,35464}, {14041,32515}, {14042,32134}, {14675,33753}, {14692,38745}, {14830,15684}, {14850,23515}, {15706,26614}, {17800,38749}, {23251,35825}, {23261,35824}, {33265,38230}

X(38732) = midpoint of X(i) and X(j) for these {i,j}: {671, 14639}, {6321, 38224}
X(38732) = reflection of X(i) in X(j) for these (i,j): (2, 38229), (3, 38224), (381, 14639), (3534, 34473), (5050, 6034), (5054, 9166), (8724, 36519), (10246, 38220), (14850, 23515), (15561, 23514), (15681, 38742), (21166, 34127), (33265, 38230), (38224, 115), (38731, 38737), (38737, 38735), (38742, 6055), (38743, 381), (38748, 5461)
X(38732) = X(38733)-Gibert-Moses centroid
X(38732) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6321, 38733), (4, 12188, 38744), (5, 148, 13188), (98, 22515, 382), (115, 6321, 3), (115, 38734, 6321), (115, 38737, 38735), (115, 38738, 20398), (6036, 38730, 3), (9880, 11632, 3830), (9880, 36523, 11632), (10723, 12042, 1657), (14061, 33813, 3526), (15561, 23514, 5055), (20398, 38738, 38739), (21166, 34127, 5054), (38224, 38731, 38737), (38731, 38737, 3), (38735, 38737, 38224), (38738, 38739, 3)

X(38733) = X(98)-CIRCUM-EULER-POINT OF X(382)

Barycentrics    3*a^8-6*(b^2+c^2)*a^6+(2*b^2-b*c+2*c^2)*(2*b^2+b*c+2*c^2)*a^4+(b^2+c^2)*(b^4-5*b^2*c^2+c^4)*a^2-2*(b^2-c^2)^4 : :
X(38733) = 3*X(3)-4*X(115) = 7*X(3)-8*X(6036) = 13*X(3)-16*X(20398) = 5*X(3)-6*X(38224) = 3*X(3)-2*X(38730) = 7*X(3)-6*X(38731) = 2*X(3)-3*X(38732) = 5*X(3)-8*X(38734) = 9*X(3)-8*X(38736) = 11*X(3)-12*X(38737) = 5*X(3)-4*X(38738) = 9*X(3)-10*X(38739) = 17*X(3)-20*X(38740) = 7*X(115)-6*X(6036) = 2*X(115)-3*X(6321) = 13*X(115)-12*X(20398) = 10*X(115)-9*X(38224) = 14*X(115)-9*X(38731) = 8*X(115)-9*X(38732) = 5*X(115)-6*X(38734) = 19*X(115)-18*X(38735) = 3*X(115)-2*X(38736) = 11*X(115)-9*X(38737) = 5*X(115)-3*X(38738) = 6*X(115)-5*X(38739) = 17*X(115)-15*X(38740)

X(38733) lies on these lines: {3,115}, {4,13188}, {5,13172}, {30,148}, {98,1657}, {99,381}, {114,3843}, {147,3627}, {382,2782}, {542,6144}, {543,3830}, {550,14651}, {620,5055}, {631,38229}, {671,3534}, {690,12902}, {999,13183}, {1656,14639}, {2482,19709}, {2783,38756}, {2784,38768}, {2785,38780}, {2793,38800}, {2794,5073}, {3023,9668}, {3027,9655}, {3295,13182}, {3526,7918}, {3543,35369}, {3845,8591}, {3851,10992}, {4027,18501}, {5054,12117}, {5070,23514}, {5076,22505}, {5186,18494}, {5461,15701}, {5969,18440}, {5984,33703}, {5989,9993}, {6034,12017}, {6054,38335}, {6055,15689}, {6248,19910}, {6319,26336}, {6320,26346}, {6407,8980}, {6408,13967}, {6722,15694}, {6777,13102}, {6778,13103}, {7687,14850}, {7983,18526}, {8596,15682}, {8724,14269}, {8782,18503}, {9166,15693}, {9654,10086}, {9669,10089}, {9860,28146}, {9875,28198}, {10769,12773}, {11599,18481}, {11623,38742}, {11632,15681}, {11646,33878}, {11648,26316}, {11711,18493}, {12121,16278}, {12702,13178}, {13173,18524}, {13174,18480}, {13179,18508}, {13180,18519}, {13181,18518}, {13189,18545}, {13190,18543}, {14830,15685}, {14849,16111}, {15041,15535}, {15452,31479}, {15484,32447}, {15695,36523}, {15696,34473}, {15703,31274}, {15716,26614}, {15720,34127}, {17800,38741}, {18510,19108}, {18512,19109}, {22514,26321}, {23251,35878}, {23261,35879}, {25154,32552}, {25164,32553}, {32521,33256}, {32819,37243}, {33511,38723}

X(38733) = midpoint of X(i) and X(j) for these {i,j}: {5984, 33703}, {8596, 15682}
X(38733) = reflection of X(i) in X(j) for these (i,j): (3, 6321), (99, 22515), (147, 3627), (382, 10723), (1657, 98), (3534, 671), (8591, 3845), (12121, 16278), (12188, 148), (12702, 13178), (12773, 10769), (13172, 5), (13174, 18480), (13188, 4), (15681, 11632), (15685, 14830), (17800, 38741), (18481, 11599), (18508, 13179), (18526, 7983), (19910, 6248), (23235, 22505), (33878, 11646), (38730, 115), (38738, 38734), (38744, 382)
X(38733) = Stammler circle-inverse of-X(2079)
X(38733) = crossdifference of every pair of points on line {X(6132), X(10567)}
X(38733) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6321, 38732), (4, 13188, 38743), (99, 22515, 381), (115, 38730, 3), (115, 38736, 38739), (6036, 38731, 3), (6321, 38224, 38734), (6321, 38730, 115), (12188, 12355, 148), (14639, 33813, 1656), (23514, 38750, 5070), (38224, 38738, 3), (38730, 38739, 38736), (38734, 38738, 38224), (38736, 38739, 3)

X(38734) = X(98)-CIRCUM-EULER-POINT OF X(546)

Barycentrics    2*a^8-4*(b^2+c^2)*a^6+(b^4+8*b^2*c^2+c^4)*a^4+2*(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^2-3*(b^2-c^2)^4 : :
X(38734) = X(3)-3*X(115) = 2*X(3)-3*X(6036) = X(3)+3*X(6321) = 5*X(3)-9*X(38224) = 7*X(3)-3*X(38730) = 13*X(3)-9*X(38731) = X(3)-9*X(38732) = 5*X(3)+3*X(38733) = 4*X(3)-9*X(38735) = 4*X(3)-3*X(38736) = 7*X(3)-9*X(38737) = 5*X(3)-3*X(38738) = 11*X(3)-15*X(38739) = 3*X(3)-5*X(38740) = 3*X(115)-2*X(20398) = 5*X(115)-3*X(38224) = 7*X(115)-X(38730) = 13*X(115)-3*X(38731) = X(115)-3*X(38732) = 5*X(115)+X(38733) = 4*X(115)-3*X(38735) = 4*X(115)-X(38736) = 7*X(115)-3*X(38737) = 5*X(115)-X(38738) = 11*X(115)-5*X(38739) = 9*X(115)-5*X(38740)

Let NA be the reflection of X(5) in the A-altitude, and define NB and NC cyclically. NANBNC is inversely similar to ABC, with similitude center X(195). X(38734) = X(187)-of-NANBNC. (see Hyacinthos #21522, 2/11/2013, Antreas Hatzipolakis) (Randy Hutson, May 31, 2020)

X(38734) lies on these lines: {2,10992}, {3,115}, {4,542}, {5,543}, {20,6055}, {30,11623}, {98,3146}, {99,3090}, {114,148}, {140,5461}, {338,32257}, {381,14981}, {382,10991}, {395,20416}, {396,20415}, {546,2782}, {547,36521}, {575,5254}, {620,3628}, {631,9166}, {632,6722}, {690,24978}, {1656,2482}, {2783,38757}, {2784,38769}, {2785,38781}, {2793,38801}, {2794,3627}, {2936,7529}, {2996,10754}, {3303,13183}, {3304,13182}, {3455,7517}, {3523,12117}, {3525,13172}, {3526,14971}, {3529,10723}, {3832,6054}, {3851,8724}, {3858,22566}, {5055,15300}, {5056,8591}, {5068,8596}, {5070,9167}, {5072,13188}, {5073,14830}, {5076,12188}, {5079,15561}, {5182,32979}, {5286,22234}, {5465,30714}, {5477,11482}, {5882,12258}, {5965,13449}, {6425,8980}, {6426,13967}, {6772,16634}, {6775,16635}, {7530,13233}, {7620,11178}, {7745,22330}, {7828,35950}, {7841,19662}, {7982,13178}, {8370,25555}, {8960,35699}, {9761,22575}, {9763,22576}, {9875,11522}, {10303,21166}, {10358,32135}, {11005,15044}, {11177,17578}, {11185,24206}, {11477,11646}, {11648,37348}, {11725,15178}, {12042,15704}, {12103,38747}, {12812,15092}, {14561,14928}, {14832,18114}, {14869,34127}, {15022,20094}, {15027,15357}, {15359,20397}, {16220,18007}, {17538,34473}, {18546,37242}, {34505,34507}

X(38734) = midpoint of X(i) and X(j) for these {i,j}: {114, 148}, {115, 6321}, {382, 10991}, {671, 9880}, {2482, 12355}, {10723, 38749}, {16001, 16002}, {25154, 31696}, {25164, 31695}, {38733, 38738}
X(38734) = reflection of X(i) in X(j) for these (i,j): (3, 20398), (99, 6721), (6036, 115), (33813, 6722), (36521, 547), (38736, 6036), (38745, 546), (38749, 35021)
X(38734) = complement of X(10992)
X(38734) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 115, 20398), (3, 20398, 6036), (99, 3090, 38751), (99, 23514, 6721), (115, 6036, 38735), (115, 38738, 38224), (148, 3091, 23235), (148, 14639, 114), (382, 11632, 10991), (3090, 38751, 6721), (3091, 23235, 114), (6321, 38224, 38733), (6321, 38732, 115), (14639, 23235, 3091), (23514, 38751, 3090), (31695, 31696, 1992), (33813, 38229, 6722), (35830, 35831, 7746), (38224, 38733, 38738), (38735, 38736, 6036)

X(38735) = X(98)-CIRCUM-EULER-POINT OF X(547)

Barycentrics    2*a^8-4*(b^2+c^2)*a^6+(9*b^4-8*b^2*c^2+9*c^4)*a^4-2*(b^2+c^2)*(6*b^4-11*b^2*c^2+6*c^4)*a^2+5*(b^2-c^2)^4 : :
X(38735) = X(3)+5*X(115) = 2*X(3)-5*X(6036) = 7*X(3)+5*X(6321) = X(3)-10*X(20398) = X(3)-5*X(38224) = 17*X(3)-5*X(38730) = 9*X(3)-5*X(38731) = 3*X(3)+5*X(38732) = 19*X(3)+5*X(38733) = 4*X(3)+5*X(38734) = 8*X(3)-5*X(38736) = 3*X(3)-5*X(38737) = 11*X(3)-5*X(38738) = 2*X(115)+X(6036) = 7*X(115)-X(6321) = X(115)+2*X(20398) = 17*X(115)+X(38730) = 9*X(115)+X(38731) = 3*X(115)-X(38732) = 19*X(115)-X(38733) = 4*X(115)-X(38734) = 8*X(115)+X(38736) = 3*X(115)+X(38737) = 11*X(115)+X(38738) = 13*X(115)+5*X(38739) = 7*X(115)+5*X(38740)

X(38735) lies on these lines: {3,115}, {4,35021}, {98,3832}, {99,3533}, {114,5056}, {542,3545}, {543,11539}, {547,2782}, {620,16239}, {632,35022}, {671,15702}, {690,38725}, {2482,15723}, {2783,38758}, {2784,38770}, {2785,38782}, {2793,38802}, {2794,3845}, {3543,6055}, {3850,11623}, {5067,6721}, {5079,14692}, {5102,6034}, {5306,14160}, {5965,33228}, {9300,14162}, {9880,11001}, {11632,36519}, {11725,33179}, {11812,36523}, {14971,15561}, {15092,38745}, {15708,21166}, {16200,38220}, {33703,38749}

X(38735) = midpoint of X(i) and X(j) for these {i,j}: {115, 38224}, {671, 38748}, {6055, 14639}, {9880, 34473}, {11632, 36519}, {14651, 23514}, {38732, 38737}
X(38735) = reflection of X(i) in X(j) for these (i,j): (6036, 38224), (38224, 20398), (38746, 547)
X(38735) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115, 6036, 38734), (115, 20398, 6036), (115, 38737, 38732), (115, 38740, 6321), (6036, 38734, 38736), (9166, 14651, 23514), (38224, 38732, 38737)

X(38736) = X(98)-CIRCUM-EULER-POINT OF X(548)

Barycentrics    6*a^8-12*(b^2+c^2)*a^6+(11*b^4+8*b^2*c^2+11*c^4)*a^4-2*(b^2+c^2)*(2*b^4-b^2*c^2+2*c^4)*a^2-(b^2-c^2)^4 : :
X(38736) = 3*X(3)-X(115) = 5*X(3)-X(6321) = 5*X(3)-2*X(20398) = 7*X(3)-3*X(38224) = 3*X(3)+X(38730) = X(3)+3*X(38731) = 11*X(3)-3*X(38732) = 9*X(3)-X(38733) = 4*X(3)-X(38734) = 8*X(3)-3*X(38735) = 5*X(3)-3*X(38737) = 9*X(3)-5*X(38739) = 11*X(3)-5*X(38740) = 2*X(115)-3*X(6036) = 5*X(115)-3*X(6321) = 5*X(115)-6*X(20398) = 7*X(115)-9*X(38224) = X(115)+9*X(38731) = 11*X(115)-9*X(38732) = 3*X(115)-X(38733) = 4*X(115)-3*X(38734) = 8*X(115)-9*X(38735) = 5*X(115)-9*X(38737) = X(115)+3*X(38738) = 3*X(115)-5*X(38739) = 11*X(115)-15*X(38740)

X(38736) lies on these lines: {3,115}, {4,6721}, {20,114}, {30,620}, {69,74}, {98,3522}, {140,15092}, {148,6055}, {381,31274}, {382,36519}, {516,11724}, {543,8703}, {548,2782}, {549,6722}, {550,2794}, {576,1285}, {631,10723}, {671,19708}, {690,38726}, {1003,19130}, {1569,9821}, {1657,15561}, {2482,3534}, {2783,38759}, {2784,38771}, {2785,38783}, {2793,38803}, {3314,32152}, {3523,14639}, {3524,9880}, {3528,13172}, {3529,38751}, {3552,9993}, {3830,9167}, {3845,22247}, {4226,5972}, {5186,37931}, {5461,12100}, {5473,6778}, {5474,6777}, {5477,33878}, {6409,8980}, {6410,13967}, {6560,8997}, {6561,13989}, {6723,35922}, {6781,35002}, {7690,35945}, {7692,35944}, {7737,9737}, {7970,9778}, {8598,19924}, {8724,15689}, {9166,15698}, {9541,19108}, {10352,33244}, {10991,13188}, {11495,22504}, {11623,33923}, {11632,14093}, {11676,29317}, {11711,31730}, {11725,13624}, {12103,38745}, {12121,15357}, {12188,15688}, {13178,35242}, {13335,15048}, {14651,21735}, {14830,15300}, {14971,15693}, {14981,15696}, {15326,15452}, {15683,23234}, {15690,36521}, {15704,20399}, {15711,26614}, {15712,34127}, {15759,36523}, {19662,35955}, {19710,22566}, {25565,35954}, {31670,35927}, {33517,36756}, {33518,36755}

X(38736) = midpoint of X(i) and X(j) for these {i,j}: {3, 38738}, {20, 114}, {98, 10992}, {99, 38749}, {115, 38730}, {550, 33813}, {1569, 9821}, {2482, 3534}, {5477, 33878}, {6055, 12117}, {6781, 35002}, {10991, 13188}, {11711, 31730}, {12121, 15357}, {14830, 15300}, {14981, 38741}, {15704, 22505}, {19710, 22566}
X(38736) = reflection of X(i) in X(j) for these (i,j): (4, 6721), (3845, 22247), (5461, 12100), (6036, 3), (6321, 20398), (11725, 13624), (22505, 20399), (22515, 6722), (38734, 6036), (38747, 548)
X(38736) = complement of X(39809)
X(38736) = circumcircle-inverse of-X(35453)
X(38736) = crossdifference of every pair of points on line {X(6132), X(14398)}
X(38736) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6321, 38737), (3, 35453, 34866), (3, 38730, 115), (3, 38731, 38738), (3, 38733, 38739), (4, 38748, 6721), (20, 21166, 114), (99, 376, 38749), (115, 38738, 38730), (382, 38750, 36519), (549, 22515, 6722), (631, 10723, 23514), (3528, 13172, 34473), (6036, 38734, 38735), (6321, 38737, 20398), (10304, 12117, 6055), (13188, 38742, 10991), (20398, 38737, 6036), (38730, 38739, 38733), (38733, 38739, 115)

X(38737) = X(98)-CIRCUM-EULER-POINT OF X(549)

Barycentrics    4*a^8-8*(b^2+c^2)*a^6+(3*b^2+4*b*c+3*c^2)*(3*b^2-4*b*c+3*c^2)*a^4-2*(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^2+(b^2-c^2)^4 : :
X(38737) = 2*X(3)+X(115) = X(3)+2*X(6036) = 5*X(3)+X(6321) = 5*X(3)+4*X(20398) = 7*X(3)-X(38730) = 3*X(3)-X(38731) = 3*X(3)+X(38732) = 11*X(3)+X(38733) = 7*X(3)+2*X(38734) = 3*X(3)+2*X(38735) = 5*X(3)-2*X(38736) = 4*X(3)-X(38738) = X(3)+5*X(38739) = 4*X(3)+5*X(38740) = X(115)-4*X(6036) = 5*X(115)-2*X(6321) = 5*X(115)-8*X(20398) = 7*X(115)+2*X(38730) = 3*X(115)+2*X(38731) = 3*X(115)-2*X(38732) = 11*X(115)-2*X(38733) = 7*X(115)-4*X(38734) = 3*X(115)-4*X(38735) = 5*X(115)+4*X(38736) = 2*X(115)+X(38738) = X(115)-10*X(38739) = 2*X(115)-5*X(38740) = 2*X(34473)+X(36519)

X(38737) lies on these lines: {2,2794}, {3,115}, {4,6722}, {5,38749}, {20,14061}, {30,5215}, {40,11725}, {98,620}, {99,3523}, {114,140}, {147,10303}, {148,15717}, {165,38220}, {182,5477}, {230,18860}, {371,13967}, {372,8980}, {376,5461}, {381,38742}, {542,5054}, {543,3524}, {548,22515}, {549,2482}, {671,15692}, {690,38727}, {754,21445}, {1078,32458}, {1506,13335}, {1569,13334}, {1656,38741}, {2023,5188}, {2783,38760}, {2784,38772}, {2785,38784}, {2787,21154}, {2793,38804}, {3090,10722}, {3398,9698}, {3522,10723}, {3525,7914}, {3526,6033}, {3530,33813}, {3564,10256}, {3627,15092}, {3628,22505}, {3785,8781}, {5152,37455}, {5346,10983}, {5969,21167}, {6034,31884}, {6054,15702}, {6680,35385}, {6683,38383}, {6684,11710}, {6699,15357}, {6781,15980}, {7619,25486}, {7755,9737}, {7794,10104}, {7862,36998}, {7940,9863}, {7987,13178}, {8150,12176}, {8703,9880}, {8721,32989}, {8722,21843}, {8724,15701}, {9166,10304}, {9540,19055}, {9756,11288}, {9864,31423}, {10124,22566}, {10168,11261}, {10257,12095}, {10299,13172}, {10352,33001}, {10519,14645}, {10992,15712}, {11177,15721}, {11632,15300}, {12117,15698}, {12188,15720}, {12243,15719}, {12355,15716}, {13085,22712}, {13935,19056}, {14830,15694}, {14869,38751}, {15359,16163}, {15709,23234}, {18332,38728}, {23235,35022}

X(38737) = midpoint of X(i) and X(j) for these {i,j}: {2, 34473}, {3, 38224}, {165, 38220}, {376, 14639}, {381, 38742}, {6034, 31884}, {6055, 38748}, {8703, 38229}, {9166, 10304}, {14651, 21166}, {14830, 38743}, {21156, 21157}, {38731, 38732}
X(38737) = reflection of X(i) in X(j) for these (i,j): (115, 38224), (2482, 38748), (6054, 38746), (9167, 5054), (9880, 38229), (14639, 5461), (23514, 34127), (36519, 2), (38224, 6036), (38732, 38735), (38746, 22247), (38748, 549)
X(38737) = X(115)-Gibert-Moses centroid
X(38737) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 115, 38738), (3, 6036, 115), (3, 6321, 38736), (3, 38732, 38731), (3, 38739, 6036), (98, 620, 14981), (98, 631, 620), (114, 12042, 10991), (115, 6036, 38740), (140, 12042, 114), (6036, 38736, 20398), (6321, 20398, 115), (6722, 38747, 4), (10991, 31274, 114), (20398, 38736, 6321), (23514, 34127, 14971), (38224, 38731, 38732), (38224, 38732, 38735), (38732, 38735, 115), (38738, 38740, 115)

X(38738) = X(98)-CIRCUM-EULER-POINT OF X(550)

Barycentrics    4*a^8-8*(b^2+c^2)*a^6+(7*b^4+6*b^2*c^2+7*c^4)*a^4-2*(b^2+c^2)*(b^4+c^4)*a^2-(b^2-c^2)^4 : :
X(38738) =2 X(3) - X(115) = 3*X(3)-2*X(6036) = 3*X(3)-X(6321) = 7*X(3)-4*X(20398) = 5*X(3)-3*X(38224) = X(3)-3*X(38731) = 7*X(3)-3*X(38732) = 5*X(3)-X(38733) = 5*X(3)-2*X(38734) = 11*X(3)-6*X(38735) = 4*X(3)-3*X(38737) = 7*X(3)-5*X(38739) = 8*X(3)-5*X(38740) = 3*X(115)-4*X(6036) = 3*X(115)-2*X(6321) = 7*X(115)-8*X(20398) = 5*X(115)-6*X(38224) = X(115)+2*X(38730) = X(115)-6*X(38731) = 7*X(115)-6*X(38732) = 5*X(115)-2*X(38733) = 5*X(115)-4*X(38734) = 11*X(115)-12*X(38735) = X(115)-4*X(38736) = 2*X(115)-3*X(38737) = 7*X(115)-10*X(38739) = 4*X(115)-5*X(38740)

X(38738) is the orthocenter of the mid-triangle of the antipedal triangles of X(13) and X(14). (Randy Hutson, May 31, 2020)

X(38738) lies on these lines: {2,10723}, {3,115}, {4,620}, {5,31274}, {20,99}, {30,114}, {98,376}, {126,7417}, {132,4235}, {140,22515}, {148,3522}, {165,13178}, {381,6721}, {382,15561}, {511,1569}, {516,11711}, {542,1350}, {548,12042}, {549,9880}, {550,2782}, {631,6722}, {632,15092}, {671,10304}, {690,16163}, {1513,32456}, {1657,6033}, {2023,21163}, {2783,38761}, {2784,38773}, {2785,38785}, {2787,24466}, {2790,36988}, {2793,38805}, {2797,3184}, {2799,14689}, {2936,9861}, {3023,15338}, {3027,15326}, {3070,8997}, {3071,13989}, {3146,20399}, {3163,5467}, {3455,35243}, {3523,14061}, {3524,5461}, {3528,14651}, {3529,10722}, {3530,34127}, {3545,22247}, {3576,11725}, {3627,38751}, {4027,33265}, {4045,35925}, {4226,35282}, {4299,10086}, {4302,10089}, {4304,24472}, {5026,29181}, {5085,36772}, {5204,13183}, {5217,13182}, {5731,7983}, {5984,8591}, {5985,37299}, {6054,11001}, {6055,8703}, {6200,8980}, {6230,13835}, {6231,13712}, {6361,7970}, {6396,13967}, {6459,19108}, {6460,19109}, {6770,35696}, {6773,35692}, {6776,14645}, {7354,15452}, {7694,8781}, {7747,9737}, {7765,13335}, {7781,36998}, {7794,32152}, {7820,37242}, {7987,38220}, {8589,37451}, {8724,15681}, {8754,22085}, {8976,36762}, {9166,15692}, {9541,19056}, {9734,37348}, {9862,17538}, {9881,34628}, {9884,34632}, {9993,35951}, {10256,31275}, {10352,33007}, {10754,25406}, {11177,15697}, {11632,15688}, {11646,31884}, {11724,12699}, {12188,15696}, {12355,14093}, {13179,16190}, {14538,25560}, {14539,25559}, {14830,15689}, {14891,26614}, {15357,17702}, {15359,38727}, {15682,23234}, {15712,38229}, {17800,38743}, {18332,38723}, {19708,36523}, {19924,32135}, {30789,36163}, {33703,38746}, {34841,35923}

X(38738) = midpoint of X(i) and X(j) for these {i,j}: {3, 38730}, {20, 99}, {98, 13172}, {376, 12117}, {1657, 6033}, {3529, 10722}, {5473, 5474}, {6054, 11001}, {6361, 7970}, {8724, 15681}, {9862, 23235}, {9881, 34628}, {9884, 34632}, {10992, 38749}, {13188, 38741}
X(38738) = reflection of X(i) in X(j) for these (i,j): (3, 38736), (4, 620), (98, 38747), (114, 33813), (115, 3), (148, 11623), (1513, 32456), (6054, 36521), (6055, 8703), (6321, 6036), (9880, 549), (10722, 38745), (10991, 38749), (12042, 548), (12699, 11724), (14981, 99), (22515, 140), (36519, 21166), (38733, 38734), (38745, 35022), (38749, 550)
X(38738) = circumperp conjugate of X(2079)
X(38738) = complement of X(10723)
X(38738) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 115, 38737), (3, 6321, 6036), (3, 38731, 38736), (3, 38732, 38739), (3, 38733, 38224), (4, 620, 36519), (4, 21166, 620), (5, 38748, 31274), (98, 376, 38747), (98, 12117, 13172), (114, 33813, 2482), (115, 38737, 38740), (376, 13172, 98), (6036, 6321, 115), (20398, 38732, 115), (38224, 38733, 38734), (38224, 38734, 115), (38730, 38731, 3), (38730, 38736, 115), (38732, 38739, 20398)

X(38739) = X(98)-CIRCUM-EULER-POINT OF X(631)

Barycentrics    3*a^8-6*(b^2+c^2)*a^6+(7*b^4+b^2*c^2+7*c^4)*a^4-(b^2+c^2)*(5*b^4-7*b^2*c^2+5*c^4)*a^2+(b^2-c^2)^4 : :
X(38739) = 6*X(2)-X(6033) = 9*X(2)+X(9862) = 3*X(2)+2*X(12042) = 4*X(2)+X(14830) = 7*X(2)-2*X(22566) = X(2)-6*X(26614) = 3*X(3)+2*X(115) = X(3)+4*X(6036) = 4*X(3)+X(6321) = 7*X(3)+8*X(20398) = 2*X(3)+3*X(38224) = 6*X(3)-X(38730) = 8*X(3)-3*X(38731) = 7*X(3)+3*X(38732) = 9*X(3)+X(38733) = 11*X(3)+4*X(38734) = 13*X(3)+12*X(38735) = 9*X(3)-4*X(38736) = X(3)-6*X(38737) = 7*X(3)-2*X(38738) = X(3)+2*X(38740) = 3*X(6033)+2*X(9862) = X(6033)+4*X(12042) = 2*X(6033)+3*X(14830) = 7*X(6033)-12*X(22566) = X(9862)-6*X(12042) = 4*X(9862)-9*X(14830) = 7*X(9862)+18*X(22566) = 8*X(12042)-3*X(14830) = 7*X(12042)+3*X(22566) = X(12042)+9*X(26614) = 7*X(14830)+8*X(22566)

X(38739) lies on the circumconic with center X(25680) and on these lines: {2,5191}, {3,115}, {4,15092}, {5,10722}, {30,14061}, {98,140}, {99,549}, {114,3526}, {147,3525}, {148,3524}, {182,9696}, {230,35002}, {376,22515}, {381,6722}, {382,23514}, {542,3763}, {543,15693}, {548,10723}, {550,14639}, {620,5054}, {631,2782}, {632,31268}, {671,12100}, {690,38728}, {1656,2794}, {2023,9821}, {2456,15993}, {2482,15701}, {2783,38762}, {2784,38774}, {2785,38786}, {2793,38806}, {3090,22505}, {3095,7735}, {3098,6034}, {3147,12131}, {3311,13967}, {3312,8980}, {3314,10104}, {3398,3815}, {3523,14651}, {3530,21166}, {3534,5461}, {3579,38220}, {3830,14971}, {4027,33015}, {5010,13183}, {5070,36519}, {5092,11646}, {5432,10069}, {5433,10053}, {5985,17566}, {5999,14693}, {6054,11539}, {6699,18332}, {6721,10991}, {6771,6778}, {6774,6777}, {7280,13182}, {7603,13335}, {7767,8781}, {7857,9993}, {7886,37243}, {7899,32151}, {7907,14880}, {7970,38028}, {8591,15719}, {8703,9166}, {8725,9478}, {8981,19055}, {9864,11231}, {9880,15688}, {10124,23234}, {10303,14692}, {10753,38110}, {10768,34126}, {11005,34128}, {11177,15709}, {11623,13188}, {11710,26446}, {11725,12702}, {12108,23235}, {12117,17504}, {12121,15359}, {12243,15708}, {12355,15706}, {12829,31401}, {13172,15717}, {13178,13624}, {13966,19056}, {14981,35021}, {15035,15535}, {15061,15545}, {15300,15722}, {15716,36523}, {15980,38225}, {16984,37334}, {19108,35256}, {19109,35255}, {22510,36755}, {22511,36756}, {33511,38727}

X(38739) = reflection of X(i) in X(j) for these (i,j): (38740, 6036), (38750, 631)
X(38739) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12042, 6033), (3, 115, 38730), (3, 6036, 38224), (3, 6321, 38731), (3, 38224, 6321), (3, 38732, 38738), (3, 38733, 38736), (5, 34473, 38741), (98, 140, 15561), (115, 38730, 6321), (115, 38736, 38733), (548, 38229, 10723), (5054, 12188, 620), (6033, 12042, 14830), (6036, 38737, 3), (6722, 38749, 381), (20398, 38738, 38732), (23514, 38747, 382), (38224, 38730, 115), (38733, 38736, 38730)

X(38740) = X(98)-CIRCUM-EULER-POINT OF X(632)

Barycentrics    4*a^8-8*(b^2+c^2)*a^6+(11*b^4-2*b^2*c^2+11*c^4)*a^4-2*(b^2+c^2)*(5*b^4-8*b^2*c^2+5*c^4)*a^2+3*(b^2-c^2)^4 : :
X(38740) = 3*X(2)+2*X(11623) = 6*X(2)-X(14981) = 9*X(2)-4*X(20399) = 2*X(3)+3*X(115) = X(3)-6*X(6036) = 7*X(3)+3*X(6321) = X(3)+4*X(20398) = X(3)+9*X(38224) = 13*X(3)-3*X(38730) = 19*X(3)-9*X(38731) = 11*X(3)+9*X(38732) = 17*X(3)+3*X(38733) = 3*X(3)+2*X(38734) = 7*X(3)+18*X(38735) = 11*X(3)-6*X(38736) = 4*X(3)-9*X(38737) = 8*X(3)-3*X(38738) = X(3)-3*X(38739) = 4*X(11623)+X(14981) = 3*X(11623)+2*X(20399) = 3*X(14981)-8*X(20399)

X(38740) lies on these lines: {2,11623}, {3,115}, {4,5461}, {5,6055}, {20,9166}, {98,3090}, {99,10303}, {114,3628}, {140,2482}, {231,10510}, {542,1656}, {543,631}, {546,12042}, {549,10992}, {550,9880}, {575,1506}, {576,7755}, {620,3525}, {632,2782}, {671,3523}, {690,38729}, {2783,38763}, {2784,38775}, {2785,38787}, {2793,38807}, {2794,3091}, {3054,21163}, {3146,34473}, {3455,6642}, {3524,36523}, {3526,9167}, {3529,14639}, {3530,26614}, {3533,12243}, {3544,9862}, {3564,31275}, {3627,38749}, {3851,14830}, {3857,15092}, {5054,15300}, {5067,6054}, {5079,6033}, {5182,32975}, {5346,11482}, {5368,22330}, {5465,20417}, {6419,8980}, {6420,13967}, {6721,12188}, {6771,20416}, {6774,20415}, {7486,11177}, {7607,34506}, {7827,10486}, {7982,11725}, {7991,38220}, {9144,15057}, {10299,12117}, {10541,11646}, {12103,22515}, {12108,33813}, {12811,22505}, {12815,20190}, {13178,30389}, {14869,38748}, {15069,19662}, {15357,20397}, {15702,36521}, {15704,38229}, {22566,35018}

X(38740) = reflection of X(i) in X(j) for these (i,j): (38739, 6036), (38751, 632)
X(38740) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11623, 14981), (3, 20398, 115), (3, 38224, 20398), (5, 6055, 10991), (98, 3090, 38745), (98, 6722, 36519), (115, 6036, 38737), (115, 38737, 38738), (632, 38751, 31274), (3090, 38745, 36519), (3525, 14651, 23235), (3525, 23235, 620), (6036, 20398, 3), (6036, 38224, 115), (6321, 38735, 115), (6722, 38745, 3090), (10991, 14971, 5), (20397, 31854, 15357), (20397, 33511, 31854)

X(38741) = X(99)-CIRCUM-EULER-POINT OF X(20)

Barycentrics    3*a^8-4*(b^2+c^2)*a^6+(3*b^4+b^2*c^2+3*c^4)*a^4-(b^6+c^6)*a^2-(b^4+c^4)*(b^2-c^2)^2 : :
X(38741) = 3*X(3)-2*X(114) = 5*X(3)-4*X(620) = 4*X(3)-3*X(15561) = 11*X(3)-8*X(20399) = 2*X(3)-3*X(38742) = 5*X(3)-3*X(38743) = 3*X(3)-X(38744) = 7*X(3)-4*X(38745) = 17*X(3)-12*X(38746) = 3*X(3)-4*X(38747) = 7*X(3)-6*X(38748) = 6*X(3)-5*X(38750) = 13*X(3)-10*X(38751) = 5*X(114)-6*X(620) = 4*X(114)-3*X(6033) = 8*X(114)-9*X(15561) = 11*X(114)-12*X(20399) = 4*X(114)-9*X(38742) = 10*X(114)-9*X(38743) = 7*X(114)-6*X(38745) = 17*X(114)-18*X(38746) = 7*X(114)-9*X(38748) = X(114)-3*X(38749) = 4*X(114)-5*X(38750) = 13*X(114)-15*X(38751)

X(38741) lies on these lines: {2,22505}, {3,114}, {4,12042}, {5,10722}, {20,2782}, {30,98}, {35,12184}, {36,12185}, {74,15545}, {99,550}, {115,382}, {147,376}, {148,3529}, {381,6036}, {384,6287}, {542,1350}, {543,15681}, {546,14061}, {548,21166}, {690,20127}, {1503,35383}, {1513,38225}, {1576,14675}, {1656,38737}, {1657,10991}, {1916,14712}, {2023,7737}, {2076,6781}, {2482,15688}, {2549,12829}, {2777,18332}, {2784,31730}, {2786,38765}, {2787,38753}, {2790,23240}, {2792,38777}, {2967,13200}, {3023,4299}, {3027,4302}, {3091,34127}, {3095,36998}, {3146,14651}, {3455,18859}, {3524,22566}, {3526,36519}, {3579,9864}, {3627,14639}, {3830,6055}, {3832,15092}, {3843,23514}, {3851,6722}, {3853,38229}, {4027,7833}, {4226,31127}, {5054,6721}, {5071,26614}, {5073,11623}, {5076,20398}, {5152,7802}, {5182,8354}, {5191,36163}, {5461,14269}, {5976,14907}, {5985,17579}, {6054,8703}, {6284,10069}, {6449,8997}, {6450,13989}, {6771,36962}, {6774,36961}, {7171,24469}, {7354,10053}, {7970,34773}, {7983,28174}, {8356,10352}, {8980,13665}, {9166,15687}, {9167,15700}, {9880,15684}, {9996,35925}, {10086,15338}, {10089,15326}, {10242,15980}, {10483,13182}, {10733,15535}, {11001,11177}, {11005,12041}, {11007,35278}, {11171,38383}, {11599,28150}, {11710,12699}, {12100,23234}, {12103,23235}, {12117,15686}, {12131,18533}, {12243,15683}, {12252,33260}, {13178,28160}, {13202,33511}, {13335,37243}, {13785,13967}, {14981,15696}, {15041,15357}, {15707,22247}, {15720,31274}, {17800,38733}, {21156,22797}, {21157,22796}, {22664,37451}, {22793,38220}, {26316,37242}

X(38741) = midpoint of X(i) and X(j) for these {i,j}: {20, 9862}, {148, 3529}, {1657, 12188}, {5984, 13172}, {11001, 11177}, {12243, 15683}, {17800, 38733}
X(38741) = reflection of X(i) in X(j) for these (i,j): (3, 38749), (4, 12042), (99, 550), (114, 38747), (147, 33813), (382, 115), (3146, 22515), (3830, 6055), (6033, 3), (6054, 8703), (6321, 98), (7970, 34773), (8724, 376), (9864, 3579), (10722, 5), (10733, 15535), (11005, 12041), (11632, 14830), (12117, 15686), (12188, 10991), (12699, 11710), (13188, 38738), (13202, 33511), (14692, 13188), (14981, 38736), (15545, 74), (15561, 38742), (15684, 9880), (36961, 6774), (36962, 6771), (38730, 20), (38744, 114)
X(38741) = circumperp conjugate of X(34217)
X(38741) = anticomplement of X(22505)
X(38741) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 114, 38750), (3, 6033, 15561), (3, 38743, 620), (3, 38744, 114), (3, 38749, 38742), (4, 12042, 38224), (5, 34473, 38739), (20, 5984, 13172), (98, 6321, 11632), (99, 550, 38731), (114, 38744, 6033), (114, 38747, 3), (114, 38749, 38747), (114, 38750, 15561), (6033, 38742, 3), (6033, 38750, 114), (6321, 14830, 98), (9862, 13172, 5984), (10722, 34473, 5), (38744, 38747, 38750)

X(38742) = X(99)-CIRCUM-EULER-POINT OF X(376)

Barycentrics    5*a^8-8*(b^2+c^2)*a^6+(7*b^4+3*b^2*c^2+7*c^4)*a^4-3*(b^6+c^6)*a^2-(b^4+c^4)*(b^2-c^2)^2 : :
X(38742) = 5*X(3)-2*X(114) = 7*X(3)-4*X(620) = 4*X(3)-X(6033) = 17*X(3)-8*X(20399) = 2*X(3)+X(38741) = 3*X(3)-X(38743) = 7*X(3)-X(38744) = 13*X(3)-4*X(38745) = 9*X(3)-4*X(38746) = X(3)-4*X(38747) = 3*X(3)-2*X(38748) = X(3)+2*X(38749) = 8*X(3)-5*X(38750) = 19*X(3)-10*X(38751) = 7*X(114)-10*X(620) = 8*X(114)-5*X(6033) = 4*X(114)-5*X(15561) = 17*X(114)-20*X(20399) = 4*X(114)+5*X(38741) = 6*X(114)-5*X(38743) = 14*X(114)-5*X(38744) = 13*X(114)-10*X(38745) = 9*X(114)-10*X(38746) = X(114)-10*X(38747) = 3*X(114)-5*X(38748) = X(114)+5*X(38749)

X(38742) lies on these lines: {3,114}, {4,15092}, {20,6321}, {30,9166}, {98,550}, {99,548}, {115,1657}, {140,10722}, {147,3528}, {148,17538}, {376,2782}, {381,38737}, {382,6036}, {542,15041}, {543,15689}, {631,22505}, {671,15686}, {690,38788}, {2482,14093}, {2786,38766}, {2787,38754}, {2792,38778}, {3522,9862}, {3529,22515}, {3534,11632}, {3627,14061}, {3830,23514}, {3839,26614}, {3843,6722}, {4027,33275}, {4316,13182}, {4324,13183}, {5010,12184}, {5054,36519}, {5461,15684}, {6034,29317}, {6054,34200}, {6055,15681}, {6455,8997}, {6456,13989}, {6721,15720}, {7280,12185}, {8703,8724}, {9167,15706}, {9864,31663}, {9880,15685}, {10053,15326}, {10069,15338}, {10723,15704}, {10991,13188}, {11623,38733}, {12041,15545}, {12117,15690}, {12188,15696}, {12243,15697}, {14677,15342}, {14971,38335}, {15692,22566}, {15718,22247}, {16111,18332}, {17504,23234}, {19686,22681}, {28146,38220}

X(38742) = midpoint of X(i) and X(j) for these {i,j}: {20, 14651}, {14830, 38731}, {15561, 38741}, {15681, 38732}
X(38742) = reflection of X(i) in X(j) for these (i,j): (4, 34127), (381, 38737), (3830, 23514), (3839, 26614), (6033, 15561), (6321, 14651), (8724, 21166), (14651, 12042), (15561, 3), (21166, 8703), (23234, 17504), (38224, 34473), (38335, 14971), (38731, 376), (38732, 6055), (38743, 38748)
X(38742) = circumperp conjugate of X(15562)
X(38742) = X(38741)-Gibert-Moses centroid
X(38742) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6033, 38750), (3, 38741, 6033), (3, 38743, 38748), (3, 38744, 620), (3, 38749, 38741), (20, 12042, 6321), (98, 550, 38730), (3522, 9862, 33813), (10991, 38736, 13188), (12188, 15696, 38738), (38743, 38748, 15561), (38747, 38749, 3)

X(38743) = X(99)-CIRCUM-EULER-POINT OF X(381)

Barycentrics    a^8+2*(b^2+c^2)*a^6-(4*b^4+3*b^2*c^2+4*c^4)*a^4+3*(b^6+c^6)*a^2-2*(b^4+c^4)*(b^2-c^2)^2 : :
X(38743) = X(3)-4*X(114) = 5*X(3)-8*X(620) = X(3)+2*X(6033) = 7*X(3)-16*X(20399) = 5*X(3)-2*X(38741) = 3*X(3)-2*X(38742) = 2*X(3)+X(38744) = X(3)+8*X(38745) = 3*X(3)-8*X(38746) = 11*X(3)-8*X(38747) = 3*X(3)-4*X(38748) = 7*X(3)-4*X(38749) = 7*X(3)-10*X(38750) = 11*X(3)-20*X(38751) = 5*X(114)-2*X(620) = 2*X(114)+X(6033) = 7*X(114)-4*X(20399) = 10*X(114)-X(38741) = 6*X(114)-X(38742) = 8*X(114)+X(38744) = X(114)+2*X(38745) = 3*X(114)-2*X(38746) = 11*X(114)-2*X(38747) = 3*X(114)-X(38748) = 7*X(114)-X(38749) = 14*X(114)-5*X(38750) = 11*X(114)-5*X(38751)

X(38743) lies on these lines: {3,114}, {4,13188}, {5,147}, {17,6777}, {18,6778}, {98,1656}, {99,382}, {115,3851}, {140,9862}, {148,546}, {262,381}, {355,21636}, {399,11005}, {542,5050}, {543,14269}, {547,11177}, {550,7885}, {690,38789}, {999,12184}, {1482,9864}, {1506,6287}, {1513,9301}, {1657,10722}, {2482,15681}, {2777,14850}, {2784,5886}, {2786,38767}, {2787,38755}, {2792,38779}, {3023,9654}, {3027,9669}, {3090,5984}, {3295,12185}, {3447,5899}, {3526,12042}, {3530,7928}, {3534,21166}, {3545,38229}, {3627,13172}, {3767,12830}, {3830,8724}, {3843,6321}, {3845,12355}, {4027,7887}, {5038,7603}, {5054,23234}, {5066,12243}, {5070,6036}, {5073,38730}, {5076,10723}, {5079,14061}, {5182,33240}, {5976,7776}, {5986,7539}, {5989,7752}, {6055,15703}, {6721,10991}, {7579,15928}, {7777,32528}, {7785,13111}, {7866,10352}, {7901,10353}, {7912,8290}, {7941,8782}, {7970,12645}, {8591,15687}, {9167,15707}, {9655,10089}, {9668,10086}, {9772,22728}, {9860,9956}, {10053,31479}, {10753,11898}, {10768,12331}, {11178,32149}, {11632,19709}, {11724,37624}, {11842,37071}, {12121,33512}, {12177,18440}, {12308,15545}, {13102,22796}, {13103,22797}, {13174,22793}, {13331,22688}, {14830,15694}, {17800,38738}, {18503,37334}, {22515,23235}, {23251,35879}, {23261,35878}, {35464,35930}

X(38743) = midpoint of X(i) and X(j) for these {i,j}: {147, 14651}, {6033, 15561}
X(38743) = reflection of X(i) in X(j) for these (i,j): (3, 15561), (98, 34127), (3534, 21166), (5054, 23234), (11632, 23514), (12188, 14651), (14651, 5), (14830, 38737), (14849, 23515), (15561, 114), (15681, 38731), (38224, 36519), (38731, 2482), (38732, 381), (38742, 38748), (38748, 38746)
X(38743) = X(38744)-Gibert-Moses centroid
X(38743) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6033, 38744), (4, 13188, 38733), (5, 147, 12188), (99, 22505, 382), (114, 6033, 3), (114, 38745, 6033), (114, 38748, 38746), (114, 38749, 20399), (620, 38741, 3), (6054, 22566, 381), (6721, 10991, 38739), (10722, 33813, 1657), (15561, 38742, 38748), (20399, 38749, 38750), (22796, 36776, 13102), (36519, 38224, 5055), (38742, 38748, 3), (38746, 38748, 15561), (38749, 38750, 3)

X(38744) = X(99)-CIRCUM-EULER-POINT OF X(382)

Barycentrics    3*a^8-2*(b^2+c^2)*a^6-b^2*c^2*a^4+(b^6+c^6)*a^2-2*(b^4+c^4)*(b^2-c^2)^2 : :
X(38744) = 3*X(3)-4*X(114) = 7*X(3)-8*X(620) = 5*X(3)-6*X(15561) = 13*X(3)-16*X(20399) = 3*X(3)-2*X(38741) = 7*X(3)-6*X(38742) = 2*X(3)-3*X(38743) = 5*X(3)-8*X(38745) = 9*X(3)-8*X(38747) = 11*X(3)-12*X(38748) = 5*X(3)-4*X(38749) = 9*X(3)-10*X(38750) = 17*X(3)-20*X(38751) = 7*X(114)-6*X(620) = 2*X(114)-3*X(6033) = 10*X(114)-9*X(15561) = 13*X(114)-12*X(20399) = 14*X(114)-9*X(38742) = 8*X(114)-9*X(38743) = 5*X(114)-6*X(38745) = 19*X(114)-18*X(38746) = 3*X(114)-2*X(38747) = 11*X(114)-9*X(38748) = 5*X(114)-3*X(38749) = 6*X(114)-5*X(38750) = 17*X(114)-15*X(38751)

X(38744) lies on these lines: {3,114}, {4,5984}, {5,9862}, {30,147}, {98,381}, {99,1657}, {115,3843}, {148,3627}, {316,5989}, {382,2782}, {542,1351}, {543,14692}, {546,14651}, {671,38335}, {690,38790}, {999,12185}, {1316,31127}, {1539,22265}, {1656,12042}, {1916,22728}, {2023,15484}, {2482,15689}, {2549,12830}, {2784,12699}, {2786,38768}, {2787,38756}, {2792,38780}, {2936,35452}, {2967,10735}, {3023,9655}, {3027,9668}, {3095,36997}, {3295,12184}, {3526,34473}, {3534,6054}, {3543,12355}, {3832,38229}, {3845,11177}, {3851,10991}, {4027,7841}, {5054,22566}, {5055,6036}, {5064,5986}, {5070,36519}, {5072,14061}, {5073,23698}, {5076,22515}, {5079,34127}, {5152,7773}, {5985,17532}, {5987,31133}, {6055,19709}, {6226,26346}, {6227,26336}, {6407,8997}, {6408,13989}, {6721,15694}, {6777,16965}, {6778,16964}, {7687,14849}, {7823,32528}, {7843,8178}, {7898,8290}, {7924,10353}, {7970,18526}, {8724,15681}, {9167,15718}, {9654,10053}, {9669,10069}, {9860,18480}, {9864,12702}, {9880,35403}, {10352,11287}, {10358,18500}, {10620,11005}, {10768,12773}, {11632,14269}, {11710,18493}, {12131,18494}, {12176,18501}, {12178,18524}, {12181,18508}, {12182,18519}, {12183,18518}, {12189,18545}, {12190,18543}, {12212,14537}, {12243,15687}, {13174,28146}, {14850,16111}, {14981,17800}, {15693,23234}, {15696,21166}, {18332,38789}, {18481,21636}, {18510,19055}, {18512,19056}, {18541,24472}, {20094,33703}, {20428,22507}, {20429,22509}, {22504,26321}, {23251,35824}, {23261,35825}, {29012,35458}, {32447,38383}, {33512,38723}

X(38744) = midpoint of X(20094) and X(33703)
X(38744) = reflection of X(i) in X(j) for these (i,j): (3, 6033), (98, 22505), (148, 3627), (382, 10722), (1657, 99), (3534, 6054), (8178, 7843), (9860, 18480), (9862, 5), (10620, 11005), (11177, 3845), (12188, 4), (12243, 15687), (12355, 3543), (12702, 9864), (12773, 10768), (13102, 36961), (13103, 36962), (13188, 147), (15681, 8724), (17800, 38730), (18481, 21636), (18508, 12181), (18526, 7970), (22265, 1539), (38730, 14981), (38733, 382), (38741, 114), (38749, 38745)
X(38744) = Stammler circle-inverse of-X(19165)
X(38744) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6033, 38743), (4, 12188, 38732), (98, 22505, 381), (114, 38741, 3), (114, 38747, 38750), (620, 38742, 3), (6033, 15561, 38745), (6033, 38741, 114), (15561, 38749, 3), (36519, 38739, 5070), (38741, 38750, 38747), (38745, 38749, 15561), (38747, 38750, 3)

X(38745) = X(99)-CIRCUM-EULER-POINT OF X(546)

Barycentrics    2*a^8+2*(b^2+c^2)*a^6-(5*b^4+4*b^2*c^2+5*c^4)*a^4+4*(b^6+c^6)*a^2-3*(b^4+c^4)*(b^2-c^2)^2 : :
X(38745) = X(3)-3*X(114) = 2*X(3)-3*X(620) = X(3)+3*X(6033) = 5*X(3)-9*X(15561) = 7*X(3)-3*X(38741) = 13*X(3)-9*X(38742) = X(3)-9*X(38743) = 5*X(3)+3*X(38744) = 4*X(3)-9*X(38746) = 4*X(3)-3*X(38747) = 7*X(3)-9*X(38748) = 5*X(3)-3*X(38749) = 11*X(3)-15*X(38750) = 3*X(3)-5*X(38751) = 5*X(114)-3*X(15561) = 3*X(114)-2*X(20399) = 7*X(114)-X(38741) = 13*X(114)-3*X(38742) = X(114)-3*X(38743) = 5*X(114)+X(38744) = 4*X(114)-3*X(38746) = 4*X(114)-X(38747) = 7*X(114)-3*X(38748) = 5*X(114)-X(38749) = 11*X(114)-5*X(38750) = 9*X(114)-5*X(38751)

X(38745) lies on these lines: {2,10991}, {3,114}, {4,543}, {5,542}, {20,2482}, {30,36521}, {98,3090}, {99,3146}, {113,31854}, {115,147}, {381,36523}, {382,8724}, {546,2782}, {576,7838}, {625,1503}, {631,22247}, {632,6721}, {671,3832}, {690,38791}, {754,1513}, {1593,2936}, {1656,6055}, {2450,3292}, {2784,11725}, {2786,38769}, {2787,38757}, {2792,38781}, {2793,14279}, {3089,20774}, {3303,12185}, {3304,12184}, {3455,7503}, {3523,9167}, {3525,7914}, {3526,14830}, {3529,10722}, {3543,15300}, {3544,14651}, {3627,22505}, {3628,6036}, {3734,7694}, {3767,5477}, {3843,9880}, {3851,11632}, {3855,12243}, {4045,9744}, {5034,31415}, {5056,11177}, {5068,9166}, {5072,12188}, {5076,13188}, {5079,38224}, {5182,14064}, {5984,14061}, {6425,8997}, {6426,13989}, {6776,7844}, {7617,11180}, {7749,9863}, {7751,10753}, {7753,13862}, {7759,11477}, {7763,36997}, {7773,35705}, {7818,37182}, {7825,8721}, {7866,10541}, {7982,9864}, {8367,19662}, {8591,17578}, {8593,32984}, {9589,9881}, {9657,12351}, {9670,12350}, {9698,37336}, {10303,31274}, {11005,14094}, {11152,14062}, {11318,18800}, {11724,15178}, {12103,38736}, {12117,33703}, {14692,38732}, {14880,32135}, {14928,36990}, {15054,15357}, {15063,15098}, {15092,38735}, {15704,33813}, {16001,22797}, {16002,22796}, {17538,21166}, {19058,31414}, {25562,34507}, {34127,35021}, {34517,36201}

X(38745) = midpoint of X(i) and X(j) for these {i,j}: {4, 14981}, {114, 6033}, {115, 147}, {382, 10992}, {3543, 15300}, {10722, 38738}, {14928, 36990}, {38744, 38749}
X(38745) = reflection of X(i) in X(j) for these (i,j): (3, 20399), (98, 6722), (620, 114), (11623, 5), (12042, 6721), (36523, 381), (38734, 546), (38738, 35022), (38747, 620)
X(38745) = complement of X(10991)
X(38745) = X(187)-of-X(4)-Brocard-triangle
X(38745) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 114, 20399), (3, 20399, 620), (4, 6054, 14981), (5, 8550, 7817), (5, 11623, 5461), (98, 3090, 38740), (98, 36519, 6722), (114, 620, 38746), (114, 38749, 15561), (382, 8724, 10992), (3090, 38740, 6722), (6033, 15561, 38744), (6033, 38743, 114), (15561, 38744, 38749), (36519, 38740, 3090), (38746, 38747, 620)

X(38746) = X(99)-CIRCUM-EULER-POINT OF X(547)

Barycentrics    2*a^8-14*(b^2+c^2)*a^6+(19*b^4+12*b^2*c^2+19*c^4)*a^4-12*(b^6+c^6)*a^2+5*(b^4+c^4)*(b^2-c^2)^2 : :
X(38746) = X(3)+5*X(114) = 2*X(3)-5*X(620) = 7*X(3)+5*X(6033) = X(3)-5*X(15561) = X(3)-10*X(20399) = 17*X(3)-5*X(38741) = 9*X(3)-5*X(38742) = 3*X(3)+5*X(38743) = 19*X(3)+5*X(38744) = 4*X(3)+5*X(38745) = 8*X(3)-5*X(38747) = 3*X(3)-5*X(38748) = 11*X(3)-5*X(38749) = 2*X(114)+X(620) = 7*X(114)-X(6033) = X(114)+2*X(20399) = 17*X(114)+X(38741) = 9*X(114)+X(38742) = 3*X(114)-X(38743) = 19*X(114)-X(38744) = 4*X(114)-X(38745) = 8*X(114)+X(38747) = 3*X(114)+X(38748) = 11*X(114)+X(38749) = 13*X(114)+5*X(38750) = 7*X(114)+5*X(38751)

X(38746) lies on these lines: {3,114}, {4,35022}, {98,3533}, {99,3832}, {115,5056}, {147,31274}, {542,11539}, {543,3545}, {547,2782}, {632,35021}, {690,38792}, {2482,3543}, {2786,38770}, {2787,38758}, {2792,38782}, {3845,14160}, {5067,6722}, {5070,14692}, {5102,14645}, {6036,16239}, {6054,15702}, {6055,15723}, {6721,11623}, {7764,9753}, {7838,37466}, {8724,23514}, {9167,15708}, {11001,21166}, {11724,33179}, {15686,22566}, {33703,38738}

X(38746) = midpoint of X(i) and X(j) for these {i,j}: {114, 15561}, {6054, 38737}, {8724, 23514}, {14651, 14981}, {38743, 38748}
X(38746) = reflection of X(i) in X(j) for these (i,j): (620, 15561), (11623, 34127), (14651, 6722), (15561, 20399), (34127, 6721), (36523, 23514), (38735, 547), (38737, 22247)
X(38746) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (114, 620, 38745), (114, 20399, 620), (114, 38748, 38743), (114, 38751, 6033), (620, 38745, 38747), (15561, 38743, 38748)

X(38747) = X(99)-CIRCUM-EULER-POINT OF X(548)

Barycentrics    6*a^8-10*(b^2+c^2)*a^6+(9*b^4+4*b^2*c^2+9*c^4)*a^4-4*(b^6+c^6)*a^2-(b^4+c^4)*(b^2-c^2)^2 : :
X(38747) = 3*X(3)-X(114) = 5*X(3)-X(6033) = 7*X(3)-3*X(15561) = 5*X(3)-2*X(20399) = 3*X(3)+X(38741) = X(3)+3*X(38742) = 11*X(3)-3*X(38743) = 9*X(3)-X(38744) = 4*X(3)-X(38745) = 8*X(3)-3*X(38746) = 5*X(3)-3*X(38748) = 9*X(3)-5*X(38750) = 11*X(3)-5*X(38751) = 2*X(114)-3*X(620) = 5*X(114)-3*X(6033) = 7*X(114)-9*X(15561) = 5*X(114)-6*X(20399) = X(114)+9*X(38742) = 11*X(114)-9*X(38743) = 3*X(114)-X(38744) = 4*X(114)-3*X(38745) = 8*X(114)-9*X(38746) = 5*X(114)-9*X(38748) = X(114)+3*X(38749) = 3*X(114)-5*X(38750) = 11*X(114)-15*X(38751)

X(38747) lies on these lines: {3,114}, {4,6722}, {20,115}, {30,5461}, {98,376}, {99,3522}, {147,2482}, {382,23514}, {516,11725}, {542,8703}, {548,2782}, {549,6721}, {550,11623}, {631,10722}, {690,37853}, {754,18860}, {1350,14645}, {1503,32456}, {1657,38224}, {1916,22676}, {2786,38771}, {2787,38759}, {2792,38783}, {3146,14061}, {3523,7935}, {3524,22247}, {3528,9862}, {3529,14639}, {3534,6055}, {3543,14971}, {3564,14148}, {3627,34127}, {3853,15092}, {5477,25406}, {5985,36004}, {5999,6781}, {6054,19708}, {6409,8997}, {6410,13989}, {6560,8980}, {6561,13967}, {6719,7417}, {7764,36998}, {7829,13335}, {7838,9737}, {7863,9863}, {7983,9778}, {8588,37182}, {8724,14093}, {9166,15683}, {9167,15692}, {9541,19055}, {9864,35242}, {9880,15681}, {10352,33008}, {10992,12188}, {11177,15300}, {11495,22514}, {11632,15689}, {11710,31730}, {11724,13624}, {12103,38734}, {12131,37931}, {12258,34638}, {13188,14830}, {13638,35946}, {13758,35947}, {14651,17538}, {14907,32458}, {15055,15357}, {15118,35345}, {15687,26614}, {15696,38730}, {15698,23234}, {15704,20398}, {16925,36997}, {17504,22566}, {18332,38788}, {21163,38383}, {35282,35922}

X(38747) = midpoint of X(i) and X(j) for these {i,j}: {3, 38749}, {20, 115}, {98, 38738}, {99, 10991}, {114, 38741}, {550, 12042}, {3534, 6055}, {5999, 6781}, {9862, 14981}, {9880, 15681}, {10992, 12188}, {11177, 15300}, {11710, 31730}, {12258, 34638}, {15704, 22515}
X(38747) = reflection of X(i) in X(j) for these (i,j): (4, 6722), (620, 3), (3853, 15092), (6033, 20399), (11623, 12042), (11724, 13624), (14981, 35022), (22505, 6721), (22515, 20398), (36523, 6055), (38736, 548), (38745, 620)
X(38747) = complement of X(39838)
X(38747) = circumperp conjugate of X(11641)
X(38747) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6033, 38748), (3, 38741, 114), (3, 38742, 38749), (3, 38744, 38750), (4, 38737, 6722), (20, 34473, 115), (98, 376, 38738), (114, 38749, 38741), (382, 38739, 23514), (549, 22505, 6721), (620, 38745, 38746), (631, 10722, 36519), (3528, 9862, 21166), (6033, 38748, 20399), (9862, 21166, 14981), (12188, 38731, 10992), (14981, 21166, 35022), (20399, 38748, 620), (38741, 38750, 38744), (38744, 38750, 114)

X(38748) = X(99)-CIRCUM-EULER-POINT OF X(549)

Barycentrics    4*a^8-10*(b^2+c^2)*a^6+(11*b^4+6*b^2*c^2+11*c^4)*a^4-6*(b^6+c^6)*a^2+(b^4+c^4)*(b^2-c^2)^2 : :
X(38748) = 4*X(2)-X(9880) = 5*X(2)+X(12117) = 2*X(3)+X(114) = X(3)+2*X(620) = 5*X(3)+X(6033) = 5*X(3)+4*X(20399) = 7*X(3)-X(38741) = 3*X(3)-X(38742) = 3*X(3)+X(38743) = 11*X(3)+X(38744) = 7*X(3)+2*X(38745) = 3*X(3)+2*X(38746) = 5*X(3)-2*X(38747) = 4*X(3)-X(38749) = X(3)+5*X(38750) = 4*X(3)+5*X(38751) = 5*X(9880)+4*X(12117) = 3*X(9880)-4*X(14639) = X(9880)+4*X(21166) = 3*X(12117)+5*X(14639) = X(12117)-5*X(21166) = 2*X(12117)+5*X(23514) = X(14639)+3*X(21166) = 2*X(14639)-3*X(23514) = 2*X(21166)+X(23514)

X(38748) lies on these lines: {2,9734}, {3,114}, {4,6721}, {5,31274}, {30,9167}, {40,11724}, {98,3523}, {99,631}, {115,140}, {147,15717}, {148,10303}, {262,33246}, {371,13989}, {372,8997}, {381,22247}, {511,35297}, {542,3524}, {543,5054}, {548,22505}, {549,2482}, {637,12974}, {638,12975}, {671,15702}, {690,38793}, {754,38225}, {1511,15357}, {1656,38730}, {2783,21154}, {2786,38772}, {2787,38760}, {2792,38784}, {2795,28465}, {3090,10723}, {3522,10722}, {3525,13172}, {3526,6321}, {3530,12042}, {3628,22515}, {5050,14645}, {5152,14928}, {5171,10352}, {5182,10519}, {5433,15452}, {5461,15694}, {5474,36770}, {5965,7799}, {5976,13334}, {6054,15692}, {6684,11711}, {6771,14145}, {6774,14144}, {7472,16760}, {7835,24206}, {7863,10104}, {7987,9864}, {8591,15721}, {8724,15693}, {9166,15709}, {9540,19108}, {9737,9753}, {9862,10299}, {10168,31958}, {10304,23234}, {10358,32973}, {10991,15712}, {11005,15051}, {11149,22677}, {11539,14971}, {11623,13188}, {11632,15701}, {11812,15300}, {12096,16976}, {13178,31423}, {13335,32458}, {13935,19109}, {14561,33191}, {14830,15700}, {14869,38740}, {15040,15545}, {15980,32456}, {18860,37459}, {19598,32204}, {22525,26613}, {22566,34200}, {22712,33274}, {33220,38317}

X(38748) = midpoint of X(i) and X(j) for these {i,j}: {2, 21166}, {3, 15561}, {99, 14651}, {381, 38731}, {2482, 38737}, {5182, 10519}, {7799, 21445}, {10304, 23234}, {33813, 34127}, {38742, 38743}
X(38748) = reflection of X(i) in X(j) for these (i,j): (114, 15561), (115, 34127), (671, 38735), (6055, 38737), (9880, 23514), (14651, 6036), (14971, 11539), (15561, 620), (23514, 2), (34127, 140), (38732, 5461), (38737, 549), (38743, 38746)
X(38748) = complement of X(14639)
X(38748) = X(114)-Gibert-Moses centroid
X(38748) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 114, 38749), (3, 620, 114), (3, 3788, 32152), (3, 6033, 38747), (3, 38743, 38742), (3, 38750, 620), (99, 631, 6036), (114, 620, 38751), (115, 33813, 10992), (140, 33813, 115), (549, 2482, 6055), (620, 38747, 20399), (6033, 20399, 114), (6721, 38736, 4), (13188, 38739, 11623), (15561, 38742, 38743), (15561, 38743, 38746), (20399, 38747, 6033), (31274, 38738, 5), (38743, 38746, 114), (38749, 38751, 114)

X(38749) = X(99)-CIRCUM-EULER-POINT OF X(550)

Barycentrics    4*a^8-6*(b^2+c^2)*a^6+(5*b^4+2*b^2*c^2+5*c^4)*a^4-2*(b^6+c^6)*a^2-(b^4+c^4)*(b^2-c^2)^2 : :
X(38749) = 3*X(3)-2*X(620) = 3*X(3)-X(6033) = 5*X(3)-3*X(15561) = 7*X(3)-4*X(20399) = X(3)-3*X(38742) = 7*X(3)-3*X(38743) = 5*X(3)-X(38744) = 5*X(3)-2*X(38745) = 11*X(3)-6*X(38746) = 4*X(3)-3*X(38748) = 7*X(3)-5*X(38750) = 8*X(3)-5*X(38751) = 3*X(114)-4*X(620) = 3*X(114)-2*X(6033) = 5*X(114)-6*X(15561) = 7*X(114)-8*X(20399) = X(114)+2*X(38741) = X(114)-6*X(38742) = 7*X(114)-6*X(38743) = 5*X(114)-2*X(38744) = 5*X(114)-4*X(38745) = 11*X(114)-12*X(38746) = X(114)-4*X(38747) = 2*X(114)-3*X(38748) = 7*X(114)-10*X(38750) = 4*X(114)-5*X(38751)

X(38749) lies on these lines: {2,10722}, {3,114}, {4,6036}, {5,38737}, {20,98}, {30,115}, {69,74}, {125,4226}, {140,22505}, {147,3522}, {165,9864}, {381,6722}, {382,38224}, {516,11710}, {543,3534}, {546,34127}, {548,14981}, {549,7853}, {550,2782}, {631,6721}, {671,11001}, {690,16111}, {754,35002}, {1003,3818}, {1285,20423}, {1562,35912}, {1657,6321}, {2482,8703}, {2783,24466}, {2784,12512}, {2786,38773}, {2787,38761}, {2790,3184}, {2792,38785}, {2799,9409}, {2847,36207}, {2967,14900}, {3023,15326}, {3027,15338}, {3070,8980}, {3071,13967}, {3146,14639}, {3529,10723}, {3552,9873}, {3576,11724}, {3627,38740}, {3830,5461}, {3845,14971}, {3972,19130}, {4027,33260}, {4045,26316}, {4299,10053}, {4302,10069}, {5066,26614}, {5092,8356}, {5152,7470}, {5204,12185}, {5217,12184}, {5731,7970}, {5870,6231}, {5871,6230}, {5972,35278}, {5984,23235}, {5985,37256}, {6054,10304}, {6200,8997}, {6361,7983}, {6396,13989}, {6459,19055}, {6460,19056}, {6680,37243}, {7473,22104}, {7750,32458}, {7756,14880}, {7774,9737}, {7792,13335}, {7794,32151}, {7804,37345}, {7820,9996}, {7845,18860}, {7913,37242}, {8591,15697}, {8598,11645}, {8724,15688}, {8725,35464}, {8781,32006}, {9166,15682}, {9167,12100}, {9541,19109}, {9734,9744}, {9738,35946}, {9739,35947}, {9774,11151}, {9841,24469}, {10352,32965}, {10356,14001}, {10753,25406}, {11005,15055}, {11007,35282}, {11177,12117}, {11299,22797}, {11300,22796}, {11632,15681}, {11676,29012}, {11725,12699}, {12041,15357}, {12076,16220}, {12176,29317}, {12181,16190}, {12244,15342}, {12295,15359}, {13172,17538}, {13188,15696}, {13334,38383}, {14645,33878}, {15041,15545}, {15300,15690}, {15685,36523}, {15692,23234}, {15693,22247}, {15695,36521}, {17800,38732}, {18332,20127}, {18800,35955}, {21156,36962}, {21157,36961}, {21243,35926}, {24206,35925}, {25561,35954}, {33703,38735}, {36997,37466}

X(38749) = midpoint of X(i) and X(j) for these {i,j}: {3, 38741}, {20, 98}, {99, 9862}, {671, 11001}, {1657, 6321}, {3529, 10723}, {3534, 14830}, {5984, 23235}, {6361, 7983}, {10991, 38738}, {11177, 12117}, {11632, 15681}, {12188, 38730}, {12244, 15342}, {18332, 20127}
X(38749) = reflection of X(i) in X(j) for these (i,j): (3, 38747), (4, 6036), (99, 38736), (114, 3), (115, 12042), (2482, 8703), (3830, 5461), (6033, 620), (6321, 11623), (9880, 6055), (10723, 38734), (10992, 38738), (12295, 15359), (12699, 11725), (14981, 33813), (15357, 12041), (22505, 140), (22566, 12100), (23514, 34473), (33813, 548), (38383, 13334), (38734, 35021), (38738, 550), (38744, 38745)
X(38749) = circumperp conjugate of X(19165)
X(38749) = complement of X(10722)
X(38749) = crossdifference of every pair of points on line {X(14398), X(34291)}
X(38749) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 114, 38748), (3, 6033, 620), (3, 38742, 38747), (3, 38743, 38750), (3, 38744, 15561), (4, 6036, 23514), (4, 34473, 6036), (99, 376, 38736), (114, 38748, 38751), (115, 12042, 6055), (140, 22505, 36519), (376, 9862, 99), (376, 14907, 3098), (620, 6033, 114), (12100, 22566, 9167), (15561, 38744, 38745), (15561, 38745, 114), (20399, 38743, 114), (35278, 35922, 5972), (38741, 38742, 3), (38741, 38747, 114), (38743, 38750, 20399)

X(38750) = X(99)-CIRCUM-EULER-POINT OF X(631)

Barycentrics    3*a^8-8*(b^2+c^2)*a^6+(9*b^4+5*b^2*c^2+9*c^4)*a^4-5*(b^6+c^6)*a^2+(b^4+c^4)*(b^2-c^2)^2 : :
X(38750) = 6*X(2)-X(6321) = 9*X(2)+X(13172) = 3*X(2)+2*X(33813) = 3*X(3)+2*X(114) = X(3)+4*X(620) = 4*X(3)+X(6033) = 2*X(3)+3*X(15561) = 7*X(3)+8*X(20399) = 6*X(3)-X(38741) = 8*X(3)-3*X(38742) = 7*X(3)+3*X(38743) = 9*X(3)+X(38744) = 11*X(3)+4*X(38745) = 13*X(3)+12*X(38746) = 9*X(3)-4*X(38747) = X(3)-6*X(38748) = 7*X(3)-2*X(38749) = X(3)+2*X(38751) = 3*X(6321)+2*X(13172) = X(6321)+4*X(33813) = X(13172)-6*X(33813)

X(38750) lies on these lines: {2,6321}, {3,114}, {4,38731}, {5,10723}, {98,549}, {99,140}, {115,3526}, {147,3524}, {148,3525}, {325,38225}, {376,22505}, {381,6721}, {382,36519}, {499,15452}, {542,15040}, {543,15694}, {547,12117}, {548,10722}, {618,22507}, {619,22509}, {625,10242}, {631,2782}, {632,14061}, {671,11539}, {690,38794}, {1511,15545}, {1656,23698}, {2023,19910}, {2080,10352}, {2482,5054}, {2786,38774}, {2787,38762}, {2792,38786}, {3090,22515}, {3095,16925}, {3147,5186}, {3311,13989}, {3312,8997}, {3523,5984}, {3530,34473}, {3628,14639}, {4027,33274}, {5010,12185}, {5055,22247}, {5067,15092}, {5070,23514}, {5432,10089}, {5433,10086}, {5463,25559}, {5464,25560}, {5976,11171}, {6034,36782}, {6054,12100}, {6055,15701}, {6722,10992}, {7280,12184}, {7622,16508}, {7891,10104}, {7983,38028}, {8591,15709}, {8703,23234}, {8981,19108}, {9166,10124}, {9862,15717}, {9864,13624}, {9880,15703}, {10256,15980}, {10303,14651}, {10304,22566}, {10754,38110}, {10769,34126}, {10796,33246}, {11177,15719}, {11231,13178}, {11272,33225}, {11711,26446}, {11724,12702}, {12188,15720}, {12243,15721}, {12355,14971}, {12829,21843}, {13966,19109}, {14869,23235}, {15357,32609}, {16239,38229}, {17005,35950}, {18332,38793}, {19055,35256}, {19056,35255}, {33512,38727}, {35002,37459}

X(38750) = reflection of X(i) in X(j) for these (i,j): (1656, 31274), (14061, 632), (38739, 631), (38751, 620)
X(38750) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 33813, 6321), (3, 114, 38741), (3, 620, 15561), (3, 6033, 38742), (3, 15561, 6033), (3, 38743, 38749), (3, 38744, 38747), (5, 21166, 38730), (98, 8724, 14692), (99, 140, 38224), (114, 38741, 6033), (114, 38747, 38744), (148, 3525, 34127), (620, 38748, 3), (6721, 38738, 381), (9167, 38738, 6721), (15561, 38741, 114), (20399, 38749, 38743), (36519, 38736, 382), (38744, 38747, 38741)

X(38751) = X(99)-CIRCUM-EULER-POINT OF X(632)

Barycentrics    4*a^8-14*(b^2+c^2)*a^6+(17*b^4+10*b^2*c^2+17*c^4)*a^4-10*(b^6+c^6)*a^2+3*(b^4+c^4)*(b^2-c^2)^2 : :
X(38751) = 9*X(2)-4*X(20398) = 9*X(2)+X(23235) = 2*X(3)+3*X(114) = X(3)-6*X(620) = 7*X(3)+3*X(6033) = X(3)+9*X(15561) = X(3)+4*X(20399) = 13*X(3)-3*X(38741) = 19*X(3)-9*X(38742) = 11*X(3)+9*X(38743) = 17*X(3)+3*X(38744) = 3*X(3)+2*X(38745) = 7*X(3)+18*X(38746) = 11*X(3)-6*X(38747) = 4*X(3)-9*X(38748) = 8*X(3)-3*X(38749) = X(3)-3*X(38750) = X(114)+4*X(620) = 7*X(114)-2*X(6033) = X(114)-6*X(15561) = 3*X(114)-8*X(20399) = 13*X(114)+2*X(38741) = 19*X(114)+6*X(38742) = 11*X(114)-6*X(38743) = 17*X(114)-2*X(38744) = 9*X(114)-4*X(38745) = 7*X(114)-12*X(38746) = 11*X(114)+4*X(38747) = 2*X(114)+3*X(38748) = 4*X(114)+X(38749) = X(114)+2*X(38750) = 4*X(20398)+X(23235)

X(38751) lies on these lines: {2,20398}, {3,114}, {5,2482}, {20,23234}, {98,10303}, {99,3090}, {115,3628}, {140,6055}, {542,631}, {543,1656}, {546,33813}, {547,15300}, {548,22566}, {549,10991}, {575,7807}, {576,7763}, {632,2782}, {671,5067}, {690,38795}, {1078,32135}, {2786,38775}, {2787,38763}, {2792,38787}, {2936,7393}, {3091,23698}, {3146,21166}, {3455,7516}, {3523,6054}, {3525,6036}, {3526,8724}, {3529,38736}, {3544,13172}, {3627,38738}, {3832,12117}, {5055,36521}, {5070,5461}, {5079,6321}, {5182,32989}, {5609,15357}, {5972,31854}, {6034,31492}, {6419,8997}, {6420,13989}, {6722,13188}, {7486,8591}, {7752,35951}, {7813,14693}, {7982,11724}, {9624,9881}, {9864,30389}, {10358,32829}, {11005,15020}, {11307,20416}, {11308,20415}, {11482,14645}, {12042,12108}, {12103,22505}, {12811,22515}, {13449,32459}, {14639,15022}, {14869,38737}, {14928,24206}, {15039,15545}, {15703,36523}, {18800,34507}, {20397,33512}

X(38751) = reflection of X(i) in X(j) for these (i,j): (38740, 632), (38750, 620)
X(38751) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 23235, 20398), (3, 15561, 20399), (3, 20399, 114), (5, 2482, 10992), (5, 10992, 9880), (99, 3090, 38734), (99, 6721, 23514), (114, 620, 38748), (114, 38748, 38749), (140, 14981, 6055), (620, 15561, 114), (620, 20399, 3), (3090, 38734, 23514), (3526, 8724, 11623), (6033, 38746, 114), (6721, 38734, 3090), (9167, 14981, 140), (11623, 22247, 3526), (31274, 38740, 632)

X(38752) = X(100)-CIRCUM-EULER-POINT OF X(2)

Barycentrics    a^7-(b+c)*a^6-(3*b^2-b*c+3*c^2)*a^5+3*(b+c)*(b^2+c^2)*a^4+(b^2+b*c+c^2)*(3*b^2-8*b*c+3*c^2)*a^3-(b^2-c^2)*(b-c)*(3*b^2+4*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2-4*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :
X(38752) = X(3)+2*X(119) = X(3)-4*X(3035) = 2*X(3)+X(10742) = X(3)+8*X(20400) = 4*X(3)-X(38753) = 5*X(3)+X(38756) = 5*X(3)+4*X(38757) = X(3)+4*X(38758) = 7*X(3)-4*X(38759) = 5*X(3)-2*X(38761) = 2*X(3)-5*X(38762) = X(3)-10*X(38763) = X(119)+2*X(3035) = 4*X(119)-X(10742) = X(119)-4*X(20400) = 8*X(119)+X(38753) = 4*X(119)+X(38754) = 10*X(119)-X(38756) = 5*X(119)-2*X(38757) = 7*X(119)+2*X(38759) = 5*X(119)+X(38761) = 4*X(119)+5*X(38762) = X(119)+5*X(38763)

Let Q be the cyclic quadrilateral ABCX(100). Taking the vertices 3 at a time yields four triangles whose nine-point centers are the vertices of a cyclic quadrilateral homothetic to Q at X(38752). (Randy Hutson, May 31, 2020)

X(38752) lies on these lines: {2,952}, {3,119}, {4,33814}, {5,100}, {8,19907}, {10,6265}, {11,498}, {12,10090}, {20,22799}, {30,34474}, {35,12764}, {36,12763}, {40,12611}, {80,2646}, {104,140}, {149,3090}, {153,631}, {214,355}, {381,5840}, {382,24466}, {499,1317}, {528,5055}, {546,10724}, {547,10707}, {549,10711}, {550,10728}, {999,10956}, {1125,12737}, {1145,1482}, {1319,12749}, {1320,5901}, {1376,6980}, {1385,12751}, {1387,3085}, {1484,3628}, {1512,35459}, {1519,35460}, {1532,12775}, {1537,6834}, {1698,6326}, {1737,12739}, {1768,31423}, {1862,3542}, {2771,5660}, {2783,38224}, {2787,15561}, {2800,10176}, {2801,38122}, {2802,5886}, {2805,38796}, {2932,3560}, {3036,26363}, {3045,32046}, {3086,12735}, {3091,13199}, {3311,13922}, {3312,13991}, {3523,12248}, {3526,6713}, {3541,12138}, {3545,38141}, {3579,34789}, {3584,5919}, {3624,6264}, {3634,10265}, {3851,10993}, {3887,38764}, {4187,37621}, {4193,32141}, {4996,6924}, {5054,21154}, {5056,20095}, {5070,6667}, {5079,6154}, {5432,7489}, {5433,10074}, {5445,5694}, {5541,8227}, {5587,15015}, {5690,6949}, {5779,10427}, {5791,9946}, {5805,6594}, {5818,6224}, {5854,10247}, {5856,38107}, {6684,12515}, {6824,9945}, {6832,12690}, {6861,12019}, {6862,10609}, {6882,18524}, {6884,9963}, {6889,13257}, {6914,17100}, {6944,18493}, {6952,18357}, {6958,18525}, {6971,11499}, {6979,22791}, {6989,13226}, {7529,13222}, {7583,19112}, {7584,19113}, {7741,13274}, {7951,13273}, {7993,34595}, {8068,11507}, {8674,14643}, {8715,13271}, {8981,19082}, {9024,14561}, {9624,12653}, {9780,12247}, {9803,19877}, {9955,14217}, {10039,12740}, {10073,17606}, {10104,13194}, {10197,32557}, {10200,37624}, {10267,31246}, {10528,25416}, {10576,35883}, {10577,35882}, {10755,18583}, {10778,20304}, {10826,12743}, {10827,18976}, {10942,13747}, {11272,32454}, {11374,12736}, {11570,24914}, {12119,18480}, {12532,31835}, {13369,17661}, {13966,19081}, {17566,32153}, {17757,22765}, {18254,26066}, {18526,26492}, {20104,31399}, {20107,33812}, {21155,28443}, {25413,37828}, {26726,33179}, {35204,37230}, {37718,38182}

X(38752) = midpoint of X(i) and X(j) for these {i,j}: {3, 38755}, {119, 38760}, {3035, 38758}, {5587, 15015}, {10742, 38754}
X(38752) = reflection of X(i) in X(j) for these (i,j): (3, 38760), (11, 38319), (119, 38758), (10246, 34123), (10742, 38755), (16173, 11230), (37718, 38182), (38753, 38754), (38754, 3), (38755, 119), (38758, 20400), (38760, 3035)
X(38752) = anticomplement of X(34126)
X(38752) = X(10742)-Gibert-Moses centroid
X(38752) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 119, 10742), (3, 3035, 38762), (3, 10742, 38753), (3, 38756, 38761), (5, 100, 10738), (10, 6265, 19914), (40, 15017, 12611), (119, 3035, 3), (119, 38761, 38757), (119, 38762, 38753), (119, 38763, 3035), (140, 11698, 104), (1656, 12331, 11), (3035, 20400, 119), (6713, 31235, 3526), (9956, 22935, 80), (10742, 38762, 3), (20400, 38763, 3), (38755, 38760, 38754), (38756, 38757, 10742), (38757, 38761, 38756), (38758, 38760, 38755)

X(38753) = X(100)-CIRCUM-EULER-POINT OF X(20)

Barycentrics    3*a^7-3*(b+c)*a^6-(5*b^2-11*b*c+5*c^2)*a^5+(b+c)*(5*b^2-8*b*c+5*c^2)*a^4+(b^4+c^4-b*c*(7*b^2-10*b*c+7*c^2))*a^3-(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a^2+(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :
X(38753) = 3*X(3)-2*X(119) = 5*X(3)-4*X(3035) = 11*X(3)-8*X(20400) = 4*X(3)-3*X(38752) = 2*X(3)-3*X(38754) = 5*X(3)-3*X(38755) = 3*X(3)-X(38756) = 7*X(3)-4*X(38757) = 17*X(3)-12*X(38758) = 3*X(3)-4*X(38759) = 7*X(3)-6*X(38760) = 6*X(3)-5*X(38762) = 13*X(3)-10*X(38763) = 5*X(119)-6*X(3035) = 4*X(119)-3*X(10742) = 11*X(119)-12*X(20400) = 8*X(119)-9*X(38752) = 4*X(119)-9*X(38754) = 10*X(119)-9*X(38755) = 7*X(119)-6*X(38757) = 17*X(119)-18*X(38758) = 7*X(119)-9*X(38760) = X(119)-3*X(38761) = 4*X(119)-5*X(38762) = 13*X(119)-15*X(38763)

X(38753) lies on these lines: {2,22799}, {3,119}, {4,38141}, {5,10728}, {11,382}, {20,952}, {30,104}, {35,12763}, {36,12764}, {80,1155}, {100,550}, {149,3529}, {153,376}, {214,16128}, {355,38213}, {381,6713}, {515,12515}, {516,12737}, {517,26726}, {528,15681}, {529,25438}, {546,31272}, {548,11698}, {1317,4302}, {1320,20067}, {1385,34789}, {1387,4293}, {1482,37002}, {1537,6938}, {1656,21154}, {1657,5840}, {2771,12119}, {2783,38730}, {2787,38741}, {2800,18481}, {2805,38797}, {2828,23240}, {3091,34126}, {3146,22938}, {3530,32633}, {3534,6244}, {3576,12611}, {3579,12751}, {3655,25485}, {3843,23513}, {3851,6667}, {3887,38765}, {4190,34122}, {4294,12735}, {4297,6265}, {4324,7972}, {4325,16173}, {5072,38319}, {5073,20418}, {5441,26201}, {5533,12953}, {5691,12619}, {5731,19907}, {5790,6948}, {6174,15688}, {6284,10074}, {6449,13922}, {6450,13991}, {6833,38135}, {6869,13226}, {6872,34123}, {6923,18515}, {7354,10058}, {8068,12943}, {8666,13271}, {8674,20127}, {8703,10711}, {10087,15338}, {10090,15326}, {10265,28164}, {10483,13273}, {10679,34698}, {10698,34773}, {11219,28168}, {11715,12699}, {11849,32213}, {12138,18533}, {12531,28224}, {12740,21578}, {12749,37568}, {12767,34628}, {12775,37621}, {13665,13913}, {13785,13977}, {14217,28146}, {15696,35250}, {15720,31235}, {16117,30264}, {17661,31837}, {17800,37726}, {18492,38182}, {18493,38032}, {18525,33899}, {19709,38069}, {20076,25416}, {21630,28150}, {22560,34620}, {30283,34745}

X(38753) = midpoint of X(i) and X(j) for these {i,j}: {20, 12248}, {149, 3529}, {1657, 12773}
X(38753) = reflection of X(i) in X(j) for these (i,j): (3, 38761), (100, 550), (119, 38759), (153, 33814), (382, 11), (3146, 22938), (5691, 12619), (6265, 4297), (10698, 34773), (10711, 8703), (10724, 1484), (10728, 5), (10738, 104), (10742, 3), (11698, 548), (12331, 24466), (12699, 11715), (12751, 3579), (13271, 8666), (16128, 214), (17661, 31837), (19914, 12515), (34789, 1385), (38752, 38754), (38756, 119)
X(38753) = anticomplement of X(22799)
X(38753) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 119, 38762), (3, 10742, 38752), (3, 38755, 3035), (3, 38756, 119), (3, 38761, 38754), (104, 10724, 1484), (119, 38756, 10742), (119, 38759, 3), (119, 38761, 38759), (119, 38762, 38752), (153, 376, 33814), (548, 11698, 34474), (1484, 10724, 10738), (3534, 12331, 24466), (10742, 38754, 3), (10742, 38762, 119), (38756, 38759, 38762)

X(38754) = X(100)-CIRCUM-EULER-POINT OF X(376)

Barycentrics    5*a^7-5*(b+c)*a^6-(9*b^2-17*b*c+9*c^2)*a^5+3*(b+c)*(3*b^2-4*b*c+3*c^2)*a^4+(3*b^4+3*c^4-b*c*(13*b^2-14*b*c+13*c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2-4*b*c+3*c^2)*a^2+(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :
X(38754) = 5*X(3)-2*X(119) = 7*X(3)-4*X(3035) = 4*X(3)-X(10742) = 17*X(3)-8*X(20400) = 2*X(3)+X(38753) = 3*X(3)-X(38755) = 7*X(3)-X(38756) = 13*X(3)-4*X(38757) = 9*X(3)-4*X(38758) = X(3)-4*X(38759) = 3*X(3)-2*X(38760) = X(3)+2*X(38761) = 8*X(3)-5*X(38762) = 19*X(3)-10*X(38763) = 7*X(119)-10*X(3035) = 8*X(119)-5*X(10742) = 17*X(119)-20*X(20400) = 4*X(119)-5*X(38752) = 4*X(119)+5*X(38753) = 6*X(119)-5*X(38755) = 14*X(119)-5*X(38756) = 13*X(119)-10*X(38757) = 9*X(119)-10*X(38758) = X(119)-10*X(38759) = 3*X(119)-5*X(38760) = X(119)+5*X(38761)

X(38754) lies on these lines: {3,119}, {4,34126}, {11,1657}, {20,10738}, {100,548}, {104,550}, {140,10728}, {149,17538}, {153,3528}, {376,952}, {381,21154}, {382,6713}, {528,15689}, {631,22799}, {1484,12103}, {2771,38723}, {2783,38731}, {2787,38742}, {2800,38778}, {2805,38798}, {3522,12248}, {3529,22938}, {3534,5840}, {3543,38141}, {3627,31272}, {3830,23513}, {3843,6667}, {3887,38766}, {4297,12515}, {4316,13273}, {4324,13274}, {5010,12763}, {6174,14093}, {6455,13922}, {6456,13991}, {6948,18515}, {7280,12764}, {7958,13743}, {7987,12611}, {8674,38788}, {8703,34474}, {9812,38044}, {10058,15326}, {10074,15338}, {10707,15686}, {10711,34200}, {10724,15704}, {11698,33923}, {12737,31730}, {12751,31663}, {12773,15696}, {13624,34789}, {14269,38069}, {15015,31142}, {16173,28146}, {18481,19914}, {28186,36004}

X(38754) = midpoint of X(38752) and X(38753)
X(38754) = reflection of X(i) in X(j) for these (i,j): (4, 34126), (381, 21154), (3543, 38141), (3830, 23513), (9812, 38044), (10742, 38752), (14269, 38069), (34474, 8703), (38752, 3), (38755, 38760)
X(38754) = X(38753)-Gibert-Moses centroid
X(38754) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10742, 38762), (3, 38753, 10742), (3, 38755, 38760), (3, 38756, 3035), (3, 38761, 38753), (3522, 12248, 33814), (12773, 15696, 24466), (38755, 38760, 38752), (38759, 38761, 3)

X(38755) = X(100)-CIRCUM-EULER-POINT OF X(381)

Barycentrics    a^7-(b+c)*a^6+7*b*c*a^5-6*b*c*(b+c)*a^4-(3*b^4+3*c^4-b*c*(b^2+10*b*c+c^2))*a^3+(b^2-c^2)*(b-c)*(3*b^2+8*b*c+3*c^2)*a^2+2*(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-2*(b^2-c^2)^3*(b-c) : :
X(38755) = X(3)-4*X(119) = 5*X(3)-8*X(3035) = X(3)+2*X(10742) = 7*X(3)-16*X(20400) = 5*X(3)-2*X(38753) = 3*X(3)-2*X(38754) = 2*X(3)+X(38756) = X(3)+8*X(38757) = 3*X(3)-8*X(38758) = 11*X(3)-8*X(38759) = 3*X(3)-4*X(38760) = 7*X(3)-4*X(38761) = 7*X(3)-10*X(38762) = 11*X(3)-20*X(38763) = 5*X(119)-2*X(3035) = 2*X(119)+X(10742) = 7*X(119)-4*X(20400) = 10*X(119)-X(38753) = 6*X(119)-X(38754) = 8*X(119)+X(38756) = X(119)+2*X(38757) = 3*X(119)-2*X(38758) = 11*X(119)-2*X(38759) = 3*X(119)-X(38760) = 7*X(119)-X(38761) = 14*X(119)-5*X(38762) = 11*X(119)-5*X(38763)

X(38755) lies on these lines: {3,119}, {4,11698}, {5,153}, {10,16128}, {11,3851}, {80,10895}, {100,382}, {104,1656}, {140,12248}, {149,546}, {354,37718}, {355,21635}, {381,952}, {528,14269}, {999,12763}, {1145,11415}, {1317,9669}, {1385,15017}, {1482,12611}, {1484,3091}, {1537,8148}, {1657,10728}, {1768,9956}, {2771,5587}, {2783,38732}, {2787,38743}, {2800,5790}, {2801,38107}, {2802,11236}, {2805,38799}, {2932,11681}, {2950,22792}, {3295,12764}, {3534,34474}, {3627,13199}, {3830,5840}, {3843,10738}, {3887,38767}, {5055,38319}, {5070,6713}, {5076,10724}, {5079,31272}, {5531,18492}, {5541,22793}, {5691,22935}, {5818,9809}, {6174,15681}, {6246,10894}, {6264,9955}, {6265,18525}, {6326,12747}, {6767,10956}, {6797,9612}, {6911,35451}, {7972,10896}, {8674,38789}, {9655,10090}, {9668,10087}, {9897,11011}, {10058,31479}, {10698,12645}, {10759,11898}, {10826,17660}, {10827,17638}, {10893,25485}, {11219,38182}, {11237,16173}, {11729,18545}, {12247,18357}, {12702,34789}, {12737,18493}, {13273,37541}, {15015,28160}, {15694,21154}, {17528,34122}, {17577,38138}, {17800,24466}, {18526,19907}, {19709,23513}, {23251,35883}, {23261,35882}, {28224,37375}, {38057,38121}

X(38755) = midpoint of X(10742) and X(38752)
X(38755) = reflection of X(i) in X(j) for these (i,j): (3, 38752), (104, 34126), (3534, 34474), (10707, 38141), (11219, 38182), (37718, 38140), (38752, 119), (38754, 38760), (38760, 38758)
X(38755) = X(38756)-Gibert-Moses centroid
X(38755) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10742, 38756), (4, 11698, 12331), (5, 153, 12773), (100, 22799, 382), (119, 10742, 3), (119, 38757, 10742), (119, 38760, 38758), (119, 38761, 20400), (3035, 38753, 3), (6326, 18480, 12747), (10728, 33814, 1657), (12611, 12751, 1482), (20400, 38761, 38762), (38752, 38754, 38760), (38754, 38760, 3), (38758, 38760, 38752), (38761, 38762, 3)

X(38756) = X(100)-CIRCUM-EULER-POINT OF X(382)

Barycentrics    3*a^7-3*(b+c)*a^6-(4*b^2-13*b*c+4*c^2)*a^5+2*(2*b-c)*(b-2*c)*(b+c)*a^4-(b^4+c^4+b*c*(5*b^2-14*b*c+5*c^2))*a^3+(b^2-c^2)*(b-c)*(b^2+8*b*c+c^2)*a^2+2*(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-2*(b^2-c^2)^3*(b-c) : :
X(38756) = 3*X(3)-4*X(119) = 7*X(3)-8*X(3035) = 13*X(3)-16*X(20400) = 5*X(3)-6*X(38752) = 3*X(3)-2*X(38753) = 7*X(3)-6*X(38754) = 2*X(3)-3*X(38755) = 5*X(3)-8*X(38757) = 9*X(3)-8*X(38759) = 11*X(3)-12*X(38760) = 5*X(3)-4*X(38761) = 9*X(3)-10*X(38762) = 17*X(3)-20*X(38763) = 7*X(119)-6*X(3035) = 2*X(119)-3*X(10742) = 13*X(119)-12*X(20400) = 10*X(119)-9*X(38752) = 14*X(119)-9*X(38754) = 8*X(119)-9*X(38755) = 5*X(119)-6*X(38757) = 19*X(119)-18*X(38758) = 3*X(119)-2*X(38759) = 11*X(119)-9*X(38760) = 5*X(119)-3*X(38761) = 6*X(119)-5*X(38762) = 17*X(119)-15*X(38763)

X(38756) lies on these lines: {3,119}, {4,1484}, {5,12248}, {11,3843}, {20,11698}, {30,153}, {80,12943}, {100,1657}, {104,381}, {149,3627}, {382,952}, {515,16128}, {528,15684}, {535,22560}, {999,12764}, {1317,9668}, {1482,34789}, {1537,10247}, {1768,18480}, {2771,5691}, {2783,38733}, {2787,38744}, {2800,18525}, {2801,31671}, {2805,38800}, {2932,5080}, {3149,35451}, {3295,12763}, {3534,10711}, {3830,10738}, {3887,38768}, {5055,6713}, {5072,31272}, {5073,5840}, {5076,22938}, {5079,34126}, {5541,28146}, {5790,12515}, {6174,15689}, {6224,28186}, {6264,22793}, {6326,28160}, {6407,13922}, {6408,13991}, {6797,9579}, {7972,12953}, {8674,38790}, {9654,10058}, {9657,16173}, {9669,10074}, {10246,12611}, {10698,18526}, {10707,38335}, {10731,21664}, {11715,18493}, {12138,18494}, {12199,18501}, {12332,18524}, {12499,18503}, {12702,12751}, {12736,18541}, {12752,18508}, {12753,26336}, {12754,26346}, {12761,18519}, {12762,18518}, {12775,18545}, {12776,18543}, {13253,28204}, {13624,15017}, {15681,24466}, {15696,34474}, {17800,37725}, {18481,21635}, {18510,19081}, {18512,19082}, {20095,33703}, {22728,32454}, {22775,26321}, {23251,35856}, {23261,35857}, {26333,34698}, {32636,37718}

X(38756) = midpoint of X(20095) and X(33703)
X(38756) = reflection of X(i) in X(j) for these (i,j): (3, 10742), (20, 11698), (104, 22799), (149, 3627), (382, 10728), (1482, 34789), (1657, 100), (1768, 18480), (3534, 10711), (6264, 22793), (12248, 5), (12331, 153), (12702, 12751), (12747, 5691), (12773, 4), (18481, 21635), (18508, 12752), (18526, 10698), (38753, 119), (38761, 38757)
X(38756) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10742, 38755), (104, 22799, 381), (119, 38753, 3), (119, 38759, 38762), (3035, 38754, 3), (10742, 38752, 38757), (10742, 38753, 119), (38752, 38761, 3), (38753, 38762, 38759), (38757, 38761, 38752), (38759, 38762, 3)

X(38757) = X(100)-CIRCUM-EULER-POINT OF X(546)

Barycentrics    2*a^7-2*(b+c)*a^6-(b^2-12*b*c+c^2)*a^5+(b+c)*(b^2-10*b*c+c^2)*a^4-4*(b^4-4*b^2*c^2+c^4)*a^3+4*(b^2-c^2)*(b-c)*(b^2+3*b*c+c^2)*a^2+3*(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-3*(b^2-c^2)^3*(b-c) : :
X(38757) = X(3)-3*X(119) = 2*X(3)-3*X(3035) = X(3)+3*X(10742) = 5*X(3)-9*X(38752) = 7*X(3)-3*X(38753) = 13*X(3)-9*X(38754) = X(3)-9*X(38755) = 5*X(3)+3*X(38756) = 4*X(3)-9*X(38758) = 4*X(3)-3*X(38759) = 7*X(3)-9*X(38760) = 5*X(3)-3*X(38761) = 11*X(3)-15*X(38762) = 3*X(3)-5*X(38763) = 3*X(119)-2*X(20400) = 5*X(119)-3*X(38752) = 7*X(119)-X(38753) = 13*X(119)-3*X(38754) = X(119)-3*X(38755) = 5*X(119)+X(38756) = 4*X(119)-3*X(38758) = 4*X(119)-X(38759) = 7*X(119)-3*X(38760) = 5*X(119)-X(38761) = 11*X(119)-5*X(38762) = 9*X(119)-5*X(38763)

X(38757) lies on these lines: {3,119}, {4,528}, {5,10199}, {10,22792}, {11,153}, {12,6912}, {20,6174}, {80,5665}, {100,3146}, {104,3090}, {381,37726}, {382,10993}, {515,5087}, {529,1532}, {546,946}, {942,2801}, {962,13996}, {1145,7991}, {1466,5229}, {1484,3857}, {1512,17768}, {1537,5854}, {1768,34122}, {1837,12831}, {2771,36253}, {2783,38734}, {2787,38745}, {2800,3036}, {2805,38801}, {2886,6982}, {3303,10956}, {3525,12248}, {3529,10728}, {3627,5840}, {3628,6713}, {3816,12115}, {3829,6968}, {3832,10707}, {3855,38077}, {3887,38769}, {5070,38069}, {5072,12773}, {5076,12331}, {5080,36002}, {5290,38055}, {5434,6945}, {5531,12690}, {5537,17757}, {5552,37001}, {5587,5851}, {5660,5691}, {5768,25558}, {5881,34640}, {6154,10724}, {6264,38038}, {6425,13922}, {6426,13991}, {6702,12436}, {6830,34697}, {6835,9656}, {6850,9711}, {6925,31141}, {6932,34606}, {6953,9657}, {6957,11237}, {6969,11194}, {6978,12114}, {7680,18516}, {7989,11219}, {8674,38791}, {10303,31235}, {10698,10912}, {10778,15044}, {11570,17661}, {11729,15178}, {12247,38156}, {12736,13227}, {12738,37533}, {12812,38319}, {13600,18483}, {13729,15888}, {15017,25522}, {15022,31272}, {15704,33814}, {16189,25416}, {17538,34474}, {18357,31803}, {19647,30960}, {21031,37437}, {31160,37374}

X(38757) = midpoint of X(i) and X(j) for these {i,j}: {4, 37725}, {11, 153}, {80, 13257}, {119, 10742}, {382, 10993}, {962, 13996}, {1145, 34789}, {1537, 12751}, {5531, 12690}, {5691, 10609}, {6154, 10724}, {10728, 24466}, {11570, 17661}, {11698, 22799}, {12736, 13227}, {38756, 38761}
X(38757) = reflection of X(i) in X(j) for these (i,j): (3, 20400), (104, 6667), (3035, 119), (12019, 19925), (13226, 6702), (20418, 5), (24466, 35023), (38759, 3035)
X(38757) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 119, 20400), (3, 20400, 3035), (119, 3035, 38758), (119, 38761, 38752), (5660, 5691, 10609), (10742, 38752, 38756), (10742, 38755, 119), (18542, 37821, 18242), (38752, 38756, 38761), (38758, 38759, 3035)

X(38758) = X(100)-CIRCUM-EULER-POINT OF X(547)

Barycentrics    2*a^7-2*(b+c)*a^6-(9*b^2+4*b*c+9*c^2)*a^5+3*(b+c)*(3*b^2+2*b*c+3*c^2)*a^4+4*(3*b^4+3*c^4-4*b*c*(b^2+b*c+c^2))*a^3-4*(b^2-c^2)*(b-c)*(3*b^2+5*b*c+3*c^2)*a^2-5*(b^2-c^2)^2*(b^2-4*b*c+c^2)*a+5*(b^2-c^2)^3*(b-c) : :
X(38758) = X(3)+5*X(119) = 2*X(3)-5*X(3035) = 7*X(3)+5*X(10742) = X(3)-10*X(20400) = X(3)-5*X(38752) = 17*X(3)-5*X(38753) = 9*X(3)-5*X(38754) = 3*X(3)+5*X(38755) = 19*X(3)+5*X(38756) = 4*X(3)+5*X(38757) = 8*X(3)-5*X(38759) = 3*X(3)-5*X(38760) = 11*X(3)-5*X(38761) = 2*X(119)+X(3035) = 7*X(119)-X(10742) = X(119)+2*X(20400) = 17*X(119)+X(38753) = 9*X(119)+X(38754) = 3*X(119)-X(38755) = 19*X(119)-X(38756) = 4*X(119)-X(38757) = 8*X(119)+X(38759) = 3*X(119)+X(38760) = 11*X(119)+X(38761) = 13*X(119)+5*X(38762) = 7*X(119)+5*X(38763)

X(38758) lies on these lines: {3,119}, {4,35023}, {5,25439}, {11,5056}, {100,3832}, {104,3533}, {153,31235}, {528,3545}, {547,551}, {1145,11531}, {1698,13257}, {2771,38725}, {2783,38735}, {2787,38746}, {2800,3740}, {2805,38802}, {3091,6154}, {3543,6174}, {3634,13226}, {3845,5840}, {3887,38770}, {5067,6667}, {5660,34122}, {5854,16200}, {6713,16239}, {7989,12690}, {8674,38792}, {9945,19925}, {10197,38043}, {10711,15702}, {11001,34474}, {11698,20418}, {11729,33179}, {13243,19877}, {24466,33703}, {30392,34123}, {38123,38179}

X(38758) = midpoint of X(i) and X(j) for these {i,j}: {119, 38752}, {5660, 34122}, {10711, 21154}, {11698, 34126}, {38755, 38760}
X(38758) = reflection of X(i) in X(j) for these (i,j): (3035, 38752), (20418, 34126), (38752, 20400)
X(38758) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (119, 3035, 38757), (119, 20400, 3035), (119, 38760, 38755), (119, 38763, 10742), (3035, 38757, 38759), (38752, 38755, 38760)

X(38759) = X(100)-CIRCUM-EULER-POINT OF X(548)

Barycentrics    6*a^7-6*(b+c)*a^6-(11*b^2-20*b*c+11*c^2)*a^5+(b+c)*(11*b^2-14*b*c+11*c^2)*a^4+4*(b^4+c^4-4*b*c*(b^2-b*c+c^2))*a^3-4*(b^3+c^3)*(b-c)^2*a^2+(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :
X(38759) = 3*X(3)-X(119) = 5*X(3)-X(10742) = 5*X(3)-2*X(20400) = 7*X(3)-3*X(38752) = 3*X(3)+X(38753) = X(3)+3*X(38754) = 11*X(3)-3*X(38755) = 9*X(3)-X(38756) = 4*X(3)-X(38757) = 8*X(3)-3*X(38758) = 5*X(3)-3*X(38760) = 9*X(3)-5*X(38762) = 11*X(3)-5*X(38763) = 2*X(119)-3*X(3035) = 5*X(119)-3*X(10742) = 5*X(119)-6*X(20400) = 7*X(119)-9*X(38752) = X(119)+9*X(38754) = 11*X(119)-9*X(38755) = 3*X(119)-X(38756) = 4*X(119)-3*X(38757) = 8*X(119)-9*X(38758) = 5*X(119)-9*X(38760) = X(119)+3*X(38761) = 3*X(119)-5*X(38762) = 11*X(119)-15*X(38763)

X(38759) lies on these lines: {3,119}, {4,6667}, {11,20}, {30,6713}, {40,5854}, {100,3522}, {104,376}, {153,6174}, {165,1145}, {382,23513}, {515,3036}, {516,1387}, {517,15528}, {529,2077}, {546,38319}, {548,952}, {549,22799}, {550,1484}, {631,10728}, {971,18254}, {1317,5731}, {1320,9778}, {1537,3576}, {1768,9841}, {2771,31805}, {2783,38736}, {2787,38747}, {2800,9943}, {2801,9945}, {2802,12512}, {2805,38803}, {2886,6948}, {3146,31272}, {3523,31235}, {3528,12248}, {3534,10738}, {3627,34126}, {3651,18861}, {3816,6938}, {3826,6955}, {3830,38069}, {3887,38771}, {4316,8068}, {4324,5533}, {4996,7411}, {4999,31775}, {5172,6909}, {5217,10956}, {5691,34122}, {5732,5851}, {5918,17638}, {6244,25438}, {6265,12520}, {6409,13922}, {6410,13991}, {6560,13913}, {6561,13977}, {6690,6950}, {6702,28164}, {6714,7427}, {6827,12761}, {6836,13273}, {6921,37001}, {6925,12764}, {6966,12943}, {7580,10090}, {7742,10058}, {7987,34123}, {7989,17583}, {7991,25416}, {7993,12732}, {8674,37853}, {8703,33814}, {9541,19081}, {10167,11570}, {10711,19708}, {10884,12739}, {10993,12773}, {11219,12690}, {11220,12532}, {11715,31730}, {11729,13624}, {12138,37931}, {12331,15688}, {12607,37002}, {12611,17502}, {12641,34716}, {12699,38032}, {12751,35242}, {13257,15015}, {15682,38077}, {15696,35252}, {15704,22938}, {16174,28150}, {17613,21578}, {18525,38128}, {22676,32454}, {26285,32213}, {28158,33709}, {30264,37403}, {31670,38119}, {31671,38124}, {31672,38131}, {31673,38133}, {33697,38182}, {33699,38084}

X(38759) = midpoint of X(i) and X(j) for these {i,j}: {3, 38761}, {11, 20}, {104, 24466}, {119, 38753}, {1768, 10609}, {4316, 37374}, {6909, 15326}, {7991, 25416}, {7993, 12732}, {10993, 12773}, {11715, 31730}, {12248, 37725}, {15704, 22938}, {17613, 21578}
X(38759) = reflection of X(i) in X(j) for these (i,j): (4, 6667), (3035, 3), (10742, 20400), (11729, 13624), (37725, 35023), (38757, 3035)
X(38759) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10742, 38760), (3, 38753, 119), (3, 38754, 38761), (3, 38756, 38762), (4, 21154, 6667), (119, 38761, 38753), (3035, 38757, 38758), (3528, 12248, 34474), (7987, 34789, 34123), (10742, 38760, 20400), (12248, 34474, 37725), (20400, 38760, 3035), (34474, 37725, 35023), (38753, 38762, 38756), (38756, 38762, 119)

X(38760) = X(100)-CIRCUM-EULER-POINT OF X(549)

Barycentrics    4*a^7-4*(b+c)*a^6-(9*b^2-10*b*c+9*c^2)*a^5+3*(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+2*(3*b^4+3*c^4-b*c*(7*b^2-2*b*c+7*c^2))*a^3-2*(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2-4*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :
X(38760) = 2*X(3)+X(119) = X(3)+2*X(3035) = 5*X(3)+X(10742) = 5*X(3)+4*X(20400) = 7*X(3)-X(38753) = 3*X(3)-X(38754) = 3*X(3)+X(38755) = 11*X(3)+X(38756) = 7*X(3)+2*X(38757) = 3*X(3)+2*X(38758) = 5*X(3)-2*X(38759) = 4*X(3)-X(38761) = X(3)+5*X(38762) = 4*X(3)+5*X(38763) = X(119)-4*X(3035) = 5*X(119)-2*X(10742) = 5*X(119)-8*X(20400) = 7*X(119)+2*X(38753) = 3*X(119)+2*X(38754) = 3*X(119)-2*X(38755) = 11*X(119)-2*X(38756) = 7*X(119)-4*X(38757) = 3*X(119)-4*X(38758) = 5*X(119)+4*X(38759) = 2*X(119)+X(38761) = X(119)-10*X(38762) = 2*X(119)-5*X(38763) = X(23513)+2*X(34474) = 3*X(23513)-4*X(38319) = 3*X(34474)+2*X(38319)

X(38760) lies on these lines: {2,5840}, {3,119}, {5,24466}, {11,35}, {36,10956}, {40,11729}, {80,31423}, {100,631}, {104,3523}, {149,10303}, {153,15717}, {214,6684}, {371,13991}, {372,13922}, {517,34123}, {528,5054}, {547,38141}, {548,22799}, {549,952}, {1006,17100}, {1145,1385}, {1317,5690}, {1387,1697}, {1484,6154}, {1537,3579}, {1698,12119}, {2771,38727}, {2783,38737}, {2787,38748}, {2800,10164}, {2801,38130}, {2802,10165}, {2805,38804}, {3090,10724}, {3522,10728}, {3525,13199}, {3526,6667}, {3530,37725}, {3624,14217}, {3628,22938}, {3634,6246}, {3887,38772}, {4187,26086}, {4995,38028}, {4996,6940}, {5298,5844}, {5432,14793}, {5854,10246}, {5856,38122}, {5886,17564}, {6068,31657}, {6175,38163}, {6691,11849}, {6735,18857}, {6880,12775}, {6883,10058}, {6921,11248}, {6961,11499}, {6966,18491}, {7987,12751}, {8068,37438}, {8674,38793}, {9024,38119}, {9540,19112}, {10090,26357}, {10172,38161}, {10299,12248}, {10427,31658}, {10526,19537}, {10609,12619}, {10707,15702}, {10711,15692}, {10955,14800}, {11014,19907}, {11112,38109}, {11230,38038}, {11231,34122}, {11374,24465}, {11570,31837}, {11698,15712}, {12331,15720}, {12515,37560}, {12611,31663}, {12665,13369}, {12738,13226}, {12758,31838}, {13747,26285}, {13935,19113}, {15017,16192}, {15178,25416}, {15670,38182}, {15699,38077}, {16174,19862}, {16202,25438}, {17757,23961}, {25440,26470}, {28465,38133}, {34789,35242}, {38042,38156}, {38043,38052}, {38147,38317}, {38152,38171}, {38159,38318}

X(38760) = midpoint of X(i) and X(j) for these {i,j}: {2, 34474}, {3, 38752}, {6174, 21154}, {33814, 34126}, {38754, 38755}
X(38760) = reflection of X(i) in X(j) for these (i,j): (11, 34126), (119, 38752), (21154, 549), (23513, 2), (34122, 11231), (34126, 140), (38032, 10165), (38038, 11230), (38069, 5054), (38077, 15699), (38124, 38122), (38128, 26446), (38141, 547), (38147, 38317), (38152, 38171), (38156, 38042), (38159, 38318), (38161, 10172), (38752, 3035), (38755, 38758)
X(38760) = anticomplement of X(38319)
X(38760) = X(119)-Gibert-Moses centroid
X(38760) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 119, 38761), (3, 3035, 119), (3, 10742, 38759), (3, 38755, 38754), (3, 38762, 3035), (11, 33814, 10993), (100, 631, 6713), (100, 6713, 37726), (119, 3035, 38763), (140, 33814, 11), (3035, 38759, 20400), (3525, 13199, 31272), (3526, 10738, 6667), (10742, 20400, 119), (20400, 38759, 10742), (20418, 35023, 12331), (24466, 31235, 5), (38752, 38754, 38755), (38752, 38755, 38758), (38755, 38758, 119), (38761, 38763, 119)

X(38761) = X(100)-CIRCUM-EULER-POINT OF X(550)

Barycentrics    4*a^7-4*(b+c)*a^6-7*(b-c)^2*a^5+(b+c)*(7*b^2-10*b*c+7*c^2)*a^4+2*(b^2-4*b*c+c^2)*(b^2-b*c+c^2)*a^3-2*(b^2-c^2)*(b-c)^3*a^2+(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :
X(38761) = 3*X(3)-2*X(3035) = 3*X(3)-X(10742) = 7*X(3)-4*X(20400) = 5*X(3)-3*X(38752) = X(3)-3*X(38754) = 7*X(3)-3*X(38755) = 5*X(3)-X(38756) = 5*X(3)-2*X(38757) = 11*X(3)-6*X(38758) = 4*X(3)-3*X(38760) = 7*X(3)-5*X(38762) = 8*X(3)-5*X(38763) = 3*X(119)-4*X(3035) = 3*X(119)-2*X(10742) = 7*X(119)-8*X(20400) = 5*X(119)-6*X(38752) = X(119)+2*X(38753) = X(119)-6*X(38754) = 7*X(119)-6*X(38755) = 5*X(119)-2*X(38756) = 5*X(119)-4*X(38757) = 11*X(119)-12*X(38758) = X(119)-4*X(38759) = 2*X(119)-3*X(38760) = 7*X(119)-10*X(38762) = 4*X(119)-5*X(38763)

X(38761) lies on these lines: {2,10728}, {3,119}, {4,6713}, {5,21154}, {11,30}, {20,104}, {35,10956}, {40,550}, {100,376}, {140,22799}, {153,3522}, {165,12751}, {355,38128}, {381,6667}, {411,18861}, {515,15863}, {516,11715}, {517,3937}, {528,3534}, {529,35000}, {546,34126}, {548,33814}, {549,31235}, {946,38032}, {956,35249}, {993,28458}, {1145,3579}, {1317,5697}, {1320,6361}, {1385,1537}, {1387,1420}, {1484,15704}, {1512,12619}, {1519,18857}, {1532,23961}, {1657,10738}, {2550,6948}, {2771,3650}, {2783,38738}, {2787,38749}, {2800,4297}, {2802,31730}, {2805,38805}, {2828,3184}, {2831,14689}, {2886,18515}, {2932,35250}, {3036,18525}, {3070,13913}, {3071,13977}, {3091,38319}, {3529,10724}, {3576,11729}, {3651,4996}, {3655,7962}, {3656,4312}, {3853,38141}, {3887,38773}, {4190,18761}, {4299,10058}, {4302,10074}, {4304,5083}, {4311,15558}, {5204,12764}, {5217,12763}, {5267,37401}, {5450,26470}, {5480,38119}, {5731,10698}, {5732,6265}, {5805,38124}, {5854,12702}, {6174,8703}, {6200,13922}, {6246,28164}, {6396,13991}, {6459,19081}, {6460,19082}, {6702,31673}, {6841,17009}, {6928,12761}, {6938,10269}, {6959,37001}, {6985,10090}, {7280,37406}, {8068,10483}, {8674,16111}, {9541,19113}, {9897,34628}, {9945,12738}, {10073,31515}, {10267,12775}, {10304,10711}, {10572,12832}, {10707,11001}, {10759,25406}, {11194,13271}, {11248,37002}, {11570,13369}, {11826,32153}, {12331,15696}, {12611,13624}, {12665,31837}, {12752,16190}, {12758,21578}, {13199,17538}, {13205,34620}, {13257,22935}, {13278,20076}, {14893,38084}, {15521,33970}, {15528,24474}, {15687,38077}, {15688,35023}, {16117,35451}, {16118,38063}, {16371,18516}, {17100,37403}, {17768,35459}, {18480,34122}, {18482,38205}, {18483,32557}, {19925,38133}, {22793,38038}, {25438,35448}, {37290,37561}

X(38761) = midpoint of X(i) and X(j) for these {i,j}: {3, 38753}, {20, 104}, {100, 12248}, {1320, 6361}, {1484, 15704}, {1657, 10738}, {1768, 12119}, {3529, 10724}, {10707, 11001}, {12515, 18481}
X(38761) = reflection of X(i) in X(j) for these (i,j): (3, 38759), (4, 6713), (119, 3), (1145, 3579), (1317, 34773), (1519, 18857), (1532, 23961), (1537, 1385), (6174, 8703), (6841, 17009), (10738, 20418), (10742, 3035), (10993, 24466), (11570, 13369), (12611, 13624), (12665, 31837), (12699, 1387), (12738, 9945), (13257, 22935), (15521, 33970), (18525, 3036), (22799, 140), (24466, 550), (24474, 15528), (31673, 6702), (33814, 548), (34789, 11729), (37725, 33814), (37726, 104), (38756, 38757)
X(38761) = complement of X(10728)
X(38761) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 119, 38760), (3, 10742, 3035), (3, 38754, 38759), (3, 38755, 38762), (3, 38756, 38752), (4, 6713, 23513), (119, 38760, 38763), (153, 3522, 34474), (376, 12248, 100), (3035, 10742, 119), (3576, 34789, 11729), (12611, 13624, 34123), (20400, 38755, 119), (38752, 38756, 38757), (38752, 38757, 119), (38753, 38754, 3), (38753, 38759, 119), (38755, 38762, 20400)

X(38762) = X(100)-CIRCUM-EULER-POINT OF X(631)

Barycentrics    3*a^7-3*(b+c)*a^6-7*(b^2-b*c+c^2)*a^5+(b+c)*(7*b^2-4*b*c+7*c^2)*a^4+(5*b^4+5*c^4-b*c*(11*b^2-2*b*c+11*c^2))*a^3-(b^2-c^2)*(b-c)*(5*b^2+4*b*c+5*c^2)*a^2-(b^2-c^2)^2*(b^2-4*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :
X(38762) = 6*X(2)-X(10738) = 9*X(2)+X(13199) = 3*X(2)+2*X(33814) = 11*X(2)-6*X(38084) = 3*X(3)+2*X(119) = X(3)+4*X(3035) = 4*X(3)+X(10742) = 7*X(3)+8*X(20400) = 2*X(3)+3*X(38752) = 6*X(3)-X(38753) = 8*X(3)-3*X(38754) = 7*X(3)+3*X(38755) = 9*X(3)+X(38756) = 11*X(3)+4*X(38757) = 13*X(3)+12*X(38758) = 9*X(3)-4*X(38759) = X(3)-6*X(38760) = 7*X(3)-2*X(38761) = X(3)+2*X(38763) = 3*X(10738)+2*X(13199) = X(10738)+4*X(33814) = X(13199)-6*X(33814) = 11*X(33814)+9*X(38084)

X(38762) lies on these lines: {2,10738}, {3,119}, {5,10724}, {11,3526}, {80,11231}, {100,140}, {104,549}, {149,3525}, {153,3524}, {165,12611}, {214,19914}, {376,22799}, {381,24466}, {528,15694}, {548,10728}, {631,952}, {632,31272}, {1145,10246}, {1156,38113}, {1320,38028}, {1385,37829}, {1387,5218}, {1482,6921}, {1537,6880}, {1656,5840}, {1862,3147}, {2771,38728}, {2783,38739}, {2787,38750}, {2800,38786}, {2805,38806}, {2932,6883}, {3090,22938}, {3311,13991}, {3312,13922}, {3530,11698}, {3654,25485}, {3887,38774}, {5010,12764}, {5054,6174}, {5056,38141}, {5070,23513}, {5432,10090}, {5433,10087}, {5550,38044}, {5657,19907}, {5854,37624}, {6265,6684}, {6594,38122}, {6667,10993}, {6878,12690}, {6910,12747}, {6961,18525}, {7280,12763}, {7288,12735}, {8674,38794}, {8981,19112}, {9956,12119}, {10164,12515}, {10165,12737}, {10707,11539}, {10711,12100}, {10755,38110}, {10769,34127}, {10778,34128}, {11230,14217}, {11729,12702}, {11849,13747}, {12248,15717}, {12531,38112}, {12619,15015}, {12749,37605}, {12751,13624}, {12773,15720}, {13966,19113}, {15017,35242}, {19081,35256}, {19082,35255}, {31663,34789}, {32213,37535}, {33812,38127}, {35023,37726}

X(38762) = midpoint of X(15017) and X(35242)
X(38762) = reflection of X(i) in X(j) for these (i,j): (1656, 31235), (31272, 632), (38763, 3035)
X(38762) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 33814, 10738), (3, 119, 38753), (3, 3035, 38752), (3, 10742, 38754), (3, 38752, 10742), (3, 38755, 38761), (3, 38756, 38759), (119, 38753, 10742), (119, 38759, 38756), (149, 3525, 34126), (214, 26446, 19914), (3035, 38760, 3), (5054, 12331, 6713), (6174, 6713, 12331), (12773, 15720, 21154), (15015, 31423, 12619), (20400, 38761, 38755), (38752, 38753, 119), (38756, 38759, 38753)

X(38763) = X(100)-CIRCUM-EULER-POINT OF X(632)

Barycentrics    4*a^7-4*(b+c)*a^6-(11*b^2-6*b*c+11*c^2)*a^5+(b+c)*(11*b^2-2*b*c+11*c^2)*a^4+2*(5*b^4+5*c^4-b*c*(9*b^2+2*b*c+9*c^2))*a^3-2*(b^2-c^2)*(b-c)*(5*b^2+6*b*c+5*c^2)*a^2-3*(b^2-c^2)^2*(b^2-4*b*c+c^2)*a+3*(b^2-c^2)^3*(b-c) : :
X(38763) = 6*X(2)-X(37726) = 2*X(3)+3*X(119) = X(3)-6*X(3035) = 7*X(3)+3*X(10742) = X(3)+4*X(20400) = X(3)+9*X(38752) = 13*X(3)-3*X(38753) = 19*X(3)-9*X(38754) = 11*X(3)+9*X(38755) = 17*X(3)+3*X(38756) = 3*X(3)+2*X(38757) = 7*X(3)+18*X(38758) = 11*X(3)-6*X(38759) = 4*X(3)-9*X(38760) = 8*X(3)-3*X(38761) = X(3)-3*X(38762) = X(119)+4*X(3035) = 7*X(119)-2*X(10742) = 3*X(119)-8*X(20400) = X(119)-6*X(38752) = 13*X(119)+2*X(38753) = 19*X(119)+6*X(38754) = 11*X(119)-6*X(38755) = 17*X(119)-2*X(38756) = 9*X(119)-4*X(38757) = 7*X(119)-12*X(38758) = 11*X(119)+4*X(38759) = 2*X(119)+3*X(38760) = 4*X(119)+X(38761) = X(119)+2*X(38762)

X(38763) lies on these lines: {2,37726}, {3,119}, {5,6174}, {10,32905}, {11,3628}, {100,3090}, {104,10303}, {140,37725}, {149,38319}, {528,1656}, {546,33814}, {632,952}, {936,6265}, {997,38129}, {1145,24680}, {1317,18395}, {2771,38729}, {2783,38740}, {2787,38751}, {2800,38787}, {2805,38807}, {3091,5840}, {3146,34474}, {3523,10711}, {3525,6713}, {3526,20418}, {3544,13199}, {3601,12019}, {3627,24466}, {3634,24299}, {3887,38775}, {5067,10707}, {5079,10738}, {5326,37525}, {5563,10956}, {5660,31423}, {5901,13996}, {6419,13922}, {6420,13991}, {6667,12331}, {6946,27529}, {6978,11499}, {7982,11729}, {8674,38795}, {9956,10609}, {11698,14869}, {12103,22799}, {12690,38182}, {12738,18443}, {12751,30389}, {12811,22938}, {15178,24982}, {22935,34122}

X(38763) = reflection of X(38762) in X(3035)
X(38763) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 20400, 119), (3, 38752, 20400), (5, 6174, 10993), (119, 3035, 38760), (119, 38760, 38761), (3035, 20400, 3), (3035, 38752, 119), (3526, 20418, 38069), (10742, 38758, 119)

X(38764) = X(101)-CIRCUM-EULER-POINT OF X(2)

Barycentrics    a^8-(b+c)*a^7-(3*b^2-b*c+3*c^2)*a^6+3*(b+c)*(b^2+c^2)*a^5+(2*b^4+2*c^4-b*c*(3*b^2+4*b*c+3*c^2))*a^4-(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)*a^3-(b^4+c^4-b*c*(b+c)^2)*(b-c)^2*a^2-(b^2-c^2)^3*(b-c)*a+(b^2-c^2)^2*(b-c)*(b^3-c^3) : :
X(38764) = X(3)+2*X(118) = X(3)-4*X(6710) = 2*X(3)+X(10741) = X(3)+8*X(20401) = 4*X(3)-X(38765) = 5*X(3)+X(38768) = 5*X(3)+4*X(38769) = X(3)+4*X(38770) = 7*X(3)-4*X(38771) = 5*X(3)-2*X(38773) = 2*X(3)-5*X(38774) = X(3)-10*X(38775) = X(118)+2*X(6710) = 4*X(118)-X(10741) = X(118)-4*X(20401) = 8*X(118)+X(38765) = 4*X(118)+X(38766) = 10*X(118)-X(38768) = 5*X(118)-2*X(38769) = 7*X(118)+2*X(38771) = 5*X(118)+X(38773) = 4*X(118)+5*X(38774) = X(118)+5*X(38775)

Let Q be the cyclic quadrilateral ABCX(101). Taking the vertices 3 at a time yields four triangles whose nine-point centers are the vertices of a cyclic quadrilateral homothetic to Q at X(38764). (Randy Hutson, May 31, 2020)

X(38764) lies on these lines: {2,2808}, {3,118}, {5,101}, {103,140}, {116,1656}, {150,3090}, {152,631}, {355,11712}, {498,3022}, {499,1362}, {544,5055}, {546,10725}, {547,10708}, {549,10710}, {550,10727}, {946,28346}, {1282,8227}, {1482,11728}, {2772,15061}, {2774,14643}, {2784,10175}, {2786,15561}, {2801,38030}, {2807,38776}, {2809,5886}, {2810,14561}, {2813,38796}, {3041,26363}, {3046,32046}, {3526,6712}, {3542,5185}, {3628,31273}, {3851,33520}, {3887,38752}, {5056,20096}, {5690,10697}, {5805,28345}, {5901,10695}, {10756,18583}, {10772,33814}, {11028,11374}, {11375,18413}, {15720,33521}, {15735,38042}

X(38764) = midpoint of X(i) and X(j) for these {i,j}: {3, 38767}, {118, 38772}, {6710, 38770}, {10741, 38766}
X(38764) = reflection of X(i) in X(j) for these (i,j): (3, 38772), (118, 38770), (10741, 38767), (38765, 38766), (38766, 3), (38767, 118), (38770, 20401), (38772, 6710)
X(38764) = X(10741)-Gibert-Moses centroid
X(38764) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 118, 10741), (3, 6710, 38774), (3, 10741, 38765), (3, 38768, 38773), (5, 101, 10739), (118, 6710, 3), (118, 38773, 38769), (118, 38774, 38765), (118, 38775, 6710), (6710, 20401, 118), (10741, 38774, 3), (20401, 38775, 3), (38767, 38772, 38766), (38768, 38769, 10741), (38769, 38773, 38768), (38770, 38772, 38767)

X(38765) = X(101)-CIRCUM-EULER-POINT OF X(20)

Barycentrics    3*a^8-3*(b+c)*a^7-(b^2-3*b*c+c^2)*a^6-(b+c)*(3*b^2-8*b*c+3*c^2)*a^5+(2*b-c)*(b-2*c)*(b^2+c^2)*a^4+(b^2-c^2)*(b-c)*(5*b^2+4*b*c+5*c^2)*a^3-(3*b^4+3*c^4+5*b*c*(b+c)^2)*(b-c)^2*a^2+(b^2-c^2)^3*(b-c)*a-(b^2-c^2)^2*(b-c)*(b^3-c^3) : :
X(38765) = 3*X(3)-2*X(118) = 5*X(3)-4*X(6710) = 11*X(3)-8*X(20401) = 4*X(3)-3*X(38764) = 2*X(3)-3*X(38766) = 5*X(3)-3*X(38767) = 3*X(3)-X(38768) = 7*X(3)-4*X(38769) = 17*X(3)-12*X(38770) = 3*X(3)-4*X(38771) = 7*X(3)-6*X(38772) = 6*X(3)-5*X(38774) = 13*X(3)-10*X(38775) = 5*X(118)-6*X(6710) = 4*X(118)-3*X(10741) = 11*X(118)-12*X(20401) = 8*X(118)-9*X(38764) = 4*X(118)-9*X(38766) = 10*X(118)-9*X(38767) = 7*X(118)-6*X(38769) = 17*X(118)-18*X(38770) = 7*X(118)-9*X(38772) = X(118)-3*X(38773) = 4*X(118)-5*X(38774) = 13*X(118)-15*X(38775)

X(38765) lies on these lines: {3,118}, {5,10727}, {20,2808}, {30,103}, {101,550}, {116,382}, {150,3529}, {152,376}, {381,6712}, {544,15681}, {546,31273}, {1362,4302}, {1657,33521}, {2772,12121}, {2774,20127}, {2784,38730}, {2786,38741}, {2807,38777}, {2813,38797}, {2822,23240}, {3022,4299}, {3887,38753}, {8703,10710}, {10695,28174}, {10697,34773}, {11714,12699}

X(38765) = midpoint of X(150) and X(3529)
X(38765) = reflection of X(i) in X(j) for these (i,j): (3, 38773), (101, 550), (118, 38771), (382, 116), (10697, 34773), (10710, 8703), (10727, 5), (10739, 103), (10741, 3), (12699, 11714), (38764, 38766), (38768, 118)
X(38765) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 118, 38774), (3, 10741, 38764), (3, 38767, 6710), (3, 38768, 118), (3, 38773, 38766), (118, 38768, 10741), (118, 38771, 3), (118, 38773, 38771), (118, 38774, 38764), (10741, 38766, 3), (10741, 38774, 118), (38768, 38771, 38774)

X(38766) = X(101)-CIRCUM-EULER-POINT OF X(376)

Barycentrics    5*a^8-5*(b+c)*a^7-(3*b^2-5*b*c+3*c^2)*a^6-3*(b+c)*(b^2-4*b*c+c^2)*a^5+(4*b^4+4*c^4-b*c*(9*b^2-4*b*c+9*c^2))*a^4+(b^2-c^2)*(b-c)*(7*b^2+4*b*c+7*c^2)*a^3-(5*b^4+5*c^4+7*b*c*(b+c)^2)*(b-c)^2*a^2+(b^2-c^2)^3*(b-c)*a-(b^2-c^2)^2*(b-c)*(b^3-c^3) : :
X(38766) = 5*X(3)-2*X(118) = 7*X(3)-4*X(6710) = 4*X(3)-X(10741) = 17*X(3)-8*X(20401) = 2*X(3)+X(38765) = 3*X(3)-X(38767) = 7*X(3)-X(38768) = 13*X(3)-4*X(38769) = 9*X(3)-4*X(38770) = X(3)-4*X(38771) = 3*X(3)-2*X(38772) = X(3)+2*X(38773) = 8*X(3)-5*X(38774) = 19*X(3)-10*X(38775) = 7*X(118)-10*X(6710) = 8*X(118)-5*X(10741) = 17*X(118)-20*X(20401) = 4*X(118)-5*X(38764) = 4*X(118)+5*X(38765) = 6*X(118)-5*X(38767) = 14*X(118)-5*X(38768) = 13*X(118)-10*X(38769) = 9*X(118)-10*X(38770) = X(118)-10*X(38771) = 3*X(118)-5*X(38772) = X(118)+5*X(38773)

X(38766) lies on these lines: {3,118}, {20,10739}, {101,548}, {103,550}, {116,1657}, {140,10727}, {150,17538}, {152,3528}, {376,2808}, {382,6712}, {544,15689}, {2772,38723}, {2774,38788}, {2784,38731}, {2786,38742}, {2807,38778}, {2813,38798}, {3627,31273}, {3887,38754}, {10708,15686}, {10710,34200}, {10725,15704}

X(38766) = midpoint of X(38764) and X(38765)
X(38766) = reflection of X(i) in X(j) for these (i,j): (10741, 38764), (38764, 3), (38767, 38772)
X(38766) = X(38765)-Gibert-Moses centroid
X(38766) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10741, 38774), (3, 38765, 10741), (3, 38767, 38772), (3, 38768, 6710), (3, 38773, 38765), (38767, 38772, 38764), (38771, 38773, 3)

X(38767) = X(101)-CIRCUM-EULER-POINT OF X(381)

Barycentrics    a^8-(b+c)*a^7+(3*b^2+b*c+3*c^2)*a^6-6*(b^3+c^3)*a^5-(b^4-8*b^2*c^2+c^4)*a^4+(b^2-c^2)*(b-c)*(5*b^2+8*b*c+5*c^2)*a^3-(b^4+c^4+5*b*c*(b+c)^2)*(b-c)^2*a^2+2*(b^2-c^2)^3*(b-c)*a+(b^2-c^2)^2*(b-c)*(-2*b^3+2*c^3) : :
X(38767) = X(3)-4*X(118) = 5*X(3)-8*X(6710) = X(3)+2*X(10741) = 7*X(3)-16*X(20401) = 5*X(3)-2*X(38765) = 3*X(3)-2*X(38766) = 2*X(3)+X(38768) = X(3)+8*X(38769) = 3*X(3)-8*X(38770) = 11*X(3)-8*X(38771) = 3*X(3)-4*X(38772) = 7*X(3)-4*X(38773) = 7*X(3)-10*X(38774) = 11*X(3)-20*X(38775) = 5*X(118)-2*X(6710) = 2*X(118)+X(10741) = 7*X(118)-4*X(20401) = 10*X(118)-X(38765) = 6*X(118)-X(38766) = 8*X(118)+X(38768) = X(118)+2*X(38769) = 3*X(118)-2*X(38770) = 11*X(118)-2*X(38771) = 3*X(118)-X(38772) = 7*X(118)-X(38773) = 14*X(118)-5*X(38774) = 11*X(118)-5*X(38775)

X(38767) lies on these lines: {3,118}, {4,20096}, {5,152}, {101,382}, {103,1656}, {116,3851}, {150,546}, {381,2808}, {544,14269}, {1282,22793}, {1362,9669}, {1657,10727}, {2772,38724}, {2774,38789}, {2784,38732}, {2786,38743}, {2807,38779}, {2813,38799}, {3022,9654}, {3843,10739}, {3887,38755}, {5070,6712}, {5076,10725}, {5079,31273}, {10697,12645}, {10758,11898}, {10772,12331}, {11728,37624}

X(38767) = midpoint of X(10741) and X(38764)
X(38767) = reflection of X(i) in X(j) for these (i,j): (3, 38764), (38764, 118), (38766, 38772), (38772, 38770)
X(38767) = X(38768)-Gibert-Moses centroid
X(38767) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10741, 38768), (118, 10741, 3), (118, 38769, 10741), (118, 38772, 38770), (118, 38773, 20401), (6710, 38765, 3), (20401, 38773, 38774), (38764, 38766, 38772), (38766, 38772, 3), (38770, 38772, 38764), (38773, 38774, 3)

X(38768) = X(101)-CIRCUM-EULER-POINT OF X(382)

Barycentrics    3*a^8-3*(b+c)*a^7+(b^2+3*b*c+c^2)*a^6-2*(b+c)*(3*b^2-5*b*c+3*c^2)*a^5+(b^4+c^4-4*b*c*(b-c)^2)*a^4+(b^2-c^2)*(b-c)*(7*b^2+8*b*c+7*c^2)*a^3-(3*b^4+3*c^4+7*b*c*(b+c)^2)*(b-c)^2*a^2+2*(b^2-c^2)^3*(b-c)*a+(b^2-c^2)^2*(b-c)*(-2*b^3+2*c^3) : :
X(38768) = 3*X(3)-4*X(118) = 7*X(3)-8*X(6710) = 13*X(3)-16*X(20401) = 5*X(3)-6*X(38764) = 3*X(3)-2*X(38765) = 7*X(3)-6*X(38766) = 2*X(3)-3*X(38767) = 5*X(3)-8*X(38769) = 9*X(3)-8*X(38771) = 11*X(3)-12*X(38772) = 5*X(3)-4*X(38773) = 9*X(3)-10*X(38774) = 17*X(3)-20*X(38775) = 7*X(118)-6*X(6710) = 2*X(118)-3*X(10741) = 13*X(118)-12*X(20401) = 10*X(118)-9*X(38764) = 14*X(118)-9*X(38766) = 8*X(118)-9*X(38767) = 5*X(118)-6*X(38769) = 19*X(118)-18*X(38770) = 3*X(118)-2*X(38771) = 11*X(118)-9*X(38772) = 5*X(118)-3*X(38773) = 6*X(118)-5*X(38774) = 17*X(118)-15*X(38775)

X(38768) lies on these lines: {3,118}, {30,152}, {101,1657}, {103,381}, {116,3843}, {150,3627}, {382,2808}, {544,15684}, {1282,28146}, {1362,9668}, {2772,12902}, {2774,38790}, {2784,38733}, {2786,38744}, {2807,18499}, {2813,38800}, {3022,9655}, {3534,10710}, {3830,10739}, {3851,33521}, {3887,38756}, {5055,6712}, {5072,31273}, {10697,18526}, {10708,38335}, {10772,12773}, {11028,18541}, {11714,18493}, {20096,33703}

X(38768) = midpoint of X(20096) and X(33703)
X(38768) = reflection of X(i) in X(j) for these (i,j): (3, 10741), (150, 3627), (382, 10727), (1657, 101), (3534, 10710), (12773, 10772), (18526, 10697), (38765, 118), (38773, 38769)
X(38768) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10741, 38767), (118, 38765, 3), (118, 38771, 38774), (6710, 38766, 3), (10741, 38764, 38769), (10741, 38765, 118), (38764, 38773, 3), (38765, 38774, 38771), (38769, 38773, 38764), (38771, 38774, 3)

X(38769) = X(101)-CIRCUM-EULER-POINT OF X(546)

Barycentrics    2*a^8-2*(b+c)*a^7+2*(2*b^2+b*c+2*c^2)*a^6-(b+c)*(9*b^2-10*b*c+9*c^2)*a^5-(b^4+c^4+b*c*(b^2-12*b*c+c^2))*a^4+4*(b^2-c^2)*(b-c)*(2*b^2+3*b*c+2*c^2)*a^3-2*(b^4+c^4+4*b*c*(b+c)^2)*(b-c)^2*a^2+3*(b^2-c^2)^3*(b-c)*a+(b^2-c^2)^2*(b-c)*(-3*b^3+3*c^3) : :
X(38769) = X(3)-3*X(118) = 2*X(3)-3*X(6710) = X(3)+3*X(10741) = 5*X(3)-9*X(38764) = 7*X(3)-3*X(38765) = 13*X(3)-9*X(38766) = X(3)-9*X(38767) = 5*X(3)+3*X(38768) = 4*X(3)-9*X(38770) = 4*X(3)-3*X(38771) = 7*X(3)-9*X(38772) = 5*X(3)-3*X(38773) = 11*X(3)-15*X(38774) = 3*X(3)-5*X(38775) = 3*X(118)-2*X(20401) = 5*X(118)-3*X(38764) = 7*X(118)-X(38765) = 13*X(118)-3*X(38766) = X(118)-3*X(38767) = 5*X(118)+X(38768) = 4*X(118)-3*X(38770) = 4*X(118)-X(38771) = 7*X(118)-3*X(38772) = 5*X(118)-X(38773) = 11*X(118)-5*X(38774) = 9*X(118)-5*X(38775)

X(38769) lies on these lines: {2,33521}, {3,118}, {4,544}, {101,3146}, {103,3090}, {116,152}, {382,33520}, {546,2808}, {2772,36253}, {2774,38791}, {2784,38734}, {2786,38745}, {2801,18483}, {2807,38781}, {2813,38801}, {3529,10727}, {3628,6712}, {3832,10708}, {3887,38757}, {4845,5229}, {11728,15178}, {15022,31273}

X(38769) = midpoint of X(i) and X(j) for these {i,j}: {116, 152}, {118, 10741}, {382, 33520}, {38768, 38773}
X(38769) = reflection of X(i) in X(j) for these (i,j): (3, 20401), (6710, 118), (38771, 6710)
X(38769) = complement of X(33521)
X(38769) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 118, 20401), (3, 20401, 6710), (118, 6710, 38770), (118, 38773, 38764), (10741, 38764, 38768), (10741, 38767, 118), (38764, 38768, 38773), (38770, 38771, 6710)

X(38770) = X(101)-CIRCUM-EULER-POINT OF X(547)

Barycentrics    2*a^8-2*(b+c)*a^7-2*(6*b^2-b*c+6*c^2)*a^6+3*(b+c)*(5*b^2-2*b*c+5*c^2)*a^5+(7*b^4+7*c^4-b*c*(9*b^2+20*b*c+9*c^2))*a^4-4*(b^2-c^2)*(b-c)*(2*b+c)*(b+2*c)*a^3-2*(b^4+c^4-4*b*c*(b+c)^2)*(b-c)^2*a^2-5*(b^2-c^2)^3*(b-c)*a+(b^2-c^2)^2*(b-c)*(5*b^3-5*c^3) : :
X(38770) = X(3)+5*X(118) = 2*X(3)-5*X(6710) = 7*X(3)+5*X(10741) = X(3)-10*X(20401) = X(3)-5*X(38764) = 17*X(3)-5*X(38765) = 9*X(3)-5*X(38766) = 3*X(3)+5*X(38767) = 19*X(3)+5*X(38768) = 4*X(3)+5*X(38769) = 8*X(3)-5*X(38771) = 3*X(3)-5*X(38772) = 11*X(3)-5*X(38773) = 2*X(118)+X(6710) = 7*X(118)-X(10741) = X(118)+2*X(20401) = 17*X(118)+X(38765) = 9*X(118)+X(38766) = 3*X(118)-X(38767) = 19*X(118)-X(38768) = 4*X(118)-X(38769) = 8*X(118)+X(38771) = 3*X(118)+X(38772) = 11*X(118)+X(38773) = 13*X(118)+5*X(38774) = 7*X(118)+5*X(38775)

X(38770) lies on these lines: {3,118}, {4,35024}, {101,3832}, {103,3533}, {116,5056}, {544,3545}, {547,2808}, {2772,38725}, {2774,38792}, {2784,38735}, {2786,38746}, {2807,38782}, {2813,38802}, {3887,38758}, {6712,16239}, {10710,15702}, {11728,33179}

X(38770) = midpoint of X(i) and X(j) for these {i,j}: {118, 38764}, {38767, 38772}
X(38770) = reflection of X(i) in X(j) for these (i,j): (6710, 38764), (38764, 20401)
X(38770) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (118, 6710, 38769), (118, 20401, 6710), (118, 38772, 38767), (118, 38775, 10741), (6710, 38769, 38771), (38764, 38767, 38772)

X(38771) = X(101)-CIRCUM-EULER-POINT OF X(548)

Barycentrics    6*a^8-6*(b+c)*a^7-2*(2*b^2-3*b*c+2*c^2)*a^6-(b+c)*(3*b^2-14*b*c+3*c^2)*a^5+(5*b^4+5*c^4-b*c*(11*b^2-4*b*c+11*c^2))*a^4+4*(b^2-c^2)*(b-c)*(2*b^2+b*c+2*c^2)*a^3-2*(3*b^4+3*c^4+4*b*c*(b+c)^2)*(b-c)^2*a^2+(b^2-c^2)^3*(b-c)*a-(b^2-c^2)^2*(b-c)*(b^3-c^3) : :
X(38771) = 3*X(3)-X(118) = 5*X(3)-X(10741) = 5*X(3)-2*X(20401) = 7*X(3)-3*X(38764) = 3*X(3)+X(38765) = X(3)+3*X(38766) = 11*X(3)-3*X(38767) = 9*X(3)-X(38768) = 4*X(3)-X(38769) = 8*X(3)-3*X(38770) = 5*X(3)-3*X(38772) = 9*X(3)-5*X(38774) = 11*X(3)-5*X(38775) = 2*X(118)-3*X(6710) = 5*X(118)-3*X(10741) = 5*X(118)-6*X(20401) = 7*X(118)-9*X(38764) = X(118)+9*X(38766) = 11*X(118)-9*X(38767) = 3*X(118)-X(38768) = 4*X(118)-3*X(38769) = 8*X(118)-9*X(38770) = 5*X(118)-9*X(38772) = X(118)+3*X(38773) = 3*X(118)-5*X(38774) = 11*X(118)-15*X(38775)

X(38771) lies on these lines: {3,118}, {20,116}, {30,6712}, {101,3522}, {103,376}, {152,10304}, {516,11726}, {548,2808}, {631,10727}, {2772,38726}, {2774,37853}, {2784,38736}, {2786,38747}, {2807,38783}, {2809,12512}, {2813,38803}, {3146,31273}, {3528,35024}, {3534,10739}, {3887,38759}, {9778,10695}, {10710,19708}, {11714,31730}, {11728,13624}

X(38771) = midpoint of X(i) and X(j) for these {i,j}: {3, 38773}, {20, 116}, {101, 33521}, {118, 38765}, {11714, 31730}
X(38771) = reflection of X(i) in X(j) for these (i,j): (6710, 3), (10741, 20401), (11728, 13624), (38769, 6710)
X(38771) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10741, 38772), (3, 38765, 118), (3, 38766, 38773), (3, 38768, 38774), (118, 38773, 38765), (6710, 38769, 38770), (10741, 38772, 20401), (20401, 38772, 6710), (38765, 38774, 38768), (38768, 38774, 118)

X(38772) = X(101)-CIRCUM-EULER-POINT OF X(549)

Barycentrics    4*a^8-4*(b+c)*a^7-2*(3*b^2-2*b*c+3*c^2)*a^6+3*(b+c)^3*a^5+(b^2+b*c+c^2)*(5*b^2-14*b*c+5*c^2)*a^4+2*(b^2-c^2)*(b-c)^3*a^3-2*(2*b^4+2*c^4+b*c*(b+c)^2)*(b-c)^2*a^2-(b^2-c^2)^3*(b-c)*a+(b^2-c^2)^2*(b-c)*(b^3-c^3) : :
X(38772) = 2*X(3)+X(118) = X(3)+2*X(6710) = 5*X(3)+X(10741) = 5*X(3)+4*X(20401) = 7*X(3)-X(38765) = 3*X(3)-X(38766) = 3*X(3)+X(38767) = 11*X(3)+X(38768) = 7*X(3)+2*X(38769) = 3*X(3)+2*X(38770) = 5*X(3)-2*X(38771) = 4*X(3)-X(38773) = X(3)+5*X(38774) = 4*X(3)+5*X(38775) = X(118)-4*X(6710) = 5*X(118)-2*X(10741) = 5*X(118)-8*X(20401) = 7*X(118)+2*X(38765) = 3*X(118)+2*X(38766) = 3*X(118)-2*X(38767) = 11*X(118)-2*X(38768) = 7*X(118)-4*X(38769) = 3*X(118)-4*X(38770) = 5*X(118)+4*X(38771) = 2*X(118)+X(38773) = X(118)-10*X(38774) = 2*X(118)-5*X(38775)

X(38772) lies on these lines: {3,118}, {40,11728}, {101,631}, {103,3523}, {116,140}, {150,10303}, {152,15717}, {544,5054}, {549,2808}, {2772,38727}, {2774,38793}, {2784,38737}, {2786,38748}, {2801,21154}, {2807,38784}, {2809,10165}, {2813,38804}, {3090,10725}, {3522,10727}, {3525,31273}, {3526,10739}, {3887,38760}, {6684,11712}, {10708,15702}, {10710,15692}, {11714,28346}, {15712,33521}, {15720,35024}, {17044,31852}

X(38772) = midpoint of X(i) and X(j) for these {i,j}: {3, 38764}, {38766, 38767}
X(38772) = reflection of X(i) in X(j) for these (i,j): (118, 38764), (38764, 6710), (38767, 38770)
X(38772) = X(118)-Gibert-Moses centroid
X(38772) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 118, 38773), (3, 6710, 118), (3, 10741, 38771), (3, 38767, 38766), (3, 38774, 6710), (101, 631, 6712), (118, 6710, 38775), (6710, 38771, 20401), (10741, 20401, 118), (20401, 38771, 10741), (38764, 38766, 38767), (38764, 38767, 38770), (38767, 38770, 118), (38773, 38775, 118)

X(38773) = X(101)-CIRCUM-EULER-POINT OF X(550)

Barycentrics    4*a^8-4*(b+c)*a^7-2*(b-c)^2*a^6-(b-3*c)*(3*b-c)*(b+c)*a^5+(3*b^4+3*c^4-b*c*(7*b^2-4*b*c+7*c^2))*a^4+2*(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^3-2*(2*b^2+b*c+c^2)*(b^2+b*c+2*c^2)*(b-c)^2*a^2+(b^2-c^2)^3*(b-c)*a-(b^2-c^2)^2*(b-c)*(b^3-c^3) : :
X(38773) = 3*X(3)-2*X(6710) = 3*X(3)-X(10741) = 7*X(3)-4*X(20401) = 5*X(3)-3*X(38764) = X(3)-3*X(38766) = 7*X(3)-3*X(38767) = 5*X(3)-X(38768) = 5*X(3)-2*X(38769) = 11*X(3)-6*X(38770) = 4*X(3)-3*X(38772) = 7*X(3)-5*X(38774) = 8*X(3)-5*X(38775) = 3*X(118)-4*X(6710) = 3*X(118)-2*X(10741) = 7*X(118)-8*X(20401) = 5*X(118)-6*X(38764) = X(118)+2*X(38765) = X(118)-6*X(38766) = 7*X(118)-6*X(38767) = 5*X(118)-2*X(38768) = 5*X(118)-4*X(38769) = 11*X(118)-12*X(38770) = X(118)-4*X(38771) = 2*X(118)-3*X(38772) = 7*X(118)-10*X(38774) = 4*X(118)-5*X(38775)

X(38773) lies on these lines: {2,10727}, {3,118}, {4,6712}, {20,103}, {30,116}, {101,376}, {152,3522}, {516,1565}, {544,3534}, {550,2808}, {1362,15338}, {1657,10739}, {2772,16163}, {2774,16111}, {2784,38738}, {2786,38749}, {2801,24466}, {2807,38785}, {2809,31730}, {2813,38805}, {2822,3184}, {2825,14689}, {3022,15326}, {3529,10725}, {3576,11728}, {3887,38761}, {5731,10697}, {6361,10695}, {10304,10710}, {10708,11001}, {10758,25406}, {11726,12699}, {15688,35024}

X(38773) = midpoint of X(i) and X(j) for these {i,j}: {3, 38765}, {20, 103}, {1657, 10739}, {3529, 10725}, {6361, 10695}, {10708, 11001}
X(38773) = reflection of X(i) in X(j) for these (i,j): (3, 38771), (4, 6712), (118, 3), (10741, 6710), (12699, 11726), (38768, 38769)
X(38773) = complement of X(10727)
X(38773) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 118, 38772), (3, 10741, 6710), (3, 38766, 38771), (3, 38767, 38774), (3, 38768, 38764), (118, 38772, 38775), (550, 33521, 33520), (6710, 10741, 118), (20401, 38767, 118), (38764, 38768, 38769), (38764, 38769, 118), (38765, 38766, 3), (38765, 38771, 118), (38767, 38774, 20401)

X(38774) = X(101)-CIRCUM-EULER-POINT OF X(631)

Barycentrics    3*a^8-3*(b+c)*a^7-(5*b^2-3*b*c+5*c^2)*a^6+(b+c)*(3*b^2+4*b*c+3*c^2)*a^5+(4*b^4+4*c^4-b*c*(7*b^2+4*b*c+7*c^2))*a^4+(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a^3-(3*b^2+4*b*c+3*c^2)*(b^2-b*c+c^2)*(b-c)^2*a^2-(b^2-c^2)^3*(b-c)*a+(b^2-c^2)^2*(b-c)*(b^3-c^3) : :
X(38774) = 6*X(2)-X(10739) = 3*X(3)+2*X(118) = X(3)+4*X(6710) = 4*X(3)+X(10741) = 7*X(3)+8*X(20401) = 2*X(3)+3*X(38764) = 6*X(3)-X(38765) = 8*X(3)-3*X(38766) = 7*X(3)+3*X(38767) = 9*X(3)+X(38768) = 11*X(3)+4*X(38769) = 13*X(3)+12*X(38770) = 9*X(3)-4*X(38771) = X(3)-6*X(38772) = 7*X(3)-2*X(38773) = X(3)+2*X(38775) = X(118)-6*X(6710) = 8*X(118)-3*X(10741) = 7*X(118)-12*X(20401) = 4*X(118)-9*X(38764) = 4*X(118)+X(38765) = 16*X(118)+9*X(38766) = 14*X(118)-9*X(38767) = 6*X(118)-X(38768) = 11*X(118)-6*X(38769) = 13*X(118)-18*X(38770) = 3*X(118)+2*X(38771) = X(118)+9*X(38772) = 7*X(118)+3*X(38773) = X(118)-3*X(38775)

X(38774) lies on these lines: {2,10739}, {3,118}, {5,10725}, {101,140}, {103,549}, {116,3526}, {150,3525}, {152,3524}, {544,15694}, {548,10727}, {631,2808}, {632,31273}, {2772,38728}, {2774,38794}, {2784,38739}, {2786,38750}, {2807,38786}, {2813,38806}, {3147,5185}, {3887,38762}, {5054,6712}, {10165,28346}, {10695,38028}, {10708,11539}, {10710,12100}, {10756,38110}, {10770,34126}, {11712,26446}, {11728,12702}, {28345,38122}

X(38774) = reflection of X(i) in X(j) for these (i,j): (31273, 632), (38775, 6710)
X(38774) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 118, 38765), (3, 6710, 38764), (3, 10741, 38766), (3, 38764, 10741), (3, 38767, 38773), (3, 38768, 38771), (118, 38765, 10741), (118, 38771, 38768), (6710, 38772, 3), (20401, 38773, 38767), (38764, 38765, 118), (38768, 38771, 38765)

X(38775) = X(101)-CIRCUM-EULER-POINT OF X(632)

Barycentrics    4*a^8-4*(b+c)*a^7-2*(5*b^2-2*b*c+5*c^2)*a^6+(b+c)*(9*b^2+2*b*c+9*c^2)*a^5+(7*b^4+7*c^4-b*c*(11*b^2+12*b*c+11*c^2))*a^4-2*(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)*a^3-2*(2*b^2+3*b*c+2*c^2)*(b-c)^4*a^2-3*(b^2-c^2)^3*(b-c)*a+(b^2-c^2)^2*(b-c)*(3*b^3-3*c^3) : :
X(38775) = 2*X(3)+3*X(118) = X(3)-6*X(6710) = 7*X(3)+3*X(10741) = X(3)+4*X(20401) = X(3)+9*X(38764) = 13*X(3)-3*X(38765) = 19*X(3)-9*X(38766) = 11*X(3)+9*X(38767) = 17*X(3)+3*X(38768) = 3*X(3)+2*X(38769) = 7*X(3)+18*X(38770) = 11*X(3)-6*X(38771) = 4*X(3)-9*X(38772) = 8*X(3)-3*X(38773) = X(3)-3*X(38774) = X(118)+4*X(6710) = 7*X(118)-2*X(10741) = 3*X(118)-8*X(20401) = X(118)-6*X(38764) = 13*X(118)+2*X(38765) = 19*X(118)+6*X(38766) = 11*X(118)-6*X(38767) = 17*X(118)-2*X(38768) = 9*X(118)-4*X(38769) = 7*X(118)-12*X(38770) = 11*X(118)+4*X(38771) = 2*X(118)+3*X(38772) = 4*X(118)+X(38773) = X(118)+2*X(38774)

X(38775) lies on these lines: {3,118}, {5,33520}, {101,3090}, {103,10303}, {116,3628}, {544,1656}, {549,33521}, {632,2808}, {2772,38729}, {2774,38795}, {2784,38740}, {2786,38751}, {2801,19862}, {2807,38787}, {2813,38807}, {3523,10710}, {3525,6712}, {3887,38763}, {5067,10708}, {5079,10739}, {7982,11728}, {9780,15735}

X(38775) = reflection of X(38774) in X(6710)
X(38775) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 20401, 118), (3, 38764, 20401), (118, 6710, 38772), (118, 38772, 38773), (6710, 20401, 3), (6710, 38764, 118), (10741, 38770, 118)

X(38776) = X(102)-CIRCUM-EULER-POINT OF X(2)

Barycentrics    a^10-(b+c)*a^9-(2*b^2-3*b*c+2*c^2)*a^8+2*(b+c)*(2*b^2-3*b*c+2*c^2)*a^7-(b^4+c^4+6*b*c*(b-c)^2)*a^6-2*(b^2-c^2)*(b-c)*(3*b^2-2*b*c+3*c^2)*a^5+(5*b^4+5*c^4+6*b*c*(2*b^2+b*c+2*c^2))*(b-c)^2*a^4+2*(b^2-c^2)*(b-c)^3*(2*b^2+b*c+2*c^2)*a^3-2*(b+c)*(b^2-c^2)*(b^3-c^3)*(2*b^2-3*b*c+2*c^2)*a^2-(b^2-c^2)^3*(b-c)^3*a+(b^2-c^2)^3*(b-c)*(b^3+c^3) : :
X(38776) = X(3)+2*X(124) = X(3)-4*X(6711) = 2*X(3)+X(10747) = 4*X(3)-X(38777) = 5*X(3)+X(38780) = 5*X(3)+4*X(38781) = X(3)+4*X(38782) = 7*X(3)-4*X(38783) = 5*X(3)-2*X(38785) = 2*X(3)-5*X(38786) = X(3)-10*X(38787) = X(124)+2*X(6711) = 4*X(124)-X(10747) = 8*X(124)+X(38777) = 4*X(124)+X(38778) = 10*X(124)-X(38780) = 5*X(124)-2*X(38781) = 7*X(124)+2*X(38783) = 5*X(124)+X(38785) = 4*X(124)+5*X(38786) = X(124)+5*X(38787)

Let Q be the cyclic quadrilateral ABCX(102). Taking the vertices 3 at a time yields four triangles whose nine-point centers are the vertices of a cyclic quadrilateral homothetic to Q at X(38776). (Randy Hutson, May 31, 2020)

X(38776) lies on these lines: {2,2818}, {3,124}, {5,102}, {109,140}, {117,1656}, {151,3090}, {355,11713}, {498,1361}, {499,1364}, {546,10726}, {547,10709}, {549,10716}, {550,10732}, {631,33650}, {1385,13532}, {1482,11734}, {1795,5433}, {1845,11375}, {2773,15061}, {2779,14643}, {2785,38224}, {2792,15561}, {2800,10176}, {2807,38764}, {2816,3817}, {2817,5886}, {2819,38796}, {3040,26364}, {3042,26363}, {3526,6718}, {5690,10703}, {5901,10696}, {10757,18583}, {10777,33814}

X(38776) = midpoint of X(i) and X(j) for these {i,j}: {3, 38779}, {124, 38784}, {6711, 38782}, {10747, 38778}
X(38776) = reflection of X(i) in X(j) for these (i,j): (3, 38784), (124, 38782), (10747, 38779), (38777, 38778), (38778, 3), (38779, 124), (38784, 6711)
X(38776) = X(10747)-Gibert-Moses centroid
X(38776) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 124, 10747), (3, 6711, 38786), (3, 10747, 38777), (3, 38780, 38785), (5, 102, 10740), (124, 6711, 3), (124, 38785, 38781), (124, 38786, 38777), (124, 38787, 6711), (10747, 38786, 3), (38779, 38784, 38778), (38780, 38781, 10747), (38781, 38785, 38780), (38782, 38784, 38779)

X(38777) = X(102)-CIRCUM-EULER-POINT OF X(20)

Barycentrics    3*a^10-3*(b+c)*a^9-3*(2*b^2-3*b*c+2*c^2)*a^8+2*(b+c)*(4*b^2-5*b*c+4*c^2)*a^7+(b^4+c^4-2*b*c*(7*b^2-10*b*c+7*c^2))*a^6-2*(b^2-c^2)*(b-c)*(3*b^2-2*b*c+3*c^2)*a^5+(3*b^4+3*c^4+2*b*c*(4*b^2+b*c+4*c^2))*(b-c)^2*a^4-2*(b^2-c^2)*(b-c)^3*b*c*a^3+2*(b^2-c^2)^2*b*c*(b^2-3*b*c+c^2)*a^2+(b^2-c^2)^3*(b-c)^3*a-(b^2-c^2)^3*(b-c)*(b^3+c^3) : :
X(38777) = 3*X(3)-2*X(124) = 5*X(3)-4*X(6711) = 4*X(3)-3*X(38776) = 2*X(3)-3*X(38778) = 5*X(3)-3*X(38779) = 3*X(3)-X(38780) = 7*X(3)-4*X(38781) = 17*X(3)-12*X(38782) = 3*X(3)-4*X(38783) = 7*X(3)-6*X(38784) = 6*X(3)-5*X(38786) = 13*X(3)-10*X(38787) = 5*X(124)-6*X(6711) = 4*X(124)-3*X(10747) = 8*X(124)-9*X(38776) = 4*X(124)-9*X(38778) = 10*X(124)-9*X(38779) = 7*X(124)-6*X(38781) = 17*X(124)-18*X(38782) = 7*X(124)-9*X(38784) = X(124)-3*X(38785) = 4*X(124)-5*X(38786) = 13*X(124)-15*X(38787)

X(38777) lies on these lines: {3,124}, {5,10732}, {20,2818}, {30,109}, {102,550}, {117,382}, {151,3529}, {355,14690}, {376,33650}, {381,6718}, {1361,4299}, {1364,4302}, {1795,6284}, {2773,12121}, {2779,20127}, {2785,38730}, {2792,38741}, {2800,18481}, {2807,38765}, {2819,38797}, {2846,23240}, {3579,13532}, {8703,10716}, {10696,28174}, {10703,34773}, {11700,12699}

X(38777) = midpoint of X(151) and X(3529)
X(38777) = reflection of X(i) in X(j) for these (i,j): (3, 38785), (102, 550), (124, 38783), (355, 14690), (382, 117), (10703, 34773), (10716, 8703), (10732, 5), (10740, 109), (10747, 3), (12699, 11700), (13532, 3579), (38776, 38778), (38780, 124)
X(38777) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 124, 38786), (3, 10747, 38776), (3, 38779, 6711), (3, 38780, 124), (3, 38785, 38778), (124, 38780, 10747), (124, 38783, 3), (124, 38785, 38783), (124, 38786, 38776), (10747, 38778, 3), (10747, 38786, 124), (38780, 38783, 38786)

X(38778) = X(102)-CIRCUM-EULER-POINT OF X(376)

Barycentrics    5*a^10-5*(b+c)*a^9-5*(2*b^2-3*b*c+2*c^2)*a^8+2*(b+c)*(7*b^2-9*b*c+7*c^2)*a^7+(b^4+c^4-12*b*c*(2*b^2-3*b*c+2*c^2))*a^6-4*(b^2-c^2)*(b-c)*(3*b^2-2*b*c+3*c^2)*a^5+(7*b^4+7*c^4+6*b*c*(3*b^2+b*c+3*c^2))*(b-c)^2*a^4+2*(b^3+c^3)*(b-c)^4*a^3-2*(b^2-c^2)^2*(b^4+c^4-b*c*(2*b-c)*(b-2*c))*a^2+(b^2-c^2)^3*(b-c)^3*a-(b^2-c^2)^3*(b-c)*(b^3+c^3) : :
X(38778) = 5*X(3)-2*X(124) = 7*X(3)-4*X(6711) = 4*X(3)-X(10747) = 2*X(3)+X(38777) = 3*X(3)-X(38779) = 7*X(3)-X(38780) = 13*X(3)-4*X(38781) = 9*X(3)-4*X(38782) = X(3)-4*X(38783) = 3*X(3)-2*X(38784) = X(3)+2*X(38785) = 8*X(3)-5*X(38786) = 19*X(3)-10*X(38787) = 7*X(124)-10*X(6711) = 8*X(124)-5*X(10747) = 4*X(124)-5*X(38776) = 4*X(124)+5*X(38777) = 6*X(124)-5*X(38779) = 14*X(124)-5*X(38780) = 13*X(124)-10*X(38781) = 9*X(124)-10*X(38782) = X(124)-10*X(38783) = 3*X(124)-5*X(38784) = X(124)+5*X(38785)

X(38778) lies on these lines: {3,124}, {20,10740}, {102,548}, {109,550}, {117,1657}, {140,10732}, {151,17538}, {376,2818}, {382,6718}, {1795,15338}, {2773,38723}, {2779,38788}, {2785,38731}, {2792,38742}, {2800,38754}, {2807,38766}, {2819,38798}, {3528,33650}, {10709,15686}, {10716,34200}, {10726,15704}, {13532,31663}, {14690,18481}

X(38778) = midpoint of X(38776) and X(38777)
X(38778) = reflection of X(i) in X(j) for these (i,j): (10747, 38776), (38776, 3), (38779, 38784)
X(38778) = X(38777)-Gibert-Moses centroid
X(38778) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10747, 38786), (3, 38777, 10747), (3, 38779, 38784), (3, 38780, 6711), (3, 38785, 38777), (38779, 38784, 38776), (38783, 38785, 3)

X(38779) = X(102)-CIRCUM-EULER-POINT OF X(381)

Barycentrics    a^10-(b+c)*a^9-(2*b^2-3*b*c+2*c^2)*a^8+(b+c)*(b^2+c^2)*a^7+(2*b^4+2*c^4-3*b*c*(b^2+c^2))*a^6+(b^2-c^2)*(b-c)*(3*b^2-2*b*c+3*c^2)*a^5-(4*b^4+4*c^4+3*b*c*(3*b^2+2*b*c+3*c^2))*(b-c)^2*a^4-(b^2-c^2)*(b-c)^3*(5*b^2+4*b*c+5*c^2)*a^3+(b^2-c^2)^2*(5*b^4+5*c^4-b*c*(b+c)^2)*a^2+2*(b^2-c^2)^3*(b-c)^3*a+(b^2-c^2)^3*(b-c)*(-2*b^3-2*c^3) : :
X(38779) = X(3)-4*X(124) = 5*X(3)-8*X(6711) = X(3)+2*X(10747) = 5*X(3)-2*X(38777) = 3*X(3)-2*X(38778) = 2*X(3)+X(38780) = X(3)+8*X(38781) = 3*X(3)-8*X(38782) = 11*X(3)-8*X(38783) = 3*X(3)-4*X(38784) = 7*X(3)-4*X(38785) = 7*X(3)-10*X(38786) = 11*X(3)-20*X(38787) = 5*X(124)-2*X(6711) = 2*X(124)+X(10747) = 10*X(124)-X(38777) = 6*X(124)-X(38778) = 8*X(124)+X(38780) = X(124)+2*X(38781) = 3*X(124)-2*X(38782) = 11*X(124)-2*X(38783) = 3*X(124)-X(38784) = 7*X(124)-X(38785) = 14*X(124)-5*X(38786) = 11*X(124)-5*X(38787)

X(38779) lies on these lines: {3,124}, {5,33650}, {102,382}, {109,1656}, {117,3851}, {151,546}, {381,2818}, {1361,9654}, {1364,9669}, {1482,13532}, {1657,10732}, {2773,38724}, {2779,38789}, {2785,38732}, {2792,38743}, {2800,5790}, {2807,38767}, {2819,38799}, {3843,10740}, {5070,6718}, {5076,10726}, {10703,12645}, {10764,11898}, {10777,12331}, {11734,37624}

X(38779) = midpoint of X(10747) and X(38776)
X(38779) = reflection of X(i) in X(j) for these (i,j): (3, 38776), (38776, 124), (38778, 38784), (38784, 38782)
X(38779) = X(38780)-Gibert-Moses centroid
X(38779) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10747, 38780), (124, 10747, 3), (124, 38781, 10747), (124, 38784, 38782), (6711, 38777, 3), (38776, 38778, 38784), (38778, 38784, 3), (38782, 38784, 38776), (38785, 38786, 3)

X(38780) = X(102)-CIRCUM-EULER-POINT OF X(382)

Barycentrics    3*a^10-3*(b+c)*a^9-3*(2*b^2-3*b*c+2*c^2)*a^8+(b+c)*(7*b^2-8*b*c+7*c^2)*a^7+(2*b^4+2*c^4-b*c*(13*b^2-16*b*c+13*c^2))*a^6-(b^2-c^2)*(b-c)*(3*b^2-2*b*c+3*c^2)*a^5+b*c*(b-c)^4*a^4-(b^2-c^2)*(b-c)^3*(3*b^2+4*b*c+3*c^2)*a^3+(b^2-c^2)^2*(3*b^4+3*c^4+b*c*(b^2-6*b*c+c^2))*a^2+2*(b^2-c^2)^3*(b-c)^3*a+(b^2-c^2)^3*(b-c)*(-2*b^3-2*c^3) : :
X(38780) = 3*X(3)-4*X(124) = 7*X(3)-8*X(6711) = 5*X(3)-6*X(38776) = 3*X(3)-2*X(38777) = 7*X(3)-6*X(38778) = 2*X(3)-3*X(38779) = 5*X(3)-8*X(38781) = 9*X(3)-8*X(38783) = 11*X(3)-12*X(38784) = 5*X(3)-4*X(38785) = 9*X(3)-10*X(38786) = 17*X(3)-20*X(38787) = 7*X(124)-6*X(6711) = 2*X(124)-3*X(10747) = 10*X(124)-9*X(38776) = 14*X(124)-9*X(38778) = 8*X(124)-9*X(38779) = 5*X(124)-6*X(38781) = 19*X(124)-18*X(38782) = 3*X(124)-2*X(38783) = 11*X(124)-9*X(38784) = 5*X(124)-3*X(38785) = 6*X(124)-5*X(38786) = 17*X(124)-15*X(38787)

X(38780) lies on these lines: {3,124}, {30,33650}, {102,1657}, {109,381}, {117,3843}, {151,3627}, {382,2818}, {1361,9655}, {1364,9668}, {1795,9669}, {2773,12902}, {2779,38790}, {2785,38733}, {2792,38744}, {2800,18525}, {2807,18499}, {2819,38800}, {3534,10716}, {3830,10740}, {5055,6718}, {10703,18526}, {10709,38335}, {10777,12773}, {11700,18493}, {12702,13532}

X(38780) = reflection of X(i) in X(j) for these (i,j): (3, 10747), (151, 3627), (382, 10732), (1657, 102), (3534, 10716), (12702, 13532), (12773, 10777), (18526, 10703), (38777, 124), (38785, 38781)
X(38780) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10747, 38779), (124, 38777, 3), (124, 38783, 38786), (6711, 38778, 3), (10747, 38776, 38781), (10747, 38777, 124), (38776, 38785, 3), (38777, 38786, 38783), (38781, 38785, 38776), (38783, 38786, 3)

X(38781) = X(102)-CIRCUM-EULER-POINT OF X(546)

Barycentrics    2*a^10-2*(b+c)*a^9-2*(2*b^2-3*b*c+2*c^2)*a^8+(b+c)*(3*b^2-2*b*c+3*c^2)*a^7+(3*b^4+3*c^4-b*c*(7*b^2-4*b*c+7*c^2))*a^6+(b^2-c^2)*(b-c)*(3*b^2-2*b*c+3*c^2)*a^5-(5*b^4+5*c^4+b*c*(11*b^2+8*b*c+11*c^2))*(b-c)^2*a^4-(b^2-c^2)*(b-c)^3*(7*b^2+6*b*c+7*c^2)*a^3+(b^2-c^2)^2*(7*b^4+7*c^4-b*c*(b^2+4*b*c+c^2))*a^2+3*(b^2-c^2)^3*(b-c)^3*a+(b^2-c^2)^3*(b-c)*(-3*b^3-3*c^3) : :
X(38781) = X(3)-3*X(124) = 2*X(3)-3*X(6711) = X(3)+3*X(10747) = 5*X(3)-9*X(38776) = 7*X(3)-3*X(38777) = 13*X(3)-9*X(38778) = X(3)-9*X(38779) = 5*X(3)+3*X(38780) = 4*X(3)-9*X(38782) = 4*X(3)-3*X(38783) = 7*X(3)-9*X(38784) = 5*X(3)-3*X(38785) = 11*X(3)-15*X(38786) = 3*X(3)-5*X(38787) = 5*X(124)-3*X(38776) = 7*X(124)-X(38777) = 13*X(124)-3*X(38778) = X(124)-3*X(38779) = 5*X(124)+X(38780) = 4*X(124)-3*X(38782) = 4*X(124)-X(38783) = 7*X(124)-3*X(38784) = 5*X(124)-X(38785) = 11*X(124)-5*X(38786) = 9*X(124)-5*X(38787)

X(38781) lies on these lines: {3,124}, {4,10716}, {102,3146}, {109,3090}, {117,3091}, {546,2818}, {2773,36253}, {2779,38791}, {2785,38734}, {2792,38745}, {2800,3036}, {2807,38769}, {2819,38801}, {3529,10732}, {3628,6718}, {3832,10709}, {5908,18483}, {7982,13532}, {11734,15178}

X(38781) = midpoint of X(i) and X(j) for these {i,j}: {117, 33650}, {124, 10747}, {38780, 38785}
X(38781) = reflection of X(i) in X(j) for these (i,j): (6711, 124), (38783, 6711)
X(38781) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (124, 6711, 38782), (124, 38785, 38776), (10747, 38776, 38780), (10747, 38779, 124), (38776, 38780, 38785), (38782, 38783, 6711)

X(38782) = X(102)-CIRCUM-EULER-POINT OF X(547)

Barycentrics    2*a^10-2*(b+c)*a^9-2*(2*b^2-3*b*c+2*c^2)*a^8+(b+c)*(11*b^2-18*b*c+11*c^2)*a^7-(5*b^4+5*c^4+3*b*c*(5*b^2-12*b*c+5*c^2))*a^6-7*(b^2-c^2)*(b-c)*(3*b^2-2*b*c+3*c^2)*a^5+(19*b^4+19*c^4+3*b*c*(15*b^2+8*b*c+15*c^2))*(b-c)^2*a^4+(b^2-c^2)*(b-c)^3*(17*b^2+10*b*c+17*c^2)*a^3-(b^2-c^2)^2*(17*b^4+17*c^4-b*c*(7*b^2-4*b*c+7*c^2))*a^2-5*(b^2-c^2)^3*(b-c)^3*a+(b^2-c^2)^3*(b-c)*(5*b^3+5*c^3) : :
X(38782) = X(3)+5*X(124) = 2*X(3)-5*X(6711) = 7*X(3)+5*X(10747) = X(3)-5*X(38776) = 17*X(3)-5*X(38777) = 9*X(3)-5*X(38778) = 3*X(3)+5*X(38779) = 19*X(3)+5*X(38780) = 4*X(3)+5*X(38781) = 8*X(3)-5*X(38783) = 3*X(3)-5*X(38784) = 11*X(3)-5*X(38785) = 2*X(124)+X(6711) = 7*X(124)-X(10747) = 17*X(124)+X(38777) = 9*X(124)+X(38778) = 3*X(124)-X(38779) = 19*X(124)-X(38780) = 4*X(124)-X(38781) = 8*X(124)+X(38783) = 3*X(124)+X(38784) = 11*X(124)+X(38785) = 13*X(124)+5*X(38786) = 7*X(124)+5*X(38787)

X(38782) lies on these lines: {3,124}, {102,3832}, {109,3533}, {117,5056}, {547,2818}, {2773,38725}, {2779,38792}, {2785,38735}, {2792,38746}, {2800,3740}, {2807,38770}, {2819,38802}, {6718,16239}, {10716,15702}, {11734,33179}

X(38782) = midpoint of X(i) and X(j) for these {i,j}: {124, 38776}, {38779, 38784}
X(38782) = reflection of X(6711) in X(38776)
X(38782) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (124, 6711, 38781), (124, 38784, 38779), (124, 38787, 10747), (6711, 38781, 38783), (38776, 38779, 38784)

X(38783) = X(102)-CIRCUM-EULER-POINT OF X(548)

Barycentrics    6*a^10-6*(b+c)*a^9-6*(2*b^2-3*b*c+2*c^2)*a^8+(b+c)*(17*b^2-22*b*c+17*c^2)*a^7+(b^4+c^4-b*c*(29*b^2-44*b*c+29*c^2))*a^6-5*(b^2-c^2)*(b-c)*(3*b^2-2*b*c+3*c^2)*a^5+(9*b^4+9*c^4+b*c*(23*b^2+8*b*c+23*c^2))*(b-c)^2*a^4+(b^2-c^2)*(b-c)^3*(3*b^2-2*b*c+3*c^2)*a^3-(b^2-c^2)^2*(3*b^4+3*c^4-b*c*(5*b^2-12*b*c+5*c^2))*a^2+(b^2-c^2)^3*(b-c)^3*a-(b^2-c^2)^3*(b-c)*(b^3+c^3) : :
X(38783) = 3*X(3)-X(124) = 5*X(3)-X(10747) = 7*X(3)-3*X(38776) = 3*X(3)+X(38777) = X(3)+3*X(38778) = 11*X(3)-3*X(38779) = 9*X(3)-X(38780) = 4*X(3)-X(38781) = 8*X(3)-3*X(38782) = 5*X(3)-3*X(38784) = 9*X(3)-5*X(38786) = 11*X(3)-5*X(38787) = 2*X(124)-3*X(6711) = 5*X(124)-3*X(10747) = 7*X(124)-9*X(38776) = X(124)+9*X(38778) = 11*X(124)-9*X(38779) = 3*X(124)-X(38780) = 4*X(124)-3*X(38781) = 8*X(124)-9*X(38782) = 5*X(124)-9*X(38784) = X(124)+3*X(38785) = 3*X(124)-5*X(38786) = 11*X(124)-15*X(38787)

X(38783) lies on these lines: {3,124}, {20,117}, {30,6718}, {102,3522}, {109,376}, {516,11727}, {548,2818}, {631,10732}, {2773,38726}, {2779,37853}, {2785,38736}, {2792,38747}, {2800,9943}, {2807,38771}, {2817,12512}, {2819,38803}, {3534,10740}, {4297,14690}, {9778,10696}, {10304,33650}, {10716,19708}, {11700,31730}, {11734,13624}, {13532,35242}

X(38783) = midpoint of X(i) and X(j) for these {i,j}: {3, 38785}, {20, 117}, {124, 38777}, {4297, 14690}, {11700, 31730}
X(38783) = reflection of X(i) in X(j) for these (i,j): (6711, 3), (11734, 13624), (38781, 6711)
X(38783) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10747, 38784), (3, 38777, 124), (3, 38778, 38785), (3, 38780, 38786), (124, 38785, 38777), (6711, 38781, 38782), (38777, 38786, 38780), (38780, 38786, 124)

X(38784) = X(102)-CIRCUM-EULER-POINT OF X(549)

Barycentrics    4*a^10-4*(b+c)*a^9-4*(2*b^2-3*b*c+2*c^2)*a^8+(b+c)*(13*b^2-18*b*c+13*c^2)*a^7-(b^4+c^4+3*b*c*(7*b^2-12*b*c+7*c^2))*a^6-5*(b^2-c^2)*(b-c)*(3*b^2-2*b*c+3*c^2)*a^5+(11*b^4+11*c^4+3*b*c*(9*b^2+4*b*c+9*c^2))*(b-c)^2*a^4+(b^2-c^2)*(b-c)^3*(7*b^2+2*b*c+7*c^2)*a^3-(b^2-c^2)^2*(7*b^4+7*c^4-b*c*(5*b^2-8*b*c+5*c^2))*a^2-(b^2-c^2)^3*(b-c)^3*a+(b^2-c^2)^3*(b-c)*(b^3+c^3) : :
X(38784) = 2*X(3)+X(124) = X(3)+2*X(6711) = 5*X(3)+X(10747) = 7*X(3)-X(38777) = 3*X(3)-X(38778) = 3*X(3)+X(38779) = 11*X(3)+X(38780) = 7*X(3)+2*X(38781) = 3*X(3)+2*X(38782) = 5*X(3)-2*X(38783) = 4*X(3)-X(38785) = X(3)+5*X(38786) = 4*X(3)+5*X(38787) = X(124)-4*X(6711) = 5*X(124)-2*X(10747) = 7*X(124)+2*X(38777) = 3*X(124)+2*X(38778) = 3*X(124)-2*X(38779) = 11*X(124)-2*X(38780) = 7*X(124)-4*X(38781) = 3*X(124)-4*X(38782) = 5*X(124)+4*X(38783) = 2*X(124)+X(38785) = X(124)-10*X(38786) = 2*X(124)-5*X(38787)

X(38784) lies on these lines: {3,124}, {40,11734}, {102,631}, {109,3523}, {117,140}, {151,10303}, {549,2818}, {2773,38727}, {2779,38793}, {2785,38737}, {2792,38748}, {2800,10164}, {2807,38772}, {2817,10165}, {2819,38804}, {3090,10726}, {3522,10732}, {3526,10740}, {3738,21154}, {6684,11713}, {7987,13532}, {10709,15702}, {10716,15692}, {15717,33650}

X(38784) = midpoint of X(i) and X(j) for these {i,j}: {3, 38776}, {38778, 38779}
X(38784) = reflection of X(i) in X(j) for these (i,j): (124, 38776), (38776, 6711), (38779, 38782)
X(38784) = X(124)-Gibert-Moses centroid
X(38784) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 124, 38785), (3, 6711, 124), (3, 10747, 38783), (3, 38779, 38778), (3, 38786, 6711), (102, 631, 6718), (124, 6711, 38787), (38776, 38778, 38779), (38776, 38779, 38782), (38779, 38782, 124), (38785, 38787, 124)

X(38785) = X(102)-CIRCUM-EULER-POINT OF X(550)

Barycentrics    4*a^10-4*(b+c)*a^9-4*(2*b^2-3*b*c+2*c^2)*a^8+(b+c)*(11*b^2-14*b*c+11*c^2)*a^7+(b^4+c^4-b*c*(19*b^2-28*b*c+19*c^2))*a^6-3*(b^2-c^2)*(b-c)*(3*b^2-2*b*c+3*c^2)*a^5+(b^2+3*b*c+c^2)*(5*b^2-2*b*c+5*c^2)*(b-c)^2*a^4+(b^2-c^2)*(b-c)^5*a^3-(b^2-c^2)^2*(b^4+c^4-b*c*(3*b^2-8*b*c+3*c^2))*a^2+(b^2-c^2)^3*(b-c)^3*a-(b^2-c^2)^3*(b-c)*(b^3+c^3) : :
X(38785) = 3*X(3)-2*X(6711) = 3*X(3)-X(10747) = 5*X(3)-3*X(38776) = X(3)-3*X(38778) = 7*X(3)-3*X(38779) = 5*X(3)-X(38780) = 5*X(3)-2*X(38781) = 11*X(3)-6*X(38782) = 4*X(3)-3*X(38784) = 7*X(3)-5*X(38786) = 8*X(3)-5*X(38787) = 3*X(124)-4*X(6711) = 3*X(124)-2*X(10747) = 5*X(124)-6*X(38776) = X(124)+2*X(38777) = X(124)-6*X(38778) = 7*X(124)-6*X(38779) = 5*X(124)-2*X(38780) = 5*X(124)-4*X(38781) = 11*X(124)-12*X(38782) = X(124)-4*X(38783) = 2*X(124)-3*X(38784) = 7*X(124)-10*X(38786) = 4*X(124)-5*X(38787)

X(38785) lies on these lines: {2,10732}, {3,124}, {4,6718}, {20,109}, {30,117}, {102,376}, {165,13532}, {515,14690}, {516,11700}, {550,2818}, {1361,15326}, {1364,15338}, {1657,10740}, {1795,4302}, {2773,16163}, {2779,16111}, {2785,38738}, {2792,38749}, {2800,4297}, {2807,38773}, {2817,31730}, {2819,38805}, {2846,3184}, {2853,14689}, {3522,33650}, {3529,10726}, {3576,11734}, {3738,24466}, {4304,12016}, {5731,10703}, {6361,10696}, {10304,10716}, {10709,11001}, {10764,25406}, {11727,12699}

X(38785) = midpoint of X(i) and X(j) for these {i,j}: {3, 38777}, {20, 109}, {1657, 10740}, {3529, 10726}, {6361, 10696}, {10709, 11001}
X(38785) = reflection of X(i) in X(j) for these (i,j): (3, 38783), (4, 6718), (124, 3), (10747, 6711), (12699, 11727), (38780, 38781)
X(38785) = complement of X(10732)
X(38785) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 124, 38784), (3, 10747, 6711), (3, 38778, 38783), (3, 38779, 38786), (3, 38780, 38776), (124, 38784, 38787), (6711, 10747, 124), (38776, 38780, 38781), (38776, 38781, 124), (38777, 38778, 3), (38777, 38783, 124)

X(38786) = X(102)-CIRCUM-EULER-POINT OF X(631)

Barycentrics    3*a^10-3*(b+c)*a^9-3*(2*b^2-3*b*c+2*c^2)*a^8+2*(b+c)*(5*b^2-7*b*c+5*c^2)*a^7-(b^4+c^4+4*b*c*(4*b^2-7*b*c+4*c^2))*a^6-4*(b^2-c^2)*(b-c)*(3*b^2-2*b*c+3*c^2)*a^5+(9*b^4+9*c^4+2*b*c*(11*b^2+5*b*c+11*c^2))*(b-c)^2*a^4+2*(b^2-c^2)*(b-c)^3*(3*b^2+b*c+3*c^2)*a^3-2*(b^2-c^2)^2*(3*b^4+3*c^4-b*c*(2*b^2-3*b*c+2*c^2))*a^2-(b^2-c^2)^3*(b-c)^3*a+(b^2-c^2)^3*(b-c)*(b^3+c^3) : :
X(38786) = 6*X(2)-X(10740) = 3*X(3)+2*X(124) = X(3)+4*X(6711) = 4*X(3)+X(10747) = 2*X(3)+3*X(38776) = 6*X(3)-X(38777) = 8*X(3)-3*X(38778) = 7*X(3)+3*X(38779) = 9*X(3)+X(38780) = 11*X(3)+4*X(38781) = 13*X(3)+12*X(38782) = 9*X(3)-4*X(38783) = X(3)-6*X(38784) = 7*X(3)-2*X(38785) = X(3)+2*X(38787) = X(124)-6*X(6711) = 8*X(124)-3*X(10747) = 4*X(124)-9*X(38776) = 4*X(124)+X(38777) = 16*X(124)+9*X(38778) = 14*X(124)-9*X(38779) = 6*X(124)-X(38780) = 11*X(124)-6*X(38781) = 13*X(124)-18*X(38782) = 3*X(124)+2*X(38783) = X(124)+9*X(38784) = 7*X(124)+3*X(38785) = X(124)-3*X(38787)

X(38786) lies on these lines: {2,10740}, {3,124}, {5,10726}, {102,140}, {109,549}, {117,3526}, {151,3525}, {548,10732}, {631,2818}, {2773,38728}, {2779,38794}, {2785,38739}, {2792,38750}, {2800,38762}, {2807,38774}, {2816,19862}, {2819,38806}, {3524,33650}, {5054,6718}, {10696,38028}, {10709,11539}, {10716,12100}, {10757,38110}, {10771,34126}, {11713,26446}, {11734,12702}, {13532,13624}

X(38786) = reflection of X(38787) in X(6711)
X(38786) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 124, 38777), (3, 6711, 38776), (3, 10747, 38778), (3, 38776, 10747), (3, 38779, 38785), (3, 38780, 38783), (124, 38777, 10747), (124, 38783, 38780), (6711, 38784, 3), (38776, 38777, 124), (38780, 38783, 38777)

X(38787) = X(102)-CIRCUM-EULER-POINT OF X(632)

Barycentrics    4*a^10-4*(b+c)*a^9-4*(2*b^2-3*b*c+2*c^2)*a^8+(b+c)*(15*b^2-22*b*c+15*c^2)*a^7-(3*b^4+3*c^4+b*c*(23*b^2-44*b*c+23*c^2))*a^6-7*(b^2-c^2)*(b-c)*(3*b^2-2*b*c+3*c^2)*a^5+(17*b^4+17*c^4+b*c*(41*b^2+20*b*c+41*c^2))*(b-c)^2*a^4+(b^2-c^2)*(b-c)^3*(13*b^2+6*b*c+13*c^2)*a^3-(b^2-c^2)^2*(13*b^4+13*c^4-b*c*(7*b^2-8*b*c+7*c^2))*a^2-3*(b^2-c^2)^3*(b-c)^3*a+(b^2-c^2)^3*(b-c)*(3*b^3+3*c^3) : :
X(38787) = 2*X(3)+3*X(124) = X(3)-6*X(6711) = 7*X(3)+3*X(10747) = X(3)+9*X(38776) = 13*X(3)-3*X(38777) = 19*X(3)-9*X(38778) = 11*X(3)+9*X(38779) = 17*X(3)+3*X(38780) = 3*X(3)+2*X(38781) = 7*X(3)+18*X(38782) = 11*X(3)-6*X(38783) = 4*X(3)-9*X(38784) = 8*X(3)-3*X(38785) = X(3)-3*X(38786) = X(124)+4*X(6711) = 7*X(124)-2*X(10747) = X(124)-6*X(38776) = 13*X(124)+2*X(38777) = 19*X(124)+6*X(38778) = 11*X(124)-6*X(38779) = 17*X(124)-2*X(38780) = 9*X(124)-4*X(38781) = 7*X(124)-12*X(38782) = 11*X(124)+4*X(38783) = 2*X(124)+3*X(38784) = 4*X(124)+X(38785) = X(124)+2*X(38786)

X(38787) lies on these lines: {3,124}, {102,3090}, {109,10303}, {117,3628}, {632,2818}, {2773,38729}, {2779,38795}, {2785,38740}, {2792,38751}, {2800,38763}, {2807,38775}, {2819,38807}, {3523,10716}, {3525,6718}, {5067,10709}, {5079,10740}, {7982,11734}, {13532,30389}

X(38787) = reflection of X(38786) in X(6711)
X(38787) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (124, 6711, 38784), (124, 38784, 38785), (6711, 38776, 124), (10747, 38782, 124)

X(38788) = X(110)-CIRCUM-EULER-POINT OF X(376)

Barycentrics    5*a^10-8*(b^2+c^2)*a^8-(5*b^4-27*b^2*c^2+5*c^4)*a^6+(b^2+c^2)*(13*b^4-29*b^2*c^2+13*c^4)*a^4-2*(b^2-c^2)^2*(2*b^4+7*b^2*c^2+2*c^4)*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(38788) = 5*X(3)-2*X(113) = 7*X(3)-4*X(5972) = 4*X(3)-X(7728) = X(3)+2*X(16111) = 2*X(3)+X(20127) = X(3)-4*X(37853) = 3*X(3)-X(38789) = 7*X(3)-X(38790) = 13*X(3)-4*X(38791) = 9*X(3)-4*X(38792) = 3*X(3)-2*X(38793) = 8*X(3)-5*X(38794) = 19*X(3)-10*X(38795) = 7*X(113)-10*X(5972) = 8*X(113)-5*X(7728) = 4*X(113)-5*X(14643) = X(113)+5*X(16111) = 4*X(113)+5*X(20127) = X(113)-10*X(37853) = 6*X(113)-5*X(38789) = 14*X(113)-5*X(38790) = 13*X(113)-10*X(38791) = 9*X(113)-10*X(38792) = 3*X(113)-5*X(38793)

X(38788) lies on these lines: {2,34584}, {3,113}, {4,15088}, {20,265}, {30,14644}, {74,550}, {110,548}, {125,1657}, {140,10721}, {146,3528}, {376,5663}, {381,38727}, {382,6699}, {399,10990}, {541,15688}, {542,15689}, {631,1539}, {690,38742}, {974,6243}, {1511,3522}, {1533,37958}, {1656,13202}, {2771,5918}, {2774,38766}, {2779,38778}, {2794,14850}, {2854,38798}, {3146,20304}, {3448,17538}, {3521,10226}, {3529,10113}, {3534,15041}, {3627,15059}, {3830,23515}, {3843,6723}, {4297,12898}, {4316,12903}, {4324,12904}, {4549,10293}, {5010,12373}, {5054,36518}, {5059,15081}, {5073,7687}, {5642,14093}, {5655,6030}, {5894,9934}, {6455,8998}, {6456,13990}, {7280,12374}, {7492,32227}, {8674,38754}, {9140,15686}, {10065,15326}, {10081,15338}, {10264,12103}, {10272,15036}, {10575,22584}, {10610,14861}, {10620,15696}, {10625,17855}, {10628,14855}, {10706,34200}, {10733,15027}, {11799,20725}, {11801,15057}, {12219,13491}, {12292,35503}, {12295,17800}, {12358,18439}, {12368,31663}, {12512,12778}, {12893,13564}, {12900,15720}, {12902,20417}, {13198,37495}, {13353,15472}, {14641,21650}, {14805,35485}, {14810,14982}, {14849,23698}, {15040,15063}, {15046,15693}, {15051,33923}, {15054,34153}, {15647,20427}, {15681,38724}, {17854,18436}, {18332,38747}, {18565,23325}, {18859,32607}, {19140,33751}, {29317,35452}, {30552,37484}

X(38788) = midpoint of X(i) and X(j) for these {i,j}: {3534, 15041}, {14643, 20127}, {15681, 38724}
X(38788) = reflection of X(i) in X(j) for these (i,j): (4, 34128), (381, 38727), (3830, 23515), (5655, 15035), (7728, 14643), (14643, 3), (15035, 8703), (15061, 15055), (20126, 15041), (38723, 376), (38789, 38793)
X(38788) = X(20127)-Gibert-Moses centroid
X(38788) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 7728, 38794), (3, 16111, 20127), (3, 20127, 7728), (3, 38789, 38793), (3, 38790, 5972), (20, 12041, 265), (74, 550, 12121), (548, 14677, 110), (3522, 12244, 1511), (10620, 15696, 16163), (10620, 16163, 23236), (10990, 38726, 399), (16111, 37853, 3), (38789, 38793, 14643)

X(38789) = X(110)-CIRCUM-EULER-POINT OF X(381)

Barycentrics    a^10+2*(b^2+c^2)*a^8-(10*b^4-9*b^2*c^2+10*c^4)*a^6+(b^2+c^2)*(8*b^4-13*b^2*c^2+8*c^4)*a^4+(b^2-c^2)^2*(b^4-10*b^2*c^2+c^4)*a^2-2*(b^4-c^4)*(b^2-c^2)^3 : :
X(38789) = 2*X(2)-3*X(15046) = X(3)-4*X(113) = 5*X(3)-8*X(5972) = X(3)+2*X(7728) = 7*X(3)-4*X(16111) = 5*X(3)-2*X(20127) = 11*X(3)-8*X(37853) = 3*X(3)-2*X(38788) = 2*X(3)+X(38790) = X(3)+8*X(38791) = 3*X(3)-8*X(38792) = 3*X(3)-4*X(38793) = 7*X(3)-10*X(38794) = 11*X(3)-20*X(38795) = 5*X(113)-2*X(5972) = 2*X(113)+X(7728) = 7*X(113)-X(16111) = 10*X(113)-X(20127) = 11*X(113)-2*X(37853) = 6*X(113)-X(38788) = 8*X(113)+X(38790) = X(113)+2*X(38791) = 3*X(113)-2*X(38792) = 3*X(113)-X(38793) = 14*X(113)-5*X(38794) = 11*X(113)-5*X(38795) = X(15041)-3*X(15046)

X(38789) lies on these lines: {2,15041}, {3,113}, {4,195}, {5,146}, {20,10272}, {30,32609}, {64,32743}, {74,1656}, {110,382}, {125,3851}, {140,12244}, {143,12284}, {154,18561}, {265,3527}, {381,5640}, {541,5055}, {542,5093}, {546,3448}, {568,22971}, {631,14677}, {690,38743}, {999,12373}, {1351,14982}, {1482,12368}, {1498,19506}, {1503,18403}, {1511,1657}, {1514,18325}, {1553,20957}, {1568,35452}, {1699,2771}, {1986,37197}, {2774,38767}, {2779,38779}, {2854,38799}, {2931,18378}, {2948,22793}, {3024,9654}, {3028,9669}, {3043,35490}, {3091,10264}, {3146,15039}, {3167,3830}, {3295,12374}, {3522,15042}, {3526,12041}, {3534,15035}, {3627,12383}, {3818,32306}, {3832,11801}, {3850,15081}, {5054,15055}, {5070,6699}, {5072,15054}, {5073,12121}, {5076,5609}, {5079,15059}, {5339,36208}, {5340,36209}, {5448,12302}, {5621,38317}, {5642,15681}, {5656,18404}, {5691,11699}, {5876,7731}, {5878,23315}, {5898,15800}, {6053,12295}, {6241,11561}, {6243,11807}, {6759,11597}, {7507,12292}, {7517,12168}, {7574,32111}, {7579,11472}, {7722,35488}, {7727,10895}, {7978,12645}, {8674,38755}, {9143,15687}, {9655,10091}, {9668,10088}, {9703,15463}, {9781,13358}, {9904,9956}, {9955,33535}, {9970,18440}, {10065,31479}, {10113,14094}, {10224,33541}, {10263,12273}, {10628,18435}, {10752,11898}, {10826,11670}, {10896,19470}, {10990,12900}, {11001,11694}, {11424,15089}, {11439,15102}, {11557,34783}, {11562,18439}, {11563,32608}, {11591,13201}, {11723,37624}, {12103,22251}, {12112,18572}, {12375,23261}, {12376,23251}, {13392,15704}, {13417,18436}, {14627,16657}, {15140,19140}, {15342,22505}, {15694,38727}, {16010,19130}, {16163,17800}, {16658,31724}, {17838,36749}, {18332,38744}, {18358,32247}, {18504,32138}, {19709,20126}, {25320,38136}, {29317,37949}

X(38789) = midpoint of X(7728) and X(14643)
X(38789) = reflection of X(i) in X(j) for these (i,j): (3, 14643), (74, 34128), (3534, 15035), (5621, 38317), (14643, 113), (15041, 2), (15061, 36518), (15681, 38723), (20126, 23515), (25320, 38136), (38723, 5642), (38724, 381), (38788, 38793), (38793, 38792)
X(38789) = X(38790)-Gibert-Moses centroid
X(38789) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 7728, 38790), (4, 399, 12902), (5, 146, 10620), (20, 10272, 15040), (110, 1539, 382), (113, 7728, 3), (113, 38791, 7728), (113, 38793, 38792), (265, 15063, 12308), (1511, 10721, 1657), (3843, 12308, 265), (5972, 20127, 3), (12121, 13202, 5073), (14643, 38788, 38793), (14982, 32271, 1351), (15041, 15046, 2), (15061, 36518, 5055), (16111, 38794, 3), (20125, 34153, 15039), (38788, 38793, 3), (38792, 38793, 14643)

X(38790) = X(110)-CIRCUM-EULER-POINT OF X(382)

Barycentrics    3*a^10-2*(b^2+c^2)*a^8-(10*b^4-19*b^2*c^2+10*c^4)*a^6+(b^2+c^2)*(12*b^4-23*b^2*c^2+12*c^4)*a^4-(b^2-c^2)^2*(b^4+14*b^2*c^2+c^4)*a^2-2*(b^4-c^4)*(b^2-c^2)^3 : :
X(38790) = 3*X(3)-4*X(113) = 7*X(3)-8*X(5972) = 5*X(3)-6*X(14643) = 5*X(3)-4*X(16111) = 3*X(3)-2*X(20127) = 9*X(3)-8*X(37853) = 7*X(3)-6*X(38788) = 2*X(3)-3*X(38789) = 5*X(3)-8*X(38791) = 11*X(3)-12*X(38793) = 9*X(3)-10*X(38794) = 17*X(3)-20*X(38795) = 7*X(113)-6*X(5972) = 2*X(113)-3*X(7728) = 10*X(113)-9*X(14643) = 5*X(113)-3*X(16111) = 3*X(113)-2*X(37853) = 14*X(113)-9*X(38788) = 8*X(113)-9*X(38789) = 5*X(113)-6*X(38791) = 19*X(113)-18*X(38792) = 11*X(113)-9*X(38793) = 6*X(113)-5*X(38794) = 17*X(113)-15*X(38795)

X(38790) lies on these lines: {2,14677}, {3,113}, {4,10264}, {5,12244}, {20,20125}, {30,146}, {64,19506}, {74,381}, {110,1657}, {125,3843}, {140,15046}, {265,541}, {376,10272}, {382,5663}, {542,6144}, {550,13392}, {568,11807}, {690,38744}, {999,12374}, {1511,3534}, {1533,2931}, {1656,12041}, {1885,14627}, {2771,3901}, {2774,38768}, {2779,38780}, {2781,18440}, {2854,38800}, {2930,29317}, {2948,28146}, {3024,9655}, {3028,9668}, {3146,32423}, {3295,12373}, {3448,3627}, {3526,15055}, {3529,34153}, {3543,12317}, {3580,31726}, {3845,15081}, {3851,10990}, {5050,32271}, {5055,6699}, {5070,36518}, {5072,15059}, {5073,12164}, {5076,10113}, {5079,15021}, {5621,19130}, {5642,15689}, {5655,15681}, {5667,11251}, {5876,13201}, {5878,18562}, {6053,15685}, {6407,8998}, {6408,13990}, {7687,14269}, {7722,35490}, {7725,26336}, {7726,26346}, {7727,12943}, {7978,18526}, {8674,38756}, {8703,15042}, {9140,38335}, {9654,10065}, {9669,10081}, {9704,15463}, {9904,18480}, {9909,32227}, {9984,18503}, {10152,34334}, {10255,20427}, {10606,23043}, {10628,18439}, {11455,15100}, {11561,15072}, {11709,18493}, {11820,19377}, {12083,12168}, {12103,22250}, {12121,15063}, {12133,18494}, {12173,12292}, {12192,18501}, {12273,13391}, {12327,18524}, {12368,12702}, {12369,18508}, {12371,18519}, {12372,18518}, {12381,18545}, {12382,18543}, {12407,33697}, {12900,15694}, {12953,19470}, {13203,18404}, {13417,34783}, {14683,33703}, {14982,33878}, {15035,15696}, {15039,15704}, {15051,15688}, {15057,15088}, {15101,32137}, {15311,18403}, {15473,18535}, {16534,38723}, {17812,18561}, {17855,37481}, {18510,19059}, {18512,19060}, {18560,19504}, {22583,26321}, {22793,33535}, {23251,35826}, {23261,35827}, {32306,36990}

X(38790) = midpoint of X(i) and X(j) for these {i,j}: {5073, 12308}, {14683, 33703}
X(38790) = reflection of X(i) in X(j) for these (i,j): (3, 7728), (64, 19506), (74, 1539), (265, 13202), (382, 10721), (399, 146), (1657, 110), (3448, 3627), (3529, 34153), (3534, 10706), (5925, 13293), (9904, 18480), (9919, 11744), (10117, 22802), (10606, 23043), (10620, 4), (12121, 15063), (12244, 5), (12284, 10263), (12407, 33697), (12702, 12368), (12902, 382), (13201, 5876), (15054, 10113), (15101, 32137), (15681, 5655), (16111, 38791), (17800, 12121), (18508, 12369), (18526, 7978), (20127, 113), (20427, 23315), (32306, 36990), (33535, 22793), (33878, 14982), (34783, 13417)
X(38790) = anticomplement of X(14677)
X(38790) = Stammler circle-inverse of-X(14703)
X(38790) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 7728, 38789), (4, 10620, 38724), (5, 12244, 15041), (74, 1539, 381), (113, 20127, 3), (113, 37853, 38794), (265, 13202, 3830), (5972, 38788, 3), (7728, 14643, 38791), (7728, 20127, 113), (14643, 16111, 3), (16111, 38791, 14643), (20127, 38794, 37853), (36518, 38728, 5070), (37853, 38794, 3)

X(38791) = X(110)-CIRCUM-EULER-POINT OF X(546)

Barycentrics    2*a^10+2*(b^2+c^2)*a^8-(15*b^4-16*b^2*c^2+15*c^4)*a^6+(b^2+c^2)*(13*b^4-22*b^2*c^2+13*c^4)*a^4+(b^2-c^2)^2*(b^4-16*b^2*c^2+c^4)*a^2-3*(b^4-c^4)*(b^2-c^2)^3 : :
X(38791) = 9*X(2)-5*X(15021) = 9*X(2)-13*X(15029) = X(3)-3*X(113) = 2*X(3)-3*X(5972) = X(3)+3*X(7728) = 5*X(3)-9*X(14643) = 5*X(3)-3*X(16111) = 7*X(3)-3*X(20127) = 4*X(3)-3*X(37853) = 13*X(3)-9*X(38788) = X(3)-9*X(38789) = 5*X(3)+3*X(38790) = 4*X(3)-9*X(38792) = 7*X(3)-9*X(38793) = 11*X(3)-15*X(38794) = 3*X(3)-5*X(38795) = 5*X(113)-3*X(14643) = 5*X(113)-X(16111) = 7*X(113)-X(20127) = 4*X(113)-X(37853) = 13*X(113)-3*X(38788) = X(113)-3*X(38789) = 5*X(113)+X(38790) = 4*X(113)-3*X(38792) = 7*X(113)-3*X(38793) = 11*X(113)-5*X(38794) = 9*X(113)-5*X(38795) = 3*X(10990)-5*X(15021) = 3*X(10990)-13*X(15029) = 5*X(15021)-13*X(15029)

Let NA be the reflection of X(5) in the A-altitude, and define NB and NC cyclically. NANBNC is inversely similar to ABC, with similitude center X(195). X(38791) = X(10564)-of-NANBNC. (see Hyacinthos #21522, 2/11/2013, Antreas Hatzipolakis) (Randy Hutson, May 31, 2020)

X(38791) lies on these lines: {2,10990}, {3,113}, {4,542}, {5,541}, {20,5642}, {30,15152}, {74,3090}, {110,3146}, {125,146}, {185,12824}, {381,15027}, {382,5655}, {389,546}, {399,5076}, {511,1514}, {632,12041}, {690,38745}, {974,15012}, {1112,16625}, {1181,34155}, {1493,11805}, {1511,15704}, {1531,29012}, {1533,29317}, {1539,3627}, {1561,23698}, {1568,7464}, {2774,38769}, {2779,38781}, {2781,5893}, {2854,38801}, {2883,6593}, {3303,12374}, {3304,12373}, {3448,15044}, {3522,15023}, {3525,12244}, {3529,10721}, {3628,6699}, {3830,23236}, {3832,9140}, {3850,20379}, {3851,20126}, {3857,10264}, {3860,13393}, {5056,15057}, {5066,20396}, {5072,10620}, {5079,15061}, {5159,15125}, {5448,15115}, {5465,10991}, {5621,11479}, {5943,16270}, {6000,10297}, {6425,8998}, {6426,13990}, {6696,15113}, {6759,23043}, {7530,9932}, {7982,12368}, {8674,38757}, {9143,17578}, {9826,17855}, {10151,13148}, {10272,12103}, {10303,15055}, {10574,17853}, {10594,15473}, {10605,18418}, {10733,24981}, {11456,25556}, {11477,14982}, {11541,20125}, {11693,15681}, {11723,15178}, {12102,32423}, {12111,14448}, {12121,15039}, {12308,37493}, {12811,20304}, {12825,13417}, {12828,37197}, {13598,14984}, {14499,15156}, {14500,15157}, {14542,34802}, {14677,14869}, {14862,18563}, {15022,15059}, {15035,17538}, {15046,38728}, {15088,38725}, {15128,32305}, {16223,17854}, {16657,22330}, {18323,18400}, {18388,31861}, {20190,34664}, {22800,34563}, {32233,34725}, {32417,36169}

X(38791) = midpoint of X(i) and X(j) for these {i,j}: {4, 15063}, {110, 13202}, {113, 7728}, {125, 146}, {382, 30714}, {399, 12295}, {1531, 32111}, {3627, 5609}, {10721, 16163}, {10733, 24981}, {11061, 32250}, {12111, 14448}, {12825, 13417}, {16111, 38790}
X(38791) = reflection of X(i) in X(j) for these (i,j): (74, 6723), (5972, 113), (12041, 12900), (15115, 5448), (17855, 9826), (20379, 3850), (20417, 5), (36253, 546), (37853, 5972), (38726, 10272)
X(38791) = complement of X(10990)
X(38791) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 10706, 15063), (74, 3090, 38729), (74, 36518, 6723), (113, 5972, 38792), (113, 16111, 14643), (146, 3091, 15054), (382, 5655, 30714), (546, 36253, 7687), (1539, 5609, 3627), (3090, 38729, 6723), (3091, 15054, 125), (3529, 15034, 16163), (7728, 14643, 38790), (7728, 38789, 113), (10721, 15034, 3529), (14643, 38790, 16111), (15021, 15029, 2), (36518, 38729, 3090), (37853, 38792, 5972)

X(38792) = X(110)-CIRCUM-EULER-POINT OF X(547)

Barycentrics    2*a^10-14*(b^2+c^2)*a^8+25*(b^4+c^4)*a^6-(b^2+c^2)*(11*b^4-10*b^2*c^2+11*c^4)*a^4-(b^2-c^2)^2*(7*b^4-16*b^2*c^2+7*c^4)*a^2+5*(b^4-c^4)*(b^2-c^2)^3 : :
X(38792) = X(3)+5*X(113) = 2*X(3)-5*X(5972) = 7*X(3)+5*X(7728) = X(3)-5*X(14643) = 11*X(3)-5*X(16111) = 17*X(3)-5*X(20127) = 8*X(3)-5*X(37853) = 9*X(3)-5*X(38788) = 3*X(3)+5*X(38789) = 19*X(3)+5*X(38790) = 4*X(3)+5*X(38791) = 3*X(3)-5*X(38793) = 2*X(113)+X(5972) = 7*X(113)-X(7728) = 11*X(113)+X(16111) = 17*X(113)+X(20127) = 8*X(113)+X(37853) = 9*X(113)+X(38788) = 3*X(113)-X(38789) = 19*X(113)-X(38790) = 4*X(113)-X(38791) = 3*X(113)+X(38793) = 13*X(113)+5*X(38794) = 7*X(113)+5*X(38795)

X(38792) lies on these lines: {3,113}, {5,6053}, {74,3533}, {110,3832}, {125,5056}, {403,5965}, {541,11539}, {542,3545}, {547,5663}, {690,38746}, {1568,29317}, {2774,38770}, {2779,38782}, {2854,38802}, {3091,24981}, {3448,15029}, {3543,5642}, {3845,17702}, {3850,7687}, {3853,10272}, {5059,13202}, {5067,6723}, {5655,15046}, {6699,16239}, {8674,38758}, {10706,15702}, {11001,15035}, {11723,33179}, {12900,20417}, {14982,32300}, {15055,15708}, {15113,31978}, {15690,34584}, {16163,33703}, {19140,32251}, {32609,38335}

X(38792) = midpoint of X(i) and X(j) for these {i,j}: {113, 14643}, {5655, 23515}, {10706, 38727}, {38789, 38793}
X(38792) = reflection of X(i) in X(j) for these (i,j): (5972, 14643), (20417, 34128), (34128, 12900), (38725, 547)
X(38792) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (113, 5972, 38791), (113, 38793, 38789), (113, 38795, 7728), (5655, 15046, 23515), (5972, 38791, 37853), (14643, 38789, 38793)

X(38793) = X(110)-CIRCUM-EULER-POINT OF X(549)

Barycentrics    4*a^10-10*(b^2+c^2)*a^8+(5*b^4+18*b^2*c^2+5*c^4)*a^6+(b^2+c^2)*(5*b^4-16*b^2*c^2+5*c^4)*a^4-(b^2-c^2)^2*(5*b^4+4*b^2*c^2+5*c^4)*a^2+(b^4-c^4)*(b^2-c^2)^3 : :
X(38793) = 2*X(3)+X(113) = X(3)+2*X(5972) = 5*X(3)+X(7728) = 4*X(3)-X(16111) = 7*X(3)-X(20127) = 5*X(3)-2*X(37853) = 3*X(3)-X(38788) = 3*X(3)+X(38789) = 11*X(3)+X(38790) = 7*X(3)+2*X(38791) = 3*X(3)+2*X(38792) = X(3)+5*X(38794) = 4*X(3)+5*X(38795) = X(113)-4*X(5972) = 5*X(113)-2*X(7728) = 2*X(113)+X(16111) = 7*X(113)+2*X(20127) = 5*X(113)+4*X(37853) = 3*X(113)+2*X(38788) = 3*X(113)-2*X(38789) = 11*X(113)-2*X(38790) = 7*X(113)-4*X(38791) = 3*X(113)-4*X(38792) = X(113)-10*X(38794) = 2*X(113)-5*X(38795) = X(14644)+3*X(15035) = 2*X(14644)-3*X(23515) = 2*X(15035)+X(23515)

X(38793) lies on these lines: {2,14644}, {3,113}, {4,12900}, {5,12295}, {20,15036}, {30,36518}, {40,11723}, {52,9826}, {74,3523}, {98,33512}, {99,33511}, {110,631}, {125,128}, {141,32275}, {146,15717}, {182,5181}, {186,14156}, {265,3526}, {371,13990}, {372,8998}, {381,38723}, {399,15720}, {468,10564}, {511,16222}, {523,31378}, {541,3524}, {542,5054}, {548,1539}, {549,5642}, {550,13202}, {569,5504}, {632,20304}, {690,38748}, {1092,12228}, {1112,10625}, {1216,1986}, {1351,32300}, {1495,15122}, {1503,10257}, {1533,37950}, {1568,15646}, {1656,7687}, {1657,15042}, {2070,29317}, {2482,11656}, {2771,11227}, {2774,38772}, {2779,38784}, {2781,21167}, {2854,38804}, {2931,15115}, {3043,27866}, {3090,10733}, {3147,15472}, {3448,10303}, {3515,15473}, {3522,10721}, {3525,12383}, {3529,15023}, {3530,10272}, {3533,15081}, {3534,15046}, {3548,11750}, {3589,33851}, {3620,32234}, {3628,10113}, {3763,32233}, {3819,10628}, {3917,16223}, {5418,10820}, {5420,10819}, {5447,11557}, {5562,14708}, {5609,12108}, {5651,18580}, {5655,15041}, {5965,22115}, {6053,10620}, {6640,23325}, {6644,22109}, {6684,11720}, {6803,12319}, {7393,19457}, {7395,12302}, {7399,23306}, {7471,31379}, {7503,12901}, {7514,32607}, {7722,11444}, {7723,11793}, {7987,12368}, {7999,12219}, {8552,32119}, {8674,38760}, {8703,34584}, {9033,26451}, {9140,15702}, {9143,15721}, {9181,16760}, {9540,19110}, {10020,25487}, {10192,14855}, {10193,18435}, {10264,14869}, {10299,12244}, {10706,15692}, {10990,15712}, {11064,32110}, {11232,26879}, {11561,14448}, {11562,12358}, {11597,32348}, {11598,16252}, {11694,11812}, {11695,11800}, {11735,12778}, {11801,16239}, {11807,13348}, {12022,12038}, {12068,25641}, {12140,37119}, {12162,23328}, {12827,18475}, {12828,15463}, {12893,17928}, {13160,33547}, {13198,13336}, {13211,31423}, {13391,16532}, {13416,25711}, {13857,18579}, {13935,19111}, {14049,21230}, {14499,35231}, {14500,35232}, {14810,32271}, {14934,22104}, {14984,38110}, {15029,17538}, {15054,20125}, {15462,19131}, {15694,38724}, {15701,20126}, {16165,35283}, {16196,20771}, {16238,16657}, {16278,33813}, {17704,17855}, {18332,38750}, {18350,25563}, {18358,32250}, {18488,20773}, {19456,37514}, {32223,37477}, {32274,34573}, {32743,37452}, {35486,37478}

X(38793) = midpoint of X(i) and X(j) for these {i,j}: {2, 15035}, {3, 14643}, {381, 38723}, {1511, 34128}, {3917, 16223}, {5642, 38727}, {5655, 15041}, {15061, 32609}, {38788, 38789}
X(38793) = reflection of X(i) in X(j) for these (i,j): (113, 14643), (125, 34128), (9140, 38725), (14643, 5972), (23515, 2), (34128, 140), (38727, 549), (38789, 38792)
X(38793) = complement of X(14644)
X(38793) = X(113)-Gibert-Moses centroid
X(38793) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 113, 16111), (3, 5972, 113), (3, 7728, 37853), (3, 38789, 38788), (3, 38794, 5972), (4, 15051, 38726), (5, 16163, 12295), (110, 631, 6699), (110, 6699, 16003), (110, 15057, 12317), (113, 5972, 38795), (125, 1511, 30714), (140, 1511, 125), (265, 3526, 6723), (3526, 15040, 265), (12900, 38726, 4), (14643, 38788, 38789), (14643, 38789, 38792), (14869, 22251, 10264), (16111, 38795, 113), (38789, 38792, 113)

X(38794) = X(110)-CIRCUM-EULER-POINT OF X(631)

Barycentrics    3*a^10-8*(b^2+c^2)*a^8+(5*b^4+13*b^2*c^2+5*c^4)*a^6+(b^2+c^2)*(3*b^4-11*b^2*c^2+3*c^4)*a^4-2*(b^2-c^2)^2*(2*b^4+b^2*c^2+2*c^4)*a^2+(b^4-c^4)*(b^2-c^2)^3 : :
X(38794) = 6*X(2)-X(265) = 3*X(2)+2*X(1511) = 9*X(2)+X(12383) = 9*X(2)-4*X(20304) = 3*X(3)+2*X(113) = X(3)+4*X(5972) = 4*X(3)+X(7728) = 2*X(3)+3*X(14643) = 7*X(3)-2*X(16111) = 6*X(3)-X(20127) = 9*X(3)-4*X(37853) = 8*X(3)-3*X(38788) = 7*X(3)+3*X(38789) = 9*X(3)+X(38790) = 11*X(3)+4*X(38791) = 13*X(3)+12*X(38792) = X(3)-6*X(38793) = X(3)+2*X(38795) = X(265)+4*X(1511) = 3*X(265)+2*X(12383) = 3*X(265)-8*X(20304) = 6*X(1511)-X(12383) = 2*X(1511)+X(15081) = 3*X(1511)+2*X(20304) = X(12383)+3*X(15081) = X(12383)+4*X(20304) = 3*X(15081)-4*X(20304)

X(38794) lies on these lines: {2,265}, {3,113}, {4,38723}, {5,10733}, {10,12898}, {30,15051}, {74,549}, {110,140}, {125,3526}, {141,11597}, {146,3524}, {376,1539}, {381,12900}, {382,36518}, {399,5054}, {402,34297}, {468,15472}, {541,15693}, {542,3763}, {548,10721}, {550,15036}, {567,5504}, {568,9826}, {590,10820}, {615,10819}, {620,14850}, {631,5663}, {632,15027}, {690,38750}, {895,38110}, {1112,3147}, {1125,12778}, {1216,16223}, {1533,18859}, {1568,37955}, {1656,15040}, {1657,15046}, {1986,23039}, {2070,14156}, {2771,25917}, {2774,38774}, {2779,38786}, {2781,31267}, {2854,38806}, {3090,10113}, {3311,13990}, {3312,8998}, {3448,3525}, {3522,34584}, {3523,12041}, {3530,15055}, {3534,13202}, {3580,12228}, {3581,11064}, {3618,14984}, {3624,12261}, {3628,14644}, {3851,12295}, {3917,11557}, {4413,12334}, {5010,12374}, {5050,5181}, {5055,7687}, {5067,15088}, {5070,12902}, {5092,14982}, {5093,32300}, {5094,12140}, {5432,10091}, {5433,10088}, {5447,13417}, {5609,10303}, {5648,9976}, {5891,22584}, {5892,21649}, {5898,12584}, {6036,14849}, {6053,15701}, {6243,16222}, {6697,10249}, {6723,30714}, {7280,12373}, {7484,12412}, {7722,11591}, {7731,7998}, {7808,12201}, {7914,12501}, {7984,38028}, {8674,38762}, {8981,19110}, {9140,11539}, {9143,15709}, {9934,10192}, {10018,15463}, {10170,21650}, {10201,25487}, {10257,10540}, {10620,15720}, {10706,12100}, {10778,34126}, {11202,32743}, {11231,13211}, {11561,12219}, {11562,11793}, {11720,26446}, {11723,12702}, {12068,14934}, {12103,15023}, {12108,15054}, {12188,33512}, {12244,15717}, {12273,15045}, {12308,20417}, {12310,15115}, {12317,15702}, {12368,13624}, {12790,15184}, {12890,24953}, {12905,26364}, {12906,26363}, {13188,33511}, {13198,37471}, {13915,32785}, {13966,19111}, {13979,32786}, {14094,14869}, {14499,28447}, {14500,28448}, {14561,33851}, {14683,20379}, {14708,18436}, {14851,31379}, {14993,22104}, {15029,15704}, {15039,20397}, {15041,15063}, {15116,23041}, {16238,37472}, {16241,36209}, {16242,36208}, {17506,18442}, {18281,20773}, {19059,35256}, {19060,35255}, {23042,32264}, {23306,37347}, {24206,32233}, {29317,37923}, {31884,32271}, {32223,37496}, {32607,34864}

X(38794) = midpoint of X(i) and X(j) for these {i,j}: {632, 22251}, {1656, 15040}, {15034, 15059}
X(38794) = reflection of X(i) in X(j) for these (i,j): (265, 15081), (15027, 15059), (15034, 22251), (15059, 632), (38728, 631), (38795, 5972)
X(38794) = complement of X(15081)
X(38794) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1511, 265), (2, 12383, 20304), (3, 113, 20127), (3, 5972, 14643), (3, 7728, 38788), (3, 14643, 7728), (3, 38789, 16111), (3, 38790, 37853), (5, 15035, 12121), (74, 10272, 5655), (113, 20127, 7728), (113, 37853, 38790), (140, 13392, 10264), (549, 10272, 74), (1511, 20304, 12383), (3526, 32609, 125), (5642, 6699, 399), (5972, 38793, 3), (10264, 13392, 110), (11539, 11694, 9140), (12383, 20304, 265), (12900, 16163, 381), (14643, 20127, 113), (16534, 38727, 10620), (31379, 36193, 14851), (36518, 38726, 382), (37853, 38790, 20127)

X(38795) = X(110)-CIRCUM-EULER-POINT OF X(632)

Barycentrics    4*a^10-14*(b^2+c^2)*a^8+(15*b^4+14*b^2*c^2+15*c^4)*a^6-(b^2+c^2)*(b^4+8*b^2*c^2+c^4)*a^4-(b^2-c^2)^2*(7*b^4-4*b^2*c^2+7*c^4)*a^2+3*(b^4-c^4)*(b^2-c^2)^3 : :
X(38795) = 9*X(2)+X(14094) = 6*X(2)-X(16003) = 3*X(2)+2*X(16534) = 9*X(2)-4*X(20397) = 2*X(3)+3*X(113) = X(3)-6*X(5972) = 7*X(3)+3*X(7728) = X(3)+9*X(14643) = 8*X(3)-3*X(16111) = 13*X(3)-3*X(20127) = 11*X(3)-6*X(37853) = 19*X(3)-9*X(38788) = 11*X(3)+9*X(38789) = 17*X(3)+3*X(38790) = 3*X(3)+2*X(38791) = 7*X(3)+18*X(38792) = 4*X(3)-9*X(38793) = X(3)-3*X(38794) = 2*X(14094)+3*X(16003) = X(14094)-6*X(16534) = X(14094)+4*X(20397) = X(16003)+4*X(16534) = 3*X(16003)-8*X(20397) = 3*X(16534)+2*X(20397)

X(38795) lies on these lines: {2,14094}, {3,113}, {4,15020}, {5,5642}, {74,10303}, {110,569}, {125,3628}, {140,15063}, {265,5079}, {376,15023}, {381,11693}, {399,6723}, {541,631}, {542,1656}, {546,1511}, {549,10990}, {576,5181}, {632,5663}, {690,38751}, {1209,6593}, {1216,12824}, {1539,12103}, {1568,7575}, {2774,38775}, {2779,38787}, {2854,38807}, {3091,15034}, {3146,15035}, {3523,10706}, {3525,6699}, {3526,5655}, {3529,38726}, {3533,15057}, {3544,12383}, {3549,9970}, {3627,16163}, {3850,11694}, {3857,34153}, {5055,23236}, {5067,9140}, {5072,7687}, {5076,15040}, {5651,15132}, {5891,25711}, {6053,15061}, {6419,8998}, {6420,13990}, {7464,14156}, {7486,9143}, {7505,12828}, {7982,11723}, {8674,38763}, {10113,12811}, {10539,15462}, {10541,14982}, {11591,14448}, {11656,14981}, {12041,12108}, {12121,15046}, {12317,38725}, {12368,30389}, {12812,32423}, {13202,15704}, {13857,16619}, {14216,15113}, {14499,30524}, {14500,30525}, {14644,15022}, {14869,38727}, {15051,17538}, {15059,20125}, {15303,34507}, {15699,20396}, {16222,16625}, {18583,32114}, {20304,24981}, {20399,31854}, {23061,37943}, {23235,33512}, {25641,31945}, {30734,32227}, {31378,36169}

X(38795) = midpoint of X(i) and X(j) for these {i,j}: {110, 15081}, {3091, 15034}, {15059, 20125}
X(38795) = reflection of X(i) in X(j) for these (i,j): (38729, 632), (38794, 5972)
X(38795) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 14094, 20397), (2, 16534, 16003), (5, 5642, 30714), (110, 3090, 36253), (110, 12900, 23515), (113, 5972, 38793), (113, 38793, 16111), (1511, 36518, 12295), (3090, 36253, 23515), (3525, 15054, 6699), (3526, 5655, 20417), (3628, 5609, 125), (3628, 10272, 5609), (5079, 15039, 265), (5972, 14643, 113), (7728, 38792, 113), (12900, 36253, 3090), (14094, 20397, 16003), (15020, 15029, 4), (16534, 20397, 14094)

X(38796) = X(111)-CIRCUM-EULER-POINT OF X(2)

Barycentrics    a^10-4*(b^2+c^2)*a^8+(5*b^4+11*b^2*c^2+5*c^4)*a^6+(b^2+c^2)*(3*b^4-17*b^2*c^2+3*c^4)*a^4-2*(3*b^4-4*b^2*c^2+3*c^4)*(b^4-4*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(b^4-4*b^2*c^2+c^4) : :
X(38796) = X(3)+2*X(5512) = X(3)-4*X(6719) = 2*X(3)+X(22338) = 4*X(3)-X(38797) = 5*X(3)+X(38800) = 5*X(3)+4*X(38801) = X(3)+4*X(38802) = 7*X(3)-4*X(38803) = 5*X(3)-2*X(38805) = 2*X(3)-5*X(38806) = X(3)-10*X(38807) = X(5512)+2*X(6719) = 4*X(5512)-X(22338) = 8*X(5512)+X(38797) = 4*X(5512)+X(38798) = 10*X(5512)-X(38800) = 5*X(5512)-2*X(38801) = 7*X(5512)+2*X(38803) = 5*X(5512)+X(38805) = 4*X(5512)+5*X(38806) = X(5512)+5*X(38807)

Let Q be the cyclic quadrilateral ABCX(111). Taking the vertices 3 at a time yields four triangles whose nine-point centers are the vertices of a cyclic quadrilateral homothetic to Q at X(38796). (Randy Hutson, May 31, 2020)

X(38796) lies on these lines: {2,33962}, {3,5512}, {4,14650}, {5,111}, {126,1656}, {140,1296}, {265,9129}, {355,11721}, {381,9172}, {498,6019}, {499,3325}, {543,5055}, {546,10734}, {547,10717}, {1352,28662}, {2072,15560}, {2780,15061}, {2793,38224}, {2805,38752}, {2813,38764}, {2819,38776}, {2854,14561}, {3048,32046}, {3090,14360}, {3091,14654}, {3545,32424}, {3564,36696}, {5056,20099}, {5901,10704}, {6699,35447}, {7506,14657}, {9179,20957}, {10765,18583}, {15702,37749}, {24206,36883}

X(38796) = midpoint of X(i) and X(j) for these {i,j}: {3, 38799}, {5512, 38804}, {6719, 38802}, {22338, 38798}
X(38796) = reflection of X(i) in X(j) for these (i,j): (3, 38804), (5512, 38802), (22338, 38799), (38797, 38798), (38798, 3), (38799, 5512), (38804, 6719)
X(38796) = X(22338)-Gibert-Moses centroid
X(38796) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5512, 22338), (3, 6719, 38806), (3, 22338, 38797), (3, 38800, 38805), (5, 111, 10748), (381, 9172, 14666), (1656, 11258, 126), (5512, 6719, 3), (5512, 38805, 38801), (5512, 38806, 38797), (5512, 38807, 6719), (22338, 38806, 3), (38799, 38804, 38798), (38800, 38801, 22338), (38801, 38805, 38800), (38802, 38804, 38799)

X(38797) = X(111)-CIRCUM-EULER-POINT OF X(20)

Barycentrics    3*a^10-12*(b^2+c^2)*a^8-(5*b^4-73*b^2*c^2+5*c^4)*a^6+(b^2+c^2)*(13*b^4-59*b^2*c^2+13*c^4)*a^4+2*(b^4-4*b^2*c^2+c^4)*(b^4-8*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2)*(b^4-4*b^2*c^2+c^4) : :
X(38797) = 3*X(3)-2*X(5512) = 5*X(3)-4*X(6719) = 4*X(3)-3*X(38796) = 2*X(3)-3*X(38798) = 5*X(3)-3*X(38799) = 3*X(3)-X(38800) = 7*X(3)-4*X(38801) = 17*X(3)-12*X(38802) = 3*X(3)-4*X(38803) = 7*X(3)-6*X(38804) = 6*X(3)-5*X(38806) = 13*X(3)-10*X(38807) = 5*X(5512)-6*X(6719) = 4*X(5512)-3*X(22338) = 8*X(5512)-9*X(38796) = 4*X(5512)-9*X(38798) = 10*X(5512)-9*X(38799) = 7*X(5512)-6*X(38801) = 17*X(5512)-18*X(38802) = 7*X(5512)-9*X(38804) = X(5512)-3*X(38805) = 4*X(5512)-5*X(38806) = 13*X(5512)-15*X(38807)

X(38797) lies on these lines: {3,5512}, {20,14654}, {30,1296}, {111,550}, {126,382}, {376,14650}, {543,15681}, {1657,23699}, {2780,12121}, {2793,38730}, {2805,38753}, {2813,38765}, {2819,38777}, {2854,20127}, {3070,11835}, {3071,11836}, {3325,4302}, {3529,14360}, {3534,11258}, {4299,6019}, {9129,38723}, {9172,15688}, {9529,23240}, {10704,28174}, {11001,32424}, {12083,14657}, {14688,31670}, {17702,35447}, {29012,36883}

X(38797) = midpoint of X(i) and X(j) for these {i,j}: {3529, 14360}, {11001, 37749}
X(38797) = reflection of X(i) in X(j) for these (i,j): (3, 38805), (111, 550), (382, 126), (5512, 38803), (10748, 1296), (14666, 3534), (22338, 3), (31670, 14688), (38796, 38798), (38800, 5512)
X(38797) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5512, 38806), (3, 22338, 38796), (3, 38799, 6719), (3, 38800, 5512), (3, 38805, 38798), (5512, 38800, 22338), (5512, 38803, 3), (5512, 38805, 38803), (5512, 38806, 38796), (22338, 38798, 3), (22338, 38806, 5512), (38800, 38803, 38806)

X(38798) = X(111)-CIRCUM-EULER-POINT OF X(376)

Barycentrics    5*a^10-20*(b^2+c^2)*a^8-5*(b^2+5*b*c+c^2)*(b^2-5*b*c+c^2)*a^6+(b^2+c^2)*(21*b^4-97*b^2*c^2+21*c^4)*a^4-20*b^2*c^2*(b^4-4*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2)*(b^4-4*b^2*c^2+c^4) : :
X(38798) = 5*X(3)-2*X(5512) = 7*X(3)-4*X(6719) = 4*X(3)-X(22338) = 2*X(3)+X(38797) = 3*X(3)-X(38799) = 7*X(3)-X(38800) = 13*X(3)-4*X(38801) = 9*X(3)-4*X(38802) = X(3)-4*X(38803) = 3*X(3)-2*X(38804) = X(3)+2*X(38805) = 8*X(3)-5*X(38806) = 19*X(3)-10*X(38807) = 7*X(5512)-10*X(6719) = 8*X(5512)-5*X(22338) = 4*X(5512)-5*X(38796) = 4*X(5512)+5*X(38797) = 6*X(5512)-5*X(38799) = 14*X(5512)-5*X(38800) = 13*X(5512)-10*X(38801) = 9*X(5512)-10*X(38802) = X(5512)-10*X(38803) = 3*X(5512)-5*X(38804) = X(5512)+5*X(38805)

X(38798) lies on these lines: {3,5512}, {20,10748}, {111,548}, {126,1657}, {376,14666}, {543,15689}, {550,1296}, {2780,38723}, {2793,38731}, {2805,38754}, {2813,38766}, {2819,38778}, {2854,38788}, {3522,14650}, {3534,23699}, {9172,14093}, {10717,15686}, {10734,15704}, {14360,17538}, {16163,35447}

X(38798) = midpoint of X(38796) and X(38797)
X(38798) = reflection of X(i) in X(j) for these (i,j): (22338, 38796), (38796, 3), (38799, 38804)
X(38798) = X(38797)-Gibert-Moses centroid
X(38798) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 22338, 38806), (3, 38797, 22338), (3, 38799, 38804), (3, 38800, 6719), (3, 38805, 38797), (38799, 38804, 38796), (38803, 38805, 3)

X(38799) = X(111)-CIRCUM-EULER-POINT OF X(381)

Barycentrics    a^10-4*(b^2+c^2)*a^8-(10*b^4-41*b^2*c^2+10*c^4)*a^6+(b^2+c^2)*(6*b^4-23*b^2*c^2+6*c^4)*a^4+(3*b^2+2*b*c-3*c^2)*(3*b^2-2*b*c-3*c^2)*(b^4-4*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(-2*b^4+8*b^2*c^2-2*c^4) : :
X(38799) = X(3)-4*X(5512) = 5*X(3)-8*X(6719) = X(3)+2*X(22338) = 5*X(3)-2*X(38797) = 3*X(3)-2*X(38798) = 2*X(3)+X(38800) = X(3)+8*X(38801) = 3*X(3)-8*X(38802) = 11*X(3)-8*X(38803) = 3*X(3)-4*X(38804) = 7*X(3)-4*X(38805) = 7*X(3)-10*X(38806) = 11*X(3)-20*X(38807) = 5*X(5512)-2*X(6719) = 2*X(5512)+X(22338) = 10*X(5512)-X(38797) = 6*X(5512)-X(38798) = 8*X(5512)+X(38800) = X(5512)+2*X(38801) = 3*X(5512)-2*X(38802) = 11*X(5512)-2*X(38803) = 3*X(5512)-X(38804) = 7*X(5512)-X(38805) = 14*X(5512)-5*X(38806) = 11*X(5512)-5*X(38807)

X(38799) lies on these lines: {3,5512}, {4,11258}, {111,382}, {126,3851}, {381,10717}, {543,14269}, {546,14360}, {547,37749}, {1296,1656}, {1657,14650}, {2780,38724}, {2793,38732}, {2805,38755}, {2813,38767}, {2819,38779}, {2854,38789}, {3325,9669}, {3627,14654}, {3830,23699}, {3843,10748}, {5076,10734}, {6019,9654}, {9172,15681}, {14657,18378}, {14666,15684}, {14856,18346}, {24206,37751}, {32424,38335}

X(38799) = midpoint of X(22338) and X(38796)
X(38799) = reflection of X(i) in X(j) for these (i,j): (3, 38796), (38796, 5512), (38798, 38804), (38804, 38802)
X(38799) = X(38800)-Gibert-Moses centroid
X(38799) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 22338, 38800), (5512, 22338, 3), (5512, 38801, 22338), (5512, 38804, 38802), (6719, 38797, 3), (38796, 38798, 38804), (38798, 38804, 3), (38802, 38804, 38796), (38805, 38806, 3)

X(38800) = X(111)-CIRCUM-EULER-POINT OF X(382)

Barycentrics    3*a^10-12*(b^2+c^2)*a^8-(10*b^4-83*b^2*c^2+10*c^4)*a^6+(b^2+c^2)*(14*b^4-61*b^2*c^2+14*c^4)*a^4+(b^4-4*b^2*c^2+c^4)*(7*b^4-26*b^2*c^2+7*c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(-2*b^4+8*b^2*c^2-2*c^4) : :
X(38800) = 3*X(3)-4*X(5512) = 7*X(3)-8*X(6719) = 5*X(3)-6*X(38796) = 3*X(3)-2*X(38797) = 7*X(3)-6*X(38798) = 2*X(3)-3*X(38799) = 5*X(3)-8*X(38801) = 9*X(3)-8*X(38803) = 11*X(3)-12*X(38804) = 5*X(3)-4*X(38805) = 9*X(3)-10*X(38806) = 17*X(3)-20*X(38807) = 7*X(5512)-6*X(6719) = 2*X(5512)-3*X(22338) = 10*X(5512)-9*X(38796) = 14*X(5512)-9*X(38798) = 8*X(5512)-9*X(38799) = 5*X(5512)-6*X(38801) = 19*X(5512)-18*X(38802) = 3*X(5512)-2*X(38803) = 11*X(5512)-9*X(38804) = 5*X(5512)-3*X(38805) = 6*X(5512)-5*X(38806) = 17*X(5512)-15*X(38807)

X(38800) lies on these lines: {3,5512}, {30,11258}, {111,1657}, {126,3843}, {381,1296}, {382,10734}, {543,14692}, {2780,12902}, {2793,38733}, {2805,38756}, {2813,38768}, {2819,38780}, {2854,38790}, {3325,9668}, {3534,14650}, {3627,14360}, {3818,37751}, {3830,10748}, {3845,37749}, {5073,23699}, {5899,14657}, {6019,9655}, {8976,11835}, {9172,15689}, {10717,38335}, {11836,13951}, {14666,15685}, {20099,33703}, {35447,38724}

X(38800) = midpoint of X(20099) and X(33703)
X(38800) = reflection of X(i) in X(j) for these (i,j): (3, 22338), (1657, 111), (14360, 3627), (15685, 14666), (37749, 3845), (37751, 3818), (38797, 5512), (38805, 38801)
X(38800) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 22338, 38799), (5512, 38797, 3), (5512, 38803, 38806), (6719, 38798, 3), (22338, 38796, 38801), (22338, 38797, 5512), (38796, 38805, 3), (38797, 38806, 38803), (38801, 38805, 38796), (38803, 38806, 3)

X(38801) = X(111)-CIRCUM-EULER-POINT OF X(546)

Barycentrics    2*a^10-8*(b^2+c^2)*a^8-3*(5*b^4-24*b^2*c^2+5*c^4)*a^6+11*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^4+(b^4-4*b^2*c^2+c^4)*(13*b^4-34*b^2*c^2+13*c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(-3*b^4+12*b^2*c^2-3*c^4) : :
X(38801) = X(3)-3*X(5512) = 2*X(3)-3*X(6719) = X(3)+3*X(22338) = 5*X(3)-9*X(38796) = 7*X(3)-3*X(38797) = 13*X(3)-9*X(38798) = X(3)-9*X(38799) = 5*X(3)+3*X(38800) = 4*X(3)-9*X(38802) = 4*X(3)-3*X(38803) = 7*X(3)-9*X(38804) = 5*X(3)-3*X(38805) = 11*X(3)-15*X(38806) = 3*X(3)-5*X(38807) = 5*X(5512)-3*X(38796) = 7*X(5512)-X(38797) = 13*X(5512)-3*X(38798) = X(5512)-3*X(38799) = 5*X(5512)+X(38800) = 4*X(5512)-3*X(38802) = 4*X(5512)-X(38803) = 7*X(5512)-3*X(38804) = 5*X(5512)-X(38805) = 11*X(5512)-5*X(38806) = 9*X(5512)-5*X(38807)

X(38801) lies on these lines: {3,5512}, {4,543}, {20,9172}, {111,3146}, {126,3091}, {546,33962}, {1296,3090}, {2780,36253}, {2793,38734}, {2805,38757}, {2813,38769}, {2819,38781}, {2854,38791}, {3627,23699}, {3832,10717}, {3853,32424}, {5056,37749}, {5073,14666}, {5076,11258}, {14645,14856}, {14650,15704}

X(38801) = midpoint of X(i) and X(j) for these {i,j}: {5512, 22338}, {38800, 38805}
X(38801) = reflection of X(i) in X(j) for these (i,j): (6719, 5512), (38803, 6719)
X(38801) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5512, 6719, 38802), (5512, 38805, 38796), (22338, 38796, 38800), (22338, 38799, 5512), (38796, 38800, 38805), (38802, 38803, 6719)

X(38802) = X(111)-CIRCUM-EULER-POINT OF X(547)

Barycentrics    2*a^10-8*(b^2+c^2)*a^8+(25*b^4-8*b^2*c^2+25*c^4)*a^6+(b^2+c^2)*(3*b^4-28*b^2*c^2+3*c^4)*a^4-(b^4-4*b^2*c^2+c^4)*(27*b^4-46*b^2*c^2+27*c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(5*b^4-20*b^2*c^2+5*c^4) : :
X(38802) = X(3)+5*X(5512) = 2*X(3)-5*X(6719) = 7*X(3)+5*X(22338) = X(3)-5*X(38796) = 17*X(3)-5*X(38797) = 9*X(3)-5*X(38798) = 3*X(3)+5*X(38799) = 19*X(3)+5*X(38800) = 4*X(3)+5*X(38801) = 8*X(3)-5*X(38803) = 3*X(3)-5*X(38804) = 11*X(3)-5*X(38805) = 2*X(5512)+X(6719) = 7*X(5512)-X(22338) = 17*X(5512)+X(38797) = 9*X(5512)+X(38798) = 3*X(5512)-X(38799) = 19*X(5512)-X(38800) = 4*X(5512)-X(38801) = 8*X(5512)+X(38803) = 3*X(5512)+X(38804) = 11*X(5512)+X(38805) = 13*X(5512)+5*X(38806) = 7*X(5512)+5*X(38807)

X(38802) lies on these lines: {3,5512}, {111,3832}, {126,5056}, {543,3545}, {547,33962}, {1296,3533}, {2780,38725}, {2793,38735}, {2805,38758}, {2813,38770}, {2819,38782}, {2854,38792}, {3543,9172}, {3845,23699}

X(38802) = midpoint of X(i) and X(j) for these {i,j}: {5512, 38796}, {38799, 38804}
X(38802) = reflection of X(6719) in X(38796)
X(38802) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5512, 6719, 38801), (5512, 38804, 38799), (5512, 38807, 22338), (6719, 38801, 38803), (38796, 38799, 38804)

X(38803) = X(111)-CIRCUM-EULER-POINT OF X(548)

Barycentrics    6*a^10-24*(b^2+c^2)*a^8-(5*b^4-136*b^2*c^2+5*c^4)*a^6+(b^2+c^2)*(25*b^4-116*b^2*c^2+25*c^4)*a^4-(b^4-4*b^2*c^2+c^4)*(b^4+22*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2)*(b^4-4*b^2*c^2+c^4) : :
X(38803) = 3*X(3)-X(5512) = 5*X(3)-X(22338) = 7*X(3)-3*X(38796) = 3*X(3)+X(38797) = X(3)+3*X(38798) = 11*X(3)-3*X(38799) = 9*X(3)-X(38800) = 4*X(3)-X(38801) = 8*X(3)-3*X(38802) = 5*X(3)-3*X(38804) = 9*X(3)-5*X(38806) = 11*X(3)-5*X(38807) = 2*X(5512)-3*X(6719) = 5*X(5512)-3*X(22338) = 7*X(5512)-9*X(38796) = X(5512)+9*X(38798) = 11*X(5512)-9*X(38799) = 3*X(5512)-X(38800) = 4*X(5512)-3*X(38801) = 8*X(5512)-9*X(38802) = 5*X(5512)-9*X(38804) = X(5512)+3*X(38805) = 3*X(5512)-5*X(38806) = 11*X(5512)-15*X(38807)

X(38803) lies on these lines: {3,5512}, {20,126}, {98,376}, {111,3522}, {548,33962}, {550,23699}, {2780,38726}, {2793,38736}, {2805,38759}, {2813,38771}, {2819,38783}, {2847,34808}, {2854,37853}, {3534,10748}, {8703,14650}, {9172,10304}, {9778,10704}, {11258,15688}, {14657,35243}, {14666,15695}, {15690,32424}, {35447,38723}

X(38803) = midpoint of X(i) and X(j) for these {i,j}: {3, 38805}, {20, 126}, {5512, 38797}
X(38803) = reflection of X(i) in X(j) for these (i,j): (6719, 3), (38801, 6719)
X(38803) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 22338, 38804), (3, 38797, 5512), (3, 38798, 38805), (3, 38800, 38806), (5512, 38805, 38797), (6719, 38801, 38802), (38797, 38806, 38800), (38800, 38806, 5512)

X(38804) = X(111)-CIRCUM-EULER-POINT OF X(549)

Barycentrics    4*a^10-16*(b^2+c^2)*a^8+(5*b^4+74*b^2*c^2+5*c^4)*a^6+(b^2+c^2)*(15*b^4-74*b^2*c^2+15*c^4)*a^4-(9*b^4-2*b^2*c^2+9*c^4)*(b^4-4*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(b^4-4*b^2*c^2+c^4) : :
X(38804) = 2*X(3)+X(5512) = X(3)+2*X(6719) = 5*X(3)+X(22338) = 7*X(3)-X(38797) = 3*X(3)-X(38798) = 3*X(3)+X(38799) = 11*X(3)+X(38800) = 7*X(3)+2*X(38801) = 3*X(3)+2*X(38802) = 5*X(3)-2*X(38803) = 4*X(3)-X(38805) = X(3)+5*X(38806) = 4*X(3)+5*X(38807) = X(5512)-4*X(6719) = 5*X(5512)-2*X(22338) = 7*X(5512)+2*X(38797) = 3*X(5512)+2*X(38798) = 3*X(5512)-2*X(38799) = 11*X(5512)-2*X(38800) = 7*X(5512)-4*X(38801) = 3*X(5512)-4*X(38802) = 5*X(5512)+4*X(38803) = 2*X(5512)+X(38805) = X(5512)-10*X(38806) = 2*X(5512)-5*X(38807)

X(38804) lies on these lines: {2,23699}, {3,5512}, {111,631}, {126,140}, {543,5054}, {549,9172}, {1296,3523}, {2780,38727}, {2793,38737}, {2805,38760}, {2813,38772}, {2819,38784}, {2830,21154}, {2854,38793}, {3090,10734}, {3525,14654}, {3526,10748}, {6684,11721}, {6699,9129}, {9179,31379}, {10303,14360}, {10519,36696}, {10717,15702}, {11258,15720}, {11539,32424}, {14666,15694}

X(38804) = midpoint of X(i) and X(j) for these {i,j}: {3, 38796}, {10519, 36696}, {38798, 38799}
X(38804) = reflection of X(i) in X(j) for these (i,j): (5512, 38796), (38796, 6719), (38799, 38802)
X(38804) = X(5512)-Gibert-Moses centroid
X(38804) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5512, 38805), (3, 6719, 5512), (3, 22338, 38803), (3, 38799, 38798), (3, 38806, 6719), (140, 14650, 126), (5512, 6719, 38807), (38796, 38798, 38799), (38796, 38799, 38802), (38799, 38802, 5512), (38805, 38807, 5512)

X(38805) = X(111)-CIRCUM-EULER-POINT OF X(550)

Barycentrics    4*a^10-16*(b^2+c^2)*a^8-(5*b^4-94*b^2*c^2+5*c^4)*a^6+(b^2+c^2)*(17*b^4-78*b^2*c^2+17*c^4)*a^4+(b^4-4*b^2*c^2+c^4)*(b^2-4*b*c-c^2)*(b^2+4*b*c-c^2)*a^2-(b^4-c^4)*(b^2-c^2)*(b^4-4*b^2*c^2+c^4) : :
X(38805) = 3*X(3)-2*X(6719) = 3*X(3)-X(22338) = 5*X(3)-3*X(38796) = X(3)-3*X(38798) = 7*X(3)-3*X(38799) = 5*X(3)-X(38800) = 5*X(3)-2*X(38801) = 11*X(3)-6*X(38802) = 4*X(3)-3*X(38804) = 7*X(3)-5*X(38806) = 8*X(3)-5*X(38807) = 3*X(5512)-4*X(6719) = 3*X(5512)-2*X(22338) = 5*X(5512)-6*X(38796) = X(5512)+2*X(38797) = X(5512)-6*X(38798) = 7*X(5512)-6*X(38799) = 5*X(5512)-2*X(38800) = 5*X(5512)-4*X(38801) = 11*X(5512)-12*X(38802) = X(5512)-4*X(38803) = 2*X(5512)-3*X(38804) = 7*X(5512)-10*X(38806) = 4*X(5512)-5*X(38807)

X(38805) lies on these lines: {3,5512}, {20,1296}, {30,126}, {111,376}, {543,3534}, {548,14650}, {550,33962}, {1657,10748}, {2780,16163}, {2793,38738}, {2805,38761}, {2813,38773}, {2819,38785}, {2830,24466}, {2854,16111}, {3184,9529}, {3325,15338}, {3529,10734}, {6019,15326}, {6361,10704}, {6560,11835}, {6561,11836}, {8703,9172}, {9129,38726}, {10717,11001}, {11258,15696}, {11414,14657}, {12121,35447}, {14654,17538}, {14666,15689}, {14688,29181}, {15686,32424}, {20099,37749}

X(38805) = midpoint of X(i) and X(j) for these {i,j}: {3, 38797}, {20, 1296}, {1657, 10748}, {3529, 10734}, {6361, 10704}, {10717, 11001}, {12121, 35447}
X(38805) = reflection of X(i) in X(j) for these (i,j): (3, 38803), (5512, 3), (9129, 38726), (9172, 8703), (14650, 548), (22338, 6719), (38800, 38801)
X(38805) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5512, 38804), (3, 22338, 6719), (3, 38798, 38803), (3, 38799, 38806), (3, 38800, 38796), (5512, 38804, 38807), (6719, 22338, 5512), (38796, 38800, 38801), (38796, 38801, 5512), (38797, 38798, 3), (38797, 38803, 5512)

X(38806) = X(111)-CIRCUM-EULER-POINT OF X(631)

Barycentrics    3*a^10-12*(b^2+c^2)*a^8+(5*b^4+53*b^2*c^2+5*c^4)*a^6+11*(b^2+c^2)*(b^4-5*b^2*c^2+c^4)*a^4-4*(b^4-4*b^2*c^2+c^4)*(2*b^4-b^2*c^2+2*c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(b^4-4*b^2*c^2+c^4) : :
X(38806) = 6*X(2)-X(10748) = 3*X(2)+2*X(14650) = 9*X(2)+X(14654) = 4*X(2)+X(14666) = 3*X(3)+2*X(5512) = X(3)+4*X(6719) = 4*X(3)+X(22338) = 2*X(3)+3*X(38796) = 6*X(3)-X(38797) = 8*X(3)-3*X(38798) = 7*X(3)+3*X(38799) = 9*X(3)+X(38800) = 11*X(3)+4*X(38801) = 13*X(3)+12*X(38802) = 9*X(3)-4*X(38803) = X(3)-6*X(38804) = 7*X(3)-2*X(38805) = X(3)+2*X(38807) = X(10748)+4*X(14650) = 3*X(10748)+2*X(14654) = 2*X(10748)+3*X(14666) = 6*X(14650)-X(14654) = 8*X(14650)-3*X(14666) = 4*X(14654)-9*X(14666)

X(38806) lies on these lines: {2,10748}, {3,5512}, {5,10734}, {111,140}, {126,3526}, {543,15694}, {549,1296}, {631,33962}, {1656,23699}, {2780,38728}, {2793,38739}, {2805,38762}, {2813,38774}, {2819,38786}, {2854,38794}, {3525,14360}, {5054,9172}, {9129,15061}, {10704,38028}, {10717,11539}, {10765,38110}, {10779,34126}, {11721,26446}, {15719,37749}, {35447,38727}

X(38806) = reflection of X(38807) in X(6719)
X(38806) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 14650, 10748), (3, 5512, 38797), (3, 6719, 38796), (3, 22338, 38798), (3, 38796, 22338), (3, 38799, 38805), (3, 38800, 38803), (5512, 38797, 22338), (5512, 38803, 38800), (6719, 38804, 3), (10748, 14650, 14666), (38796, 38797, 5512), (38800, 38803, 38797)

X(38807) = X(111)-CIRCUM-EULER-POINT OF X(632)

Barycentrics    4*a^10-16*(b^2+c^2)*a^8+3*(5*b^4+18*b^2*c^2+5*c^4)*a^6+(b^2+c^2)*(13*b^4-70*b^2*c^2+13*c^4)*a^4-(19*b^4-22*b^2*c^2+19*c^4)*(b^4-4*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(3*b^4-12*b^2*c^2+3*c^4) : :
X(38807) = 2*X(3)+3*X(5512) = X(3)-6*X(6719) = 7*X(3)+3*X(22338) = X(3)+9*X(38796) = 13*X(3)-3*X(38797) = 19*X(3)-9*X(38798) = 11*X(3)+9*X(38799) = 17*X(3)+3*X(38800) = 3*X(3)+2*X(38801) = 7*X(3)+18*X(38802) = 11*X(3)-6*X(38803) = 4*X(3)-9*X(38804) = 8*X(3)-3*X(38805) = X(3)-3*X(38806) = X(5512)+4*X(6719) = 7*X(5512)-2*X(22338) = X(5512)-6*X(38796) = 13*X(5512)+2*X(38797) = 19*X(5512)+6*X(38798) = 11*X(5512)-6*X(38799) = 17*X(5512)-2*X(38800) = 9*X(5512)-4*X(38801) = 7*X(5512)-12*X(38802) = 11*X(5512)+4*X(38803) = 2*X(5512)+3*X(38804) = 4*X(5512)+X(38805) = X(5512)+2*X(38806)

X(38807) lies on these lines: {3,5512}, {5,9172}, {111,3090}, {126,3628}, {543,1656}, {546,14650}, {632,33962}, {1296,10303}, {2780,38729}, {2793,38740}, {2805,38763}, {2813,38775}, {2819,38787}, {2854,38795}, {3091,23699}, {3544,14654}, {3851,14666}, {5067,10717}, {5079,10748}, {9129,36253}

X(38807) = reflection of X(38806) in X(6719)
X(38807) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5512, 6719, 38804), (5512, 38804, 38805), (6719, 38796, 5512), (22338, 38802, 5512)

X(38808) =  ISOGONAL CONJUGATE OF X(8798)

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) (3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4) (a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4) : :
Barycentrics    SB SC (S^2+SA SC) (-S^2+2 SB SC) (-2 S^2+(SA+SB) SC) : :
X(38808) = X(4)-3*X(3462),X(4)-6*X(33549),8*X(140)-3*X(15319)

See Kadir Altintas and Ercole Suppa, Euclid 904 .

X(38808) lies on the cubics K361, K618 and these lines: {3,14371}, {4,54}, {95,253}, {96,10018}, {97,3346}, {107,10282}, {140,6760}, {154,14249}, {185,19209}, {186,18349}, {550,933}, {648,7488}, {1075,11202}, {1093,9707}, {1141,33640}, {1249,33629}, {1656,19176}, {1990,14533}, {2052,19357}, {2167,5882}, {2190,3668}, {2883,10152}, {3515,9307}, {3516,16030}, {3517,19173}, {4993,5068}, {5056,19188}, {5059,16251}, {6146,14165}, {7691,35311}, {7768,18831}, {8794,37070}, {8795,14528}, {10990,19208}, {14615,35602}, {14863,15712}, {16252,34170}, {19185,32534}, {19192,35477}, {19193,30714}

X(38808) = isogonal conjugate of X(8798)
X(38808) = reflection of X(3462) in X(33549)
X(38808) = polar conjugate of X(13157)
X(38808) = X(95)-Ceva conjugate of X(275)
X(38808) = X(154)-cross conjugate of X(33629)
X(38808) = X(i)-isoconjugate of X(j) for these (i,j): (5,19614), (48,13157), (51,19611), (216,2184), (343,2155)
X(38808)= X(i)-reciprocal conjugate of X(j) for these {i,j}: {6,8798}, {20,343}, {54,1073}, {95,34403}, {97,15394}
X(38808) = X(4)-vertex conjugate of X(14371)
X(38808) = cevapoint of X(154) and X(1249)
X(38808) = barycentric product X(i)*X(j) for these (i,j): (20,275), (54,15466), (95,1249), (97,14249), (154,276)
X(38808) = barycentric quotient X(i)/X(j) for these {i,j}: {20,343}, {54,1073}, {95,34403}, {97,15394}, {154,216}
X(38808) = trilinear product X(i)*X(j) for these (i,j): (20,2190), (54,1895), (92,33629), (95,204), (275,610)
X(38808) = trilinear quotient X(i)/X(j) for these (i,j): (54,19614), (95,19611), (204,51), (275,2184), (610,216)
X(38808) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (54,8884,275), (275,8884,19169)

leftri

Vu cevian tangential perspectors: X(38809)-X(38824)

rightri

This preamble is based on notes from Vu Thanh Tung, May 31, 2020.

Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC, and let

A1B1C1 = cevian triangle of P
ta = line tangent to the circle (AB'C') at A, and define tb and tc cyclically
A' = tb∩tc, and define B' and C' cyclically.

The triangle A'B'C' is the anticevian triangle of the point

V(P) = a^2/(p q + p r) : b^2/(q r + q p) : c^2/(r p + r q),

which is the perspector of ABC and A'B'C'.

See Vu Cevian Tangential Perspector.

The appearance of (i,j) in the following list means that V(X(i)) = X(j):
(1,81), (2,6), (3,275), (4,2), (5,288), (6,83), (7,1), (8,57), (9,1170), (10,1171), (11,38809), (31, 38810), (32, 38811), (75,58), (76,251), (560,38812), (561, 38813)

The triangle A'B'C' is also perspective to A1B1C1 , the perspector, denoted by V'(P), being the P-Ceva-conjugate of V(P):

V'(P) = a^2/(p(q+r)) * (b^2/(q^2(r+p)) + c^2/(r^2(p+q)) - a^2/(p^2(q+r))) : :

The appearance of (i,j) in the following list means that V'(X(i)) = X(j):
(1,38814), (2,3), (3,38815), (4,193), (5,38816), (6,38817), (7,57), (8,34488), (9,38818), (10,38819), (31,38820), (32,38821)

V(P) is the cevapoint of X(6) and P', where P' is the isogonal conjugate of P. (Randy Hutson, May 31, 2020)

X(38809) = VU CEVIAN TANGENTIAL PERSPECTOR OF X(11)

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

X(38809) lies on these lines: {59,3271}, {1086,1275}

X(38809) = isogonal conjugate of the complement of X(4998)
X(38809) = isogonal conjugate of the isotomic conjugate of X(31619)
X(38809) = complement of the anticomplementary conjugate of X(4998)
X(38809) = barycentric product X(i)*X(j) for these {i, j}: {6, 31619}, {651, 31628}
X(38809) = barycentric quotient X(i)/X(j) for these (i, j): (59, 3035), (109, 21105), (692, 11124), (2149, 17439)
X(38809) = trilinear product X(i)*X(j) for these {i, j}: {31, 31619}, {109, 31628}
X(38809) = trilinear quotient X(i)/X(j) for these (i, j): (59, 17439), (101, 11124), (651, 21105), (2149, 20958)
X(38809) = 1st Saragossa point of X(3271)
X(38809) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(1086)}} and {{A, B, C, X(59), X(651)}}
X(38809) = cevapoint of X(6) and X(59)
X(38809) = X(6)-cross conjugate of-X(18771)
X(38809) = X(i)-isoconjugate-of-X(j) for these {i,j}: {11, 17439}, {514, 11124}, {650, 21105}, {2170, 3035}
X(38809) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (59, 3035), (109, 21105), (692, 11124), (2149, 17439)

X(38810) = VU CEVIAN TANGENTIAL PERSPECTOR OF X(31)

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

X(38810) lies on these lines: {6,706}, {75,560}, {239,7104}, {274,7132}, {314,983}, {717,9063}, {723,789}, {894,1922}, {1178,33296}, {7304,7307}, {17033,17743}

X(38810) = isogonal conjugate of X(16584)
X(38810) = isotomic conjugate of X(3721)
X(38810) = complement of the anticomplementary conjugate of X(561)
X(38810) = barycentric product X(i)*X(j) for these {i, j}: {37, 7307}, {58, 7034}, {86, 7033}, {274, 17743}, {310, 983}, {313, 7305}
X(38810) = barycentric quotient X(i)/X(j) for these (i, j): (1, 3778), (9, 20684), (10, 7237), (21, 3056), (32, 21751), (55, 4531)
X(38810) = trilinear product X(i)*X(j) for these {i, j}: {42, 7307}, {76, 38813}, {81, 7033}, {86, 17743}, {190, 7255}, {274, 983}
X(38810) = trilinear quotient X(i)/X(j) for these (i, j): (2, 3778), (8, 20684), (9, 4531), (21, 20665), (31, 21751), (32, 8022)
X(38810) = 1st Saragossa point of X(560)
X(38810) = trilinear pole of the line {814, 7255}
X(38810) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(4362)}} and {{A, B, C, X(6), X(560)}}
X(38810) = cevapoint of X(i) and X(j) for these {i,j}: {6, 75}, {333, 33296}
X(38810) = X(31)-Ceva conjugate of X(38820)
X(38810) = X(6)-cross conjugate of X(38813)
X(38810) = X(i)-isoconjugate-of X(j) for these {i,j}: {6, 3778}, {25, 20727}, {32, 2887}, {37, 7032}
X(38810) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 3778), (9, 20684), (10, 7237), (21, 3056)
X(38810) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 1502, 33776), (75, 38813, 38840), (38813, 38837, 560)

X(38811) = VU CEVIAN TANGENTIAL PERSPECTOR OF X(2321)

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

X(38811) lies on these lines: {391,34016}, {1334,1412}

X(38811) = isogonal conjugate of X(38930)
X(38811) = isogonal conjugate of the complement of X(1434)
X(38811) = isogonal conjugate of the complementary conjugate of X(17050)
X(38811) = complement of the anticomplementary conjugate of X(1434)
X(38811) = barycentric product X(1434)*X(38825)
X(38811) = barycentric quotient X(i)/X(j) for these (i, j): (42, 21673), (56, 4854), (1412, 3946)
X(38811) = trilinear product X(1014)*X(38825)
X(38811) = trilinear quotient X(i)/X(j) for these (i, j): (37, 21673), (57, 4854), (1014, 3946)
X(38811) = 1st Saragossa point of X(1334)
X(38811) = trilinear pole of the line {2605, 8653}
X(38811) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(391)}} and {{A, B, C, X(35), X(57)}}
X(38811) = cevapoint of X(6) and X(1412)
X(38811) = X(6)-cross conjugate of X(38825)
X(38811) = X(i)-isoconjugate-of X(j) for these {i,j}: {9, 4854}, {81, 21673}, {210, 3946}
X(38811) = X(i)-reciprocal conjugate of X(j) for these (i,j): (42, 21673), (56, 4854), (1412, 3946)

X(38812) = VU CEVIAN TANGENTIAL PERSPECTOR OF X(560)

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

X(38812) lies on these lines: {561,1917}, {9065,23626}

X(38812) = isogonal conjugate of the complement of X(1928)
X(38812) = complement of the anticomplementary conjugate of X(1928)
X(38812) = barycentric product X(1928)*X(38827)
X(38812) = barycentric quotient X(i)/X(j) for these (i, j): (75, 23626), (76, 21324), (304, 22404), (310, 18168), (561, 21235), (1502, 21409)
X(38812) = trilinear product X(1502)*X(38827)
X(38812) = trilinear quotient X(i)/X(j) for these (i, j): (76, 23626), (305, 22404), (561, 21324), (1502, 21235), (1928, 21409)
X(38812) = 1st Saragossa point of X(1917)
X(38812) = trilinear pole of the line {792, 23403}
X(38812) = cevapoint of X(6) and X(561)
X(38812) = X(560)-Ceva conjugate of X(38823)
X(38812) = X(6)-cross conjugate of X(38827)
X(38812) = Vu cevian-circles perspector of X(1928)
X(38812) = X(i)-isoconjugate-of X(j) for these {i,j}: {32, 23626}, {560, 21324}, {1501, 21235}, {1917, 21409}
X(38812) = X(i)-reciprocal conjugate of X(j) for these (i,j): (75, 23626), (76, 21324), (304, 22404), (310, 18168)
X(38812) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (561, 38827, 38843), (38827, 38839, 1917)

X(38813) = VU CEVIAN TANGENTIAL PERSPECTOR OF X(561)

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

X(38813) lies on these lines: {6,6660}, {21,976}, {28,7132}, {58,1469}, {60,3736}, {75,560}, {256,2210}, {261,7305}, {284,2273}, {291,7122}, {723,825}, {759,8685}, {1014,7204}, {1178,2194}, {1333,2311}, {2206,13588}, {3778,38831}, {4492,8424}

X(38813) = isogonal conjugate of X(2887)
X(38813) = anticomplement of the complementary conjugate of X(6679)
X(38813) = complement of the anticomplementary conjugate of X(31)
X(38813) = barycentric product X(i)*X(j) for these {i, j}: {21, 7132}, {31, 38810}, {37, 7305}, {58, 17743}, {81, 983}, {101, 7255}
X(38813) = barycentric quotient X(i)/X(j) for these (i, j): (1, 20234), (31, 3721), (32, 3778), (42, 16886), (55, 4136), (56, 16888)
X(38813) = trilinear product X(i)*X(j) for these {i, j}: {32, 38810}, {42, 7305}, {58, 983}, {284, 7132}, {692, 7255}, {788, 33514}
X(38813) = trilinear quotient X(i)/X(j) for these (i, j): (2, 20234), (6, 3721), (9, 4136), (21, 3705), (31, 3778), (32, 16584)
X(38813) = 1st Saragossa point of X(75)
X(38813) = trilinear pole of the line {3250, 7252}
X(38813) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(976)}} and {{A, B, C, X(6), X(75)}}
X(38813) = cevapoint of X(6) and X(560)
X(38813) = X(561)-Ceva conjugate of X(38824)
X(38813) = X(i)-cross conjugate of X(j) for these (i,j): (6, 38810), (814, 15440)
X(38813) = X(i)-isoconjugate-of X(j) for these {i,j}: {2, 3721}, {6, 20234}, {9, 16888}, {10, 982}
X(38813) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 20234), (31, 3721), (32, 3778), (42, 16886)
X(38813) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (75, 560, 33772), (560, 38810, 38837), (38810, 38840, 75)

X(38814) = X(1)-CEVA CONJUGATE OF X(81)

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

X(38814) lies on the cubic K774 and these lines: {1,1326}, {2,5110}, {6,1931}, {21,238}, {37,662}, {42,6157}, {48,28606}, {58,9277}, {60,7193}, {81,593}, {86,1086}, {99,894}, {142,24617}, {192,27958}, {214,4653}, {229,2646}, {239,261}, {284,3512}, {409,1958}, {501,3743}, {643,4689}, {645,17261}, {992,19237}, {1014,1429}, {1255,21353}, {1442,1950}, {1621,1964}, {1654,6626}, {1790,25058}, {2076,28369}, {2269,17209}, {2287,4469}, {2651,4414}, {2669,17000}, {2905,17084}, {3552,17103}, {3733,4068}, {4307,35915}, {4360,19623}, {4560,25255}, {4645,35916}, {4850,37791}, {5153,32911}, {5196,33112}, {5235,32851}, {5333,19786}, {7058,38000}, {8298,8935}, {15988,36790}, {16466,37029}, {16666,33766}, {16706,25536}, {16865,27640}, {17363,34016}, {19642,29639}, {22370,37574}, {24454,33761}, {27569,27954}

X(38814) = isogonal conjugate of the antitomic conjugate of X(9278)
X(38814) = barycentric product X(i)*X(j) for these {i, j}: {1, 6626}, {21, 17084}, {58, 17762}, {81, 1654}, {86, 846}, {274, 18755}
X(38814) = barycentric quotient X(i)/X(j) for these (i, j): (48, 15377), (58, 13610), (81, 6625), (846, 10), (1333, 2248), (1654, 321)
X(38814) = trilinear product X(i)*X(j) for these {i, j}: {6, 6626}, {27, 22139}, {58, 1654}, {60, 27691}, {81, 846}, {86, 18755}
X(38814) = trilinear quotient X(i)/X(j) for these (i, j): (3, 15377), (58, 2248), (81, 13610), (86, 6625), (846, 37), (1333, 18757)
X(38814) = intersection, other than A,B,C, of conics {{A, B, C, X(28), X(17689)}} and {{A, B, C, X(81), X(256)}}
X(38814) = cevapoint of X(846) and X(18755)
X(38814) = crossdifference of every pair of points on line {X(4705), X(9279)}
X(38814) = crosspoint of X(1) and X(846)
X(38814) = crosssum of X(i) and X(j) for these {i,j}: {1, 13610}, {512, 21823}, {661, 21833}
X(38814) = X(1)-Ceva conjugate of X(81)
X(38814) = X(846)-cross conjugate of X(6626)
X(38814) = X(i)-Hirst inverse of X(j) for these {i,j}: {6, 1931}, {1931, 6}
X(38814) = X(i)-isoconjugate-of X(j) for these {i,j}: {4, 15377}, {10, 2248}, {37, 13610}, {42, 6625}
X(38814) = X(i)-reciprocal conjugate of X(j) for these (i,j): (48, 15377), (58, 13610), (81, 6625), (846, 10)
X(38814) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (593, 17011, 81), (757, 1100, 81), (1100, 16702, 757), (2185, 3666, 81), (17011, 17190, 593)

X(38815) = X(3)-CEVA CONJUGATE OF X(275)

Barycentrics    (a^2 + b^2 - c^2)(a^2 - b^2 + c^2) (a^4 + b^4 - b^2c^2 - a^2(2b^2 + c^2))(a^4 - b^2c^2 + c^4 - a^2(b^2 + 2c^2))(-(b^4c^4(b^2 - c^2)^6) - a^2b^2c^2(b^2 - c^2)^6(b^2 + c^2) + a^16(b^4 + 3b^2c^2 + c^4) - a^14(6b^6 + 11b^4c^2 + 11b^2c^4 + 6c^6) + a^4(b^2 - c^2)^4(b^8 + 5b^6c^2 + 7b^4c^4 + 5b^2c^6 + c^8) + 3a^8(b^2 - c^2)^2(5b^8 + 5b^6c^2 + 3b^4c^4 + 5b^2c^6 + 5c^8) + a^12(15b^8 + 11b^6c^2 + 21b^4c^4 + 11b^2c^6 + 15c^8) - a^6(b^2 - c^2)^2(6b^10 + 5b^8c^2 - 3b^6c^4 - 3b^4c^6 + 5b^2c^8 + 6c^10) - a^10(20b^10 -5b^8c^2 + 9b^6c^4 + 9b^4c^6 - 5b^2c^8 + 20c^10)) : :

X(38815) lies on these lines: {95,37871}, {184,21449}, {389,8884}, {26902,38808}

X(38815) = X(3)-Ceva conjugate of X(275)

X(38816) = X(5)-CEVA CONJUGATE OF X(288)

Barycentrics    a^2(a^4 + b^4 - b^2c^2 - a^2(2b^2 + c^2)) (a^4 - b^2c^2 + c^4 - a^2(b^2 + 2c^2))(a^4 + 2b^4 - 3b^2c^2 + c^4 - a^2(3b^2 + 2c^2))(a^4 + b^4 - 3b^2c^2 + 2c^4 - a^2(2b^2 + 3c^2)) (a^26 - 11a^24(b^2 + c^2) - (b^2 - c^2)^12(b^2 + c^2) + a^22(54b^4 + 91b^2c^2 + 54c^4) - 2a^20(79b^6 + 162b^4c^2 + 162b^2c^4 + 79c^6) + a^18(313b^8 + 654b^6c^2 + 801b^4c^4 + 654b^2c^6 + 313c^8) - a^4(b^2 - c^2)^6(54b^10 + 4b^8c^2 - 59b^6c^4 - 59b^4c^6 + 4b^2c^8 + 54c^10) - 3a^16(153b^10 + 275b^8c^2 + 349b^6c^4 + 349b^4c^6 + 275b^2c^8 + 153c^10) + a^2(b^2 - c^2)^6 (11b^12 - 25b^10c^2 + b^8c^4 + 25b^6c^6 + b^4c^8 - 25b^2c^10 + 11c^12) + a^6(b^2 -c^2)^4(158b^12 + 12b^10c^2 - 103b^8c^4 - 147b^6c^6 - 103b^4c^8 + 12b^2c^10 + 158c^12) + a^10(b^2 - c^2)^2(459b^12 + 531b^10c^2 + 586b^8c^4 + 597b^6c^6 + 586b^4c^8 + 531b^2c^10 + 459c^12) + a^14(540b^12 + 641b^10c^2 + 737b^8c^4 + 781b^6c^6 + 737b^4c^8 + 641b^2c^10 + 540c^12) - 5a^12(108b^14 + 32b^12c^2 + 37b^10c^4 + 39b^8c^6 + 39b^6c^8 + 37b^4c^10 + 32b^2c^12 + 108c^14) - a^8(b^2 - c^2)^2(313b^14 - 75b^12c^2 - 112b^10c^4 - 150b^8c^6 - 150b^6c^8 - 112b^4c^10 - 75b^2c^12 + 313c^14)) : :

X(38816) lies on these lines: {}

X(38816) = X(5)-Ceva conjugate of X(288)

X(38817) = X(6)-CEVA CONJUGATE OF X(83)

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

X(38817) lies on the cubic K739 and these lines: {51,14970}, {83,1207}, {251,4027}, {385,1194}, {689,3051}, {894,3112}, {1186,19562}, {1915,4577}, {1916,8856}, {7760,13511}, {14885,16985}

X(38817) = barycentric product X(308)*X(3499)
X(38817) = crosspoint of X(6) and X(3499)
X(38817) = X(6)-Ceva conjugate of X(83)

X(38818) = X(9)-CEVA CONJUGATE OF X(1170)

Barycentrics    a*(a^2-(2*b+c)*a+(b-c)*b)*(a+b-c)*(a^2-(b+2*c)*a-(b-c)*c)*(a-b+c)*(a^8-(c+b)*a^7-(8*b^2+3*b*c+8*c^2)*a^6+(c+b)*(21*b^2-10*b*c+21*c^2)*a^5-(20*b^4+20*c^4+(7*b^2-6*b*c+7*c^2)*b*c)*a^4+3*(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^3-(4*b^4+4*c^4+(b^2+6*b*c+c^2)*b*c)*(b-c)^2*a^2+(b^2-c^2)*(b-c)^3*(3*b^2+2*b*c+3*c^2)*a-(b^2+3*b*c+c^2)*(b-c)^6) : :

X(38818) lies on these lines: {}

X(38818) = barycentric product X(2942)*X(21453)
X(38818) = barycentric quotient X(2942)/X(4847)
X(38818) = trilinear product X(1170)*X(2942)
X(38818) = trilinear quotient X(2942)/X(1212)
X(38818) = X(9)-Ceva conjugate of X(1170)

X(38819) = X(10)-CEVA CONJUGATE OF X(1171)

Barycentrics    a^2*(a^8+5*(c+b)*a^7+3*(3*b^2+7*b*c+3*c^2)*a^6+3*(5*b^2+11*b*c+5*c^2)*a^4*b*c+(c+b)*(7*b^2+24*b*c+7*c^2)*a^5-(c+b)*(7*b^4+7*c^4+(8*b^2-3*b*c+8*c^2)*b*c)*a^3-(9*b^6+9*c^6+(31*b^4+31*c^4+(37*b^2+29*b*c+37*c^2)*b*c)*b*c)*a^2-(5*b^4+5*c^4+2*(3*b^2-b*c+3*c^2)*b*c)*(c+b)^3*a-(b^4+c^4+(b^2-b*c+c^2)*b*c)*(c+b)^4)*(c+a)*(c+a+2*b)*(a+b)*(b+2*c+a) : :

X(38819) lies on these lines: {}

X(38819) = X(10)-Ceva conjugate of X(1171)

X(38820) = X(31)-CEVA CONJUGATE OF X(38810)

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

X(38820) lies on these lines: {}

X(38820) = X(31)-Ceva conjugate of X(38810)

X(38821) = X(32)-CEVA CONJUGATE OF X(38830)

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

X(38821) lies on these lines: {}

X(38821) = X(32)-Ceva conjugate of X(38830)

X(38822) = X(75)-CEVA CONJUGATE OF X(58)

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

X(38822) lies on these lines: {1,229}, {27,86}, {110,5285}, {284,17011}, {306,662}, {593,16470}, {1412,1443}, {1817,16586}, {2360,4511}, {2915,23130}, {14213,37793}, {17866,36022}

X(38822) = barycentric product X(i)*X(j) for these {i, j}: {27, 23130}, {58, 21287}, {81, 21376}, {86, 2915}, {1333, 21595}
X(38822) = barycentric quotient X(2915)/X(10)
X(38822) = trilinear product X(i)*X(j) for these {i, j}: {28, 23130}, {58, 21376}, {81, 2915}, {1333, 21287}, {2206, 21595}
X(38822) = trilinear quotient X(2915)/X(37)
X(38822) = intersection, other than A,B,C, of conics {{A, B, C, X(27), X(267)}} and {{A, B, C, X(58), X(21595)}}
X(38822) = crosspoint of X(75) and X(21595)
X(38822) = crosssum of X(661) and X(21046)
X(38822) = X(75)-Ceva conjugate of X(58)

X(38823) = X(560)-CEVA CONJUGATE OF X(38812)

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

X(38823) lies on these lines: {}

X(38823) = X(560)-Ceva conjugate of X(38812)

X(38824) = X(561)-CEVA CONJUGATE OF X(38813)

Barycentrics    a^2 (a + b) (a^2 - a b + b^2) (a + c) (a^2 - a c + c^2) (a^11 - b^11 + a^5 b^3 c^3 - b^8 c^3 - b^3 c^8 - c^11 + a^8 (b^3 + c^3) - a^3 (b^8 + b^5 c^3 + b^3 c^5 + c^8)) : :

X(38824) lies on these lines: {}

X(38824) = X(561)-Ceva conjugate of X(38813)

leftri

Vu cevian-circles perspectors and related cyclocevian conjugates: X(38825)-X(38843)

rightri

This preamble is based on notes from Vu Thanh Tung, June 1-3, 2020.

Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC, but not on any of the medians (i.e., the lines AG, BG, CG), and let

A1B1C1 = cevian triangle of P
A2 = the point, other than A, in which the circles (ABC) and (AB1C1) intersect, and define B2 and C2 cyclically
A' = BB2∩CC2, and define B' and C' cyclically

The triangles ABC and A'B'C' are perspective, and the perspector is the point

V(P) = a^2*(p + q)*(p + r) : b^2*(q + r)*(q + p) : c^2*(r + p)*(r + q).

Also, V(P) = complement of the anticomplementary conjugate of P (César Lozada, June 4, 2020)

Also, V(P) is the isogonal conjugate of the complement of P. (Randy Hutson, June 9, 2020)

A'B'C' is the anticevian triangle of V(P), given by

A' = -a^2*(p + q)*(p + r) : b^2*(q + r)*(q + p) : c^2*(r + p)*(r + q).

See Vu Cevian-Circles Perspector.

The triangle A'B'C' is here named the Vu cevian-circles triangle of P.

The appearance of (i,j) in the following list means that V(X(i)) = X(j):
(1,58), (3,54), (4,4), (5,1173), (6,251), (7,57), (8,1), (9,1174), (10,1126), (11,18771), (21,38825), (22, 38828), (23, 38831), (32,38826), (75,81), (76,83), (560,38827), (1501,38829), (1502,38830), (1928,38843)

Let V'(P) = cyclocevian conjugate of V(P) wrt A'B'C', which is also the perspector of A'B'C' and A2B2C2, given by

V'(P) = a^2*(p + q)*(p + r)*(p*q + p*r - q*r) : :

V'(P) is the barycentric product V(P)*[P-Ceva conjugate of X(2)] = V(P)*[anticompliment of isotomic conjugate of P]. (Randy Hutson, June 9, 2020)

The appearance of (i,j) in the following list means that V'(X(i)) = X(j):
(1,38832), (3, 26887), (4, 6353), (5,38833), (6,38834), (7,1420), (8,165), (9,38835), (10,38836), (31,38837), (32,38838), (75,21), (76,1799), (560,38839), (561,38840), (1501, 38841), (1502, 38842)

The triangle A2B2C2 is here named the Vu perspectivities triangle of P; see the preamble just before X(38848). Barycentrics for A2 are given by

A2 = a^2*(p + q)*(p + r) : b^2*(q - r)*(p + q): c^2*(r - q)*(p + r) : :

X(38825) = VU CEVIAN-CIRCLES PERSPECTOR OF X(2321)

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

X(38825) lies on these lines: {1334,1412}, {1449,3870}, {1617,2256}, {4350,16577}, {29574,33770}

X(38825) = isogonal conjugate of X(3946)
X(38825) = complement of the anticomplementary conjugate of X(2321)
X(38825) = barycentric product X(2321)*X(38811)
X(38825) = barycentric quotient X(i)/X(j) for these (i, j): (42, 4854), (56, 10521), (649, 23729)
X(38825) = trilinear product X(210)*X(38811)
X(38825) = trilinear quotient X(i)/X(j) for these (i, j): (37, 4854), (57, 10521), (513, 23729)
X(38825) = 1st Saragossa point of X(1412)
X(38825) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(218)}} and {{A, B, C, X(2), X(2291)}}
X(38825) = cevapoint of X(6) and X(1334)
X(38825) = X(6)-cross conjugate of X(38811)
X(38825) = X(i)-isoconjugate-of X(j) for these {i,j}: {9, 10521}, {81, 4854}
X(38825) = X(i)-reciprocal conjugate of X(j) for these (i,j): (42, 4854), (56, 10521), (649, 23729)

X(38826) = VU CEVIAN-CIRCLES PERSPECTOR OF X(32)

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

X(38826) lies on the circumconic with center X(3124), on the circumconic with center X(36471), and on these lines: {6,33717}, {39,5012}, {76,1501}, {141,1078}, {251,27375}, {711,827}, {1629,27376}, {1843,10312}, {1915,7828}, {14970,33515}

X(38826) = isogonal conjugate of X(626)
X(38826) = anticomplement of the complementary conjugate of X(6680)
X(38826) = complement of the anticomplementary conjugate of X(32)
X(38826) = barycentric product X(i)*X(j) for these {i, j}: {31, 38847}, {32, 38830}, {711, 16985}
X(38826) = barycentric quotient X(i)/X(j) for these (i, j): (1, 20627), (3, 4121), (31, 4118), (32, 20859), (39, 16893), (42, 16894)
X(38826) = trilinear product X(i)*X(j) for these {i, j}: {32, 38847}, {560, 38830}, {1923, 3115}, {2084, 33515}
X(38826) = trilinear quotient X(i)/X(j) for these (i, j): (2, 20627), (6, 4118), (9, 4178), (31, 20859), (32, 2085), (37, 16894)
X(38826) = 1st Saragossa point of X(76)
X(38826) = trilinear pole of the line {3005, 3050}
X(38826) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(7832)}} and {{A, B, C, X(4), X(2065)}}
X(38826) = cevapoint of X(6) and X(1501)
X(38826) = X(i)-cross conjugate of X(j) for these (i,j): (6, 38830), (804, 2715)
X(38826) = X(i)-isoconjugate-of X(j) for these {i,j}: {2, 4118}, {6, 20627}, {9, 7217}, {10, 18167}
X(38826) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 20627), (3, 4121), (31, 4118), (32, 20859)
X(38826) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (76, 1501, 33773), (1501, 38830, 38838), (38830, 38842, 76)

X(38827) = VU CEVIAN-CIRCLES PERSPECTOR OF X(560)

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

X(38827) lies on this line:i {561,1917}

X(38827) = isogonal conjugate of X(21235)
X(38827) = complement of the anticomplementary conjugate of X(560)
X(38827) = barycentric product X(560)*X(38812)
X(38827) = barycentric quotient X(i)/X(j) for these (i, j): (1, 21409), (31, 21324), (32, 23626), (184, 22404), (251, 18090), (1333, 18168)
X(38827) = trilinear product X(1501)*X(38812)
X(38827) = trilinear quotient X(i)/X(j) for these (i, j): (2, 21409), (6, 21324), (31, 23626), (48, 22404), (58, 18168), (82, 18090)
X(38827) = 1st Saragossa point of X(561)
X(38827) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(561)}} and {{A, B, C, X(251), X(9075)}}
X(38827) = cevapoint of X(6) and X(1917)
X(38827) = X(6)-cross conjugate of X(38812)
X(38827) = X(i)-isoconjugate-of X(j) for these {i,j}: {2, 21324}, {6, 21409}, {10, 18168}, {38, 18090}
X(38827) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 21409), (31, 21324), (32, 23626), (184, 22404)
X(38827) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1917, 38812, 38839), (38812, 38843, 561)

X(38828) = VU CEVIAN-CIRCLES PERSPECTOR OF X(3676)

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

X(38828) lies on the circumconic with center X(478) and on these lines: {77,27819}, {109,1293}, {190,5382}, {223,8056}, {651,23704}, {1419,1462}, {1461,34080}, {2415,4552}, {3445,10571}, {3680,9372}, {4551,31343}, {6557,18623}, {16079,38296}

X(38828) = isogonal conjugate of X(4521)
X(38828) = complement of the anticomplementary conjugate of X(3676)
X(38828) = barycentric product X(i)*X(j) for these {i, j}: {7, 1293}, {57, 27834}, {85, 34080}, {100, 19604}, {101, 27818}, {109, 4373}
X(38828) = barycentric quotient X(i)/X(j) for these (i, j): (31, 4162), (55, 4546), (56, 3667), (57, 4462), (65, 4404), (101, 3161)
X(38828) = trilinear product X(i)*X(j) for these {i, j}: {7, 34080}, {56, 27834}, {57, 1293}, {101, 19604}, {190, 16945}, {664, 38266}
X(38828) = trilinear quotient X(i)/X(j) for these (i, j): (6, 4162), (7, 4462), (9, 4546), (56, 4394), (57, 3667), (65, 14321)
X(38828) = 1st Saragossa point of X(3939)
X(38828) = trilinear pole of the line {56, 1149}
X(38828) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(3939)}} and {{A, B, C, X(57), X(190)}}
X(38828) = cevapoint of X(i) and X(j) for these {i,j}: {101, 1293}, {649, 32577}
X(38828) = crossdifference of every pair of points on line {X(4534), X(4953)}
X(38828) = X(101)-cross conjugate of X(109)
X(38828) = X(i)-isoconjugate-of X(j) for these {i,j}: {2, 4162}, {8, 4394}, {9, 3667}, {21, 14321}
X(38828) = X(i)-reciprocal conjugate of X(j) for these (i,j): (31, 4162), (55, 4546), (56, 3667), (57, 4462)
X(38828) = X(40)-Zayin conjugate of X(4394)

X(38829) = VU CEVIAN-CIRCLES PERSPECTOR OF X(1501)

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

X(38829) lies on these lines: {1176,3094}, {1502,9233}, {1799,3314}, {9229,14602}

X(38829) = isogonal conjugate of the complement of X(1501)
X(38829) = isogonal conjugate of the complementary conjugate of X(8265)
X(38829) = complement of the anticomplementary conjugate of X(1501)
X(38829) = barycentric quotient X(i)/X(j) for these (i, j): (251, 5025), (1501, 14820)
X(38829) = trilinear quotient X(i)/X(j) for these (i, j): (82, 5025), (560, 14820)
X(38829) = 1st Saragossa point of X(1502)
X(38829) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(1502)}} and {{A, B, C, X(251), X(1176)}}
X(38829) = cevapoint of X(6) and X(9233)
X(38829) = X(i)-isoconjugate-of X(j) for these {i,j}: {38, 5025}, {561, 14820}, {1930, 3981}
X(38829) = X(i)-reciprocal conjugate of X(j) for these (i,j): (251, 5025), (1501, 14820)

X(38830) = VU CEVIAN-CIRCLES PERSPECTOR OF X(1502)

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

X(38830) lies on the circumconic with center X(339), on the circumconic with center X(35078), and on these lines: {6,35530}, {76,1501}, {385,1627}, {419,1235}, {689,711}, {3051,3978}, {4039,38847}, {14603,34945}

X(38830) = isogonal conjugate of X(8265)
X(38830) = isotomic conjugate of X(20859)
X(38830) = complement of the anticomplementary conjugate of X(1502)
X(38830) = barycentric product X(i)*X(j) for these {i, j}: {66, 38842}, {75, 38847}, {141, 3115}, {1502, 38826}
X(38830) = barycentric quotient X(i)/X(j) for these (i, j): (1, 2085), (3, 4173), (39, 3118), (69, 20819), (75, 4118), (76, 626)
X(38830) = trilinear product X(i)*X(j) for these {i, j}: {2, 38847}, {38, 3115}, {561, 38826}, {2156, 38842}
X(38830) = trilinear quotient X(i)/X(j) for these (i, j): (2, 2085), (38, 3118), (48, 23209), (63, 4173), (76, 4118), (86, 16717)
X(38830) = 1st Saragossa point of X(1501)
X(38830) = trilinear pole of the line {804, 5152}
X(38830) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(385)}} and {{A, B, C, X(4), X(16951)}}
X(38830) = cevapoint of X(i) and X(j) for these {i,j}: {6, 76}, {782, 35078}
X(38830) = X(32)-Ceva conjugate of X(38821)
X(38830) = X(6)-cross conjugate of X(38826)
X(38830) = X(i)-isoconjugate-of X(j) for these {i,j}: {6, 2085}, {19, 4173}, {32, 4118}, {42, 16717}
X(38830) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 2085), (3, 4173), (39, 3118), (69, 20819)
X(38830) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (76, 38826, 38842), (38826, 38838, 1501)
X(38830) = Vu cevian-tangential perspector of X(32)

X(38831) = VU CEVIAN-CIRCLES PERSPECTOR OF X(2887)

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

X(38831) lies on these lines: {6,33772}, {75,33776}, {984,3924}, {3778,38813}, {17189,30966}

X(38831) = isogonal conjugate of X(6679)
X(38831) = complement of the anticomplementary conjugate of X(2887)
X(38831) = barycentric quotient X(55)/X(4168)
X(38831) = trilinear quotient X(9)/X(4168)
X(38831) = 1st Saragossa point of X(38813)
X(38831) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(3924)}} and {{A, B, C, X(6), X(75)}}
X(38831) = cevapoint of X(6) and X(3778)
X(38831) = X(57)-isoconjugate-of X(4168)
X(38831) = X(55)-reciprocal conjugate of X(4168)

X(38832) = CYCLOCEVIAN CONJUGATE OF V(X(1)) WRT ANTICEVIAN TRIANGLE OF V(X(1))

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

X(38832) lies on these lines: {1,21}, {2,4279}, {6,16058}, {8,27660}, {32,1613}, {42,4476}, {43,2209}, {55,3736}, {86,171}, {101,715}, {109,741}, {110,727}, {190,22024}, {238,333}, {274,3980}, {284,893}, {310,24259}, {314,983}, {386,22076}, {501,5161}, {580,10441}, {602,10476}, {748,5235}, {750,5333}, {851,28368}, {902,4184}, {940,5156}, {985,29644}, {995,4276}, {1010,5255}, {1011,5145}, {1014,9316}, {1043,37588}, {1104,18178}, {1155,16700}, {1185,4251}, {1191,4267}, {1201,4225}, {1279,18165}, {1326,1790}, {1331,1999}, {1333,2162}, {1724,10449}, {1754,10446}, {1764,13329}, {1961,20964}, {2176,20760}, {2194,5009}, {2299,3512}, {2361,21334}, {3011,17167}, {3052,3286}, {3550,13588}, {3769,30939}, {3786,3961}, {3936,25689}, {4039,28660}, {4199,28369}, {4281,16466}, {4418,30599}, {4640,16696}, {4660,33730}, {4697,25124}, {4970,7304}, {5192,5278}, {5264,25526}, {6327,30984}, {7009,8750}, {7290,18163}, {8025,17126}, {9315,33628}, {9352,16753}, {10453,16704}, {11358,28365}, {12545,37570}, {14009,28356}, {16056,28350}, {16690,27164}, {16750,24283}, {17017,25060}, {17122,25507}, {17139,33144}, {17173,29681}, {17174,29665}, {17202,29634}, {19278,28619}, {19767,34281}, {21214,37442}, {23439,24525}, {24896,35466}, {25059,29821}, {25645,29984}, {26860,30652}, {27631,30942}, {28242,29637}, {28375,33109}, {29972,30104}, {30965,33085}, {32922,36862}, {37303,37617}

X(38832) = isogonal conjugate of the isotomic conjugate of X(33296)
X(38832) = barycentric product X(i)*X(j) for these {i, j}: {1, 27644}, {6, 33296}, {21, 1423}, {27, 20760}, {28, 22370}, {31, 31008}, {58,192}
X(38832) = barycentric quotient X(i)/X(j) for these (i, j): (21, 27424), (31, 16606), (32, 23493), (43, 321), (58, 330), (81, 6384)
X(38832) = trilinear product X(i)*X(j) for these {i, j}: {6, 27644}, {21, 1403}, {28, 20760}, {31, 33296}, {32, 31008}, {43, 58}
X(38832) = trilinear quotient X(i)/X(j) for these (i, j): (6, 16606), (21, 7155), (31, 23493), (32, 21759), (42, 7148), (43, 10)
X(38832) = trilinear pole of the line {16695, 20979}
X(38832) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(43)}} and {{A, B, C, X(21), X(741)}}
X(38832) = cevapoint of X(31) and X(1613)
X(38832) = crossdifference of every pair of points on line {X(2533), X(20486)}
X(38832) = crosspoint of X(110) and X(4600)
X(38832) = crosssum of X(i) and X(j) for these {i,j}: {10, 4135}, {523, 3122}
X(38832) = X(1333)-Ceva conjugate of X(58)
X(38832) = X(i)-isoconjugate-of X(j) for these {i,j}: {2, 16606}, {10, 87}, {37, 330}, {42, 6384}
X(38832) = X(i)-reciprocal conjugate of X(j) for these (i,j): (21, 27424), (31, 16606), (32, 23493), (43, 321)
X(38832) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 17185, 35623), (21, 81, 18169), (31, 81, 58), (31, 3915, 8616), (81, 1621, 10458), (902, 17187, 4184), (1621, 10458, 4653), (3550, 18792, 13588), (8616, 18169, 21)

X(38833) = CYCLOCEVIAN CONJUGATE OF V(X(5)) WRT ANTICEVIAN TRIANGLE OF V(X(5))

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

X(38833) lies on these lines: {51,54}, {418,31626}

X(38833) = barycentric product X(1173)*X(17035)
X(38833) = {X(51), X(288)}-harmonic conjugate of X(1173)

X(38834) = CYCLOCEVIAN CONJUGATE OF V(X(6)) WRT ANTICEVIAN TRIANGLE OF V(X(6))

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

X(38834) lies on these lines: {1,23496}, {2,32}, {110,6579}, {112,733}, {182,14134}, {308,384}, {695,1176}, {699,827}, {1186,5012}, {1501,33786}, {1968,32085}, {3203,38854}, {3575,10549}, {7750,16890}, {7839,33875}, {9292,33632}, {10312,32581}

X(38834) = barycentric product X(i)*X(j) for these {i, j}: {82, 1740}, {83, 1613}, {194, 251}, {689, 9491}, {827, 23301}, {1176, 3186}
X(38834) = barycentric quotient X(i)/X(j) for these (i, j): (82, 18832), (194, 8024), (251, 2998), (827, 3222), (1501, 19606), (1613, 141)
X(38834) = trilinear product X(i)*X(j) for these {i, j}: {82, 1613}, {251, 1740}
X(38834) = trilinear quotient X(i)/X(j) for these (i, j): (82, 2998), (83, 18832), (194, 1930), (251, 3223), (560, 19606), (1424, 3665)
X(38834) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(194)}} and {{A, B, C, X(6), X(7787)}}
X(38834) = cevapoint of X(32) and X(33786)
X(38834) = X(i)-isoconjugate-of X(j) for these {i,j}: {38, 2998}, {39, 18832}, {141, 3223}, {561, 19606}
X(38834) = X(i)-reciprocal conjugate of X(j) for these (i,j): (82, 18832), (194, 8024), (251, 2998), (827, 3222)
X(38834) = {X(32), X(83)}-harmonic conjugate of X(251)

X(38835) = CYCLOCEVIAN CONJUGATE OF V(X(9)) WRT ANTICEVIAN TRIANGLE OF V(X(9))

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

X(38835) lies on these lines: {41,57}, {101,10482}, {2346,9310}, {9439,33634}, {14827,35215}

X(38835) = barycentric product X(i)*X(j) for these {i, j}: {1174, 3177}, {1742, 2346}
X(38835) = barycentric quotient X(1742)/X(20880)
X(38835) = trilinear product X(i)*X(j) for these {i, j}: {1174, 1742}, {2346, 20995}
X(38835) = trilinear quotient X(1742)/X(142)
X(38835) = cevapoint of X(41) and X(35215)
X(38835) = X(1742)-reciprocal conjugate of X(20880)
X(38835) = {X(41), X(1170)}-harmonic conjugate of X(1174)

X(38836) = CYCLOCEVIAN CONJUGATE OF V(X(10)) WRT ANTICEVIAN TRIANGLE OF V(X(10))

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

X(38836) lies on these lines: {35,42}, {100,1255}, {893,33635}, {1030,2248}, {1500,35216}, {4028,31013}, {4272,17735}, {5524,32635}, {6626,21085}, {8701,28482}

X(38836) = barycentric product X(i)*X(j) for these {i, j}: {846, 1255}, {1126, 1654}, {1171, 21085}, {1268, 18755}, {1796, 4213}
X(38836) = barycentric quotient X(i)/X(j) for these (i, j): (846, 4359), (1126, 6625), (1654, 1269)
X(38836) = trilinear product X(i)*X(j) for these {i, j}: {846, 1126}, {1171, 21879}, {1255, 18755}, {1654, 28615}
X(38836) = trilinear quotient X(i)/X(j) for these (i, j): (846, 1125), (1126, 13610), (1255, 6625), (1654, 4359)
X(38836) = intersection, other than A,B,C, of conics {{A, B, C, X(42), X(21085)}} and {{A, B, C, X(58), X(846)}}
X(38836) = cevapoint of X(42) and X(35216)
X(38836) = X(i)-isoconjugate-of X(j) for these {i,j}: {1100, 6625}, {1125, 13610}, {1269, 18757}
X(38836) = X(i)-reciprocal conjugate of X(j) for these (i,j): (846, 4359), (1126, 6625), (1654, 1269)
X(38836) = {X(42), X(1171)}-harmonic conjugate of X(1126)

X(38837) = CYCLOCEVIAN CONJUGATE OF V(X(31)) WRT ANTICEVIAN TRIANGLE OF V(X(31))

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

X(38837) lies on this line: {75,560}

X(38837) = barycentric product X(17486)*X(38813)
X(38837) = {X(560), X(38810)}-harmonic conjugate of X(38813)

X(38838) = CYCLOCEVIAN CONJUGATE OF V(X(32)) WRT ANTICEVIAN TRIANGLE OF V(X(32))

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

X(38838) lies on this line:i {76,1501}

X(38838) = barycentric product X(8264)*X(38826)
X(38838) = {X(1501), X(38830)}-harmonic conjugate of X(38826)

X(38839) = CYCLOCEVIAN CONJUGATE OF V(X(560)) WRT ANTICEVIAN TRIANGLE OF V(X(560))

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

X(38839) lies on this line:i {561,1917}

X(38839) = {X(1917), X(38812)}-harmonic conjugate of X(38827)

X(38840) = CYCLOCEVIAN CONJUGATE OF V(X(561)) WRT ANTICEVIAN TRIANGLE OF V(X(561))

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

X(38840) lies on these lines: {75,560}, {1502,33801}

X(38840) = barycentric quotient X(i)/X(j) for these (i, j): (1631, 16584), (1759, 3778)
X(38840) = trilinear product X(1759)*X(38810)
X(38840) = trilinear quotient X(1759)/X(16584)
X(38840) = cevapoint of X(i) and X(j) for these {i,j}: {75, 33801}, {1631, 20444}
X(38840) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1631, 16584), (1759, 3778)
X(38840) = {X(75), X(38813)}-harmonic conjugate of X(38810)

X(38841) = CYCLOCEVIAN CONJUGATE OF V(X(1501)) WRT ANTICEVIAN TRIANGLE OF V(X(1501))

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

X(38841) lies on this line: {1502,9233}

X(38842) = CYCLOCEVIAN CONJUGATE OF V(X(1502)) WRT ANTICEVIAN TRIANGLE OF V(X(1502))

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

Let A'B'C' be the circumcevian triangle of X(76). Let A" be the cevapoint of B' and C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(38842). (Randy Hutson, June 9, 2020)

X(38842) lies on these lines: {76,1501}, {17984,27369}

X(38842) = cevapoint of X(76) and X(33802)
X(38842) = barycentric product X(315)*X(38830)
X(38842) = barycentric quotient X(i)/X(j) for these (i, j): (22, 8265), (315, 20859), (1760, 2085)
X(38842) = trilinear product X(i)*X(j) for these {i, j}: {315, 38847}, {1760, 38830}
X(38842) = trilinear quotient X(i)/X(j) for these (i, j): (315, 2085), (1760, 8265)
X(38842) = intersection, other than A,B,C, of conics {{A, B, C, X(22), X(1501)}} and {{A, B, C, X(315), X(17984)}}
X(38842) = X(i)-reciprocal conjugate of X(j) for these (i,j): (22, 8265), (315, 20859), (1760, 2085)
X(38842) = {X(76), X(38826)}-harmonic conjugate of X(38830)

X(38843) = CYCLOCEVIAN CONJUGATE OF V(X(1928)) WRT ANTICEVIAN TRIANGLE OF V(X(1928))

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

X(38843) lies on this line: {561,1917}

X(38843) = barycentric product X(21275)*X(38812)
X(38843) = {X(561), X(38827)}-harmonic conjugate of X(38812)

X(38844) = ANTICOMPLEMENT OF X(2085)

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

X(38844) lies on these lines: {2, 2085}, {766, 27801}, {976, 3923}, {3730, 3952}, {6327, 17492}, {9941, 17794}, {17165, 28598}, {20863, 21435}

X(38844) = isotomic conjugate of X(38845)
X(38844) = anticomplement of X(2085)
X(38844) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {3115, 76}, {33515, 826}, {38826, 194}, {38830, 69}

X(38845) = ISOTOMIC CONJUGATE OF X(38844)

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

X(38845 lies on these lines: (none)

X(38845) = isogonal conjugate of X(38846)
X(38845) = isotomic conjugate of X(38844)

X(38846) = ISOGONAL CONJUGATE OF X(38845)

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

X(38846 lies on these lines: {55, 4094}, {321, 8629}, {1631, 23847}

X(38846) = isotomic conjugate of X(38844)

X(38847) = VU CEVIAN-TANGENTIAL PERSPECTOR OF X(38845)

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

Vu cevian-tangential perspectors are introduced in the preamble just before X(38809).

X(38847) lies on these lines: {31, 1928}, {75, 1917}, {1580, 1930}, {1923, 1966}, {2085, 4593}, {3115, 18099}, {4039, 38830}

X(38847) = isogonal conjugate of X(2085)
X(38847) = isotomic conjugate of X(4118)
X(38847) = X(i)-cross conjugate of X(j) for these (i,j): {2084, 4593}, {20953, 799}, {21613, 668}
X(38847) = cevapoint of X(31) and X(75)
X(38847) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2085}, {2, 8265}, {4, 4173}, {6, 20859}, {25, 20819}, {31, 4118}, {32, 626}, {37, 16717}, {83, 3118}, {213, 18167}, {264, 23209}, {560, 20627}, {710, 14946}, {1397, 4178}, {1502, 8023}, {1918, 16891}, {1974, 4121}, {2175, 7217}, {2206, 16894}, {3051, 16890}, {8039, 9233}, {21110, 32739}
X(38847) = barycentric product X(i)*X(j) for these {i,j}: {1, 38830}, {38, 3115}, {561, 38826}, {2156, 38842}
X(38847) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 20859}, {2, 4118}, {6, 2085}, {31, 8265}, {48, 4173}, {58, 16717}, {63, 20819}, {75, 626}, {76, 20627}, {85, 7217}, {86, 18167}, {274, 16891}, {304, 4121}, {312, 4178}, {321, 16894}, {693, 21110}, {1917, 8023}, {1928, 8039}, {1930, 16893}, {1964, 3118}, {3112, 16890}, {3115, 3112}, {9247, 23209}, {33515, 4599}, {38826, 31}, {38830, 75}, {38842, 20641}
X(38847) = {X(31),X(1928)}-harmonic conjugate of X(33777)

leftri

Vu tangential transforms: X(38848)-X(38887)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, June 7, 2020.

Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC. The Vu perspectivies triangle of P, here denoted by T(P), is introduced in the preamble just before X(38825). The A-vertex is given by

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

T(P) is perspective to the anticomplementary triangle for every P on the de Longchamps axis (a^2 x + b^2 y + c^2 z = 0).

T(P) is perspective to the anticomplementary triangle for every P on the following conic:

a^2(b^2 - c^2)x^2 + b^2(c^2 - a^2)y^2 + c^2(a^2 - b^2)z^2 = 0, this being the anticomplement of the conic {{A, B, C, X(2), X(6)}}, which passes through X(i) for these i: 2, 69, 75, 1272, 1369, 1370, 6527, 14360, 17135, 17149, 18133, 18750, 19583, 20245, 20351, 20934, 22339, 22340, 25332, 30698, 32747.

T(P) is perspective to the excentral triangle for every P on the anticomplement of the excentral triangle.

T(P) is perspective to the excentral triangle for every P on the following conic: anticomplement of the conic {{A, B, C, X(1), X(2)}}, which passes through X(i) for these i: 2, 8, 329, 556, 1655, 2895, 8055, 17794, 18297, 20344, 21219, 30578, 30695.

T(P) is perspective to the tangential triangle for every P. The perspector, here named the Vu tangential transform of P, denoted by VT(P), is given by

VT(P) = a^2*(p^2 + p*q + p*r - q*r) : b^2*(q^2 + q*r + q*p - r*p) : c^2*(r^2 + r*p + r*q - p*q).

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

The VT(P)) = isogonal conjugate of the perspector of ABC and the reflection of the cevian triangle of P in the centroid of ABCP (or the complement of the complement of P). It is also the perspector T1(X(6),P), as defined in the preamble just before X(33760). (Randy Hutson, June 9, 2020)

If P lies on the circumcircle, then VT(P) = isogonal conjugate of P wrt its cevian triangle. If P lies on the line at infinity, then VT(P) = isogonal conjugate of P. (Randy Hutson, June 9, 2020)

The Vu tangential transform of the Euler line is a conic centered at X(15647) and passing through X(i) for these i: 6, 24, 74, 1498, 1614, 38848, 38850, 38851, 38852, 38867, 38879, 38885. (Randy Hutson, June 9, 2020)

From Peter Moses (June 10, 2020): Suppose that P lies on a line u x + v y + w z = 0. Then VT(P) lies on the following conic:

4*b^4*c^4*u^2*(v - w)^2*x^2 - a^4*b^2*c^2*(u^4 - 2*u^2*v^2 + v^4 - 4*u^2*v*w + 8*u*v^2*w - 4*v^3*w - 2*u^2*w^2 + 8*u*v*w^2 - 2*v^2*w^2 - 4*v*w^3 + w^4)*y*z + (cyclic) = 0.

Examples:
conic VT(Euler line) passes through X(i) for these i: 6, 24, 74, 1498, 1614, 1620, 35217, 35218, 35219
conic VT(X(1)X(3)) passes through X(i) for these i: 104, 595, 1614
conic VT(Brocard axis) passes through X(i) for these i: 98, 1614, 1627, 33773
conic VT(Soddy line) passes through X(i) for these i: 103, 595, 1498, 1617, 1619, 1622
conic VT(Nagel line) passes through X(i) for these i: 3, 6, 106, 595, 1616, 23374, 33771, 33804, 35223
conic VT(X(2)X(6)) passes through X(i) for these i: 6, 22, 111, 1611, 1627, 33774, 35212, 35216
VT(circumcircle) is the circular circumquartic given by

2*b^2*c^2*(a^8 - 2*a^6*b^2 - 4*a^4*b^4 - 2*a^2*b^6 + b^8 - 2*a^6*c^2 - 4*a^4*b^2*c^2 - 6*a^2*b^4*c^2 - 4*a^4*c^4 - 6*a^2*b^2*c^4 - 2*b^4*c^4 - 2*a^2*c^6 + c^8)*x^2*y*z + a^4*(a^8 - 2*a^4*b^4 + b^8 - 4*a^4*b^2*c^2 - 8*a^2*b^4*c^2 - 4*b^6*c^2 - 2*a^4*c^4 - 8*a^2*b^2*c^4 - 2*b^4*c^4 - 4*b^2*c^6 + c^8)*y^2*z^2 - 4*a^2*b^2*c^2*x*(c^2*(a^2 + c^2)^2*y^3 + b^2*(a^2 + b^2)^2*z^3) + (cyclic) = 0,

which has singular focus X(5899) and passes through ABC, the vertices of the Aries triangle (cf. X(5596)), and X(i) for these i: 1618, 33803, 38861.

X(38848) = VU TANGENTIAL TRANSFORM OF X(5)

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

X(38848) lies on these lines: {3, 11451}, {4, 74}, {5, 7691}, {6, 26882}, {22, 15024}, {23, 5462}, {24, 9781}, {25, 1614}, {26, 5640}, {51, 54}, {52, 13595}, {110, 143}, {140, 15107}, {154, 11423}, {156, 13321}, {186, 10110}, {378, 1620}, {382, 15053}, {389, 14157}, {403, 11745}, {428, 26879}, {436, 4994}, {476, 10223}, {1147, 11002}, {1154, 18369}, {1181, 31860}, {1199, 1495}, {1498, 5890}, {1576, 25044}, {1597, 11468}, {1598, 6241}, {1656, 10545}, {1994, 9705}, {1995, 11412}, {2070, 10095}, {2937, 15026}, {3060, 7506}, {3066, 7509}, {3078, 21265}, {3098, 3533}, {3517, 11464}, {3574, 37943}, {3581, 3850}, {3839, 7689}, {3843, 11440}, {3845, 15062}, {3853, 13445}, {3854, 4550}, {5012, 37440}, {5020, 7999}, {5056, 37478}, {5059, 37470}, {5198, 11455}, {5422, 9714}, {5480, 10018}, {5622, 35218}, {5643, 7555}, {5889, 13861}, {5899, 12006}, {5943, 7512}, {5944, 15038}, {5946, 18378}, {6000, 26863}, {6030, 37471}, {6102, 7545}, {6240, 15873}, {6756, 25739}, {6995, 11457}, {7387, 15045}, {7487, 12289}, {7517, 15043}, {7529, 11459}, {7530, 10574}, {7714, 18916}, {7715, 34224}, {7716, 12283}, {8537, 19136}, {8549, 35219}, {8567, 35502}, {8718, 9730}, {9706, 14627}, {9707, 9777}, {9729, 37925}, {9786, 12290}, {9969, 19128}, {10117, 17823}, {10301, 16659}, {10539, 14002}, {10540, 16881}, {10564, 12002}, {11430, 14483}, {11591, 21308}, {12038, 13482}, {12106, 15034}, {12112, 13382}, {12163, 16261}, {12242, 12380}, {12834, 13353}, {13336, 37913}, {13363, 13564}, {13451, 37472}, {14269, 32138}, {14449, 23061}, {14643, 20424}, {14788, 32269}, {14853, 31267}, {15019, 32046}, {15036, 37814}, {15047, 37956}, {15054, 32137}, {15058, 37489}, {15305, 37490}, {15360, 23410}, {15801, 18350}, {15806, 30551}, {16835, 32062}, {17712, 20063}, {18394, 18494}, {18874, 34864}, {18912, 37122}, {22462, 32142}, {26881, 36753}, {32223, 32396}, {35264, 37493}

X(38849) = VU TANGENTIAL TRANSFORM OF X(9)

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

X(38849) lies on these lines: {3, 35599}, {41, 57}, {55, 101}, {109, 14827}, {284, 4228}, {354, 4251}, {572, 2348}, {672, 4210}, {910, 1630}, {991, 5452}, {1055, 1202}, {1190, 1617}, {1200, 2078}, {1477, 22769}, {1604, 15804}, {1615, 37541}, {2246, 34544}, {6602, 35445}, {9310, 10389}, {16788, 26040}, {21059, 33634}

X(38850) = VU TANGENTIAL TRANSFORM OF X(21)

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

X(38850) lies on these lines: {1, 60}, {28, 65}, {35, 1819}, {54, 5721}, {58, 1042}, {74, 34435}, {184, 387}, {501, 15931}, {1612, 20986}, {1614, 5706}, {1630, 2150}, {1714, 5012}, {1780, 4225}, {1858, 2074}, {2360, 37583}, {3072, 23692}, {3332, 6759}, {6061, 12514}, {11337, 16471}

X(38851) = VU TANGENTIAL TRANSFORM OF X(23)

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

X(38851) lies on these lines: {5, 15133}, {6, 110}, {24, 2781}, {25, 15141}, {67, 468}, {74, 34437}, {125, 206}, {182, 23515}, {184, 15118}, {542, 10539}, {578, 30714}, {1176, 15059}, {1205, 1495}, {1352, 19138}, {1498, 5621}, {1614, 5622}, {2393, 34470}, {3043, 14853}, {3047, 25320}, {5095, 19136}, {5157, 6723}, {5181, 9306}, {5480, 15463}, {6698, 19127}, {8262, 32244}, {9786, 25711}, {9970, 12106}, {9971, 15140}, {9973, 13248}, {10117, 35217}, {12228, 14561}, {12294, 17701}, {15106, 32262}, {15116, 32239}, {15647, 35219}, {19140, 19161}, {19153, 32251}, {19457, 23041}, {25335, 26958}, {26883, 36201}, {27085, 37760}, {32165, 38405}, {32260, 35370}

X(38852) = VU TANGENTIAL TRANSFORM OF X(27)

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

X(38852) lies on these lines: {1, 2189}, {24, 37499}, {27, 20291}, {37, 943}, {71, 1474}, {74, 34440}, {112, 5301}, {1301, 2155}, {1621, 2326}, {7054, 13739}

X(38853) = VU TANGENTIAL TRANSFORM OF X(37)

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

X(38853) lies on these lines: {6, 5284}, {31, 101}, {81, 213}, {100, 7109}, {995, 36808}, {1185, 1621}, {1206, 3230}, {3294, 10458}, {17027, 32911}, {21753, 21788}

X(38854) = VU TANGENTIAL TRANSFORM OF X(39)

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

X(38854) lies on these lines: {32, 110}, {83, 3051}, {99, 3499}, {1078, 1186}, {1207, 3231}, {3203, 38834}, {7760, 18899}, {12150, 33786}

X(38855) = VU TANGENTIAL TRANSFORM OF X(57)

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

X(38855) lies on these lines: {9, 604}, {56, 573}, {109, 21769}, {284, 34489}, {319, 2985}, {572, 1319}, {1100, 3660}, {1108, 1630}, {1400, 5193}, {1412, 1444}, {1436, 2291}, {1604, 37519}, {1617, 2256}, {1813, 8732}, {34880, 37508}

X(38856) = VU TANGENTIAL TRANSFORM OF X(72)

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

X(38856) lies on these lines: {3, 5273}, {4, 198}, {24, 197}, {28, 228}, {40, 101}, {55, 17562}, {72, 1817}, {100, 2915}, {104, 23361}, {201, 2939}, {329, 37063}, {387, 2178}, {631, 15509}, {910, 37528}, {942, 11349}, {944, 1622}, {1260, 37408}, {1602, 9798}, {1604, 5657}, {1834, 19297}, {2182, 26878}, {2975, 37058}, {3185, 36009}, {3198, 6198}, {3207, 5706}, {3487, 11347}, {3488, 13737}, {3556, 12250}, {4294, 15494}, {5658, 37413}, {5759, 9122}, {7520, 11517}, {11500, 37441}, {31445, 37402}

X(38857) = VU TANGENTIAL TRANSFORM OF X(78)

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

X(38857) lies on these lines: {3, 951}, {6, 10306}, {20, 1331}, {31, 6769}, {34, 40}, {58, 37531}, {73, 2077}, {101, 6759}, {106, 11249}, {109, 1035}, {255, 6282}, {517, 580}, {581, 11248}, {1068, 5759}, {1181, 3190}, {1253, 10268}, {1260, 1498}, {1451, 7982}, {1616, 22770}, {1754, 5758}, {1766, 5317}, {2328, 30733}, {2361, 7957}, {3730, 26938}, {3939, 11500}, {5709, 13329}, {6260, 23693}, {9316, 10270}, {11827, 18340}, {13348, 22161}, {23832, 38559}

X(38858) = VU TANGENTIAL TRANSFORM OF X(81)

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

X(38858) lies on these lines: {19, 112}, {21, 37}, {32, 966}, {58, 71}, {110, 16685}, {111, 3444}, {248, 8044}, {333, 32025}, {464, 577}, {593, 28606}, {595, 2150}, {1030, 1187}, {1326, 22054}, {1627, 5275}, {2220, 5546}, {2300, 5006}, {5301, 7054}, {7031, 35192}, {13739, 36420}, {16785, 33635}, {17735, 33774}

X(38859) = VU TANGENTIAL TRANSFORM OF X(85)

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

X(38859) lies on these lines: {1, 23839}, {3, 7}, {22, 17093}, {28, 1847}, {32, 1462}, {36, 10481}, {41, 57}, {56, 105}, {58, 269}, {65, 38459}, {77, 3333}, {85, 2975}, {100, 6604}, {251, 1427}, {348, 5253}, {404, 9436}, {651, 4253}, {664, 20247}, {938, 37412}, {942, 36012}, {956, 31994}, {999, 3160}, {1088, 4228}, {1119, 17562}, {1188, 20995}, {1323, 5563}, {1434, 4225}, {1442, 5045}, {1446, 1447}, {1612, 3668}, {1621, 33765}, {2260, 34028}, {2346, 30502}, {3188, 3673}, {3295, 5543}, {4258, 5228}, {4292, 37048}, {5022, 6180}, {5435, 37272}, {5687, 32003}, {7175, 20459}, {7247, 7465}, {32636, 34855}, {37117, 38461}

X(38860) = VU TANGENTIAL TRANSFORM OF X(92)

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

X(38860) lies on these lines: {1, 19}, {3, 281}, {4, 198}, {7, 37378}, {9, 37305}, {25, 1604}, {34, 2270}, {53, 19297}, {56, 1249}, {57, 2331}, {92, 1817}, {104, 1436}, {108, 393}, {228, 37386}, {273, 11349}, {278, 7011}, {579, 1783}, {608, 22124}, {910, 1841}, {1033, 1617}, {1055, 7120}, {1400, 2202}, {1420, 7129}, {1603, 14017}, {1753, 2324}, {1826, 4219}, {1870, 2262}, {1880, 1951}, {1903, 12262}, {1990, 21773}, {2182, 9119}, {2257, 7156}, {2322, 2975}, {3079, 20991}, {4224, 5089}, {4254, 34231}, {7490, 15509}, {8755, 37583}, {8761, 32726}, {16054, 17913}, {18610, 34429}, {26030, 37278}, {27382, 37258}

X(38861) = VU TANGENTIAL TRANSFORM OF X(110)

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

X(38861) lies on these lines: {110, 924}, {112, 2485}, {206, 37918}, {249, 34968}, {250, 523}, {691, 827}, {842, 3447}, {933, 1304}, {1316, 14575}, {2071, 34146}, {2693, 2935}, {2715, 3050}, {2892, 13573}, {5189, 21458}, {10419, 15463}, {10547, 36157}, {13198, 34950}, {14560, 20188}

X(38861) = isogonal conjugate of X(110) wrt the cevian triangle of X(110)

X(38862) = VU TANGENTIAL TRANSFORM OF X(141)

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

X(38862) lies on these lines: {2, 99}, {3, 1180}, {6, 34572}, {22, 5013}, {23, 31652}, {25, 15302}, {39, 251}, {83, 35929}, {110, 8041}, {112, 8792}, {187, 34482}, {570, 10313}, {1078, 8267}, {1194, 15246}, {1196, 7496}, {1369, 7830}, {1506, 37349}, {1611, 7485}, {1915, 6030}, {2076, 11205}, {2548, 20062}, {3094, 5012}, {3162, 35477}, {3815, 34603}, {5116, 20859}, {5304, 10979}, {5354, 8589}, {5913, 7734}, {6032, 34609}, {7394, 31401}, {7500, 31400}, {7519, 31450}, {7735, 14806}, {7748, 37353}, {7782, 16949}, {7783, 8024}, {7786, 16932}, {7808, 16952}, {8878, 33260}, {9698, 20063}, {10007, 10328}, {12055, 15107}, {12834, 20977}, {14153, 23061}, {22352, 34945}, {29815, 31451}, {31088, 33651}, {33774, 33863}

X(38863) = VU TANGENTIAL TRANSFORM OF X(149)

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

X(38863) lies on these lines: {3, 37815}, {6, 59}, {36, 1279}, {513, 3446}, {910, 2078}, {1319, 3827}, {1486, 38530}, {4057, 20999}, {5172, 20470}, {6129, 10016}, {7669, 23388}

X(38864) = VU TANGENTIAL TRANSFORM OF X(226)

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

X(38864) lies on these lines: {19, 108}, {56, 573}, {65, 37508}, {109, 2305}, {226, 8822}, {284, 1400}, {478, 4257}, {604, 34544}, {1182, 1630}, {1617, 36744}, {7098, 21078}

X(38865) = VU TANGENTIAL TRANSFORM OF X(239)

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

X(38865) lies on the ellipse described at X(9306) and on these lines: {3, 1015}, {6, 3939}, {32, 101}, {39, 33771}, {106, 187}, {109, 1979}, {292, 1438}, {665, 2440}, {1017, 21309}, {1331, 1977}, {1333, 1634}, {1384, 8649}, {1616, 3053}, {3052, 20672}, {5023, 33804}, {9575, 37552}, {10987, 37586}, {12194, 19557}, {13589, 26278}, {21495, 24625}

X(38865) = crossdifference of every pair of points on line X(120)X(2977) (the tangent to the nine-point circle at X(120))

X(38866) = VU TANGENTIAL TRANSFORM OF X(279)

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

X(38866) lies on these lines: {40, 269}, {63, 220}, {64, 103}, {221, 34855}, {279, 20070}, {595, 7053}, {738, 34040}, {934, 1616}, {1191, 7177}, {1498, 4341}, {6180, 6554}, {16483, 17106}

X(38867) = VU TANGENTIAL TRANSFORM OF X(297)

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

X(38867) lies on these lines: {24, 112}, {25, 9475}, {74, 1987}, {232, 248}, {393, 1632}, {1301, 20998}, {1498, 3269}, {2409, 6531}

X(38868) = VU TANGENTIAL TRANSFORM OF X(306)

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

X(38868) lies on these lines: {3, 2256}, {9, 4222}, {71, 1474}, {100, 610}, {106, 29014}, {284, 3453}, {579, 595}, {1616, 37500}, {3430, 3990}, {8193, 15830}, {30269, 36080}

X(38869) = VU TANGENTIAL TRANSFORM OF X(312)

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

X(38869) lies on these lines: {2, 1696}, {3, 3161}, {9, 604}, {21, 3731}, {37, 82}, {63, 26669}, {75, 26265}, {100, 198}, {329, 6349}, {404, 17355}, {573, 644}, {651, 29497}, {664, 20248}, {1444, 17336}, {1760, 28978}, {2270, 14923}, {2324, 3869}, {2345, 26258}, {3196, 17362}, {3436, 27508}, {3496, 21809}, {3553, 34195}, {3729, 11349}, {3746, 4098}, {3871, 3950}, {3973, 8666}, {3974, 15494}, {3986, 5047}, {4370, 5124}, {4578, 12329}, {4869, 24328}, {5057, 21068}, {5176, 20262}, {5253, 5749}, {5260, 5296}, {6557, 19517}, {7172, 20991}, {7229, 16412}, {8055, 16435}, {8686, 11194}, {11683, 25243}, {17074, 29529}, {17247, 21516}, {17261, 21511}, {17339, 21495}, {17340, 19297}, {21214, 33589}, {26703, 30555}, {28395, 37680}, {31995, 37272}

X(38870) = VU TANGENTIAL TRANSFORM OF X(318)

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

X(38870) lies on these lines: {4, 11}, {25, 1622}, {33, 84}, {55, 18283}, {281, 1604}, {318, 6909}, {406, 12667}, {451, 18242}, {515, 1610}, {958, 37410}, {1012, 7952}, {1013, 9799}, {1490, 2360}, {1861, 6705}, {1872, 34862}, {2975, 37420}, {4219, 6245}, {5450, 37305}, {6001, 6198}, {6796, 37289}, {7046, 10310}, {7071, 12330}, {11500, 37441}, {12672, 15500}

X(38871) = VU TANGENTIAL TRANSFORM OF X(321)

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

X(38871) lies on these lines: {1, 4264}, {2, 2178}, {3, 2345}, {6, 2975}, {8, 36744}, {9, 48}, {21, 37}, {24, 281}, {35, 2321}, {36, 5750}, {55, 17314}, {63, 3553}, {71, 2329}, {75, 21511}, {100, 594}, {105, 16684}, {190, 27958}, {198, 958}, {213, 1778}, {284, 21061}, {313, 26243}, {321, 27174}, {344, 16367}, {346, 4189}, {404, 17303}, {579, 16788}, {825, 15168}, {835, 26227}, {894, 1444}, {943, 22021}, {956, 4254}, {965, 3207}, {992, 21008}, {1014, 4670}, {1213, 5260}, {1449, 8666}, {1612, 37317}, {1621, 16777}, {1622, 6554}, {1633, 8424}, {1761, 2171}, {1766, 6906}, {1812, 3219}, {1817, 31993}, {1826, 4231}, {1953, 3496}, {2092, 5291}, {2174, 2287}, {2199, 24806}, {2277, 4426}, {2294, 3509}, {2295, 2305}, {2324, 31424}, {3247, 5248}, {3294, 4877}, {3686, 5258}, {3713, 37504}, {3728, 18266}, {3739, 11349}, {3871, 17299}, {4000, 11343}, {4007, 8715}, {4529, 23226}, {4640, 21871}, {5006, 21810}, {5124, 5303}, {5204, 26039}, {5251, 5257}, {5253, 17398}, {5267, 17355}, {5749, 36743}, {6645, 26110}, {10445, 11012}, {10934, 13615}, {11340, 19822}, {16058, 23159}, {16706, 21516}, {16713, 31039}, {16916, 26107}, {16946, 16975}, {17281, 17549}, {17289, 21495}, {17371, 21540}, {17541, 25505}, {19308, 28604}, {22008, 24632}, {25946, 28653}, {26242, 37325}, {30576, 33761}

X(38872) = VU TANGENTIAL TRANSFORM OF X(323)

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

X(38872) lies on these lines: {4, 36423}, {6, 74}, {22, 2493}, {30, 50}, {32, 6128}, {111, 1291}, {115, 571}, {566, 18570}, {577, 3018}, {1627, 31133}, {3163, 5063}, {7736, 36415}, {8882, 15664}, {11062, 37970}

X(38873) = VU TANGENTIAL TRANSFORM OF X(325)

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

X(38873) lies on these lines: {2, 35387}, {6, 14060}, {22, 110}, {69, 1632}, {74, 11634}, {99, 25046}, {111, 694}, {237, 35383}, {287, 4226}, {511, 1976}, {1611, 3124}, {1627, 3060}, {2871, 4558}, {3094, 5012}, {3098, 36213}, {3796, 20976}, {5191, 33878}, {15595, 36163}, {18286, 20998}

X(38874) = VU TANGENTIAL TRANSFORM OF X(335)

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

X(38874) lies on these lines: {31, 3252}, {238, 36906}, {292, 1438}, {672, 1911}, {741, 813}, {2109, 2382}, {4562, 33295}, {18034, 26247}

X(38875) = VU TANGENTIAL TRANSFORM OF X(345)

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

X(38875) lies on these lines: {6, 34195}, {63, 4341}, {100, 3197}, {219, 608}, {1332, 4329}, {1621, 2256}, {3211, 4511}, {3990, 4463}, {19350, 27396}

X(38876) = VU TANGENTIAL TRANSFORM OF X(346)

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

X(38876) lies on these lines: {6, 3726}, {9, 17625}, {63, 220}, {100, 1615}, {101, 154}, {644, 9778}, {1190, 3870}, {2284, 7074}, {4513, 17784}, {5732, 24771}, {7580, 35341}

X(38877) = VU TANGENTIAL TRANSFORM OF X(348)

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

X(38877) lies on these lines: {7, 1068}, {77, 1158}, {221, 934}, {222, 607}, {269, 15932}, {279, 1406}, {7177, 34043}

X(38878) = VU TANGENTIAL TRANSFORM OF X(350)

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

X(38878) lies on these lines: {1, 20663}, {2, 11}, {6, 23560}, {86, 1634}, {238, 660}, {739, 3231}, {932, 9259}, {1613, 1977}, {1740, 3123}, {1979, 21001}, {15485, 18794}, {16484, 21352}, {17379, 21320}, {24486, 25050}

X(38879) = VU TANGENTIAL TRANSFORM OF X(377)

Barycentrics    a^2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 4*a^6*b*c - 4*a^5*b^2*c + 4*a^2*b^5*c + 4*a*b^6*c - 2*a^6*c^2 - 4*a^5*b*c^2 - 2*a^4*b^2*c^2 - 8*a^3*b^3*c^2 - 6*a^2*b^4*c^2 + 4*a*b^5*c^2 + 2*b^6*c^2 - 8*a^3*b^2*c^3 - 16*a^2*b^3*c^3 - 8*a*b^4*c^3 - 6*a^2*b^2*c^4 - 8*a*b^3*c^4 - 2*b^4*c^4 + 4*a^2*b*c^5 + 4*a*b^2*c^5 + 2*a^2*c^6 + 4*a*b*c^6 + 2*b^2*c^6 - c^8): :

X(38879) lies on these lines: {1, 22}, {3, 3332}, {24, 5706}, {25, 387}, {105, 11365}, {1486, 1612}, {1621, 4295}, {1714, 1995}, {1754, 17928}, {3682, 37576}, {5721, 10594}, {5800, 30733}, {14017, 37538}, {19310, 19843}

X(38880) = VU TANGENTIAL TRANSFORM OF X(385)

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

X(38880) lies on these lines: {6, 2987}, {22, 3124}, {32, 36213}, {110, 699}, {111, 8627}, {112, 9431}, {237, 694}, {1611, 20998}, {3506, 5106}, {4611, 9427}, {5116, 20859}, {14602, 16385}

X(38881) = VU TANGENTIAL TRANSFORM OF X(388)

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

X(38881) lies on these lines: {3, 4307}, {22, 56}, {197, 1183}, {959, 1036}, {961, 5716}, {1612, 11365}, {1617, 3145}, {1621, 3485}, {25524, 27006}, {34036, 37583}

X(38882) = VU TANGENTIAL TRANSFORM OF X(529)

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

X(38882) lies on the circumcircle and these lines: {6, 8687}, {8, 8707}, {100, 960}, {101, 2269}, {108, 1829}, {109, 1193}, {110, 4267}, {831, 997}, {934, 24471}, {1603, 38273}, {2743, 5529}, {3920, 9058}

X(38882) = isogonal conjugate of X(529)
X(38882) = Λ(X(2), X(12))

X(38883) = VU TANGENTIAL TRANSFORM OF X(534)

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

X(38883) lies on the circumcircle and these lines: {6, 32691}, {63, 1310}, {100, 5227}, {101, 7085}, {108, 2285}, {109, 2286}, {346, 835}, {934, 28606}

X(38883) = isogonal conjugate of X(534)

X(38884) = VU TANGENTIAL TRANSFORM OF X(544)

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

X(38884) lies on the circumcircle and these lines: {6, 32682}, {100, 5011}, {101, 674}, {109, 5030}, {110, 14964}, {514, 675}, {573, 2742}, {919, 4262}, {29067, 33844}

X(38884) = isogonal conjugate of X(544)
X(38884) = Λ(X(2), X(101))

X(38885) = VU TANGENTIAL TRANSFORM OF X(858)

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

X(38885) lies on the circumcircle and these lines: {6, 15647}, {20, 1632}, {23, 895}, {24, 5622}, {30, 23296}, {67, 74}, {110, 159}, {125, 36851}, {154, 6593}, {206, 13248}, {468, 15128}, {511, 9934}, {524, 32264}, {542, 9833}, {1112, 19459}, {1495, 34470}, {1498, 2781}, {1614, 2904}, {1619, 32262}, {1620, 5621}, {1843, 13198}, {1853, 6698}, {2854, 9924}, {2935, 33851}, {3357, 32257}, {3564, 12419}, {3827, 7984}, {4232, 15118}, {5596, 14683}, {6759, 9970}, {7387, 14984}, {7691, 16789}, {7716, 11746}, {10282, 15462}, {10535, 32290}, {10752, 19149}, {11061, 11206}, {11579, 13289}, {11598, 31884}, {11663, 34788}, {11744, 29181}, {12219, 34146}, {15034, 15582}, {15035, 15577}, {15036, 35228}, {15059, 23300}, {15116, 16063}, {15585, 23315}, {22467, 29959}, {26888, 32289}, {31383, 32239}, {32233, 34782}, {32247, 34781}, {35371, 37777}

X(38886) = VU TANGENTIAL TRANSFORM OF X(908)

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

X(38886) lies on these lines: {9, 34256}, {36, 909}, {101, 102}, {109, 23980}, {165, 1633}, {610, 2265}, {910, 2161}, {1604, 36743}, {1615, 3196}, {2170, 2270}, {2182, 12034}, {3207, 17455}, {4266, 22767}

X(38887) = VU TANGENTIAL TRANSFORM OF X(984)

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

X(38887) lies on these lines: {6, 741}, {386, 5277}, {869, 985}, {1078, 3216}, {1185, 5156}, {1613, 37502}, {1627, 5009}, {1964, 8300}, {3736, 5276}

X(38888) = X(2)X(32)∩X(827)X(22329)

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

X(38888) lies on the cubic K1153 and these lines: {2, 32}, {827, 22329}, {4577, 11054}, {4628, 37854}

X(38889) = X(2)X(51)∩X(6094)X(37858)

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

X(38889) lies on the cubic K1153 and these lines: {2, 51}, {6094, 37858}, {22329, 26714}

X(38890) = X(2)X(512)∩X(729)X(22329)

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

X(38890) lies on the cubic K1153 and these lines: {2, 512}, {729, 22329}, {3228, 6094}

X(38891) = X(2)X(513)∩X(739)X(22329)

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

X(38891) lies on the cubic K1153 and these lines: {2, 513}, {739, 22329}, {3227, 11054}, {6094, 37857}

X(38892) = X(2)X(514)∩X(106)X(22329)

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

X(38892) lies on the cubic K1153 and these lines: {2, 514}, {106, 22329}, {543, 17969}, {903, 11054}, {4674, 37857}, {6094, 37854}, {8752, 37855}

X(38893) = X(2)X(522)∩X(1121)X(11054)

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

X(38893) lies on the cubic K1153 and these lines: {2, 522}, {1121, 11054}, {2291, 22329}, {6094, 37856}

X(38894) = X(2)X(525)∩X(74)X(22329)

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

X(38894) lies on the cubic K1153 and these lines: {2, 525}, {74, 22329}, {671, 9139}, {1494, 11054}, {6094, 8749}, {8598, 32640}

X(38895) = X(2)X(650)∩X(105)X(22329)

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

X(38895) lies on the cubic K1153 and these lines: {2, 650}, {105, 22329}, {666, 671}, {2481, 11054}, {8751, 37855}, {18785, 37854}

X(38895) = X(672)-isoconjugate of X(2721)
X(38895) = barycentric product X(2481)*X(2805)
X(38895) = barycentric quotient X(i)/X(j) for these {i,j}: {105, 2721}, {2805, 518}

X(38896) = X(5)X(523)∩X(265)X(1154)

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

X(38896) lies on the cubic K1154 and these lines: {5, 523}, {265, 1154}, {476, 1141}, {6344, 14993}

X(38896) = isogonal conjugate of X(38897)
X(38896) = X(6149)-isoconjugate of X(14979)
X(38896) = barycentric product X(94)*X(32423)
X(38896) = barycentric quotient X(i)/X(j) for these {i,j}: {1989, 14979}, {32423, 323}
X(38896) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {265, 14980, 3153}, {476, 1141, 2070}, {14993, 15392, 37943}

X(38897) = X(54)X(526)∩X(110)X(1154)

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

X(38897) lies on the cubic K1154 and these lines: {54, 526}, {110, 1154}, {1141, 33565}

X(38897) = isogonal conjugate of X(38896)
X(38897) = X(i)-isoconjugate of X(j) for these (i,j): {1, 38896}, {2166, 32423}
X(38897) = barycentric product X(323)*X(14979)
X(38897) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 38896}, {50, 32423}, {14979, 94}

X(38898) = X(3)X(7731)∩X(4)X(94)

Barycentrics    a^2*(a^12*b^2 - 4*a^10*b^4 + 5*a^8*b^6 - 5*a^4*b^10 + 4*a^2*b^12 - b^14 + a^12*c^2 - 2*a^10*b^2*c^2 + a^8*b^4*c^2 - 3*a^6*b^6*c^2 + 8*a^4*b^8*c^2 - 7*a^2*b^10*c^2 + 2*b^12*c^2 - 4*a^10*c^4 + a^8*b^2*c^4 + 4*a^6*b^4*c^4 - 4*a^4*b^6*c^4 + 3*a^2*b^8*c^4 + 5*a^8*c^6 - 3*a^6*b^2*c^6 - 4*a^4*b^4*c^6 - b^8*c^6 + 8*a^4*b^2*c^8 + 3*a^2*b^4*c^8 - b^6*c^8 - 5*a^4*c^10 - 7*a^2*b^2*c^10 + 4*a^2*c^12 + 2*b^2*c^12 - c^14) : :
X(38898) = 3 X[3] - X[13201], 3 X[51] - 2 X[11801], 2 X[125] - 3 X[5946], 2 X[140] - 3 X[16223], 3 X[381] - X[12281], 3 X[568] - X[3448], 3 X[568] - 2 X[13358], 3 X[2979] - 5 X[15040], 3 X[3060] - X[12902], 3 X[3060] + X[15102], 5 X[3567] - X[15100], 5 X[3567] - 3 X[38724], X[5876] + 2 X[14448], 3 X[5890] - X[10620], 3 X[5946] - X[15101], 4 X[5972] - 3 X[15067], X[6101] - 4 X[25711], X[7723] - 3 X[12824], 2 X[7723] - 3 X[15060], X[7731] + 2 X[11561], 3 X[7731] + X[13201], 4 X[9826] - 3 X[34128], 4 X[10095] - 3 X[14644], 5 X[10574] - 3 X[15041], 2 X[10627] - 3 X[15035], X[11412] - 3 X[32609], 6 X[11561] - X[13201], 2 X[11591] - 3 X[14643], 4 X[12006] - 3 X[15061], X[12111] - 3 X[38789], X[12219] - 3 X[14643], 6 X[13363] - 5 X[15059], 5 X[15026] - 6 X[16222], 5 X[15026] - 4 X[20304], 9 X[15046] - 7 X[15056], X[15100] - 3 X[38724], 3 X[16222] - 2 X[20304], X[18438] - 3 X[25321], 4 X[32142] - 5 X[38794]

S(38898) lies on these lines: {3, 7731}, {4, 94}, {5, 10628}, {30, 11562}, {49, 2914}, {51, 11801}, {52, 32196}, {74, 13434}, {110, 1154}, {113, 5876}, {125, 5946}, {140, 16223}, {182, 2781}, {185, 11560}, {381, 12281}, {382, 12270}, {389, 10264}, {399, 5889}, {511, 25329}, {546, 21650}, {1092, 1511}, {1173, 11559}, {1624, 14670}, {1994, 15089}, {2771, 31732}, {2777, 13491}, {2979, 15040}, {3043, 32171}, {3060, 12902}, {3567, 15100}, {3627, 11807}, {5480, 11806}, {5562, 10272}, {5609, 15083}, {5890, 10620}, {5944, 10274}, {5961, 8157}, {5972, 15067}, {6000, 11692}, {6241, 38790}, {6243, 12383}, {6639, 12358}, {6644, 17847}, {7488, 11597}, {7526, 17835}, {7723, 10254}, {9826, 15131}, {9970, 19138}, {10095, 14644}, {10263, 17702}, {10574, 15041}, {10610, 12228}, {10627, 15035}, {11271, 23236}, {11412, 32609}, {11563, 13754}, {11591, 12219}, {12006, 15061}, {12111, 38789}, {12121, 13391}, {12161, 12412}, {12165, 32139}, {12284, 12308}, {12319, 31815}, {13363, 15059}, {13621, 32339}, {14049, 15532}, {15026, 16222}, {15046, 15056}, {15055, 18364}, {18428, 22804}, {18438, 25321}, {18562, 34584}, {25338, 32269}, {32142, 38794}

X(38898) = midpoint of X(i) and X(j) for these {i,j}: {3, 7731}, {113, 14448}, {146, 34783}, {382, 12270}, {399, 5889}, {6241, 38790}, {6243, 12383}, {7722, 7728}, {11562, 13417}, {12284, 12308}, {12902, 15102}
X(38898) = reflection of X(i) in X(j) for these {i,j}: {3, 11561}, {5, 11557}, {74, 13630}, {265, 143}, {1511, 25711}, {3448, 13358}, {3627, 11807}, {5562, 10272}, {5876, 113}, {6101, 1511}, {6102, 1986}, {10113, 1112}, {10264, 389}, {12041, 14708}, {12219, 11591}, {15060, 12824}, {15101, 125}, {15532, 14049}, {21650, 546}
X(38898) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {568, 3448, 13358}, {3060, 15102, 12902}, {3567, 15100, 38724}, {5946, 15101, 125}, {12219, 14643, 11591}, {16222, 20304, 15026}

X(38899) = X(4)X(11671)∩X(137)X(25043)

Barycentrics    (a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(-(a^4*b^2) + 2*a^2*b^4 - b^6 + a^5*c - a^3*b^2*c + a^2*b^2*c^2 + 2*b^4*c^2 - 2*a^3*c^3 - a*b^2*c^3 - b^2*c^4 + a*c^5)*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^5*c - a^3*b^2*c - a^2*b^2*c^2 - 2*b^4*c^2 - 2*a^3*c^3 - a*b^2*c^3 + b^2*c^4 + a*c^5)*(a^5*b - 2*a^3*b^3 + a*b^5 - a^4*c^2 - a^3*b*c^2 + a^2*b^2*c^2 - a*b^3*c^2 - b^4*c^2 + 2*a^2*c^4 + 2*b^2*c^4 - c^6)*(a^5*b - 2*a^3*b^3 + a*b^5 + a^4*c^2 - a^3*b*c^2 - a^2*b^2*c^2 - a*b^3*c^2 + b^4*c^2 - 2*a^2*c^4 - 2*b^2*c^4 + c^6) : :
X(38899) = 4 X[18016] - 3 X[38706]

X(38899) lies on the conic {{A,B,C,X(4),X(5)}} and these lines: {4, 11671}, {137, 25043}, {930, 15345}, {1141, 1154}, {1263, 14140}, {1487, 32551}, {13512, 18807}, {14072, 22335}, {15619, 32423}, {18016, 38706} X(38899) = reflection of X(i) in X(j) for these {i,j}: {930, 15345}, {13512, 18807}, {25043, 137}
X(38899) = antigonal image of X(25043)
X(38899) = symgonal image of X(15345)
X(38899) = X(16336)-cross conjugate of X(5)
X(38899) = X(25149)-isoconjugate of X(36134)
X(38899) = cevapoint of X(1154) and X(15345)
X(38899) = trilinear pole of line {12077, 34520}
X(38899) = Kirikami concurrent circles image of X(5)
X(38899) = barycentric quotient X(i)/X(j) for these {i,j}: {233, 24147}, {12077, 25149}

leftri

Vu PCC perspectors: X(38900)-X(38929)

rightri

This preamble is contributed by Vu Thanh Tung, June 22, 2020.

Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC, not on any of the lines AO, BO, CO, where O = X(3) = circumcenter. Let

A1B1C1 = pedal triangle of P
A2 = the point, other than A, where the circles (ABC) and (AB1C1) meet, and define B2 and C2 cyclically
A3 = BB2∩CC2, define B3 and C3 cyclically
A4B4C4 = circumcevian triangle of P.

Then A4B4C4 is perspective to A2B2C2, and the perspector is the point V(P), here named the Vu 1st PCC perspector, given by

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

Also, A4B4C4 is perspective to A3B3C3, and the perspector is the point T(P), here named the Vu 2nd PCC perspector, given by

T(P) = -2 a^6 c^2 p^3 q + 2 a^4 b^2 c^2 p^3 q + 2 a^2 b^4 c^2 p^3 q - 2 b^6 c^2 p^3 q + 2 a^4 c^4 p^3 q - 4 a^2 b^2 c^4 p^3 q + 2 b^4 c^4 p^3 q + 2 a^2 c^6 p^3 q + 2 b^2 c^6 p^3 q - 2 c^8 p^3 q - 4 a^6 c^2 p^2 q^2 + 8 a^4 b^2 c^2 p^2 q^2 - 4 a^2 b^4 c^2 p^2 q^2 + 4 a^2 c^6 p^2 q^2 - 2 a^6 b^2 p^3 r + 2 a^4 b^4 p^3 r + 2 a^2 b^6 p^3 r - 2 b^8 p^3 r + 2 a^4 b^2 c^2 p^3 r - 4 a^2 b^4 c^2 p^3 r + 2 b^6 c^2 p^3 r + 2 a^2 b^2 c^4 p^3 r + 2 b^4 c^4 p^3 r - 2 b^2 c^6 p^3 r - 3 a^8 p^2 q r + 2 a^6 b^2 p^2 q r + 4 a^4 b^4 p^2 q r - 2 a^2 b^6 p^2 q r - b^8 p^2 q r + 2 a^6 c^2 p^2 q r - 8 a^4 b^2 c^2 p^2 q r + 2 a^2 b^4 c^2 p^2 q r + 4 b^6 c^2 p^2 q r + 4 a^4 c^4 p^2 q r + 2 a^2 b^2 c^4 p^2 q r - 6 b^4 c^4 p^2 q r - 2 a^2 c^6 p^2 q r + 4 b^2 c^6 p^2 q r - c^8 p^2 q r - 4 a^8 p q^2 r + 8 a^6 b^2 p q^2 r - 4 a^4 b^4 p q^2 r - 4 a^6 c^2 p q^2 r + 4 a^2 b^4 c^2 p q^2 r + 4 a^4 c^4 p q^2 r - 8 a^2 b^2 c^4 p q^2 r + 4 a^2 c^6 p q^2 r - 4 a^6 b^2 p^2 r^2 + 4 a^2 b^6 p^2 r^2 + 8 a^4 b^2 c^2 p^2 r^2 - 4 a^2 b^2 c^4 p^2 r^2 - 4 a^8 p q r^2 - 4 a^6 b^2 p q r^2 + 4 a^4 b^4 p q r^2 + 4 a^2 b^6 p q r^2 + 8 a^6 c^2 p q r^2 - 8 a^2 b^4 c^2 p q r^2 - 4 a^4 c^4 p q r^2 + 4 a^2 b^2 c^4 p q r^2 - 4 a^8 q^2 r^2 + 4 a^4 b^4 q^2 r^2 - 8a a^4 b^2 c^2 q^2 r^2 + 4 a^4 c^4 q^2 r^2 : :

See Vu Pedal Circles Circumcevian Perspector.

The appearance of (i,j) in the following list means that V(X(i)) = X(j):
(1,56),(2,1995), (4,24), (5,13621), (6,1384), (7,38900), (8,38901), (9,38902), (10,38903), (31,38904), (32,38905), (75,38906), (76,38907), (83,38908), (141,38909), (560,38910), (561,38911), (1501,38912), (1502,38913), (1928,38914), (2321,38915), (2887,38916), (3676,38917)

The appearance of (i,j) in the following list means that T(X(i)) = X(j):
(1,1394), (2,38918), (4,4), (5,38919), (6,38920), (8,38922), (9,38923), (10,38924), (31,38925), (32,38926), (75,38927) (76,38928), (83,38929)

V(P) is the isogonal conjugate of the pedal antipodal perspector of P. (Randy Hutson, June 17, 2020)

X(38900) = VU 1ST PCC PERSPECTOR OF X(7)

X(38900) lies on these lines: {3,7}, {24,38461}, {55,23839}, {56,38459}, {279,37579}, {479,934}, {1384,38466}, {1602,23397}, {1847,14017}, {1995,37761}, {4293,37048}, {4306,38877}, {4341,7177}, {4350,37583}, {9436,37300}, {10481,36152}, {13621,38464}, {37272,37797}, {38468,38901}

X(38900) = pole of the trilinear polar of X(34521) wrt circumcircle
X(38900) = {X(7), X(32624)}-harmonic conjugate of X(3)

X(38901) = VU 1ST PCC PERSPECTOR OF X(8)

X(38901) lies on these lines: {1,10094}, {2,26476}, {3,8}, {22,1603}, {24,38462}, {35,997}, {55,3890}, {56,8668}, {78,1158}, {145,1470}, {377,10321}, {404,3086}, {474,11680}, {1012,11681}, {1376,5086}, {1384,38467}, {1466,3873}, {1604,38869}, {1737,25440}, {1995,37762}, {2098,13205}, {3435,37577}, {3436,6909}, {3560,27529}, {3869,10310}, {3871,8071}, {3872,37561}, {3880,34880}, {3885,22767}, {4188,5265}, {4421,37564}, {4511,11248}, {4861,10269}, {4881,11510}, {5010,31424}, {5172,37301}, {5175,37249}, {5176,12114}, {5217,20846}, {5440,5887}, {5450,6735}, {5537,11682}, {5552,6906}, {5704,37282}, {5730,35000}, {6911,11928}, {6940,10527}, {6948,10522}, {7742,13587}, {8715,14793}, {9342,37244}, {9709,19525}, {10058,26364}, {10571,35281}, {10914,32612}, {10916,36152}, {10950,15813}, {11509,34772}, {12758,30144}, {13621,38465}, {17615,34862}, {22837,39776}, {25304,36741}, {26357,37293}, {34195,37541}, {37307,37578}, {38468,38900}

X(38901) = anticomplement of X(26476)
X(38901) = pole of the trilinear polar of X(34523) wrt circumcircle
X(38901) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 17100, 3), (56, 8668, 38460), (1376, 22760, 25005)

X(38902) = VU 1ST PCC PERSPECTOR OF X(9)

X(38902) lies on these lines: {3,9}, {55,101}, {56,169}, {57,23840}, {218,1466}, {220,10310}, {474,8568}, {956,5144}, {1384,21002}, {1415,5452}, {1995,37763}, {2174,2256}, {3207,32561}, {3576,15288}, {3911,37272}, {4251,22153}, {4253,33590}, {5540,22767}, {5657,36012}, {5744,11349}, {5819,22753}, {6554,12114}, {7368,11248}, {8732,38859}, {9605,11505}, {16788,33848}, {20853,35342}

X(38902) = barycentric product X(100)*X(30235)
X(38902) = trilinear product X(101)*X(30235)
X(38902) = intersection, other than A,B,C, of conics {{A, B, C, X(84), X(2291)}} and {{A, B, C, X(282), X(4845)}}
X(38902) = pole of the trilinear polar of X(34525) wrt circumcircle
X(38902) = crossdifference of every pair of points on line {X(1638), X(6129)}
X(38902) = {X(9), X(32625)}-harmonic conjugate of X(3)

X(38903) = VU 1ST PCC PERSPECTOR OF X(10)

X(38903) lies on these lines: {3,10}, {8,37259}, {24,17927}, {25,318}, {35,20834}, {45,36744}, {55,5293}, {56,24174}, {58,23841}, {100,3145}, {101,1500}, {109,29958}, {169,4386}, {404,20999}, {956,20842}, {999,17054}, {1995,37764}, {2550,19548}, {2886,36558}, {2915,3437}, {3295,30115}, {3434,28077}, {3436,37397}, {3507,37576}, {3871,23858}, {4245,17734}, {4413,16422}, {5251,20849}, {5253,24183}, {5399,36942}, {5552,13733}, {5687,11334}, {6211,10310}, {7083,11490}, {7141,26704}, {7742,16059}, {8069,28383}, {8669,11365}, {9780,37247}, {11329,27321}, {11507,20760}, {11509,20805}, {16286,24931}, {16414,24880}, {16453,25446}, {16824,37257}, {17757,37227}, {20741,23619}, {20836,20989}, {23853,37034}, {24443,36560}

X(38903) = pole of the trilinear polar of X(34527) wrt circumcircle
X(38903) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 18525, 35455), (10, 1324, 3), (958, 2933, 3), (1376, 23843, 3), (23850, 25440, 3)

X(38904) = VU 1ST PCC PERSPECTOR OF X(31)

X(38904) lies on these lines: {3,31}, {81,17521}, {110,21769}, {902,3781}, {1331,16946}, {1495,37819}, {1501,21775}, {1627,20995}, {1724,17977}, {1974,32691}, {3285,21793}, {4588,28583}, {5078,16790}, {15440,33772}, {17025,17595}, {17104,38832}, {17126,35984}

X(38904) = barycentric product X(101)*X(30183)
X(38904) = trilinear product X(692)*X(30183)
X(38904) = {X(31), X(5161)}-harmonic conjugate of X(3)

X(38905) = VU 1ST PCC PERSPECTOR OF X(32)

X(38905) lies on these lines: {3,6}, {83,7773}, {112,699}, {1078,7868}, {1285,7791}, {1501,11328}, {1613,14602}, {1625,33786}, {1627,20885}, {2715,32540}, {3314,7767}, {3406,13860}, {3407,7770}, {3933,39141}, {4027,7754}, {5026,6309}, {5976,6179}, {6287,7746}, {6656,7787}, {7737,18501}, {7758,13196}, {7776,10349}, {7778,39603}, {7779,10351}, {7788,7870}, {7805,35700}, {7837,35297}, {7859,10348}, {7876,10346}, {7879,10333}, {7881,10334}, {7906,10353}, {8150,15271}, {10104,37071}, {10131,31859}, {10359,37450}, {10547,38834}, {11057,11287}, {11333,16985}, {19597,32661}, {21778,33772}, {32134,37242}, {33235,39652}

X(38905) = barycentric product X(110)*X(30217)
X(38905) = barycentric quotient X(1576)/X(30254)
X(38905) = trilinear product X(163)*X(30217)
X(38905) = trilinear quotient X(163)/X(30254)
X(38905) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(699)}} and {{A, B, C, X(98), X(30270)}}
X(38905) = circumcircle-inverse of-X(16385)
X(38905) = X(6)-Hirst inverse of-X(16385)
X(38905) = X(1577)-isoconjugate-of-X(30254)
X(38905) = X(1576)-reciprocal conjugate of-X(30254)
X(38905) = X(512)-vertex conjugate of-X(16385)
X(38905) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32, 1691, 3), (32, 3398, 30435), (32, 5033, 13356), (1379, 1380, 16385), (1687, 1688, 30270), (5033, 13356, 12054), (12054, 13356, 5024), (36759, 36760, 1350)

X(38906) = VU 1ST PCC PERSPECTOR OF X(75)

X(38906) lies on these lines: {3,75}, {24,38457}, {76,23843}, {99,1626}, {183,1324}, {350,11334}, {1078,2933}, {1310,1621}, {1975,23850}, {4441,37311}, {4713,21004}, {11329,33129}, {16367,32779}, {16681,33801}

X(38906) = barycentric product X(75)*X(30878)
X(38906) = trilinear product X(2)*X(30878)
X(38906) = {X(75), X(39556)}-harmonic conjugate of X(3)

X(38907) = VU 1ST PCC PERSPECTOR OF X(76)

X(38907) lies on these lines: {3,76}, {24,17984}, {157,1502}, {315,5999}, {316,9873}, {384,3767}, {385,31981}, {789,23852}, {1003,14568}, {1352,7763}, {2023,10349}, {3148,3978}, {3329,10351}, {3552,32815}, {3763,5116}, {4048,7807}, {5017,6179}, {5149,7746}, {5162,7751}, {7470,14907}, {7752,13860}, {7770,7828}, {7793,10997}, {7795,7824}, {7796,15069}, {7851,11356}, {7907,8290}, {7930,31268}, {8178,32452}, {8289,10131}, {17932,23128}, {18906,35424}, {20023,37183}, {24273,33217}, {33801,33802}, {34827,35894}

X(38907) = isotomic conjugate of the isogonal conjugate of X(2001)
X(38907) = barycentric product X(76)*X(2001)
X(38907) = barycentric quotient X(2001)/X(6)
X(38907) = trilinear product X(75)*X(2001)
X(38907) = trilinear quotient X(2001)/X(31)
X(38907) = X(2001)-reciprocal conjugate of-X(6)
X(38907) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (76, 5152, 3), (7828, 10000, 7770), (12215, 37334, 7763)

X(38908) = VU 1ST PCC PERSPECTOR OF X(83)

X(38908) lies on these lines: {3,83}, {32,733}, {251,3117}, {1078,35222}, {3398,14247}, {14880,38946}

X(38908) = {X(83), X(39557)}-harmonic conjugate of X(3)

X(38909) = VU 1ST PCC PERSPECTOR OF X(141)

X(38909) lies on these lines: {3,66}, {907,38862}, {3734,11641}, {7669,7820}, {7779,11327}, {10317,14913}

X(38909) = {X(141), X(5938)}-harmonic conjugate of X(3)

X(38910) = VU 1ST PCC PERSPECTOR OF X(560)

X(38910) lies on the line {3,560}

X(38911) = VU 1ST PCC PERSPECTOR OF X(561)

X(38911) lies on the line {3,561}

X(38912) = VU 1ST PCC PERSPECTOR OF X(1501)

X(38912) lies on the line {3,1501}

X(38913) = VU 1ST PCC PERSPECTOR OF X(1502)

X(38913) lies on the line {3,1502}

X(38914) = VU 1ST PCC PERSPECTOR OF X(1928)

X(38914) lies on the line {3,1928}

X(38915) = VU 1ST PCC PERSPECTOR OF X(2321)

X(38915) lies on the line {3,2321}

X(38916) = VU 1ST PCC PERSPECTOR OF X(2887)

X(38916) lies on the line {3,2887}

X(38917) = VU 1ST PCC PERSPECTOR OF X(3676)

X(38917) lies on the line {3,3676}

X(38918) = VU 2ND PCC PERSPECTOR OF X(2)

X(38918) lies on these lines: {2,154}, {20,35278}, {22,1661}, {25,9748}, {110,37668}, {232,4232}, {441,15428}, {6194,10565}, {6353,9755}, {7493,15589}, {7735,15448}, {10130,21458}

X(38919) = VU 2ND PCC PERSPECTOR OF X(5)

X(38919) lies on these lines: {5,5944}, {1624,7488}

X(38920) = VU 2ND PCC PERSPECTOR OF X(6)

X(38920) lies on these lines: {6,1597}, {112,26864}, {1384,1495}, {3172,11456}, {3284,11820}, {3331,21309}, {8778,11464}, {15905,35237}

X(38920) = X(i)-Hirst inverse of-X(j) for these {i,j}: {1384, 1495}, {1495, 1384}

X(38921) = VU 2ND PCC PERSPECTOR OF X(7)

X(38921) lies on these lines: {7,3427}, {5731,14256}

X(38922) = VU 2ND PCC PERSPECTOR OF X(8)

X(38922) lies on these lines: {8,6001}, {5423,14155}

X(38923) = VU 2ND PCC PERSPECTOR OF X(9)

X(38923) lies on the line {9,3197}

X(38924) = VU 2ND PCC PERSPECTOR OF X(10)

X(38924) lies on the line {10,1503}

X(38925) = VU 2ND PCC PERSPECTOR OF X(31)

X(38925) lies on these lines: {}

X(38926) = VU 2ND PCC PERSPECTOR OF X(32)

X(38926) lies on these lines: {}

X(38927) = VU 2ND PCC PERSPECTOR OF X(75)

X(38927) lies on these lines: {}

X(38928) = VU 2ND PCC PERSPECTOR OF X(76)

X(38928) lies on the line {76,1503}

X(38929) = VU 2ND PCC PERSPECTOR OF X(83)

X(38929) lies on the line {83,1503}

X(38930) = COMPLEMENT OF X(1434)

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

X(38930) lies on these lines: {2, 1434}, {6, 5084}, {9, 46}, {10, 17747}, {11, 3691}, {37, 21075}, {72, 21049}, {115, 13540}, {141, 17671}, {144, 25004}, {220, 1834}, {274, 25682}, {429, 7079}, {430, 1855}, {452, 4258}, {594, 3697}, {661, 23764}, {857, 1211}, {910, 12572}, {960, 1146}, {965, 6836}, {966, 3091}, {1014, 24952}, {1212, 3452}, {1334, 21031}, {2082, 4679}, {2324, 4272}, {2328, 33306}, {2329, 23905}, {2478, 37658}, {3061, 10026}, {3294, 17757}, {3501, 9711}, {3649, 21921}, {3704, 3985}, {3730, 3820}, {3816, 21384}, {3932, 4109}, {3943, 4006}, {3947, 5257}, {3948, 30854}, {4015, 21090}, {4187, 16552}, {4205, 18250}, {4253, 17527}, {4415, 16583}, {4520, 6735}, {5044, 5179}, {5233, 27523}, {5234, 17514}, {5241, 26035}, {5254, 37673}, {5283, 37662}, {5286, 37679}, {5737, 36662}, {5742, 6991}, {5743, 7377}, {5795, 6603}, {6707, 25679}, {7379, 26244}, {9310, 34606}, {9709, 17732}, {13407, 25086}, {15048, 17749}, {16842, 17398}, {17330, 17556}, {17499, 33034}, {21044, 21677}, {21258, 30946}, {21879, 21965}, {24045, 31419}, {27039, 27049}, {27065, 27068}, {27541, 31043}, {37421, 37499}

X(38930) = complement of X(1434)
X(38930) = isogonal conjugate of X(38811)
X(38930) = complement of the isogonal conjugate of X(1334)
X(38930) = complement of the isotomic conjugate of X(2321)
X(38930) = medial isogonal conjugate of X(17050)
X(38930) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 3946}, {37206, 523}
X(38930) = X(i)-isoconjugate of X(j) for these (i,j): {1, 38811}, {1014, 38825}
X(38930) = X(i)-reciprocal conjugate of X(j) for these (i,j): {6, 38811}, {7, 4854}, {10, 21673}, {1334, 38825}, {1434, 3946}, {17096, 23729} X(38930) = crosspoint of X(2) and X(2321)
X(38930) = crosssum of X(6) and X(1412)
X(38930) = crossdifference of every pair of points on line {2605, 8653}
X(38930) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 17050}, {6, 3742}, {8, 21240}, {9, 3741}, {10, 17046}, {31, 3946}, {33, 34830}, {37, 2886}, {41, 1125}, {42, 142}, {55, 3739}, {65, 21258}, {71, 34822}, {72, 18639}, {100, 17066}, {181, 18635}, {200, 21246}, {210, 141}, {213, 1}, {220, 960}, {228, 17073}, {321, 17047}, {512, 4904}, {607, 942}, {644, 512}, {646, 23301}, {657, 34589}, {661, 17059}, {663, 17761}, {692, 17069}, {756, 17052}, {762, 34829}, {798, 3756}, {872, 17056}, {1018, 17072}, {1253, 5745}, {1334, 10}, {1400, 11019}, {1402, 4000}, {1500, 442}, {1802, 34851}, {1824, 16608}, {1918, 3752}, {1924, 16614}, {2175, 3666}, {2194, 17045}, {2200, 17102}, {2205, 17053}, {2318, 18589}, {2321, 2887}, {2328, 21233}, {2333, 1210}, {3063, 244}, {3690, 18642}, {3694, 1368}, {3700, 21252}, {3701, 626}, {3709, 11}, {3939, 4369}, {3971, 20547}, {3985, 20542}, {4041, 116}, {4069, 3835}, {4079, 8286}, {4082, 21244}, {4171, 124}, {4433, 20333}, {4515, 1329}, {4524, 26932}, {4557, 4885}, {4559, 3900}, {6057, 21245}, {7064, 1211}, {7071, 6708}, {7079, 34831}, {7109, 2092}, {8641, 4858}, {18098, 17049}, {20683, 17060}, {20691, 20338}, {21759, 17063}, {21814, 17055}, {21871, 20307}, {23493, 20257}, {23990, 34977}, {28658, 17051}, {30713, 21235}, {30730, 21260}, {33635, 27798}, {34074, 4843}
X(38930) = barycentric product X(i)*X(j) for these {i,j}: {8, 4854}, {86, 21673}, {2321, 3946}, {4082, 10521}, {23729, 30730}
X(38930) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 38811}, {1334, 38825}, {3946, 1434}, {4854, 7}, {21673, 10}, {23729, 17096}
X(38930) = {X(3697),X(21073)}-harmonic conjugate of X(594)

X(38931) = X(3)X(299)∩X(4)X(3440)

Barycentrics    (Sqrt[3]*a^2 + 2*S)/(Sqrt[3]*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 2*(a^2 - b^2 - c^2)*S) : :
X(38931) = 3 X[15768] - 2 X[15784]

X(38931) lies on the cubics K005 and K438) and these lines: {3, 299}, {4, 3440}, {5, 11119}, {17, 8839}, {62, 8919}, {398, 523}, {532, 15768}, {3180, 15793}, {3462, 3489}

X(38931) = isogonal conjugate of X(39262)
X(38931) = Kosnita(X(616),X(3)) point
X(38931) = X(14372)-Ceva conjugate of X(3480)
X(38931) = barycentric product X(i)*X(j) for these {i,j}: {396, 2992}, {14372, 14922}
X(38931) = barycentric quotient X(i)/X(j) for these {i,j}: {396, 621}, {3438, 2981}, {19294, 14368}, {23714, 11093}
X(38931) = {X(2992),X(3438)}-harmonic conjugate of X(3480)

X(38932) = X(3)X(298)∩X(4)X(3441)

Barycentrics    (Sqrt[3]*a^2 - 2*S)/(Sqrt[3]*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 2*(a^2 - b^2 - c^2)*S) : :
X(38932) = 3 X[15769] - 2 X[15785]

X(38932) lies on the cubics K005 and K438b and these lines: {3, 298}, {4, 3441}, {5, 11120}, {18, 8837}, {61, 8918}, {397, 523}, {533, 15769}, {3181, 15794}, {3462, 3490}

X(38932) = isogonal conjugate of X(39261)
X(38932) = Kosnita(X(617),X(3)) point
X(38932) = X(14373)-Ceva conjugate of X(3479)
X(38932) = barycentric product X(395)*X(2993)
X(38932) = barycentric quotient X(i)/X(j) for these {i,j}: {395, 622}, {3439, 6151}, {19295, 14369}, {23715, 11094}
X(38932) = {X(2993),X(3439)}-harmonic conjugate of X(3479)

X(38933) = X(3)X(2132)∩X(4)X(74)

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

X(38933) lies on the cubic K005 and these lines: {3, 2132}, {4, 74}, {5, 34298}, {54, 3470}, {195, 3463}, {389, 34329}, {1304, 10282}, {1498, 34185}, {1552, 5893}, {3336, 3469}, {6146, 17986}, {7395, 35910}, {7592, 14264}, {7691, 36831}, {10421, 21659}, {10982, 35908}, {12079, 14895}, {14059, 14919}

X(38933) = X(i)-Ceva conjugate of X(j) for these (i,j): {5, 3470}, {34298, 2132}
X(38933) = X(11587)-cross conjugate of X(3484)
X(38933) = barycentric product X(i)*X(j) for these {i,j}: {6761, 14919}, {15781, 16080}
X(38933) = barycentric quotient X(i)/X(j) for these {i,j}: {11587, 14920}, {15781, 11064}

X(38934) = X(3)X(3464)∩X(4)X(5680)

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

X(38934) lies on the cubic K005 and these lines: {3, 3464}, {4, 5680}, {5, 34299}, {3460, 3471}, {3462, 3467}, {6191, 8918}, {6192, 8919}, {15775, 15795}

X(38934) = X(34299)-Ceva conjugate of X(3466)
X(38934) = X(i)-isoconjugate of X(j) for these (i,j): {3065, 3465}, {7343, 34301}
X(38934) = barycentric product X(3466)*X(17484)
X(38934) = barycentric quotient X(i)/X(j) for these {i,j}: {3466, 21739}, {11076, 34301}, {19297, 3465}

X(38935) = X(3)X(1263)∩X(4)X(5671)

Barycentrics    (a^6 - a^4*b^2 - a^2*b^4 + b^6 - 3*a^4*c^2 + a^2*b^2*c^2 - 3*b^4*c^2 + 3*a^2*c^4 + 3*b^2*c^4 - c^6)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - a^4*c^2 + a^2*b^2*c^2 + 3*b^4*c^2 - a^2*c^4 - 3*b^2*c^4 + c^6)*(2*a^16 - 11*a^14*b^2 + 25*a^12*b^4 - 31*a^10*b^6 + 25*a^8*b^8 - 17*a^6*b^10 + 11*a^4*b^12 - 5*a^2*b^14 + b^16 - 11*a^14*c^2 + 34*a^12*b^2*c^2 - 31*a^10*b^4*c^2 - 4*a^8*b^6*c^2 + 31*a^6*b^8*c^2 - 38*a^4*b^10*c^2 + 27*a^2*b^12*c^2 - 8*b^14*c^2 + 25*a^12*c^4 - 31*a^10*b^2*c^4 - 5*a^6*b^6*c^4 + 34*a^4*b^8*c^4 - 51*a^2*b^10*c^4 + 28*b^12*c^4 - 31*a^10*c^6 - 4*a^8*b^2*c^6 - 5*a^6*b^4*c^6 - 14*a^4*b^6*c^6 + 29*a^2*b^8*c^6 - 56*b^10*c^6 + 25*a^8*c^8 + 31*a^6*b^2*c^8 + 34*a^4*b^4*c^8 + 29*a^2*b^6*c^8 + 70*b^8*c^8 - 17*a^6*c^10 - 38*a^4*b^2*c^10 - 51*a^2*b^4*c^10 - 56*b^6*c^10 + 11*a^4*c^12 + 27*a^2*b^2*c^12 + 28*b^4*c^12 - 5*a^2*c^14 - 8*b^2*c^14 + c^16) : :

X(38935) lies on the cubic K005 and these lines: {3, 1263}, {4, 5671}, {5, 34302}, {54, 3471}, {1291, 15787}, {1510, 10095}, {8902, 32639}, {10285, 27357}, {13582, 21230}

X(38935) = X(i)-Ceva conjugate of X(j) for these (i,j): {5, 3471}, {34302, 1263}
X(38935) = barycentric product X(13582)*X(24385)
X(38935) = barycentric quotient X(24385)/X(37779)

X(38936) = X(3)X(10420)∩X(4)X(110)

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

X(38936) lies on the cubics Kl028 and K613 and these lines: {3, 10420}, {4, 110}, {24, 250}, {54, 15328}, {76, 18878}, {340, 10411}, {378, 10419}, {578, 18279}, {1986, 15468}, {3431, 35373}, {5962, 22115}, {7577, 12028}, {8743, 32708}, {10312, 14910}, {14385, 15470}, {14911, 35481}

X(38936) = isogonal conjugate of X(39170)
X(38936) = isogonal conjugate of the complement of X(15454)
X(38936) = X(i)-cross conjugate of X(j) for these (i,j): {186, 1300}, {526, 10420}, {1511, 186}
X(38936) = X(i)-isoconjugate of X(j) for these (i,j): {94, 2315}, {265, 1725}, {686, 32680}, {2166, 13754}, {6334, 32678}
X(38936) = cevapoint of X(186) and X(3043)
X(38936) = crosssum of X(i) and X(j) for these {i,j}: {18558, 20975}, {21731, 39021}
X(38936) = trilinear pole of line {50, 15470}
X(38936) = barycentric product X(i)*X(j) for these {i,j}: {186, 2986}, {323, 1300}, {340, 14910}, {526, 687}, {648, 15470}, {3268, 32708}, {4558, 14222}, {5504, 14165}, {10419, 14920}, {14590, 15328}, {18879, 35235}, {32679, 36114}
X(38936) = barycentric quotient X(i)/X(j) for these {i,j}: {50, 13754}, {186, 3580}, {526, 6334}, {687, 35139}, {1300, 94}, {2986, 328}, {3043, 34834}, {14222, 14618}, {14270, 686}, {14591, 15329}, {14910, 265}, {15328, 14592}, {15470, 525}, {32708, 476}, {34397, 3003}, {36114, 32680}, {36423, 1986}
X(38936) = trilinear product X(i)*X(j) for these {i,j}: {162, 15470}, {186, 36053}, {526, 36114}, {687, 2624}, {1300, 6149}, {4575, 14222}, {10419, 35201}, {32679, 32708}
X(38936) = {X(4),X(15454)}-harmonic conjugate of X(1300)

X(38937) = X(3)X(1304)∩X(4)X(74)

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

X(38937) lies on the cubics Kl028 and K928 and these lines: {3, 1304}, {4, 74}, {5, 1552}, {20, 13573}, {64, 15384}, {76, 16077}, {378, 10419}, {1105, 6368}, {1593, 35908}, {1885, 12079}, {1968, 8749}, {3470, 35475}, {3516, 9717}, {3520, 14385}, {4240, 13445}, {8743, 32695}, {10421, 35481}, {11250, 14059}, {11440, 36831}, {12292, 15468}, {15072, 32715}, {18560, 34150}

X(38937) = X(11598)-cross conjugate of X(2071)
X(38937) = cevapoint of X(2071) and X(12825)
X(38937) = barycentric product X(i)*X(j) for these {i,j}: {1494, 15262}, {2071, 16080}, {14919, 34170}
X(38937) = barycentric quotient X(i)/X(j) for these {i,j}: {2071, 11064}, {8749, 11744}, {15262, 30}, {32695, 22239}

X(38938) = X(3)X(759)∩X(4)X(6003)

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

X(38938) lies on the cubic K028 and these lines: {3, 759}, {4, 6003}, {8, 80}, {10, 36815}, {76, 14616}, {1168, 2099}, {1411, 10571}, {2161, 3730}, {2222, 37579}, {4193, 6740}, {4251, 23903}, {11114, 24624}, {17556, 27739}, {24883, 36171}

X(38938) = isogonal conjugate of X(39166)
X(38938) = X(15906)-cross conjugate of X(30117)
X(38938) = crosssum of X(i) and X(j) for these (i,j): {36, 35204}, {3724, 35069}
X(38938) = barycentric product X(i)*X(j) for these {i,j}: {80, 33129}, {1731, 18815}, {18359, 30117}
X(38938) = barycentric quotient X(i)/X(j) for these {i,j}: {1731, 4511}, {5146, 17923}, {13589, 4585}, {15906, 16586}, {30117, 3218}, {33129, 320}
X(38938) = trilinear product X(i)*X(j) for these {i,j}: {80, 30117}, {1731, 2006}, {1807, 5146}, {2161, 33129}
X(38938) = {X(24880),X(38511)}-harmonic conjugate of X(3)

X(38939) = X(3)X(691)∩X(4)X(690)

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

X(38939) lies on the cubic K028 and these lines: {3, 691}, {4, 690}, {76, 5641}, {8743, 32708}, {14247, 38539}, {18333, 23108}

X(38939) = X(14984)-cross conjugate of X(842)
X(38939) = crosssum of X(5191) and X(23967)
X(38939) = barycentric product X(i)*X(j) for these {i,j}: {2493, 5641}, {7468, 14223}, {14221, 14998}
X(38939) = barycentric quotient X(i)/X(j) for these {i,j}: {2493, 542}, {7468, 14999}

X(38940) = X(2)X(6)∩X(20)X(1499)

Barycentrics    3*a^6 - 5*a^4*b^2 + 3*a^2*b^4 - b^6 - 5*a^4*c^2 + 5*a^2*b^2*c^2 - b^4*c^2 + 3*a^2*c^4 - b^2*c^4 - c^6 : : ?
X(38940) = 3 X[2] - 4 X[5108], 5 X[2] - 4 X[9169], X[3146] - 4 X[15098], 5 X[5108] - 3 X[9169], 2 X[5108] - 3 X[14916], 5 X[6792] - 6 X[9169], X[6792] - 3 X[14916], 2 X[9169] - 5 X[14916]

X(3940) lies on the cubic K072 and these lines: {2, 6}, {20, 1499}, {30, 18346}, {99, 9143}, {111, 14645}, {315, 6328}, {542, 9146}, {842, 6082}, {843, 2858}, {2396, 14907}, {2482, 10554}, {2709, 2857}, {2753, 2856}, {2770, 10425}, {3146, 15098}, {3448, 4563}, {3926, 14357}, {4226, 32817}, {4576, 14683}, {5182, 10552}, {5969, 20099}, {7763, 9716}, {7850, 22254}, {9999, 10553}, {10411, 11003}, {11061, 36792}, {14929, 36194}, {32815, 36181}, {32833, 34245}

X(38940) = anticomplement of X(6792)
X(38940) = reflection of X(i) in X(j) for these {i,j}: {2, 14916}, {6792, 5108}, {14360, 9146}
X(38940) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(325)
X(38940) = deLongchamps-circle-inverse of X(6563)
X(38940) = psi-transform of X(620)
X(38940) = orthogonal projection of X(20) on line X(2)X(6)
X(38940) = barycentric product X(4590)*X(15357)
X(38940) = barycentric quotient X(15357)/X(115)
X(38940) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 5468, 2}, {5108, 6792, 2}, {6792, 14916, 5108}

X(38941) = X(1)X(7)∩X(2)X(514)

Barycentrics    a^4 - a^3*b - 2*a^2*b^2 + a*b^3 + b^4 - a^3*c + 5*a^2*b*c - a*b^2*c - 3*b^3*c - 2*a^2*c^2 - a*b*c^2 + 4*b^2*c^2 + a*c^3 - 3*b*c^3 + c^4 : :
X(38941) = 4 X[140] - X[18329], X[150] + 2 X[664], X[150] - 4 X[1565], X[664] + 2 X[1565], 4 X[1323] - X[5088], 8 X[1323] + X[5195], 5 X[3091] - 2 X[18328], X[3146] - 4 X[31851], 7 X[3523] - 4 X[31852], X[3732] - 4 X[17044], 2 X[5088] + X[5195]

X(38941) lies on the cubic K165 and these lines: {1, 7}, {2, 514}, {69, 6790}, {85, 5886}, {86, 3109}, {140, 18329}, {144, 1023}, {150, 664}, {348, 5657}, {499, 17090}, {517, 17078}, {905, 24499}, {927, 953}, {934, 38693}, {1111, 16173}, {1358, 24203}, {2087, 4000}, {3091, 18328}, {3146, 31851}, {3328, 5218}, {3523, 31852}, {3732, 17044}, {3777, 24488}, {4389, 14260}, {4618, 36944}, {4872, 28160}, {5074, 29569}, {5179, 17244}, {5532, 10589}, {5587, 9312}, {5603, 17079}, {5845, 35110}, {6516, 34474}, {6788, 23816}, {9311, 20269}, {9436, 28234}, {10222, 32007}, {11231, 17095}, {16820, 17257}, {17084, 37701}, {17379, 24279}, {19943, 20006}

X(38941) = anticomplement of X(101)-of-orthocentroidal-triangle
X(38941) = incircle-inverse of X(4887)
X(38941) = deLongchamps-circle-inverse of X(3007)
X(38941) = crossdifference of every pair of points on line {657, 902}
X(38941) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 4089, 7}, {664, 1565, 150}

X(38942) = X(2)X(9927)∩X(3)X(9544)

Barycentrics    a^2*(4*a^8 - 10*a^6*b^2 + 6*a^4*b^4 + 2*a^2*b^6 - 2*b^8 - 10*a^6*c^2 + 17*a^4*b^2*c^2 - 8*a^2*b^4*c^2 + b^6*c^2 + 6*a^4*c^4 - 8*a^2*b^2*c^4 + 2*b^4*c^4 + 2*a^2*c^6 + b^2*c^6 - 2*c^8) : :
X(38942) = 3 X[3] - X[11999], 3 X[11270] - 2 X[11999]

Let HAHBHC be as defined at X(7666). HAHBHC is inversely similar to ABC with similitude center X(38942). (Randy Hutson, June 17, 2020)

X(38942) lies on the cubic K850 and these lines: {2, 9927}, {3, 9544}, {4, 1511}, {20, 10282}, {110, 3357}, {140, 3431}, {155, 37941}, {186, 2904}, {195, 15040}, {323, 32534}, {376, 18442}, {394, 38448}, {550, 7712}, {576, 15020}, {578, 12834}, {631, 2888}, {1071, 5694}, {1092, 7691}, {1147, 15035}, {1204, 15051}, {1350, 7488}, {1498, 2071}, {3516, 15052}, {3522, 4549}, {3523, 3620}, {3528, 5944}, {3533, 14805}, {3543, 15751}, {5050, 17928}, {5056, 11430}, {5059, 10564}, {5876, 23040}, {6102, 9716}, {6640, 12383}, {7592, 9545}, {8546, 10541}, {8567, 11441}, {10255, 34153}, {10257, 34799}, {10304, 11821}, {10539, 15034}, {11002, 38638}, {11202, 38435}, {11250, 32609}, {11413, 11820}, {12085, 35265}, {12111, 35493}, {12162, 35494}, {12289, 14156}, {13352, 38848}, {13620, 37486}, {14118, 32620}, {15043, 15516}, {15717, 18475}, {17818, 19357}, {18281, 18432}, {18350, 35475}, {21844, 22115}, {22555, 22951}, {23294, 30714}, {26917, 38793}, {32139, 37948}, {32233, 34118}, {32330, 37444}, {37498, 37940}

X(38942) = reflection of X(11270) in X(3)
X(38942) = anticomplement of X(11704)
X(38942) = X(33641)-isoconjugate of X(36119)
X(38942) = barycentric quotient X(3284)/X(33641)
X(38942) = {X(10564),X(26882)}-harmonic conjugate of X(5059)

X(38943) = X(4)X(13)∩X(5)X(8837)

Barycentrics    (Sqrt[3]*(a^2 - b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 2*(3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 2*b^2*c^2 - c^4)*S)*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 4*S*(Sqrt[3]*a^2 + S)) : :
X(38943) = 3 X[11094] - 2 X[33501]

X(38943) lies on the cubic K060 and these lines: {4, 13}, {5, 8837}, {14, 1606}, {15, 6113}, {30, 5668}, {79, 17405}, {398, 11083}, {621, 19772}, {1141, 5994}, {3166, 19773}, {3457, 5334}, {5321, 11081}, {5339, 11142}, {5627, 11600}, {7731, 36981}, {8918, 11244}, {11094, 33501}, {11586, 37974}, {11601, 34295}, {13202, 19107}, {16965, 21659}, {35469, 36967}

X(38943) = reflection of X(5669) in X(6110)
X(38943) = polar circle inverse of X(31687)
X(38943) = antigonal image of X(8174)
X(38943) = symgonal image of X(33496)
X(38943) = X(265)-Ceva conjugate of X(13)
X(38943) = X(2151)-isoconjugate of X(19775)
X(38943) = cevapoint of X(5668) and X(8837)
X(38943) = barycentric product X(i)*X(j) for these {i,j}: {13, 19773}, {94, 3166}, {300, 11244}, {8174, 8838}, {8918, 11078}
X(38943) = barycentric quotient X(i)/X(j) for these {i,j}: {13, 19775}, {3166, 323}, {8918, 11092}, {11083, 8471}, {11244, 15}, {19773, 298}, {36303, 470}
X(38943) = {X(4),X(36296)}-harmonic conjugate of X(13)

X(38944) = X(4)X(14)∩X(5)X(8839)

Barycentrics    (Sqrt[3]*(a^2 - b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 2*(3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 2*b^2*c^2 - c^4)*S)*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 4*(Sqrt[3]*a^2 - S)*S) : :
X(38944) = 3 X[11093] - 2 X[33499]

X(38944) lies on the cubic K060 and these lines: {4, 14}, {5, 8839}, {13, 1605}, {16, 6112}, {30, 5669}, {79, 17406}, {397, 11088}, {622, 19773}, {1141, 5995}, {3165, 19772}, {3458, 5335}, {5318, 11086}, {5340, 11141}, {5627, 11601}, {7731, 36979}, {8919, 11243}, {11093, 33499}, {11600, 34296}, {13202, 19106}, {15743, 37975}, {16964, 21659}, {35470, 36968}

X(38944) = reflection of X(5668) in X(6111)
X(38944) = polar circle inverse of X(31688)
X(38944) = antigonal image of X(8175)
X(38944) = symgonal image of X(33497)
X(38944) = X(265)-Ceva conjugate of X(14)
X(38944) = X(2152)-isoconjugate of X(19774)
X(38944) = cevapoint of X(5669) and X(8839)
X(38944) = barycentric product X(i)*X(j) for these {i,j}: {14, 19772}, {94, 3165}, {301, 11243}, {8175, 8836}, {8919, 11092}
X(38944) = barycentric quotient X(i)/X(j) for these {i,j}: {14, 19774}, {3165, 323}, {8919, 11078}, {11088, 8479}, {11243, 16}, {19772, 299}, {36302, 471}
X(38944) = {X(4),X(36297)}-harmonic conjugate of X(14)

X(38945) = X(1)X(4)∩X(30)X(109)

Barycentrics    (a + b - c)*(a - b + c)*(a^5 - 2*a^4*b + a^2*b^3 - a*b^4 + b^5 - 2*a^4*c + a^3*b*c + b^4*c + 2*a*b^2*c^2 - 2*b^3*c^2 + a^2*c^3 - 2*b^2*c^3 - a*c^4 + b*c^4 + c^5) : :
X(38945) = 3 X[1785] - 2 X[16869]

X(38945) lies on the cubic K025 and these lines: {1, 4}, {12, 4653}, {20, 34030}, {30, 109}, {53, 1630}, {80, 1725}, {115, 17966}, {221, 382}, {222, 12943}, {227, 18480}, {284, 8736}, {316, 664}, {355, 35194}, {522, 17950}, {603, 10483}, {671, 37856}, {759, 859}, {968, 9578}, {1300, 35187}, {1393, 37702}, {1411, 34172}, {1455, 28160}, {1464, 13273}, {2475, 37558}, {2689, 36195}, {4306, 18961}, {4551, 5080}, {4559, 5134}, {5523, 32674}, {9655, 34046}, {10570, 17555}, {12953, 34040}, {21935, 37583}, {28164, 34050}, {34170, 36127}

X(38945) = reflection of X(2689) in X(36195)
X(38945) = antigonal image of X(7424)
X(38945) = symgonal image of X(36195)
X(38945) = crossdifference of every pair of points on line {652, 21748}
X(38945) = trilinear product X(65)*(7424)
X(38945) = barycentric product X(226)*X(7424)
X(38945) = barycentric quotient X(7424)/X(333)

X(38946) = X(4)X(83)∩X(23)X(1287)

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

X(38946) lies on the cubics K025 and K940 and these lines: {4, 83}, {23, 1287}, {30, 827}, {251, 5355}, {316, 4577}, {671, 38888}, {826, 14318}, {1263, 34175}, {1799, 6031}, {4628, 5134}, {5899, 12188}, {7747, 14885}, {7756, 9481}, {14880, 38908}, {19552, 34174}

X(38946) = reflection of X(1287) in X(23)
X(38946) = reflection of X(827) in line X(23)X(385)
X(38946) = antigonal image of X(5189)
X(38946) = symgonal image of X(23)
X(38946) = X(38)-isoconjugate of X(34437)
X(38946) = cevapoint of X(5189) and X(19596)
X(38946) = barycentric product X(i)*X(j) for these {i,j}: {82, 20916}, {83, 5189}, {308, 19596}, {3112, 16546}
X(38946) = barycentric quotient X(i)/X(j) for these {i,j}: {251, 34437}, {5189, 141}, {16546, 38}, {18627, 3665}, {19596, 39}, {20916, 1930}, {21064, 15523}, {21176, 16892}, {22121, 3917}
X(38946) = {X(4),X(14247)}-harmonic conjugate of X(83)

X(38947) = X(4)X(147)∩X(30)X(805)

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

X(38947) lies on the cubic K025 and these lines: {4, 147}, {30, 805}, {98, 446}, {99, 18896}, {115, 9468}, {316, 18829}, {694, 804}, {1503, 34238}, {2450, 15391}, {2794, 17970}, {5475, 18872}, {8842, 35705}, {9467, 29012}, {34175, 36897}

X(38947) = isogonal conjugate of X(40077)
X(38947) = polar-circle-inverse of X(39931)
X(38947) = trilinear product X(1316)*X(1581)
X(38947) = antigonal image of X(1316)
X(38947) = symgonal image of X(11007)
X(38947) = X(1580)-isoconjugate of X(9513)
X(38947) = barycentric product X(1316)*X(1916)
X(38947) = barycentric quotient X(i)/X(j) for these {i,j}: {694, 9513}, {1316, 385}

X(38948) = X(4)X(7)∩X(30)X(934)

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

X(3948) lies on the cubic K025 and these lines: {4, 7}, {30, 934}, {316, 4569}, {1020, 5134}, {1375, 37797}, {1817, 13853}, {3160, 6850}, {3900, 17896}, {4566, 5080}, {5088, 34550}, {5523, 32714}, {5762, 15725}, {7177, 9579}, {24604, 32625}, {34059, 37437}, {34529, 38459}

X(38949) = X(4)X(65)∩X(30)X(108)

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

X(38949) lies on the cubic K and these lines: {4, 65}, {30, 108}, {316, 18026}, {1068, 18961}, {1300, 2720}, {2766, 37799}, {5172, 37979}, {6923, 7952}, {10431, 37769}

X(38949) = reflection of X(2766) in X(37982)

X(38949) = polar-circle-inverse of X(1858)
X(38949) = symgonal image of X(37982)

X(38950) = X(4)X(145)∩X(11)X(106)

Barycentrics    (a + b - 2*c)*(a - 2*b + c)*(2*a^5 - 2*a^4*b + a^2*b^3 - 2*a*b^4 + b^5 - 2*a^4*c + 2*a^3*b*c - a^2*b^2*c + b^4*c - a^2*b*c^2 + 4*a*b^2*c^2 - 2*b^3*c^2 + a^2*c^3 - 2*b^2*c^3 - 2*a*c^4 + b*c^4 + c^5) : :
X(38950) = X[901] - 3 X[36590]

X(38950) lies on the cubics K025 and K275 and these lines: {4, 145}, {11, 106}, {30, 901}, {80, 900}, {88, 12019}, {100, 13744}, {150, 903}, {190, 1145}, {316, 4555}, {1168, 34172}, {1387, 24865}, {1478, 34230}, {2829, 36058}, {4013, 25436}, {4792, 10777}, {4997, 10609}, {7354, 16944}, {7428, 17100}, {14193, 26073}, {26139, 34123}

X(38950) = midpoint of X(12747) and X(18342)
X(38950) = antigonal image of X(3109)
X(38950) = symgonal image of X(36155)
X(38950) = crossdifference of every pair of points on line {17455, 22086}
X(38950) = barycentric product X(3109)*X(4080)
X(38950) = barycentric quotient X(3109)/X(16704)

X(38951) = MIDPOINT OF X(3543) AND X(5971)

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

X(38951) lies on the cubic K025 and these lines: {2, 38532}, {4, 524}, {30, 1296}, {115, 17968}, {265, 34169}, {316, 35179}, {325, 2418}, {381, 5913}, {1499, 8352}, {1995, 32133}, {3543, 5971}, {10297, 34320}, {11645, 17979}, {12505, 16509}, {15406, 37855}, {16092, 32648}

X(38951) = midpoint of X(3543) and X(5971)
X(38951) = reflection of X(i) in X(j) for these {i,j}: {5913, 381}, {34320, 10297}
X(38951) = antigonal image of X(7426)
X(38951) = X(5505)-isoconjugate of X(36277)
X(38951) = barycentric product X(5485)*X(7426)
X(38951) = barycentric quotient X(i)/X(j) for these {i,j}: {7426, 1992}, {15303, 27088}, {21448, 5505}
X(38951) = {X(4),X(34165)}-harmonic conjugate of X(14262)

X(38952) = X(4)X(513)∩X(30)X(104)

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

X(38952) lies on the cubic K025 and these lines: {4, 513}, {30, 104}, {265, 5080}, {316, 18816}, {1300, 2720}, {2250, 5134}, {17737, 32641}

X(38952) = antigonal image of X(7477)

X(38953) = X(2)X(38613)∩X(30)X(98)

Barycentrics    a^14 - a^12*b^2 - a^10*b^4 + a^8*b^6 - a^6*b^8 + a^4*b^10 + a^2*b^12 - b^14 - a^12*c^2 + 3*a^8*b^4*c^2 - 4*a^6*b^6*c^2 + 3*a^4*b^8*c^2 - 5*a^2*b^10*c^2 + 4*b^12*c^2 - a^10*c^4 + 3*a^8*b^2*c^4 + 3*a^6*b^4*c^4 - 3*a^4*b^6*c^4 + 5*a^2*b^8*c^4 - 6*b^10*c^4 + a^8*c^6 - 4*a^6*b^2*c^6 - 3*a^4*b^4*c^6 - 2*a^2*b^6*c^6 + 3*b^8*c^6 - a^6*c^8 + 3*a^4*b^2*c^8 + 5*a^2*b^4*c^8 + 3*b^6*c^8 + a^4*c^10 - 5*a^2*b^2*c^10 - 6*b^4*c^10 + a^2*c^12 + 4*b^2*c^12 - c^14 : :
X(38953) = 4 X[140] - 3 X[38704], 3 X[186] - 4 X[14693], 3 X[249] - 2 X[34153], 3 X[381] - 2 X[5099], 3 X[381] - X[38583], 4 X[546] - X[38680], 2 X[550] - 3 X[38702], 5 X[1656] - 4 X[16760], 2 X[3627] + X[38679], 2 X[7575] - 3 X[38227], 2 X[10295] - 3 X[38225], 2 X[13449] - 3 X[18403], 3 X[15561] - 4 X[36170], 2 X[36166] - 3 X[38224]

X(38953) lies on the cubic K301 and these lines: {2, 38613}, {3, 16188}, {4, 38552}, {5, 842}, {20, 38611}, {30, 98}, {67, 265}, {140, 38704}, {186, 14693}, {249, 34153}, {316, 18572}, {381, 2453}, {382, 38582}, {476, 868}, {512, 7728}, {523, 6033}, {546, 38680}, {550, 38702}, {858, 35002}, {1478, 6027}, {1479, 6023}, {1550, 15545}, {1551, 8724}, {1656, 16760}, {2782, 36173}, {3627, 38679}, {7472, 38730}, {7575, 38227}, {9181, 12121}, {9821, 36187}, {10295, 38225}, {10297, 10745}, {10749, 13449}, {11799, 13556}, {15561, 36170}, {18325, 22338}, {22515, 36174}, {36166, 38224}

X(38953) = midpoint of X(382) and X(38582)
X(38953) = reflection of X(i) in X(j) for these {i,j}: {3, 16188}, {20, 38611}, {316, 18572}, {842, 5}, {8724, 1551}, {12121, 9181}, {14830, 16092}, {15545, 1550}, {35002, 858}, {36174, 22515}, {38583, 5099}, {38730, 7472}
X(38953) = reflection of X(6033) in the Euler line
X(38953) = anticomplement of X(38613)
X(38953) = X(842)-of-Johnson-triangle
X(38953) = {X(381),X(38583)}-harmonic conjugate of X(5099)

X(38954) = X(2)X(38617)∩X(3)X(2222)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^6 - a^5*b - a^4*b^2 + 2*a^3*b^3 - a^2*b^4 - a*b^5 + b^6 - a^5*c + 5*a^4*b*c - 4*a^3*b^2*c - 3*a^2*b^3*c + 5*a*b^4*c - 2*b^5*c - a^4*c^2 - 4*a^3*b*c^2 + 10*a^2*b^2*c^2 - 4*a*b^3*c^2 - b^4*c^2 + 2*a^3*c^3 - 3*a^2*b*c^3 - 4*a*b^2*c^3 + 4*b^3*c^3 - a^2*c^4 + 5*a*b*c^4 - b^2*c^4 - a*c^5 - 2*b*c^5 + c^6) : :
X(38954) = 3 X[3] - 4 X[22102], 4 X[140] - 3 X[38707], 3 X[381] - 2 X[3259], 3 X[381] - X[38586], 4 X[546] - X[38682], 2 X[550] - 3 X[38705], 3 X[5587] - X[34464], 2 X[22102] - 3 X[31841]

X(38954) lies on the cubics K275 and K684 and these lines: {2, 38617}, {3, 2222}, {5, 953}, {20, 38614}, {30, 901}, {80, 517}, {140, 38707}, {150, 35174}, {265, 5080}, {381, 3259}, {382, 38584}, {513, 10742}, {546, 38682}, {550, 38705}, {1478, 3025}, {1479, 13756}, {1482, 14584}, {1484, 14511}, {5587, 34464}, {5722, 24201}, {6073, 12331}, {6075, 12773}, {10247, 34232}, {10739, 18330}, {11698, 14513}, {18357, 20957}, {22938, 31512}, {28204, 36909}

X(38954) = midpoint of X(382) and X(38584)
X(38954) = reflection of X(i) in X(j) for these {i,j}: {3, 31841}, {20, 38614}, {953, 5}, {12331, 6073}, {12773, 6075}, {14511, 1484}, {14513, 11698}, {31512, 22938}, {38586, 3259}
X(38954) = anticomplement of X(38617)
X(38954) = X(953)-of-Johnson-triangle
X(38954) = {X(381),X(38586)}-harmonic conjugate of X(3259)

X(38955) = ISOGONAL CONJUGATE OF X(859)

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

X(38955) lies on the Jerabek circumhyperbola, the cubic K275, and these lines: {1, 26095}, {2, 34586}, {3, 8}, {4, 151}, {6, 281}, {10, 73}, {65, 17869}, {68, 3436}, {69, 150}, {71, 1018}, {72, 3701}, {74, 1309}, {80, 3738}, {92, 1243}, {109, 37043}, {145, 27506}, {153, 37781}, {248, 5291}, {265, 5080}, {291, 2401}, {758, 15065}, {879, 18003}, {895, 5380}, {914, 6735}, {1177, 14776}, {1330, 18123}, {1439, 1441}, {1771, 1795}, {2800, 24026}, {3036, 16506}, {3434, 4846}, {3611, 7046}, {3679, 36819}, {3754, 6757}, {3936, 17757}, {4674, 4707}, {4858, 12736}, {5082, 15740}, {5086, 34800}, {5554, 24537}, {5906, 10526}, {6001, 38462}, {6740, 16704}, {10099, 13576}, {10449, 34259}, {11105, 20306}, {12016, 24034}, {12115, 26871}, {12647, 16499}, {12649, 27378}, {14964, 35321}, {15635, 35059}, {21060, 21087}, {23528, 37562}, {23661, 34339}

X(38955) = reflection of X(i) in X(j) for these {i,j}: {1, 34589}, {4551, 10}
X(38955) = isogonal conjugate of X(859)
X(38955) = isotomic conjugate of X(17139)
X(38955) = isotomic conjugate of the anticomplement of X(2245)
X(38955) = anticomplement of X(34586)
X(38955) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {80, 153}, {104, 6224}, {2006, 36918}
X(38955) = X(34234)-Ceva conjugate of X(2250)
X(38955) = X(2245)-cross conjugate of X(2)
X(38955) = cevapoint of X(i) and X(j) for these (i,j): {10, 758}, {125, 6370}, {210, 3943}, {526, 6741}, {3738, 34589}
X(38955) = crosspoint of X(18816) and X(34234)
X(38955) = trilinear pole of line {37, 647}
X(38955) = crossdifference of every pair of points on line {3310, 8677}
X(38955) = X(i)-isoconjugate of X(j) for these (i,j): {1, 859}, {21, 1457}, {28, 22350}, {31, 17139}, {58, 517}, {81, 2183}, {110, 1769}, {162, 8677}, {163, 10015}, {283, 1875}, {284, 1465}, {593, 21801}, {662, 3310}, {692, 23788}, {741, 15507}, {759, 34586}, {811, 23220}, {849, 17757}, {908, 1333}, {1019, 2427}, {1408, 6735}, {1437, 1785}, {1459, 4246}, {1576, 36038}, {1790, 14571}, {2194, 22464}, {2206, 3262}, {3737, 23981}, {7252, 24029}, {14010, 24027}, {16586, 34079}, {23189, 23706}
X(38955) = barycentric product X(i)*X(j) for these {i,j}: {10, 34234}, {37, 18816}, {65, 36795}, {72, 16082}, {75, 2250}, {104, 321}, {306, 36123}, {313, 909}, {349, 2342}, {523, 13136}, {525, 1309}, {850, 32641}, {1577, 36037}, {2401, 3952}, {2423, 27808}, {3267, 14776}, {3701, 34051}, {4080, 36944}, {4086, 37136}, {27801, 34858}, {30588, 36921}
X(38955) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 17139}, {6, 859}, {10, 908}, {37, 517}, {42, 2183}, {65, 1465}, {71, 22350}, {104, 81}, {226, 22464}, {321, 3262}, {512, 3310}, {514, 23788}, {523, 10015}, {594, 17757}, {647, 8677}, {661, 1769}, {756, 21801}, {758, 16586}, {909, 58}, {1146, 14010}, {1309, 648}, {1400, 1457}, {1577, 36038}, {1783, 4246}, {1795, 1790}, {1809, 1812}, {1824, 14571}, {1826, 1785}, {1880, 1875}, {2238, 15507}, {2245, 34586}, {2250, 1}, {2321, 6735}, {2342, 284}, {2401, 7192}, {2423, 3733}, {2720, 4565}, {3049, 23220}, {3700, 2804}, {3943, 1145}, {3952, 2397}, {4120, 23757}, {4551, 24029}, {4557, 2427}, {4559, 23981}, {13136, 99}, {14578, 1437}, {14776, 112}, {15501, 1817}, {15635, 16726}, {16082, 286}, {17757, 26611}, {18816, 274}, {21044, 35015}, {21801, 24028}, {21933, 1532}, {24005, 1519}, {32641, 110}, {34051, 1014}, {34234, 86}, {34858, 1333}, {35321, 2617}, {36037, 662}, {36123, 27}, {36795, 314}, {36819, 18206}, {36921, 5235}, {36944, 16704}, {37136, 1414}
X(38955) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 14266, 36944}, {10, 37558, 37154}, {34234, 36921, 36944}

X(38956) = REFLECTION OF X(122) IN X(4)

Barycentrics    (a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 - 3*c^4)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)^2*(a^4 + 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 2*b^2*c^2 + c^4) : :
X(38956) = 3 X[4] - X[1294], 3 X[122] - 2 X[1294], 3 X[133] - 2 X[38605], 3 X[381] - 2 X[34842], 3 X[382] + X[38577], 4 X[546] - 3 X[36520], 3 X[1699] - 2 X[11732], 5 X[3091] - 3 X[38714], 3 X[3184] - 4 X[38605], X[3529] - 3 X[23239], 3 X[3543] - X[10152], 3 X[3543] + X[34549], 3 X[3830] - X[10745], 5 X[5076] - X[38591], X[5667] + 3 X[15682], 3 X[9812] - X[10701], 5 X[17578] - X[34186], 3 X[22337] - X[38577], 3 X[36520] - 2 X[38621]

X(38956) lies on the cubic K427 and these lines: {4, 122}, {20, 6716}, {30, 133}, {64, 265}, {107, 3146}, {253, 317}, {381, 34842}, {459, 5667}, {546, 36520}, {1073, 3830}, {1539, 20123}, {1699, 11732}, {3091, 38714}, {3529, 23239}, {3627, 8798}, {5073, 23240}, {5076, 38591}, {6525, 16251}, {9033, 13202}, {9812, 10701}, {11718, 28164}, {13157, 33699}, {13611, 36162}, {14572, 15640}, {14703, 18534}, {17578, 34186}, {18577, 31726}

X(38956) = midpoint of X(i) and X(j) for these {i,j}: {107, 3146}, {382, 22337}, {5073, 23240}, {10152, 34549}
X(38956) = reflection of X(i) in X(j) for these {i,j}: {20, 6716}, {122, 4}, {3184, 133}, {38621, 546}
X(38956) = X(16240)-cross conjugate of X(3163)
X(38956) = X(i)-isoconjugate of X(j) for these (i,j): {2349, 15291}, {10152, 35200}
X(38956) = crosssum of X(154) and X(15291)
X(38956) = barycentric product X(i)*X(j) for these {i,j}: {64, 36789}, {253, 3163}, {459, 16163}, {1073, 34334}, {1099, 2184}, {16240, 34403}
X(38956) = barycentric quotient X(i)/X(j) for these {i,j}: {253, 31621}, {1099, 18750}, {1301, 34568}, {1354, 18623}, {1495, 15291}, {1990, 10152}, {3163, 20}, {3233, 36841}, {6062, 27382}, {9408, 154}, {11589, 14919}, {14401, 8057}, {16163, 37669}, {16240, 1249}, {34334, 15466}, {36789, 14615}
X(38956) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {546, 38621, 36520}, {3543, 34549, 10152}

leftri

Crosssums of circumcircle-P-antipodes and line intercepts: X(38957)-X(39102)

rightri

Let P = p : q : r (barycentrics) be a point in the plane of triangle ABC. Let P' be the isogonal conjugate of P. Then the locus of the crosssum of circumcircle-P-antipodes is the bicevian conic of X(2) and P', with center 2 a^2/p + b^2/q + c^2/r : :. For example, the locus of the crosssum of circumcircle antipodes is the nine-point circle, which is the bicevian conic of X(2) and X(4), and the locus of the crosssum of circumcircle-X(6)-antipodes is the Steiner inellipse, which is the limit of the bicevian conic of X(2) and Q as Q approaches X(2).

Let L be a line. The crosssum of the (real or nonreal) circumcircle-intercepts of L is the X(2)-Ceva conjugate of the crossdifference of every pair of points on L, and is the center of the circumconic that is the isogonal conjugate of L. If L passes through X(3), then the crosssum is also the orthopole of L, and lies on the cevian circle of the isogonal conjugate of every point on L.

Contributed by Randy Hutson, June 17, 2020.

X(38957) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(34)

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

X(38957) lies on the nine-point circle, the cevian circle of X(78), and these lines: {33, 25640}, {117, 226}, {123, 7004}, {124, 26956}, {3259, 3270}, {5511, 35015}

X(38957) = crosssum of circumcircle-intercepts of line X(3)X(34)
X(38957) = center of rectangular hyperbola {{A,B,C,X(4),X(78)}}
X(38957) = orthopole of line X(3)X(34)

X(38958) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(38)

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

X(38958) lies on the nine-point circle, the cevian circle of X(82), and these lines: {867, 15611}, {1283, 5051}

X(38958) = complement of trilinear pole of line X(6)X(3874)
X(38958) = crosssum of circumcircle-intercepts of line X(3)X(38)
X(38958) = center of rectangular hyperbola {{A,B,C,X(4),X(82)}}
X(38958) = orthopole of line X(3)X(38)

X(38959) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(41)

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

X(38959) lies on the nine-point circle, the cevian circle of X(85), and these lines: {116, 1111}, {118, 13576}, {124, 3119}, {1086, 5519}, {1826, 20621}, {5219, 31844}, {17059, 38980}, {21617, 36905}

X(38959) = complement of trilinear pole of line X(6)X(142)
X(38959) = crosssum of circumcircle-intercepts of line X(3)X(41)
X(38959) = center of rectangular hyperbola {{A,B,C,X(4),X(85)}}
X(38959) = orthopole of line X(3)X(41)

X(38960) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(42)

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

X(38960) lies on the nine-point circle, the cevian circle of X(86), and these lines: {11, 16726}, {116, 3120}, {121, 9780}, {2486, 5515}, {2969, 5139}

X(38960) = complement of trilinear pole of line X(6)X(1125)
X(38960) = crosssum of circumcircle-intercepts of line X(3)X(42)
X(38960) = center of rectangular hyperbola {{A,B,C,X(4),X(86)}}
X(38960) = orthopole of line X(3)X(42)

X(38961) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(43)

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

X(38961) lies on the nine-point circle, the cevian circle of X(87), and these lines: {11, 3248}, {121, 3841}, {3120, 5518}

X(38961) = complement of trilinear pole of line X(6)X(978)
X(38961) = crosssum of circumcircle-intercepts of line X(3)X(43)
X(38961) = center of rectangular hyperbola {{A,B,C,X(4),X(87)}}
X(38961) = orthopole of line X(3)X(43)

X(38962) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(44)

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

X(38962) lies on the nine-point circle, the cevian circle of X(88), and these lines: {11, 2087}, {120, 7951}, {1015, 5516}, {1826, 20619}

X(38962) = complement of trilinear pole of line X(6)X(3306)
X(38962) = crosssum of circumcircle-intercepts of line X(3)X(44)
X(38962) = center of rectangular hyperbola {{A,B,C,X(4),X(88)}}
X(38962) = orthopole of line X(3)X(44)
X(38962) = intersection, other than X(5516), of nine-point circle and cevian circle of X(88)

X(38963) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(45)

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

X(38963) lies on the nine-point circle, the cevian circle of X(89), and these lines: {2, 13396}, {1213, 31845}

X(38963) = complement of X(13396)
X(38963) = crosssum of circumcircle-intercepts of line X(3)X(45)
X(38963) = center of rectangular hyperbola {{A,B,C,X(4),X(89)}}
X(38963) = orthopole of line X(3)X(45)

X(38964) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(47)

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

X(38964) lies on the nine-point circle, the cevian circle of X(91), and these lines: {4, 36076}, {117, 5348}, {1086, 15608}, {1155, 31841}

X(38964) = complement of trilinear pole of line X(6)X(1737)
X(38964) = crosssum of circumcircle-intercepts of line X(3)X(47)
X(38964) = center of rectangular hyperbola {{A,B,C,X(4),X(91)}}
X(38964) = orthopole of line X(3)X(47)
X(38964) = polar-circle-inverse of X(36076)

X(38965) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(53)

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

X(38965) lies on the nine-point circle, the cevian circle of X(97), and these lines: {114, 17814}, {136, 3269}, {137, 1562}

X(38965) = crosssum of circumcircle-intercepts of line X(3)X(53)
X(38965) = center of rectangular hyperbola {{A,B,C,X(4),X(97)}}
X(38965) = orthopole of line X(3)X(53)
X(38965) = polar-circle-inverse of trilinear pole of line X(6)X(14569)

X(38966) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(77)

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

X(38966) lies on the nine-point circle, the cevian circle of X(33), and these lines: {4, 934}, {33, 20623}, {117, 1827}, {119, 1863}, {225, 20622}, {1566, 8735}, {2310, 5517}, {4081, 5514}, {5521, 34969}

X(38966) = crosssum of circumcircle-intercepts of line X(3)X(77)
X(38966) = center of rectangular hyperbola {{A,B,C,X(4),X(33)}}
X(38966) = X(2)-Ceva conjugate of polar conjugate of X(4569)
X(38966) = orthopole of line X(3)X(77)
X(38966) = polar-circle-inverse of X(934)

X(38967) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(81)

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

X(38967) lies on the nine-point circle, the cevian circle of X(37), and these lines: {4, 36077}, {115, 3121}, {125, 2486}, {5521, 8754}, {8286, 15614}, {15607, 34969}

X(38967) = complement of trilinear pole of line X(6)X(4658)
X(38967) = crosssum of circumcircle-intercepts of line X(3)X(81)
X(38967) = center of rectangular hyperbola {{A,B,C,X(4),X(37)}}
X(38967) = orthopole of line X(3)X(81)
X(38967) = polar-circle-inverse of X(36077)

X(38968) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(86)

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

X(38968) lies on the nine-point circle, the cevian circle of X(42), and these lines: {114, 7380}, {115, 3122}, {5190, 8754}, {5518, 8287}

X(38968) = complement of trilinear pole of line X(6)X(16058)
X(38968) = crosssum of circumcircle-intercepts of line X(3)X(86)
X(38968) = center of rectangular hyperbola {{A,B,C,X(4),X(42)}}
X(38968) = orthopole of line X(3)X(86)

X(38969) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(92)

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

X(38969) lies on the nine-point circle, the cevian circle of X(48), and these lines: {4, 681}, {3270, 20620}, {7117, 38976}

X(38969) = crosssum of circumcircle-intercepts of line X(3)X(92)
X(38969) = center of rectangular hyperbola {{A,B,C,X(4),X(48)}}
X(38969) = orthopole of line X(3)X(92)
X(38969) = polar-circle-inverse of X(681)

X(38970) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(248)

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

X(38970) lies on the nine-point circle, the cevian circle of X(297), and these lines: {4, 2715}, {114, 232}, {115, 16229}, {125, 2501}, {132, 230}, {136, 647}, {427, 38975}, {3580, 36426}

X(38970) = complement of trilinear pole of line X(6)X(15595)
X(38970) = crosssum of circumcircle-intercepts of line X(3)X(248)
X(38970) = center of rectangular hyperbola {{A,B,C,X(4),X(297)}}
X(38970) = X(2)-Ceva conjugate of X(16230)
X(38970) = orthopole of line X(3)X(248)
X(38970) = Moses-radical-circle-inverse of X(136)
X(38970) = polar-circle-inverse of X(2715)

X(38971) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(1177)

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

X(38971) lies on the nine-point circle, the cevian circle of X(850), circle {{X(3),X(5),X(114),X(182)}}, and these lines: {2, 935}, {3, 16188}, {4, 2697}, {23, 37801}, {30, 132}, {113, 1503}, {114, 2072}, {115, 2485}, {125, 525}, {126, 5159}, {127, 523}, {133, 11799}, {136, 36189}, {339, 23285}, {381, 18809}, {691, 35923}, {858, 1560}, {868, 16221}, {1368, 31655}, {2088, 36472}, {2453, 37073}, {3150, 3258}, {5099, 18311}, {5139, 14120}, {5664, 16177}, {6794, 18911}, {7574, 18402}, {8705, 13249}, {9967, 33330}, {10257, 31842}, {10316, 36187}, {10415, 14364}, {14675, 18403}, {18314, 20625}, {20621, 30447}

X(38971) = reflection of X(127) in Euler line
X(38971) = complement of X(935)
X(38971) = crosssum of circumcircle-intercepts of line X(3)X(1177)
X(38971) = center of rectangular hyperbola {{A,B,C,X(4),X(850)}}
X(38971) = X(1297)-of-[reflection of Euler triangle in Euler line]
X(38971) = orthopole of line X(3)X(1177)
X(38971) = intersection, other than X(114) of nine-point circle and circle {{X(3),X(5),X(114),X(182)}}
X(38971) = polar-circle-inverse of X(10423)
X(38971) = barycentric product X(125)*X(316)*X(858)

X(38972) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(1415)

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

X(38972) lies on the nine-point circle, the cevian circle of X(4391), and these lines: {4, 32688}, {11, 6588}, {116, 14837}, {123, 650}, {124, 3239}, {127, 1577}, {7348, 14606}

X(38972) = complement of trilinear pole of line X(6)X(26932)
X(38972) = crosssum of circumcircle-intercepts of line X(3)X(1415)
X(38972) = center of rectangular hyperbola {{A,B,C,X(4),X(4391)}}
X(38972) = orthopole of line X(3)X(1415)
X(38972) = Stevanovic-circle-inverse of X(123)
X(38972) = polar-circle-inverse of X(32688)

X(38973) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(1615)

Barycentrics    (b - c)^2 (a - b - c)^3 (a (b + c) - (b - c)^2) (4 a^5 - 9 a^4 (b + c) + 4 a^3 (b + c)^2 + 2 a^2 (b - c)^2 (b + c) - (b - c)^4 (b + c)) : :
Barycentrics    b^2/(sec^4(C/2) - sec^4(A/2)) + c^2/(sec^4(A/2) - sec^4(B/2)) : :

X(38973) lies on the nine-point circle, the cevian circle of X(9), and these lines: {11, 35508}, {116, 13609}, {3119, 15607}, {5514, 17426}

X(38973) = complement of trilinear pole of line X(6)X(279)
X(38973) = complementary conjugate of X(6607)
X(38973) = crosssum of circumcircle-intercepts of line X(3)X(1615)
X(38973) = orthopole of line X(3)X(1615)
X(38973) = intersection, other than X(11), of nine-point circle and cevian circle of X(9)
X(38973) = point of concurrence of cevian circles of the vertices of the anticevian triangle of X(9)

X(38974) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(1625)

Barycentrics    SA (SA^2 - SB SC) (SB^2 - SC^2) (SB (SB^2 - SC SA) (SC^2 - SA^2) + SC (SC^2 - SA SB) (SA^2 - SB^2)) : :
Barycentrics    (csc B)/(sec C sin^3 A - sec A sin^3 C) + (csc C)/(sec A sin^3 B - sec B sin^3 A) : :

X(38974) lies on the nine-point circle, the cevian circle of X(401), and these lines: {2, 22456}, {3, 33330}, {113, 11672}, {125, 647}, {127, 35071}, {129, 1971}, {130, 3269}, {132, 232}, {216, 16188}, {684, 39000}, {1562, 2524}, {2715, 15920}, {15526, 39009}, {18809, 33843}

X(38974) = complement of X(22456)
X(38974) = crosssum of circumcircle-intercepts of line X(3)X(1625)
X(38974) = center of rectangular hyperbola {{A,B,C,X(4),X(401)}}
X(38974) = orthopole of line X(3)X(1625)
X(38974) = Moses-radical-circle-inverse of X(125)
X(38974) = barycentric product X(684)*X(6130)
X(38974) = barycentric quotient X(6130)/X(22456)

X(38975) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(3569)

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

X(38975) lies on the nine-point circle, the cevian circle of X(2966), and these lines: {2, 2715}, {51, 2679}, {114, 647}, {115, 2450}, {125, 230}, {127, 441}, {132, 2501}, {136, 232}, {427, 38970}, {460, 5139}, {1316, 5099}, {1555, 5512}, {3258, 3815}, {6792, 9753}, {10311, 16221}, {11610, 36830}

X(38975) = complement of X(2857)
X(38975) = complementary conjugate of X(2871)
X(38975) = crosssum of circumcircle-intercepts of line X(3)X(3569)
X(38975) = center of rectangular hyperbola {{A,B,C,X(4),X(2966)}}
X(38975) = orthopole of line X(3)X(3569)
X(38975) = Moses-radical-circle-inverse of X(114)
X(38975) = orthoptic-circle-of-Steiner-inellipse-inverse of X(2715)

X(38976) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(8612)

Barycentrics    a^2 (a^2 - b^2 - c^2)^2 (b^2 - c^2)^2 (a^8 (b^2 + c^2) - 4 a^6 ((b^2 + c^2)^2 - b^2 c^2) + 2 a^4 (b^2 + c^2) (3 b^4 - 2 b^2 c^2 + 3 c^4) - 4 a^2 (b^2 - c^2)^2 ((b^2 + c^2)^2 - b^2 c^2) + (b^2 - c^2)^4 (b^2 + c^2)) (a^16 - 6 a^14 (b^2 + c^2) + a^12 (15 b^4 + 23 b^2 c^2 + 15 c^4) - 10 a^10 (b^2 + c^2) (2 b^4 + b^2 c^2 + 2 c^4) + a^8 (15 b^8 + 11 b^6 c^2 + 12 b^4 c^4 + 11 b^2 c^6 + 15 c^8) - 6 a^6 (b^2 - c^2)^2 (b^2 + c^2) (b^4 + c^4) + a^4 (b^2 - c^2)^4 ((b^2 + c^2)^2 - b^2 c^2) - 2 a^2 b^2 c^2 (b^2 - c^2)^4 (b^2 + c^2) + b^2 c^2 (b^2 - c^2)^6) : :
Barycentrics    (sec^2 B)/(cos^2 2C - cos^2 2A) + (sec^2 C)/(cos^2 2A - cos^2 2B) : :

X(38976) lies on the nine-point circle, the cevian circle of X(3), and these lines: {3, 129}, {114, 10600}, {125, 35071}, {133, 233}, {216, 8439}, {7117, 38969}, {17434, 35579}

X(38976) = complement of trilinear pole of line X(6)X(275)
X(38976) = crosssum of circumcircle-intercepts of line X(3)X(8612)
X(38976) = center of rectangular hyperbola {{A,B,C,X(4),X(8613)}}
X(38976) = orthopole of line X(3)X(3569)
X(38976) = intersection, other than X(125), of the nine-point circle and the cevian circle of X(3)
X(38976) = point of concurrence of the cevian circles of the vertices of the anticevian triangle of X(3)
X(38976) = circumcircle-inverse of X(99)-of-tangential-triangle
X(38976) = polar-circle-inverse of trilinear pole of line X(6)X(8612)

X(38977) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(3)X(14529)

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

X(38977) lies on the nine-point circle and these lines: {2, 26704}, {3, 117}, {113, 37836}, {124, 34588}, {125, 2968}, {136, 3137}, {5521, 14010}, {6842, 25640}

X(38977) = complement of X(26704)
X(38977) = crosssum of circumcircle-intercepts of line X(3)X(14529)
X(38977) = orthopole of line X(3)X(14529)
X(38977) = nine-point circle antipode of polar-circle-inverse of X(102)

X(38978) = CROSSSUM OF X(99) AND X(741)

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

X(38978) lies on the bicevian ellipse of X(1) and X(2) (with center X(1125)) and these lines: {1, 4589}, {11, 23301}, {350, 740}, {512, 1015}, {888, 39011}, {4117, 8054}

X(38978) = crosssum of X(99) and X(741)
X(38978) = crosspoint of X(512) and X(740)
X(38978) = crossdifference of every pair of points on line X(3570)X(4584)
X(38978) = center of hyperbola {{A,B,C,X(1),X(512)}}

X(38979) = CROSSSUM OF X(100) AND X(106)

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

X(38979) lies on the bicevian ellipse of X(1) and X(2) (with center X(1125)) and these lines: {11, 31946}, {214, 12746}, {244, 513}, {519, 3992}, {661, 1015}, {678, 899}, {1149, 34586}, {1647, 4448}, {2605, 8054}, {3251, 14434}, {3835, 17761}, {4728, 35119}, {4938, 17793}, {31855, 37680}

X(38979) = crosssum of X(100) and X(106)
X(38979) = crosspoint of X(513) and X(519)
X(38979) = center of hyperbola {{A,B,C,X(1),X(44)}}
X(38979) = X(2)-Ceva conjugate of X(1635)
X(38979) = trilinear product X(i)*X(j) for these {i,j}: {900, 4491}, {1635, 21385}, {1960, 21297}, {2087, 37680}
X(38979) = trilinear quotient X(i)/X(j) for these (i,j): (4491, 901), (21297, 4555), (21385, 3257), (37680, 5376)
X(38979) = barycentric product X(i)*X(j) for these {i,j}: {900, 21385}, {1635, 21297}, {1647, 37680}, {1960, 21606}, {2087, 17160}, {3762, 4491}
X(38979) = barycentric quotient X(i)/X(j) for these (i,j): (4491, 3257), (21385, 4555), (33882, 9268)

X(38980) = CROSSSUM OF X(101) AND X(105)

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

X(38980) lies on the bicevian ellipse of X(1) and X(2) (with center X(1125)) and these lines: {11,661}, {244,38375}, {514,1111}, {518,3693}, {2238,39046}, {2254,35505}, {3119,4521}, {3911,24582}, {5519,35094}, {6544,34590}, {8776,9502}, {11998,14714}, {17059,38959}, {17464,34587}

X(38980) = crosssum of X(101) and X(105)
X(38980) = crosspoint of X(514) and X(518)
X(38980) = center of hyperbola {{A,B,C,X(1),X(514)}}
X(38980) = center of the circumconic {{A,B,C,X(1),X(85)}}
X(38980) = X(2)-Ceva conjugate of X(2254)
X(38980) = X(31)-complementary conjugate of-X(2254)

X(38981) = CROSSSUM OF X(104) AND X(109)

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

X(38981) lies on the bicevian ellipse of X(1) and X(2) (with center X(1125)) and these lines: {11, 656}, {244, 6129}, {517, 1457}, {522, 7004}, {650, 34591}, {851, 39046}, {1735, 3911}, {1769, 3326}, {2605, 38983}, {7658, 17761}, {8299, 9371}, {14010, 35015}, {22465, 23710}, {23757, 34590}

X(38981) = crosssum of X(104) and X(109)
X(38981) = crosspoint of X(517) and X(522)
X(38981) = center of hyperbola {{A,B,C,X(1),X(517)}}

X(38982) = CROSSSUM OF X(110) AND X(759)

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

X(38982) lies on the bicevian ellipse of X(1) and X(2) (with center X(1125)), the bicevian ellipse of X(2) and X(12) (with center X(6668)), and these lines: {11, 523}, {36, 214}, {244, 31947}, {526, 3025}, {647, 1015}, {1649, 3675}, {2632, 38983}, {3259, 6089}, {4988, 21044}, {12081, 18455}, {14417, 35094}, {15325, 31945}

X(38982) = crosssum of X(110) and X(759)
X(38982) = crosspoint of X(523) and X(758)
X(38982) = crossdifference of every pair of points on line X(1983)X(13589)
X(38982) = center of hyperbola {{A,B,C,X(1),X(12)}}
X(38982) = X(2)-Ceva conjugate of X(2610)

X(38983) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(1)X(4)

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

X(38983) lies on the bicevian ellipse of X(1) and X(2) (with center X(1125)), the bicevian conic of X(2) and X(3) (with center X(140)), the rectangular bicevian hyperbola of X(2) and X(102) (with center X(6711)), and these lines: {11, 3137}, {48, 35326}, {73, 1385}, {212, 23067}, {244, 2638}, {1364, 1459}, {1457, 11713}, {2605, 38981}, {2632, 38982}, {2972, 3270}, {7004, 34588}, {11998, 34591}, {30144, 34587}

X(38983) = complement of isogonal conjugate of X(39199)
X(38983) = crosssum of circumcircle-intercepts of line X(1)X(4)
X(38983) = center of hyperbola {{A,B,C,X(1),X(3)}}
X(38983) = X(2)-Ceva conjugate of X(652)

X(38984) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(1)X(5)

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

X(38984) lies on the bicevian ellipse of X(1) and X(2) (with center X(1125)), the bicevian conic of X(2) and X(54) (with center X(6689)), the rectangular bicevian hyperbola of X(2) and X(953), and these lines: {11, 2605}, {215, 12835}, {244, 1459}, {1319, 34586}, {2594, 12266}

X(38984) = crosssum of circumcircle-intercepts of line X(1)X(5)
X(38984) = center of hyperbola {{A,B,C,X(1),X(36)}}
X(38984) = X(2)-Ceva conjugate of X(654)

X(38985) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(1)X(29)

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

X(38985) lies on the bicevian ellipse of X(1) and X(2) (with center X(1125)) and these lines: {214, 820}, {1364, 2638}, {2632, 34588}

X(38985) = crosssum of circumcircle-intercepts of line X(1)X(29)
X(38985) = center of hyperbola {{A,B,C,X(1),X(48)}}
X(38985) = X(2)-Ceva conjugate of X(822)

X(38986) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(1)X(75)

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

X(38986) lies on the bicevian ellipse of X(1) and X(2) (with center X(1125)), the bicevian ellipse of X(2) and X(31) (with center X(6679)), the rectangular bicevian hyperbola of X(2) and X(741), and these lines: {1, 668}, {11, 5518}, {43, 4595}, {87, 32039}, {214, 995}, {244, 4117}, {551, 34832}, {798, 9427}, {932, 20332}, {1015, 1960}, {1193, 4161}, {1201, 39046}, {1911, 9259}, {1927, 29055}, {2275, 24513}, {2310, 17419}, {2667, 34587}, {3223, 6384}, {4602, 18826}, {6197, 34063}, {8054, 25569}, {16604, 23493}

X(38986) = isogonal conjugate of isotomic conjugate of X(3123)
X(38986) = crosssum of circumcircle-intercepts of line X(1)X(75)
X(38986) = crossdifference of every pair of points on line X(190)X(4598)
X(38986) = center of hyperbola {{A,B,C,X(1),X(31)}}
X(38986) = X(2)-Ceva conjugate of X(798)
X(38986) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 5383}, {87, 7035}, {330, 1016}, {668, 932}, {765, 6384}, {2162, 31625}, {4998, 7155}
X(38986) = trilinear product X(i)*X(j) for these {i,j}: {2, 21762}, {6, 6377}, {31, 3123}, {32, 21138}, {43, 3248}, {192, 1977}, {213, 16742}, {244, 2209}, {512, 16695}, {649, 20979}, {667, 4083}, {669, 17217}, {798, 18197}, {1015, 2176}, {1403, 3271}, {1918, 23824}, {1919, 3835}, {1980, 20906}, {3249, 4595}
X(38986) = trilinear quotient X(i)/X(j) for these (i,j): (6, 5383), (43, 7035), (192, 31625), (244, 6384), (649, 4598), (667, 932), (1015, 330), (1086, 6383), (1403, 4998), (1977, 2162), (2170, 27424), (2176, 1016), (2209, 765), (3123, 75), (3248, 87), (3271, 7155), (3835, 1978), (4083, 668), (6377, 2), (16695, 99), (16742, 274), (17217, 670), (18197, 799), (20906, 6386), (21138, 76), (21762, 6), (23824, 310)
X(38986) = barycentric product X(i)*X(j) for these {i,j}: {1, 6377}, {6, 3123}, {31, 21138}, {42, 16742}, {43, 1015}, {75, 21762}, {192, 3248}, {213, 23824}, {244, 2176}, {512, 18197}, {661, 16695}, {667, 3835}, {798, 17217}, {1086, 2209}, {1403, 2170}, {1423, 3271}, {1919, 20906}, {1977, 6376}, {3249, 36863}, {4595, 8027}
X(38986) = barycentric quotient X(i)/X(j) for these (i,j): (31, 5383), (43, 31625), (244, 6383), (667, 4598), (1015, 6384), (1919, 932), (1977, 87), (2176, 7035), (2209, 1016), (3123, 76), (3248, 330), (3271, 27424), (3835, 6386), (6377, 75), (16695, 799), (16742, 310), (17217, 4602), (18197, 670), (21138, 561), (21762, 1), (23824, 6385)

X(38987) = CROSSSUM OF X(98) AND X(110)

Trilinears    sec(B + ω) csc(A - B) + sec(C + ω) csc(C - A) : :
Barycentrics    a^2 (b^2 - c^2)^2 (b^4 + c^4 - a^2 b^2 - a^2 c^2) (a^6 - 2 a^4 (b^2 + c^2) + a^2 (b^4 - b^2 c^2 + c^4) + b^2 c^2 (b^2 + c^2)) : :

X(38987) lies on the bicevian ellipse of X(2) and X(6) (with center X(3589)), the rectangular bicevian hyperbola of X(2) and X(842) (with center X(16760)), and these lines: {125, 3005}, {237, 511}, {338, 523}, {468, 2967}, {647, 3124}, {868, 33752}, {1649, 16186}, {3003, 9475}, {5099, 21731}, {5201, 6593}, {5640, 33927}, {15000, 18114}

X(38987) = crosssum of X(98) and X(110)
X(38987) = crosspoint of X(511) and X(523)
X(38987) = crossdifference of every pair of points on line X(2395)X(4226)
X(38987) = center of hyperbola {{A,B,C,X(6),X(232)}}
X(38987) = X(2)-Ceva conjugate of X(3569)

X(38988) = CROSSSUM OF X(99) AND X(111)

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

X(38988) lies on the bicevian ellipse of X(2) and X(6) (with center X(3589)), the rectangular bicevian hyperbola of X(2) and X(843), and these lines: {6, 36827}, {512, 3124}, {524, 3266}, {1084, 3005}, {1648, 11183}, {2030, 3231}, {7668, 8288}, {8623, 38998}, {9148, 35078}, {37085, 38996}

X(38988) = crosssum of X(99) and X(111)
X(38988) = crosspoint of X(512) and X(524)
X(38988) = crossdifference of every pair of points on line X(5468)X(34290)
X(38988) = center of hyperbola {{A,B,C,X(6),X(187)}}, which is the locus of barycentric product of circumcircle-X(690)-antipodes
X(38988) = X(2)-Ceva conjugate of X(351)

X(38989) = CROSSSUM OF X(100) AND X(105)

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

X(38989) lies on the bicevian ellipse of X(2) and X(6) (with center X(3589)), the bicevian ellipse of X(2) and X(7) (with center X(142)), the rectangular bicevian hyperbola of X(2) and X(840), and these lines: {2, 660}, {7, 34085}, {11, 3835}, {238, 1284}, {244, 661}, {320, 39044}, {512, 6547}, {513, 1086}, {518, 3717}, {891, 35092}, {1155, 9471}, {1357, 3323}, {1362, 3660}, {1463, 39063}, {1647, 14434}, {2679, 15611}, {3123, 6615}, {3662, 36294}, {3675, 17435}, {3834, 20333}, {3836, 20340}, {4389, 24482}, {4871, 16594}, {5091, 36086}

X(38989) = complement of X(660)
X(38989) = complementary conjugate of X(3837)
X(38989) = crosssum of X(100) and X(105)
X(38989) = crosspoint of X(512) and X(517)
X(38989) = crossdifference of every pair of points on line X(660)X(2284)
X(38989) = center of hyperbola {{A,B,C,X(6),X(7)}}
X(38989) = X(2)-Ceva conjugate of X(665)
X(38989) = X(685)-of-intouch-triangle
X(38989) = X(i)-isoconjugate of X(j) for these {i,j}: {105, 5378}, {291, 5377}, {660, 36086}, {666, 813}, {919, 4562)
X(38989) = trilinear product X(i)*X(j) for these {i,j}: {238, 3675}, {244, 8299}, {518, 27846}, {659, 2254}, {665, 812}, {672, 27918}, {918, 8632}, {1015, 17755}, {1429, 17435}, {1458, 4124}, {2170, 34253}
X(38989) = trilinear quotient X(i)/X(j) for these (i,j): (238, 5377), (518, 5378), (659, 36086), (665, 813), (812, 666), (918, 4562), (2254, 660), (3675, 291), (4124, 14942), (8299, 765), (8632, 919), (17435, 4876), (17755, 1016), (27846, 105), (27918, 673), (34253, 4564)
X(38989) = barycentric product X(i)*X(j) for these {i,j}: {11, 34253}, {239, 3675}, {241, 4124}, {244, 17755}, {518, 27918}, {659, 918}, {665, 3766}, {812, 2254}, {1086, 8299}, {1447, 17435}, {3912, 27846}
X(38989) = barycentric quotient X(i)/X(j) for these (i,j): (659, 666), (665, 660), (672, 5378), (918, 4583), (1914, 5377), (2254, 4562), (3675, 292), (3766, 36803), (4124, 36796), (8299, 1016), (17435, 4518), (17755, 7035), (27846, 673), (27918, 2481), (34253, 4998)

X(38990) = CROSSSUM OF X(101) AND X(675)

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

X(38990) lies on the bicevian ellipse of X(2) and X(6) (with center X(3589)) and these lines: {514, 20974}, {661, 7668}, {674, 2225}

X(38990) = crosssum of X(101) and X(675)
X(38990) = crosspoint of X(514) and X(674)
X(38990) = center of hyperbola {{A,B,C,X(6),X(514)}}

X(38991) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(2)X(7)

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

X(38991) lies on the bicevian ellipse of X(2) and X(6) (with center X(3589)), the bicevian ellipse of X(2) and X(9) (with center X(6666)), the rectangular bicevian hyperbola of X(2) and X(2291), and these lines: {41, 4557}, {649, 4014}, {657, 3271}, {1200, 2246}, {2082, 8257}, {2170, 21343}, {2347, 17792}, {3124, 5075}, {21748, 36213}

X(38991) = complement of isogonal conjugate of X(23865)
X(38991) = complement of isotomic conjugate of X(21302)
X(38991) = complementary conjugate of X(9613)
X(38991) = crosssum of circumcircle-intercepts of line X(2)X(7)
X(38991) = crossdifference of every pair of points on line X(17136)X(35338)
X(38991) = center of hyperbola {{A,B,C,X(6),X(9)}}
X(38991) = X(2)-Ceva conjugate of X(663)
X(38991) = trilinear product X(i)*X(j) for these {i,j}: {650, 23865}, {663, 21390}, {3063, 21302}
X(38991) = trilinear quotient X(i)/X(j) for these (i,j): (21302, 4554), (21390, 664), (21611, 4572), (23865, 651)
X(38991) = barycentric product X(i)*X(j) for these {i,j}: {522, 23865}, {650, 21390}, {663, 21302}, {3063, 21611}
X(38991) = barycentric quotient X(i)/X(j) for these (i,j): (21302, 4572), (21390, 4554), (23865, 664)

X(38992) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(2)X(12)

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

X(38992) lies on the bicevian ellipse of X(2) and X(6) (with center X(3589)) and these lines: {2170, 3124}, {2968, 3271}, {6371, 15611}, {7668, 8286}, {8300, 36213}, {12623, 36524}

X(38992) = crosssum of circumcircle-intercepts of line X(2)X(12)
X(38992) = center of hyperbola {{A,B,C,X(6),X(60)}}

X(38993) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(2)X(13)

Trilinears    a sin(B - C) sin(A + π/3) (a^2 sin(B - C) sin(A + π/3) - b^2 sin(C - A) sin(B + π/3) - c^2 sin(A - B) sin(C + π/3)) : :
Barycentrics    a^2 (b^2 - c^2)^2 (3 (a^2 b^2 + a^2 c^2 - b^4 - c^4) - 2 Sqrt[3] (2 a^2 - b^2 - c^2) S) (3 (a^6 - 2 a^4 (b^2 + c^2) + a^2 (b^4 - b^2 c^2 + c^4) + b^2 c^2 (b^2 + c^2)) + 2 Sqrt[3] (a^4 + a^2 (b^2 + c^2) - 3 b^2 c^2) S) : :

X(38993) lies on the bicevian ellipse of X(2) and X(6) (with center X(3589)), the bicevian conic of X(2) and X(14) (with center X(6670)), the bicevian conic of X(2) and X(15) (with center X(6671)), the rectangular bicevian hyperbola of X(2) and X(2378), and these lines: {618, 14921}, {3005, 38994}, {3124, 6138}, {7668, 30468}

X(38993) = crosssum of circumcircle-intercepts of line X(2)X(13)
X(38993) = center of hyperbola {{A,B,C,X(6),X(14),X(15)}}
X(38993) = X(2)-Ceva conjugate of X(6137)

X(38994) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(2)X(14)

Trilinears    a sin(B - C) sin(A - π/3) (a^2 sin(B - C) sin(A - π/3) - b^2 sin(C - A) sin(B - π/3) - c^2 sin(A - B) sin(C - π/3)) : :
Barycentrics    a^2 (b^2 - c^2)^2 (3 (a^2 b^2 + a^2 c^2 - b^4 - c^4) + 2 Sqrt[3] (2 a^2 - b^2 - c^2) S) (3 (a^6 - 2 a^4 (b^2 + c^2) + a^2 (b^4 - b^2 c^2 + c^4) + b^2 c^2 (b^2 + c^2)) - 2 Sqrt[3] (a^4 + a^2 (b^2 + c^2) - 3 b^2 c^2) S) : :

X(38994) lies on the bicevian ellipse of X(2) and X(6) (with center X(3589)), the bicevian conic of X(2) and X(13) (with center X(6669)), the bicevian conic of X(2) and X(16) (with center X(6672)), the rectangular bicevian hyperbola of X(2) and X(2379), and these lines: {619, 14922}, {3005, 38993}, {3124, 6137}, {7668, 30465}

X(38994) = crosssum of circumcircle-intercepts of line X(2)X(14)
X(38994) = center of hyperbola {{A,B,C,X(6),X(13),X(16)}}
X(38994) = X(2)-Ceva conjugate of X(6138)

X(38995) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(2)X(31)

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

X(38995) lies on the bicevian ellipse of X(2) and X(6) (with center X(3589)), the rectangular bicevian hyperbola of X(2) and X(753), and these lines: {1086, 7668}, {1193, 36213}, {3122, 27846}, {3123, 4858}, {3124, 6377}

X(38995) = crosssum of circumcircle-intercepts of line X(2)X(31)
X(38995) = center of hyperbola {{A,B,C,X(6),X(75)}}
X(38995) = X(2)-Ceva conjugate of X(3250)

X(38996) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(2)X(39)

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

X(38996) lies on the bicevian ellipse of X(2) and X(6) (with center X(3589)), the bicevian ellipse of X(2) and X(32) (with center X(6680)), the rectangular bicevian hyperbola of X(2) and X(729), and these lines: {6, 4576}, {690, 31317}, {2086, 7668}, {3051, 20976}, {3124, 5027}, {3225, 4609}, {5007, 38998}, {8290, 34482}, {20965, 39080}, {37085, 38988}

X(38996) = complement of isogonal conjugate of X(21006)
X(38996) = complement of trilinear pole of line X(39)X(698)
X(38996) = crosssum of circumcircle-intercepts of line X(2)X(39)
X(38996) = center of hyperbola {{A,B,C,X(6),X(32)}}, which is the locus of barycentric product of circumcircle-X(512)-antipodes
X(38996) = X(2)-Ceva conjugate of X(669)
X(38996) = trilinear product X(i)*X(j) for these {i,j}: {798, 21006}, {1084, 33760}, {1919, 22322}, {4117, 7760}, {9426, 20953}, {9427, 18064}
X(38996) = trilinear quotient X(i)/X(j) for these (i,j): (1627, 24037), (20953, 4609), (21006, 799), (22322, 1978), (33760, 34537)
X(38996) = barycentric product X(i)*X(j) for these {i,j}: {512, 21006}, {1084, 7760}, {1627, 3124}, {1924, 20953}, {2489, 22159}, {4117, 18064}, {9780, 22322}
X(38996) = barycentric quotient X(i)/X(j) for these (i,j): (1084, 6664), (1627, 34537), (21006, 670), (22322, 28626)

X(38997) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(2)X(51)

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

X(38997) lies on the bicevian ellipse of X(2) and X(6) (with center X(3589)), the rectangular bicevian hyperbola of X(2) and X(98) (with center X(6036)), and these lines: {575, 34236}, {868, 7668}, {1995, 9755}

X(38997) = crosssum of circumcircle-intercepts of line X(2)X(51)
X(38997) = center of hyperbola {{A,B,C,X(6),X(95),X(98)}}
X(38997) = X(2)-Ceva conjugate of X(3288)

X(38998) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(2)X(512)

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

X(38998) lies on the bicevian ellipse of X(2) and X(6) (with center X(3589)), the rectangular bicevian hyperbola of X(2) and X(99) (with center X(620)), and these lines: {2, 1634}, {23, 8290}, {39, 373}, {99, 34087}, {125, 6292}, {237, 2482}, {351, 36213}, {669, 5026}, {887, 9148}, {1084, 20965}, {3001, 34834}, {3231, 6786}, {3266, 5976}, {5007, 38996}, {5664, 14420}, {6337, 23181}, {8623, 38988}, {10291, 37916}, {11147, 37184}, {11165, 11328}, {14096, 15810}, {35298, 39100}

X(38998) = crosssum of circumcircle-intercepts of line X(2)X(512)
X(38998) = center of hyperbola {{A,B,C,X(6),X(99)}}
X(38998) = X(2)-Ceva conjugate of X(3231)

X(38999) = CROSSSUM OF X(74) AND X(107)

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

X(38999) lies on the bicevian conic of X(2) and X(3) (with center X(140)), the rectangular bicevian hyperbola of X(2) and X(2693), and these lines: {3, 36831}, {30, 34334}, {520, 2972}, {17434, 35071}

X(38999) = crosssum of X(74) and X(107)
X(38999) = crosspoint of X(30) and X(520)
X(38999) = center of hyperbola {{A,B,C,X(3),X(30)}}
X(38999) = X(2)-Ceva conjugate of X(1636)

X(39000) = CROSSSUM OF X(98) AND X(112)

Trilinears    sec(B + ω) tan C csc(A - B) + sec(C + ω) tan B csc(C - A) : :
Barycentrics    a^2 (b^2 - c^2)^2 (a^2 - b^2 - c^2) (a^2 (b^2 + c^2) - b^4 - c^4) (a^8 - a^6 (b^2 + c^2) - a^4 (b^4 + b^2 c^2 + c^4) + a^2 (b^2 + c^2) (b^4 + c^4) + b^2 c^2 (b^2 - c^2)^2) : :

X(39000) lies on the bicevian conic of X(2) and X(3) (with center X(140)), the rectangular bicevian hyperbola of X(2) and X(2710), and these lines: {125, 17434}, {232, 511}, {339, 525}, {647, 2972}, {684, 38974}, {3124, 6587}, {7664, 11064}

X(39000) = crosssum of X(98) and X(112)
X(39000) = crosspoint of X(511) and X(525)
X(39000) = crossdifference of every pair of points on line X(879)X(2409)
X(39000) = center of hyperbola {{A,B,C,X(3),X(511)}}
X(39000) = X(2)-Ceva conjugate of X(684)

X(39001) = CROSSSUM OF X(99) AND X(3563)

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

X(39001) lies on the bicevian conic of X(2) and X(3) (with center X(140)), the rectangular bicevian hyperbola of X(2) and X(23700), and these lines: {512, 2971}, {2974, 3564}

X(39001) = crosssum of X(99) and X(3563)
X(39001) = crosspoint of X(512) and X(3564)
X(39001) = center of hyperbola {{A,B,C,X(3),X(512)}}

X(39002) = CROSSSUM OF X(100) AND X(915)

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

X(39002) lies on the bicevian conic of X(2) and X(3) (with center X(140)) and these lines: {513, 2969}, {912, 914}

X(39002) = crosssum of X(100) and X(915)
X(39002) = crosspoint of X(513) and X(912)
X(39002) = center of hyperbola {{A,B,C,X(3),X(513)}}

X(39003) = CROSSSUM OF X(101) AND X(917)

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

X(39003) lies on the bicevian conic of X(2) and X(3) (with center X(140)) and these lines: {514, 2973}, {916, 2253}

X(39003) = crosssum of X(101) and X(917)
X(39003) = crosspoint of X(514) and X(916)
X(39003) = center of hyperbola {{A,B,C,X(3),X(514)}}

X(39004) = CROSSSUM OF X(104) AND X(108)

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

X(39004) lies on the bicevian conic of X(2) and X(3) (with center X(140)), the bicevian ellipse of X(2) and X(8) (with center X(10)), the rectangular bicevian hyperbola of X(2) and X(2745), and these lines: {517, 1361}, {521, 1364}, {656, 2972}, {1146, 14298}, {3756, 33646}, {8677, 10017}

X(39004) = crosssum of X(104) and X(108)
X(39004) = crosspoint of X(517) and X(521)
X(39004) = center of hyperbola {{A,B,C,X(3),X(8)}}

X(39005) = CROSSSUM OF X(110) AND X(1300)

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

X(39005) lies on the bicevian conic of X(2) and X(3) (with center X(140)), the rectangular bicevian hyperbola of X(2) and X(32710), and these lines: {523, 2970}, {6102, 14670}, {13754, 34333}

X(39005) = crosssum of X(110) and X(1300)
X(39005) = crosspoint of X(523) and X(13754)
X(39005) = center of hyperbola {{A,B,C,X(3),X(523)}}
X(39005) = X(2)-Ceva conjugate of X(686)

X(39006) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(4)X(9)

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

X(39006) lies on the bicevian conic of X(2) and X(3) (with center X(140)), the bicevian ellipse of X(2) and X(57) (with center X(6692)), the rectangular bicevian hyperbola of X(2) and X(103) (with center X(6712)), and these lines: {603, 4559}, {649, 1146}, {652, 7117}, {654, 2170}, {1565, 4091}, {1765, 2250}, {2972, 3937}, {3138, 6506}, {8676, 14714}

X(39006) = crosssum of circumcircle-intercepts of line X(4)X(9)
X(39006) = center of hyperbola {{A,B,C,X(3),X(57),X(103)})
X(39006) = X(2)-Ceva conjugate of X(1459)
X(39006) = trilinear product X(20293)*X(22383)
X(39006) = barycentric product X(1459)*X(20293)

X(39007) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(4)X(12)

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

X(39007) lies on the bicevian conic of X(2) and X(3) (with center X(140)), the bicevian ellipse of X(2) and X(7) (with center X(142)), and these lines: {11, 15616}, {1086, 1364}, {2972, 7004}

X(39007) = crosssum of circumcircle-intercepts of line X(4)X(12)
X(39007) = center of hyperbola {{A,B,C,X(3),X(7)}}

X(39008) = CROSSSUM OF X(74) AND X(112)

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

X(39008) lies on the Steiner inellipse, the rectangular bicevian hyperbola of X(2) and X(2697), and these lines: {2, 16077}, {3, 23967}, {30, 1990}, {113, 11672}, {115, 6587}, {187, 3184}, {393, 32646}, {401, 14920}, {441, 2482}, {525, 15526}, {577, 32662}, {647, 1562}, {1650, 14401}, {2088, 39021}, {2420, 18508}, {3269, 17434}, {5664, 35088}, {6794, 32640}, {15075, 15454}, {15351, 23582}, {30227, 37188}, {35087, 35923}

X(39008) = complement of X(16077)
X(39008) = complementary conjugate of complement of X(9409)
X(39008) = crosssum of X(i) and X(j) for these {i,j}: {6, 1304}, {74, 112}
X(39008) = crosspoint of X(i) and X(j) for these {i,j}: {2, 9033}, {30, 525}
X(39008) = crossdifference of every pair of points on line X(1304)X(5502) (the tangent to the circumcircle at X(1304))
X(39008) = center of hyperbola {{A,B,C,X(30),X(525)}}
X(39008) = X(2)-Ceva conjugate of X(9033)
X(39008) = barycentric square of X(9033)
X(39008) = trilinear product X(2631)*X(9033)

X(39009) = CROSSSUM OF X(98) AND X(26714)

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

X(39009) lies on the Steiner inellipse and these lines: {511, 11672}, {5661, 10168}, {15526, 38974}

X(39009) = complement of isogonal conjugate of X(9420)
X(39009) = crosssum of X(i) and X(j) for these {i,j}: {6, 6037}, {98, 26714}
X(39009) = crossdifference of every pair of points on the tangent to the circumcircle at X(6037)
X(39009) = center of hyperbola {{A,B,C,X(2),X(511)}}
X(39009) = X(2)-Ceva conjugate of isogonal conjugate of X(6037)
X(39009) = barycentric square of isogonal conjugate of X(6037)

X(39010) = CROSSSUM OF X(99) AND X(729)

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

X(39010) lies on the Steiner inellipse, the rectangular bicevian hyperbola of X(2) and X(5970), and these lines: {2, 886}, {39, 35077}, {115, 9151}, {512, 1084}, {538, 30736}, {2482, 3229}, {3117, 30229}, {14700, 35146}

X(39010) = complement of X(886)
X(39010) = complementary conjugate of complement of X(887)
X(39010) = crosssum of X(i) and X(j) for these {i,j}: {6, 886}, {99, 729}
X(39010) = crosspoint of X(i) and X(j) for these {i,j}: {2, 888}, {512, 538}
X(39010) = crossdifference of every pair of points on line X(9150)X(23342) (the tangent to the circumcircle at X(9150))
X(39010) = center of hyperbola {{A,B,C,X(2),X(512)}}
X(39010) = X(2)-Ceva conjugate of X(888)
X(39010) = barycentric square of X(888)
X(39010) = trilinear product X(i)*X(j) for these {i,j}: {1645, 2234}, {4117, 35073}
X(39010) = trilinear quotient X(1645)/X(37132)
X(39010) = barycentric product X(i)*X(j) for these {i,j}: {538, 1645}, {887, 9148}, {888, 888}, {1084, 35073}
X(39010) = barycentric quotient X(i)/X(j) for these (i,j): (887, 9150), (888, 886), (1645, 3228)

X(39011) = CROSSSUM OF X(100) AND X(739)

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

X(39011) lies on the Steiner inellipse, the rectangular bicevian hyperbola of X(2) and X(9081), and these lines: {2, 889}, {37, 35123}, {115, 31946}, {350, 35073}, {513, 1015}, {536, 6381}, {661, 1084}, {888, 38978}, {1086, 3835}, {1575, 4370}, {1646, 14434}, {2276, 24338}, {3666, 35089}, {3943, 20532}, {4364, 35126}, {9295, 31625}, {14441, 33917}, {27854, 35119}

X(39011) = complement of X(889)
X(39011) = complementary conjugate of complement of X(890)
X(39011) = crosssum of X(i) and X(j) for these {i,j}: {6, 898}, {100, 739}
X(39011) = crosspoint of X(i) and X(j) for these {i,j}: {2, 891}, {513, 536}
X(39011) = crossdifference of every pair of points on line X(898)X(5381) (the tangent to the circumcircle at X(898))
X(39011) = center of hyperbola {{A,B,C,X(2),X(513)}}
X(39011) = X(2)-Ceva conjugate of X(891)
X(39011) = barycentric square of X(891)
X(39011) = X(i)-isoconjugate of X(j) for these {i,j}: {889, 34075}, {898, 4607}, {5381, 37129}
X(39011) = trilinear product X(i)*X(j) for these {i,j}: {649, 14434}, {890, 4728}, {891, 3768}, {899, 1646}, {3230, 19945}, {3248, 13466}
X(39011) = trilinear quotient X(i)/X(j) for these (i,j): (890, 34075), (891, 4607), (899, 5381), (1646, 37129), (3768, 898), (4728, 889), (13466, 7035), (14434, 190), (19945, 3227)
X(39011) = barycentric product X(i)*X(j) for these {513, 14434}, {536, 1646}, {891, 891}, {899, 19945}, {1015, 13466}, {3768, 4728}
X(39011) = barycentric quotient X(i)/X(j) for these (i,j): (890, 898), (891, 889), (1646, 3227), (3230, 5381), (3768, 4607), (13466, 31625), (14434, 668), (19945, 31002)

X(39012) = CROSSSUM OF X(105) AND X(8693)

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

X(39012) lies on the Steiner inellipse and these lines: {518, 6184}, {3789, 35026}, {5701, 35113}, {17435, 39014}

X(39012) = complement of trilinear pole of line X(2)X(2481)
X(39012) = crosssum of X(105) and X(8693)
X(39012) = crosspoint of X(518) and X(4762)
X(39012) = center of hyperbola {{A,B,C,X(2),X(518),X(4762)}}

X(39013) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(5)X(6)

Barycentrics    a^4 (b^2 - c^2)^2 (a^4 - 2 a^2 b^2 - 2 a^2 c^2 + b^4 + c^4)^2 : :
Barycentrics    (sec 2B - sec 2C)^2 : :

X(39013) lies on the Steiner inellipse, the bicevian conic of X(2) and X(24) (with center X(16238)), the bicevian conic of X(2) and X(54) (with center X(6689)), the rectangular bicevian hyperbola of X(2) and X(3563), and these lines: {32, 1147}, {39, 35067}, {115, 136}, {187, 12095}, {647, 39021}, {800, 3163}, {2088, 35071}

X(39013) = isotomic conjugate of polar conjugate of X(6754)
X(39013) = complement of polar conjugate of X(6753)
X(39013) = complementary conjugate of complement of X(34952)
X(39013) = crosspoint of X(2) and X(924)
X(39013) = crosssum of X(6) and X(925)
X(39013) = crosssum of circumcircle-intercepts of line X(5)X(6)
X(39013) = crossdifference of every pair of points on line X(925)X(6380) (the tangent to the circumcircle at X(925))
X(39013) = center of hyperbola {{A,B,C,X(2),X(24)}}
X(39013) = X(2)-Ceva conjugate of X(924)
X(39013) = barycentric square of X(924)

X(39014) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(6)X(7)

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

X(39014) lies on the Steiner inellipse, the bicevian ellipse of X(2) and X(55) (with center X(6690)), and these lines: {32, 32642}, {39, 35093}, {115, 1566}, {1086, 6586}, {1212, 35120}, {8608, 23972}, {16588, 35113}, {17435, 39012}

X(39014) = complement of isotomic conjugate of X(926)
X(39014) = complement of trilinear pole of line X(2)X(4554)
X(39014) = complementary conjugate of complement of X(8638)
X(39014) = crosspoint of X(2) and X(926)
X(39014) = crosssum of X(6) and X(927)
X(39014) = crosssum of circumcircle-intercepts of line X(6)X(7)
X(39014) = crossdifference of every pair of points on the tangent to the circumcircle at X(927)
X(39014) = center of hyperbola {{A,B,C,X(2),X(55)}}
X(39014) = X(2)-Ceva conjugate of X(926)
X(39014) = barycentric square of X(926)
X(39014) = trilinear quotient X(926)/X(34085)

X(39015) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(6)X(8)

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

X(39015) lies on the Steiner inellipse, the bicevian ellipse of X(2) and X(56) (with center X(6691)), the rectangular bicevian hyperbola of X(2) and X(28479), and these lines: {115, 15611}, {1084, 20982}, {1146, 6377}, {16742, 35119}

X(39015) = complement of isotomic conjugate of X(6371)
X(39015) = complement of trilinear pole of line X(2)X(1240)
X(39015) = crosspoint of X(2) and X(6371)
X(39015) = crosssum of X(6) and X(8707)
X(39015) = crosssum of circumcircle-intercepts of line X(6)X(8)
X(39015) = crossdifference of every pair of points on the tangent to the circumcircle at X(8707)
X(39015) = center of hyperbola {{A,B,C,X(2),X(56)}}
X(39015) = X(2)-Ceva conjugate of X(6371)
X(39015) = barycentric square of X(6371)

X(39016) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(6)X(10)

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

X(39016) lies on the Steiner inellipse, the bicevian ellipse of X(2) and X(58) (with center X(6693)), the rectangular bicevian hyperbola of X(2) and X(28476), and these lines: {115, 5515}, {1015, 18191}, {8978, 15015}, {16614,35076}

X(39016) = complement of isogonal conjugate of X(8637)
X(39016) = complement of trilinear pole of line X(2)X(313)
X(39016) = complementary conjugate of complement of X(8637)
X(39016) = crosspoint of X(2) and X(834)
X(39016) = crosssum of X(6) and X(835)
X(39016) = crosssum of circumcircle-intercepts of line X(6)X(10)
X(39016) = crossdifference of every pair of points on the tangent to the circumcircle at X(835)
X(39016) = center of hyperbola {{A,B,C,X(2),X(58)}}
X(39016) = center of the circumconic {{A,B,C,X(2), X(58)}}
X(39016) = X(2)-Ceva conjugate of X(834)
X(39016) = barycentric square of X(834)
X(39016) = barycentric product X(834)*X(834)
X(39016) = trilinear quotient X(834)/X(37218)
X(39016) = X(i)-complementary conjugate of-X(j) for these (i,j): (31, 834), (386, 21260), (560, 6590), (834, 2887)
X(39016) = X(835)-isoconjugate-of-X(37218)

X(39017) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(6)X(11)

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

X(39017) lies on the Steiner inellipse, the bicevian ellipse of X(2) and X(59) (with center X(6667)), and these lines: {115, 15612}, {1086, 6589}, {1146, 6586}, {7113, 11672}, {8607, 23972}, {8608, 23986}

X(39017) = complement of trilinear pole of line X(2)X(13006)
X(39017) = crosspoint of X(2) and X(928)
X(39017) = crosssum of X(6) and X(929)
X(39017) = crosssum of circumcircle-intercepts of line X(6)X(11)
X(39017) = crossdifference of every pair of points on the tangent to the circumcircle at X(929)
X(39017) = center of hyperbola {{A,B,C,X(2),X(59)}}
X(39017) = X(2)-Ceva conjugate of X(928)
X(39017) = barycentric square of X(928)

X(39018) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF NAPOLEON AXIS

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

Let A'B'C' be the orthic triangle. Let LA be the Napoleon axis of triangle AB'C', and define LB, LC cyclically. Let A" = LB∩LC, and define B", C" cyclically. Triangle A"B"C" is inversely similar to ABC, with similicenter X(39018).

X(39018) lies on the Steiner inellipse, the bicevian conic of X(2) and X(61) (with center X(6694), the bicevian conic of X(2) and X(62) (with center X(6695), the rectangular bicevian hyperbola of X(2) and X(5966), and these lines: {32, 14586}, {39, 15345}, {115, 137}, {187, 6150}, {1493, 5007}, {1506, 31376}, {10413, 39019}, {34833, 35067}

X(39018) = complement of trilinear pole of line X(2)X(1225)
X(39018) = complementary conjugate of X(39512)
X(39018) = crosspoint of X(2) and X(1510)
X(39018) = crosssum of X(6) and X(930)
X(39018) = crosssum of circumcircle-intercepts of Napoleon axis
X(39018) = center of hyperbola {{A,B,C,X(2),X(61),X(62)}}
X(39018) = X(2)-Ceva conjugate of X(1510)
X(39018) = barycentric square of X(1510)

X(39019) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(6)X(24)

Barycentrics    (b^2 - c^2)^2 (a^2 - b^2 - c^2)^2 (a^2 (b^2 + c^2) - (b^2 - c^2)^2)^2 : :
Barycentrics    (cot 2B - cot 2C)^2 : :
Barycentrics    sin^2 2A sin^2(2B - 2C) : :
Barycentrics    csc 2B csc(2C - 2A) + csc 2C csc(2A - 2B) : :

X(39019) lies on the Steiner inellipse, the bicevian conic of X(2) and X(5) (with center X(3628)), and these lines: {2, 18831}, {5, 35318}, {68, 36433}, {115, 8902}, {125, 35071}, {216, 34520}, {233, 547}, {339, 18314}, {577, 11077}, {1209, 11672}, {3269, 17434}, {5007, 23976}, {10317, 23967}, {10413, 39018}, {10600, 35067}, {14391, 35442}, {16573, 35128}

X(39019) = isotomic conjugate of polar conjugate of X(24862)
X(39019) = complement of X(18831)
X(39019) = crosspoint of X(2) and X(6368)
X(39019) = crosssum of X(6) and X(933)
X(39019) = complementary conjugate of complement of X(15451)
X(39019) = crosssum of circumcircle-intercepts of line X(6)X(24) (or of circle {{X(4),X(15),X(16)}} (V(X(4)))
X(39019) = crossdifference of every pair of points on the tangent to the circumcircle at X(933)
X(39019) = center of hyperbola {{A,B,C,X(2),X(5)}}
X(39019) = X(2)-Ceva conjugate of X(6368)
X(39019) = barycentric square of X(6368)

X(39020) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(6)X(64)

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

X(39020) lies on the Steiner inellipse, the bicevian conic of X(2) and X(20) (with center X(3)), the rectangular bicevian hyperbola of X(2) and X(34168), and these lines: {3, 23976}, {20, 14944}, {112, 376}, {115, 13611}, {122, 1562}, {393, 3344}, {2883, 11672}, {3269, 15526}, {11589, 16318}, {12096, 15341}, {16573, 35508}, {31377, 35067}, {35075, 36908}

X(39020) = complement of isotomic conjugate of X(8057)
X(39020) = complement of polar conjugate of X(6587)
X(39020) = complement of trilinear pole of line X(2)X(253)
X(39020) = crosspoint of X(2) and X(8057)
X(39020) = crosssum of X(6) and X(1301)
X(39020) = crosssum of circumcircle-intercepts of line X(6)X(64)
X(39020) = crossdifference of every pair of points on line X(1301)X(32713) (the tangent to the circumcircle at X(1301))
X(39020) = center of hyperbola {{A,B,C,X(2),X(20)}}
X(39020) = X(2)-Ceva conjugate of X(8057)
X(39020) = barycentric square of X(8057)

X(39021) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(6)X(1511)

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

X(39021) lies on the Steiner inellipse and these lines: {2, 18878}, {32, 23967}, {115, 2501}, {131, 187}, {647, 39013}, {2088, 39008}, {3163, 16310}, {10413, 18334}

X(39021) = complement of X(18878)
X(39021) = complementary conjugate of complement of X(21731)
X(39021) = crosssum of X(6) and X(10420)
X(39021) = crosssum of circumcircle-intercepts of line X(6)X(1511)
X(39021) = crossdifference of every pair of points on the tangent to the circumcircle at X(10420)
X(39021) = pole wrt polar circle of line X(687)X(2407)
X(39021) = center of hyperbola {{A,B,C,X(2),X(403)}}
X(39021) = X(2)-Ceva conjugate of isogonal conjugate of X(10420)
X(39021) = barycentric square of isogonal conjugate of X(10420)

X(39022) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(6)X(5638)

Trilinears    (sin B) (e cos C + cos(C + ω)) + (sin C) (e cos B + cos(B + ω)) : :
Barycentrics    2 a^2 - b^2 - c^2 + 2 Sqrt[a^4 + b^4 + c^4 - a^2 b^2 - a^2 c^2 - b^2 c^2] : :
Barycentrics    1/((SA^2 - SB SC + SA Sqrt[SA^2 + SB^2 + SC^2 - S^2]))^2 : :
Barycentrics    SB^2 + SC^2 + (SB + SC) (Sqrt[SW^2 - 3 S^2] - SA) : :
Barycentrics    (csc B) (e cos B - cos(B + ω) + (csc C) (e cos C - cos(C + ω)) : :
Barycentrics    (csc A) (sin A + e sin(A + ω) - sin(A + 2ω)) : :
Barycentrics    (sin^2 A)/(e cos A + cos(A + ω))^2 : :

Let RA be the radical axis of the nine-point circles of P(118)BC and U(118)BC; define RB, RC cyclically. The lines RA, RB, RC concur in X(39022). (Hyacinthos #21938/21940, Apr 12, 2013, Antreas Hatzipolakis/Randy Hutson)

X(39022) lies on the Steiner inellipse, the ellipse described at X(6784), and these lines: {2, 6}, {5, 3557}, {30, 1379}, {32, 6178}, {115, 2029}, {140, 14631}, {376, 35913}, {523, 13722}, {549, 1341}, {1340, 15048}, {1348, 7818}, {1503, 6040}, {2482, 3413}, {5305, 14630}, {6177, 7888}, {7668, 39067}, {7794, 14633}, {7801, 19660}

X(39022) = reflection of X(39023) in X(2)
X(39022) = isotomic conjugate of isogonal conjugate of X(2029)
X(39022) = complement of X(6189)
X(39022) = complementary conjugate of complement of X(5639)
X(39022) = crosssum of circumcircle-intercepts of line X(6)X(5638)
X(39022) = crosssum of X(6) and X(1380)
X(39022) = crosspoint of X(2) and X(3414)
X(39022) = crossdifference of every pair of points on line X(512)X(1380)
X(39022) = center of hyperbola {{A,B,C,X(2),X(6190),F1",F2"}}, where F1", F2" are the isotomic conjugates of PU(118) (the real foci of the Steiner inellipse)
X(39022) = X(2)-Ceva conjugate of X(3414)
X(39022) = X(36953)-Ceva conjugate of X(39023)
X(39022) = X(31644)-cross conjugate of X(39023)
X(39022) = barycentric square of X(3414)
X(39022) = centroid of (degenerate) pedal triangle of X(1379)
X(39022) = trilinear quotient X(1109)/X(39023)
X(39022) = barycentric quotient X(115)/X(39023)
X(39022) = {X(i),X(j)}-harmonic conjugate of X(39023) for these {i,j}: {6, 230}, {141, 325}, {395, 396}, {597, 22329}, {599, 22110}
X(39022) = insimilicenter of pedal circles of X(13) and X(14) (which are also the pedal circles of X(15) and X(16))

X(39023) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(6)X(5639)

Trilinears    (sin B) (e cos C - cos(C + ω)) + (sin C) (e cos B - cos(B + ω)) : :
Barycentrics    2 a^2 - b^2 - c^2 - 2 Sqrt[a^4 + b^4 + c^4 - a^2 b^2 - a^2 c^2 - b^2 c^2] : :
Barycentrics    1/((SA^2 - SB SC - SA Sqrt[SA^2 + SB^2 + SC^2 - S^2]))^2 : :
Barycentrics    SB^2 + SC^2 - (SB + SC) (Sqrt(SW^2 - 3 S^2) + SA) : :
Barycentrics    (csc B) (e cos B + cos(B + ω)) + (csc C) (e cos C + cos(C + ω)) : :
Barycentrics    (csc A) (sin A - e sin(A + ω) - sin(A + 2ω)) : :
Barycentrics    (sin^2 A)/(e cos A - cos(A + ω))^2 : :

Let RA be the radical axis of the nine-point circles of P(119)BC and U(119)BC; define RB, RC cyclically. The lines RA, RB, RC concur in X(39023).

X(39023) lies on the Steiner inellipse, the ellipse described at X(6784), and these lines: {2, 6}, {5, 3558}, {30, 1380}, {32, 6177}, {115, 2028}, {140, 14630}, {376, 35914}, {523, 13636}, {549, 1340}, {1341, 15048}, {1349, 7818}, {1503, 6039}, {2482, 3414}, {5305, 14631}, {6178, 7888}, {7668, 39068}, {7794, 14632}, {7801, 19659}

X(39023) = reflection of X(39022) in X(2)
X(39023) = isotomic conjugate of isogonal conjugate of X(2028)
X(39023) = complement of X(6190)
X(39023) = complementary conjugate of complement of X(5638)
X(39023) = crosssum of circumcircle-intercepts of line X(6)X(5639)
X(39023) = crosssum of X(6) and X(1379)
X(39023) = crosspoint of X(2) and X(3413)
X(39023) = crossdifference of every pair of points on line X(512)X(1379)
X(39023) = center of hyperbola {{A,B,C,X(2),X(6189),F1",F2"}}, where F1", F2" are the isotomic conjugates of PU(119) (the imaginary foci of the Steiner inellipse)
X(39023) = X(2)-Ceva conjugate of X(3413)
X(39023) = X(36953)-Ceva conjugate of X(39022)
X(39023) = X(31644)-cross conjugate of X(39022)
X(39023) = barycentric square of X(3413)
X(39023) = centroid of (degenerate) pedal triangle of X(1380)
X(39023) = trilinear quotient X(1109)/X(39022)
X(39023) = barycentric quotient X(115)/X(39022)
X(39023) = {X(i),X(j)}-harmonic conjugate of X(39022) for these {i,j}: {6, 230}, {141, 325}, {395, 396}, {597, 22329}, {599, 22110}
X(39023) = exsimilicenter of pedal circles of X(13) and X(14) (which are also the pedal circles of X(15) and X(16))

X(39024) = CROSSSUM OF X(39022) AND X(39023)

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

X(39024) lies on these lines: {2, 2987}, {6, 110}, {23, 1692}, {32, 11002}, {39, 2981}, {51, 251}, {99, 25047}, {112, 1112}, {115, 3448}, {125, 6792}, {247, 6794}, {323, 1570}, {371, 7599}, {372, 7598}, {576, 9463}, {694, 3108}, {1084, 23584}, {1180, 5422}, {1194, 34545}, {1506, 7605}, {1627, 3060}, {1648, 15059}, {1691, 15107}, {2030, 13192}, {3231, 5111}, {3767, 37644}, {3981, 5012}, {4558, 7600}, {4576, 10754}, {5032, 10554}, {5033, 7492}, {5052, 5354}, {5106, 7772}, {5116, 20859}, {5182, 10330}, {5191, 30435}, {5359, 9777}, {5477, 14683}, {5622, 38356}, {5943, 34945}, {5972, 6791}, {6034, 9140}, {6531, 35360}, {7708, 22112}, {8627, 35006}, {9131, 11596}, {9138, 14398}, {9155, 9605}, {9544, 34481}, {9759, 14848}, {10601, 36790}, {11647, 13857}, {13207, 33875}, {16187, 22111}, {23589, 23991}

X(39024) = isogonal conjugate of X(36953)
X(39024) = polar conjugate of isotomic conjugate of X(14060)
X(39024) = crosssum of X(i) and X(j) for these {i,j}: {1648, 2482}, {39022, 39023}
X(39024) = crossdifference of every pair of points on line X(690)X(24981) (the radical axis of the antipedal circles of X(13) and X(14))
X(39024) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 36953}, {63, 14052}, {662, 36955}
X(39024) = trilinear product X(i)*X(j) for these {i,j}: {19, 14060}, {31, 14061}, {661, 33803}, {669, 33809}, {798, 33799}, {1101, 31644}
X(39024) = trilinear quotient X(i)/X(j) for these (i,j): (1, 36953), (19, 14052), (661, 36955), (14060, 63), (14061, 75), (31644, 1109), (33799, 799), (33803, 662), (33809, 670)
X(39024) = barycentric product X(i)*X(j) for these {i,j}: {4, 14060}, {6, 14061}, {249, 31644}, {250, 34953}, {512, 33799}, {523, 33803}, {798, 33809}
X(39024) = barycentric quotient X(i)/X(j) for these (i,j): (6, 36953), (25, 14052), (512, 36955), (14060, 69), (14061, 76), (31644, 338), (33799, 670), (33803, 99), (33809, 4602), (34953, 339)

X(39025) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF LINE X(7)X(8)

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

X(39025) lies on the bicevian ellipse of X(2) and X(55) (with center X(6690)), the bicevian ellipse of X(2) and X(56) (with center X(6691)), the rectangular bicevian hyperbola of X(2) and X(105) (with center X(6714)), and these lines: {3271, 20975}, {7083, 8069}

X(39025) = anticomplement of isotomic conjugate of X(3796)
X(39025) = crosssum of circumcircle-intercepts of line X(7)X(8)
X(39025) = center of hyperbola {{A,B,C,X(55),X(56)}}
X(39025) = X(2)-Ceva conjugate of X(3063)

X(39026) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF FEUERBACH TANGENT LINE

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

X(39026) lies on the bicevian ellipse of X(2) and X(59), the rectangular bicevian hyperbola of X(2) and X(901) (with center X(22102)), and these lines: {58, 33649}, {59, 1459}, {100, 656}, {101, 6586}, {109, 513}, {238, 1737}, {517, 1279}, {521, 3939}, {595, 6788}, {653, 7012}, {663, 1618}, {692, 2605}, {765, 1331}, {1633, 6615}, {1757, 6149}, {1936, 3011}, {2245, 17735}, {2690, 6577}, {8578, 14887}

X(39026) = complement of isotomic conjugate of X(150)
X(39026) = complementary conjugate of complement of X(20999)
X(39026) = crosssum of circumcircle-intercepts of Feuerbach tangent line (or PU(121), line X(11)X(244))
X(39026) = center of hyperbola {{A,B,C,X(59),X(677)}}
X(39026) = X(2)-Ceva conjugate of X(101)
X(39026) = trilinear product X(101)*X(16560)
X(39026) = trilinear quotient X(16569)/X(514)
X(39026) = barycentric product X(101)*X(150)
X(39026) = barycentric quotient X(150)/X(3261)

X(39027) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(7)

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

X(39027) lies on the bicevian conic of X(2) and X(54) (with center X(6689)), the rectangular bicevian hyperbola of X(2) and X(1296), and these lines: {95, 8901}, {574, 3981}

X(39027) = crosssum of circumcircle-intercepts of PU(7) (line X(5)X(1499))
X(39027) = center of conic {{A,B,C,X(54),X(1296)}}
X(39027) = X(2)-Ceva conjugate of X(9225)

X(39028) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(9)

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

X(39028) lies on these lines: {2, 3121}, {10, 274}, {75, 3789}, {120, 325}, {350, 740}, {538, 13466}, {874, 4368}, {1015, 36812}, {1211, 18037}, {1655, 2276}, {1966, 2238}, {3452, 7018}, {8299, 39044}, {9263, 31997}, {14434, 21297}, {16589, 27020}, {30571, 32020}

X(39028) = complement of X(39925)
X(39028) = complement of isogonal conjugate of X(21788)
X(39028) = complement of isotomic conjugate of X(17759)
X(39028) = complementary conjugate of complement of X(21788)
X(39028) = crosssum of circumcircle-intercepts of PU(9) (line X(213)X(667))
X(39028) = center of hyperbola {{A,B,C,X(274),X(668),X(874),PU(10)}}
X(39028) = X(2)-Ceva conjugate of X(350)
X(39028) = X(i)-isoconjugate of X(j) for these {i,j}: {741, 2107}, {1911, 2665}
X(39028) = trilinear product X(i)*X(j) for these {i,j}: {190, 27854}, {239, 17759}, {350, 2664}, {740, 2669}, {1921, 21788}, {2106, 3948}
X(39028) = trilinear quotient X(i)/X(j) for these (i,j): (350, 2665), (740, 2107), (2106, 18268), (2664, 1911), (2669, 741), (17759, 292), (21788, 1922), (27854, 649)
X(39028) = barycentric product X(i)*X(j) for these {i,j}: {350, 17759}, {668, 27854}, {1921, 2664}, {2106, 35544}, {2669, 3948}, {18891, 21788}
X(39028) = barycentric quotient X(i)/X(j) for these (i,j): (239, 2665), (2106, 741), (2238, 2107), (2664, 292), (2669, 37128), (17759, 291), (21788, 1911), (27854, 513)

X(39029) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(10)

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

X(39029) lies on the bicevian ellipse of X(2) and X(58) (with center X(6693)), the rectangular bicevian hyperbola of X(2) and X(101) (with center X(6710)), and these lines: {81, 105}, {172, 1913}, {2140, 11263}, {2503, 2653}, {4974, 7193}

X(39029) = complement of isogonal conjugate of X(20872)
X(39029) = complement of isotomic conjugate of X(20553)
X(39029) = complementary conjugate of complement of X(20872)
X(39029) = crosssum of circumcircle-intercepts of PU(10) (line X(10)X(514))
X(39029) = center of hyperbola {{A,B,C,X(58),X(101),PU(9)}}
X(39029) = X(2)-Ceva conjugate of X(1914)
X(39029) = trilinear product X(i)*X(j) for these {i,j}: {238, 20872}, {1914, 20602}, {2210, 20553}, {14599, 20643}
X(39029) = trilinear quotient X(i)/X(j) for these (i,j): (20553, 334), (20602, 335), (20643, 18895), (20872, 291)
X(39029) = barycentric product X(i)*X(j) for these {i,j}: {238, 20602}, {239, 20872}, {1914, 20553}, {2210, 20643}
X(39029) = barycentric quotient X(i)/X(j) for these (i,j): (20553, 18895), (20602, 334), (20872, 335)

X(39030) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(13)

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

X(39030) lies on the line {1925, 9238}

X(39030) = crosssum of circumcircle-intercepts of PU(13) (line X(1923)X(1924))
X(39030) = center of conic {{A,B,C,X(4602),PU(14)}}
X(39030) = X(2)-Ceva conjugate of X(1926)

X(39031) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(14)

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

X(39031) lies on the line {1932, 9235}

X(39031) = crosssum of circumcircle-intercepts of PU(14) (line X(1577)X(1930))
X(39031) = center of conic {{A,B,C,X(163),PU(13)}}
X(39031) = X(2)-Ceva conjugate of X(1933)

X(39032) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(15)

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

X(39032) lies on the bicevian ellipse of X(2) and X(21) (with center X(6675)) and these lines: {101, 226}, {243, 2202}, {284, 20262}, {1935, 9241}

X(39032) = crosssum of circumcircle-intercepts of PU(15) (line X(65)X(650))
X(39032) = center of hyperbola {A,B,C,X(21),X(651),PU(16)}
X(39032) = X(2)-Ceva conjugate of X(1936)

X(39033) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(16)

Barycentrics    (a^2 (a^2 - b^2 - c^2)^2 - b c (b^2 - c^2 - a^2) (c^2 - a^2 - b^2)) (a^9 - a^7 (b^2 + b c + c^2) - a^6 (b + c) (3 b^2 - 5 b c + 3 c^2) + a^5 (b + c)^2 (b^2 - b c + c^2) + a^4 (b - c)^2 (b + c) (5 b^2 + b c + 5 c^2) - a^3 (b^2 - c^2)^2 (3 b^2 - b c + 3 c^2) - a^2 (b - c)^2 (b + c) (b^4 - b^3 c + 4 b^2 c^2 - b c^3 + c^4) + a (b^2 - c^2)^2 (2 b^4 - b^3 c + 2 b^2 c^2 - b c^3 + 2 c^4) - (b - c)^4 (b + c)^3 (b^2 + b c + c^2))/(a^2 - b^2 - c^2) : :
Barycentrics    (tan A) (cos^2 A - cos B cos C) ((tan A) (cos^2 A - cos B cos C) - (tan B) (cos^2 B - cos C cos A) - (tan C) (cos^2 C - cos A cos B)) : :

X(39033) lies on the bicevian conic of X(2) and X(29) and these lines: {57, 18679}, {1939, 1940}

X(39033) = crosssum of circumcircle-intercepts of PU(16) (line X(73)X(652))
X(39033) = center of conic {{A,B,C,X(29),X(653),PU(15)}}
X(39033) = X(2)-Ceva conjugate of X(243)

X(39034) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(17)

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

X(39034) lies on these lines: {1941, 9243}, {14165, 15595}

X(39034) = crosssum of circumcircle-intercepts of PU(17) (line X(185)X(647))
X(39034) = center of conic {A,B,C,X(648),X(1105),PU(17)}
X(39034) = X(2)-Ceva conjugate of X(450)

X(39035) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(18)

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

X(39035) lies on these lines: {2, 9317}, {239, 16586}, {333, 664}, {524, 35086}, {1010, 25536}, {1813, 7176}, {1944, 5088}, {2785, 5075}, {4511, 39046}, {6505, 17778}, {17044, 17056}, {35102, 36905}

X(39035) = complement of X(17947)
X(39035) = crosssum of circumcircle-intercepts of PU(18) (line X(663)X(1400))
X(39035) = center of conic {{A,B,C,X(333),X(664)}}
X(39035) = X(2)-Ceva conjugate of X(1944)
X(39035) = X(i)-isoconjugate of X(j) for these {i,j}: {1937, 17963}, {1945, 2648}
X(39035) = trilinear product X(i)*X(j) for these {i,j}: {1758, 1944}, {1936, 17950}
X(39035) = trilinear quotient X(i)/X(j) for these (i,j): (1758, 1945), (1936, 17963), (1944, 2648), (17950, 1937)
X(39035) = barycentric product X(1944)*X(17950)
X(39035) = barycentric quotient X(i)/X(j) for these (i,j): (1758, 1937), (1936, 2648), (1944, 17947), (1951, 17963), (17950, 1952), (17966, 1945)

X(39036) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(19)

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

X(39036) lies on these lines: {226, 1947}, {1948, 8680}

X(39036) = crosssum of circumcircle-intercepts of PU(19) (line X(1409)X(1946))
X(39036) = center of conic {{A,B,C,X(18026),X(31623),PU(20)}}
X(39036) = X(2)-Ceva conjugate of X(1948)
X(39036) = X(1949)-isoconjugate of X(2656)
X(39036) = trilinear product X(1948)*X(2655)
X(39036) = trilinear quotient X(i)/X(j) for these (i,j): (1948, 2656), (2655, 1949)
X(39036) = barycentric quotient X(243)/X(2656)

X(39037) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(20)

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

X(39037) lies on the rectangular bicevian hyperbola of X(2) and X(109) (with center X(6718)) and the line {65, 692}

X(39037) = crosssum of circumcircle-intercepts of PU(20) (line X(226)X(522))
X(39037) = center of conic {{A,B,C,X(109),X(284),PU(19)}}
X(39037) = X(2)-Ceva conjugate of X(1951)

X(39038) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(21)

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

X(39038) lies on the line {2549, 24185}

X(39038) = crosssum of circumcircle-intercepts of PU(21) (line X(656)X(1953))
X(39038) = center of hyperbola {{A,B,C,X(162),X(2167)}}
X(39038) = X(2)-Ceva conjugate of X(1955)

X(39039) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(22)

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

X(39039) lies on the line {1957, 9257}

X(39039) = crosssum of circumcircle-intercepts of PU(22) (line X(48)X(656))
X(39039) = center of conic {{A,B,C,X(92),X(162),PU(23)}}
X(39039) = X(2)-Ceva conjugate of X(240)

X(39040) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(23)

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

X(39040) lies on these lines: {244, 5249}, {1086, 5949}, {1958, 9254}, {1959, 16591}, {4357, 4858}

X(39040) = complement of trilinear pole of line X(10)X(3907)
X(39040) = crosssum of circumcircle-intercepts of PU(23) (line X(31)X(661))
X(39040) = center of hyperbola {{A,B,C,X(75),X(662),PU(22)}}
X(39040) = X(2)-Ceva conjugate of X(1959)
X(39040) = X(1910)-isoconjugate of X(7095)
X(39040) = trilinear product X(1756)*X(1959)
X(39040) = trilinear quotient X(i)/X(j) for these (i,j): (1756, 1910), (1959, 7095)
X(39040) = barycentric product X(325)*X(1756)
X(39040) = barycentric quotient X(1756)/X(98)

X(39041) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(31)

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

X(39041) lies on the rectangular bicevian hyperbola of X(2) and X(100) (with center X(3035)) and these lines: {101, 3647}, {142, 16826}, {292, 2092}, {442, 20531}, {1961, 9280}, {3294, 13089}, {5053, 6594}, {6600, 21010}

X(39041) = complement of trilinear pole of line X(650)X(4500)
X(39041) = crosssum of circumcircle-intercepts of PU(31) (line X(513)X(1100))
X(39041) = center of conic {{A,B,C,X(100),X(1255),PU(32)}}
X(39041) = X(2)-Ceva conjugate of X(1757)

X(39042) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(32)

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

X(39042) lies on these lines: {86, 5949}, {662, 1100}, {5249, 19642}

X(39042) = complement of isogonal conjugate of X(20472)
X(39042) = complement of isotomic conjugate of X(20349)
X(39042) = complement of trilinear pole of line X(3743)X(14838)
X(39042) = crosssum of circumcircle-intercepts of PU(32) (line X(661)X(1962))
X(39042) = center of conic {{A,B,C,X(662),PU(31)}}
X(39042) = X(2)-Ceva conjugate of X(1931)
X(39042) = trilinear product X(i)*X(j) for these {i,j}: {1326, 20349}, {1931, 20369}, {17731, 20472}
X(39042) = trilinear quotient X(i)/X(j) for these (i,j): (20349, 11599), (20369, 9278), (20472, 2054)
X(39042) = barycentric product X(i)*X(j) for these {i,j}: {1326, 20450}, {1931, 20349}, {17731, 20369}
X(39042) = barycentric quotient X(20369)/X(11599)

X(39043) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(35)

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

X(39043) lies on the bicevian ellipse of X(2) and X(82), the rectangular bicevian hyperbola of X(2) and X(30670), and these lines: {1582, 9287}, {17755, 17941}

X(39043) = complement of trilinear pole of line X(9237)X(14838)
X(39043) = crosssum of circumcircle-intercepts of PU(35) (line X(38)X(661))
X(39043) = center of conic {A,B,C,X(82),X(662),PU(36)}
X(39043) = X(2)-Ceva conjugate of X(1580)

X(39044) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(36)

Barycentrics    b c (a^2 - b c)^2 : :

X(39044) lies on these lines: {1, 668}, {2, 292}, {9, 75}, {76, 16825}, {86, 2665}, {238, 1921}, {239, 350}, {244, 799}, {274, 1111}, {320, 38989}, {336, 34591}, {341, 17144}, {561, 748}, {614, 17149}, {659, 3766}, {756, 3112}, {872, 4360}, {874, 4432}, {1089, 4986}, {1909, 16823}, {1920, 17123}, {2112, 4586}, {3747, 35544}, {3762, 27939}, {3768, 27854}, {3802, 4368}, {3923, 10009}, {4011, 6382}, {4388, 30631}, {4441, 16816}, {4974, 30940}, {5224, 20542}, {5272, 6384}, {7018, 7261}, {7033, 27538}, {7035, 18822}, {8026, 30568}, {8299, 39028}, {9359, 10436}, {16815, 20331}, {17493, 30643}, {18863, 23962}, {20353, 30632}

X(39044) = isogonal conjugate of trilinear square of X(292)
X(39044) = isotomic conjugate of X(30663)
X(39044) = complement of X(30669)
X(39044) = crosssum of circumcircle-intercepts of PU(36) (line X(513)X(1834))
X(39044) = center of conic {{A,B,C,X(799),X(3112),PU(35)}}
X(39044) = X(2)-Ceva conjugate of X(1966)
X(39044) = trilinear square of X(239)
X(39044) = X(i)-isoconjugate of X(j) for these {i,j}: {31, 30663}, {291, 1911}, {292, 292}, {334, 14598}, {335, 1922}, {660, 875}, {813, 3572}, {876, 34067}, {893, 30657}, {1967, 18787}, {9468, 30669}, {18895, 18897}
X(39044) = trilinear product X(i)*X(j) for these {i,j}: {2, 4366}, {75, 8300}, {86, 4368}, {100, 27855}, {190, 4375}, {238, 350}, {239, 239}, {385, 17493}, {659, 874}, {812, 3570}, {1914, 1921}, {1966, 18786}, {2210, 18891}, {3573, 3766}, {8632, 27853}
X(39044) = trilinear quotient X(i)/X(j) for these (i,j): (75, 30663), (238, 1911), (239, 292), (350, 291), (659, 875), (812, 3572), (874, 660), (894, 30657), (1914, 1922), (1921, 335), (1966, 18787), (2210, 14598), (3570, 813), (3573, 34067), (3766, 876), (3978, 30669), (4366, 6), (4368, 42), (4375, 649), (8300, 31), (14599, 18897), (17493, 694), (18786, 1967), (18891, 334), (20769, 2196), (27853, 4562), (27855, 513)
X(39044) = barycentric product X(i)*X(j) for these {i,j}: {75, 4366}, {76, 8300}, {190, 27855}, {238, 1921}, {239, 350}, {274, 4368}, {659, 27853}, {668, 4375}, {812, 874}, {1914, 18891}, {1966, 17493}, {3570, 3766}, {3978, 18786}
X(39044) = barycentric quotient X(i)/X(j) for these (i,j): (2, 30663), (171, 30657), (238, 292), (239, 291), (242, 20769), (350, 335), (385, 18787), (659, 3572), (812, 876), (874, 4562), (1914, 1911), (1920, 30642), (1921, 334), (1966, 30669), (2210, 1922), (3570, 660), (3573, 813), (4366, 1), (4368, 37), (4375, 513), (7193, 2196), (8300, 6), (8632, 875), (14599, 14598), (17493, 1581), (18786, 694), (18891, 18895), (18892, 18897), (20769, 295), (27853, 4583), (27855, 514)

X(39045) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(38)

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

X(39045) lies on the bicevian conic of X(2) and X(54) (with center X(6689)), the rectangular bicevian hyperbola of X(2) and X(112) (with center X(6720)), and the line {98, 275}

X(39045) = complement of trilinear pole of line X(570)X(2485)
X(39045) = crosssum of circumcircle-intercepts of PU(38) (line X(5)X(525))
X(39045) = center of conic {{A,B,C,X(54),X(112)}}
X(39045) = X(2)-Ceva conjugate of X(1971)

X(39046) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(47)

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

X(39046) lies on the bicevian ellipse of X(1) and X(2) (with center X(1125)), the rectangular bicevian hyperbola of X(2) and X(101) (with center X(6710)), and these lines: {1, 2140}, {2, 4551}, {11, 3136}, {42, 244}, {73, 1212}, {663, 11712}, {664, 18031}, {672, 1362}, {851, 38981}, {899, 34590}, {1015, 1193}, {1201, 38986}, {1214, 34588}, {1457, 35110}, {1818, 4966}, {2238, 38980}, {2254, 8299}, {2646, 14714}, {3160, 10571}, {4368, 14432}, {4511, 39035}, {5400, 26102}, {6505, 34036}, {9310, 9438}, {14413, 34586}, {26031, 31330}

X(39046) = complement of isogonal conjugate of X(20470)
X(39046) = complement of isotomic conjugate of X(20347)
X(39046) = complement of trilinear pole of line X(37)X(522)
X(39046) = complementary conjugate of complement of X(20470)
X(39046) = crosssum of circumcircle-intercepts of PU(47) (line X(1)X(514))
X(39046) = crossdifference of every pair of points on line X(885)X(18785)
X(39046) = center of conic {{A,B,C,X(1),X(101),PU(93)}}
X(39046) = X(2)-Ceva conjugate of X(672)
X(39046) = trilinear product X(i)*X(j) for these {i,j}: {518, 20470}, {672, 20367}, {2223, 20347}, {5089, 20744}, {9454, 20448}
X(39046) = trilinear quotient X(i)/X(j) for these (i,j): (20347, 2481), (20367, 673), (20448, 18031), (20470, 105), (20744, 1814)
X(39046) = barycentric product X(i)*X(j) for these {i,j}: {518, 20367}, {672, 20347}, {1861, 20744}, {2223, 20448}, {2284, 20520}, {3912, 20470}
X(39046) = barycentric quotient X(i)/X(j) for these (i,j): (20347, 18031), (20367, 2481), (20470, 673), (20744, 31637)

X(39047) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(49)

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

X(39047) lies on these lines: {1, 18343}, {105, 9471}, {651, 36905}

X(39047) = crosssum of circumcircle-intercepts of PU(49) (line X(663)X(672))
X(39047) = center of conic {{A,B,C,X(664),X(673)}}
X(39047) = X(2)-Ceva conjugate of X(9318)

X(39048) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(56)

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

X(39048) lies on the bicevian ellipse of X(2) and X(57) (with center X(6692)), the rectangular bicevian hyperbola of X(2) and X(100) (with center X(3035)), and these lines: {1, 41}, {3, 24036}, {9, 1633}, {10, 30618}, {142, 3589}, {659, 3126}, {910, 6184}, {1145, 3234}, {1486, 4557}, {1743, 28017}, {2182, 10427}, {2348, 3008}, {2976, 8659}, {4222, 7719}, {11349, 20602}

X(39048) = complement of isotomic conjugate of X(32850)
X(39048) = complement of trilinear pole of line X(650)X(3752)
X(39048) = crosssum of circumcircle-intercepts of PU(56) (line X(9)X(513))
X(39048) = center of hyperbola {{A,B,C,X(57),X(100)}}
X(39048) = X(2)-Ceva conjugate of X(1279)
X(39048) = trilinear product X(100)*X(14425)
X(39048) = trilinear quotient X(14425)/X(513)
X(39048) = barycentric product X(i)*X(j) for these {i,j}: {190, 14425}, {1279, 32850}
X(39048) = barycentric quotient X(14425)/X(514)

X(39049) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(59)

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

X(39049) lies on the line {9363, 9434}

X(39049) = crosssum of circumcircle-intercepts of PU(59) (line X(650)X(3057))
X(39049) = center of conic {{A,B,C,X(651),X(1476),PU(92)}}
X(39049) = X(2)-Ceva conjugate of X(9364)

X(39050) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(60)

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

X(39050) lies on the bicevian ellipse of X(2) and X(8) (with center X(10)) and these lines: {6, 3756}, {9, 2968}, {11, 20262}, {226, 4904}, {497, 1146}, {4697, 37996}

X(39050) = complement of trilinear pole of line X(9)X(905)
X(39050) = crosssum of circumcircle-intercepts of PU(60) (line X(56)X(650))
X(39050) = center of hyperbola {{A,B,C,X(8),X(651)}}
X(39050) = X(2)-Ceva conjugate of X(9371)
X(39050) = trilinear product X(8074)*X(9371)
X(39050) = trilinear quotient X(8074)/X(9372)

X(39051) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(70)

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

X(39051) lies on the line {2606, 2615}

X(39051) = complement of trilinear pole of line X(3700)X(35466)
X(39051) = crosssum of circumcircle-intercepts of PU(70) (line X(2245)X(2605))
X(39051) = center of conic {{A,B,C,X(6742),X(24624),PU(71)}}
X(39051) = X(2)-Ceva conjugate of X(2607)

X(39052) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(75)

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

X(39052) lies on these lines: {162, 23964}, {2630, 2633}

X(39052) = crosssum of circumcircle-intercepts of PU(75) (line X(2631)X(2632))
X(39052) = center of conic {{A,B,C,X(5379),X(24000),PU(74)}}
X(39052) = X(2)-Ceva conjugate of X(162)

X(39053) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(77)

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

X(39053) lies on these lines: {101, 7128}, {653, 14837}, {905, 36118}, {1785, 7541}, {2639, 9393}

X(39053) = complement of trilinear pole of line X(15252)X(24030)
X(39053) = crosssum of circumcircle-intercepts of PU(77) (line X(2637)X(2638))
X(39053) = center of conic {{A,B,C,X(24032),PU(76)}}
X(39053) = X(2)-Ceva conjugate of X(653)

X(39054) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(79)

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

X(39054) lies on the rectangular bicevian hyperbola of X(2) and X(36066) and these lines: {238, 5127}, {249, 662}, {1414, 7178}, {1931, 2641}, {9218, 9508}

X(39054) = complement of isogonal conjugate of X(21004)
X(39054) = complement of isotomic conjugate of X(21221)
X(39054) = complement of trilinear pole of line X(4458)X(6370)
X(39054) = crosssum of circumcircle-intercepts of PU(79) (line X(2642)X(2643))
X(39054) = center of conic {{A,B,C,X(4567),X(4600),X(4622),X(36066),X(36085),PU(78)}}
X(39054) = X(2)-Ceva conjugate of X(662)
X(39054) = trilinear product X(i)*X(j) for these {i,j}: {99, 21004}, {110, 21221}, {163, 20951}, {662, 21381}
X(39054) = trilinear quotient X(i)/X(j) for these (i,j): (20951, 1577), (21004, 512), (21221, 523), (21381, 661)
X(39054) = barycentric product X(i)*X(j) for these {i,j}: {99, 21381}, {110, 20951}, {662, 21221}, {799, 21004}
X(39054) = barycentric quotient X(i)/X(j) for these (i,j): (20951, 850), (21004, 661), (21221, 1577), (21381, 523)

X(39055) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(80)

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

X(39055) lies on these lines: {150, 226}, {2647, 2653}

X(39055) = crosssum of circumcircle-intercepts of PU(80) (line X(650)X(2646))
X(39055) = center of conic {{A,B,C,X(651),PU(81)}}
X(39055) = X(2)-Ceva conjugate of X(1758)

X(39056) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(84)

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

X(39056) lies on the bicevian hyperbola of X(2) and X(190) (with center X(4422)) and these lines: {37, 24578}, {291, 9401}, {2663, 2670}, {2664, 21788}, {3508, 17755}, {6376, 34566}, {31336, 32009}

X(39056) = complement of trilinear pole of line X(514)X(25092)
X(39056) = crosssum of circumcircle-intercepts of PU(84) (line X(649)X(2666))
X(39056) = center of conic {{A,B,C,X(190),PU(85)}}
X(39056) = X(2)-Ceva conjugate of X(2664)
X(39056) = trilinear product X(2664)*X(24727)
X(39056) = trilinear quotient X(24727)/X(2665)
X(39056) = barycentric product X(17759)*X(24727)

X(39057) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(85)

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

X(39057) lies on these lines: {1107, 18827}, {2666, 2668}, {17669, 30966}, {17731, 39044}

X(39057) = crosssum of circumcircle-intercepts of PU(85) (line X(798)X(2667))
X(39057) = center of conic {{A,B,C,X(799),PU(84)}}
X(39057) = X(2)-Ceva conjugate of X(2669)

X(39058) = CROSSSUM OF PU(89)

Barycentrics    b^2 c^2 (a^8 (b^4 + 3 b^2 c^2 + c^4) - a^6 (b^2 + c^2) (2 b^4 + b^2 c^2 + 2 c^4) + a^4 (b^8 + b^6 c^2 + b^4 c^4 + b^2 c^6 + c^8) - a^2 b^2 c^2 (b^2 - c^2)^2 (b^2 + c^2) - b^4 c^4 (b^2 - c^2)^2)/(a^2 b^2 + a^2 c^2 - b^4 - c^4) : :
Barycentrics    csc A sec(A + ω) (csc A sec(A + ω) - csc B sec(B + ω) - csc C sec(C + ω)) : :

X(39058) lies on the rectangular bicevian hyperbola of X(2) and X(22456) and these lines: {6, 14382}, {290, 511}, {297, 694}, {325, 14941}

X(39058) = complement of isotomic conjugate of X(39355)
X(39058) = complement of trilinear pole of line X(6130)X(24284)
X(39058) = crosssum of PU(89)
X(39058) = crosspoint of PU(177)
X(39058) = center of hyperbola {{A,B,C,X(22456),PU(177)}}
X(39058) = X(2)-Ceva conjugate of X(290)

X(39059) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(92)

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

X(39059) lies on the bicevian hyperbola of X(2) and X(190) (with center X(4422)) and these lines: {37, 25531}, {3705, 15487}, {3977, 6651}, {5029, 6544}, {9367, 9369}

X(39059) = complement of isotomic conjugate of X(5211)
X(39059) = complement of trilinear pole of line X(514)X(17355)
X(39059) = crosssum of circumcircle-intercepts of PU(92) (line X(649)X(1201))
X(39059) = center of conic {{A,B,C,X(190),X(1222),PU(59)}}
X(39059) = X(2)-Ceva conjugate of X(5205)
X(39059) = barycentric product X(5205)*X(5211)

X(39060) = CROSSSUM OF PU(101)

Barycentrics    (a^8 (b^2 - 3 b c + c^2) - a^7 (b - 2 c) (2 b - c) (b + c) - a^6 (b^4 - 5 b^3 c + 9 b^2 c^2 - 5 b c^3 + c^4) + a^5 (b - c)^2 (b + c) (4 b^2 - b c + 4 c^2) - a^4 (b - c)^2 (b^4 + 3 b^3 c - b^2 c^2 + 3 b c^3 + c^4) - a^3 (b - c)^2 (b + c) (2 b^4 + b^3 c + 6 b^2 c^2 + b c^3 + 2 c^4) + a^2 (b^2 - c^2)^2 (b^4 - b^3 c + 5 b^2 c^2 - b c^3 + c^4) + a b c (b - c) (b^2 - c^2)^3 - b^2 c^2 (b - c)^4 (b + c)^2)/(a (b - c) (a - b - c) (a^2 - b^2 - c^2)) : :
Barycentrics    1/((sec A - sec B) (sec A - sec C)) - 1/(sec B - sec C)^2 : :

X(39060) lies on the line {521, 18026}

X(39060) = crosssum of PU(101)
X(39060) = X(2)-Ceva conjugate of X(18026)
X(39060) = trilinear product X(2636)*X(18026)

X(39061) = CROSSSUM OF PU(107)

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

X(39061) lies on these lines: {3, 8914}, {99, 35087}, {316, 524}, {543, 31998}, {897, 18201}, {3291, 8859}, {5159, 16092}, {8542, 14246}, {9165, 14061}, {9166, 23991}, {16093, 32479}, {17983, 38294}

X(39061) = complement of isotomic conjugate of X(8591)
X(39061) = complement of trilinear pole of line X(690)X(5461)
X(39061) = crosssum of PU(107)
X(39061) = crosspoint of PU(180)
X(39061) = center of hyperbola {{A,B,C,X(892),PU(180)}}
X(39061) = X(2)-Ceva conjugate of X(671)
X(39061) = trilinear product X(897)*X(8591)
X(39061) = trilinear quotient X(8591)/X(896)
X(39061) = barycentric product X(671)*X(8591)
X(39061) = barycentric quotient X(8591)/X(524)

X(39062) = CROSSSUM OF PU(109)

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

X(39062) lies on the rectangular bicevian hyperbola of X(2) and X(22456) and these lines: {2, 15351}, {30, 30716}, {107, 16229}, {264, 34360}, {525, 648}, {685, 3566}, {6528, 18314}, {9217, 14382}, {17120, 20068}, {17907, 30227}

X(39062) = complement of X(15351)
X(39062) = crosssum of PU(109)
X(39062) = center of hyperbola {{A,B,C,X(16077),X(18020),X(22456)}} (the circumconic through the polar conjugates of PU(40))
X(39062) = X(2)-Ceva conjugate of X(648)
X(39062) = X(i)-isoconjugate of X(j) for these {i,j}: {6, 9392}, {647, 9390}
X(39062) = trilinear product X(i)*X(j) for these {i,j}: {2, 2633}, {648, 2629}
X(39062) = trilinear quotient X(i)/X(j) for these (i,j): (2, 9392), (648, 9390), (2629, 647), (2633, 6)
X(39062) = barycentric product X(i)*X(j) for these {i,j}: {75, 2633}, {811, 2629}
X(39062) = barycentric quotient X(i)/X(j) for these (i,j): (1, 9392), (162, 9390), (2629, 656), (2633, 1)

X(39063) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(112)

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

X(39063) is the antipode of X(9) in the hyperbola described at X(36949).

X(39063) lies on the bicevian ellipse of X(2) and X(7) (with center X(142)), the rectangular bicevian hyperbola of X(2) and X(6183), and these lines: {2, 658}, {6, 7}, {9, 1020}, {11, 118}, {57, 30623}, {65, 20455}, {142, 1439}, {223, 1040}, {516, 1456}, {518, 1861}, {583, 1445}, {664, 20533}, {883, 4437}, {918, 16593}, {934, 17044}, {1122, 14524}, {1146, 4566}, {1211, 34846}, {1463, 38989}, {2283, 20776}, {3321, 3323}, {3738, 10427}, {4327, 38046}, {4569, 36796}, {5290, 5808}, {5452, 34048}, {6554, 14256}, {8287, 18635}, {10134, 13389}, {10135, 13388}, {15725, 31852}, {16595, 18591}, {30275, 36918}

X(39063) = reflection of X(9) in X(36949)
X(39063) = complement of X(36101)
X(39063) = crosssum of circumcircle-intercepts of PU(112) (line X(55)X(650))
X(39063) = center of hyperbola {{A,B,C,X(7),X(651),PU(46)}}
X(39063) = X(2)-Ceva conjugate of X(241)
X(39063) = X(i)-isoconjugate of X(j) for these {i,j}: {55, 9503}, {103, 294}, {677, 1024}, {911, 14942}, {2195, 36101}
X(39063) = trilinear product X(i)*X(j) for these {i,j}: {7, 9502}, {241, 516}, {676, 1025}, {910, 9436}, {1456, 3912}, {1458, 30807}, {1876, 26006}
X(39063) = trilinear quotient X(i)/X(j) for these (i,j): (7, 9503), (241, 103), (516, 294), (518, 2338), (676, 1024), (910, 2195), (1025, 677), (1456, 1438), (1458, 911), (2283, 36039), (9436, 36101), (9502, 55), (27818, 18025), (30807, 14942), (35517, 36796)
X(39063) = barycentric product X(i)*X(j) for these {i,j}: {85, 9502}, {241, 30807}, {516, 9436}, {676, 883}, {910, 27818}, {1456, 3263}, {1458, 35517}, {5236, 26006}
X(39063) = barycentric quotient X(i)/X(j) for these (i,j): (57, 9503), (241, 36101), (516, 14942), (672, 2338), (676, 885), (910, 294), (1456, 105), (1458, 103), (2283, 677), (9436, 18025), (9502, 9), (30807, 36796)

X(39064) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(113)

Barycentrics    a (a^2 + a b + a c + b^2 + 3 b c + c^2) (a^2 + a b + a c - b^2 - b c - c^2) (a^7 + 3 a^6 (b + c) + a^5 (4 b^2 + 11 b c + 4 c^2) + 2 a^4 (b^3 + 9 b^2 c + 9 b c^2 + c^3) - a^3 (2 b^4 - 5 b^3 c - 17 b^2 c^2 - 5 b c^3 + 2 c^4) - a^2 (b + c) (4 b^4 + 6 b^3 c - 3 b^2 c^2 + 6 b c^3 + 4 c^4) - a (3 b^6 + 9 b^5 c + 8 b^4 c^2 + 3 b^3 c^3 + 8 b^2 c^4 + 9 b c^5 + 3 c^6) - (b + c) (b^2 + b c + c^2) (b^4 + b^3 c - 2 b^2 c^2 + b c^3 + c^4))/(b + c) : :

X(39064) lies on the line {6156, 6160}

X(39064) = crosssum of circumcircle-intercepts of PU(113) (line X(4983)X(6155))
X(39064) = center of conic {{A,B,C,X(4596),PU(113)}}
X(39064) = X(2)-Ceva conjugate of X(6157)

X(39065) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(114)

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

X(39065) lies on the line {6162, 6166}

X(39065) = crosssum of circumcircle-intercepts of PU(114) (line X(2087)X(6161))
X(39065) = center of conic {{A,B,C,PU(114)}}
X(39065) = X(2)-Ceva conjugate of X(6163)

X(39066) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(115)

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

X(39066) lies on the line {6167, 6171}

X(39066) = crosssum of circumcircle-intercepts of PU(115) (line X(663)X(2082))
X(39066) = center of conic {{A,B,C,X(664),X(7131),X(9312),PU(115)}}
X(39066) = X(2)-Ceva conjugate of X(6168)

X(39067) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(116)

Barycentrics    (sin(A - ω)/(e cos A + cos(A + ω))) (sin(A - ω)/(e cos A + cos(A + ω)) - sin(B - ω)/(e cos B + cos(B + ω)) - sin(C - ω)/(e cos C + cos(C + ω))) : :
Barycentrics    (SB+SC)*(8*S^2+5*SA^2-2*SB*SC-3*SW^2+(SA-3*SW)*sqrt(SW^2-3*S^2)) : :

X(39067) lies on the bicevian ellipse of X(2) and X(6) (with center X(3589)), the rectangular bicevian hyperbola of X(2) and X(1379), and these lines: {2, 3557}, {1379, 15107}, {2029, 3124}, {5007, 8623}, {5638, 36213}, {7668, 39022}

X(39067) = complement of trilinear pole of line X(39)X(3413)
X(39067) = crosssum of circumcircle-intercepts of PU(116) (line X(2)X(1341))
X(39067) = center of hyperbola {{A,B,C,X(6),X(1379),X(6190),PU(118)}}
X(39067) = X(2)-Ceva conjugate of X(5639)
X(39067) = {X(8623),X(20965)}-harmonic conjugate of X(39068)

X(39068) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(117)

Barycentrics    (sin(A - ω)/(e cos A - cos(A + ω))) (sin(A - ω)/(e cos A - cos(A + ω)) - sin(B - ω)/(e cos B - cos(B + ω)) - sin(C - ω)/(e cos C - cos(C + ω))) : :
Barycentrics    (SB+SC)*(8*S^2+5*SA^2-2*SB*SC-3*SW^2-(SA-3*SW)*sqrt(SW^2-3*S^2)) : :

X(39068) lies on the bicevian ellipse of X(2) and X(6) (with center X(3589)), the rectangular bicevian hyperbola of X(2) and X(1380), and these lines: {2, 3558}, {1380, 15107}, {2028, 3124}, {5007, 8623}, {5639, 36213}, {7668, 39023}

X(39068) = complement of trilinear pole of line X(39)X(3414)
X(39068) = crosssum of circumcircle-intercepts of PU(117) (line X(2)X(1340))
X(39068) = center of hyperbola {{A,B,C,X(6),X(1380),X(6189),PU(119)}}
X(39068) = X(2)-Ceva conjugate of X(5638)
X(39068) = {X(8623),X(20965)}-harmonic conjugate of X(39067)

X(39069) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(128)

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

X(39069) lies on the bicevian conic of X(2) and X(19) and these lines: {}

X(39069) = complement of trilinear pole of line X(14838)X(16583)
X(39069) = crosssum of circumcircle-intercepts of PU(128) (line X(63)X(661))
X(39069) = center of conic {{A,B,C,X(19),X(662)}}
X(39069) = X(2)-Ceva conjugate of X(8772)

X(39070) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(129)

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

X(39070) lies on the bicevian conic of X(2) and X(19), the rectangular bicevian hyperbola of X(2) and X(109) (with center X(6718)), and these lines: {65, 17463}, {1195, 14936}

X(39070) = complement of trilinear pole of X(6589)X(16583)
X(39070) = crosssum of circumcircle-intercepts of PU(129) (line X(63)X(522))
X(39070) = center of conic {{A,B,C,X(19),X(109)}}
X(39070) = X(2)-Ceva conjugate of X(8776)

X(39071) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(131)

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

X(39071) lies on the bicevian conic of X(2) and X(3) (with center X(140)), the rectangular bicevian hyperbola of X(2) and X(112) (with center X(6720)), and these lines: {32, 9475}, {184, 2972}, {3162, 34131}, {13367, 35071}, {13558, 17810}

X(39071) = complement of trilinear pole of line X(216)X(2485)
X(39071) = crosssum of circumcircle-intercepts of PU(131) (line X(4)X(525))
X(39071) = center of conic {{A,B,C,X(3),X(112)}}
X(39071) = X(2)-Ceva conjugate of X(8779)

X(39072) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(132)

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

X(39072) lies on the bicevian conic of X(2) and X(25) (with center X(6677)), the rectangular bicevian hyperbola of X(2) and X(110) (with center X(5972)), and these lines: {5, 2794}, {6, 157}, {25, 32654}, {206, 34990}, {230, 460}, {1147, 2909}, {5027, 6132}, {10288, 32394}, {11326, 17423}, {31635, 35278}

X(39072) = complement of isogonal conjugate of X(37183)
X(39072) = complement of trilinear pole of line X(647)X(1196)
X(39072) = crosssum of circumcircle-intercepts of PU(132) (line X(69)X(523))
X(39072) = center of conic {{A,B,C,X(25),X(110)}}
X(39072) = X(2)-Ceva conjugate of X(1692)

X(39073) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(135)

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

X(39073) lies on the rectangular bicevian hyperbola of X(2) and X(2715) and these lines: {1692, 8779}, {2445, 7418}, {3569, 34146}, {11734, 38061}

X(39073) = complement of trilinear pole of line X(11672)X(23976)
X(39073) = crosssum of circumcircle-intercepts of PU(135) (line X(98)X(1297))
X(39073) = center of hyperbola {{A,B,C,X(511),X(1503),X(2715)}}
X(39073) = X(2)-Ceva conjugate of X(9475)

X(39074) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(137)

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

X(39074) lies on the bicevian ellipse of X(2) and X(39) (with center X(6683)), the rectangular bicevian hyperbola of X(2) and X(827), and the line {5007, 14990}

X(39074) = crosssum of circumcircle-intercepts of PU(137) (line X(83)X(826))
X(39074) = center of hyperbola {{A,B,C,X(39),X(827)}}
X(39074) = X(2)-Ceva conjugate of X(9482)

X(39075) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(138)

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

X(39075) lies on the bicevian ellipse of X(2) and X(6) (with center X(3589)), the rectangular bicevian hyperbola of X(2) and X(2709), and these lines: {3124, 5107}, {11186, 36213}

X(39075) = complement of trilinear pole of line X(39)X(11615)
X(39075) = crosssum of circumcircle-intercepts of PU(138) (line X(2)X(2793))
X(39075) = center of hyperbola {{A,B,C,X(6),X(2709),X(9124)}}
X(39075) = X(2)-Ceva conjugate of X(9486)

X(39076) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(141)

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

X(39076) lies on the bicevian ellipse of X(2) and X(39) (with center X(6683)) and these lines: {39, 9495}, {511, 39082}, {3934, 14990}

X(39076) = complement of antitomic conjugate of X(39)
X(39076) = crosssum of circumcircle-intercepts of PU(141) (line X(83)X(9494))
X(39076) = X(2)-Ceva conjugate of X(9496)

X(39077) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(142)

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

X(39077) lies on the rectangular bicevian hyperbola of X(2) and X(2736) and the line {971, 2254}

X(39077) = complement of trilinear pole of line X(6184)X(23972)
X(39077) = crosssum of circumcircle-intercepts of PU(142) (line X(103)X(105))
X(39077) = crosssum of X(103) and X(105)
X(39077) = crosspoint of X(516) and X(518)
X(39077) = center of hyperbola {{A,B,C,X(516),X(518),X(2736),X(36086)}}
X(39077) = X(2)-Ceva conjugate of X(9502)

X(39078) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(145)

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

X(39078) lies on the rectangular bicevian hyperbola of X(2) and X(98) (with center X(6036)), the rectangular bicevian hyperbola of X(2) and X(98) (with center X(6036)), and these lines: {6, 868}, {1249, 35922}, {2079, 17423}, {2395, 34156}, {3163, 36194}, {6795, 8429}, {14401, 34810}

X(39078) = complement of isotomic conjugate of X(36163)
X(39078) = complement of trilinear pole of line X(230)X(525)
X(39078) = crosssum of circumcircle-intercepts of PU(145) (line X(511)X(647))
X(39078) = center of conic {{A,B,C,X(98),X(648)}}
X(39078) = X(2)-Ceva conjugate of X(1316)
X(39078) = barycentric product X(1316)*X(36163)

X(39079) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(147)

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

X(39079) lies on the bicevian ellipse of X(2) and X(6) (with center X(3589)) and these lines: {1691, 8623}, {2679, 3005}, {7668, 15449}, {15573, 39080}

X(39079) = complement of trilinear pole of line X(39)X(83)
X(39079) = crosssum of circumcircle-intercepts of PU(147) (line X(2)X(4048))
X(39079) = center of hyperbola {{A,B,C,X(6),X(1031),X(2076),X(34214),X(35511)}}
X(39079) = X(2)-Ceva conjugate of X(5113)
X(39079) = barycentric product X(6)*X(115)*X(141)*X(385)*X(7779)
X(39079) = barycentric product X(i)*X(j)*X(k) for these {i,j,k}: {39, 115, 8290}, {141, 2086, 7779}, {141, 3124, 8290}, {385, 826, 5113}, {512, 826, 8290}, {523, 3005, 8290}, {804, 826, 2076}, {826, 5027, 7779}

X(39080) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(148)

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

X(39080) lies on the bicevian ellipse of X(2) and X(6) (with center X(3589)), the rectangular bicevian hyperbola of X(2) and X(25424), and these lines: {2, 694}, {6, 670}, {39, 10161}, {69, 32746}, {126, 2679}, {141, 7668}, {239, 1966}, {287, 14382}, {524, 35073}, {698, 3229}, {706, 30736}, {732, 3978}, {804, 4107}, {1084, 3589}, {1107, 17755}, {1368, 2882}, {1613, 4563}, {1691, 16985}, {3618, 25054}, {5031, 21531}, {5103, 21536}, {6656, 37890}, {15573, 39079}, {20339, 20340}, {20965, 38996}, {32748, 35524}, {34021, 35068}

X(39080) = isotomic conjugate of trilinear pole of line X(804)X(881)
X(39080) = complement of X(694)
X(39080) = crosssum of circumcircle-intercepts of PU(148) (line X(2)X(669))
X(39080) = crossdifference of every pair of points on line X(694)X(5027)
X(39080) = center of hyperbola {{A,B,C,X(6),X(670),X(25424)}}
X(39080) = X(2)-Ceva conjugate of X(3229)
X(39080) = X(i)-isoconjugate of X(j) for these {i,j}: {699, 1581}, {1967, 3225}
X(39080) = trilinear product X(i)*X(j) for these {i,j}: {385, 2227}, {698, 1580}, {1933, 35524}, {1966, 3229}
X(39080) = trilinear quotient X(i)/X(j) for these (i,j): (698, 1581), (1580, 699), (1966, 3225), (2227, 694), (3229, 1967), (35524, 1934)
X(39080) = barycentric product X(i)*X(j) for these {i,j}: {385, 698}, {1691, 35524}, {1966, 2227}, {3229, 3978}
X(39080) = barycentric quotient X(i)/X(j) for these (i,j): (385, 3225), (698, 1916), (1691, 699), (2227, 1581), (3229, 694), (35524, 18896)

X(39081) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(157)

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

X(39081) lies on the bicevian conic of X(2) and X(95) and these lines: {2, 35311}, {6, 17035}, {95, 216}, {233, 23583}, {287, 575}, {401, 32428}, {1463, 28180}, {3163, 36426}, {5965, 15595}, {6304, 36827}, {6709, 15526}

X(39081) = complement of trilinear pole of line X(140)X(525)
X(39081) = crosssum of circumcircle-intercepts of PU(157) (line X(51)X(647))
X(39081) = center of conic {{A,B,C,X(95),X(648)}}
X(39081) = X(2)-Ceva conjugate of X(401)

X(39082) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(159)

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

X(39082) lies on these lines: {2, 14946}, {511, 39076}, {3934, 37890}, {5976, 37891}

X(39082) = complement of X(14946)
X(39082) = crosssum of circumcircle-intercepts of PU(159) (line X(76)X(9494))

X(39083) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(162)

Barycentrics    a^2 (8 a^6 - 11 a^4 (b^2 + c^2) - 2 a^2 (b^4 - 9 c^2 b^2 + c^4) + 5 (b^2 - c^2)^2 (b^2 + c^2)) (4 a^8 - 8 a^6 (b^2 + c^2) + 14 a^4 b^2 c^2 + a^2 (b^2 + c^2) (8 b^4 - 17 b^2 c^2 + 8 c^4) - (b - c)^2 (b + c)^2 (4 b^4 + 5 b^2 c^2 + 4 c^4)) : :

X(39083) lies on the rectangular bicevian hyperbola of X(2) and X(110) (with center X(5972)) and these lines: {110, 20421}, {1147, 15020}, {4550, 15035}, {15034, 33556}

X(39083) = complement of isogonal conjugate of X(18571)
X(39083) = complement of trilinear pole of line X(647)X(18573)
X(39083) = crosssum of circumcircle-intercepts of PU(162) (line X(523)X(3830))
X(39083) = center of conic {{A,B,C,X(110),X(20421)}}

X(39084) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(163)

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

X(39084) lies on the rectangular bicevian hyperbola of X(2) and X(110) (with center X(5972)) and these lines: {6, 15051}, {110, 3532}, {2883, 37853}, {4550, 20773}, {5663, 33556}, {13416, 34472}, {15036, 33537}

X(39084) = complement of isogonal conjugate of X(37941)
X(39084) = crosssum of circumcircle-intercepts of PU(163) (line X(523)X(3146))
X(39084) = center of conic {{A,B,C,X(110),X(3532)}}
X(39084) = X(2)-Ceva conjugate of X(34569)

X(39085) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(169)

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

X(39085) lies on the rectangular bicevian hyperbola of X(2) and X(2715) and these lines: {232, 1692}, {248, 34137}, {1503, 1976}, {3564, 17974}, {6389, 34156}, {14355, 14912}, {15407, 36212}

X(39085) = complement of isogonal conjugate of X(19165)
X(39085) = crosssum of circumcircle-intercepts of PU(169) (line X(114)X(132))
X(39085) = center of conic {{A,B,C,X(685),X(2065),X(2715),X(15407),X(17932)}}
X(39085) = X(2)-Ceva conjugate of X(248)

X(39086) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(170)

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

X(39086) lies on the rectangular bicevian hyperbola of X(2) and X(112) (with center X(6720)) and these lines: {32, 13558}, {427, 1576}

X(39086) = crosssum of circumcircle-intercepts of PU(170) (line X(427)X(525))
X(39086) = center of conic {{A,B,C,X(112),X(1176)}}

X(39087) = CROSSDIFFERENCE OF PU(181)

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

X(39087) lies on these lines: {6, 694}, {99, 737}, {110, 707}, {688, 3050}, {1691, 21444}, {3124, 3329}, {3736, 17429}, {5017, 9431}, {5801, 35937}, {7766, 9998}, {8289, 11654}, {9427, 12212}, {19571, 23642}

X(39087) = crossdifference of every pair of points on PU(181) (line X(804)X(24256))
X(39087) = X(2)-Ceva conjugate of X(39088)

X(39088) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(181)

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

X(39088) lies on the rectangular bicevian hyperbola of X(2) and X(805) (with center X(22103)) and these lines: {}

X(39088) = crosssum of circumcircle-intercepts of PU(181) (line X(804)X(24256))
X(39088) = X(2)-Ceva conjugate of X(39087)

X(39089) = CROSSDIFFERENCE OF PU(182)

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

Let P be a point on the circumcircle. Let D and E be the contact points of tangents to the 2nd Brocard circle from P. The locus of the crossdifference of D and E as P varies is a conic centered at X(39089).

X(39089) lies on these lines: {2, 6}, {39, 34885}, {76, 10334}, {83, 736}, {98, 35377}, {538, 33694}, {698, 8290}, {1503, 11606}, {1691, 1916}, {1692, 4027}, {1976, 36897}, {2030, 14931}, {3095, 3406}, {3972, 13085}, {5989, 35006}, {7839, 32515}, {7878, 10345}, {8295, 35375}, {8569, 9468}, {9865, 10352}, {9983, 10349}, {10353, 12215}, {12054, 32476}, {20088, 37336}

X(39089) = crossdifference of PU(182)
X(39089) = X(2)-Ceva conjugate of X(39090)

X(39090) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(182)

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

X(39090) lies on the rectangular bicevian hyperbola of X(2) and X(99) (with center X(620)) and these lines: {39, 4027}, {5976, 10997}, {5989, 10335}

X(39090) = crosssum of circumcircle-intercepts of PU(182)
X(39090) = X(2)-Ceva conjugate of X(39089)

X(39091) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(183)

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

X(39091) lies on the rectangular bicevian hyperbola of X(2) and X(99) (with center X(620)) and these lines: {39, 8782}, {99, 6292}, {147, 14810}, {1125, 1281}, {2076, 7779}, {2482, 4027}, {6337, 7766}, {7710, 7897}, {7876, 20094}, {10353, 12212}, {11165, 33246}, {12215, 39094}, {13172, 37336}

X(39091) = complement of trilinear pole of line X(523)X(34573)
X(39091) = crosssum of circumcircle-intercepts of PU(183) (line X(512)X(5007))
X(39091) = X(2)-Ceva conjugate of X(7779)

X(39092) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(185)

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

X(39092) lies on the rectangular bicevian hyperbola of X(2) and X(805) (with center X(22103)) and these lines: {2, 14251}, {237, 805}, {290, 325}, {297, 17980}, {511, 694}, {626, 3493}, {881, 34290}, {1916, 22735}, {8623, 9468}, {9467, 36213}, {16068, 36212}

X(39092) = complement of isogonal conjugate of X(3511)
X(39092) = complement of isotomic conjugate of X(25332)
X(39092) = complement of trilinear pole of line X(804)X(2023)
X(39092) = crosssum of circumcircle-intercepts of PU(185) (line X(804)X(4107))
X(39092) = center of conic {{A,B,C,X(805),X(18829),X(36897)}}
X(39092) = X(2)-Ceva conjugate of X(694)
X(39092) = trilinear product X(1967)*X(25332)
X(39092) = trilinear quotient X(25332)/X(1966)
X(39092) = barycentric product X(694)*X(25332)
X(39092) = barycentric quotient X(25332)/X(3978)

X(39093) = CROSSDIFFERENCE OF PU(188)

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

X(39093) lies on these lines: {2, 6}, {76, 10796}, {99, 9301}, {316, 12188}, {511, 5989}, {732, 36849}, {736, 7754}, {1916, 15514}, {1975, 32515}, {2076, 4027}

X(39093) = crossdifference of PU(188)
X(39093) = X(2)-Ceva conjugate of X(39094)

X(39094) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(188)

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

X(39094) lies on the rectangular bicevian hyperbola of X(2) and X(99) (with center X(620)) and these lines: {5989, 15819}, {12215, 39091}

X(39094) = crosssum of circumcircle-intercepts of PU(188)
X(39094) = X(2)-Ceva conjugate of X(39093)

X(39095) = CROSSDIFFERENCE OF PU(189)

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

X(39095) lies on these lines: {2, 6}, {32, 262}, {83, 13357}, {98, 1692}, {232, 419}, {736, 7808}, {1691, 2023}, {1976, 9418}, {2021, 11676}, {2024, 10352}, {3094, 37455}, {3734, 13085}, {3767, 9744}, {5215, 5503}, {6036, 35377}, {7745, 37336}, {7786, 13356}, {7824, 34870}, {8569, 11672}, {9605, 32515}, {9751, 37512}, {9774, 11648}, {9888, 32456}, {12191, 13586}, {18907, 37345}, {38383, 39096}

X(39095) = crossdifference of PU(189)
X(39095) = X(2)-Ceva conjugate of X(39096)

X(39096) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(189)

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

X(39096) lies on the rectangular bicevian hyperbola of X(2) and X(99) (with center X(620)) and these lines: {39, 98}, {1503, 8290}, {2794, 6292}, {5976, 5999}, {5989, 6337}, {9478, 37336}, {9756, 10335}, {35374, 38227}

X(39096) = crosssum of circumcircle-intercepts of PU(189) (line X(512)X(5188))
X(39096) = X(2)-Ceva conjugate of X(39095)

X(39097) = CROSSDIFFERENCE OF PU(190)

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

X(39097) lies on these lines: {2, 6}, {384, 22521}, {511, 4027}, {538, 12191}, {736, 7760}, {1916, 5111}, {3407, 13330}, {5107, 14931}, {5989, 15514}, {8289, 8586}, {8290, 13196}, {8782, 12215}, {9301, 13586}, {9821, 10131}, {9865, 35388}, {10352, 35377}, {12188, 14041}, {13111, 14042}, {21445, 33004}

X(39097) = crossdifference of PU(190)
X(39097) = X(2)-Ceva conjugate of X(39098)

X(39098) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(190)

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

X(39098) lies on the rectangular bicevian hyperbola of X(2) and X(99) (with center X(620)) and the line {114, 10997}

X(39098) = crosssum of circumcircle-intercepts of PU(190)
X(39098) = X(2)-Ceva conjugate of X(39097)

X(39099) = CROSSDIFFERENCE OF PU(191)

Barycentrics    a^6 - 3 a^4 (b^2 + c^2) + a^2 (2 b^4 - b^2 c^2 + 2 c^4) + b^2 c^2 (b^2 + c^2) : :
Barycentrics    csc A (2 cos A - cos(A - 2ω)) : :

X(39099) lies on these lines: {2, 6}, {23, 35356}, {76, 576}, {99, 511}, {182, 7771}, {187, 5182}, {264, 8541}, {290, 892}, {308, 1232}, {311, 33769}, {316, 542}, {339, 18449}, {340, 5095}, {350, 8540}, {384, 13330}, {538, 5107}, {575, 1078}, {625, 7879}, {698, 15514}, {732, 5111}, {736, 5028}, {754, 5477}, {1003, 11173}, {1235, 8537}, {1351, 18906}, {1353, 14929}, {1799, 13366}, {1909, 19369}, {1975, 11477}, {2076, 13196}, {2080, 39100}, {2456, 12042}, {2854, 13207}, {3266, 9146}, {3284, 6394}, {3564, 5207}, {3734, 22486}, {3849, 8593}, {5026, 5104}, {5034, 15482}, {5052, 7804}, {5097, 14994}, {5112, 32220}, {5201, 35298}, {5641, 35138}, {5939, 5999}, {5969, 8586}, {6031, 11003}, {6390, 9301}, {6393, 34380}, {7370, 13492}, {7664, 15360}, {7750, 8550}, {7752, 34507}, {7773, 15069}, {7809, 19905}, {7813, 14645}, {8597, 9830}, {11161, 31173}, {11179, 14907}, {11180, 32827}, {11185, 20423}, {11416, 30737}, {11646, 14041}, {13857, 30786}, {14928, 19924}, {14984, 18322}, {15574, 32621}, {16276, 21969}, {31998, 32127}, {33651, 34984}, {35146, 35179}

X(39099) = isotomic conjugate of isogonal conjugate of X(2080)
X(39099) = isotomic conjugate of antigonal conjugate of X(262)
X(39099) = isotomic conjugate of trilinear pole of line X(523)X(3815)
X(39099) = anticomplement of X(15993)
X(39099) = crossdifference of every pair of points on PU(191) (line X(512)X(5052))
X(39099) = X(2)-Ceva conjugate of X(39100)
X(39099) = trilinear product X(75)*X(2080)
X(39099) = trilinear quotient X(2080)/X(31)
X(39099) = barycentric product X(76)*X(2080)
X(39099) = barycentric quotient X(2080)/X(6)

X(39100) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(191)

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

X(39100) lies on the rectangular bicevian hyperbola of X(2) and X(99) (with center X(620)) and these lines: {39, 10754}, {99, 15819}, {114, 316}, {2080, 39099}, {2482, 12215}, {7879, 15810}, {8290, 32459}, {35298, 38998}

X(39100) = crosssum of circumcircle-intercepts of PU(191) (line X(512)X(5052))
X(39100) = X(2)-Ceva conjugate of X(39099)

X(39101) = CROSSDIFFERENCE OF PU(192)

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

X(39101) lies on these lines: {2, 6}, {39, 10350}, {287, 36897}, {384, 3095}, {401, 38382}, {511, 10352}, {736, 7772}, {1570, 36849}, {1916, 12215}, {2456, 4027}, {3398, 7824}, {5103, 12830}, {5111, 5976}, {5989, 13196}, {6033, 14041}, {7470, 10131}, {7770, 32515}, {8356, 34682}, {9301, 35297}, {9468, 36212}, {10256, 33206}, {10353, 10997}, {12188, 33228}, {13111, 19687}, {13586, 35002}, {32458, 35388}

X(39101) = crossdifference of PU(192)
X(39101) = X(2)-Ceva conjugate of X(39102)

X(39102) = CROSSSUM OF CIRCUMCIRCLE-INTERCEPTS OF PU(192)

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

X(39102) lies on the rectangular bicevian hyperbola of X(2) and X(99) (with center X(620)) and these lines: {39, 36849}, {114, 35375}, {4027, 21445}, {10997, 39096}

X(39102) = crosssum of circumcircle-intercepts of PU(192)
X(39102) = X(2)-Ceva conjugate of X(39101)

leftri

Points on the self-dual permutation ellipse: X(39103)-X(39108)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, June 20, 2020.

Suppose that Γ is a conic. The dual of Γ is the conic consisting of points P = p : q : r such that the line px + qy + rz = 0 is tangent to Γ. See, for example, Paul Yiu's Introduction to the Geometry of the Triangle, p. 125.

The dual of a permutation ellipse (defined in the preamble just before X(34341)), is also a permutation ellipse. There exists a unique self-dual permutation ellipse, given by the equation

x^2 + y^2 + z^2 + 4(y z + z x + x y) = 0.

If U = u : v : w is a point on the Steiner circumellipse, then the point

D(U) = 3 u + (sqrt(2) - 1)(u + v + w) : :

is on the self-dual permutation ellipse. The appearance of (i,j) in the following list means that D(X(i)) = X(j): (99,39103), (190,39104), (671,39105), (903,39106), (6189,39107), (6190,39108).

X(39103) = D(X(99)) ON THE SELF-DUAL PERMUTATION ELLIPSE

Barycentrics    3*(a - b)*(a + b)*(a - c)*(a + c) + (-1 + Sqrt[2])*((a - b)*(a + b)*(a - c)*(a + c) + (-a + b)*(a + b)*(b - c)*(b + c) + (-a + c)*(a + c)*(-b + c)*(b + c)) : :

X(39103) lies on the cubic K398 and these lines: {2, 99}, {10992, 14783}, {14784, 38734}

X(39103) = reflection of X(39105) in X(2)
X(39103) = antipode of X(39105) in self-dual permutation ellipse

X(39104) = D(X(190)) ON THE SELF-DUAL PERMUTATION ELLIPSE

Barycentrics    3*(a - b)*(a - c) + (-1 + Sqrt[2])*((a - b)*(a - c) - (a - b)*(b - c) - (a - c)*(-b + c)) : :

X(39104) lies on the cubic K399 and this line: {2,45}

X(39104) = reflection of X(39106) in X(2)
X(39104) = antipode of X(39106) in self-dual permutation ellipse

X(39105) = D(X(671)) ON THE SELF-DUAL PERMUTATION ELLIPSE

Barycentrics    3*(a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2) + (-1 + Sqrt[2])*(-((a^2 + b^2 - 2*c^2)*(2*a^2 - b^2 - c^2)) + (a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2) - (2*a^2 - b^2 - c^2)*(a^2 - 2*b^2 + c^2)) : :

X(39105) lies on the cubic K398 and these lines: {2, 99}, {10992, 14782}, {14785, 38734}

X(39105) = reflection of X(39103) in X(2)
X(39105) = antipode of X(39103) in self-dual permutation ellipse

X(39106) = D(X(903)) ON THE SELF-DUAL PERMUTATION ELLIPSE

Barycentrics    3*(a + b - 2*c)*(a - 2*b + c) - (-1 + Sqrt[2])*((a + b - 2*c)*(2*a - b - c) - (a + b - 2*c)*(a - 2*b + c) + (2*a - b - c)*(a - 2*b + c)) : :

X(39106) lies on the cubic K399 and this line: {2, 45}

X(39106) = reflection of X(39104) in X(2)
X(39106) = antipode of X(39104) in self-dual permutation ellipse

X(39107) = D(X(6189)) ON THE SELF-DUAL PERMUTATION ELLIPSE

Barycentrics    3*(-1 + Sqrt[2])*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] + 3*(-2*a^2 + b^2 + c^2 + Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]) : :

X(39107) lies on the cubic K068 and this line: {2, 6}

X(39107) = reflection of X(39108) in X(2)
X(39107) = antipode of X(39108) in self-dual permutation ellipse

X(39108) = D(X(6190)) ON THE SELF-DUAL PERMUTATION ELLIPSE

Barycentrics    3*(-1 + Sqrt[2])*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] - 3*(-2*a^2 + b^2 + c^2 - Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]) : :

X(39108) lies on the cubic K068 and this line: {2, 6}

X(39108) = reflection of X(39107) in X(2)
X(39108) = antipode of X(39107) in self-dual permutation ellipse

X(39109) = X(3)X(34428)∩X(4)X(155)

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

X(39109) lies on the cubic K176 and these lines: {3, 34428}, {4, 155}, {5, 32132}, {6, 14593}, {22, 3563}, {25, 571}, {136, 394}, {1406, 1426}, {2971, 23606}, {6524, 8745}, {7494, 15517}, {8946, 10132}, {8948, 10133}, {11402, 14248}, {13557, 18534}

X(39109) = isogonal conjugate of the anticomplement of X(2165)
X(39109) = isogonal conjugate of the isotomic conjugate of X(254)
X(39109) = X(34756)-Ceva conjugate of X(6)
X(39109) = X(184)-cross conjugate of X(25)
X(39109) = X(i)-isoconjugate of X(j) for these (i,j): {3, 33808}, {63, 6515}, {69, 920}, {75, 155}, {92, 6503}, {304, 1609}, {326, 3542}, {8883, 18695}
X(39109) = cevapoint of X(2971) and X(3049)
X(39109) = crosssum of X(155) and X(6503)
X(39109) = trilinear pole of line {2489, 34952}
X(39109) = barycentric product of circumcircle-intercepts of Dou circle
X(39109) = barycentric product X(i)*X(j) for these {i,j}: {6, 254}, {19, 921}, {25, 6504}, {393, 15316}, {2165, 34756}, {2501, 13398}, {8745, 32132}, {8800, 8882}, {14910, 16172}
X(39109) = barycentric quotient X(i)/X(j) for these {i,j}: {19, 33808}, {25, 6515}, {32, 155}, {184, 6503}, {254, 76}, {921, 304}, {1973, 920}, {1974, 1609}, {2207, 3542}, {6504, 305}, {8800, 28706}, {13398, 4563}, {15316, 3926}, {34756, 7763}
X(39109) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 254, 8800}, {254, 34756, 15316}

X(39110) = X(4)X(14518)∩X(25)X(52)

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

X(39110) lies on the cubic K176 and these lines: {4, 14518}, {25, 52}, {68, 135}, {1092, 6754}

X(39110) = X(14518)-Ceva conjugate of X(34428)
X(39110) = X(184)-cross conjugate of X(571)
X(39110) = X(91)-isoconjugate of X(6193)
X(39110) = cevapoint of X(6754) and X(30451)
X(39110) = crosspoint of X(14517) and X(34756)
X(39110) = crosssum of X(8906) and X(34853)
X(39110) = barycentric product X(1993)*X(34428)
X(39110) = barycentric quotient X(i)/X(j) for these {i,j}: {571, 6193}, {34428, 5392}

X(39111) = X(3)X(68)∩X(4)X(14518)

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

X(39111) lies on the cubic K176 and these lines: {3, 68}, {4, 14518}, {6, 14593}, {96, 18925}, {155, 8906}, {184, 2165}, {925, 6515}, {1147, 32132}, {5392, 6776}, {23291, 37802}

X(39111) = orthic isogonal conjugate of X(2165)
X(39111) = X(4)-Ceva conjugate of X(2165)
X(39111) = barycentric product X(2165)*X(6193)
X(39111) = barycentric quotient X(6193)/X(7763)

X(39112) = X(184)-CROSS CONJUGATE OF X(155)

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

X(39112) lies on the cubic K176 and these lines: {4, 14517}, {155, 571}

X(39112) = X(184)-cross conjugate of X(155)
X(39112) = barycentric product X(14517)*X(34853)

X(39113) = ISOTOMIC CONJUGATE OF X(96)

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

X(39113) lies on the hyperbola {{A,B,C,X(5),X(6)}}|, the cubic K674, and these lines: {2, 6}, {4, 8905}, {5, 311}, {24, 317}, {52, 34835}, {53, 14570}, {76, 1238}, {95, 1166}, {99, 2383}, {114, 1843}, {264, 847}, {315, 7509}, {340, 10018}, {467, 14576}, {850, 38380}, {1232, 3933}, {1585, 13430}, {1586, 13441}, {1975, 7544}, {3001, 23333}, {3147, 32001}, {3260, 11585}, {3432, 7488}, {3926, 7401}, {3964, 6642}, {4558, 8882}, {6331, 8795}, {6337, 7487}, {6643, 32816}, {7393, 7776}, {7750, 37126}, {7771, 38434}, {7773, 20477}, {7814, 14615}, {15233, 34392}, {15234, 34391}, {18354, 32819}, {34336, 34338}

X(39113) = isotomic conjugate of X(96)
X(39113) = isotomic conjugate of the anticomplement of X(34835)
X(39113) = isotomic conjugate of the isogonal conjugate of X(52)
X(39113) = isotomic conjugate of the polar conjugate of X(467)
X(39113) = X(i)-Ceva conjugate of X(j) for these (i,j): {264, 311}, {34405, 5562}
X(39113) = X(i)-cross conjugate ofX(j) for these (i,j): {52, 467}, {34835, 2}
X(39113) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2168}, {31, 96}, {560, 34385}, {661, 32692}, {1820, 8882}, {2148, 2165}, {2169, 14593}, {2190, 2351}, {2616, 32734}, {2623, 36145}, {14573, 20571}
X(39113) = cevapoint of X(343) and X(8905)
X(39113) = crosspoint of X(264) and X(317)
X(39113) = crosssum of X(184) and X(2351)
X(39113) = barycentric product X(i)*X(j) for these {i,j}: {5, 7763}, {24, 28706}, {52, 76}, {69, 467}, {305, 14576}, {311, 1993}, {317, 343}, {324, 9723}, {561, 2180}, {1273, 18883}, {1748, 18695}, {6563, 14570}
X(39113) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2168}, {2, 96}, {5, 2165}, {24, 8882}, {47, 2148}, {52, 6}, {53, 14593}, {76, 34385}, {110, 32692}, {216, 2351}, {311, 5392}, {317, 275}, {324, 847}, {343, 68}, {467, 4}, {491, 16032}, {492, 16037}, {924, 2623}, {1147, 14533}, {1273, 37802}, {1625, 32734}, {1748, 2190}, {1993, 54}, {2180, 31}, {2617, 36145}, {3133, 571}, {5961, 11077}, {6563, 15412}, {7763, 95}, {9723, 97}, {11547, 8884}, {14213, 91}, {14570, 925}, {14576, 25}, {14918, 5962}, {18883, 1141}, {27362, 3767}, {28706, 20563}
X(39113) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {302, 303, 14389}, {311, 28706, 1225}, {317, 7763, 9723}, {491, 492, 1993}

X(39114) = X(4)X(155)∩X(97)X(13398)

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

X(39114) lies on the cubic K674 and these lines: {4, 155}, {97, 13398}

X(39114) = X(34835)-cross conjugate of X(317)
X(39114) = X(i)-isoconjugate of X(j) for these (i,j): {155, 2168}, {1820, 8883}, {2148, 34853}
X(39114) = barycentric product X(i)*X(j) for these {i,j}: {311, 34756}, {317, 8800}, {467, 6504}
X(39114) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 34853}, {24, 8883}, {52, 155}, {254, 96}, {467, 6515}, {8800, 68}, {14576, 1609}, {27361, 15242}, {34756, 54}

X(39115) = X(34853)-CROSS CONJUGATE OF X(6515)

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

X(39115) lies on the cubic K674 and this line: {4, 8905}

X(39115) = X(34853)-cross conjugate of X(6515)
X(39115) = barycentric quotient X(6515)/X(6193)

X(39116) = X(2)X(311)∩X(4)X(8906)

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

X(39116) lies on the cubic K555 and K674 and these lines: {2, 311}, {4, 8906}, {5, 847}, {25, 925}, {68, 5891}, {96, 7503}, {2052, 30450}, {3541, 32132}, {3542, 34853}, {5962, 18531}, {6997, 14593}

X(39116) = polar conjugate of X(34756)
X(39116) = polar conjugate of the isogonal conjugate of X(34853)
X(39116) = X(264)-Ceva conjugate of X(847)
X(39116) = X(i)-isoconjugate of X(j) for these (i,j): {48, 34756}, {254, 563}, {571, 921}
X(39116) = cevapoint of X(2165) and X(8906)
X(39116) = crosssum of X(30451) and X(39013)
X(39116) = barycentric product X(i)*X(j) for these {i,j}: {91, 33808}, {264, 34853}, {920, 20571}, {3542, 20563}, {5392, 6515}
X(39116) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 34756}, {68, 15316}, {91, 921}, {135, 34338}, {155, 1147}, {847, 254}, {920, 47}, {925, 13398}, {1609, 571}, {3542, 24}, {5392, 6504}, {6515, 1993}, {15242, 15317}, {27087, 12095}, {34853, 3}

X(39117) = X(2)X(254)∩X(311)X(8800)

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

X(39117) lies on the cubic K674 and these lines: {2, 254}, {311, 8800}

X(39117) = barycentric quotient X(8905)/X(6503)

X(39118) = ANTICOMPLEMENT OF X(5961)

Barycentrics    a^16 - 3*a^14*b^2 + 2*a^12*b^4 + a^10*b^6 - a^6*b^10 - 2*a^4*b^12 + 3*a^2*b^14 - b^16 - 3*a^14*c^2 + 5*a^12*b^2*c^2 + a^10*b^4*c^2 - 7*a^8*b^6*c^2 + 5*a^6*b^8*c^2 + 5*a^4*b^10*c^2 - 11*a^2*b^12*c^2 + 5*b^14*c^2 + 2*a^12*c^4 + a^10*b^2*c^4 - 2*a^8*b^4*c^4 - 6*a^4*b^8*c^4 + 15*a^2*b^10*c^4 - 10*b^12*c^4 + a^10*c^6 - 7*a^8*b^2*c^6 + 6*a^4*b^6*c^6 - 7*a^2*b^8*c^6 + 11*b^10*c^6 + 5*a^6*b^2*c^8 - 6*a^4*b^4*c^8 - 7*a^2*b^6*c^8 - 10*b^8*c^8 - a^6*c^10 + 5*a^4*b^2*c^10 + 15*a^2*b^4*c^10 + 11*b^6*c^10 - 2*a^4*c^12 - 11*a^2*b^2*c^12 - 10*b^4*c^12 + 3*a^2*c^14 + 5*b^2*c^14 - c^16 : :
X(39118) = 4 X[20304] - 3 X[34310]

X(39118) lies on these lines: {2, 5961}, {4, 110}, {5, 13558}, {20, 13496}, {127, 131}, {264, 18474}, {265, 338}, {925, 37444}, {2931, 35235}, {2970, 15133}, {3153, 14731}, {6033, 13556}, {6643, 34844}, {7574, 38583}, {10745, 18404}, {18420, 34840}, {20304, 34310}

X(39118) = reflection of X(i) in X(j) for these {i,j}: {4, 22823}, {20, 13496}, {13558, 5}
X(39118) = anticomplement of X(5961)
X(39118) = polar-circle inverse of X(12140)
X(39118) = circumcircle-of-anticomplementary-triangle-inverse of X(12383)
X(39118) = anticomplement of the isogonal conjugate of X(5962)
X(39118) = X(13558)-of-Johnson-triangle
X(39118) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {91, 3153}, {5962, 8}, {37802, 4329}

X(39119) = REFLECTION OF X(15478) IN X(5)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^14 - 4*a^12*b^2 + 6*a^10*b^4 - 3*a^8*b^6 - 3*a^6*b^8 + 6*a^4*b^10 - 4*a^2*b^12 + b^14 - 4*a^12*c^2 + 13*a^10*b^2*c^2 - 16*a^8*b^4*c^2 + 13*a^6*b^6*c^2 - 13*a^4*b^8*c^2 + 10*a^2*b^10*c^2 - 3*b^12*c^2 + 6*a^10*c^4 - 16*a^8*b^2*c^4 + 8*a^6*b^4*c^4 + 3*a^4*b^6*c^4 - 8*a^2*b^8*c^4 + 3*b^10*c^4 - 3*a^8*c^6 + 13*a^6*b^2*c^6 + 3*a^4*b^4*c^6 + 4*a^2*b^6*c^6 - b^8*c^6 - 3*a^6*c^8 - 13*a^4*b^2*c^8 - 8*a^2*b^4*c^8 - b^6*c^8 + 6*a^4*c^10 + 10*a^2*b^2*c^10 + 3*b^4*c^10 - 4*a^2*c^12 - 3*b^2*c^12 + c^14) : :

X(39119) lies on these lines: {4, 12825}, {5, 15478}, {25, 114}, {858, 13398}, {930, 1299}, {8906, 18569}, {18403, 21268}, {30771, 34843}

X(39119) = reflection of X(15478) in X(5)
X(39119) = X(15478)-of-Johnson-triangle

X(39120) = X(4)X(542)∩X(13)X(2903)

Barycentrics    a^12 - 3*a^10*b^2 + 3*a^8*b^4 - 3*a^4*b^8 + 3*a^2*b^10 - b^12 - 3*a^10*c^2 + 7*a^8*b^2*c^2 - 6*a^6*b^4*c^2 + 6*a^4*b^6*c^2 - 9*a^2*b^8*c^2 + 5*b^10*c^2 + 3*a^8*c^4 - 6*a^6*b^2*c^4 - 2*a^4*b^4*c^4 + 6*a^2*b^6*c^4 - 11*b^8*c^4 + 6*a^4*b^2*c^6 + 6*a^2*b^4*c^6 + 14*b^6*c^6 - 3*a^4*c^8 - 9*a^2*b^2*c^8 - 11*b^4*c^8 + 3*a^2*c^10 + 5*b^2*c^10 - c^12 : :
X(39120) = 3 X[671] + X[20774]

X(39120) lies on these lines: {4, 542}, {13, 2903}, {14, 2902}, {98, 7391}, {114, 6997}, {115, 577}, {147, 37349}, {148, 17035}, {543, 18420}, {1157, 14651}, {1370, 6055}, {2782, 11818}, {2790, 22515}, {3537, 38736}, {6036, 7386}, {6054, 7394}, {6321, 30258}, {6643, 20398}, {6815, 10992}, {7528, 14981}, {7544, 23235}, {11623, 14790}, {11632, 31723}, {18403, 38732}

X(39121) = X(2)X(1488)∩X(8)X(188)

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

X(39121) lies on the cubic K1077 and these lines: {2, 1488}, {8, 188}, {9, 258}, {236, 8056}, {8078, 12646}

X(39121) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 7028}, {5430, 21244}, {8078, 141}
X(39121) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 7028}, {1488, 188}
X(39121) = X(56)-isoconjugate of X(24158)
X(39121) = barycentric product X(i)*X(j) for these {i,j}: {1488, 5430}, {7048, 12646}
X(39121) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 24158}, {8078, 2089}, {12646, 7057}

X(39122) = X(57)X(558)∩X(174)X(13390)

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

X(39122) lies on the cubic K965 and these lines: {57, 558}, {174, 13390}, {178, 189}, {557, 15891}, {13389, 16664}

X(39122) = X(31)-complementary conjugate of X(558)
X(39122) = X(2)-Ceva conjugate of X(558)

X(39123) = ISOGONAL CONJUGATE OF X(7963)

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

X(39123) lies on the cubic K201 and these lines: {1, 24151}, {8, 24150}, {145, 8055}, {1420, 7963}, {1743, 2136}, {2137, 8056}

X(39123) = isogonal conjugate of X(7963)
X(39123) = X(3680)-cross conjugate of X(1)
X(39123) = X(i)-isoconjugate of X(j) for these (i,j): {1, 7963}, {56, 8834}
X(39123) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 7963}, {9, 8834}

X(39124) = X(8)X(24158)∩X(145)X(2089)

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

X(39124) lies on the cubic K201 and these lines: {8, 24158}, {145, 2089}, {173, 2136}, {188, 8834}, {7028, 24242}

X(39124) = X(3680)-cross conjugate of X(12644)
X(39124) = X(56)-isoconjugate of X(12643)
X(39124) = barycentric quotient X(9)/X(12643)

X(39125) = MIDPOINT OF X(206) AND X(34777)

Barycentrics    a^2*(2*a^6 - 3*a^4*b^2 - 2*a^2*b^4 + 3*b^6 - 3*a^4*c^2 + 6*a^2*b^2*c^2 - 3*b^4*c^2 - 2*a^2*c^4 - 3*b^2*c^4 + 3*c^6) : :
X(39125) = 11 X[6] - 3 X[154], 5 X[6] - X[159], 3 X[6] - X[206], 9 X[6] - X[9924], X[6] + 3 X[11216], 5 X[6] + 3 X[17813], 13 X[6] - 5 X[19132], 7 X[6] - 3 X[19153], 3 X[6] + X[34777], X[66] + 3 X[1992], X[141] - 3 X[10169], 15 X[154] - 11 X[159], 9 X[154] - 11 X[206], 27 X[154] - 11 X[9924], X[154] + 11 X[11216], 5 X[154] + 11 X[17813], 39 X[154] - 55 X[19132], 7 X[154] - 11 X[19153], 9 X[154] + 11 X[34777], 3 X[159] - 5 X[206], 9 X[159] - 5 X[9924], X[159] + 15 X[11216], X[159] + 3 X[17813], 13 X[159] - 25 X[19132], 7 X[159] - 15 X[19153], 3 X[159] + 5 X[34777], X[193] + 3 X[23327], 3 X[206] - X[9924], X[206] + 9 X[11216], 5 X[206] + 9 X[17813], 13 X[206] - 15 X[19132], 7 X[206] - 9 X[19153]

X(39125) lies on these lines: {6, 25}, {66, 1992}, {69, 11443}, {141, 10169}, {193, 23327}, {511, 11250}, {524, 6697}, {576, 34146}, {1351, 34778}, {1353, 20424}, {1503, 3853}, {3313, 11416}, {3564, 32155}, {3589, 12039}, {3629, 23300}, {5032, 5596}, {5041, 15270}, {5092, 37968}, {5093, 8549}, {5480, 22833}, {5965, 20300}, {6144, 32127}, {6776, 11458}, {8537, 19161}, {8584, 15583}, {9967, 32368}, {10250, 37517}, {10836, 39024}, {11232, 18381}, {11264, 18383}, {11482, 19149}, {14831, 17835}, {15520, 34117}, {15577, 34788}, {15826, 17710}, {18382, 23048}, {21850, 36201}

X(39125) = midpoint of X(i) and X(j) for these {i,j}: {206, 34777}, {3629, 23300}, {15577, 34788}
X(39125) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 8541, 9969}, {6, 12167, 19136}, {6, 17813, 159}, {6, 34777, 206}, {3629, 23326, 23300}

X(39126) = ISOTOMIC CONJUGATE OF X(3680)

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

X(39126) lies on these lines: {7, 8}, {37, 31225}, {56, 3685}, {57, 312}, {77, 4360}, {86, 7190}, {144, 20905}, {190, 1445}, {192, 241}, {226, 5233}, {239, 6180}, {269, 664}, {273, 903}, {278, 19796}, {279, 4452}, {307, 4389}, {309, 18816}, {314, 1434}, {321, 21454}, {344, 8732}, {347, 17078}, {348, 3672}, {497, 9801}, {536, 1418}, {651, 3759}, {740, 4334}, {894, 5228}, {982, 3663}, {1014, 30939}, {1088, 4373}, {1111, 4902}, {1215, 7274}, {1229, 4454}, {1266, 3668}, {1400, 20923}, {1407, 1999}, {1419, 16834}, {1423, 17755}, {1427, 3210}, {1442, 17393}, {1447, 30758}, {1471, 4676}, {1721, 14942}, {1788, 27549}, {2263, 32922}, {2481, 3062}, {3161, 5435}, {3263, 3598}, {3339, 4385}, {3600, 4673}, {3644, 4552}, {3673, 4862}, {3739, 26125}, {3758, 28968}, {3769, 9316}, {3886, 4321}, {3911, 25101}, {4008, 24231}, {4327, 5263}, {4328, 8583}, {4346, 17863}, {4355, 4647}, {4357, 8582}, {4398, 22464}, {4406, 30181}, {4440, 20171}, {4441, 7196}, {4569, 23062}, {4687, 17077}, {4848, 4899}, {4852, 6610}, {4887, 17861}, {5018, 32921}, {5088, 30283}, {5905, 20921}, {7176, 17144}, {7201, 30097}, {7269, 17394}, {7271, 9312}, {7365, 30699}, {8545, 17277}, {9282, 10001}, {9856, 17753}, {9965, 17862}, {10106, 34282}, {10444, 10860}, {10447, 35613}, {10521, 33937}, {11678, 20347}, {12848, 37788}, {16706, 28739}, {17086, 17301}, {17095, 17321}, {17234, 30379}, {17276, 17950}, {17298, 20881}, {17335, 29007}, {17336, 37787}, {17370, 28780}, {18227, 30946}, {18625, 19831}, {19788, 33146}, {20059, 30807}, {21059, 24409}, {26059, 31269}, {26567, 31300}, {37756, 37800}

X(39126) = isotomic conjugate of X(3680)
X(39126) = isotomic conjugate of the anticomplement of X(12640)
X(39126) = isotomic conjugate of the isogonal conjugate of X(1420)
X(39126) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1476, 329}, {3451, 144}, {6613, 20295}
X(39126) = X(i)-Ceva conjugate of X(j) for these (i,j): {1088, 85}, {7035, 664}, {18811, 75}
X(39126) = X(i)-cross conjugate of X(j) for these (i,j): {145, 18743}, {4848, 5435}, {4939, 4462}, {12640, 2}, {18743, 85}
X(39126) = X(i)-isoconjugate of X(j) for these (i,j): {9, 38266}, {31, 3680}, {32, 6557}, {41, 8056}, {55, 3445}, {200, 16945}, {650, 34080}, {657, 38828}, {663, 1293}, {667, 31343}, {1253, 19604}, {1397, 6556}, {1639, 32645}, {2175, 4373}, {3052, 33963}, {3063, 27834}, {4895, 36042}, {14827, 27818}
X(39126) = cevapoint of X(i) and X(j) for these (i,j): {7, 4452}, {57, 36846}, {145, 5435}, {4462, 4939}
X(39126) = crosspoint of X(4998) and X(6613)
X(39126) = trilinear pole of line {4462, 4521}
X(39126) = barycentric product X(i)*X(j) for these {i,j}: {7, 18743}, {75, 5435}, {76, 1420}, {85, 145}, {274, 4848}, {331, 4855}, {349, 16948}, {664, 4462}, {668, 30719}, {1088, 3161}, {1231, 4248}, {1275, 4939}, {1743, 6063}, {3052, 20567}, {3212, 27496}, {3667, 4554}, {4394, 4572}, {4404, 4573}, {4521, 4569}, {4625, 14321}, {4899, 34018}, {4925, 34085}, {6555, 23062}
X(39126) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3680}, {7, 8056}, {56, 38266}, {57, 3445}, {75, 6557}, {85, 4373}, {109, 34080}, {145, 9}, {190, 31343}, {279, 19604}, {312, 6556}, {651, 1293}, {664, 27834}, {934, 38828}, {1088, 27818}, {1407, 16945}, {1420, 6}, {1441, 4052}, {1743, 55}, {2403, 23838}, {3052, 41}, {3158, 220}, {3161, 200}, {3667, 650}, {3756, 2170}, {3950, 210}, {4162, 657}, {4248, 1172}, {4374, 27831}, {4394, 663}, {4404, 3700}, {4452, 24151}, {4462, 522}, {4487, 2325}, {4504, 3287}, {4521, 3900}, {4534, 2310}, {4546, 4130}, {4729, 3709}, {4848, 37}, {4849, 1334}, {4855, 219}, {4856, 3683}, {4881, 2323}, {4884, 33299}, {4891, 3691}, {4898, 3715}, {4899, 3693}, {4918, 21033}, {4936, 480}, {4939, 1146}, {4953, 3119}, {4998, 5382}, {5226, 10563}, {5435, 1}, {6049, 1743}, {6555, 728}, {6604, 27819}, {7056, 27832}, {8056, 33963}, {8643, 3063}, {14321, 4041}, {14425, 4895}, {15519, 4936}, {16948, 284}, {18743, 8}, {20818, 212}, {21950, 4516}, {23703, 2429}, {27496, 7155}, {30719, 513}, {30720, 4578}, {31182, 4162}, {31227, 1320}, {33628, 2194}
X(39126) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 75, 85}, {7, 3212, 1122}, {7, 32003, 21296}, {75, 320, 322}, {75, 16284, 20895}, {144, 20905, 30854}, {269, 3875, 664}, {7271, 17151, 9312}, {9965, 17862, 18750}, {20895, 21296, 16284}

X(39127) = X(6)X(6337)∩X(69)X(1368)

Barycentrics    (a^2 - b^2 - c^2)*(5*a^6 - 5*a^4*b^2 - 9*a^2*b^4 + b^6 - 5*a^4*c^2 + 14*a^2*b^2*c^2 + 3*b^4*c^2 - 9*a^2*c^4 + 3*b^2*c^4 + c^6) : :
X(39127) = 5 X[3618] - 4 X[6387]

X(39127) lies on the cubic K707 and these lines: {6, 6337}, {69, 1368}, {193, 19583}, {3618, 6387}, {5866, 37491}, {10765, 11008}

X(39127) = reflection of X(i) in X(j) for these {i,j}: {69, 6338}, {6339, 6}
X(39127) = isotomic conjugate of the polar conjugate of X(18287)
X(39127) = X(i)-Ceva conjugate of X(j) for these (i,j): {193, 69}, {19583, 6337}
X(39127) = X(14248)-isoconjugate of X(19214)
X(39127) = barycentric product X(69)*X(18287)
X(39127) = barycentric quotient X(18287)/X(4)

X(39128) = ISOGONAL CONJUGATE OF X(18287)

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

X(39128) lies on the cubic K707 and these lines: {69, 6342}, {193, 19583}, {1611, 19118}, {3053, 19588}, {8770, 15369}

X(39128) = isogonal conjugate of X(18287)
X(39128) = isogonal conjugate of the anticomplement of X(6340)
X(39128) = X(6391)-cross conjugate of X(6)
X(39128) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18287}, {193, 19213}
X(39128) = trilinear pole of line {2519, 8651}
X(39128) = barycentric product X(8769)*X(19214)
X(39128) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 18287}, {19214, 18156}, {38252, 19213}

X(39129) = ISOGONAL CONJUGATE OF X(8793)

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

X(39129) lies on these lines: {6, 14376}, {22, 69}, {66, 34427}, {141, 19595}, {1249, 18840}, {3313, 3933}, {18018, 23300}

X(39129) = reflection of X(19595) in X(141)
X(39129) = isogonal conjugate of X(8793)
X(39129) = X(1843)-cross conjugate of X(141)
X(39129) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8793}, {82, 159}, {251, 18596}, {3162, 34055}
X(39129) = crosssum of X(206) and X(20993)
X(39129) = barycentric product X(i)*X(j) for these {i,j}: {141, 13575}, {8024, 34207}
X(39129) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8793}, {38, 18596}, {39, 159}, {141, 1370}, {1843, 3162}, {1930, 21582}, {3665, 18629}, {3917, 23115}, {3933, 28419}, {13575, 83}, {34207, 251}

X(39130) = ISOGONAL CONJUGATE OF X(2360)

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

X(39130) = 5 X[3698] - 3 X[11214]

X(39130) lie on the cubic K033 and these lines: {1, 281}, {4, 2184}, {8, 20}, {10, 227}, {28, 26702}, {65, 7157}, {72, 1903}, {92, 946}, {268, 958}, {285, 3193}, {293, 5247}, {304, 309}, {306, 857}, {318, 6260}, {405, 7367}, {442, 13853}, {519, 2192}, {950, 7008}, {956, 1436}, {996, 1413}, {1422, 9623}, {1433, 10570}, {1440, 3160}, {1490, 7046}, {2167, 5882}, {2322, 2360}, {2349, 2816}, {2695, 6081}, {2968, 6245}, {3191, 21096}, {3341, 18391}, {3668, 6355}, {3698, 11214}, {4081, 12688}, {4300, 23529}, {4848, 38955}, {5271, 24580}, {5691, 33893}, {6350, 6684}, {6757, 12609}, {7090, 31528}, {8074, 13737}, {14121, 31529}, {15065, 21077}, {17102, 20205}, {21620, 37448}, {30032, 30059}

X(39130) = reflection of X(5930) in X(10)
X(39130) = isogonal conjugate of X(2360)
X(39130) = isotomic conjugate of X(8822)
X(39130) = isotomic conjugate of the anticomplement of X(1901)
X(39130) = isotomic conjugate of the isogonal conjugate of X(2357)
X(39130) = X(189)-Ceva conjugate of X(1903)
X(39130) = X(i)-cross conjugate of X(j) for these (i,j): {65, 10}, {1826, 226}, {1901, 2}, {1903, 8808}, {21933, 321}
X(39130) = crosspoint of X(i) and X(j) for these (i,j): {92, 253}, {189, 309}, {280, 7020}
X(39130) = crosssum of X(i) and X(j) for these (i,j): {48, 154}, {198, 2187}, {221, 7114}
X(39130) = trilinear pole of line {656, 3700}
X(39130) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2360}, {3, 3194}, {6, 1817}, {21, 221}, {28, 7078}, {29, 7114}, {31, 8822}, {34, 1819}, {40, 58}, {60, 227}, {81, 198}, {86, 2187}, {110, 6129}, {163, 14837}, {196, 2193}, {208, 283}, {223, 284}, {322, 2206}, {329, 1333}, {333, 2199}, {347, 2194}, {593, 21871}, {604, 27398}, {849, 21075}, {859, 15501}, {1014, 7074}, {1172, 7011}, {1408, 7080}, {1412, 2324}, {1437, 7952}, {1444, 3195}, {1576, 17896}, {1790, 2331}, {1812, 3209}, {2287, 6611}, {2299, 7013}, {4565, 14298}
X(39130) = barycentric product X(i)*X(j) for these {i,j}: {8, 8808}, {10, 189}, {37, 309}, {65, 34404}, {75, 1903}, {76, 2357}, {84, 321}, {226, 280}, {282, 1441}, {285, 6358}, {307, 7003}, {313, 1436}, {349, 2192}, {850, 36049}, {1043, 13853}, {1214, 7020}, {1231, 7008}, {1413, 30713}, {1422, 3701}, {1440, 2321}, {1577, 13138}, {2208, 27801}, {2322, 6355}, {2358, 3718}, {4086, 37141}, {7129, 20336}, {7157, 27398}, {20948, 32652}, {24005, 34413}
X(39130) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1817}, {2, 8822}, {6, 2360}, {8, 27398}, {10, 329}, {19, 3194}, {37, 40}, {42, 198}, {65, 223}, {71, 7078}, {73, 7011}, {84, 81}, {189, 86}, {210, 2324}, {213, 2187}, {219, 1819}, {225, 196}, {226, 347}, {268, 283}, {271, 1812}, {280, 333}, {282, 21}, {285, 2185}, {309, 274}, {321, 322}, {523, 14837}, {594, 21075}, {661, 6129}, {756, 21871}, {1042, 6611}, {1214, 7013}, {1334, 7074}, {1400, 221}, {1402, 2199}, {1409, 7114}, {1413, 1412}, {1422, 1014}, {1433, 1790}, {1436, 58}, {1440, 1434}, {1577, 17896}, {1824, 2331}, {1826, 7952}, {1880, 208}, {1903, 1}, {2171, 227}, {2188, 2193}, {2192, 284}, {2208, 1333}, {2250, 15501}, {2321, 7080}, {2333, 3195}, {2357, 6}, {2358, 34}, {3668, 14256}, {3700, 8058}, {4041, 14298}, {7003, 29}, {7008, 1172}, {7020, 31623}, {7118, 2194}, {7129, 28}, {7151, 1474}, {7154, 2299}, {7157, 8808}, {7367, 2328}, {8059, 4565}, {8808, 7}, {13138, 662}, {13156, 17169}, {13853, 3668}, {21044, 38357}, {21871, 1103}, {21933, 6260}, {32652, 163}, {34404, 314}, {36049, 110}, {37141, 1414}
X(39130) = {X(189),X(280)}-harmonic conjugate of X(84)

X(39131) = POINT PULCHERRIMA

Barycentrics    Sqrt[a*(b + c)] : :

X(39131) lies on the Jerabek circumhyperbola of the excentral triangle, the cubics K033, K345, K372, K720, K880, the curve Q066, and no lines X(i)X(j) for 1 ≤ i < j ≤ 39130.

X(39132) = SYMGONAL IMAGE OF X(13)

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

X(39132) lies on the cubic K060 and these lines: {5, 11119}, {13, 2380}, {14, 8014}, {30, 5674}, {265, 11117}, {622, 5627}, {1337, 11078}, {14451, 39133}, {34302, 39135}

X(39132) = antigonal image of X(616)
X(39132) = symgonal image of X(13)
X(39132) = X(6149)-isoconjugate of X(34296)
X(39132) = barycentric quotient X(i)/X(j) for these {i,j}: {616, 14922}, {1989, 34296}, {11084, 3440}

X(39133) = SYMGONAL IMAGE OF X(14)

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

X(39133) lies on the cubic K060 and these lines: {5, 11120}, {13, 8015}, {14, 2381}, {30, 5675}, {265, 11118}, {621, 5627}, {1338, 11092}, {14451, 39132}, {34302, 39134}

X(39133) = antigonal image of X(617)
X(39133) = symgonal image of X(14)
X(39133) = X(6149)-isoconjugate of X(34295)
X(39133) = barycentric quotient X(i)/X(j) for these {i,j}: {1989, 34295}, {11089, 3441}

X(39134) = ANTIGONAL IMAGE OF X(3479)

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

X(39134) lies on the cubics K060 and K066a and these lines: {5, 302}, {30, 8437}, {61, 18813}, {5318, 30215}, {6671, 15802}, {6761, 34296}, {11581, 38943}, {14372, 34304}, {34302, 39133}

X(39134) = antigonal image of X(3479)
X(39134) = X(i)-cross conjugate of X(j) for these (i,j): {4, 34296}, {36304, 396}
X(39134) = cevapoint of X(3489) and X(8437)
X(39134) = barycentric product X(396)*X(19712)
X(39134) = barycentric quotient X(i)/X(j) for these {i,j}: {396, 627}, {3489, 2981}

X(39135) = ANTIGONAL IMAGE OF X(3480)

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

X(39135) lies on the cubics K060 and K066b and these lines: {5, 303}, {30, 8438}, {62, 18814}, {5321, 30216}, {6672, 15778}, {6761, 34295}, {11582, 38944}, {14373, 34304}, {34302, 39132}

X(39135) = antigonal image of X(3480)
X(39135) = X(i)-cross conjugate of X(j) for these (i,j): {4, 34295}, {36305, 395}
X(39135) = cevapoint of X(3490) and X(8438)
X(39135) = barycentric product X(395)*X(19713)
X(39135) = barycentric quotient X(i)/X(j) for these {i,j}: {395, 628}, {3490, 6151}

X(39136) = X(79)X(476)∩X(80)X(758)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^5 - a^3*b^2 - a^2*b^3 + b^5 + a^2*b^2*c - 2*b^4*c - a^3*c^2 + a^2*b*c^2 + a*b^2*c^2 + b^3*c^2 - a^2*c^3 + b^2*c^3 - 2*b*c^4 + c^5) : :
X(39136) = 2 X[2222] - 3 X[34309]

X(39136) lies on the curve Q155 and these lines: on lines {79, 476}, {80, 758}, {226, 2222}, {1836, 36815}, {2607, 5520}, {2651, 5057}, {6003, 34789}, {17768, 34311}

X(39136) = reflection of X(80) in X(34172)

X(39137) = X(110)-CROSS CONJUGATE OF X(1)

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

X(39137) lies on these lines: {1, 2612}, {12, 2595}, {484, 4645}, {523, 2606}, {2614, 3737}, {3509, 4053}, {34997, 35550}

X(39137) = isogonal conjugate of X(21381)
X(39137) = isotomic conjugate of X(20951)
X(39137) = X(110)-cross conjugate of X(1)
X(39137) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21381}, {2, 21004}, {4, 22156}, {6, 21221}, {31, 20951}, {58, 21098}, {81, 21890}, {101, 21209}, {661, 39054}
X(39137) = cevapoint of X(523) and X(5954)
X(39137) = trilinear pole of line {2610, 21341}
X(39137) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 21221}, {2, 20951}, {6, 21381}, {31, 21004}, {37, 21098}, {42, 21890}, {48, 22156}, {110, 39054}, {513, 21209}

X(39138) = X(513)-CEVA CONJUGATE OF X(110)

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^10 - 2*a^8*b^2 + a^6*b^4 - a^4*b^6 + 2*a^2*b^8 - b^10 - 2*a^8*c^2 + 4*a^6*b^2*c^2 - a^4*b^4*c^2 - 4*a^2*b^6*c^2 + 2*b^8*c^2 + a^6*c^4 - a^4*b^2*c^4 + 5*a^2*b^4*c^4 - b^6*c^4 - a^4*c^6 - 4*a^2*b^2*c^6 - b^4*c^6 + 2*a^2*c^8 + 2*b^2*c^8 - c^10) : :
X(39138) = 2 X[5] - 3 X[34365]

X(39138) lies on the cubic K1071 and these lines: {1, 2612}, {3, 9218}, {5, 34365}, {20, 31990}, {32, 6792}, {512, 14366}, {2613, 2617}, {4226, 14884}, {6328, 36163}, {7785, 9514}, {13434, 18114}

X(39138) = reflection of X(14366) in the Brocard axis
X(39138) = circumcircle-inverse of X(9218)
X(39138) = X(523)-Ceva conjugate of X(110)

X(39139) = X(265)X(22265)∩X(9293)X(14559)

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

X(39139) lies on the curve Q155 and these lines: {265, 22265}, {9293, 14559}

X(39140) = POINT RASALGETHI

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

X(39140) lies on the curve Q155 and these lines: {}

leftri

Perspectors related to the obverse triangle of X(69): X(39144)-X(39146)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, June 25, 2020.

The obverse triangle A'B'C' of X(69) is the central triangle given by

A' = -a^2 + b^2 + c^2 : a^2 + b^2 - c^2 : a^2 - b^2 + c^2.

See the preamble just before X(24307) for the definition of obverse triangle.

The appearance of (T,i) in the following list means that A'B'C' is perspective to triangle T and the perspector is X(i):

(1st Brocard,2), (2nd Brocard,2), (anti-1st-Brocard)

(ABC,4), (orthic,4), (Euler,4), (circumorthic,4), (half-altitude,4), (reflection of ABC in X(4),4), (orthocentroidal,4), (2nd extouch,4), (3rd extouch,4), (1st orthosymmedial,4), (anti-1st-Euler,4), (infinite altitude,4), (Ehrmann vertex-triangle,4), (Moses-Steiner-osculatory triangle,4), (Moses-Steiner osculating triangle, 4)

(medial,6337), (anticomplementary,194), (3rd Brocard,194), (symmedial,194), (6th Brocard, 7791), (2nd Conway,33867), (reflected 1st Brocard,194), (anti-Artzt,1992), (Gemini 107,9741), (Gemini 111,20088), (anti-6th-Brocard,29141), (Gemini 109,39142), (Gemini 110,39143)

X(39141) = PERSPECTOR OF THESE TRIANGLES: OBVERSE OF X(69) AND ANTI-6TH BROCARD

Barycentrics    2*a^6 - a^4*b^2 + a^2*b^4 - a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 : :
X(39141) = X[193] + 4 X[7863]

X(39141) lies on these lines: {2, 98}, {4, 2456}, {6, 194}, {20, 10350}, {32, 193}, {69, 1691}, {76, 1692}, {83, 2996}, {99, 5028}, {141, 7907}, {315, 2458}, {439, 5171}, {511, 3552}, {524, 33246}, {611, 6645}, {613, 4366}, {637, 2462}, {638, 2461}, {1003, 1351}, {1078, 3620}, {1350, 13586}, {1353, 8369}, {1503, 5025}, {1570, 7816}, {1992, 12151}, {1993, 16951}, {1994, 16949}, {2030, 14994}, {2080, 32985}, {3094, 5026}, {3098, 33014}, {3292, 35275}, {3398, 14001}, {3407, 7774}, {3564, 7807}, {3589, 7932}, {3618, 5038}, {3619, 33000}, {3763, 16923}, {3818, 32966}, {3933, 38905}, {3972, 5052}, {5007, 6309}, {5017, 39099}, {5032, 12150}, {5039, 33201}, {5050, 7770}, {5085, 7824}, {5092, 33004}, {5093, 32134}, {5097, 22486}, {5207, 7912}, {5422, 16950}, {5477, 7835}, {5480, 11361}, {6337, 34870}, {6393, 7891}, {6658, 31670}, {6800, 26257}, {7766, 9865}, {7791, 10131}, {7808, 32987}, {7887, 18440}, {7892, 8550}, {8177, 35006}, {8370, 18583}, {8722, 35287}, {9967, 35952}, {10104, 32970}, {10341, 30226}, {10358, 32979}, {10516, 32967}, {10519, 32964}, {10653, 12214}, {10654, 12213}, {10753, 35387}, {10796, 14033}, {11285, 12017}, {11288, 11898}, {11324, 11402}, {12007, 14036}, {12054, 16043}, {12110, 32981}, {12203, 32974}, {12216, 20088}, {13193, 25321}, {14035, 14853}, {14041, 36990}, {14064, 14880}, {14561, 16044}, {14927, 33017}, {15066, 35288}, {15069, 33245}, {15988, 16915}, {16055, 35265}, {16932, 34545}, {17508, 33022}, {18358, 33249}, {19120, 32528}, {19130, 33018}, {19131, 37186}, {19686, 20423}, {19687, 21850}, {20080, 33205}, {24248, 32115}, {25320, 32242}, {26864, 35301}, {29012, 33019}, {29181, 33257}, {31884, 33276}, {32990, 37479}, {32992, 38110}, {33002, 38317}, {33198, 33748}, {33235, 33878}, {35431, 39097}

X(39141) = crossdifference of every pair of points on line {3221, 3569}
X(39141) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 4048, 18906}, {6, 12215, 194}, {69, 1691, 7793}, {182, 12177, 6776}, {4027, 10334, 2}, {5182, 12177, 4027}, {10131, 10333, 7791}.

X(39142) = PERSPECTOR OF THESE TRIANGLES: OBVERSE OF X(69) AND GEMINI 109

Barycentrics    15*a^4 - 16*a^2*b^2 + 9*b^4 - 16*a^2*c^2 - 2*b^2*c^2 + 9*c^4 : :

X(39142) lies on these lines: {2, 32822}, {4, 620}, {39, 33231}, {99, 32958}, {141, 631}, {194, 32970}, {574, 32953}, {1285, 16925}, {1992, 7763}, {3524, 3788}, {3525, 3934}, {3528, 37690}, {3533, 7789}, {3767, 9167}, {3923, 19862}, {5013, 33195}, {6337, 32959}, {7736, 33236}, {7769, 14039}, {7778, 10299}, {7782, 33292}, {7784, 15698}, {7786, 14069}, {7795, 15702}, {7830, 15715}, {7835, 32957}, {7874, 33230}, {7891, 32977}, {7918, 32951}, {7940, 33285}, {8368, 18841}, {10256, 15428}, {10303, 18840}, {10654, 36770}, {11163, 32887}, {11286, 32871}, {11288, 32835}, {11541, 32456}, {15815, 33196}, {16832, 24268}, {20080, 32818}, {31268, 32960}, {31400, 33197}, {32459, 33703}, {32817, 33233}, {32823, 35297}, {32829, 33191}, {33232, 37512}.

X(39143) = PERSPECTOR OF THESE TRIANGLES: OBVERSE OF X(69) AND GEMINI 110

Barycentrics    3*a^4 - 4*a^2*b^2 + 9*b^4 - 4*a^2*c^2 - 14*b^2*c^2 + 9*c^4 : :
X(30143) = 3 X[2] + X[38259]

X(39143) lies on these lines: {2, 15815}, {4, 6036}, {5, 3618}, {69, 13881}, {99, 32958}, {115, 6337}, {194, 1007}, {230, 32980}, {1078, 33292}, {1992, 3767}, {2549, 32976}, {3054, 33023}, {3090, 6248}, {3091, 9756}, {3533, 7847}, {3545, 7828}, {3619, 3934}, {3634, 3923}, {5025, 34229}, {5056, 7851}, {5067, 7790}, {5068, 7792}, {5071, 7803}, {5254, 32988}, {5418, 6118}, {5420, 6119}, {6669, 10654}, {6670, 10653}, {6704, 7844}, {6722, 32970}, {7615, 7874}, {7735, 9478}, {7736, 32963}, {7738, 32967}, {7746, 16041}, {7748, 32977}, {7749, 33238}, {7750, 23055}, {7763, 9166}, {7776, 8355}, {7795, 18362}, {7834, 14762}, {7841, 23053}, {7861, 32978}, {7864, 33270}, {7886, 14033}, {7919, 32957}, {7923, 33009}, {7932, 33005}, {7940, 32822}, {8359, 32883}, {11008, 32816}, {11165, 32887}, {11174, 15022}, {11185, 32955}, {11287, 32867}, {11318, 16509}, {14039, 15031}, {14568, 32823}, {14971, 32985}, {16634, 37172}, {16635, 37173}, {16831, 24268}, {16984, 32995}, {16989, 33011}, {17004, 33290}, {29583, 30828}, {31168, 32832}, {32006, 33228}, {32838, 33184}, {32982, 37637}, {33200, 37688}, {33277, 37690}

X(39143) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {115, 32969, 6337}, {5254, 32988, 34803}, {13881, 32972, 69}.

X(39144) = ISOGONAL CONJUGATE OF X(32622)

Barycentrics    1/((a^2 - b^2 - c^2)*Sqrt[-a^2 - b^2 - c^2 + 2*(a*b + a*c + b*c)] + 2*(a - b - c)*S) : :

The obverse triangle of X(69) is the anticomplementary triangle of the 1st Brocard triangle (which is the obverse triangle of X(6)). (Randy Hutson, August 7, 2020)

X(39144) lies on the Feuerbach circumhyperbola, the curves K352 and Q030, and these lines: {9, 32623}, {11, 57}, {516, 32622}, {971, 3513}

X(39144) = reflection of X(39145) in X(11)
X(39144) = isogonal conjugate of X(32622)
X(39144) = antigonal image of X(39145)
X(39144) = X(i)-cross conjugate of X(j) for these (i,j): {971, 39145}, {3513, 1}
X(39144) = X(1)-isoconjugate of X(32622)
X(39144) = barycentric quotient X(6)/X(32622)
X(39144) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {57, 11372, 39145}, {1699, 4312, 39145}, {1709, 15299, 39145), {1836, 5805, 39145}, {3358, 30223, 39145}

X(39145) = ISOGONAL CONJUGATE OF X(32623)

Barycentrics    1/((a^2 - b^2 - c^2)*Sqrt[-a^2 - b^2 - c^2 + 2*(a*b + a*c + b*c)] - 2*(a - b - c)*S) : :

X(39145) lies on the on the Feuerbach circumhyperbola, the curves K352 and Q030, and these lines: {9, 32622}, {11, 57}, {516, 32623}, {971, 3514}

X(39145) = reflection of X(39144) in X(11)
X(39145) = isogonal conjugate of X(32623)
X(39145) = antigonal image of X(39144)
X(39145) = X(i)-cross conjugate of X(j) for these (i,j): {971, 39144}, {3514, 1}
X(39145) = X(1)-isoconjugate of X(32623)
X(39145) = barycentric quotient X(6)/X(32623)
X(39145) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {57, 11372, 39144}, {1699, 4312, 39144}, {1709, 15299, 39144}, {1836, 5805, 39144}, {3358, 30223, 39144}

X(39146) = ISOGONAL CONJUGATE OF X(8072)

Barycentrics    a^2/(2*a^4 - a^3*b + a^2*b^2 - 3*a*b^3 + b^4 - a^3*c - 2*a^2*b*c + 3*a*b^2*c + a^2*c^2 + 3*a*b*c^2 - 2*b^2*c^2 - 3*a*c^3 + c^4 + 2*(2*a^3 - a^2*b - b^3 - a^2*c + b^2*c + b*c^2 - c^3)*Sqrt[s^2 - 4*R*(r + R)]) : :

X(39146) lies on the curves K206, K269, K352, K436, Q039, and these lines: {9, 48}, {57, 8073}, {515, 8072}

X(39146) = isogonal conjugate of X(8072)
X(39146) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8072}, {1785, 39147}
X(39146) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8072}, {14578, 39147}

X(39147) = ISOGONAL CONJUGATE OF X(8073)

Barycentrics    a^2/(2*a^4 - a^3*b + a^2*b^2 - 3*a*b^3 + b^4 - a^3*c - 2*a^2*b*c + 3*a*b^2*c + a^2*c^2 + 3*a*b*c^2 - 2*b^2*c^2 - 3*a*c^3 + c^4 - 2*(2*a^3 - a^2*b - b^3 - a^2*c + b^2*c + b*c^2 - c^3)*Sqrt[s^2 - 4*R*(r + R)]) : :

X(39147) lies on the curves K206, K269, K352, K436, Q039, and these lines: {9, 48}, {57, 8072}, {515, 8073}

X(39147) = isogonal conjugate of X(8073)
X(39147) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8073}, {1785, 39146}
X(39147) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8073}, {14578, 39146}

X(39148) = X(36)X(106)∩X(44)X(517)

Barycentrics    a^2*(a + b - 2*c)*(a - 2*b + c)*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 5*a*b*c + 3*b^2*c - a*c^2 + 3*b*c^2 - c^3) : :
X(39148) = X[4792] - 4 X[14190]

X(39148) lies on the cubic K206 and these lines: {36, 106}, {44, 517}, {55, 14260}, {88, 484}, {109, 13756}, {519, 3257}, {758, 1320}, {999, 16944}, {2342, 10428}, {2802, 6163}, {3583, 36590}, {3814, 19634}, {4080, 5180}, {4997, 11813}, {5048, 10703}, {6767, 34230}, {19636, 28174}

X(39148) = reflection of X(i) in X(j) for these {i,j}: {1168, 14190}, {4792, 1168}
X(39148) = X(1)-Ceva conjugate of X(106)
X(39148) = X(44)-isoconjugate of X(8046)
X(39148) = crosspoint of X(1) and X(5541)
X(39148) = barycentric product X(i)*X(j) for these {i,j}: {88, 5541}, {106, 30578}, {901, 21198}, {903, 3196}, {6336, 22141}, {9456, 20937}
X(39148) = barycentric quotient X(i)/X(j) for these {i,j}: {106, 8046}, {3196, 519}, {5541, 4358}, {22141, 3977}, {30578, 3264}
X(39148) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {36, 1318, 106}, {901, 1318, 36}, {2226, 16489, 106}

X(39149) = X(1)X(229)∩X(2)X(6757)

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

X(39149) lies on the cubic K206 and these lines: {1, 229}, {2, 6757}, {12, 502}, {30, 3743}, {323, 758}, {523, 8043}, {1029, 1478}, {5127, 39137}, {7799, 35550}, {16307, 25081}

X(39149) = X(36)-cross conjugate of X(758)
X(39149) = X(i)-isoconjugate of X(j) for these (i,j): {80, 501}, {191, 759}, {1030, 24624}, {1807, 2906}, {2895, 34079}, {6740, 8614}
X(39149) = trilinear pole of line {526, 2610}
X(39149) = barycentric product X(i)*X(j) for these {i,j}: {267, 3936}, {320, 21353}, {502, 3218}, {758, 1029}, {3444, 35550}
X(39149) = barycentric quotient X(i)/X(j) for these {i,j}: {267, 24624}, {502, 18359}, {758, 2895}, {1029, 14616}, {2245, 191}, {3444, 759}, {3724, 1030}, {3936, 20932}, {4053, 21081}, {7113, 501}, {21353, 80}, {21828, 31947}

X(39150) = X(1)X(61)∩X(10)X(14)

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

X(39150) lies on the cubics K040, K058, K206, K261b, K588, K662 and these lines: {1, 61}, {10, 14}, {15, 214}, {16, 1276}, {18, 5445}, {44, 517}, {58, 11072}, {80, 14358}, {202, 37772}, {501, 15788}, {519, 36930}, {3179, 19373}, {3376, 33654}, {3639, 3911}, {5357, 6126}, {9324, 11789}

X(39150) = reflection of X(39151) in X(44)
X(39150) = reflection of X(39151) in the anti-Orthic axis
X(39150) = isogonal conjugate of X(39151)
X(39150) = X(i)-isoconjugate of X(j) for these (i,j): {1, 39151}, {13, 5353}, {36, 14358}, {80, 203}, {2307, 36933}, {5239, 33655}, {7026, 7051}, {19551, 37773}
X(39150) = cevapoint of X(36) and X(5357)
X(39150) = crosssum of X(36) and X(5353)
X(39150) = barycentric product X(i)*X(j) for these {i,j}: {202, 18359}, {298, 11072}, {559, 36932}, {3218, 14359}, {7043, 37772}
X(39150) = barycentric quotient X(i)/X(j) for these {i,j}: {6, X(39151)}, {202, 3218}, {1251, 36933}, {2151, 5353}, {2161, 14358}, {3458, 11073}, {7113, 203}, {7126, 7026}, {11072, 13}, {14359, 18359}, {19373, 37773}
X(39150) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 7150, 7126}, {4792, 12034, 39151}, {7052, 7126, 1}

X(39151) = X(1)X(62)∩X(10)X(13)

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

X(39151) lies on the cubics K040, K058, K206, K261b, K588, K662 and these lines: {1, 62}, {10, 13}, {15, 1277}, {16, 214}, {17, 5445}, {44, 517}, {58, 11073}, {80, 7126}, {203, 37773}, {501, 15789}, {519, 36931}, {2306, 3383}, {3638, 3911}, {5353, 6126}, {9324, 11752}

X(39151) = reflection of X(39150) in X(44)
X(39151) = reflection of X(39150) in the anti-orthic axis
X(39151) = isogonal conjugate of X(39150)
X(39151) = X(i)-isoconjugate of X(j) for these (i,j): {1, 39150}, {14, 5357}, {36, 14359}, {80, 202}, {5240, 7052}, {7043, 19373}, {7126, 37772}
X(39151) = cevapoint of X(36) and X(5353)
X(39151) = crosssum of X(36) and X(5357)
X(39151) = barycentric product X(i)*X(j) for these {i,j}: {203, 18359}, {299, 11073}, {1082, 36933}, {3218, 14358}, {7026, 37773}
X(39151) = barycentric quotient X(i)/X(j) for these {i,j}: {6, X(39150)}, {203, 3218}, {2152, 5357}, {2161, 14359}, {3457, 11072}, {7051, 37772}, {7113, 202}, {7127, 5240}, {11073, 14}, {14358, 18359}, {19551, 7043}, {33653, 36932}
X(39151) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4792, 12034, 39150}, {19551, 33655, 1}

X(39152) = X(1)X(16)∩X(14)X(226)

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

X(39152) lies on the cubics K206 and K261b and these lines: {1, 16}, {13, 34301}, {14, 226}, {15, 18593}, {18, 16038}, {61, 1652}, {79, 1251}, {106, 11073}, {202, 758}, {203, 29821}, {284, 501}, {1125, 5239}, {1276, 5018}, {3376, 7052}, {5357, 6126}, {7006, 20116}, {7073, 36738}, {14358, 33655}

X(39152) = crosssum of X(35) and X(5357)
X(39152) = X(36)-cross conjugate of X(39150)
X(39152) = X(i)-isoconjugate of X(j) for these (i,j): {13, 35}, {16, 80}, {299, 6187}, {319, 3457}, {559, 19551}, {2152, 18359}, {2153, 3219}, {5357, 14358}, {5995, 7265}, {20566, 34395}
X(39152) = barycentric product X(i)*X(j) for these {i,j}: {14, 3218}, {15, 30690}, {298, 2160}, {301, 7113}, {320, 2154}, {470, 7100}, {554, 5240}, {2151, 20565}, {3458, 20924}, {13486, 23870}, {36932, 37773}
X(39152) = barycentric quotient X(i)/X(j) for these {i,j}: {14, 18359}, {15, 3219}, {298, 33939}, {1870, 471}, {2151, 35}, {2154, 80}, {2160, 13}, {3218, 299}, {3458, 2161}, {6186, 2153}, {7113, 16}, {8739, 6198}, {11073, 14358}, {13486, 23895}, {19373, 559}, {30690, 300}, {33653, 7026}, {34394, 2174}, {36297, 1807}
X(39152) = {X(942),X(1100)}-harmonic conjugate of X(39153)

X(39153) = X(1)X(15)∩X(13)X(226)

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

X(39353) lies on the cubics K206 and K261a and these lines: {1, 15}, {13, 226}, {14, 34301}, {16, 18593}, {17, 37701}, {62, 1653}, {79, 10651}, {106, 11072}, {202, 29821}, {203, 758}, {284, 501}, {1125, 5240}, {1277, 5018}, {3383, 33655}, {5353, 6126}, {7005, 20116}, {7052, 14359}, {7073, 36737}

X(39353) = crosssum of X(35) and X(5353)
X(39353) = X(36)-cross conjugate of X(39151)
X(39353) = X(i)-isoconjugate of X(j) for these (i,j): {14, 35}, {15, 80}, {298, 6187}, {319, 3458}, {1082, 7126}, {2151, 18359}, {2154, 3219}, {2307, 7043}, {5353, 14359}, {5994, 7265}, {20566, 34394}
X(39353) = barycentric product X(i)*X(j) for these {i,j}: {13, 3218}, {16, 30690}, {299, 2160}, {300, 7113}, {320, 2153}, {471, 7100}, {1081, 5239}, {2152, 20565}, {3457, 20924}, {13486, 23871}, {36933, 37772}
X(39353) = barycentric quotient X(i)/X(j) for these {i,j}: {13, 18359}, {16, 3219}, {299, 33939}, {1251, 7043}, {1870, 470}, {2152, 35}, {2153, 80}, {2160, 14}, {3218, 298}, {3457, 2161}, {6186, 2154}, {7051, 1082}, {7113, 15}, {8740, 6198}, {11072, 14359}, {13486, 23896}, {30690, 301}, {34395, 2174}, {36296, 1807}
X(39353) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3179, 1251}, {942, 1100, 39152}, {1251, 2306, 3179}

X(39154) = X(44)X(517)∩X(100)X(513)

Barycentrics    a*(a - b)*(a + b - 2*c)*(a - c)*(a - 2*b + c)*(2*a^3 - 2*a^2*b + b^3 - 2*a^2*c + 2*a*b*c - b^2*c - b*c^2 + c^3) : :
X(39154) = 3 X[765] - X[14513]

X(39154) lies on the cubic K662 and these lines: {44, 517}, {100, 513}, {484, 17960}, {23344, 23352}

X(39154) = midpoint of X(901) and X(3257)
X(39154) = barycentric product X(i)*X(j) for these {i,j}: {4567, 18011}, {4601, 17998}
X(39154) = barycentric quotient X(i)/X(j) for these {i,j}: {17998, 3125}, {18011, 16732}

X(39155) = X(100)X(4730)∩X(214)X(1635)

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

X(39155) lies on the buic K662 and these lines: {100, 4730}, {214, 1635}, {244, 3960}, {513, 3218}

X(39155) = X(i)-isoconjugate of X(j) for these (i,j): {4570, 18011}, {4600, 17998}
X(39155) = barycentric quotient X(i)/X(j) for these {i,j}: {3121, 17998}, {3125, 18011}

X(39156) = STEINER POINT OF EXCENTRAL TRIANGLE

Barycentrics    a*(a^6 - a^5*b + a^4*b^2 - 6*a^3*b^3 + 7*a^2*b^4 - a*b^5 - b^6 - a^5*c - a^4*b*c + 6*a^3*b^2*c - 2*a^2*b^3*c - 5*a*b^4*c + 3*b^5*c + a^4*c^2 + 6*a^3*b*c^2 - 10*a^2*b^2*c^2 + 6*a*b^3*c^2 - 3*b^4*c^2 - 6*a^3*c^3 - 2*a^2*b*c^3 + 6*a*b^2*c^3 + 2*b^3*c^3 + 7*a^2*c^4 - 5*a*b*c^4 - 3*b^2*c^4 - a*c^5 + 3*b*c^5 - c^6)::
X(39156) =3 X[1] - 2 X[10697], 3 X[1] - 4 X[11714], 2 X[101] - 3 X[165], 3 X[103] - X[10697], 3 X[103] - 2 X[11714], 4 X[116] - 3 X[1699], 4 X[118] - 5 X[1698], 3 X[3576] - 4 X[38601], 7 X[3624] - 8 X[6712], 3 X[5587] - 2 X[10741], 5 X[7987] - 4 X[11712], 5 X[7987] - 6 X[38692], 9 X[7988] - 10 X[31273], X[7991] + 2 X[38668], 3 X[9778] - X[20096], 4 X[9956] - 3 X[38767], 2 X[10710] - 3 X[19875], 2 X[10772] - 3 X[37718], 2 X[11712] - 3 X[38692], 8 X[11728] - 9 X[25055], 7 X[16192] - 6 X[38690], 7 X[31423] - 6 X[38764], 5 X[35242] - 4 X[38599].

X(39156) lies on the Bevan circle and these lines: {1, 103}, {10, 152}, {20, 2784}, {40, 170}, {43, 34457}, {57, 3022}, {64, 7281}, {101, 165}, {116, 1699}, {118, 1698}, {150, 516}, {517, 5527}, {658, 24009}, {1044, 9899}, {1046, 2825}, {1054, 2821}, {1362, 1697}, {1695, 3033}, {1706, 3041}, {1721, 2093}, {1742, 2807}, {1768, 3887}, {2772, 2938}, {2774, 9904}, {2786, 9860}, {2801, 2951}, {2809, 7991}, {2820, 5540}, {2942, 34925}, {2958, 3309}, {3046, 9586}, {3339, 11028}, {3464, 5902}, {3576, 38601}, {3579, 38572}, {3624, 6712}, {4845, 8917}, {5587, 10741}, {7987, 11712}, {7988, 31273}, {9357, 15599}, {9518, 12408}, {9778, 20096}, {9956, 38767}, {10695, 11531}, {10710, 19875}, {10772, 37718}, {10980, 14760}, {11010, 38502}, {11728, 25055}, {15730, 31508}, {15856, 16192}, {18480, 38768}, {24010, 36101}, {27000, 34848}, {31423, 38764}, {35242, 38599}

X(39156) = reflection of X(i) in X(j) for these {i,j}: {1, 103}, {152, 10}, {1282, 40}, {10697, 11714}, {11531, 10695}, {38572, 3579}, {38768, 18480}
X(39156) = excentral-isogonal conjugate of X(514)
X(39156) = X(3900)-Ceva conjugate of X(1)
X(39156) = X(99)-of-excentral-triangle
X(39156) = X(3022)-of-tangential-of-excentral-triangle
X(39156) = Bevan-circle-antipode of X(1282)
X(39156) = trilinear pole, wrt excentral triangle, of line X(9)X(165)
X(39156) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {103, 10697, 11714}, {10697, 11714, 1}, {11712, 38692, 7987}

X(39157) = ISOGONAL CONJUGATE OF X(13493)

Barycentrics    (a^2 + b^2 - 3*b*c + c^2)*(a^2 + b^2 + 3*b*c + c^2)*(a^4 - 4*a^2*b^2 + b^4 - c^4)*(a^4 - b^4 - 4*a^2*c^2 + c^4) : :
X(39157) = 4 X[468] - 3 X[10102]

X(39157) lies on the cubic K008 and thesse lines: {2, 12505}, {69, 3266}, {316, 34166}, {468, 10102}, {671, 858}, {8593, 37860}, {9225, 35188}, {13574, 34163}

X(39157) = isogonal conjugate of X(13493)
X(39157) = isotomic conjugate of X(34166)
X(39157) = isotomic conjugate of the anticomplement of X(10354)
X(39157) = antigonal image of X(13492)
X(39157) = orthoptic-circle-of-Steinerinellipse-inverse of X(12506)
X(39157) = psi-transform of X(31959)
X(39157) = X(897)-anticomplementary conjugate of X(34165)
X(39157) = X(i)-cross conjugate of X(j) for these (i,j): {9872, 11054}, {10354, 2}
X(39157) = X(i)-isoconjugate of X(j) for these (i,j): {1, 13493}, {31, 34166}
X(39157) = barycentric product X(5486)*X(11054)
X(39157) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 34166}, {6, 13493}, {5486, 34898}, {9872, 8542}, {11054, 11185}, {11580, 1995}, {13492, 14262}, {13608, 34581}
X(39157) = {X(13608),X(32133)}-harmonic conjugate of X(2)

leftri

Foci of the real and imaginary Steiner ellipses: X(39158)-X(39161)

rightri

This preamble and centers X(39158)-X(39161) were contributed by Peter Moses, July 2-3, 2020.

The foci of the Steiner circumellipse and the Steiner inellipse are all given by the following form for 1st barycentric:

2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - sgn1*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] + const*S*sgn2*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + sgn1*2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]

The Steiner circumellipse has const = 1. Specifically, X(39158) is given by (sgn1,sgn2) = (1,1)
X(39159) is given by (sgn1,sgn2) = (1,-1)
X(39160) is given by (sgn1,sgn2) = (-1,1)
X(39161) is given by (sgn1,sgn2) = (-1,-1)

The Steiner inellipse has const = 2. Specifically, X(39162) is given by (sgn1,sgn2) = (1,1)
X(39163) is given by (sgn1,sgn2) = (1,-1)
X(39164) is given by (sgn1,sgn2) = (-1,1)
X(39165) is given by (sgn1,sgn2) = (-1,-1)

See the preamble just before X(39202) for vertices of the two ellipses.

X(39158) = 1ST REAL FOCUS OF THE STEINER CIRCUMELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] + Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]*S : :
Barycentrics    (S^2-3*SB*SC)*sqrt(SW^2-3*S^2)+3*(SA^2-SB*SC-SW^2)*SA+S^2*(6*SA+SW+sqrt(T)) : : , where T=2*(9*R^2-2*SW)*sqrt(SW^2-3*S^2)-3*S^2-18*SW*R^2+5*SW^2

X(39158) = X[39159] - 4 X[39162], 3 X[39159] - 4 X[39163], 3 X[39162] - X[39163]

The foci of the Steiner circumellipse (X(39158) and X(39159)) are also known as the Bickart points. The first barycentric for the 1st focus has " + Sqrt[-2*a^8 +" where the 2nd has " - Sqrt[-2*a^8 +". The two points are triangle centers that are also a bicentric pair, PU(116). They are the equicevian points of a triangle; i.e., the points X such that the lengths of segments AA', BB', CC', where A'B'C' is the cevian triangle of X, are equal; see Sadi Abu-Saymeh, Mowaffaq Hajja, and Hellmuth Stachel, "Equicevian Points of a Triangle", Amer. Math. Monthly 122 (2015) 995-1000.

Since X(39158) is on K007 (Lucas cubic), the lines perpendicular to the sides of ABC through the respective cevian traces of X(39158) concur in a point, namely X(40851), on the K004 (Darboux cubic). (César Lozada, December 31, 2020).

X(39158) lies on the cubics K007, K025, K309, K310, K347, K348, K715 and this line: {2,1341}

X(39158) = reflection of X(i) in X(j) for these {i,j}: {2, 39162}, {39159, 2}
X(39158) = anticomplement of X(39163)
X(39158) = anticomplement of the isogonal conjugate of X(39162)
X(39158) = anticomplementary isogonal conjugate of X(39159)
X(39158) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 39159}, {39162, 8}
X(39158) = X(3146)-cross conjugate of X(39159)

X(39159) = 2ND REAL FOCUS OF THE STEINER CIRCUMELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] - Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]*S : :
Barycentrics    (S^2-3*SB*SC)*sqrt(SW^2-3*S^2)+3*(SA^2-SB*SC-SW^2)*SA+S^2*(6*SA+SW-sqrt(T)) : : where T=2*(9*R^2-2*SW)*sqrt(SW^2-3*S^2)-3*S^2-18*SW*R^2+5*SW^2
X(39159) = 3 X[39158] - 4 X[39162], X[39158] - 4 X[39163], X[39162] - 3 X[39163]

The foci of the Steiner circumellipse (X(39158) and X(39159)) are also known as the Bickart points. The first barycentric for the 1st focus has " + Sqrt[-2*a^8 +" where the 2nd has " - Sqrt[-2*a^8 +". The two points are triangle centers that are also a bicentric pair, PU(116). They are the equicevian points of a triangle; i.e., the points X such that the lengths of segments AA', BB', CC', where A'B'C' is the cevian triangle of X, are equal; see Sadi Abu-Saymeh, Mowaffaq Hajja, and Hellmuth Stachel, "Equicevian Points of a Triangle", Amer. Math. Monthly 122 (2015) 995-1000.

Since X(39159) is on K007 (Lucas cubic), the lines perpendicular to the sides of ABC through the respective cevian traces of X(39159) concur in a point, namely X(40852), on the K004 (Darboux cubic). (César Lozada, December 31, 2020).

X(39159) lies on the cubics K007, K025, K309, K310, K347, K348, K715 and these lines: {2, 1341}, {30, 32443}

X(39159) = reflection of X(i) in X(j) for these {i,j}: {2, 39163}, {39158, 2}
X(39159) = anticomplement of X(39162)
X(39159) = anticomplement of the isogonal conjugate of X(39163)
X(39159) = anticomplementary isogonal conjugate of X(39158)
X(39159) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 39158}, {39163, 8}
X(39159) = X(3146)-cross conjugate of X(39158)

X(39160) = 1ST IMAGINARY FOCUS OF THE STEINER CIRCUMELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 + (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] + Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 - 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]*S : :
Barycentrics    (S^2-3*SB*SC)*sqrt(SW^2-3*S^2)-3*(SA^2-SB*SC-SW^2)*SA-(6*SA+SW+sqrt(T))*S^2 : : where T=-2*(9*R^2-2*SW)*sqrt(SW^2-3*S^2)-3*S^2-18*R^2*SW+5*SW^2
X(39160) = X[39161] - 4 X[39164], 3 X[39161] - 4 X[39165], 3 X[39164] - X[39165]

X(39160) lies on the cubics K007, K025, K309, K310, K347, K348, K715 and this line: {2, 1340}

X(39160) = reflection of X(i) in X(j) for these {i,j}: {2, 39164}, {39161, 2}
X(39160) = anticomplement of X (39165)
X(39160) = anticomplement of the isogonal conjugate of X (39164)
X(39160) = anticomplementary isogonal conjugate of X (39161)

X(39161) = 2ND IMAGINARY FOCUS OF THE STEINER CIRCUMELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 + (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] - Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 - 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]*S : :
Barycentrics    (S^2-3*SB*SC)*sqrt(SW^2-3*S^2)-3*(SA^2-SB*SC-SW^2)*SA-(6*SA+SW-sqrt(T))*S^2 : : where T=-2*(9*R^2-2*SW)*sqrt(SW^2-3*S^2)-3*S^2-18*R^2*SW+5*SW^2
X(39161) = 3 X[39160] - 4 X[39164], X[39160] - 4 X[39165], X[39164] - 3 X[39165]

X(39161 lies on the cubics K007, K025, K309, K310, K347, K348, K715 and this line: {2, 1340}

X(39161) = reflection of X(i) in X(j) for these {i,j}: {2, 39165}, {39160, 2}
X(39161) = anticomplement of X(39164)
X(39161) = anticomplement of the isogonal conjugate of X(39165)
X(39161) = anticomplementary isogonal conjugate of X(39160)

X(39162) = 1ST REAL FOCUS OF THE STEINER INELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] + 2*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]*S : :
X(39162) = 3 X[39158] + X[39159], 2 X[39158] + X[39163], 2 X[39159] - 3 X[39163]

With reference to X(39162) and X(39163), let M be a point in the plane of a reference triangle T (red), and let T' (green) be the triangle whose vertices are the circumcenters of the triangles MBC, MCA, MAB. If M = X(6), then T' is a focus of the Steiner inellipse of T'; i.e., either X(39162) of T' or X(39163) of T'. (Dan Reznik, April 2, 2021).

Continuing, if M = X(6), the green triangle A'B'C' is the first Ehrmann triangle (as in the preamble just before X(8537)). The foci of the Steiner inellipse of the first Ehrmann triangle are X(3) and X(6). The foci of the Steiner circumellipse of the first Ehrmann triangle are X(3098) and X(576). The A-vertex, A', of the triangle T', for arbitrary P = p : q : r, is given (Peter Moses, April 3, 2021) by

A' = a^2*((a^2 - b^2 - c^2)*p^2 + (a^2 - b^2 + c^2)*p*q + (a^2 + b^2 - c^2)*p*r + 2*a^2*q*r)
: -(b^2*(a^2 - b^2 + c^2)*p^2) - (a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*p*q - 2*a^2*b^2*p*r - a^2*(a^2 + b^2 - c^2)*q*r
: -(c^2*(a^2 + b^2 - c^2)*p^2) - 2*a^2*c^2*p*q - (a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*p*r - a^2*(a^2 - b^2 + c^2)*q*r

X(39162) lies on the curves K002, K018, K038, K057a, K086, K187, K248, K287, K352, K353, K358, K463, K705, K706, K727, K729, K758, K800, K812, K816, K833, K847, K893, K911, K1092, Q016, Q017, Q019, Q030, Q037, Q039, Q054, Q059, Q091, Q094, Q115, Q116, Q117, Q118, Q119, Q135, Q137, Q138, Q139, and this line: {2, 1341}

X(39162) = midpoint of X(2) and X(39158)
X(39162) = reflection of X(39163) in X(2)
X(39162) = isogonal conjugate of X(39163)
X(39162) = complement of X(39159)
X(39162) = isogonal conjugate of the complement of X(39158)
X(39162) = Thomson-isogonal conjugate of X(39163)
X(39162) = psi-transform of X(39162)
X(39162) = X(6177)-Ceva conjugate of X(39163)
X(39162) = X(3557)-cross conjugate of X(39163)
X(39162) = barycentric quotient X(6)/X(39163)
X(39162) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1341, 5638, 39163}, {13722, 31863, 39163}, {14899, 35607, 39163}

X(39163) = 2ND REAL FOCUS OF THE STEINER INELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] - 2*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]*S : :
X(39163) = X[39158] + 3 X[39159], 2 X[39158] - 3 X[39162], 2 X[39159] + X[39162]

X(39162) lies on the curves K002, K018, K038, K057b, K086, K187, K248, K287, K352, K353, K358, K463, K705, K706, K727, K729, K758, K800, K812, K816, K833, K847, K893, K911, K1092, Q016, Q017, Q019, Q030, Q037, Q039, Q054, Q059, Q091, Q094, Q115, Q116, Q117, Q118, Q119, Q135, Q137, Q138, Q139, and this line: {2, 1341}

X(39163) = midpoint of X(2) and X(39159)
X(39163) = reflection of X(39162) in X(2)
X(39163) = isogonal conjugate of X(39162)
X(39163) = complement of X(39158)
X(39163) = isogonal conjugate of the complement of X(39159)
X(39163) = Thomson-isogonal conjugate of X(39162)
X(39163) = psi-transform of X(39163)
X(39163) = X(6177)-Ceva conjugate of X(39162)
X(39163) = X(3557)-cross conjugate of X(39162)
X(39163) = barycentric quotient X(6)/X(39162)
X(39163) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1341, 5638, 39162}, {13722, 31863, 39162}, {14899, 35607, 39162}

X(39164) = 1ST IMAGINARY FOCUS OF THE STEINER INELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 + (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] + 2*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 - 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]*S : :
Barycentrics    (S^2-3*SB*SC)*sqrt(SW^2-3*S^2)-3*(SA^2-SB*SC-SW^2)*SA-(6*SA+SW+2*sqrt(T))*S^2 : : where T=-2*(9*R^2-2*SW)*sqrt(SW^2-3*S^2)-3*S^2-18*R^2*SW+5*SW^2
X(39164) = 3 X[39160] + X[39161], 2 X[39160] + X[39165], 2 X[39161] - 3 X[39165]

X(39164) lies on the curves K002, K018, K038, K086, K187, K248, K287, K352, K353, K358, K463, K705, K706, K727, K729, K758, K800, K812, K816, K833, K847, K893, K911, K1092, Q016, Q017, Q019, Q030, Q037, Q039, Q054, Q059, Q091, Q094, Q115, Q116, Q117, Q118, Q119, Q135, Q137, Q138, Q139 and this line:: {2, 1340}

X(39164) = complement of X(39161)
X(39164) = midpoint of X(2) and X(39160)
X(39164) = reflection of X(39165) in X(2)
X(39164) = isogonal conjugate of X(39165)
X(39164) = isogonal conjugate of the complement of X(39160)
X(39164) = Thomson-isogonal conjugate of X(39165)
X(39164) = psi-transform of X(39164)
X(39164) = barycentric quotient X(6)/X(39165)

X(39165) = 2ND IMAGINARY FOCUS OF THE STEINER INELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 + (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] - 2*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 - 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]*S : :
Barycentrics    (S^2-3*SB*SC)*sqrt(SW^2-3*S^2)-3*(SA^2-SB*SC-SW^2)*SA-(6*SA+SW-2*sqrt(T))*S^2 : : where T=-2*(9*R^2-2*SW)*sqrt(SW^2-3*S^2)-3*S^2-18*R^2*SW+5*SW^2 d
X(39165) = X[39160] + 3 X[39161], 2 X[39160] - 3 X[39164], 2 X[39161] + X[39164]

X(39165) lies on the curves K002, K018, K038, K086, K187, K248, K287, K352, K353, K358, K463, K705, K706, K727, K729, K758, K800, K812, K816, K833, K847, K893, K911, K1092, Q016, Q017, Q019, Q030, Q037, Q039, Q054, Q059, Q091, Q094, Q115, Q116, Q117, Q118, Q119, Q135, Q137, Q138, Q139 and this line:: {2, 1340}

X(39165) = midpoint of X(2) and X(39161)
X(39165) = reflection of X(39164) in X(2)
X(39165) = isogonal conjugate of X(39164)
X(39165) = complement of X(39160)
X(39165) = isogonal conjugate of the complement of X(39161)
X(39165) = Thomson-isogonal conjugate of X(39164)
X(39165) = psi-transform of X(39165)
X(39165) = barycentric quotient X(6)/X(39164)

X(39166) = ISOGONAL CONJUGATE OF X(38938)

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

X(39166) lies on the cubic K009 and these lines: {3, 3733}, {4, 6011}, {32, 35069}, {56, 214}, {849, 4257}, {2718, 6790}, {4188, 25650}, {4996, 16944}, {5124, 34877}, {18120, 25440}, {28607, 33863}

X(39166) = isogonal conjugate of X(38938)
X(39166) = cevapoint of X(i) and X(j) for these (i,j): {36, 35204}, {3724, 35069}
X(39166) = X(i)-isoconjugate of X(j) for these (i,j): {1, 38938}, {80, 30117}, {1731, 2006}, {1807, 5146}, {2161, 33129}
X(39166) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 38938}, {36, 33129}, {1983, 13589}, {2361, 1731}, {7113, 30117}, {35069, 31845}

X(39167) = ISOGONAL CONJUGATE OF X(14257)

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

X(39167) lies on the cubic K009 and these lines: {3, 960}, {4, 123}, {21, 34277}, {32, 35072}, {56, 34588}, {268, 4254}, {2968, 22760}, {6337, 34160}, {15261, 34159}, {20846, 37741}

X(39167) = isogonal conjugate of X(14257)
X(39167) = isogonal conjugate of the polar conjugate of X(34277)
X(39167) = X(i)-cross conjugate of X(j) for these (i,j): {184, 219}, {20967, 2193}
X(39167) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14257}, {4, 21147}, {34, 3436}, {75, 17408}, {92, 478}, {108, 21186}, {123, 24033}, {197, 273}, {205, 331}, {225, 16049}, {278, 1766}, {608, 20928}, {653, 6588}, {1396, 21074}, {5307, 34263}
X(39167) = cevapoint of X(1946) and X(35072)
X(39167) = crosssum of X(478) and X(17408)
X(39167) = barycentric product X(i)*X(j) for these {i,j}: {3, 34277}, {219, 8048}, {345, 3435}, {15385, 23983}, {34259, 34279}
X(39167) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 14257}, {32, 17408}, {48, 21147}, {78, 20928}, {184, 478}, {212, 1766}, {219, 3436}, {652, 21186}, {1946, 6588}, {2193, 16049}, {2318, 21074}, {3435, 278}, {6056, 22132}, {8048, 331}, {15385, 23984}, {34277, 264}, {35072, 123}

X(39168) = X(2)X(34110)∩X(3)X(143)

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

X(39168) lies on the cubic K009 and these lines: {2, 34110}, {3, 143}, {4, 11792}, {1487, 17500}, {3523, 26862}, {14381, 34156}

X(39168) = barycentric product X(i)*X(j) for these {i,j}: {288, 10095}, {1173, 34545}, {31626, 34484}
X(39168) = barycentric quotient X(34545)/X(1232)

X(39169) = X(2)X(10422)∩X(3)X(895)

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

X(39169) lies on the cubic K009 and these lines: {2, 10422}, {3, 895}, {4, 691}, {1995, 15899}, {3053, 36830}, {5968, 17928}, {8869, 11188}, {14357, 34161}, {14385, 34157}, {16092, 38323}

X(39169) = X(10422)-Ceva conjugate of X(895)
X(39169) = barycentric product X(i)*X(j) for these {i,j}: {111, 5866}, {895, 37784}, {14908, 37803}
X(39169) = barycentric quotient X(i)/X(j) for these {i,j}: {5866, 3266}, {37777, 37778}

X(39170) = X(3)X(125)∩X(4)X(476)

Barycentrics    (a^2 - b^2 - c^2)*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^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(39170) lies on the cubic K009 and these lines: {2, 5627}, {3, 125}, {4, 476}, {5, 523}, {32, 23967}, {113, 14264}, {381, 14583}, {847, 6344}, {1640, 18558}, {1989, 13881}, {2072, 16186}, {3003, 34104}, {3818, 15295}, {5158, 10217}, {6663, 13406}, {7752, 35139}, {9717, 12900}, {14357, 34157}, {14559, 16534}, {23515, 33927}, {30510, 37943}

X(39170) = complement of X(15454)
X(39170) = isogonal conjugate of X(38936)
X(39170) = complement of the isogonal conjugate of X(14264)
X(39170) = X(i)-complementary conjugate of X(j) for these (i,j): {1725, 113}, {2159, 11064}, {14264, 10}, {35200, 10257}, {36119, 13754}
X(39170) = X(i)-Ceva conjugate of X(j) for these (i,j): {265, 13754}, {5627, 265}
X(39170) = X(i)-cross conjugate of X(j) for these (i,j): {21731, 32662}, {34333, 13754}
X(39170) = X(i)-isoconjugate of X(j) for these (i,j): {1, 38936}, {162, 15470}, {186, 36053}, {526, 36114}, {687, 2624}, {1300, 6149}, {4575, 14222}, {10419, 35201}, {32679, 32708}
X(39170) = cevapoint of X(i) and X(j) for these (i,j): {18558, 20975}, {21731, 39021}
X(39170) = crosssum of X(186) and X(3043)
X(39170) = crossdifference of every pair of points on line {50, 15470}
X(39170) = barycentric product X(i)*X(j) for these {i,j}: {94, 13754}, {265, 3580}, {328, 3003}, {476, 6334}, {686, 35139}, {14592, 15329}
X(39170) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 38936}, {113, 14920}, {265, 2986}, {403, 14165}, {476, 687}, {647, 15470}, {686, 526}, {1989, 1300}, {2315, 6149}, {2501, 14222}, {3003, 186}, {3580, 340}, {6334, 3268}, {11079, 10419}, {13754, 323}, {14560, 32708}, {14582, 15328}, {15329, 14590}, {32662, 10420}, {32678, 36114}, {34333, 34834}, {39021, 16221}
X(39170) = trilinear product X(i)*X(j) for these {i,j}: {94, 2315}, {265, 1725}, {686, 32680}, {2166, 13754}, {6334, 32678}
X(39170) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 14254, 14356}, {5, 34209, 14254}, {381, 14993, 14583}

X(39171) = X(3)X(539)∩X(4)X(128)

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

X(29171) lies on the cubic K009 and these lines: {2, 1487}, {3, 539}, {4, 128}, {140, 25043}, {252, 3523}, {1656, 18370}, {1657, 19552}, {3851, 31392}, {6592, 11803}, {14378, 34157}, {25044, 32637}, {33923, 35888}

X(29171) = X(36134)-complementary conjugate of X(20184)
X(29171) = X(i)-Ceva conjugate of X(j) for these (i,j): {930, 20184}, {1487, 3519}
X(29171) = {X(18370),X(21975)}-harmonic conjugate of X(1656)

X(39172) = X(3)X(206)∩X(4)X(34168)

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

X(39172) lies on the cubic K009 and these lines: {3, 206}, {4, 34168}, {1660, 23172}, {7494, 13575}, {15261, 34158}, {17409, 19615}, {19126, 39129}, {34156, 34853}

X(39172) = X(i)-cross conjugate of X(j) for these (i,j): {184, 206}, {20968, 10316}
X(39172) = X(i)-isoconjugate of X(j) for these (i,j): {75, 17407}, {13854, 21582}
X(39172) = barycentric product X(i)*X(j) for these {i,j}: {10316, 13575}, {20806, 34207}
X(39172) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 17407}, {10316, 1370}, {20968, 3162}, {22075, 159}

X(39173) = X(3)X(513)∩X(4)X(100)

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

X(39173) lies on the cubic K009 and these lines: {3, 513}, {4, 100}, {32, 23980}, {56, 215}, {953, 6099}, {957, 2990}, {1482, 34434}, {1875, 23981}, {3428, 34431}, {10269, 10428}, {10310, 17101}, {22758, 36944}

X(39173) = isogonal conjugate of X(14266)
X(39173) = X(915)-Ceva conjugate of X(517)
X(39173) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14266}, {104, 1737}, {912, 36123}, {2252, 16082}, {8609, 34234}
X(39173) = crosssum of X(i) and X(j) for these (i,j): {912, 34332}, {1737, 11570}
X(39173) = Cundy-Parry Phi transform of X(513)
X(39173) = Cundy-Parry Psi transform of X(100)
X(39173) = barycentric product X(i)*X(j) for these {i,j}: {517, 2990}, {908, 36052}, {3262, 32655}, {6099, 10015}, {15381, 26611}, {22350, 37203}
X(39173) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 14266}, {913, 36123}, {915, 16082}, {2183, 1737}, {2990, 18816}, {6099, 13136}, {22350, 914}, {23980, 119}, {32655, 104}, {32698, 1309}, {36052, 34234}

X(39174) = X(3)X(520)∩X(4)X(74)

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

X(39174) lies on the cubic K009 and these lines: {3, 520}, {4, 74}, {20, 36831}, {32, 18877}, {184, 14385}, {185, 14264}, {389, 35908}, {1147, 14379}, {1181, 9717}, {1304, 6759}, {1552, 22802}, {10605, 34329}, {12079, 36179}, {14059, 34783}, {16836, 35910}, {18909, 36875}

X(39174) = X(36053)-complementary conjugate of X(6000)
X(39174) = X(74)-Ceva conjugate of X(6000)
X(39174) = X(i)-isoconjugate of X(j) for these (i,j): {1294, 1784}, {9033, 36043}
X(39174) = crosssum of X(30) and X(34334)
X(39174) = crossdifference of every pair of points on line {1636, 1990}
X(39174) = Cundy-Parry Phi transform of X(520)
X(39174) = Cundy-Parry Psi transform of X(107)
X(39174) = barycentric product X(6000)*X(14919)
X(39174) = barycentric quotient X(i)/X(j) for these {i,j}: {18877, 1294}, {32715, 32646}, {36131, 36043}
X(39174) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {74, 38933, 38937}, {74, 38937, 3357}

X(39175) = X(3)X(521)∩X(4)X(11)

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

X(39175) lies on the cubic K009 and these lines: {3, 521}, {4, 11}, {32, 14578}, {1035, 36110}, {1433, 1795}, {10310, 36037}, {11517, 14379}, {12675, 36819}

X(39175) = X(i)-complementary conjugate of X(j) for these (i,j): {32655, 34050}, {36052, 6001}
X(39175) = X(104)-Ceva conjugate of X (6001)
X(39175) = X(i)-isoconjugate of X(j) for these (i,j): {1295, 1785}, {2804, 36044}
X(39175) = crosssum of X(517) and X (21664)
X(39175) = Cundy-Parry Phi transform of X(521)
X(39175) = Cundy-Parry Psi transform of X(108)
X(39175) = barycentric quotient X (i)/X(j) for these {i,j}: {14578, 1295}, {32669, 36044}

X(39176) = BARYCENTRIC PRODUCT X(30)*X(186)

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

X(39176) lies on the cubics K496 and K1049 and these lines: {4, 1989}, {6, 74}, {25, 1576}, {30, 1990}, {50, 186}, {53, 3018}, {115, 6748}, {187, 16328}, {216, 32902}, {230, 1560}, {232, 37969}, {250, 34370}, {427, 5306}, {526, 2081}, {566, 35473}, {577, 7502}, {933, 11077}, {1104, 1844}, {1464, 7113}, {2420, 15469}, {2871, 8541}, {2967, 5467}, {4230, 6593}, {5063, 17409}, {5158, 18570}, {6148, 14920}, {8428, 9609}, {8739, 11086}, {8740, 11081}, {8746, 12083}, {8882, 14579}, {9607, 14836}, {10151, 16310}, {10311, 15364}, {11063, 37970}, {11070, 16240}, {13366, 34980}, {14591, 34210}, {23964, 32663}

X(39176) = isogonal conjugate of the isotomic conjugate of X(14920)
X(39176) = polar conjugate of the isotomic conjugate of X(1511)
X(39176) = orthic isogonal conjugate of X(1495)
X(39176) = X(i)-Ceva conjugate of X(j) for these (i,j): {{4, 1495}, {6, 11062}, {250, 23347}, {933, 9409}, {14920, 1511}, {23964, 14591}, {38936, 34397}}
X(39176) = X(i)-isoconjugate of X(j) for these (i,j): {{63, 5627}, {75, 11079}, {94, 35200}, {265, 2349}, {328, 2159}, {2166, 14919}, {2394, 36061}, {14380, 32680}, {14592, 36034}, {15395, 20902}, {32678, 34767}}
X(39176) = crosspoint of X(i) and X(j) for these (i,j): {{4, 14165}, {250, 14590}, {8739, 8740}}
X(39176) = crosssum of X(i) and X(j) for these (i,j): {{6, 13289}, {125, 14582}}
X(39176) = crossdifference of every pair of points on line {265, 6334}
X(39176) = barycentric product X(i)*X(j) for these {i,j}: {{1, 35201}, {4, 1511}, {6, 14920}, {15, 6111}, {16, 6110}, {25, 6148}, {30, 186}, {112, 5664}, {113, 38936}, {250, 3258}, {323, 1990}, {340, 1495}, {526, 4240}, {1637, 14590}, {1784, 6149}, {1986, 15454}, {2624, 24001}, {2914, 3471}, {3043, 14254}, {3260, 34397}, {3268, 23347}, {3284, 14165}, {7799, 14581}, {11251, 34210}, {14385, 34334}}
X(39176) = barycentric quotient X(i)/X(j) for these {i,j}: {{25, 5627}, {30, 328}, {32, 11079}, {50, 14919}, {186, 1494}, {526, 34767}, {1495, 265}, {1511, 69}, {1637, 14592}, {1990, 94}, {3258, 339}, {4240, 35139}, {5664, 3267}, {6110, 301}, {6111, 300}, {6148, 305}, {8739, 36311}, {8740, 36308}, {14270, 14380}, {14398, 14582}, {14401, 18557}, {14581, 1989}, {14920, 76}, {16240, 14254}, {19627, 18877}, {23347, 476}, {34397, 74}, {35201, 75}}

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Vietnamese points: X(39177)-X(39201)

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This preamble and centers X(39177)-X(39201) were contributed by Vu Thanh Tung, July 3, 2020.

Suppose that T1 = U1V1W1 and T2 = U2V2W2 are triangles and that T1 is not perspective to T2. Let UVW be the vertex triangle of T1 and T2. Let LU = radical axis to the circles (UV1W1) and (UV2W2), and define LV and LW cyclically.

The lines LU, LV, LW concur in a point Vn(T1,T2), here named the Vietnamese point of T1 and T2, denoted by Vn(T1, T2). This result is based on Problem 1 in the IMO (International Mathematics Olympiad) 1995 Vietnam TST (Team Selection Test).

See Vietnamese Point.

Next, suppose that P = p : q : r and U = u : v : w are points, and let T1 = cevian triangle of P, and T2 = cevian triangle of U. Then

Vn(T1, T2) = a^2 (p + q) (p + r) (u + v) (u + w) (r v - q w) : :

Let T3 = anticevian triangle of P, and T4 = anticevian triangle of U. Then

Vn(T3, T4) = (p - q - r) (-q u + p v) (u - v - w) (-r u + p w) (b^2 (-(q - r)^3 u^2 w + p^3 u w (-v + w) + p^2 (r u^2 (v - 2 w) - r (v - w)^3 + q u^2 w) + p u (q^2 (v - w) w + r^2 (u^2 - v^2 + 3 v w - 2 w^2) + q r (-u^2 + v^2 - 4 v w + 3 w^2))) + c^2 ((q - r)^3 u^2 v + p^3 u v (v - w) + p u (r^2 v (-v + w) + q^2 (u^2 - 2 v^2 + 3 v w - w^2) + q r (-u^2 + 3 v^2 - 4 v w + w^2)) + p^2 (r u^2 v + q ((v - w)^3 + u^2 (-2 v + w)))) + a^2 (p^3 u v w + (q - r) (r^2 v (-u^2 + v (v - w)) + 2 q r v w (-v + w) + q^2 w (u^2 + (v - w) w)) - p u (q^2 v w + r^2 v w + q r (-u^2 + v^2 - 4 v w + w^2)) - p^2 (q w (u^2 + (v - w) w) + r v (u^2 + v (-v + w))))) : :

Let T5 = circumcevian triangle of P, and T6 = circumcevian triangle of U. Then

Vn(T5, T6) = a^2 (b^2 (r u - p w) + c^2 (p v - q u) - a^2 (q w - r v)) : : , this being the pole wrt the circumcircle of the line PU

Let O = X(3) = circumcenter of ABC. If O, P, U are collinear, then Vn(T5, T6) is the infinite point on the line OPU.

The appearance of (i,j,k) in the following list means that the Vietnamese point of the cevian triangle of X(i) and the cevian triangle of X(j) is X(k): (1,2,3733), (1,3,39177), (1,4,4560), (1,5,39178), (1,6,39179), (2,3,23286), (2,4,523), (2,5,39180), (2,6,18105), (3,4,15412), (3,5,39181), (3,6,39182), (4,5,39183), (4,6,4580), (5,6,39184)

The appearance of (i,j,k) in the following list means that the Vietnamese point of the anticevian triangle of X(i) and the anticevian triangle of X(j) is X(k): (1,2,39185), (1,3,39186), (1,4,39187), (1,5,39188), (1,6,39189), (2,3,39190), (2,4,39192), (2,5,29192), (2,6,39193), (3,5,39194), (3,5,39195), (3,6,38861), (4,5,39196), (4,6,39197), (5,6,39198)

The appearance of (i,j,k) in the following list means that the Vietnamese point of the circumcevian triangle of X(i) and the circumcevian triangle of X(j) is X(k): (1,2,4057), (1,3,513), (1,4,39199), (1,5,39200), (1,6,667), (2,3,523), (2,4,523), (2,5,523), (2,6,669), (3,4,523), (3,5,523), (3,6,512), (4,5,523), (4,6,39201), (5,6,34952)

X(39177) = VIETNAMESE POINT OF THESE TRIANGLES: CEVIAN OF X(1) AND CEVIAN OF X(3)

X(39177) lies on these lines: {662,18315}, {933,2728}, {2167,4560}, {3733,23286}, {23696,35196}, {39178,39181}, {39179,39182}

X(39177) = isogonal conjugate of X(35307)
X(39177) = barycentric product X(i)*X(j) for these {i, j}: {54, 18155}, {95, 3737}, {261, 2616}, {693, 35196}, {933, 17880}, {2167, 4560}
X(39177) = barycentric quotient X(i)/X(j) for these (i, j): (11, 2618), (54, 4551), (60, 2617), (270, 35360), (650, 21011), (663, 21807)
X(39177) = trilinear product X(i)*X(j) for these {i, j}: {11, 18315}, {54, 4560}, {60, 15412}, {95, 7252}, {261, 2623}, {275, 23189}
X(39177) = trilinear quotient X(i)/X(j) for these (i, j): (11, 12077), (54, 4559), (60, 1625), (95, 4552), (97, 23067), (261, 14570)
X(39177) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(18662)}} and {{A, B, C, X(60), X(1262)}}
X(39177) = X(i)-isoconjugate-of-X(j) for these {i,j}: {5, 4559}, {12, 1625}, {51, 4552}, {53, 23067}
X(39177) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (11, 2618), (54, 4551), (60, 2617), (270, 35360)

X(39178) = VIETNAMESE POINT OF THESE TRIANGLES: CEVIAN OF X(1) AND CEVIAN OF X(5)

X(39178) lies on these lines: {3733,39180}, {4560,39183}, {39177,39181}, {39179,39184}

X(39178) = isogonal conjugate of X(35308)
X(39178) = barycentric quotient X(654)/X(21012)
X(39178) = X(654)-reciprocal conjugate of-X(21012)

X(39179) = VIETNAMESE POINT OF THESE TRIANGLES: CEVIAN OF X(1) AND CEVIAN OF X(6)

X(39179) lies on these lines: {82,876}, {514,1919}, {662,4562}, {765,36081}, {827,1308}, {1980,16757}, {3257,4599}, {4560,4580}, {4593,34075}, {18070,18111}, {34072,36146}, {39177,39182}, {39178,39184}

X(39179) = isogonal conjugate of X(35309)
X(39179) = barycentric product X(i)*X(j) for these {i, j}: {81, 10566}, {82, 7192}, {83, 1019}, {86, 18108}, {244, 4577}, {251, 7199}
X(39179) = barycentric quotient X(i)/X(j) for these (i, j): (58, 4553), (81, 4568), (82, 3952), (83, 4033), (244, 826), (251, 1018)
X(39179) = trilinear product X(i)*X(j) for these {i, j}: {58, 10566}, {81, 18108}, {82, 1019}, {83, 3733}, {244, 4599}, {251, 7192}
X(39179) = trilinear quotient X(i)/X(j) for these (i, j): (81, 4553), (82, 1018), (83, 3952), (86, 4568), (251, 4557), (308, 27808)
X(39179) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(16679)}} and {{A, B, C, X(59), X(593)}}
X(39179) = crossdifference of every pair of points on line {X(3954), X(20969)}
X(39179) = X(i)-isoconjugate-of-X(j) for these {i,j}: {37, 4553}, {38, 1018}, {39, 3952}, {42, 4568}
X(39179) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (58, 4553), (81, 4568), (82, 3952), (83, 4033)

X(39180) = VIETNAMESE POINT OF THESE TRIANGLES: CEVIAN OF X(2) AND CEVIAN OF X(5)


X(39180) = 2*X(15451)+X(23286)

X(39180) lies on these lines: {520,34983}, {523,37943}, {647,11063}, {1157,1510}, {1173,3470}, {1487,10412}, {3733,39178}, {18105,39184}, {33631,34212}

X(39180) = isogonal conjugate of X(35311)
X(39180) = anticomplement of the complementary conjugate of X(35442)
X(39180) = barycentric product X(i)*X(j) for these {i, j}: {3, 39183}, {5, 39181}, {288, 6368}, {520, 39284}, {523, 31626}, {525, 1173}
X(39180) = barycentric quotient X(i)/X(j) for these (i, j): (51, 35318), (184, 35324), (288, 18831), (512, 6748), (525, 1232), (647, 140)
X(39180) = trilinear product X(i)*X(j) for these {i, j}: {48, 39183}, {656, 1173}, {661, 31626}, {822, 39284}, {1953, 39181}
X(39180) = trilinear quotient X(i)/X(j) for these (i, j): (48, 35324), (525, 20879), (647, 17438), (656, 140), (661, 6748), (810, 13366)
X(39180) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(1138)}} and {{A, B, C, X(4), X(14157)}}
X(39180) = Cevapoint of X(647) and X(15451)
X(39180) = crossdifference of every pair of points on line {X(140), X(233)}
X(39180) = crosspoint of X(933) and X(34567)
X(39180) = crosssum of X(i) and X(j) for these {i,j}: {523, 12242}, {525, 6709}
X(39180) = X(647)-cross conjugate of-X(39181)
X(39180) = X(i)-isoconjugate-of-X(j) for these {i,j}: {92, 35324}, {112, 20879}, {140, 162}, {648, 17438}
X(39180) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (51, 35318), (184, 35324), (288, 18831), (512, 6748)

X(39181) = VIETNAMESE POINT OF THESE TRIANGLES: CEVIAN OF X(3) AND CEVIAN OF X(5)

X(39181) lies on these lines: {1157,1510}, {12077,15412}, {39177,39178}, {39182,39184}

X(39181) = isogonal conjugate of X(35318)
X(39181) = barycentric product X(i)*X(j) for these {i, j}: {95, 39180}, {97, 39183}, {288, 525}, {520, 39286}, {647, 31617}, {850, 20574}
X(39181) = barycentric quotient X(i)/X(j) for these (i, j): (54, 35311), (288, 648), (523, 14978), (647, 233), (1173, 35360)
X(39181) = trilinear product X(i)*X(j) for these {i, j}: {288, 656}, {810, 31617}, {822, 39286}, {1577, 20574}
X(39181) = trilinear quotient X(i)/X(j) for these (i, j): (288, 162), (656, 233), (822, 32078), (1577, 14978), (2167, 35311)
X(39181) = intersection, other than A,B,C, of conics {{A, B, C, X(96), X(28724)}} and {{A, B, C, X(97), X(1157)}}
X(39181) = Cevapoint of X(647) and X(23286)
X(39181) = crossdifference of every pair of points on line {X(233), X(3078)}
X(39181) = crosssum of X(233) and X(35441)
X(39181) = X(647)-cross conjugate of-X(39180)
X(39181) = X(i)-isoconjugate-of-X(j) for these {i,j}: {162, 233}, {163, 14978}, {823, 32078}
X(39181) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (54, 35311), (288, 648), (523, 14978), (647, 233)

X(39182) = VIETNAMESE POINT OF THESE TRIANGLES: CEVIAN OF X(3) AND CEVIAN OF X(6)

X(39182) lies on these lines: {2799,4580}, {3049,39283}, {4577,18315}, {18105,23286}, {39177,39179}, {39181,39184}

X(39182) = isogonal conjugate of X(35319)
X(39182) = barycentric product X(i)*X(j) for these {i, j}: {83, 15412}, {275, 4580}, {308, 2623}, {523, 39287}, {2167, 18070}
X(39182) = barycentric quotient X(i)/X(j) for these (i, j): (54, 1634), (82, 2617), (83, 14570), (95, 4576), (98, 35362), (251, 1625)
X(39182) = trilinear product X(i)*X(j) for these {i, j}: {54, 18070}, {82, 15412}, {83, 2616}, {661, 39287}, {2190, 4580}
X(39182) = trilinear quotient X(i)/X(j) for these (i, j): (82, 1625), (83, 2617), (798, 27374), (1821, 35362)
X(39182) = trilinear pole of the line {868, 8901}
X(39182) = intersection, other than A,B,C, of conics {{A, B, C, X(250), X(34536)}} and {{A, B, C, X(338), X(523)}}
X(39182) = Cevapoint of X(523) and X(3050)
X(39182) = X(i)-isoconjugate-of-X(j) for these {i,j}: {38, 1625}, {39, 2617}, {799, 27374}, {1634, 1953}
X(39182) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (54, 1634), (82, 2617), (83, 14570), (95, 4576)

X(39183) = VIETNAMESE POINT OF THESE TRIANGLES: CEVIAN OF X(4) AND CEVIAN OF X(5)


X(39183) = 4*X(12077)-X(15412)

X(39183) lies on these lines: {288,2413}, {523,37943}, {525,15340}, {879,1173}, {2394,39284}, {2966,33513}, {4560,39178}, {4580,39184}, {12077,15412}, {14592,31610}, {15421,31626}

X(39183) = isogonal conjugate of X(35324)
X(39183) = polar conjugate of X(35311)
X(39183) = barycentric product X(i)*X(j) for these {i, j}: {125, 33513}, {264, 39180}, {288, 18314}, {324, 39181}, {525, 39284}, {826, 39289}
X(39183) = barycentric quotient X(i)/X(j) for these (i, j): (4, 35311), (53, 35318), (288, 18315), (512, 13366), (514, 17168), (523, 140)
X(39183) = trilinear product X(i)*X(j) for these {i, j}: {92, 39180}, {288, 2618}, {656, 39284}, {1173, 1577}
X(39183) = trilinear quotient X(i)/X(j) for these (i, j): (92, 35311), (288, 36134), (523, 17438), (656, 22052), (661, 13366), (693, 17168)
X(39183) = Gibert-Simson transform of X(1173)
X(39183) = trilinear pole of the line {125, 137}
X(39183) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(37779)}} and {{A, B, C, X(4), X(39454)}}
X(39183) = Cevapoint of X(i) and X(j) for these {i,j}: {523, 12077}, {647, 1510}
X(39183) = crossdifference of every pair of points on line {X(13366), X(22052)}
X(39183) = X(1487)-anticomplementary conjugate of-X(21294)
X(39183) = X(115)-cross conjugate of-X(34110)
X(39183) = X(i)-isoconjugate-of-X(j) for these {i,j}: {48, 35311}, {110, 17438}, {140, 163}, {162, 22052}
X(39183) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (4, 35311), (53, 35318), (288, 18315), (512, 13366)

X(39184) = VIETNAMESE POINT OF THESE TRIANGLES: CEVIAN OF X(5) AND CEVIAN OF X(6)

X(39184) lies on these lines: {4580,39183}, {18105,39180}, {39178,39179}, {39181,39182}

X(39184) = barycentric product X(924)*X(39289)

X(39185) = VIETNAMESE POINT OF THESE TRIANGLES: ANTICEVIAN OF X(1) AND ANTICEVIAN OF X(2)


X(39185) = 4*X(2325)-X(5057) = 2*X(4542)+X(20095)

X(39185) lies on these lines: {2,7336}, {36,726}, {59,4552}, {100,522}, {190,660}, {192,5091}, {346,21280}, {514,6163}, {517,3685}, {523,765}, {901,4427}, {1618,2397}, {2325,4071}, {3257,4977}, {3259,17777}, {3326,27542}, {3667,36237}, {3717,5176}, {3952,4076}, {4036,36804}, {4542,20095}, {5376,30580}, {5990,33889}, {6550,32028}, {17262,38530}, {17280,24250}, {25253,38568}, {32939,34583}

X(39185) = reflection of X(i) in X(j) for these (i,j): (5176, 3717), (36236, 765)
X(39185) = anticomplement of X(7336)
X(39185) = barycentric product X(190)*X(17719)
X(39185) = trilinear product X(100)*X(17719)
X(39185) = X(1110)-anticomplementary conjugate of-X(17036)

X(39186) = VIETNAMESE POINT OF THESE TRIANGLES: ANTICEVIAN OF X(1) AND ANTICEVIAN OF X(3)

X(39186) lies on these lines: {1461,3669}, {7128,7178}

X(39186) = barycentric product X(1119)*X(39189)
X(39186) = trilinear product X(1435)*X(39189)

X(39187) = VIETNAMESE POINT OF THESE TRIANGLES: ANTICEVIAN OF X(1) AND ANTICEVIAN OF X(4)

X(39187) lies on these lines: {}

X(39188) = VIETNAMESE POINT OF THESE TRIANGLES: ANTICEVIAN OF X(1) AND ANTICEVIAN OF X(5)

X(39188) lies on these lines: {}

X(39189) = VIETNAMESE POINT OF THESE TRIANGLES: ANTICEVIAN OF X(1) AND ANTICEVIAN OF X(6)

X(39189) lies on these lines: {59,23067}, {521,1331}

X(39189) = barycentric product X(1265)*X(39186)

X(39190) = VIETNAMESE POINT OF THESE TRIANGLES: ANTICEVIAN OF X(2) AND ANTICEVIAN OF X(3)

X(39190) lies on these lines: {112,7472}, {523,32713}, {16177,39221}, {16237,39193}, {39191,39194}, {39192,39195}

X(39191) = VIETNAMESE POINT OF THESE TRIANGLES: ANTICEVIAN OF X(2) AND ANTICEVIAN OF X(4)

X(39191) lies on these lines: {648,15384}, {39190,39194}, {39192,39196}

X(39192) = VIETNAMESE POINT OF THESE TRIANGLES: ANTICEVIAN OF X(2) AND ANTICEVIAN OF X(5)

X(39192) lies on these lines: {1625,2451}, {16177,17847}, {39190,39195}, {39191,39196}

X(39193) = VIETNAMESE POINT OF THESE TRIANGLES: ANTICEVIAN OF X(2) AND ANTICEVIAN OF X(6)

X(39193) lies on these lines: {99,935}, {250,14570}, {476,930}, {523,4558}, {2453,9723}, {4226,38861}, {14060,34978}, {14480,31941}, {16237,39190}, {28437,38971}, {30717,36189}, {38552,39803}

X(39194) = VIETNAMESE POINT OF THESE TRIANGLES: ANTICEVIAN OF X(3) AND ANTICEVIAN OF X(4)

X(39194) lies on these lines: {112,20580}, {39190,39191}, {39195,39196}

X(39195) = VIETNAMESE POINT OF THESE TRIANGLES: ANTICEVIAN OF X(3) AND ANTICEVIAN OF X(5)

X(39195) lies on these lines: {39190,39192}, {39194,39196}

X(39196) = VIETNAMESE POINT OF THESE TRIANGLES: ANTICEVIAN OF X(4) AND ANTICEVIAN OF X(5)

X(39196) lies on these lines: {39191,39192}, {39194,39195}

X(39197) = VIETNAMESE POINT OF THESE TRIANGLES: ANTICEVIAN OF X(4) AND ANTICEVIAN OF X(6)

X(39197) lies on these lines: {}

X(39198) = VIETNAMESE POINT OF THESE TRIANGLES: ANTICEVIAN OF X(5) AND ANTICEVIAN OF X(6)

X(39198) lies on these lines: {}

X(39199) = VIETNAMESE POINT OF THESE TRIANGLES: CIRCUMCEVIAN OF X(1) AND CIRCUMCEVIAN OF X(4)

X(39199) lies on these lines: {3,522}, {36,238}, {56,1459}, {186,523}, {514,39225}, {579,23146}, {900,18861}, {958,20316}, {1604,14330}, {1769,23226}, {1946,6129}, {2178,6586}, {2605,8677}, {2975,20293}, {3667,39476}, {3738,23187}, {4017,8648}, {4458,23093}, {7662,23864}, {8672,39480}, {9000,22769}, {10016,20999}, {11249,32475}, {11340,27486}

X(39199) = reflection of X(i) in X(j) for these (i,j): (3, 39226), (3733, 34948), (4057, 39200)
X(39199) = isogonal conjugate of the anticomplement of X(38983)
X(39199) = barycentric product X(i)*X(j) for these {i, j}: {653, 38983}, {905, 7412}
X(39199) = trilinear product X(i)*X(j) for these {i, j}: {108, 38983}, {1459, 7412}
X(39199) = circumcircle-inverse of-X(31866)
X(39199) = pole of the trilinear polar of X(653) wrt circumcircle
X(39199) = crossdifference of every pair of points on line {X(37), X(216)}
X(39199) = crosspoint of X(i) and X(j) for these {i,j}: {54, 109}, {58, 108}, {100, 1167}
X(39199) = crosssum of X(i) and X(j) for these {i,j}: {10, 521}, {117, 39471}, {513, 1210}, {514, 16608}
X(39199) = X(653)-Ceva conjugate of-X(6)
X(39199) = X(515)-vertex conjugate of-X(31866)

X(39200) = VIETNAMESE POINT OF THESE TRIANGLES: CIRCUMCEVIAN OF X(1) AND CIRCUMCEVIAN OF X(5)


X(39200) = 3*X(26275)-X(39534)

X(39200) lies on these lines: {3,900}, {21,26144}, {25,26275}, {36,238}, {404,26078}, {522,39225}, {523,2070}, {665,2178}, {1011,4800}, {1324,35013}, {1486,8638}, {1769,8648}, {1960,8677}, {2815,11249}, {3145,28114}, {3667,39226}, {4189,27545}, {4435,36744}, {6006,39476}, {6085,34139}, {6089,14270}, {6615,23226}, {7485,31131}, {13738,28284}, {14667,15914}, {16286,24959}, {16414,24920}, {16419,30792}, {23067,23703}, {25923,37248}, {28348,28396}

X(39200) = midpoint of X(4057) and X(39199)
X(39200) = reflection of X(3) in X(39478)
X(39200) = isogonal conjugate of the anticomplement of X(38984)
X(39200) = barycentric product X(655)*X(38984)
X(39200) = trilinear product X(2222)*X(38984)
X(39200) = pole of the trilinear polar of X(655) wrt circumcircle
X(39200) = crossdifference of every pair of points on line {X(37), X(570)}
X(39200) = crosspoint of X(i) and X(j) for these {i,j}: {58, 2222}, {100, 36052}, {1166, 36078}
X(39200) = crosssum of X(i) and X(j) for these {i,j}: {10, 3738}, {513, 1737}, {522, 3814}, {1209, 6369}
X(39200) = X(655)-Ceva conjugate of-X(6)

X(39201) = VIETNAMESE POINT OF THESE TRIANGLES: CIRCUMCEVIAN OF X(4) AND CIRCUMCEVIAN OF X(6)

Barycentrics    a^4*(b - c)*(b + c)*(-a^2 + b^2 + c^2)^2 : :
Trilinears    sin^2 2A sin(B - C) : :
Trilinears    (sin^2 A) (cos A) (sin 2B - sin 2C) : :
Trilinears    (sin^2 A) (cos A) (tan B - tan C) : :
Trilinears    (sin 2A) (b sin(A - B) - c sin(A - C)) : :
X(39201) = 3*X(381)-4*X(39510) = 2*X(647)+X(9409) = X(669)-4*X(14270) = X(22089)-4*X(39228)

For a given point P, the following trilinear polars concur:
  trilinear polar of the isogonal conjugate of the isotomic conjugate of P;
  trilinear polar of the polar conjugate of the isotomic conjugate of P;
  trilinear polar of the isogonal conjugate of the polar conjugate of P;
  trilinear polar of the isotomic conjugate of the polar conjugate of P.
The point of concurrence, Q, is the crossdifference of the isotomic conjugate of P and the polar conjugate of P, which is the barycentric product P*X(647). If P = X(3), the point of concurrence is X(39201). See also X(15451). (Randy Hutson, September 30, 2020)

X(39201) is the perspector of the vertex-triangle and side-triangle of the circumanticevian triangles of X(3) and X(4). (Randy Hutson, September 30, 2020)

The trilinear polar of X(39201) passes through X(34980).

X(39201) lies on these lines: {3,525}, {25,9209}, {107,34538}, {110,2713}, {157,21397}, {186,523}, {187,237}, {381,39510}, {418,34983}, {520,4091}, {684,8673}, {826,39477}, {852,22264}, {1141,32439}, {1576,32640}, {1624,37937}, {1640,37457}, {2351,8029}, {2394,35473}, {2485,17994}, {3049,23200}, {3398,39517}, {3431,14380}, {3800,5926}, {6130,14618}, {7485,30474}, {7502,18556}, {7927,39481}, {11171,39500}, {14379,23103}, {20975,34982}, {23878,25644}

X(39201) = midpoint of X(9409) and X(15451)
X(39201) = reflection of X(i) in X(j) for these (i,j): (3, 39228), (669, 34952), (14618, 6130), (15451, 647), (17994, 2485), (22089, 3), (34952, 14270)
X(39201) = isogonal conjugate of X(6528)
X(39201) = Gibert circumtangential conjugate of X(107)
X(39201) = isotomic conjugate of the polar conjugate of X(3049)
X(39201) = polar conjugate of the isotomic conjugate of X(32320)
X(39201) = anticomplement of the complementary conjugate of X(35071)
X(39201) = barycentric product X(i)*X(j) for these {i, j}: {1, 822}, {3, 647}, {4, 32320}, {6, 520}, {31, 24018}, {32, 3265}
X(39201) = barycentric quotient X(i)/X(j) for these (i, j): (3, 6331), (25, 15352), (31, 823), (32, 107), (48, 811), (163, 23999)
X(39201) = trilinear product X(i)*X(j) for these {i, j}: {3, 810}, {6, 822}, {19, 32320}, {31, 520}, {32, 24018}, {42, 23224}
X(39201) = trilinear quotient X(i)/X(j) for these (i, j): (3, 811), (6, 823), (19, 15352), (25, 36126), (31, 107), (32, 24019)
X(39201) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(237)}} and {{A, B, C, X(6), X(3331)}}
X(39201) = circumcircle-inverse of-X(18338)
X(39201) = pole of the trilinear polar of X(107) wrt circumcircle
X(39201) = pole of the trilinear polar of X(23286) wrt Jerabek hyperbola
X(39201) = crossdifference of every pair of points on line {X(2), X(216)}
X(39201) = crosspoint of X(i) and X(j) for these {i,j}: {3, 32661}, {6, 107}, {54, 112}, {184, 1576}
X(39201) = crosssum of X(i) and X(j) for these {i,j}: {2, 520}, {4, 14618}, {5, 525}, {107, 648}
X(39201) = X(i)-Ceva conjugate of-X(j) for these (i,j): (3, 3269), (48, 39687), (99, 39643), (107, 6)
X(39201) = X(i)-isoconjugate-of-X(j) for these {i,j}: {2, 823}, {4, 811}, {19, 6331}, {27, 6335}
X(39201) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (3, 6331), (25, 15352), (31, 823), (32, 107)
X(39201) = X(i)-vertex conjugate of-X(j) for these {i,j}: {6, 3331}, {1503, 18338}

leftri

Vertices of the real and imaginary Steiner ellipses: X(39202)-X(39209)

rightri

This preamble and centers X(39202)-X(39209) were contributed by Peter Moses, July 3, 2020.

The vertices of the Steiner circumellipse and the Steiner inellipse are all given by the following form for 1st barycentric:

2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - sgn1*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] + 2*const*S*sgn2*Sqrt[Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]/(a^2 + b^2 + c^2 + 2*sgn1*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4])]*Sqrt[sgn1*(-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*sgn1*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2)]

The Steiner circumellipse has const = 1. Specifically, X(39202) is given by (sgn1,sgn2) = (1,1)
X(39203) is given by (sgn1,sgn2) = (1,-1)
X(39204) is given by (sgn1,sgn2) = (-1,1)
X(39205) is given by (sgn1,sgn2) = (-1,-1)

The Steiner inellipse has const = 2. Specifically, X(39206) is given by (sgn1,sgn2) = (1,1)
X(39207) is given by (sgn1,sgn2) = (1,-1)
X(39208) is given by (sgn1,sgn2) = (-1,1)
X(39209) is given by (sgn1,sgn2) = (-1,-1)

The points X(39202) and X(39203) are endpoints of the major axis of the Steiner circumellipse, and X(39204) and X(39205) lies on the minor axis. The points X(39206) and X(39207) are endpoints of the major axis of the Steiner inellipse, and X(39208) and X(39209) lies on the minor axis.

See the preamble just before X(39158) for foci of the two ellipses.

X(39202) = 1ST VERTEX OF THE STEINER CIRCUMELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] + 2*S*Sqrt[Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]/(a^2 + b^2 + c^2 + 2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4])]*Sqrt[(-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2)]
X(39202) = X[39203] - 4 X[39206], 3 X[39203] - 4 X[39207], 3 X[39206] - X[39207]

X(39202) lies on curves Q059 and Q135 and the Steiner circumellipse and this line: {2, 1341}

X(39202) = reflection of X(i) in X(j) for these {i,j}: {2, 39206}, {39203, 2}
X(39202) = anticomplement of X(39207)
X(39202) = X(3414)-cross conjugate of X(39203)
X(39202) = cevapoint of X(3414) and X(39206)
X(39202) = barycentric quotient X(6189)/X(39203)

X(39203) = 2ND VERTEX OF THE STEINER CIRCUMELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] - 2*SSqrt[Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]/(a^2 + b^2 + c^2 + 2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4])]*Sqrt[(-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2)]
X(39203) = 3 X[39202] - 4 X[39206], X[39202] - 4 X[39207], X[39206] - 3 X[39207]

X(39203) lies on curves Q059 and Q135 and the Steiner circumellipse and this line: {2, 1341}

X(39203) = reflection of X(i) in X(j) for these {i,j}: {2, 39207}, {39202, 2}
X(39203) = anticomplement of X (39206)
X(39203) = X(3414)-cross conjugate of X (39202)
X(39203) = cevapoint of X(3414) and X (39207)
X(39203) = barycentric quotient X (6189)/X(39202)

X(39204) = 3RD VERTEX OF THE STEINER CIRCUMELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 +(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] + 2*S*Sqrt[Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]/(a^2 + b^2 + c^2 - 2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4])]*Sqrt[-(-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 - 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2)]
X(39204) = X[39205] - 4 X[39208], 3 X[39205] - 4 X[39209], 3 X[39208] - X[39209]

X(39204) lies on curves Q059 and Q135 and the Steiner circumellipse and this line: {2, 1341}

X(39204) = reflection of X(i) in X(j) for these {i,j}: {2, 39208}, {39205, 2}
X(39204) = anticomplement of X(39209)
X(39204) = X(3413)-cross conjugate of X(39205)
X(39204) = cevapoint of X(3413) and X(39208)
X(39204) = barycentric quotient X(6190)/X(39205)

X(39205) = 4TH VERTEX OF THE STEINER CIRCUMELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 + (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] - 2*S*Sqrt[Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]/(a^2 + b^2 + c^2 - 2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4])]*Sqrt[-(-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 - 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2)]
X(39205) = 3 X[39204] - 4 X[39208], X[39204] - 4 X[39209], X[39208] - 3 X[39209]

X(39205) lies on curves Q059 and Q135 and the Steiner circumellipse and this line: {2, 1341}

X(39205) = reflection of X(i) in X(j) for these {i,j}: {2, 39209}, {39204, 2}
X(39205) = anticomplement of X (39208)
X(39205) = X(3413)-cross conjugate of X(39204)
X(39205) = cevapoint of X(3413) and X(39209)
X(39205) = barycentric quotient X (6190)/X(39204)

X(39206) = 1ST VERTEX OF THE STEINER INELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] + 4*S*Sqrt[Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]/(a^2 + b^2 + c^2 + 2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4])]*Sqrt[(-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2)]
X(39206) = 3 X[39202] + X[39203], 2 X[39202] + X[39207], 2 X[39203] - 3 X[39207]

X(39206) lies on the Steiner inellipse and this line: {2, 1341}

X(39206) = complement of X(39203)
X(39206) = midpoint of X(2) and X(39202)
X(39206) = reflection of X(39207) in X(2)
X(39206) = X(39202)-Ceva conjugate of X(3414)
X(39206) = barycentric quotient X(39022)/X(39207)

X(39207) = 2ND VERTEX OF THE STEINER INELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] - 4*S*Sqrt[Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]/(a^2 + b^2 + c^2 + 2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4])]*Sqrt[(-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2)]
X(39207) = X[39202] + 3 X[39203], 2 X[39202] - 3 X[39206], 2 X[39203] + X[39206]}

X(39207) X lies on the Steiner inellipse and this line: {2, 1341}

X(39207) = complement of X (39202)
X(39207) = midpoint of X(2) and X (39203)
X(39207) = reflection of X(39206) in X (2)
X(39207) = X(39203)-Ceva conjugate of X (3414)
X(39207) = barycentric quotient X (39022)/X(39206)

X(39208) = 3RD VERTEX OF THE STEINER INELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 + (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] +4*S*Sqrt[Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]/(a^2 + b^2 + c^2 - 2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4])]*Sqrt[-(-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 - 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2)]
X(39208) = 3 X[39204] + X[39205], 2 X[39204] + X[39209], 2 X[39205] - 3 X[39209]

X(39208) lies on the Steiner inellipse and this line: {2, 1340}

X(39208) = complement of X(39205)
X(39208) = midpoint of X(2) and X(39204)
X(39208) = reflection of X(39209) in X(2)
X(39208) = X(39204)-Ceva conjugate of X(3413)
X(39208) = barycentric quotient X(39023)/X(39209)

X(39209) = 4TH VERTEX OF THE STEINER INELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 + (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] - 4*S*Sqrt[Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]/(a^2 + b^2 + c^2 - 2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4])]*Sqrt[-(-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 - 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2)]
X(39209) = X[39204] + 3 X[39205], 2 X[39204] - 3 X[39208], 2 X[39205] + X[39208]

X(39209) lies on the Steiner inellipse and this line: {2, 1340}

X(39209) = complement of X(39204)
X(39209) = midpoint of X(2) and X(39205)
X(39209) = reflection of X(39208) in X(2)
X(39209) = X(39205)-Ceva conjugate of X(3413)
X(39209) = barycentric quotient X(39023)/X(39208)

leftri

Centers of Vietnamese circles: X(39210)-X(39228)

rightri

This preamble and centers X(39210)-X(39228) were contributed by Vu Thanh Tung, July 4, 2020.

Suppose that T1 = U1V1W1 and T2 = U2V2W2 are triangles and that T1 is not perspective to T2. Let UVW be the vertex triangle of T1 and T2. Let LU = radical axis to the circles (UV1W1) and (UV2W2), and define LV and LW cyclically.

The lines LU, LV, LW concur in a point Vn(T1,T2), the Vietnamese point of T1 and T2, denoted by Vn(T1, T2), introduced in the preamble just before X(39177). Continuing, let U0 be the point, other than U, in which the circles (UV1W1) and (UV2W2) intersect and define V0 and W0 cyclically. Let Mi be the Miquel point of Ti for i = 1, 2; e.g.,

M1 = (UV1W1)∩(UV1W1)∩(U1V1W).

The six points U0,V0,W0,M1,M2,Vn lie on a circle, here named the Vietnamese circle of T1 and T2, denoted by v(T1,T2). The center of this circle is denoted by Vo(T1,T2).

See Vietnamese Circle.

If P = p : q : r and U = u : v : w are distinct with distinct circumcevian triangles T1, T2 then v(T1,T2) passes through the circumcenter, O, and is the circumcircle-inverse of the line PU. If P, U, O are collinear then v(T1,T2) is the circle of infinite radius - that is, the line at infinity, and Vo(T1,T2) = O.

If T1 = cevian triangle of P, and T2 = cevian triangle of U, then the appearance of (i,j,k) in the following list means that the center of the Vietnamese circle of the cevian triangle of X(i) and the cevian triangle of X(j) is X(k): (2,3,37084), (3,4,15451), (2,4,523)

The appearance of (i,j,k) in the following list means that center of the Vietnamese circle of the anticevian triangle of X(i) and the anticevian triangle of X(j) is X(k): (1,2,2957), (2,6,30715)

The appearance of (i,j,k) in the following list means that the center of the Vietnamese circle of the circumcevian triangle of X(i) and the circumcevian triangle of X(j) is X(k): (1,3,513), (2,3,523), (2,4,523), (2,6,5926), (3,4,523), (3,6,523)

X(39210) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: CEVIAN OF X(1) AND CEVIAN OF X(2)

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

X(39210) lies on these lines: {3,3733}, {36,4017}, {104,2758}, {140,31946}, {514,23224}, {522,34948}, {572,798}, {573,20981}, {667,3667}, {1385,4132}, {1511,38614}, {2070,39212}, {2827,4057}, {2975,4404}, {3737,23226}, {5428,28217}, {8648,21173}, {14838,23189}, {23961,39225}, {24006,37117}

X(39210) = midpoint of X(3) and X(3733)
X(39210) = reflection of X(i) in X(j) for these (i,j): (31946, 140), (39225, 39227)
X(39210) = crosssum of X(661) and X(7069)

X(39211) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: CEVIAN OF X(1) AND CEVIAN OF X(3)

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

X(39211) lies on these lines: {}

X(39212) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: CEVIAN OF X(1) AND CEVIAN OF X(4)

Barycentrics    a*(b - c)*(b^5 - b^3*c^2 - b^2*c^3 + c^5 + a^3*(b^2 + c^2) - a^2*(b^3 + c^3) - a*(b^4 + c^4)) : :
X(39212) = X(4804)-4*X(9955) = 2*X(4913)+X(12699) = 4*X(9956)-3*X(21052)

X(39212) lies on these lines: {3,14838}, {4,4560}, {5,1577}, {355,3907}, {517,4041}, {810,5396}, {830,38324}, {905,8760}, {946,4151}, {1491,3309}, {2070,39210}, {2530,2826}, {3737,30212}, {4490,28537}, {4705,28473}, {4804,9955}, {4833,6003}, {4913,12699}, {5398,21761}, {6002,36716}, {6362,6841}, {6589,23782}, {7254,36742}, {9956,21052}, {14663,38583}, {23882,39536}

X(39212) = midpoint of X(4) and X(4560)
X(39212) = reflection of X(i) in X(j) for these (i,j): (3, 14838), (1577, 5)
X(39212) = trilinear product X(522)*X(20122)

X(39213) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: CEVIAN OF X(1) AND CEVIAN OF X(6)

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

X(39213) lies on these lines: {}

X(39214) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: CEVIAN OF X(2) AND CEVIAN OF X(6)

Barycentrics    a^2*(b - c)*(b + c)*(a^8 - 2*b^4*c^4 - a^6*(b^2 + c^2) - a^4*(b^4 + 3*b^2*c^2 + c^4) + a^2*(b^6 + c^6)) : :
X(39214) = 2*X(5092)+X(14318) = 4*X(5926)-X(11616)

X(39214) lies on these lines: {3,18105}, {182,688}, {512,25644}, {523,5926}, {669,9148}, {882,39750}, {924,7630}, {1510,39495}, {2793,21006}, {5092,14318}, {11620,39216}, {23878,34952}

X(39214) = midpoint of X(3) and X(18105)
X(39214) = reflection of X(39216) in X(11620)

X(39215) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: CEVIAN OF X(3) AND CEVIAN OF X(6)

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

X(39215) lies on these lines: {}

X(39216) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: CEVIAN OF X(4) AND CEVIAN OF X(6)

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

X(39216) lies on these lines: {4,4580}, {5,23285}, {523,11799}, {2485,30209}, {2514,32472}, {2799,15451}, {3005,14420}, {6249,33752}, {11615,21006}, {11620,39214}, {16229,17994}

X(39216) = midpoint of X(4) and X(4580)
X(39216) = reflection of X(i) in X(j) for these (i,j): (21006, 11615), (23285, 5), (39214, 11620)

X(39217) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: ANTICEVIAN OF X(1) AND ANTICEVIAN OF X(3)

Barycentrics    a^2*(a^14 + 14*a^12*b*c - 3*a^13*(b + c) - b^2*(b - c)^6*c^2*(b + c)^4 + a^11*(8*b^3 - 15*b^2*c - 15*b*c^2 + 8*c^3) - a^10*(7*b^4 + 16*b^3*c - 47*b^2*c^2 + 16*b*c^3 + 7*c^4) - a^9*(3*b^5 - 40*b^4*c + 32*b^3*c^2 + 32*b^2*c^3 - 40*b*c^4 + 3*c^5) + a^2*(b - c)^4*(b + c)^2*(3*b^6 - 6*b^5*c + 2*b^4*c^2 + 20*b^3*c^3 + 2*b^2*c^4 - 6*b*c^5 + 3*c^6) + a^6*(b - c)^2*(3*b^6 + 34*b^5*c - 9*b^4*c^2 - 114*b^3*c^3 - 9*b^2*c^4 + 34*b*c^5 + 3*c^6) + a^8*(8*b^6 - 24*b^5*c - 45*b^4*c^2 + 114*b^3*c^3 - 45*b^2*c^4 - 24*b*c^5 + 8*c^6) + a^5*(b - c)^2*(7*b^7 - 25*b^6*c - 53*b^5*c^2 + 66*b^4*c^3 + 66*b^3*c^4 - 53*b^2*c^5 - 25*b*c^6 + 7*c^7) - a^7*(8*b^7 + 2*b^6*c - 94*b^5*c^2 + 83*b^4*c^3 + 83*b^3*c^4 - 94*b^2*c^5 + 2*b*c^6 + 8*c^7) + a^3*b*(b - c)^2*c*(17*b^7 - 21*b^6*c - 48*b^5*c^2 + 44*b^4*c^3 + 44*b^3*c^4 - 48*b^2*c^5 - 21*b*c^6 + 17*c^7) - 2*a^4*(b - c)^2*(4*b^8 + 3*b^7*c - 29*b^6*c^2 - 9*b^5*c^3 + 42*b^4*c^4 - 9*b^3*c^5 - 29*b^2*c^6 + 3*b*c^7 + 4*c^8) - a*(b - c)^4*(b^9 + 2*b^8*c - 2*b^7*c^2 + 3*b^6*c^3 + 20*b^5*c^4 + 20*b^4*c^5 + 3*b^3*c^6 - 2*b^2*c^7 + 2*b*c^8 + c^9)) : :

X(39217) lies on these lines: {}

X(39218) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: ANTICEVIAN OF X(1) AND ANTICEVIAN OF X(4)

Barycentrics    a*(3*a^23-8*a^22*(b+c)-2*(4*b^2-23*b*c+4*c^2)*a^21+(b+c)*(43*b^2-94*b*c+43*c^2)*a^20-3*(11*b^4+11*c^4+(38*b^2-91*b*c+38*c^2)*b*c)*a^19-(b+c)*(48*b^4+48*c^4-(359*b^2-604*b*c+359*c^2)*b*c)*a^18+(158*b^6+158*c^6-(214*b^4+214*c^4+(549*b^2-1190*b*c+549*c^2)*b*c)*b*c)*a^17-(b+c)*(139*b^6+139*c^6+(203*b^4+203*c^4-(1647*b^2-2608*b*c+1647*c^2)*b*c)*b*c)*a^16-2*(101*b^6+101*c^6-(318*b^4+318*c^4+(309*b^2-620*b*c+309*c^2)*b*c)*b*c)*(b-c)^2*a^15+4*(b^2-c^2)*(b-c)*(112*b^6+112*c^6-(15*b^4+15*c^4+(339*b^2-575*b*c+339*c^2)*b*c)*b*c)*a^14-4*(2*b^8+2*c^8+(323*b^6+323*c^6-(192*b^4+192*c^4+(269*b^2-744*b*c+269*c^2)*b*c)*b*c)*b*c)*(b-c)^2*a^13-2*(b^2-c^2)*(b-c)*(241*b^8+241*c^8-2*(225*b^6+225*c^6-(33*b^4+33*c^4+(626*b^2-1195*b*c+626*c^2)*b*c)*b*c)*b*c)*a^12+2*(115*b^10+115*c^10+(452*b^8+452*c^8-(722*b^6+722*c^6-(878*b^4+878*c^4+(823*b^2-2612*b*c+823*c^2)*b*c)*b*c)*b*c)*b*c)*(b-c)^2*a^11+2*(b^2-c^2)*(b-c)*(88*b^10+88*c^10-(403*b^8+403*c^8-2*(394*b^6+394*c^6-(139*b^4+139*c^4+5*(178*b^2-293*b*c+178*c^2)*b*c)*b*c)*b*c)*b*c)*a^10-2*(86*b^12+86*c^12+(70*b^10+70*c^10-(129*b^8+129*c^8-2*(631*b^6+631*c^6-(541*b^4+541*c^4+(698*b^2-1125*b*c+698*c^2)*b*c)*b*c)*b*c)*b*c)*b*c)*(b-c)^2*a^9+2*(b^2-c^2)*(b-c)^3*(29*b^10+29*c^10+(119*b^8+119*c^8-(193*b^6+193*c^6-2*(216*b^4+216*c^4+(256*b^2-411*b*c+256*c^2)*b*c)*b*c)*b*c)*b*c)*a^8+(b^2-c^2)^2*(b-c)^2*(7*b^10+7*c^10-(82*b^8+82*c^8-(451*b^6+451*c^6+2*(204*b^4+204*c^4-(341*b^2-1498*b*c+341*c^2)*b*c)*b*c)*b*c)*b*c)*a^7-4*(b^2-c^2)*(b-c)^3*(14*b^12+14*c^12-(b^10+c^10-(31*b^8+31*c^8+(162*b^6+162*c^6-(68*b^4+68*c^4-(31*b^2+462*b*c+31*c^2)*b*c)*b*c)*b*c)*b*c)*b*c)*a^6+2*(b^2-c^2)^2*(b-c)^2*(16*b^12+16*c^12+(7*b^10+7*c^10-(10*b^8+10*c^8-(151*b^6+151*c^6-2*(96*b^4+96*c^4+(175*b^2-186*b*c+175*c^2)*b*c)*b*c)*b*c)*b*c)*b*c)*a^5+(b^2-c^2)^3*(b-c)*(7*b^12+7*c^12-(20*b^10+20*c^10-(146*b^8+146*c^8-(320*b^6+320*c^6-(345*b^4+345*c^4+4*(37*b^2-89*b*c+37*c^2)*b*c)*b*c)*b*c)*b*c)*b*c)*a^4-(b^2-c^2)^3*(b-c)^3*(9*b^10+9*c^10-(4*b^8+4*c^8+(55*b^6+55*c^6+2*(12*b^4+12*c^4+(89*b^2+132*b*c+89*c^2)*b*c)*b*c)*b*c)*b*c)*a^2*b*c-(5*b^8+5*c^8-(6*b^6+6*c^6-7*(15*b^4+15*c^4-22*(b-c)^2*b*c)*b*c)*b*c)*(b^2-c^2)^6*a^3-(b^2-c^2)^6*(b-c)^2*(2*b^8+2*c^8-(6*b^6+6*c^6+(b^2+12*b*c+c^2)*(b+c)^2*b*c)*b*c)*a+(b^2-c^2)^7*(b-c)^3*(b^6+c^6+(b^2+b*c+c^2)*(b+c)^2*b*c))*(a^2-b^2+c^2)*(a^2+b^2-c^2): :

X(39218) lies on these lines: {}

X(39219) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: ANTICEVIAN OF X(1) AND ANTICEVIAN OF X(6)

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

X(39219) lies on these lines:{}

X(39220) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: ANTICEVIAN OF X(2) AND ANTICEVIAN OF X(3)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^16 - 2*a^14*(b^2 + c^2) - 2*a^12*(b^4 - 5*b^2*c^2 + c^4) - 2*a^6*(b^2 - c^2)^2* (3*b^6 - 4*b^4*c^2 - 4*b^2*c^4 + 3*c^6) + a^10*(6*b^6 - 8*b^4*c^2 - 8*b^2*c^4 + 6*c^6) + a^8*(-16*b^6*c^2 + 33*b^4*c^4 - 16*b^2*c^6) - (b^2 - c^2)^4*(b^8 + 2*b^6*c^2 + 2*b^2*c^6 + c^8) + a^4*(b^2 - c^2)^2*(2*b^8 + 8*b^6*c^2 - 13*b^4*c^4 + 8*b^2*c^6 + 2*c^8) + 2*a^2*(b^2 - c^2)^2*(b^10 - 3*b^8*c^2 + b^6*c^4 + b^4*c^6 - 3*b^2*c^8 + c^10)) : :

X(39220) lies on these lines: {}

X(39221) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: ANTICEVIAN OF X(2) AND ANTICEVIAN OF X(4)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)* (a^22 - 2*a^20*(b^2 + c^2) + a^18*(-8*b^4 + 22*b^2*c^2 - 8*c^4) - 9*a^12*(b^2 - c^2)^2*(2*b^6 - 17*b^4*c^2 - 17*b^2*c^4 + 2*c^6) + a^16*(29*b^6 - 31*b^4*c^2 - 31*b^2*c^4 + 29*c^6) - a^14*(24*b^8 + 70*b^6*c^2 - 189*b^4*c^4 + 70*b^2*c^6 + 24*c^8) + a^10*(b^2 - c^2)^2*(38*b^8 - 24*b^6*c^2 - 289*b^4*c^4 - 24*b^2*c^6 + 38*c^8) - (b^2 - c^2)^6*(b^10 + 4*b^8*c^2 + 5*b^6*c^4 + 5*b^4*c^6 + 4*b^2*c^8 + c^10) + a^4*(b^2 - c^2)^4*(4*b^10 + 21*b^8*c^2 - 21*b^6*c^4 - 21*b^4*c^6 + 21*b^2*c^8 + 4*c^10) - a^8*(b^2 - c^2)^2*(12*b^10 + 91*b^8*c^2 - 173*b^6*c^4 - 173*b^4*c^6 + 91*b^2*c^8 + 12*c^10) + a^2*(b^2 - c^2)^4*(2*b^12 - 6*b^10*c^2 - 13*b^8*c^4 + 2*b^6*c^6 - 13*b^4*c^8 - 6*b^2*c^10 + 2*c^12) - a^6*(b^2 - c^2)^2*(9*b^12 -48*b^10*c^2 - 40*b^8*c^4 + 190*b^6*c^6 - 40*b^4*c^8 - 48*b^2*c^10 + 9*c^12)) : :

X(39221) lies on these lines: {}

X(39222) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: ANTICEVIAN OF X(3) AND ANTICEVIAN OF X(4)

Barycentrics    = -(a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^22 - 9*a^20*(b^2 + c^2) + a^18*(-9*b^4 + 64*b^2*c^2 - 9*c^4) + a^16*(63*b^6 - 95*b^4*c^2 - 95*b^2*c^4 + 63*c^6) - a^14*(70*b^8 + 128*b^6*c^2 - 447*b^4*c^4 + 128*b^2*c^6 + 70*c^8) - a^12*(22*b^10 - 434*b^8*c^2 + 423*b^6*c^4 + 423*b^4*c^6 - 434*b^2*c^8 + 22*c^10) - 2*a^8*(b^2 - c^2)^2*(23*b^10 + 101*b^8*c^2 - 240*b^6*c^4 - 240*b^4*c^6 + 101*b^2*c^8 + 23*c^10) - a^2*(b^2 - c^2)^4*(b^12 + 24*b^10*c^2 + 13*b^8*c^4 - 40*b^6*c^6 + 13*b^4*c^8 + 24*b^2*c^10 + c^12) - a^6*(b^2 - c^2)^2*(13*b^12 - 158*b^10*c^2 - 48*b^8*c^4 + 530*b^6*c^6 - 48*b^4*c^8 - 158*b^2*c^10 + 13*c^12) + 2*a^10*(45*b^12 - 146*b^10*c^2 - 205*b^8*c^4 + 614*b^6*c^6 - 205*b^4*c^8 - 146*b^2*c^10 + 45*c^12) - (b^2 - c^2)^4*(b^14 - b^12*c^2 - 9*b^10*c^4 + b^8*c^6 + b^6*c^8 - 9*b^4*c^10 - b^2*c^12 + c^14) + a^4*(b^2 - c^2)^2*(15*b^14 - 3*b^12*c^2 - 164*b^10*c^4 + 160*b^8*c^6 + 160*b^6*c^8 - 164*b^4*c^10 - 3*b^2*c^12 + 15*c^14))) : :

X(39222) lies on these lines: {}

X(39223) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: ANTICEVIAN OF X(3) AND ANTICEVIAN OF X(6)

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

X(39223) lies on these lines: {}

X(39224) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: ANTICEVIAN OF X(4) AND ANTICEVIAN OF X(6)

Barycentrics    a^2*(3*a^30 - 7*a^28*(b^2 + c^2) + a^26*(-21*b^4 + 68*b^2*c^2 - 21*c^4) + a^24*(73*b^6 - 85*b^4*c^2 - 85*b^2*c^4 + 73*c^6) - a^22*(b^8 + 328*b^6*c^2 - 669*b^4*c^4 + 328*b^2*c^6 + c^8) + a^18*(b^2 - c^2)^2*(151*b^8 + 426*b^6*c^2 - 1150*b^4*c^4 + 426*b^2*c^6 + 151*c^8) + a^16*(b^2 - c^2)^2*(189*b^10 - 973*b^8*c^2 + 638*b^6*c^4 + 638*b^4*c^6 - 973*b^2*c^8 + 189*c^10) - a^20*(195*b^10 - 747*b^8*c^2 + 553*b^6*c^4 + 553*b^4*c^6 - 747*b^2*c^8 + 195*c^10) + a^10*(b^2 - c^2)^4*(193*b^12 - 32*b^10*c^2 - 909*b^8*c^4 + 136*b^6*c^6 - 909*b^4*c^8 - 32*b^2*c^10 + 193*c^12) - a^14*(b^2 - c^2)^2*(279*b^12 - 2*b^10*c^2 - 1441*b^8*c^4 + 2088*b^6*c^6 - 1441*b^4*c^8 - 2*b^2*c^10 + 279*c^12) - (b^2 - c^2)^8*(b^14 + b^12*c^2 - 11*b^10*c^4 - 23*b^8*c^6 - 23*b^6*c^8 - 11*b^4*c^10 + b^2*c^12 + c^14) - a^12*(b^2 - c^2)^2*(21*b^14 - 989*b^12*c^2 + 1519*b^10*c^4 - 519*b^8*c^6 - 519*b^6*c^8 + 1519*b^4*c^10 - 989*b^2*c^12 + 21*c^14) - a^8*(b^2 - c^2)^4*(69*b^14 + 543*b^12*c^2 - 163*b^10*c^4 + 95*b^8*c^6 + 95*b^6*c^8 - 163*b^4*c^10 + 543*b^2*c^12 + 69*c^14) - a^2*(b^2 - c^2)^6*(3*b^16 + 46*b^14*c^2 + 72*b^12*c^4 + 2*b^10*c^6 + 10*b^8*c^8 + 2*b^6*c^10 + 72*b^4*c^12 + 46*b^2*c^14 + 3*c^16) - a^6*(b^2 - c^2)^4*(43*b^16 - 236*b^14*c^2 - 455*b^12*c^4 + 104*b^10*c^6 - 448*b^8*c^8 + 104*b^6*c^10 - 455*b^4*c^12 - 236*b^2*c^14 + 43*c^16) + a^4*(b^2 - c^2)^4*(31*b^18 + 49*b^16*c^2 - 263*b^14*c^4 - 39*b^12*c^6 - 34*b^10*c^8 - 34*b^8*c^10 - 39*b^6*c^12 - 263*b^4*c^14 + 49*b^2*c^16 + 31*c^18)) : :

X(39224) lies on these lines: {}

X(39225) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: CIRCUMCEVIAN OF X(1) AND CIRCUMCEVIAN OF X(2)

Barycentrics    a^2*(b - c)*(a^5 + a^2*b*c*(b + c) - a^3*(2*b^2 + b*c + 2*c^2) - b*c*(b^3 + c^3) + a*(b^4 + b^3*c - b^2*c^2 + b*c^3 + c^4)) : :
X(39225) = X(39476)-4*X(39478)

X(39225) lies on these lines: {2,39508}, {3,3667}, {24,7649}, {25,16231}, {514,39199}, {522,39200}, {523,5926}, {572,20979}, {573,1919}, {667,6003}, {1995,39490}, {2776,8643}, {3738,34948}, {4786,11340}, {5592,20838}, {7488,20294}, {8648,21189}, {23961,39210}

X(39225) = midpoint of X(3) and X(4057)
X(39225) = reflection of X(i) in X(j) for these (i,j): (39210, 39227), (39226, 39478), (39476, 39226)
X(39225) = anticomplement of X(39508)
X(39225) = center of circumcircle-inverse of Nagel line
X(39225) = crossdifference of every pair of points on line {X(566), X(8610)}
X(39225) = crosssum of X(513) and X(17606)
X(39225) = {X(572), X(20979)}-harmonic conjugate of X(39525)

X(39226) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: CIRCUMCEVIAN OF X(1) AND CIRCUMCEVIAN OF X(4)

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

X(39226) lies on these lines: {3,522}, {36,1459}, {523,15646}, {993,20316}, {3667,39200}, {3738,23224}, {4057,6006}, {4458,22388}, {6003,34948}, {7280,21173}, {8648,23800}, {21189,23226}, {23146,37500}, {23961,39210}, {26286,32475}

X(39226) = midpoint of X(i) and X(j) for these {i,j}: {3, 39199}, {39225, 39476}
X(39226) = reflection of X(39225) in X(39478)
X(39226) = center of circumcircle-inverse of line X(1)X(4)

X(39227) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: CIRCUMCEVIAN OF X(1) AND CIRCUMCEVIAN OF X(6)

Barycentrics    a^2*(b - c)*(2*a^4 - 2*a^3*(b + c) + a^2*(-2*b^2 + 3*b*c - 2*c^2) - b*c*(b^2 + c^2) + 2*a*(b^3 + c^3)) : :
X(39227) = 5*X(631)-X(21301) = 7*X(3523)+X(31291) = 7*X(3526)-5*X(31251) = 3*X(3576)+X(4063) = X(4782)+2*X(13624) = 3*X(5054)-X(31149)

X(39227) lies on these lines: {3,667}, {5,31288}, {24,18344}, {28,39536}, {32,39519}, {35,4162}, {36,3669}, {140,21260}, {182,9010}, {474,25901}, {512,5926}, {521,34948}, {574,39502}, {631,21301}, {905,8648}, {993,20317}, {1385,4083}, {2775,30234}, {2826,4401}, {3523,31291}, {3526,31251}, {3576,4063}, {4057,30198}, {4367,28537}, {4782,13624}, {4905,7280}, {5054,31149}, {5428,29150}, {6050,8760}, {7501,17924}, {9320,38599}, {13731,28255}, {19270,24561}, {19514,28373}, {19649,26249}, {23961,39210}

X(39227) = midpoint of X(i) and X(j) for these {i,j}: {3, 667}, {39210, 39225}
X(39227) = reflection of X(i) in X(j) for these (i,j): (5, 31288), (21260, 140)

X(39227) = center of circumcircle-inverse of line X(1)X(6)
X(39227) = circumcircle-inverse of-X(14661)
X(39227) = crossdifference of every pair of points on line {X(3290), X(37637)}
X(39227) = X(518)-vertex conjugate of-X(14661)

X(39228) = CENTER OF THE VIETNAMESE CIRCLE OF THESE TRIANGLES: CIRCUMCEVIAN OF X(4) AND CIRCUMCEVIAN OF X(6)

Barycentrics    a^2*(b - c)*(b + c)*(a^2 - b^2 - c^2)*(2*a^4 + b^2*c^2 - 2*a^2*(b^2 + c^2)) : :
X(39228) = 3*X(3)-X(22089) = X(5926)-4*X(39477) = 3*X(5926)-4*X(39481) = X(22089)+3*X(39201) = 3*X(39477)-X(39481)

X(39228) lies on these lines: {3,525}, {4,39510}, {22,9209}, {32,39517}, {378,39491}, {512,5926}, {523,15646}, {574,39500}, {1499,34952}, {2394,35493}, {3566,14270}, {8552,8673}, {9210,37184}, {9218,32662}, {10298,18556}, {14566,18570}, {15246,30474}

X(39228) = midpoint of X(3) and X(39201)
X(39228) = reflection of X(4) in X(39510)
X(39228) = barycentric product X(i)*X(j) for these {i, j}: {525, 9544}, {647, 7782}
X(39228) = barycentric quotient X(647)/X(13481)
X(39228) = trilinear product X(i)*X(j) for these {i, j}: {656, 9544}, {810, 7782}
X(39228) = trilinear quotient X(656)/X(13481)
X(39228) = crossdifference of every pair of points on line {X(232), X(13481)}
X(39228) = X(162)-isoconjugate-of-X(13481)
X(39228) = X(647)-reciprocal conjugate of-X(13481)
X(39228) = center of circumcircle-inverse of van Aubel line

X(39229) = POINT RASALGETHI ONE

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

X(39229) lies on the curves K273, K303a, K303b, K305, K800, K854, Q016, and on this line: {3, 6}

X(39229) = reflection of X(39230) in X(2080)
X(39229) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 576, 39230}, {6, 574, 39230}, {39, 5038, 39230}, {182, 32447, 39230}, {187, 8586, 39230}

X(39230) = POINT RASALGETHI TWO

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

X(39230) lies on the curves K273, K303a, K303b, K305, K800, K854, Q016, and on this line: {3, 6}

X(39230) = reflection of X(39229) in X(2080)
X(39230) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 576, 39229}, {6, 574, 39229}, {39, 5038, 39229}, {182, 32447, 39229}, {187, 8586, 39229}

X(39231) = X(3)X(9019)∩X(6)X(30534)

Barycentrics    a^4*(a^4 - 3*a^2*b^2 + 2*b^4 - 3*a^2*c^2 + b^2*c^2 + 2*c^4) : :
X(39231) = 4 X[187] - X[9142], 2 X[2080] + X[9145]

X(39231) lies on these lines: {3, 9019}, {6, 30534}, {50, 237}, {95, 3613}, {160, 184}, {187, 2393}, {524, 2080}, {577, 19136}, {1157, 1510}, {1384, 32621}, {1609, 10602}, {2076, 2871}, {2965, 20775}, {3001, 35296}, {3003, 21639}, {4558, 5201}, {5191, 19596}, {5421, 34565}, {5467, 21460}, {5965, 11811}, {6593, 23164}, {6641, 17810}, {7608, 15464}, {8705, 38225}, {9792, 15033}, {9969, 22052}, {9971, 37457}, {10510, 22087}, {11063, 20975}, {22151, 35298}

X(39231) = isogonal conjugate of the isotomic conjugate of X(23061)
X(39231) = X(691)-Ceva conjugate of X(10562)
X(39231) = crosspoint of X(i) and X(j) for these (i,j): {111, 1173}, {691, 23357}
X(39231) = crosssum of X(i) and X(j) for these (i,j): {23, 34545}, {140, 524}, {338, 690}
X(39231) = crossdifference of every pair of points on line {233, 13162}
X(39231) = barycentric product X(i)*X(j) for these {i,j}: {6, 23061}, {110, 39232}, {524, 10558}, {5467, 10562}, {Z1, 39230}
X(39231) = barycentric quotient X(i)/X(j) for these {i,j}: {10558, 671}, {23061, 76}, {39232, 850}
X(39231) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {50, 237, 1576}, {50, 18374, 23200}, {237, 23200, 18374}, {18374, 23200, 1576}

X(39232) = X(6)X(526)∩X(50)X(647)

Barycentrics    a^2*(b - c)*(b + c)*(a^4 - 3*a^2*b^2 + 2*b^4 - 3*a^2*c^2 + b^2*c^2 + 2*c^4) : :
X(39232) = 3 X[6] - 4 X[2492], 5 X[6] - 6 X[14398], 3 X[599] - 2 X[35522], 4 X[647] - 3 X[3050], 2 X[2492] - 3 X[3569], 10 X[2492] - 9 X[14398], 5 X[3569] - 3 X[14398], 5 X[3763] - 4 X[24284]

X(39232) lies on these lines: {6, 526}, {50, 647}, {67, 690}, {512, 5104}, {599, 35522}, {1625, 1634}, {1987, 14380}, {2502, 9138}, {2930, 9517}, {3738, 4016}, {3763, 24284}, {7669, 9409}, {8430, 8586}, {9003, 14273}, {10561, 20977}, {10562, 23061}, {11163, 22734}

X(39232) = reflection of X(i) in X(j) for these {i,j}: {6, 3569}, {8586, 8430}
X(39232) = X(36142)-anticomplementary conjugate of X(8266)
X(39232) = X(31)-complementary conjugate of X(39233)
X(39232) = X(2)-Ceva conjugate of X(39233)
X(39232) = crosspoint of X(i) and X(j) for these (i,j): {110, 671}, {275, 935}, {5466, 39183}
X(39232) = crosssum of X(i) and X(j) for these (i,j): {187, 523}, {216, 9517}, {5467, 35324}
X(39232) = crossdifference of every pair of points on line {5, 542}
X(39232) = barycentric product X(i)*X(j) for these {i,j}: {523, 23061}, {524, 10562}, {850, 39231}, {10558, 35522}
X(39232) = barycentric quotient X(i)/X(j) for these {i,j}: {10558, 691}, {10562, 671}, {23061, 99}, {39231, 110}

X(39233) = X(2080)X(22115)∩X(8901)X(9213)

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

X(39233) lies on these lines: {2080, 22115}, {8901, 9213}

X(39233) = X(31)-complementary conjugate of X(39232)
X(39233) = X(2)-Ceva conjugate of X(39232)

X(39234) =  EULER LINE INTERCEPT OF X(9033)X(15115)

Barycentrics    (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) (a^12-3 a^10 b^2+a^8 b^4+6 a^6 b^6-9 a^4 b^8+5 a^2 b^10-b^12-3 a^10 c^2+11 a^8 b^2 c^2-12 a^6 b^4 c^2+4 a^4 b^6 c^2-a^2 b^8 c^2+b^10 c^2+a^8 c^4-12 a^6 b^2 c^4+14 a^4 b^4 c^4-4 a^2 b^6 c^4+5 b^8 c^4+6 a^6 c^6+4 a^4 b^2 c^6-4 a^2 b^4 c^6-10 b^6 c^6-9 a^4 c^8-a^2 b^2 c^8+5 b^4 c^8+5 a^2 c^10+b^2 c^10-c^12) : :
Barycentrics    (S^2-3 SB SC) (-36 R^4+S^2-24 R^2 SA+2 SA^2+12 R^2 SW+4 SA SW-SW^2) : :

As a point on the Euler line, X(39234) has Shinagawa coefficients (e^2-28 e f+52 f^2+4 S^2,-3 e^2+12 e f-12 f^2+4 S^2).

See Kadir Altintas and Ercole Suppa, Euclid 985 .

X(39234) lies on these lines: {2,3}, {9033,15115}

X(39234) = complement of X(34104)

X(39235) =  EULER LINE INTERCEPT OF X(94)X(10264)

Barycentrics    a^14 b^2-5 a^12 b^4+9 a^10 b^6-5 a^8 b^8-5 a^6 b^10+9 a^4 b^12-5 a^2 b^14+b^16+a^14 c^2-8 a^8 b^6 c^2+15 a^6 b^8 c^2-18 a^4 b^10 c^2+16 a^2 b^12 c^2-6 b^14 c^2-5 a^12 c^4+12 a^8 b^4 c^4-8 a^6 b^6 c^4+3 a^4 b^8 c^4-18 a^2 b^10 c^4+16 b^12 c^4+9 a^10 c^6-8 a^8 b^2 c^6-8 a^6 b^4 c^6+12 a^4 b^6 c^6+7 a^2 b^8 c^6-26 b^10 c^6-5 a^8 c^8+15 a^6 b^2 c^8+3 a^4 b^4 c^8+7 a^2 b^6 c^8+30 b^8 c^8-5 a^6 c^10-18 a^4 b^2 c^10-18 a^2 b^4 c^10-26 b^6 c^10+9 a^4 c^12+16 a^2 b^2 c^12+16 b^4 c^12-5 a^2 c^14-6 b^2 c^14+c^16 : :
Barycentrics    -9 R^2 SB SC (3 R^2-SW)+S^2 (27 R^4-2 SB SC-15 R^2 SW+2 SW^2) : :
X(39235) = X(4)-3*X(18867),2*X(5)-3*X(13448),X(265)+2*X(18780),2*X(10264)+X(18781)

As a point on the Euler line, X(39235) has Shinagawa coefficients (e^2-4 e f-32 f^2,-9 e^2-36 e f+32 S^2).

See Kadir Altintas and Ercole Suppa, Euclid 985 .

X(39235) lies on these lines: {2,3}, {94,10264}, {136,7723}, {265,18780}, {568,18121}, {3580,14254}, {10733,18576}, {12028,35465}, {12091,16221}

X(39236) =  ISOGONAL CONJUGATE OF X(9741)

Barycentrics    a^2*(a^4 - 16*a^2*b^2 + 7*b^4 + 14*a^2*c^2 - 16*b^2*c^2 + c^4)*(a^4 + 14*a^2*b^2 + b^4 - 16*a^2*c^2 - 16*b^2*c^2 + 7*c^4) : :

X(39236) lies on the cubic K314 and these lines: {6, 17979}, {1384, 35259}, {1992, 11159}, {18842, 32130}

X(39236) = isogonal conjugate of X(9741)
X(39236) = X(22111)-cross conjugate of X(6)
X(39236) = cevapoint of X(6) and X(33979)
X(39236) = trilinear pole of line {8644, 14328}
X(39236) = barycentric product X(1384)*X(32130)
X(39236) = barycentric quotient X(6)/X(9741)

X(39237) =  X(2)X(2418)∩X(3)X(17968)

Barycentrics    a^2*(a^2 + b^2 - 5*c^2)*(a^2 - 5*b^2 + c^2)*(a^4 + 14*a^2*b^2 + b^4 + 14*a^2*c^2 - 22*b^2*c^2 + c^4) : :

X(39237) lies on the cubic K314 and these lines: {2, 2418}, {3, 17968}, {6, 17979}, {1296, 1384}, {9605, 14262}, {22253, 35179}

X(39238) =  ISOGONAL CONJUGATE OF X(11059)

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

X(39238) lies on these lines: {6, 373}, {32, 23200}, {39, 33929}, {51, 38532}, {83, 5485}, {182, 729}, {184, 32740}, {575, 14262}, {576, 5166}, {597, 9516}, {1974, 14567}, {2422, 9171}, {2489, 21905}, {3224, 5038}, {3225, 35179}, {5063, 32654}, {5476, 38951}, {5486, 6791}, {5967, 6531}, {8546, 28662}, {10354, 32154}

X(39238) = isogonal conjugate of X(11059)
X(39238) = isogonal conjugate of the isotomic conjugate of X(21448)
X(39238) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11059}, {75, 1992}, {76, 36277}, {99, 14207}, {304, 4232}, {561, 1384}, {668, 4786}, {799, 1499}, {1978, 30234}, {2408, 24039}, {4602, 8644}, {6791, 24037}, {33805, 35266}
X(39238) = barycentric product X(i)*X(j) for these {i,j}: {6, 21448}, {32, 5485}, {512, 1296}, {669, 35179}, {690, 32648}, {798, 37216}, {2434, 9178}, {2642, 36045}, {2709, 17999}, {19136, 32133}
X(39238) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 11059}, {32, 1992}, {560, 36277}, {669, 1499}, {798, 14207}, {1084, 6791}, {1296, 670}, {1501, 1384}, {1919, 4786}, {1974, 4232}, {1980, 30234}, {5485, 1502}, {9407, 35266}, {9426, 8644}, {14567, 27088}, {21448, 76}, {32648, 892}, {35179, 4609}, {37216, 4602}

X(39239) =  X(3)X(74)∩X(4)X(12003)

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

X(39239) lies on the cubic K762 and these lines: {3, 74}, {4, 12003}, {547, 12079}, {3543, 14731}, {3545, 5627}, {3581, 30510}, {3845, 34150}, {5890, 14685}, {12112, 16186}, {14380, 14483}, {35478, 38937}
X(39239) = crosssum of X(30) and X(11694)
X(39239) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {74, 3470, 9717}, {74, 9717, 14385}, {3470, 14264, 14385}, {9717, 14264, 74}

X(39240) =  X(4)X(2575)∩X(125)X(136)

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

X(39240) lies on the cubic K238, the X-parabola of ABC (see X(12065), and these lines: {4, 2575}, {125, 136}, {468, 9174}, {476, 1114}, {523, 1313}, {892, 15165}, {1007, 22340}, {1344, 15928}, {2395, 8105}, {2574, 15328}, {2592, 5466}

X(39240) = X(i)-Ceva conjugate of X(j) for these (i,j): {2593, 2501}, {15165, 2592}
X(39240) = X(125)-cross conjugate of X(1313)
X(39240) = X(i)-isoconjugate of X(j) for these (i,j): {110, 1822}, {163, 8115}, {249, 2579}, {250, 2585}, {1101, 2575}, {1113, 4575}, {1823, 15461}, {2576, 4558}, {2580, 32661}, {2583, 23357}, {22340, 23995}
X(39240) = crosspoint of X(i) and X(j) for these (i,j): {850, 2593}, {2592, 15165}
X(39240) = barycentric product X(i)*X(j) for these {i,j}: {115, 15165}, {338, 1114}, {523, 2592}, {850, 8105}, {1109, 2581}, {1313, 2593}, {1577, 2588}, {2501, 22339}, {2574, 14618}, {2577, 23994}, {2582, 24006}, {2587, 20902}, {2970, 8116}
X(39240) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 2575}, {338, 22340}, {523, 8115}, {661, 1822}, {1109, 2583}, {1114, 249}, {1313, 8116}, {2501, 1113}, {2574, 4558}, {2577, 1101}, {2578, 4575}, {2581, 24041}, {2582, 4592}, {2588, 662}, {2592, 99}, {2643, 2579}, {2970, 2593}, {3708, 2585}, {8105, 110}, {8106, 15461}, {8754, 8106}, {14618, 15164}, {15165, 4590}, {22339, 4563}, {24006, 2580}
X(39240) = {X(2970),X(35235)}-harmonic conjugate of X(39241)

X(39241) =  X(4)X(2574)∩X(125)X(136)

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

X(39241) lies on the cubic K238, the X-parabola of ABC (see X(12065), and these lines: {4, 2574}, {125, 136}, {468, 9173}, {476, 1113}, {523, 1312}, {892, 15164}, {1007, 22339}, {1345, 15928}, {2395, 8106}, {2575, 15328}, {2593, 5466}

X(39241) = X(i)-Ceva conjugate of X(j) for these (i,j): {2592, 2501}, {15164, 2593}
X(39241) = X(125)-cross conjugate of X(1312)
X(39241) = X(i)-isoconjugate of X(j) for these (i,j): {110, 1823}, {163, 8116}, {249, 2578}, {250, 2584}, {1101, 2574}, {1114, 4575}, {1822, 15460}, {2577, 4558}, {2581, 32661}, {2582, 23357}, {22339, 23995}
X(39241) = crosspoint of X(i) and X(j) for these (i,j): {850, 2592}, {2593, 15164}
X(39241) = barycentric product X(i)*X(j) for these {i,j}: {115, 15164}, {338, 1113}, {523, 2593}, {850, 8106}, {1109, 2580}, {1312, 2592}, {1577, 2589}, {2501, 22340}, {2575, 14618}, {2576, 23994}, {2583, 24006}, {2586, 20902}, {2970, 8115}
X(39241) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 2574}, {338, 22339}, {523, 8116}, {661, 1823}, {1109, 2582}, {1113, 249}, {1312, 8115}, {2501, 1114}, {2575, 4558}, {2576, 1101}, {2579, 4575}, {2580, 24041}, {2583, 4592}, {2589, 662}, {2593, 99}, {2643, 2578}, {2970, 2592}, {3708, 2584}, {8105, 15460}, {8106, 110}, {8754, 8105}, {14618, 15165}, {15164, 4590}, {22340, 4563}, {24006, 2581}
X(39241) = {X(2970),X(35235)}-harmonic conjugate of X(39240)

X(39242) =  X(2)X(14644)∩X(3)X(6)

Barycentrics    a^2*(3*a^8 - 7*a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 - 2*b^8 - 7*a^6*c^2 + 10*a^4*b^2*c^2 - a^2*b^4*c^2 - 2*b^6*c^2 + 3*a^4*c^4 - a^2*b^2*c^4 + 8*b^4*c^4 + 3*a^2*c^6 - 2*b^2*c^6 - 2*c^8) : :
X(39242) = X[184] + 2 X[18570], X[378] + 2 X[18475]

X(39242) lies on the cubic K882 and these lines: {2, 14644}, {3, 6}, {30, 13394}, {51, 18324}, {54, 7689}, {74, 11003}, {110, 3431}, {125, 18580}, {154, 16194}, {184, 5663}, {186, 5640}, {373, 6644}, {378, 6800}, {381, 11202}, {1092, 15067}, {1147, 11459}, {1204, 32046}, {1209, 12118}, {1352, 30714}, {1495, 31861}, {1511, 5651}, {1658, 11424}, {3516, 10575}, {3520, 15072}, {3541, 11750}, {3567, 38448}, {3796, 14855}, {4549, 37645}, {4846, 16111}, {5012, 15055}, {5067, 33556}, {5422, 35472}, {5446, 38444}, {5462, 32534}, {5576, 34785}, {5650, 7514}, {5891, 6090}, {5892, 15078}, {5944, 26883}, {6030, 11001}, {6639, 13403}, {6699, 18911}, {6759, 14130}, {6776, 16003}, {7464, 8717}, {7503, 10170}, {7526, 10539}, {7527, 11464}, {7550, 33879}, {7575, 34417}, {7699, 10296}, {7706, 10295}, {7998, 35921}, {8254, 15332}, {9306, 32609}, {9545, 15083}, {9818, 35259}, {9833, 18488}, {10298, 11002}, {10610, 10984}, {10619, 32140}, {10628, 11204}, {11449, 35500}, {11454, 15032}, {11472, 26864}, {12162, 19357}, {12279, 35478}, {13363, 37814}, {13434, 21844}, {13596, 26881}, {14169, 35469}, {14170, 35470}, {15018, 37952}, {15040, 16187}, {15043, 17506}, {15045, 37941}, {16657, 34351}, {16776, 35228}, {18364, 34783}, {18583, 37934}, {18912, 20191}, {20791, 37948}, {37458, 38136}

X(39242) = midpoint of X(i) and X(j) for these {i,j}: {378, 6800}, {15055, 15463}
X(39242) = reflection of X(i) in X(j) for these {i,j}: {19131, 5085}, {6800, 18475}, {35268, 34513}
X(39242) = Brocard-circle-inverse of X(32110)
X(39242) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 32110}, {3, 182, 37470}, {3, 567, 11438}, {3, 11425, 52}, {3, 11430, 13352}, {3, 13352, 37478}, {3, 14805, 182}, {3, 37477, 3098}, {3, 37506, 9730}, {4846, 35485, 16111}, {6090, 32620, 5891}, {7464, 15080, 8717}, {7526, 13367, 10539}, {7527, 35265, 16261}, {9730, 37506, 569}, {10295, 14389, 7706}, {10610, 11250, 10984}, {11464, 16261, 35265}

X(39243) =  ISOGONAL CONJUGATE OF X(30102)

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

See Kadir Altintas and Ercole Suppa, Euclid 988 .

X(39243) lies on these lines: {3,143}, {20,8796}, {22,14577}, {95,30506}, {97,110}, {577,5012}, {1993,26865}, {1994,26907}, {5640,37068}, {6503,33523}, {7494,14494}, {9706,19210}, {11422,26898}, {12161,26896}, {15107,22052}, {26876,34148}

X(39243) = isogonal conjugate of X(30102)
X(39243) = X(6)-reciprocal conjugate of X(30102)
X(39243) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (97,418,110), (577,26874,5012), (1993,26865,26895)

X(39244) =  CENTER OF TC(0 : b : c)

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

The trilinear permutation conic TC(0:b:c) passes through points identified in the preamble just before X(36256); they are the points with trilinear coordinates 0:b:c, 0:c:b, c:0:a, a:0:c, a:b:0, b:a:0.

X(39244) lies on these lines: {1, 1390}, {2, 257}, {6, 21033}, {9, 604}, {10, 2170}, {11, 21029}, {21, 2112}, {37, 1201}, {38, 2275}, {39, 2292}, {41, 997}, {56, 5282}, {72, 1475}, {73, 1212}, {75, 30099}, {78, 2280}, {191, 5030}, {205, 2267}, {220, 36476}, {244, 3721}, {274, 18061}, {392, 1334}, {404, 3496}, {517, 25068}, {518, 17474}, {551, 3970}, {583, 3958}, {612, 9575}, {672, 960}, {748, 16968}, {756, 1107}, {762, 2087}, {896, 33863}, {936, 2082}, {976, 16502}, {1015, 3954}, {1018, 3884}, {1055, 17614}, {1100, 3949}, {1125, 21808}, {1193, 21840}, {1213, 22073}, {1574, 4695}, {1575, 3727}, {1953, 17303}, {2171, 5750}, {2225, 22065}, {2246, 33950}, {2294, 17398}, {2312, 37247}, {2329, 17439}, {2345, 17452}, {2478, 24247}, {2650, 24512}, {3119, 5316}, {3208, 3890}, {3244, 4006}, {3263, 30038}, {3290, 28352}, {3294, 24036}, {3501, 3877}, {3509, 5253}, {3617, 4051}, {3691, 5044}, {3735, 24443}, {3739, 20593}, {3740, 4875}, {3780, 21805}, {3786, 24727}, {3816, 21928}, {3869, 17754}, {3876, 21384}, {3878, 16549}, {3938, 16781}, {3942, 17237}, {3985, 26770}, {4187, 21044}, {4253, 5692}, {4414, 5013}, {4515, 5919}, {4975, 21070}, {5299, 30115}, {5525, 9327}, {5701, 16699}, {5745, 24583}, {5837, 8568}, {6537, 20982}, {7187, 31004}, {7308, 25930}, {9317, 17681}, {10176, 16552}, {16503, 34772}, {16583, 27627}, {16783, 22836}, {16788, 30144}, {17062, 26548}, {17353, 26689}, {17355, 21809}, {17369, 21801}, {17541, 24291}, {17756, 37598}, {17760, 26234}, {17868, 28604}, {18055, 31997}, {18230, 26658}, {18671, 24943}, {20196, 23058}, {20247, 27146}, {20258, 24547}, {20363, 22167}, {20706, 24325}, {20911, 29991}, {21139, 26563}, {21214, 26242}, {21272, 27025}, {21674, 37661}, {21764, 37539}, {21811, 25078}, {24484, 24697}, {25079, 27040}, {25615, 38375}, {25838, 25856}, {25887, 28351}, {27221, 27225}, {27340, 30946}, {30036, 30758}, {33944, 33946}

X(39245) =  CENTER OF TC(0 : b^2 : c^2)

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

The trilinear permutation conic TC(0:b^2:c^2) passes through points identified in the preamble just before X(36256).

X(39245) lies on these lines: {37, 42}, {125, 21918}, {2643, 21954}, {3981, 4118}, {20599, 31089}, {21329, 25760}, {21807, 21923}

X(39246) =  CENTER OF TC(0 : c^2+a^2-b^2 : a^2+b^2-c^2)

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

The trilinear permutation conic passes through points identified in the preamble just before X(36256).

X(39246) lies on these lines: {37, 20684}, {1402, 36197}, {1864, 9454}

X(39247) =  CENTER OF TC(0 : a+b : a+c)

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

The trilinear permutation conic passes through points identified in the preamble just before X(36256).

X(39247) lies on these lines: {1, 41}, {672, 37548}, {756, 3780}, {1015, 6155}, {1100, 2650}, {1107, 1962}, {1449, 3869}, {1475, 3931}, {1500, 14439}, {1959, 17011}, {2292, 20963}, {3061, 17018}, {3666, 17474}, {3691, 6051}, {3720, 21921}, {4136, 29835}, {4642, 24512}, {4875, 15569}, {10460, 37593}, {16503, 17016}, {16605, 30950}, {16781, 17017}, {17450, 20271}, {25089, 31855}

X(39248) =  CENTER OF TC(-a : b : c)

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

The trilinear permutation conic passes through points identified in the preamble just before X(36256).

X(39248) lies on these lines: {1, 6}, {10, 1572}, {19, 992}, {32, 997}, {36, 21771}, {39, 12514}, {40, 1575}, {57, 16604}, {63, 2275}, {78, 1914}, {172, 19861}, {230, 25681}, {614, 3721}, {748, 17451}, {893, 2339}, {936, 4386}, {970, 20606}, {978, 3496}, {993, 9619}, {1201, 5282}, {1265, 5839}, {1697, 20691}, {1707, 33863}, {1716, 3094}, {1722, 3959}, {2082, 2238}, {2241, 3811}, {2270, 21892}, {2276, 5250}, {3509, 21214}, {3556, 36743}, {3661, 14555}, {3726, 28011}, {3767, 21616}, {3780, 4517}, {3815, 26066}, {3869, 33854}, {3878, 9620}, {3915, 33299}, {4640, 5013}, {5011, 17749}, {5087, 13881}, {5254, 24703}, {5272, 20271}, {5275, 25917}, {5698, 7738}, {5743, 17308}, {5791, 31466}, {5794, 7745}, {6703, 29603}, {6734, 9599}, {6765, 20693}, {7737, 17647}, {7991, 21888}, {9592, 31424}, {9596, 24987}, {11512, 36643}, {12701, 21956}, {14823, 18786}, {14974, 25066}, {16605, 37679}, {17125, 21921}, {17397, 38000}, {17474, 32912}, {17798, 20991}, {18904, 27626}, {20665, 22065}, {21608, 39044}, {21793, 37552}, {26998, 27640}, {31445, 31449}

X(39249) =  CENTER OF TC(-a^2 : b^2 : c^2)

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

The trilinear permutation conic passes through points identified in the preamble just before X(36256).

X(39249) lies on these lines: {9, 1613}, {37, 1184}, {44, 394}, {3051, 36405}, {15487, 18904}

X(39250) =  CENTER OF TC(-b c : c a : a b)

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

The trilinear permutation conic passes through points identified in the preamble just before X(36256).

X(39250) lies on these lines: {1, 3495}, {9, 55}, {42, 36406}, {63, 4366}, {238, 21387}, {304, 3500}, {312, 2319}, {649, 24756}, {982, 16557}, {1707, 24727}, {1740, 30646}, {2083, 3496}, {2162, 20363}, {2179, 3501}, {3508, 3749}, {3550, 20372}, {3685, 7075}, {7262, 21384}, {8616, 21369}, {9315, 30567}, {9360, 16569}, {17155, 38346}, {18194, 21345}, {21257, 22439}, {23415, 24349}, {28798, 30827}>

X(39251) =  CENTER OF TC(-2a : b : c)

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

The trilinear permutation conic passes through points identified in the preamble just before X(36256).

X(39251) lies on these lines: {6, 896}, {31, 36404}, {38, 5332}, {44, 678}, {238, 2246}, {244, 2243}, {1054, 33854}, {1386, 21764}, {1914, 3722}, {2220, 21033}, {2280, 3751}, {2292, 5007}, {3218, 16786}, {3242, 5282}, {3315, 3509}, {3707, 35263}, {3795, 37657}, {3977, 4700}, {7031, 33299}, {16670, 35258}

X(39252) =  CENTER OF TC(b c (a^2 - b c) : c a (b^2 - c a) : a b (c^2 - a b) )

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

The trilinear permutation conic passes through points identified in the preamble just before X(36256).

X(39252) lies on these lines: {1, 6}, {190, 25428}, {239, 672}, {292, 2238}, {579, 8866}, {666, 2311}, {869, 37657}, {870, 2279}, {1400, 7176}, {1475, 16823}, {1707, 21387}, {2223, 3684}, {2235, 9359}, {2280, 23407}, {3691, 16830}, {4253, 16825}, {4876, 20683}, {5257, 26110}, {5276, 20985}, {5364, 37652}, {5750, 16819}, {9468, 18786}, {16557, 16570}, {16826, 25427}, {16831, 37632}, {21010, 37658}, {37686, 39044}

X(39253) =  CENTER OF TC(a : a + b : a + c)

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

The trilinear permutation conic passes through points identified in the preamble just before X(36256).

X(39253) lies on these lines: {1, 9346}, {6, 78}, {81, 16968}, {1453, 24512}, {1468, 16972}, {2238, 37554}, {2275, 16475}, {3780, 5269}, {5256, 33863}

X(39254) =  CENTER OF TC(-a : a + b : a + c)

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

The trilinear permutation conic passes through points identified in the preamble just before X(36256).

X(39254) lies on these lines: {1, 32}, {6, 4689}, {9, 10987}, {44, 55}, {896, 2280}, {902, 36404}, {1051, 5332}, {2238, 15601}, {2276, 16670}, {3744, 16672}, {3749, 16676}, {4384, 4760}, {16502, 37599}, {16786, 17601}, {16973, 36263}, {20331, 35445}

X(39255) =  CENTER OF TC(a : a - b : a - c)

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

The trilinear permutation conic passes through points identified in the preamble just before X(36256).

X(39255) lies on these lines: {1, 574}, {37, 5217}, {78, 17735}, {100, 16968}, {165, 3721}, {187, 17742}, {2176, 4855}, {2275, 3749}, {2276, 37552}, {2295, 3601}, {3053, 3693}, {3158, 3780}, {3670, 31422}, {3726, 15803}, {3744, 5013}, {3870, 33863}, {5248, 25089}, {5266, 31448}, {5440, 14974}, {9619, 37610}, {16780, 20331}, {16969, 35262}, {30701, 32985}, {31443, 37549}, {31477, 37539}

X(39256) =  CENTER OF TC(0 : a^2 - b^2 : a^2 - c^2)

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

The trilinear permutation conic passes through points identified in the preamble just before X(36256).

X(39256) lies on these lines: {6, 17467}, {9, 662}, {37, 2054}, {45, 9509}, {125, 21948}, {897, 16676}, {2183, 17462}, {2247, 5282}, {2640, 3731}, {3700, 6174}, {3709, 6184}, {5257, 24086}, {5296, 26081}, {6594, 19557}, {9508, 14439}, {10868, 15587}, {16598, 21383}, {17476, 21341}

X(39257) =  CENTER OF TC(b c : a^2 : a^2)

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

X(39257) lies on these lines: {32, 4676}, {609, 4432}, {1333, 30939}, {1914, 3758}, {2235, 21793}, {4386, 17335}, {4672, 7031}, {21764, 24696}

X(39258) =  CENTER OF TC(0: - a b , a c)

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

X(39258) lies on these lines: {9, 4676}, {37, 65}, {41, 15624}, {55, 5364}, {57, 21453}, {75, 3501}, {101, 2711}, {181, 21795}, {190, 1966}, {192, 20247}, {213, 872}, {220, 34247}, {292, 21788}, {512, 798}, {517, 20593}, {518, 672}, {692, 19554}, {740, 1018}, {813, 30663}, {902, 2225}, {984, 3730}, {1055, 20780}, {1279, 20459}, {1500, 2667}, {1755, 2076}, {2223, 9454}, {2238, 21897}, {2284, 3252}, {3052, 20665}, {3219, 27495}, {3248, 21760}, {3294, 3842}, {3305, 31322}, {3725, 7109}, {3769, 7075}, {3938, 36808}, {4432, 20372}, {7083, 14974}, {16549, 24325}, {20173, 27659}, {20448, 21232}, {20684, 22276}, {21813, 21936}, {22173, 22278}, {22197, 22300}, {22230, 22325}, {24578, 32922}

X(39259) =  CENTER OF TC(0: - b^2 , c^2)

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

X(39259) lies on these lines: {2, 39245}, {3094, 4118}, {21026, 21339}, {21329, 25957}

X(39260) =  CENTER OF TC(1 : 2 : 3)

Barycentrics    a*(4*a + 7*b + 7*c) : :

X(39260) lies on these lines: {1, 6}, {2, 4727}, {86, 4718}, {89, 27789}, {354, 21864}, {536, 29570}, {551, 3943}, {594, 19862}, {1213, 3625}, {1255, 17012}, {1962, 21338}, {2321, 15808}, {3622, 17281}, {3626, 17388}, {3635, 17330}, {3636, 17369}, {3739, 29595}, {3752, 17021}, {4007, 19872}, {4360, 29578}, {4393, 4755}, {4657, 29583}, {4670, 29580}, {4681, 17394}, {4686, 17319}, {4688, 16826}, {4690, 29588}, {4698, 16816}, {4708, 17389}, {4852, 16815}, {4887, 17392}, {4889, 17248}, {4896, 17246}, {4908, 38314}, {4909, 17334}, {4969, 16590}, {5311, 17782}, {5550, 17314}, {8162, 37503}, {9780, 17299}, {15513, 37589}, {15515, 37599}, {16602, 20182}, {17019, 37520}, {17045, 29596}, {17067, 17395}, {17229, 29608}, {17231, 29601}, {17237, 29574}, {17275, 20050}, {17301, 29624}, {17305, 29625}, {17315, 25498}, {17318, 29597}, {17325, 29602}, {17359, 29586}, {17374, 29585}, {17382, 29569}, {17384, 29579}, {22034, 37869}, {28329, 29576}, {31652, 37592}

X(39261) =  ISOGONAL CONJUGATE OF X(38932)

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

X(39261) lies on the cubic K005 and these lines: {3, 3441}, {4, 617}, {5, 11120}, {17, 8930}, {61, 16460}, {62, 110}, {195, 3489}, {636, 36185}, {3130, 15794}, {3470, 18114}

X(39261) = isogonal conjugate of X(38932)
X(39261) = X(i)-Ceva conjugate of X(j) for these (i,j): {11120, 6151}, {39133, 5675}
X(39261) = X(14369)-cross conjugate of X(1337)
X(39261) = barycentric product X(622)*X(6151)
X(39261) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 38932}, {3130, 395}, {6151, 2993}, {11089, 14373}

X(39262) =  ISOGONAL CONJUGATE OF X(38931)

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

X(39262) lies on the cubic K005 and these lines: {3, 3440}, {4, 616}, {5, 11119}, {18, 8929}, {61, 110}, {62, 16459}, {195, 3490}, {635, 36186}, {3129, 15793}, {3470, 18114}

X(39262) = isogonal conjugate of X(38931)
X(39262) = X(i)-Ceva conjugate of X(j) for these (i,j): {11119, 2981}, {39132, 5674}
X(39262) = X(14368)-cross conjugate of X(1338)
X(39262) = barycentric product X(621)*X(2981)
X(39262) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 38931}, {2981, 2992}, {3129, 396}, {11084, 14372}, {14368, 14922}

X(39263) =  X(3)X(1302)∩X(4)X(4846)

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

X(39263) lies on the cubics K028 and K923 and these lines: {3, 1302}, {4, 4846}, {76, 1494}, {5286, 34288}, {14262, 14265}, {14266, 14268}

X(39263) = X(9007)-cross conjugate of X(1302)
X(39263) = barycentric product X(376)*X(34289)
X(39263) = barycentric quotient X(i)/X(j) for these {i,j}: {376, 15066}, {9209, 8675}, {26864, 5063}, {34288, 3426}, {34289, 36889}

X(39264) =  X(3)X(106)∩X(4)X(2457)

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

X(39264) lies on the cubic K028 and these lines: {3, 106}, {4, 2457}, {8, 596}, {10, 36814}, {32, 17969}, {76, 903}, {995, 14260}, {5883, 24429}, {8743, 8752}, {18600, 36887}

X(39264) = X(23644)-cross conjugate of X(8610)
X(39264) = X(4738)-isoconjugate of X(15383)
X(39264) = cevapoint of X(8610) and X(23644)
X(39264) = crosspoint of X(903) and X(2226)
X(39264) = crosssum of X(902) and X(4370)
X(39264) = crossdifference of every pair of points on line {14425, 22356}
X(39264) = barycentric product X(i)*X(j) for these {i,j}: {88, 1739}, {121, 2226}, {679, 17465}, {903, 8610}, {4674, 16753}
X(39264) = barycentric quotient X(i)/X(j) for these {i,j}: {121, 36791}, {1739, 4358}, {8610, 519}, {16753, 30939}, {17465, 4738}, {23644, 4370}

X(39265) =  X(3)X(112)∩X(4)X(525)

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

X(39265) lies on the cubic K028 and these lines: {3, 112}, {4, 525}, {6, 15407}, {24, 32649}, {76, 6528}, {249, 35602}, {458, 9476}, {877, 6393}, {2207, 2710}, {4230, 36212}, {5895, 9289}, {6330, 11331}, {6530, 34138}, {14246, 38937}, {33752, 35908}

X(39265) = isogonal conjugate of X(34156)
X(39265) = X(35140)-Ceva conjugate of X(297)
X(39265) = X(i)-cross conjugate of X(j) for these (i,j): {511, 1297}, {17994, 32687}
X(39265) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34156}, {98, 8766}, {287, 2312}, {293, 1503}, {441, 1910}, {1821, 8779}, {15407, 24023}
X(39265) = cevapoint of X(511) and X(2967)
X(39265) = trilinear pole of line {232, 684}
X(39265) = Cundy-Parry Phi transform of X(112)
X(39265) = Cundy-Parry Psi transform of X(525)
X(39265) = barycentric product X(i)*X(j) for these {i,j}: {232, 35140}, {297, 1297}, {511, 6330}, {877, 34212}, {1959, 8767}, {2967, 9476}, {6333, 32687}, {15407, 36426}
X(39265) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 34156}, {232, 1503}, {237, 8779}, {297, 30737}, {511, 441}, {1297, 287}, {1755, 8766}, {2967, 15595}, {4230, 34211}, {5968, 36894}, {6330, 290}, {8767, 1821}, {32649, 2715}, {32687, 685}, {34212, 879}, {34854, 16318}, {34859, 2445}, {36046, 36084}

X(39266) =  MIDPOINT OF X(76) AND X(316)

Barycentrics    a^6*b^2 - a^2*b^6 + a^6*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 + a^2*b^2*c^4 + 2*b^4*c^4 - a^2*c^6 - 2*b^2*c^6 : :
X(39266) = X[1916] - 3 X[14041], 2 X[2023] - 3 X[33228], X[2080] - 3 X[7697], 3 X[5031] - 2 X[10007], 3 X[5103] - X[32449], 3 X[5207] + X[18906], 4 X[6683] - 5 X[31275], 2 X[8590] - 3 X[32149], X[14712] - 5 X[31276]

X(39266) lies on these lines: {2, 2021}, {4, 69}, {30, 5976}, {39, 625}, {83, 1692}, {99, 5999}, {115, 736}, {148, 9865}, {183, 2080}, {187, 384}, {194, 14063}, {290, 13137}, {325, 2782}, {512, 14295}, {538, 671}, {1007, 7709}, {1570, 7760}, {1691, 7770}, {1915, 33301}, {1975, 35002}, {2022, 3767}, {2023, 33228}, {2024, 7803}, {2025, 5286}, {2031, 6179}, {3094, 7841}, {3095, 7773}, {3407, 7804}, {3734, 5162}, {3785, 14035}, {3849, 7811}, {4576, 14957}, {5031, 6656}, {5052, 7812}, {5103, 5254}, {5111, 7754}, {5188, 7802}, {5475, 33300}, {5969, 8352}, {6683, 7901}, {7747, 18806}, {7748, 8149}, {7757, 9770}, {7763, 11257}, {7769, 13334}, {7771, 15819}, {7776, 13108}, {7780, 10631}, {7786, 14064}, {7825, 32452}, {7828, 13357}, {7892, 31239}, {8370, 24256}, {8590, 32149}, {10358, 35377}, {13085, 33006}, {13857, 22254}, {14039, 26613}, {14045, 32450}, {14693, 37688}, {14907, 22712}, {16068, 18896}, {20023, 33873}, {20025, 34236}, {20081, 32996}, {22503, 37668}, {32522, 32829}, {32832, 38227}

X(39266) = midpoint of X(i) and X(j) for these {i,j}: {76, 316}, {148, 9865}
X(39266) = reflection of X(i) in X(j) for these {i,j}: {39, 625}, {187, 3934}
X(39266) = anticomplement of X(2021)
X(39266) = crossdifference of every pair of points on line {3049, 18899}
X(39266) = {X(4),X(5207)}-harmonic conjugate of X(316)

X(39267) =  ISOGONAL CONJUGATE OF X(11517)

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

X(39267) l;ies on the cubics K028 and K680 and these lines: {3, 915}, {4, 912}, {19, 46}, {28, 1851}, {34, 4306}, {278, 1612}, {1118, 5146}, {4219, 11248}, {5521, 7040}, {8743, 8751}, {14249, 14266}, {14257, 38938}

X(39267) = isogonal conjugate of X(11517)
X(39267) = polar conjugate of X(17776)
X(39267) = polar conjugate of the isotomic conjugate of X(15474)
X(39267) = X(i)-cross conjugate of X(j) for these (i,j): {6, 278}, {513, 13397}, {942, 28}
X(39267) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11517}, {3, 3811}, {8, 3215}, {9, 3173}, {48, 17776}, {63, 2911}, {72, 1780}, {78, 37579}, {219, 1708}, {1260, 4341}, {1331, 15313}, {1794, 14054}, {3682, 30733}
X(39267) = cevapoint of X(513) and X(2969)
X(39267) = trilinear pole of line {6591, 23770}
X(39267) = barycentric product X(i)*X(j) for these {i,j}: {4, 15474}, {27, 23604}, {13397, 17924}
X(39267) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 17776}, {6, 11517}, {19, 3811}, {25, 2911}, {34, 1708}, {56, 3173}, {604, 3215}, {608, 37579}, {1435, 4341}, {1474, 1780}, {1841, 14054}, {5317, 30733}, {6591, 15313}, {13397, 1332}, {15474, 69}, {23604, 306}, {28787, 3998}

X(39268) =  X(3)X(1301)∩X(4)X(64)

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

X(39268) lies on the cubic K028 and these lines: {3, 1301}, {4, 64}, {24, 11589}, {378, 5879}, {1073, 1593}, {3146, 33584}, {8798, 35502}, {10606, 28785}, {11413, 17510}, {14248, 39265}, {14264, 34756}, {14390, 36982}, {15394, 32000}, {35490, 38956}

X(39268) = isotomic conjugate of the isogonal conjugate of X(17510)
X(39268) = polar conjugate of the isogonal conjugate of X(14390)
X(39268) = X(264)-Ceva conjugate of X(1073)
X(39268) = X(30211)-cross conjugate of X(1301)
X(39268) = cevapoint of X(14390) and X(17510)
X(39268) = barycentric product X(i)*X(j) for these {i,j}: {76, 17510}, {264, 14390}, {459, 11413}, {2063, 6526}, {15466, 33583}
X(39268) = barycentric quotient X(i)/X(j) for these {i,j}: {1660, 15905}, {11413, 37669}, {14091, 2883}, {14390, 3}, {17510, 6}, {30211, 20580}, {33583, 1073}
X(39268) = {X(1593),X(31942)}-harmonic conjugate of X(1073)

X(39269) =  X(3)X(935)∩X(4)X(67)

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

X(39269) lies on the cubic K028 and these lines: {3, 935}, {4, 67}, {264, 14246}, {847, 38939}, {850, 1235}, {5094, 10415}, {14254, 39265}, {14265, 38936}, {14357, 37118}, {18281, 34897}

X(39269) = X(15116)-cross conjugate of X(858)
X(39269) = barycentric product X(i)*X(j) for these {i,j}: {1236, 8791}, {5523, 18019}
X(39269) = barycentric quotient X (i)/X(j) for these {i,j}: {67, 18876}, {858, 22151}, {1236, 37804}, {1560, 6593}, {2393, 10317}, {5523, 23}, {8791, 1177}, {14580, 18374}, {20410, 36415}

X(39270) =  X(3)X(2222)∩X(4)X(80)

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

X(39270) lies on the cubic K028 and these lines: {3, 2222}, {4, 80}, {76, 35174}, {1168, 2099}, {1478, 14266}, {2476, 21307}, {10570, 11681}, {13744, 14882}, {17579, 36590}, {23981, 38945}

X(39270) = crosspoint of X(23592) and X(35174)
X(39270) = crosssum of X(8648) and X(35128)
X(39270) = barycentric product X(2006)*X(5176)
X(39270) = barycentric quotient X(5176)/X(32851)

X(39271) =  X(1)X(3)∩X(5)X(51)

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

X(39271) is the point T in Une Maille Remarquable de la Voilette de la Dame Géométrie, by Jean-Louis Ayme, where it is proved that T lies on the line X(355)X(32128).

X(39271) lies on these lines: {1,3}, {5,51}, {140,22076}, {185,37356}, {355,32128}, {389,6882}, {442,1216}, {511,6842}, {568,6971}, {1532,5446}, {2476,11412}, {2779,13605}, {2979,6937}, {3060,6941}, {3567,4193}, {3814,31760}, {3822,31738}, {3917,37438}, {4187,5462}, {4197,7999}, {5396,18178}, {5499,10627}, {5640,6975}, {5752,6863}, {5889,6830}, {5890,6943}, {5907,6841}, {6102,34462}, {6243,6980}, {6265,34434}, {6828,11459}, {6829,11444}, {6831,13754}, {6845,12111}, {6881,11793}, {6907,10625}, {6922,9730}, {6923,37482}, {6940,33852}, {6945,9781}, {6963,15043}, {6990,15056}, {7491,15488}, {8727,12162}, {10883,15058}, {10942,16980}, {15644,37401}, {20305,24220}, {21479,36754}, {25639,34825}, {31825,31847}

X(39271) = barycentric product of X(343) and X(37117)
X(39271) = crossdifference of every pair of points on line X(650)-X(2623)

X(39272) =  TRILINEAR POLE OF X(105)X(518)

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

X(39272) lies on these lines: {2, 6185}, {1025, 5377}, {1026, 23704}, {2402, 2414}, {3573, 6078}, {3675, 28071}, {31637, 35160}

X(39272) = isotomic conjugate of the complement of X(2402)
X(39272) = X(i)-cross conjugate of X(j) for these (i,j): {1, 5377}, {644, 666}, {3309, 2481}, {4162, 294}
X(39272) = X(i)-isoconjugate of X(j) for these (i,j): {514, 20662}, {649, 16593}, {665, 3008}, {672, 6084}, {1019, 20680}, {1279, 2254}, {3912, 8659}, {7649, 20749}
X(39272) = cevapoint of X(i) and X(j) for these (i,j): {1, 35355}, {2, 2402}, {100, 36086}, {650, 28071}, {1814, 24562}
X(39272) = trilinear pole of line {105, 518}
X(39272) = barycentric product X(i)*X(j) for these {i,j}: {666, 1280}, {2481, 6078}, {36086, 36807}
X(39272) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 16593}, {105, 6084}, {692, 20662}, {906, 20749}, {919, 1279}, {1280, 918}, {3309, 5519}, {4557, 20680}, {6078, 518}, {36086, 3008}

X(39273) =  TRILINEAR POLE OF X(663)X(905)

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

X(39273) lies on these lines: {1, 1814}, {6, 77}, {7, 19}, {9, 69}, {41, 7131}, {55, 63}, {57, 7056}, {81, 2299}, {85, 673}, {169, 17758}, {189, 5809}, {284, 1444}, {286, 8748}, {333, 18157}, {514, 1024}, {954, 34381}, {991, 3751}, {1174, 1708}, {2161, 5845}, {2291, 6183}, {2316, 8257}, {3306, 26007}, {3358, 6776}, {5686, 10327}, {5728, 37492}, {6169, 9311}, {6600, 25083}, {7077, 22116}, {9001, 23351}, {10394, 36101}, {13386, 13427}, {13387, 13456}, {14828, 27475}

X(39273) = reflection of X(7675) in X(36740)
X(39273) = isotomic conjugate of the anticomplement of X(37597)
X(39273) = X(i)-cross conjugate of X(j) for these (i,j): {2280, 1}, {5728, 7}, {37597, 2}
X(39273) = X(i)-isoconjugate of X(j) for these (i,j): {2, 37580}, {6, 2550}, {9, 2263}, {42, 16054}, {55, 948}, {57, 28043}, {651, 6182}, {7071, 23603}
X(39273) = cevapoint of X(i) and X(j) for these (i,j): {6, 22769}, {949, 3423}, {2170, 4724}
X(39273) = trilinear pole of line {663, 905}
X(39273) = barycentric product X(i)*X(j) for these {i,j}: {75, 3423}, {85, 949}, {522, 6183}
X(39273) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2550}, {31, 37580}, {55, 28043}, {56, 2263}, {57, 948}, {81, 16054}, {663, 6182}, {949, 9}, {3423, 1}, {6183, 664}, {7177, 23603}

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Trilinear poles of Vietnamese lines: X(39274)-X(39289)

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This preamble and centers X(39274)-X(39289) were contributed by Vu Thanh Tung, July 19, 2020.

Suppose that T = U1V1W1, T = U2V2W2, T = U3V3W3 are triangles such that U1,U2,U3 are collinear, V1,V2,V3 are collinear, and W1, W2, W3 are collinear, so that the three pairs {T2,T3}, {T3,T1}, {T1,T2} have the same vertex triangle.

Theorem: The Vietnamese points (defined in the preamble just before X(39177)) of the three pairs {T2,T3}, {T3,T1}, {T1,T2} are collinear.

Corollary: Let TP, TU, TX be the cevian triangles of points P = p : q : r, U = u : v : w, X = x : y : z (barycentrics). Then the three Vietnamese points of the three pairs {TP,TU}, {TU,TX}, {TX,TP}} lie on a line L'(P,U,X), here named the Vietnamese line of {P,U,X}, of which the trilinear pole is given by

L(P,U,X) = a^2 (p + q) (p + r) (u + v) (u + w) (x + y) (x + z) (p r v^2 x y + q r v^2 x y - q^2 u w x y - q r u w x y - q^2 v w x y + p r v w x y - p r u v y^2 - q r u v y^2 + p q u w y^2 - q r u w y^2 + p q v w y^2 + p r v w y^2 + q^2 u v x z + q r u v x z - p q v^2 x z + q r v^2 x z - p q v w x z - q^2 v w x z + q^2 u v y z - p r u v y z - p q v^2 y z - p r v^2 y z + p q u w y z + q^2 u w y z) (-q r u w x y - r^2 u w x y + p r v w x y + r^2 v w x y + p r w^2 x y - q r w^2 x y + q r u v x z + r^2 u v x z - p q v w x z + r^2 v w x z - p q w^2 x z - q r w^2 x z - p r u v y z - r^2 u v y z + p q u w y z - r^2 u w y z + p q w^2 y z + p r w^2 y z - p r u v z^2 + q r u v z^2 + p q u w z^2 + q r u w z^2 - p q v w z^2 - p r v w z^2) : :

See Vietnamese Line.
See Vietnamese Point.

The appearance of (i,j,k,v) in the following list means that L(X(i),X(j),X(k)) = X(v):
(1,2,3,39274), (1,2,4,14534), (1,2,5,39275), (1,2,6,39276), (1,3,4,39277), (1,3,5,39278), (1,3,6,39279), (1,4,5,39280), (1,4,6,39281), (1,5,6,39282), (2,3,4,275), (2,3,5,288), (2,3,6,39283), (2,4,5,39284), (2,4,6,83), (2,5,6,39285), (3,4,5,39286), (3,4,6,39287), (3,5,6,39288), (4,5,6,39289)

X(39274) =  TRILINEAR POLE OF THE VIETNAMESE LINE OF {X(1), X(2), X(3)}

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

X(39274) lies on these lines: {97,593}, {275,36419}, {1509,34386}

X(39274) = barycentric quotient X(54)/X(37225)
X(39274) = trilinear quotient X(2167)/X(37225)
X(39274) = trilinear pole of the line {3733, 23286}
X(39274) = intersection, other than A,B,C, of conics {{A, B, C, X(54), X(97)}} and {{A, B, C, X(58), X(81)}}
X(39274) = X(54)-reciprocal conjugate of-X(37225)

X(39275) =  TRILINEAR POLE OF THE VIETNAMESE LINE OF {X(1), X(2), X(5)}

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

X(39275) lies on these lines: {593,31626}, {36419,39284}

X(39275) = trilinear pole of the line {3733, 39178}
X(39275) = intersection, other than A,B,C, of conics {{A, B, C, X(58), X(81)}} and {{A, B, C, X(288), X(1173)}}

X(39276) =  TRILINEAR POLE OF THE VIETNAMESE LINE OF {X(1), X(2), X(6)}

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

X(39276) lies on these lines: {58,291}, {81,335}, {82,33766}, {83,1509}, {251,593}, {733,36066}, {741,18267}, {36800,39281}

X(39276) = isogonal conjugate of the complement of X(30940)
X(39276) = complement of the anticomplementary conjugate of X(30940)
X(39276) = barycentric product X(i)*X(j) for these {i, j}: {82, 18827}, {83, 37128}, {251, 40017}, {308, 18268}, {689, 875}, {733, 8033}
X(39276) = barycentric quotient X(i)/X(j) for these (i, j): (31, 4093), (82, 740), (83, 3948), (251, 2238), (291, 15523), (292, 3954)
X(39276) = trilinear product X(i)*X(j) for these {i, j}: {82, 37128}, {83, 741}, {251, 18827}, {660, 39179}, {733, 17103}, {805, 18111}
X(39276) = trilinear quotient X(i)/X(j) for these (i, j): (6, 4093), (82, 2238), (83, 740), (251, 3747), (291, 3954), (292, 21035)
X(39276) = trilinear pole of the line {82, 876}
X(39276) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(24512)}} and {{A, B, C, X(58), X(81)}}
X(39276) = Cevapoint of X(81) and X(33854)
X(39276) = X(875)-cross conjugate of-X(36066)
X(39276) = X(i)-isoconjugate-of-X(j) for these {i,j}: {2, 4093}, {38, 2238}, {39, 740}, {141, 3747}
X(39276) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (31, 4093), (82, 740), (83, 3948), (251, 2238)

X(39277) =  TRILINEAR POLE OF THE VIETNAMESE LINE OF {X(1), X(3), X(4)}

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

X(39277) lies on these lines: {95,261}, {275,24624}, {286,18815}, {1141,6578}, {18021,34384}

X(39277) = barycentric product X(95)*X(24624)
X(39277) = barycentric quotient X(i)/X(j) for these (i, j): (54, 2245), (80, 21011), (95, 3936), (275, 860), (759, 1953), (1141, 8818)
X(39277) = trilinear product X(i)*X(j) for these {i, j}: {54, 14616}, {95, 759}, {655, 39177}
X(39277) = trilinear quotient X(i)/X(j) for these (i, j): (54, 3724), (80, 21807), (95, 758), (655, 35307), (759, 51)
X(39277) = trilinear pole of the line {2167, 4560}
X(39277) = intersection, other than A,B,C, of conics {{A, B, C, X(86), X(261)}} and {{A, B, C, X(95), X(275)}}
X(39277) = Cevapoint of X(81) and X(6905)
X(39277) = X(i)-isoconjugate-of-X(j) for these {i,j}: {5, 3724}, {36, 21807}, {51, 758}, {654, 35307}
X(39277) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (54, 2245), (80, 21011), (95, 3936), (275, 860)

X(39278) =  TRILINEAR POLE OF THE VIETNAMESE LINE OF {X(1), X(3), X(5)}

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

X(39278) lies on theser lines: {}

X(39279) =  TRILINEAR POLE OF THE VIETNAMESE LINE OF {X(1), X(3), X(6)}

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

X(39279) lies on these lines: {}

X(39279) = trilinear pole of the line {39177, 39179}

X(39280) =  TRILINEAR POLE OF THE VIETNAMESE LINE OF {X(1), X(4), X(5)}

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

X(39280) lies on theser lines: {}

X(39281) =  TRILINEAR POLE OF THE VIETNAMESE LINE OF {X(1), X(4), X(6)}

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

X(39281) lies on these lines: {261,1799}, {36800,39276}

X(39281) = barycentric quotient X(83)/X(5051)
X(39281) = trilinear pole of the line {4560, 4580}
X(39281) = intersection, other than A,B,C, of conics {{A, B, C, X(83), X(1799)}} and {{A, B, C, X(86), X(261)}}
X(39281) = Cevapoint of X(82) and X(27067)
X(39281) = X(83)-reciprocal conjugate of-X(5051)

X(39282) =  TRILINEAR POLE OF THE VIETNAMESE LINE OF {X(1), X(5), X(6)}

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

X(39282) lies on theser lines: {}

X(39283) =  TRILINEAR POLE OF THE VIETNAMESE LINE OF {X(2), X(3), X(6)}

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

X(39283) lies on these lines: {54,9418}, {83,3289}, {97,251}, {262,14575}, {275,2211}, {3049,39182}

X(39283) = barycentric product X(262)*X(39287)
X(39283) = barycentric quotient X(i)/X(j) for these (i, j): (54, 14096), (95, 14994)
X(39283) = trilinear product X(2186)*X(39287)
X(39283) = trilinear quotient X(2167)/X(14096)
X(39283) = trilinear pole of the line {18105, 23286}
X(39283) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(2211)}} and {{A, B, C, X(54), X(97)}}
X(39283) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (54, 14096), (95, 14994)

X(39284) =  TRILINEAR POLE OF THE VIETNAMESE LINE OF {X(2), X(4), X(5)}

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

X(39284) lies on the Kiepert hyperbola and these lines: {2,10979}, {4,1173}, {17,473}, {18,472}, {25,7607}, {53,275}, {83,39289}, {96,7576}, {98,428}, {107,38833}, {262,5064}, {297,10159}, {324,11140}, {381,13599}, {427,7608}, {468,10185}, {470,10188}, {471,10187}, {648,13582}, {671,33513}, {1487,3518}, {1585,10195}, {1586,10194}, {2394,39183}, {3839,31363}, {7612,7714}, {8884,20574}, {31617,35937}, {36419,39275}

X(39284) = isogonal conjugate of X(22052)
X(39284) = polar conjugate of X(140)
X(39284) = isotomic conjugate of the isogonal conjugate of X(33631)
X(39284) = complement of the anticomplementary conjugate of X(32002)
X(39284) = barycentric product X(i)*X(j) for these {i, j}: {5, 39286}, {53, 31617}, {76, 33631}, {264, 1173}, {275, 31610}, {288, 324}
X(39284) = barycentric quotient X(i)/X(j) for these (i, j): (4, 140), (19, 17438), (25, 13366), (27, 17168), (51, 32078), (53, 233)
X(39284) = trilinear product X(i)*X(j) for these {i, j}: {75, 33631}, {92, 1173}, {158, 31626}, {162, 39183}, {661, 33513}, {823, 39180}
X(39284) = trilinear quotient X(i)/X(j) for these (i, j): (4, 17438), (19, 13366), (92, 140), (158, 6748), (162, 35324), (264, 20879)
X(39284) = trilinear pole of the line {523, 37943}
X(39284) = intersection, other than A,B,C, of Kiepert hyperbola and conic {{A, B, C, X(6), X(10979)}}
X(39284) = Cevapoint of X(i) and X(j) for these {i,j}: {4, 53}, {6, 3518}, {472, 473}, {1173, 33631}
X(39284) = X(i)-cross conjugate of-X(j) for these (i,j): (4, 39286), (6, 1487)
X(39284) = X(i)-isoconjugate-of-X(j) for these {i,j}: {3, 17438}, {48, 140}, {63, 13366}, {184, 20879}
X(39284) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (4, 140), (19, 17438), (25, 13366), (27, 17168)
X(39284) = {X(288), X(39286)}-harmonic conjugate of X(275)

X(39285) =  TRILINEAR POLE OF THE VIETNAMESE LINE OF {X(2), X(5), X(6)}

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

X(39285) lies on this line: {251,31626}

X(39285) = trilinear pole of the line {18105, 39180}
X(39285) = intersection, other than A,B,C, of conics {{A, B, C, X(83), X(251)}} and {{A, B, C, X(288), X(1173)}}

X(39286) =  TRILINEAR POLE OF THE VIETNAMESE LINE OF {X(3), X(4), X(5)}

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

X(39286) lies on these lines: {4,20574}, {5,95}, {53,275}, {264,25043}, {311,34384}, {436,38833}, {472,36300}, {473,36301}, {1173,4994}, {1263,18831}, {4993,8613}, {6748,16813}, {17500,39287}

X(39286) = isogonal conjugate of X(32078)
X(39286) = polar conjugate of X(233)
X(39286) = isotomic conjugate of the anticomplement of X(12242)
X(39286) = isotomic conjugate of the complement of X(15801)
X(39286) = anticomplement of the complementary conjugate of X(10184)
X(39286) = barycentric product X(i)*X(j) for these {i, j}: {4, 31617}, {95, 39284}, {264, 288}, {276, 1173}
X(39286) = barycentric quotient X(i)/X(j) for these (i, j): (4, 233), (53, 3078), (54, 22052), (107, 35318), (275, 140), (276, 1232)
X(39286) = trilinear product X(i)*X(j) for these {i, j}: {19, 31617}, {92, 288}, {823, 39181}, {2167, 39284}
X(39286) = trilinear quotient X(i)/X(j) for these (i, j): (92, 233), (275, 17438), (276, 20879), (288, 48), (823, 35318), (1577, 35441)
X(39286) = trilinear pole of the line {12077, 15412}
X(39286) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(5)}} and {{A, B, C, X(6), X(1298)}}
X(39286) = Cevapoint of X(i) and X(j) for these {i,j}: {2, 15801}, {4, 275}, {1173, 39284}
X(39286) = X(i)-cross conjugate of-X(j) for these (i,j): (4, 39284), (288, 31617), (523, 16813), (1173, 288)
X(39286) = X(i)-isoconjugate-of-X(j) for these {i,j}: {48, 233}, {163, 35441}, {216, 17438}, {217, 20879}
X(39286) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (4, 233), (53, 3078), (54, 22052), (107, 35318)
X(39286) = {X(275), X(39284)}-harmonic conjugate of X(288)

X(39287) =  TRILINEAR POLE OF THE VIETNAMESE LINE OF {X(3), X(4), X(6)}

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

X(39287) lies on these lines: {54,34384}, {83,275}, {95,325}, {97,251}, {182,18022}, {184,327}, {308,34385}, {1176,8795}, {4993,39668}, {8882,37876}, {17500,39286}, {28724,37872}

X(39287) = polar conjugate of X(27371)
X(39287) = isotomic conjugate of the complement of X(5012)
X(39287) = barycentric product X(i)*X(j) for these {i, j}: {54, 308}, {83, 95}, {99, 39182}, {183, 39283}, {251, 34384}, {275, 1799}
X(39287) = barycentric quotient X(i)/X(j) for these (i, j): (4, 27371), (32, 27374), (54, 39), (82, 1953), (83, 5), (95, 141)
X(39287) = trilinear product X(i)*X(j) for these {i, j}: {54, 3112}, {82, 95}, {83, 2167}, {275, 34055}, {308, 2148}, {662, 39182}
X(39287) = trilinear quotient X(i)/X(j) for these (i, j): (31, 27374), (54, 1964), (82, 51), (83, 1953), (92, 27371), (95, 38)
X(39287) = trilinear pole of the line {2799, 4580}
X(39287) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(297)}} and {{A, B, C, X(4), X(7499)}}
X(39287) = Cevapoint of X(i) and X(j) for these {i,j}: {2, 5012}, {54, 95}, {83, 1176}
X(39287) = X(i)-isoconjugate-of-X(j) for these {i,j}: {5, 1964}, {38, 51}, {39, 1953}, {48, 27371}
X(39287) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (4, 27371), (32, 27374), (54, 39), (82, 1953)

X(39288) =  TRILINEAR POLE OF THE VIETNAMESE LINE OF {X(3), X(5), X(6)}

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

X(39288) lies on these lines: {}

X(39288) = trilinear pole of the line {39181, 39182}

X(39289) =  TRILINEAR POLE OF THE VIETNAMESE LINE OF {X(4), X(5), X(6)}

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

X(39289) lies on these lines: {83,39284}, {1799,37439}, {17500,39286}, {31626,39668}, {33631,37876}

X(39289) = barycentric product X(i)*X(j) for these {i, j}: {308, 1173}, {1799, 39284}
X(39289) = barycentric quotient X(i)/X(j) for these (i, j): (82, 17438), (83, 140), (251, 13366), (288, 16030), (308, 1232), (827, 35324)
X(39289) = trilinear product X(1173)*X(3112)
X(39289) = trilinear quotient X(i)/X(j) for these (i, j): (82, 13366), (83, 17438), (308, 20879), (1173, 1964)
X(39289) = trilinear pole of the line {4580, 39183}
X(39289) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(37688)}} and {{A, B, C, X(4), X(37439)}}
X(39289) = Cevapoint of X(83) and X(17500)
X(39289) = X(i)-isoconjugate-of-X(j) for these {i,j}: {38, 13366}, {39, 17438}, {140, 1964}, {1232, 1923}
X(39289) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (82, 17438), (83, 140), (251, 13366), (288, 16030)

X(39290) =  TRILINEAR POLE OF THE LINE X(30)X(74)

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

X(39290) lies on these lines: {94, 14919}, {265, 16075}, {476, 1304}, {525, 30528}, {648, 34568}, {1494, 1989}, {2132, 23097}, {2394, 2407}, {3545, 5627}, {14264, 15928}, {14559, 15395}, {14590, 16077}, {14592, 15459}, {16080, 18883}

X(39290) = isotomic conjugate of X(5664)
X(39290) = isotomic conjugate of the anticomplement of X(14566)
X(39290) = isotomic conjugate of the complement of X(2394)
X(39290) = X(i)-cross conjugate of X(j) for these (i,j): {110, 16077}, {523, 1494}, {525, 94}, {9209, 18316}, {11079, 15395}, {14566, 2}, {14611, 99}, {18558, 265}, {38340, 32680}
X(39290) = X(i)-isoconjugate of X(j) for these (i,j): {30, 2624}, {31, 5664}, {50, 36035}, {163, 3258}, {186, 2631}, {526, 2173}, {647, 35201}, {656, 39176}, {661, 1511}, {798, 6148}, {810, 14920}, {1495, 32679}, {1637, 6149}, {3268, 9406}, {14206, 14270}
X(39290) = cevapoint of X(i) and X(j) for these (i,j): {2, 2394}, {110, 32662}, {265, 18558}, {523, 1989}, {525, 14919}, {647, 21650}, {2433, 14264}, {23895, 23896}
X(39290) = trilinear pole of line {30, 74} (the antipedal line of X(74))
X(39290) = barycentric product X(i)*X(j) for these {i,j}: {74, 35139}, {99, 5627}, {265, 16077}, {328, 1304}, {476, 1494}, {850, 15395}, {2349, 32680}, {6331, 11079}, {20573, 32640}, {23895, 36311}, {23896, 36308}, {32678, 33805}
X(39290) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 5664}, {74, 526}, {99, 6148}, {110, 1511}, {112, 39176}, {162, 35201}, {265, 9033}, {476, 30}, {523, 3258}, {648, 14920}, {1304, 186}, {1494, 3268}, {1989, 1637}, {2159, 2624}, {2166, 36035}, {2349, 32679}, {2433, 2088}, {3470, 8562}, {5627, 523}, {6742, 6739}, {9139, 9213}, {10419, 15470}, {11060, 14398}, {11079, 647}, {14380, 16186}, {14559, 5642}, {14560, 1495}, {14611, 31378}, {14919, 8552}, {15395, 110}, {15459, 14165}, {16077, 340}, {18558, 39008}, {18808, 35235}, {32640, 50}, {32662, 3284}, {32678, 2173}, {32680, 14206}, {32715, 34397}, {35139, 3260}, {35189, 15469}, {36034, 6149}, {36064, 1464}, {36129, 1784}, {36306, 6111}, {36308, 23871}, {36309, 6110}, {36311, 23870}, {36831, 1154}

X(39291) =  TRILINEAR POLE OF THE LINE X(98)X(385)

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

X(39291) lies on the hyperbola {{A, B, C, X(2), X(110)}} and these lines: {2, 14382}, {83, 3493}, {110, 18858}, {287, 694}, {290, 9468}, {685, 4230}, {805, 4226}, {850, 30530}, {868, 38947}, {880, 2395}, {1316, 14251}, {1975, 9476}, {2421, 2422}, {5967, 34238}, {5968, 9154}, {11328, 32540}, {31636, 36214}

X(39291) = isotomic conjugate of the complement of X(2395)
X(39291) = X(18858)-Ceva conjugate of X(2966)
X(39291) = X(i)-cross conjugate of X(j) for these (i,j): {99, 2966}, {512, 290}, {647, 694}, {2524, 248}, {10352, 4590}, {24782, 37128}, {37137, 37134}
X(39291) = cevapoint of Jerabek-hyperbola-intercepts of PU(1) (line X(39)X(512))
X(39291) = X(i)-isoconjugate of X(j) for these (i,j): {661, 36213}, {662, 2679}, {798, 5976}, {804, 1755}, {1580, 3569}, {1933, 2799}, {1959, 5027}, {1966, 2491}, {4107, 5360}, {9417, 14295}
X(39291) = cevapoint of X(i) and X(j) for these (i,j): {2, 2395}, {99, 18829}, {287, 647}, {512, 9468}, {2422, 32540}
X(39291) = trilinear pole of line {98, 385} (the antipedal line of X(98))
X(39291) = barycentric product X(i)*X(j) for these {i,j}: {98, 18829}, {99, 36897}, {290, 805}, {325, 18858}, {670, 34238}, {1581, 36036}, {1821, 37134}, {1916, 2966}, {1934, 36084}, {2715, 18896}, {6037, 8842}, {6331, 15391}, {17938, 18024}, {22456, 36214}
X(39291) = barycentric quotient X(i)/X(j) for these {i,j}: {98, 804}, {99, 5976}, {110, 36213}, {287, 24284}, {290, 14295}, {512, 2679}, {685, 419}, {694, 3569}, {805, 511}, {1916, 2799}, {1976, 5027}, {2422, 2086}, {2715, 1691}, {2966, 385}, {5967, 11183}, {9468, 2491}, {15391, 647}, {17932, 12215}, {17938, 237}, {17980, 17994}, {18829, 325}, {18858, 98}, {22456, 17984}, {34238, 512}, {36036, 1966}, {36065, 1284}, {36084, 1580}, {36214, 684}, {36897, 523}, {37134, 1959}, {37137, 16591}, {38947, 31953}

X(39292) =  TRILINEAR POLE OF THE LINE X(99)X(512)

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

X(39292) lies on the circumhyperbola {{A,B,C,X(2),X(6)}} and these lines: {2, 34537}, {6, 4590}, {25, 18020}, {37, 4601}, {42, 4600}, {99, 14606}, {111, 1916}, {249, 18898}, {251, 31614}, {694, 10754}, {733, 38278}, {805, 5468}, {880, 2395}, {892, 9178}, {1400, 4620}, {1989, 18896}, {2142, 23099}, {3228, 9468}, {3572, 37134}, {6035, 14998}, {9146, 9170}, {14948, 18872}, {16098, 36214}, {17938, 17941}, {34204, 34245}

X(39292) = isogonal conjugate of X(2086)
X(39292) = isotomic conjugate of the anticomplement of X(11052)
X(39292) = isotomic conjugate of the complement of X(2396)
X(39292) = X(i)-cross conjugate of X(j) for these (i,j): {385, 99}, {511, 670}, {1916, 18829}, {3229, 110}, {9468, 805}, {10754, 892}, {11052, 2}, {36849, 2966}, {37128, 37134}
X(39292) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2086}, {115, 1933}, {238, 21823}, {661, 5027}, {740, 21755}, {798, 804}, {1084, 1966}, {1109, 14602}, {1580, 3124}, {1691, 2643}, {1910, 2679}, {1914, 21725}, {1924, 14295}, {1926, 9427}, {1967, 35078}, {2238, 4128}, {3121, 4039}, {3747, 16592}, {3978, 4117}, {4079, 4164}, {4155, 20981}, {7234, 21832}, {18902, 23994}
X(39292) = cevapoint of X(i) and X(j) for these (i,j): {2, 2396}, {99, 385}, {805, 9468}, {1916, 18829}, {2086, 14824}, {4563, 36212}
X(39292) = trilinear pole of line {99, 512} (the antipedal line of X(99))
X(39292) = barycentric product X(i)*X(j) for these {i,j}: {99, 18829}, {249, 18896}, {670, 805}, {694, 34537}, {799, 37134}, {1581, 24037}, {1916, 4590}, {1934, 24041}, {4584, 7260}, {4589, 4594}, {4603, 4639}, {4609, 17938}, {36806, 37137}
X(39292) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 2086}, {99, 804}, {110, 5027}, {249, 1691}, {291, 21725}, {292, 21823}, {385, 35078}, {511, 2679}, {670, 14295}, {694, 3124}, {741, 4128}, {805, 512}, {881, 23099}, {882, 22260}, {1101, 1933}, {1581, 2643}, {1916, 115}, {1927, 4117}, {1934, 1109}, {3903, 4155}, {4563, 24284}, {4589, 2533}, {4590, 385}, {4594, 4010}, {4600, 4039}, {4603, 21832}, {4610, 4107}, {4623, 14296}, {5378, 21803}, {5468, 11183}, {7303, 27846}, {8789, 9427}, {9468, 1084}, {14970, 34294}, {16068, 6071}, {17938, 669}, {17980, 2971}, {18020, 419}, {18268, 21755}, {18829, 523}, {18872, 21906}, {18896, 338}, {23357, 14602}, {23963, 18902}, {24037, 1966}, {24041, 1580}, {31614, 17941}, {34238, 15630}, {34537, 3978}, {36066, 4367}, {36214, 20975}, {37128, 16592}, {37134, 661}

X(39293) =  TRILINEAR POLE OF THE LINE X(101)X(514)

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

X(39293) lines on the circumconic {{A,B,C,X(2),X(7)}} and these lines: {2, 1252}, {7, 59}, {75, 765}, {86, 4570}, {273, 7012}, {310, 4600}, {335, 1462}, {664, 30573}, {666, 28132}, {677, 883}, {901, 927}, {903, 9268}, {1024, 1025}, {1088, 7045}, {1416, 31002}, {2481, 18815}, {4564, 27475}, {14887, 17753}, {34075, 36146}, {34085, 36086}

X(39293) = isotomic conjugate of the anticomplement of X(24980)
X(39293) = isotomic conjugate of the complement of X(2398)
X(39293) = X(i)-cross conjugate of X(j) for these (i,j): {238, 651}, {516, 190}, {673, 34085}, {1416, 36146}, {1738, 653}, {3008, 658}, {4872, 668}, {6996, 799}, {9436, 664}, {9441, 100}, {13329, 662}, {14942, 666}, {20367, 4610}, {24618, 4615}, {24715, 655}, {24781, 13149}, {24980, 2}
X(39293) = X(i)-isoconjugate of X(j) for these (i,j): {6, 17435}, {11, 2223}, {55, 3675}, {241, 14936}, {244, 2340}, {294, 35505}, {513, 926}, {518, 3271}, {650, 665}, {663, 2254}, {672, 2170}, {693, 8638}, {884, 3126}, {911, 1566}, {918, 3063}, {1015, 3693}, {1458, 2310}, {1876, 3270}, {2356, 7004}, {2481, 15615}, {3022, 34855}, {3248, 3717}, {3286, 4516}, {4858, 9454}, {5089, 7117}, {7077, 38989}, {7252, 24290}, {8735, 20752}, {9455, 34387}, {17197, 39258}, {18191, 20683}, {18210, 37908}, {34018, 39014}
X(39293) = cevapoint of X(i) and X(j) for these (i,j): {1, 1025}, {2, 2398}, {109, 1429}, {190, 32850}, {651, 7677}, {664, 9436}, {666, 14942}, {673, 36086}, {1331, 26006}, {1416, 36146}, {20715, 35310}
X(39293) = trilinear pole of line {101, 514} (the antipedal line of X(101))
X(39293) = barycentric product X(i)*X(j) for these {i,j}: {59, 18031}, {85, 5377}, {100, 34085}, {109, 36803}, {190, 927}, {658, 36802}, {664, 666}, {668, 36146}, {673, 4998}, {765, 34018}, {919, 4572}, {1275, 14942}, {1416, 31625}, {1462, 7035}, {1978, 32735}, {2481, 4564}, {4554, 36086}, {4620, 13576}, {7045, 36796}
X(39293) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17435}, {57, 3675}, {59, 672}, {101, 926}, {105, 2170}, {109, 665}, {294, 2310}, {516, 1566}, {651, 2254}, {664, 918}, {666, 522}, {673, 11}, {765, 3693}, {919, 663}, {927, 514}, {1016, 3717}, {1025, 3126}, {1252, 2340}, {1262, 1458}, {1275, 9436}, {1416, 1015}, {1429, 38989}, {1438, 3271}, {1458, 35505}, {1462, 244}, {1814, 7004}, {2149, 2223}, {2195, 14936}, {2481, 4858}, {3212, 23773}, {4551, 24290}, {4552, 4088}, {4564, 518}, {4573, 23829}, {4619, 2283}, {4620, 30941}, {4998, 3912}, {5377, 9}, {6559, 4081}, {6654, 4124}, {7012, 5089}, {7045, 241}, {7115, 2356}, {7128, 1876}, {7677, 38980}, {9436, 35094}, {9454, 15615}, {13576, 21044}, {14942, 1146}, {18031, 34387}, {18785, 4516}, {28071, 3119}, {28132, 23615}, {31615, 1026}, {31637, 26932}, {32660, 23225}, {32666, 3063}, {32735, 649}, {32739, 8638}, {34018, 1111}, {34085, 693}, {36057, 7117}, {36086, 650}, {36124, 8735}, {36146, 513}, {36796, 24026}, {36802, 3239}, {36803, 35519}

X(39294) =  TRILINEAR POLE OF THE LINE X(109)X(522)

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

X(39294) lies on these lines: {2, 1262}, {8, 59}, {92, 7128}, {312, 4564}, {1309, 14733}, {1981, 36146}, {2399, 2406}, {4620, 28660}, {14942, 36123}, {16082, 18359}

X(39294) = polar conjugate of X(35015)
X(39294) = isotomic conjugate of the complement of X(2406)
X(39294) = X(i)-cross conjugate of X(j) for these (i,j): {515, 664}, {3911, 653}, {5053, 162}, {5081, 18026}, {13532, 35174}
X(39294) = X(i)-isoconjugate of X(j) for these (i,j): {6, 35014}, {48, 35015}, {517, 7117}, {521, 3310}, {650, 8677}, {652, 1769}, {1364, 14571}, {1409, 14010}, {1411, 38353}, {1457, 34591}, {1465, 3270}, {1875, 35072}, {1946, 10015}, {2170, 22350}, {2183, 7004}, {2804, 22383}, {3326, 14578}, {4391, 23220}, {10017, 32677}
X(39294) = cevapoint of X(i) and X(j) for these (i,j): {2, 2406}, {653, 17923}, {1783, 37305}, {1877, 32674}, {34234, 37136}
X(39294) = trilinear pole of line {109, 522} (the antipedal line of X(109))
X(39294) = barycentric product X(i)*X(j) for these {i,j}: {653, 13136}, {664, 1309}, {668, 36110}, {1809, 24032}, {1978, 32702}, {4564, 16082}, {4572, 14776}, {4998, 36123}, {6335, 37136}, {7012, 18816}, {7128, 36795}, {18026, 36037}
X(39294) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 35014}, {4, 35015}, {29, 14010}, {59, 22350}, {104, 7004}, {108, 1769}, {109, 8677}, {515, 10017}, {653, 10015}, {909, 7117}, {1309, 522}, {1457, 35012}, {1785, 3326}, {1795, 1364}, {1809, 24031}, {1877, 3259}, {1897, 2804}, {2323, 38353}, {2342, 3270}, {2720, 1459}, {7012, 517}, {7115, 2183}, {7128, 1465}, {13136, 6332}, {14776, 663}, {15500, 35065}, {15742, 6735}, {16082, 4858}, {18026, 36038}, {18816, 17880}, {24033, 1875}, {32641, 652}, {32669, 22383}, {32674, 3310}, {32702, 649}, {34051, 3942}, {34234, 26932}, {36037, 521}, {36110, 513}, {36123, 11}, {37136, 905}, {37305, 38981}

X(39295) =  TRILINEAR POLE OF THE LINE X(110)X(476)

Barycentrics    (a^2 - b^2)^2*(a^2 - c^2)^2*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2) : :
X(39295) = 3 X[2] - 4 X[36597]

X(39295) lies on the Kiepert circumhyperbola and these lines: {2, 249}, {4, 250}, {10, 4570}, {76, 4590}, {94, 23588}, {98, 265}, {99, 5649}, {110, 32731}, {321, 4567}, {476, 691}, {648, 32711}, {671, 1989}, {687, 14590}, {1916, 11060}, {2052, 23582}, {2394, 2407}, {2421, 2437}, {2966, 14592}, {2986, 18879}, {3972, 34289}, {4049, 4591}, {4235, 32697}, {4444, 32678}, {4558, 30528}, {5475, 7578}, {9180, 14559}, {9273, 24624}, {11078, 36317}, {11092, 36316}, {13582, 30529}, {14560, 32717}, {14566, 14999}, {23105, 39138}, {32680, 37140}

X(39295) = isogonal conjugate of X(2088)
X(39295) = polar conjugate of X(35235)
X(39295) = isotomic conjugate of the anticomplement of X(24975)
X(39295) = isotomic conjugate of the complement of X(2407)
X(39295) = X(i)-cross conjugate of X(j) for these (i,j): {30, 99}, {50, 110}, {74, 18878}, {265, 35139}, {395, 36839}, {396, 36840}, {1989, 476}, {3580, 648}, {9140, 892}, {10733, 16077}, {11078, 23895}, {11092, 23896}, {14910, 1304}, {15107, 4577}, {15360, 35138}, {16310, 107}, {24624, 32680}, {24975, 2}, {35345, 31614}, {35466, 38340}, {38872, 10420}
X(39295) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2088}, {19, 16186}, {36, 21824}, {48, 35235}, {50, 1109}, {115, 6149}, {186, 3708}, {323, 2643}, {512, 32679}, {523, 2624}, {526, 661}, {758, 20982}, {798, 3268}, {1577, 14270}, {2081, 2616}, {2151, 30468}, {2152, 30465}, {2159, 3258}, {2166, 18334}, {2245, 2611}, {2290, 8901}, {2605, 2610}, {2642, 9213}, {3724, 8287}, {7113, 21054}, {19627, 23994}, {20902, 34397}
X(39295) = cevapoint of X(i) and X(j) for these (i,j): {2, 2407}, {6, 15329}, {50, 110}, {99, 7809}, {112, 403}, {265, 32662}, {476, 1989}, {4558, 11064}, {11078, 23895}, {11092, 23896}, {24624, 37140}
X(39295) = trilinear pole of line {110, 476} (the antipedal line of X(110))
X(39295) = barycentric product X(i)*X(j) for these {i,j}: {94, 249}, {99, 476}, {110, 35139}, {250, 328}, {265, 18020}, {662, 32680}, {670, 14560}, {799, 32678}, {811, 36061}, {892, 14559}, {1989, 4590}, {2166, 24041}, {2410, 30528}, {3260, 15395}, {4592, 36129}, {6035, 23968}, {6331, 32662}, {7799, 23588}, {11060, 34537}, {15455, 37140}, {15475, 31614}, {20573, 23357}, {23895, 23896}
X(39295) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 16186}, {4, 35235}, {6, 2088}, {13, 30468}, {14, 30465}, {30, 3258}, {50, 18334}, {80, 21054}, {94, 338}, {99, 3268}, {110, 526}, {163, 2624}, {249, 323}, {250, 186}, {265, 125}, {328, 339}, {403, 16221}, {476, 523}, {662, 32679}, {691, 9213}, {759, 2611}, {1101, 6149}, {1141, 8901}, {1576, 14270}, {1625, 2081}, {1989, 115}, {2161, 21824}, {2166, 1109}, {2407, 5664}, {4558, 8552}, {4590, 7799}, {5627, 12079}, {5994, 6137}, {5995, 6138}, {6344, 2970}, {6740, 6741}, {6742, 6370}, {7799, 23965}, {9274, 17104}, {10412, 23105}, {10420, 15470}, {11060, 3124}, {14356, 868}, {14559, 690}, {14560, 512}, {14616, 17886}, {15395, 74}, {15475, 8029}, {18020, 340}, {18384, 8754}, {20123, 19223}, {20573, 23962}, {23357, 50}, {23582, 14165}, {23588, 1989}, {23895, 23871}, {23896, 23870}, {23963, 19627}, {23966, 11060}, {23968, 1640}, {23969, 14998}, {24624, 8287}, {30466, 30470}, {30469, 30467}, {30528, 2411}, {32662, 647}, {32678, 661}, {32680, 1577}, {34079, 20982}, {34209, 6070}, {35049, 18593}, {35139, 850}, {35189, 15453}, {36061, 656}, {36069, 2605}, {36129, 24006}, {36210, 30463}, {36211, 30460}, {36839, 23283}, {36840, 23284}, {37140, 14838}

X(39296) =  TRILINEAR POLE OF THE LINE X(111)X(524)

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

X(39296) lies on these lines: {2, 10630}, {99, 34574}, {671, 34581}, {691, 5468}, {2408, 2418}, {9139, 36890}, {10754, 34898}, {30786, 34166}, {31614, 34205}

X(39296) = isotomic conjugate of the complement of X(2408)
X(39296) = X(i)-cross conjugate of X(j) for these (i,j): {1499, 671}, {9146, 82}
X(39296) = X(i)-isoconjugate of X(j) for these (i,j): {896, 6088}, {2642, 11580}
X(39296) = cevapoint of X(i) and X(j) for these (i,j): {2, 2408}, {1499, 34581}
X(39296) = trilinear pole of line {111, 524} (the antipedal line of X(111))
X(39296) = barycentric product X(i)*X(j) for these {i,j}: {671, 6082}, {892, 34898}
X(39296) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 6088}, {691, 11580}, {892, 11054}, {1296, 10354}, {1499, 31654}, {6082, 524}, {32583, 9872}, {34581, 9125}, {34898, 690}, {37210, 16597}

X(39297) =  TRILINEAR POLE OF THE LINE X(112)X(525)

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

X(39297) lies on these lines: {2, 23964}, {69, 250}, {253, 15384}, {264, 32230}, {305, 18020}, {1304, 2867}, {2409, 2419}, {5379, 20336}

X(39297) = isotomic conjugate of the complement of X(2409)
X(39297) = X(i)-cross conjugate of X(j) for these (i,j): {1297, 2966}, {1503, 648}
X(39297) = X(i)-isoconjugate of X(j) for these (i,j): {656, 2881}, {2631, 15292}
X(39297) = cevapoint of X(i) and X(j) for these (i,j): {2, 2409}, {110, 441}
X(39297) = trilinear pole of line {112, 525} (the antipedal line of X(112))
X(39297) = barycentric product X(648)*X(2867)
X(39297) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 2881}, {1304, 15292}, {1503, 33504}, {2867, 525}, {37202, 34846}

X(39298) =  TRILINEAR POLE OF THE LINE X(110)X(1113)

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

X(39298) lies on these lines: {3, 16071}, {99, 112}, {182, 15461}, {249, 8116}, {250, 1114}, {691, 1113}, {2580, 37140}, {2966, 15164}

X(39298) = polar conjugate of X(39240)
X(39298) = X(18020)-Ceva conjugate of X(15461)
X(39298) = X(i)-cross conjugate of X(j) for these (i,j): {1114, 99}, {2575, 15164}, {8106, 1113}, {8116, 648}, {15461, 18020}
X(39298) = X(i)-isoconjugate of X(j) for these (i,j): {48, 39240}, {115, 1823}, {125, 2577}, {512, 2582}, {523, 2578}, {647, 2588}, {656, 8105}, {661, 2574}, {798, 22339}, {810, 2592}, {1114, 3708}, {1313, 2579}, {2501, 2584}, {2581, 20975}, {2587, 3269}, {2589, 15166}, {2643, 8116}
X(39298) = cevapoint of X(i) and X(j) for these (i,j): {112, 1114}, {1113, 8106}, {4558, 8116}
X(39298) = trilinear pole of line {110, 1113} (the antipedal line of X(1113))
X(39298) = barycentric product X(i)*X(j) for these {i,j}: {99, 1113}, {110, 15164}, {249, 2593}, {250, 22340}, {648, 8115}, {662, 2580}, {799, 2576}, {811, 1822}, {2575, 18020}, {2585, 23999}, {2586, 4592}, {2589, 24041}, {4590, 8106}, {15165, 15461}
X(39298) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 39240}, {99, 22339}, {110, 2574}, {112, 8105}, {162, 2588}, {163, 2578}, {249, 8116}, {250, 1114}, {648, 2592}, {662, 2582}, {1101, 1823}, {1113, 523}, {1114, 1313}, {1822, 656}, {2575, 125}, {2576, 661}, {2579, 3708}, {2580, 1577}, {2583, 20902}, {2585, 2632}, {2586, 24006}, {2589, 1109}, {2593, 338}, {4575, 2584}, {8106, 115}, {8115, 525}, {15164, 850}, {15461, 2575}, {18020, 15165}, {22340, 339}, {24000, 2587}, {39241, 23105}
X(39298) = {X(112),X(4235)}-harmonic conjugate of X(39299)

X(39299) =  TRILINEAR POLE OF THE LINE X(110)X(1114)

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

X(39299) lies on these lines: {3, 16070}, {99, 112}, {182, 15460}, {249, 8115}, {250, 1113}, {691, 1114}, {2581, 37140}, {2966, 15165}

X(39299) = polar conjugate of X(39241)
X(39299) = X(18020)-Ceva conjugate of X(15460)
X(39299) = X(i)-cross conjugate of X(j) for these (i,j): {1113, 99}, {2574, 15165}, {8105, 1114}, {8115, 648}, {15460, 18020}
X(39299) = X(i)-isoconjugate of X(j) for these (i,j): {48, 39241}, {115, 1822}, {125, 2576}, {512, 2583}, {523, 2579}, {647, 2589}, {656, 8106}, {661, 2575}, {798, 22340}, {810, 2593}, {1113, 3708}, {1312, 2578}, {2501, 2585}, {2580, 20975}, {2586, 3269}, {2588, 15167}, {2643, 8115}
X(39299) = cevapoint of X(i) and X(j) for these (i,j): {112, 1113}, {1114, 8105}, {4558, 8115}
X(39299) = trilinear pole of line {110, 1114} (the antipedal line of X(1114))
X(39299) = barycentric product X(i)*X(j) for these {i,j}: {99, 1114}, {110, 15165}, {249, 2592}, {250, 22339}, {648, 8116}, {662, 2581}, {799, 2577}, {811, 1823}, {2574, 18020}, {2584, 23999}, {2587, 4592}, {2588, 24041}, {4590, 8105}, {15164, 15460}
X(39299) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 39241}, {99, 22340}, {110, 2575}, {112, 8106}, {162, 2589}, {163, 2579}, {249, 8115}, {250, 1113}, {648, 2593}, {662, 2583}, {1101, 1822}, {1113, 1312}, {1114, 523}, {1823, 656}, {2574, 125}, {2577, 661}, {2578, 3708}, {2581, 1577}, {2582, 20902}, {2584, 2632}, {2587, 24006}, {2588, 1109}, {2592, 338}, {4575, 2585}, {8105, 115}, {8116, 525}, {15165, 850}, {15460, 2574}, {18020, 15164}, {22339, 339}, {24000, 2586}, {39240, 23105}
X(39299) = {X(112),X(4235)}-harmonic conjugate of X(39298)

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Centers of osculating circles to Steiner ellipses: X(39300)-X(39307)

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This preamble and centers X(39300)-X(39307) were contributed by Peter Moses, July 23-24, 2020.

The centers of osculating circles at the four vertices of the Steiner circumellipse and the Steiner inellipse are all given by the following form for 1st barycentric:

2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*sgn1 + const*S*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2*sgn1]*sgn2*Sqrt[2 - ((a^2 + b^2 + c^2)*sgn3)/Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]]

The Steiner circumellipse has const = 1/2. Specifically, X(39300) is given by (sgn1,sgn2,sgn3) = (1,1,-1)
X(39301) is given by (sgn1,sgn2,sgn3) = (1,-1,-1)
X(39302) is given by (sgn1,sgn2,sgn3) = (-1,1,1)
X(39303) is given by (sgn1,sgn2,sgn3) = (-1,-1,1)

The Steiner inellipse has const = 1. Specifically, X(39304) is given by (sgn1,sgn2,sgn3) = (1,1,-1)
X(39305) is given by (sgn1,sgn2,sgn3) = (1,-1,-1)
X(39306) is given by (sgn1,sgn2,sgn3) = (-1,1,1)
X(39307) is given by (sgn1,sgn2,sgn3) = (-1,-1,1)

For the placement of the circles within the ellipses, see .

Steiner Cirumellipse Osculating Circles
Steiner Inellipse Osculating Circles

See also the preambles just before X(39202 (vertices) and X(39158) (foci).

X(39300) =  CENTER OF OSCULATING CIRCLE OF VERTEX (1,1,-1) OF STEINER CIRCUMELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] + (Sqrt[2 + (a^2 + b^2 + c^2)/Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]]*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]*S)/2 : :
X(39300) = X[39301] - 4 X[39304], 3 X[39301] - 4 X[39305], 3 X[39304] - X[39305]

X(39300) lies on this line: {2, 1341}

X(39300) = reflection of X(i) in X(j) for these {i,j}: {2, 39304}, {39301, 2}
X(39300) = anticomplement of X(39305)
X(39300) = {X(39158),X(39159)}-harmonic conjugate of X(39202)

X(39301) =  CENTER OF OSCULATING CIRCLE OF VERTEX (1,-1,-1) OF STEINER CIRCUMELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] - (Sqrt[2 + (a^2 + b^2 + c^2)/Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]]*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]*S)/2 : :
X(39301) = 3 X[39300] - 4 X[39304], X[39300] - 4 X[39305], X[39304] - 3 X[39305]

X(39301 lies on this line: {2, 1341}

X(30301) = reflection of X(i) in X(j) for these {i,j}: {2, 39305}, {39300, 2}
X(30301) = anticomplement of X(39304)
X(30301) = {X(39158),X(39159)}-harmonic conjugate of X(39203)

X(39302) =  CENTER OF OSCULATING CIRCLE OF VERTEX (-1,1,1) OF STEINER CIRCUMELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 + (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] + (Sqrt[2 - (a^2 + b^2 + c^2)/Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]]*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 - 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]*S)/2 : :
X(39302) = X[39303] - 4 X[39306], 3 X[39303] - 4 X[39307], 3 X[39306] - X[39307]

X(39302) lies on this line: {2, 1340}

X(39302) = reflection of X(i) in X(j) for these {i,j}: {2, 39306}, {39303, 2}
X(39302) = anticomplement of X(39307)

X(39303) =  CENTER OF OSCULATING CIRCLE OF VERTEX (-1,-1,1) OF STEINER CIRCUMELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 + (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] - (Sqrt[2 - (a^2 + b^2 + c^2)/Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]]*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 - 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]*S)/2 : :
X(39303) = 3 X[39302] - 4 X[39306], X[39302] - 4 X[39307], X[39306] - 3 X[39307]

X(39303) lies on this line: {2, 1340}

X(30303) = reflection of X(i) in X(j) for these {i,j}: {2, 39307}, {39302, 2}
X(30303) = anticomplement of X(39306)

X(39304) =  CENTER OF OSCULATING CIRCLE OF VERTEX (1,1,-1) OF STEINER INELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] + Sqrt[2 + (a^2 + b^2 + c^2)/Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]]*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]*S : :
X(39304) = 3 X[39300] + X[39301], 2 X[39300] + X[39305], 2 X[39301] - 3 X[39305]

X(39304) lies on this line: {2, 1341}

X(39304) = midpoint of X(2) and X(39300)
X(39304) = reflection of X(39305) in X(2)
X(39304) = complement of X(39301)
X(39304) = psi-transform of X(39206)
X(39304) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13722, 35913, 39305}, {39162, 39163, 39206}

X(39305) =  CENTER OF OSCULATING CIRCLE OF VERTEX (1,-1,-1) OF STEINER INELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] - Sqrt[2 + (a^2 + b^2 + c^2)/Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]]*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]*S : :
X(39305) = X[39300] + 3 X[39301], 2 X[39300] - 3 X[39304], 2 X[39301] + X[39304]

X(39305) lies on this line: {2, 1341}

X(39305) = midpoint of X(2) and X(39301)
X(39305) = reflection of X(39304) in X(2)
X(39305) = complement of X(39300)
X(39305) = psi-transform of X(39207)
X(39305) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13722, 35913, 39304}, {39162, 39163, 39207}

X(39306) =  CENTER OF OSCULATING CIRCLE OF VERTEX (-1,1,1) OF STEINER INELLIPSE

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

X(39306) = 3 X[39302] + X[39303], 2 X[39302] + X[39307], 2 X[39303] - 3 X[39307]

X(39306) lies on this lines: {2, 1340}

X(39306) = midpoint of X(2) and X(39302)
X(39306) = reflection of X(39307) in X(2)
X(39306) = complement of X(39303)
X(39306) = psi-transform of X(39208)
X(39306) = {X(13636),X(35914)}-harmonic conjugate of X(39307)

X(39307) =  CENTER OF OSCULATING CIRCLE OF VERTEX (-1,-1,1) OF STEINER INELLIPSE

Barycentrics    2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 + (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] - Sqrt[2 - (a^2 + b^2 + c^2)/Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]]*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 - 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]*S : :
X(39307) = X[39302] + 3 X[39303], 2 X[39302] - 3 X[39306], 2 X[39303] + X[39306]

X(39307) lies on this line: {2, 1340}

X(39307) = midpoint of X(2) and X(39303)
X(39307) = reflection of X(39306) in X(2)
X(39307) = complement of X(39302)
X(39307) = psi-transform of X(39209)
X(39307) = {X(13636),X(35914)}-harmonic conjugate of X(39306)

X(39308) =  X(100)X(518)∩X(650)X(3246)

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

X(39308) lies on these lines: {100,518}, {650,3246}, {1319,37139}, {5126,14077}

X(39309) =  X(105)X(518)∩X(1054)X(9441)

Barycentrics    a (4 a^8 - 13 a^7 b + 22 a^6 b^2 - 28 a^5 b^3 + 25 a^4 b^4 - 17 a^3 b^5 + 12 a^2 b^6 - 6 a b^7 + b^8 - 13 a^7 c + 18 a^6 b c - 3 a^5 b^2 c - 17 a^4 b^3 c + 29 a^3 b^4 c - 28 a^2 b^5 c + 19 a b^6 c - 5 b^7 c + 22 a^6 c^2 - 3 a^5 b c^2 - 2 a^4 b^2 c^2 - 4 a^3 b^3 c^2 + 16 a^2 b^4 c^2 - 29 a b^5 c^2 + 16 b^6 c^2 - 28 a^5 c^3 - 17 a^4 b c^3 - 4 a^3 b^2 c^3 - 8 a^2 b^3 c^3 + 16 a b^4 c^3 - 27 b^5 c^3 + 25 a^4 c^4 + 29 a^3 b c^4 + 16 a^2 b^2 c^4 + 16 a b^3 c^4 + 30 b^4 c^4 - 17 a^3 c^5 - 28 a^2 b c^5 - 29 a b^2 c^5 - 27 b^3 c^5 + 12 a^2 c^6 + 19 a b c^6 + 16 b^2 c^6 - 6 a c^7 - 5 b c^7 + c^8) : :

X(39309) lies on these lines: {105,518}, {1054,9441}, {1083,4936}

X(39310) =  X(1)X(1646)∩X(244)X(518)

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

X(39310) lies on these lines: {1,1646}, {244,518}, {649,3246}}

leftri

Points associated with Paasche triangles: X(39311)-X(39234

rightri

This preamble was contributed by Dasari Naga Vijay Krishna, July 24, 2020.

In the preambles just before X(37994) and X(38487) we define points labeled as Ab, Ac, Bc, Ba, Ca, Cb , A'b, A'c, B'c, B'a, C'a, C'b . Here, we define six more points, A''b, A''c, B''c, B''a, C''a, C''b as the respective reflections of Ab, Ac, Bc, Ba, Ca, Cb in the vertices A,B,C.; e.g.,

A''b = Ab in B, and A''c = reflection of Ac in C, etc.

Barycentrics for the 6 points are as follows:

A"b = 0 : 2R+2c : -c
A"c = 0 : -b : 2R+2b
B"c= -a : 0 : 2R+2a
B"a = 2R+2c : 0 : -c
C''a = 2R+2b : -b : 0
C''b = -a : 2R+2a : 0

It is easy to verify that the 6 points, A"b , A"c, B"c, B"a, C''a, C''b, lie on an ellipse, here named the Paasche Reflection Ellipse (PRE), given by the barycentric equation

2 b c (R + a) x^2 + 2 c a (R + b) y^2 + 2 a b (R + c) z^2 + a (4 R^2 + 4 R b + 4 R c + 5 b c) y z + b (4 R^2 + 4 R c + 4 R c + 5 c a) x z + c (4 R^2 + 4 R a + 4 R b + 5 a b ) x y = 0.

The center of the PRE is

X(39311) = a(16R^4(b+c-a)+16R^3(b+c)(-a+b+c)+4R^2bc(-4a+5b+5c)+8Rb^2c^2+ab^2c^2) : :

The PRE is a "Kimberling generalized Paasche conic" given by (p,q,r) = (2(R+a),2(R+b),2(R+c)) and (u,v,w) = (-a,-b,-c); see the preamble just before X(38310).

Define 54 points (18 triangles) by the following intersections:

A12= B''cA"c∩C''bA''b, B12= B''cA"c∩C''aB''a, C12= C''aB''a∩C''bA''b
A13=A''bB''a∩A"cC"a, B13=A"bB"a∩C"bB"c, C13=A"cC"a∩B"cC"b
A14 =A"bB"c∩C''bA''c, B14=B"cC"a∩A"cB"a, C14=C"aA"b∩B"aC"b
A15 =B"aA"c∩C"aA"b, B15=B"aC"b∩A"b∩B"c, C15 = B"cC"a∩A"cBa
A16 = BB"c∩CC"b, B16 = CC"a∩AA''c, C16 =AA"b∩BB"a
A17 = AbB"c∩AcC''b, B17= BcC"a∩BaA''c, C17=CaA''b∩CbB''a
A18 = A'bB''c∩A'cC''b B18 = B'aA''c∩B'cC''a C18=C'aA''b∩C'bB''a
A19 = BC13∩CB13, B19 = CA13∩AC13, C19= BA13∩AB13
A20 = AbC13∩AcB13, B20 = BcA13∩BaC13, C20 = CaB13∩CbA13
A21 = A'bC13∩A'cB13, B21 = B'cA13∩B'aC13, C21 = C'aB13∩C'bA13
A22 = A''bC13∩A''cB13, B22 = B''cA13∩B''aC13, C22 = C''aB13∩C''bA13
A23 =C"aB13∩B"aC13, B23 =C"bA13∩A"bC13, C23 =A"cB13∩B"cA13
A24 = A14A16A17A18A25∩A19A20A21A22∩BC
B24 = B14B16B17B18B25∩B19B20B21B22∩AC
C24 = C14C16C17C18C25∩C19C20C21C22∩AB
A25 =BC15∩CB15, B25 =CA15∩AC15, C25 =AB15∩BA15
A26 = AbC15∩AcB15, B26 =BcA15∩BaC15, C26 =CaB15∩CbA15
A27 = A'bC15∩A'cB15, B27=B'cA15∩B'aC15, C27 = C'aB15∩C'bA15
A28 = A''bC15∩A''cB15, B28=B''cA15∩B''aC15, C28 = C''aB15∩C''bA15
A29 = B''aC15∩C''aB15, B29=C''bA15∩A''bC15, C29 = A''cB15∩B''cA15
A30 = midpoint of C″bB″c, B30 = midpoint of A″cC″a, C30 = midpoint of A″bB″a
A31 = midpoint of B″aC″a, B31 = midpoint of A″bC″b, C31 = midpoint of A″cB″c

Barycentrics for A-vertices of the 18 triangles follow:

A12 = -a (4 R^2 + 4 R b + 4 R c + 3 b c) : 2 b (R + a) (2 R + c) : 2 c (R + a) (2 R + b)
A13 = -(4 R^2 + 4 R b + 4 R c + 3 b c) : b (2 R + c) : c (2 R + b)
A14 = -a (4 R^2 + 4 R b + 4 R c + 3 b c) : 4 (R + a) (2 R + b) (R + c) : 4 (R + a) (R + b) (2 R + c)
A15 = -2 (R + b) (R + c) (4 R^2 + 4 R b + 4 R c + 3 b c) : b c (2 R + b) (R + c) : b c (2R + c) (R + b)
A16 = -a : 2 (R + a) : 2 (R + a)
A17 = -a (4 R^2 - b c) : 4 R (R + a) (2 R + b) : 4 R (R + a) (2 R + c)
A18 = -a R (4 R + b + c) : (R + a) (2 R + b) (4 R + c) : (R + a) (2 R + c) (4 R + b)
A19 = -a (2 R + b) (2 R + c) : (2 R + b) (4 R^2 + 4 R a + 4 R c + 3 a c) : (2 R + c) (4 R^2 + 4 R a + 4 R b + 3 a b)
A20 = -a (2 R + b ) (2 R + c) (4 R^2 - b c) : (32 R^5 + 32 R^4 a + 32 R^4 b + 32 R^4 c + 32 R^3 a b + 32 R^3 b c + 24 R^3 a c + 16 R^3 b^2 + 16 R^2 b^2 c + 12 R^2 a b^2 + 24 R^2 a b c + 10 R a b^2 c + 2R b^2 c^2 + a b^2 c^2) : (32 R^5 + 32 R^4 a + 32 R^4 b + 32 R^4 c + 32 R^3 a c + 32 R^3 b c + 24 R^3 a b + 16 R^3 c^2 + 16 R^2 b c^2 + 12 R^2 a c^2 + 24 R^2 a b c + 10 R a b c^2 + 2 R b^2 c^2 + a b^2 c^2)
A21 = -R a (2 R + b) (2 R + c) (4 R + b + c) : (32 R^5 + 32 R^4 a + 32 R^4 b + 40 R^4 c + 32 R^3 a b + 32 R^3 a c + 40 R^3 b c + 12 R^3 b^2 + 8 R^3 c^2 + 10 R^2 a b^2 + 6R^2 a c^2 + 14 R^2 b^2 c + 8R^2 b c^2 + 32 R^2 a b c + 10 R a b^2 c + 6 R a b c^2 + 3 R b^2 c^2 + 2 a b^2 c^2) : (32 R^5 + 32 R^4 a + 32 R^4 c + 40 R^4 b + 32 R^3 a b + 32 R^3 a c + 40 R^3 b c + 12 R^3 c^2 + 8 R^3 b^2 + 10 R^2 a c^2 + 6 R^2 a b^2 + 14 R^2 b c^2 + 8 R^2 b^2 c + 32 R^2 a b c + 10 R a b c^2 + 6 R a b^2 c + 3 R b^2 c^2 + 2 a b^2 c^2)
A22= -a (2 R + c) (2 R + b) (4 R^2 + 4 R b + 4 R c + 3 b c) : (32 R^5 + 32 R^4 a + 32 R^4 b + 64 R^4 c + 32 R^3 a b + 56 R^3 a c + 64 R^3 b c + 32 R^3 c^2 + 4 R^2 a b^2 + 24 R^2 a c^2 + 8 R^2 b^2 c + 32 R^2 b c^2 + 56 R^2 a b c + 10 R a b^2 c + 24 R a b c^2 + 6 R b^2 c^2 + 5 a b^2 c^2) : (32 R^5 + 32 R^4 a + 32 R^4 c + 64 R^4 b + 32 R^3 a c + 56 R^3 a b + 64 R^3 b c + 32 R^3 b^2 + 4 R^2 a c^2 + 24 R^2 a b^2 + 8 R^2 b c^2 + 32 R^2 b^2 c + 56 R^2 a b c + 10 R a b c^2 + 24 R a b^2 c + 6R b^2 c^2 + 5 a b^2 c^2)
A23 = -(64 R^6 + 128 R^5 a + 128 R^5 b + 128 R^5 c + 64 R^4 a^2 + 64 R^4 b^2 + 64 R^4 c^2 + 224 R^4 a b + 224 R^4 a c + 240 R^4 b c + 96 R^3 a b^2 + 96 R^3 a c^2 + 96 R^3 a^2 c + 96 R^3 a^2 b + 112 R^3 b c^2 + 112 R^3 b^2 c + 384 R^3 a b c + 36 R^2 a^2 b^2 + 36 R^2 a^2 c^2 + 48 R^2 b^2 c^2 + 160 R^2 a b c^2 + 160 R^2 a b^2 c +148 R^2 a^2 b c + 56 R a^2 b^2 c + 56 R a^2 b c^2 + 64 R a b^2 c^2 + 21 a^2 b^2 c^2) : b c (2 R + a) (2 R + b) (4 R^2 + 4 R a + 4 R c + 3 a c) : b c (2 R + a) (2 R + c) (4 R^2 + 4 R a + 4 R b + 3 a b) A24 = 0 : b (2 R + c) : c (2 R + b)
A25 = -a b c (2 R + a) : 2 b (R + a) (4 R^2 + 4 R a + 4 R c + 3 a c ) : 2 c (R + a) (4 R^2 + 4 R a + 4 R b + 3 a b)
A26 = -a b c (2 R + a) (R + b) (R + c) (4 R^2 - b c) : b (R + a) (32 R^6 + 32 R^5 a + 32 R^5 b + 80 R^5 c + 32 R^4 a b + 72 R^4 a c + 96 R^4 b c + 48 R^4 c^2 + 88 R^3 a b c + 16 R^3 b^2 c + 64 R^3 b c^2 + 40 R^3 a c^2 + 16 R^2 b^2 c^2 + 16 R^2 a b^2 c + 56 R^2 a b c^2 + 15 R a b^2 c^2 + 2 R a b c^3 + a b^2 c^3) : c (R + a) (32 R^6 + 32 R^5 a + 32 R^5 c + 80 R^5 b + 32 R^4 a c + 72 R^4 a b + 96 R^4 b c + 48 R^4 b^2 + 88 R^3 a b c + 16 R^3 b c^2 + 64 R^3 b^2 c + 40 R^3 a b^2 + 16 R^2 b^2 c^2 + 16 R^2 a b c^2 + 56 R^2 a b^2 c + 15 R a b^2 c^2 + 2 R a b^3 c + a b^3 c^2)
A27 = -4 R a b c (2 R + a) (R + b) (R + c) (4 R + b + c) : b (R + a) (128 R^6 + 128 R^5 a + 160 R^5 b + 320 R^5 c + 160 R^4 a b + 288 R^4 a c + 424 R^4 b c + 32 R^4 b^2 + 232 R^4 c^2 + 384 R^3 a b c + 32 R^3 a b^2 + 192 R^3 a c^2 + 104 R^3 b^2 c + 320 R^3 b c^2 + 40 R^3 c^3 + 88 R^2 b^2 c^2 + 96 R^2 a b^2 c + 268 R^2 a b c^2 + 32 R^2 a c^3 + 56 R^2 b c^3 + 75 R a b^2 c^2 + 47 R a b c^3 + 16 R b^2 c^3 + 14 a b^2 c^3) : c (R + a) (128 R^6 + 128 R^5 a + 160 R^5 c + 320 R^5 b + 160 R^4 a c + 288 R^4 a b + 424 R^4 b c + 32 R^4 c^2 + 232 R^4 b^2 + 384 R^3 a b c + 32 R^3 a c^2 + 192 R^3 a b^2 + 104 R^3 b c^2 + 320 R^3 b^2 c + 40 R^3 b^3 + 88 R^2 b^2 c^2 + 96 R^2 a b c^2 + 268 R^2 a b^2 c + 32 R^2 a b^3 + 56 R^2 b^3 c + 75 R a b^2 c^2 + 47 R a b^3 c + 16 R b^3 c^2 + 14 a b^3 c^2)
A28 = -a b c (2 R + a) (R + b) (R + c) (4 R^2 + 4 R b + 4 R c + 3 b c) : b (R + a) (R + c) (32 R^5 + 32 R^4 a + 64 R^4 b + 48 R^4 c + 64 R^3 a b + 96 R^3 b c + 40 R^3 a c + 32 R^3 b^2 + 16 R^3 c^2 + 32 R^2 a b^2 + 8 R^2 a c^2 + 32 R^2 b c^2 + 48 R^2 b^2 c + 80 R^2 a b c + 16 R b^2 c^2 + 20 R a b c^2 + 40 R a b^2 c + 11 a b^2 c^2) : c (R + a) (R + b) (32 R^5 + 32 R^4 a + 64 R^4 c + 48 R^4 b + 64 R^3 a c + 96 R^3 b c + 40 R^3 a b + 32 R^3 c^2 + 16 R^3 b^2 + 32 R^2 a c^2 + 8 R^2 a b^2 + 32 R^2 b^2 c + 48 R^2 b c^2 + 80 R^2 a b c + 16 R b^2 c^2 + 20 R a b^2 c + 40 R a b c^2 + 11 a b^2 c^2)
A29 = -(R + b) (R + c) (16 R^4 + 32 R^3 a + 16 R^3 b + 16 R^3 c + 16 R^2 a^2 + 32 R^2 a b + 32 R^2 a c + 16 R^2 b c + 28 R a b c +16 R a^2 b + 16 R a^2 c + 13 a^2 b c) : a b c (R + a) (2 R + b) (R + c) : a b c (R + a) (R + b) (2 R + c)
A30 = -a : R + a : R + a
A31 = -2(4 R^2 + 3 R b + 3 R c + 2 b c) : b (2 R + c) : c (2 R + b)

Steps for obtaining B-vertices from the A-vertices in the preceding list:

As an example we start with A12 = -a (4 R^2 + 4 R b + 4 R c + 3 b c) : 2 b (R + a)( 2 R + c) : 2 c (R + a)(2 R + b). To get the coordinates of B12 we follow these steps:

Step1. Copy the x coordinate of A12 as y coordinate of B12 and apply a→b, b→a, c→c, so that the y coordinate of B12 is - b (4 R^2 + 4 R a + 4 R c + 3 a c).
Step 2. Copy the y coordinate of A12 as x coordinate of B12 and apply a→b, b→a, c→c, so that x coordinate of B12 is 2 a (R + b)(2 R + c).
Step 3. Copy the z coordinate of A12 as z coordinate of B12 and apply a→b, b→a, c→c, so that z coordinate of B12 is 2 c (R + b)(2 R + a).

We now have B12 = 2 a (R + b)(2 R + c) : -b (4 R^2 + 4 R a + 4 R c + 3 a c) : 2 c (R + b)(2 R + a)

Steps for obtaining C-vertices from the A-vertices:

Step 1. Copy the x coordinate of A12 as z coordinate of C12 and apply a→c, c→a, b→b, so that z coordinate of C12 is -c (4 R^2 + 4 R b + 4 R a + 3 a b).
Step 2. Copy the Z coordinate of A12 as x coordinate of C12 and apply a→c, c→a, b→b, so that x coordinate of C12 is 2 a (R + c)( 2R + b).
Step 3. Copy the y coordinate of A12 as y coordinate of C12 and apply a→c, c→a, b→b, so that y coordinate of C12 is 2 b (R + c)(2 R + a).

We now have C12 = 2 a (R + c)(2 R + b) : 2 b (R + c)(2 R + a) : -c (4 R^2 + 4 R a + 4 R b + 3 a b).

The 18 triangles are here given names as follows:

A12B12C12 = 12th Vijay-Paasche-Hutson triangle
A13B13C13 = 13th Vijay-Paasche-Hutson triangle
A14B14C14= 14th Vijay-Paasche-Hutson triangle
A15B15C15 = 15th Vijay-Paasche-Hutson triangle
A16B16C16= 16th Vijay-Paasche-Hutson triangle
A17B17C17 = 17th Vijay-Paasche-Hutson triangle
A18B18C18 = 18th Vijay-Paasche-Hutson triangle
A19B19C19 = 19th Vijay-Paasche-Hutson triangle
A20B20C20 = 20th Vijay-Paasche-Hutson triangle
A21B21C21 = 21th Vijay-Paasche-Hutson triangle
A22B22C22 = 22th Vijay-Paasche-Hutson triangle
A23B23C23 = 23th Vijay-Paasche-Hutson triangle
A24B24C24= 24th Vijay-Paasche-Hutson triangle
A25B25C25 = 25th Vijay-Paasche-Hutson triangle
A26B26C26 = 26th Vijay-Paasche-Hutson triangle
A27B27C27= 27th Vijay-Paasche-Hutson triangle
A28B28C28 = 28th Vijay-Paasche-Hutson triangle
A29B29C29 = 29th Vijay-Paasche-Hutson triangle
A30B30C30 = 30th Vijay-Paasche-Hutson triangle
A31B31C31 = 31th Vijay-Paasche-Hutson triangle

Collinearities:

A, A12, A13, A24, A31 are collinear.
A, A14, A15, A29 are collinear.
A, A16, A30 are collinear.
A12, A14, A30, X(39311) are collinear.
A14, A16, A17, A18, A24, A25 are collinear.
A19, A20, A21, A22, A24 are collinear.
A, A19, A23 are collinear.
A12, A15, A28 are collinear.
A13, A28, A29 are collinear.
A15, A22, A23 are collinear.

Perspectors:

X(1123) = ANaVaL'aLaHaA12A13A24∩BNbVbL'bLbHbB12B13B24∩CNcVcL'cLcHcC12C13C24 = a/2R+a : b/2R+b : c/2R+c = Paasche point
X(2) = AA16A30∩BB16B30∩CC16C30 = 1 : 1 : 1 = centroid
X(3083) = AA17∩BB17∩CC17 = AMaKa ∩ BMbKb ∩ CMcKc = 2R+a : 2R+b : 2R+c = X(1)X(2)∩X(37)X(494)
X(38488 )= AA18∩ BB18∩CC18 = AK'a ∩ BK'b ∩ CK'c = (2R+a) (4R+b)(4R+c) : (2R+b)(4R+c)(4R+a) : (2R+c)(4R+b)(4R+a) = perspector of these triangles: ABC and 6th Vijay-Paasche-Hudson
X(1123) = ANaVaL'aLaHaA12A13A24A31∩BNbVbL'bLbHbB12B13B24B31∩CNcVcL'cLcHcC12C13C24C31 = a/2R+a : b/2R+b : c/2R+c = Paasche point
X(39311) = A12A14∩ B12B14∩C12C14 = center of reflected Paasche ellipse
X(39312) = AA14A15A29∩BB14B15B29∩CC14C15C29 = A24A28∩B24B28∩C24C28
X(39313) = AA19A23∩BB19B23∩CC19C23
X(39314) = AA25∩BB25∩CC25
X(39315) =A12A15A28∩B12B15B28∩C12C15C28 = A29A31∩B29B31∩C29C31
X(39316) =A12A16∩B12B16∩C12C16
X(39317) =A12A19∩B12B19∩C12C19
X(39318) =A12A22∩B12B22∩C12C22
X(39319) = A12A25∩B12B25∩C12C25
X(39320) = A13A14∩ B13B14∩C13C14 = A15A22A23∩B15B22B23∩C15C22C23
X(39321) = A13A15∩B13B15∩C13C15
X(39322) =A13A19∩B13B19∩C13C19
X(39323) =A13A22∩B13B22∩C13C22
X(39324) = A13A23∩B13B23∩C13C23
X(39325) =A13A28A29∩B13B28B29∩C13C28C29
X(39326) = A14A19∩B14B19∩C14C19
X(39327) = A15A19∩B15B19∩C15C19
X(39328) = A15A25∩B15B25∩C15C25
X(39329) = A19A29∩B19B29∩C19C29
X(39330) = A23A24∩B23B24∩C23C24
X(39331) = A29A30∩B29B30∩C29C30
X(39332) = A30A31∩B30B31∩C30C31
X(39333)= A14A31∩B14B31∩C14C31
X(39334)= A16A31∩B16B31∩C16C31
X(39369)= A19A31∩B19B31∩C19C31
X(39370)= A22A31∩B22B31∩C22C31

X(39311) = CENTER OF PAASCHE REFLECTION ELLIPSE

Barycentrics    a (16 R^3 (b + c - a) (b + c + R) + 4 R^2 b c (5 b + 5 c - 4 a) + 8 R b^2 c^2 + a b^2 c^2) : :

X(39311) is the perspector of 12th and 14th Vijay-Paasche-Hutson triangles.

See X(39311). (Dasari Naga Vijay Krishna)

X(39311) lies on these lines: {1123,3086}, {39312,39316}, {39318,39320}

X(39311) = {X(1123), X(39319)}-harmonic conjugate of X(39314)

X(39312) = PERSPECTOR OF ABC AND 14TH VIJAY-PAAASCHE-HUTSON TRIANGLE

Barycentrics    (R + b) (2 R + a) (R + c) : :

X(39312) is the perspector of each pair of the following triangles: ABC, 14th, 15th, and 29th Vijay-Paasche-Hutson. X(39312) is also the perspector of the 24th and 28th Vijay-Paasche-Hutson triangles.

See X(39312). (Dasari Naga Vijay Krishna)

X(39312) lies on these lines: {2,3300}, {1123,39315}, {1267,39313}, {3302,11140}, {6347,34216}, {39311,39316}, {39321,39322}

X(39312) = isotomic conjugate of the complement of X(39322)
X(39312) = barycentric product X(1267)*X(3300)
X(39312) = barycentric quotient X(i)/X(j) for these (i, j): (1124, 3299), (1267, 32791)
X(39312) = Cevapoint of X(2) and X(39322)
X(39312) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1124, 3299), (1267, 32791)
X(39312) = {X(3300), X(39328)}-harmonic conjugate of X(39314)

X(39313) = PERSPECTOR OF ABC AND 19TH VIJAY-PAASCHE-HUTSON TRIANGLE

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

X(39313) is the perspector of each pair of these Vijay-Paasche-Hutson triangles: ABC, 19th, and 23rd .

See X(39313). (Dasari Naga Vijay Krishna)

X(39313) lies on these lines: {2,586}, {1267,39312}, {39314,39320}, {39316,39317}, {39319,39327}

X(39313) = barycentric product X(1267)*X(39314)
X(39313) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 39322, 1123), (1123, 39322, 39323)

X(39314) = PERSPECTOR OF ABC AND 25th- VIJAY-PAASCHE-HUTSON TRIANGLE

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

See X(39314). (Dasari Naga Vijay Krishna)

X(39314) lies on these lines: {1,13941}, {2,3300}, {1123,3086}, {3085,6351}, {7133,10591}, {39313,39320}

X(39314) = barycentric product X(1123)*X(39313)
X(39314) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1123, 39319, 39311), (3300, 39328, 39312)

X(39315) = PERSPECTOR OF THE 12TH AND 15TH VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    a (R + b) (R + c) (4 R^2 + 4 R b + 4 R c + 3 b c) (16 R^4 + 32 R^3 a + 8 R^3 b + 8 R^3 c + 16 R^2 a b + 16 R^2 a^2 + 16 R^2 a c + 8 R a^2 b + 8 R a^2 c + 4 R a b c + 3 a^2 b c) : :

X(39315) is the perspector of each pair of the following triangles: 12th, 15th, and 28th Vijay-Paasche-Hutson. X(39315) is also the perspector of the 29th and 31st Vijay- Paasche-Hutson triangles.

See X(39315). (Dasari Naga Vijay Krishna)

X(39315) lies on these lines: {1123,39312}, {39316,39319}, {39317,39326}, {39322,39327}

X(39315) = {X(39316), X(39328)}-harmonic conjugate of X(39319)

X(39316) = PERSPECTOR OF THE: 12TH AND 16TH VIJAY-PAASCHE-HUTSON TRIANGLES

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

See X(39316). (Dasari Naga Vijay Krishna)

X(39316) lies on these lines: {499,1123}, {39311,39312}, {39313,39317}, {39315,39319}

X(39316) = {X(39315), X(39319)}-harmonic conjugate of X(39328)

X(39317) = PERSPECTOR OF THE 12TH AND 19TH VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    a (49 a^3 b^4 c^4 + 364 R a^2 b^4 c^4 + 756 R^2 a b^4 c^4 + 480 R^3 b^4 c^4 + 182 R a^3 b^3 c^4 + 1604 R^2 a^2 b^3 c^4 + 3480 R^3 a b^3 c^4 + 2224 R^4 b^3 c^4 + 216 R^2 a^3 b^2 c^4 + 2632 R^3 a^2 b^2 c^4 + 6080 R^4 a b^2 c^4 + 3936 R^5 b^2 c^4 + 80 R^3 a^3 b c^4 + 1952 R^4 a^2 b c^4 + 4864 R^5 a b c^4 + 3200 R^6 b c^4 + 576 R^5 a^2 c^4 + 1536 R^6 a c^4 + 1024 R^7 c^4 + 182 R a^3 b^4 c^3 + 1604 R^2 a^2 b^4 c^3 + 3480 R^3 a b^4 c^3 + 2224 R^4 b^4 c^3 + 380 R^2 a^3 b^3 c^3 + 5968 R^3 a^2 b^3 c^3 + 14272 R^4 a b^3 c^3 + 9280 R^5 b^3 c^3 - 320 R^3 a^3 b^2 c^3 + 7584 R^4 a^2 b^2 c^3 + 21760 R^5 a b^2 c^3 + 14656 R^6 b^2 c^3 - 1104 R^4 a^3 b c^3 + 3744 R^5 a^2 b c^3 + 14976 R^6 a b c^3 + 10624 R^7 b c^3 - 576 R^5 a^3 c^3 + 576 R^6 a^2 c^3 + 4096 R^7 a c^3 + 3072 R^8 c^3 + 216 R^2 a^3 b^4 c^2 + 2632 R^3 a^2 b^4 c^2 + 6080 R^4 a b^4 c^2 + 3936 R^5 b^4 c^2 - 320 R^3 a^3 b^3 c^2 + 7584 R^4 a^2 b^3 c^2 + 21760 R^5 a b^3 c^2 + 14656 R^6 b^3 c^2 - 3248 R^4 a^3 b^2 c^2 + 4736 R^5 a^2 b^2 c^2 + 27392 R^6 a b^2 c^2 + 20224 R^7 b^2 c^2 - 4832 R^5 a^3 b c^2 - 2816 R^6 a^2 b c^2 + 14208 R^7 a b c^2 + 12544 R^8 b c^2 - 2112 R^6 a^3 c^2 - 2560 R^7 a^2 c^2 + 2560 R^8 a c^2 + 3072 R^9 c^2 + 80 R^3 a^3 b^4 c + 1952 R^4 a^2 b^4 c + 4864 R^5 a b^4 c + 3200 R^6 b^4 c - 1104 R^4 a^3 b^3 c + 3744 R^5 a^2 b^3 c + 14976 R^6 a b^3 c + 10624 R^7 b^3 c - 4832 R^5 a^3 b^2 c - 2816 R^6 a^2 b^2 c + 14208 R^7 a b^2 c + 12544 R^8 b^2 c -6208 R^6 a^3 b c - 9216 R^7 a^2 b c + 3072 R^8 a b c + 6144 R^9 b c - 2560 R^7 a^3 c -4608 R^8 a^2 c - 1024 R^9 a c + 1024 R^10 c + 576 R^5 a^2 b^4 + 1536 R^6 a b^4 + 1024 R^7 b^4 - 576 R^5 a^3 b^3 + 576 R^6 a^2 b^3 + 4096 R^7 a b^3 + 3072 R^8 b^3 - 2112 R^6 a^3 b^2 - 2560 R^7 a^2 b^2 + 2560 R^8 a b^2 + 3072 R^9 b^2 - 2560 R^7 a^3 b - 4608 R^8 a^2 b - 1024 R^9 a b +1024 R^10 b - 1024 R^8 a^3 - 2048 R^9 a^2 - 1024 R^10 a) : :

See X(39317). (Dasari Naga Vijay Krishna)

X(39317) lies on these lines: {39313,39316}, {39315,39326}, {39318,39322}

X(39318) = PERSPECTOR OF THE 12TH AND 22ND VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    a (4 R^2 + 4 R b + 4 R c + 3 b c) (3*a^3*b^3*c^3 + 40*R*a^2*b^3*c^3 + 96*R^2*a*b^3*c^3 + 64*R^3*b^3*c^3 + 2*R*a^3*b^2*c^3 + 140*R^2*a^2*b^2*c^3 + 376*R^3*a*b^2*c^3 + 256*R^4*b^2*c^3 - 4*R^2*a^3*b*c^3 + 216*R^3*a^2*b*c^3 + 608*R^4*a*b*c^3 + 416*R^5*b*c^3 + 144*R^4*a^2*c^3 + 384*R^5*a*c^3 + 256*R^6*c^3 + 2*R*a^3*b^3*c^2 + 140*R^2*a^2*b^3*c^2 + 376*R^3*a*b^3*c^2 + 256*R^4*b^3*c^2 - 88*R^2*a^3*b^2*c^2 + 224*R^3*a^2*b^2*c^2 + 1040*R^4*a*b^2*c^2 + 768*R^5*b^2*c^2 - 232*R^3*a^3*b*c^2 + 32*R^4*a^2*b*c^2 + 1184*R^5*a*b*c^2 + 960*R^6*b*c^2 - 144*R^4*a^3*c^2 + 640*R^6*a*c^2 + 512*R^7*c^2 - 4*R^2*a^3*b^3*c + 216*R^3*a^2*b^3*c + 608*R^4*a*b^3*c + 416*R^5*b^3*c - 232*R^3*a^3*b^2*c + 32*R^4*a^2*b^2*c + 1184*R^5*a*b^2*c + 960*R^6*b^2*c - 592*R^4*a^3*b*c - 832*R^5*a^2*b*c + 512*R^6*a*b*c + 768*R^7*b*c - 384*R^5*a^3*c - 640*R^6*a^2*c + 256*R^8*c + 144*R^4*a^2*b^3 + 384*R^5*a*b^3 + 256*R^6*b^3 - 144*R^4*a^3*b^2 + 640*R^6*a*b^2 + 512*R^7*b^2 - 384*R^5*a^3*b - 640*R^6*a^2*b + 256*R^8*b - 256*R^6*a^3 - 512*R^7*a^2 - 256*R^8*a ) : :

See X(39318). (Dasari Naga Vijay Krishna)

X(39318) lies on these lines: {1123,39324}, {39311,39320}, {39317,39322}

X(39319) = PERSPECTOR OF THE 12TH AND 25TH VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    a (2 R + a) (8 R^4 + 16 R^3 b + 16 R^3 c + 8 R^2 b^2 + 8 R^2 c^2 + 26 R^2 b c + 11 R b^2 c + 11 R b c^2 + 4 b^2 c^2) : :

See X(39319). (Dasari Naga Vijay Krishna)

X(39319) lies on these lines: {1123,3086}, {39313,39327}, {39315,39316}

X(39319) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39311, 39314, 1123), (39316, 39328, 39315)

X(39320) = PERSPECTOR OF THE 13TH AND 14TH VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    (4 R^2 + 4 R b + 4 R c + 3 b c) (16 R^4 + 16 R^3 a + 16 R^3 b + 16 R^3 c + 16 R^2 a b + 16 R^2 b c + 16 R^2 c a + 2 R a^2 b + 2 R a^2 c + 16 R a b c + 3 a^2 b c) : :

X(39320) is also the perspector of each pair of the following triangles: 15th, 22th, and 23th Vijay-Paasche-Hutson.

See X(39320). (Dasari Naga Vijay Krishna)

X(39320) lies on these lines: {1123,39312}, {39311,39318}, {39313,39314}, {39321,39323}, {39322,39326}, {39328,39329}

X(39321) = PERSPECTOR OF THE 13TH AND 15TH VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    a (16 R^5 + 16 R^4 (a + b + c) + 4 R^3 (5 a b + 5 a c + 4 b c) + 4 R^2 (a b^2 + a c^2 + 6 a b c) + R b c (5 a b + 5 a c - b c) + a b^2 c^2) : :

See X(39321). (Dasari Naga Vijay Krishna)

X(39321) lies on these lines: {1123,3086}, {39312,39322}, {39320,39323}

X(39321) = {X(1123), X(39326)}-harmonic conjugate of X(39314)

X(39322) = PERSPECTOR OF THE: 13TH AND 19TH VIJAY-PAASCHE-HUTSON TRIANGLES

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

See X(39322). (Dasari Naga Vijay Krishna)

X(39322) lies on these lines: {2,586}, {1267,37883}, {39312,39321}, {39315,39327}, {39317,39318}, {39320,39326}, {39325,39329}

X(39322) = anticomplement of the isotomic conjugate of X(39312)
X(39322) = pole of the trilinear polar of X(39312) wrt Steiner circumellipse
X(39322) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1123, 39313, 2), (39313, 39323, 1123)

X(39323) = PERSPECTOR OF THE 13TH AND 22ND VIJAY-PAASCHE-HUTSON TRTIANGLES

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

See X(39323). (Dasari Naga Vijay Krishna)

X(39323) lies on these lines: X(39323) lies on these lines: {2,586}, {39320,39321}

X(39323) = {X(1123), X(39322)}-harmonic conjugate of X(39313)

X(39324) = PERSPECTOR OF THE 13TH AND 23RD VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    a (4 R^2 + 4 R b + 4 R c + 3 b c) (64 R^6 + 64 R^5 (b + c + 2a) + 16 R^4 (4 a^2 + 5 b c + 8 a b + 8 a c) + 8 R^3 (8 a^2 (b + c) + 2 a (b^2 + c^2) + b c (b + c) + 18 a b c) + 4 R^2 (3 a^2 (b^2 + c^2) + a b c (6 b + 6 c + 17 a) ) + 2 R a b c (7 a b + 7 a c + 2 b c) + 3 a^2 b^2 c^2) : :

See X(39324). (Dasari Naga Vijay Krishna)

X(39324) lies on these lines: X(39324) lies on these lines: {1123,39318}, {39320,39321}

X(39325) = PERSPECTOR OF THE 13TH AND 28TH VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    a (R + b) (R + c) (32 R^6 + 16 R^5 (4 a + 3 b + 3 c) + 16 R^4 (2 a^2 + b^2 + c^2 + 6 a b + 6 a c + 4 b c) + 16 R^3 (3 a^2 (b + c) + 2 a (b^2 + c^2) + b c (b + c) + 8 a b c) + 2 R^2 (8 a^2 b^2 + 8 a^2 c^2 + b^2 c^2 + 2 a b c (16 a + 9 b + 9 c) ) + R a b c (19 a b + 19 a c + 8 b c) + 5 a^2 b^2 c^2) : :

X(39325) is the perspector of each pair of these Vijay-Paasche-Hutson triangles: 13th, 28th, and 29th.

See X(39325). (Dasari Naga Vijay Krishna)

X(39325) lies on these lines: {1123,39312}, {39322,39329}, {39326,39328

X(39326) = PERSPECTOR OF THE: 14TH AND 19TH VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    a (2 R + a) (8 R^3 (b + c) + 4 R^2 (2 b^2 + 2 c^2 + 5 b c) + 12 R b c (b + c) + 5 b^2 c^2) : :

See X(39326). (Dasari Naga Vijay Krishna)

X(39326) lies on these lines: {1123,3086}, {1267,39312}, {39315,39317}, {39320,39322}, {39325,39328}

X(39326) = {X(39314), X(39321)}-harmonic conjugate of X(1123)

X(39327) = PERSPECTOR OF THE 15TH AND 19TH VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    (2 R + a) (289*a^3*b^4*c^4 + 1071*R*a^2*b^4*c^4 + 1344*R^2*a*b^4*c^4 + 576*R^3*b^4*c^4 + 1479*R*a^3*b^3*c^4 + 5334*R^2*a^2*b^3*c^4 + 6496*R^3*a*b^3*c^4 +2688*R^4*b^3*c^4+ 2838*R^2*a^3*b^2*c^4 + 9960*R^3*a^2*b^2*c^4 + 11744*R^4*a*b^2*c^4 + 4672*R^5*b^2*c^4 + 2416*R^3*a^3*b*c^4 + 8256*R^4*a^2*b*c^4 + 9408*R^5*a*b*c^4 + 3584*R^6*b*c^4 + 768*R^4*a^3*c^4 + 2560*R^5*a^2*c^4 + 2816*R^6*a*c^4 + 1024*R^7*c^4 + 1479*R*a^3*b^4*c^3 + 5334*R^2*a^2*b^4*c^3 + 6496*R^3*a*b^4*c^3 + 2688*R^4*b^4*c^3 + 7264*R^2*a^3*b^3*c^3 + 25452*R^3*a^2*b^3*c^3 + 30080*R^4*a*b^3*c^3 + 12032*R^5*b^3*c^3 + 13368*R^3*a^3*b^2*c^3 + 45528*R^4*a^2*b^2*c^3 + 52128*R^5*a*b^2*c^3 + 20096*R^6*b^2*c^3 + 10912*R^4*a^3*b*c^3 + 36160*R^5*a^2*b*c^3 + 40064*R^6*a*b*c^3 + 14848*R^7*b*c^3 + 3328*R^5*a^3*c^3 + 10752*R^6*a^2*c^3 + 11520*R^7*a*c^3 + 4096*R^8*c^3 + 2838*R^2*a^3*b^4*c^2 + 9960*R^3*a^2*b^4*c^2 + 11744*R^4*a*b^4*c^2 + 4672*R^5*b^4*c^2 + 13368*R^3*a^3*b^3*c^2 + 45528*R^4*a^2*b^3*c^2 + 52128*R^5*a*b^3*c^2 + 20096*R^6*b^3*c^2 + 23568*R^4*a^3*b^2*c^2 + 78000*R^5*a^2*b^2*c^2 + 86656*R^6*a*b^2*c^2 + 32320*R^7*b^2*c^2 + 18416*R^5*a^3*b*c^2 + 59328*R^6*a^2*b*c^2 + 63936*R^7*a*b*c^2 + 23040*R^8*b*c^2 + 5376*R^6*a^3*c^2 + 16896*R^7*a^2*c^2 + 17664*R^8*a*c^2 + 6144*R^9*c^2 + 2416*R^3*a^3*b^4*c + 8256*R^4*a^2*b^4*c + 9408*R^5*a*b^4*c + 3584*R^6*b^4*c + 10912*R^4*a^3*b^3*c + 36160*R^5*a^2*b^3*c + 40064*R^6*a*b^3*c + 14848*R^7*b^3*c + 18416*R^5*a^3*b^2*c + 59328*R^6*a^2*b^2*c + 63936*R^7*a*b^2*c + 23040*R^8*b^2*c + 13760*R^6*a^3*b*c + 43200*R^7*a^2*b*c + 45312*R^8*a*b*c + 15872*R^9*b*c + 3840*R^7*a^3*c + 11776*R^8*a^2*c + 12032*R^9*a*c + 4096*R^10*c + 768*R^4*a^3*b^4 + 2560*R^5*a^2*b^4 + 2816*R^6*a*b^4 + 1024*R^7*b^4 + 3328*R^5*a^3*b^3 + 10752*R^6*a^2*b^3 + 11520*R^7*a*b^3 + 4096*R^8*b^3 + 5376*R^6*a^3*b^2 + 16896*R^7*a^2*b^2 + 17664*R^8*a*b^2 + 6144*R^9*b^2 + 3840*R^7*a^3*b + 11776*R^8*a^2*b + 12032*R^9*a*b + 4096*R^10*b + 1024*R^8*a^3 + 3072*R^9*a^2 + 3072*R^10*a + 1024*R^11 ) : :

See X(39327). (Dasari Naga Vijay Krishna)

X(39327) lies on these lines: {39313,39319}, {39315,39322}

X(39328) = PERSPECTOR OF THE 15TH AND 25ND VIJAY-PAASCHE-HUTSON TRIANGLES

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

X(39328) = perspector of each pair of these Vijay-Paasche-Hutson triangles: 15th, 22nd, 23rd.

See X(39328). (Dasari Naga Vijay Krishna)

X(39328) lies on these lines: {2,3300}, {39315,39316}, {39320,39329}, {39325,39326}

X(39328) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39312, 39314, 3300), (39315, 39319, 39316)

X(39329) = PERSPECTOR OF THE 19TH AND 29TH VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    (2 R + a) (R + b) (R + c) (32 R^6 + 64 R^5 (a + b + c) + 8 R^4 (4 (a^2 + b^2 + c^2) + 16 a (b + c) + 15 b c) + 8 R^3 (8 a^2 (b + c) + 8 a (b^2 + c^2) + 7 b c (b + c) + 29 a b c) + 8 R^2 (4 a^2 (b^2 + c^2) + 3 b^2 c^2 + a b c (14 a + 13 b + 13 c) ) + 7 R a b c (7 a b + 7 a c + 6 b c) + 19 a^2 b^2 c^2) : :

See X(39329). (Dasari Naga Vijay Krishna)

X(39329) lies on these lines: {1267,39312}, {39320,39328}, {39322,39325}

X(39330) = PERSPECTOR OF THE 23RD AND 24TH VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    a (2 R + b) (2 R + c) (4 R^2 + 4 R b + 4 R c + 3 b c) (16 R^4 + 16 R^3 (2 a + b + c) + 4 R^2 (4 a^2 + 7 a (b + c) + 2 b c) + 4 R a (3 a b + 3 a c + 4 b c) + 7 a^2 b c) (64 R^6 + 128 R^5 (a + b + c) + 16 R^4 (14 a (b + c) + 4 (a^2 + b^2 + c^2) + 15 b c) + 16 R^3 (6 a^2 (b + c) + 6 a (b^2 + c^2) + b c (24 a + 7 b + 7 c) ) + 4 R^2 (9 a^2 b^2 + 9 a^2 c^2 + 12 b^2 c^2 + a b c (37 a + 40 b + 40 c) ) + 8 R a b c (7 a b + 7 a c + 8 b c) + 21 a^2 b^2 c^2) : :

See X(39330). (Dasari Naga Vijay Krishna)

X(39330) lies on these lines: {}

X(39331) = PERSPECTOR OF THE 29TH AND 30TH VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    (R + b) (R + c) (144*a^4*b^3*c^3 + 648*R*a^3*b^3*c^3 + 1112*R^2*a^2*b^3*c^3 + 864*R^3*a*b^3*c^3 + 256*R^4*b^3*c^3 + 528*R*a^4*b^2*c^3 + 2293*R^2*a^3*b^2*c^3 + 3764*R^3*a^2*b^2*c^3 + 2768*R^4*a*b^2*c^3 + 768*R^5*b^2*c^3 + 640*R^2*a^4*b*c^3 + 2672*R^3*a^3*b*c^3 + 4192*R^4*a^2*b*c^3 + 2928*R^5*a*b*c^3 + 768*R^6*b*c^3 + 256*R^3*a^4*c^3 + 1024*R^4*a^3*c^3 + 1536*R^5*a^2*c^3 + 1024*R^6*a*c^3 + 256*R^7*c^3 + 528*R*a^4*b^3*c^2 + 2293*R^2*a^3*b^3*c^2 + 3764*R^3*a^2*b^3*c^2 + 2768*R^4*a*b^3*c^2 + 768*R^5*b^3*c^2 + 1807*R^2*a^4*b^2*c^2 + 7648*R^3*a^3*b^2*c^2 + 12176*R^4*a^2*b^2*c^2 + 8640*R^5*a*b^2*c^2 + 2304*R^6*b^2*c^2 + 2048*R^3*a^4*b*c^2 + 8432*R^4*a^3*b*c^2 + 13024*R^5*a^2*b*c^2 + 8944*R^6*a*b*c^2 + 2304*R^7*b*c^2 + 768*R^4*a^4*c^2 + 3072*R^5*a^3*c^2 + 4608*R^6*a^2*c^2 + 3072*R^7*a*c^2 + 768*R^8*c^2 + 640*R^2*a^4*b^3*c + 2672*R^3*a^3*b^3*c + 4192*R^4*a^2*b^3*c + 2928*R^5*a*b^3*c + 768*R^6*b^3*c + 2048*R^3*a^4*b^2*c + 8432*R^4*a^3*b^2*c + 13024*R^5*a^2*b^2*c + 8944*R^6*a*b^2*c + 2304*R^7*b^2*c + 2176*R^4*a^4*b*c + 8832*R^5*a^3*b*c + 13440*R^6*a^2*b*c + 9088*R^7*a*b*c + 2304*R^8*b*c + 768*R^5*a^4*c + 3072*R^6*a^3*c + 4608*R^7*a^2*c + 3072*R^8*a*c + 768*R^9*c + 256*R^3*a^4*b^3 + 1024*R^4*a^3*b^3 + 1536*R^5*a^2*b^3 + 1024*R^6*a*b^3 + 256*R^7*b^3 + 768*R^4*a^4*b^2 + 3072*R^5*a^3*b^2 + 4608*R^6*a^2*b^2 + 3072*R^7*a*b^2 + 768*R^8*b^2 + 768*R^5*a^4*b + 3072*R^6*a^3*b + 4608*R^7*a^2*b + 3072*R^8*a*b + 768*R^9*b + 256*R^6*a^4 + 1024*R^7*a^3 + 1536*R^8*a^2 + 1024*R^9*a + 256*R^10) : :

See X(39331). (Dasari Naga Vijay Krishna)

X(39331) lies on these lines: {}

X(39332) = PERSPECTOR OF THE 30TH AND 31ST VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    (9*a^2*b^2*c^2 + 36*R*a*b^2*c^2 + 32*R^2*b^2*c^2 + 24*R*a^2*b*c^2 + 94*R^2*a*b*c^2 + 80*R^3*b*c^2 + 16*R^2*a^2*c^2 + 60*R^3*a*c^2 + 48*R^4*c^2 + 24*R*a^2*b^2*c + 94*R^2*a*b^2*c + 80*R^3*b^2*c + 64*R^2*a^2*b*c + 240*R^3*a*b*c + 192*R^4*b*c + 44*R^3*a^2*c + 152*R^4*a*c + 112*R^5*c + 16*R^2*a^2*b^2 + 60*R^3*a*b^2 + 48*R^4*b^2 + 44*R^3*a^2*b + 152*R^4*a*b + 112*R^5*b + 32*R^4*a^2 + 96*R^5*a + 64*R^6) : :

See X(39332). (Dasari Naga Vijay Krishna)

X(39332) lies on these lines: {}

X(39333) = PERSPECTOR OF THE 14TH AND 31ST VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    (13*a^2*b^2*c^2 + 64*R*a*b^2*c^2 + 64*R^2*b^2*c^2 + 30*R*a^2*b*c^2 + 160*R^2*a*b*c^2 + 160*R^3*b*c^2 + 16*R^2*a^2*c^2 + 96*R^3*a*c^2 + 96*R^4*c^2 + 30*R*a^2*b^2*c + 160*R^2*a*b^2*c + 160*R^3*b^2*c + 64*R^2*a^2*b*c + 384*R^3*a*b*c + 384*R^4*b*c + 32*R^3*a^2*c + 224*R^4*a*c + 224*R^5*c + 16*R^2*a^2*b^2 + 96*R^3*a*b^2 + 96*R^4*b^2 + 32*R^3*a^2*b + 224*R^4*a*b + 224*R^5*b + 16*R^4*a^2 + 128*R^5*a + 128*R^6 ) : :

See X(39333). (Dasari Naga Vijay Krishna)

X(39333) lies on these lines: {}

X(39334) = PERSPECTOR OF THE 16TH AND 31ST VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    (49*a^2*b^2*c^2 + 168*R*a*b^2*c^2 + 128*R^2*b^2*c^2 + 126*R*a^2*b*c^2 + 428*R^2*a*b*c^2 + 320*R^3*b*c^2 + 80*R^2*a^2*c^2 + 264*R^3*a*c^2 + 192*R^4*c^2 + 126*R*a^2*b^2*c + 428*R^2*a*b^2*c + 320*R^3*b^2*c + 316*R^2*a^2*b*c + 1056*R^3*a*b*c + 768*R^4*b*c + 200*R^3*a^2*c + 640*R^4*a*c + 448*R^5*c + 80*R^2*a^2*b^2 + 264*R^3*a*b^2 + 192*R^4*b^2 + 200*R^3*a^2*b + 640*R^4*a*b + 448*R^5*b + 128*R^4*a^2 + 384*R^5*a + 256*R^6 ) : :

See X(39334). (Dasari Naga Vijay Krishna)

X(39334) lies on these lines: {}

leftri

Points on the conic TC(X(1054)): X(39335)-X(39344)

rightri

This preamble and centers X(36335)-X(39344) were contributed by Clark Kimberling and Peter Moses, July 24-25, 2020.

This preamble extends the preamble just before X(36256), in which, for a point P = p : q : r (trilinears), the trilinear permutation conic denoted by TC(P), is defined as the conic that passes through the six points

p : q : r,    q : r : p,    r : p : q,    p : r : q,    q : p : r,   r : q : p.

An equation for TC(P) is (q r + r p + p q)(x^2 + y^2 + z^2) - (p^2 + q^2 + r^2)(y z + z x + x y) = 0.

Thus, TC(P) is analogous, and symbolically identical, to the permutation ellipse E(P), defined in the preamble just before X(34341).

***** Suppose that P = p : q : r (trilinears). For the rest of this paragraph, p:q:r are trilinears, but all other coordinates, and the equation for TC(P), are in barycentric coordinates. The conic TC(P) passes through the six points

ap : bq : cr,    aq : br : cp,    ar : bp : cq,    ap : br : cq,    aq : bp : cr,   ar : bq : cp.

(Note that these six points are not six permutations of ap : bq : cr.)

An equation for TC(P) is (q r + r p + p q)(b^2 c^2 x^2 + c^2 a^2 y^2 + a^2 b^2 z^2) - abc(p^2 + q^2 + r^2)(ayz + bzx + cxy) = 0. ****

The conic TC(X(1054)), given by the trilinear equation x^2 + 3 y z + (cyclic) = 0. Define f(x,y,z) = (x-y)(x-z) - (y-z)^2. If X = x : y : z (trilinears) is a point other than X(1) = 1:1:1 then the point F(X) = f(x,y,z) : f(y,z,x) : f(z,x,y) is on TC(X(1054)).

The appearance of (i,j) in the following list means that the point F(X(i)) = X(j) is on TC(X(1054)):

(2,9359), (4,2636), (6,1054), (19,2639), (31,2640), (100,9324), (171,9355), .

Let f(x : y : z) = (x - y)(x - z) - (y - z)^2. If X = x : y : z and X' = x' : y ' : z' are collinear with X(1), then f(X') = f(X), on TC(X(1054)).

If P is on the anti-orthic axis, (the line X(44) X(513)), then the P-Ceva conjugate of X(1) is on TC(X(1054).

The excenters, with trilinears -1:1:1, 1:-1:1, 1:1:-1, also lie on TC(X(1054)). Let EC denote the circumconic of ABC that passes through the excenters, so that EC is given by the barycentric equation

g(a,b,c) y z + g(b,c,a) z x + g(c,a,b) yz = 0, where g(a,b,c) = (a^2 + b^2 - c^2)(a^2 - b^2 + c^2).

EC passes through X(i) for i = 107, 648, 653, 685, 687, 1897, 6330, 6331, 6335, 6336, 8764, 13149, 15352, 15459, 16080, 16081, 16082, 16813, 17983, 30450, 36306, 36309, 38342. Six examples follow:

X(1054) = X(107)-of-excentral-triangle
X(2636) = X(30450)-of-excentral-triangle
X(2640) = X(16813)-of-excentral-triangle
X(9324) = X(16080)-of-excentral-triangle
X(9355) = X(648)-of-excentral-triangle
X(9359) = X(15352)-of-excentral-triangle

X(39335) = X(1)X(1492)∩X(1054)X(21382)

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

X(39335) lies on the conic TC(X(1054)) and these lines: {1, 1492}, {1054, 21382}, {2640, 16565}, {3497, 18754}, {3512, 18794}, {9359, 16560}, {18787, 20373}

X(39336) = X(1)X(4599)∩X(31)X(17957)

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

X(39336) lies on the conic TC(X(1054)) and these lines: {1, 4599}, {31, 17957}, {1958, 16556}, {2156, 33782}, {9359, 21382}

X(39337) = X(1)X(799)∩X(2)X(2107)

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

X(39337) lies on the conic TC(X(1054)) and these lines: {1, 799}, {2, 2107}, {63, 1967}, {1045, 3240}, {1054, 5539}, {1707, 33782}, {1740, 2617}, {1755, 18272}, {2665, 3121}, {3306, 14823}, {6377, 35040}, {13174, 18793}, {18056, 38275}

X(39338) = X(1)X(37133)∩X(7168)X(17738)

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

X(39338) lies on the conic TC(X(1054)) and these lines: {1, 37133}, {7168, 17738}, {7346, 17739}

X(39339) = X(1)X(662)∩X(3)X(7609)

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

X(39339) lies on the conic TC(X(1054)) and these lines: {1, 662}, {3, 7609}, {6, 1054}, {44, 9509}, {610, 2247}, {1740, 2617}, {1958, 16556}, {2173, 16559}, {2652, 37606}, {2930, 5524}, {3216, 9359}, {3634, 26081}, {7067, 9881}, {9324, 9508}

X(39339) = trilinear quotient X(39061)/X(671)

X(39340) = X(1)X(37212)∩X(171)X(1051)

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

X(39340) lies on the conic TC(X(1054)) and these lines: {1, 37212}, {171, 1051}, {9355, 34460}

X(39341) = X(1)X(673)∩X(9)X(1742)

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

X(39341) lies on the conic TC(X(1054)) and these lines: {1, 673}, {9, 1742}, {43, 57}, {165, 25577}, {171, 20229}, {200, 24586}, {238, 2110}, {610, 2636}, {1026, 17738}, {1045, 1781}, {1376, 20995}, {1740, 1743}, {1958, 4579}, {2663, 3664}, {4447, 24727}, {5268, 10382}, {5540, 9319}, {8580, 32916}, {18788, 20605}, {28850, 36796}

X(39345) = X(672)-Ceva conjugate of X(1)
X(39345) = X(6185)-isoconjugate of X(33700)
X(39345) = barycentric product X(i)*X(j) for these {i,j}: {92, 20795}, {672, 33675}
X(39345) = barycentric quotient X (i)/X(j) for these {i,j}: {20795, 63}, {33675, 18031}
X(39345) = {X(7077),X(34253)}-harmonic conjugate of X(3751)

X(39342) = X(1)X(1821)∩X(19)X(1581)

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

X(39342) lies on the conic TC(X(1054)) and these lines: {1, 1821}, {19, 1581}, {63, 1956}, {573, 2636}, {579, 9359}, {1054, 1730}, {1580, 9417}, {1707, 33782}, {1765, 9355}

X(39342) = trilinear quotient X(39058)/X(290)

X(39343) = X(1)X(3257)∩X(44)X(9324)

Barycentrics    a*(a^4 - 2*a^3*b - 5*a^2*b^2 + 6*a*b^3 - b^4 - 2*a^3*c + 16*a^2*b*c - 8*a*b^2*c - 2*b^3*c - 5*a^2*c^2 - 8*a*b*c^2 + 7*b^2*c^2 + 6*a*c^3 - 2*b*c^3 - c^4) : :

X(39343) lies on the conic TC(X(1054)) and these lines: {1, 3257}, {44, 9324}, {238, 5126}, {513, 1052}, {518, 13541}, {1053, 3309}, {1757, 5541}, {3722, 6163}, {4040, 9359}, {9025, 16561}

X(39344) = X(1)X(36086)∩X(57)X(38989)

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

X(39344) lies on the conic TC(X(1054)) and these lines: {1, 36086}, {57, 38989}, {63, 660}, {238, 241}, {513, 2957}, {514, 1053}, {518, 9441}, {650, 1054}, {1052, 9359}, {1155, 1282}, {1768, 37998}, {3836, 37774}, {17596, 24499}

X(39344) = X(2254)-Ceva conjugate of X(1)

leftri

Points on the permutation ellipse E(X(4440)): X(39345)-X(39368)

rightri

This preamble and centers X(36345)-X(39368) were contributed by Clark Kimberling and Peter Moses, July 25-28, 2020.

This preamble extends the preamble just before X(34341), in which, for a point P = p : q : r (barycentrics), the barycentric permutation ellipse denoted by E(P), is defined as the ellipse that passes through these six points:

p : q : r,    q : r : p,    r : p : q,    p : r : q,    q : p : r,   r : q : p.

An equation for E(P) is (q r + r p + p q)(x^2 + y^2 + z^2) - (p^2 + q^2 + r^2)(y z + z x + x y) = 0.

Thus, TC(P) is analogous, and symbolically identical to, the trilinear permutation conic TC(P), defined and discussed in the preambles just before X(36256) and X(39335).

The ellipse E(X(4440)), is given by the barycentric equation x^2 + 3 y z + (cyclic) = 0. Define f(x,y,z) = (x-y)(x-z) - (y-z)^2. If X = x : y : z (trilinears) is a point other than X(2) = 1:1:1 then the point F(X) = f(x,y,z) : f(y,z,x) : f(z,x,y) is on E(X(4440)).

The appearance of (i,j) in the following list means that the point F(X(i)) = X(j) is on E(X(4440)):

(1,4440), (3,39352), (6,148), (11,39353), (31,39345), (32,39346), (63, 39351), (75,9263), (76,25054), (192,9263), (561,39347), (514,17486), (523,8591), (594,39348), (649,39354), (1086,39349), (693,39350)

The vertices -1:1:1, 1:-1:1, 1:1:-1 of the anticomplementary triangle also lie on E(X(4440)), so that this ellipse is the anticomplement of the Steiner circumellipse.

Let f(x : y : z) = (x - y)(x - z) - (y - z)^2. If X = x : y : z and X' = x' : y ' : z' are collinear with X(2), then f(X') = f(X), on E(X(4440)). Example: if X is on the Euler line (but is not X(2)), then f(X) = X(39352).

If P is on the line at infinity, then the P-Ceva conjugate of X(2), which is also the anticomplement of the isotomic conjugate of P, is on E(X(4440).

X(39345) = X(2)X(4586)∩X(148)X(21294)

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

X(39345) lies on the ellipse E(X(4440)) and these lines: {2, 4586}, {148, 21294}, {150, 9263}, {4440, 21293}, {7224, 30661}, {7261, 30667}, {7357, 17486}, {20353, 30669}, {21221, 25054}

X(39345) = anticomplement of X(4586)
X(39345) = anticomplementary conjugate of anticomplement of X(3250)

X(39346) = X(2)X(4577)∩X(6)X(1031)

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

X(39346) lies on the ellipse E(X(4440)) and these lines: {2, 4577}, {6, 1031}, {20, 14718}, {22, 16102}, {66, 8264}, {141, 15588}, {148, 9479}, {338, 32528}, {599, 1975}, {2892, 10340}, {3448, 25054}, {7837, 8878}, {8272, 9483}, {9263, 21293}, {20079, 33785}, {31374, 31375}

X(39346) = anticomplement of X(4577)
X(39346) = anticomplementary conjugate of isogonal conjugate of X(1634)
X(39346) = anticomplementary conjugate of isotomic conjugate of X(4576)
X(39346) = anticomplementary conjugate of anticomplement of X(3005)
X(39346) = anticomplementary conjugate of trilinear pole of line X(115)X(804) (the tangent to the nine-point circle at X(115))

X(39347) = X(794)X(39345)∩X(3300)X(11234)

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

X(39347) lies on the ellipse E(X(4440)) and these lines: {794, 39345}, {3300, 11234}, {6327, 14945}, {13176, 17097}, {21224, 39362} {6327, 14945}

X(39347) = anticomplement of isogonal conjugate of X(8630)
X(39347) = anticomplement of isotomic conjugate of X(788)
X(39347) = anticomplement of trilinear pole of line X(2)X(561)

X(39348) = X(2)X(6540)∩X(148)X(17154)

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

X(39348) lies on the ellipse E(X(4440)) and these lines: {2, 6540}, {148, 17154}, {551, 894}

X(39348) = anticomplement of X(6540)

X(39349) = X(2)X(4555)∩X(148)X(31290)

Barycentrics    a^4 - 2*a^3*b - 5*a^2*b^2 + 6*a*b^3 - b^4 - 2*a^3*c + 16*a^2*b*c - 8*a*b^2*c - 2*b^3*c - 5*a^2*c^2 - 8*a*b*c^2 + 7*b^2*c^2 + 6*a*c^3 - 2*b*c^3 - c^4 : :

X(39349) lies on the ellipse E(X(4440)) and these lines: {2, 4555}, {148, 31290}, {239, 30577}, {514, 4440}, {519, 4480}, {4422, 6631}, {4468, 17036}, {4473, 6633}, {6542, 30578}, {8046, 17495}, {8591, 20016}, {9263, 17494}, {20042, 33920}, {20055, 30225}, {21290, 32847}, {29569, 35962}

X(39349) = anticomplement of X(4555)
X(39349) = anticomplementary conjugate of anticomplement of X(1960)

X(39350) = X(2)X(2481)∩X(7)X(192)

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

X(39350) lies on the ellipse E(X(4440)) and these lines: {2, 2481}, {7, 192}, {8, 3177}, {20, 2808}, {144, 4499}, {145, 194}, {148, 1655}, {149, 14947}, {153, 14732}, {190, 36216}, {193, 7674}, {239, 672}, {894, 2293}, {1334, 9317}, {1742, 3729}, {1975, 4513}, {2340, 10025}, {2897, 18666}, {3208, 9312}, {3779, 25050}, {5773, 37416}, {6542, 25257}, {17244, 30949}, {17248, 21914}, {17784, 21218}, {27523, 32034}

X(39350) = polar conjugate of isogonal conjugate of X(20795)
X(39350) = anticomplement of X(2481)
X(39350) = anticomplementary conjugate of X(20556)

X(39351) = X(2)X(664)∩X(7)X(21139)

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

X(39351) lies on the ellipse E(X(4440)) and these lines: {2, 664}, {7, 21139}, {8, 3177}, {63, 8591}, {92, 17778}, {116, 38941}, {144, 528}, {145, 10405}, {148, 2785}, {149, 6366}, {150, 514}, {152, 18328}, {189, 3210}, {192, 5942}, {257, 5484}, {519, 10025}, {668, 30225}, {673, 4534}, {908, 17310}, {918, 4440}, {934, 25954}, {952, 3732}, {1837, 9311}, {1952, 2994}, {2975, 8301}, {3436, 20535}, {3621, 20111}, {3632, 30625}, {4462, 6630}, {4468, 17036}, {4530, 9317}, {4561, 27546}, {4568, 21290}, {5554, 27340}, {6542, 30807}, {6604, 20089}, {9312, 26531}, {11681, 20531}, {11998, 24499}, {20100, 36171}, {20533, 21272}, {23058, 25716}, {25718, 26658}, {25726, 26006}, {29582, 30852}, {30577, 34234}, {31994, 36905}

X(39351) = reflection of X(39357) in X(2)
X(39351) = isotomic conjugate of isogonal conjugate of crosspoint of PU(103)
X(39351) = anticomplement of X(664)
X(39351) = anticomplementary conjugate of X(21302)
X(39351) = trilinear pole, wrt anticomplementary triangle, of line X(2)X(7)

X(39352) = X(2)X(648)∩X(20)X(542)

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

X(39352) lies on the ellipse E(X(4440)) and these lines: {2, 648}, {20, 542}, {69, 1972}, {125, 11596}, {148, 2799}, {193, 253}, {264, 17035}, {401, 524}, {523, 14721}, {525, 15351}, {858, 7840}, {1297, 5984}, {1634, 37918}, {2373, 7665}, {2450, 38294}, {2895, 18667}, {2897, 18666}, {3146, 9530}, {3151, 17487}, {3448, 9033}, {3506, 19121}, {3620, 15595}, {4558, 35520}, {5032, 11348}, {6330, 17037}, {6527, 20080}, {7779, 30737}, {7946, 37444}, {8878, 18018}, {9035, 25053}, {10620, 35241}

X(39352) = reflection of X(2) in X(1494)
X(39352) = reflection of X(39358) in X(2)
X(39352) = isotomic conjugate of X(15351)
X(39352) = anticomplement of X(648)
X(39352) = anticomplementary conjugate of X(850)

X(39353) = X(2)X(666)∩X(69)X(4562)

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

X(39353) lies on the ellipse E(X(4440)) and these lines: {2, 666}, {69, 4562}, {150, 514}, {239, 9436}, {522, 4440}, {527, 17310}, {693, 17036}, {3912, 10025}, {6542, 25257}, {6604, 35167}, {8047, 17494}, {8591, 20067}, {9263, 17496}, {13577, 21218}, {17365, 35080}, {17950, 36918}, {31300, 34361}

X(39353) = reflection of X(2) in X(18821)
X(39353) = reflection of X(39363) in X(2)
X(39353) = anticomplement of X(666)
X(39353) = anticomplementary conjugate of X(3766)

X(39354) = X(2)X(3226)∩X(8)X(291)

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

X(39354) lies on the ellipse E(X(4440)) and these lines: {2, 3226}, {8, 291}, {69, 1278}, {148, 1330}, {192, 25311}, {350, 6542}, {1654, 25054}, {1909, 20899}, {3436, 20535}, {8050, 21224}, {20936, 24524}, {21290, 32847}, {23354, 30667}

X(39354) = anticomplement of X(3226)
X(39354) = anticomplementary conjugate of X(20352)

X(39355) = X(2)X(290)∩X(4)X(147)

Barycentrics    a^8*b^4 - 2*a^6*b^6 + a^4*b^8 + 3*a^8*b^2*c^2 - 3*a^6*b^4*c^2 + a^4*b^6*c^2 - a^2*b^8*c^2 + a^8*c^4 - 3*a^6*b^2*c^4 + a^4*b^4*c^4 + a^2*b^6*c^4 - b^8*c^4 - 2*a^6*c^6 + a^4*b^2*c^6 + a^2*b^4*c^6 + 2*b^6*c^6 + a^4*c^8 - a^2*b^2*c^8 - b^4*c^8 : :
X(39355) = 3 X[2] - 4 X[11672]

X(39355) lies on the ellipse E(X(4440)), Jerabek circumhyperbola of the anticomplementary triangle, the cubic K355, and these lines: {2, 290}, {4, 147}, {69, 1972}, {193, 8264}, {217, 384}, {237, 385}, {401, 3289}, {1625, 10684}, {1975, 32445}, {2892, 14721}, {3448, 9513}, {3868, 9263}, {4440, 17148}, {6193, 20065}, {7777, 37988}, {7793, 37114}, {9143, 31296}, {9308, 20794}, {9512, 20775}, {12383, 14712}, {21226, 39351}, {31859, 32444}

X(39355) = reflection of X(i) in X(j) for these {i,j}: {290, 11672}, {3448, 9513}
X(39355) = anticomplement of X(290)
X(39355) = anticomplement of the isogonal conjugate of X(237)
X(39355) = anticomplement of the isotomic conjugate of X(511)
X(39355) = isotomic conjugate of the anticomplement of X(39058)
X(39355) = anticomplementary isogonal conjugate of X(14957)
X(39355) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 14957}, {31, 511}, {48, 30737}, {232, 21270}, {237, 8}, {240, 11442}, {325, 21275}, {511, 6327}, {560, 385}, {662, 14295}, {1755, 69}, {1910, 290}, {1927, 1916}, {1959, 315}, {1967, 20021}, {2211, 5905}, {2396, 21305}, {2421, 17217}, {2491, 21221}, {3289, 4329}, {3569, 21294}, {4230, 21300}, {5360, 1330}, {5968, 21298}, {9247, 401}, {9417, 2}, {9418, 192}, {14966, 7192}, {17209, 17137}, {17453, 34137}, {23995, 4226}, {23997, 512}, {34854, 5906}
X(39355) = X(511)-Ceva conjugate of X(2)
X(39355) = X(39058)-cross conjugate of X(2)
X(39355) = barycentric product X(i)*X(j) for these {i,j}: {75, 39342}, {511, 39058}
X(39355) = barycentric quotient X(i)/X(j) for these {i,j}: {39058, 290}, {39342, 1}
X(39355) = {X(290),X(11672)}-harmonic conjugate of X(2)

X(39356) = X(2)X(892)∩X(148)X(523)

Barycentrics    a^8 - 2*a^6*b^2 - 5*a^4*b^4 + 6*a^2*b^6 - b^8 - 2*a^6*c^2 + 16*a^4*b^2*c^2 - 8*a^2*b^4*c^2 - 2*b^6*c^2 - 5*a^4*c^4 - 8*a^2*b^2*c^4 + 7*b^4*c^4 + 6*a^2*c^6 - 2*b^2*c^6 - c^8 : :
X(39356) = 3 X[2] - 4 X[23992], 5 X[2] - 4 X[35087], X[148] - 3 X[35511], 4 X[620] - 3 X[31998], X[892] - 3 X[18823], 5 X[892] - 6 X[35087], 4 X[5461] - 3 X[39061], 3 X[18823] - 2 X[23992], 5 X[18823] - 2 X[35087], 5 X[23992] - 3 X[35087]

X(39356) lies on the ellipse E(X(4440)) and these lines: {2, 892}, {148, 523}, {385, 7426}, {524, 8591}, {620, 31998}, {858, 16103}, {4440, 17161}, {4590, 31372}, {5461, 39061}, {6563, 39352}, {7779, 14360}, {17004, 36207}, {17487, 20536}, {25054, 31296}

X(39356) = reflection of X(i) in X(j) for these {i,j}: {2, 18823}, {858, 16103}, {892, 23992}, {31372, 4590}
X(39356) = anticomplement of X(892)
X(39356) = anticomplement of the isogonal conjugate of X(351)
X(39356) = anticomplement of the isotomic conjugate of X(690)
X(39356) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {31, 690}, {42, 30709}, {163, 5468}, {187, 7192}, {351, 8}, {468, 21300}, {512, 17491}, {523, 21298}, {524, 17217}, {661, 316}, {667, 17162}, {669, 17497}, {690, 6327}, {798, 524}, {810, 858}, {896, 512}, {922, 523}, {923, 5466}, {1101, 33919}, {1648, 21294}, {1967, 34290}, {1973, 9979}, {2642, 69}, {3266, 21305}, {4062, 21301}, {4750, 17137}, {5467, 21295}, {14273, 21270}, {14419, 17135}, {14567, 4560}, {16702, 17159}, {21839, 20295}, {21906, 21221}, {22105, 21278}, {23889, 4576}, {24039, 670}, {35522, 21275}, {36142, 892}, {38252, 9134}
X(39356) = anticomplementary conjugate of anticomplement of X(351)
X(39356) = X(690)-Ceva conjugate of X(2)
X(39356) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {892, 18823, 23992}, {892, 23992, 2}

X(39357) = X(2)X(664)∩X(145)X(528)

Barycentrics    5*a^4 - 5*a^3*b - 6*a^2*b^2 + 7*a*b^3 - b^4 - 5*a^3*c + 17*a^2*b*c - 7*a*b^2*c - 5*b^3*c - 6*a^2*c^2 - 7*a*b*c^2 + 12*b^2*c^2 + 7*a*c^3 - 5*b*c^3 - c^4 : :
X(39357) = 5 X[2] - 4 X[1146], 7 X[2] - 8 X[17044], 11 X[2] - 10 X[31640], 3 X[2] - 4 X[35110], 3 X[664] - X[1121], 5 X[664] - 2 X[1146], 7 X[664] - 4 X[17044], 11 X[664] - 5 X[31640], 3 X[664] - 2 X[35110], 4 X[664] - X[39351], 5 X[1121] - 6 X[1146], 7 X[1121] - 12 X[17044], 11 X[1121] - 15 X[31640], 4 X[1121] - 3 X[39351], 7 X[1146] - 10 X[17044], 22 X[1146] - 25 X[31640], 3 X[1146] - 5 X[35110], 8 X[1146] - 5 X[39351], 2 X[10708] - 3 X[38941], 44 X[17044] - 35 X[31640], 6 X[17044] - 7 X[35110], 16 X[17044] - 7 X[39351], 15 X[31640] - 22 X[35110], 20 X[31640] - 11 X[39351], 8 X[35110] - 3 X[39351]

X(39357) lies on the ellipse E(X(4440)) and these lines: {2, 664}, {145, 528}, {148, 17389}, {239, 30577}, {519, 39353}, {918, 17487}, {2094, 3210}, {2785, 8591}, {3152, 39352}, {3177, 4552}, {3227, 34342}, {3241, 28850}, {5484, 17254}, {6173, 9312}, {10708, 38941}, {21222, 39350}, {26136, 29621}, {27295, 29575}

X(39357) = reflection of X(i) in X(j) for these {i,j}: {2, 664}, {1121, 35110}, {39351, 2}
X(39357) = anticomplement of X(1121)
X(39357) = anticomplement of the isogonal conjugate of X(1055)
X(39357) = anticomplement of the isotomic conjugate of X(527)
X(39357) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 5057}, {31, 527}, {527, 6327}, {604, 26015}, {692, 30565}, {1055, 8}, {1155, 69}, {1323, 21285}, {1415, 6366}, {1638, 21293}, {1911, 24712}, {6139, 37781}, {6510, 1370}, {6603, 3436}, {6610, 3434}, {6745, 21286}, {9456, 10707}, {14413, 150}, {23346, 693}, {23710, 21270}, {23890, 21302}, {24685, 20554}, {30574, 21294}, {30806, 315}, {34068, 1121}, {35293, 20552}, {37780, 21280}, {37805, 11442}
X(39357) = anticomplementary conjugate of anticomplement of X(1055)
X(39357) = X(527)-Ceva conjugate of X(2)
X(39357) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {664, 1121, 35110}, {1121, 35110, 2}

X(39358) = X(2)X(648)∩X(146)X(148)

Barycentrics    5*a^8 - 5*a^6*b^2 - 6*a^4*b^4 + 7*a^2*b^6 - b^8 - 5*a^6*c^2 + 17*a^4*b^2*c^2 - 7*a^2*b^4*c^2 - 5*b^6*c^2 - 6*a^4*c^4 - 7*a^2*b^2*c^4 + 12*b^4*c^4 + 7*a^2*c^6 - 5*b^2*c^6 - c^8 : :
X(39358) = 3 X[2] - 4 X[3163], 5 X[2] - 4 X[15526], 7 X[2] - 8 X[23583], 2 X[287] - 3 X[5032], 3 X[648] - X[1494], 3 X[648] - 2 X[3163], 5 X[648] - 2 X[15526], 7 X[648] - 4 X[23583], 4 X[648] - X[39352], 5 X[1494] - 6 X[15526], 7 X[1494] - 12 X[23583], 4 X[1494] - 3 X[39352], 5 X[3163] - 3 X[15526], 7 X[3163] - 6 X[23583], 8 X[3163] - 3 X[39352], 7 X[15526] - 10 X[23583], 8 X[15526] - 5 X[39352], 16 X[23583] - 7 X[39352]

X(39358) lies on the ellipse E(X(4440)), the cubic K860, and these lines: {2, 648}, {146, 148}, {287, 5032}, {340, 18487}, {376, 3164}, {381, 9308}, {385, 7426}, {1272, 2407}, {2799, 8591}, {3180, 11092}, {3181, 11078}, {3187, 4440}, {7837, 8878}, {9033, 9143}, {9530, 15683}, {15687, 27377}, {25045, 25054}, {31621, 36435}

X(39358) = reflection of X(i) in X(j) for these {i,j}: {2, 648}, {340, 18487}, {1494, 3163}, {31621, 36435}, {39352, 2}
X(39358) = anticomplement of X(1494)
X(39358) = anticomplement of the isogonal conjugate of X(1495)
X(39358) = anticomplement of the isotomic conjugate of X(30)
X(39358) = isotomic conjugate of the anticomplement of X(9410)
X(39358) = isotomic conjugate of the isogonal conjugate of X(9412)
X(39358) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {19, 340}, {30, 6327}, {31, 30}, {32, 18668}, {163, 3268}, {923, 9140}, {1333, 18661}, {1495, 8}, {1637, 21294}, {1784, 11442}, {1973, 3580}, {1990, 21270}, {2159, 1494}, {2173, 69}, {2407, 17217}, {2420, 7192}, {2631, 13219}, {3260, 21275}, {3284, 4329}, {4240, 21300}, {6357, 21285}, {7359, 21286}, {9214, 21298}, {9406, 2}, {9407, 192}, {11125, 21293}, {14206, 315}, {14398, 21221}, {14399, 150}, {14581, 5905}, {18653, 17137}, {23347, 7253}, {32676, 9033}, {36131, 34767}
X(39358) = X(i)-Ceva conjugate of X(j) for these (i,j): {30, 2}, {9410, 34582}
X(39358) = X(i)-cross conjugate of X(j) for these (i,j): {9410, 2}, {34582, 9410}
X(39358) = crosspoint of X(30) and X(34582)
X(39358) = barycentric product X(i)*X(j) for these {i,j}: {30, 9410}, {76, 9412}, {1494, 34582}
X(39358) = barycentric quotient X(i)/X(j) for these {i,j}: {9410, 1494}, {9412, 6}, {34582, 30}
X(39358) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {648, 1494, 3163}, {1494, 3163, 2}
X(39358) = anticomplementary conjugate of anticomplement of X(1495)

X(39359) = X(2)X(2966)∩X(30)X(147)

Barycentrics    a^12 - 2*a^10*b^2 + 2*a^8*b^4 - 2*a^6*b^6 + 2*a^2*b^10 - b^12 - 2*a^10*c^2 + 2*a^8*b^2*c^2 - 2*a^2*b^8*c^2 + 2*b^10*c^2 + 2*a^8*c^4 + a^4*b^4*c^4 - 4*b^8*c^4 - 2*a^6*c^6 + 6*b^6*c^6 - 2*a^2*b^2*c^8 - 4*b^4*c^8 + 2*a^2*c^10 + 2*b^2*c^10 - c^12 : :
X(39359) = 5 X[2] - 4 X[23967], 3 X[2] - 4 X[35088], 4 X[441] - 5 X[7925], X[2966] - 3 X[5641], 5 X[2966] - 6 X[23967], 5 X[5641] - 2 X[23967], 3 X[5641] - 2 X[35088], 3 X[23967] - 5 X[35088]

X(39359) lies on the ellipse E(X(4440)) and these lines: {2, 2966}, {30, 147}, {148, 525}, {297, 385}, {315, 18829}, {325, 401}, {441, 7925}, {523, 14721}, {7785, 10684}, {7898, 35923}, {11361, 34360}, {13485, 31296}, {30227, 33017}

X(39359) = reflection of X(i) in X(j) for these {i,j}: {2, 5641}, {385, 297}, {401, 325}, {2966, 35088}
X(39359) = anticomplement of X(2966)
X(39359) = anticomplement of the isogonal conjugate of X(3569)
X(39359) = anticomplement of the isotomic conjugate of X(2799)
X(39359) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {31, 2799}, {75, 14295}, {162, 877}, {163, 4226}, {232, 7253}, {237, 4560}, {240, 850}, {297, 21300}, {325, 17217}, {511, 7192}, {656, 30737}, {661, 511}, {684, 4329}, {798, 385}, {810, 401}, {868, 21294}, {1577, 14957}, {1755, 523}, {1910, 879}, {1959, 512}, {1967, 2395}, {2211, 17498}, {2421, 21295}, {2491, 192}, {2799, 6327}, {3569, 8}, {5360, 514}, {8430, 17491}, {9417, 31296}, {14966, 6758}, {16230, 21270}, {17209, 17166}, {17994, 5905}, {23997, 99}, {32676, 34211}, {36046, 685}, {37134, 805}
X(39359) = anticomplementary conjugate of anticomplement of X(3569)
X(39359) = X(2799)-Ceva conjugate of X(2)
X(39359) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2966, 5641, 35088}, {2966, 35088, 2}

X(39360) = X(2)X(668)∩X(8)X(537)

Barycentrics    a^2*b^2 - 7*a^2*b*c + 5*a*b^2*c + a^2*c^2 + 5*a*b*c^2 - 5*b^2*c^2 : :
X(39360) = 5 X[2] - 4 X[1015], 3 X[2] - 4 X[13466], 7 X[2] - 8 X[27076], 11 X[2] - 10 X[27195], 5 X[2] - 8 X[36524], 5 X[668] - 2 X[1015], 3 X[668] - X[3227], 4 X[668] - X[9263], 3 X[668] - 2 X[13466], 7 X[668] - 4 X[27076], 11 X[668] - 5 X[27195], 2 X[668] + X[31298], 5 X[668] - 4 X[36524], 6 X[1015] - 5 X[3227], 8 X[1015] - 5 X[9263], 3 X[1015] - 5 X[13466], 7 X[1015] - 10 X[27076], 22 X[1015] - 25 X[27195], 4 X[1015] + 5 X[31298], 4 X[3227] - 3 X[9263], 7 X[3227] - 12 X[27076], 11 X[3227] - 15 X[27195], 2 X[3227] + 3 X[31298], 5 X[3227] - 12 X[36524], 3 X[9263] - 8 X[13466], 7 X[9263] - 16 X[27076], 11 X[9263] - 20 X[27195], X[9263] + 2 X[31298], 5 X[9263] - 16 X[36524], 7 X[13466] - 6 X[27076], 22 X[13466] - 15 X[27195], 4 X[13466] + 3 X[31298], 5 X[13466] - 6 X[36524], 4 X[17793] - 3 X[38314], 44 X[27076] - 35 X[27195], 8 X[27076] + 7 X[31298], 5 X[27076] - 7 X[36524], 10 X[27195] + 11 X[31298], 25 X[27195] - 44 X[36524], 5 X[31298] + 8 X[36524]

X(39360) lies on the ellipse E(X(4440)) and these lines: {2, 668}, {8, 537}, {148, 2895}, {194, 25296}, {329, 39351}, {519, 17794}, {812, 17487}, {1655, 4664}, {1909, 4688}, {2787, 8591}, {2810, 11160}, {3570, 21781}, {6542, 30578}, {9457, 27912}, {14839, 31145}, {17793, 38314}, {20344, 39353}, {26840, 39348} X(39360) = midpoint of X(2) and X(31298)
X(39360) = reflection of X(i) in X(j) for these {i,j}: {2, 668}, {1015, 36524}, {3227, 13466}, {9263, 2}
X(39360) = anticomplement of X(3227)
X(39360) = anticomplement of the isogonal conjugate of X(3230)
X(39360) = anticomplement of the isotomic conjugate of X(536)
X(39360) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 29824}, {31, 536}, {101, 891}, {536, 6327}, {890, 4440}, {891, 150}, {899, 69}, {1461, 30704}, {3230, 8}, {3768, 149}, {3994, 21287}, {4009, 21286}, {4465, 20554}, {4526, 33650}, {4728, 21293}, {6381, 315}, {6632, 33917}, {14404, 21221}, {14431, 21294}, {23343, 20295}, {23891, 21301}, {35543, 21275}, {36816, 20556}, {38266, 3999} X(39360) = X(536)-Ceva conjugate of X(2)
X(39360) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {668, 3227, 13466}, {668, 31298, 9263}, {3227, 13466, 2}
X(39360) = anticomplementary conjugate of anticomplement of X(3230)

X(39361) = X(2)X(670)∩X(69)X(148)

Barycentrics    a^4*b^4 - 7*a^4*b^2*c^2 + 5*a^2*b^4*c^2 + a^4*c^4 + 5*a^2*b^2*c^4 - 5*b^4*c^4 : :
X(39361) = 5 X[2] - 4 X[1084], 11 X[2] - 10 X[31639], 3 X[2] - 4 X[35073], 7 X[2] - 8 X[36950], 4 X[597] - 3 X[25318], 5 X[670] - 2 X[1084], 3 X[670] - X[3228], 4 X[670] - X[25054], 11 X[670] - 5 X[31639], 3 X[670] - 2 X[35073], 7 X[670] - 4 X[36950], 2 X[694] - 3 X[21356], 6 X[1084] - 5 X[3228], 8 X[1084] - 5 X[25054], 22 X[1084] - 25 X[31639], 3 X[1084] - 5 X[35073], 7 X[1084] - 10 X[36950], 2 X[1992] - 3 X[25319], 4 X[3228] - 3 X[25054], 11 X[3228] - 15 X[31639], 7 X[3228] - 12 X[36950], 11 X[25054] - 20 X[31639], 3 X[25054] - 8 X[35073], 7 X[25054] - 16 X[36950], 15 X[31639] - 22 X[35073], 35 X[31639] - 44 X[36950], 7 X[35073] - 6 X[36950]

X(39361) lies on the ellipse E(X(4440)) and these lines: {2, 670}, {69, 148}, {75, 7200}, {524, 25332}, {597, 25318}, {694, 21356}, {804, 8591}, {1369, 39346}, {1370, 39352}, {1992, 25319}, {4440, 17135}, {7779, 14360}, {9230, 9466}, {11160, 34383}, {15534, 25327}, {20245, 39351}

X(39361) = reflection of X(i) in X(j) for these {i,j}: {2, 670}, {3228, 35073}, {15534, 25327}, {25054, 2}
X(39361) = anticomplement of X(3228)
X(39361) = anticomplement of the isogonal conjugate of X(3231)
X(39361) = anticomplement of the isotomic conjugate of X(538)
X(39361) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {31, 538}, {163, 9147}, {538, 6327}, {662, 888}, {887, 21220}, {888, 21221}, {2234, 69}, {3231, 8}, {5118, 7192}, {9148, 21294}, {14609, 17491}, {23342, 17217}, {30736, 21275}, {30938, 17138}, {33875, 192}
X(39361) = anticomplementary conjugate of anticomplement of X(3231)
X(39361) = X(538)-Ceva conjugate of X(2)
X(39361) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {670, 3228, 35073}, {3228, 35073, 2}

X(39362) = X(2)X(4562)∩X(75)X(24510)

Barycentrics    a^4*b^2 + a^3*b^3 - a^2*b^4 - 3*a^4*b*c + a^3*b^2*c - 3*a^2*b^3*c + a*b^4*c + a^4*c^2 + a^3*b*c^2 + 2*a^2*b^2*c^2 + a*b^3*c^2 + b^4*c^2 + a^3*c^3 - 3*a^2*b*c^3 + a*b^2*c^3 - 3*b^3*c^3 - a^2*c^4 + a*b*c^4 + b^2*c^4 : :
X(39362) = 3 X[2] - 4 X[35119], 5 X[2] - 4 X[35123], 4 X[1575] - 5 X[29590], X[4562] - 3 X[18822], 5 X[4562] - 6 X[35123], 3 X[18822] - 2 X[35119], 5 X[18822] - 2 X[35123], 5 X[35119] - 3 X[35123]

X(39362) lies on the ellipse E(X(4440)) and these lines: {2, 4562}, {75, 24510}, {149, 39345}, {239, 672}, {350, 6542}, {513, 4440}, {514, 9263}, {519, 17794}, {536, 17487}, {1278, 36216}, {1575, 29590}, {4360, 35148}, {4462, 6630}, {4740, 36222}, {9294, 9295}, {17143, 35173}, {20016, 30668}, {21221, 39346}, {21224, 39347}

X(39362) = reflection of X(i) in X(j) for these {i,j}: {2, 18822}, {4562, 35119}, {6542, 350}, {17759, 239}
X(39362) = anticomplement of X(4562)
X(39362) = anticomplement of the isogonal conjugate of X(8632)
X(39362) = anticomplement of the isotomic conjugate of X(812)
X(39362) = anticomplementary isogonal conjugate of X(21303)
X(39362) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 21303}, {31, 812}, {101, 23354}, {110, 874}, {238, 20295}, {239, 21301}, {350, 21304}, {513, 20553}, {593, 4155}, {604, 2254}, {649, 4645}, {659, 69}, {667, 6542}, {812, 6327}, {875, 6653}, {919, 660}, {1428, 693}, {1429, 21302}, {1914, 513}, {1919, 17759}, {1980, 19565}, {2201, 20293}, {2210, 514}, {3573, 668}, {3716, 21286}, {3733, 30941}, {3766, 315}, {4010, 21287}, {4164, 30660}, {4375, 20554}, {4435, 3436}, {4455, 2895}, {5009, 7192}, {8632, 8}, {14599, 17494}, {17940, 4589}, {18268, 876}, {18892, 21225}, {21832, 1330}, {22384, 4329}, {27846, 150}, {27918, 21293}, {31905, 21300}, {33295, 17217}, {34067, 4562}, {34248, 23656}, {38367, 39354}
X(39362) = X(812)-Ceva conjugate of X(2)
X(39362) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4562, 18822, 35119}, {4562, 35119, 2}

X(39363) = X(2)X(666)∩X(144)X(36236)

Barycentrics    5*a^6 - 10*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 4*a^2*b^4 + 2*a*b^5 - b^6 - 10*a^5*c + 10*a^4*b*c - 2*a*b^4*c + 2*b^5*c + 10*a^4*c^2 - 3*a^2*b^2*c^2 - 8*b^4*c^2 - 10*a^3*c^3 + 14*b^3*c^3 + 4*a^2*c^4 - 2*a*b*c^4 - 8*b^2*c^4 + 2*a*c^5 + 2*b*c^5 - c^6 : :
X(39363) = 5 X[2] - 4 X[35094], 3 X[2] - 4 X[35113], 3 X[666] - X[18821], 5 X[666] - 2 X[35094], 3 X[666] - 2 X[35113], 4 X[666] - X[39353], 5 X[18821] - 6 X[35094], 4 X[18821] - 3 X[39353], 3 X[35094] - 5 X[35113], 8 X[35094] - 5 X[39353], 8 X[35113] - 3 X[39353]

X(39363) lies on the ellipse E(X(4440)) and these lines: {2, 666}, {144, 36236}, {145, 39349}, {239, 527}, {519, 10025}, {522, 17487}, {894, 36220}, {1992, 18822}, {10707, 17036}, {17494, 39350}

X(39363) = reflection of X(i) in X(j) for these {i,j}: {2, 666}, {18821, 35113}, {39353, 2}
X(39363) = anticomplement of X(18821)
X(39363) = anticomplement of the isotomic conjugate of X(528)
X(39363) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {31, 528}, {528, 6327}, {1642, 20552}, {1643, 150}, {2246, 69}, {5723, 21285}, {14190, 21282}
X(39363) = anticomplementary conjugate of anticomplement of Parry-isodynamic-circle-inverse of X(5098)
X(39363) = X(528)-Ceva conjugate of X(2)
X(39363) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {666, 18821, 35113}, {18821, 35113, 2}

X(39364) = X(2)X(4597)∩X(8)X(17487)

Barycentrics    4*a^4 - 2*a^3*b - 5*a^2*b^2 + 6*a*b^3 - 4*b^4 - 2*a^3*c + 7*a^2*b*c - 5*a*b^2*c - 2*b^3*c - 5*a^2*c^2 - 5*a*b*c^2 + 13*b^2*c^2 + 6*a*c^3 - 2*b*c^3 - 4*c^4 : :
X(39364) = 5 X[2] - 4 X[35124], 5 X[4597] - 6 X[35124], X[4597] - 3 X[35170], 2 X[35124] - 5 X[35170]

X(39364) lies on the ellipse E(X(4440)) and these lines: {2, 4597}, {8, 17487}, {149, 39349}, {8591, 30564}, {9263, 20042}, {21291, 39354}

X(39364) = reflection of X(2) in X(35170)
X(39364) = anticomplement of X(4597)
X(39364) = anticomplement of the isogonal conjugate of X(4775)
X(39364) = anticomplement of the isotomic conjugate of X(4777)
X(39364) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {31, 4777}, {45, 20295}, {513, 21283}, {667, 3241}, {692, 4781}, {798, 37635}, {1405, 693}, {2099, 21302}, {2177, 513}, {3063, 5744}, {3679, 21301}, {3733, 17146}, {4273, 7192}, {4653, 512}, {4671, 21304}, {4752, 668}, {4770, 1330}, {4775, 8}, {4777, 6327}, {4791, 315}, {4800, 20554}, {4814, 3436}, {4825, 21291}, {4833, 17135}, {4893, 69}, {4931, 21287}, {4944, 21286}, {5235, 17217}, {23352, 21282}, {34073, 4597}
X(39364) = anticomplementary conjugate of anticomplement of X(4775)
X(39364) = X(4777)-Ceva conjugate of X(2)

X(39365) = X(2)X(6)∩X(148)X(3414)

Barycentrics    2*(2*a^2 - b^2 - c^2) + Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] : :
X(39365) = X[6189] - 3 X[6190]

X(39365) lies on the ellipse E(X(4440)) and these lines: {2, 6}, {148, 3414}, {194, 13325}, {2558, 7793}, {3413, 8591}, {3564, 6040}, {20794, 21032}

X(39365) = reflection of X(2) in X(6190)
X(39365) = reflection of X(39366) in X(2)
X(39365) = anticomplement of X(6189)
X(39365) = anticomplement of the isogonal conjugate of X(5639)
X(39365) = anticomplement of the isotomic conjugate of X(3414)
X(39365) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {31, 3414}, {163, 30509}, {1379, 7192}, {3414, 6327}, {5639, 8}, {6190, 17217}, {13722, 21294}
X(39365) = anticomplementary conjugate of anticomplement of X(5639)
X(39365) = X(3414)-Ceva conjugate of X(2)
X(39365) = crosspoint of X(4590) and X(6190)
X(39365) = crosssum of X(3124) and X(5638)
X(39365) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 7779, 39366}, {6189, 6190, 39022}, {6189, 39022, 2}

X(39366) = X(2)X(6)∩X(148)X(3413)

Barycentrics    2*(2*a^2 - b^2 - c^2) - Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] : :
X(39366) = 3 X[6189] - X[6190]

X(39366) lies on the ellipse E(X(4440)) and these lines: {2, 6}, {148, 3413}, {194, 13326}, {2559, 7793}, {3414, 8591}, {3564, 6039}, {20794, 21036}

X(39366) = reflection of X(2) in X(6189)
X(39366) = reflection of X(39365) in X(2)
X(39366) = anticomplement of X(6190)
X(39366) = anticomplement of the isogonal conjugate of X(5638)
X(39366) = anticomplement of the isotomic conjugate of X(3413)
X(39366) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {31, 3413}, {163, 30508}, {1380, 7192}, {3413, 6327}, {5638, 8}, {6189, 17217}, {13636, 21294}
X(39366) = anticomplementary conjugate of anticomplement of X(5638)
X(39366) = X(3413)-Ceva conjugate of X(2)
X(39366) = crosspoint of X(4590) and X(6189)
X(39366) = crosssum of X(3124) and X(5639)
X(39366) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 7779, 39365}, {6189, 6190, 39023}, {6190, 39023, 2}

X(39367) = X(1)X(1655)∩X(2)X(18827)

Barycentrics    a^4*b^2 - a^3*b^3 - a^2*b^4 + 3*a^4*b*c + a^3*b^2*c - 3*a^2*b^3*c - a*b^4*c + a^4*c^2 + a^3*b*c^2 - 2*a^2*b^2*c^2 + a*b^3*c^2 + b^4*c^2 - a^3*c^3 - 3*a^2*b*c^3 + a*b^2*c^3 + 3*b^3*c^3 - a^2*c^4 - a*b*c^4 + b^2*c^4 : :
X(39367) = 3 X[2] - 4 X[35068]

X(39367) lies on the ellipse E(X(4440)) and these lines: {1, 1655}, {2, 18827}, {8, 148}, {75, 1654}, {192, 25054}, {385, 1931}, {537, 24505}, {1909, 21879}, {2895, 39345}, {3509, 20609}, {3570, 9509}, {3869, 39351}, {3952, 21220}, {4037, 6542}, {4088, 39350}, {4329, 39352}, {4876, 24504}, {17475, 36269}, {20016, 30668}, {20529, 29569}, {21289, 39346}, {21699, 28604}

X(39367) = reflection of X(18827) in X(35068)
X(39367) = isotomic conjugate of X(39719)
X(39367) = anticomplement of X(18827)
X(39367) = anticomplement of the isogonal conjugate of X(3747)
X(39367) = anticomplement of the isotomic conjugate of X(740)
X(39367) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 30941}, {31, 740}, {37, 20553}, {41, 1959}, {42, 4645}, {110, 4155}, {213, 6542}, {238, 17135}, {239, 17137}, {242, 20242}, {251, 30940}, {350, 17138}, {740, 6327}, {862, 4}, {1018, 21303}, {1252, 874}, {1284, 3434}, {1428, 3873}, {1429, 20244}, {1914, 75}, {1918, 17759}, {2201, 17220}, {2205, 19565}, {2209, 2227}, {2210, 1}, {2238, 69}, {3570, 17217}, {3573, 512}, {3684, 20245}, {3747, 8}, {3948, 315}, {3985, 21286}, {4010, 21293}, {4037, 21287}, {4093, 21289}, {4155, 3448}, {4368, 20554}, {4433, 3436}, {4455, 149}, {5009, 17140}, {6651, 20560}, {7193, 20243}, {8298, 20351}, {14599, 17147}, {16609, 21285}, {18268, 18827}, {18786, 17153}, {18892, 17148}, {20769, 18659}, {21832, 150}, {27853, 21305}, {34067, 876}, {35544, 21275}
X(39367) = anticomplementary conjugate of anticomplement of X(3747)
X(39367) = anticomplementary conjugate of isotomic conjugate of isogonal conjugate of X(23398)
X(39367) = X(740)-Ceva conjugate of X(2)
X(39367) = {X(18827),X(35068)}-harmonic conjugate of X(2)

X(39368) = X(2)X(35080)∩X(148)X(514)

Barycentrics    a^6 - a^5*b - 2*a^4*b^2 - 3*a^3*b^3 + 2*a^2*b^4 + 3*a*b^5 - b^6 - a^5*c + 3*a^4*b*c + 5*a^3*b^2*c - 3*a^2*b^3*c - a*b^4*c - b^5*c - 2*a^4*c^2 + 5*a^3*b*c^2 + a^2*b^2*c^2 - 3*a*b^3*c^2 - 3*a^3*c^3 - 3*a^2*b*c^3 - 3*a*b^2*c^3 + 5*b^3*c^3 + 2*a^2*c^4 - a*b*c^4 + 3*a*c^5 - b*c^5 - c^6 : :
X(39368) = 3 X[2] - 4 X[35080], 5 X[2] - 4 X[35085], 5 X[35080] - 3 X[35085], 2 X[35080] - 3 X[35153], 6 X[35085] - 5 X[35148], 2 X[35085] - 5 X[35153], X[35148] - 3 X[35153]

X(39368) lies on the ellipse E(X(4440)) and these lines: {2, 35080}, {148, 514}, {519, 8591}, {523, 4440}, {524, 17487}, {1654, 35960}, {4037, 6542}, {4560, 6630}, {6631, 21711}, {7779, 39354}, {17731, 20016}, {20294, 39352}, {21221, 35511}, {21225, 25054}

X(39368) = reflection of X(i) in X(j) for these {i,j}: {2, 35153}, {20016, 17731}, {20536, 6542}, {35148, 35080}
X(39368) = anticomplement of X(35148)
X(39368) = anticomplement of the isogonal conjugate of X(5029)
X(39368) = anticomplement of the isotomic conjugate of X(2786)
X(39368) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 4010}, {31, 2786}, {423, 21300}, {512, 20349}, {604, 4458}, {661, 20558}, {667, 20016}, {692, 3570}, {798, 20536}, {875, 6650}, {1326, 7192}, {1757, 20295}, {1911, 4444}, {1931, 512}, {2786, 6327}, {5029, 8}, {6542, 21301}, {9508, 69}, {17731, 17217}, {17735, 513}, {17990, 2895}, {18004, 21287}, {18266, 514}, {20947, 21304}, {27929, 20554}, {38348, 20345}
X(39368) = anticomplementary conjugate of anticomplement of X(5029)
X(39368) = X(2786)-Ceva conjugate of X(2)
X(39368) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {35080, 35148, 2}, {35148, 35153, 35080}

X(39369) = PERSPECTOR OF THE 19TH AND 31ST VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    (11*a^2*b^2*c^2 + 48*R*a*b^2*c^2 + 48*R^2*b^2*c^2 + 30*R*a^2*b*c^2 + 136*R^2*a*b*c^2 + 136*R^3*b*c^2 + 20*R^2*a^2*c^2 + 96*R^3*a*c^2 + 96*R^4*c^2 + 30*R*a^2*b^2*c + 136*R^2*a*b^2*c + 136*R^3*b^2*c + 72*R^2*a^2*b*c + 352*R^3*a*b*c + 352*R^4*b*c + 40*R^3*a^2*c + 224*R^4*a*c + 224*R^5*c + 20*R^2*a^2*b^2 + 96*R^3*a*b^2 + 96*R^4*b^2 + 40*R^3*a^2*b + 224*R^4*a*b + 224*R^5*b + 16*R^4*a^2 + 128*R^5*a + 128*R^6) : :

See X(39369). (Dasari Naga Vijay Krishna)

X(39369) lies on these lines: {2,586}, {39311,39317}, {39315,39329}, {39316,39318}, {39319,39320}, {39326,39333}, {39611,39623}

X(39370) = PERSPECTOR OF THE 22ND AND 31ST VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    (289*a^4*b^4*c^4 + 2788*R*a^3*b^4*c^4 + 8460*R^2*a^2*b^4*c^4 + 10464*R^3*a*b^4*c^4 + 4608*R^4*b^4*c^4 + 1326*R*a^4*b^3*c^4 + 14180*R^2*a^3*b^3*c^4 + 44112*R^3*a^2*b^3*c^4 + 54608*R^4*a*b^3*c^4 + 23808*R^5*b^3*c^4 + 2112*R^2*a^4*b^2*c^4 + 26616*R^3*a^3*b^2*c^4 + 85632*R^4*a^2*b^2*c^4 + 106240*R^5*a*b^2*c^4 + 45824*R^6*b^2*c^4 + 1288*R^3*a^4*b*c^4 + 21760*R^4*a^3*b*c^4 + 73280*R^5*a^2*b*c^4 + 91264*R^6*a*b*c^4 + 38912*R^7*b*c^4 + 192*R^4*a^4*c^4 + 6496*R^5*a^3*c^4 + 23296*R^6*a^2*c^4 + 29184*R^7*a*c^4 + 12288*R^8*c^4 + 1326*R*a^4*b^4*c^3 + 14180*R^2*a^3*b^4*c^3 + 44112*R^3*a^2*b^4*c^3 + 54608*R^4*a*b^4*c^3 + 23808*R^5*b^4*c^3 + 5228*R^2*a^4*b^3*c^3 + 67888*R^3*a^3*b^3*c^3 + 219792*R^4*a^2*b^3*c^3 + 273088*R^5*a*b^3*c^3 + 117760*R^6*b^3*c^3 + 6216*R^3*a^4*b^2*c^3 + 118784*R^4*a^3*b^2*c^3 + 406880*R^5*a^2*b^2*c^3 + 508672*R^6*a*b^2*c^3 + 216832*R^7*b^2*c^3 + 1248*R^4*a^4*b*c^3 + 89248*R^5*a^3*b*c^3 + 331392*R^6*a^2*b*c^3 + 418176*R^7*a*b*c^3 + 176128*R^8*b*c^3 - 1184*R^5*a^4*c^3 + 23936*R^6*a^3*c^3 + 100096*R^7*a^2*c^3 + 128000*R^8*a*c^3 + 53248*R^9*c^3 + 2112*R^2*a^4*b^4*c^2 + 26616*R^3*a^3*b^4*c^2 + 85632*R^4*a^2*b^4*c^2 + 106240*R^5*a*b^4*c^2 + 45824*R^6*b^4*c^2 + 6216*R^3*a^4*b^3*c^2 + 118784*R^4*a^3*b^3*c^2 + 406880*R^5*a^2*b^3*c^2 + 508672*R^6*a*b^3*c^2 + 216832*R^7*b^3*c^2 + 1584*R^4*a^4*b^2*c^2 + 190784*R^5*a^3*b^2*c^2 + 717120*R^6*a^2*b^2*c^2 + 907008*R^7*a*b^2*c^2 + 381952*R^8*b^2*c^2 - 8832*R^5*a^4*b*c^2 + 128192*R^6*a^3*b*c^2 + 555264*R^7*a^2*b*c^2 + 713984*R^8*a*b*c^2 + 296960*R^9*b*c^2 - 6528*R^6*a^4*c^2 + 29184*R^7*a^3*c^2 + 159232*R^8*a^2*c^2 + 209408*R^9*a*c^2 + 86016*R^10*c^2 + 1288*R^3*a^4*b^4*c + 21760*R^4*a^3*b^4*c + 73280*R^5*a^2*b^4*c + 91264*R^6*a*b^4*c + 38912*R^7*b^4*c + 1248*R^4*a^4*b^3*c + 89248*R^5*a^3*b^3*c + 331392*R^6*a^2*b^3*c + 418176*R^7*a*b^3*c + 176128*R^8*b^3*c - 8832*R^5*a^4*b^2*c + 128192*R^6*a^3*b^2*c + 555264*R^7*a^2*b^2*c + 713984*R^8*a*b^2*c + 296960*R^9*b^2*c - 17856*R^6*a^4*b*c + 72704*R^7*a^3*b*c + 408320*R^8*a^2*b*c + 538624*R^9*a*b*c + 221184*R^10*b*c - 9216*R^7*a^4*c + 11776*R^8*a^3*c + 111104*R^9*a^2*c + 151552*R^10*a*c + 61440*R^11*c + 192*R^4*a^4*b^4 + 6496*R^5*a^3*b^4 + 23296*R^6*a^2*b^4 + 29184*R^7*a*b^4 + 12288*R^8*b^4 - 1184*R^5*a^4*b^3 + 23936*R^6*a^3*b^3 + 100096*R^7*a^2*b^3 + 128000*R^8*a*b^3 + 53248*R^9*b^3 - 6528*R^6*a^4*b^2 + 29184*R^7*a^3*b^2 + 159232*R^8*a^2*b^2 + 209408*R^9*a*b^2 + 86016*R^10*b^2 - 9216*R^7*a^4*b + 11776*R^8*a^3*b + 111104*R^9*a^2*b + 151552*R^10*a*b + 61440*R^11*b - 4096*R^8*a^4 + 28672*R^10*a^2 + 40960*R^11*a + 16384*R^12) : :

See X(39370). (Dasari Naga Vijay Krishna)

X(39370) lies on these lines: {}

leftri

Points associated with the Cip transform: X(39371)-X(39372)

rightri

This preamble is contributed by Peter Moses, August 3, 2020.

The Cip transform is introduced in Suren, Moses, and Kimberling, "The Circumcevian-Inversion Perspector of Two Triangles", to appear in Journal for Geometry and Graphics, as follows: In the plane of a triangle ABC, let P be a point that is not on one of the sidelines, BC, CA, AB. Let Γ be the circumcircle of ABC, and let DEF be the circumcevian triangle of P; that is, D is the point, other than A, in which the line AP meets Γ, and likewise for the vertices E and F. Let X,Y,Z be the reflections of P in D,E,F, respectively, and let X',Y',Z' be the Γ-inverse of X,Y,Z, respectively. Let P' = BY'∩CZ'. Then ABC and X'Y'Z' are perspective triangles, and P' is the circumcevian-inversion perspector of X'Y'Z' and ABC. The mapping P → P' is the Cip transform.

If Λ is a conic that passes through the circumcenter, O, then it is the Cip transform of a circular cubic. If you have Geogebra, you can view seven examples, named according to the choice of Λ:

CIP_ABCIO.
CIP_BrocardCircle.
CIP_FeuerbachOfTangential .
CIP_Jerabek.
CIP_KiepertOfMedial.
CIP_Lester.
CIP_Thomson-Gibert-Moses hyperbola .

Names for the seven circular cubics are introduced here as ABCIO circular cubic, Brocard circular cubic, Feuerbach-of-tangential circular cubic, Jerabek circular cubic, Kiepert-of-medial circular cubic, Lester circular cubic, and Thomson-Gibert-Moses-hyperbola circular cubic. For each of these, O is the Cip-image of O and every point on the infinity line. Aside from O, two of the seven circular cubics have Cip-fixed points: X(102) on the ABCIO circular cubic, and X(110) on the Thomson-Gibert-Moses-hyperbola circular cubic.

Another point on the ABCIO circular cubic is X(38599), and two on the Jerabek circular cubic are X(35372) and X(35465);

If Λ is a conic that does not pass through O, then the curve having Λ as Cip image has formal degree 4. An example is the Steiner circumquartic, see CIP_SteinerCircum, which passes through X(12188).

X(39371) = X(23)X(15786)∩X(30)X(15454)

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

X(39371) lies on these lines: {23, 15786}, {30, 15454}, {74, 323}, {186, 38936}, {526, 15470}, {2986, 12028}, {3284, 14910}, {6148, 31378}

X(39371) = X(i)-Ceva conjugate of X(j) for these (i,j): {10420, 15470}, {38936, 1511}
X(39371) = (i)-cross conjugate of X(j) for these (i,j): {1511, 15454}, {3284, 323}, {14270, 2420}
X(39371) = X(i)-isoconjugate of X(j) for these (i,j): {1725, 5627}, {2166, 14264}, {36119, 39170}
X(39371) = barycentric product X(i)*X(j) for these {i,j}: {323, 15454}, {1511, 2986}, {2407, 15470}, {3258, 18879}, {5504, 14920}, {5664, 10420}, {6148, 14910}, {11064, 38936}
X(39371) = barycentric quotient X(i)/X(j) for these {i,j}: {50, 14264}, {1511, 3580}, {3284, 39170}, {10420, 39290}, {14910, 5627}, {15454, 94}, {15470, 2394}, {38936, 16080}, {39176, 403}

X(39372) = X(74)X(38936)∩X(265)X(15454)

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

X(39372) lies on the Jerabek circumhyperbola these lines: {74, 38936}, {265, 15454}, {14380, 15470}

X(39372) = X(1725)-isoconjugate of X(10733)
X(39372) = barycentric quotient X(14910)/X(10733)

X(39373) = X(74)X(35465)∩X(265)X(13556)

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

X(39373) lies on the Jerabek circular cubic and these lines: {74, 35465}, {265, 13556}, {1299, 35189}, {5961, 13754}

X(39374) = X(187)X(5961)∩X(265)X(290)

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

X(39374) lies on the cubic K724 and these lines: {187, 5961}, {265, 690}, {6344, 35139}, {9155, 34157}

X(39374) = X(i)-isoconjugate of X(j) for these (i,j): {1733, 14355}, {6149, 14265}
X(39374) = barycentric product X(i)*X(j) for these {i,j}: {94, 34157}, {2987, 14356}
X(39374) = barycentric quotient X (i)/X(j) for these {i,j}: {1989, 14265}, {32654, 14355}, {34157, 323}, {34370, 34174}

X(39375) = X(74)X(94)∩X(186)X(476)

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

X(39375) lies on the cubic K724 and these lines: {74, 94}, {186, 476}, {265, 526}, {1511, 14254}, {5504, 34210}, {34334, 39176}

X(39375) = X(30)-cross conjugate of X(15454)
X(39375) = X(i)-isoconjugate of X(j) for these (i,j): {1725, 14385}, {1986, 35200}, {2159, 34834}, {6149, 14264}
X(39375) = cevapoint of X(30) and X(14254)
X(39375) = barycentric product X(i)*X(j) for these {i,j}: {94, 15454}, {2986, 14254}
X(39375) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 34834}, {1989, 14264}, {1990, 1986}, {12028, 14919}, {14254, 3580}, {14583, 3003}, {14910, 14385}, {15454, 323}

X(39376) = X(74)X(2132)∩X(265)X(6334)

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

X(39376) lies on the cubic K724 and these lines: {74, 2132}, {265, 6334}, {5627, 6344}, {5961, 11589}, {7687, 14264}, {10152, 34298}, {11060, 11079}

X(39376) = X(1294)-isoconjugate of X(35201)
X(39376) = barycentric product X(94)*X(39174)
X(39376) = barycentric quotient X(i)/X(j) for these {i,j}: {6000, 14920}, {11079, 1294}, {39174, 323}

X(39377) = ISOGONAL CONJUGATE OF X(6110)

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

X(39377) lies on the cubics K114 and K261a and these lines: {13, 470}, {14, 34298}, {15, 74}, {62, 38933}, {532, 1494}, {3284, 11079}, {5627, 11601}, {6699, 10217}, {8452, 36211}, {14919, 38414}

X(39377) = isogonal conjugate of X(6110)
X(39377) = isogonal conjugate of the polar conjugate of X(36308)
X(39377) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6110}, {14, 35201}, {15, 1784}, {470, 2173}, {2154, 14920}, {6137, 24001}, {8739, 14206}
X(39377) = trilinear pole of line {14380, 36296}
X(39377) = barycentric product X(i)*X(j) for these {i,j}: {3, 36308}, {13, 14919}, {299, 11079}, {300, 18877}, {1494, 36296}, {2394, 38414}, {5995, 34767}, {14380, 23895}
X(39377) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6110}, {16, 14920}, {74, 470}, {2152, 35201}, {2153, 1784}, {3457, 1990}, {5995, 4240}, {11079, 14}, {11081, 6111}, {14380, 23870}, {14919, 298}, {18877, 15}, {34395, 39176}, {36296, 30}, {36299, 34334}, {36308, 264}, {38414, 2407}
X(39377) = {X(3284),X(34329)}-harmonic conjugate of X(39378)

X(39378) = ISOGONAL CONJUGATE OF X(6111)

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

X(39378) lies on the cubics K114, K261b, and these lines: {13, 34298}, {14, 471}, {16, 74}, {61, 38933}, {533, 1494}, {3284, 11079}, {5627, 11600}, {6699, 10218}, {8462, 36210}, {14919, 38413}

X(39378) = isogonal conjugate of X(6111)
X(39378) = isogonal conjugate of the polar conjugate of X(36311)
X(39378) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6111}, {13, 35201}, {16, 1784}, {471, 2173}, {2153, 14920}, {6138, 24001}, {8740, 14206}
X(39378) = trilinear pole of line {14380, 36297}
X(39378) = barycentric product X(i)*X(j) for these {i,j}: {3, 36311}, {14, 14919}, {298, 11079}, {301, 18877}, {1494, 36297}, {2394, 38413}, {5994, 34767}, {14380, 23896}
X(39378) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6111}, {15, 14920}, {74, 471}, {2151, 35201}, {2154, 1784}, {3458, 1990}, {5994, 4240}, {11079, 13}, {11086, 6110}, {14380, 23871}, {14919, 299}, {18877, 16}, {34394, 39176}, {36297, 30}, {36298, 34334}, {36311, 264}, {38413, 2407}
X(39378) = {X(3284),X(34329)}-harmonic conjugate of X(39377)

X(39379) = ISOGONAL CONJUGATE OF X(34104)

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

X(39379) lies on the Jerabek circumhyperbola and these lines: {3, 10419}, {6, 38936}, {265, 2986}, {1300, 11744}, {4846, 15454}

X(39379) = isogonal conjugate of X(34104)
X(39379) = isogonal conjugate of the anticomplement of X(39234)
X(39379) = X(5504)-cross conjugate of X(10419)
X(39379) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34104}, {113, 1725}, {1784, 34333}
X(39379) = trilinear pole of line {647, 15470}
X(39379) = barycentric product X(2986)*X(10419)
X(39379) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 34104}, {10419, 3580}, {14910, 113}, {18877, 34333}

X(39380) = X(3)(39377)∩X(4)X(11080)

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

X(39380) lies on the Jerabek circumhyperbola and these lines: {3, 39377}, {4, 11080}, {74, 11081}, {265, 39378}, {2993, 36308}, {3458, 11074}, {4846, 10217}, {8749, 11083}, {11079, 36297}, {15328, 23283}, {18877, 36296}

X(39380) = X(36296)-cross conjugate of X(11079)
X(39380) = X(i)-isoconjugate of X(j) for these (i,j): {1784, 11131}, {11092, 35201}
X(39380) = barycentric product X(i)*X(j) for these {i,j}: {13, 39377}, {74, 10217}, {11078, 11079}, {11080, 14919}, {14380, 36839}, {36211, 39378}, {36296, 36308}
X(39380) = barycentric quotient X(i)/X(j) for these {i,j}: {3457, 6110}, {10217, 3260}, {11079, 11092}, {11081, 14920}, {14919, 11129}, {18877, 11131}, {39377, 298}

X(39381) = X(3)(39378)∩X(4)X(11085)

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

X(39381) lies on the Jerabek circumhyperbola and these lines: {3, 39378}, {4, 11085}, {74, 11086}, {265, 39377}, {2992, 36311}, {3457, 11074}, {4846, 10218}, {8749, 11088}, {11079, 36296}, {15328, 23284}, {18877, 36297}

X(39381) = X(36297)-cross conjugate of X(11079)
X(39381) = X(i)-isoconjugate of X(j) for these (i,j): {1784, 11130}, {11078, 35201}
X(39381) = barycentric product X(i)*X(j) for these {i,j}: {14, 39378}, {74, 10218}, {11079, 11092}, {11085, 14919}, {14380, 36840}, {36210, 39377}, {36297, 36311}
X(39381) = barycentric quotient X(i)/X(j) for these {i,j}: {3458, 6111}, {10218, 3260}, {11079, 11078}, {11086, 14920}, {14919, 11128}, {18877, 11130}, {39378, 299}

leftri

Circumcenters of circumcevian polar triangles: X(39382)-X(39384) and X(39628)-X(39640)

rightri

This preamble is based on notes from Suren and Peter Moses (August-September, 2020).

Let A'B'C' be the circumcevian triangle of a point P in the plane of a triangle ABC. Let La be the polar of A' wrt the circumcircle of BPC, and define Lb and Lc cyclically. Let A'' = Lb∩Lc, and define B'' and C'' cyclically. The triangle A''B''C'' is here named the circumcevian polar triangle of P.

If P = p : q : r (barycentrics), then the A-vertex of A"B"C" is given by u : v : w, where

u = 2*b^2*c^4*p^4*q^2 + c^4*(a^2 + 2*b^2 - 2*c^2)*p^3*q^3 + a^2*c^4*p^2*q^4 - b^2*c^2*(a^2 - 2*b^2 - 2*c^2)*p^4*q*r + b^2*c^2*(a^2 + 2*b^2 - 2*c^2)*p^3*q^2*r + a^2*c^2*(a^2 + 2*b^2 - 4*c^2)*p^2*q^3*r + a^4*c^2*p*q^4*r + 2*b^4*c^2*p^4*r^2 + b^2*c^2*(a^2 - 2*b^2 + 2*c^2)*p^3*q*r^2 - 6*a^2*b^2*c^2*p^2*q^2*r^2 - 3*a^4*c^2*p*q^3*r^2 + b^4*(a^2 - 2*b^2 + 2*c^2)*p^3*r^3 + a^2*b^2*(a^2 - 4*b^2 + 2*c^2)*p^2*q*r^3 - 3*a^4*b^2*p*q^2*r^3 - a^6*q^3*r^3 + a^2*b^4*p^2*r^4 + a^4*b^2*p*q*r^4,

v = b^2*(c^2*p^2*q + b^2*p^2*r + 2*a^2*p*q*r + a^2*q^2*r + a^2*q*r^2)*(-(c^2*p^2*q) - c^2*p*q^2 + b^2*p*r^2 + a^2*q*r^2),

w = c^2*(c^2*p*q^2 - b^2*p^2*r + a^2*q^2*r - b^2*p*r^2)*(c^2*p^2*q + b^2*p^2*r + 2*a^2*p*q*r + a^2*q^2*r + a^2*q*r^2).

The circumcenter of A''B''C'', denoted by MSV(P), is given by

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

Theorems:
MSV(P) = circumcircle-antipode of the Vu circumcircle point of P (defined in the preamble just before X(38451).
If P lies on the cubic K269, then MSV(P) = X(109).
If P lies on the Neuberg, then MSV(P) = X(110).
If P lies on the cubic K270, then MSV(P) = X(112).
If P lies on the cubic K1156, then MSV(P) = X(1296).

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

(2,1296), (3,110), (4,110), (5,1291), (6,1296), (7,20219), (8,39628), (9,28291), (13,110), (14,110), (15,110), (16,110), (17,36514), (18,36515), (19,36516), (20,112), (30,110), (32,39629), (35,39630), (36,109), (39,36517), (40,109), (43,39631), (46,36082), (54,1291), (55,20219), (57,28291), (61,36514), (62,36515), (63,36516), (64,112), (74,110), (80,109), (83,36517), (84,109), (90,36082), (98,99), (99,98), (100,104), (101,103), (102,109), (103,101), (104,100), (105,1292), (106,1293), (107,1294), (108,1295), (109,102), (110,74), (111,1296), (112,1297), (147,112), (155,13398), (159,39382), (164,3659), (165,14074), (182,39632), (186,12092), (187,8600), (191,39633), (194,30254), (195,33639), (238,39634), (254,13398), (265,12092), (371,39383), (372,39384), (529,39635), (532,39636), (533,39637), (535,39638), (538,39639), (544,39640)

X(39382) = CIRCUMCENTER OF CIRCUMCEVIAN POLAR TRIANGLE OF X(159)

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

X(39382) lies on the circumcircle and these lines: {4, 2373}, {24, 111}, {74, 6403}, {98, 18533}, {376, 34168}, {378, 1297}, {403, 2770}, {925, 4235}, {1294, 35485}, {1302, 2409}, {1352, 2366}, {2374, 3542}, {2697, 10295}, {2706, 11674}, {4227, 26703}, {4244, 9058}, {6240, 9076}, {6325, 35480}, {6353, 9084}, {7473, 16167}, {7482, 10420}, {9060, 37937}, {9061, 30733}, {10102, 37777}, {10423, 32713}, {11634, 13398}

X(39382) = polar-circle-inverse of X(14672)
X(39382) = X(30209)-cross conjugate of X(4)
X(39382) = cevapoint of X(i) and X(j) for these (i,j): {3, 30213}, {647, 8541}
X(39382) = trilinear pole of line {6, 14580}
X(39382) = X(i)-isoconjugate of X(j) for these (i,j): {656, 7493}, {14208, 19153}
X(39382) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 7493}, {35325, 16789}

X(39383) = CIRCUMCENTER OF CIRCUMCEVIAN POLAR TRIANGLE OF X(371)

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

X(39383) lies on the circumcircle and these lines: {3, 13030}, {6, 32422}, {74, 6200}, {98, 485}, {111, 8577}, {372, 3563}, {1297, 11825}, {1299, 5412}, {1300, 6560}, {1306, 4558}, {1504, 32437}, {1576, 1625}, {2367, 34391}, {2373, 11090}, {2374, 8944}, {2460, 23700}, {8982, 13440}, {9732, 10665}

X(39383) = Collings transform of X(5062)
X(39383) = X(i)-cross conjugate of X(j) for these (i,j): {3155, 250}, {10533, 23964}, {12968, 249}
X(39383) = cevapoint of X(i) and X(j) for these (i,j): {512, 5062}, {647, 21640}
X(39383) = crosssum of X(523) and X(17431)
X(39383) = trilinear pole of line {6, 3156}
X(39383) = X(i)-isoconjugate of X(j) for these (i,j): {371, 1577}, {492, 661}, {656, 1585}, {5408, 24006}, {5413, 14208}
X(39383) = barycentric product X(i)*X(j) for these {i,j}: {99, 8577}, {110, 485}, {112, 11090}, {372, 925}, {491, 32734}, {648, 6413}, {1414, 13455}, {1576, 34391}, {1625, 16032}, {3565, 8944}, {26920, 30450}
X(39383) = barycentric quotient X (i)/X(j) for these {i,j}: {110, 492}, {112, 1585}, {372, 6563}, {485, 850}, {925, 34392}, {1576, 371}, {6413, 525}, {6423, 14325}, {8577, 523}, {11090, 3267}, {13455, 4086}, {32661, 5408}, {32692, 16037}, {32734, 486}
X(39383) = {X(1576),X(1625)}-harmonic conjugate of X(39384)

X(39384) = CIRCUMCENTER OF CIRCUMCEVIAN POLAR TRIANGLE OF X(372)

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

X(39384) lies on the circumcircle and these lines: {3, 13032}, {6, 32420}, {74, 6396}, {98, 486}, {111, 8576}, {371, 3563}, {1297, 11824}, {1299, 5413}, {1300, 6561}, {1307, 4558}, {1505, 32434}, {1576, 1625}, {2367, 34392}, {2373, 11091}, {2374, 8940}, {2459, 23700}, {9733, 10666}, {13429, 26441}

X(39384) = Collings transform of X(5058)
X(39384) = X(i)-cross conjugate of X(j) for these (i,j): {3156, 250}, {10534, 23964}, {12963, 249}
X(39384) = cevapoint of X(i) and X(j) for these (i,j): {512, 5058}, {647, 21641}
X(39384) = crosssum of X(523) and X(17432)
X(39384) = trilinear pole of line {6, 3155}
X(39384) = X(i)-isoconjugate of X(j) for these (i,j): {372, 1577}, {491, 661}, {656, 1586}, {4017, 13461}, {5409, 24006}, {5412, 14208}
X(39384) = barycentric product X(i)*X(j) for these {i,j}: {99, 8576}, {107, 26922}, {110, 486}, {112, 11091}, {371, 925}, {492, 32734}, {648, 6414}, {1576, 34392}, {1625, 16037}, {3565, 8940}, {8911, 30450}
X(39384) = barycentric quotient X (i)/X(j) for these {i,j}: {110, 491}, {112, 1586}, {371, 6563}, {486, 850}, {925, 34391}, {1576, 372}, {5546, 13461}, {6414, 525}, {6424, 14326}, {8576, 523}, {11091, 3267}, {26922, 3265}, {32661, 5409}, {32692, 16032}, {32734, 485}
X(39384) = {X(1576),X(1625)}-harmonic conjugate of X(39383)

X(39385) = X(3)X(6)∩X(940)X(21909)

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

X(39285) lies on these lines: {3, 6}, {940, 21909}, {1124, 37607}, {1335, 37573}, {1702, 10476}, {3299, 37608}, {3301, 37574}, {4383, 21992}, {5706, 36715}, {7969, 37529}, {11292, 37676}, {21991, 37633}, {35631, 35775}

X(39386) = ISOGONAL CONJUGATE OF X(8698)

Barycentrics    (b-c)*(6*a-b-c) : :

X(39386) lies on these lines: {30, 511}, {2487, 4940}, {2505, 7659}, {2527, 14321}, {2976, 4806}, {4057, 8660}, {4927, 20295}, {5557, 23838}, {6161, 24099}, {6615, 14315}, {21115, 23729}, {23836, 32635}, {34502, 39771}

X(39386) = isogonal conjugate of X(8698)
X(39386) = crossdifference of every pair of points on line {X(6), X(9327)}
X(39386) = crosssum of X(513) and X(31792)
X(39386) = barycentric product X(i)*X(j) for these {i, j}: {514, 3635}, {693, 16671}
X(39386) = trilinear product X(i)*X(j) for these {i, j}: {513, 3635}, {514, 16671}

X(39387) = EULER LINE INTERCEPT OF X(39)X(13758)

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

As a point on the Euler line, X(39387) has Shinagawa coefficients (E+F+2*S, -S)

Let A*B*C* be the inner- (or outer-) Vecten triangle of ABC. Let a* be the circle through A centered at A*, and define b* and c* cyclically. Let {A', A"} = b*∩c* and define {B', B"} and {C', C"} cyclically, with A′, B′, C′ farthest to the radical center of a*, b*, c*. Then AA', BB', CC' concur in X(4) and AA", BB", CC" concur in a point, Q, on the Euler line of ABC (Dao Thanh Oai, August 17, 2020).

Continuing, if A*B*C* is the outer-Vecten triangle of ABC then Q=X(39837), and if A*B*C* is the innerr-Vecten triangle of ABC then Q=X(39838). (César Lozada, August 17, 2020).

Let DEF be the outer Vecten triangle and let V be the outer Vecten circle. Let U be the circle having diameter BC, and let A' be the point of intersection, other than D, of U and V. Define B' and C' cyclically. The triangle A'B'C' is perspective to ABC, and the perspector is X(39387). For the "inner version", see X(39388). See also Triangle centers around Vecten circles. (Stanisław Majchrzak, June 24, 2021)

If you have GeoGebra, you can view X(39387) and X(39388). (Stanisław Majchrzak, June 26, 2021)

X(39387) lies on these lines: {2,3}, {39,13758}, {69,9540}, {99,13873}, {230,32497}, {371,492}, {485,490}, {486,32807}, {487,5590}, {488,3068}, {489,639}, {491,1078}, {524,31454}, {590,638}, {591,3592}, {637,1151}, {1270,32818}, {1588,15883}, {3069,6422}, {3103,5420}, {3618,13935}, {6118,6564}, {6179,35685}, {6289,26441}, {6453,32419}, {6459,26361}, {7796,32808}, {7870,9680}, {7881,35255}, {8960,32421}, {9541,12322}, {9600,32786}, {9606,32788}, {10312,26912}, {12222,13886}, {12257,26516}, {12297,26330}, {13637,35812}, {13701,35822}, {13847,31492}, {23312,32789}, {33340,33346}, {33350,33360}, {33352,33358}, {33366,33440}, {33368,33442}

X(39387) = reflection of X(32489) in X(32491)
X(39387) = anticomplement of X(32491)
X(39387) = complement of X(32489)
X(39387) = barycentric product X(i)*X(j) for these {i, j}: {99, 14325}, {488, 1585}, {492, 3068}
X(39387) = barycentric quotient X(i)/X(j) for these (i, j): (371, 493), (488, 11090), (492, 5490), (1585, 24244)
X(39387) = trilinear product X(i)*X(j) for these {i, j}: {662, 14325}, {1585, 19215}
X(39387) = trilinear quotient X(1585)/X(19218)
X(39387) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(488)}} and {{A, B, C, X(3), X(24246)}}
X(39387) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (371, 493), (488, 11090), (492, 5490), (1585, 24244)
X(39387) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 631, 39388), (2, 1599, 1585), (2, 11294, 5), (2, 32488, 1656), (3, 7807, 39388), (3, 11315, 2), (3, 36656, 20), (3, 37466, 6813), (4, 32989, 39388), (632, 32490, 2), (140, 11285, 39388), (3525, 7376, 2), (3530, 8366, 39388), (11294, 32964, 3)

X(39388) = EULER LINE INTERCEPT OF X(39)X(13638)

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

As a point on the Euler line, X(39388) has Shinagawa coefficients (E+F-2*S, S)

See X(39387).

Let DEF be the inner Vecten triangle and let V be the inner Vecten circle. Let U be the circle having diameter BC, and let A'' be the point of intersection, other than D, of U and V. Define B'' and C'' cyclically. The triangle A''B''C'' is perspective to ABC, and the perspector is X(39388). For the "outer version", see X(39387). See also Triangle centers around Vecten circles. (Stanisław Majchrzak, June 24, 2021)

X(39388) lies on these lines: {2,3}, {39,13638}, {69,13935}, {99,13926}, {230,32494}, {372,491}, {486,489}, {487,3069}, {488,5591}, {490,640}, {492,1078}, {597,31454}, {615,637}, {638,1152}, {639,32807}, {1271,32818}, {1587,15884}, {1991,3594}, {3068,6421}, {3102,5418}, {3618,9540}, {6119,6565}, {6179,35684}, {6290,8982}, {6454,32421}, {6460,26362}, {7796,32809}, {7881,35256}, {9606,32787}, {12221,13939}, {12256,26521}, {12296,26331}, {13757,35813}, {13821,35823}, {13846,31492}, {23311,32790}, {33341,33347}, {33351,33361}, {33353,33359}, {33367,33443}, {33369,33441}

X(39388) = reflection of X(32488) in X(32490)
X(39388) = anticomplement of X(32490)
X(39388) = complement of X(32488)
X(39388) = barycentric product X(i)*X(j) for these {i, j}: {99, 14326}, {487, 1586}, {491, 3069}
X(39388) = barycentric quotient X(i)/X(j) for these (i, j): (372, 494), (487, 11091), (491, 5491), (1586, 24243)
X(39388) = trilinear product X(i)*X(j) for these {i, j}: {662, 14326}, {1586, 19216}
X(39388) = trilinear quotient X(1586)/X(19217)
X(39388) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(487)}} and {{A, B, C, X(3), X(24245)}}
X(39388) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (372, 494), (487, 11091), (491, 5491), (1586, 24243)
X(39388) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 631, 39387), (2, 1600, 1586), (2, 11293, 5), (2, 32489, 1656), (3, 7807, 39387), (3, 11316, 2), (3, 36655, 20), (3, 37466, 6811), (4, 32989, 39387), (140, 11285, 39387), (632, 32491, 2), (3525, 7375, 2), (3530, 8366, 39387), (3526, 11313, 2)

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Circlecevian Perspectors: X(39389)-X(39401)

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This preamble and centers X(39389)-X(39401) are contributed by Vu Thanh Tung, Auguist 21, 2020.

The circlecevian triangle of a point is defined in the preamble just before X(37841), as follows:

Let P = p:q:r (barycentrics) be a point in the plane of a triangle ABC. Let A' be the point, other than P, in which the line AP meets the circle (PBC), and define B' and C' cyclically; the triangle A'B'C' is called the circlecevian triangle of P with respect to triangle ABC by Floor van Lamoen (Hyacinthos #10039); see the preambles just before X(34520) and X(34892).

Let A1B1C1 be the circlecevian triangle of P. Let A2 be the point, other than A, of intersection of the circles (ABC) and AB1C1), and define B2 and C2 cyclically. Let A3 = BB2∩CC2, and define B3 and C3 cyclically. Theb ABC is perspective to A3B3C3, and the perspector, here named the circlecevian perspector of P, is given by

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

Let P* be the isogonal conjugate of P; then V(P) = V(P*).

See Circlecevian Perspector. (Vu Thanh Tung)

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

(1,1),
(2,39389), (6,39389),
(3,54), (4,54),
(5,39390), (54,39390),
(7,39391), (55,39391),
(8,39392), (56,39392),
(9,39393), (57,39393),
(10,39394), (58,39394),
(31,39395), (75,39395),
(32,39396), (76,39396),
(39,39397), (83,39397),
(141,39398), (251,39398),
(560,39399), (561,39399),
(1501,39400), (1502,39401),
(1917,39401), (1928,39401)

X(39389) = CIRCLECEVIAN PERSPECTOR OF X(2)

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

X(39389) lies on these lines:

X(39389) = isogonal conjugate of X(597)
X(39389) = cevapoint of X(6) and X(574)
X(39389) = trilinear pole of line X(512)X(5104)

X(39390) = CIRCLECEVIAN PERSPECTOR OF X(5)

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

X(39390) lies on these lines:

X(39391) = CIRCLECEVIAN PERSPECTOR OF X(7)

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

X(39391) lies on these lines:

X(39392) = CIRCLECEVIAN PERSPECTOR OF X(8)

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

X(39392) lies on these lines:

X(39393) = CIRCLECEVIAN PERSPECTOR OF X(9)

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

X(39393) lies on these lines:

X(39394) = CIRCLECEVIAN PERSPECTOR OF X(10)

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

X(39394) lies on these lines:

X(39395) = CIRCLECEVIAN PERSPECTOR OF X(31)

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

X(39395) lies on these lines:

X(39396) = CIRCLECEVIAN PERSPECTOR OF X(32)

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

X(39396) lies on these lines:

X(39397) = CIRCLECEVIAN PERSPECTOR OF X(39)

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

X(39397) lies on these lines:

X(39398) = CIRCLECEVIAN PERSPECTOR OF X(141)

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

X(39398) lies on these lines:

X(39399) = CIRCLECEVIAN PERSPECTOR OF X(560)

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

X(39399) lies on these lines:

X(39400) = CIRCLECEVIAN PERSPECTOR OF X(1501)

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

X(39400) lies on these lines:

X(39401) = CIRCLECEVIAN PERSPECTOR OF X(1917)

Barycentrics    a^2*(a^5*b^12 + a^12*c^5 + 2*a^5*b^7*c^5 + b^12*c^5 + a^5*c^12)*(a^12*b^5 + a^5*b^12 + 2*a^5*b^5*c^7 + a^5*c^12 + b^5*c^12) : :

X(39401) lies on these lines:

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Vu Isodynamic Perspectors: X(39402)-X(39411)

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This preamble is contributed by Vu Thanh Tung, August 21, 2020.

Let P be a point in the plane of a triangle ABC. Let A1 be the 1st isodnamic point, X(15), of triangle PBC, and define B1 and C1 cyclically. Let A2 be the point of intersection, other than A, of the circles (ABC) and (AB1C1), and define B2 and C2 cyclically. Let A3 = BB2∩CC2, and define B3 and C3 cyclically. The triangle A3B3C3 is perpsective to ABC, and the perspector, V(P), is here named the 1st Vu isodynamic perspector. Note that A3B3C3 is the anticevian triangle of V(P).

If the above construction is carried out with the 2nd isodynamic point, X(16), instead of X(15), the resulting perspector, T(P), is the 2nd Vu isodynamic perspector.

Barycentrics were found by Peter Moses (August 22, 2020):

V(P) = a^2/(Sqrt[3]*a^2*((a^2 - b^2 - c^2)*q*r - b^2*r^2 - c^2*q^2) - 2*(a^2*q*r + b^2*r*p + c^2*p*q)*S) : :
T(P) = a^2/(Sqrt[3]*a^2*((a^2 - b^2 - c^2)*q*r - b^2*r^2 - c^2*q^2) + 2*(a^2*q*r + b^2*r*p + c^2*p*q)*S) : :

The appearance of (i,j) in the following list means that V(X(i)) = X(j): (1,39402), (2,39404), (3,2981), (4,39406), (5,39408), (6,39410)

The appearance of (i,j) in the following list means that T(X(i)) = X(j): (1,39403), (2,39405), (3,6151), (4,39407), (5,39409), (6,39411)

See Isodynamic Perspector. (Vu Thanh Tung)

X(39402) = 1ST VU ISODYNAMIC PERSPECTOR OF X(1)

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

X(39402) lies on these lines: X(39402) lies on these lines: {9,5367}, {16,55}, {1475,39403}, {39151,39152}

X(39402) = cevapoint of X(6) and X(19373)
X(39402) = X(9)-isoconjugate-of-X(3638)
X(39402) = X(56)-reciprocal conjugate of-X(3638)
X(39402) = X(2195)-vertex conjugate of-X(39403)
X(39402) = barycentric product X(7)*X(36737)
X(39402) = barycentric quotient X(56)/X(3638)
X(39402) = trilinear product X(57)*X(36737)
X(39402) = trilinear quotient X(57)/X(3638)
X(39402) = trilinear pole of the line {663, 7051}
X(39402) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(9)}} and {{A, B, C, X(16), X(58)}}

X(39403) = 2ND VU ISODYNAMIC PERSPECTOR OF X(1)

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

X(39403) lies on these lines: {9,5362}, {15,55}, {109,2307}, {1475,39402}, {39150,39153}

X(39403) = cevapoint of X(6) and X(7051)
X(39403) = X(9)-isoconjugate-of-X(3639)
X(39403) = X(56)-reciprocal conjugate of-X(3639)
X(39403) = X(2195)-vertex conjugate of-X(39402)
X(39403) = barycentric product X(7)*X(36738)
X(39403) = barycentric quotient X(56)/X(3639)
X(39403) = trilinear product X(57)*X(36738)
X(39403) = trilinear quotient X(57)/X(3639)
X(39403) = trilinear pole of the line {663, 19373}
X(39403) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(9)}} and {{A, B, C, X(15), X(58)}}

X(39404) = 1ST VU ISODYNAMIC PERSPECTOR OF X(2)

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

X(39404) lies on these lines: {6,25221}, {187,11146}, {11581,31710}, {13410,39405}

X(39404) = cevapoint of X(6) and X(37776)
X(39404) = trilinear pole of the line {351, 37775}
X(39404) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(15)}} and {{A, B, C, X(6), X(187)}}

X(39405) = 2ND VU ISODYNAMIC PERSPECTOR OF X(2)

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

X(39405) lies on these lines: {6,25222}, {187,11145}, {11582,31709}, {13410,39404}

X(39405) = cevapoint of X(6) and X(37775)
X(39405) = trilinear pole of the line {351, 37776}
X(39405) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(16)}} and {{A, B, C, X(6), X(187)}}

X(39406) = 1ST VU ISODYNAMIC PERSPECTOR OF X(4)

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

X(39406) lies on these lines: {16,20412}, {216,11145}, {6117,11601}

X(39406) = cevapoint of X(6) and X(10633)
X(39406) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(16)}} and {{A, B, C, X(6), X(51)}}

X(39407) = 2ND VU ISODYNAMIC PERSPECTOR OF X(4)

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

X(39407) lies on these lines: {15,20411}, {216,11146}, {6116,11600}

X(39407) = cevapoint of X(6) and X(10632)
X(39407) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(15)}} and {{A, B, C, X(6), X(51)}}

X(39408) = 1ST VU ISODYNAMIC PERSPECTOR OF X(5)

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

X(39408) lies on these lines: {}

X(39408) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(11538)}} and {{A, B, C, X(13), X(10633)}}

X(39409) = 2ND VU ISODYNAMIC PERSPECTOR OF X(5)

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

X(39409) lies on these lines: {}

X(39409) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(11538)}} and {{A, B, C, X(14), X(10632)}}

X(39410) = 1ST VU ISODYNAMIC PERSPECTOR OF X(6)

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

X(39410) lies on these lines: {2,5472}, {6,14170}, {187,2981}, {5008,39024}, {21448,37776}

X(39410) = isogonal conjugate of the complement of X(37786)
X(39410) = complement of the anticomplementary conjugate of X(37786)
X(39410) = X(6)-vertex conjugate of-X(2981)
X(39410) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(6)}} and {{A, B, C, X(74), X(14170)}}

X(39411) = 2ND VU ISODYNAMIC PERSPECTOR OF X(6)

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

X(39411) lies on these lines: {2,5471}, {6,14169}, {187,6151}, {5008,39024}, {21448,37775}

X(39411) = isogonal conjugate of the complement of X(37785)
X(39411) = complement of the anticomplementary conjugate of X(37785)
X(39411) = X(6)-vertex conjugate of-X(6151)
X(39411) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(6)}} and {{A, B, C, X(74), X(14169)}}

X(39412) = EIGENCENTER OF CIRCUMCEVIAN TRIANGLE OF X(21)

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

X(39412) lies on the circumcircle and these lines: {98, 3931}, {8707, 21859}

X(39413) = EIGENCENTER OF CIRCUMCEVIAN TRIANGLE OF X(23)

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

X(39413) lies on the circumcircle and these lines: {111, 3455}, {115, 2770}, {671, 2373}, {842, 10630}, {892, 35569}, {2374, 8791}, {6082, 17708}, {11636, 34574}, {11646, 36833}

X(39413) = trilinear pole of line X(6)X(10558)
X(39413) = Ψ(X(6), X(10558))
X(39413) = Ψ(X(23), X(111))

X(39414) = EIGENCENTER OF CIRCUMCEVIAN TRIANGLE OF X(44)

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

X(39414) lies on the circumcircle and these lines: {100, 4618}, {101, 4638}, {679, 2718}, {840, 1318}, {1015, 2226}, {4555, 6079}

X(39414) = isogonal conjugate of X(33922)
X(39414) = trilinear pole of line X(6)X(2226)
X(39414) = Λ(X(3251), X(4543)) (line X(3251)X(4543) is the trilinear polar of X(4370))
X(39414) = Λ(X(4542), X(33922)) (line X(4542)X(33922) is the trilinear polar of X(6544))
X(39414) = Ψ(X(6), X(2226))

X(39415) = EIGENCENTER OF CIRCUMCEVIAN TRIANGLE OF X(58)

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

X(39415) lies on the circumcircle and these lines: {110, 4115}, {190, 6578}, {4103, 8701}

X(39415) = trilinear pole of line X(6)X(8013)
X(39415) = Ψ(X(6), X(8013))

X(39416) = EIGENCENTER OF CIRCUMCEVIAN TRIANGLE OF X(155)

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

X(39416) lies on the circumcircle and these lines: {96, 23233}, {99, 30450}, {254, 3563}, {1297, 32132}, {1299, 2165}, {5962, 9721}, {36082, 36145}

X(39416) = trilinear pole of line X(6)X(14593)
X(39416) = Ψ(X(6), X(14593))

X(39417) = EIGENCENTER OF CIRCUMCEVIAN TRIANGLE OF X(159)

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

X(39417) lies on the circumcircle and these lines: {4, 34168}, {24, 1297}, {74, 34207}, {98, 3542}, {378, 5897}, {403, 2697}, {842, 37951}, {925, 2409}, {1286, 16237}, {1289, 32713}, {1294, 18533}, {1614, 15324}, {1974, 2366}, {2373, 6353}, {4230, 13398}, {4244, 13397}, {10420, 37937}, {16167, 31510}, {26703, 30733}

X(39417) = trilinear pole of line X(6)X(17409)
X(39417) = X(159)-isoconjugate of X(14208)
X(39417) = Ψ(X(6), X(17409))
X(39417) = Ψ(X(69), X(22))

X(39418) = EIGENCENTER OF CIRCUMCEVIAN TRIANGLE OF X(184)

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

X(39418) lies on the circumcircle and these lines: {98, 34449}, {933, 6528}, {1141, 18817}, {30450, 32692}

X(39418) = trilinear pole of line X(6)X(324)
X(39418) = Ψ(X(6), X(324))

X(39419) = EIGENCENTER OF CIRCUMCEVIAN TRIANGLE OF X(195)

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

X(39419) lies on the circumcircle and these lines: {1141, 34433}, {1298, 12291}, {2963, 33643}, {3459, 5966}, {10121, 15109}

X(39420) = EIGENCENTER OF CIRCUMCEVIAN TRIANGLE OF X(238)

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

X(39420) lies on the circumcircle and these lines: {292, 727}, {2109, 2382}, {2702, 34067}, {4562, 8709}

X(39420) = trilinear pole of line X(6)X(2109)
X(39420) = Ψ(X(6), X(2109))

X(39421) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(7)

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

X(39421) lies on the circumcircle and these lines: {55, 927}, {100, 10025}, {101, 9441}, {103, 15599}, {105, 8641}, {165, 813}, {919, 14827}, {934, 2223}, {5091, 14733}

X(39421) = Λ(X(7), X(2481)) (line X(7)X(2481) is the tangent to hyperbola {{A,B,C,X(6),X(7)}} at X(7))

X(39422) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(13)

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

X(39422) lies on the circumcircle and these lines: {15, 476}, {16, 9160}, {50, 5995}, {512, 39423}, {5618, 11081}, {5994, 32761}, {9202, 37477}

X(39422) = Λ(X(13), X(94)) (line X(13)X(94) is the tangent to hyperbola {{A,B,C,X(6),X(13)}} at X(13))

X(39423) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(14)

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

X(39423) lies on the circumcircle and these lines: {15, 9160}, {16, 476}, {50, 5994}, {512, 39422}, {5619, 11086}, {5995, 32761}, {9203, 37477}

X(39423) = Λ(X(14), X(94)) (line X(14)X(94) is the tangent to hyperbola {{A,B,C,X(6),X(14)}} at X(14))

X(39424) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(15)

Barycentrics    1/((Sqrt[3]*(a^2 - b^2 - c^2) - 2*S)*(Sqrt[3]*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6) + 2*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*S)) : :

X(39424) lies on the circumcircle and these lines: {13, 110}, {74, 5916}, {99, 300}, {112, 8737}, {115, 11082}, {476, 6105}, {530, 10410}, {691, 11586}, {805, 33958}, {930, 37848}, {933, 35714}, {1989, 5994}, {5618, 11142}, {5995, 11080}, {10409, 11119}, {11139, 16806}, {36515, 36967}

X(39424) = trilinear pole of line X(6)X(20578)
X(39424) = Λ(X(15), X(323)) (line X(15)X(323) is the tangent to hyperbola {{A,B,C,X(6),X(15)}} at X(15))
X(39424) = Ψ(X(6), X(20578))

X(39425) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(16)

Barycentrics    1/((Sqrt[3]*(a^2 - b^2 - c^2) + 2*S)*(Sqrt[3]*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6) - 2*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*S)) : :

X(39425) lies on the circumcircle and these lines: {14, 110}, {74, 5917}, {99, 301}, {112, 8738}, {115, 11087}, {476, 6104}, {531, 10409}, {691, 15743}, {805, 33957}, {930, 37850}, {933, 35715}, {1989, 5995}, {5619, 11141}, {5994, 11085}, {10410, 11120}, {11138, 16807}, {36514, 36968}

X(39425) = trilinear pole of line X(6)X(20579)
X(39425) = Λ(X(16), X(323)) (line X(16)X(323) is the tangent to hyperbola {{A,B,C,X(6),X(16)}} at X(16))
X(39425) = Ψ(X(6), X(20579))

X(39426) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(34)

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

X(39426) lies on the circumcircle and these lines: {78, 108}, {107, 1043}, {109, 1259}, {112, 2327}, {326, 934}, {1305, 6527}, {2720, 7111}, {9056, 10327}, {15394, 36079}

X(39426) = Λ(X(34), X(207)) (line X(34)X(207) is the tangent to hyperbola {{A,B,C,X(1),X(6)}} at X(34))

X(39427) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(39)

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

X(39427) lies on the circumcircle and these lines: {83, 110}, {99, 308}, {101, 18082}, {109, 18097}, {112, 32085}, {689, 1078}, {805, 14970}, {827, 12150}, {1799, 9066}, {9102, 33651}

X(39427) = Λ(X(39), X(3051)) (line X(39)X(3051) is the tangent to hyperbola {{A,B,C,X(6),X(39)}} at X(39))

X(39428) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(45)

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

X(39428) lies on the circumcircle and these lines: {6, 28911}, {89, 100}, {101, 2163}, {2177, 6014}

X(39428) = Λ(X(45), X(3679)) (line X(45)X(3679) is the tangent to hyperbola {{A,B,C,X(6),X(45)}} at X(45))

X(39429) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(48)

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

X(39429) lies on the circumcircle and these lines: {92, 109}, {100, 7017}, {101, 318}, {108, 2052}, {110, 31623}, {112, 1896}, {331, 934}, {2720, 16082}, {2864, 16089}

X(39429) = Λ(X(48), X(577)) (line X(48)X(577) is the tangent to hyperbola {{A,B,C,X(6),X(48)}} at X(48))

X(39430) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(50)

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

X(39430) lies on the circumcircle and these lines: {94, 110}, {99, 20573}, {112, 6344}, {265, 9160}, {930, 23217}, {933, 2970}, {1141, 14270}

X(39430) = trilinear pole of line X(6)X(10412)
X(39430) = Λ(X(50), X(18334)) (line X(50)X(18334) is the tangent to hyperbola {{A,B,C,X(6),X(50)}} at X(50))
X(39430) = Ψ(X(6), X(10412))

X(39431) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(54)

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

X(39431) lies on the circumcircle and these lines: {3, 30248}, {4, 6799}, {5, 933}, {20, 33639}, {30, 13863}, {107, 13621}, {110, 2888}, {112, 36412}, {930, 7488}, {1291, 3153}, {1304, 10096}, {3518, 20626}, {12225, 20185}

X(39431) = cevapoint of X(5) and X(2070)
X(39431) = Λ(X(54), X(186)) (line X(54)X(186) is the tangent to the Jerabek hyperbola at X(54))

X(39432) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(61)

Barycentrics    1/((a^2 - b^2 - c^2 - 2*Sqrt[3]*S)*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6 + 2*Sqrt[3]*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*S)) : :

X(39432) lies on the circumcircle and these lines: {17, 110}, {99, 34389}, {112, 8741}, {623, 10409}, {2963, 16807}, {5472, 16806}, {5994, 11087}, {5995, 11139}, {8172, 36514}

X(39432) = Λ(X(61), X(1994)) (line X(61)X(1994) is the tangent to hyperbola {{A,B,C,X(6),X(61)}} at X(61))

X(39433) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(62)

Barycentrics    1/((a^2 - b^2 - c^2 + 2*Sqrt[3]*S)*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6 - 2*Sqrt[3]*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*S)) : :

X(39433) lies on the circumcircle and these lines: {18, 110}, {99, 34390}, {112, 8742}, {624, 10410}, {2963, 16806}, {5471, 16807}, {5994, 11138}, {5995, 11082}, {8173, 36515}

X(39433) = Λ(X(62), X(1994)) (line X(62)X(1994) is the tangent to hyperbola {{A,B,C,X(6),X(62)}} at X(62))

X(39434) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(64)

Barycentrics    (a^14 + 3*a^12*b^2 - 15*a^10*b^4 + 11*a^8*b^6 + 11*a^6*b^8 - 15*a^4*b^10 + 3*a^2*b^12 + b^14 - 4*a^12*c^2 + 10*a^10*b^2*c^2 + 28*a^8*b^4*c^2 - 68*a^6*b^6*c^2 + 28*a^4*b^8*c^2 + 10*a^2*b^10*c^2 - 4*b^12*c^2 + 5*a^10*c^4 - 39*a^8*b^2*c^4 + 34*a^6*b^4*c^4 + 34*a^4*b^6*c^4 - 39*a^2*b^8*c^4 + 5*b^10*c^4 + 28*a^6*b^2*c^6 - 56*a^4*b^4*c^6 + 28*a^2*b^6*c^6 - 5*a^6*c^8 + 5*a^4*b^2*c^8 + 5*a^2*b^4*c^8 - 5*b^6*c^8 + 4*a^4*c^10 - 6*a^2*b^2*c^10 + 4*b^4*c^10 - a^2*c^12 - b^2*c^12)*(a^14 - 4*a^12*b^2 + 5*a^10*b^4 - 5*a^6*b^8 + 4*a^4*b^10 - a^2*b^12 + 3*a^12*c^2 + 10*a^10*b^2*c^2 - 39*a^8*b^4*c^2 + 28*a^6*b^6*c^2 + 5*a^4*b^8*c^2 - 6*a^2*b^10*c^2 - b^12*c^2 - 15*a^10*c^4 + 28*a^8*b^2*c^4 + 34*a^6*b^4*c^4 - 56*a^4*b^6*c^4 + 5*a^2*b^8*c^4 + 4*b^10*c^4 + 11*a^8*c^6 - 68*a^6*b^2*c^6 + 34*a^4*b^4*c^6 + 28*a^2*b^6*c^6 - 5*b^8*c^6 + 11*a^6*c^8 + 28*a^4*b^2*c^8 - 39*a^2*b^4*c^8 - 15*a^4*c^10 + 10*a^2*b^2*c^10 + 5*b^4*c^10 + 3*a^2*c^12 - 4*b^2*c^12 + c^14) : :

X(39434) lies on the circumcircle and these lines: {3, 30249}, {20, 1301}, {107, 11413}, {110, 6225}, {112, 36413}, {1289, 21312}, {1304, 16386}, {1370, 9064}, {2071, 22239}

X(39434) = cevapoint of X(20) and X(2071)
X(39434) = Λ(X(25), X(64)) (line X(25)X(64) is the tangent to the Jerabek hyperbola at X(64))

X(39435) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(65)

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

X(39435) lies on the circumcircle and these lines: {3, 30250}, {20, 6011}, {21, 108}, {22, 9070}, {99, 17134}, {100, 1792}, {101, 2327}, {107, 11101}, {109, 283}, {110, 3869}, {112, 7054}, {934, 1444}, {1289, 14015}, {1300, 14127}, {1301, 13739}, {1325, 2766}, {1789, 26700}, {1793, 2222}, {2689, 10538}, {2701, 3100}, {2733, 7253}, {3563, 7427}, {4221, 26706}, {4228, 9107}, {10101, 37960}, {15952, 32704}, {17515, 36067}, {17521, 32691}, {21312, 30257}, {26704, 37227}, {26705, 36011}, {36077, 37032}

X(39435) = anticomplement of polar-circle-inverse of X(759)
X(39435) = cevapoint of X(21) and X(1325)
X(39435) = Λ(X(65), X(225)) (line X(65)X(225) is the tangent to the Jerabek hyperbola at X(65))

X(39436) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(66)

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

X(39436) lies on the circumcircle and these lines: {3, 30251}, {20, 39382}, {22, 1289}, {101, 21079}, {107, 26283}, {110, 2892}, {112, 1370}, {858, 10423}, {935, 37929}, {1296, 30552}, {1301, 7493}, {1304, 16387}, {1995, 30249}, {10098, 16386}, {11413, 30247}, {22239, 37980}

X(39436) = cevapoint of X(22) and X(858)
X(39436) = Λ(X(66), X(1843)) (line X(66)X(1843) is the tangent to the Jerabek hyperbola at X(66))

X(39437) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(68)

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

X(39437) lies on the circumcircle and these lines: {4, 13398}, {24, 925}, {69, 39115}, {100, 31385}, {110, 3542}, {112, 36416}, {403, 10420}, {476, 37951}, {3565, 18533}, {6240, 20185}, {13397, 31384}, {16167, 37777}

X(39437) = cevapoint of X(24) and X(403)
X(39437) = Λ(X(68), X(5562)) (line X(68)X(5562) is the tangent to the Jerabek hyperbola at X(68))
X(39437) = polar-circle-inverse of complement of X(1299)

X(39438) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(71)

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

X(39438) lies on the circumcircle and these lines: {27, 101}, {100, 286}, {110, 17220}, {112, 36419}, {1305, 4184}, {2073, 2690}

X(39438) = cevapoint of X(27) and X(2073)
X(39438) = trilinear pole of line X(6)X(17925)
X(39438) = Λ(X(71), X(228)) (line X(71)X(228) is the tangent to the Jerabek hyperbola at X(71))
X(39438) = Ψ(X(6), X(17925))

X(39439) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(72)

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

X(39439) lies on the circumcircle and these lines: {4, 6011}, {21, 13397}, {24, 30250}, {25, 9070}, {28, 100}, {29, 33637}, {71, 29014}, {99, 14015}, {101, 1474}, {108, 30733}, {109, 1780}, {110, 3868}, {112, 5301}, {242, 2690}, {378, 30257}, {835, 17520}, {901, 17515}, {925, 11101}, {1290, 2074}, {1292, 4227}, {1294, 14127}, {1297, 7427}, {1305, 1441}, {2222, 37168}, {2691, 37961}, {3565, 17512}, {4233, 9058}, {4247, 6012}, {4653, 29305}

X(39439) = cevapoint of X(28) and X(2074)
X(39439) = Λ(X(72), X(306)) (line X(72)X(306) is the tangent to the Jerabek hyperbola at X(72))

X(39440) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(73)

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

X(39440) lies on the circumcircle and these lines: {29, 109}, {101, 2322}, {107, 23383}, {108, 1896}, {112, 36421}, {286, 934}, {2075, 2689}

X(39440) = cevapoint of X(29) and X(2075)
X(39440) = trilinear pole of line X(6)X(17926)
X(39440) = Λ(X(73), X(228)) (line X(73)X(228) is the tangent to the Jerabek hyperbola at X(73))
X(39440) = Ψ(X(6), X(17926))

X(39441) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(86)

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

X(39441 lies on the circumcircle and these lines: {42, 99}, {100, 872}, {101, 7109}, {110, 1918}, {689, 18082}, {1911, 36066}, {2375, 8632}, {34594, 38814}

X(39441) = Λ(X(86), X(310)) (line X(86)X(310) is the tangent to hyperbola {{A,B,C,X(1),X(6)}} at X(86))

X(39442) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(99)

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

X(39442) lies on the circumcircle and these lines: {99, 888}, {110, 887}, {512, 9150}, {729, 1084}

X(39442) = reflection of X(99) in line X(3)X(5106)
X(39442) = trilinear pole of line X(6)X(1645)
X(39442) = Λ(X(99), X(669)) (line X(99)X(669) is the tangent to hyperbola {{A,B,C,X(6),X(99)}} at X(99))
X(39442) = Ψ(X(6), X(1645))

X(39443) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(100)

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

X(39443) lies on the circumcircle and these lines: {36, 2382}, {100, 891}, {101, 3768}, {104, 29349}, {513, 898}, {517, 29348}, {739, 1015}

X(39443) = reflection of X(100) in line X(3)X(16686)
X(39443) = trilinear pole of line X(6)X(1646)
X(39443) = Λ(X(100), X(667)) (line X(100)X(667) is the tangent to hyperbola {{A,B,C,X(6),X(100)}} at X(100))
X(39443) = Ψ(X(6), X(1646))

X(39444) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(101)

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

X(39444) lies on the circumcircle and these lines: {100, 3762}, {101, 900}, {102, 24466}, {103, 952}, {105, 10058}, {106, 1086}, {109, 30725}, {190, 6551}, {514, 901}, {516, 953}, {840, 21578}, {1311, 17100}, {8701, 15343}

X(39444) = reflection of X(101) in line X(3)X(8)
X(39444) = trilinear pole of line X(6)X(1647)
X(39444) = Λ(X(101), X(649)) (line X(101)X(649) is the tangent to hyperbola {{A,B,C,X(6),X(101)}} at X(101))
X(39444) = Ψ(X(6), X(1647))

X(39445) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(106)

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

X(39445) lies on the circumcircle and these lines: {100, 4738}, {101, 4370}, {106, 900}, {109, 1317}, {519, 901}, {952, 1293}, {953, 3667}, {6014, 36924}, {6789, 15343}

X(39445) = reflection of X(106) in line X(3)X(8)
X(39445) = trilinear pole of line X(6)X(6544)
X(39445) = Λ(X(36), X(106)) (line X(36)X(106) is the tangent to hyperbola {{A,B,C,X(1),X(6)}} at X(106))
X(39445) = Ψ(X(6), X(6544))

X(39446) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(111)

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

X(39446) lies on the circumcircle and these lines: {99, 2930}, {110, 2482}, {111, 690}, {112, 5095}, {524, 691}, {542, 1296}, {842, 1499}, {3565, 32244}, {5108, 15342}, {5182, 11636}, {5912, 9136}, {5971, 9080}, {5987, 9066}, {5994, 30455}, {5995, 30454}, {6792, 15357}, {14444, 32740}

X(39446) = reflection of X(111) in line X(3)X(67)
X(39446) = trilinear pole of line X(6)X(1649)
X(39446) = Λ(X(23), X(111)) (line X(23)X(111) is the tangent to hyperbola {{A,B,C,X(2),X(6)}} at X(111))
X(39446) = Ψ(X(6), X(1649))

X(39447) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(112)

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

X(39447) lies on the circumcircle and these lines: {74, 3184}, {98, 23239}, {112, 9033}, {476, 18557}, {525, 1304}, {842, 1514}, {1294, 1632}, {1297, 2777}, {1503, 2693}, {5897, 16163}

X(39447) = reflection of X(112) in line X(3)X(113)
X(39447) = trilinear pole of line X(6)X(1650)
X(39447) = Λ(X(112), X(647)) (line X(112)X(647) is the tangent to hyperbola {{A,B,C,X(6),X(112)}} at X(112))
X(39447) = Ψ(X(6), X(1650))

X(39448) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(115)

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

X(39448) lies on the circumcircle and these lines: {98, 32423}, {110, 6140}, {249, 476}, {511, 14979}, {512, 1291}, {842, 37496}, {1141, 12383}, {3563, 15463}, {5966, 32609}, {10420, 33803}

X(39448) = Λ(X(115), X(12077)) (line X(115)X(12077) is the tangent to hyperbola {{A,B,C,X(6),X(115)}} at X(115))

X(39449) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(141)

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

X(39449) lies on the circumcircle and these lines: {2, 6573}, {99, 251}, {827, 1627}, {3222, 10130}, {11636, 38834}

X(39449) = trilinear pole of line X(6)X(8711)
X(39449) = Λ(X(141), X(6665)) (line X(141)X(6665) is the tangent to hyperbola {{A,B,C,X(6),X(39)}} at X(141))
X(39449) = Ψ(X(6), X(8711))

X(39450) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(187)

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

X(39450) lies on the circumcircle and these lines: {99, 18023}, {110, 671}, {112, 17983}, {691, 39231}, {2715, 9154}, {5994, 36310}, {5995, 36307}, {11636, 18818}

X(39450) = trilinear pole of line X(6)X(5466)
X(39450) = Λ(X(187), X(3292)) (line X(187)X(3292) is the tangent to hyperbola {{A,B,C,X(6),X(187)}} at X(187))
X(39450) = Ψ(X(6), X(5466))

X(39451) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(198)

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

X(39451) lies on the circumcircle and these lines: {100, 34404}, {101, 280}, {109, 189}, {934, 20220}

X(39451) = Λ(X(198), X(221)) (line X(198)X(221) is the tangent to hyperbola {{A,B,C,X(6),X(198)}} at X(198))

X(39452) = EIGENCENTER OF CIRCUMANTICEVIAN TRIANGLE OF X(216)

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

X(39452) lies on the circumcircle and these lines: {99, 276}, {107, 8794}, {110, 275}, {112, 8884}, {933, 1629}

X(39452) = Λ(X(216), X(217)) (line X(216)X(217) is the tangent to the X(4)-Altintas hyperbola, {{A,B,C,X(6),X(51)}}, at X(216))

leftri

Vu Isogonal Perspectors: X(39453)-X(39460)

rightri

This preamble is contributed by Vu Thanh Tung, August 26, 2020.

In the plane of a triangle ABC, let P be a point such that P and its isogonal conjugate, P', are distinct finite points. Let A1B1C1 be the pedal triangle of P, and let A'1B'1C'1 be the pedal triangle of P'. Let A2 be the point of intersection, other than A, of the circles (ABC) and (AA1A'1), and define B2 and C2 cyclically. Let A3 = BB2∩CC2, and define B3 and C3 cyclically. The triangle A3B3C3 is perpsective to ABC, and the perspector, V(P), is here named the Vu isogonal perspector of P. Note that V(P') = V(P), and A3B3C3 is the anticevian triangle of V(P).

If P = p : q : r (barycentrics), then

V(P) = 1/(p*(b^2*r + SA*q)*(c^2q + SA*r)) : :

The appearance of (i,j) in the following list means that V(X(i)) = X(j): (2,39353), (3,4), (4,4), (5,39454), (6,39353), (7,39455), (8,39456), (9,39457), (10,39458), (31,39459), (32,39460), (75,39459), (76, 39460)

Generalization (Vu Thanh Tung, August 28, 2020):

Theorem: In the plane of a triangle ABC, let A1 and A'1 be points on BC such that B, C, A1, and A'1 are distinct. Define B1 and C1 cyclically, and define B'1 and C'1 cyclically. Let A2 be the point of intersection, other than A, of the circles (ABC) and (AA1A'1), and define B2 and C2 cyclically. Let A3 = BB2∩CC2, and define B3 and C3 cyclically. Then A3B3C3 is perspective to ABC if and only if the points A1, A'1, B1, B'1, C1, C'1 lie on a conic.

Corollary: Let A1B1C1 be the cevian triangle of P, and let A'1B'1C'1 be the cevian triangle of P'. Then the six vertices lie on the bicevian conic of P and P', and the perspector of the triangles ABC and A3B3C3 is the barycentric product P*P'.

See Isogonal Perspector and Six Points Conic. (Vu Thanh Tung)

Let Oa be the center of the involution {B, C}, {A1, A'1} (as in the preamble just before X(14782)) and define Ob and Oc cyclically. The points Oa, Ob, Oc are collinear on a line whose trilinear pole is the Vu-isogonal perspector of P or P'. (César Lozada, March 3, 2021.)

X(39453) = VU ISOGONAL PERSPECTOR OF X(2)

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

X(39453) lies on the circumconic with center X(3258) and on these lines: {30,5050}, {1990,5304}, {3260,34229}

X(39453) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(30)}} and {{A, B, C, X(6), X(5050)}}

X(39454) = VU ISOGONAL PERSPECTOR OF X(5)

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

X(39454) lies on these lines: {}

X(39455) = VU ISOGONAL PERSPECTOR OF X(7)

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

X(39455) lies on these lines: {}

X(39455) = intersection, other than A,B,C, of conics {{A, B, C, X(75), X(10509)}} and {{A, B, C, X(279), X(331)}}

X(39456) = VU ISOGONAL PERSPECTOR OF X(8)

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

X(39456) lies on these lines: {}

X(39456) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(2401)}} and {{A, B, C, X(85), X(1222)}}

X(39457) = VU ISOGONAL PERSPECTOR OF X(9)

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

X(39457) lies on the circumconic with center X(10017) and on the line {515,5809}

X(39457) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(280)}} and {{A, B, C, X(9), X(1440)}}

X(39458) = VU ISOGONAL PERSPECTOR OF X(10)

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

X(39458) lies on these lines: {}

X(39458) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(2695)}} and {{A, B, C, X(29), X(20028)}}

X(39459) = VU ISOGONAL PERSPECTOR OF X(31)

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

X(39459) lies on these lines: {}

X(39460) = VU ISOGONAL PERSPECTOR OF X(32)

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

X(39460) lies on the line {5939,38947}

X(39460) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(1316)}} and {{A, B, C, X(83), X(3972)}}

X(39461) = X(186)X(3003)∩X(523)X(32715)

Barycentrics    (SB+SC)*(SA-SB)*(SA-SC)*(S^2-3*SA*SB)*(S^2-3*SA*SC)*(2*S^4+2*(3*R^2-SW)*SB*SC*SW+(72*R^4-(9*SA+25*SW)*R^2+2*SA^2+2*SW^2)*S^2) : :

Let P be a point on the circumcircle of ABC. The perpendicular through P to BC cuts AC, AB at Ab, Ac, respectively. Define {Bc, Ba}, {Ca, Cb} cyclically. Then these six points lie on a conic K(P). (César Lozada, Aug. 28, 2020).

The appearance of (i, j) in the following list means that the center of K( X(i) ) is X(j): (74, 39461), (98, 2966), (99, 39462), (100, 39463), (107, 39464), (110, 39465).

X(39461) lies on these lines: {186, 3003}, {523, 32715}, {1301, 1304}, {1503, 34150}, {7471, 8057}

X(39461) = reflection of X(39465) in X(32715)

X(39462) = X(98)X(16316)∩X(250)X(523)

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

See X(39461).

X(39462) lies on these lines: {98, 16316}, {250, 523}, {14721, 35088}

X(39462) = reflection of X(i) in X(j) for these (i,j): (2966, 685), (14721, 35088)

X(39463) = X(514)X(2720)∩X(522)X(1618)

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

See X(39461).

X(39463) lies on these lines: {514, 2720}, {522, 1618}, {523, 14776}, {1415, 1783}

X(39464) = X(107)X(8057)∩X(6529)X(6587)

Barycentrics    SB^2*SC^2*(SA-SB)(SA-SC)*(S^2-2*SA*SB)*(S^2-2*SA*SC)*(S^4-(3*SB+2*SC)*SA*S^2+3*(SB+SC)*SA^2*SB)*(S^4-(2*SB+3*SC)*SA*S^2+3*(SB+SC)*SA^2*SC) : :

See X(39461).

X(39464) lies on the circumconic centered at X(6523) and these lines: {107, 8057}, {6529, 6587}

X(39464) = barycentric product X(i)*X(j) for these {i, j}: {253, 32646}, {2184, 36043}
X(39464) = barycentric quotient X(1294)/X(20580)
X(39464) = trilinear product X(i)*X(j) for these {i, j}: {64, 36043}, {2184, 32646}
X(39464) = trilinear pole of the line {393, 1562}
X(39464) = X(1637)-cross conjugate of-X(459)
X(39464) = X(1294)-reciprocal conjugate of-X(20580)

X(39465) = X(112)X(647)∩X(523)X(32715)

Barycentrics    (SB+SC)*(SA-SB)*(SA-SC)*(S^2-3*SA*SB)*(S^2-3*SA*SC)*(2*S^4+(SB+SC)*(9*R^2-2*SA-2*SW)*S^2-2*(6*R^2-SW)*SB*SC*SW) : :

See X(39461).

X(39465) lies on these lines: {112, 647}, {523, 32715}, {525, 14611}, {1561, 14989}

X(39465) = reflection of X(39461) in X(32715)

X(39466) = X(6)X(2909)∩X(206)X(626)

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

Let ABC be a triangle, P = x : y : z (barycentrics) a point, A'B'C' the anticevian triangle of P and A", B", C" the reflections of P in A, B, C, respectively. Then the six points A', B', C', A", B", C" lie on a conic with center O(P) = (x^3-y^3-z^3+(y+z)*(x-y)*(x-z))*x : :. O(P)=P'-Ceva conjugate-of-P, where P'=Dilation(P, X(2), 3). (César Lozada, Aug. 28, 2020).

The appearance of (i, j) in the following list means that O( X(i) ) = X(j): (1, 40), (2, 2), (3, 155), (4, 3183), (5, 15912), (6, 159), (7, 15913), (8, 8834), (9, 3174), (10, 3159), (11, 15914), (32, 39466), (37, 22271), (69, 39127), (75, 39467), (76, 39468), (115, 12076), (188, 12646), (650, 11934), (14078, 514), (14079, 513), (14086, 523), (14090, 512). If P lies in the infinity then O(P)=P.

X(39466) lies on these lines: {6, 2909}, {206, 626}, {394, 20993}

X(39466) = X(315)-Ceva conjugate of-X(32)

X(39467) = X(37)X(2998)∩X(75)X(982)

Barycentrics    b*c*((b^2-c^2)*(b-c)*a^3+(b+c)^2*b*c*a^2-(b+c)*b^2*c^2*a-b^3*c^3) : :
X(39467) = 5*X(4687)-4*X(6375)

See X(39466)..

X(39467) lies on these lines: {37, 2998}, {75, 982}, {192, 17149}, {561, 22167}, {698, 17762}, {4043, 22039}, {4110, 19567}, {4664, 6379}, {4687, 6375}, {8026, 21080}, {18149, 30090}

X(39467) = midpoint of X(192) and X(32747)
X(39467) = reflection of X(i) in X(j) for these (i,j): (75, 6374), (2998, 37)
X(39467) = barycentric product X(1969)*X(23177)
X(39467) = barycentric quotient X(192)/X(15967)
X(39467) = trilinear product X(264)*X(23177)
X(39467) = X(192)-Ceva conjugate of-X(75)
X(39467) = X(192)-reciprocal conjugate of-X(15967)
X(39467) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17786, 24732, 6376), (18832, 22028, 6376)

X(39468) = X(39)X(6374)∩X(76)X(3981)

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

See X(39466)

X(39468) lies on these lines: {39, 6374}, {76, 3981}, {194, 19562}

X(39468) = X(194)-Ceva conjugate of-X(76)

leftri

YIU INFINITY POINTS: X(39469)-X(39474)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, August 30, 2020.

Suppose that F = f : g : h (barycentrics) is a point on the infinity line, L. The orthopoint (a.k.a. orthogonal conjugate) of F, denoted here by F' = f' : g' : h' also lies on L. In Introduction to the Geometry of the Triangle,, Paul Yiu notes that

SA*f*f' + SB*g*g' + SC*h*h' = 0.

Equialvently, the point F'' = SA*f*f' : SB*g*g' : SC*h*h'

also lies on L. The point F'' is here named the Yiu infinity point of F. Note that F'' = F*F'/H, where H = X(4), the orthocenter.

The appearance of (i,j,k) in the following list means that X(i) is on L, X(j) = orthopoint of X(i), and X(k) = Yiu infinity point of X(i):

(30, 523, 9033)
(511, 512, 39469)
(512, 511, 39469)
(513, 517, 8677)
(514, 516, 39470)
(515, 522, 39471)
(516, 514, 39470)
(517, 513, 8677)
(519, 3667, 39472)
(522, 515, 39471)
(523, 30, 9033)
(525, 1503, 39473)
(542, 690, 39574)
(690, 542, 39474)
(1503, 525, 39475)

There is a connection between Yiu infinity point and the Psi mapping (TCCT, p. 80): If Q is the the Yiu infinite point of a point P and the orthopoint of P), and if U is a point on the line of X(3) and the isogonal conjugate of P, then Psi(U) is the isogonal conjugate of Q. (Peter Moses, January 27, 2021)

X(39469) = YIU INFINITY POINT OF X(511)

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

X(39469) lies on these lines: {6, 2507}, {30, 511}, {51, 1637}, {182, 25644}, {351, 1636}, {669, 32320}, {805, 17932}, {878, 17970}, {1576, 14270}, {2491, 9419}, {2979, 3268}, {3005, 17434}, {3049, 23200}, {3060, 9979}, {3269, 9409}, {3569, 17994}, {3917, 14417}, {8029, 14391}, {9208, 33876}, {9426, 34952}, {12077, 21646}, {12824, 14697}, {13302, 14406}, {13303, 23610}, {14396, 14428}, {14424, 23616}, {14908, 18877}

X(39469) = isogonal conjugate of X(22456)
X(39469) = crossdifference of every pair of points on line X(6)X(264)
X(39469) = ideal point of PU(89)
X(39469) = ideal point of PU(109)

X(39470) = YIU INFINITY POINT OF X(514)

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

X(39470) lies on these lines: {30, 511}, {440, 2631}, {651, 653}, {652, 17094}, {1565, 3942}, {1638, 14395}, {2504, 22086}, {3904, 37781}, {5927, 30692}, {6587, 23724}, {7178, 20980}, {7192, 23090}, {17930, 17932}, {21104, 36054}, {21107, 22383}

X(39470) = isogonal conjugate of X(40116)
X(39470) = crosspoint of X(i) and X(j) for these {i,j}: {648, 37202}, {651, 1814}
X(39470) = crosssum of X(i) and X(j) for these {i,j}: {647, 39690}, {650, 5089}
X(39470) = crossdifference of every pair of points on line X(6)X(3270)
X(39470) = trilinear product X(i)*X(j) for these {i,j}: {63, 676}, {513, 26006}, {516, 905}, {656, 14953}, {910, 4025}, {1456, 6332}, {1459, 30807}, {1886, 4131}, {2398, 3942}, {3270, 24015}, {22383, 35517}, {23696, 39063}, {23973, 34591}

X(39471) = YIU INFINITY POINT OF X(515)

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

X(39471) lies on these lines: {1, 14312}, {30, 511}, {651, 1897}, {1639, 14395}, {1807, 37628}, {2968, 7004}, {3700, 36054}, {14302, 21172}, {14304, 24034}, {14414, 14429}, {15252, 24030}, {17931, 17932}

X(39471) = isogonal conjugate of X(36067)
X(39471) = crossdifference of every pair of points on line X(6)X(3209)
X(39471) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 36067}, {2, 32667}, {4, 36040}, {92, 32643}, {102, 108}, {109, 36121}, {653, 32677}, {2432, 7128}, {10571, 36108}, {17080, 32700}, {32674, 36100}, {36055, 36127}
X(39471) = trilinear product X(i)*X(j) for these {i,j}: {3, 14304}, {515, 521}, {1946, 35516}, {2182, 6332}, {2406, 34591}, {23987, 24031}, {24035, 35072}

X(39472) = YIU INFINITY POINT OF X(519)

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

X(39472) lies on these lines: {1, 25923}, {30, 511}, {239, 26568}, {3241, 25020}, {3679, 25996}, {26078, 34619}, {26144, 34625}

X(39472) = crossdifference of every pair of points on line X(6)X(8752)

X(39473) = YIU INFINITY POINT OF X(525)

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

X(39473) lies on these lines: {2, 14345}, {30, 511}, {287, 2419}, {648, 2404}, {1636, 14417}, {2394, 15291}, {2525, 32320}, {3269, 15526}, {4143, 14638}, {6334, 22146}

X(39473) = isogonal conjugate of X(32687)
X(39473) = crossdifference of every pair of points on line X(6)X(32713)

X(39474) = YIU INFINITY POINT OF X(542)

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

X(39474) lies on these lines: {30, 511}, {1648, 14401}, {3292, 14417}, {3580, 14697}, {5468, 17932}, {8030, 23616}, {38239, 38240}

leftri

CENTERS OF CIRCUMCIRCLE-INVERSION CIRCLES: X(39475)-X(39481)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, September 1, 2020.

Suppose that Γ is the circle with powers (u,v,w) -- that is, u = power of A with respect to the circle. and v and w are defined cyclically. Suppose that L is a line, given by p x + q y + r z = 0 (barycentrics), that does not pass through X(3). The Γ-inverse of L is a circle that passes through X(3). This circle is here named the Γ-inversion circle of L, denoted by ((Γ, L)). The center of ((Γ, L)) is given by

f(a,b,c,u,v,w,p,q,r) : f(b,c,a,v,w,u,q,r,p) : f(c,a,b,w,u,v,r,p,q), where

f(a,b,c,u,v,w,p,q,r) = r*(a^2*c^2*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4) + 2*a^2*c^2*(a^2 + b^2 - c^2)*u + a^2*(a^2 - b^2 + c^2)*u^2 + (-a^6 + 3*a^4*b^2 - 3*a^2*b^4 + b^6 + a^4*c^2 - 2*a^2*b^2*c^2 - 3*b^4*c^2 + a^2*c^4 + 3*b^2*c^4 - c^6)*v - 2*a^2*(a^2 - b^2 - c^2)*u*v + (a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4)*v^2 - 2*a^2*c^2*(a^2 - b^2 - c^2)*w - 4*a^2*c^2*u*w + 2*c^2*(a^2 + b^2 - c^2)*v*w + c^2*(a^2 - b^2 + c^2)*w^2) + q*(a^2*b^2*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2) + 2*a^2*b^2*(a^2 - b^2 + c^2)*u + a^2*(a^2 + b^2 - c^2)*u^2 - 2*a^2*b^2*(a^2 - b^2 - c^2)*v - 4*a^2*b^2*u*v + b^2*(a^2 + b^2 - c^2)*v^2 + (-a^6 + a^4*b^2 + a^2*b^4 - b^6 + 3*a^4*c^2 - 2*a^2*b^2*c^2 + 3*b^4*c^2 - 3*a^2*c^4 - 3*b^2*c^4 + c^6)*w - 2*a^2*(a^2 - b^2 - c^2)*u*w + 2*b^2*(a^2 - b^2 + c^2)*v*w + (a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*w^2) + p*(-(a^4*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4)) - 2*a^4*(a^2 - b^2 - c^2)*u - 2*a^4*u^2 + 2*a^2*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4)*v + 2*a^2*(a^2 + b^2 - c^2)*u*v + (-a^4 - b^4 + 2*a^2*c^2 + 2*b^2*c^2 - c^4)*v^2 + 2*a^2*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*w + 2*a^2*(a^2 - b^2 + c^2)*u*w - 2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*v*w + (-a^4 + 2*a^2*b^2 - b^4 + 2*b^2*c^2 - c^4)*w^2).

In particular, the center of ((circumcircle, L)) is the point

a^2*(a^2*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4)*p - b^2*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*q - c^2*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4)*r) : : , and the squared radius of ((circumcircle, L)) is

(a^4*b^4*c^4*(a^2*p^2 + (-a^2 - b^2 + c^2)*p*q + b^2*q^2 + (-a^2 + b^2 - c^2)*p*r + (a^2 - b^2 - c^2)*q*r + c^2*r^2))/((a + b - c)*(a - b + c)*(-a + b + c)*(a + b + c)*(a^2*(a^2 - b^2 - c^2)*p - b^2*(a^2 - b^2 + c^2)*q - c^2*(a^2 + b^2 - c^2)*r)^2).

If P and P' are distinct points on L, then ((circumcircle, L)) is the Vietnamese circle of the circumcevian triangles of P and P'; see the preamble just before X(39210) and centers X(39219)-X(39228).

Examples: ((circumcircle, L)), for selected lines L:

1. L = orthic axis = X(230)X(232); center of ((circumcircle, L)) is X(6644)
2. L = anti-orthic axis = X(44)X(513); center of ((circumcircle, L)) is X(32613)
3. L = Lemoine axis = X(187)X(237); center of ((circumcircle, L)) is X(182)
4. L = de Longchamnps axis = X(325)X(523); center of ((circumcircle, L)) is X(7502)
5. L = Nagel line = X(1)X(2); center of ((circumcircle, L)) is X(39225)
6. L = Van Aubel line = X(4)X(6); center of ((circumcircle, L)) is X(39228)
7. L = X(2)X(6); center of ((circumcircle, L)) is X(5926)
8. L = X(1)X(6), center of ((circumcircle, L)) is X(39227)
9. L = Gergonne line = X(241)X(514), center of ((circumcircle, L)) is X(39475)
10. L = Soddy line = X(1)X(7), center of ((circumcircle, L)) is X(39476)
11. L = Fermat line = X(6)X(13), center of ((circumcircle, L)) is X(39477)
12. L = X(1)X(5), center of ((circumcircle, L)) is X(39478)
13. L = Sherman line = X(3259)X(3326), center of ((circumcircle, L)) is X(39479)
14. L = Apollonius line = X(1)X(181), center of ((circumcircle, L)) is X(39280)
15. L = Napoleon axis = X(6)X(17), center of ((circumcircle, L)) is X(39281)

See the preamble just before X(39486).

X(39475) = CENTER OF ((CIRCUMCIRCLE, GERGONNE LINE))

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

X(39475) lies on these lines: {1,1775}, {3,142}, {19,24}, {22,15931}, {35,4319}, {36,2263}, {55,11028}, {140,23305}, {182,2876}, {534,14070}, {631,11677}, {991,2195}, {1006,15177}, {1385,3827}, {1602,3220}, {1836,18590}, {2385,37812}, {2772,35273}, {2801,24320}, {2822,14673}, {2835,10269}, {3668,7742}, {3825,21484}, {4297,13730}, {4329,7488}, {4331,36152}, {5259,7503}, {6642,6796}, {6644,32613}, {6986,37557}, {7412,10198}, {7509,31261}, {7580,20988}, {8273,20833}, {10164,37577}, {11507,25065}, {13329,37576}

X(39475) = midpoint of X(3) and X(1486)
X(39475) = center of Vietnamese circle of the circumcevian triangles of X(214) and X(514)

X(39476) = CENTER OF ((CIRCUMCIRCLE, SODDY LINE))

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

X(39476) lies on these lines: {3,514}, {35,4449}, {36,663}, {58,22090}, {513,23961}, {523,34152}, {993,17072}, {1946,3960}, {2774,4091}, {3667,39199}, {4040,7280}, {4147,25440}, {4184,4379}, {4210,4893}, {4252,22154}, {4794,5204}, {4905,8648}, {6003,23224}, {6006,39200}, {14793,21185}, {14838,22091}, {16064,21204}, {21761,22437}, {22089,23879}, {23226,23800}}.

X(39476) = center of Vietnamese circle of the circumcevian triangles of X(1) and X(7)

X(39477) = CENTER OF ((CIRCUMCIRCLE, FERMAT LINE))

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

X(39477) lies on these lines: {3,690}, {23,9189}, {186,16230}, {187,2491}, {511,14271}, {512,5926}, {523,18571}, {669,9125}, {826,39201}, {5466,11643}, {7492,9185}, {7771,14295}, {9033,12893}, {9208,37184}, {11616,25644}, {11621,20403}, {32478,34952}

X(39477) = midpoint of X(3) and X(14270)
X(39477) = center of Vietnamese circle of the circumcevian triangles of X(1) and X(7)

X(39478) = CENTER OF ((CIRCUMCIRCLE, X(1)X(5)))

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

X(39478) lies on these lines: {3,900}, {22,26275}, {186,523}, {513,23961}, {676,14667}, {1030,4435}, {1631,8638}, {2815,26286}, {4057,27086}, {4184,4800}, {4188,26078}, {4189,26144}, {4225,28284}, {6370,14270}, {7484,30792}, {15246,31131}, {16287,24959}, {16453,24920}, {17548,27545}, {20836,27620}, {27728,37247}

X(39478) = midpoint of X(3) and X(39200)
X(39478) = center of Vietnamese circle of the circumcevian triangles of X(1) and X(5)

X(39479) = CENTER OF ((CIRCUMCIRCLE, SHERMAN LINE))

Barycentrics    a^2*(a^14 - 2*a^13*b - 3*a^12*b^2 + 8*a^11*b^3 + a^10*b^4 - 10*a^9*b^5 + 5*a^8*b^6 - 5*a^6*b^8 + 10*a^5*b^9 - a^4*b^10 - 8*a^3*b^11 + 3*a^2*b^12 + 2*a*b^13 - b^14 - 2*a^13*c + 8*a^12*b*c - 2*a^11*b^2*c - 22*a^10*b^3*c + 20*a^9*b^4*c + 10*a^8*b^5*c - 20*a^7*b^6*c + 20*a^6*b^7*c - 10*a^5*b^8*c - 20*a^4*b^9*c + 22*a^3*b^10*c + 2*a^2*b^11*c - 8*a*b^12*c + 2*b^13*c - 3*a^12*c^2 - 2*a^11*b*c^2 + 21*a^10*b^2*c^2 - 6*a^9*b^3*c^2 - 33*a^8*b^4*c^2 + 26*a^7*b^5*c^2 + a^6*b^6*c^2 - 22*a^5*b^7*c^2 + 30*a^4*b^8*c^2 - 18*a^2*b^10*c^2 + 4*a*b^11*c^2 + 2*b^12*c^2 + 8*a^11*c^3 - 22*a^10*b*c^3 - 6*a^9*b^2*c^3 + 44*a^8*b^3*c^3 - 10*a^7*b^4*c^3 - 28*a^6*b^5*c^3 + 20*a^5*b^6*c^3 + 6*a^4*b^7*c^3 - 26*a^3*b^8*c^3 + 6*a^2*b^9*c^3 + 14*a*b^10*c^3 - 6*b^11*c^3 + a^10*c^4 + 20*a^9*b*c^4 - 33*a^8*b^2*c^4 - 10*a^7*b^3*c^4 + 26*a^6*b^4*c^4 + 2*a^5*b^5*c^4 - 13*a^4*b^6*c^4 + 2*a^3*b^7*c^4 + 19*a^2*b^8*c^4 - 14*a*b^9*c^4 - 10*a^9*c^5 + 10*a^8*b*c^5 + 26*a^7*b^2*c^5 - 28*a^6*b^3*c^5 + 2*a^5*b^4*c^5 - 4*a^4*b^5*c^5 + 10*a^3*b^6*c^5 - 8*a^2*b^7*c^5 - 4*a*b^8*c^5 + 6*b^9*c^5 + 5*a^8*c^6 - 20*a^7*b*c^6 + a^6*b^2*c^6 + 20*a^5*b^3*c^6 - 13*a^4*b^4*c^6 + 10*a^3*b^5*c^6 - 8*a^2*b^6*c^6 + 6*a*b^7*c^6 - b^8*c^6 + 20*a^6*b*c^7 - 22*a^5*b^2*c^7 + 6*a^4*b^3*c^7 + 2*a^3*b^4*c^7 - 8*a^2*b^5*c^7 + 6*a*b^6*c^7 - 4*b^7*c^7 - 5*a^6*c^8 - 10*a^5*b*c^8 + 30*a^4*b^2*c^8 - 26*a^3*b^3*c^8 + 19*a^2*b^4*c^8 - 4*a*b^5*c^8 - b^6*c^8 + 10*a^5*c^9 - 20*a^4*b*c^9 + 6*a^2*b^3*c^9 - 14*a*b^4*c^9 + 6*b^5*c^9 - a^4*c^10 + 22*a^3*b*c^10 - 18*a^2*b^2*c^10 + 14*a*b^3*c^10 - 8*a^3*c^11 + 2*a^2*b*c^11 + 4*a*b^2*c^11 - 6*b^3*c^11 + 3*a^2*c^12 - 8*a*b*c^12 + 2*b^2*c^12 + 2*a*c^13 + 2*b*c^13 - c^14) : :

X(39479) lies on these lines: {3,2222}, {24,3259}, {186,953}, {901,7488}, {7502,38614}, {7512,38705}, {22467,38707}, {37814,38617}

X(39479) = midpoint of X(3) and X(10016)
X(39479) = center of Vietnamese circle of the circumcevian triangles of X(3259) and X(3326)

X(39480) = CENTER OF ((CIRCUMCIRCLE, APOLLONIUS LINE))

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

X(39480) lies on these lines: {513,23961}, {656,667}, {8672,39199}

X(39480) = center of Vietnamese circle of the circumcevian triangles of X(1) and X(181)

X(39481) = CENTER OF ((CIRCUMCIRCLE, NAPOLEON AXIS))

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

X(39481) lies on these lines: {3,18336}, {512,5926}, {690,22089}, {826,14270}, {1116,6644}, {6292,9494}, {7927,39201}, {18308,18570}

X(39481) = center of Vietnamese circle of the circumcevian triangles of X(6) and X(17)

X(39482) = CENTER OF ((ORTHOCENTROIDAL CIRCLE, BROCARD AXIS))

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

X(39482) lies on these lines: {115,35582}, {381,512}, {523,5066}, {804,11620}, {1116,3566}, {3091,23105}, {4108,7533}, {6041,18362}, {7577,16229}, {8675,25561}, {19709,34291}

X(39483) = CENTER OF ((ORTHOCENTROIDAL CIRCLE, X(1)X(3)))

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

X(39483) lies on these lines: {381,513}, {523,5066}, {5587,8702}, {7577,16228}

X(39484) = CENTER OF ((ORTHOCENTROIDAL CIRCLE, ORTHIC AXIS))

Barycentrics    a^10 + 6*a^8*b^2 - 16*a^6*b^4 + 2*a^4*b^6 + 15*a^2*b^8 - 8*b^10 + 6*a^8*c^2 - 4*a^6*b^2*c^2 - 4*a^4*b^4*c^2 - 22*a^2*b^6*c^2 + 24*b^8*c^2 - 16*a^6*c^4 - 4*a^4*b^2*c^4 + 14*a^2*b^4*c^4 - 16*b^6*c^4 + 2*a^4*c^6 - 22*a^2*b^2*c^6 - 16*b^4*c^6 + 15*a^2*c^8 + 24*b^2*c^8 - 8*c^10 : :

X(39484) lies on these lines: {2,3}, {3066,15088}, {7699,9140}, {7703,10706}, {8176,14356}, {14852,15534}, {15928,22566}

X(39484) = midpoint of X(381) and X(5094)
X(39484) = center of circle {{X(381),X(5094),PU(4)}}

X(39485) = CENTER OF ((ORTHOCENTROIDAL CIRCLE, ANTI-ORTHIC AXIS))

Barycentrics    a^9*b - 6*a^5*b^5 + 8*a^3*b^7 - 3*a*b^9 + a^9*c + a^8*b*c + a^7*b^2*c - 4*a^6*b^3*c - 4*a^5*b^4*c + a^4*b^5*c + a^3*b^6*c + 6*a^2*b^7*c + a*b^8*c - 4*b^9*c + a^7*b*c^2 - a^5*b^3*c^2 - 8*a^3*b^5*c^2 + 8*a*b^7*c^2 - 4*a^6*b*c^3 - a^5*b^2*c^3 - 4*a^4*b^3*c^3 - a^3*b^4*c^3 - 6*a^2*b^5*c^3 + 16*b^7*c^3 - 4*a^5*b*c^4 - a^3*b^3*c^4 - 6*a*b^5*c^4 - 6*a^5*c^5 + a^4*b*c^5 - 8*a^3*b^2*c^5 - 6*a^2*b^3*c^5 - 6*a*b^4*c^5 - 24*b^5*c^5 + a^3*b*c^6 + 8*a^3*c^7 + 6*a^2*b*c^7 + 8*a*b^2*c^7 + 16*b^3*c^7 + a*b*c^8 - 3*a*c^9 - 4*b*c^9 : :

X(39485) lies on this line: {210,381}

leftri

CENTERS OF CIRCLE-INVERSION CIRCLES: X(39486)-X(39552)

rightri

This preamble, a sequel to the preamble just before X(39475), is contributed by Clark Kimberling and Peter Moses, September 2, 2020.

Suppose that Γ is the circle with powers (u,v,w) -- that is, u = power of A with respect to the circle. and v and w are defined cyclically. Suppose that L is a line, given by p x + q y + r z = 0 (barycentrics). The A-power of the circle ((Γ, L)) is given by

(r*u*(c^2*(a^2 + b^2 - c^2) + (a^2 - b^2 + c^2)*u + (-a^2 + b^2 + c^2)*v - 2*c^2*w) + q*u*(b^2*(a^2 - b^2 + c^2) + (a^2 + b^2 - c^2)*u - 2*b^2*v - (a^2 - b^2 - c^2)*w) + p*(a^2*b^2*c^2 - a^2*u^2 - b^2*(a^2 - b^2 + c^2)*v + b^2*v^2 - c^2*(a^2 + b^2 - c^2)*w + (a^2 - b^2 - c^2)*v*w + c^2*w^2))/(r*(c^2*(a^2 + b^2 - c^2) + (a^2 - b^2 + c^2)*u - (a^2 - b^2 - c^2)*v - 2*c^2*w) + q*(b^2*(a^2 - b^2 + c^2) + (a^2 + b^2 - c^2)*u - 2*b^2*v - (a^2 - b^2 - c^2)*w) - p*(a^2*(a^2 - b^2 - c^2) + 2*a^2*u - (a^2 + b^2 - c^2)*v - (a^2 - b^2 + c^2)*w)),

and the B- and C- powers are defined cyclically. These powers are useful for finding the center, radius, and equation for the circle ((Γ, L)).

Circle Γ Line L numbers i such that X(i) is on the circle ((Γ, L))
circumcircle orthic axis 3, 25, 5941, 15551, 35901
circumcircle Lemoine axis (Brocard circle)
circumcircle Soddy line 3, 36, 2071, 14878, 32624
circumcircle Nagerl line 3, 23, 36, 1324, 4057, 17100, 32759
circumcircle Fermat line 3, 187, 1511, 2079, 5961, 6104, 6105, 7575, 12042, 14270, 14702, 14703, 15550, 34010, 34217
circumcircle van Aubel line 3, 186, 187, 12096, 21397, 39201
circumcircle X(2)X(6) 3, 23, 187, 353, 669, 5866, 5867, 5937, 5938, 5939, 5940, 5941, 6031, 32531, 34014
circumcircle X(1)X(5) 3, 36, 2070, 10260, 14667, 32626, 39200
circumcircle Napoleon axis 3, 187, 6150, 15551, 32627, 32628, 37936
incircle Euler line 1, 1314, 1315, 5533, 5570
nine-point circle Brocard axis 5, 2072, 6112, 6113, 34964
nine-point circle Fermat line 5, 113, 115, 6106, 6107, 11799, 15367
nine-point circle X(2)X(6) 5, 625, 858, 6032, 23301, 34512
Brocard circle Euler line 3, 110, 182, 1316, 2698, 8723
Brocard circle van Aubel line 6, 182, 5622, 18338
Brocard circle X(2)X(6) 6, 110, 182, 5108, 5970
Spieker circle X(1)X(6) 10, 5123, 5199, 20317
orthocentroidal circle Brocard axis 5, 115, 381, 15538, 25641
orthocentroidal circle van Aubel line 4, 115, 381, 6794, 18809
orthocentroidal circle X(2)X(6) 2, 115, 381, 6792, 8371, 16188
orthocentroidal circle Napoleon axis 115, 140, 381, 1116
Furhmann circle Euler line 4, 355, 3448, 36154
Furhmann circle Nagel line 8, 80, 355, 6788
Parry circle Eulerl line 2, 23, 351, 9129
Parry circle Brocard axis 15, 16, 351, 2502, 9129, 10166
Parry circle X(2)X(6) 2, 351, 352, 5106
Bevan circle X(1)X(6) 40, 484, 4063, 5011, 5184
Moses circle orthic axis 39, 14537, 15544, 15548
Stevanovic circle Euler line 650, 5000, 5001, 5089
Stevanovic circle X(1)X(3) 650, 910, 32622, 32623
Evans circle X(1)X(6) 40, 484, 4063, 5011, 5184
Lester circle Euler line 3, 5, 1116, 10414
Lester circle Fermat line 13, 14, 1116, 10413
Lester circle Napoleon axis 115, 140, 381, 1116
Conway circle Euler line 1, 5214, 38473, 38474
Spieker radical circle Brocard axis 10, 3814, 4129, 5179, 14873
Spieker radical circle X(1)X(5) 10, 3030, 34459, 38471
polar circle Brocard axis 4, 403, 5523, 5962, 6116, 6117, 6761, 14618, 35718
polar circle orthic axis (orthocentroidal circle)
polar circle Nagel line 4, 242, 468, 1785, 1878, 7649
polar circle Fermat line 4, 112, 1300, 1986, 5523, 5667, 6110, 6111, 10295
polar circle X(2)X(6) 4, 468, 2501, 5140, 5203, 5523
polar circle X(1)X(5) 4, 108, 186, 915, 1785, 1845
polar circle X(1)X(6) 4, 1785, 5146, 5523, 17924, 37989
Dao-Moses-Telv circle Euler line 1637, 5000, 5001, 6103
Dao-Moses-Telv circle Fermat line 13, 14, 125, 1637, 11657, 14847, 34310
orthoptic circle of Steiner inellipse Brocard axis 2, 858, 5913, 5996, 6032, 6114, 6115, 32526
orthoptic circle of Steiner inellipse orthic axis (orthocentroidal circle)
orthoptic circle of Steiner inellipse Lemoine axis 2, 262, 6032, 34235, 36183
orthoptic circle of Steiner inellipse de Longchamps axis 2, 3, 3111, 5108, 6789, 9153, 9828, 32531, 34583
orthoptic circle of Steiner inellipse Fermat line 2, 111, 1302, 5913, 6108, 6109, 7426, 9185
orthoptic circle of Steiner inellipse van Aubel 2, 468, 5913, 9209
orthoptic circle of Steiner inellipse X(1)X(5) 2, 23, 105, 5121, 9058
orthoptic circle of Steiner inellipse Napoleon axis 2, 5913, 5987, 37900
orthoptic circle of Steiner circumellipse de Longchamps axis 2, 20, 6790, 38940, 38941
orthoptic circle of Steiner circumellipse Fermat line 2, 9999, 20099, 37901
orthoptic circle of Steiner circumellipse X(1)X(5) 2, 5211, 20063, 20097
Moses radical circle Euler line 232, 647, 5000, 5001, 34235, 35901
Moses radical circle Brocard axis 15, 16, 647, 1495, 14685, 16319, 35901
de Longchamps circle de Longchamps axis 2, 20, 6790, 38940, 38941
de Longchamps circle Nagel line 20, 858, 10538, 20294
nine-point circle of medial triangle X(2)X(6) 140, 468, 10160, 10163
Schoutte circle Euler line 6, 187, 5912, 15925
2nd Brocard circle van Aubel line 3, 39, 37991, 38642
2nd Brocard circle X(2)X(6) 3, 39, 14824, 36182
McCay circumcircle X(2)X(6) 111, 7610, 7617, 9169
Stammler circle Fermat line 3, 399, 5938, 9301, 11641, 12188, 34106, 35463, 37924
circumcircle of inner Napoleon triangle Brocard axis 2, 14, 14182, 34313, 36185
circumcircle of outer Napoleon triangle Brocard axis 2, 13, 14178, 34314, 36186
Moses-Parry circle Brocard axis 6, 187, 2492, 3018
Moses-Parry circle Lemoine axis 187, 1637, 2492, 3569
Moses-Parry circle Fermat line 6, 115, 2492, 3003
Moses-Parry circle X(2)X(6) 6, 2492, 2493, 5913
Hutson-Parry circle Euler line 2, 125, 373, 5459, 5460, 8371, 9169, 9172, 10162, 31655
Hutson-Parry circle Fermat line 13, 14, 1648, 8371, 9169
Hutson-Parry circle X(2)X(6) 2, 115, 381, 6792, 8371, 16188
orthosymmedial circle Euler line 4, 1112, 1316, 5480
orthosymmedial circle Fermat line 6, 5480, 18907, 19161
orthosymmedial circle X(2)X(6) 6, 5480, 6792, 11746
Parry isodynamic circle Brocard axis 15, 16, 351, 2502, 9129, 10166
Parry isodynamic circle X(2)X(6) 23, 353, 2502, 5027, 9129

X(39486) = CENTER OF ((ORTHOCENTROIDAL CIRCLE, LEMOINE AXIS))

Barycentrics    a^10*b^2 - 6*a^6*b^6 + 8*a^4*b^8 - 3*a^2*b^10 + a^10*c^2 + 3*a^8*b^2*c^2 - 9*a^6*b^4*c^2 - 6*a^4*b^6*c^2 + 15*a^2*b^8*c^2 - 4*b^10*c^2 - 9*a^6*b^2*c^4 - 6*a^4*b^4*c^4 - 12*a^2*b^6*c^4 + 16*b^8*c^4 - 6*a^6*c^6 - 6*a^4*b^2*c^6 - 12*a^2*b^4*c^6 - 24*b^6*c^6 + 8*a^4*c^8 + 15*a^2*b^2*c^8 + 16*b^4*c^8 - 3*a^2*c^10 - 4*b^2*c^10 : :

X(39486) lies on these lines: {5,11594}, {114,5169}, {381,511}, {566,7579}

X(39486) = midpoint of X(381) and X(18575)

X(39487) = CENTER OF ((ORTHOCENTROIDAL CIRCLE, DE LONGCHAMPS AXIS))

Barycentrics    a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 - 4*a^6*b^2*c^2 - 13*a^4*b^4*c^2 + 23*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 13*a^4*b^2*c^4 - 40*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + 23*a^2*b^2*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10 : :

X(39487) lies on these lines: {2,3}, {2854,5476}, {4550,15361}, {5640,5655}, {10545,10706}, {15068,15534}, {16261,20126}, {32267,34513}

X(39487) = midpoint of X(381) and X(1995)

X(39488) = CENTER OF ((ORTHOCENTROIDAL CIRCLE, GERGONNE LINE))

Barycentrics    a^9 + a^8*b + 5*a^7*b^2 - 5*a^6*b^3 - 11*a^5*b^4 - a^4*b^5 + 3*a^3*b^6 + 13*a^2*b^7 + 2*a*b^8 - 8*b^9 + a^8*c - 3*a^6*b^2*c + 3*a^4*b^4*c - a^2*b^6*c + 5*a^7*c^2 - 3*a^6*b*c^2 + 2*a^5*b^2*c^2 - 4*a^4*b^3*c^2 - 3*a^3*b^4*c^2 - 13*a^2*b^5*c^2 - 8*a*b^6*c^2 + 24*b^7*c^2 - 5*a^6*c^3 - 4*a^4*b^2*c^3 + a^2*b^4*c^3 + 8*b^6*c^3 - 11*a^5*c^4 + 3*a^4*b*c^4 - 3*a^3*b^2*c^4 + a^2*b^3*c^4 + 12*a*b^4*c^4 - 24*b^5*c^4 - a^4*c^5 - 13*a^2*b^2*c^5 - 24*b^4*c^5 + 3*a^3*c^6 - a^2*b*c^6 - 8*a*b^2*c^6 + 8*b^3*c^6 + 13*a^2*c^7 + 24*b^2*c^7 + 2*a*c^8 - 8*c^9 : :

X(39488) lies on this line: {381,516}

X(39489) = CENTER OF ((ORTHOCENTROIDAL CIRCLE, SODDY LINE))

Barycentrics    (b - c)*(4*a^6 - 5*a^5*b + 6*a^4*b^2 - 5*a^3*b^3 - 9*a^2*b^4 + 10*a*b^5 - b^6 - 5*a^5*c + 5*a^4*b*c - 5*a^3*b^2*c + 5*a^2*b^3*c + 10*a*b^4*c - 10*b^5*c + 6*a^4*c^2 - 5*a^3*b*c^2 + 17*a^2*b^2*c^2 - 20*a*b^3*c^2 + b^4*c^2 - 5*a^3*c^3 + 5*a^2*b*c^3 - 20*a*b^2*c^3 + 20*b^3*c^3 - 9*a^2*c^4 + 10*a*b*c^4 + b^2*c^4 + 10*a*c^5 - 10*b*c^5 - c^6) : :

X(39489) lies on this line: {381,514}

X(39490) = CENTER OF ((ORTHOCENTROIDAL CIRCLE, NAGEL LINE))

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

X(39490) lies on these lines: {5,523}, {381,3667}, {1995,39225}, {5094,16231}

X(39491) = CENTER OF ((ORTHOCENTROIDAL CIRCLE, VAN AUBEL LINE))

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

X(39491) lies on these lines: {30,14566}, {378,39228}, {381,525}, {523,3845}, {2394,3839}, {3543,18556}, {3545,5664}, {3843,5489}

X(39492) = CENTER OF ((ORTHOCENTROIDAL CIRCLE, X(2)X(6)))

Barycentrics    (b^2 - c^2)*(a^8 + 6*a^6*b^2 - 14*a^4*b^4 + 6*a^2*b^6 + b^8 + 6*a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 - 12*b^6*c^2 - 14*a^4*c^4 + a^2*b^2*c^4 + 22*b^4*c^4 + 6*a^2*c^6 - 12*b^2*c^6 + c^8) : :

X(39492) lies on these lines: {5,523}, {114,18007}, {381,1499}, {525,7615}, {1637,32228}, {1649,5055}, {1995,5926}, {2793,5461}, {3091,10280}, {3545,5466}, {3845,10189}, {3851,10279}, {5056,32204}, {5066,10278}, {5071,9168}, {8029,19709}, {10109,10190}, {11615,37742}, {14644,14846}

X(39492) = midpoint of X(381) and X(8371)

X(39493) = CENTER OF ((ORTHOCENTROIDAL CIRCLE, X(1)X(5)))

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

X(39493) lies on these lines: {2,523}, {381,900}, {526,5902}, {1995,26275}, {5094,30792}, {5169,31131}, {16763,35055}

X(39494) = CENTER OF ((ORTHOCENTROIDAL CIRCLE, NAPOLEON AXIS))

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

X(39494) lies on these lines: {381,1116}, {523,10109}, {3545,18308}, {5055,15475}, {15543,38071}, {27363,32904}

X(39494) = midpoint of X(381) and X(1116)

X(39495) = CENTER OF ((BROCARD CIRCLE, EULER LINE))

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

X(39495) lies on these lines: {110, 1649}, {182, 523}, {184, 10190}, {206, 3566}, {512, 5092}, {526, 1511}, {669, 15080}, {804, 4107}, {1510, 39214}, {3049, 3094}, {3050, 5116}, {5012, 11123}, {5118, 35345}, {5926, 34513}, {9168, 11003}, {11422, 30219}, {13339, 16220}, {15920, 34291}, {32046, 32204}

X(39495) = midpoint of X(182) and X(8723)

X(39496) = CENTER OF ((BROCARD CIRCLE, X(1)X(3)))

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

X(39496) lies on these lines: {182, 513}, {512, 5092}, {3063, 3094}, {4132, 8723}, {5116, 21007}

X(39497) = CENTER OF ((BROCARD CIRCLE, ORTHIC AXIS))

Barycentrics    a^2*(a^12 - 3*a^10*b^2 + 4*a^8*b^4 - 4*a^6*b^6 + 3*a^4*b^8 - a^2*b^10 - 3*a^10*c^2 + 2*a^8*b^2*c^2 + 8*a^6*b^4*c^2 - 12*a^4*b^6*c^2 + 5*a^2*b^8*c^2 + 4*a^8*c^4 + 8*a^6*b^2*c^4 - 18*a^4*b^4*c^4 - 6*a^2*b^6*c^4 - 2*b^8*c^4 - 4*a^6*c^6 - 12*a^4*b^2*c^6 - 6*a^2*b^4*c^6 + 4*b^6*c^6 + 3*a^4*c^8 + 5*a^2*b^2*c^8 - 2*b^4*c^8 - a^2*c^10) : :

X(39497) lies on these lines: {30, 182}, {574, 1511}, {2871, 8546}, {2909, 32516}, {3148, 6800}

X(39498) = CENTER OF ((BROCARD CIRCLE, LEMOINE AXIS))

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

X(39498) lies on these lines: {3, 6}, {543, 7606}, {3618, 35951}, {3734, 32149}, {4048, 20398}, {5476, 11676}, {7790, 38317}, {7824, 34507}, {8859, 37455}, {9129, 10166}, {11261, 12177}, {11645, 13860}, {15920, 32305}, {20399, 24206}, {32429, 38664}, {35925, 38064}

X(39498) = midpoint of X(182) and X(574)
X(39498) = X(182)-of-2nd-Brocard-triangle
X(39498) = X(182)-of-X(3)PU(1)

X(39499) = CENTER OF ((BROCARD CIRCLE, FERMAT LINE))

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

X(39499) lies on these lines: {54, 826}, {182, 690}, {512, 575}, {526, 32154}, {888, 8546}, {5027, 9485}, {11171, 14270}

X(39500) = CENTER OF ((BROCARD CIRCLE, VAN AUBEL LINE))

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

X(39500) lies on these lines: {182, 525}, {512, 575}, {574, 39228}, {1640, 5012}, {11171, 39201}, {22089, 26316}

X(39501) = CENTER OF ((BROCARD CIRCLE, X(2)X(6)))

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

X(39501) lies on these lines: {182, 1499}, {512, 575}, {525, 1147}, {574, 5926}, {669, 11171}, {2780, 6593}, {3398, 14824}

X(39502) = CENTER OF ((BROCARD CIRCLE, X(1)X(6)))

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

X(39502) lies on these lines: {182, 3309}, {512, 575}, {574, 39227}, {667, 11171}

X(39503) = CENTER OF ((NINE-POINT CIRCLE, BROCARD AXIS))

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

X(39503) lies on these lines: {5, 512}, {115, 2524}, {924, 34963}, {1594, 16229}, {5996, 37353}, {7577, 14618}, {10255, 23105}, {13881, 22159}, {18117, 18314}, {23301, 32478}

X(39503) = midpoint of X(5) and X(34964)

X(39504) = CENTER OF ((NINE-POINT CIRCLE, ORTHIC AXIS))

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

X(39504) lies on these lines: {2, 3}, {52, 34826}, {115, 15070}, {125, 5946}, {143, 5449}, {146, 11559}, {156, 15806}, {184, 34514}, {264, 34900}, {265, 15033}, {389, 13561}, {567, 25739}, {568, 23293}, {1154, 21243}, {1209, 6101}, {1568, 15060}, {2781, 6697}, {3448, 11804}, {3574, 6102}, {3613, 14356}, {3818, 10272}, {5462, 32767}, {5480, 13451}, {5663, 18388}, {5889, 20424}, {5890, 7703}, {5902, 15095}, {6288, 34148}, {7699, 15102}, {7706, 23315}, {8157, 34845}, {8254, 10274}, {8371, 15099}, {9019, 24206}, {10116, 12242}, {10610, 11750}, {10627, 13565}, {10796, 14676}, {11412, 21230}, {11430, 30522}, {11560, 21650}, {11801, 18390}, {12026, 23319}, {12161, 18356}, {13403, 18379}, {13630, 20299}, {14072, 14769}, {15019, 15027}, {15038, 38724}, {15053, 15061}, {18474, 32423}, {18583, 20300}, {19127, 38317}, {23294, 37481}, {25738, 32165}, {34799, 36966}

X(39504) = midpoint of X(5) and X(427)
X(39504) = center of circle {{X(5),X(427),PU(4)}}

X(39505) = CENTER OF ((NINE-POINT CIRCLE, ANTI-ORTHIC AXIS))

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

X(39505) lies on these lines: {5, 10}, {1656, 16678}, {1985, 32613}, {33862, 37357}

X(39506) = CENTER OF ((NINE-POINT CIRCLE, LEMOINE AXIS))

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

X(39506) lies on these lines: {5, 141}, {114, 137}, {182, 37988}, {311, 7752}, {570, 1506}, {575, 34845}, {1656, 8266}, {7668, 15516}

X(39506) = midpoint of X(5) and X(3613)

X(39507) = CENTER OF ((NINE-POINT CIRCLE, GERGONNE LINE))

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

X(39507) lies on these lines: {5, 516}, {1594, 1826}, {1631, 1656}, {4336, 7741}, {9018, 24206}

X(39508) = CENTER OF ((NINE-POINT CIRCLE, NAGEL LINE))

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

X(39508) lies on these lines: {2, 39225}, {5, 3667}, {427, 16231}, {1594, 7649}, {1656, 4057}, {6003, 21260}, {11585, 20315}, {21262, 24220}

X(39509) = CENTER OF ((NINE-POINT CIRCLE, FERMAT LINE))

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

X(39509) lies on these lines: {4, 14270}, {5, 690}, {115, 2491}, {403, 16230}, {512, 6130}, {523, 19918}, {804, 11620}, {858, 9189}, {5169, 9185}, {6334, 10024}, {9144, 31858}, {9208, 37988}, {14271, 29012}, {15423, 16229}, {20184, 34963}, {21731, 23105}, {32478, 34964}

X(39510) = CENTER OF ((NINE-POINT CIRCLE, VAN AUBEL LINE))

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

X(39510) lies on these lines: {4, 39228}, {5, 525}, {381, 39201}, {1499, 34964}, {1656, 22089}, {5133, 9209}, {10254, 14566}

X(39511) = CENTER OF ((NINE-POINT CIRCLE, X(2)X(6)))

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

X(39511) lies on these lines: {2, 5926}, {3, 31279}, {5, 1499}, {525, 34964}, {547, 25423}, {669, 1656}, {1594, 2501}, {5055, 31176}, {7486, 31299}, {7579, 8371}, {9009, 24206}, {10189, 13413}, {10224, 10279}, {11585, 32204}

X(39511) = midpoint of X(5) and X(23301)

X(39512) = CENTER OF ((NINE-POINT CIRCLE, NAPOLEON AXIS))

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

X(39512) lies on these lines: {5, 32478}, {512, 625}, {690, 34964}, {15475, 31074}, {20184, 34967}, {31279, 34952}

X(39512) = complementary conjugate of X(39018)

X(39513) = CENTER OF ((1ST LEMOINE CIRCLE, EULER LINE))

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

X(39513) lies on these lines: {182, 523}, {804, 9426}, {1691, 2451}, {3049, 5038}

X(39514) = CENTER OF ((1ST LEMOINE CIRCLE, ORTHIC AXIS))

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

X(39514) lies on these lines: {30, 182}, {32, 5946}, {575, 14917}, {3148, 3398}, {10104, 13567}, {11163, 22115}

X(39515) = CENTER OF ((1ST LEMOINE CIRCLE, LEMOINE AXIS))

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

X(39515) lies on these lines: {3, 6}, {10359, 11185}, {12177, 14061}, {13196, 38110}, {33020, 39141}

X(39515) = midpoint of X(182) and X(5034)

X(39516) = CENTER OF ((1ST LEMOINE CIRCLE, DE LONGCHAMPS AXIS))

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

X(39516) lies on these lines: {30, 182}, {49, 7922}, {3060, 3398}, {7759, 32046}, {7878, 13353}, {14880, 18474}

X(39517) = CENTER OF ((1ST LEMOINE CIRCLE, VAN AUBEL LINE))

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

X(39517) lies on these lines: {32, 39228}, {182, 525}, {3398, 39201}, {5012, 9210}, {12054, 22089}, {25644, 37085}

X(39518) = CENTER OF ((1ST LEMOINE CIRCLE, X(2)X(6)))

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

X(39518) lies on these lines: {32, 5926}, {182, 1499}, {669, 3398}, {5012, 11186}

X(39519) = CENTER OF ((1ST LEMOINE CIRCLE, X(1)X(6)))

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

X(39519) lies on these lines: {32, 39227}, {182, 3309}, {667, 3398}, {10104, 31288}, {10359, 21301}

X(39520) = CENTER OF ((2ND LEMOINE CIRCLE, EULER LINE))

Barycentrics    a^2*(b^2 - c^2)*(2*a^4 - 2*a^2*b^2 - 2*a^2*c^2 + 3*b^2*c^2) : :

X(39520) lies on these lines: {6, 523}, {520, 2492}, {526, 2489}, {6753, 14346}, {7651, 9517}

X(39520) = midpoint of X(6) and X(2451)

X(39521) = CENTER OF ((2ND LEMOINE CIRCLE, X(1)X(3)))

Barycentrics    a^2*(b - c)*(2*a^2 - 2*a*b - 2*a*c + 3*b*c) : :

X(39521) lies on these lines: {6, 513}, {44, 21348}, {650, 21758}, {1919, 23650}, {2451, 4132}, {2515, 6371}, {3287, 4802}, {3758, 20906}, {3768, 23472}, {4277, 22095}, {4776, 37685}, {16670, 21390}

X(39521) = midpoint of X(6) and X(20980)

X(39522) = CENTER OF ((2ND LEMOINE CIRCLE, ORTHIC AXIS))

Barycentrics    a^2*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + 8*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - 6*b^6*c^2 + 6*a^4*c^4 + 2*a^2*b^2*c^4 + 10*b^4*c^4 - 4*a^2*c^6 - 6*b^2*c^6 + c^8) : :

X(39522) lies on these lines: {3, 143}, {4, 1994}, {5, 394}, {6, 30}, {20, 36753}, {22, 567}, {24, 37472}, {26, 578}, {49, 10594}, {51, 6644}, {52, 7526}, {54, 7517}, {140, 17825}, {155, 546}, {156, 1598}, {184, 7530}, {186, 11002}, {195, 3843}, {323, 3545}, {376, 34545}, {378, 568}, {381, 1993}, {382, 7592}, {389, 12084}, {399, 14269}, {511, 7514}, {539, 3818}, {547, 17811}, {548, 37514}, {549, 10601}, {550, 36752}, {569, 22352}, {575, 33532}, {576, 8548}, {1147, 10110}, {1154, 1351}, {1181, 3627}, {1192, 10226}, {1199, 3146}, {1493, 5198}, {1498, 3853}, {1539, 19456}, {1593, 6102}, {1595, 13292}, {1597, 5093}, {1658, 11425}, {1995, 22115}, {3088, 18951}, {3098, 21852}, {3431, 37940}, {3517, 32171}, {3520, 37490}, {3524, 15018}, {3527, 6642}, {3530, 15805}, {3534, 15037}, {3543, 15032}, {3830, 11456}, {3839, 11004}, {3845, 18451}, {3850, 17814}, {5012, 12083}, {5020, 13364}, {5050, 35243}, {5054, 15360}, {5055, 15066}, {5064, 34514}, {5066, 37672}, {5097, 6000}, {5447, 13154}, {5448, 18418}, {5462, 13346}, {5480, 11818}, {5651, 14845}, {5654, 34966}, {5876, 12160}, {5890, 13445}, {5892, 37480}, {5899, 6800}, {5944, 9714}, {6101, 7395}, {6243, 7503}, {6288, 7566}, {6985, 36750}, {7387, 11426}, {7393, 10627}, {7485, 13340}, {7502, 33586}, {7506, 9781}, {7509, 37484}, {7516, 10625}, {7525, 37476}, {7545, 9703}, {7564, 9927}, {7689, 16625}, {8717, 15516}, {9545, 34484}, {9707, 18378}, {9715, 10610}, {9730, 15004}, {9786, 11250}, {9820, 15873}, {10113, 18386}, {10201, 23292}, {10323, 13353}, {10546, 14483}, {11003, 37925}, {11402, 18534}, {11413, 37481}, {11422, 14157}, {11430, 18324}, {11432, 12085}, {11449, 38848}, {11479, 11591}, {11801, 17847}, {11819, 19467}, {12022, 31723}, {12102, 15811}, {12106, 13451}, {12118, 31830}, {12173, 35603}, {12228, 16165}, {12236, 15472}, {12241, 18569}, {12605, 31815}, {12699, 16473}, {13567, 18281}, {13568, 34350}, {13598, 37505}, {14449, 17834}, {14787, 37636}, {14853, 18420}, {14865, 16880}, {14915, 15520}, {15058, 15801}, {15559, 25738}, {16226, 37470}, {16472, 18481}, {17702, 34155}, {17928, 37495}, {18400, 34117}, {18494, 30522}, {18570, 37489}, {18952, 23335}, {19347, 32136}, {19357, 37440}, {21969, 37478}, {32196, 32333}, {34783, 35502}, {35921, 37494}

X(39523) = CENTER OF ((2ND LEMOINE CIRCLE, ANTI-ORTHIC AXIS))

Barycentrics    a^2*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5 + a^4*c - 3*a^2*b^2*c + 2*b^4*c - 2*a^3*c^2 - 3*a^2*b*c^2 - 3*b^3*c^2 - 2*a^2*c^3 - 3*b^2*c^3 + a*c^4 + 2*b*c^4 + c^5) : :

X(39523) lies on these lines: {1, 7161}, {3, 5312}, {6, 517}, {40, 36750}, {42, 5398}, {58, 26285}, {65, 15317}, {81, 26446}, {386, 26286}, {387, 10526}, {392, 5422}, {580, 1126}, {912, 4663}, {940, 11231}, {1191, 33179}, {1203, 1482}, {1385, 36754}, {1451, 5399}, {1468, 32612}, {1993, 3753}, {2003, 36279}, {2323, 9708}, {2334, 16202}, {3057, 16472}, {3157, 31794}, {3339, 23070}, {3579, 36742}, {3877, 34545}, {4252, 26086}, {4383, 11230}, {5313, 22765}, {5315, 10247}, {5657, 37685}, {5706, 18480}, {5707, 9956}, {5886, 32911}, {9856, 10982}, {10222, 16466}, {10246, 16474}, {11529, 23071}, {12432, 32047}, {13624, 36745}, {14627, 25413}, {18412, 18455}, {31663, 36746}, {31786, 36752}, {31787, 37498}, {31788, 36747}, {36749, 37562}

X(39524) = CENTER OF ((2ND LEMOINE CIRCLE, DE LONGCHAMPS AXIS))

Barycentrics    a^2*(a^8 - 2*a^6*b^2 + 2*a^4*b^4 - 2*a^2*b^6 + b^8 - 2*a^6*c^2 - 3*a^2*b^4*c^2 - 3*b^6*c^2 + 2*a^4*c^4 - 3*a^2*b^2*c^4 + 4*b^4*c^4 - 2*a^2*c^6 - 3*b^2*c^6 + c^8) : :

X(39524) lies on these lines: {6, 30}, {32, 13391}, {323, 33219}, {381, 34945}, {394, 8360}, {567, 1180}, {1003, 34545}, {1154, 5028}, {1627, 13340}, {1993, 33184}, {1994, 7841}, {3981, 12106}, {5422, 8369}, {7502, 20859}, {8365, 17825}, {8368, 10601}, {9465, 22115}, {11002, 37898}, {11003, 37902}, {11004, 33251}, {15018, 33220}, {17811, 33213}

X(39525) = CENTER OF ((2ND LEMOINE CIRCLE, NAGEL LINE))

Barycentrics    a^2*(b - c)*(a^5 - 2*a^4*b + 2*a^2*b^3 - a*b^4 - 2*a^4*c + 5*a^3*b*c - 3*a^2*b^2*c - a*b^3*c + b^4*c - 3*a^2*b*c^2 + 7*a*b^2*c^2 - 2*b^3*c^2 + 2*a^2*c^3 - a*b*c^3 - 2*b^2*c^3 - a*c^4 + b*c^4) : :

X(39525) lies on these lines: {6, 3667}, {572, 20979}, {2827, 3063}, {4786, 5422}, {6003, 20980}

X(39526) = CENTER OF ((PARRY CIRCLE, EULER LINE))

Barycentrics    a^2*(b^2 - c^2)*(2*a^10*b^2 - 3*a^8*b^4 - a^6*b^6 + 3*a^4*b^8 - a^2*b^10 + 2*a^10*c^2 - 9*a^8*b^2*c^2 + 11*a^6*b^4*c^2 - 8*a^4*b^6*c^2 + 3*a^2*b^8*c^2 - 3*a^8*c^4 + 11*a^6*b^2*c^4 - 2*a^2*b^6*c^4 - 2*b^8*c^4 - a^6*c^6 - 8*a^4*b^2*c^6 - 2*a^2*b^4*c^6 + 5*b^6*c^6 + 3*a^4*c^8 + 3*a^2*b^2*c^8 - 2*b^4*c^8 - a^2*c^10) : :

X(39526) lies on these lines: {351, 523}, {2422, 20998}, {2782, 11176}, {2854, 9188}

X(39527) = CENTER OF ((PARRY CIRCLE, BROCARD AXIS))

Barycentrics    a^2*(b^2 - c^2)*(5*a^8 - 10*a^6*b^2 + 5*a^2*b^6 - b^8 - 10*a^6*c^2 + 30*a^4*b^2*c^2 - 15*a^2*b^4*c^2 - b^6*c^2 - 15*a^2*b^2*c^4 + 9*b^4*c^4 + 5*a^2*c^6 - b^2*c^6 - c^8) : :

X(39527) lies on these lines: {111, 9171}, {187, 237}, {888, 11631}, {2854, 9188}, {9178, 20998}, {9218, 34539}, {9830, 11176}

X(39527) = midpoint of X(351) and X(2502)
X(39527) = X(351)-of-2nd-Brocard-triangle
X(39527) = center of circle {{X(15),X(16),X(351),X(2502)}}

X(39528) = CENTER OF ((PARRY CIRCLE, ORTHIC AXIS))

Barycentrics    a^2*(2*a^12*b^4 - 5*a^10*b^6 + 2*a^8*b^8 + 4*a^6*b^10 - 4*a^4*b^12 + a^2*b^14 - 5*a^10*b^4*c^2 + 8*a^8*b^6*c^2 - 5*a^6*b^8*c^2 + a^4*b^10*c^2 + 4*a^2*b^12*c^2 - 3*b^14*c^2 + 2*a^12*c^4 - 5*a^10*b^2*c^4 + 20*a^8*b^4*c^4 - 19*a^6*b^6*c^4 + 4*a^4*b^8*c^4 - 10*a^2*b^10*c^4 + 10*b^12*c^4 - 5*a^10*c^6 + 8*a^8*b^2*c^6 - 19*a^6*b^4*c^6 + 18*a^4*b^6*c^6 + 3*a^2*b^8*c^6 - 15*b^10*c^6 + 2*a^8*c^8 - 5*a^6*b^2*c^8 + 4*a^4*b^4*c^8 + 3*a^2*b^6*c^8 + 16*b^8*c^8 + 4*a^6*c^10 + a^4*b^2*c^10 - 10*a^2*b^4*c^10 - 15*b^6*c^10 - 4*a^4*c^12 + 4*a^2*b^2*c^12 + 10*b^4*c^12 + a^2*c^14 - 3*b^2*c^14) : :

Let A15 be the intersection, other than X(15), of line AX(15) and the Parry circle, and define B15 and C15 cyclically. Let A16 be the intersection, other than X(16), of line AX(16) and the Parry circle, and define B16 and C16 cyclically. Let SASBSC be the side-triangle of A15B15C15 and A16B16C16. Let VAVBVC be the vertex-triangle of A15B15C15 and A16B16C16. SASBSC and VAVBVC are perspective at X(39528). (Randy Hutson, September 30, 2020)

X(39528) lies on these lines: {2, 2782}, {30, 351}, {511, 9208}, {542, 9175}, {1511, 2502}, {5191, 7468}, {5653, 5663}, {9129, 31861}, {9130, 33532}, {14984, 14998}

X(39529) = CENTER OF ((POLAR CIRCLE, ANTI-ORTHIC AXIS))

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^5*b - 2*a^3*b^3 + a*b^5 + a^5*c - 2*a^2*b^3*c - a*b^4*c + 2*b^5*c - 2*a^3*c^3 - 2*a^2*b*c^3 - 4*b^3*c^3 - a*b*c^4 + a*c^5 + 2*b*c^5) : :

X(39529) lies on these lines: {1, 7524}, {3, 6708}, {4, 8}, {5, 1214}, {29, 1385}, {104, 17519}, {119, 1826}, {158, 942}, {243, 24929}, {278, 5886}, {281, 26446}, {286, 6528}, {354, 1784}, {412, 3579}, {495, 1785}, {515, 7510}, {912, 1859}, {944, 7518}, {1013, 32613}, {1148, 31794}, {1598, 23853}, {1842, 7511}, {1844, 24475}, {1852, 37290}, {1857, 5722}, {1865, 30444}, {1895, 5045}, {1940, 37582}, {1957, 5398}, {3091, 6360}, {3652, 31902}, {5125, 9956}, {5307, 7497}, {5805, 36876}, {7330, 7534}, {7531, 13624}, {7541, 38140}, {7551, 11230}, {8680, 15762}, {10269, 37393}, {10746, 12918}, {15763, 26470}, {23961, 37304}, {32612, 37253}, {34339, 37235}

X(39529) = midpoint of X(4) and X(92)

X(39530) = CENTER OF ((POLAR CIRCLE, LEMOINE AXIS))

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - a^2*b^2 - a^2*c^2 - 2*b^2*c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) : :

In the plane of a triangle ABC, let
N=X(5), DEF = orthic triangle, A'B'C = Euler triangle, A"=ND∩B'C', and define B' and C' cyclically. The lines A'A", B'B", C'C" concur in X(39530). (Angel Montesdeoca, August 2, 2021)

See X(39530).

X(39530) lies on these lines: {2, 26895}, {3, 14767}, {4, 69}, {5, 53}, {51, 324}, {52, 14978}, {114, 136}, {133, 14672}, {182, 458}, {275, 34986}, {297, 24206}, {338, 19161}, {343, 6755}, {381, 30258}, {393, 14561}, {418, 11197}, {472, 5617}, {473, 5613}, {575, 36794}, {576, 9308}, {648, 5097}, {1353, 6749}, {1368, 37873}, {1990, 18583}, {2052, 5943}, {2207, 10358}, {2782, 33843}, {3091, 3164}, {3098, 37200}, {3564, 6748}, {3575, 32152}, {5092, 37124}, {5133, 6747}, {5965, 27377}, {6530, 19130}, {6688, 15466}, {9742, 35710}, {10282, 37127}, {10796, 14581}, {13334, 37337}, {13450, 17500}, {14516, 35717}, {14810, 35474}, {16264, 29012}, {17907, 38317}

X(39530) = midpoint of X(4) and X(264)

X(39531) = CENTER OF ((POLAR CIRCLE, GERGONNE LINE))

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^5 - 2*a^2*b^3 - a*b^4 + 2*b^5 + 2*a*b^2*c^2 - 2*b^3*c^2 - 2*a^2*c^3 - 2*b^2*c^3 - a*c^4 + 2*c^5) : :

X(39531) lies on these lines: {4, 9}, {5, 17073}, {29, 5691}, {33, 118}, {92, 1699}, {123, 8727}, {158, 342}, {204, 17902}, {226, 1857}, {243, 5219}, {273, 38150}, {278, 3817}, {347, 1838}, {412, 1698}, {653, 4312}, {857, 30265}, {946, 10002}, {993, 37380}, {1598, 36641}, {1754, 7076}, {1785, 10590}, {1844, 12528}, {1859, 5927}, {1867, 37197}, {1872, 21867}, {1895, 5290}, {1940, 9579}, {3176, 3671}, {3576, 7551}, {3634, 37417}, {3947, 7952}, {4297, 7498}, {5125, 7989}, {5174, 37714}, {5251, 37258}, {5307, 37372}, {5732, 37448}, {7046, 21060}, {7524, 18480}, {7988, 17923}, {10171, 17917}, {28164, 37028}

X(39531) = midpoint of X(4) and X(281)

X(39532) = CENTER OF ((POLAR CIRCLE, SODDY LINE))

Barycentrics    (b - c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^2 - 4*a*b + b^2 - 4*a*c + 4*b*c + c^2) : :

X(39532) lies on these lines: {4, 514}, {318, 4546}, {522, 16228}, {3887, 18344}, {4151, 16229}, {4962, 7649}

X(39533) = CENTER OF ((POLAR CIRCLE, X(2)X(6)))

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-3*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 - 6*b^2*c^2 + c^4) : :
Trilinears    (csc 2A) ((2 cos B - cos C cos A) (sin C - 2 cos C cot ω) - (2 cos C - cos A cos B) (sin B - 2 cos B cot ω)) : :

X(39533) lies on these lines: {3, 14341}, {4, 1499}, {25, 5926}, {132, 25641}, {523, 16231}, {525, 16229}, {669, 1598}, {1595, 23301}, {1885, 10280}, {2489, 2510}, {2793, 16183}, {3091, 6563}, {3800, 14618}, {5512, 8754}, {6753, 11615}, {10982, 30451}

X(39533) = midpoint of X(4) and X(2501)
X(39533) = pole wrt polar circle of trilinear polar of line X(376)X(524)

X(39534) = CENTER OF ((POLAR CIRCLE, X(1)X(5)))

Barycentrics    (b - c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3) : :
Trilinears    (b - c) (sec A) (cos B + cos C - 1) : :

X(39534) lies on these lines: {4, 900}, {11, 2969}, {19, 22108}, {25, 26275}, {108, 676}, {119, 2804}, {403, 523}, {513, 1835}, {522, 16228}, {665, 1841}, {1769, 1846}, {1785, 35013}, {1845, 8677}, {2849, 21180}, {4194, 26144}, {4200, 26078}, {4207, 4800}, {6370, 16230}, {7378, 31131}, {8819, 24006}, {8889, 30792}, {11109, 25996}, {17555, 25923}, {17981, 18002}, {23345, 36123}, {28076, 28114}, {28284, 37384}

X(39534) = polar conjugate of X(13136)
X(39534) = pole wrt polar circle of trilinear polar of X(13136) (line X(3)X(8))

X(39535) = CENTER OF ((POLAR CIRCLE, SHERMAN LINE))

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(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)*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 - 2*a^5*b^2*c + 2*a^4*b^3*c + 4*a^3*b^4*c - 4*a^2*b^5*c - 2*a*b^6*c + 2*b^7*c + a^6*c^2 - 2*a^5*b*c^2 + 4*a^4*b^2*c^2 - 4*a^3*b^3*c^2 - 3*a^2*b^4*c^2 + 6*a*b^5*c^2 - 2*b^6*c^2 + 2*a^4*b*c^3 - 4*a^3*b^2*c^3 + 8*a^2*b^3*c^3 - 4*a*b^4*c^3 - 2*b^5*c^3 - 3*a^4*c^4 + 4*a^3*b*c^4 - 3*a^2*b^2*c^4 - 4*a*b^3*c^4 + 6*b^4*c^4 - 4*a^2*b*c^5 + 6*a*b^2*c^5 - 2*b^3*c^5 + 3*a^2*c^6 - 2*a*b*c^6 - 2*b^2*c^6 + 2*b*c^7 - c^8) : :

X(39535) lies on these lines: {2, 2734}, {4, 953}, {5, 10017}, {11, 1785}, {117, 522}, {123, 6882}, {124, 515}, {125, 860}, {225, 13999}, {2808, 35590}, {7952, 35580}, {16177, 36155}, {16228, 25640}, {31841, 35013}

X(39535) = midpoint of X(4) and X(1309)
X(39535) = polar-circle-inverse of X(953)

X(39536) = CENTER OF ((POLAR CIRCLE, X(1)X(6))

Barycentrics    (b - c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^5 - 2*a^4*b + 2*a^2*b^3 - a*b^4 - 2*a^4*c + 2*a^3*b*c - 2*a*b^3*c + 2*b^4*c + 2*a*b^2*c^2 - 2*b^3*c^2 + 2*a^2*c^3 - 2*a*b*c^3 - 2*b^2*c^3 - a*c^4 + 2*b*c^4) : :

X(39536) lies on these lines: {4, 885}, {28, 39227}, {522, 16228}, {525, 16229}, {667, 7497}, {1838, 3669}, {1871, 4083}, {3064, 28473}, {5511, 8735}, {23595, 30199}, {23882, 39212}

X(39536) = midpoint of X(4) and X(17924)

X(39537) = CENTER OF ((TANGENTIAL CIRCLE, BROCARD AXIS))

Barycentrics    a^2*(b^2 - c^2)*(a^12 - 3*a^10*b^2 + 2*a^8*b^4 + 2*a^6*b^6 - 3*a^4*b^8 + a^2*b^10 - 3*a^10*c^2 + 5*a^8*b^2*c^2 - 2*a^6*b^4*c^2 + a^2*b^8*c^2 - b^10*c^2 + 2*a^8*c^4 - 2*a^6*b^2*c^4 + 2*b^8*c^4 + 2*a^6*c^6 - 2*b^6*c^6 - 3*a^4*c^8 + a^2*b^2*c^8 + 2*b^4*c^8 + a^2*c^10 - b^2*c^10) : :

X(39537) lies on these lines: {24, 16229}, {26, 512}, {2524, 9700}, {7502, 34964}

X(39538) = CENTER OF ((TANGENTIAL CIRCLE, ORTHIC AXIS))

Barycentrics    a^2*(a^20 - 4*a^18*b^2 + 3*a^16*b^4 + 8*a^14*b^6 - 14*a^12*b^8 + 14*a^8*b^12 - 8*a^6*b^14 - 3*a^4*b^16 + 4*a^2*b^18 - b^20 - 4*a^18*c^2 + 10*a^16*b^2*c^2 - 2*a^14*b^4*c^2 - 10*a^12*b^6*c^2 + 6*a^10*b^8*c^2 - 6*a^8*b^10*c^2 + 10*a^6*b^12*c^2 + 2*a^4*b^14*c^2 - 10*a^2*b^16*c^2 + 4*b^18*c^2 + 3*a^16*c^4 - 2*a^14*b^2*c^4 - 8*a^12*b^4*c^4 + 10*a^10*b^6*c^4 + 2*a^8*b^8*c^4 - 14*a^6*b^10*c^4 + 8*a^4*b^12*c^4 + 6*a^2*b^14*c^4 - 5*b^16*c^4 + 8*a^14*c^6 - 10*a^12*b^2*c^6 + 10*a^10*b^4*c^6 - 4*a^8*b^6*c^6 + 4*a^6*b^8*c^6 - 10*a^4*b^10*c^6 + 2*a^2*b^12*c^6 - 14*a^12*c^8 + 6*a^10*b^2*c^8 + 2*a^8*b^4*c^8 + 4*a^6*b^6*c^8 + 6*a^4*b^8*c^8 - 2*a^2*b^10*c^8 + 6*b^12*c^8 - 6*a^8*b^2*c^10 - 14*a^6*b^4*c^10 - 10*a^4*b^6*c^10 - 2*a^2*b^8*c^10 - 8*b^10*c^10 + 14*a^8*c^12 + 10*a^6*b^2*c^12 + 8*a^4*b^4*c^12 + 2*a^2*b^6*c^12 + 6*b^8*c^12 - 8*a^6*c^14 + 2*a^4*b^2*c^14 + 6*a^2*b^4*c^14 - 3*a^4*c^16 - 10*a^2*b^2*c^16 - 5*b^4*c^16 + 4*a^2*c^18 + 4*b^2*c^18 - c^20) : :

X(39538) lies on this line: {2,3}

X(39538) = midpoint of X(26) and X(21213)
X(39538) = center of circle {{X(26),X(21213),PU(4)}}

X(39539) = CENTER OF ((TANGENTIAL CIRCLE, DE LONGCHAMPS AXIS))

Barycentrics    a^2*(a^20 - 4*a^18*b^2 + 3*a^16*b^4 + 8*a^14*b^6 - 14*a^12*b^8 + 14*a^8*b^12 - 8*a^6*b^14 - 3*a^4*b^16 + 4*a^2*b^18 - b^20 - 4*a^18*c^2 + 8*a^16*b^2*c^2 + 4*a^14*b^4*c^2 - 12*a^12*b^6*c^2 - 4*a^10*b^8*c^2 + 4*a^8*b^10*c^2 + 12*a^6*b^12*c^2 - 4*a^4*b^14*c^2 - 8*a^2*b^16*c^2 + 4*b^18*c^2 + 3*a^16*c^4 + 4*a^14*b^2*c^4 - 16*a^12*b^4*c^4 + 8*a^10*b^6*c^4 + 2*a^8*b^8*c^4 - 12*a^6*b^10*c^4 + 16*a^4*b^12*c^4 - 5*b^16*c^4 + 8*a^14*c^6 - 12*a^12*b^2*c^6 + 8*a^10*b^4*c^6 + 4*a^6*b^8*c^6 - 16*a^4*b^10*c^6 + 8*a^2*b^12*c^6 - 14*a^12*c^8 - 4*a^10*b^2*c^8 + 2*a^8*b^4*c^8 + 4*a^6*b^6*c^8 + 14*a^4*b^8*c^8 - 4*a^2*b^10*c^8 + 6*b^12*c^8 + 4*a^8*b^2*c^10 - 12*a^6*b^4*c^10 - 16*a^4*b^6*c^10 - 4*a^2*b^8*c^10 - 8*b^10*c^10 + 14*a^8*c^12 + 12*a^6*b^2*c^12 + 16*a^4*b^4*c^12 + 8*a^2*b^6*c^12 + 6*b^8*c^12 - 8*a^6*c^14 - 4*a^4*b^2*c^14 - 3*a^4*c^16 - 8*a^2*b^2*c^16 - 5*b^4*c^16 + 4*a^2*c^18 + 4*b^2*c^18 - c^20) : :

X(39539) lies on this line: {2,3}

X(39540) = CENTER OF ((INCIRCLE, EULER LINE)

Barycentrics    (b - c)*(-4*a^4 + a^3*b + 3*a^2*b^2 - a*b^3 + b^4 + a^3*c + 2*a^2*b*c - 3*a*b^2*c + 3*a^2*c^2 - 3*a*b*c^2 - 2*b^2*c^2 - a*c^3 + c^4) : :

X(39540) lies on these lines: {1, 523}, {354, 5583}, {521, 676}, {1459, 6362}, {3900, 21172}, {4897, 17096}, {4977, 21185}, {4990, 23874}, {5045, 34954}, {6003, 34958}, {6366, 7649}, {15252, 33562}

X(39541) = CENTER OF ((INCIRCLE, BROCARD AXIS))

Barycentrics    a^2*(b - c)*(a^3*b - a^2*b^2 - a*b^3 + b^4 + a^3*c + 2*a^2*b*c - 3*a*b^2*c - a^2*c^2 - 3*a*b*c^2 + 2*b^2*c^2 - a*c^3 + c^4) : :

X(39541) lies on these lines: {1, 512}, {354, 7178}, {514, 2488}, {520, 2605}, {905, 926}, {918, 2494}, {942, 28473}, {1210, 21051}, {2495, 28846}, {2821, 4162}, {3803, 6371}, {4091, 8641}, {4129, 11019}, {4524, 14838}, {5045, 34958}, {6003, 7250}, {6372, 21185}, {12915, 29126}

X(39542) = CENTER OF ((INCIRCLE, ORTHIC AXIS))

Barycentrics    2*a^3*b + a^2*b^2 - 2*a*b^3 - b^4 + 2*a^3*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(39542) lies on these lines: {1, 30}, {2, 36279}, {3, 3474}, {4, 15911}, {5, 65}, {7, 104}, {8, 5714}, {10, 3838}, {11, 5902}, {12, 5690}, {36, 7508}, {40, 5763}, {46, 140}, {55, 5719}, {56, 5901}, {57, 5886}, {72, 25006}, {191, 24953}, {221, 5707}, {226, 495}, {329, 9708}, {354, 11551}, {355, 3340}, {376, 37606}, {377, 5730}, {381, 1159}, {382, 3486}, {388, 1482}, {392, 5249}, {405, 11415}, {442, 3869}, {484, 5432}, {496, 942}, {497, 15934}, {498, 37567}, {499, 5221}, {515, 37728}, {516, 8255}, {546, 1837}, {548, 3612}, {549, 1155}, {550, 1770}, {551, 5126}, {653, 7551}, {758, 2886}, {908, 3753}, {912, 5173}, {920, 16617}, {938, 9669}, {944, 4323}, {950, 22793}, {952, 1478}, {954, 36976}, {956, 5905}, {960, 8728}, {962, 3295}, {993, 17768}, {995, 1086}, {997, 5880}, {1006, 8543}, {1056, 6925}, {1058, 10431}, {1060, 2263}, {1118, 7524}, {1125, 4640}, {1191, 24159}, {1210, 9955}, {1279, 26728}, {1319, 10283}, {1329, 3754}, {1385, 4292}, {1388, 4317}, {1428, 38040}, {1452, 21841}, {1470, 11729}, {1479, 12433}, {1483, 11011}, {1484, 11570}, {1519, 7956}, {1538, 7682}, {1596, 1905}, {1621, 5180}, {1656, 1788}, {1657, 4305}, {1699, 5722}, {1709, 3333}, {1727, 3337}, {1739, 37663}, {1745, 37698}, {1854, 5878}, {2093, 5219}, {2292, 29682}, {2362, 7584}, {2550, 3940}, {2771, 18389}, {2800, 7680}, {2975, 14450}, {3057, 13407}, {3085, 12702}, {3086, 5708}, {3244, 13463}, {3245, 3584}, {3336, 5433}, {3338, 11376}, {3339, 8227}, {3361, 9624}, {3428, 5762}, {3454, 5835}, {3475, 6767}, {3476, 10247}, {3488, 9668}, {3576, 4312}, {3579, 13411}, {3583, 5425}, {3585, 10950}, {3600, 6938}, {3614, 18395}, {3616, 5303}, {3627, 10572}, {3628, 24914}, {3654, 31434}, {3678, 9710}, {3812, 17527}, {3813, 3874}, {3814, 3919}, {3816, 5883}, {3825, 33815}, {3826, 10176}, {3850, 10826}, {3853, 37724}, {3861, 37721}, {3868, 24390}, {3872, 31164}, {3877, 31019}, {3878, 11263}, {3892, 21630}, {3911, 11230}, {3916, 24541}, {3918, 9711}, {3925, 5692}, {3927, 19843}, {3944, 37715}, {3947, 11362}, {4004, 24982}, {4018, 6734}, {4023, 4714}, {4084, 25639}, {4245, 15507}, {4293, 10246}, {4298, 13464}, {4299, 34471}, {4301, 9957}, {4302, 28178}, {4304, 28146}, {4311, 15178}, {4315, 25405}, {4325, 24926}, {4333, 12103}, {4415, 30116}, {4424, 5718}, {4511, 11112}, {4848, 9956}, {4860, 10072}, {5045, 10391}, {5049, 5542}, {5057, 11113}, {5082, 20015}, {5119, 17718}, {5122, 10165}, {5131, 5444}, {5195, 14828}, {5226, 5657}, {5228, 15251}, {5229, 18525}, {5248, 11281}, {5252, 5844}, {5253, 19525}, {5261, 12245}, {5270, 10944}, {5290, 7982}, {5298, 38022}, {5327, 36011}, {5587, 11545}, {5665, 5715}, {5691, 37739}, {5694, 15556}, {5697, 15888}, {5698, 16418}, {5703, 6361}, {5761, 10306}, {5790, 10590}, {5791, 12526}, {5805, 12560}, {5812, 31799}, {5836, 21077}, {6261, 20420}, {6675, 12514}, {6738, 18483}, {6797, 21635}, {6830, 38039}, {6831, 33899}, {6911, 37541}, {6922, 34339}, {6924, 11509}, {6974, 21454}, {7288, 37545}, {7583, 16232}, {7681, 31870}, {7686, 12608}, {7951, 38042}, {8703, 37600}, {9613, 37727}, {9614, 11518}, {9623, 28609}, {10044, 22768}, {10052, 18967}, {10106, 10222}, {10129, 17530}, {10386, 37080}, {10573, 10895}, {10609, 17579}, {10943, 24475}, {10980, 37704}, {11010, 37731}, {11041, 13257}, {11108, 28629}, {11237, 12647}, {11573, 35631}, {12432, 20117}, {12611, 12736}, {12709, 24474}, {12943, 28186}, {13369, 16193}, {13375, 25414}, {13405, 28194}, {13462, 38041}, {13750, 37356}, {14110, 37424}, {15326, 37525}, {15338, 37571}, {15908, 37625}, {16763, 38063}, {16821, 33066}, {17139, 19259}, {17638, 33593}, {17724, 37610}, {17728, 23708}, {17753, 33949}, {17757, 31053}, {18243, 31673}, {18398, 37722}, {18419, 38038}, {20328, 30949}, {21049, 24045}, {21629, 21848}, {21740, 37468}, {24473, 26015}, {26437, 32153}, {28224, 37740}, {28458, 35459}, {31777, 37531}, {32857, 37617}

X(39542) = midpoint of X(1) and X(1836)

X(39543) = CENTER OF ((INCIRCLE, LEMOINE AXIS))

Barycentrics    a^2*(a^2*b^2 - b^4 + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 4*b^2*c^2 - c^4) : :

X(39543) lies on these lines: {1, 256}, {37, 9052}, {42, 5943}, {43, 6688}, {51, 17018}, {65, 29309}, {181, 3750}, {182, 37580}, {354, 29349}, {373, 3240}, {496, 24220}, {497, 35612}, {516, 942}, {518, 17332}, {573, 3295}, {674, 15569}, {740, 17049}, {991, 999}, {1001, 4260}, {1058, 10446}, {1206, 21813}, {1386, 2876}, {1486, 5138}, {1742, 3333}, {2293, 37609}, {3271, 4649}, {3555, 4416}, {3664, 5045}, {3720, 3819}, {3881, 17770}, {3889, 17364}, {3917, 29814}, {3931, 12109}, {3993, 14839}, {4032, 29073}, {4263, 23841}, {5049, 29353}, {5092, 37576}, {6007, 24325}, {9729, 37529}, {9957, 29311}, {10110, 37698}, {10122, 29097}, {10219, 16569}, {13334, 18758}, {13476, 17768}, {15082, 30950}, {15254, 22277}, {15489, 37573}, {15516, 20958}, {20961, 21849}, {34379, 34791}

X(39543) = midpoint of X(1) and X(21746)

X(39544) = CENTER OF ((INCIRCLE, DE LONGCHAMPS AXIS))

Barycentrics    2*a^4 + a^2*b^2 + 4*a*b^3 + b^4 + a^2*c^2 - 2*b^2*c^2 + 4*a*c^3 + c^4 : :

X(39544) lies on these lines: {1, 30}, {5, 37549}, {37, 26728}, {72, 26723}, {140, 3670}, {376, 4346}, {495, 30448}, {545, 551}, {547, 37691}, {549, 17595}, {758, 17061}, {942, 34937}, {982, 15325}, {999, 4310}, {1086, 30115}, {2292, 29689}, {3333, 21375}, {3487, 19542}, {3616, 32933}, {3663, 24929}, {3666, 5719}, {3677, 5886}, {3721, 5305}, {3744, 28174}, {3745, 11551}, {3940, 4000}, {4234, 4440}, {4353, 29069}, {4415, 30117}, {4419, 16418}, {4424, 17724}, {4653, 17246}, {5306, 36283}, {5902, 17602}, {6691, 24167}, {8359, 30113}, {8368, 30886}, {8728, 24159}, {9669, 36583}, {10593, 36574}, {11036, 19645}, {11112, 33146}, {11113, 33151}, {11545, 37716}, {13161, 37730}, {17054, 17527}, {17276, 37817}, {17721, 38034}, {17757, 33153}, {24470, 37539}, {28212, 37610}, {33152, 37715}, {37592, 37737}

X(39544) = midpoint of X(1) and X(3782)

X(39545) = CENTER OF ((INCIRCLE, X(2)X(6))

Barycentrics    (b - c)*(-4*a^3 - a^2*b + b^3 - a^2*c - 4*a*b*c + b^2*c + b*c^2 + c^3) : :

X(39545) lies on these lines: {1, 1499}, {10, 2487}, {525, 4367}, {676, 29148}, {1019, 3800}, {1125, 14321}, {3600, 31603}, {3669, 28481}, {3676, 28475}, {3798, 14077}, {4106, 28306}, {4160, 17069}, {4458, 29126}, {4468, 30234}, {4504, 28473}, {4927, 28312}, {5926, 16678}, {6002, 34958}, {7178, 28533}, {7180, 37592}, {28569, 30719}

X(39545) = midpoint of X(1) and X(4897)

X(39546) = CENTER OF ((INCIRCLE, SHERMAN LINE)

Barycentrics    2*a^10 - 4*a^9*b - 2*a^8*b^2 + 10*a^7*b^3 - 7*a^6*b^4 - 6*a^5*b^5 + 13*a^4*b^6 - 2*a^3*b^7 - 7*a^2*b^8 + 2*a*b^9 + b^10 - 4*a^9*c + 16*a^8*b*c - 14*a^7*b^2*c - 16*a^6*b^3*c + 40*a^5*b^4*c - 18*a^4*b^5*c - 22*a^3*b^6*c + 20*a^2*b^7*c - 2*b^9*c - 2*a^8*c^2 - 14*a^7*b*c^2 + 48*a^6*b^2*c^2 - 34*a^5*b^3*c^2 - 37*a^4*b^4*c^2 + 54*a^3*b^5*c^2 - 6*a^2*b^6*c^2 - 6*a*b^7*c^2 - 3*b^8*c^2 + 10*a^7*c^3 - 16*a^6*b*c^3 - 34*a^5*b^2*c^3 + 84*a^4*b^3*c^3 - 30*a^3*b^4*c^3 - 20*a^2*b^5*c^3 - 2*a*b^6*c^3 + 8*b^7*c^3 - 7*a^6*c^4 + 40*a^5*b*c^4 - 37*a^4*b^2*c^4 - 30*a^3*b^3*c^4 + 26*a^2*b^4*c^4 + 6*a*b^5*c^4 + 2*b^6*c^4 - 6*a^5*c^5 - 18*a^4*b*c^5 + 54*a^3*b^2*c^5 - 20*a^2*b^3*c^5 + 6*a*b^4*c^5 - 12*b^5*c^5 + 13*a^4*c^6 - 22*a^3*b*c^6 - 6*a^2*b^2*c^6 - 2*a*b^3*c^6 + 2*b^4*c^6 - 2*a^3*c^7 + 20*a^2*b*c^7 - 6*a*b^2*c^7 + 8*b^3*c^7 - 7*a^2*c^8 - 3*b^2*c^8 + 2*a*c^9 - 2*b*c^9 + c^10 : :

X(39546) lies on these lines: {1, 3319}, {515, 12735}, {517, 15252}, {522, 1387}, {999, 2716}, {1785, 5048}, {2222, 3295}, {10271, 25405}

X(39546) = midpoint of X(1) and X(3326)

X(39547) = CENTER OF ((CONWAY CIRCLE, EULER LINE)

Barycentrics    (b - c)*(a^4 + a*b^3 + a^2*b*c + 3*a*b^2*c + b^3*c + 3*a*b*c^2 + 2*b^2*c^2 + a*c^3 + b*c^3) : :

X(39547) lies on these lines: {1, 523}, {314, 4374}, {513, 4170}, {521, 7662}, {522, 4367}, {693, 832}, {784, 1459}, {1577, 38469}, {1999, 4789}, {2533, 35057}, {3063, 6590}, {3287, 22044}, {4024, 20981}, {4139, 4581}, {4378, 28623}, {4467, 17212}, {4705, 8062}, {7253, 8672}, {9013, 30591}, {10472, 17066}, {23752, 29094}, {34954, 35620}

X(39547) = midpoint of X(1) and X(5214)

X(39548) = CENTER OF ((CONWAY CIRCLE, BROCARD AXIS))

Barycentrics    a^2*(b - c)*(a^2*b^2 + a*b^3 + a^2*b*c + 3*a*b^2*c + b^3*c + a^2*c^2 + 3*a*b*c^2 + b^2*c^2 + a*c^3 + b*c^3) : :

X(39548) lies on these lines: {1, 512}, {513, 4170}, {514, 2978}, {667, 834}, {788, 14349}, {3741, 4129}, {3805, 7265}, {4040, 6371}, {4079, 21763}, {5214, 8672}, {7178, 10473}, {7254, 16874}, {9040, 10477}, {10441, 28473}, {10479, 21051}, {29126, 35645}, {34958, 35620}

X(39548) = midpoint of X(1) and X(5216)

X(39549) = CENTER OF ((CONWAY CIRCLE, ORTHIC AXIS))

Barycentrics    a^7 + 2*a^6*b + 3*a^5*b^2 + 2*a^4*b^3 - 3*a^3*b^4 - 4*a^2*b^5 - a*b^6 + 2*a^6*c + 8*a^5*b*c - 4*a^3*b^3*c - 4*a*b^5*c - 2*b^6*c + 3*a^5*c^2 + 2*a^3*b^2*c^2 + a*b^4*c^2 - 2*b^5*c^2 + 2*a^4*c^3 - 4*a^3*b*c^3 + 8*a*b^3*c^3 + 4*b^4*c^3 - 3*a^3*c^4 + a*b^2*c^4 + 4*b^3*c^4 - 4*a^2*c^5 - 4*a*b*c^5 - 2*b^2*c^5 - a*c^6 - 2*b*c^6 : :

X(39549) lies on these lines: {1, 30}, {515, 32946}, {946, 29644}, {1012, 10444}, {1709, 10476}, {4362, 28194}, {6001, 10441}, {6907, 10888}, {6914, 10882}, {6938, 10465}, {10391, 35620}, {10442, 35638}, {12435, 14988}

X(39550) = CENTER OF ((CONWAY CIRCLE, ANTI-ORTHIC AXIS))

Barycentrics    a*(a^4*b^2 + a^3*b^3 - a^2*b^4 - a*b^5 - a^4*b*c + 3*a^3*b^2*c + 2*a^2*b^3*c - 3*a*b^4*c - b^5*c + a^4*c^2 + 3*a^3*b*c^2 - 4*a^2*b^2*c^2 + 2*a*b^3*c^2 + a^3*c^3 + 2*a^2*b*c^3 + 2*a*b^2*c^3 + 2*b^3*c^3 - a^2*c^4 - 3*a*b*c^4 - a*c^5 - b*c^5) : :

X(39550) lies on these lines: {1, 3}, {376, 29309}, {511, 5603}, {551, 29311}, {944, 15488}, {946, 33064}, {970, 3616}, {1125, 10440}, {2801, 35638}, {3293, 19549}, {3741, 10175}, {5752, 5901}, {6176, 29814}, {8679, 34647}, {9052, 34625}, {10176, 35628}, {10446, 29349}, {10454, 28186}, {10478, 38034}, {10479, 38042}, {11194, 20718}, {12109, 14986}, {12545, 28150}, {15172, 31774}, {15310, 31162}, {22791, 37482}, {28236, 35633}

X(39550) = midpoint of X(1) and X(10439)

X(39551) = CENTER OF ((CONWAY CIRCLE, LEMOINE AXIS))

Barycentrics    a^2*(a^4*b^3 + a^3*b^4 - a^2*b^5 - a*b^6 + 2*a^3*b^3*c + a^2*b^4*c - 2*a*b^5*c - b^6*c + 3*a^3*b^2*c^2 + a^2*b^3*c^2 - 2*b^5*c^2 + a^4*c^3 + 2*a^3*b*c^3 + a^2*b^2*c^3 + b^4*c^3 + a^3*c^4 + a^2*b*c^4 + b^3*c^4 - a^2*c^5 - 2*a*b*c^5 - 2*b^2*c^5 - a*c^6 - b*c^6) : :

X(39551) lies on these lines: {1, 256}, {320, 10446}, {516, 10441}, {517, 30273}, {991, 37620}, {1742, 10476}, {3244, 29311}, {3664, 35620}, {3741, 37521}, {3784, 10473}, {10439, 29349}, {10449, 24523}, {12435, 29309}, {26892, 35614}, {29097, 35637}

X(39552) = CENTER OF ((CONWAY CIRCLE, DE LONGCHAMPS AXIS))

Barycentrics    a^7 + a^6*b + a^5*b^2 + a^4*b^3 - 2*a^3*b^4 - 2*a^2*b^5 + a^6*c + 4*a^5*b*c - 2*a^3*b^3*c - 2*a*b^5*c - b^6*c + a^5*c^2 - 2*a^3*b^2*c^2 - b^5*c^2 + a^4*c^3 - 2*a^3*b*c^3 + 4*a*b^3*c^3 + 2*b^4*c^3 - 2*a^3*c^4 + 2*b^3*c^4 - 2*a^2*c^5 - 2*a*b*c^5 - b^2*c^5 - b*c^6 : :

X(39552) lies on these lines: {1, 30}, {515, 33064}, {993, 3923}, {1999, 3868}, {5791, 18229}, {6763, 10476}, {10444, 29243}, {29073, 30269}

X(39553) = CENTER OF ((CONWAY CIRCLE, GERGONNE LINE)

Barycentrics    a^6 + a^5*b + 4*a^4*b^2 - 5*a^2*b^4 - a*b^5 + a^5*c + 4*a^4*b*c - 2*a^3*b^2*c + 2*a^2*b^3*c - 3*a*b^4*c - 2*b^5*c + 4*a^4*c^2 - 2*a^3*b*c^2 - 2*a^2*b^2*c^2 + 4*a*b^3*c^2 + 2*a^2*b*c^3 + 4*a*b^2*c^3 + 4*b^3*c^3 - 5*a^2*c^4 - 3*a*b*c^4 - a*c^5 - 2*b*c^5 : :

X(39553) lies on these lines: {1, 7}, {9, 10476}, {40, 24325}, {518, 5786}, {971, 35631}, {1001, 37620}, {1125, 10443}, {1699, 27184}, {2801, 10439}, {3741, 10888}, {3751, 6996}, {3775, 5587}, {4362, 12555}, {5208, 12669}, {5223, 11679}, {5272, 9535}, {5572, 35620}, {5728, 10473}, {5850, 17733}, {5851, 35638}, {7982, 32921}, {8245, 11522}, {8581, 21334}, {10394, 35617}, {10453, 36991}, {11021, 20116}, {12547, 35632}, {36682, 38146}

X(39553) = midpoint of X(1) and X(10442)
X(39553) = X(182)-of-3rd-Conway-triangle

X(39554) = CIRCUMCIRCLE-INVERSE OF X(61)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 - c^4 + 2*Sqrt[3]*(2*a^2 - b^2 - c^2)*S) : :

X(39554) = X(15) + 2*X(16)

X(39554) lies on these lines: {3, 6}, {5, 10617}, {13, 38230}, {14, 33389}, {17, 622}, {18, 20428}, {23, 14705}, {30, 5469}, {230, 23005}, {316, 6672}, {382, 16630}, {396, 6779}, {530, 16267}, {531, 16963}, {532, 16529}, {623, 14712}, {624, 11307}, {625, 11311}, {636, 33225}, {691, 2380}, {1003, 25167}, {1080, 5478}, {2912, 3489}, {3181, 14144}, {3412, 14138}, {3457, 16461}, {3479, 34219}, {3564, 22998}, {3849, 11298}, {5184, 11708}, {5965, 9115}, {5995, 6104}, {6109, 36968}, {6297, 37341}, {6582, 35917}, {6695, 33021}, {6781, 23004}, {6783, 25235}, {7684, 16965}, {7685, 36992}, {8595, 22495}, {9218, 33957}, {12205, 22691}, {14139, 33464}, {14170, 34394}, {14693, 20429}, {16961, 33518}, {31939, 36514}, {33381, 33387}

X(39554) = reflection of X(39555) in X(187)
X(39554) = isogonal conjugate of X(11602)
X(39554) = {X(16),X(187)}-harmonic conjugate of X(15)
X(39554) = radical trace of circumcircle and 7th Lozada circle
X(39554) = center of circle {{X(16),PU(2)}}
X(39554) = Brocard axis intercept, other than X(15), of circle {{X(15),PU(2)}}

X(39555) = CIRCUMCIRCLE-INVERSE OF X(62)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 - c^4 - 2*Sqrt[3]*(2*a^2 - b^2 - c^2)*S) : :
X(39555) = 2*X(15) + X(16)

X(39555) lies on these lines: {3, 6}, {5, 10616}, {13, 33388}, {14, 38230}, {17, 20429}, {18, 621}, {23, 14704}, {30, 5470}, {230, 23004}, {316, 6671}, {382, 16631}, {383, 5479}, {395, 6780}, {530, 16962}, {531, 16268}, {533, 16530}, {623, 11308}, {624, 14712}, {625, 11312}, {635, 33225}, {691, 2381}, {1003, 25157}, {2913, 3490}, {3180, 14145}, {3411, 14139}, {3458, 16462}, {3480, 34220}, {3564, 22997}, {3849, 11297}, {5184, 11707}, {5965, 9117}, {5979, 36782}, {5994, 6105}, {6108, 36967}, {6295, 35918}, {6296, 37340}, {6694, 33021}, {6781, 23005}, {6782, 25236}, {7684, 36994}, {7685, 16964}, {8594, 22496}, {9218, 33958}, {12204, 22692}, {14138, 33465}, {14169, 34395}, {14693, 20428}, {16960, 33517}, {31940, 36515}, {33380, 33386}

X(39555) = reflection of X(39554) in X(187)
X(39555) = isogonal conjugate of X(11603)
X(39555) = {X(15),X(187)}-harmonic conjugate of X(16)
X(39555) = radical trace of circumcircle and 6th Lozada circle
X(39555) = center of circle {{X(15),PU(2)}}
X(39555) = Brocard axis intercept, other than X(16), of circle {{X(16),PU(2)}}

X(39556) = CIRCUMCIRCLE-INVERSE OF X(75)

Barycentrics    a*(a^6 - a^4*b^2 + a^3*b^3 - a*b^5 - a^4*c^2 + a^3*c^3 + b^3*c^3 - a*c^5) : :

X(39556) lies on these lines: {3, 75}, {186, 38457}, {350, 37311}, {814, 7255}, {1447, 5172}, {6376, 23843}, {6626, 37296}, {11334, 30963}, {19308, 33129}

X(39557) = CIRCUMCIRCLE-INVERSE OF X(83)

Barycentrics    a^2*(a^2 + b^2)*(a^2 + c^2)*(2*a^4*b^4 - 2*a^2*b^6 + 5*a^4*b^2*c^2 - a^2*b^4*c^2 - b^6*c^2 + 2*a^4*c^4 - a^2*b^2*c^4 - b^4*c^4 - 2*a^2*c^6 - b^2*c^6) : :

X(39557) lies on these lines: {3, 83}, {32, 14247}, {98, 38946}, {187, 733}, {827, 2080}, {1078, 26192}, {9494, 18105}

X(39558) = CIRCUMCIRCLE-INVERSE OF X(84)

Barycentrics    a^2*(a^3 - a^2*b - a*b^2 + b^3 + a^2*c + 2*a*b*c + b^2*c - a*c^2 - b*c^2 - c^3)*(a^3 + a^2*b - a*b^2 - b^3 - a^2*c + 2*a*b*c - b^2*c - a*c^2 + b*c^2 + c^3)*(a^4 - 2*a^2*b^2 + b^4 - a^2*b*c + 2*a*b^2*c - b^3*c - 2*a^2*c^2 + 2*a*b*c^2 - b*c^3 + c^4) : :

X(39558) lies on these lines: {3, 9}, {35, 1433}, {36, 29374}, {40, 1804}, {59, 2077}, {165, 1422}, {2078, 8059}, {2192, 15931}, {3900, 23224}, {6244, 6612}, {7677, 37141}, {10902, 37741}

leftri

CENTERS OF CIRCLE-INVERSION CIRCLES: X(39559)-X(39606)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, September 4, 2020.

Suppose that Γ is the circle with powers (u,v,w) -- that is, u = power of A with respect to the circle, and v and w are defined cyclically. Suppose that Γ* is the circle with powers (p,q,r). The Γ-inverse of Γ*, denoted by ((Γ, Γ*)), is the circle having powers (f,g,h) given by

f = (r*u*(c^2*(a^2 + b^2 - c^2) + (a^2 - b^2 + c^2)*u + (-a^2 + b^2 + c^2)*v - 2*c^2*w) + q*u*(b^2*(a^2 - b^2 + c^2) + (a^2 + b^2 - c^2)*u - 2*b^2*v - (a^2 - b^2 - c^2)*w) + p*(a^2*b^2*c^2 - a^2*u^2 - b^2*(a^2 - b^2 + c^2)*v + b^2*v^2 - c^2*(a^2 + b^2 - c^2)*w + (a^2 - b^2 - c^2)*v*w + c^2*w^2))/(r*(c^2*(a^2 + b^2 - c^2) + (a^2 - b^2 + c^2)*u - (a^2 - b^2 - c^2)*v - 2*c^2*w) + q*(b^2*(a^2 - b^2 + c^2) + (a^2 + b^2 - c^2)*u - 2*b^2*v - (a^2 - b^2 - c^2)*w) - p*(a^2*(a^2 - b^2 - c^2) + 2*a^2*u - (a^2 + b^2 - c^2)*v - (a^2 - b^2 + c^2)*w)),

where g and h are definined cyclically.
Circle Γ Circle Γ* center of ((Γ, Γ*))
Spieker circle circumcircle X(39559)
1st Lemoine circle 2nd Lemoine circle X(39560)
2nd Lemoine circle 3rd Lemoine circle (Ehrmann) X(39561)
2nd Lemoine circle orthocentroidal circle X(39562)
Moses circle orthocentroidal circle X(39563)
Spieker circle Apollonious circle X(39564)
nine-point circle Moses circle X(39565)
Spieker radical circle circum circle X(39566)
orthoptic circle of Steiner circumellipse incircle X(39567)
Stammler circle nine-point X(39568)
polar circle Brocard circle X(39569)
orthoptic circle of Steiner inellipse Spieker circle X(39570)
polar circle Taylor circle X(39571)
orthoptic circle of Steiner circumellipse Furhmann circle X(39572)
Spieker radical circle Bevan circle X(39573)
polar circle Bevan circle X(39574)
polar circle Moses circle X(39575)
orthoptic circle of Steiner inellipse Moses circle X(39576)
circumcircle Evans circle X(39577)
circumcircle Conway circle X(39578)
polar Conway circle X(39579)
orthoptic circle of Steiner inellipse Conway circle X(39580)
orthoptic circle of Steiner circumellipse Conway circle X(39581)
circumcircle Spieker radical circle X(39582)
nine-point circle Spieker radical circle X(39583)
Conway circle Spieker radical circle X(39584)
polar circle Spieker radical circle X(39585)
orthoptic circle of Steiner inellipse Spieker radical circle X(39586)
orthoptic circle of Steiner circumellipse Spieker radical circle X(39587)
2nd Lemoine circle polar circle X(39588)
Spieker circle polar circle X(39589)
Moses circle polar circle X(39590)
Spieker radical circle polar circle X(39591)
Bevan circle orthoptic circle of Steiner inellipse X(39592)
Moses circle orthoptic circle of Steiner inellipse X(39593)
Conway circle orthoptic circle of Steiner inellipse X(39594)
incircle orthoptic circle of Steiner circumellipse X(39595)
orthoptic circle of Steiner circumellipse Bevan circle X(39596)
Spieker radical circle orthoptic circle of Steiner circumellipse X(39597)
Conway circle 2nd Droz-Farney circle X(39598)
incircle 1st Droz-Farney circle X(39599)
Stammler circle 1st Droz-Farney circle X(39600)
orthocentroidal circle Schoutte circle X(39601)
orthoptic circle of Steiner inellipse Schoutte circle X(39602)
circumcircle 1st Neuberg circle X(39603)
polar circle 1st Lemoine X(39604)
orthoptic circle of Steiner inellipse Bevan circle X(39605)
polar circle Lester circle X(39606)

X(39559) = CENTER OF ((SPIEKER CIRCLE, CIRCUMCIRCLE))

Barycentrics    2*a^4 + 3*a^3*b - a^2*b^2 + a*b^3 + 3*b^4 + 3*a^3*c + 2*a^2*b*c + 3*a*b^2*c + 4*b^3*c - a^2*c^2 + 3*a*b*c^2 + 2*b^2*c^2 + a*c^3 + 4*b*c^3 + 3*c^4 : :
X(39559) = 5 X[1698] - X[13161], 5 X[1698] + 3 X[33167], X[13161] + 3 X[33167]

X(39559) lise on these lines: {2, 3710}, {3, 10}, {5, 17355}, {519, 6693}, {527, 3454}, {726, 3634}, {942, 20106}, {966, 31446}, {1210, 32777}, {1698, 13161}, {1738, 36499}, {1782, 36504}, {2321, 5292}, {2345, 5705}, {2885, 3828}, {3923, 18483}, {3946, 20083}, {3977, 5051}, {4967, 25446}, {4968, 25982}, {5530, 32780}, {5708, 21255}, {5847, 8258}, {6734, 32779}, {9843, 17279}, {17303, 31396}, {21075, 33163}, {24443, 30768}, {24850, 28150}, {28557, 36250}

X(39559) = complement of X(34937)
X(39559) = Spieker-circle-inverse of X(1324)
X(39559) = {X(1698),X(33167)}-harmonic conjugate of X(13161)

X(39560) = CENTER OF ((1ST LEMOINE CIRCLE, 2ND LEMOINE CIRCLE))

Barycentrics    a^2*(2*a^4 - a^2*b^2 - a^2*c^2 - 3*b^2*c^2) : :
Trilinears    sin A - 2 sin(A - 2ω) : :

X(39560) lies on these lines: {2, 14567}, {3, 6}, {69, 13196}, {76, 5026}, {83, 598}, {98, 7607}, {141, 7907}, {183, 4027}, {184, 9225}, {217, 18371}, {524, 7793}, {542, 7749}, {597, 7787}, {599, 1078}, {692, 16969}, {729, 33638}, {1153, 7815}, {1176, 30496}, {1503, 37446}, {1506, 10168}, {1613, 5012}, {1975, 10131}, {1976, 32540}, {2056, 36650}, {2548, 38064}, {2896, 10351}, {3124, 15080}, {3203, 38303}, {3224, 19127}, {3231, 11003}, {3297, 12839}, {3298, 12838}, {3314, 10353}, {3407, 11174}, {3589, 5025}, {3618, 6655}, {3763, 7930}, {3981, 22352}, {4048, 17128}, {5476, 18501}, {5622, 14585}, {5640, 8627}, {5695, 32115}, {5969, 7782}, {6034, 7748}, {6329, 33275}, {6593, 38523}, {6800, 20998}, {7492, 20977}, {7606, 33013}, {7708, 14002}, {7745, 10359}, {7746, 11646}, {7747, 25555}, {7778, 10352}, {7784, 10349}, {7808, 11318}, {7868, 10334}, {7887, 7943}, {7998, 20976}, {8177, 12215}, {8550, 15993}, {8587, 8860}, {8859, 33683}, {10104, 12177}, {12150, 35955}, {12151, 15533}, {12829, 22712}, {13881, 14880}, {15561, 32135}, {18374, 38297}, {18502, 38072}, {18800, 34506}, {20986, 21780}, {28343, 38525}, {28662, 38524}, {32217, 36182}, {32531, 32748}, {36615, 39238}, {36990, 38228}

X(39560) = circumcircle-inverse of X(10631)
X(39560) = 1st-Lemoine-circle-inverse of X(5111)
X(39560) = circle-{{X(371),X(372),PU(1),PU(39)}}-inverse of X(32447)
X(39560) = circle-{{X(1687),X(1688),PU(1),PU(2)}}-inverse of X(576)
X(39560) = center of ((circle {{X(1687),X(1688),PU(1),PU(2)}}, 2nd Lemoine circle))
X(39560) = harmonic center of 1st Lemoine circle and circle {{X(1687),X(1688),PU(1),PU(2)}}
X(39560) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 1350, 15514}, {6, 5085, 5116}, {32, 182, 5038}, {32, 5038, 6}, {32, 12054, 5013}, {182, 1691, 6}, {182, 5033, 1691}, {182, 8590, 11842}, {187, 575, 13330}, {187, 8590, 1691}, {187, 10485, 6}, {371, 372, 32447}, {575, 13330, 6}, {576, 5206, 5104}, {1342, 1343, 574}, {1379, 1380, 10631}, {1662, 1663, 5111}, {1687, 1688, 576}, {1691, 5038, 32}, {1691, 10485, 11842}, {1692, 3094, 6}, {1692, 5092, 3094}, {2030, 20190, 39}, {2558, 2559, 32}, {5017, 5050, 6}, {5034, 12212, 6}, {5116, 35006, 6}, {10104, 12177, 15069}, {10485, 13330, 575}, {19145, 19146, 35431}, {34870, 37479, 15815}, {35229, 35230, 3}

X(39561) = CENTER OF ((2ND LEMOINE CIRCLE, 3RD LEMOINE CIRCLE))

Barycentrics    a^2*(3*a^4 - 5*a^2*b^2 + 2*b^4 - 5*a^2*c^2 - 6*b^2*c^2 + 2*c^4) : :
X(39561) = X[3] + 5 X[6], 2 X[3] - 5 X[182], X[3] - 10 X[575], 4 X[3] + 5 X[576], 11 X[3] - 5 X[1350], 7 X[3] + 5 X[1351], 8 X[3] - 5 X[3098], X[3] - 5 X[5050], 3 X[3] - 5 X[5085], 7 X[3] - 10 X[5092], 3 X[3] + 5 X[5093], -X[3] + 2 X[5097], X[5] - 4 X[6329], X[5] + 2 X[12007], 2 X[6] + X[182], X[6] + 2 X[575], 4 X[6] - X[576], 11 X[6] + X[1350], 7 X[6] - X[1351], 8 X[6] + X[3098], 3 X[6] + X[5085], 7 X[6] + 2 X[5092], 3 X[6] - X[5093], 5 X[6] - 2 X[5097], 5 X[6] - X[5102], 5 X[69] - 17 X[3533], 2 X[140] + X[3629], 5 X[141] - 8 X[16239], 2 X[143] + X[17710], X[182] - 4 X[575]

X(39561) lies on these lines: {2, 5965}, {3, 6}, {5, 6329}, {22, 34565}, {51, 6800}, {69, 3533}, {110, 5645}, {114, 16989}, {140, 3629}, {141, 16239}, {143, 17710}, {184, 5640}, {206, 15580}, {323, 22112}, {373, 5422}, {394, 15082}, {524, 11539}, {542, 3545}, {547, 597}, {549, 20583}, {599, 15723}, {632, 3631}, {1147, 9977}, {1199, 11459}, {1352, 5056}, {1353, 3589}, {1386, 33179}, {1428, 37587}, {1503, 3845}, {1974, 34484}, {1992, 10168}, {1993, 5650}, {1994, 7998}, {2393, 23042}, {2854, 9813}, {3167, 6688}, {3292, 16187}, {3525, 11008}, {3543, 11179}, {3618, 5067}, {3734, 13196}, {3796, 21849}, {3818, 3850}, {3832, 6776}, {3853, 5480}, {3917, 34566}, {5012, 11002}, {5032, 10519}, {5059, 31670}, {5355, 37348}, {5462, 32366}, {5651, 11422}, {5663, 34155}, {5943, 11402}, {5946, 11202}, {6036, 7736}, {6090, 10601}, {6593, 9976}, {6771, 37641}, {6774, 37640}, {7592, 15030}, {7709, 12150}, {7739, 23698}, {7760, 10359}, {7787, 32467}, {7878, 10358}, {8541, 19128}, {8584, 11812}, {8705, 37936}, {9544, 12834}, {9545, 9972}, {10170, 12161}, {10282, 34777}, {11001, 20423}, {11003, 15019}, {11180, 25565}, {11216, 23041}, {11424, 15072}, {11579, 25556}, {11645, 14848}, {11649, 37940}, {12045, 17825}, {13570, 32063}, {13596, 19124}, {14482, 38736}, {14614, 15819}, {15032, 16261}, {15067, 19150}, {15074, 32191}, {15577, 34788}, {15686, 29181}, {15988, 17535}, {16200, 16475}, {16477, 31394}, {18581, 20415}, {18582, 20416}, {20398, 31415}, {30392, 38029}

X(39561) = midpoint of X(i) and X(j) for these {i,j}: {3, 5102}, {6, 5050}, {5032, 38064}, {5085, 5093}, {11179, 14853}, {11216, 23041}, {14561, 14912}, {20423, 25406}
X(39561) = reflection of X(i) in X(j) for these {i,j}: {182, 5050}, {5050, 575}, {5102, 5097}, {11178, 38317}, {38317, 597}
X(39561) = isogonal conjugate of X(11669)
X(39561) = Brocard-circle-inverse of X(5097)
X(39561) = 2nd-Lemoine-circle-inverse of X(8586)
X(39561) = circle-{{X(371),X(372),PU(1),PU(39)}}-inverse of X(37512)
X(39561) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 5097}, {3, 5097, 37517}, {6, 182, 576}, {6, 575, 182}, {6, 1350, 11482}, {6, 1351, 22330}, {6, 1692, 5039}, {6, 5038, 5028}, {6, 5085, 5093}, {6, 15516, 22234}, {182, 576, 3098}, {182, 22234, 6}, {182, 37517, 3}, {371, 372, 37512}, {575, 15516, 6}, {575, 22234, 576}, {1353, 3589, 34507}, {1666, 1667, 8586}, {1668, 1669, 5116}, {3398, 7772, 9737}, {5012, 11002, 35268}, {5050, 5093, 5085}, {5050, 15520, 17508}, {5092, 22330, 1351}, {5097, 37517, 576}, {5422, 13366, 9306}, {6329, 12007, 5}, {8550, 18583, 3818}, {11003, 15019, 34417}, {11422, 15018, 5651}, {11477, 12017, 14810}, {14853, 33748, 11179}, {15004, 35268, 11002}, {15577, 39125, 34788}, {19130, 33749, 6776}, {35770, 35771, 5041}, {36752, 37505, 13346}, {36757, 36758, 5034}

X(39562) = CENTER OF ((2ND LEMOINE CIRCLE, ORTHOCENTROIDAL CIRCLE))

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + 5*a^4*b^2*c^2 - 3*a^2*b^4*c^2 + 6*b^6*c^2 - 3*a^2*b^2*c^4 - 10*b^4*c^4 + 2*a^2*c^6 + 6*b^2*c^6 - c^8) : :
X(39562) = X[3] + 2 X[895], 4 X[6] - X[399], X[6] + 2 X[9976], 5 X[6] - 2 X[19140], 7 X[6] - 4 X[25556], 3 X[6] - 2 X[34155], 4 X[125] - X[11898], 8 X[182] - 5 X[15040], X[193] + 2 X[10264], X[399] + 8 X[9976], 5 X[399] - 8 X[19140], 7 X[399] - 16 X[25556], 3 X[399] - 8 X[34155], 4 X[575] - X[2930], 16 X[575] - 7 X[15039], 2 X[576] + X[16010]

X(39562) lies on the cubic K943 and these lines: {3, 895}, {6, 13}, {30, 37784}, {49, 575}, {54, 32248}, {67, 15317}, {74, 38263}, {110, 5020}, {125, 394}, {155, 32272}, {182, 15040}, {193, 10264}, {195, 15141}, {246, 2987}, {382, 8549}, {427, 1353}, {511, 5621}, {526, 7663}, {568, 8541}, {576, 16010}, {1147, 19361}, {1154, 11416}, {1177, 7517}, {1205, 6243}, {1351, 2781}, {1503, 31726}, {1597, 5093}, {1656, 15118}, {1992, 18917}, {1993, 9140}, {2070, 2393}, {2072, 3564}, {2854, 5050}, {2931, 6467}, {3526, 5181}, {3548, 23296}, {5052, 18373}, {5092, 15042}, {5095, 36749}, {5504, 6391}, {5642, 10601}, {5889, 11255}, {5890, 11443}, {5921, 11801}, {5972, 17825}, {6102, 8537}, {6403, 13358}, {6593, 32254}, {6699, 19348}, {6776, 12902}, {6997, 14683}, {7506, 32246}, {7545, 19136}, {8538, 18436}, {8681, 22115}, {9143, 34545}, {9714, 38885}, {9777, 12824}, {9970, 11482}, {9977, 33749}, {10116, 19362}, {10117, 11800}, {10250, 13352}, {10982, 15063}, {11061, 14627}, {11432, 25711}, {11470, 18439}, {11477, 32305}, {11511, 23039}, {11806, 12302}, {12161, 32234}, {12164, 15738}, {12167, 12236}, {12283, 19154}, {12310, 13198}, {12359, 19378}, {13754, 18449}, {14561, 15046}, {14853, 38789}, {14912, 18420}, {14961, 35463}, {15002, 18125}, {15027, 32275}, {15069, 20301}, {15115, 15316}, {15132, 32240}, {15436, 32255}, {15826, 32599}, {16003, 17822}, {16266, 20379}, {17813, 37489}, {18438, 19457}, {18531, 18919}, {19138, 19459}, {20304, 26206}, {20417, 37498}, {21850, 38790}, {23236, 36753}, {26944, 32285}, {30714, 36752}, {32114, 38794}, {32127, 37477}, {32249, 34148}, {32264, 34780}, {32284, 37472}, {34382, 34982}

X(39562) = midpoint of X(895) and X(5622)
X(39562) = reflection of X(i) in X(j) for these {i,j}: {3, 5622}, {18449, 21639}, {32609, 5050}, {38724, 25320}, {38789, 14853}
X(39562) = 2nd-Lemoine-circle-inverse of X(115)
X(39562) = crossdifference of every pair of points on line {526, 14273}
X(39562) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 12099, 5020}, {265, 19456, 399}, {1351, 11579, 10620}, {1993, 9140, 15106}, {5889, 11458, 11255}, {6102, 32155, 8537}

X(39563) = CENTER OF ((MOSES CIRCLE, ORTHOCENTROIDAL CIRCLE))

Barycentrics    2*a^4 - a^2*b^2 - 4*b^4 - a^2*c^2 + 8*b^2*c^2 - 4*c^4 : :
X(39563) = 2 X[99] - 5 X[31275], 4 X[115] - X[187], 5 X[115] - 2 X[230], 7 X[115] - X[6781], X[148] + 2 X[625], 5 X[187] - 8 X[230], 7 X[187] - 4 X[6781], 14 X[230] - 5 X[6781], X[316] + 2 X[32457], 3 X[671] + X[7809], 5 X[671] + X[7840], 2 X[671] + X[31173]

X(39563) lies on these lines: {2, 7748}, {3, 18362}, {4, 5007}, {5, 31652}, {6, 14269}, {30, 115}, {32, 3830}, {39, 381}, {99, 31275}, {148, 625}, {316, 19570}, {376, 7746}, {382, 35007}, {395, 35020}, {396, 35019}, {511, 38732}, {538, 671}, {542, 1570}, {543, 33228}, {546, 7765}, {549, 7756}, {574, 5055}, {754, 8352}, {1196, 31133}, {1506, 5066}, {1569, 22566}, {1692, 6034}, {1989, 3284}, {2549, 3545}, {2996, 7855}, {3053, 15684}, {3054, 17504}, {3070, 13687}, {3071, 13807}, {3094, 25561}, {3163, 34569}, {3291, 10989}, {3530, 12815}, {3534, 13881}, {3543, 3767}, {3544, 31450}, {3627, 7755}, {3734, 33219}, {3815, 38071}, {3839, 5475}, {3843, 7772}, {3845, 5041}, {3849, 14568}, {3850, 9698}, {3857, 9606}, {3858, 9607}, {3926, 38259}, {3934, 7924}, {5008, 38335}, {5013, 19709}, {5023, 15685}, {5025, 7880}, {5034, 38072}, {5054, 8589}, {5056, 31457}, {5064, 14580}, {5071, 31455}, {5107, 11646}, {5206, 15681}, {5215, 9166}, {5277, 15679}, {5305, 12101}, {5306, 7747}, {5461, 35297}, {5569, 33207}, {6000, 15544}, {6128, 18487}, {6175, 16589}, {6179, 19569}, {6321, 18860}, {6661, 7852}, {6683, 15031}, {7615, 15810}, {7745, 14893}, {7749, 8703}, {7759, 32996}, {7760, 14044}, {7761, 33278}, {7780, 11057}, {7788, 7825}, {7801, 16041}, {7804, 7884}, {7811, 7842}, {7817, 11361}, {7818, 14711}, {7821, 14063}, {7822, 33223}, {7828, 19686}, {7837, 7843}, {7841, 7865}, {7844, 33220}, {7853, 11185}, {7854, 32982}, {7856, 14066}, {7872, 31239}, {7873, 33229}, {7874, 32819}, {7886, 33246}, {7909, 33289}, {7925, 15301}, {8353, 34506}, {8355, 15300}, {8588, 15689}, {9619, 30308}, {9651, 10072}, {9664, 10056}, {11063, 37949}, {11614, 15709}, {13349, 22891}, {13350, 22846}, {14061, 32456}, {14160, 32447}, {15048, 23046}, {15515, 15694}, {15688, 37637}, {15703, 15815}, {15820, 31105}, {15980, 38734}, {18353, 22052}, {19876, 31422}, {30435, 35403}, {31398, 38076}, {31443, 38083}, {32832, 33263}, {32837, 32980}, {34866, 37955}

X(39563) = midpoint of X(i) and X(j) for these {i,j}: {148, 7799}, {316, 19570}, {671, 14041}
X(39563) = reflection of X(i) in X(j) for these {i,j}: {1692, 6034}, {5215, 9166}, {7799, 625}, {19570, 32457}, {31173, 14041}, {35297, 5461}
X(39563) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 5309, 14537}, {381, 11648, 39}, {2549, 18424, 7603}, {3839, 7739, 5475}, {3845, 5254, 7753}, {5306, 15687, 7747}, {5309, 14537, 5007}, {7818, 34505, 14711}, {7841, 18546, 9466}

X(39564) = CENTER OF ((SPIEKER CIRCLE, APOLLONIUS CIRCLE))

Barycentrics    a^3*b + 4*a^2*b^2 + 3*a*b^3 + a^3*c + 6*a^2*b*c + 9*a*b^2*c + 4*b^3*c + 4*a^2*c^2 + 9*a*b*c^2 + 8*b^2*c^2 + 3*a*c^3 + 4*b*c^3 : :
X(39564) = 3 X[2] + X[5295], 5 X[1698] - X[3931], 9 X[19875] - X[37598]

X(39564) lies on these lines: {2, 5295}, {3, 18229}, {5, 10}, {141, 3824}, {740, 3634}, {942, 10479}, {1698, 3931}, {1999, 19280}, {2049, 11679}, {2345, 5791}, {3454, 17239}, {3714, 19858}, {3741, 5045}, {3831, 27798}, {3841, 3844}, {4358, 9780}, {4688, 24046}, {4739, 24176}, {5232, 5714}, {5292, 17303}, {5704, 5936}, {5707, 5783}, {5708, 25590}, {5737, 31445}, {5955, 11231}, {8580, 37529}, {10472, 37536}, {16456, 17022}, {16832, 16853}, {17021, 17551}, {17289, 25446}, {17385, 20083}, {19277, 37554}, {19875, 37598}, {31330, 34790}, {31663, 32916}, {37582, 37660}

X(39564) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2049, 11679, 37594}, {10479, 31993, 942}

X(39565) = CENTER OF ((NINE-POINT CIRCLE, MOSES CIRCLE))

Barycentrics    a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4 : :
X(39565) = 3 X[7752] - X[7906]

X(39565) lies on these lines: {2, 7748}, {4, 187}, {5, 39}, {6, 3851}, {30, 7749}, {32, 381}, {76, 625}, {83, 7817}, {98, 38228}, {99, 32967}, {140, 7756}, {148, 7769}, {183, 7825}, {194, 32457}, {216, 1879}, {217, 3016}, {230, 546}, {232, 16868}, {235, 27371}, {315, 31173}, {316, 7780}, {382, 5206}, {384, 7886}, {385, 7843}, {403, 3199}, {427, 35723}, {498, 9664}, {499, 9651}, {538, 7752}, {550, 3054}, {567, 9697}, {570, 18353}, {574, 1656}, {575, 11646}, {620, 32819}, {626, 9466}, {671, 7783}, {800, 9722}, {1015, 7741}, {1078, 7842}, {1194, 37353}, {1196, 5133}, {1352, 1570}, {1500, 7951}, {1573, 25639}, {1574, 3814}, {1594, 33843}, {1657, 8588}, {1692, 3818}, {1968, 35488}, {1971, 18383}, {1975, 7862}, {2072, 22401}, {2079, 14130}, {2241, 10896}, {2242, 10895}, {2476, 16589}, {2489, 18313}, {2548, 3545}, {2549, 3090}, {2996, 34511}, {3003, 9220}, {3018, 15860}, {3053, 3843}, {3055, 35018}, {3091, 3767}, {3096, 3934}, {3104, 16631}, {3105, 16630}, {3146, 21843}, {3229, 37988}, {3291, 5169}, {3526, 15515}, {3544, 7736}, {3614, 31476}, {3627, 6781}, {3734, 7874}, {3763, 33241}, {3788, 11185}, {3830, 5023}, {3832, 7737}, {3849, 7793}, {3850, 5008}, {3855, 7735}, {3857, 18907}, {3933, 14711}, {3972, 33018}, {4045, 32992}, {4403, 17181}, {5013, 5055}, {5028, 10516}, {5033, 36990}, {5052, 19130}, {5056, 31401}, {5058, 6565}, {5062, 6564}, {5066, 5305}, {5068, 5286}, {5070, 15815}, {5071, 7738}, {5072, 7772}, {5073, 5210}, {5079, 31489}, {5103, 14994}, {5107, 34507}, {5141, 5283}, {5164, 15488}, {5188, 15980}, {5215, 33007}, {5277, 17577}, {5306, 38071}, {5318, 22832}, {5319, 34571}, {5321, 22831}, {5355, 12811}, {5461, 6680}, {5523, 35487}, {5569, 33192}, {6034, 25561}, {6292, 33184}, {6310, 14962}, {6375, 34845}, {6656, 31239}, {6683, 7790}, {6722, 7807}, {7486, 31457}, {7514, 9700}, {7547, 10311}, {7615, 7801}, {7617, 7815}, {7697, 32452}, {7739, 31404}, {7751, 7773}, {7754, 7775}, {7761, 14063}, {7763, 32963}, {7770, 7844}, {7771, 33019}, {7776, 17131}, {7777, 32450}, {7778, 17130}, {7785, 7805}, {7786, 33002}, {7795, 32972}, {7797, 33024}, {7800, 16041}, {7802, 14062}, {7803, 32962}, {7804, 7828}, {7808, 7851}, {7809, 7882}, {7810, 37350}, {7814, 20081}, {7820, 8361}, {7822, 14064}, {7830, 33229}, {7834, 16924}, {7848, 7885}, {7849, 7934}, {7854, 32828}, {7855, 32816}, {7857, 11361}, {7859, 33020}, {7867, 11318}, {7872, 11285}, {7880, 7899}, {7901, 7915}, {7902, 11174}, {7905, 19570}, {7907, 32456}, {7911, 14045}, {7914, 33219}, {7928, 33289}, {7935, 15271}, {7944, 14046}, {7988, 9619}, {7989, 9620}, {8253, 9674}, {8352, 34506}, {8369, 14971}, {8571, 13754}, {9300, 11737}, {9598, 31501}, {9605, 19709}, {9675, 23261}, {9696, 18350}, {9699, 13861}, {9756, 36997}, {10104, 10631}, {10159, 33286}, {10255, 14961}, {12963, 35787}, {12968, 35786}, {13335, 38224}, {13851, 14585}, {14001, 39143}, {14023, 32827}, {14162, 32447}, {14639, 37334}, {14644, 14901}, {14907, 32996}, {15022, 31400}, {15030, 15544}, {15575, 16194}, {17006, 33256}, {18581, 37825}, {18582, 37824}, {23903, 37693}, {26019, 31198}, {31168, 33291}, {31417, 37665}, {31703, 36756}, {31704, 36755}, {32189, 39266}, {32479, 33274}, {32815, 32988}, {32838, 32982}, {32867, 33023}, {32884, 38259}, {33046, 36812}

X(39565) = complement of X(7782)
X(39565) = reflection of X(15513) in X(7749)
X(39565) = complement of the isotomic conjugate of X(13481)
X(39565) = X(13481)-complementary conjugate of X(2887)
X(39565) = crosspoint of X(2) and X(13481)
X(39565) = crosssum of X(6) and X(9544)
X(39565) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7748, 37512}, {4, 7746, 187}, {5, 39, 7603}, {5, 115, 39}, {5, 5254, 1506}, {32, 18362, 13881}, {76, 625, 7821}, {76, 7912, 7895}, {76, 32966, 625}, {115, 1506, 5254}, {140, 7756, 8589}, {183, 7825, 7873}, {230, 546, 7747}, {230, 7747, 35007}, {235, 27371, 33842}, {381, 13881, 32}, {382, 37637, 5206}, {384, 14061, 7886}, {625, 7895, 7912}, {1078, 14041, 7842}, {1506, 5254, 39}, {2009, 2010, 32448}, {2548, 5309, 5041}, {2549, 3090, 31455}, {2549, 31455, 31652}, {3091, 3767, 5475}, {3291, 5169, 15820}, {3734, 7887, 7874}, {3767, 5475, 5007}, {3788, 32961, 31275}, {3815, 7765, 39}, {3934, 5025, 7853}, {5068, 5286, 31415}, {7615, 32984, 7801}, {7745, 7755, 5008}, {7746, 18424, 4}, {7751, 7773, 7845}, {7770, 7844, 7852}, {7785, 14568, 7805}, {7790, 16921, 6683}, {7809, 17129, 7882}, {7828, 16044, 7804}, {7862, 18546, 1975}, {7895, 7912, 7821}, {7899, 17128, 7880}, {7934, 31276, 7849}, {9166, 33013, 7817}, {9698, 15048, 39}, {9722, 36412, 800}, {11185, 32961, 3788}, {14061, 15031, 384}, {14062, 17004, 7802}, {14063, 32832, 7761}, {32819, 33249, 620}, {33229, 37688, 7830}

X(39566) = CENTER OF ((SPIEKER RADICAL CIRCLE, CIRCUMCIRCLE))

Barycentrics    a^6*b + 2*a^5*b^2 - a^4*b^3 - 2*a^3*b^4 + a^2*b^5 - b^7 + a^6*c + 2*a^5*b*c - a^4*b^2*c + a^2*b^4*c - 2*a*b^5*c - b^6*c + 2*a^5*c^2 - a^4*b*c^2 + 2*a^2*b^3*c^2 + b^5*c^2 - a^4*c^3 + 2*a^2*b^2*c^3 + 4*a*b^3*c^3 + b^4*c^3 - 2*a^3*c^4 + a^2*b*c^4 + b^3*c^4 + a^2*c^5 - 2*a*b*c^5 + b^2*c^5 - b*c^6 - c^7 : :

X(39566) lies on these lines: {3, 10}, {4, 345}, {5, 37}, {12, 3666}, {40, 32778}, {43, 17857}, {72, 970}, {306, 10441}, {386, 37700}, {517, 3704}, {573, 5810}, {726, 12610}, {912, 10974}, {946, 29671}, {1259, 5130}, {1352, 5227}, {1791, 6905}, {1867, 3998}, {3436, 17740}, {3454, 29069}, {3579, 29020}, {5812, 10445}, {7683, 29016}, {8227, 29657}, {9548, 33167}, {10950, 37539}, {11374, 34937}, {15496, 37419}, {18517, 30172}, {21081, 29311}, {23537, 30448}, {24982, 37255}, {26118, 29641}, {29347, 36250}, {32777, 37415}, {32779, 37399}, {37360, 37528}

X(39566) = Spieker radical circle inverse of X(1324)
X(39566) = {X(355),X(5791)}-harmonic conjugate of X (5788)

X(39567) = CENTER OF ((ORTHOPTIC CIRCLE OF STEINER CIRCUMELLIPSE, INCIRCLE))

Barycentrics    7*a^3 - 3*a^2*b + 5*a*b^2 - b^3 - 3*a^2*c - 6*a*b*c - 3*b^2*c + 5*a*c^2 - 3*b*c^2 - c^3 : :
X(39567) = 3 X[2] - 4 X[16020]

X(39567) lies on these lines: {1, 2}, {20, 20097}, {105, 6553}, {346, 1279}, {390, 4452}, {391, 3242}, {516, 4373}, {527, 15590}, {536, 4779}, {675, 28295}, {1104, 1219}, {1266, 30332}, {1447, 6049}, {1616, 4513}, {3177, 17480}, {3598, 4308}, {3600, 7195}, {3875, 8236}, {4310, 4655}, {4349, 30712}, {4402, 5853}, {4673, 31130}, {4864, 5839}, {4869, 5846}, {4875, 26242}, {5542, 32093}, {6555, 37679}, {7390, 7967}, {7407, 10595}, {7613, 17766}, {9053, 10005}, {9740, 26273}, {11106, 25237}, {17597, 37655}, {24231, 33800}, {30331, 32105}, {32857, 36606}

X(39567) = reflection of X(10005) in X(37650)
X(39567) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 3616, 17284}, {8, 31189, 10}, {145, 3622, 17316}, {145, 24599, 8}, {390, 32922, 4452}, {614, 7172, 2}, {5211, 26245, 2}

X(39568) = CENTER OF ((STAMMLER CIRCLE, NINE-POINT CIRCLE))

Barycentrics    a^2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 - 12*a^4*b^2*c^2 + 6*a^2*b^4*c^2 + 8*b^6*c^2 + 6*a^2*b^2*c^4 - 14*b^4*c^4 + 2*a^2*c^6 + 8*b^2*c^6 - c^8) : :
X(39568) = 3 X[3] - 4 X[26], 7 X[3] - 8 X[1658], 7 X[26] - 6 X[1658], 3 X[154] - 2 X[13346], 2 X[155] - 3 X[32063], 4 X[1147] - 5 X[14530], 3 X[3167] - 4 X[6759], 3 X[3167] - 2 X[37498], X[3913] - 3 X[34724], 9 X[5093] - 8 X[11255], X[5493] - 3 X[34642], 4 X[7689] - 3 X[35450], 4 X[8666] - 3 X[34723], 4 X[8715] - 3 X[34703], 4 X[10282] - 3 X[37497], X[12513] - 3 X[34702], 4 X[13464] - 3 X[34643], 8 X[25487] - 9 X[38638]

X(39568) lies on these lines: {2, 3}, {6, 13598}, {40, 24320}, {68, 34780}, {84, 37581}, {141, 16656}, {154, 13346}, {155, 32063}, {159, 2883}, {161, 5895}, {185, 33586}, {394, 26883}, {511, 1498}, {515, 9910}, {516, 9798}, {534, 24328}, {962, 8192}, {971, 37547}, {999, 4320}, {1092, 8780}, {1147, 14530}, {1151, 13889}, {1152, 13943}, {1181, 1351}, {1350, 5907}, {1352, 16621}, {1495, 35602}, {1503, 9914}, {1578, 35765}, {1579, 35764}, {1619, 17845}, {2207, 15905}, {2777, 12310}, {2794, 12413}, {2829, 13222}, {3053, 34809}, {3167, 6759}, {3172, 10313}, {3295, 4319}, {3426, 37478}, {3527, 36752}, {3564, 12311}, {3796, 11424}, {3913, 34724}, {3964, 32006}, {4297, 11365}, {4299, 10046}, {4302, 10037}, {5050, 10982}, {5093, 7592}, {5254, 8573}, {5446, 11432}, {5462, 8717}, {5493, 34642}, {5562, 33878}, {5691, 8193}, {5706, 37492}, {5840, 9913}, {5889, 12174}, {5893, 15577}, {6000, 17834}, {6199, 11265}, {6284, 16541}, {6395, 11266}, {6459, 19006}, {6460, 19005}, {6465, 10669}, {6466, 10673}, {6767, 8144}, {6776, 13142}, {7083, 37570}, {7354, 10833}, {7373, 32047}, {7689, 35450}, {8666, 34723}, {8715, 34703}, {8743, 38292}, {8744, 33636}, {9683, 22644}, {9695, 13886}, {9729, 17810}, {9861, 23698}, {9908, 9919}, {9921, 12303}, {9922, 12304}, {10110, 37514}, {10282, 37497}, {10306, 20760}, {10539, 37483}, {10575, 10938}, {10625, 18451}, {10721, 12168}, {10733, 13171}, {10790, 12203}, {10831, 12943}, {10832, 12953}, {11438, 33534}, {11456, 12160}, {12162, 37486}, {12163, 13093}, {12233, 31670}, {12279, 15107}, {12309, 12315}, {12513, 34702}, {13347, 17825}, {13348, 17811}, {13391, 32139}, {13464, 34643}, {14927, 18945}, {15574, 32819}, {15644, 17814}, {16655, 18440}, {18405, 29323}, {18439, 37494}, {19347, 36747}, {19459, 31802}, {25487, 38638}, {26864, 34148}, {26928, 37532}, {26937, 32269}, {29012, 37488}, {31884, 33537}, {36990, 37485}

X(39568) = reflection of X(i) in X(j) for these {i,j}: {34938, 5}, {12085, 26}, {34780, 68}, {12164, 1498}, {37498, 6759}, {3, 7387}, {12410, 9911}, {12303, 9921}, {12304, 9922}, {13093, 12163}, {12084, 17714}, {37972, 37924}
X(39568) = circumcircle-inverse of X(16976)
X(39568) = Stammler-circle-inverse of X(2072)
X(39568) = X(4)-of-3rd-antipedal-triangle-of-X(3)
X(39568) = center of ((Stammler circle, 1st Droz-Farney circle))
X(39568) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 33524, 37198}, {2, 37198, 3}, {3, 4, 11479}, {3, 5, 16419}, {3, 382, 1597}, {3, 1598, 5020}, {3, 5899, 9714}, {3, 7387, 9909}, {3, 7517, 3517}, {3, 9909, 16195}, {3, 10244, 14070}, {3, 10245, 1658}, {3, 11484, 140}, {3, 18534, 1598}, {3, 18535, 5}, {3, 20850, 24}, {4, 20, 12362}, {4, 7399, 381}, {4, 7400, 5}, {4, 10323, 7395}, {4, 11414, 3}, {4, 12082, 11414}, {5, 34938, 34609}, {5, 35243, 3}, {20, 25, 3}, {20, 6816, 7667}, {22, 1593, 3}, {22, 3146, 1593}, {23, 5059, 11413}, {23, 11413, 3515}, {24, 3529, 21312}, {24, 21312, 3}, {26, 12084, 15331}, {26, 12085, 3}, {26, 15331, 14070}, {378, 9715, 3}, {378, 12088, 9715}, {382, 1657, 18563}, {382, 12083, 3}, {550, 6642, 3}, {550, 7530, 6642}, {1113, 1114, 16976}, {1350, 15811, 5907}, {1657, 7517, 3}, {1657, 37924, 7517}, {1906, 6816, 381}, {1906, 7667, 6816}, {2043, 2044, 10691}, {2071, 15750, 3}, {3146, 12087, 22}, {3515, 11413, 3}, {3516, 7488, 3}, {3529, 37925, 24}, {3534, 7506, 3}, {3543, 7503, 11403}, {3832, 16661, 7485}, {5198, 33524, 3}, {5198, 37198, 2}, {5899, 17800, 3}, {6284, 18954, 16541}, {6759, 37498, 3167}, {6909, 37257, 3}, {6995, 10996, 9825}, {7387, 12084, 10244}, {7387, 12085, 26}, {7387, 14070, 17714}, {7387, 14790, 10243}, {7395, 10323, 3}, {7395, 11414, 10323}, {7420, 37287, 3}, {7487, 35513, 31829}, {7500, 37201, 3575}, {7512, 15682, 35502}, {7580, 13730, 3}, {9924, 32602, 1498}, {10244, 17714, 9909}, {10982, 10984, 5050}, {11410, 38444, 3}, {12082, 34621, 7387}, {12084, 14070, 3}, {12084, 17714, 14070}, {12086, 37913, 38444}, {12086, 38444, 11410}, {12088, 33703, 378}, {14070, 17714, 10244}, {15154, 15155, 2072}, {15331, 17714, 26}, {15681, 18378, 3}, {18535, 35243, 16419}, {20838, 35987, 3}, {21312, 37925, 20850}, {26283, 37196, 3}, {34726, 37924, 7387}, {37022, 37034, 3}, {37195, 37250, 3}, {37899, 37945, 37924}

X(39569) = CENTER OF ((POLAR CIRCLE, BROCARD CIRCLE))

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) : :
X(39569) = X[30716] - 3 X[37943]

X(39569) lies on these lines: {2, 6747}, {4, 83}, {5, 53}, {49, 33549}, {51, 467}, {52, 15897}, {107, 32223}, {110, 37766}, {114, 232}, {133, 11799}, {136, 468}, {264, 24206}, {297, 511}, {311, 13450}, {317, 576}, {343, 14569}, {393, 1352}, {403, 23514}, {450, 5972}, {458, 38317}, {542, 37765}, {648, 5965}, {1351, 15274}, {1594, 39506}, {1968, 32152}, {1990, 3564}, {2052, 21243}, {3818, 33971}, {4230, 14356}, {5097, 27377}, {6248, 27376}, {6368, 18314}, {6748, 18583}, {7577, 39486}, {8746, 19139}, {8747, 37823}, {9306, 11547}, {9308, 34507}, {9880, 37855}, {10002, 31670}, {11197, 23607}, {13434, 35717}, {14129, 30506}, {14853, 37174}, {14918, 35360}, {15014, 23698}, {29317, 35474}, {30716, 37943}

X(39569) = midpoint of X(297) and X(6530)
X(39569) = X(4230)-Ceva conjugate of X(16230)
X(39569) = X(i)-isoconjugate of X(j) for these (i,j): {54, 293}, {97, 1910}, {98, 2169}, {248, 2167}, {287, 2148}, {879, 36134}, {1821, 14533}, {2190, 17974}, {19210, 36120}, {23286, 36084}
X(39569) = crossdifference of every pair of points on line {14533, 23286}
X(39569) = barycentric product X(i)*X(j) for these {i,j}: {5, 297}, {53, 325}, {232, 311}, {240, 14213}, {324, 511}, {343, 6530}, {877, 12077}, {2421, 23290}, {2799, 35360}, {4230, 18314}, {6393, 14569}, {13450, 36212}, {14356, 14918}, {14570, 16230}, {20022, 27371}, {28706, 34854}
X(39569) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 287}, {51, 248}, {53, 98}, {216, 17974}, {232, 54}, {237, 14533}, {240, 2167}, {297, 95}, {324, 290}, {325, 34386}, {343, 6394}, {467, 31635}, {511, 97}, {1755, 2169}, {1953, 293}, {2181, 1910}, {3199, 1976}, {3289, 19210}, {3569, 23286}, {4230, 18315}, {6333, 15414}, {6530, 275}, {11062, 14355}, {12077, 879}, {13450, 16081}, {14213, 336}, {14569, 6531}, {14570, 17932}, {14966, 15958}, {16230, 15412}, {17994, 2623}, {27371, 20021}, {34854, 8882}, {35360, 2966}
X(39569) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 17907, 182}, {5, 53, 39530}, {5, 10003, 233}, {450, 14165, 5972}

X(39570) = CENTER OF ((ORTHOPTIC CIRCLE OF STEINER INELLIPSE, SPIEKER CIRCLE))

Barycentrics    a^3 - a^2*b + 3*a*b^2 - 3*b^3 - a^2*c + 6*a*b*c - b^2*c + 3*a*c^2 - b*c^2 - 3*c^3 : :

X(39570) lies on these lines: {1, 2}, {4, 10743}, {7, 3717}, {69, 5686}, {85, 341}, {100, 37254}, {105, 3913}, {120, 6552}, {142, 4901}, {144, 4645}, {226, 5423}, {344, 390}, {346, 2550}, {391, 3416}, {452, 5300}, {516, 3161}, {518, 4869}, {528, 4779}, {726, 4373}, {728, 1706}, {1447, 32003}, {1699, 8055}, {1738, 4452}, {2321, 38200}, {2481, 18135}, {2899, 3832}, {3290, 21896}, {3339, 32098}, {3475, 30615}, {3598, 6604}, {3672, 4429}, {3699, 30828}, {3701, 5177}, {3703, 26040}, {3790, 4461}, {3817, 6557}, {3823, 4000}, {3836, 4310}, {3871, 16048}, {3883, 18230}, {3925, 3974}, {3992, 10590}, {4188, 31073}, {4208, 4385}, {4223, 5687}, {4307, 4672}, {4312, 4488}, {4344, 17353}, {4454, 5880}, {4518, 10405}, {4646, 26242}, {4899, 17298}, {4968, 37436}, {5015, 5129}, {5141, 31084}, {5154, 31126}, {5223, 21296}, {5274, 18743}, {5281, 33116}, {5290, 32086}, {5342, 7378}, {5657, 7390}, {5772, 10436}, {5818, 7407}, {5828, 33864}, {5846, 37650}, {6555, 25568}, {6559, 27541}, {7179, 31994}, {8165, 30854}, {9053, 17265}, {9812, 30568}, {10712, 11236}, {11038, 17234}, {11522, 28661}, {14552, 33078}, {16284, 25280}, {17277, 38048}, {17296, 24393}, {17776, 17784}, {31995, 38052}, {33118, 37666}

X(39570) = midpoint of X(4869) and X(10005)
X(39570) = reflection of X(4373) in X(7613)
X(39570) = anticomplement of X(16020)
X(39570) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1698, 31191}, {2, 10327, 7172}, {8, 7080, 28057}, {8, 9780, 4384}, {8, 29627, 1}, {344, 32850, 390}, {2550, 3932, 346}, {3416, 38057, 391}, {3617, 29616, 8}, {4645, 27549, 144}, {5205, 29641, 30741}, {5205, 30741, 2}, {9780, 16830, 2}

X(39571) = CENTER OF ((POLAR CIRCLE, TAYLOR CIRCLE))

Barycentrics    a^10 - 3*a^8*b^2 + 4*a^6*b^4 - 4*a^4*b^6 + 3*a^2*b^8 - b^10 - 3*a^8*c^2 + 4*a^6*b^2*c^2 + 4*a^4*b^4*c^2 - 8*a^2*b^6*c^2 + 3*b^8*c^2 + 4*a^6*c^4 + 4*a^4*b^2*c^4 + 10*a^2*b^4*c^4 - 2*b^6*c^4 - 4*a^4*c^6 - 8*a^2*b^2*c^6 - 2*b^4*c^6 + 3*a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(39571) = X[4] + 3 X[18950], X[18909] - 3 X[18950]

X(349571) lies on these lines: {2, 578}, {3, 12241}, {4, 51}, {5, 6}, {20, 11438}, {24, 12022}, {25, 6146}, {30, 9786}, {32, 10600}, {52, 18531}, {54, 7505}, {64, 13488}, {66, 3527}, {69, 6804}, {70, 1173}, {125, 3541}, {140, 11425}, {143, 18569}, {154, 21841}, {182, 3547}, {184, 3542}, {235, 1181}, {265, 11746}, {343, 7395}, {378, 26879}, {381, 11432}, {382, 13568}, {393, 6750}, {403, 7592}, {427, 10982}, {468, 19357}, {498, 11429}, {499, 19365}, {511, 6643}, {546, 32140}, {548, 37487}, {550, 1192}, {567, 6639}, {568, 18404}, {569, 3549}, {631, 11430}, {1147, 22529}, {1199, 16868}, {1209, 14786}, {1243, 20029}, {1368, 13142}, {1478, 19366}, {1479, 11436}, {1498, 1596}, {1503, 1598}, {1593, 16657}, {1595, 1853}, {1597, 6247}, {1620, 8703}, {1656, 11426}, {1871, 5928}, {1885, 10605}, {1974, 3089}, {1995, 14516}, {2072, 36749}, {2453, 14896}, {2888, 5056}, {3060, 37444}, {3088, 20299}, {3090, 3292}, {3091, 11442}, {3147, 13367}, {3153, 32411}, {3357, 18913}, {3410, 5068}, {3448, 3832}, {3517, 34782}, {3546, 13346}, {3548, 13352}, {3574, 15004}, {3575, 18396}, {3580, 7503}, {3850, 18356}, {4846, 13630}, {5020, 12429}, {5085, 16197}, {5198, 16655}, {5422, 13160}, {5446, 14790}, {5462, 9815}, {5562, 6515}, {5640, 7544}, {5702, 8888}, {5812, 9119}, {5889, 37644}, {5907, 11411}, {5943, 7401}, {6193, 9306}, {6353, 10282}, {6531, 18855}, {6622, 14912}, {6623, 22533}, {6640, 37472}, {6644, 12118}, {6676, 37476}, {6756, 17810}, {6803, 11695}, {6823, 37514}, {6995, 13419}, {7386, 15644}, {7399, 10601}, {7400, 37515}, {7404, 21243}, {7487, 18400}, {7507, 9777}, {7528, 18474}, {7529, 12134}, {7542, 37506}, {8550, 16252}, {8889, 32767}, {9544, 21451}, {9715, 32269}, {9818, 12359}, {9970, 36253}, {10024, 36753}, {10055, 37696}, {10071, 37697}, {10095, 11818}, {10114, 14683}, {10201, 32046}, {10255, 14627}, {10263, 14791}, {10594, 31383}, {10628, 18933}, {10937, 22833}, {10961, 35837}, {10963, 35836}, {10996, 16836}, {11470, 14853}, {11487, 34507}, {11585, 36747}, {11745, 18494}, {11800, 12319}, {11807, 13203}, {11808, 32346}, {12160, 16072}, {12162, 18917}, {12293, 31833}, {12362, 17834}, {12586, 13374}, {12605, 37489}, {13321, 31724}, {13371, 39522}, {13595, 34799}, {13598, 34938}, {13754, 18951}, {15033, 26917}, {15760, 36752}, {15807, 32138}, {16195, 21970}, {16196, 37497}, {16621, 18535}, {16881, 18377}, {17809, 37942}, {17845, 37458}, {18379, 32533}, {18382, 32191}, {18533, 21659}, {18563, 37490}, {22750, 22953}, {31829, 37475}, {32369, 38443}, {34417, 37122}
midpoint of X(i) and X(j) for these {i,j}: {4, 18909}, {7487, 18945}, {18933, 18947}

X(39571) = reflection of X(i) in X(j) for these {i,j}: {1598, 15873}, {17814, 5}
X(39571) = crosssum of X(3) and X(6642)
X(39571) = crossdifference of every pair of points on line {924, 32320}
X(39571) = X(936)-of-orthic-triangle if ABC is acute
X(39571) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 185, 5878}, {4, 1899, 14216}, {4, 11433, 389}, {4, 12324, 13474}, {4, 18912, 1899}, {4, 18916, 185}, {4, 18918, 18383}, {4, 18950, 18909}, {5, 68, 1352}, {5, 12161, 5654}, {5, 13292, 155}, {5, 32358, 15068}, {24, 12022, 19467}, {25, 6146, 9833}, {69, 6804, 11793}, {125, 11424, 3541}, {235, 11245, 1181}, {378, 26879, 26937}, {381, 11432, 12233}, {389, 18390, 4}, {1368, 13142, 37498}, {1596, 18914, 1498}, {1597, 26944, 6247}, {1656, 11426, 23292}, {1885, 10605, 20427}, {3088, 23291, 20299}, {3089, 6776, 6759}, {5446, 14790, 31670}, {5462, 9927, 18420}, {5462, 18420, 9815}, {6353, 18925, 10282}, {6515, 6816, 5562}, {6644, 12370, 12118}, {6803, 18928, 11695}, {8550, 16252, 19347}, {9306, 10112, 6193}, {9781, 25739, 4}, {10110, 18381, 4}, {10594, 34224, 31383}, {11411, 18537, 5907}, {11425, 26958, 140}, {11438, 13403, 20}, {12241, 13567, 3}, {15033, 26917, 37119}, {15068, 32358, 9936}, {18535, 34780, 16621}, {21841, 31804, 154}

X(39572) = CENTER OF ((ORTHOPTIC CIRCLE OF STEINER CIRCUMELLIPSE, FUHRMANN CIRCLE))

Barycentrics    2*a^6 - 2*a^5*b + 2*a^3*b^3 - 2*a^2*b^4 - 2*a^5*c + 4*a^4*b*c - 3*a^3*b^2*c + a^2*b^3*c + a*b^4*c - b^5*c - 3*a^3*b*c^2 + 2*a^2*b^2*c^2 - a*b^3*c^2 + 2*a^3*c^3 + a^2*b*c^3 - a*b^2*c^3 + 2*b^3*c^3 - 2*a^2*c^4 + a*b*c^4 - b*c^5 : :

X(39572) lies on these lines: {2, 355}, {3, 4968}, {4, 26228}, {5, 26230}, {20, 26245}, {98, 1290}, {104, 37449}, {105, 2731}, {140, 26251}, {388, 1455}, {515, 3011}, {517, 20045}, {902, 29057}, {952, 3006}, {1352, 33122}, {1447, 17895}, {1503, 17724}, {2177, 24257}, {2792, 32856}, {2975, 4696}, {3576, 29828}, {3701, 13732}, {3757, 4220}, {3920, 7413}, {4192, 26237}, {5205, 38669}, {5266, 15971}, {5297, 21554}, {5587, 29855}, {5881, 29857}, {5882, 29639}, {5886, 29831}, {6211, 32927}, {6996, 26247}, {7465, 11491}, {14679, 33849}, {15178, 29823}, {19540, 26238}, {19541, 26246}, {29832, 37727}

X(39572) = reflection of X(8229) in X(3011)

X(39573) = CENTER OF ((SPIEKER RADICAL CIRCLE, BEVAN CIRCLE))

Barycentrics    a^6*b + 2*a^5*b^2 + a^4*b^3 - a^2*b^5 - 2*a*b^6 - b^7 + a^6*c + 2*a^5*b*c + 3*a^4*b^2*c - a^2*b^4*c - 2*a*b^5*c - 3*b^6*c + 2*a^5*c^2 + 3*a^4*b*c^2 + 10*a^2*b^3*c^2 + 2*a*b^4*c^2 - b^5*c^2 + a^4*c^3 + 10*a^2*b^2*c^3 + 4*a*b^3*c^3 + 5*b^4*c^3 - a^2*b*c^4 + 2*a*b^2*c^4 + 5*b^3*c^4 - a^2*c^5 - 2*a*b*c^5 - b^2*c^5 - 2*a*c^6 - 3*b*c^6 - c^7 : :

X(39573) lies on these lines: {4, 9}, {12, 3666}, {210, 970}, {312, 946}, {355, 36754}, {386, 17857}, {515, 1220}, {1698, 10856}, {1837, 5710}, {2899, 3091}, {3687, 4385}, {3714, 5799}, {4260, 14872}, {5777, 10822}, {5786, 38047}, {6684, 19808}, {29671, 36651}, {32778, 36671}

X(39573) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 10445, 40}

X(39574) = CENTER OF ((POLAR CIRCLE, BEVAN CIRCLE))

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^5*b + a^4*b^2 - 2*a^3*b^3 - 2*a^2*b^4 + a*b^5 + b^6 + a^5*c - 2*a^2*b^3*c - a*b^4*c + 2*b^5*c + a^4*c^2 - b^4*c^2 - 2*a^3*c^3 - 2*a^2*b*c^3 - 4*b^3*c^3 - 2*a^2*c^4 - a*b*c^4 - b^2*c^4 + a*c^5 + 2*b*c^5 + c^6) : :

X(39574) lies on these lines: {1, 1857}, {4, 9}, {5, 1214}, {12, 1785}, {29, 515}, {33, 2910}, {92, 946}, {158, 226}, {210, 1872}, {235, 1867}, {243, 13411}, {278, 8227}, {318, 21075}, {355, 7524}, {412, 6684}, {580, 7076}, {912, 1844}, {1013, 6796}, {1118, 9612}, {1125, 7551}, {1148, 3671}, {1784, 13407}, {1856, 7952}, {1859, 5777}, {1895, 21620}, {1902, 7140}, {1940, 4292}, {1957, 37530}, {2184, 5715}, {3176, 11529}, {3194, 8755}, {3576, 7498}, {4183, 10902}, {4297, 7531}, {5125, 10175}, {5146, 7559}, {5450, 37253}, {7510, 18480}, {11012, 37380}, {12114, 37393}, {12608, 30687}, {12616, 37235}, {18242, 37321}, {31423, 37417}, {37276, 37526}

X(39574) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 281, 40}, {5, 39529, 1838}

X(39575) = CENTER OF ((POLAR CIRCLE, MOSES CIRCLE))

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 - b^2*c^2 - c^4) : :
Barycentrics    (tan A) ((b^2 + c^2) sin 2A + (c^2 - a^2) sin 2B + (b^2 - a^2) sin 2C) : :

X(39575) lies on these lines: {2, 1235}, {3, 112}, {4, 39}, {5, 5523}, {6, 24}, {20, 14961}, {22, 23115}, {25, 1180}, {26, 10313}, {32, 186}, {53, 9606}, {110, 23128}, {115, 16868}, {127, 26170}, {140, 16318}, {156, 22146}, {187, 21844}, {206, 34137}, {216, 631}, {217, 5890}, {230, 10018}, {235, 15048}, {251, 21213}, {264, 7786}, {325, 28710}, {340, 7905}, {376, 22401}, {378, 2207}, {393, 570}, {403, 5254}, {420, 3117}, {427, 31406}, {451, 5283}, {468, 5305}, {566, 1990}, {574, 1968}, {577, 7512}, {648, 1078}, {1172, 4261}, {1194, 6353}, {1196, 38282}, {1384, 15750}, {1506, 7577}, {1593, 5024}, {1594, 3815}, {1614, 38867}, {1625, 12111}, {1658, 10317}, {1783, 2975}, {1870, 2275}, {1971, 26882}, {2211, 3094}, {2276, 6198}, {2332, 4256}, {2493, 13881}, {2937, 22121}, {3003, 35486}, {3053, 32534}, {3087, 5421}, {3147, 7735}, {3162, 7485}, {3192, 4253}, {3269, 6241}, {3284, 7556}, {3289, 11412}, {3331, 12290}, {3515, 30435}, {3518, 7772}, {3535, 8962}, {3542, 5286}, {3767, 7505}, {4235, 7782}, {5023, 35472}, {5094, 15302}, {5158, 15262}, {5206, 17506}, {5309, 37943}, {5354, 37977}, {6103, 7749}, {6143, 31455}, {6240, 7745}, {6636, 17409}, {6749, 13337}, {6759, 13509}, {6793, 10182}, {7071, 31461}, {7482, 38526}, {7487, 37665}, {7488, 10316}, {7576, 9300}, {7737, 35471}, {7746, 14940}, {7747, 34797}, {7753, 18559}, {7763, 17907}, {7783, 15014}, {7832, 18797}, {7881, 11331}, {7952, 13006}, {7998, 35325}, {8589, 23040}, {8749, 14385}, {8750, 24047}, {8779, 10282}, {9308, 11285}, {9698, 27371}, {9715, 15905}, {10295, 16308}, {10766, 34117}, {11386, 27369}, {11449, 32661}, {11455, 38297}, {11459, 22416}, {12173, 15484}, {12220, 14965}, {14581, 35473}, {15149, 24598}, {15340, 18381}, {15341, 16252}, {15815, 35477}, {16195, 38292}, {20410, 37808}, {22332, 35502}, {26636, 37448}, {28407, 30737}, {30904, 37168}, {31401, 37119}, {31652, 35475}, {32000, 32960}, {32818, 36212}

X(39575) = crosspoint of X(250) and X(6331)
X(39575) = crosssum of X(i) and X(j) for these (i,j): {6, 1853}, {125, 3049}
X(39575) = polar conjugate of the isotomic conjugate of X(2979)
X(39575) = polar conjugate of the isogonal conjugate of X(160)
X(39575) = X(34412)-complementary conjugate of X(21235)
X(39575) = X(i)-Ceva conjugate of X(j) for these (i,j): {276, 4}, {15388, 112}
X(39575) = X(i)-cross conjugate of X(j) for these (i,j): {160, 2979}, {15897, 4}
X(39575) = X(i)-isoconjugate of X(j) for these (i,j): {63, 2980}, {27366, 34055}
X(39575) = barycentric product X(i)*X(j) for these {i,j}: {4, 2979}, {25, 7796}, {160, 264}, {1670, 16246}, {3202, 18022}, {5379, 18188}, {15897, 34384}
X(39575) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 2980}, {160, 3}, {1843, 27366}, {2979, 69}, {3202, 184}, {7796, 305}, {15897, 51}
X(39575) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 8743, 112}, {6, 24, 10312}, {24, 10312, 10986}, {26, 22120, 10313}, {39, 232, 4}, {264, 7786, 37125}, {393, 31400, 3541}, {574, 1968, 3520}, {2207, 5013, 378}, {3269, 32445, 6241}, {3520, 8744, 1968}, {3815, 27376, 1594}, {5421, 14576, 3087}, {10880, 10881, 19128}, {15355, 26216, 5}

X(39576) = CENTER OF ((ORTHOPTIC CIRCLE OF STEINER INELLIPSE, MOSES CIRCLE))

Barycentrics    a^2*(a^2*b^2 + b^4 + a^2*c^2 - 7*b^2*c^2 + c^4) : :

X(39576) lies on these lines: {2, 39}, {3, 111}, {5, 5913}, {6, 5643}, {23, 5206}, {32, 8585}, {140, 16317}, {187, 14002}, {251, 1611}, {352, 576}, {353, 20190}, {373, 9463}, {511, 8617}, {566, 3054}, {575, 7708}, {647, 5466}, {1078, 16055}, {1383, 35007}, {1384, 30734}, {1560, 16868}, {1613, 11451}, {1627, 5020}, {1648, 38397}, {1656, 9745}, {1691, 10546}, {1968, 37962}, {1995, 3053}, {2493, 18573}, {2979, 20977}, {3003, 15355}, {3055, 13337}, {3060, 21001}, {3094, 33879}, {3124, 7998}, {3231, 5640}, {4277, 37675}, {5017, 10545}, {5033, 35265}, {5108, 14515}, {5166, 8542}, {5971, 7751}, {6792, 34507}, {7485, 8770}, {7496, 15515}, {7664, 7907}, {7665, 33259}, {7749, 10418}, {7781, 9870}, {7793, 26276}, {9172, 9829}, {9225, 11422}, {10717, 34505}, {11178, 11647}, {11284, 30435}, {15066, 39024}, {15080, 20998}, {15246, 34481}, {16020, 26278}, {16419, 38862}, {23297, 33002}, {33798, 35294}

X(39576) = complement of isotomic conjugate of X(40103)
X(39576) = crosssum of X(6) and X(15534)
X(39576) = crossdifference of every pair of points on line {669, 9125}
X(39576) = perspector of 3rd Parry triangle and unary cofactor triangle of 1st Parry triangle
X(39576) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3291, 9465}, {2, 9465, 15302}, {3, 14262, 38688}, {3, 14263, 38524}, {32, 8585, 16042}, {9465, 15302, 1180}, {11580, 16042, 32}, {20481, 21448, 111}

X(39577) = CENTER OF ((CIRCUMCIRCLE, EVANS CIRCLE))

Barycentrics    a^2*(b - c)*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c - b^3*c - a^2*c^2 - b^2*c^2 + a*c^3 - b*c^3) : :

X(39577) lies on these lines: {3, 1019}, {21, 4129}, {35, 512}, {36, 4367}, {100, 4807}, {187, 14991}, {667, 1734}, {830, 8648}, {1946, 14349}, {3746, 4879}, {5251, 21051}, {7178, 37583}, {11012, 28473}, {16785, 22229}, {23864, 29487}

X(39577) = crosssum of X(513) and X(33105)

X(39578) = CENTER OF ((CIRCUMCIRCLE, CONWAY CIRCLE))

Barycentrics    a^2*(a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 + a^4*c + 2*a^3*b*c - 2*a*b^3*c - b^4*c + a^3*c^2 - a*b^2*c^2 - b^3*c^2 - a^2*c^3 - 2*a*b*c^3 - b^2*c^3 - a*c^4 - b*c^4) : :

X(39578) lies on these lines: {1, 3}, {10, 16451}, {24, 5307}, {31, 4278}, {43, 19338}, {191, 3185}, {199, 23850}, {228, 5904}, {404, 19858}, {958, 37058}, {993, 4225}, {1001, 37057}, {1011, 5259}, {1125, 16452}, {1203, 19762}, {1468, 4276}, {1479, 37400}, {1631, 9591}, {1698, 16453}, {1724, 35206}, {2915, 16682}, {3624, 16287}, {3899, 23846}, {4184, 5248}, {4192, 7741}, {4210, 25440}, {4216, 5267}, {4299, 19262}, {4897, 5926}, {5132, 5312}, {5251, 13738}, {5263, 37288}, {5271, 11340}, {5691, 7420}, {5692, 35289}, {6097, 12699}, {7488, 17134}, {7793, 17148}, {7951, 13731}, {8583, 16294}, {9840, 10483}, {16058, 25542}, {16286, 34595}, {16289, 25512}, {16297, 19872}, {16298, 19881}, {16466, 19759}, {16690, 18792}, {17524, 23383}, {18655, 39475}, {19339, 26102}, {19341, 25502}, {19343, 32783}, {19854, 37264}, {19859, 37282}, {20988, 37292}

X(39578) = circumcircle-inverse of X(38474)
X(39578) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 16678, 1}, {35, 36, 37522}, {1381, 1382, 38474}, {16287, 20470, 3624}

X(39579) = CENTER OF ((POLAR CIRCLE, CONWAY CIRCLE))

Barycentrics    (b + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2 + 2*a*b + b^2 + 2*a*c + c^2) : :

X(39579) lies on these lines: {1, 4}, {10, 429}, {19, 406}, {25, 5248}, {35, 4231}, {92, 4044}, {142, 14018}, {318, 469}, {407, 12609}, {427, 1900}, {430, 1867}, {431, 3822}, {442, 18589}, {516, 37194}, {519, 5130}, {860, 1869}, {993, 26377}, {1089, 1826}, {1125, 4185}, {1210, 12610}, {1426, 3671}, {1825, 4848}, {1829, 1904}, {1861, 5142}, {1890, 4222}, {2333, 3294}, {3817, 37368}, {4101, 11682}, {4194, 5250}, {4292, 21621}, {5155, 11396}, {5706, 5928}, {7414, 31730}, {8666, 11401}, {8736, 10408}, {10165, 37117}, {11323, 12572}, {11363, 35016}, {17171, 31902}, {17555, 30687}

X(39579) = X(4194)-Ceva conjugate of X(3931)
X(39579) = barycentric product X(i)*X(j) for these {i,j}: {92, 3931}, {225, 14555}, {226, 4194}, {321, 7713}, {1826, 17321}
X(39579) = barycentric quotient X(i)/X(j) for these {i,j}: {3931, 63}, {4194, 333}, {4254, 283}, {5250, 1812}, {5256, 1444}, {7713, 81}, {14555, 332}, {16466, 1790}, {17321, 17206}
X(39579) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 1068, 5307}, {4, 6198, 1891}, {429, 1824, 10}

X(39580) = CENTER OF ((ORTHOPTIC CIRCLE OF STEINER INELLIPSE, CONWAY CIRCLE))

Barycentrics    5*a^2*b + 4*a*b^2 + b^3 + 5*a^2*c + 12*a*b*c + 5*b^2*c + 4*a*c^2 + 5*b*c^2 + c^3 : :

X(39580) lies on these lines: {1, 2}, {99, 11110}, {726, 3986}, {958, 19321}, {966, 34379}, {993, 19309}, {1376, 19319}, {1386, 6707}, {1738, 4751}, {3501, 3646}, {3576, 7410}, {3663, 25354}, {3817, 7380}, {4026, 31238}, {4078, 4698}, {4133, 4967}, {4297, 6998}, {4472, 15254}, {4708, 25557}, {5248, 16849}, {5257, 24325}, {5267, 19310}, {5847, 15668}, {7390, 28164}, {7407, 12571}, {8728, 20544}, {9746, 12512}, {10164, 21554}, {10436, 24695}, {12558, 37360}, {16589, 37592}, {17248, 24231}, {17398, 38049}, {17723, 19749}, {19313, 25440}, {24178, 37039}, {24295, 38059}, {25590, 28526}

X(39580) = orthoptic circle of the Steiner inellipe inverse of X(38473)
X(39580) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 24603, 10}, {10, 19862, 29571}

X(39581) = CENTER OF ((ORTHOPTIC CIRCLE OF STEINER CIRCUMELLIPSE, CONWAY CIRCLE))

Barycentrics    a^3 - 3*a^2*b - a*b^2 - b^3 - 3*a^2*c - 6*a*b*c - 3*b^2*c - a*c^2 - 3*b*c^2 - c^3 : :

X(39581) lies on these lines: {1, 2}, {7, 4655}, {20, 26260}, {21, 26241}, {100, 19314}, {105, 405}, {120, 9710}, {141, 38053}, {238, 5749}, {329, 32771}, {377, 20556}, {391, 3751}, {497, 31993}, {515, 7390}, {516, 25590}, {518, 966}, {740, 32087}, {944, 6998}, {946, 7407}, {956, 19309}, {958, 4223}, {962, 7379}, {984, 5296}, {993, 37254}, {999, 16852}, {1001, 2345}, {1010, 4339}, {1191, 2295}, {1211, 3475}, {1213, 3242}, {1215, 18228}, {1279, 17303}, {1281, 20094}, {1334, 31435}, {1447, 3600}, {1621, 19822}, {1655, 26274}, {2550, 3739}, {2551, 34852}, {2975, 19310}, {3189, 19288}, {3246, 26039}, {3295, 16849}, {3303, 4433}, {3416, 4648}, {3486, 37149}, {3598, 4298}, {3673, 13725}, {3780, 5275}, {3846, 5226}, {3883, 4307}, {3886, 4967}, {3923, 7229}, {3945, 5847}, {3966, 5712}, {3980, 9778}, {3991, 6051}, {4000, 4026}, {4008, 24993}, {4297, 9746}, {4309, 20097}, {4310, 4357}, {4356, 17151}, {4363, 5698}, {4385, 28809}, {4447, 25524}, {4517, 25917}, {4647, 31130}, {4663, 37654}, {4673, 30758}, {4684, 17270}, {4733, 28635}, {4751, 32850}, {4968, 37314}, {5015, 37153}, {5232, 11038}, {5257, 7174}, {5258, 26258}, {5260, 16048}, {5303, 19326}, {5435, 32916}, {5542, 17272}, {5603, 7380}, {5657, 21554}, {5687, 19313}, {5734, 30740}, {5737, 24477}, {5743, 25568}, {5744, 32917}, {5748, 25960}, {5750, 7290}, {5772, 18230}, {5846, 15668}, {5936, 8236}, {6172, 32935}, {7222, 17768}, {7410, 7967}, {7613, 24199}, {9776, 26034}, {9779, 25385}, {9785, 28644}, {11037, 36854}, {12513, 26244}, {12618, 38037}, {13464, 30757}, {13728, 26978}, {15569, 17314}, {16342, 19758}, {16454, 19761}, {16466, 33854}, {17049, 35628}, {17116, 24280}, {17257, 24349}, {17259, 38057}, {17260, 27549}, {17321, 32922}, {17359, 38025}, {17398, 38315}, {19825, 32929}, {19888, 19942}, {21258, 28629}, {21283, 27812}, {21296, 33082}, {24248, 31995}, {24695, 35578}, {24697, 31178}, {28581, 28634}, {30754, 37050}, {37650, 38047}

X(39581) = orthoptic circle of the Steiner circumellipe inverse of X(38473)
X(39581) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1698, 19868}, {1, 19857, 19866}, {2, 145, 16830}, {2, 7172, 5268}, {2, 16823, 16020}, {8, 3616, 17316}, {10, 1125, 17284}, {10, 31211, 1698}, {10, 36479, 8}, {3883, 10436, 4307}

X(39582) = CENTER OF ((CIRCUMCIRCLE, SPIEKER RADICAL CIRCLE))

Barycentrics    a^2*(a^5 + a^4*b - a*b^4 - b^5 + a^4*c + 2*a^3*b*c - 2*a*b^3*c - b^4*c - 2*a*b^2*c^2 - 2*a*b*c^3 - a*c^4 - b*c^4 - c^5) : :

Let OA be the circle ((circumcircle, A-excircle)), and define OB and OC cyclically. X(39582) is the radical center of OA, OB, OC. (Randy Hutson, September 30, 2020)

X(39582) lies on these lines: {1, 11337}, {3, 10}, {19, 24}, {21, 8185}, {22, 35}, {25, 5248}, {26, 32613}, {28, 10198}, {55, 2915}, {100, 37557}, {198, 2922}, {199, 4362}, {222, 23156}, {386, 5329}, {404, 19784}, {405, 20989}, {498, 37231}, {997, 1603}, {1125, 37034}, {1478, 16049}, {1479, 35996}, {1714, 16451}, {1995, 5259}, {2077, 10323}, {3085, 7520}, {3145, 12567}, {3647, 24320}, {3678, 7085}, {3811, 5285}, {3814, 37415}, {3822, 4185}, {3825, 37366}, {3871, 37546}, {3874, 37581}, {4191, 25453}, {4225, 5230}, {4294, 35988}, {4385, 19842}, {5078, 16466}, {5217, 20833}, {5264, 20847}, {5271, 11340}, {5752, 20986}, {6097, 9958}, {6636, 10327}, {6644, 9895}, {7742, 11350}, {8069, 37250}, {8192, 8666}, {8193, 8715}, {9571, 36744}, {11490, 23868}, {11491, 15177}, {12410, 25439}, {12558, 37387}, {15078, 15940}, {15931, 17928}, {16453, 20083}, {16777, 35212}, {19544, 25639}, {19844, 28612}, {26063, 37120}, {26364, 37431}, {33771, 37576}

X(39582) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 197, 10}, {3, 9798, 993}, {3, 22654, 5267}, {24, 10902, 39475}, {55, 27802, 3743}

X(39583) = CENTER OF ((NINE-POINT CIRCLE, SPIEKER RADICAL CIRCLE))

Barycentrics    (b + c)*(a^4*b^2 + a^3*b^3 - a^2*b^4 - a*b^5 + a^3*b^2*c + a^2*b^3*c - a*b^4*c - b^5*c + a^4*c^2 + a^3*b*c^2 + a^2*b^2*c^2 + 2*a*b^3*c^2 + a^3*c^3 + a^2*b*c^3 + 2*a*b^2*c^3 + 2*b^3*c^3 - a^2*c^4 - a*b*c^4 - a*c^5 - b*c^5) : :

Let OA be the circle ((nine-point circle, A-excircle)), and define OB and OC cyclically. X(39583) is the radical center of OA, OB, OC. (Randy Hutson, September 30, 2020)

X(39583) lies on these lines: {5, 10}, {35, 14008}, {42, 7741}, {71, 24045}, {313, 7752}, {403, 1869}, {1594, 1826}, {1985, 5248}, {2476, 16828}, {3136, 3841}, {3142, 3822}, {4129, 34964}, {5141, 19874}, {5154, 26115}, {6757, 21318}, {14321, 39511}, {16837, 21011}, {21051, 39503}

X(39583) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2540, 2541, 970}, {9956, 22299, 10}

X(39584) = CENTER OF ((CONWAY CIRCLE, SPIEKER RADICAL CIRCLE))

Barycentrics    a^4 + 3*a^3*b + a^2*b^2 - a*b^3 + 3*a^3*c + 10*a^2*b*c - a*b^2*c - 2*b^3*c + a^2*c^2 - a*b*c^2 - 4*b^2*c^2 - a*c^3 - 2*b*c^3 : :

Let OA be the circle ((Conway circle, A-excircle)), and define OB and OC cyclically. X(39584) is the radical center of OA, OB, OC. (Randy Hutson, September 30, 2020)

X(39584) lies on these lines: {1, 2}, {740, 3333}, {958, 4891}, {971, 35631}, {1058, 5847}, {2334, 30818}, {3813, 4851}, {3868, 38478}, {3875, 3976}, {3886, 37607}, {4252, 4702}, {5250, 32919}, {9614, 32946}, {9841, 10476}, {9848, 21334}, {10439, 12544}, {10862, 31871}, {25524, 28581}, {31435, 32853}, {32941, 37554}

X(39584) = Conway-circle-inverse of X(38471)
X(39584) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 35613, 10}, {145, 38475, 1}, {35631, 35635, 39553}

X(39585) = CENTER OF ((POLAR CIRCLE, SPIEKER RADICAL CIRCLE))

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^3 - a*b^2 - 2*a*b*c - 2*b^2*c - a*c^2 - 2*b*c^2) : :
Barycentrics    a + (a + b + c) sec A : :

Let OA be the circle ((polar circle, A-excircle)), and define OB and OC cyclically. X(39585) is the radical center of OA, OB, OC. (Randy Hutson, September 30, 2020)

Let OA, OB, OC be the centers of the Odehnal tritangent circles. X(39585) is the trilinear product OA*OB*OC. (Randy Hutson, September 30, 2020)

X(39585) lies on these lines: {1, 29}, {2, 1076}, {3, 6708}, {4, 9}, {8, 7518}, {12, 37321}, {25, 1867}, {27, 31424}, {28, 993}, {33, 2901}, {34, 5136}, {35, 1013}, {36, 37253}, {46, 37235}, {56, 37393}, {57, 1940}, {58, 1957}, {72, 1859}, {165, 412}, {191, 1748}, {196, 3671}, {200, 318}, {225, 406}, {226, 1118}, {243, 3601}, {278, 1125}, {286, 10436}, {355, 7510}, {380, 8748}, {405, 1882}, {461, 7952}, {517, 7524}, {653, 3339}, {936, 7513}, {937, 36123}, {946, 6523}, {950, 1857}, {958, 7497}, {960, 1871}, {968, 1785}, {990, 20320}, {1074, 27505}, {1148, 11529}, {1214, 7532}, {1430, 37522}, {1698, 5125}, {1724, 7076}, {1841, 2049}, {1844, 3868}, {1848, 21616}, {1852, 11113}, {1865, 4205}, {1870, 30143}, {1884, 5130}, {1888, 3753}, {2181, 2292}, {2886, 15763}, {2975, 17519}, {3176, 6738}, {3185, 11500}, {3576, 7531}, {3624, 17923}, {3647, 31902}, {3679, 5174}, {3682, 27413}, {3771, 4213}, {3814, 5142}, {3931, 14571}, {4183, 5248}, {4200, 19855}, {4219, 25440}, {4297, 37028}, {4384, 11341}, {5090, 7140}, {5155, 37226}, {5267, 7501}, {5342, 11109}, {5709, 34831}, {5791, 15762}, {6743, 7046}, {6824, 34851}, {6826, 34823}, {6851, 34822}, {7009, 37055}, {7080, 32929}, {7102, 28076}, {7280, 37304}, {7337, 27802}, {7466, 29828}, {7515, 37695}, {7534, 31445}, {7541, 7989}, {7551, 8227}, {9816, 37093}, {10164, 37417}, {12047, 30687}, {12436, 37276}, {12609, 30686}, {16832, 37389}, {17917, 19862}, {18687, 36250}, {20317, 39536}, {25639, 37372}, {31993, 37377}

X(39585) = polar conjugate of the isotomic conjugate of X(5271)
X(39585) = X(i)-isoconjugate of X(j) for these (i,j): {63, 2215}, {222, 2335}, {520, 36077}, {905, 36080}
X(39585) = crossdifference of every pair of points on line {822, 1459}
X(39585) = barycentric product X(i)*X(j) for these {i,j}: {4, 5271}, {27, 5295}, {92, 405}, {318, 37543}, {333, 1882}, {1451, 7017}, {1897, 23882}, {1969, 5320}
X(39585) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 2215}, {33, 2335}, {405, 63}, {1451, 222}, {1882, 226}, {5271, 69}, {5295, 306}, {5320, 48}, {8750, 36080}, {23882, 4025}, {24019, 36077}, {37543, 77}
X(39585) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 242, 7713}, {4, 281, 10}, {29, 92, 1}, {278, 7498, 1125}, {1826, 1842, 4}, {3176, 34231, 6738}

X(39586) = CENTER OF ((ORTHOPTIC CIRCLE OF STEINER INELLIPSE, SPIEKER RADICAL CIRCLE))

Barycentrics    a^3 + 2*a^2*b + 3*a*b^2 + 2*a^2*c + 6*a*b*c + 2*b^2*c + 3*a*c^2 + 2*b*c^2 : :

X(39586) lies on these lines: {1, 2}, {9, 3842}, {35, 19310}, {36, 19314}, {37, 5695}, {40, 6998}, {55, 19309}, {56, 19313}, {86, 3751}, {120, 8728}, {148, 5988}, {171, 19311}, {210, 19701}, {213, 5275}, {274, 4385}, {341, 31997}, {405, 2223}, {474, 37575}, {516, 3986}, {518, 15668}, {726, 25590}, {740, 3247}, {958, 16849}, {966, 5847}, {980, 16458}, {984, 10436}, {986, 26240}, {999, 19319}, {1001, 4698}, {1213, 3416}, {1215, 7322}, {1376, 16852}, {1386, 17259}, {1447, 3339}, {1699, 7385}, {1738, 17321}, {1743, 33682}, {1759, 2198}, {2049, 5283}, {2279, 16552}, {2345, 4078}, {3263, 32092}, {3290, 3931}, {3295, 19321}, {3305, 32772}, {3550, 19312}, {3576, 21554}, {3681, 5333}, {3685, 27268}, {3696, 16777}, {3729, 24342}, {3731, 3923}, {3743, 26242}, {3745, 19732}, {3746, 19316}, {3775, 17296}, {3790, 28604}, {3821, 38052}, {3823, 25498}, {3826, 4657}, {3836, 17306}, {3844, 17327}, {3929, 4697}, {3932, 17303}, {3945, 34379}, {3993, 16673}, {4034, 17772}, {4042, 37595}, {4085, 38200}, {4104, 5712}, {4160, 30765}, {4220, 12511}, {4223, 5248}, {4307, 5296}, {4327, 17077}, {4364, 5880}, {4429, 17322}, {4645, 17248}, {4660, 25354}, {4670, 5220}, {4682, 5737}, {4687, 5263}, {4733, 17299}, {4751, 32922}, {5047, 23407}, {5204, 19323}, {5217, 19322}, {5235, 9347}, {5241, 17723}, {5257, 16970}, {5259, 16048}, {5261, 7176}, {5266, 16844}, {5290, 7179}, {5437, 6682}, {5563, 19320}, {5587, 7380}, {5657, 7410}, {5691, 7379}, {5725, 37661}, {5750, 16517}, {7174, 24325}, {7308, 25496}, {7407, 19925}, {8258, 31446}, {10180, 37553}, {11108, 14535}, {11110, 37552}, {13161, 37153}, {13634, 35242}, {16456, 30748}, {16475, 17277}, {16973, 17398}, {17270, 32846}, {18206, 25526}, {19329, 37586}, {21530, 30779}, {24220, 39553}, {25539, 31252}, {26234, 28612}, {30782, 37719}, {37650, 38049}

X(39586) = orthoptic-circle-of-Steiner-inellipe-inverse of X(5212)
X(39586) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1698, 4384}, {1, 16832, 16825}, {2, 5297, 29828}, {2, 16830, 1}, {8, 16826, 1}, {40, 6998, 9746}, {1125, 3634, 31191}, {1125, 36480, 1}, {3634, 16825, 16832}, {19856, 29674, 17308}, {25501, 29652, 10582}

X(39587) = CENTER OF ((ORTHOPTIC CIRCLE OF STEINER CIRCUMLLIPSE, SPIEKER RADICAL CIRCLE))

Barycentrics    3*a^3 + a^2*b + 5*a*b^2 - b^3 + a^2*c + 6*a*b*c + b^2*c + 5*a*c^2 + b*c^2 - c^3 : :

X(39587) lies on these lines: {1, 2}, {6, 5686}, {7, 7174}, {9, 4344}, {37, 390}, {38, 21454}, {55, 37254}, {65, 3598}, {105, 3303}, {144, 984}, {220, 5276}, {241, 3600}, {277, 37436}, {279, 388}, {346, 5263}, {355, 7407}, {377, 20344}, {452, 16601}, {517, 7390}, {518, 3945}, {948, 5261}, {964, 26770}, {966, 5846}, {1212, 5716}, {1386, 37681}, {1447, 5543}, {1449, 24393}, {2292, 20070}, {2550, 3672}, {3090, 15251}, {3160, 7179}, {3242, 4648}, {3247, 5853}, {3295, 4223}, {3416, 5232}, {3717, 5749}, {3745, 37666}, {3871, 19310}, {3883, 5296}, {3946, 38200}, {4318, 8232}, {4339, 11106}, {4346, 5880}, {4349, 5223}, {4353, 38052}, {4656, 9812}, {4675, 30340}, {4682, 24477}, {4901, 5750}, {4909, 4924}, {4968, 31130}, {4981, 14552}, {5141, 31126}, {5266, 17558}, {5269, 5273}, {5275, 36007}, {5717, 5815}, {5724, 34522}, {5992, 35369}, {6904, 37597}, {6998, 12245}, {7222, 28582}, {7226, 9965}, {7290, 18230}, {7322, 18228}, {7967, 21554}, {8158, 19544}, {9053, 15668}, {12632, 26242}, {13161, 37161}, {16487, 38059}, {17158, 26234}, {17301, 38092}, {17321, 32850}, {17572, 31073}, {17580, 37592}, {17599, 26040}, {17784, 28606}, {21226, 27340}, {25242, 31087}, {25303, 30758}, {31420, 36250}, {37650, 38315}

X(39587) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(5212)
X(39587) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 10, 5222}, {1, 29571, 3616}, {8, 16830, 2}, {145, 29624, 1}, {984, 4307, 144}, {1386, 38057, 37681}, {3242, 4648, 11038}, {3621, 29588, 145}, {4901, 5750, 5772}

X(39588) = CENTER OF ((2ND LEMOINE CIRCLE, POLAR CIRCLE))

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 + 4*a^2*b^2*c^2 + 3*b^4*c^2 + 3*a^2*c^4 + 3*b^2*c^4 - c^6) : :
X(39588) = X[6] - X[11456], X[378] + 2 X[8541]

X(39588) lies on these lines: {3, 6403}, {4, 6}, {5, 26206}, {22, 19131}, {24, 182}, {25, 5012}, {26, 19129}, {51, 10250}, {54, 12283}, {69, 3541}, {74, 32251}, {141, 37119}, {155, 5921}, {159, 9707}, {186, 5085}, {193, 3088}, {232, 5034}, {235, 18583}, {378, 511}, {394, 8889}, {403, 14561}, {427, 1993}, {468, 38110}, {475, 15988}, {575, 1974}, {576, 12294}, {578, 6467}, {611, 1870}, {613, 6198}, {895, 15472}, {1092, 14913}, {1204, 21851}, {1350, 3520}, {1351, 1593}, {1352, 1594}, {1353, 1595}, {1495, 23042}, {1570, 33843}, {1597, 5093}, {1598, 19118}, {1611, 7612}, {1614, 19125}, {1692, 10311}, {1885, 21850}, {1968, 5052}, {1986, 11579}, {1994, 7378}, {2854, 15463}, {2930, 3043}, {3066, 37777}, {3098, 35477}, {3186, 37124}, {3515, 12017}, {3516, 7691}, {3518, 7716}, {3542, 3618}, {3548, 26156}, {3589, 7505}, {3763, 6143}, {3818, 7547}, {4232, 15018}, {5092, 32534}, {5094, 15066}, {5095, 9976}, {5097, 11470}, {5102, 13596}, {5359, 9755}, {6353, 10601}, {6756, 36753}, {6759, 21637}, {6995, 33748}, {7387, 19121}, {7487, 36752}, {7503, 9967}, {7507, 11441}, {7509, 11574}, {7517, 19154}, {7526, 18438}, {7576, 11179}, {7577, 10516}, {7687, 34470}, {7722, 16010}, {8739, 36758}, {8740, 36757}, {9818, 11416}, {9924, 19357}, {9970, 12292}, {9973, 15577}, {10151, 38136}, {10323, 19126}, {10602, 15033}, {10880, 19145}, {10881, 19146}, {11245, 15809}, {11381, 34779}, {11403, 11482}, {11413, 37511}, {11457, 26926}, {11473, 35841}, {11474, 35840}, {11477, 14865}, {11649, 37970}, {11898, 16266}, {12061, 35228}, {12161, 34514}, {12272, 34148}, {13352, 34382}, {13860, 14965}, {14157, 19123}, {15038, 18535}, {15073, 34777}, {17508, 35472}, {17825, 38282}, {18449, 31861}, {18533, 25406}, {18560, 31670}, {19130, 35488}, {19140, 32250}, {19504, 32234}, {20190, 35479}, {29012, 35480}, {29181, 35481}, {31884, 35473}

X(39588) = midpoint of X(8541) and X(19124)
X(39588) = reflection of X(i) in X(j) for these {i,j}: {22, 19131}, {378, 19124}, {6800, 5050}
X(39588) = 2nd-Lemoine-circle-inverse of X(5523)
X(39588) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 12167, 6403}, {6, 6776, 7592}, {6, 8549, 6776}, {6, 36990, 34117}, {25, 5050, 19128}, {54, 12283, 19459}, {182, 1843, 24}, {6776, 36851, 34224}, {18440, 19139, 11441}

X(39589) = CENTER OF ((SPIEKER CIRCLE, POLAR CIRCLE))

Barycentrics    2*a^4 + 5*a^3*b - a^2*b^2 - a*b^3 + 3*b^4 + 5*a^3*c - 2*a^2*b*c - 3*a*b^2*c + 4*b^3*c - a^2*c^2 - 3*a*b*c^2 + 2*b^2*c^2 - a*c^3 + 4*b*c^3 + 3*c^4 : :

X(39589) lies on these lines: {4, 9}, {72, 20106}, {387, 3950}, {519, 1104}, {1125, 4438}, {1834, 2325}, {3952, 25982}, {4082, 5230}, {12575, 29673}, {17132, 23537}, {17279, 24391}, {24036, 37528}, {24564, 33170}, {24987, 33166}

X(39589) = Spieker-circle-inverse of X(242)

X(39590) = CENTER OF ((MOSES CIRCLE, POLAR CIRCLE))

Barycentrics    2*a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4 : :
X(39590) = X[1078] - 3 X[33013], 3 X[7785] - X[7905]

X(39590) lies on these lines: {2, 7842}, {3, 7603}, {4, 39}, {5, 187}, {6, 3843}, {20, 8589}, {30, 1506}, {32, 381}, {76, 7843}, {83, 7861}, {99, 14042}, {115, 546}, {140, 6781}, {148, 7858}, {216, 18404}, {217, 13851}, {230, 3850}, {315, 9466}, {316, 2896}, {382, 574}, {384, 625}, {385, 15031}, {397, 5471}, {398, 5472}, {538, 7785}, {548, 3055}, {598, 7787}, {620, 19687}, {626, 8370}, {671, 7839}, {1003, 7862}, {1015, 3585}, {1078, 3849}, {1194, 37349}, {1196, 7394}, {1478, 9665}, {1479, 9650}, {1500, 3583}, {1504, 23261}, {1505, 23251}, {1531, 22416}, {1569, 22515}, {1570, 5480}, {1572, 18492}, {1656, 5206}, {1657, 15515}, {1692, 19130}, {1968, 7547}, {1975, 7775}, {1995, 15820}, {2021, 37243}, {2079, 18369}, {2207, 18386}, {2241, 10895}, {2242, 10896}, {2275, 18513}, {2276, 18514}, {3053, 3851}, {3070, 6251}, {3071, 6250}, {3091, 7737}, {3146, 31401}, {3291, 7533}, {3329, 14044}, {3331, 11572}, {3363, 7810}, {3526, 8588}, {3543, 31404}, {3627, 3815}, {3734, 7773}, {3767, 3832}, {3788, 14035}, {3818, 5052}, {3830, 5013}, {3839, 5309}, {3845, 5041}, {3853, 9698}, {3858, 7755}, {3861, 7765}, {3972, 7886}, {4045, 33229}, {4232, 15880}, {4302, 31501}, {5023, 5055}, {5025, 7804}, {5034, 36990}, {5038, 11645}, {5046, 16589}, {5056, 21843}, {5058, 6564}, {5062, 6565}, {5070, 5210}, {5072, 37637}, {5073, 15815}, {5116, 29323}, {5167, 27375}, {5188, 37348}, {5225, 31409}, {5277, 37375}, {5306, 23046}, {5355, 34571}, {6032, 14002}, {6284, 31476}, {6561, 31481}, {6655, 6683}, {6656, 6704}, {6658, 7769}, {6680, 33228}, {6749, 34569}, {7526, 9699}, {7530, 9700}, {7752, 7816}, {7754, 18546}, {7759, 11185}, {7760, 32457}, {7761, 16924}, {7763, 14068}, {7764, 32819}, {7770, 7825}, {7771, 33002}, {7772, 15484}, {7776, 17130}, {7777, 14066}, {7780, 7823}, {7786, 33019}, {7790, 14062}, {7795, 32827}, {7800, 32983}, {7801, 32816}, {7802, 16921}, {7803, 32996}, {7805, 7812}, {7807, 31275}, {7808, 7841}, {7809, 7895}, {7822, 32971}, {7828, 32993}, {7830, 32992}, {7831, 33020}, {7834, 14063}, {7835, 14034}, {7848, 7860}, {7849, 7885}, {7854, 32006}, {7855, 14711}, {7867, 11286}, {7872, 11174}, {7880, 7912}, {7889, 33184}, {7911, 31268}, {7915, 7934}, {7919, 14045}, {7926, 20081}, {8176, 33007}, {9300, 14893}, {9596, 9664}, {9599, 9651}, {9605, 11648}, {9608, 31861}, {9696, 37472}, {9697, 10540}, {10311, 35488}, {10631, 12110}, {12953, 31451}, {13330, 18553}, {13335, 14160}, {14162, 38225}, {14568, 20088}, {14712, 33024}, {14845, 15575}, {14907, 32962}, {15482, 33234}, {15602, 31457}, {15687, 31406}, {16306, 23323}, {16310, 36412}, {16950, 30785}, {17005, 19696}, {17578, 31400}, {18383, 32445}, {18403, 18429}, {20576, 23514}, {22615, 31463}, {22856, 22900}, {23047, 27371}, {23259, 31411}, {32826, 34511}, {32832, 32995}, {33030, 36812}, {33250, 37647}

X(39590) = reflection of X(37512) in X(1506)
X(39590) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 2548, 7748}, {4, 5475, 39}, {5, 7747, 187}, {20, 31415, 31455}, {20, 31455, 8589}, {76, 7843, 7845}, {76, 7900, 7882}, {83, 14041, 7861}, {115, 7745, 5007}, {148, 7858, 32450}, {316, 3934, 7873}, {316, 16044, 3934}, {384, 625, 7874}, {546, 7745, 115}, {1657, 31489, 15515}, {2548, 7748, 39}, {3091, 7737, 7746}, {3627, 3815, 7756}, {3734, 7773, 7821}, {3767, 3832, 18424}, {3815, 7756, 31652}, {3972, 32966, 7886}, {5025, 7804, 7852}, {5254, 7753, 5041}, {5475, 7748, 2548}, {6658, 7769, 32456}, {7737, 7746, 35007}, {7752, 11361, 7816}, {7761, 16924, 31239}, {7770, 7825, 7853}, {7809, 17128, 7895}, {7843, 7882, 7900}, {7860, 31276, 7848}, {7882, 7900, 7845}, {32827, 32979, 7795}

X(39591) = CENTER OF ((SPIEKER RADICAL CIRCLE, POLAR CIRCLE))

Barycentrics    a^6*b + 4*a^5*b^2 + a^4*b^3 - 4*a^3*b^4 - a^2*b^5 - b^7 + a^6*c + 6*a^5*b*c + a^4*b^2*c - 4*a^3*b^3*c - a^2*b^4*c - 2*a*b^5*c - b^6*c + 4*a^5*c^2 + a^4*b*c^2 + 2*a^2*b^3*c^2 + b^5*c^2 + a^4*c^3 - 4*a^3*b*c^3 + 2*a^2*b^2*c^3 + 4*a*b^3*c^3 + b^4*c^3 - 4*a^3*c^4 - a^2*b*c^4 + b^3*c^4 - a^2*c^5 - 2*a*b*c^5 + b^2*c^5 - b*c^6 - c^7 : :

X(39591) lies on these lines: {4, 9}, {20, 26065}, {37, 5799}, {43, 1490}, {72, 970}, {201, 1848}, {212, 1891}, {226, 986}, {386, 18446}, {515, 580}, {517, 3695}, {950, 5255}, {962, 29641}, {1071, 4260}, {1695, 32778}, {1698, 10888}, {1750, 6048}, {2050, 5791}, {3912, 10441}, {5294, 37399}, {5480, 15852}, {5746, 9593}, {5758, 9535}, {6001, 10822}, {6684, 7413}, {6832, 19858}, {6846, 19853}, {6848, 27539}, {6905, 38856}, {7070, 7718}, {9549, 11523}, {12691, 34458}, {13442, 35203}, {17353, 37415}, {23537, 29069}, {33137, 35635}

X(39591) = Spieker-radical-circle-inverse of X(242)
X(39591) = {X(10),X(10445)}-harmonic conjugate of X(4)
;

X(39592) = CENTER OF ((BEVAN, ORTHOPTIC CIRCLE OF STEINER INELLIPSE))

Barycentrics    a*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5 + 3*a^4*c + 4*a^3*b*c - 2*a^2*b^2*c - 4*a*b^3*c - b^4*c + 2*a^3*c^2 - 2*a^2*b*c^2 + 6*a*b^2*c^2 + 2*b^3*c^2 - 2*a^2*c^3 - 4*a*b*c^3 + 2*b^2*c^3 - 3*a*c^4 - b*c^4 - c^5) : :

X(39592) lies on these lines: {1, 11350}, {2, 40}, {9, 19822}, {27, 4384}, {46, 1203}, {57, 77}, {63, 391}, {484, 23511}, {517, 37269}, {1427, 38866}, {1445, 3101}, {1453, 16049}, {1697, 5287}, {1730, 16368}, {1763, 20367}, {1998, 3198}, {3218, 20211}, {3333, 17011}, {3359, 16435}, {3576, 11340}, {3587, 21483}, {5011, 21370}, {5119, 17022}, {5271, 24633}, {5587, 7382}, {5709, 11347}, {5732, 33586}, {10476, 11329}, {17019, 31393}, {17810, 30271}, {26001, 37185}

X(39593) = CENTER OF ((MOSES CIRCLE, ORTHOPTIC CIRCLE OF STEINER INELLIPSE))

Barycentrics    2*a^4 + 5*a^2*b^2 + 2*b^4 + 5*a^2*c^2 - 4*b^2*c^2 + 2*c^4 : :
X(39593) = X[2] - 3 X[7827], X[5007] + 2 X[7765], 2 X[7760] + X[7873]

X(39593) lies on these lines: {2, 39}, {6, 3830}, {30, 5007}, {32, 3534}, {115, 5066}, {187, 5306}, {230, 11812}, {376, 5319}, {381, 7772}, {547, 9698}, {549, 7755}, {574, 15693}, {1285, 2549}, {1506, 10109}, {1570, 8584}, {3053, 15695}, {3329, 32457}, {3845, 5041}, {4045, 37671}, {4677, 9620}, {5013, 15701}, {5028, 15534}, {5304, 15697}, {5305, 12100}, {5346, 7738}, {5368, 15690}, {6321, 15516}, {6661, 7829}, {7603, 18362}, {7735, 8589}, {7737, 15640}, {7745, 12101}, {7747, 33699}, {7748, 15682}, {7749, 15713}, {7754, 7865}, {7756, 19710}, {7758, 33223}, {7759, 33251}, {7760, 7873}, {7766, 11057}, {7781, 33220}, {7788, 7798}, {7790, 7837}, {7805, 7811}, {7809, 7839}, {7816, 7920}, {7821, 7902}, {7842, 7894}, {7856, 33246}, {7882, 7918}, {7895, 7923}, {7913, 22253}, {8597, 12156}, {8667, 15810}, {9605, 19709}, {9606, 15699}, {14901, 34945}, {15515, 15716}, {15602, 21843}, {15685, 30435}, {15689, 22331}, {15694, 22332}, {15702, 31457}, {15709, 31450}, {15723, 31492}, {18424, 37665}

X(39593) = midpoint of X(7760) and X(7924)
X(39593) = reflection of X(i) in X(j) for these {i,j}: {6661, 7829}, {7873, 7924}
X(39593) = Moses-circle-inverse of X(32526)
X(39593) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 11648, 14537}, {194, 7884, 7880}, {5286, 7739, 5309}, {5309, 7739, 39}, {5346, 7738, 15513}, {5355, 15048, 187}, {7755, 9607, 31652}, {7797, 32450, 7874}, {7880, 7884, 7852}

X(39594) = CENTER OF ((CONWAY CIRCLE, ORTHOPTIC CIRCLE OF STEINER INELLIPSE))

Barycentrics    a^3 + 2*a^2*b - a*b^2 + 2*a^2*c + 2*a*b*c - 2*b^2*c - a*c^2 - 2*b*c^2 : :

X(39594) lies on these lines: {1, 2}, {9, 32853}, {57, 740}, {63, 32915}, {69, 24210}, {171, 3886}, {312, 3751}, {314, 8033}, {329, 34379}, {430, 5155}, {497, 5847}, {516, 12555}, {518, 35645}, {524, 24703}, {752, 9580}, {940, 3706}, {968, 1150}, {982, 3875}, {1001, 4891}, {1376, 28581}, {1449, 25496}, {1699, 32946}, {1707, 3685}, {1738, 18141}, {1743, 4011}, {1757, 30568}, {1864, 10477}, {1889, 1900}, {2801, 10439}, {2886, 4851}, {2887, 17296}, {3052, 4702}, {3158, 4434}, {3185, 15571}, {3210, 18193}, {3243, 32920}, {3305, 32864}, {3306, 32860}, {3474, 28580}, {3586, 38456}, {3677, 32921}, {3696, 37674}, {3729, 32913}, {3742, 4361}, {3749, 3769}, {3772, 4966}, {3784, 6007}, {3791, 7290}, {3842, 25430}, {3879, 26098}, {3928, 32934}, {3971, 5223}, {4038, 10436}, {4387, 4641}, {4425, 17272}, {4650, 4693}, {4684, 33144}, {4716, 17063}, {4865, 24392}, {5220, 35652}, {5269, 32941}, {5737, 15569}, {5839, 26105}, {6001, 10441}, {7075, 24727}, {7996, 10444}, {8167, 17348}, {9345, 21020}, {9965, 28526}, {10473, 17625}, {10980, 17151}, {13425, 26300}, {13458, 26301}, {14829, 17594}, {15733, 35892}, {16475, 32942}, {16496, 32926}, {17064, 18134}, {17163, 26627}, {17233, 33121}, {17274, 33154}, {17282, 33132}, {17286, 32780}, {17298, 17889}, {17314, 24477}, {17377, 33071}, {24217, 32861}, {24231, 30699}, {25527, 33087}, {32846, 33141}, {32863, 33134}, {32916, 37553}, {32929, 37639}, {32932, 37684}, {37593, 37660}

X(39594) = Conway-circle-inverse of X(5121)
X(39594) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 35613, 3741}, {1999, 10453, 1}, {1999, 38473, 10453}, {3187, 29824, 614}, {3685, 37683, 1707}, {10441, 35635, 12544}, {10980, 17151, 24165}, {17733, 35633, 1}, {20037, 38475, 1}, {32858, 33142, 29857}, {32915, 32919, 63}, {33087, 33135, 25527}

X(39595) = CENTER OF ((INCIRCLE, ORTHOPTIC CIRCLE OF STEINER CIRCUMELLIPSE))

Barycentrics    2*a^3 + a^2*b + b^3 + a^2*c + 4*a*b*c - b^2*c - b*c^2 + c^3 : :
X(39595) = 3 X[2] + X[1999], 3 X[171] + X[33095], 3 X[24210] - X[33095]

X(39595) lies on these lines: {1, 2}, {4, 37554}, {5, 5717}, {6, 3452}, {9, 37642}, {11, 3745}, {27, 1785}, {37, 5745}, {45, 5325}, {56, 16435}, {57, 1766}, {58, 12572}, {63, 4656}, {81, 908}, {142, 3772}, {171, 516}, {181, 21334}, {222, 226}, {312, 17355}, {345, 3950}, {354, 17602}, {497, 5269}, {515, 37715}, {527, 4415}, {553, 3782}, {750, 3914}, {942, 34937}, {946, 2050}, {948, 2124}, {950, 19542}, {959, 12435}, {982, 4353}, {999, 19517}, {1076, 4292}, {1100, 37662}, {1104, 30847}, {1155, 4854}, {1323, 1427}, {1376, 3755}, {1386, 3816}, {1400, 1764}, {1449, 30827}, {1453, 5084}, {1468, 12527}, {1699, 4307}, {1736, 11031}, {1738, 17122}, {1743, 18228}, {1997, 3618}, {2183, 18163}, {2269, 21363}, {2325, 35652}, {2886, 4682}, {3305, 24597}, {3306, 19785}, {3333, 7397}, {3666, 3911}, {3672, 5435}, {3686, 5743}, {3717, 33121}, {3718, 17353}, {3731, 5273}, {3742, 17061}, {3749, 30331}, {3751, 21060}, {3752, 3946}, {3769, 3883}, {3817, 4349}, {3846, 5847}, {3879, 4417}, {3928, 4419}, {3931, 6684}, {3944, 37604}, {3945, 5226}, {3947, 36662}, {3977, 3995}, {4000, 5437}, {4001, 26580}, {4035, 4851}, {4038, 17719}, {4078, 4438}, {4082, 33163}, {4104, 32853}, {4192, 37609}, {4298, 6996}, {4304, 37419}, {4310, 10980}, {4314, 36698}, {4340, 9612}, {4344, 5274}, {4356, 10164}, {4357, 14829}, {4358, 5294}, {4383, 5316}, {4416, 37683}, {4644, 28609}, {4648, 25525}, {4653, 37265}, {4654, 4896}, {4684, 33126}, {4719, 6691}, {4848, 37614}, {4856, 5233}, {4862, 21454}, {4883, 17724}, {4906, 17051}, {4909, 5718}, {5045, 19512}, {5218, 37553}, {5219, 5712}, {5247, 18250}, {5249, 33133}, {5255, 12575}, {5257, 5737}, {5264, 10624}, {5328, 16667}, {5432, 37593}, {5542, 33144}, {5710, 12053}, {5716, 9581}, {5725, 10175}, {5750, 6703}, {5788, 11374}, {5850, 32913}, {6176, 24929}, {6260, 36746}, {6358, 24209}, {6510, 17052}, {6588, 23799}, {6678, 11018}, {6690, 15569}, {7174, 24477}, {7290, 26105}, {7365, 10481}, {7490, 7952}, {7536, 17102}, {8069, 11350}, {8609, 16579}, {8756, 18677}, {9345, 33127}, {9347, 11680}, {9776, 23681}, {10171, 17717}, {10391, 16870}, {10443, 0889}, {12047, 37559}, {12436, 23537}, {12512, 37603}, {12688, 35672}, {14873, 15524}, {14996, 31053}, {16602, 17366}, {17010, 27174}, {17124, 33128}, {17132, 32939}, {17151, 31325}, {17272, 37655}, {17278, 37682}, {17298, 26132}, {17354, 20942}, {17598, 24216}, {17599, 17728}, {17716, 24217}, {18141, 21255}, {18678, 37276}, {19722, 30824}, {20359, 21746}, {21246, 22020}, {24231, 33152}, {25072, 35466}, {25074, 25092}, {26065, 30568}, {27003, 33155}, {27131, 37685}, {27184, 37684}, {32851, 34064}, {35650, 37613}, {37416, 37608}, {37631, 37691}

X(39595) = midpoint of X(i) and X(j) for these {i,j}: {171, 24210}, {181, 21334}, {1999, 3687}
X(39595) = complement of X(3687)
X(39595) = incircle-inverse of X(5211)
X(39595) = X(i)-complementary conjugate of X(j) for these (i,j): {961, 141}, {1169, 960}, {1220, 21244}, {2298, 1329}, {2359, 34823}, {2363, 21246}, {6648, 21260}, {8687, 513}, {31643, 626}, {32736, 20317}, {36098, 3835}
X(39595) = X(i)-Ceva conjugate of X(j) for these (i,j): {6648, 514}, {24993, 5795}
X(39595) = crosssum of X(6) and X(2269)
X(39595) = barycentric product X(i)*X(j) for these {i,j}: {1, 24993}, {7, 5795}
X(39595) = barycentric quotient X(i)/X(j) for these {i,j}: {5795, 8}, {24993, 75}
X(39595) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1999, 3687}, {2, 3912, 20106}, {2, 5222, 23511}, {2, 11679, 10}, {2, 17022, 29571}, {2, 29841, 17023}, {5, 37594, 5717}, {37, 37646, 5745}, {226, 940, 3664}, {553, 3782, 4887}, {612, 11269, 4847}, {940, 17720, 226}, {975, 5292, 10}, {2534, 2535, 3687}, {3306, 19785, 24177}, {3666, 37634, 3911}, {3720, 29683, 3011}, {3772, 37674, 142}, {3782, 37520, 553}, {3817, 4349, 26098}, {3840, 29645, 1125}, {3946, 6692, 3752}, {4000, 5437, 24175}, {4356, 10164, 17594}, {5268, 33137, 10}, {5297, 33142, 25006}, {5311, 29662, 29639}, {5393, 5405, 10}, {13161, 37607, 4298}, {17122, 33135, 1738}, {18141, 25527, 21255}, {18228, 37666, 1743}, {26580, 37639, 4001}, {29635, 29649, 10}, {29668, 29842, 551}, {29687, 29863, 30768}, {33133, 37633, 5249}

X(39596) = CENTER OF ((BEVAN CIRCLE, ORTHOPTIC CIRCLE OF STEINER CIRCUMELLIPSE))

Barycentrics    a*(a^5 + 2*a^4*b + a^3*b^2 - a^2*b^3 - 2*a*b^4 - b^5 + 2*a^4*c + a^3*b*c - 3*a*b^3*c + a^3*c^2 + 2*a*b^2*c^2 + b^3*c^2 - a^2*c^3 - 3*a*b*c^3 + b^2*c^3 - 2*a*c^4 - c^5) : :

X(39596) lies on these lines: {1, 199}, {2, 40}, {46, 29821}, {57, 17017}, {1697, 5311}, {1730, 37061}, {1761, 3966}, {1961, 5119}, {2270, 5282}, {3169, 3509}, {3895, 20069}, {4191, 18788}, {4847, 5011}, {8301, 24310}, {12699, 34119}, {24468, 33325}

X(39597) = CENTER OF ((SPIEKER RADICAL CIRCLE, ORTHOPTIC CIRCLE OF STEINER CIRCUMELLIPSE))

Barycentrics    a^2*b + 3*b^3 + a^2*c + 2*a*b*c + 3*b^2*c + 3*b*c^2 + 3*c^3 : :

X(39597) lies on these lines: {1, 2}, {141, 21342}, {497, 17286}, {1211, 3717}, {1738, 28595}, {2321, 32773}, {3619, 3677}, {3686, 33118}, {3703, 4357}, {3773, 24210}, {3790, 4656}, {3883, 32777}, {3914, 4431}, {3925, 4967}, {3966, 17353}, {3977, 33083}, {4001, 33170}, {4054, 25958}, {4104, 33165}, {4388, 17355}, {4416, 33163}, {4480, 4683}, {4665, 21949}, {4884, 17237}, {4899, 33162}, {5294, 33075}, {5750, 33073}, {5847, 32780}, {24199, 25957}

X(39597) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 32778, 3687}

X(39598) = CENTER OF ((CONWAY CIRCLE, 2ND DROZ-FARNEY CIRCLE))

Barycentrics    a*(a^6 + a^4*b^2 + 2*a^3*b^3 - 3*a^2*b^4 - 2*a*b^5 + b^6 + 2*a^4*b*c + 2*a^3*b^2*c - 2*a*b^4*c - 2*b^5*c + a^4*c^2 + 2*a^3*b*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 + 2*a^3*c^3 + 4*b^3*c^3 - 3*a^2*c^4 - 2*a*b*c^4 - b^2*c^4 - 2*a*c^5 - 2*b*c^5 + c^6) : :

X(39598) lies on these lines: {1, 3}, {28, 1812}, {33, 15488}, {34, 511}, {63, 13733}, {78, 27622}, {141, 5794}, {224, 851}, {343, 5130}, {377, 3662}, {442, 25527}, {579, 5336}, {960, 25514}, {997, 28258}, {1352, 1891}, {1448, 3784}, {1708, 13732}, {1828, 33586}, {1944, 16066}, {3868, 37231}, {3869, 4224}, {3924, 28274}, {4185, 10477}, {4192, 10393}, {4362, 24391}, {4511, 27621}, {5208, 16049}, {5272, 28275}, {5752, 37697}, {6851, 39549}, {7009, 10449}, {10461, 37227}, {10884, 37409}, {11415, 28104}, {12635, 15509}

X(39598) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {}

X(39599) = CENTER OF ((INCIRCLE, 1ST DROZ-FARNEY CIRCLE))

Barycentrics    a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 + a^6*c - 4*a^5*b*c + 3*a^4*b^2*c + 4*a^3*b^3*c - 5*a^2*b^4*c + b^6*c + 3*a^4*b*c^2 - 4*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + 3*b^5*c^2 - 3*a^4*c^3 + 4*a^3*b*c^3 + 2*a^2*b^2*c^3 - 3*b^4*c^3 - 5*a^2*b*c^4 - 3*b^3*c^4 + 3*a^2*c^5 + 3*b^2*c^5 + b*c^6 - c^7 : :

X(39599) lies on these lines: {1, 4}, {8, 10524}, {10, 8068}, {11, 5887}, {21, 16153}, {40, 10320}, {46, 6891}, {65, 6882}, {404, 12609}, {411, 32760}, {496, 1858}, {499, 12514}, {517, 10523}, {908, 10573}, {912, 26475}, {920, 3086}, {962, 10321}, {1125, 14793}, {1210, 4084}, {1319, 7491}, {1320, 5086}, {1387, 37290}, {1737, 3869}, {1770, 6909}, {1836, 22766}, {1898, 17660}, {2476, 10039}, {3057, 6842}, {3452, 18395}, {3560, 11376}, {3612, 6948}, {3817, 8070}, {3825, 12736}, {4295, 6890}, {4870, 28452}, {5119, 6825}, {5176, 21077}, {5187, 18391}, {5886, 8071}, {6824, 23708}, {6838, 30305}, {6867, 10827}, {6868, 37618}, {6911, 11375}, {6944, 37692}, {6973, 10826}, {6985, 11508}, {7288, 31515}, {7702, 10269}, {7743, 10948}, {8069, 12699}, {10165, 14792}, {11373, 22760}, {11374, 11501}, {13750, 37356}, {14986, 17483}, {16193, 26201}, {26437, 37826}, {28628, 37249}

X(39599) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {226, 12053, 5882}, {3086, 11415, 920}, {5603, 10629, 1}, {10572, 12047, 12608}, {12047, 30384, 4}, {12053, 12608, 10572}

X(39600) = CENTER OF ((CIRCUMCIRCLE, SPIEKER CIRCLE))

Barycentrics    a^2*(3*a^5 + a^4*b - 2*a^3*b^2 + 2*a^2*b^3 - a*b^4 - 3*b^5 + a^4*c + 2*a^3*b*c - 2*a*b^3*c - b^4*c - 2*a^3*c^2 - 2*a*b^2*c^2 + 4*b^3*c^2 + 2*a^2*c^3 - 2*a*b*c^3 + 4*b^2*c^3 - a*c^4 - b*c^4 - 3*c^5) : :

X(39600) lies on these lines: {2, 34936}, {3, 10}, {22, 280}, {24, 1604}, {25, 7952}, {56, 34039}, {198, 10902}, {1603, 1622}, {1617, 37034}, {2182, 17857}, {3085, 37052}, {3220, 10310}, {3556, 10306}, {5266, 7713}, {7742, 37269}, {8069, 8185}, {8192, 37259}, {8679, 23072}, {13737, 20989}, {20831, 26309}, {20999, 37257}, {33582, 36641}

X(39600) = complement of X(34936)
X(39600) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {197, 23843, 3}, {958, 1376, 39559}, {1324, 9798, 3}, {2933, 22654, 3}, {8069, 8185, 37260}, {20989, 37579, 13737}

X(39601) = CENTER OF ((ORTHOCENTROIDAL CIRCLE, SCHOUTTE CIRCLE))

Barycentrics    3*a^2*b^2 - 4*b^4 + 3*a^2*c^2 + 8*b^2*c^2 - 4*c^4 : :

X(39601) lies on these lines: {2, 8589}, {4, 11668}, {5, 39}, {6, 18362}, {32, 3851}, {76, 33011}, {183, 7617}, {187, 381}, {216, 9220}, {230, 5066}, {325, 14711}, {373, 8288}, {403, 33842}, {543, 37647}, {546, 7749}, {547, 15602}, {549, 11614}, {574, 5055}, {625, 3314}, {647, 39482}, {671, 17005}, {1007, 7615}, {1153, 8597}, {1196, 37353}, {1570, 10516}, {1656, 37512}, {2080, 14160}, {2482, 20112}, {2548, 3544}, {2549, 5071}, {3054, 3845}, {3055, 10109}, {3090, 7748}, {3091, 7737}, {3199, 16868}, {3291, 6032}, {3329, 9166}, {3363, 14971}, {3545, 5008}, {3628, 7756}, {3734, 31275}, {3767, 5068}, {3788, 32963}, {3830, 8588}, {3839, 21843}, {3843, 5206}, {3849, 17004}, {3850, 7747}, {3858, 12815}, {3934, 7934}, {5007, 5072}, {5025, 31239}, {5056, 31455}, {5070, 15515}, {5079, 31652}, {5107, 11178}, {5133, 5913}, {5141, 16589}, {5210, 14269}, {5215, 11317}, {5309, 31415}, {5355, 14892}, {5461, 7792}, {5585, 15684}, {6683, 33002}, {6722, 8370}, {7577, 33843}, {7745, 12811}, {7753, 11737}, {7761, 33006}, {7769, 20094}, {7774, 8176}, {7777, 32457}, {7786, 33010}, {7804, 14061}, {7816, 15031}, {7822, 32972}, {7828, 33024}, {7834, 32962}, {7842, 32993}, {7852, 16924}, {7853, 33228}, {7861, 16921}, {7873, 32832}, {7874, 32961}, {7886, 16044}, {9993, 22678}, {10255, 22401}, {10486, 10723}, {11648, 31489}, {11742, 15700}, {13701, 32790}, {13821, 32789}, {15022, 31401}, {15810, 37350}, {15993, 19130}, {26235, 31132}, {32819, 35022}, {33061, 36812}, {37334, 38228}

X(39601) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 115, 7603}, {115, 1506, 15048}, {115, 7603, 39}, {3054, 3845, 6781}, {14061, 33013, 7804}, {15031, 32967, 7816}

X(39602) = CENTER OF ((ORTHOPTIC CIRCLE OF STEINER INELLIPSE, SCHOUTTE CIRCLE))

Barycentrics    3*a^4*b^2 + a^2*b^4 - 2*b^6 + 3*a^4*c^2 - 8*a^2*b^2*c^2 + 2*b^4*c^2 + a^2*c^4 + 2*b^2*c^4 - 2*c^6 : :

X(39601) lies on these lines: {2, 187}, {6, 13857}, {30, 10418}, {50, 5094}, {111, 10989}, {114, 9193}, {115, 858}, {125, 15993}, {126, 325}, {230, 6128}, {232, 1560}, {352, 9140}, {381, 8585}, {385, 30786}, {511, 1648}, {574, 9745}, {647, 804}, {1184, 22121}, {1194, 1368}, {1196, 31101}, {1513, 1561}, {2502, 11645}, {3003, 3815}, {3231, 8288}, {3284, 7735}, {3314, 30749}, {3787, 26913}, {5104, 32225}, {5107, 6792}, {5971, 7845}, {6781, 7426}, {7664, 32456}, {7703, 8617}, {7778, 11336}, {7842, 16055}, {8352, 9172}, {9380, 30802}, {10317, 30771}, {16984, 30777}, {17964, 34320}, {18424, 31105}, {31125, 32457}

X(39601) = reflection of X(10418) in X(24855)
X(39601) = complement of X(26276)
X(39601) = crosssum of X(6) and X(32217)
X(39601) = crossdifference of every pair of points on line {11328, 17414}
X(39601) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6032, 7603}, {2, 7771, 10163}, {115, 5913, 3291}, {858, 5913, 115}, {7603, 15820, 6032}

X(39603) = CENTER OF ((CIRCUMCIRCLE, 1ST NEUBERG CIRCLE))

Barycentrics    a^8 + a^4*b^4 - a^2*b^6 - a^2*b^4*c^2 - b^6*c^2 + a^4*c^4 - a^2*b^2*c^4 - a^2*c^6 - b^2*c^6 : :
X(39603) = 2 X[6680] - 3 X[34506], 3 X[7775] - 5 X[7867]

X(39603) lies on these lines: {2, 32}, {3, 736}, {69, 2458}, {76, 5162}, {98, 7751}, {99, 37004}, {182, 7764}, {183, 18806}, {262, 35386}, {385, 32452}, {511, 7780}, {538, 14880}, {1691, 3788}, {2794, 5171}, {3398, 7759}, {3407, 7832}, {4027, 7796}, {4045, 13356}, {4769, 11364}, {5152, 9983}, {7761, 34870}, {7778, 38905}, {7781, 12203}, {7799, 10131}, {7821, 10349}, {7826, 32458}, {7843, 10796}, {7888, 10352}, {7909, 10334}, {7922, 10333}, {22712, 35385}, {35428, 38317}

X(39603) = midpoint of X(7751) and X(30270)
X(39603) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {32, 7815, 6680}, {32, 7818, 10350}, {32, 7867, 83}, {315, 7793, 32}, {7751, 8178, 31981}, {7780, 32189, 8177}

X(39604) = CENTER OF ((POLAR CIRCLE, 1ST LEMOINE CIRCLE))

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6*b^4 - a^4*b^6 - a^2*b^8 + b^10 + 4*a^6*b^2*c^2 - a^4*b^4*c^2 - 2*a^2*b^6*c^2 - b^8*c^2 + a^6*c^4 - a^4*b^2*c^4 - 2*a^2*b^4*c^4 - a^4*c^6 - 2*a^2*b^2*c^6 - a^2*c^8 - b^2*c^8 + c^10) : :

X(39604) lies on these lines: {3, 132}, {4, 83}, {5, 232}, {51, 297}, {112, 32152}, {114, 39575}, {458, 6747}, {626, 2967}, {1235, 24206}, {1968, 37242}, {2794, 20968}, {5117, 15466}, {5286, 39571}, {5523, 6248}, {6103, 10104}, {6530, 6656}, {7784, 15274}, {7859, 37124}, {9967, 27373}, {16868, 23514}

X(39605) = CENTER OF ((ORTHOPTIC CIRCLE OF STEINER INELLIPSE, BEVAN CIRCLE))

Barycentrics    a^5*b - a^4*b^2 - 4*a^3*b^3 + 3*a*b^5 + b^6 + a^5*c + 4*a^4*b*c - 6*a^3*b^2*c - 2*a^2*b^3*c + a*b^4*c + 2*b^5*c - a^4*c^2 - 6*a^3*b*c^2 - 4*a^2*b^2*c^2 - 4*a*b^3*c^2 - b^4*c^2 - 4*a^3*c^3 - 2*a^2*b*c^3 - 4*a*b^2*c^3 - 4*b^3*c^3 + a*b*c^4 - b^2*c^4 + 3*a*c^5 + 2*b*c^5 + c^6 : :

X(39605) lies on these lines: {2, 40}, {4, 39586}, {5, 29365}, {10, 262}, {496, 3931}, {515, 7379}, {516, 6998}, {581, 612}, {1125, 21554}, {1738, 27633}, {2223, 15972}, {2782, 5988}, {3428, 16849}, {3815, 4646}, {4301, 39580}, {5089, 39574}, {5257, 6210}, {5587, 7407}, {5711, 11374}, {5750, 6211}, {5881, 39587}, {6361, 7410}, {7179, 21620}, {7385, 18483}, {7982, 39581}, {9624, 16020}, {10310, 16852}, {11496, 19309}, {12512, 13634}, {13464, 16823}, {14520, 14872}, {17259, 38035}, {17381, 38118}, {18788, 19856}, {19313, 22753}, {37360, 37528}

X(39605) = midpoint of X(7379) and X(16830)
X(39605) = {X(6361),X(7410)}-harmonic conjugate of X(9746)

X(39606) = CENTER OF ((POLAR CIRCLE, LESTER CIRCLE))

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(2*a^10 - 7*a^8*b^2 + 8*a^6*b^4 - 2*a^4*b^6 - 2*a^2*b^8 + b^10 - 7*a^8*c^2 + 12*a^6*b^2*c^2 - 9*a^4*b^4*c^2 + 7*a^2*b^6*c^2 - 3*b^8*c^2 + 8*a^6*c^4 - 9*a^4*b^2*c^4 - 4*a^2*b^4*c^4 + 2*b^6*c^4 - 2*a^4*c^6 + 7*a^2*b^2*c^6 + 2*b^4*c^6 - 2*a^2*c^8 - 3*b^2*c^8 + c^10) : :

X(39606) lies on these lines: {4, 1116}, {24, 15475}, {98, 23096}, {107, 1291}, {125, 20184}, {230, 231}, {235, 15543}, {16868, 18308}

leftri

Points associated with Vijay-Paasche-Hutson triangles: X(39607)-X(39623)

rightri

This preamble is based on notes from Dasari Naga Vijay Krishna, September 10, 2020.

In the preambles just before X(37994), X(38387), and X(39311), 18 points, denoted by Ab, Ac, Bc, Ba, Ca, Cb , A′b, A′c, B′c, B′a, C′a, C′b , A″b, A″c, B″c, B″a, C″a, C″b, are defined, along with 31 Vijay-Paasche-Hutson triangles and liests of collinearities and perspectivities. This preamble introduces further such results. The points La, Pa, Ka, L′a, P′a, K′a, Ma, Na, Va, Ra, Qa defined in the preambles just before X(37994) and X(38487) are here relabled as A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, respectively.

The following sets of points are collinear:

{A1,A2,A3}
{A,A16,A30}
{A,A19,A23}
{A13,A28,A29}
{A15,A22,A23}
{A12,A15,A28}
{A4,A5,A6}
{A6,A9,A11}
{A2,A8,A20}
{A6,A7,A24}
{A11,A18,A31}
{A12,A15,A28}
{A,A3,A7,A17}
{A,A14,A15,A29}
{A12,A14,A30,X(39311)}
{A19,A20,A21,A22,A24}
{A14,A16,A17,A18,A24,A25}
{A,A1,A4,A8,A9,A12,A13,A24,A31}

Perspectors:of triangles:

X(1123) = Paasche point = AA1A4A8A9A12A13A24A31∩BB1B4B8B9B12B13B24B31∩CC1C4C8C9C12C13C24C31
X(1123) = a/(a + 2R) : b/(b+2R) : c/(c + 2R) = (sin A)/(1+ sin A) : (sin B)/(1 + sin B) : (sin C)/(1 + sin C)
X(3083) = AA3A7A17∩BB3B7B17∩CC3C7C17= 2R+a : 2R+b : 2R+c
X(39607)= A1A14∩B1B14∩C1C14
X(39608)= A3A14∩B3B14∩C3C14
X(39609)= A4A14∩B4B14∩C4C14
X(39610) = A8A14∩B8B14∩C8C14
X(39611) = A3A19∩B3B19∩C3C19
X(39612) = A4A19∩B4B19∩C4C19
X(39613) = A8A19∩B8B19∩C8C19
X(39614) = A9A19∩B9B19∩C9C19
X(39615) = A4A21∩B4B21∩C4C21
X(39616) = A9A21∩B9B21∩C9C21
X(39617) = A2A24∩B2B24∩C2C24
X(39618) = A5A24∩B5B24∩C5C24
X(39619) = A4A27∩B4B27∩C4C27
X(39620) = A3A29∩B3B29∩C3C29
X(39621) = A4A29∩B4B29∩C4C29
X(39622) = A8A29∩B8B29∩C8C29
X(39623) = A3A31∩B3B31∩C3C31

X(39607) = PERSPECTOR OF IST AND 14TH VIJAY-PAAASCHE-HUTSON TRIANGLE

Barycentrics    (64*R^6 + 64*R^5*(a + b + c) + 16*R^4*(2*a^2 + 3*b*c + 4*a*b + 4*a *c) + 8*R^3*(5*a^2*(b + c) + 2*b*c*(3*a - b - c)) + 4*R^2*(2*a^2*(b^2 + c^2) + a*b*c*(11*a - 4*b - 4*c) - 4*b^2*c^2) + 2*R*a*b*c*(3*a*b + 3*a*c - 8*b*c) - a^2*b^2*c^2) : :

See X39607.

. X(39607) lies on these lines: {}

X(39608) = PERSPECTOR OF 3rd AND 14TH VIJAY-PAAASCHE-HUTSON TRIANGLE

Barycentrics    (a*(2*R + a)*(8*R^4 + 8*R^3*(b + c) + 6*R^2*b*c - R*b*c*(b + c) - b^2*c^2)) : :

See X39608.

. X(39608) lies on these lines: {}

X(39609) = PERSPECTOR OF 4TH AND 14TH VIJAY-PAASCHE-HUTSON TRIANGLE

Barycentrics    (3*a^2*b^2*c^2+18*R*a^2*b*c^2+32*R^2*a*b*c^2+32*R^3*b*c^2+16*R^2*a^2*c^2+32*R^3*a*c^2+32*R^4*c^2+18*R*a^2*b^2*c+32*R^2*a*b^2*c+32*R^3*b^2*c+80*R^2*a^2*b*c+192*R^3*a*b*c+192*R^4*b*c+64*R^3*a^2*c+160*R^4*a*c+160*R^5*c+16*R^2*a^2*b^2+32*R^3*a*b^2+32*R^4*b^2+64*R^3*a^2*b+160*R^4*a*b+160*R^5*b+48*R^4*a^2+128*R^5*a+128*R^6) : :

See X39609.

. X(39609) lies on these lines: {}

X(39610) = PERSPECTOR OF 8th AND 14TH VIJAY-PAASCHE-HUTSON TRIANGLE

Barycentrics    (7*a^2*b^2*c^2+16*R*a*b^2*c^2+16*R^2*b^2*c^2+30*R*a^2*b*c^2+80*R^2*a*b*c^2+80*R^3*b*c^2+24*R^2*a^2*c^2+64*R^3*a*c^2+64*R^4*c^2+30*R*a^2*b^2*c+80*R^2*a*b^2*c+80*R^3*b^2*c+116*R^2*a^2*b*c+336*R^3*a*b*c+336*R^4*b*c+88*R^3*a^2*c+256*R^4*a*c+256*R^5*c+24*R^2*a^2*b^2+64*R^3*a*b^2+64*R^4*b^2+88*R^3*a^2*b+256*R^4*a*b+256*R^5*b+64*R^4*a^2+192*R^5*a+192*R^6) : :

See X39610.

. X(39610) lies on these lines: {}

X(39611) = PERSPECTOR OF THE 3rd, AND 19th- VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    (a*(2*R + a)*(32*R^4 + 24*R^3*(b + c) + 12*R^2*b*c - 4*R*b*c*(b + c) - 3*b^2*c^2)) : :

See X39611.

. X(39611) lies on these lines: {}

X(39612) = PERSPECTOR OF THE: 4TH AND 19th VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    (a^2*b^2*c^2+10*R*a^2*b*c^2+24*R^2*a*b*c^2+24*R^3*b*c^2+12*R^2*a^2*c^2+32*R^3*a*c^2+32*R^4*c^2+10*R*a^2*b^2*c+24*R^2*a*b^2*c+24*R^3*b^2*c+56*R^2*a^2*b*c+160*R^3*a*b*c+160*R^4*b*c+56*R^3*a^2*c+160*R^4*a*c+160*R^5*c+12*R^2*a^2*b^2+32*R^3*a*b^2+32*R^4*b^2+56*R^3*a^2*b+160*R^4*a*b+160*R^5*b+48*R^4*a^2+128*R^5*a+128*R^6) : :

See X39612.

. X(39612) lies on these lines: {}

X(39613) = PERSPECTOR OF THE 8th AND 19th VIJAY-PAASCHE-HUTSON TRIANGLE

Barycentrics    (a^2*b^2*c^2+3*R*a*b^2*c^2+3*R^2*b^2*c^2+5*R*a^2*b*c^2+16*R^2*a*b*c^2+16*R^3*b*c^2+5*R^2*a^2*c^2+16*R^3*a*c^2+16*R^4*c^2+5*R*a^2*b^2*c+16*R^2*a*b^2*c+16*R^3*b^2*c+22*R^2*a^2*b*c+72*R^3*a*b*c+72*R^4*b*c+20*R^3*a^2*c+64*R^4*a*c+64*R^5*c+5*R^2*a^2*b^2+16*R^3*a*b^2+16*R^4*b^2+20*R^3*a^2*b+64*R^4*a*b+64*R^5*b+16*R^4*a^2+48*R^5*a+48*R^6) : :

See X39613.

. X(39613) lies on these lines: {}

X(39614) = PERSPECTOR OF THE 9th AND 19th VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    a*(a^3*b^4*c^4-8*R*a^2*b^4*c^4-72*R^2*a*b^4*c^4-96*R^3*b^4*c^4+2*R*a^3*b^3*c^4+20*R^2*a^2*b^3*c^4-216*R^3*a*b^3*c^4-416*R^4*b^3*c^4 -12*R^2*a^3*b^2*c^4 + 280*R^3*a^2*b^2*c^4 + 320*R^4*a*b^2*c^4 - 288*R^5*b^2*c^4 - 16*R^3*a^3*b*c^4 + 752*R^4*a^2*b*c^4 + 1792*R^5*a*b*c^4 + 896*R^6*b*c^4 + 576*R^5*a^2*c^4 + 1536*R^6*a*c^4 + 1024*R^7*c^4 + 2*R*a^3*b^4*c^3 + 20*R^2*a^2*b^4*c^3 - 216*R^3*a*b^4*c^3 - 416*R^4*b^4*c^3 - 28*R^2*a^3*b^3*c^3 + 736*R^3*a^2*b^3*c^3 + 640*R^4*a*b^3*c^3 - 1088*R^5*b^3*c^3 - 416*R^3*a^3*b^2*c^3 + 2544*R^4*a^2*b^2*c^3 + 6016*R^5*a*b^2*c^3 + 1408*R^6*b^2*c^3 - 912*R^4*a^3*b*c^3 + 3936*R^5*a^2*b*c^3 + 13056*R^6*a*b*c^3 + 7552*R^7*b*c^3 -576*R^5*a^3*c^3 + 2304*R^6*a^2*c^3 + 8704*R^7*a*c^3 + 6144*R^8*c^3 - 12*R^2*a^3*b^4*c^2 + 280*R^3*a^2*b^4*c^2 + 320*R^4*a*b^4*c^2 - 288*R^5*b^4*c^2 - 416*R^3*a^3*b^3*c^2 + 2544*R^4*a^2*b^3*c^2 + 6016*R^5*a*b^3*c^2 + 1408*R^6*b^3*c^2 - 3056*R^4*a^3*b^2*c^2 + 3008*R^5*a^2*b^2*c^2 + 17024*R^6*a*b^2*c^2 + 7936*R^7*b^2*c^2 - 6176*R^5*a^3*b*c^2 -1472*R^6*a^2*b*c^2 + 21120*R^7*a*b*c^2 + 14848*R^8*b*c^2 - 3840*R^6*a^3*c^2 - 2560*R^7*a^2*c^2 + 10240*R^8*a*c^2 + 9216*R^9*c^2 - 16*R^3*a^3*b^4*c + 752*R^4*a^2*b^4*c + 1792*R^5*a*b^4*c + 896*R^6*b^4*c - 912*R^4*a^3*b^3*c + 3936*R^5*a^2*b^3*c + 13056*R^6*a*b^3*c + 7552*R^7*b^3*c - 6176*R^5*a^3*b^2*c - 1472*R^6*a^2*b^2*c + 21120*R^7*a*b^2*c + 14848*R^8*b^2*c - 11968*R^6*a^3*b*c - 16128*R^7*a^2*b*c + 9216*R^8*a*b*c + 12288*R^9*b*c - 7168*R^7*a^3*c - 12288*R^8*a^2*c - 1024*R^9*a*c + 4096*R^10*c + 576*R^5*a^2*b^4 + 1536*R^6*a*b^4 + 1024*R^7*b^4 - 576*R^5*a^3*b^3 + 2304*R^6*a^2*b^3 + 8704*R^7*a*b^3 + 6144*R^8*b^3 - 3840*R^6*a^3*b^2 - 2560*R^7*a^2*b^2 + 10240*R^8*a*b^2 + 9216*R^9*b^2 - 7168*R^7*a^3*b - 12288*R^8*a^2*b - 1024*R^9*a*b + 4096*R^10*b - 4096*R^8*a^3 - 8192*R^9*a^2 - 4096*R^10*a) : :

See X39614.

. X(39614) lies on these lines: {}

X(39615) = PERSPECTOR OF THE 4TH AND 21st VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    (5*a^2*b^2*c^2 + 6*R*a*b^2*c^2 + 24*R*a^2*b*c^2 + 44*R^2*a*b*c^2 + 24*R^3*b*c^2 + 24*R^2*a^2*c^2 + 48*R^3*a*c^2 + 32*R^4*c^2 + 24*R*a^2*b^2*c + 44*R^2*a*b^2*c + 24*R^3*b^2*c + 104*R^2*a^2*b*c + 224*R^3*a*b*c + 160*R^4*b*c + 96*R^3*a^2*c + 208*R^4*a*c + 160*R^5*c + 24*R^2*a^2*b^2 + 48*R^3*a*b^2 + 32*R^4*b^2 + 96*R^3*a^2*b + 208*R^4*a*b + 160*R^5*b + 80*R^4*a^2 + 160*R^5*a + 128*R^6) : :

See X39615.

. X(39615) lies on these lines: {}

X(39616) = PERSPECTOR OF THE 9th AND 21st VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    (a*(9*a^3*b^4*c^4 + 60*R*a^2*b^4*c^4 + 108*R^2*a*b^4*c^4 + 48*R^3*b^4*c^4 + 54*R*a^3*b^3*c^4 + 436*R^2*a^2*b^3*c^4 + 840*R^3*a*b^3*c^4 + 400*R^4*b^3*c^4 + 80*R^2*a^3*b^2*c^4 + 1016*R^3*a^2*b^2*c^4 + 2176*R^4*a*b^2*c^4 + 1120*R^5*b^2*c^4 + 32*R^3*a^3*b*c^4 + 1152*R^4*a^2*b*c^4 + 2816*R^5*a*b*c^4 + 1664*R^6*b*c^4 + 576*R^5*a^2*c^4 + 1536*R^6*a*c^4 + 1024*R^7*c^4 + 54*R*a^3*b^4*c^3 + 436*R^2*a^2*b^4*c^3 + 840*R^3*a*b^4*c^3 + 400*R^4*b^4*c^3 + 268*R^2*a^3*b^3*c^3 + 3056*R^3*a^2*b^3*c^3 + 6416*R^4*a*b^3*c^3 + 3264*R^5*b^3*c^3 + 48*R^3*a^3*b^2*c^3 + 6304*R^4*a^2*b^2*c^3 + 15488*R^5*a*b^2*c^3 + 8448*R^6*b^2*c^3 -704*R^4*a^3*b*c^3 + 5792*R^5*a^2*b*c^3 + 17920*R^6*a*b*c^3 + 11136*R^7*b*c^3 -576*R^5*a^3*c^3 + 2304*R^6*a^2*c^3 + 8704*R^7*a*c^3 + 6144*R^8*c^3 + 80*R^2*a^3*b^4*c^2 + 1016*R^3*a^2*b^4*c^2 + 2176*R^4*a*b^4*c^2 + 1120*R^5*b^4*c^2 + 48*R^3*a^3*b^3*c^2 + 6304*R^4*a^2*b^3*c^2 + 15488*R^5*a*b^3*c^2 + 8448*R^6*b^3*c^2 - 2448*R^4*a^3*b^2*c^2 + 8448*R^5*a^2*b^2*c^2 + 31296*R^6*a*b^2*c^2 + 18432*R^7*b^2*c^2 - 5984*R^5*a^3*b*c^2 + 832*R^6*a^2*b*c^2 + 27776*R^7*a*b*c^2 + 19712*R^8*b*c^2 - 3840*R^6*a^3*c^2 - 2560*R^7*a^2*c^2 + 10240*R^8*a*c^2 + 9216*R^9*c^2 + 32*R^3*a^3*b^4*c + 1152*R^4*a^2*b^4*c + 2816*R^5*a*b^4*c + 1664*R^6*b^4*c - 704*R^4*a^3*b^3*c + 5792*R^5*a^2*b^3*c + 17920*R^6*a*b^3*c + 11136*R^7*b^3*c - 5984*R^5*a^3*b^2*c + 832*R^6*a^2*b^2*c + 27776*R^7*a*b^2*c + 19712*R^8*b^2*c - 11968*R^6*a^3*b*c - 15360*R^7*a^2*b*c + 12032*R^8*a*b*c + 14336*R^9*b*c - 7168*R^7*a^3*c - 12288*R^8*a^2*c - 1024*R^9*a*c + 4096*R^10*c + 576*R^5*a^2*b^4 + 1536*R^6*a*b^4 + 1024*R^7*b^4 - 576*R^5*a^3*b^3 + 2304*R^6*a^2*b^3 + 8704*R^7*a*b^3 + 6144*R^8*b^3 - 3840*R^6*a^3*b^2 - 2560*R^7*a^2*b^2 + 10240*R^8*a*b^2 + 9216*R^9*b^2 - 7168*R^7*a^3*b - 12288*R^8*a^2*b -1024*R^9*a*b + 4096*R^10*b - 4096*R^8*a^3 - 8192*R^9*a^2 - 4096*R^10*a)) : :

See X39616.

. X(39616) lies on these lines: {}

X(39617) = PERSPECTOR OF THE 2nd AND 24TH VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    (a*(2*R + b)*(2*R + c)*(4*R^2 - b*c)*(16*R^4 - a^2*b*c)) : :

X(39617) lies on these lines: {}

See X39617.

.

X(39618) = PERSPECTOR OF THE 5th AND 24TH VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    ( a*(2*R + b)*(2*R + c)*(4*R + b + c)*(((4*R + a)*(16*R^2 + 4*R*a + 4*R*b + 4*R*c + a*b + b*c + c*a)) +a*b*c)) : :

See X39618.

. X(39618) lies on these lines: {}

X(39619) = PERSPECTOR OF THE 4TH AND 27th VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    (a*(16*a^2*b^3*c^3 + 34*R*a*b^3*c^3 + 16*R^2*b^3*c^3 + 81*R*a^2*b^2*c^3 + 205*R^2*a*b^2*c^3 + 120*R^3*b^2*c^3 + 100*R^2*a^2*b*c^3 + 286*R^3*a*b*c^3 + 184*R^4*b*c^3 + 32*R^3*a^2*c^3 + 112*R^4*a*c^3 + 80*R^5*c^3 + 81*R*a^2*b^3*c^2 + 205*R^2*a*b^3*c^2 + 120*R^3*b^3*c^2 + 406*R^2*a^2*b^2*c^2 + 1148*R^3*a*b^2*c^2 + 736*R^4*b^2*c^2 + 510*R^3*a^2*b*c^2 + 1592*R^4*a*b*c^2 + 1080*R^5*b*c^2 + 176*R^4*a^2*c^2 + 640*R^5*a*c^2 + 464*R^6*c^2 + 100*R^2*a^2*b^3*c + 286*R^3*a*b^3*c + 184*R^4*b^3*c + 510*R^3*a^2*b^2*c + 1592*R^4*a*b^2*c + 1080*R^5*b^2*c + 624*R^4*a^2*b*c + 2160*R^5*a*b*c + 1536*R^6*b*c + 208*R^5*a^2*c + 848*R^6*a*c + 640*R^7*c + 32*R^3*a^2*b^3 + 112*R^4*a*b^3 + 80*R^5*b^3 + 176*R^4*a^2*b^2 + 640*R^5*a*b^2 + 464*R^6*b^2 + 208*R^5*a^2*b + 848*R^6*a*b + 640*R^7*b + 64*R^6*a^2 + 320*R^7*a + 256*R^8)) : :

See X39619

. X(39619) lies on these lines: {}

X(39620) = PERSPECTOR OF THE 3RD AND 29TH VIJAY-PAASCHE-HUTSON TRTIANGLES

Barycentrics    ((R + b)*(R + c)*(2*R + a)*(a^2*b^2*c^2 - 56*R^2*a^2*b*c - 112*R^3*a*b*c - 64*R^4*b*c - 64*R^3*a^2*c - 128*R^4*a*c - 64*R^5*c - 64*R^3*a^2*b - 128*R^4*a*b - 64*R^5*b - 64*R^4*a^2 - 128*R^5*a - 64*R^6)) : :

See X39620.

. X(39620) lies on these lines: {}

X(39621) = PERSPECTOR OF THE 4TH AND 29th VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    (a*(R + b)*(R + c)*(13*a^2*b^2*c^2 + 28*R*a*b^2*c^2 + 16*R^2*b^2*c^2 + 44*R*a^2*b*c^2 + 96*R^2*a*b*c^2 + 56*R^3*b*c^2 + 32*R^2*a^2*c^2 + 64*R^3*a*c^2 + 32*R^4*c^2 + 44*R*a^2*b^2*c + 96*R^2*a*b^2*c + 56*R^3*b^2*c + 140*R^2*a^2*b*c + 304*R^3*a*b*c + 176*R^4*b*c + 96*R^3*a^2*c + 192*R^4*a*c + 96*R^5*c + 32*R^2*a^2*b^2 + 64*R^3*a*b^2 + 32*R^4*b^2 + 96*R^3*a^2*b + 192*R^4*a*b + 96*R^5*b + 64*R^4*a^2 + 128*R^5*a + 64*R^6) ) : :

See X39621.

. X(396201) lies on these lines: {}

X(39622) = PERSPECTOR OF THE 8TH AND 29TH VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    ( a*(R + b)*(R + c)*(7*a^2*b^2*c^2 + 16*R*a*b^2*c^2 + 10*R^2*b^2*c^2 + 23*R*a^2*b*c^2 + 52*R^2*a*b*c^2 + 32*R^3*b*c^2 + 16*R^2*a^2*c^2 + 32*R^3*a*c^2 + 16*R^4*c^2 + 23*R*a^2*b^2*c + 52*R^2*a*b^2*c + 32*R^3*b^2*c + 72*R^2*a^2*b*c + 160*R^3*a*b*c + 96*R^4*b*c + 48*R^3*a^2*c + 96*R^4*a*c + 48*R^5*c + 16*R^2*a^2*b^2 + 32*R^3*a*b^2 + 16*R^4*b^2 + 48*R^3*a^2*b + 96*R^4*a*b + 48*R^5*b + 32*R^4*a^2 + 64*R^5*a + 32*R^6)) : :

See X39622.

. X(39622) lies on these lines: {}

X(39623) = PERSPECTOR OF THE: 3RD AND 31ST VIJAY-PAASCHE-HUTSON TRIANGLES

Barycentrics    (a^2*b^2*c^2 + 2*R*a^2*b*c^2 + 2*R*a^2*b^2*c - 16*R^2*a^2*b*c - 64*R^3*a*b*c - 64*R^4*b*c - 32*R^3*a^2*c - 96*R^4*a*c - 96*R^5*c - 32*R^3*a^2*b - 96*R^4*a*b - 96*R^5*b - 48*R^4*a^2 - 128*R^5*a - 128*R^6) : :

See X39623.

. X(39623) lies on these lines: {}

X(39624) = SUREN-SHOURYA MAJOR INFINITY POINT

Barycentrics    3A - π : 3B - π 3C - π
Trilinears    (3A - π) csc A : (3B - π) csc B : (3C - π) csc C

Theorem: X(39624) is the unique major center on the line at infinity.

Proof (by Shourya Pandey):

First, suppose that f(A) : f(B) : f(C) is a major center on the line at infinity. Then f(A) + f(B) + f(C) = 0. Define g(A) = f(A + π/3), or equivalently, g(A - π/3) = f(A). Let x = A - π//3, y = B - π/3, z = C - π/3, so that x,y,z are real variables such that x + y + z = 0, and

g(x) + g(y) = -g(-x - y)        (*)

Putting x = y = 0 in (*) gives g(0) = 0, and putting y = -x in (*) shows that g is an odd function, so that

g(x) + g(y) = g(x + y)        (**)

Equation (**) is the Cauchy functional equation, for which the only continuous solutions are g(x) = k*x. Thus, f(A) = k(A - π/3), so that

f(A) : f(B) : f(C) = 3A - π : 3B - π : 3C - π .

The method of proof shows that this solution is unique.

See Euclid #1069

The domain of X(39624) consists of all non-equilateral triangles.

X(39624) lies on these lines: {2,360}, {30,511}

X(39624) = isogonal conjugate of X(39625)
X(39624) = isotomic conjugate of X(39626)

X(39625) = SUREN-SHOURYA MAJOR CIRCUMCIRCLE POINT

Barycentrics    (sin^2 A)/(3A - π); : (sin^2 B)/(3B - π); : (sin^2 B)/(3B - π);
Trilinears    (sin A)/(3A - π) : (sin B)/(3B - π) : (sin C)/(3C - π)

It follows from the theorem at X(39607) that X(39608) is the unique continuous major center on the circumcircle. The domain of X(39608) consists of all non-equilateral triangles.

X(39625) lies on the circumcircle and these lines: {99, 39626}, {359, 3067}

X(39625) = isogonal conjugate of X(39624)
X(39625) = Gibert circumtangential conjugate of X(39627)
X(39625) = intersection, other than A,B,C, of circumcircle and circumconic {{A, B, C, X(6), X(359)}}
X(39625) = barycentric product X(6)*X(39626)
X(39625) = barycentric quotient X(32)/X(39627)
X(39625) = trilinear product X(31)*X(39626)
X(39625) = trilinear quotient X(31)/X(39627)

X(39626) = SUREN-SHOURYA MAJOR STEINER CIRCUMELLIPSE POINT

Barycentrics    1/(3A - π); : 1/(3B - π); : 1/(3B - π);
Trilinears    (csc A)/(3A - π) : (csc B)/(3B - π) : (csc C)/(3C - π)

It follows from the theorem at X(39624) that X(39626) is the unique continuous major center on the Steiner circumellipse. The domain of X(39626) consists of all non-equilateral triangles.

X(39626) lies on the Steiner circumellipse and this line: {99, 39625}

X(39626) = isogonal conjugate of X(39627)
X(39626) = isotomic conjugate of X(39624)
X(39626) = barycentric product X(76)*X(39625)
X(39626) = trilinear product X(75)*X(39625)

X(39627) = ISOGONAL CONJUGATE OF X(39626)

Barycentrics    (sin^2 A)(3A - π) : (sin^2 B)(3B - π) : (sin^2 C)(3C - π)
Trilinears    (sin A)(3A - π) : (sin B)(3B - π) : (sin C)(3C - π)

X(39627) lies on this line: {187, 237}

X(39627) = isogonal conjugate of X(39626)
X(39627) = Gibert circumtangential conjugate of X(39625)
X(39627) = pole of the trilinear polar of X(39625) wrt circumcircle
X(39627) = barycentric product X(6)*X(39624)
X(39627) = barycentric quotient X(32)/X(39625)
X(39627) = trilinear product X(31)*X(39624)
X(39627) = trilinear quotient X(31)/X(39625)

X(39628) = CIRCUMCENTER OF CIRCUMCEVIAN POLAR TRIANGLE OF X(8)

Barycentrics    a^2*(a - b)*(a - c)*(a^4 - 2*a^2*b^2 + b^4 - 3*a^3*c + 5*a^2*b*c + 5*a*b^2*c - 3*b^3*c + a^2*c^2 - 12*a*b*c^2 + b^2*c^2 + 3*a*c^3 + 3*b*c^3 - 2*c^4)*(a^4 - 3*a^3*b + a^2*b^2 + 3*a*b^3 - 2*b^4 + 5*a^2*b*c - 12*a*b^2*c + 3*b^3*c - 2*a^2*c^2 + 5*a*b*c^2 + b^2*c^2 - 3*b*c^3 + c^4) : :

See the preamble just before X(38383).

X(39628) lies on these lines: {3, 38452}, {104, 11260}, {106, 37561}, {8686, 24928}

X(39629) = CIRCUMCENTER OF CIRCUMCEVIAN POLAR TRIANGLE OF X(32)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(-2*a^4*b^4 - a^2*b^6 - b^8 + a^6*c^2 - a^4*b^2*c^2 - 2*a^2*b^4*c^2 - b^6*c^2 - a^2*b^2*c^4 - 2*b^4*c^4 + a^2*c^6)*(a^6*b^2 + a^2*b^6 - a^4*b^2*c^2 - a^2*b^4*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 - a^2*c^6 - b^2*c^6 - c^8) : :

X(39629) lies on these lines: {3, 737}, {98, 8177}, {729, 26316}, {733, 35422}

X(39630) = CIRCUMCENTER OF CIRCUMCEVIAN POLAR TRIANGLE OF X(35)

Barycentrics    a^2*(a - b)*(a - c)*(a^4 - 2*a^2*b^2 + b^4 - a^2*b*c - a*b^2*c - 2*a^2*c^2 - 3*a*b*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - a^2*b*c - 3*a*b^2*c - 2*a^2*c^2 - a*b*c^2 - 2*b^2*c^2 + c^4) : :

X(39630) lies on these lines: {104, 15910}, {109, 21784}, {759, 2646}, {2291, 17454}, {2687, 5538}

X(39631) = CIRCUMCENTER OF CIRCUMCEVIAN POLAR TRIANGLE OF X(43)

Barycentrics    a*(a - b)*(a - c)*(a^4*b + a^3*b^2 + a^2*b^3 + a*b^4 - a^4*c - a^3*b*c + 2*a^2*b^2*c - a*b^3*c - b^4*c + a^3*c^2 - 4*a^2*b*c^2 - 4*a*b^2*c^2 + b^3*c^2 + a^2*c^3 + a*b*c^3 + b^2*c^3 - a*c^4 - b*c^4)*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4 - a^4*c + a^3*b*c + 4*a^2*b^2*c - a*b^3*c + b^4*c - a^3*c^2 - 2*a^2*b*c^2 + 4*a*b^2*c^2 - b^3*c^2 - a^2*c^3 + a*b*c^3 - b^2*c^3 - a*c^4 + b*c^4) : : lies on these lines:

X(39631) lies on these lines: {3, 9082}, {376, 15323}, {741, 4221}, {9061, 17522}

X(39632) = CIRCUMCENTER OF CIRCUMCEVIAN POLAR TRIANGLE OF X(182)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(2*a^4*b^2 + b^6 + a^4*c^2 + 2*a^2*b^2*c^2 + a^2*c^4 + 2*b^2*c^4)*(a^4*b^2 + a^2*b^4 + 2*a^4*c^2 + 2*a^2*b^2*c^2 + 2*b^4*c^2 + c^6) : :

X(39632) lies on these lines: {6, 737}, {98, 7697}, {707, 5106}, {6325, 11655}, {14509, 32694}

X(39633) = CIRCUMCENTER OF CIRCUMCEVIAN POLAR TRIANGLE OF X(191)

Barycentrics    a^2*(a - b)*(a - c)*(a^4 - 2*a^2*b^2 + b^4 + a^2*b*c + a*b^2*c - 2*a^2*c^2 - 3*a*b*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 + a^2*b*c - 3*a*b^2*c - 2*a^2*c^2 + a*b*c^2 - 2*b^2*c^2 + c^4) : :

X(39633) lies on these lines: {40, 5951}, {74, 11012}, {104, 6597}, {477, 12119}, {501, 759}, {952, 2372}, {953, 2392}, {2801, 15168}, {6265, 12030}, {15931, 28471}

X(39634) = CIRCUMCENTER OF CIRCUMCEVIAN POLAR TRIANGLE OF X(238)

Barycentrics    a*(a - b)*(a - c)*(a^5*b - 2*a^4*b^2 + 2*a^3*b^3 - 2*a^2*b^4 + a*b^5 - a^4*b*c + a^3*b^2*c + 2*a^2*b^3*c - 3*a*b^4*c + b^5*c - a^4*c^2 - a^2*b^2*c^2 + 2*a*b^3*c^2 - 2*b^4*c^2 + 2*a^3*c^3 + a*b^2*c^3 + 2*b^3*c^3 - a^2*c^4 - a*b*c^4 - 2*b^2*c^4 + b*c^5)*(-(a^4*b^2) + 2*a^3*b^3 - a^2*b^4 + a^5*c - a^4*b*c - a*b^4*c + b^5*c - 2*a^4*c^2 + a^3*b*c^2 - a^2*b^2*c^2 + a*b^3*c^2 - 2*b^4*c^2 + 2*a^3*c^3 + 2*a^2*b*c^3 + 2*a*b^2*c^3 + 2*b^3*c^3 - 2*a^2*c^4 - 3*a*b*c^4 - 2*b^2*c^4 + a*c^5 + b*c^5) : :

X(39634) lies on these lines: {105, 9317}, {518, 12032}, {812, 1292}, {813, 3309}, {919, 9323}

X(39635) = CIRCUMCENTER OF CIRCUMCEVIAN POLAR TRIANGLE OF X(529)

Barycentrics    a^2*(a - b)*(a - c)*(a^4 - 2*a^2*b^2 + b^4 - a^3*c + 3*a^2*b*c + 3*a*b^2*c - b^3*c - a^2*c^2 - 8*a*b*c^2 - b^2*c^2 + a*c^3 + b*c^3)*(a^4 - a^3*b - a^2*b^2 + a*b^3 + 3*a^2*b*c - 8*a*b^2*c + b^3*c - 2*a^2*c^2 + 3*a*b*c^2 - b^2*c^2 - b*c^3 + c^4) : :

X(39635) lies on these lines: {3,38882}, {104, 9943}, {105, 11227}, {1350, 29206}

X(39636) = CIRCUMCENTER OF CIRCUMCEVIAN POLAR TRIANGLE OF X(532)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(a^4 - 5*a^2*b^2 - 8*b^4 - 5*a^2*c^2 + 40*b^2*c^2 - 8*c^4 - 2*Sqrt[3]*(a^2 - 2*b^2 - 2*c^2)*S) : :

X(39636) lies on these lines: {3, 2380}, {74, 14540}, {98, 5474}, {111, 10645}, {1141, 22843}, {2378, 36755}, {5238, 8460}, {11612, 12121}, {14658, 39555}, {22739, 34374}

X(39637) = CIRCUMCENTER OF CIRCUMCEVIAN POLAR TRIANGLE OF X(533)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(a^4 - 5*a^2*b^2 - 8*b^4 - 5*a^2*c^2 + 40*b^2*c^2 - 8*c^4 + 2*Sqrt[3]*(a^2 - 2*b^2 - 2*c^2)*S) : :

X(39637) lies on these lines: {3, 2381}, {74, 14541}, {98, 5473}, {111, 10646}, {1141, 22890}, {2379, 36756}, {5237, 8450}, {11613, 12121}, {14658, 39554}, {22738, 34376}

X(39638) = CIRCUMCENTER OF CIRCUMCEVIAN POLAR TRIANGLE OF X(535)

Barycentrics    a^2*(a - b)*(a - c)*(a^4 - 2*a^2*b^2 + b^4 - a^3*c + 2*a^2*b*c + 2*a*b^2*c - b^3*c - a^2*c^2 - 5*a*b*c^2 - b^2*c^2 + a*c^3 + b*c^3)*(a^4 - a^3*b - a^2*b^2 + a*b^3 + 2*a^2*b*c - 5*a*b^2*c + b^3*c - 2*a^2*c^2 + 2*a*b*c^2 - b^2*c^2 - b*c^3 + c^4) : :

X(39638) lies on these lines: {105, 5197}, {106, 37469}, {572, 2291}, {753, 991}, {1350, 29045}, {1983, 32722}

X(39639) = CIRCUMCENTER OF CIRCUMCEVIAN POLAR TRIANGLE OF X(538)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(a^2*b^2 + 3*b^4 - 2*a^2*c^2 + b^2*c^2)*(2*a^2*b^2 - a^2*c^2 - b^2*c^2 - 3*c^4) : :

X(39639) lies on these lines: {3, 729}, {55, 7333}, {56, 6022}, {74, 30270}, {98, 1350}, {99, 34203}, {111, 3098}, {112, 5118}, {182, 699}, {511, 5970}, {733, 11654}, {843, 35002}, {2698, 18860}, {4563, 9150}, {4576, 9066}, {5104, 9136}, {9084, 21766}, {10620, 39446}, {14659, 35383}

X(39640) = CIRCUMCENTER OF CIRCUMCEVIAN POLAR TRIANGLE OF X(544)

Barycentrics    a^2*(a - b)*(a - c)*(a^5 - a^3*b^2 - a^2*b^3 + b^5 - a^4*c + 2*a^2*b^2*c - b^4*c - 3*a^3*c^2 - 2*a^2*b*c^2 - 2*a*b^2*c^2 - 3*b^3*c^2 + 5*a^2*c^3 + 4*a*b*c^3 + 5*b^2*c^3 - 2*a*c^4 - 2*b*c^4)*(a^5 - a^4*b - 3*a^3*b^2 + 5*a^2*b^3 - 2*a*b^4 - 2*a^2*b^2*c + 4*a*b^3*c - 2*b^4*c - a^3*c^2 + 2*a^2*b*c^2 - 2*a*b^2*c^2 + 5*b^3*c^2 - a^2*c^3 - 3*b^2*c^3 - b*c^4 + c^5) : :

X(39640) lies on these lines: {3, 38884}, {103, 674}, {516, 675}, {840, 991}, {2291, 13329}, {2426, 32682}

X(39641) = 1ST IMAGINARY FOCUS OF BROCARD INELLIPSE

Barycentrics    Sin[A]*(Sin[A] + i*Cos[A]) : :
Barycentrics    a^2*(i*(b^4 + c^4 - a^2*(b^2 + c^2)) + 2*(b^2 + c^2)*S) : :
X(39641) = (1 + i)*X[39] - X[3102] = (1 - i)*X[39] - X[3103] = X[3102] - i*X[3103]

Contributed by Peter Moses, September 11, 2020.

The real foci of the Brocard inellipse are the Brocard points. They are a bicentric pair, with barycentrics 1/b^2 : 1/c^2 : 1/a^2 and 1/c^2 : 1/b^2 : 1/a^2, denoted by PU(1) in Bicentric Pairs of Points.

In contrast to the real foci, the imaginary foci of the Brocard inellipse are major triangle centers. They both lie on the Brocard axis, X(3)X(6), and on the Kiepert circumhyperbola, and on the following curves: K003, K019, K049, K102, K115, K166, K248, K268, K326, K359, K369, K373, K444, K512, K516, K587, K588, K643, K688, K708, K731, K763, K801, K828, K833, K1096, K1098, K1103, K1113, K1139, Q019, Q039, Q065, Q073, Q088, Q094, Q138.

X(39641) = isogonal conjugate of X(39642)
X(39641) = Brocard-circle-inverse of X(39642)

X(39642) = 2ND IMAGINARY FOCUS OF BROCARD INELLIPSE

Barycentrics    Sin[A]*(Sin[A] - i*Cos[A]) : :
Barycentrics    a^2*(i*(b^4 + c^4 - a^2*(b^2 + c^2)) - 2*(b^2 + c^2)*S) : :
X(39642) = (1 + i)*X[39] - X[3103] = (1 - i)*X[39] - X[3102] = X[3103] - i*X[3102]

See X(39641). Contributed by Peter Moses, September 11, 2020.

In contrast to the real foci, the imaginary foci of the Brocard inellipse are major triangle centers. They both lie on the Brocard axis, X(3)X(6), and on the Kiepert circumhyperbola, and on the following curves: K003, K019, K049, K102, K115, K166, K248, K268, K326, K359, K369, K373, K444, K512, K516, K587, K588, K643, K688, K708, K731, K763, K801, K828, K833, K1096, K1098, K1103, K1113, K1139, Q019, Q039, Q065, Q073, Q088, Q094, Q138.

X(39642) = isogonal conjugate of X(39641)
X(39642) = Brocard-circle-inverse of X(39641)

leftri

Points associated with the 7th Brocard triangle: X(39643)-X(39663)

rightri

This preamble is based on notes from Dan Reznik and by Peter Moses, September 14, 2020.

In the plane of a triangle ABC, let O = X(3), the circumcenter. Let A' be the point, other than O, in which the line AO meets the Brocard circle, and define B' and C' cyclically. The triangle A'B'C', here named the 7th Brocard triangle; the first six Brocard triangles are defined by Bernard Gibert at Brocard triangles and related cubics.

The 7th Brocard triangle is perspective to the following triangles, with perspector O: ABC, Lucas central, 2nd Hyacinth, and the Lucas antipodal and other Lucas triangles. Also, A'B'C' is perspective to the 2nd Brocard triangle, with perspector X(184), and to the symmedial triangle, with perspector X(39643).

The A vertex of the 7th Brocard triangle is given by

A' = a^4 + b^4 + c^4 - 2*b^2*c^2 : b^2*(a^2 - b^2 + c^2) : c^2*(a^2 + b^2 - c^2).

Let KK denote the cubic pK(X(39644),X(39645)). If P is a point on KK, then the cevian triangle of P is perspective to A'B'C'. The cubic KK passes through the points X(i) for i = 3, 6, 230, 5253, 6530, 8770.

The triangle A'B'C' under the Brocard porism (in which the circumcircle and Brocard inellipse are fixed; see X(39)) can be viewed at

Loci of Centers of Ellipse-Mounted Triangles.) (Dan Reznik, September 14, 2020).

See also Brocard Porism: locus of 1st, 2nd, 5th, and 7th Brocard Triangles Vertices are Circles.) (Dan Reznik, September 21, 2020).

Let A'' = circumcircle-inverse of A', and define B'' and C'' cyclically. The triangle A''B''C'', here named the 8th Brocard triangle, is given by

A'' = 2*a^4 : b^2*(-a^2 + b^2 - c^2) : c^2*(-a^2 - b^2 + c^2)
B'' = a^2*(a^2 - b^2 - c^2) : 2*b^4 : c^2*(-a^2 - b^2 + c^2)
C'' = a^2*(a^2 - b^2 - c^2) : b^2*(-a^2 + b^2 - c^2) : 2*c^4

Note that A"B"C" are collinear (on the Lemoine axis), so that A"B"C" is a degenerate triangle.

Let A* = isogonal conjugate of A'', and define B* and C* cyclically. The triangle A*B*C* is here named the 9th Brocard triangle. Barycentrics are given by

A* = -((a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)) : 2*a^2*(a^2 + b^2 - c^2) : 2*a^2*(a^2 - b^2 + c^2)
B* = 2*b^2*(a^2 + b^2 - c^2) : -((a^2 + b^2 - c^2)*(-a^2 + b^2 + c^2)) : 2*b^2*(-a^2 + b^2 + c^2)
C* = 2*c^2*(a^2 - b^2 + c^2) : 2*c^2*(-a^2 + b^2 + c^2) : -((a^2 - b^2 + c^2)*(-a^2 + b^2 + c^2))

The vertices A*, B*, C* lies on the Steiner circumellipse.

The 7th Brocard triangle is the dual-of-orthic triangle of the 1st Brocard triangle, and also the anticomplement of the orthic-of-1st-Brocard triangle. (Randy Hutson, September 30, 2020)

The 9th Brocard triangle is the Steiner-orthic triangle (the Steiner circumellipse counterpart to the circumorthic triangle). (Randy Hutson, September 30, 2020)

X(39643) = PERSPECTOR OF THESE TRIANGLES: 7TH BROCARD AND SYMMEDIAL

Barycentrics    a^2*(a^2 - b^2 - c^2)^2*(a^4 + b^4 - 2*b^2*c^2 + c^4) : :

X(39643) lies on the cubic K1065 and these lines: {3, 248}, {4, 6}, {24, 1971}, {32, 185}, {39, 184}, {76, 287}, {99, 9289}, {112, 6241}, {125, 7746}, {147, 11441}, {155, 3289}, {187, 1204}, {194, 401}, {216, 10984}, {232, 6759}, {378, 1970}, {389, 10311}, {394, 3926}, {574, 13367}, {577, 5562}, {1092, 22401}, {1147, 14961}, {1409, 12089}, {1425, 2242}, {1501, 22135}, {1504, 19032}, {1505, 19033}, {1562, 7748}, {1614, 38867}, {1625, 32139}, {1899, 2450}, {1968, 6000}, {2241, 3270}, {2549, 19467}, {3053, 10605}, {3172, 12174}, {3199, 26883}, {4173, 34980}, {5012, 26216}, {5013, 19357}, {5028, 6467}, {5034, 21637}, {5206, 21663}, {5305, 18914}, {5422, 7797}, {5622, 38520}, {5889, 10313}, {5890, 10312}, {6421, 19356}, {6422, 19355}, {6752, 27374}, {7124, 23137}, {7735, 18909}, {7738, 18925}, {7836, 14972}, {9605, 19347}, {10316, 13754}, {10317, 34783}, {10766, 15073}, {11424, 33843}, {11550, 27371}, {12118, 15075}, {12164, 15905}, {12215, 20806}, {14264, 18877}, {14965, 19139}, {15048, 31804}, {16502, 19354}, {18445, 22120}, {19459, 23642}, {23123, 23150}, {23620, 23640}, {26206, 39141}

X(39643) = isogonal conjugate of the isotomic conjugate of X(6389)
X(39643) = isogonal conjugate of the polar conjugate of X(1899)
X(39643) = polar conjugate of the isotomic conjugate of X(426)
X(39643) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 39201}, {6389, 426}, {9289, 3}, {32734, 647}
X(39643) = X(6751)-cross conjugate of X(1899)
X(39643) = X(19)-isoconjugate of X(34405)
X(39643) = crosspoint of X(i) and X(j) for these (i,j): {6, 394}, {68, 14376}, {1899, 6389}
X(39643) = crosssum of X(i) and X(j) for these (i,j): {2, 393}, {24, 8743}, {216, 467}, {232, 36426}
X(39643) = crossdifference of every pair of points on line {520, 16230}
X(39643) = barycentric product X(i)*X(j) for these {i,j}: {3, 1899}, {4, 426}, {6, 6389}, {63, 2083}, {95, 6751}, {255, 17871}, {394, 3767}, {520, 1632}, {2450, 17974}
X(39643) = barycentric quotient X (i)/X(j) for these {i,j}: {3, 34405}, {426, 69}, {1632, 6528}, {1899, 264}, {2083, 92}, {3767, 2052}, {6389, 76}, {6751, 5}
X(39643) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 22146, 23128}, {6, 1181, 217}, {6, 1498, 2207}, {6, 17849, 393}, {6, 19149, 2211}, {6, 32445, 8743}, {155, 23115, 3289}, {185, 8779, 32}, {1498, 2207, 3331}, {1562, 21659, 7748}, {3269, 14585, 3}, {6146, 15341, 5254}, {6776, 34137, 6}, {8743, 11456, 32445}

X(39644) = X(3)X(114)∩X(22)X(32458)

Barycentrics    a^2*(a^6 + b^6 - a^4*c^2 - a^2*b^2*c^2 - b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6)*(a^6 - a^4*b^2 + a^2*b^4 - b^6 - a^2*b^2*c^2 + b^4*c^2 - b^2*c^4 + c^6) : :

X(39644) lies on these lines: {3, 114}, {22, 32458}, {23, 36849}, {115, 2353}, {157, 6036}, {184, 5028}, {1692, 14600}, {2351, 14713}, {2386, 14908}, {5938, 6321}, {8781, 37183}, {9909, 14645}, {17970, 35374}

X(39644) = isogonal conjugate of the anticomplement of X(1692)
X(39644) = X(75)-isoconjugate of X(37183)
X(39644) = barycentric quotient X(32)/X(37183)

X(39645) = X(2)X(38652)∩X(297)X(2987)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 + b^6 - a^4*c^2 - a^2*b^2*c^2 - b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6)*(a^6 - a^4*b^2 + a^2*b^4 - b^6 - a^2*b^2*c^2 + b^4*c^2 - b^2*c^4 + c^6) : :

X(39645) lies on the conic {{A,B,C,X(2),X(6)}} and these lines: {2, 38652}, {297, 2987}, {460, 1976}, {2395, 2508}, {2970, 13854}, {3228, 37765}

X(39645) = X(868)-cross conjugate of X(2501)
X(39645) = X(63)-isoconjugate of X(37183)
X(39645) = cevapoint of X(230) and X(5254)
X(39645) = barycentric quotient X(25)/X(37183)

X(39646) = PERSPECTOR OF THESE TRIANGLES: 7TH BROCARD AND 9TH BROCARD

Barycentrics    a^8 + a^6*b^2 - a^4*b^4 - a^2*b^6 + a^6*c^2 - 4*a^4*b^2*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 - a^4*c^4 + a^2*b^2*c^4 + 4*b^4*c^4 - a^2*c^6 - 2*b^2*c^6 : :

X(39646) lies on these lines: {3, 76}, {4, 6}, {5, 7803}, {20, 385}, {30, 14614}, {32, 9755}, {39, 13860}, {64, 19222}, {83, 5050}, {114, 7887}, {125, 11331}, {147, 5025}, {154, 419}, {182, 6248}, {184, 458}, {185, 37200}, {194, 5999}, {262, 9605}, {264, 19459}, {297, 1899}, {315, 3564}, {317, 26926}, {376, 8667}, {381, 7827}, {394, 37190}, {420, 26958}, {489, 12257}, {490, 12256}, {511, 7754}, {538, 30270}, {542, 7841}, {575, 10358}, {631, 7789}, {648, 10602}, {698, 1350}, {1003, 13335}, {1351, 7760}, {1352, 6656}, {1513, 3767}, {1656, 7859}, {1657, 9301}, {1853, 5117}, {1993, 14957}, {2456, 39266}, {2794, 7748}, {2854, 19221}, {3053, 11676}, {3091, 3329}, {3146, 7766}, {3186, 9924}, {3398, 35930}, {3406, 39560}, {3524, 8556}, {3534, 11054}, {3788, 14981}, {3830, 12156}, {3934, 37479}, {4048, 5085}, {5013, 7709}, {5023, 8719}, {5041, 22682}, {5093, 7894}, {5188, 7751}, {5201, 12082}, {5305, 9753}, {5476, 6249}, {5921, 32974}, {5984, 6655}, {6036, 33233}, {6054, 11318}, {6194, 17129}, {6222, 33371}, {6322, 12505}, {6399, 33370}, {6467, 9308}, {6620, 11206}, {6759, 9418}, {6770, 11303}, {6773, 11304}, {6795, 36165}, {7400, 26870}, {7467, 20023}, {7503, 26214}, {7697, 12054}, {7710, 39095}, {7746, 11623}, {7756, 10991}, {7757, 10983}, {7768, 11898}, {7773, 15980}, {7780, 8722}, {7781, 18860}, {7784, 15069}, {7790, 18440}, {7795, 37450}, {7797, 13862}, {7815, 21163}, {7824, 32522}, {7828, 37071}, {7833, 11177}, {7864, 37336}, {7879, 34507}, {8266, 10323}, {8370, 11179}, {9737, 31859}, {9752, 15428}, {10349, 12177}, {10356, 18553}, {10605, 35474}, {11180, 33190}, {11185, 35429}, {11285, 13334}, {11648, 36997}, {12110, 30435}, {12122, 33706}, {12243, 12252}, {13732, 19768}, {13881, 14651}, {14063, 39101}, {14510, 38527}, {14568, 34624}, {14615, 19588}, {17928, 26164}, {18322, 34783}, {19125, 36794}, {19357, 37124}, {22265, 38520}, {24256, 35423}, {29012, 35431}, {31276, 37455}, {32006, 39099}, {32152, 33234}, {32832, 37451}, {33019, 39097}, {35824, 35830}, {35825, 35831}, {36670, 37686}

X(39647) = PERSPECTOR OF THESE TRIANGLES: 8TH BROCARD AND 1ST NEUBERG

Barycentrics    (a^2 - b^2 - c^2)*(7*a^6 - 5*a^4*b^2 + 5*a^2*b^4 + b^6 - 5*a^4*c^2 - 10*a^2*b^2*c^2 - b^4*c^2 + 5*a^2*c^4 - b^2*c^4 + c^6) : :

X(39647) lies on these lines: {2, 13335}, {3, 69}, {4, 230}, {20, 98}, {32, 14853}, {114, 32989}, {147, 32964}, {182, 32990}, {193, 9737}, {315, 8781}, {376, 8667}, {439, 5921}, {485, 13921}, {486, 13880}, {542, 35287}, {631, 7778}, {1352, 32973}, {1503, 5023}, {1992, 10983}, {2351, 37183}, {2896, 3523}, {3522, 6194}, {5013, 14912}, {5188, 31981}, {5206, 8721}, {5286, 34870}, {5480, 22331}, {5596, 8553}, {5984, 33014}, {6036, 32972}, {6228, 6281}, {6229, 6278}, {6248, 32981}, {6289, 11291}, {6290, 11292}, {6308, 8722}, {6312, 12123}, {6316, 12124}, {6462, 9738}, {6463, 9739}, {7505, 13200}, {7684, 33421}, {7685, 33420}, {7694, 7749}, {7738, 9755}, {7751, 38747}, {7938, 10303}, {8550, 15815}, {8669, 12512}, {8982, 12222}, {9540, 32497}, {9863, 16925}, {9888, 14645}, {9991, 33341}, {9992, 33340}, {10104, 32828}, {11155, 11179}, {11177, 33208}, {11180, 32985}, {12007, 22332}, {12221, 26441}, {12251, 38642}, {13334, 33684}, {13449, 32980}, {13708, 32419}, {13828, 32421}, {13935, 32494}, {14023, 18860}, {15270, 31381}, {16312, 37934}, {18935, 36748}, {19119, 36751}, {23291, 37188}, {32152, 32974}, {32838, 37348}

X(39648) = PERSPECTOR OF THESE TRIANGLES: 8TH BROCARD AND LUCAS TANGENTS

Barycentrics    a^2*(a^4 - b^4 - c^4 - 2*b^2*c^2 - 4*a^2*S) : :
Trilinears    sin A + (1 - tan ω) cos A : :
Trilinears    cos A cos ω + sin (A + ω) : :

X(39648) lies on these lines: {3, 6}, {4, 13638}, {237, 10132}, {325, 39387}, {382, 35831}, {485, 6222}, {487, 14001}, {489, 1588}, {491, 3785}, {590, 6290}, {1285, 36701}, {1975, 35938}, {3068, 21736}, {3071, 37342}, {5152, 8304}, {5191, 7598}, {5337, 16433}, {5409, 37344}, {5418, 7800}, {6033, 13873}, {6561, 36714}, {6564, 36711}, {7388, 7750}, {7389, 7792}, {7735, 21737}, {7738, 35944}, {7778, 11315}, {7784, 11314}, {7889, 10577}, {8576, 10601}, {8577, 33586}, {8960, 8980}, {10133, 34396}, {11293, 16989}, {13644, 36656}, {18510, 35833}, {31454, 32497}, {32556, 37552}, {35821, 36712}

X(39648) = radical trace of Lucas circles radical circle and circle {{X(1687),X(1688),PU(1),PU(2)}}
X(39648) = {X(3),X(32)}-harmonic conjugate of X(39679)

X(39649) = PERSPECTOR OF THESE TRIANGLES: 8TH BROCARD AND LUCAS INNER

Barycentrics    a^2*(2*(a^2 - b^2 - c^2)*(a^2 + b^2 + c^2) - (a^2 + 3 b^2 + 3 c^2)*S) : :

X(39649) lies on these lines: {3, 6}, {20, 13644}, {237, 32564}, {590, 36712}, {3933, 9543}, {5152, 8308}, {5191, 7601}, {9540, 36714}, {9541, 36709}, {14227, 21736}, {15048, 36703}, {35255, 37342}, {36717, 37665}

X(39650) = PERSPECTOR OF THESE TRIANGLES: 8TH BROCARD AND 1ST SHARYGIN

Barycentrics    a*(a^6 - a^4*b^2 + 2*a^3*b^3 + 2*a^4*b*c - a^4*c^2 + a^2*b^2*c^2 + b^4*c^2 + 2*a^3*c^3 + b^2*c^4) : :

X(39650) lies on these lines: {3, 256}, {32, 1580}, {172, 8844}, {1384, 8296}, {3053, 8424}, {3550, 8931}, {12725, 37552}, {16362, 18265}

X(39651) = PERSPECTOR OF THESE TRIANGLES: 8TH BROCARD AND 2ND SHARYGIN

Barycentrics    a*(a^6 - a^4*b^2 + 2*a^3*b^3 - 4*a^4*b*c + 2*a^2*b^3*c - a^4*c^2 - a^2*b^2*c^2 - b^4*c^2 + 2*a^3*c^3 + 2*a^2*b*c^3 - b^2*c^4) : :

X(39651) lies on these lines: {1, 38865}, {3, 291}, {32, 8300}, {172, 8299}, {244, 21511}, {1384, 8297}, {1580, 2223}, {3053, 8301}, {3751, 3939}, {18266, 21495}

X(39652) = PERSPECTOR OF THESE TRIANGLES: 8TH BROCARD AND ANTI-1ST-BROCARD

Barycentrics    -a^8 + 2*a^6*b^2 - 2*a^4*b^4 + 2*a^6*c^2 - 2*a^4*c^4 + b^4*c^4 : :

X(39652) lies on these lines: {2, 12191}, {3, 1916}, {20, 98}, {32, 99}, {39, 39098}, {83, 620}, {114, 7785}, {115, 1078}, {147, 20065}, {182, 10754}, {187, 5152}, {192, 10086}, {193, 12177}, {237, 16985}, {330, 10089}, {384, 5976}, {385, 2080}, {538, 10631}, {543, 19570}, {571, 8264}, {577, 2998}, {618, 12204}, {619, 12205}, {671, 33264}, {1003, 3407}, {1384, 8289}, {1691, 5969}, {2021, 39089}, {2023, 7824}, {2482, 12150}, {2794, 14712}, {3053, 5989}, {3398, 7783}, {3972, 5149}, {4558, 25054}, {5026, 12212}, {5032, 5039}, {5106, 17941}, {5206, 8178}, {6033, 7823}, {6308, 11606}, {6321, 10104}, {6392, 20094}, {7470, 12042}, {7754, 13188}, {7774, 10788}, {7777, 10796}, {7787, 10352}, {7789, 10333}, {7802, 39603}, {7815, 7933}, {7820, 10347}, {7836, 10350}, {7837, 8724}, {7891, 10349}, {8290, 12206}, {8591, 35927}, {8722, 34473}, {8789, 9431}, {9301, 35464}, {10345, 14001}, {10799, 15452}, {11152, 14614}, {11711, 12194}, {11842, 31859}, {12188, 36864}, {12203, 38738}, {14651, 17008}, {14880, 38730}, {17004, 38224}, {17006, 34127}, {17970, 38527}, {18993, 19109}, {18994, 19108}, {25332, 38880}, {33214, 35369}, {33235, 38905}

X(39653) = PERSPECTOR OF THESE TRIANGLES: 8TH BROCARD AND ARA

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 - 14*a^2*b^2*c^2 - b^4*c^2 + 3*a^2*c^4 - b^2*c^4 + c^6) : :

X(39653) lies on these lines: {3, 69}, {22, 18287}, {25, 1611}, {159, 5023}, {5171, 11414}, {6339, 37491}, {6461, 8681}, {8276, 8996}, {8573, 11326}, {9737, 10608}, {15815, 32621}, {27802, 37552}

X(39654) = PERSPECTOR OF THESE TRIANGLES: 8TH BROCARD AND LUCAS-BROCARD

Barycentrics    a^2*(a^2*(a^2 + b^2 + c^2) - 2*(2*a^2 - b^2 - c^2)*S) : :

X(39654) lies on these lines: {3, 6}, {1506, 11316}, {2548, 39388}, {2549, 35946}, {6813, 7737}, {7747, 36655}, {7749, 11314}, {8576, 37457}, {13637, 31411}

X(39654) = {X(32),X(187)}-harmonic conjugate of X(39655)

X(39655) = PERSPECTOR OF THESE TRIANGLES: 8TH BROCARD AND LUCAS(-1)-BROCARD

Barycentrics    a^2*(a^2*(a^2 + b^2 + c^2) + 2*(2*a^2 - b^2 - c^2)*S) : :

X(39655) lies on these lines: {3, 6}, {490, 31411}, {1506, 11315}, {2548, 39387}, {2549, 35947}, {6811, 7737}, {7747, 36656}, {7749, 11313}, {8577, 37457}, {13757, 35306}, {13882, 31481}

X(39655) = {X(32),X(187)}-harmonic conjugate of X(39654)

X(39656) = PERSPECTOR OF THESE TRIANGLES: 8TH BROCARD AND ARTZT

Barycentrics    -3*a^8 + 9*a^6*b^2 - 5*a^4*b^4 - a^2*b^6 + 9*a^6*c^2 + 4*a^4*b^2*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 - 5*a^4*c^4 + a^2*b^2*c^4 + 4*b^4*c^4 - a^2*c^6 - 2*b^2*c^6 : :

X(39656) lies on these lines: {3, 83}, {4, 230}, {5, 9754}, {6, 7709}, {20, 7864}, {25, 35278}, {30, 9753}, {32, 9755}, {51, 35941}, {98, 1384}, {99, 1351}, {112, 33971}, {183, 2080}, {187, 13860}, {316, 37071}, {376, 597}, {381, 8860}, {384, 6194}, {394, 35919}, {511, 1003}, {576, 31859}, {631, 15491}, {1285, 6776}, {1350, 35925}, {1499, 2422}, {1513, 7694}, {1656, 7831}, {1975, 32515}, {2076, 22678}, {2782, 14614}, {3545, 15597}, {3552, 39101}, {3830, 14639}, {5017, 38654}, {5023, 37334}, {5050, 12150}, {5093, 7757}, {5102, 8716}, {5171, 7770}, {5476, 35955}, {6530, 18533}, {7606, 38072}, {7773, 37466}, {7782, 10983}, {7804, 8722}, {7851, 20576}, {8356, 14561}, {8598, 20423}, {9301, 10000}, {9732, 33435}, {9733, 33434}, {9737, 33235}, {9744, 18907}, {9862, 36990}, {10104, 22681}, {10256, 16925}, {10295, 16324}, {10358, 11285}, {10519, 14039}, {10753, 11173}, {11002, 35933}, {11163, 37461}, {11257, 30435}, {11286, 22712}, {11288, 34733}, {11477, 35951}, {13188, 32469}, {13862, 14712}, {22253, 23235}, {22693, 39554}, {22694, 39555}, {32833, 34380}, {33586, 35926}

X(39657) = PERSPECTOR OF THESE TRIANGLES: 8TH BROCARD AND LUCAS INNER TANGENTIAL

Barycentrics    a^2*(2*(a^2 - b^2 - c^2)*(a^2 + b^2 + c^2) - (5*a^2 + b^2 + c^2)*S) : :

See also X(39664).

X(39657) lies on these lines: {3,6}, {3146,8974}, {5306,36703}, {8308,39652}, {8324,39650}, {8340,39651}, {10843,39656}, {13846,36709}, {19388,39659}, {21736,31454}

X(39657) = {X(3),X(9605)}-harmonic conjugate of X(39658)

X(39658) = PERSPECTOR OF THESE TRIANGLES: 8TH BROCARD AND LUCAS(-1)-INNER

Barycentrics    a^2*(2*(a^2 - b^2 - c^2)*(a^2 + b^2 + c^2) + (a^2 + 3*b^2 + 3*c^2)*S) : :

X(39658) lies on these lines: {3, 6}, {20, 13763}, {237, 32571}, {615, 36711}, {5152, 8309}, {5191, 7602}, {13935, 36709}, {15048, 36701}, {15484, 21737}, {35256, 37343}, {36702, 37665}

X(39658) = {X(3),X(9605)}-harmonic conjugate of X(39657)

X(39659) = PERSPECTOR OF THESE TRIANGLES: 8TH BROCARD AND ANTI-AOA

Barycentrics    a^2*(a^14 - 3*a^12*b^2 + a^10*b^4 + 5*a^8*b^6 - 5*a^6*b^8 - a^4*b^10 + 3*a^2*b^12 - b^14 - 3*a^12*c^2 + 10*a^10*b^2*c^2 - 7*a^8*b^4*c^2 - 5*a^6*b^6*c^2 + 10*a^4*b^8*c^2 - 5*a^2*b^10*c^2 + a^10*c^4 - 7*a^8*b^2*c^4 + 11*a^6*b^4*c^4 - 6*a^4*b^6*c^4 - 3*a^2*b^8*c^4 + 2*b^10*c^4 + 5*a^8*c^6 - 5*a^6*b^2*c^6 - 6*a^4*b^4*c^6 + 10*a^2*b^6*c^6 - b^8*c^6 - 5*a^6*c^8 + 10*a^4*b^2*c^8 - 3*a^2*b^4*c^8 - b^6*c^8 - a^4*c^10 - 5*a^2*b^2*c^10 + 2*b^4*c^10 + 3*a^2*c^12 - c^14) : :

For the anti-AOA triangle, see X(15015).

X(39659) lies on these lines: {3, 67}, {32, 15140}, {1384, 19379}, {3053, 15141}, {5013, 32251}, {5210, 15106}

X(39660) = PERSPECTOR OF THESE TRIANGLES: 9TH BROCARD AND INNER VECTEN

Barycentrics    (a^2+b^2+c^2)*(-a^2+b^2+c^2)*a^2+2*((b^2+c^2)*a^2+(b^2-c^2)^2)*S : :

X(39660) lies on these lines: {3, 230}, {4, 371}, {76, 486}, {99, 13873}, {193, 638}, {372, 5286}, {460, 10132}, {487, 32974}, {488, 6392}, {590, 15883}, {1352, 3071}, {1585, 18289}, {2041, 36251}, {2042, 36252}, {2459, 12124}, {3070, 19145}, {3127, 8280}, {3311, 7745}, {5200, 8854}, {5418, 7389}, {5860, 35793}, {6200, 12123}, {6248, 37343}, {6396, 26294}, {6420, 26456}, {6560, 12203}, {6565, 10515}, {6620, 8970}, {6806, 8855}, {6811, 11257}, {7375, 10577}, {7709, 9758}, {7795, 11314}, {7828, 39388}, {8370, 13789}, {8396, 18509}, {8721, 36656}, {11158, 33274}, {14568, 35949}, {15682, 35873}, {18539, 23261}, {23273, 35833}, {26300, 35775}, {26306, 35777}, {26314, 35783}, {26324, 35785}, {26339, 35795}, {26355, 35809}, {26369, 35763}, {26375, 35765}, {26429, 35767}, {26435, 35769}, {26444, 35789}, {26449, 35791}, {26462, 35771}, {26473, 35803}, {26479, 35801}, {26485, 35799}, {26490, 35797}, {26512, 35773}, {26514, 35811}, {26519, 35819}, {26520, 35817}, {32471, 33372}, {33346, 33432}, {33348, 33452}, {35813, 38426}

X(39660) = {X(3),X(5254)}-harmonic conjugate of X(39661)

X(39661) = PERSPECTOR OF THESE TRIANGLES: 9TH BROCARD AND OUTER VECTEN

Barycentrics    (a^2+b^2+c^2)*(-a^2+b^2+c^2)*a^2-2*((b^2+c^2)*a^2+(b^2-c^2)^2)*S : :

X(39661) lies on these lines: {3, 230}, {4, 372}, {76, 485}, {99, 13926}, {193, 637}, {371, 5286}, {460, 10133}, {487, 6392}, {488, 32974}, {615, 15884}, {1352, 3070}, {1586, 18290}, {2041, 36252}, {2042, 36251}, {2460, 12123}, {3071, 19146}, {3128, 8281}, {3312, 7745}, {5420, 7388}, {5861, 35792}, {6200, 26295}, {6248, 37342}, {6396, 12124}, {6419, 26463}, {6561, 12203}, {6564, 10514}, {6805, 8854}, {6813, 11257}, {7376, 10576}, {7709, 9757}, {7795, 11313}, {7828, 39387}, {8370, 13669}, {8416, 18511}, {8721, 36655}, {11157, 33274}, {14568, 35948}, {15682, 35874}, {23251, 26438}, {23267, 35832}, {26301, 35774}, {26307, 35776}, {26315, 35782}, {26325, 35784}, {26340, 35794}, {26356, 35808}, {26370, 35762}, {26376, 35764}, {26430, 35766}, {26436, 35768}, {26445, 35788}, {26450, 35790}, {26457, 35770}, {26474, 35802}, {26480, 35800}, {26486, 35798}, {26491, 35796}, {26513, 35772}, {26515, 35810}, {26524, 35818}, {26525, 35816}, {32470, 33373}, {33347, 33433}, {33349, 33453}, {35812, 38425}

X(39661) = {X(3),X(5254)}-harmonic conjugate of X(39660)

X(39662) = PERSPECTOR OF THESE TRIANGLES: 9TH BROCARD AND 5TH EULER

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 - 7*a^4*b^2 + 7*a^2*b^4 - b^6 - 7*a^4*c^2 + 26*a^2*b^2*c^2 + b^4*c^2 + 7*a^2*c^4 + b^2*c^4 - c^6) : :

X(39662) lies on these lines: {2, 33843}, {4, 3815}, {112, 33872}, {376, 6748}, {378, 1609}, {393, 5421}, {468, 15433}, {475, 17916}, {574, 6995}, {1285, 6749}, {1595, 7738}, {1597, 7736}, {1990, 14482}, {2549, 7378}, {3088, 5286}, {3269, 14853}, {3618, 35920}, {14581, 14930}, {14806, 18533}, {18907, 35501}, {35940, 36794}

X(39663) = PERSPECTOR OF THESE TRIANGLES: 9TH BROCARD AND ARTZT

Barycentrics    3*a^6*b^2 - 5*a^4*b^4 + 5*a^2*b^6 - 3*b^8 + 3*a^6*c^2 + 4*a^4*b^2*c^2 - 5*a^2*b^4*c^2 + 10*b^6*c^2 - 5*a^4*c^4 - 5*a^2*b^2*c^4 - 14*b^4*c^4 + 5*a^2*c^6 + 10*b^2*c^6 - 3*c^8 : :

X(39663) lies on these lines: {2, 10256}, {3, 9754}, {4, 230}, {5, 76}, {30, 9166}, {99, 10011}, {115, 1513}, {140, 7919}, {235, 9747}, {381, 9753}, {385, 3091}, {403, 523}, {460, 35278}, {511, 23514}, {524, 3545}, {538, 36519}, {546, 12110}, {1503, 14651}, {2080, 10242}, {3564, 14568}, {3767, 7694}, {3839, 8859}, {3850, 6287}, {3858, 18500}, {5025, 6194}, {5056, 7925}, {5068, 7779}, {5071, 22110}, {5171, 33229}, {5254, 7709}, {5480, 15993}, {6321, 37459}, {6392, 9742}, {6656, 15819}, {6722, 18860}, {7710, 39095}, {7745, 22521}, {7790, 37451}, {7792, 37348}, {7809, 34380}, {7844, 37450}, {7884, 38110}, {8598, 9880}, {9478, 15980}, {9737, 33249}, {10026, 36677}, {10519, 33285}, {11185, 37071}, {13172, 32459}, {14561, 22525}, {14693, 22515}, {14981, 32457}, {16315, 37984}, {22682, 39565}, {22712, 33184}, {23698, 35297}, {29012, 38735}, {32819, 37466}, {32962, 39101}, {33024, 39097}, {37242, 37688}

X(39663) = anticomplement of X(10256)

X(39664) = PERSPECTOR OF THESE TRIANGLES: 8TH BROCARD AND LUCAS(-1) INNER TANGENTIAL

Barycentrics    a^2*(2*(a^2 - b^2 - c^2)*(a^2 + b^2 + c^2) + (5*a^2 + b^2 + c^2)*S) : :

See also X(39657).

X(39664) lies on these lines: {3,6}, {3146,13950}, {5306,36701}, {8309,39652}, {8325,39650}, {8341,39651}, {10844,39656}, {13847,36714}, {19389,39659}}

X(39665) = ISOGONAL CONJUGATE OF X(2480)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(b^2 - c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4 - Sqrt[a^8 - a^6*b^2 - a^2*b^6 + b^8 - a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 - b^6*c^2 + a^2*b^2*c^4 - a^2*c^6 - b^2*c^6 + c^8])

X(39665) lies on the Jerabek circumhyperbola and these lines: {4, 24008}, {187, 237}, {290, 2479}

X(39665) = reflection of X(39666) in X(3569)
X(39665) = isogonal conjugate of X(2480)
X(39665) = isogonal conjugate of the anticomplement of X(2454)
X(39665) = isogonal conjugate of the polar conjugate of X(24008)
X(39665) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2480}, {662, 24007}
X(39665) = crosssum of X(525) and X(2455)
X(39665) = crossdifference of every pair of points on line {2, 24007}
X(39665) = barycentric product X(i)*X(j) for these {i,j}: {3, 24008}, {647, 2479}
X(39665) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 2480}, {512, 24007}, {2479, 6331}, {24008, 264}

X(39666) = ISOGONAL CONJUGATE OF X(2479)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(b^2 - c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4 + Sqrt[a^8 - a^6*b^2 - a^2*b^6 + b^8 - a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 - b^6*c^2 + a^2*b^2*c^4 - a^2*c^6 - b^2*c^6 + c^8]) : :

X(39666) lies on the Jerabek circumhyperbola and these lines: {4, 24007}, {187, 237}, {290, 2480}

X(39666) = reflection of X(39665) in X(3569)
X(39666) = isogonal conjugate of X(2479)
X(39666) = isogonal conjugate of the anticomplement of X(2455)
X(39666) = isogonal conjugate of the polar conjugate of X(24007)
X(39666) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2479}, {662, 24008}
X(39666) = crosssum of X(525) and X(2454)
X(39666) = crossdifference of every pair of points on line {2, 24008}
X(39666) = barycentric product X(i)*X(j) for these {i,j}: {3, 24007}, {647, 2480}
X(39666) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 2479}, {512, 24008}, {2480, 6331}, {24007, 264}

leftri

Cevian Centroid Isogonal Perspectors: X(39667)-X(39678)

rightri

This preamble is contributed by Vu Thanh Tung, September 18, 2020.

Let A1B1C1 be the cevian triangle of a point P = p : q : r in the plane of a triangle ABC. Let Ga be the centroid of PB1C1. Let A' be the isogonal conjugate of Ga with respect to AB1C1, and define B' and C' cyclically. The triangle A'B'C' is perspective to ABC, and the perspector is given by

V(P) = a^2 q r (p + q)(p + r)(p + 2q + 2r) : :

The appearance of (i,j) in the following list means that V(X(i)) = X(j): (1,4658), (2,6), (3,49094), (4,631), (5,39667), (6,39668), (7,10389), (8,3361), (9,39669), (10,39670), (31,39671), (32,39672), (75,39673), (76,39674), (83,39675), (141,39676), (560,39677), (561,39678)

See Cevian Centroid Isogonal Perspector

X(39667) = CEVIAN CENTROID ISOGONAL PERSPECTOR OF X(5)

Barycentrics    a^2*(4*a^4 + 3*(b^2 - c^2)^2 - 7*a^2*(b^2 + c^2))*(a^4 + b^4 - b^2*c^2 - a^2*(2*b^2 + c^2))*(a^4 - b^2*c^2 + c^4 - a^2*(b^2 + 2*c^2))*(a^4 + 2*b^4 - 3*b^2*c^2 + c^4 - a^2*(3*b^2 + 2*c^2))*(a^4 + b^4 - 3*b^2*c^2 + 2*c^4 - a^2*(2*b^2 + 3*c^2)) : : X(39667) lies on these lines: {51,54}, {19210,31626}

X(39667) = barycentric product X(288)*X(632)
X(39667) = {X(54), X(288)}-harmonic conjugate of X(1173)

X(39668) = CEVIAN CENTROID ISOGONAL PERSPECTOR OF X(6)

Barycentrics    (a^2 + b^2)*(a^2 + c^2)*(a^2 + 2*(b^2 + c^2)) : :

X(39668) lies on these lines: {2,32}, {5,8793}, {25,14535}, {82,17385}, {111,37876}, {308,3108}, {373,10551}, {384,38862}, {574,16953}, {733,8842}, {1176,3589}, {1180,7770}, {1194,9870}, {1370,17500}, {3934,34482}, {4993,39287}, {6636,7804}, {6683,15246}, {7378,32085}, {7539,10547}, {7786,16949}, {7834,37353}, {9148,17997}, {10191,10328}, {11174,18092}, {11205,24273}, {11324,15302}, {16277,18841}, {18098,31993}, {26209,26216}, {31626,39289}

X(39668) = barycentric product X(i)*X(j) for these {i, j}: {83, 3763}, {308, 7772}, {689, 8665}, {1799, 5064}
X(39668) = barycentric quotient X(827)/X(7954)
X(39668) = trilinear product X(82)*X(3763)
X(39668) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(3763)}} and {{A, B, C, X(32), X(3108)}}
X(39668) = X(827)-reciprocal conjugate of-X(7954)
X(39668) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 83, 251), (2, 251, 10130), (2, 1369, 6292), (3329, 8024, 3108), (3934, 37875, 34482), (6704, 21248, 2)

X(39669) = CEVIAN CENTROID ISOGONAL PERSPECTOR OF X(9)

Barycentrics    a*(a + b - c)*(a - b + c)*(a^2 + 2*(b - c)^2 - 3*a*(b + c))*(a^2 + b*(b - c) - a*(2*b + c))*(a^2 + c*(-b + c) - a*(b + 2*c)) : :

X(39669) lies on these lines: {41,57}, {354,10482}, {3911,21453}, {5173,19624}, {5437,6605}

X(39669) = barycentric product X(1170)*X(20195)
X(39669) = {X(57), X(1170)}-harmonic conjugate of X(1174)

X(39670) = CEVIAN CENTROID ISOGONAL PERSPECTOR OF X(10)

Barycentrics    a^2*(a + b)*(a + c)*(a + 2*b + c)*(a + b + 2*c)*(4*a + 3*(b + c)) : :

X(39670) lies on this line: {35,42}

X(39670) = barycentric product X(1171)*X(19862)
X(39670) = {X(58), X(1171)}-harmonic conjugate of X(1126)

X(39671) = CEVIAN CENTROID ISOGONAL PERSPECTOR OF X(31)

Barycentrics    b*(a + b)*(a^2 - a*b + b^2)*c*(a + c)*(a^2 - a*c + c^2)*(a^3 + 2*(b^3 + c^3)) : :

X(39671) lies on this line: {75,560}

X(39671) = {X(75), X(38810)}-harmonic conjugate of X(38813)

X(39672) = CEVIAN CENTROID ISOGONAL PERSPECTOR OF X(32)

Barycentrics    b^2*(a^4 + b^4)*c^2*(a^4 + c^4)*(a^4 + 2*(b^4 + c^4)) : :

X(39672) lies on the line {76,1501}

X(39672) = {X(76), X(38830)}-harmonic conjugate of X(38826)

X(39673) = CEVIAN CENTROID ISOGONAL PERSPECTOR OF X(75)

Barycentrics    a^2*(a + b)*(a + c)*(b*c + 2*a*(b + c)) : :

X(39673) lies on these lines: {1,21}, {6,4184}, {86,17127}, {171,5235}, {172,38853}, {238,5333}, {333,17126}, {580,37402}, {593,2194}, {741,8652}, {1010,5278}, {1203,4278}, {1333,2205}, {1724,14005}, {1778,4275}, {2299,14014}, {2308,3736}, {4189,5331}, {4221,5398}, {4225,4252}, {4383,35983}, {4640,25060}, {5012,5035}, {5021,16876}, {5156,13588}, {6679,30984}, {8025,30653}, {11115,37652}, {14008,37646}, {14956,37642}, {16704,30652}, {17187,21747}, {17557,37522}, {19767,37296}, {34281,37442}

X(39673) = barycentric product X(i)*X(j) for these {i, j}: {58, 4687}, {81, 17018}, {333, 16878}, {662, 6005}, {799, 8655}
X(39673) = barycentric quotient X(i)/X(j) for these (i, j): (163, 6013), (1333, 10013)
X(39673) = trilinear product X(i)*X(j) for these {i, j}: {21, 16878}, {58, 17018}, {99, 8655}, {110, 6005}, {1333, 4687}
X(39673) = trilinear quotient X(i)/X(j) for these (i, j): (58, 10013), (110, 6013)
X(39673) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(16878)}} and {{A, B, C, X(105), X(5248)}}
X(39673) = X(i)-isoconjugate-of-X(j) for these {i,j}: {10, 10013}, {523, 6013}
X(39673) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (163, 6013), (1333, 10013)
X(39673) = {X(31), X(58)}-harmonic conjugate of X(81)

X(39674) = CEVIAN CENTROID ISOGONAL PERSPECTOR OF X(76)

Barycentrics    a^2*(a^2 + b^2)*(a^2 + c^2)*(b^2*c^2 + 2*a^2*(b^2 + c^2)) : :

X(39674) lies on these lines: {2,32}, {308,6179}, {733,7954}, {1176,5039}, {1915,38854}, {10312,32085}

X(39674) = barycentric product X(251)*X(7786)
X(39674) = barycentric quotient X(251)/X(34816)
X(39674) = trilinear quotient X(82)/X(34816)
X(39674) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(7786)}} and {{A, B, C, X(6), X(7808)}}
X(39674) = X(38)-isoconjugate-of-X(34816)
X(39674) = X(251)-reciprocal conjugate of-X(34816)
X(39674) = {X(32), X(251)}-harmonic conjugate of X(83)

X(39675) = CEVIAN CENTROID ISOGONAL PERSPECTOR OF X(83)

Barycentrics    a^2*(a^2 + 2*b^2 + c^2)*(a^2 + b^2 + 2*c^2)*(a^4 + 2*b^4 + 5*b^2*c^2 + 2*c^4 + 5*a^2*(b^2 + c^2)) : :

X(39675) lies on these lines: {39,251}, {3933,10159}

X(39676) = CEVIAN CENTROID ISOGONAL PERSPECTOR OF X(141)

Barycentrics    a^2*(a^2 + b^2)*(a^2 + c^2)*(a^2 + 2*b^2 + c^2)*(a^2 + b^2 + 2*c^2)*(4*a^2 + 3*(b^2 + c^2)) : :

X(39676) lies on this line: {39,251}

X(39677) = CEVIAN CENTROID ISOGONAL PERSPECTOR OF X(560)

Barycentrics    b^3*c^3*(a + b)*(a^4 - a^3*b + a^2*b^2 - a*b^3 + b^4)*(a + c)*(a^4 - a^3*c + a^2*c^2 - a*c^3 + c^4)*(a^5 + 2*(b^5 + c^5)) : :

X(39677) lies on this line: {561,1917}

X(39677) = {X(561), X(38812)}-harmonic conjugate of X(38827)

X(39678) = CEVIAN CENTROID ISOGONAL PERSPECTOR OF X(561)

Barycentrics    a^2*(a + b)*(a^2 - a*b + b^2)*(a + c)*(a^2 - a*c + c^2)*(b^3*c^3 + 2*a^3*(b^3 + c^3)) : :

X(39678) lies on this line: {75,560}

X(39678) = {X(560), X(38813)}-harmonic conjugate of X(38810)

X(39679) = PERSPECTOR OF THESE TRIANGLES: 8TH BROCARD AND LUCAS(-1) TANGENTS

Barycentrics    a^2 (a^4 - b^4 - c^4 - 2 b^2 c^2 + 4 a^2 S) : :
Trilinears    sin A - (1 + tan ω) cos A : :
Trilinears    cos A cos ω - sin (A + ω) : :

X(39679) lies on these lines: {3, 6}, {4, 13758}, {237, 10133}, {325, 39388}, {382, 35830}, {486, 6399}, {488, 14001}, {490, 1587}, {492, 3785}, {615, 6289}, {1285, 36703}, {1975, 35939}, {3070, 37343}, {5152, 8305}, {5191, 7599}, {5337, 16432}, {5408, 37344}, {5420, 7800}, {6033, 13926}, {6460, 21736}, {6560, 36709}, {6565, 36712}, {7388, 7792}, {7389, 7750}, {7738, 35945}, {7778, 11316}, {7784, 11313}, {7889, 10576}, {8576, 33586}, {8577, 10601}, {10132, 34396}, {11294, 16989}, {13763, 36655}, {18512, 35832}, {32555, 37552}, {35820, 36711}

X(39679) = radical trace of Lucas(-1) circles radical circle and circle {{X(1687),X(1688),PU(1),PU(2)}}
X(39679) = {X(3),X(32)}-harmonic conjugate of X(39648)

X(39680) = REFLECTION OF X(881) IN X(882)

Barycentrics    a^2*(b^2 - c^2)*(-b^2 + a*c)*(b^2 + a*c)*(a*b - c^2)*(a*b + c^2)*(a^4 - a^2*b^2 - a^2*c^2 - 2*b^2*c^2) : :

X(39680) lies on the cubic K1158 and these lines: {39, 512}, {694, 804}, {805, 11634}, {2896, 15412}, {3288, 14096}, {14960, 39291}, {14994, 23878}, {30229, 36881}

X(39680) = reflection of X(881) in X(882)
X(39680) = X(i)-isoconjugate of X(j) for these (i,j): {880, 3402}, {1966, 26714}, {2186, 17941}, {5976, 36132}
X(39680) = crossdifference of every pair of points on line {385, 36213}
X(39680) = barycentric product X(i)*X(j) for these {i,j}: {183, 882}, {512, 8842}, {694, 23878}, {881, 20023}, {1916, 3288}, {6784, 18829}
X(39680) = barycentric quotient X(i)/X(j) for these {i,j}: {182, 17941}, {183, 880}, {881, 263}, {882, 262}, {3288, 385}, {6784, 804}, {8842, 670}, {9420, 36213}, {9468, 26714}, {23878, 3978}, {34238, 6037}

X(39681) = X(4)X(2896)∩X(263)X(3094)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^2 - b*c)*(a^2 + b*c)*(a^2*b^2 - b^4 + 2*a^2*c^2 + b^2*c^2)*(2*a^2*b^2 + a^2*c^2 + b^2*c^2 - c^4) : :

X(39681) lies on the cubic K1159 and these lines: {3, 83}, {99, 1625}, {327, 3734}, {385, 14382}, {805, 4226}, {2966, 14966}, {5026, 16069}, {7782, 14252}, {26613, 38889}

X(39681) = isogonal conjugate of X(39680)
X(39681) = X(i)-isoconjugate of X(j) for these (i,j): {1, 39680}, {798, 8842}, {881, 3403}, {1581, 3288}, {1967, 23878}, {6784, 37134}
X(39681) = trilinear pole of line {385, 36213}
X(39681) = X(i)-isoconjugate of X(j) for these (i,j): {1, 39680}, {798, 8842}, {881, 3403}, {1581, 3288}, {1967, 23878}, {6784, 37134}
X(39681) = barycentric product X(i)*X(j) for these {i,j}: {262, 17941}, {263, 880}, {3978, 26714}, {5976, 6037}
X(39681) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 39680}, {99, 8842}, {263, 882}, {385, 23878}, {880, 20023}, {1691, 3288}, {5027, 6784}, {6037, 36897}, {17941, 183}, {26714, 694}, {32716, 34238}

X(39682) = X(3)X(327)∩X(4)X(39)

Barycentrics    (a^2*b^2 - b^4 + 2*a^2*c^2 + b^2*c^2)*(2*a^2*b^2 + a^2*c^2 + b^2*c^2 - c^4)*(a^8 - 2*a^6*b^2 + a^4*b^4 - 2*a^6*c^2 + a^4*b^2*c^2 + b^6*c^2 + a^4*c^4 - 2*b^4*c^4 + b^2*c^6) : :

X(39682) lies on the cubic K1158 and these lines: {3, 327}, {4, 39}, {98, 237}, {217, 12110}, {263, 6776}, {511, 39355}, {512, 11674}, {1503, 1987}, {1971, 32545}, {3331, 26714}, {9833, 36998}

X(39682) = X(182)-isoconjugate of X(1956)
X(39682) = barycentric product X(i)*X(j) for these {i,j}: {262, 401}, {327, 1971}
X(39682) = barycentric quotient X(i)/X(j) for these {i,j}: {262, 1972}, {263, 1987}, {401, 183}, {1971, 182}, {2186, 1956}, {6130, 23878}

X(39683) = X(3)X(1625)∩X(83)X(37124)

Barycentrics    a^2*(a^4 - a^2*b^2 - a^2*c^2 - 2*b^2*c^2)*(a^4*b^4 - 2*a^2*b^6 + b^8 + a^6*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 - 2*a^4*c^4 + b^4*c^4 + a^2*c^6)*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + a^4*c^4 + a^2*b^2*c^4 + b^4*c^4 - 2*a^2*c^6 - 2*b^2*c^6 + c^8) : :

X(39683) lies on the cubic K1159 and these lines: {3, 1625}, {83, 37124}, {99, 1298}, {143, 11437}, {287, 401}

X(38683) = X(i)-isoconjugate of X(j) for these (i,j): {262, 1955}, {401, 2186}
X(38683) = barycentric product X(i)*X(j) for these {i,j}: {182, 1972}, {183, 1987}, {458, 14941}, {8842, 32542}
X(38683) = barycentric quotient X(i)/X(j) for these {i,j}: {182, 401}, {458, 16089}, {1972, 327}, {1987, 262}, {3288, 6130}, {34396, 1971}

X(39684) = X(4)X(2896)∩X(263)X(3094)

Barycentrics    a^4*(2*a^2*b^2 + b^4 + a^2*c^2 + 2*b^2*c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*(a^2*b^2 + 2*a^2*c^2 + 2*b^2*c^2 + c^4) : :

X(39684) lies on the cubic K1158 and these lines: {4, 2896}, {263, 3094}, {2076, 34238}, {2698, 9301}, {5976, 20022}

X(39684) = X(1821)-isoconjugate of X(3329)
X(39684) = barycentric quotient X(i)/X(j) for these {i,j}: {237, 3329}, {9418, 12212}

X(39685) = X(3)X(76)∩X(83)X(1625)

Barycentrics    b^2*c^2*(a^4 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 + 2*a^2*b^2 + 2*a^2*c^2 + b^2*c^2)*(-a^4 + a^2*b^2 + b^2*c^2 - c^4) : :

X(39685) lies on the cubic K1159 and these lines: {3, 76}, {83, 1625}, {182, 327}, {5207, 20021}

X(39685) = barycentric product X(i)*X(j) for these {i,j}: {290, 3329}, {12212, 18024}
X(39685) = barycentric quotient X(i)/X(j) for these {i,j}: {3329, 511}, {12212, 237}, {14318, 2491}
X(39685) = {X(98),X(290)}-harmonic conjugate of X(14382)

X(39686) = P(97) + U(97)

Barycentrics    a^4*(a*b - b^2 + a*c - c^2)^2 : :
Barycentrics    a^4 (a s - Sω)^2 : :

X(39686) lies on the Brocard inellipse and these lines: {2, 34253}, {6, 105}, {31, 1911}, {42, 1200}, {184, 14827}, {213, 1015}, {220, 2975}, {672, 1362}, {2284, 8299}, {3124, 20455}, {6354, 39063}, {8041, 22432}, {9454, 9455}

X(39686) = isogonal conjugate of the isotomic conjugate of X(6184)
X(39686) = polar conjugate of the isotomic conjugate of X(20776)
X(39686) = X(i)-Ceva conjugate of X(j) for these (i,j): {6, 2223}, {692, 8638}, {1275, 2283}, {6184, 20776}
X(39686) = X(i)-isoconjugate of X(j) for these (i,j): {75, 6185}, {105, 18031}, {673, 2481}, {885, 34085}, {1027, 36803}, {14942, 34018}
X(39686) = crosspoint of X(i) and X(j) for these (i,j): {6, 2223}, {241, 13476}, {1275, 2283}
X(39686) = crosssum of X(i) and X(j) for these (i,j): {2, 2481}, {294, 1621}, {885, 14936}
X(39686) = crossdifference of every pair of points on line {885, 2481}
X(39686) = bicentric sum of PU(97)
X(39686) = barycentric square of X(672)
X(39686) = barycentric product X(i)*X(j) for these {i,j}: {4, 20776}, {6, 6184}, {31, 4712}, {32, 4437}, {55, 1362}, {105, 23612}, {184, 34337}, {213, 16728}, {518, 2223}, {665, 2284}, {672, 672}, {692, 3126}, {883, 8638}, {926, 2283}, {1252, 35505}, {1275, 39014}, {1458, 2340}, {1818, 2356}, {3263, 9455}, {3286, 20683}, {3323, 6066}, {3912, 9454}, {4998, 15615}, {5089, 20752}, {8693, 33570}, {18206, 39258}, {20455, 34159}, {23990, 35094}
X(39686) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 6185}, {672, 18031}, {1362, 6063}, {2223, 2481}, {2284, 36803}, {4437, 1502}, {4712, 561}, {6184, 76}, {8638, 885}, {9454, 673}, {9455, 105}, {15615, 11}, {16728, 6385}, {20776, 69}, {23612, 3263}, {34337, 18022}, {35505, 23989}, {39014, 1146}

X(39687) = P(101) + U(101)

Barycentrics    a^4*(a - b - c)^2*(b - c)^2*(a^2 - b^2 - c^2)^2 : :
Barycentrics    a^2 (sec B - sec C)^2 : :
Barycentrics    (sin^2 2A) (cos B - cos C)^2 : :

X(39687) lies on the Brocard inellipse and these lines: {6, 108}, {41, 217}, {48, 1949}, {184, 14827}, {393, 8761}, {604, 20233}, {652, 38983}, {1364, 36054}, {2202, 3331}, {2968, 23090}, {3269, 7117}, {3270, 14936}, {7124, 23137}, {9419, 20967}, {19354, 30706}, {22096, 34980}

X(39687) = isogonal conjugate of the isotomic conjugate of X(35072)
X(39687) = isogonal conjugate of the polar conjugate of X(3270)
X(39687) = X(i)-Ceva conjugate of X(j) for these (i,j): {6, 1946}, {48, 39201}, {2192, 8641}, {7152, 667}, {8761, 663}, {32677, 23220}, {36421, 21789}
X(39687) = X(i)-isoconjugate of X(j) for these (i,j): {2, 24032}, {75, 23984}, {76, 24033}, {158, 1275}, {264, 7128}, {331, 7012}, {561, 23985}, {653, 18026}, {823, 4566}, {1020, 6528}, {1847, 15742}, {1897, 13149}, {2052, 7045}, {4554, 36127}, {6335, 36118}, {6354, 23999}, {18027, 24027}
X(39687) = crosspoint of X(i) and X(j) for these (i,j): {6, 1946}, {3270, 35072}, {6056, 36054}, {21789, 36421}
X(39687) = crosssum of X(i) and X(j) for these (i,j): {2, 18026}, {331, 13149}
X(39687) = crossdifference of every pair of points on line {4566, 18026}
X(39687) = bicentric sum of PU(101)
X(39687) = barycentric square of X(652)
X(39687) = barycentric product X(i)*X(j) for these {i,j}: {1, 2638}, {3, 3270}, {6, 35072}, {11, 6056}, {31, 24031}, {32, 23983}, {48, 34591}, {55, 1364}, {108, 23614}, {184, 2968}, {212, 7004}, {213, 16731}, {219, 7117}, {255, 2310}, {394, 14936}, {520, 21789}, {521, 1946}, {577, 1146}, {647, 23090}, {650, 36054}, {652, 652}, {657, 4091}, {822, 1021}, {1259, 3271}, {1260, 3937}, {1265, 22096}, {1802, 3942}, {1804, 3022}, {2170, 2289}, {2326, 37754}, {3049, 15411}, {3119, 7125}, {3269, 7054}, {3900, 23224}, {4081, 7335}, {4131, 8641}, {7071, 7215}, {7253, 39201}, {14585, 23978}, {17926, 32320}, {18604, 36197}, {21666, 23606}, {32726, 33572}, {35071, 36421}
X(39687) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 24032}, {32, 23984}, {560, 24033}, {577, 1275}, {1146, 18027}, {1364, 6063}, {1501, 23985}, {1946, 18026}, {2638, 75}, {2968, 18022}, {3270, 264}, {6056, 4998}, {7117, 331}, {9247, 7128}, {14585, 1262}, {14936, 2052}, {16731, 6385}, {21789, 6528}, {22096, 1119}, {22383, 13149}, {23090, 6331}, {23224, 4569}, {23614, 35518}, {23983, 1502}, {24031, 561}, {34591, 1969}, {34980, 6356}, {35072, 76}, {36054, 4554}, {39201, 4566}

X(39688) = P(106) + U(106)

Barycentrics    a^2*(b + c)*(a^2 + b^2 - 3*b*c + c^2) : :
X(39688) = 3 X[1] + X[38302]

X(39688) lies on these lines: {1, 524}, {6, 4890}, {37, 42}, {111, 36070}, {187, 21009}, {513, 663}, {614, 4854}, {674, 3230}, {714, 4368}, {899, 4933}, {922, 5168}, {995, 1001}, {1334, 4735}, {1460, 1486}, {1716, 4851}, {1918, 22172}, {2092, 4068}, {2223, 8610}, {2241, 4471}, {2245, 3122}, {2611, 21323}, {3120, 16581}, {3649, 28011}, {3730, 4484}, {3932, 31855}, {4442, 4956}, {4657, 26102}, {5396, 7611}, {6019, 10354}, {7191, 17320}, {8053, 17053}, {11553, 28082}, {16569, 17279}, {16611, 24394}, {16685, 21746}, {16686, 19596}, {17278, 21926}, {17321, 29814}, {25887, 25889}, {28358, 28360}

X(39688) = isogonal conjugate of the isotomic conjugate of X(4442)
X(39688) = X(i)-Ceva conjugate of X(j) for these (i,j): {7292, 16611}, {7316, 6}
X(39688) = X(i)-isoconjugate of X(j) for these (i,j): {81, 34892}, {86, 34893}, {2748, 7192}, {4160, 6082}, {5387, 16726}
X(39688) = crosspoint of X(i) and X(j) for these (i,j): {1, 111}, {7292, 16784}
X(39688) = crosssum of X(i) and X(j) for these (i,j): {1, 524}, {690, 17058}, {34892, 34893}
X(39688) = crossdifference of every pair of points on line {9, 1019}
X(39688) = bicentric sum of PU(106)
X(39688) = barycentric product X(i)*X(j) for these {i,j}: {1, 16611}, {6, 4442}, {10, 16784}, {37, 7292}, {42, 37756}, {57, 24394}, {92, 23230}, {111, 16597}, {1018, 2832}, {4033, 8650}, {4956, 28658}, {6088, 37210}
X(39688) = barycentric quotient X(i)/X(j) for these {i,j}: {42, 34892}, {213, 34893}, {2832, 7199}, {4442, 76}, {6088, 4789}, {7292, 274}, {8650, 1019}, {16597, 3266}, {16611, 75}, {16784, 86}, {23230, 63}, {24394, 312}, {36070, 39296}, {37756, 310}
X(39688) = {X(3122),X(3747)}-harmonic conjugate of X(2245)

X(39689) = P(107) + U(107)

Barycentrics    a^2*(2*a^2 - b^2 - c^2)^2 : :

X(39689) lies on the Brocard inellipse and these lines: {2, 8587}, {6, 110}, {23, 8586}, {32, 9486}, {126, 1641}, {184, 574}, {187, 3292}, {249, 843}, {323, 5104}, {352, 1691}, {353, 3094}, {394, 5210}, {511, 38369}, {524, 7664}, {542, 8288}, {599, 10554}, {1015, 2308}, {1017, 5029}, {1383, 9716}, {1384, 1501}, {1495, 5107}, {1648, 5477}, {1914, 17455}, {1977, 2300}, {1992, 7665}, {2028, 5639}, {2029, 5638}, {2030, 3231}, {2056, 5354}, {2434, 10354}, {2482, 8030}, {3051, 5008}, {3569, 9408}, {5026, 5468}, {5027, 38366}, {5108, 5182}, {5111, 13192}, {5147, 21747}, {5202, 16971}, {5969, 35356}, {5971, 12151}, {6199, 7598}, {6395, 7599}, {6433, 7601}, {6434, 7602}, {8585, 9306}, {8593, 30786}, {9143, 11646}, {9225, 11580}, {9544, 20859}, {9830, 31125}, {10836, 19153}, {14936, 21748}, {23992, 30454}

X(39689) = midpoint of X(7664) and X(10552)
X(39689) = isogonal conjugate of the isotomic conjugate of X(2482)
X(39689) = isogonal conjugate of the polar conjugate of X(5095)
X(39689) = X(i)-Ceva conjugate of X(j) for these (i,j): {6, 187}, {110, 351}, {249, 5467}
X(39689) = X(i)-isoconjugate of X(j) for these (i,j): {75, 10630}, {92, 15398}, {671, 897}, {892, 23894}, {923, 18023}, {1109, 34539}, {1577, 34574}, {5466, 36085}, {30786, 36128}
X(39689) = crosspoint of X(i) and X(j) for these (i,j): {6, 187}, {249, 5467}, {524, 14357}, {2482, 5095}
X(39689) = crosssum of X(i) and X(j) for these (i,j): {2, 671}, {111, 14246}, {115, 5466}, {892, 33799}, {10630, 15398}
X(39689) = crossdifference of every pair of points on line {671, 690}
X(39689) = bicentric sum of PU(107)
X(39689) = barycentric square of X(896)
X(39689) = PU(107)-harmonic conjugate of X(351)
X(39689) = barycentric product X(i)*X(j) for these {i,j}: {3, 5095}, {6, 2482}, {15, 30454}, {16, 30455}, {31, 24038}, {32, 36792}, {55, 1366}, {56, 7067}, {110, 1649}, {111, 8030}, {184, 34336}, {187, 524}, {213, 16733}, {249, 23992}, {351, 5468}, {468, 3292}, {574, 20380}, {690, 5467}, {691, 33915}, {896, 896}, {922, 14210}, {2434, 9125}, {2642, 23889}, {3266, 14567}, {5026, 18872}, {5642, 9717}, {5967, 9155}, {6593, 14357}, {16702, 21839}, {23106, 32740}
X(39689) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 10630}, {184, 15398}, {187, 671}, {351, 5466}, {524, 18023}, {922, 897}, {1366, 6063}, {1576, 34574}, {1649, 850}, {2482, 76}, {3292, 30786}, {5095, 264}, {5467, 892}, {7067, 3596}, {8030, 3266}, {14443, 23105}, {14567, 111}, {16733, 6385}, {23200, 895}, {23357, 34539}, {23992, 338}, {24038, 561}, {30454, 300}, {30455, 301}, {33915, 35522}, {34336, 18022}, {36792, 1502}
X(39689) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 110, 2502}, {6, 2502, 3124}, {110, 20976, 3124}, {2502, 20976, 6}, {5477, 5642, 1648}

X(39690) = P(108) + U(108)

Barycentrics    a^2*(b + c)*(a^4 - b^4 - a^2*b*c + b^3*c + b*c^3 - c^4) : :

X(39690) lies on these lines: {6, 25}, {9, 440}, {19, 1836}, {37, 17441}, {44, 513}, {65, 2333}, {71, 210}, {72, 4456}, {101, 14963}, {169, 442}, {190, 16085}, {198, 5452}, {218, 10974}, {220, 22076}, {573, 5776}, {579, 4383}, {607, 3556}, {857, 4872}, {859, 3002}, {906, 1324}, {966, 26052}, {992, 28259}, {1212, 37225}, {1213, 15487}, {1400, 1427}, {1781, 33097}, {1834, 2082}, {2092, 30706}, {2200, 2594}, {2264, 2354}, {2876, 20875}, {3125, 21745}, {3827, 5089}, {4548, 23847}, {5051, 33950}, {5279, 33066}, {5546, 37311}, {5746, 37388}, {7124, 22654}, {7289, 18635}, {8679, 20752}, {9310, 33587}, {9798, 22131}, {9840, 16699}, {14555, 37419}, {16583, 21744}, {17052, 24694}, {18596, 21530}, {18603, 28379}, {21034, 23624}, {22070, 23361}, {22080, 26867}, {23674, 33313}, {26068, 27021}

X(39690) = isogonal conjugate of X(37202)
X(39690) = isogonal conjugate of the isotomic conjugate of X(857)
X(39690) = X(i)-complementary conjugate of X(j) for these (i,j): {42, 20621}, {26703, 3741}
X(39690) = X(i)-Ceva conjugate of X(j) for these (i,j): {1297, 55}, {7281, 3725}, {32735, 512}, {37202, 1}
X(39690) = X(i)-isoconjugate of X(j) for these (i,j): {1, 37202}, {2, 26702}, {2867, 16612}, {3267, 32673}, {14208, 36071}
X(39690) = crosspoint of X(i) and X(j) for these (i,j): {1, 37202}, {3220, 7291}
X(39690) = crosssum of X(2) and X(14953)
X(39690) = crossdifference of every pair of points on line {1, 525}
X(39690) = bicentric sum of PU(108)
X(39690) = barycentric product X(i)*X(j) for these {i,j}: {6, 857}, {10, 3220}, {37, 7291}, {42, 4872}, {65, 3100}, {213, 7112}, {1400, 37774}
X(39690) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 37202}, {31, 26702}, {857, 76}, {3100, 314}, {3220, 86}, {4872, 310}, {7112, 6385}, {7291, 274}, {37774, 28660}
X(39690) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2200, 23619, 2594}, {2333, 23620, 65}

X(39691) = P(147) + U(147)

Barycentrics    (b - c)^2*(b + c)^2*(b^2 + c^2) : :

X(39691) lies on these lines: {2, 4048}, {5, 14822}, {6, 3448}, {11, 31003}, {39, 35319}, {110, 11646}, {111, 15059}, {115, 125}, {141, 4576}, {338, 23962}, {373, 39565}, {427, 3051}, {542, 20976}, {858, 3231}, {1213, 31098}, {1501, 11550}, {1503, 14567}, {1613, 31074}, {2076, 5189}, {2086, 7668}, {2211, 37981}, {2502, 5972}, {2623, 8901}, {3266, 5031}, {3580, 20977}, {3763, 14360}, {3981, 23293}, {4609, 18896}, {5017, 31133}, {5025, 37889}, {5133, 20965}, {5207, 19577}, {5304, 15431}, {6034, 9140}, {6723, 10418}, {7703, 9465}, {7790, 34512}, {8024, 16893}, {8287, 27918}, {8627, 29012}, {9463, 31857}, {11792, 36472}, {13410, 19130}, {14416, 14424}, {15514, 37779}, {18010, 34294}, {20337, 31041}, {20859, 21243}, {21001, 31101}, {21043, 23941}, {21817, 21939}, {23897, 31025}, {23903, 33155}, {31128, 34573}, {32225, 39563}

X(39691) = complement of X(10330)
X(39691) = complement of the isotomic conjugate of X(31065)
X(39691) = X(i)-complementary conjugate of X(j) for these (i,j): {3108, 4369}, {7953, 21254}, {31065, 2887}
X(39691) = X(i)-Ceva conjugate of X(j) for these (i,j): {83, 523}, {141, 826}, {427, 3005}, {850, 8029}, {8024, 2528}, {15321, 512}, {16277, 669}, {27366, 688}, {31125, 14424}
X(39691) = X(i)-isoconjugate of X(j) for these (i,j): {82, 249}, {83, 1101}, {99, 34072}, {110, 4599}, {163, 4577}, {250, 34055}, {251, 24041}, {308, 23995}, {662, 827}, {799, 4630}, {1576, 4593}, {3112, 23357}, {14574, 37204}, {18833, 23963}, {24000, 28724}
X(39691) = crosspoint of X(i) and X(j) for these (i,j): {2, 31065}, {83, 523}, {115, 338}, {141, 826}, {512, 1843}, {850, 8024}
X(39691) = crosssum of X(i) and X(j) for these (i,j): {39, 110}, {99, 1799}, {249, 23357}, {251, 827}
X(39691) = crossdifference of every pair of points on line {110, 827}
X(39691) = bicentric sum of PU(147)
X(39691) = PU(147)-harmonic conjugate of X(2)
X(39691) = barycentric product of the vertices of the anti-Ursa-minor triangle
X(39691) = barycentric product X(i)*X(j) for these {i,j}: {38, 1109}, {39, 338}, {83, 15449}, {115, 141}, {125, 427}, {339, 1843}, {512, 23285}, {523, 826}, {850, 3005}, {868, 20021}, {1235, 20975}, {1365, 3703}, {1577, 8061}, {1634, 23105}, {1648, 31125}, {1930, 2643}, {1964, 23994}, {2084, 20948}, {2501, 2525}, {2530, 4036}, {2970, 3917}, {3051, 23962}, {3120, 15523}, {3124, 8024}, {3665, 4092}, {3708, 20883}, {3933, 8754}, {3954, 16732}, {4024, 16892}, {4064, 21108}, {4466, 21016}, {4568, 21131}, {4576, 8029}, {5466, 14424}, {7794, 34294}, {7927, 31067}, {8288, 23297}, {9148, 35366}, {15526, 27376}, {16703, 21833}, {16887, 21043}, {17171, 21046}, {17442, 20902}, {19174, 35442}, {21035, 21207}
X(39691) = barycentric quotient X(i)/X(j) for these {i,j}: {38, 24041}, {39, 249}, {115, 83}, {125, 1799}, {141, 4590}, {338, 308}, {427, 18020}, {512, 827}, {523, 4577}, {661, 4599}, {669, 4630}, {688, 1576}, {798, 34072}, {826, 99}, {850, 689}, {868, 20022}, {1109, 3112}, {1577, 4593}, {1843, 250}, {1923, 23995}, {1930, 24037}, {1964, 1101}, {2084, 163}, {2525, 4563}, {2528, 4576}, {2643, 82}, {3005, 110}, {3051, 23357}, {3124, 251}, {3269, 28724}, {3665, 7340}, {3703, 6064}, {3708, 34055}, {3954, 4567}, {4079, 4628}, {4576, 31614}, {8024, 34537}, {8061, 662}, {8288, 10130}, {8754, 32085}, {8901, 39287}, {9494, 14574}, {14424, 5468}, {15449, 141}, {15523, 4600}, {16892, 4610}, {20948, 37204}, {20975, 1176}, {21035, 4570}, {21043, 18082}, {21123, 4556}, {21131, 10566}, {21833, 18098}, {22260, 18105}, {23285, 670}, {23994, 18833}, {27376, 23582}, {31067, 35137}, {33919, 22105}, {35366, 9150}, {35971, 1915}
X(39691) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {115, 125, 3124}, {115, 8288, 1648}, {125, 3124, 1648}, {3124, 8288, 125}

X(39692) = NINE-POINT-CIRCLE INVERSE OF X(1)

Barycentrics    a^5*b^2 - a^4*b^3 - 2*a^3*b^4 + 2*a^2*b^5 + a*b^6 - b^7 - a^4*b^2*c + 4*a^3*b^3*c - 4*a*b^5*c + b^6*c + a^5*c^2 - a^4*b*c^2 - 2*a^3*b^2*c^2 - 2*a^2*b^3*c^2 - a*b^4*c^2 + 3*b^5*c^2 - a^4*c^3 + 4*a^3*b*c^3 - 2*a^2*b^2*c^3 + 8*a*b^3*c^3 - 3*b^4*c^3 - 2*a^3*c^4 - a*b^2*c^4 - 3*b^3*c^4 + 2*a^2*c^5 - 4*a*b*c^5 + 3*b^2*c^5 + a*c^6 + b*c^6 - c^7::
X(39692) = 3 X[381] + X[35451]

X(39692) lies on these lines: {1, 5}, {2, 10058}, {3, 12764}, {4, 10090}, {10, 12758}, {35, 3035}, {36, 1532}, {40, 32554}, {46, 2950}, {55, 38752}, {56, 10742}, {65, 12611}, {79, 24465}, {90, 1768}, {100, 1479}, {104, 499}, {149, 5154}, {153, 3086}, {214, 3825}, {381, 1470}, {403, 1877}, {442, 6667}, {497, 10087}, {498, 6975}, {528, 17533}, {900, 21189}, {999, 12763}, {1145, 1329}, {1210, 11570}, {1320, 11681}, {1465, 11809}, {1478, 6945}, {1519, 1737}, {1537, 5903}, {1772, 35015}, {2077, 3583}, {2476, 10200}, {2771, 18838}, {2802, 3814}, {2841, 3259}, {2886, 34122}, {3036, 24390}, {3582, 5193}, {3586, 15015}, {3746, 20400}, {3816, 34123}, {3822, 32557}, {3847, 10609}, {3918, 6702}, {4299, 10728}, {4302, 6963}, {4996, 5046}, {5010, 38760}, {5204, 38753}, {5217, 38762}, {5225, 13199}, {5433, 15446}, {5445, 15908}, {5499, 7294}, {5540, 6506}, {5541, 9614}, {5563, 38757}, {5687, 13271}, {5704, 9809}, {5854, 17757}, {6284, 33814}, {6691, 14800}, {6701, 16120}, {6713, 6842}, {6734, 18254}, {6797, 9955}, {6830, 12775}, {6834, 36152}, {6881, 38319}, {6907, 21154}, {6922, 24466}, {6929, 14793}, {6932, 38693}, {6943, 10724}, {6971, 10738}, {6980, 10269}, {6981, 10320}, {7280, 37406}, {7288, 12248}, {7354, 22799}, {7678, 20119}, {7680, 38038}, {9669, 12331}, {10039, 15558}, {10072, 10711}, {10265, 12608}, {10395, 12691}, {10573, 10698}, {10589, 12115}, {10707, 13278}, {10893, 12332}, {10915, 21630}, {11571, 12832}, {12047, 12736}, {12515, 24914}, {12607, 25416}, {12619, 17606}, {12641, 12653}, {12743, 22935}, {12773, 18542}, {13407, 18240}, {13904, 19082}, {13962, 19081}, {14792, 37290}, {15343, 28074}, {15863, 24387}, {17527, 31235}, {17636, 23340}, {18242, 21842}, {18480, 18976}, {18514, 37356}, {25436, 28018}, {35802, 35883}, {35803, 35882}

X(39692) = midpoint of X(4) and X(18861)
X(39692) = reflection of X(5533) in X(11)
X(39692) = complement of X(17100)
X(39692) = incircle-inverse of X(496)
X(39692) = nine-point-circle-inverse of X(1)
X(39692) = complement of the isogonal conjugate of X(17101)
X(39692) = X(i)-complementary conjugate of X(j) for these (i,j): {17101, 10}, {34901, 34823}, {38544, 121}
X(39692) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 11, 8068}, {5, 26476, 1}, {11, 12, 1387}, {11, 119, 1}, {11, 1317, 496}, {80, 7741, 11}, {80, 7972, 37711}, {149, 5552, 25438}, {153, 3086, 10074}, {496, 11698, 1317}, {496, 26482, 1}, {999, 38755, 12763}, {1210, 21635, 11570}, {1484, 10593, 11}, {6326, 9581, 10073}, {7741, 7951, 23708}, {7988, 37718, 16173}, {9669, 12331, 13274}, {12749, 16173, 1}, {17606, 17638, 12619}, {23477, 23517, 10944}

leftri

Points on cubics: X(39693)-X(39749)

rightri

This preamble is contributed by Clark Kimberling, September 28, 2020.

Suppose that P = p(a,b,c) : p(b,c,a) : p(c,a,b) is an polynomial triangle center such that the point (b-c)p(a,b,c) : (c-a)p(b,c,a) : (a-b)p(c,a,b) is on the line at infinity; i.e.,

(b-c)p(a,b,c) + (c-a)p(b,c,a) + (a-b)p(c,a,b) = 0.

Let χ(P) denote the cubic given by

(b+c)p(a,b,c)(y-z)(x+y)(x+z) + (c+a)p(b,c,a)(z-x)(y+z)(y+x) + (a+b)p(c,a,b)(x-y)(z+x)(z+y) = 0.

This is the cubic pK(X(2)),P*), where P* = (b + c)*p - (a + c)*q - (a + b)*r : :

Let A'B'C' denote the anticomplementary triangle of ABC. It is easy to check that the following 9 points lie on χ(P): A, B, C, A', B', C', X(1), X(2), X(75) .

It is also easy to check that if U is a point on χ(P), then the isotomic conjugate of U also lies on χ(P). This section consists of isotomic conjugates of points on cubics χ(P) for selected points P. In column 1 of the following table, the appearance of u(a,b,c) in a row means that the points indicated in column 2 lie on the cubic

u(a,b,c)(y-z)(x+y)(x+z) + u(b,c,a)(z-x)(y+z)(y+x) + u(c,a,b)(x-y)(z+x)(z+y) = 0;

so that the point P is given by u(a,b,c) = (b+c)p(a,b,c). The appearance of {i,j} in column 2 means that {X(i),X(j)} are a pair of isotomic conjugate points that lie on χ(P).

b+c {192,330}, {3224,17149}
(b+c)^2 {596,4360}, {17147,35058}, {24068,39693}
a^2 (b+c)^2 {8049,17135}, {13476,17143}
(b+c)(b+c-a) {145,4373}, {3210,39694}, {3875,34860}
a(b+c)(b+c-a)(b^2+c^2-a^2) {12649,39695}, {30699,39696}
(b+c)^2 cos^2 A {2997,3868}
(b+c)(2a-b-c) {519,903}, {17160,39697}, {17495,39698}, {30579,39699}
(b+c)(-a+b+c) {3187,39700}
(b+c)(3a+b+c) {3617,30712}, {10436,31359}, {27835,39701}
(b+c)(-3a+b+c) {3621,36606}, {17151,39702}, {17490,39703}
(b+c)(4a+b+c) {3679,39704}, {30564,39705}, {31035,39706}
(b+c)(-4a+b+c) {3632,39707}
(b+c)(3a+2b+2c) {9780,28626}, {17394,39708}, {28626,9780}
(b+c)(-3a+2b+2c) {20050,39709}
(b+c)(2a+3b+3c) {1125,1268}
(b+c)(-2a+3b+3c) {3244,39710}
(b+c)(a+2b+2c) {3616,5936},{17393,39711}
(b+c)(-a+2b+2c) {3241,36588}
(b+c)(a^2+b^2+c^2+a(a+b+c)) {3661,14621}, {5263,39712}, {31087,39713}
(b+c)(a^2+b^2+c^2-a(a+b+c)) {239,335}, {32922,39714}
(b+c)(a^2+b^2+c^2-2a(a+b+c)) {26274,39715}
(b+c)(2(a^2+b^2+c^2)+a(a+b+c)) {29611,39716}
(b+c)(2(a^2+b^2+c^2)-a(a+b+c)) {5222,39749}
(b+c)(bc+ca+ab+a(a+b+c)) {24325,39717}
(b+c)(bc+ca+ab-a(a+b+c)) {740,18827}, {6542,6650}, {13174,39718}, {39367,39719}
(b+c)(bc+ca+ab-2a(a+b+c)) {20055,39720}
(b+c)(2(bc+ca+ab)-a(a+b+c)) {17316,39721}
(b+c)(abc+a(a^2+b^2+c^2)) {17302,39722}, {33090,39723}
(b+c)(abc-a(a^2+b^2+c^2)) {82,1930}, {17280,39724}, {20934,39725}, {21289,39726}, {21378,39727}, {33091,39728}
(b+c)(abc+2a(a^2+b^2+c^2)) {17383,39729}
(b+c)(abc-2a(a^2+b^2+c^2)) {17358,39730}
(b+c)(2abc+a(a^2+b^2+c^2)) {1219,3672}, {23051,39731}
(b+c)(2abc-a(a^2+b^2+c^2)); (K605) {19,304}, {279,346}, {2184,18750},{4329,7219}, {10327, 39732},{33091,39733}
(b+c)(abc+a(bc+ca+ab)) {37,274},{4651,39734}, {16552,39735}
(b+c)(abc-a(bc+ca+ab)) {8049,17135}, {13476,17143}
(b+c)(abc+2a(bc+ca+ab)) {27268,39736}, {32092,39737}
(b+c)(abc-2a(bc+ca+ab)) {4699,39738}, {32104,39739}
(b+c)(2abc+a(bc+ca+ab)) {4704,39740, {17038,31997}
(b+c)(2abc-a(bc+ca+ab)) {1278,38247}, {10453,39741}, {17144, 39742}
(b+c)(a^3+b^3+c^3+a(bc+ca+ab)) {31079,39743}
(b+c)(b^2c^2+c^2a^2+a^2b^2+a^2bc) {17445,39744}, {26801,39745}
(b+c)(b^2c^2+c^2a^2+a^2b^2-a^2bc) {1964,18833}, {26752,39746}
(b+c)(2a+b+c) {10,86},{3995,39747}, {18133,39748}

X(39693) = ISOTOMIC CONJUGATE OF X(24068)

Barycentrics    (a^3*b + 2*a^2*b^2 + a*b^3 - a^3*c - a^2*b*c - a*b^2*c - b^3*c - a^2*c^2 - b^2*c^2)*(a^3*b + a^2*b^2 - a^3*c + a^2*b*c - 2*a^2*c^2 + a*b*c^2 + b^2*c^2 - a*c^3 + b*c^3) : :

X(39693) lies on these lines: {1, 18133}, {6, 17147}, {75, 39748}, {1126, 4360}, {3226, 33769}

X(39694) = ISOTOMIC CONJUGATE OF X(3210)

Barycentrics    (a^2*b + a*b^2 - a^2*c + a*b*c - b^2*c - a*c^2 - b*c^2)*(a^2*b + a*b^2 - a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2) : :

X(39694) lies on these lines: {1, 979}, {2, 17786}, {57, 1999}, {81, 192}, {88, 17490}, {89, 17147}, {145, 959}, {274, 6382}, {278, 37759}, {279, 30699}, {291, 10453}, {321, 330}, {329, 9263}, {346, 39696}, {957, 20037}, {961, 3891}, {1022, 23825}, {1258, 4393}, {1432, 17778}, {1929, 4362}, {2282, 3187}, {3995, 25417}, {4358, 39703}, {4671, 35058}, {5333, 16722}, {8056, 11679}, {10027, 27659}, {17178, 28605}, {17494, 23834}, {17495, 26745}, {19684, 31999}, {19785, 39724}, {27789, 31035}, {32774, 39730}

X(39694) = isogonal conjugate of X(21769)
X(39694) = isotomic conjugate of X(3210)
X(39694) = X(19)-isoconjugate of X(20805)

X(39695) = ISOTOMIC CONJUGATE OF X(12649)

Barycentrics    (a^4 - 2*a^2*b^2 + b^4 - 2*a^3*c - 2*b^3*c + 2*a*b*c^2 + 2*a*c^3 + 2*b*c^3 - c^4)*(a^4 - 2*a^3*b + 2*a*b^3 - b^4 + 2*a*b^2*c + 2*b^3*c - 2*a^2*c^2 - 2*b*c^3 + c^4) : :

X(39695) lies on these lines: {2, 24179}, {7, 224}, {27, 5905}, {86, 34430}, {273, 5174}, {307, 7318}, {4329, 16099}, {4373, 20007}, {10446, 39732}

X(39696) = ISOTOMIC CONJUGATE OF X(30699)

Barycentrics    (a^3 + a^2*b + a*b^2 + b^3 - 3*a^2*c + b^2*c - 3*a*c^2 + b*c^2 + c^3)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3 + a^2*c + b^2*c + a*c^2 + b*c^2 + c^3) : :

X(39696) lies on these lines: {1, 26065}, {57, 3169}, {105, 37652}, {145, 961}, {278, 30699}, {279, 3210}, {346, 39694}, {959, 3873}, {3687, 8056}, {17353, 25430}, {19789, 21907}, {32029, 37669}

X(39697) = ISOTOMIC CONJUGATE OF X(17160)

Barycentrics    (a*b + b^2 - 3*a*c + b*c)*(3*a*b - a*c - b*c - c^2) : :

X(39697) lies on these lines: {1, 4427}, {10, 244}, {37, 537}, {65, 1317}, {75, 17205}, {99, 30593}, {106, 24841}, {354, 22306}, {519, 4674}, {897, 32922}, {900, 21630}, {994, 3873}, {1120, 4792}, {1125, 3952}, {1647, 4013}, {2218, 8666}, {3555, 22313}, {3874, 34434}, {3892, 13476}, {4358, 4694}, {4669, 36924}, {4681, 5049}, {4793, 24165}, {4975, 6534}, {14210, 39714}, {19862, 24003}, {21087, 25377}, {21093, 23869}

X(39697) = isogonal conjugate of X(40091)
X(39697) = isotomic conjugate of X(17160)
X(39697) = cevapoint of X(i) and X(j) for these {i,j}: {244, 900}, {519, 1125}, {523, 1647}, {1086, 21115}
X(39697) = trilinear pole of line X(661)X(1213)
X(39697) = trilinear product X(i)*X(j) for these {i,j}: {2, 39981}, {6, 39994}

X(39698) = ISOTOMIC CONJUGATE OF X(17495)

Barycentrics    (a^2*b + a*b^2 - a^2*c + 2*a*b*c - b^2*c - a*c^2 - b*c^2)*(a^2*b + a*b^2 - a^2*c - 2*a*b*c + b^2*c - a*c^2 + b*c^2) : :

X(39698) lies on these lines: {1, 3952}, {2, 4033}, {28, 1897}, {37, 16729}, {57, 4552}, {75, 39706}, {81, 190}, {88, 17160}, {89, 192}, {105, 15571}, {145, 957}, {274, 1978}, {291, 29824}, {312, 35058}, {321, 16726}, {330, 4671}, {985, 4613}, {1015, 27070}, {1022, 4080}, {1390, 29823}, {1929, 17763}, {3210, 26745}, {4595, 27664}, {6542, 17946}, {9263, 30578}, {17751, 26727}, {19743, 25417}, {21907, 37759}, {29822, 30571}, {33155, 39724}

X(39698) = isogonal conjugate of polar conjugate of isotomic conjugate of X(23169)
X(39698) = isotomic conjugate of X(17495)
X(39698) = X(19)-isoconjugate of X(23169)

X(39699) = ISOTOMIC CONJUGATE OF X(30579)

Barycentrics    (5*a^2*b + a*b^2 - 4*b^3 - 3*a^2*c - 2*a*b*c + b^2*c - 3*a*c^2 + 5*b*c^2)*(3*a^2*b + 3*a*b^2 - 5*a^2*c + 2*a*b*c - 5*b^2*c - a*c^2 - b*c^2 + 4*c^3) : :

X(39699) lies on these lines: {16704, 17160}, {30578, 31011}

X(39700) = ISOTOMIC CONJUGATE OF X(3187)

Barycentrics    (-(a*b^2) - b^3 + a^2*c - b^2*c + a*c^2)*(a^2*b + a*b^2 - a*c^2 - b*c^2 - c^3) : :

X(39700) lies on these lines: {1, 15376}, {2, 3670}, {7, 17147}, {27, 3187}, {63, 272}, {75, 17184}, {86, 18601}, {310, 33935}, {335, 32858}, {675, 29014}, {1231, 15467}, {1246, 3210}, {3151, 4440}, {6994, 11851}, {14621, 17011}

X(39700) = isogonal conjugate of X(5301)
X(39700) = isotomic conjugate of X(3187)

X(39701) = ISOTOMIC CONJUGATE OF X(27835)

Barycentrics    (3*a - b - c)*(a^2*b + a*b^2 - a^2*c + a*b*c - b^2*c - a*c^2 - b*c^2)*(a^2*b + a*b^2 - a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2) : :

X(39701) lies on these lines: {1, 979}, {2137, 7308}, {2975, 8686}, {10436, 19604}

X(39702) = ISOTOMIC CONJUGATE OF X(17151)

Barycentrics    (a*b + b^2 - 4*a*c + b*c)*(4*a*b - a*c - b*c - c^2) : :

X(39702) lies on these lines: {10, 3976}, {37, 3622}, {65, 3241}, {341, 4694}, {596, 4673}, {969, 17393}, {994, 3881}, {3621, 3999}, {3633, 4674}, {3668, 4398}, {3873, 34434}, {4096, 25055}, {8769, 32922}, {18156, 39714}

X(39703) = ISOTOMIC CONJUGATE OF X(17490)

Barycentrics    (a^2*b + a*b^2 - a^2*c + 3*a*b*c - b^2*c - a*c^2 - b*c^2)*(a^2*b + a*b^2 - a^2*c - 3*a*b*c + b^2*c - a*c^2 + b*c^2) : :

X(39703) lies on these lines: {1, 4090}, {2, 4110}, {57, 192}, {81, 17350}, {88, 3210}, {89, 3995}, {279, 30545}, {312, 330}, {1258, 32095}, {1432, 17316}, {1929, 29649}, {4358, 39694}, {4671, 39747}, {8055, 9263}, {8056, 17151}, {15474, 37759}, {17147, 26745}, {19786, 39730}, {25417, 31035}, {27268, 37870}, {28605, 39706}, {30861, 32017}, {31993, 39736}

X(39703) = isogonal conjugate of X(21785)
X(39703) = isotomic conjugate of X(17490)
X(39703) = X(19)-isoconjugate of X(23085)

X(39704) = ISOTOMIC CONJUGATE OF X(3679)

Barycentrics    (2*a + 2*b - c)*(2*a - b + 2*c) : :

X(39704) lies on these lines: {1, 903}, {2, 44}, {7, 1319}, {69, 5936}, {75, 519}, {85, 14584}, {86, 2163}, {88, 20973}, {190, 36911}, {239, 31139}, {244, 751}, {273, 1877}, {310, 4479}, {335, 545}, {513, 6548}, {524, 27483}, {527, 27475}, {536, 27494}, {551, 4389}, {664, 36595}, {673, 2364}, {675, 4588}, {750, 765}, {894, 17241}, {1240, 30596}, {1268, 10436}, {3241, 36588}, {3306, 33794}, {3644, 7228}, {3661, 10022}, {3663, 39707}, {3672, 36606}, {3679, 17360}, {3759, 26806}, {3945, 4373}, {4357, 19883}, {4360, 39710}, {4363, 17310}, {4370, 17244}, {4648, 17336}, {4687, 17333}, {4725, 31314}, {4751, 17330}, {4764, 17390}, {4821, 4889}, {4945, 37633}, {5224, 28650}, {6650, 17301}, {7222, 17315}, {7232, 17400}, {7238, 17397}, {7277, 27147}, {10446, 28198}, {14621, 17399}, {15668, 17329}, {16704, 16724}, {16826, 24441}, {17116, 17386}, {17139, 39734}, {17228, 17376}, {17240, 17281}, {17264, 35578}, {17298, 17371}, {17321, 30712}, {17322, 28626}, {17379, 17382}, {17389, 28309}, {17395, 36525}, {17487, 29569}, {21296, 28653}, {27747, 37684}, {28301, 29574}, {29597, 31332}, {30607, 31019}, {30963, 31002}, {37756, 39721}

In the plane of a tiangle ABC, let (BC) be the half-plane determined by the line BC that does not include A. Let Ab be the point in (BC) that satisfies |BAb| = |AC|, and let Ac be the point in (BC) that satisfies |BAc| = |AB|. Let La be the line of the diagonal points, other than A, of the complete quadrangle BCAbAc. Define Lb and Lc cyclically. Let A' = Lb∩Lc, and define B' and C' cyclically. Then A'B'C' is perspective to ABC, and the perspector is X(39704). (Angerl Montesdeoca, March 20, 2023)

X(39704) = isogonal conjugate of X(2177)
X(39704) = isotomic conjugate of X(3679)
X(39704) = complement of X(17488)
X(39704) = anticomplement of X(16590)

X(39705) = ISOTOMIC CONJUGATE OF X(30564)

Barycentrics    (5*a^2*b + a*b^2 - 4*b^3 + 3*a^2*c + 4*a*b*c + b^2*c + 3*a*c^2 + 5*b*c^2)*(3*a^2*b + 3*a*b^2 + 5*a^2*c + 4*a*b*c + 5*b^2*c + a*c^2 + b*c^2 - 4*c^3) : :

X(39705) lies on these lines: {3679, 31025}, {4945, 31035}, {5235, 30564}, {23598, 31019}, {37635, 39364}

X(39706) = ISOTOMIC CONJUGATE OF X(31035)

Barycentrics    (a^2*b + a*b^2 - a^2*c - 4*a*b*c - b^2*c - a*c^2 - b*c^2)*(a^2*b + a*b^2 - a^2*c + 4*a*b*c + b^2*c - a*c^2 + b*c^2) : :

X(39706) lies on these lines: {1, 17495}, {57, 19742}, {75, 39698}, {88, 17277}, {89, 24620}, {1002, 19998}, {1022, 17494}, {1255, 17147}, {2006, 17077}, {3210, 27789}, {3995, 4659}, {4359, 35058}, {17490, 25417}, {19804, 39747}, {24616, 26745}, {27107, 31025}, {28605, 39703}

X(39707) = ISOTOMIC CONJUGATE OF X(3632)

Barycentrics    (2*a + 2*b - 3*c)*(2*a - 3*b + 2*c) : :

X(39707) lies on these lines: {2, 4912}, {7, 1392}, {27, 19830}, {69, 36588}, {75, 3626}, {86, 4862}, {320, 4373}, {335, 3644}, {675, 8697}, {903, 3875}, {1268, 17250}, {3244, 4398}, {3632, 17361}, {3663, 39704}, {4106, 6548}, {4346, 17394}, {4357, 28650}, {4389, 30598}, {4440, 17241}, {4686, 27494}, {4764, 7238}, {4796, 37677}, {18815, 39126}, {20050, 39709}

X(39708) = ISOTOMIC CONJUGATE OF X(17394)

Barycentrics    (2*a*b + 2*b^2 + a*c + 2*b*c)*(a*b + 2*a*c + 2*b*c + 2*c^2) : :

X(39708) lies on these lines: {1, 4720}, {8, 30589}, {10, 28605}, {19, 191}, {37, 1698}, {38, 39711}, {63, 267}, {65, 3679}, {79, 17275}, {596, 31330}, {740, 39737}, {759, 37032}, {969, 25590}, {2214, 24342}, {2218, 37322}, {3632, 31327}, {3899, 34434}, {4647, 31359}, {4688, 18398}, {5312, 31993}, {17890, 23604}, {18827, 32092}, {19858, 27812}, {24325, 39739}, {25055, 28617}, {26066, 36910}, {27020, 31329}, {32104, 39717}

X(39709) = ISOTOMIC CONJUGATE OF X(20050)

Barycentrics    (3*a + 3*b - 7*c)*(3*a - 7*b + 3*c) : :

X(39709) lies on these lines: {7, 3244}, {8, 39710}, {903, 21296}, {1266, 36606}, {1268, 31995}, {3663, 28626}, {4346, 5936}, {4373, 32099}, {4398, 20057}, {4681, 27475}, {4862, 36588}, {20050, 39707}

X(39710) = ISOTOMIC CONJUGATE OF X(3244)

Barycentrics    (a + b - 4*c)*(a - 4*b + c) : :

X(39710) lies on these lines: {2, 4398}, {7, 10944}, {8, 39709}, {27, 19820}, {69, 36606}, {75, 32101}, {86, 1266}, {319, 903}, {335, 4686}, {675, 28218}, {1268, 3663}, {3244, 7321}, {3626, 17273}, {3632, 17361}, {3644, 27475}, {4360, 39704}, {4389, 5936}, {4467, 6548}, {4739, 27483}, {17271, 36588}, {17378, 32105}, {28626, 31995}

X(39711) = ISOTOMIC CONJUGATE OF X(17393)

Barycentrics    (2*a*b + 2*b^2 - a*c + 2*b*c)*(a*b - 2*a*c - 2*b*c - 2*c^2) : :

X(39711) lies on these lines: {10, 7226}, {19, 6763}, {37, 3624}, {38, 39708}, {65, 3632}, {740, 39739}, {969, 17151}, {994, 3901}, {4647, 34860}, {4668, 4674}, {5557, 17299}, {18827, 32104}, {24325, 39737}, {32092, 39717}

X(39712) = ISOTOMIC CONJUGATE OF X(5263)

Barycentrics    (a^2*b + b^3 + a^2*c + a*b*c + a*c^2 + b*c^2)*(a^2*b + a*b^2 + a^2*c + a*b*c + b^2*c + c^3) : :

X(39712) lies on these lines: {1, 16705}, {2, 18785}, {10, 1930}, {19, 10436}, {37, 141}, {38, 39714}, {65, 760}, {82, 86}, {304, 31359}, {1500, 7794}, {1509, 2363}, {2214, 18166}, {2218, 24549}, {3661, 31087}, {3875, 23051}, {4674, 29659}, {6063, 32773}, {8769, 25590}, {16782, 17023}, {17192, 17758}, {24170, 29633}, {24603, 30748}, {30966, 39717}, {34860, 39731}

X(39713) = ISOTOMIC CONJUGATE OF X(31087)

Barycentrics    (a^3*b + a*b^3 - a^3*c - 2*a^2*b*c - 2*a*b^2*c - b^3*c - a*c^3 - b*c^3)*(a^3*b + a*b^3 - a^3*c + 2*a^2*b*c + b^3*c + 2*a*b*c^2 - a*c^3 + b*c^3) : :

X(39713) lies on these lines: {1, 17489}, {2, 18057}, {291, 33076}, {1390, 5263}, {21226, 39722}, {26759, 30701}

X(39714) = ISOTOMIC CONJUGATE OF X(32922)

Barycentrics    (a^2*b + b^3 - a^2*c - a*b*c - a*c^2 + b*c^2)*(a^2*b + a*b^2 - a^2*c + a*b*c - b^2*c - c^3) : :

X(39714) lies on these lines: {1, 17141}, {10, 4986}, {19, 3875}, {37, 3589}, {38, 39712}, {65, 7198}, {75, 24180}, {82, 4360}, {99, 2363}, {239, 18785}, {304, 34860}, {596, 1930}, {876, 23829}, {897, 17160}, {1438, 32029}, {2217, 3905}, {3726, 3912}, {4674, 32847}, {8769, 17151}, {10436, 23051}, {14210, 39697}, {18156, 39702}, {24166, 29674}, {24841, 31637}, {27808, 33941}, {31359, 39731}

X(39715) = ISOTOMIC CONJUGATE OF X(36274)

Barycentrics    b*c*(-(a^4*b^2) + 2*a^3*b^3 - 3*a^2*b^4 + 2*a*b^5 + a^4*b*c + a^3*b^2*c + a^2*b^3*c - 3*a*b^4*c + 2*b^5*c + a^4*c^2 - a^3*b*c^2 - a^2*b^2*c^2 + a*b^3*c^2 - 3*b^4*c^2 - 2*a^3*c^3 - a^2*b*c^3 + a*b^2*c^3 + 2*b^3*c^3 + a^2*c^4 + a*b*c^4 - b^2*c^4)*(-(a^4*b^2) + 2*a^3*b^3 - a^2*b^4 - a^4*b*c + a^3*b^2*c + a^2*b^3*c - a*b^4*c + a^4*c^2 - a^3*b*c^2 + a^2*b^2*c^2 - a*b^3*c^2 + b^4*c^2 - 2*a^3*c^3 - a^2*b*c^3 - a*b^2*c^3 - 2*b^3*c^3 + 3*a^2*c^4 + 3*a*b*c^4 + 3*b^2*c^4 - 2*a*c^5 - 2*b*c^5) : :

X(39715) lies on these lines: {}

X(39716) = ISOTOMIC CONJUGATE OF X(29611)

Barycentrics    (3*a^2 + 2*a*b + 3*b^2 + c^2)*(3*a^2 + b^2 + 2*a*c + 3*c^2) : :

X(39716) lies on these lines: {1, 39749}, {2, 3883}, {7, 4747}, {75, 3618}, {239, 5772}, {335, 26626}, {673, 38048}, {894, 4373}, {903, 35578}, {1268, 17352}, {3616, 27475}, {3662, 30712}, {4307, 29598}, {4384, 5936}, {5308, 36807}, {5435, 7249}, {5550, 20135}, {5905, 39723}, {9776, 39732}, {17367, 39721}, {17368, 24599}, {28626, 29603}

X(39717) = ISOTOMIC CONJUGATE OF X(24325)

Barycentrics    (a*b^2 + a^2*c + 2*a*b*c + b^2*c + a*c^2)*(a^2*b + a*b^2 + 2*a*b*c + a*c^2 + b*c^2) : :

X(39717) lies on these lines: {1, 4093}, {10, 350}, {19, 31905}, {37, 239}, {38, 873}, {65, 1447}, {75, 24450}, {82, 3747}, {83, 3294}, {86, 13476}, {274, 596}, {756, 3112}, {870, 984}, {876, 3005}, {1581, 32010}, {3875, 17038}, {4674, 27922}, {5276, 16514}, {6654, 18785}, {9278, 25368}, {10436, 39742}, {12263, 19856}, {17144, 31359}, {17322, 25347}, {17393, 39737}, {17394, 39739}, {18833, 35544}, {21264, 29610}, {30966, 39712}, {31997, 34860}, {32092, 39711}, {32104, 39708}

X(39717) = isogonal conjugate of X(20985)
X(39717) = isotomic conjugate of X(24325)
X(39717) = X(19)-isoconjugate of X(22099)

X(39718) = ISOTOMIC CONJUGATE OF X(13174)

Barycentrics    (a^4*b + 2*a^3*b^2 + a^2*b^3 - 2*a*b^4 - b^5 + 2*a^2*b^2*c - 2*b^4*c - 2*a^3*c^2 - 3*a^2*b*c^2 + 2*a*b^2*c^2 + b^3*c^2 - 2*a^2*c^3 + 2*b^2*c^3 + b*c^4)*(-2*a^3*b^2 - 2*a^2*b^3 + a^4*c - 3*a^2*b^2*c + b^4*c + 2*a^3*c^2 + 2*a^2*b*c^2 + 2*a*b^2*c^2 + 2*b^3*c^2 + a^2*c^3 + b^2*c^3 - 2*a*c^4 - 2*b*c^4 - c^5) : :

X(39718) lies on these lines: {594, 35960}, {740, 13174}, {4037, 6542}

X(39719) = ISOTOMIC CONJUGATE OF X(39367)

Barycentrics    (a^4*b^2 + a^3*b^3 - a^2*b^4 + a^4*b*c + 3*a^3*b^2*c - a^2*b^3*c - 3*a*b^4*c - a^4*c^2 - a^3*b*c^2 + 2*a^2*b^2*c^2 - a*b^3*c^2 - b^4*c^2 - 3*a^3*c^3 - a^2*b*c^3 + 3*a*b^2*c^3 + b^3*c^3 - a^2*c^4 + a*b*c^4 + b^2*c^4)*(a^4*b^2 + 3*a^3*b^3 + a^2*b^4 - a^4*b*c + a^3*b^2*c + a^2*b^3*c - a*b^4*c - a^4*c^2 - 3*a^3*b*c^2 - 2*a^2*b^2*c^2 - 3*a*b^3*c^2 - b^4*c^2 - a^3*c^3 + a^2*b*c^3 + a*b^2*c^3 - b^3*c^3 + a^2*c^4 + 3*a*b*c^4 + b^2*c^4) : :

X(39719) lies on these lines: {740, 39367}, {1757, 13174}, {1931, 2669}, {3842, 24505}, {6650, 9510}

X(39720) = ISOTOMIC CONJUGATE OF X(20055)

Barycentrics    (2*a^2 + 3*a*b + 2*b^2 - a*c - b*c - 2*c^2)*(2*a^2 - a*b - 2*b^2 + 3*a*c - b*c + 2*c^2) : :

X(39720) lies on these lines: {2, 32857}, {75, 4410}, {86, 17323}, {673, 20158}, {1268, 4363}, {4373, 20016}, {5936, 6646}, {17351, 28650}, {17483, 39741}, {20055, 24692}, {26806, 28626}, {30712, 33869}

X(39721) = ISOTOMIC CONJUGATE OF X(17316)

Barycentrics    (a^2 + b^2 - 2*a*c - 2*b*c - c^2)*(a^2 - 2*a*b - b^2 - 2*b*c + c^2) : :

X(39721) lies on these lines: {1, 33039}, {2, 968}, {7, 193}, {8, 335}, {27, 1851}, {75, 966}, {86, 4000}, {329, 39741}, {673, 5698}, {675, 28847}, {903, 17346}, {962, 16827}, {1088, 10030}, {1268, 2345}, {1722, 32971}, {2550, 17316}, {3661, 39749}, {3931, 33026}, {4366, 16020}, {4373, 6646}, {4384, 24248}, {4440, 27484}, {5222, 14621}, {5296, 5936}, {5530, 33037}, {5905, 8049}, {6548, 27486}, {6650, 16816}, {7249, 17490}, {7318, 17077}, {9776, 17027}, {9791, 27483}, {16706, 30598}, {16833, 32857}, {17014, 26806}, {17289, 28650}, {17321, 20181}, {17367, 39716}, {17397, 28626}, {20162, 38053}, {26582, 29579}, {27478, 36479}, {27494, 31302}, {31317, 31995}, {37756, 39704}

X(39722) = ISOTOMIC CONJUGATE OF X(17302)

Barycentrics    (a^2 + a*b + b^2 - a*c + b*c + c^2)*(a^2 - a*b + b^2 + a*c + b*c + c^2) : :

X(39722) lies on these lines: {1, 3790}, {2, 21021}, {8, 985}, {10, 1929}, {28, 17927}, {57, 3661}, {75, 39724}, {81, 2295}, {88, 29591}, {105, 5260}, {274, 3963}, {277, 4699}, {330, 2345}, {344, 39738}, {594, 6645}, {1255, 29586}, {1432, 2171}, {2306, 36929}, {3954, 17946}, {4000, 39730}, {4071, 6625}, {4099, 34914}, {4692, 7797}, {7261, 17741}, {8056, 17308}, {17358, 30701}, {17397, 25430}, {17743, 33169}, {21226, 39713}, {21907, 31025}, {25417, 29588}, {26801, 27005}, {33654, 36928}

X(39723) = ISOTOMIC CONJUGATE OF X(33090)

Barycentrics    (a^3 + a^2*b + a*b^2 + b^3 - a^2*c + a*b*c - b^2*c + a*c^2 + b*c^2 - c^3)*(a^3 - a^2*b + a*b^2 - b^3 + a^2*c + a*b*c + b^2*c + a*c^2 - b*c^2 + c^3) : :

X(39723) lies on these lines: {1, 39728}, {2, 17744}, {7, 29815}, {75, 33091}, {335, 26842}, {673, 33150}, {4373, 20020}, {5905, 39716}, {7198, 15246}, {7264, 37349}, {14621, 17483}, {17024, 39732}, {29838, 39743}

X(39724) = ISOTOMIC CONJUGATE OF X(17280)

Barycentrics    (a^2 + a*b + b^2 - a*c - b*c + c^2)*(a^2 - a*b + b^2 + a*c - b*c + c^2) : :

X(39724) lies on these lines: {1, 2896}, {2, 16720}, {7, 7132}, {28, 31917}, {57, 7185}, {75, 39722}, {81, 21997}, {105, 5253}, {192, 30701}, {274, 27349}, {277, 27340}, {291, 24443}, {330, 4000}, {1031, 39726}, {1086, 6645}, {1111, 7797}, {1255, 18139}, {1432, 17086}, {1930, 17280}, {2345, 39729}, {3020, 10799}, {3329, 3665}, {4376, 19689}, {7187, 16706}, {7195, 16989}, {17244, 25430}, {17321, 39738}, {19785, 39694}, {20913, 30710}, {25499, 32009}, {26978, 30669}, {28604, 33933}, {33150, 35058}, {33155, 39698}

X(39725) = ISOTOMIC CONJUGATE OF X(20934)

Barycentrics    a*(a^4 + a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - c^4)*(a^4 + a^2*b^2 - b^4 + a^2*c^2 + b^2*c^2 + c^4) : :

X(39725) lies on these lines: {31, 17957}, {38, 1582}, {1031, 4388}, {1930, 1965}, {3496, 3954}

X(39726) = ISOTOMIC CONJUGATE OF X(21289)

Barycentrics    (a^5 + a^3*b^2 + a^2*b^3 + b^5 - a^2*b^2*c + a^3*c^2 + a^2*b*c^2 + a*b^2*c^2 + b^3*c^2 - a^2*c^3 - b^2*c^3 - c^5)*(a^5 + a^3*b^2 - a^2*b^3 - b^5 + a^2*b^2*c + a^3*c^2 - a^2*b*c^2 + a*b^2*c^2 - b^3*c^2 + a^2*c^3 + b^2*c^3 + c^5) : :

X(39726) lies on these lines: {1031, 39724}, {1369, 20934}, {1930, 21289}, {2896, 16566}

X(39726) = isogonal conjugate of X(20994)
X(39726) = isotomic conjugate of X(21289)
X(39726) = X(19)-isoconjugate of X(22137)

X(39727) = ISOTOMIC CONJUGATE OF X(21378)

Barycentrics    b*c*(-a^6 - a^4*b^2 + a^2*b^4 + b^6 - a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 - b^2*c^4 - c^6)*(a^6 + a^4*b^2 + a^2*b^4 + b^6 + a^4*c^2 + a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 - b^2*c^4 - c^6) : :

X(39727) lies on these lines: {1930, 16546}, {20916, 21598}, {20933, 21064}, {21289, 33091}

X(39728) = ISOTOMIC CONJUGATE OF X(33091)

Barycentrics    (a^3 + a^2*b + a*b^2 + b^3 - a^2*c - a*b*c - b^2*c + a*c^2 + b*c^2 - c^3)*(a^3 - a^2*b + a*b^2 - b^3 + a^2*c - a*b*c + b^2*c + a*c^2 - b*c^2 + c^3) : : f

X(39728) lies on these lines: {1, 39723}, {2, 16555}, {7, 17024}, {75, 1369}, {335, 17483}, {1111, 37349}, {1930, 33091}, {3598, 7318}, {3665, 6636}, {4373, 19993}, {5905, 39749}, {7195, 7394}, {8049, 33865}, {14621, 26842}, {17170, 29815}, {18139, 36807}

X(39729) = ISOTOMIC CONJUGATE OF X(17383)

Barycentrics    (2*a^2 + a*b + 2*b^2 - a*c + b*c + 2*c^2)*(2*a^2 - a*b + 2*b^2 + a*c + b*c + 2*c^2) : :

X(39729) lies on these lines: {1, 17268}, {57, 17292}, {75, 39730}, {81, 17230}, {277, 28604}, {330, 17289}, {2345, 39724}, {17279, 39738}, {17303, 39736}, {25430, 29614}

X(39730) = ISOTOMIC CONJUGATE OF X(17358)

Barycentrics    (2*a^2 + a*b + 2*b^2 - a*c - b*c + 2*c^2)*(2*a^2 - a*b + 2*b^2 + a*c - b*c + 2*c^2) : :

X(39730) lies on these lines: {1, 17232}, {57, 29630}, {75, 39729}, {330, 16706}, {1255, 29572}, {4000, 39722}, {4657, 39738}, {7132, 7225}, {17266, 25430}, {17278, 39736}, {17302, 30701}, {19786, 39703}, {27268, 32019}, {27340, 34578}, {32774, 39694}

X(39731) = ISOTOMIC CONJUGATE OF X(23051)

Barycentrics    b*c*(3*a^2 + b^2 + c^2) : :

X(39731) lies on these lines: {1, 75}, {2, 30701}, {8, 26234}, {63, 2179}, {69, 3555}, {85, 10106}, {92, 1973}, {145, 20911}, {306, 19798}, {312, 17023}, {315, 4514}, {321, 26626}, {336, 17883}, {341, 18140}, {350, 4385}, {519, 33945}, {551, 33942}, {894, 16502}, {942, 21281}, {1015, 25918}, {1107, 24357}, {1125, 30758}, {1219, 3672}, {1496, 4592}, {1698, 4986}, {1895, 1969}, {1909, 3673}, {1953, 18049}, {2172, 18042}, {2275, 24326}, {3057, 24282}, {3244, 33936}, {3263, 3616}, {3290, 27299}, {3501, 24631}, {3622, 31130}, {3674, 39126}, {3718, 17321}, {3739, 27248}, {3757, 37580}, {3758, 5299}, {3759, 5280}, {3760, 4692}, {3761, 7264}, {3868, 17152}, {3869, 17141}, {3873, 17137}, {3889, 30941}, {3912, 19804}, {4020, 19591}, {4123, 21609}, {4352, 17480}, {4359, 17316}, {4363, 16781}, {4441, 4968}, {4561, 19861}, {4687, 16818}, {4696, 18135}, {4737, 6376}, {4812, 21417}, {5045, 30962}, {5173, 6604}, {8772, 20904}, {12721, 24349}, {16284, 33934}, {16583, 26274}, {16739, 33297}, {16787, 17789}, {16788, 20927}, {16969, 35274}, {17140, 18659}, {17277, 17742}, {17322, 19784}, {17335, 17744}, {17341, 32019}, {17442, 17868}, {18689, 31637}, {18743, 29598}, {19836, 28653}, {20171, 21418}, {20947, 29646}, {21582, 34065}, {21808, 30036}, {24524, 33944}, {24589, 29579}, {24995, 33120}, {25303, 33930}, {26242, 26965}, {26807, 31087}, {29633, 30963}, {31359, 39714}, {34860, 39712}

X(39732) = ISOTOMIC CONJUGATE OF X(10327)

Barycentrics    (a^3 + a^2*b + a*b^2 + b^3 - a^2*c - 2*a*b*c - b^2*c + a*c^2 + b*c^2 - c^3)*(a^3 - a^2*b + a*b^2 - b^3 + a^2*c - 2*a*b*c + b^2*c + a*c^2 - b*c^2 + c^3) : :

The trilinear polar of X(39732) meets the line at infinity at X(514).

X(39732) lies on these lines: {2, 169}, {7, 7191}, {22, 348}, {25, 1565}, {75, 1370}, {85, 6997}, {86, 4228}, {279, 6995}, {304, 10327}, {310, 14956}, {329, 39749}, {335, 5905}, {673, 24584}, {1440, 3598}, {1447, 7318}, {1851, 23989}, {2400, 20295}, {3827, 13577}, {4295, 8049}, {5088, 7500}, {5936, 29667}, {7219, 13575}, {9776, 39716}, {10431, 14268}, {10446, 39695}, {17024, 39723}, {18043, 35516}, {18134, 36807}, {21453, 33949}, {28626, 29666}, {33867, 39741}

X(39732) = isogonal conjugate of X(12329)
X(39732) = isotomic conjugate of X(10327)
X(39732) = anticomplement of X(15487)

X(39733) = ISOTOMIC CONJUGATE OF X(18596)

Barycentrics    b*c*(a^6 - a^4*b^2 - a^2*b^4 + b^6 + a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 - b^2*c^4 - c^6)*(-a^6 - a^4*b^2 + a^2*b^4 + b^6 + a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 - b^2*c^4 - c^6) : :

X(39733) lies on these lines: {304, 1760}, {336, 33808}, {1231, 21215}, {4150, 20914}, {4329, 4463}, {20641, 21582}

X(39734) = ISOTOMIC CONJUGATE OF X(4651)

Barycentrics    (a + b)*(a + c)*(a*b - b^2 + a*c + b*c)*(a*b + a*c + b*c - c^2) : :

X(39734) lies on these lines: {1, 8049}, {2, 2350}, {7, 29814}, {42, 17205}, {75, 3873}, {81, 673}, {86, 5284}, {226, 17198}, {274, 4651}, {310, 29824}, {335, 3995}, {354, 16727}, {1088, 14956}, {1246, 3945}, {1268, 30966}, {1434, 4184}, {2296, 16705}, {3664, 20028}, {4210, 14828}, {6650, 26842}, {7192, 18165}, {8025, 14621}, {16738, 27152}, {17018, 18600}, {17139, 39704}, {17165, 18157}, {17177, 31019}, {17183, 30712}, {18164, 30949}, {20295, 33097}, {26813, 26815}, {28626, 30946}, {29822, 33947}, {31002, 31008}

X(39735) = ISOTOMIC CONJUGATE OF X(16552)

Barycentrics    b*c*(-(a^2*b) + a*b^2 - a^2*c + b^2*c - a*c^2 - b*c^2)*(a^2*b + a*b^2 + a^2*c + b^2*c - a*c^2 - b*c^2) : :

X(39735) lies on these lines: {75, 3681}, {76, 4043}, {85, 4687}, {274, 16552}, {286, 14004}, {334, 18133}, {767, 6577}, {870, 16684}, {872, 1111}, {2481, 4360}, {4828, 27471}, {6385, 18137}, {7199, 24220}

X(39736) = ISOTOMIC CONJUGATE OF X(27268)

Barycentrics    (3*a*b + a*c + 3*b*c)*(a*b + 3*a*c + 3*b*c) : :

X(39736) lies on these lines: {1, 4699}, {2, 32107}, {57, 16815}, {75, 39738}, {81, 16816}, {192, 32009}, {274, 32005}, {291, 9780}, {330, 3739}, {1002, 3617}, {1107, 39740}, {1255, 29595}, {1258, 27623}, {4740, 31996}, {4772, 32095}, {5217, 16994}, {5550, 30571}, {16819, 36871}, {16832, 36603}, {17278, 39730}, {17303, 39729}, {17358, 32019}, {17490, 37870}, {19862, 30998}, {19872, 31276}, {25430, 29578}, {27248, 34892}, {27268, 32092}, {27342, 34578}, {28604, 30701}, {31993, 39703}, {31997, 38247}

X(39737) = ISOTOMIC CONJUGATE OF X(32092)

Barycentrics    a*(2*a*b + b^2 + 2*a*c + 2*b*c)*(2*a*b + 2*a*c + 2*b*c + c^2) : :

X(39737) lies on these lines: {10, 4687}, {37, 3681}, {38, 39739}, {65, 15569}, {75, 1962}, {740, 39708}, {749, 25092}, {968, 969}, {1255, 15624}, {1621, 2214}, {2667, 17038}, {3247, 18785}, {3869, 31503}, {4751, 17592}, {13476, 28606}, {17393, 39717}, {17394, 18827}, {24325, 39711}

X(39738) = ISOTOMIC CONJUGATE OF X(4699)

Barycentrics    (3*a*b + a*c + b*c)*(a*b + 3*a*c + b*c) : :}

X(39738) lies on these lines: {1, 4991}, {2, 17144}, {8, 30571}, {37, 330}, {57, 16826}, {75, 39736}, {81, 2176}, {88, 29595}, {105, 29199}, {192, 274}, {194, 36871}, {239, 25430}, {277, 17302}, {291, 3616}, {344, 39722}, {869, 9401}, {985, 3915}, {1002, 3622}, {1107, 38247}, {1255, 4393}, {1258, 16969}, {3227, 5283}, {3303, 16993}, {4657, 39730}, {4664, 25130}, {4699, 31996}, {4704, 31997}, {8056, 16831}, {16672, 34063}, {16975, 32090}, {17279, 39729}, {17321, 39724}, {20257, 29581}, {27076, 32020}, {27255, 30998}

X(39739) = ISOTOMIC CONJUGATE OF X(32104)

Barycentrics    a*(2*a*b - b^2 + 2*a*c + 2*b*c)*(2*a*b + 2*a*c + 2*b*c - c^2) : :

X(39739) lies on these lines: {10, 4684}, {37, 3873}, {38, 39737}, {65, 17092}, {740, 39711}, {1449, 18785}, {2667, 39742}, {17393, 18827}, {17394, 39717}, {24325, 39708}

X(39740) = ISOTOMIC CONJUGATE OF X(4704)

Barycentrics    (a*b - 3*a*c - 3*b*c)*(3*a*b - a*c + 3*b*c) : :

X(39740) lies on these lines: {1, 1278}, {2, 20943}, {57, 16816}, {75, 38247}, {194, 32009}, {291, 3617}, {330, 4772}, {959, 1463}, {1002, 3621}, {1107, 39736}, {4384, 36603}, {4704, 31997}, {4821, 31999}, {5204, 16996}, {5550, 20081}, {8056, 16815}, {9780, 21219}, {16722, 39747}, {25430, 29595}, {27076, 27318}, {27304, 34578}, {27789, 30562}, {28365, 36647}, {32020, 34284}

X(39741) = ISOTOMIC CONJUGATE OF X(10453)

Barycentrics    (a^2*b + a*b^2 + a^2*c - a*b*c + b^2*c - a*c^2 - b*c^2)*(a^2*b - a*b^2 + a^2*c - a*b*c - b^2*c + a*c^2 + b*c^2) : :

X(39741) lies on these lines: {2, 1334}, {7, 42}, {8, 310}, {27, 607}, {43, 17753}, {55, 86}, {65, 1088}, {75, 210}, {273, 1824}, {329, 39721}, {335, 3210}, {346, 29976}, {673, 4383}, {1240, 4441}, {1440, 2357}, {3240, 8049}, {3666, 27475}, {4373, 20347}, {5905, 6650}, {5936, 26037}, {6384, 10453}, {9776, 17032}, {16059, 24203}, {17018, 18600}, {17483, 39720}, {18815, 34857}, {20012, 36854}, {20348, 27442}, {24190, 27264}, {27479, 27483}, {33867, 39732}

X(39741) = isogonal conjugate of X(20992)
X(39741) = isotomic conjugate of X(10453)
X(39741) = polar conjugate of X(17920)
X(39741) = X(19)-isoconjugate of X(22127)

X(39742) = ISOTOMIC CONJUGATE OF X(17144)

Barycentrics    a*(a*b - 2*b^2 + a*c + b*c)*(a*b + a*c + b*c - 2*c^2) : :

X(39742) lies on these lines: {1, 17207}, {10, 3662}, {19, 32913}, {37, 982}, {38, 17038}, {65, 4334}, {87, 20358}, {537, 30090}, {740, 34860}, {876, 4139}, {1215, 4751}, {1278, 4365}, {1743, 18785}, {2667, 39739}, {3551, 23633}, {3875, 18827}, {3894, 4674}, {4090, 27343}, {10436, 39717}, {24325, 31359}

X(39742) = isogonal conjugate of X(8616)
X(39742) = isotomic conjugate of X(17144)

X(39743) = ISOTOMIC CONJUGATE OF X(31079)

Barycentrics    (2*a^3 + a^2*b + a*b^2 + 2*b^3 - a^2*c - b^2*c + a*c^2 + b*c^2)*(2*a^3 - a^2*b + a*b^2 + a^2*c + b^2*c + a*c^2 - b*c^2 + 2*c^3) : :

X(39743) lies on these lines: {7, 29831}, {75, 20045}, {1268, 37670}, {29838, 39723}, {36807, 37633}

X(39744) = ISOTOMIC CONJUGATE OF X(17145)

Barycentrics    (a^2*b + a*b^2 + 3*a^2*c - 2*a*b*c + 3*b^2*c - 3*a*c^2 - 3*b*c^2)*(3*a^2*b - 3*a*b^2 + a^2*c - 2*a*b*c - 3*b^2*c + a*c^2 + 3*b*c^2) : :

X(39744) lies on these lines: {1621, 4436}, {3294, 16815}

X(39744) = isogonal conjugate of X(16694)
X(39744) = isotomic conjugate of X(17145)

X(39745) = ISOTOMIC CONJUGATE OF X(26801)

Barycentrics    (a^2*b^2 + a^2*b*c + a*b^2*c + a^2*c^2 - a*b*c^2 + b^2*c^2)*(a^2*b^2 + a^2*b*c - a*b^2*c + a^2*c^2 + a*b*c^2 + b^2*c^2) : :

X(39745) lies on these lines: {1, 39746}, {2, 21803}, {7, 3503}, {75, 21021}, {86, 2295}, {192, 871}, {310, 3963}, {673, 20148}, {2171, 7249}, {2209, 14621}, {2276, 6384}, {7148, 27447}, {17445, 26801}, {26048, 27483}

X(39746) = ISOTOMIC CONJUGATE OF X(26752)

Barycentrics    (a^2*b^2 - a^2*b*c + a*b^2*c + a^2*c^2 - a*b*c^2 + b^2*c^2)*(a^2*b^2 - a^2*b*c - a*b^2*c + a^2*c^2 + a*b*c^2 + b^2*c^2) : :

X(39746) lies on these lines: {1, 39745}, {2, 3510}, {7, 3495}, {75, 2275}, {86, 26149}, {335, 21330}, {673, 24610}, {1964, 26752}, {7184, 26959}, {8049, 26815}, {26113, 27475}, {26963, 30669}, {30955, 31002}, {32011, 37686}

X(39747) = ISOTOMIC CONJUGATE OF X(3995)

Barycentrics    (a + b)*(a + c)*(a*b + b^2 - a*c + b*c)*(a*b - a*c - b*c - c^2) : :

X(39747) lies on these lines: {1, 596}, {2, 3770}, {57, 16704}, {75, 26819}, {81, 17495}, {86, 1255}, {88, 333}, {89, 17490}, {105, 34594}, {192, 27789}, {291, 4651}, {310, 32020}, {321, 16726}, {957, 9965}, {959, 20615}, {1002, 20011}, {1022, 4560}, {1219, 19825}, {1224, 19874}, {1790, 2224}, {3187, 18164}, {3210, 25417}, {4080, 17182}, {4671, 39703}, {5271, 18186}, {15474, 16713}, {16602, 16729}, {16722, 39740}, {17165, 18792}, {17175, 31036}, {17178, 28605}, {17946, 26840}, {18206, 19742}, {19789, 26818}, {19804, 39706}, {20913, 27005}, {24620, 26745}, {24621, 37685}, {25430, 31035}, {27163, 30599}

X(39748) = ISOTOMIC CONJUGATE OF X(18133)

Barycentrics    a*(a^2*b + a*b^2 - a^2*c - b^2*c - a*c^2 - b*c^2)*(a^2*b + a*b^2 - a^2*c + b^2*c - a*c^2 + b*c^2) : :

X(39748) lies on these lines: {1, 3159}, {3, 34463}, {6, 3293}, {56, 1724}, {58, 404}, {75, 39693}, {86, 13741}, {87, 1698}, {106, 2975}, {191, 9359}, {238, 34444}, {292, 16552}, {405, 3445}, {596, 8054}, {964, 996}, {978, 2163}, {979, 3679}, {1089, 3248}, {1474, 4222}, {1743, 2215}, {3226, 17143}, {4040, 23355}, {4234, 5331}, {6048, 36598}, {16297, 18792}, {16569, 36602}, {19742, 36604}, {21173, 23345}

X(39748) = isogonal conjugate of X(3216)
X(39748) = isotomic conjugate of X(18133)

X(39749) = ISOTOMIC CONJUGATE OF X(5222)

Barycentrics    (a^2 + 3*b^2 - 2*a*c + c^2)*(a^2 - 2*a*b + b^2 + 3*c^2) : :

X(39749) lies on these lines: {1, 39716}, {2, 3677}, {7, 346}, {8, 673}, {27, 34255}, {75, 3619}, {86, 344}, {273, 1229}, {312, 1088}, {329, 39732}, {335, 29579}, {345, 21453}, {675, 5744}, {894, 29621}, {1016, 39293}, {1440, 28739}, {1462, 4513}, {3661, 39721}, {3662, 4373}, {3886, 29616}, {4082, 30813}, {4307, 29573}, {4310, 10322}, {4344, 14621}, {4384, 10005}, {5226, 7249}, {5905, 39728}, {5936, 17308}, {6650, 17230}, {7155, 27498}, {7318, 28780}, {15590, 36807}, {16831, 28626}, {17263, 30598}, {17264, 35578}, {17738, 30332}, {24349, 27475}, {27494, 29587}

X(39750) = MIDPOINT OF X(32) AND X(182)

Barycentrics    a^2*(a^8 - a^6*b^2 - a^6*c^2 - 4*a^4*b^2*c^2 - a^2*b^4*c^2 - b^6*c^2 - a^2*b^2*c^4 - b^2*c^6) : :
Barycentrics    Sin[A]^2*(Cot[A] + Tan[3*ω]) : :
Trilinears    cos(A - ω) (2 sin A sin(A - ω) + sin B sin(B - ω) + sin C sin(C - ω)) + sin(A - ω) (sin B cos(B - ω) + sin C cos(C - ω)) : :
X(39750) = X[5017] + 3 X[5050], 3 X[14561] + X[36998]

X(39750) lies on these lines: {3, 6}, {83, 3406}, {98, 3407}, {140, 39603}, {315, 10359}, {699, 39629}, {736, 8177}, {754, 10168}, {882, 39214}, {1353, 13196}, {1503, 20576}, {2794, 10796}, {3117, 38880}, {3506, 11328}, {3972, 12176}, {4048, 18806}, {5182, 33246}, {5476, 12150}, {5480, 32134}, {6680, 10104}, {7503, 8152}, {7787, 14561}, {7876, 10350}, {7892, 34507}, {10131, 32467}, {10335, 36849}, {10788, 31670}, {11261, 21445}, {12192, 32305}, {12201, 32273}, {12208, 19150}, {13193, 19140}, {14880, 29012}, {15917, 33695}, {18502, 36997}, {33225, 39141}, {34396, 36213}

X(39750) = midpoint of X(i) and X(j) for these {i,j}: {3, 35431}, {6, 35424}, {32, 182}, {576, 35387}, {2456, 35379}, {3098, 35389}, {35375, 35388}, {35385, 35426}
X(39750) = reflection of X(24206) in X(6680)
X(39750) = 1st-Lemoine-circle inverse of X(35377)
X(39750) = center of circle {{X(32),X(182),PU(1)}}
X(39750) = harmonic center of circles {{X(15),X(62),PU(1)}} and {{X(16),X(61),PU(1)}}
X(39750) = harmonic center of circles {{X(3364),X(3365),PU(1)}} and {{X(3389),X(3390),PU(1)}}
X(39750) = circle-{{X(371),X(372),PU(1),PU(39)}}-inverse of X(32452)
X(39750) = circle-{{X(1687),X(1688),PU(1),PU(2)}}-inverse of X(3094)
X(39750) = center of circle-{{X(1687),X(1688),PU(1),PU(2)}}-inverse-of-Brocard-circle
X(39750) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {32, 2458, 5017}, {32, 35387, 35379}, {32, 35389, 35428}, {182, 575, 39515}, {182, 576, 2456}, {182, 17508, 12054}, {182, 39561, 5038}, {371, 372, 32452}, {1342, 1343, 3095}, {1662, 1663, 35377}, {1687, 1688, 3094}, {1691, 3398, 182}, {2456, 12212, 576}, {5017, 39560, 2458}, {5050, 39560, 182}, {35766, 35767, 32}

X(39751) = MIDPOINT OF X(1325) AND X(38514)

Barycentrics    (-a + b - c)*(a + b - c)*(b + c)*(-2*a^4 - a^3*b + a^2*b^2 + a*b^3 + b^4 - a^3*c + a^2*c^2 - 2*b^2*c^2 + a*c^3 + c^4) : :

X(39751) lies on the cubic K1160 and these lines: {1, 30}, {11, 25641}, {12, 502}, {55, 36001}, {56, 1325}, {225, 37982}, {278, 2074}, {517, 3028}, {523, 656}, {858, 29664}, {1068, 37979}, {1284, 37919}, {1290, 5172}, {1319, 1365}, {1358, 6023}, {1388, 36171}, {1465, 5520}, {2070, 14667}, {5127, 6357}, {5620, 18593}, {7424, 15950}, {7426, 29681}, {10208, 15888}, {10944, 36154}, {11700, 15326}, {13407, 32167}, {37080, 37992}

X(39751) = midpoint of X(1325) and X(38514)
X(39751) = reflection of X(i) in X(j) for these {i,j}: {10149, 16272}, {31524, 1319}
X(39751) = reflection of X(10149) in the Orthic axis
X(39751) = incircle-inverse of X(3649)
X(39751) = Conway-circle-inverse of X(16124)
X(39751) = crossdifference of every pair of points on line {284, 9404}
X(39751) = X(1157)-of-intouch-triangle
X(39751) = X(1325)-of-2nd-anti-circumperp-tangential-triangle
X(39751) = barycentric product X(1441)*X(15586)
X(39751) = barycentric quotient X(15586)/X(21)

X(39752) = X(1)X(3667)∩X(214)X(519)

Barycentrics    (2*a - b - c)^2*(a + b - c)*(a - b + c)*(a^3 - 2*a^2*b - 3*a*b^2 - 2*a^2*c + 11*a*b*c - b^2*c - 3*a*c^2 - b*c^2) : :

X(39752) lies on the cubic K1160 and these lines: {1,3667}, {214,519}, {1210,13625}, {1388,6789}, {11700,37743}, {14584,23869}

X(39753) = X(1)X(2827)∩X(106)X(23703)

Barycentrics    a*(2*a - b - c)*(a + b - c)*(a - b + c)*(a^3 - 2*a^2*b - 3*a*b^2 - 2*a^2*c + 11*a*b*c - b^2*c - 3*a*c^2 - b*c^2)*(a^2*b - b^3 + a^2*c - 4*a*b*c + 2*b^2*c + 2*b*c^2 - c^3) : :

X(39753) lies on the cubic K1160 and these lines: {1, 2827}, {106, 23703}, {515, 37743}, {1319, 1357}, {4551, 10700}

X(39754) = X(1)X(3309)∩X(241)X(518)

Barycentrics    a*(a + b - c)*(a - b + c)*(a*b - b^2 + a*c - c^2)*(2*a^5 - 5*a^4*b + 7*a^3*b^2 - 7*a^2*b^3 + 3*a*b^4 - 5*a^4*c + 4*a^3*b*c - 4*a*b^3*c + b^4*c + 7*a^3*c^2 + 6*a*b^2*c^2 - b^3*c^2 - 7*a^2*c^3 - 4*a*b*c^3 - b^2*c^3 + 3*a*c^4 + b*c^4) : :

X(39754) lies on the cubic K1160 and these lines: {1, 3309}, {241, 518}, {1317, 1323}, {1319, 1362}, {6603, 35505}

X(39754) = crossdifference of every pair of points on line {1024, 2348}

X(39755) = X(1)X(2826)∩X(105)X(2283)

Barycentrics    (a + b - c)*(a - b + c)*(2*a^3 - 2*a^2*b + a*b^2 - b^3 - 2*a^2*c + b^2*c + a*c^2 + b*c^2 - c^3)*(2*a^5 - 5*a^4*b + 7*a^3*b^2 - 7*a^2*b^3 + 3*a*b^4 - 5*a^4*c + 4*a^3*b*c - 4*a*b^3*c + b^4*c + 7*a^3*c^2 + 6*a*b^2*c^2 - b^3*c^2 - 7*a^2*c^3 - 4*a*b*c^3 - b^2*c^3 + 3*a*c^4 + b*c^4)

X(39755) lies on the cubic K1160 and these lines: {1, 2826}, {105, 2283}, {515, 38386}, {528, 5723}, {1317, 15730}, {1319, 1323}

X(39756) = X(1)X(513)∩X(515)X(1317)

Barycentrics    a*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3)*(2*a^6 - 3*a^5*b - 4*a^4*b^2 + 6*a^3*b^3 + 2*a^2*b^4 - 3*a*b^5 - 3*a^5*c + 14*a^4*b*c - 7*a^3*b^2*c - 13*a^2*b^3*c + 10*a*b^4*c - b^5*c - 4*a^4*c^2 - 7*a^3*b*c^2 + 22*a^2*b^2*c^2 - 7*a*b^3*c^2 + 6*a^3*c^3 - 13*a^2*b*c^3 - 7*a*b^2*c^3 + 2*b^3*c^3 + 2*a^2*c^4 + 10*a*b*c^4 - 3*a*c^5 - b*c^5) : :

X(39756) lies on the cubic K1160 and these lines: {1, 513}, {515, 1317}, {517, 1457}, {651, 14511}, {1319, 1361}, {1877, 21664}, {15730, 33902}, {32486, 33646}

X(39757) = X(1)X(514)∩X(515)X(15730)

Barycentrics    (2*a^3 - a^2*b - b^3 - a^2*c + b^2*c + b*c^2 - c^3)*(2*a^7 - 3*a^6*b + a^5*b^2 - 2*a^4*b^3 + 5*a^2*b^5 - 3*a*b^6 - 3*a^6*c + 4*a^5*b*c + a^4*b^2*c - 5*a^2*b^4*c + 4*a*b^5*c - b^6*c + a^5*c^2 + a^4*b*c^2 - 5*a*b^4*c^2 + 3*b^5*c^2 - 2*a^4*c^3 + 8*a*b^3*c^3 - 2*b^4*c^3 - 5*a^2*b*c^4 - 5*a*b^2*c^4 - 2*b^3*c^4 + 5*a^2*c^5 + 4*a*b*c^5 + 3*b^2*c^5 - 3*a*c^6 - b*c^6) : :

X(39757) lies on the cubic K1160 and these lines: {1, 514}, {515, 15730}, {516, 1456}, {1319, 1360}, {1323, 11700}, {1566, 23972}, {3234, 9502}

X(39758) = X(1)X(900)∩X(11)X(515)

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)*(2*a^6 - 3*a^5*b - 4*a^4*b^2 + 6*a^3*b^3 + 2*a^2*b^4 - 3*a*b^5 - 3*a^5*c + 14*a^4*b*c - 7*a^3*b^2*c - 13*a^2*b^3*c + 10*a*b^4*c - b^5*c - 4*a^4*c^2 - 7*a^3*b*c^2 + 22*a^2*b^2*c^2 - 7*a*b^3*c^2 + 6*a^3*c^3 - 13*a^2*b*c^3 - 7*a*b^2*c^3 + 2*b^3*c^3 + 2*a^2*c^4 + 10*a*b*c^4 - 3*a*c^5 - b*c^5) : :

X(39758) lies on the cubic K1160 and these lines: {1, 900}, {11, 515}, {104, 23981}, {1317, 11700}, {1323, 33902}, {1387, 35015}, {2804, 36944}, {15524, 37743}

X(39758) = reflection of X(35015) in X(1387)

X(39759) = X(1)X(2254)∩X(104)X(4845)

Barycentrics    a*(a^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5 + a^4*c + a^2*b^2*c - 2*a*b^3*c - 2*a^3*c^2 + a^2*b*c^2 + b^3*c^2 - 2*a*b*c^3 + b^2*c^3 + 2*a*c^4 - c^5)*(2*a^7 - 3*a^6*b + a^5*b^2 - 2*a^4*b^3 + 5*a^2*b^5 - 3*a*b^6 - 3*a^6*c + 4*a^5*b*c + a^4*b^2*c - 5*a^2*b^4*c + 4*a*b^5*c - b^6*c + a^5*c^2 + a^4*b*c^2 - 5*a*b^4*c^2 + 3*b^5*c^2 - 2*a^4*c^3 + 8*a*b^3*c^3 - 2*b^4*c^3 - 5*a^2*b*c^4 - 5*a*b^2*c^4 - 2*b^3*c^4 + 5*a^2*c^5 + 4*a*b*c^5 + 3*b^2*c^5 - 3*a*c^6 - b*c^6) : :

X(39759) lies on the cubic K1160 and these lines: {1, 2254}, {104, 4845}, {515, 1323}, {1319, 3022}, {3675, 11028}, {11700, 15730}

X(39760) = X(1)X(30719)∩X(1279)X(3008)

Barycentrics    (2*a^2 - a*b + b^2 - a*c - 2*b*c + c^2)*(2*a^8 - 9*a^7*b + 16*a^6*b^2 - 11*a^5*b^3 - 6*a^4*b^4 + 17*a^3*b^5 - 12*a^2*b^6 + 3*a*b^7 - 9*a^7*c + 34*a^6*b*c - 52*a^5*b^2*c + 53*a^4*b^3*c - 53*a^3*b^4*c + 40*a^2*b^5*c - 14*a*b^6*c + b^7*c + 16*a^6*c^2 - 52*a^5*b*c^2 + 38*a^4*b^2*c^2 - 36*a^2*b^4*c^2 + 20*a*b^5*c^2 - 2*b^6*c^2 - 11*a^5*c^3 + 53*a^4*b*c^3 + 32*a^2*b^3*c^3 - 9*a*b^4*c^3 - b^5*c^3 - 6*a^4*c^4 - 53*a^3*b*c^4 - 36*a^2*b^2*c^4 - 9*a*b^3*c^4 + 4*b^4*c^4 + 17*a^3*c^5 + 40*a^2*b*c^5 + 20*a*b^2*c^5 - b^3*c^5 - 12*a^2*c^6 - 14*a*b*c^6 - 2*b^2*c^6 + 3*a*c^7 + b*c^7) : :

X(39760) lies on the cubic K1160 and these lines: {1, 30719}, {1279, 3008}, {1317, 16184}, {1319, 3021}, {15730, 37743}

X(39760) = incircle-inverse of X(38371)

X(39761) = X(1)X(523)∩X(30)X(6357)

Barycentrics    (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(2*a^9 - a^8*b - 3*a^7*b^2 + 3*a^6*b^3 - 3*a^5*b^4 - 3*a^4*b^5 + 7*a^3*b^6 + a^2*b^7 - 3*a*b^8 - a^8*c + 2*a^7*b*c - a^6*b^2*c - 2*a^5*b^3*c + 4*a^4*b^4*c - 2*a^3*b^5*c - a^2*b^6*c + 2*a*b^7*c - b^8*c - 3*a^7*c^2 - a^6*b*c^2 + 12*a^5*b^2*c^2 - 2*a^4*b^3*c^2 - 8*a^3*b^4*c^2 + 2*a^2*b^5*c^2 - a*b^6*c^2 + b^7*c^2 + 3*a^6*c^3 - 2*a^5*b*c^3 - 2*a^4*b^2*c^3 + 6*a^3*b^3*c^3 - 2*a^2*b^4*c^3 - 2*a*b^5*c^3 + 3*b^6*c^3 - 3*a^5*c^4 + 4*a^4*b*c^4 - 8*a^3*b^2*c^4 - 2*a^2*b^3*c^4 + 8*a*b^4*c^4 - 3*b^5*c^4 - 3*a^4*c^5 - 2*a^3*b*c^5 + 2*a^2*b^2*c^5 - 2*a*b^3*c^5 - 3*b^4*c^5 + 7*a^3*c^6 - a^2*b*c^6 - a*b^2*c^6 + 3*b^3*c^6 + a^2*c^7 + 2*a*b*c^7 + b^2*c^7 - 3*a*c^8 - b*c^8) : :

X(39761) lies on the cubic K1160 and these lines: {1, 523}, {30, 6357}, {1317, 33903}, {1319, 1354}, {11700, 15326}

X(39762) = X(1)X(522)∩X(117)X(515)

Barycentrics    (2*a^4 - a^3*b - a^2*b^2 + a*b^3 - b^4 - a^3*c + 2*a^2*b*c - a*b^2*c - a^2*c^2 - a*b*c^2 + 2*b^2*c^2 + a*c^3 - c^4)*(2*a^9 - 3*a^8*b - 3*a^7*b^2 + 9*a^6*b^3 - 3*a^5*b^4 - 9*a^4*b^5 + 7*a^3*b^6 + 3*a^2*b^7 - 3*a*b^8 - 3*a^8*c + 12*a^7*b*c - 10*a^6*b^2*c - 16*a^5*b^3*c + 28*a^4*b^4*c - 4*a^3*b^5*c - 14*a^2*b^6*c + 8*a*b^7*c - b^8*c - 3*a^7*c^2 - 10*a^6*b*c^2 + 38*a^5*b^2*c^2 - 19*a^4*b^3*c^2 - 27*a^3*b^4*c^2 + 28*a^2*b^5*c^2 - 8*a*b^6*c^2 + b^7*c^2 + 9*a^6*c^3 - 16*a^5*b*c^3 - 19*a^4*b^2*c^3 + 48*a^3*b^3*c^3 - 17*a^2*b^4*c^3 - 8*a*b^5*c^3 + 3*b^6*c^3 - 3*a^5*c^4 + 28*a^4*b*c^4 - 27*a^3*b^2*c^4 - 17*a^2*b^3*c^4 + 22*a*b^4*c^4 - 3*b^5*c^4 - 9*a^4*c^5 - 4*a^3*b*c^5 + 28*a^2*b^2*c^5 - 8*a*b^3*c^5 - 3*b^4*c^5 + 7*a^3*c^6 - 14*a^2*b*c^6 - 8*a*b^2*c^6 + 3*b^3*c^6 + 3*a^2*c^7 + 8*a*b*c^7 + b^2*c^7 - 3*a*c^8 - b*c^8) : :

X(39762) lies on the cubic K1160 and these lines: {1, 522}, {117, 515}, {1317, 15524}, {1319, 1359}, {1785, 21578}

X(39762) = crossdifference of every pair of points on line {2183, 2432}

X(39763) = X(1)X(1769)∩X(102)X(15501)

Barycentrics    a*(a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6 + a^5*c - 2*a^4*b*c + 3*a^3*b^2*c + a^2*b^3*c - 4*a*b^4*c + b^5*c - a^4*c^2 + 3*a^3*b*c^2 - 6*a^2*b^2*c^2 + 3*a*b^3*c^2 + b^4*c^2 - 2*a^3*c^3 + a^2*b*c^3 + 3*a*b^2*c^3 - 2*b^3*c^3 + 2*a^2*c^4 - 4*a*b*c^4 + b^2*c^4 + a*c^5 + b*c^5 - c^6)*(2*a^9 - 3*a^8*b - 3*a^7*b^2 + 9*a^6*b^3 - 3*a^5*b^4 - 9*a^4*b^5 + 7*a^3*b^6 + 3*a^2*b^7 - 3*a*b^8 - 3*a^8*c + 12*a^7*b*c - 10*a^6*b^2*c - 16*a^5*b^3*c + 28*a^4*b^4*c - 4*a^3*b^5*c - 14*a^2*b^6*c + 8*a*b^7*c - b^8*c - 3*a^7*c^2 - 10*a^6*b*c^2 + 38*a^5*b^2*c^2 - 19*a^4*b^3*c^2 - 27*a^3*b^4*c^2 + 28*a^2*b^5*c^2 - 8*a*b^6*c^2 + b^7*c^2 + 9*a^6*c^3 - 16*a^5*b*c^3 - 19*a^4*b^2*c^3 + 48*a^3*b^3*c^3 - 17*a^2*b^4*c^3 - 8*a*b^5*c^3 + 3*b^6*c^3 - 3*a^5*c^4 + 28*a^4*b*c^4 - 27*a^3*b^2*c^4 - 17*a^2*b^3*c^4 + 22*a*b^4*c^4 - 3*b^5*c^4 - 9*a^4*c^5 - 4*a^3*b*c^5 + 28*a^2*b^2*c^5 - 8*a*b^3*c^5 - 3*b^4*c^5 + 7*a^3*c^6 - 14*a^2*b*c^6 - 8*a*b^2*c^6 + 3*b^3*c^6 + 3*a^2*c^7 + 8*a*b*c^7 + b^2*c^7 - 3*a*c^8 - b*c^8) : :

X(39763) lies on the cubic K1160 and these lines: {1, 1769}, {102, 15501}, {515, 38357}, {1319, 1364}, {2800, 34913}, {33901, 33903}

X(39764) = X(3)X(6)∩X(69)X(7749)

Barycentrics    a^2*(3*a^4 - 3*a^2*b^2 + 2*b^4 - 3*a^2*c^2 - 4*b^2*c^2 + 2*c^4) : :
Barycentrics    Sin[A]^2*(Cot[A] - Cot[w] + 4*Tan[ω]) : :
Trilinears    cos A - (sin A) (cot ω - 4 tan ω) : :
X(39764) = 2 X[6] + X[5033], 4 X[6] + X[5206]

X(39764) lies on these lines: {3, 6}, {69, 7749}, {115, 6776}, {184, 3124}, {193, 7764}, {230, 1353}, {542, 18362}, {1196, 11402}, {1352, 5477}, {1501, 15004}, {1506, 3618}, {1569, 5182}, {2451, 8723}, {2548, 7829}, {3199, 19118}, {3564, 7746}, {3734, 39141}, {3767, 14912}, {5032, 7622}, {5286, 33748}, {5475, 18583}, {5651, 20976}, {5921, 38745}, {7735, 9754}, {7747, 14853}, {7753, 16041}, {7756, 25406}, {7790, 32982}, {8541, 19627}, {11003, 39024}, {11060, 39238}, {11179, 11648}, {11898, 37637}, {14498, 36696}, {14537, 14848}, {14567, 34417}, {18440, 39565}, {20977, 35268}, {22111, 39689}, {31455, 38110}, {31481, 32490}, {39495, 39520}, {39496, 39521}

X(39764) = reflection of X(5206) in X(5033)
X(39764) = isogonal conjugate of the isotomic conjugate of X(37637)
X(39764) = Brocard-circle inverse of X(1570)
X(39764) = crosssum of X(2) and X(1007)
X(39764) = barycentric product X(i)*X(j) for these {i,j}: {6, 37637}, {25, 11898}
X(39764) = barycentric quotient X(i)/X(j) for these {i,j}: {11898, 305}, {37637, 76}
X(39764) = intersection of tangents to Brocard circle at intersections with 2nd Lemoine circle
X(39764) = pole wrt Brocard circle of line X(512)X(1570)
X(39764) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 1570}, {6, 182, 5028}, {6, 575, 5034}, {6, 1691, 576}, {6, 1692, 32}, {6, 3053, 5093}, {6, 5017, 5097}, {6, 5034, 7772}, {6, 5050, 39}, {6, 13330, 15520}, {6, 35006, 13330}, {6, 39560, 5111}, {182, 5028, 574}, {1340, 1341, 15815}, {1668, 1669, 3}, {2025, 13354, 32452}, {2030, 5097, 5017}, {5039, 22234, 6}, {5058, 5062, 3}, {5111, 39560, 3098}, {5116, 10485, 182}, {6423, 6424, 22331}

leftri

Points on cubics: X(39765)-X(39770)

rightri

This preamble is contributed by Clark Kimberling, October 1, 2020.

Lete A"B"C" denote the anticompletmentary triangle. The cubic given by

a(b+c)^2 (b^2+c^2-a^2-bc)(y-z)(x+y)(x+z) + b(c+a)^2 (c^2+a^2-b^2-ca)(z-x)(y+z)(y+x) + c(a+b^2) (a^2+b^2-c^2-ab)(x-y)(z+x)(z+y) = 0

is a self-isotomic cubic that passes through the following points: A, B, C, A', B', C', X(2), X(7), X(8), and also the six points X(39765) - X(39770).

X(39765) = X(2)X(4053)∩X(7)X(8)

Barycentrics    a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 + a^4*c + 4*a^3*b*c - 2*a*b^3*c + b^4*c + a^3*c^2 - b^3*c^2 - a^2*c^3 - 2*a*b*c^3 - b^2*c^3 - a*c^4 + b*c^4 : :

X(39765) lies on these lines:{2, 4053}, {7, 8}, {86, 34195}, {523, 4467}, {758, 17139}, {2975, 4360}, {3875, 6629}, {3897, 17393}, {4389, 20247}, {4967, 13407}, {8609, 28606}, {10436, 12559}, {16713, 18714}, {17147, 37683}, {17377, 21271}, {20016, 31297}, {21276, 26751}, {29829, 35552}, {30941, 35550}

X(39765) = isotomic conjugate of X(39768)
X(39765) = anticomplement of X(4053)
X(39765) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {593, 6224}, {759, 2895}, {9273, 4427}, {14616, 21287}, {24624, 1330}, {32671, 17494}, {34079, 1654}, {36069, 514}, {37140, 513}
X(39765) = crosspoint of X(1268) and X(14616)
X(39765) = crosssum of X(2308) and X(3724)
X(39765) = crossdifference of every pair of points on line {3063, 20970}

X(39766) = X(1)X(2)∩X(30)X(4442)

Barycentrics    2*a^4 + a^3*b + a*b^3 + a^3*c + 2*a^2*b*c - 2*a*b^2*c - b^3*c - 2*a*b*c^2 - 2*b^2*c^2 + a*c^3 - b*c^3 : :
X(39766) = 2 X[1] + X[17162], 5 X[3616] - 2 X[4062]

X(39766) lies on these lines: {1, 2}, {30, 4442}, {36, 17495}, {58, 17164}, {385, 17497}, {523, 1325}, {540, 17491}, {758, 16704}, {956, 3891}, {993, 17147}, {999, 16429}, {1104, 3702}, {1266, 18661}, {1279, 4742}, {1319, 7235}, {1724, 25253}, {2238, 30729}, {2975, 17512}, {3120, 38456}, {3295, 16430}, {3304, 16424}, {3476, 24446}, {3743, 17588}, {3744, 3902}, {3896, 24929}, {3995, 5251}, {4293, 19789}, {4387, 11346}, {4402, 24435}, {4418, 5429}, {4434, 4695}, {4647, 11115}, {4653, 27804}, {5080, 37759}, {5247, 32938}, {5563, 16426}, {5692, 19742}, {5902, 37639}, {14450, 20077}, {17154, 21381}, {19284, 28612}, {24390, 37346}, {32924, 37617}, {32929, 37817}

X(39766) = isotomic conjugate of X(39769)
X(39766) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {759, 1330}, {849, 6224}, {9274, 4427}, {24624, 21287}, {32671, 514}, {34079, 2895}, {36069, 513}, {37140, 20295}
X(39766) = crosspoint of X(14616) and X(30710)
X(39766) = crosssum of X(2300) and X(3724)
X(39766) = crossdifference of every pair of points on line {649, 2092}
X(39766) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 27368, 8}

X(39767) = X(2)X(7)∩X(81)X(25255)

Barycentrics    2*a^6 - a^5*b - 4*a^4*b^2 + 2*a^3*b^3 + 2*a^2*b^4 - a*b^5 - a^5*c - 2*a^4*b*c - a^3*b^2*c + a^2*b^3*c + 2*a*b^4*c + b^5*c - 4*a^4*c^2 - a^3*b*c^2 + 2*a^2*b^2*c^2 - a*b^3*c^2 + 2*a^3*c^3 + a^2*b*c^3 - a*b^2*c^3 - 2*b^3*c^3 + 2*a^2*c^4 + 2*a*b*c^4 - a*c^5 + b*c^5 : :

X(39767) lies on these lines: {2, 7}, {81, 25255}, {323, 18668}, {448, 525}, {1010, 11684}, {4360, 18662}, {5857, 23541}, {8025, 16585}, {14206, 16704}, {17075, 37645}, {17151, 20223}, {18609, 28606}, {18653, 18661}

X(39767) = isotomic conjugate of X(39770)
X(39767) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {759, 2893}, {2150, 6224}, {2341, 1330}, {6740, 21287}, {9274, 17136}, {32671, 522}, {34079, 2475}, {36069, 693}, {37140, 21302}

X(39768) = ISOTOMIC CONJUGATE OF X(39765)

Barycentrics    (a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 - a^4*c + 2*a^3*b*c - 4*a*b^3*c - b^4*c + a^3*c^2 - b^3*c^2 + a^2*c^3 + 2*a*b*c^3 + b^2*c^3 - a*c^4 + b*c^4)*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4 - a^4*c - 2*a^3*b*c - 2*a*b^3*c - b^4*c - a^3*c^2 - b^3*c^2 + a^2*c^3 + 4*a*b*c^3 + b^2*c^3 + a*c^4 + b*c^4) : :

X(39768) lies on the Feuerbach circumhyperbola and these lines: {1, 24235}, {9, 4115}, {21, 4427}, {3952, 32635}, {6598, 36974}, {P2, P3}

X(39768) = isotomic conjugate of X(39765)
X(39768) = isotomic conjugate of the anticomplement of X(4053)
X(39768) = X(4053)-cross conjugate of X(2)
X(39768) = X(31)-isoconjugate of X(39764)
X(39768) = cevapoint of X(i) and X(j) for these (i,j): {11, 6370}, {758, 1125}
X(39768) = trilinear pole of line {650, 1213}
X(39768) = barycentric quotient X(2)/X(39765)

X(39769) = ISOTOMIC CONJUGATE OF X(39766)

Barycentrics    (a^3*b + a*b^3 + 2*b^4 - a^3*c - 2*a^2*b*c + 2*a*b^2*c + b^3*c - 2*a^2*c^2 - 2*a*b*c^2 - a*c^3 + b*c^3)*(a^3*b + 2*a^2*b^2 + a*b^3 - a^3*c + 2*a^2*b*c + 2*a*b^2*c - b^3*c - 2*a*b*c^2 - a*c^3 - b*c^3 - 2*c^4) : :

X(39769) lies on the conic {A, B, C, X(2), X(7)} and these lines: {2, 24195}, {1240, 27808}, {P1, P2}

X(39769) = isotomic conjugate of X(39766)
X(39769) = cevapoint of X(i) and X(j) for these (i,j): {758, 3666}, {1086, 6370}
X(39769) = trilinear pole of line {514, 1211}
X(39769) = barycentric quotient X(2)/X(39766)

X(39770) = ISOTOMIC CONJUGATE OF X(39767)

Barycentrics    (a^5*b - 2*a^4*b^2 - 2*a^3*b^3 + 4*a^2*b^4 + a*b^5 - 2*b^6 - a^5*c - 2*a^4*b*c - a^3*b^2*c + a^2*b^3*c + 2*a*b^4*c + b^5*c + a^3*b*c^2 - 2*a^2*b^2*c^2 + a*b^3*c^2 + 4*b^4*c^2 + 2*a^3*c^3 + a^2*b*c^3 - a*b^2*c^3 - 2*b^3*c^3 - 2*a*b*c^4 - 2*b^2*c^4 - a*c^5 + b*c^5)*(a^5*b - 2*a^3*b^3 + a*b^5 - a^5*c + 2*a^4*b*c - a^3*b^2*c - a^2*b^3*c + 2*a*b^4*c - b^5*c + 2*a^4*c^2 + a^3*b*c^2 + 2*a^2*b^2*c^2 + a*b^3*c^2 + 2*b^4*c^2 + 2*a^3*c^3 - a^2*b*c^3 - a*b^2*c^3 + 2*b^3*c^3 - 4*a^2*c^4 - 2*a*b*c^4 - 4*b^2*c^4 - a*c^5 - b*c^5 + 2*c^6) : :

X(39770) lies on these lines: {29, 34195}, {X(39765), X(39767)}

X(39770) = isotomic conjugate of X(39767)
X(39770) = cevapoint of X(1146) and X(6370)
X(39770) = trilinear pole of line {442, 522}
X(39770) = barycentric quotient X(2)/X(39767)

X(39771) = INCIRCLE-INVERSE OF X(39752)

Barycentrics    (b - c)*(-a + b - c)*(a + b - c)*(-2*a + b + c)^2 : :
X(39771) = 9 X[4017] - 10 X[30722], 2 X[23757] - 3 X[30573], 3 X[30572] - 4 X[30725]

X(39771) lies on Jerabek circumhyperbola of the intouch triangle, and on these lines: {1, 3667}, {7, 3676}, {65, 513}, {109, 15343}, {145, 522}, {514, 16236}, {900, 1317}, {1071, 30198}, {1647, 3259}, {2099, 23345}, {2171, 21143}, {2254, 10427}, {2827, 11570}, {3174, 4105}, {3649, 4017}, {3911, 34764}, {5586, 21201}, {6003, 13375}, {17365, 23760}, {17780, 23703}, {34502, 39386}, {35015, 38385}

X(39771) = reflection of X(21105) in X(14812)
X(39771) = incircle-inverse of X(39752)
X(39771) = X(i)-Ceva conjugate of X(j) for these (i,j): {664, 3911}, {1317, 14027}, {3676, 30725}, {14584, 1647}
X(39771) = X(i)-cross conjugate of X(j) for these (i,j): {3251, 6544}, {14027, 1317}
X(39771) = X(i)-isoconjugate of X(j) for these (i,j): {9, 4638}, {55, 4618}, {88, 5548}, {100, 1318}, {644, 2226}, {679, 3939}, {901, 1320}, {2316, 3257}, {3689, 39414}, {4582, 9456}, {4997, 32665}, {5546, 30575}, {9268, 23838}
X(39771) = crosspoint of X(i) and X(j) for these (i,j): {664, 3911}, {3676, 30725}
X(39771) = crosssum of X(i) and X(j) for these (i,j): {663, 2316}, {3939, 5548}
X(39771) = trilinear pole of line {14027, 35092}
X(39771) = crossdifference of every pair of points on line {1318, 2316}
X(39771) = X(14380)-of-intouch-triangle
X(39771) = barycentric product X(i)*X(j) for these {i,j}: {7, 6544}, {85, 3251}, {190, 14027}, {279, 4543}, {514, 1317}, {519, 30725}, {658, 4542}, {664, 35092}, {678, 24002}, {900, 3911}, {1319, 3762}, {3669, 4738}, {3676, 4370}, {4017, 16729}, {4998, 14442}, {16704, 30572}
X(39771) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 4638}, {57, 4618}, {519, 4582}, {649, 1318}, {678, 644}, {900, 4997}, {902, 5548}, {1017, 3939}, {1317, 190}, {1319, 3257}, {1404, 901}, {1635, 1320}, {1960, 2316}, {2087, 23838}, {3251, 9}, {3669, 679}, {3911, 4555}, {4017, 30575}, {4152, 6558}, {4370, 3699}, {4542, 3239}, {4543, 346}, {4738, 646}, {6544, 8}, {8028, 30731}, {14027, 514}, {14442, 11}, {16729, 7257}, {21821, 4069}, {22371, 4587}, {23703, 5376}, {30572, 4080}, {30725, 903}, {33922, 2325}, {35092, 522}

X(39772) = X(1)X(21)∩X(7)X(2475)

Barycentrics    a*(a^2*b - b^3 + a^2*c + 2*a*b*c + b^2*c + b*c^2 - c^3)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - a*b*c - a*c^2 + c^3) : :
X(39772) = X[191] + 3 X[3894], 3 X[354] - 2 X[11281], 4 X[942] - X[31938], X[3868] + 2 X[10122], 3 X[3873] - X[34195], X[3901] + 3 X[5426], X[4018] + 2 X[15174], 5 X[18398] - 3 X[26725], 3 X[28465] - 2 X[31837]

X(39772) lies on the Jerabek circumhyperbola of the intouch triangle and on these lines: {1, 21}, {7, 2475}, {20, 5884}, {30, 1071}, {57, 224}, {65, 16465}, {72, 5719}, {79, 10052}, {100, 12432}, {354, 11281}, {377, 5902}, {442, 942}, {518, 8261}, {519, 13375}, {912, 6841}, {1004, 5221}, {1259, 37308}, {1317, 34791}, {1537, 2771}, {1998, 35990}, {2478, 10399}, {2949, 6986}, {3174, 7672}, {3336, 35976}, {3436, 18412}, {3649, 10957}, {3651, 5709}, {3870, 37550}, {4018, 15174}, {4084, 4304}, {4197, 5883}, {4292, 11570}, {4757, 11015}, {4973, 14794}, {5273, 15674}, {5428, 24299}, {5693, 6837}, {5705, 31254}, {5715, 12528}, {5735, 12669}, {5904, 10198}, {6003, 23775}, {6583, 26470}, {6839, 31870}, {6884, 20117}, {9799, 9812}, {9964, 11604}, {9965, 15680}, {10267, 16139}, {10391, 10543}, {10431, 15071}, {10527, 11036}, {10883, 31803}, {10902, 31660}, {10916, 11263}, {11012, 12005}, {11249, 33858}, {11544, 17653}, {12540, 13131}, {12635, 37248}, {12704, 16132}, {14100, 16142}, {14923, 16236}, {15016, 37112}, {15556, 27086}, {15677, 28610}, {17616, 34502}, {18398, 26363}, {18977, 33667}, {22836, 37300}, {26357, 33857}, {28465, 31837}, {30329, 38052}, {31806, 37106}

X(39772) = midpoint of X(21) and X(3868)
X(39772) = reflection of X(i) in X(j) for these {i,j}: {21, 10122}, {72, 6675}, {442, 942}, {17653, 11544}, {21677, 8261}, {31938, 442}, {33592, 6583}
X(39772) = X(7)-Ceva conjugate of X(5249)
X(39772) = X(2259)-isoconjugate of X(37887)
X(39772) = intouch-isogonal conjugate of X(2646)
X(39772) = X(6145)-of-intouch-triangle
X(39772) = barycentric product X(i)*X(j) for these {i,j}: {942, 33116}, {5174, 18607}, {5249, 34772}
X(39772) = barycentric quotient X(i)/X(j) for these {i,j}: {942, 37887}, {37583, 2982}
X(39772) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {942, 14054, 6734}, {2475, 12649, 6598}, {3868, 3873, 11520}, {3868, 11020, 3869}, {3874, 18389, 3868}, {5426, 31424, 21}

X(39773) = X(1)X(69)∩X(7)X(1999)

Barycentrics    (a^2 + a*b + a*c + 2*b*c)*(a^3 + 3*a^2*b + a*b^2 - b^3 + 3*a^2*c - b^2*c + a*c^2 - b*c^2 - c^3) : :

X(39773) lies on the Jerabek circumhyperbola of the intouch triangle and on these lines: {1, 69}, {7, 1999}, {57, 39774}, {65, 3875}, {75, 35616}, {145, 5933}, {314, 8033}, {940, 3713}, {969, 10447}, {1071, 10441}, {1266, 5586}, {1449, 25898}, {1537, 35638}, {3649, 10401}, {3664, 17733}, {3718, 3751}, {4464, 16236}, {10371, 17270}, {10435, 13244}, {10446, 35635}, {10889, 14100}, {11570, 35636}, {33859, 35646}

X(39773) = X(7)-Ceva conjugate of X(10436)

X(39774) = X(1)X(75)∩X(7)X(3210)

Barycentrics    (a*b + b^2 + a*c + c^2)*(a^3 + a^2*b + a^2*c + a*b*c - b^2*c - b*c^2) : :

X(39774) lies on the Jerabek circumhyperbola of the intouch triangle and on these lines: {1, 75}, {7, 3210}, {43, 3718}, {57, 39773}, {65, 3879}, {69, 986}, {239, 21233}, {256, 7019}, {511, 1071}, {1211, 2092}, {1266, 3649}, {1317, 4464}, {1537, 2783}, {1959, 18177}, {3670, 10452}, {6007, 14100}, {16006, 29301}, {17147, 20245}, {17863, 21422}, {19791, 29967}, {20040, 21273}, {25978, 38408}, {27565, 28252}

X(39774) = complement of the isotomic conjugate of X(28630)
X(39774) = X(28630)-complementary conjugate of X(2887)
X(39774) = X(7)-Ceva conjugate of X(4357)
X(39774) = crosspoint of X(2) and X(28630)
X(39774) = crosssum of X(6) and X(28631)
X(39774) = intouch-isogonal conjugate of X(21334)
X(39774) = X(8612)-of-intouch-triangle
X(39774) = barycentric product X(i)*X(j) for these {i,j}: {1999, 4357}, {5247, 20911}
X(39774) = barycentric quotient X(i)/X(j) for these {i,j}: {1999, 1220}, {5247, 2298}, {6002, 4581}, {24560, 15420}

X(39775) = X(1)X(85)∩X(7)X(192)

Barycentrics    (a + b - c)*(a - b + c)*(a^2 - b*c)*(a*b - b^2 + a*c - c^2) : :

L X(39775) lies on the Jerabek circumhyperbola of the intouch triangle, the cubic K770, and these lines: {1, 85}, {2, 9502}, {7, 192}, {57, 7075}, {65, 7198}, {69, 4073}, {145, 3212}, {224, 3905}, {239, 385}, {241, 3693}, {350, 18033}, {740, 10030}, {1071, 2808}, {1266, 10427}, {1358, 2795}, {1959, 20347}, {3061, 3177}, {3174, 3875}, {3664, 24225}, {3716, 3766}, {3879, 15185}, {4357, 15595}, {4554, 18149}, {4875, 30038}, {4955, 34502}, {5723, 17023}, {7179, 17244}, {7204, 30962}, {10025, 25943}, {16603, 20341}, {17090, 25718}, {17095, 29637}, {17284, 31225}, {17755, 34253}, {19581, 33701}, {24549, 37523}, {29573, 32041}, {29674, 33298}

X(39775) = isotomic conjugate of X(33676)
X(39775) = X(i)-Ceva conjugate of X(j) for these (i,j): {7, 9436}, {85, 16609}
X(39775) = X(i)-cross conjugate of X(j) for these (i,j): {350, 33701}, {8299, 17755}
X(39775) = X(i)-isoconjugate of X(j) for these (i,j): {31, 33676}, {105, 7077}, {291, 2195}, {292, 294}, {660, 884}, {813, 1024}, {875, 36802}, {885, 34067}, {1438, 4876}, {1911, 14942}, {1922, 36796}, {2311, 18785}, {2481, 18265}
X(39775) = cevapoint of X(8299) and X(34253)
X(39775) = crosspoint of X(7) and X(1447)
X(39775) = crosssum of X(55) and X(7077)
X(39775) = intouch-isogonal conjugate of X(20358)
X(39775) = X(1987)-of-intouch-triangle
X(39775) = excentral-to-intouch similarity image of X(24578)
X(39775) = barycentric product X(i)*X(j) for these {i,j}: {7, 17755}, {75, 34253}, {85, 8299}, {239, 9436}, {241, 350}, {331, 20778}, {518, 10030}, {672, 18033}, {812, 883}, {1025, 3766}, {1284, 18157}, {1429, 3263}, {1447, 3912}, {1458, 1921}, {3975, 34855}, {7233, 27919}, {16609, 30941}
X(39775) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 33676}, {238, 294}, {239, 14942}, {241, 291}, {350, 36796}, {518, 4876}, {659, 1024}, {672, 7077}, {812, 885}, {883, 4562}, {1025, 660}, {1284, 18785}, {1362, 3252}, {1428, 1438}, {1429, 105}, {1447, 673}, {1458, 292}, {1914, 2195}, {2283, 813}, {3286, 2311}, {3570, 36802}, {3684, 28071}, {3685, 6559}, {3716, 28132}, {3912, 4518}, {8299, 9}, {8632, 884}, {9436, 335}, {9454, 18265}, {10030, 2481}, {16609, 13576}, {17755, 8}, {18033, 18031}, {20778, 219}, {27919, 3685}, {30941, 36800}, {34253, 1}, {38989, 2170}
X(39775) = {X(17316),X(39350)}-harmonic conjugate of X(4876)

X(39776) = X(1)X(88)∩X(8)X(153)

Barycentrics    a*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c + 5*a*b*c - 2*b^2*c - a*c^2 - 2*b*c^2 + c^3) : :
X(39776) = 3 X[8] - X[12532], 2 X[1387] - 3 X[3753], 3 X[3679] - 2 X[18254], 5 X[3698] - 4 X[6667], 3 X[3753] - X[17652], 4 X[3918] - 3 X[32557], 3 X[5902] - X[26726], 2 X[9951] - 3 X[10707], 2 X[9957] - 3 X[34123]

X(39776) lies on the Jerabek circumhyperbola of the intouch triangle and on these lines: {1, 88}, {2, 15558}, {7, 12641}, {8, 153}, {10, 12758}, {11, 5836}, {63, 2950}, {65, 5854}, {78, 10698}, {80, 3434}, {104, 3359}, {119, 517}, {145, 5083}, {149, 5554}, {200, 13253}, {224, 2136}, {519, 11570}, {528, 12743}, {942, 25416}, {952, 1071}, {956, 12515}, {1317, 3880}, {1376, 12740}, {1387, 3753}, {1470, 22560}, {1768, 4853}, {1837, 13271}, {2057, 11682}, {2077, 4996}, {2771, 16006}, {2801, 12531}, {2840, 38512}, {2886, 32198}, {2932, 10269}, {3035, 3057}, {3036, 17638}, {3174, 14151}, {3419, 12691}, {3436, 34789}, {3577, 12703}, {3632, 11571}, {3649, 10956}, {3679, 18254}, {3698, 6667}, {3813, 20118}, {3869, 14740}, {3873, 16236}, {3887, 38371}, {3893, 17660}, {3913, 12739}, {3918, 32557}, {4511, 25485}, {4861, 11715}, {4915, 12767}, {4973, 27247}, {5082, 12247}, {5086, 15863}, {5193, 37789}, {5440, 19907}, {5552, 6979}, {5687, 6265}, {5697, 26364}, {5902, 26726}, {5903, 10052}, {6224, 15528}, {6702, 11680}, {6736, 21635}, {8256, 25414}, {9945, 10202}, {9951, 10707}, {9957, 34123}, {10200, 16173}, {10270, 38693}, {10609, 32900}, {10912, 20586}, {11729, 23340}, {12619, 24390}, {12763, 32049}, {12832, 26015}, {14217, 26333}, {22837, 38901}

X(39776) = midpoint of X(i) and X(j) for these {i,j}: {100, 14923}, {3632, 11571}, {3893, 17660}
X(39776) = reflection of X(i) in X(j) for these {i,j}: {11, 5836}, {145, 5083}, {1320, 12736}, {3057, 3035}, {3869, 14740}, {12691, 19914}, {12758, 10}, {17638, 3036}, {17652, 1387}, {23340, 11729}, {25416, 942}
X(39776) = anticomplement of X(15558)
X(39776) = X(7)-Ceva conjugate of X(908)
X(39776) = crosspoint of X(7) and X(37789)
X(39776) = crossdifference of every pair of points on line {1635, 2423}
X(39776) = intouch-isogonal conjugate of X(5048)
X(39776) = X(11744)-of-intouch-triangle
X(39776) = antipode of X(145) in Jerabek hyperbola of intouch triangle
X(39776) = barycentric product X(i)*X(j) for these {i,j}: {517, 37758}, {908, 38460}, {2397, 2827}, {6735, 37789}
X(39776) = barycentric quotient X(i)/X(j) for these {i,j}: {2427, 2743}, {2827, 2401}, {5193, 34051}, {37758, 18816}, {38460, 34234}
X(39776) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5541, 25438}, {119, 1145, 6735}, {3753, 17652, 1387}, {4861, 17100, 11715}

X(39777) = X(1)X(140)∩X(7)X(10944)

Barycentrics    (2*a - 3*b - 3*c)*(4*a - b - c)*(a + b - c)*(a - b + c) : :
X(39777) = 5 X[1] - 3 X[5559], 4 X[3634] - 3 X[15862], 3 X[13375] - 4 X[31794]

X(39777) lies on the Jerabek circumhyperbola of the intouch triangle and on these lines: on lines {1, 140}, {7, 10944}, {12, 3625}, {145, 5221}, {515, 16006}, {519, 3649}, {758, 33859}, {1159, 10044}, {1317, 32636}, {1389, 10598}, {1483, 37524}, {1537, 6246}, {2099, 3621}, {2802, 33667}, {3241, 32157}, {3340, 11524}, {3579, 37734}, {3617, 15950}, {3626, 11011}, {3633, 5434}, {3634, 15862}, {3635, 5298}, {4701, 4870}, {5554, 34743}, {5560, 22791}, {8148, 9668}, {10106, 34502}, {11009, 18357}, {11237, 20053}, {13144, 37736}, {13375, 31794}, {16236, 37738}, {28473, 38371}

X(39777) = X(664)-Ceva conjugate of X(30726)
X(39777) = intouch-isogonal conjugate of X(9957)
X(39777) = X(16835)-of-intouch-triangle

X(39778) = X(1)X(149)∩X(21)X(104)

Barycentrics    a*(a^2 - b^2 + b*c - c^2)*(a^4 - 2*a^3*b + 2*a*b^3 - b^4 - 2*a^3*c + a^2*b*c - a*b^2*c - a*b*c^2 + 2*b^2*c^2 + 2*a*c^3 - c^4) : :
X(39778) = 3 X[1] + X[13146], X[80] - 3 X[26725], X[3065] - 3 X[5426], 3 X[3576] - 2 X[17009], 3 X[4881] - 2 X[27086], X[6224] + 2 X[33593], 2 X[6675] - 3 X[34123], X[10609] + 2 X[16137], X[13146] - 3 X[34600], 3 X[15015] + X[16126]

X(39778) lies on the Jerabek circumhyperbola of the intouch triangle and on these lines: {1, 149}, {2, 6326}, {4, 33594}, {7, 6596}, {11, 11281}, {21, 104}, {30, 1537}, {36, 214}, {65, 100}, {80, 3822}, {145, 18467}, {153, 18446}, {191, 30144}, {404, 5885}, {442, 952}, {942, 33598}, {1145, 3935}, {1317, 38460}, {1319, 33667}, {1320, 3957}, {1387, 29817}, {1621, 17638}, {1768, 4189}, {2320, 3065}, {2800, 10902}, {2802, 13375}, {2975, 17660}, {3035, 21677}, {3219, 12532}, {3336, 4188}, {3576, 17009}, {3651, 10698}, {3870, 16236}, {3873, 22560}, {3877, 37286}, {3897, 12773}, {4190, 10044}, {4293, 10052}, {4308, 10941}, {4861, 7972}, {5046, 21635}, {5141, 30143}, {5154, 15017}, {5253, 8261}, {5531, 19860}, {5538, 36004}, {5731, 15680}, {6175, 10031}, {6675, 34123}, {6841, 11729}, {6872, 9809}, {9897, 30147}, {10090, 27003}, {10222, 35982}, {10609, 16137}, {11114, 16128}, {11684, 37605}, {11715, 12757}, {12524, 12947}, {12635, 37301}, {12738, 31254}, {12740, 14100}, {12767, 35258}, {13199, 37533}, {14151, 15185}, {15674, 19861}, {17100, 31660}, {17653, 33657}, {17768, 18450}, {18254, 27065}, {24926, 35016}, {25005, 37700}, {37308, 37535}

X(39778) = midpoint of X(i) and X(j) for these {i,j}: {1, 34600}, {100, 34195}, {3651, 10698}, {6175, 10031}, {6224, 11604}, {6265, 33858}, {11263, 33337}
X(39778) = reflection of X(i) in X(j) for these {i,j}: {4, 33594}, {11, 11281}, {6841, 11729}, {11604, 33593}, {21677, 3035}, {35204, 214}
X(39778) = X(7)-Ceva conjugate of X(3218)
X(39778) = X(1411)-isoconjugate of X(6596)
X(39778) = crosspoint of X(7) and X(37797)
X(39778) = barycentric product X(4511)*X(37797)
X(39778) = barycentric quotient X(i)/X(j) for these {i,j}: {2323, 6596}, {37797, 18815}
X(39778) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {100, 12739, 34772}, {214, 11570, 4996}, {4996, 11570, 3218}

X(39779) = X(1)X(227)∩X(7)X(517)

Barycentrics    a*(a^2 - b^2 + 4*b*c - c^2)*(a^3*b - a^2*b^2 - a*b^3 + b^4 + a^3*c - 6*a^2*b*c + a*b^2*c - a^2*c^2 + a*b*c^2 - 2*b^2*c^2 - a*c^3 + c^4) : :
X(39779) = 3 X[354] - 2 X[14563], 3 X[5902] - X[16236]

X(39779) lies on the Jerabek circumhyperbola of the intouch triangle and on these lines: {1, 227}, {7, 517}, {65, 28234}, {72, 10941}, {145, 942}, {153, 5927}, {226, 1537}, {354, 1317}, {388, 12672}, {392, 495}, {515, 14100}, {518, 36922}, {519, 15185}, {952, 5728}, {999, 3306}, {1071, 10106}, {1159, 11526}, {1320, 3957}, {2800, 8581}, {2802, 5542}, {3057, 3649}, {3174, 3880}, {3333, 5836}, {3487, 9957}, {3600, 31788}, {3885, 11036}, {4308, 9940}, {4345, 31792}, {4900, 10980}, {5252, 18908}, {5328, 8164}, {5439, 24928}, {5586, 5903}, {5884, 9850}, {5902, 16236}, {5919, 37703}, {6147, 23340}, {6797, 24297}, {7967, 11018}, {9844, 18525}, {9848, 31673}, {10052, 12709}, {10587, 11374}, {10595, 20789}, {10866, 18483}, {11037, 14923}, {11570, 17625}, {12047, 17622}, {12128, 31794}, {12245, 37544}, {12737, 17626}, {30199, 38371}

X(39779) = midpoint of X(5903) and X(8275)
X(39779) = reflection of X(i) in X(j) for these {i,j}: {11041, 942}, {11525, 5836}, {24297, 6797}
X(39779) = incircle-inverse of X(33902)
X(39779) = crosspoint of X(7) and X(3306)
X(39779) = intouch-isogonal conjugate of X(2099)
X(39779) = X(4846)-of-intouch-triangle
X(39779) = barycentric product X(i)*X(j) for these {i,j}: {999, 26591}, {3306, 31397}

X(39780) = X(1)X(256)∩X(7)X(310)

Barycentrics    a^2*(a + b - c)*(a - b + c)*(b + c)*(a*b^2 + b^2*c + a*c^2 + b*c^2) : :

X(39780) lies on the Jerabek circumhyperbola of the intouch triangle and on these lines: {1, 256}, {6, 20864}, {7, 310}, {31, 23440}, {37, 22198}, {42, 181}, {56, 3736}, {57, 1045}, {65, 740}, {145, 35104}, {1071, 12723}, {1317, 1356}, {1458, 10475}, {1463, 3649}, {2171, 22167}, {2300, 3271}, {2309, 22389}, {2783, 11570}, {3728, 27880}, {3741, 30097}, {3963, 17751}, {4854, 21334}, {5586, 17114}, {6646, 35614}, {10381, 31964}, {10441, 24248}, {16609, 25124}, {17257, 35628}, {20683, 21061}, {21769, 39688}, {23416, 36635}

X(39780) = X(i)-Ceva conjugate of X(j) for these (i,j): {7, 30097}, {664, 7180}
X(39780) = crosspoint of X(7) and X(1400)
X(39780) = crosssum of X(i) and X(j) for these (i,j): {2, 11688}, {21, 27958}, {55, 333}
X(39780) = crossdifference of every pair of points on line {3287, 4560}
X(39780) = X(i)-isoconjugate of X(j) for these (i,j): {284, 1221}, {333, 1258}
X(39780) = intouch-isogonal conjugate of X(3666)
X(39780) = excentral-to-intouch similarity image of X(1045)
X(39780) = X(8795)-of-intouch-triangle
X(39780) = barycentric product X(i)*X(j) for these {i,j}: {7, 21838}, {42, 30097}, {56, 21024}, {57, 3728}, {65, 1107}, {181, 16738}, {225, 22065}, {226, 2309}, {331, 23212}, {1014, 22206}, {1197, 1441}, {1400, 3741}, {1402, 20891}, {1412, 21713}, {1432, 27880}, {1434, 21700}, {2171, 18169}
X(39780) = barycentric quotient X(i)/X(j) for these {i,j}: {65, 1221}, {1107, 314}, {1197, 21}, {1402, 1258}, {2309, 333}, {3728, 312}, {3741, 28660}, {16738, 18021}, {21024, 3596}, {21700, 2321}, {21713, 30713}, {21838, 8}, {22065, 332}, {22206, 3701}, {22389, 1812}, {23212, 219}, {27880, 17787}, {30097, 310}

X(39781) = X(1)X(381)∩X(7)X(1392)

Barycentrics    (3*a - 2*b - 2*c)*(4*a - b - c)*(a + b - c)*(a - b + c) : :
X(39781) = 5 X[1] - X[5560], 3 X[1392] - 7 X[20057], 5 X[3616] - 3 X[7705]

X(39781) lies on the Jerabek circumhyperbola of the intouch triangle and on these lines: {1, 381}, {7, 1392}, {10, 1317}, {56, 16236}, {65, 3635}, {145, 1319}, {354, 13375}, {1071, 5919}, {1385, 5559}, {1388, 3632}, {1537, 5882}, {2099, 5586}, {3057, 11570}, {3241, 32636}, {3616, 7705}, {3636, 10944}, {4004, 13751}, {4701, 5433}, {5048, 10052}, {7972, 17606}, {10543, 16006}, {10914, 33812}, {12616, 32905}, {14100, 17622}, {20053, 24914}, {22791, 33176}
X(39781) = crosspoint of X(7) and X(31231)
X(39781) = intouch-isogonal conjugate of X(5903)
X(39781) = X(11270)-of-intouch-triangle
X(39781) = barycentric product X(3244)*X(31231)
X(39781) = barycentric quotient X(i)/X(j) for these {i,j}: {16669, 1392}, {31231, 39710}

X(39782) = X(1)X(382)∩X(7)X(1319)

Barycentrics    (a - 2*b - 2*c)*(a + b - c)*(a - b + c)*(4*a + b + c) : :
X(39782) = 3 X[1] + X[5561], X[145] + 5 X[10129], 5 X[2320] - 9 X[38314], 3 X[3679] - 5 X[17057]

X(39782) lies on the Jerabek circumhyperbola of the intouch triangle and on these lines: {1, 382}, {7, 1319}, {11, 14563}, {12, 3625}, {56, 5586}, {57, 16558}, {65, 392}, {80, 17605}, {145, 3485}, {226, 1317}, {354, 1387}, {551, 4031}, {1000, 17718}, {1071, 13464}, {1385, 11552}, {2099, 3679}, {2646, 10044}, {3057, 5719}, {3649, 4315}, {3656, 3748}, {3671, 34502}, {4323, 9780}, {5919, 37703}, {10052, 20323}, {10283, 11551}, {11278, 37731}, {12047, 37728}, {12730, 12739}, {13407, 33176}, {15935, 30384}, {17625, 33667}, {21578, 39542}, {30305, 37080}, {31018, 34647}, {31425, 37567}

X(39782) = X(7)-Ceva conjugate of X(4031)
X(39782) = crosspoint of X(7) and X(5219)
X(39782) = crosssum of X(55) and X(2364)
X(39782) = intouch-isogonal conjugate of X(5902)
X(39782) = excentral-to-intouch similarity image of X(16558)
X(39782) = X(3431)-of-intouch-triangle
X(39782) = barycentric product X(i)*X(j) for these {i,j}: {7, 16590}, {57, 4793}, {331, 22372}, {551, 5219}, {2099, 24589}, {3679, 4031}, {4767, 30722}, {6063, 21754}
X(39782) = barycentric quotient X(i)/X(j) for these {i,j}: {551, 30608}, {4031, 39704}, {4793, 312}, {16590, 8}, {16666, 2320}, {21747, 2364}, {21754, 55}, {21822, 210}, {22372, 219}
X(39782) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2099, 5219, 36920}, {4870, 36920, 5219}

X(39783) = X(1)X(442)∩X(7)X(34195)

Barycentrics    (a - b - c)*(2*a^2 + a*b - b^2 + a*c + 2*b*c - c^2)*(2*a^4 - 3*a^3*b - a^2*b^2 + 3*a*b^3 - b^4 - 3*a^3*c - a*b^2*c - a^2*c^2 - a*b*c^2 + 2*b^2*c^2 + 3*a*c^3 - c^4) : :

X(39783) lies on the Jerabek circumhyperbola of the intouch triangle and on these lines: {1, 442}, {7, 34195}, {30, 4930}, {65, 12437}, {145, 35979}, {758, 1071}, {2136, 16236}, {2646, 5745}, {3158, 32157}, {3623, 6601}, {3880, 13375}, {4847, 37734}, {6161, 6362}, {6600, 20013}, {6831, 22836}, {10044, 11112}, {10543, 14100}, {10609, 11570}, {12563, 17647}, {12629, 22992}, {19843, 37728}

X(39783) = reflection of X(6598) in X(11281)
X(39783) = X(7)-Ceva conjugate of X(5745)
X(39783) = {X(5794),X(11281)}-harmonic conjugate of X(442)

X(39784) = X(2)X(39)∩X(83)X(7873)

Barycentrics    (2*a^2 + b^2 + c^2)*(a^2 + 2*b^2 + 2*c^2) : :

X(39784) lies on these lines: {2, 39}, {83, 7873}, {99, 16896}, {141, 5041}, {187, 5103}, {385, 31268}, {597, 7826}, {625, 7948}, {1078, 16987}, {1506, 8364}, {1656, 37479}, {3053, 15810}, {3096, 7845}, {3329, 7849}, {3526, 30270}, {3589, 5007}, {3618, 7854}, {3619, 7855}, {3763, 7772}, {4045, 32819}, {5008, 7800}, {5067, 8721}, {5188, 38317}, {5475, 32956}, {6287, 20190}, {6656, 6704}, {7603, 7866}, {7748, 16045}, {7760, 16988}, {7770, 7872}, {7773, 7808}, {7780, 7875}, {7794, 34573}, {7802, 7804}, {7805, 16986}, {7816, 16895}, {7819, 32459}, {7820, 31652}, {7821, 7914}, {7839, 10159}, {7843, 7937}, {7848, 7878}, {7861, 15031}, {7913, 39565}, {7935, 14537}, {8363, 31275}, {8589, 14001}, {14981, 34127}, {15482, 33217}, {15513, 16043}, {15515, 33237}, {19689, 32456}, {32968, 39601}

X(39784) = crosspoint of X(3763) and X(39668)
X(39784) = barycentric product X(i)*X(j) for these {i,j}: {3589, 3763}, {5064, 7767}, {6292, 39668}, {7950, 10330}
X(39784) = barycentric quotient X(i)/X(j) for these {i,j}: {3763, 10159}, {7772, 3108}, {7950, 31065}
X(39784) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6683, 7874}, {2, 7786, 7915}, {2, 7834, 31239}, {2, 7859, 3934}, {2, 7943, 7886}, {39, 7795, 39785}, {597, 7826, 34571}, {3589, 6292, 5007}, {7797, 7945, 9865}, {7886, 7943, 7852}, {7889, 8362, 187}, {7914, 11174, 7821}

X(39785) = X(2)X(39)∩X(3)X(15533)

Barycentrics    (a^2 - 2*b^2 - 2*c^2)*(2*a^2 - b^2 - c^2) : :
X(39785) = X[2] - 3 X[7799], 5 X[2] - 3 X[14568], 7 X[2] - 3 X[19570], 2 X[99] + X[7845], 3 X[99] - X[9855], 3 X[115] - 4 X[8355], 7 X[187] - 4 X[3793], X[187] - 4 X[6390], X[187] + 2 X[7813], 3 X[187] - 4 X[27088], 5 X[187] - 8 X[32459], 2 X[230] - 3 X[9167], X[316] + 2 X[15301], 3 X[325] - X[8352], X[325] + 2 X[14148], 4 X[620] - 3 X[5215], 7 X[2482] - 2 X[3793], 3 X[2482] - 2 X[27088], 5 X[2482] - 4 X[32459], X[3793] - 7 X[6390], 2 X[3793] + 7 X[7813], 3 X[3793] - 7 X[27088], 5 X[3793] - 14 X[32459], 3 X[5215] - 2 X[22329], 4 X[5461] - 5 X[31275], 2 X[6390] + X[7813], 3 X[6390] - X[27088], 5 X[6390] - 2 X[32459], 3 X[7757] - X[9887], X[7779] + 2 X[32456], 9 X[7799] - X[11054], 5 X[7799] - X[14568], 7 X[7799] - X[19570], 3 X[7809] - X[8597], 3 X[7813] + 2 X[27088], 5 X[7813] + 4 X[32459], 3 X[7840] + X[9855], 3 X[7845] + 2 X[9855], 5 X[7925] - 3 X[9166], 5 X[7925] - 2 X[32457], X[8352] + 6 X[14148], 2 X[8352] - 3 X[31173], 2 X[8355] - 3 X[22110], X[8593] - 3 X[12215], 3 X[9166] - 2 X[32457], 3 X[9865] + X[9887], 6 X[10150] - 5 X[14061], 5 X[11054] - 9 X[14568], 7 X[11054] - 9 X[19570], 3 X[12151] - X[15534], 4 X[14148] + X[31173], 7 X[14568] - 5 X[19570], X[22567] - 3 X[30471], X[22569] - 3 X[30472], 5 X[27088] - 6 X[32459], 3 X[33228] - 2 X[36523], 4 X[37350] - 3 X[39563]

X(39785) lies on these lines: {2, 39}, {3, 15533}, {30, 10992}, {32, 12151}, {39, 7795, 39784}, {69, 7618}, {99, 3849}, {115, 8355}, {183, 7622}, {187, 524}, {193, 37809}, {230, 9167}, {298, 9885}, {299, 9886}, {315, 34504}, {316, 8591}, {325, 543}, {511, 8724}, {532, 35303}, {533, 35304}, {542, 18860}, {574, 599}, {597, 7820}, {620, 5215}, {625, 671}, {754, 8598}, {1007, 7615}, {1975, 7775}, {1992, 5008}, {2549, 9741}, {3292, 8030}, {3534, 30270}, {3629, 19661}, {3734, 11163}, {3906, 4141}, {3933, 7810}, {4403, 32851}, {5007, 7863}, {5024, 21358}, {5041, 7789}, {5077, 7818}, {5148, 12351}, {5194, 12350}, {5319, 33197}, {5461, 31275}, {5475, 9770}, {6337, 7855}, {6388, 37746}, {7230, 17095}, {7603, 11184}, {7610, 17131}, {7619, 37688}, {7620, 18424}, {7748, 32818}, {7758, 32985}, {7759, 33007}, {7764, 8370}, {7765, 8360}, {7772, 33237}, {7779, 32456}, {7780, 33274}, {7781, 7821}, {7782, 7882}, {7783, 7883}, {7788, 35955}, {7794, 8359}, {7796, 7833}, {7805, 7891}, {7809, 8597}, {7812, 7816}, {7842, 7871}, {7853, 7908}, {7854, 33215}, {7861, 7947}, {7888, 11318}, {7897, 32480}, {7905, 34604}, {7925, 9166}, {8176, 11185}, {8182, 11160}, {8367, 9698}, {8593, 12215}, {8721, 11001}, {9155, 32225}, {9740, 21843}, {9766, 11159}, {9888, 11161}, {10150, 14061}, {10717, 39602}, {10983, 19709}, {11149, 33687}, {11168, 12040}, {11645, 35002}, {14023, 35287}, {14961, 34897}, {15693, 37479}, {19911, 39099}, {22567, 30471}, {22569, 30472}, {23878, 35522}, {33228, 36523}, {34505, 39565}, {37350, 39563}

X(39785) = midpoint of X(i) and X(j) for these {i,j}: {99, 7840}, {316, 8591}, {2482, 7813}, {7757, 9865}
X(39785) = reflection of X(i) in X(j) for these {i,j}: {115, 22110}, {187, 2482}, {671, 625}, {2482, 6390}, {7845, 7840}, {8591, 15301}, {8598, 36521}, {22329, 620}, {31173, 325}
X(39785) = isotomic conjugate of X(18818)
X(39785) = complement of X(11054)
X(39785) = complement of the isotomic conjugate of X(34898)
X(39785) = X(i)-complementary conjugate of X(j) for these (i,j): {798, 31654}, {34898, 2887}
X(39785) = X(2418)-Ceva conjugate of X(690)
X(39785) = X(i)-isoconjugate of X(j) for these (i,j): {31, 18818}, {598, 923}, {897, 1383}, {8599, 36142}, {11636, 23894}
X(39785) = crosspoint of X(2) and X(34898)
X(39785) = crosssum of X(6) and X(11580)
X(39785) = crossdifference of every pair of points on line {669, 1383}
X(39785) = barycentric product X(i)*X(j) for these {i,j}: {187, 9464}, {524, 599}, {574, 3266}, {690, 9146}, {3906, 5468}, {5094, 6390}, {7813, 10130}, {9145, 35522}, {13857, 36890}, {14210, 36263}
X(39785) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 18818}, {187, 1383}, {524, 598}, {574, 111}, {599, 671}, {690, 8599}, {1649, 23287}, {3906, 5466}, {3908, 5380}, {5094, 17983}, {5467, 11636}, {5468, 35138}, {7813, 23297}, {8030, 20380}, {8541, 8753}, {9145, 691}, {9146, 892}, {9464, 18023}, {10510, 14246}, {13857, 9214}, {14357, 10511}, {14444, 20382}, {17414, 9178}, {32583, 34574}, {36263, 897}
X(39785) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {194, 7870, 7817}, {574, 599, 15810}, {599, 11165, 574}, {620, 22329, 5215}, {3926, 34511, 7801}, {6337, 7855, 15513}, {6390, 7813, 187}, {7781, 32821, 7821}, {7799, 9865, 7880}, {7801, 34511, 39}, {7817, 7870, 7874}, {7836, 32450, 7852}, {7908, 31859, 7853}

X(39786) = X(2)X(874)∩X(10)X(37)

Barycentrics    a*(b - c)^2*(b + c)*(a^2 - b*c) : :

X(39786) lies on these lines: {2, 874}, {6, 36280}, {10, 37}, {11, 6377}, {39, 5701}, {75, 19974}, {115, 804}, {292, 24715}, {514, 23822}, {764, 2087}, {812, 1015}, {893, 33135}, {1211, 22184}, {1573, 4364}, {1646, 1647}, {1836, 23543}, {1966, 16706}, {2170, 3123}, {2229, 4442}, {2275, 33149}, {2642, 2643}, {2887, 21345}, {3120, 3121}, {3735, 4443}, {3782, 22199}, {3836, 20363}, {3914, 16584}, {3925, 21827}, {4389, 16975}, {4854, 21838}, {4972, 16587}, {6378, 21927}, {6650, 37128}, {8620, 33136}, {9468, 19637}, {16705, 16738}, {16829, 17320}, {17448, 36217}, {17961, 24624}, {20284, 33141}, {20593, 20861}, {21044, 21835}, {21093, 21893}, {21950, 22227}, {21963, 22215}, {23632, 33145}, {24289, 24484}, {24528, 33101}, {27846, 38989}, {30646, 33132}, {30647, 33128}, {30659, 33125}, {30666, 32776}, {35092, 39011}

X(39786) = midpoint of X(17493) and X(30940)
X(39786) = complement of X(874)
X(39786) = complement of the isogonal conjugate of X(875)
X(39786) = complement of the isotomic conjugate of X(876)
X(39786) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 27854}, {32, 27929}, {291, 21260}, {292, 3835}, {335, 21262}, {513, 20542}, {649, 20333}, {667, 17793}, {669, 35068}, {741, 512}, {813, 27076}, {875, 10}, {876, 2887}, {1911, 513}, {1919, 17755}, {1922, 514}, {1927, 3709}, {1967, 21051}, {1977, 35119}, {3248, 38989}, {3572, 141}, {4128, 2679}, {4444, 626}, {9468, 25666}, {14598, 650}, {18263, 2786}, {18265, 4521}, {18267, 665}, {18268, 4369}, {18827, 23301}, {18897, 6586}, {20981, 39080}, {34067, 24003}
X(39786) = X(i)-Ceva conjugate of X(j) for these (i,j): {740, 4155}, {876, 8034}, {1086, 38989}, {1284, 4455}, {2238, 21832}, {3948, 4010}, {6650, 513}, {13576, 512}, {17493, 812}, {27809, 523}
X(39786) = X(i)-isoconjugate of X(j) for these (i,j): {59, 36800}, {81, 5378}, {99, 813}, {100, 4584}, {101, 4589}, {110, 4562}, {163, 4583}, {291, 4567}, {292, 4600}, {335, 4570}, {660, 662}, {692, 4639}, {741, 1016}, {765, 37128}, {799, 34067}, {805, 18047}, {1018, 36066}, {1252, 18827}, {1911, 4601}, {2311, 4998}, {4565, 36801}, {4579, 37134}, {4620, 7077}, {7035, 18268}, {20964, 39292}
X(39786) = crosspoint of X(i) and X(j) for these (i,j): {2, 876}, {740, 812}, {1019, 1929}, {1284, 7212}, {2238, 21832}, {3948, 4010}, {27846, 27918}
X(39786) = crosssum of X(i) and X(j) for these (i,j): {6, 3573}, {741, 813}, {1018, 1757}, {2284, 22116}, {4562, 36800}, {4576, 30941}, {4584, 37128}
X(39786) = crossdifference of every pair of points on line {660, 662}
X(39786) = barycentric product X(i)*X(j) for these {i,j}: {10, 27846}, {11, 1284}, {37, 27918}, {65, 4124}, {238, 3120}, {239, 3125}, {242, 18210}, {244, 740}, {350, 3122}, {512, 3766}, {513, 4010}, {514, 21832}, {523, 659}, {650, 7212}, {661, 812}, {693, 4455}, {862, 1565}, {874, 8034}, {1015, 3948}, {1086, 2238}, {1109, 5009}, {1111, 3747}, {1358, 4433}, {1429, 21044}, {1447, 4516}, {1577, 8632}, {1874, 7004}, {1914, 16732}, {1921, 3121}, {2170, 16609}, {2201, 4466}, {2210, 21207}, {2632, 34856}, {2643, 33295}, {3124, 30940}, {3248, 35544}, {3708, 31905}, {3716, 4017}, {4037, 16726}, {4148, 7216}, {4155, 7192}, {4435, 7178}, {6591, 24459}, {7235, 18191}, {13576, 38989}, {16592, 17493}, {18014, 38348}, {22384, 24006}
X(39786) = barycentric quotient X(i)/X(j) for these {i,j}: {42, 5378}, {238, 4600}, {239, 4601}, {244, 18827}, {512, 660}, {513, 4589}, {514, 4639}, {523, 4583}, {649, 4584}, {659, 99}, {661, 4562}, {669, 34067}, {740, 7035}, {798, 813}, {812, 799}, {862, 15742}, {1015, 37128}, {1284, 4998}, {1429, 4620}, {1914, 4567}, {1977, 18268}, {2086, 2295}, {2170, 36800}, {2210, 4570}, {2238, 1016}, {3120, 334}, {3121, 292}, {3122, 291}, {3125, 335}, {3248, 741}, {3716, 7257}, {3733, 36066}, {3747, 765}, {3766, 670}, {3948, 31625}, {4010, 668}, {4041, 36801}, {4124, 314}, {4128, 18787}, {4148, 7258}, {4155, 3952}, {4433, 4076}, {4435, 645}, {4455, 100}, {4516, 4518}, {4560, 36806}, {5009, 24041}, {5027, 4579}, {7212, 4554}, {8034, 876}, {8632, 662}, {16592, 30669}, {16732, 18895}, {18105, 36081}, {18210, 337}, {21832, 190}, {22384, 4592}, {24193, 17205}, {27846, 86}, {27918, 274}, {30940, 34537}, {33295, 24037}, {34856, 23999}, {35119, 30940}, {38348, 17934}, {38978, 2664}, {38989, 30941}
X(39786) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3120, 3121, 16592}, {4972, 21327, 16587}

X(39787) = X(1)X(36737)∩X(7)X(80)

Barycentrics    a*(a + b - c)*(a - b + c)*(a^2 - b^2 - 2*b*c - c^2 - 2*Sqrt[3]*S)*((a + b + c)*(a*b - b^2 + a*c + 2*b*c - c^2) + 2*Sqrt[3]*(b + c)*S) : :

X(39787) lies on the Kiepert circumhyperbola of the intouch triangle and on these lines: {1, 36737}, {7, 80}, {56, 36940}, {57, 11789}, {101, 1652}, {103, 10650}, {116, 30383}, {202, 37772}, {354, 3639}, {517, 3638}, {3022, 30378}, {10648, 11712}, {10695, 30322}, {11714, 18471}, {14760, 30352}, {30357, 39156}

X(39787) = crosspoint of X(7) and X(37772)
X(39787) = crosssum of X(55) and X(19551)
X(39787) = intouch-isogonal-conjugate of X(3638)
X(39787) = X(13)-of-intouch-triangle
X(39787) = {X(7),X(5902)}-harmonic conjugate of X(39788)

X(39788) = X(1)X(36738)∩X(7)X(80)

Barycentrics    a*(a + b - c)*(a - b + c)*(a^2 - b^2 - 2*b*c - c^2 + 2*Sqrt[3]*S)*((a + b + c)*(a*b - b^2 + a*c + 2*b*c - c^2) - 2*Sqrt[3]*(b + c)*S) : :

X(39788) lies on the Kiepert circumhyperbola of the intouch triangle and on these lines: {1, 36738}, {7, 80}, {56, 36941}, {57, 11752}, {101, 1653}, {103, 10649}, {116, 30382}, {203, 37773}, {354, 3638}, {517, 3639}, {3022, 30377}, {10647, 11712}, {10695, 30321}, {11714, 18469}, {14760, 30351}, {30356, 39156}

X(39788) = crosspoint of X(7) and X(37773)
X(39788) = crosssum of X(55) and X(7126)
X(39788) = intouch-isogonal-conjugate of X(3639)
X(39788) = X(14)-of-intouch-triangle
X(39788) = {X(7),X(5902)}-harmonic conjugate of X(39787)

X(39789) = X(1)X(1362)∩X(11)X(2140)

Barycentrics    a^2*(a - b - c)*(a^2*b^2 - 2*a*b^3 + b^4 + 2*a*b^2*c - 2*b^3*c + a^2*c^2 + 2*a*b*c^2 + 2*b^2*c^2 - 2*a*c^3 - 2*b*c^3 + c^4) : :

X(39789) lies on the Kiepert circumhyperbola of the intouch triangle and on these lines: {1, 1362}, {11, 2140}, {55, 218}, {57, 170}, {65, 516}, {181, 10382}, {226, 34848}, {497, 17753}, {674, 21872}, {1212, 2389}, {1334, 2293}, {1401, 10391}, {1697, 3056}, {1742, 29957}, {2082, 3271}, {2310, 21808}, {3779, 4326}, {4014, 24796}, {4845, 9327}, {4890, 11553}, {4907, 11518}, {7084, 14935}, {10939, 17609}, {13077, 24840}, {17435, 23667}, {21346, 22440}, {36639, 37734}, {38451, 39391}

X(39789) = X(21346)-Ceva conjugate of X(23653)
X(39789) = crosspoint of X(7) and X(55)
X(39789) = crosssum of X(7) and X(55)
X(39789) = intouch-isogonal-conjugate of X(10481)
X(39789) = intouch-isotomic-conjugate of X(7)
X(39789) = X(76)-of-intouch-triangle
X(39789) = barycentric product X(i)*X(j) for these {i,j}: {8, 23653}, {9, 21346}, {41, 21436}, {55, 21258}, {281, 22440}, {284, 21931}, {3939, 23748}
X(39789) = barycentric quotient X(i)/X(j) for these {i,j}: {21258, 6063}, {21346, 85}, {21436, 20567}, {21931, 349}, {22440, 348}, {23653, 7}
X(39789) = {X(31588),X(31589)}-harmonic conjugate of X(5728)

X(39790) = X(65)X(5542)∩X(354)X(10481)

Barycentrics    a*(a*b - b^2 + a*c + 2*b*c - c^2)*(a^3*b - 2*a^2*b^2 + a*b^3 + a^3*c - 2*a^2*b*c - a*b^2*c + 2*b^3*c - 2*a^2*c^2 - a*b*c^2 - 4*b^2*c^2 + a*c^3 + 2*b*c^3) : :

X(39790) lies on the Kiepert circumhyperbola of the intouch triangle and on these lines: {65, 5542}, {354, 10481}, {942, 1362}, {1401, 7195}, {4253, 4860}, {4334, 11518}, {5571, 10489}, {14520, 18398}, {17758, 20683}, {21746, 24796}

X(39790) = X(36838)-Ceva conjugate of X(10581)
X(39790) = crosspoint of X(i) and X(j) for these (i,j): {7, 354}, {10481, 17758}
X(39790) = crosssum of X(i) and X(j) for these (i,j): {55, 2346}, {4251, 10482}
X(39790) = barycentric product X(354)*X(6706)
X(39790) = intouch-isotomic-conjugate of X(5572)
X(39790) = X(83)-of-intouch-triangle
X(39790) = trilinear pole, wrt intouch triangle, of antiorthic axis
X(39790) = {X(10481),X(15658)}-harmonic conjugate of X(354)

X(39791) = X(1)X(6000)∩X(56)X(58)

Barycentrics    a^2*(a + b - c)*(a - b + c)*(b + c)*(a^2 - b^2 - c^2)*(a^2*b - b^3 + a^2*c + 2*a*b*c + b^2*c + b*c^2 - c^3) : :

X(39791) lies on the Kiepert circumhyperbola of the intouch triangle and on these lines: {1, 6000}, {48, 3284}, {56, 58}, {57, 2939}, {65, 1439}, {73, 228}, {185, 20277}, {221, 1486}, {223, 19366}, {354, 946}, {389, 3468}, {942, 1838}, {950, 2822}, {1214, 7066}, {1361, 1363}, {1362, 1367}, {1364, 2646}, {1406, 37538}, {2260, 14597}, {3157, 6056}, {4303, 23207}, {4890, 11553}, {6357, 18180}, {7100, 13754}, {10974, 18593}, {22115, 23070}, {26888, 34043}

X(39791) = reflection of X(1844) in X(942)
X(39791) = X(222)-Ceva conjugate of X(14597)
X(39791) = X(i)-isoconjugate of X(j) for these (i,j): {29, 943}, {318, 1175}, {643, 14775}, {1794, 1896}, {2259, 31623}, {2322, 2982}
X(39791) = crosspoint of X(i) and X(j) for these (i,j): {7, 1214}, {222, 1439}, {942, 4303}
X(39791) = crosssum of X(i) and X(j) for these (i,j): {29, 11107}, {55, 1172}, {281, 4183}
X(39791) = crossdifference of every pair of points on line {3700, 17926}
X(39791) = intouch-isogonal-conjugate of X(4292)
X(39791) = X(96)-of-intouch-triangle
X(39791) = barycentric product X(i)*X(j) for these {i,j}: {7, 18591}, {65, 18607}, {73, 5249}, {77, 2294}, {222, 442}, {226, 4303}, {307, 2260}, {942, 1214}, {1441, 14597}, {1446, 23207}, {1804, 1865}, {1813, 23752}, {8021, 20618}
X(39791) = barycentric quotient X(i)/X(j) for these {i,j}: {442, 7017}, {942, 31623}, {1409, 943}, {1410, 2982}, {1841, 1896}, {2260, 29}, {2294, 318}, {4303, 333}, {7180, 14775}, {14547, 2322}, {14597, 21}, {18591, 8}, {18607, 314}, {23207, 2287}
X(39791) = {X(56),X(8614)}-harmonic conjugate of X(2194)

X(39792) = X(7)X(1002)∩X(57)X(846)

Barycentrics    a*(a + b - c)*(a - b + c)*(a^2 - a*b - a*c - 2*b*c)*(a*b^2 - b^3 + 4*a*b*c + b^2*c + a*c^2 + b*c^2 - c^3) : :

X(39792) lies on the Kiepert circumhyperbola of the intouch triangle and on these lines: {7, 1002}, {57, 846}, {65, 10521}, {170, 11518}, {354, 516}, {1401, 5173}, {2140, 3649}, {2808, 5902}, {3730, 5221}, {9436, 28600}

X(39792) = crosspoint of X(7) and X(5228)
X(39792) = intouch-isogonal-conjugate of X(5542)
X(39792) = X(262)-of-intouch-triangle
X(39792) = barycentric product X(3826)*X(5228)

X(39793) = X(1)X(37425)∩X(10)X(12)

Barycentrics    a*(a + b - c)*(a - b + c)*(b + c)*(a*b + a*c + 2*b*c) : :

X(39792) lies on the Kiepert circumhyperbola of the intouch triangle and on these lines: {1, 37425}, {7, 310}, {10, 12}, {51, 24725}, {56, 4278}, {57, 846}, {354, 3663}, {497, 17753}, {511, 33097}, {553, 1357}, {942, 24210}, {1042, 10474}, {1362, 1365}, {1397, 37543}, {1463, 3982}, {1464, 2099}, {1469, 4654}, {1836, 21746}, {3485, 27339}, {3664, 21334}, {3706, 4059}, {3720, 22060}, {3750, 29309}, {3873, 4442}, {3944, 5902}, {4111, 21020}, {4260, 17889}, {4388, 17049}, {4847, 21926}, {4862, 11021}, {4888, 10439}, {5943, 33096}, {9052, 33109}, {14839, 33073}, {17056, 20718}, {17768, 18165}, {17778, 35104}, {20256, 39542}, {20367, 21321}, {20961, 38389}, {21949, 22277}, {24248, 35612}, {33095, 39543}

X(39793) = X(i)-Ceva conjugate of X(j) for these (i,j): {7, 4059}, {4554, 7180}, {21859, 4017}
X(39793) = X(2667)-cross conjugate of X(16589)
X(39793) = X(i)-isoconjugate of X(j) for these (i,j): {284, 32009}, {3737, 8708}
X(39793) = crosspoint of X(7) and X(65)
X(39793) = crosssum of X(21) and X(55)
X(39793) = intouch-isogonal-conjugate of X(3664)
X(39793) = intouch-isotomic-conjugate of X(10391)
X(39793) = excentral-to-intouch similarity image of X(846)
X(39793) = X(275)-of-intouch-triangle
X(39793) = barycentric product X(i)*X(j) for these {i,j}: {7, 16589}, {12, 18166}, {37, 4059}, {57, 21020}, {65, 3739}, {85, 2667}, {181, 16748}, {226, 3720}, {279, 4111}, {331, 22369}, {1400, 20888}, {1427, 3706}, {1434, 21699}, {1441, 20963}, {2171, 17175}, {3668, 3691}, {4436, 7178}, {4552, 6372}, {6063, 21753}
X(39793) = barycentric quotient X(i)/X(j) for these {i,j}: {65, 32009}, {2667, 9}, {3691, 1043}, {3720, 333}, {3739, 314}, {4059, 274}, {4111, 346}, {4436, 645}, {4559, 8708}, {6372, 4560}, {16589, 8}, {16748, 18021}, {18166, 261}, {20888, 28660}, {20963, 21}, {21020, 312}, {21699, 2321}, {21753, 55}, {21820, 210}, {22060, 1812}, {22369, 219}
X(39793) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 10473, 1401}, {65, 226, 181}, {354, 4854, 4890}

X(39794) = X(1)X(16213)∩X(4)X(7)

Barycentrics    a*(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 - 4*b*c*S) : :

X(39794) lies on the Kiepert circumhyperbola of the intouch triangle and on these lines: {1, 16213}, {4, 7}, {57, 16432}, {65, 482}, {175, 5045}, {176, 517}, {218, 6204}, {222, 1124}, {241, 8953}, {354, 481}, {1362, 22107}, {1371, 5903}, {1373, 5902}, {1374, 18398}, {3057, 31538}, {5049, 17802}, {5697, 17806}, {6502, 10252}, {9957, 17805}, {10167, 31549}, {10980, 34495}, {14100, 31572}, {17609, 31539}, {21169, 31794}

X(39794) = crosspoint of X(7) and X(13389)
X(39794) = crosssum of X(55) and X(7133)
X(39794) = intouch-isogonal-conjugate of X(482)
X(39794) = intouch-isotomic-conjugate of X(30375)
X(39794) = X(485)-of-intouch-triangle
X(39794) = {X(7),X(942)}-harmonic conjugate of X(39795)

X(39795) = X(1)X(16214)∩X(4)X(7)

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*b*c*S) : :

X(39794) lies on the Kiepert circumhyperbola of the intouch triangle and on these lines: {1, 16214}, {4, 7}, {57, 16433}, {65, 481}, {175, 517}, {176, 5045}, {218, 6203}, {222, 1335}, {354, 482}, {1362, 22106}, {1372, 5903}, {1373, 18398}, {1374, 5902}, {2067, 10253}, {3057, 31539}, {5049, 17805}, {5697, 17803}, {9957, 17802}, {10167, 31550}, {10980, 34494}, {14100, 31571}, {17609, 31538}

X(39795) = crosspoint of X(7) and X(13388)
X(39795) = intouch-isogonal-conjugate of X(481)
X(39795) = intouch-isotomic-conjugate of X(30376)
X(39795) = X(486)-of-intouch-triangle
X(39795) = {X(7),X(942)}-harmonic conjugate of X(39794)

X(39796) = X(1)X(6000)∩X(3)X(1794)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^3*b^2 - a^2*b^3 - a*b^4 + b^5 + a^2*b^2*c - b^4*c + a^3*c^2 + a^2*b*c^2 + 2*a*b^2*c^2 - a^2*c^3 - a*c^4 - b*c^4 + c^5) : :

X(39794) lies on the Kiepert circumhyperbola of the intouch triangle and on these lines: {1, 6000}, {3, 1794}, {19, 26892}, {40, 15644}, {51, 2635}, {55, 103}, {57, 2947}, {64, 34046}, {65, 515}, {71, 3917}, {73, 185}, {84, 6254}, {196, 36996}, {209, 1155}, {216, 22069}, {223, 11436}, {354, 1439}, {389, 1745}, {511, 24310}, {916, 1214}, {942, 1888}, {971, 1859}, {1363, 34228}, {1367, 35504}, {1437, 4278}, {1473, 19350}, {1490, 19366}, {1836, 21746}, {2003, 11428}, {2550, 26871}, {2654, 11381}, {2808, 24430}, {3198, 8679}, {3220, 10536}, {3270, 20277}, {3611, 3937}, {3688, 18734}, {3779, 7289}, {3784, 10319}, {3925, 26932}, {4303, 5562}, {5751, 37543}, {5907, 37523}, {6237, 24467}, {7004, 37755}, {7335, 37579}, {9729, 37694}, {12428, 34800}, {18673, 23154}, {18921, 26929}, {20731, 22418}, {22070, 22411}, {22084, 22410}, {26933, 26957}, {31657, 39007}

X(39796) = isogonal conjugate of the polar conjugate of X(16608)
X(39796) = crosspoint of X(3) and X(7)
X(39796) = crosssum of X(4) and X(55)
X(39796) = intouch-isogonal-conjugate of X(3668)
X(39796) = intouch-isotomic-conjugate of X(1836)
X(39796) = excentral-to-intouch similarity image of X(2947)
X(39796) = X(5392)-of-intouch-triangle
X(39796) = barycentric product X(i)*X(j) for these {i,j}: {3, 16608}, {48, 23581}, {1331, 23726}, {1790, 21911}
X(39796) = barycentric quotient X(i)/X(j) for these {i,j}: {16608, 264}, {23581, 1969}
X(39796) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 3173, 6056}, {3611, 3937, 26934}, {3917, 22440, 22053}, {5751, 37543, 37993}

X(39797) = ISOGONAL CONJUGATE OF X(16552)

Barycentrics    a*(a^2*b + a*b^2 + a^2*c + b^2*c - a*c^2 - b*c^2)*(a^2*b - a*b^2 + a^2*c - b^2*c + a*c^2 + b*c^2) : :

X(39797) lies on the conic {{A,B,C,X(1),X(2)}} and these lines: {1, 5132}, {2, 2140}, {81, 1730}, {105, 595}, {239, 35058}, {274, 16552}, {277, 579}, {291, 3216}, {386, 1002}, {955, 37264}, {980, 19731}, {1219, 9534}, {1224, 33159}, {1400, 24790}, {1764, 37870}, {1929, 3336}, {3218, 39706}, {3227, 34063}, {6553, 20018}, {9401, 17596}, {17143, 29437}, {17946, 24897}, {18206, 19742}, {21371, 32092}, {21384, 36871}, {25430, 37555}, {29438, 29454}, {29439, 32019}, {29445, 33793}, {29487, 35348}, {29552, 30701}

X(39797) = isogonal conjugate of X(16552)
X(39797) = isotomic conjugate of X(18137)
X(39797) = isogonal conjugate of the anticomplement of X(17758)
X(39797) = isogonal conjugate of the isotomic conjugate of X(39735)
X(39797) = X(i)-cross conjugate of X(j) for these (i,j): {213, 1}, {2350, 6}, {17761, 1019}
X(39797) = X(i)-isoconjugate of X(j) for these (i,j): {1, 16552}, {2, 8053}, {3, 17911}, {4, 22126}, {6, 17135}, {31, 18137}, {42, 29767}, {55, 17077}, {58, 21070}, {81, 22271}, {101, 8714}
X(39797) = cevapoint of X(i) and X(j) for these (i,j): {244, 798}, {661, 17463}, {2238, 20356}
X(39797) = trilinear pole of line {513, 23506}
X(39797) = barycentric product X(i)*X(j) for these {i,j}: {1, 8049}, {6, 39735}, {75, 34444}, {693, 6577}
X(39797) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17135}, {2, 18137}, {6, 16552}, {19, 17911}, {31, 8053}, {37, 21070}, {42, 22271}, {48, 22126}, {57, 17077}, {81, 29767}, {513, 8714}, {6577, 100}, {8049, 75}, {34444, 1}, {39735, 76}
X(39797) = {X(274),X(33792)}-harmonic conjugate of X(16552)

X(39798) = ISOGONAL CONJUGATE OF X(32911)

Barycentrics    a*(a*b + b^2 - a*c + b*c)*(a*b - a*c - b*c - c^2) : :

X(39798) lies on the conic {{A,B,C,X(2),X(6)}} and these lines: {2, 3770}, {6, 474}, {9, 36814}, {25, 36743}, {37, 39}, {42, 1100}, {44, 583}, {75, 26963}, {111, 34594}, {141, 16726}, {251, 1333}, {274, 308}, {330, 30473}, {393, 475}, {404, 2220}, {493, 31473}, {572, 34079}, {594, 1015}, {749, 3681}, {757, 1171}, {941, 4261}, {967, 4383}, {1009, 1279}, {1107, 16828}, {1169, 5035}, {1427, 3911}, {1506, 5949}, {1574, 17362}, {1880, 8756}, {2238, 2350}, {2245, 28349}, {2275, 14624}, {2981, 5367}, {2998, 27318}, {3009, 21865}, {3108, 5276}, {3688, 22323}, {3758, 27641}, {3834, 29964}, {4007, 9336}, {4016, 39244}, {4043, 27166}, {4370, 21826}, {4670, 27633}, {5013, 19528}, {5022, 14553}, {5109, 28245}, {5124, 20833}, {5362, 6151}, {8050, 17275}, {8053, 37586}, {15988, 30535}, {16549, 16685}, {16606, 25121}, {16666, 21857}, {16669, 21892}, {16814, 28403}, {17077, 17278}, {17277, 37128}, {17337, 31198}, {17351, 28366}, {17381, 24598}, {17385, 37596}, {20456, 22271}, {26982, 29756}, {37675, 39389}

X(39798) = complement of X(18133)
X(39798) = isogonal conjugate of X(32911)
X(39798) = isotomic conjugate of X(18140)
X(39798) = complement of the isotomic conjugate of X(39748)
X(39798) = isogonal conjugate of the complement of X(32863)
X(39798) = X(i)-complementary conjugate of X(j) for these (i,j): {35058, 626}, {39748, 2887}
X(39798) = X(39747)-Ceva conjugate of X(596)
X(39798) = X(i)-cross conjugate of X(j) for these (i,j): {756, 1}, {1213, 37}, {3248, 513}, {20966, 65}, {21035, 13476}, {21936, 34434}
X(39798) = X(i)-isoconjugate of X(j) for these (i,j): {1, 32911}, {2, 595}, {6, 4360}, {31, 18140}, {57, 3871}, {58, 3995}, {63, 4222}, {75, 2220}, {81, 3293}, {100, 4063}, {101, 20295}, {110, 4129}, {190, 4057}, {593, 4075}, {662, 4132}, {692, 20949}, {727, 27044}, {1016, 8054}, {1171, 4065}, {1252, 21208}, {1331, 17922}, {1897, 22154}
X(39798) = cevapoint of X(i) and X(j) for these (i,j): {6, 5124}, {244, 2530}, {661, 1015}, {3121, 4983}
X(39798) = crosspoint of X(2) and X(39748)
X(39798) = crosssum of X(6) and X(3216)
X(39798) = trilinear pole of line {512, 4895} (the incentral-isotomic-conjugate of the incentral inellipse)
X(39798) = crossdifference of every pair of points on line {4057, 4063}
X(39798) = barycentric product X(i)*X(j) for these {i,j}: {1, 596}, {8, 20615}, {37, 39747}, {513, 8050}, {523, 34594}, {661, 37205}
X(39798) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4360}, {2, 18140}, {6, 32911}, {25, 4222}, {31, 595}, {32, 2220}, {37, 3995}, {42, 3293}, {55, 3871}, {244, 21208}, {512, 4132}, {513, 20295}, {514, 20949}, {596, 75}, {649, 4063}, {661, 4129}, {667, 4057}, {756, 4075}, {1575, 27044}, {1962, 4065}, {3248, 8054}, {6591, 17922}, {8050, 668}, {20615, 7}, {22383, 22154}, {34594, 99}, {37205, 799}, {39747, 274}
X(39798) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 16710, 18143}, {39, 17398, 37}, {583, 992, 44}, {1100, 1575, 21858}, {5153, 24512, 1100}, {8610, 17355, 37}, {17053, 17369, 37}

X(39799) = X(233)X(13162) ∩ X(647)X(39503)

Barycentrics    (2*(b^2+c^2)*a^6-(3*b^4+2*b^2*c^2+3*c^4)*a^4-(b^2-c^2)^2*b^2*c^2+(b^4-c^4)*(b^2-c^2)*a^2)*(b^2-c^2) : :

Let K be a conic or circle and A, B, C three distinct and non-collinear points external to K. Let A', A" be the intersections of the line BC with the tangent lines to K through A. Build B', B" and C', C" cyclically. Then these six points lie on another conic K' (César Lozada, September 23, 2020)

If ABC is acute and K is the nine-point-circle of ABC, then K' is a hyperbola through X(3005) and X(15451) and having center X(39799).
If K is the Spieker circle of ABC (i.e., the incircle of its medial triangle) then the center of K' is X(39800).

X(39799) lies on these lines: {233,13162}, {647,39503}, {804,34964}, {826,1506}, {3569,22889}, {7755,37085}

X(39800) = X(2)X(8834) ∩ X(8)X(3158)

Barycentrics    (3*a^3+13*(b+c)*a^2+(b+c)^2*a-(b+c)*(9*b^2-14*b*c+9*c^2))*(-a+b+c) : :

See X(39799).

X(39800) lies on these lines: {2,8834}, {8,3158}, {10,3161}, {2551,31722}, {3616,3756}, {6557,9780}, {9746,39570}, {19877,28661}, {21896,25082}

leftri

More centers related to 7th-, 8th- and 9th- Brocard triangles: X(39801)-X(39913)

rightri

This preamble and centers X(39801)-X(39913) were contributed by César Eliud Lozada, October 3, 2020.

7th-, 8th- and 9th- Brocard triangles are defined in the preamble just before X(38643). A list of related centers may be seen here.

X(39801) = PERSPECTOR OF THESE TRIANGLES: 7th BROCARD AND LUCAS REFLECTION

Barycentrics
SA*(SB+SC)*(4*S^6-4*(2*R^2-SA+3*SW)*S^5+(32*R^4-8*R^2*SA-28*R^2*SW+8*SA^2-8*SB*SC-21*SW^2)*S^4+2*(SA^2-2*SB*SC-9*SW^2)*SW*S^3+SW*(-SW*(SA^2+9*SW^2)-4*(SA+SW)*(2*SA-3*SW)*R^2+8*R^4*SW)*S^2+2*(4*R^2*SW-SA^2-SW^2)*SW^3*S-(2*R^2-SW)*(2*R^2*SW-SA^2)*SW^3) : :

X(39801) lies on these lines: {5028,11984}, {6401,6776}

X(39802) = PERSPECTOR OF THESE TRIANGLES: 7th BROCARD AND LUCAS(-1) REFLECTION

Barycentrics
SA*(SB+SC)*(4*S^6+4*(2*R^2-SA+3*SW)*S^5+(32*R^4-8*R^2*SA-28*R^2*SW+8*SA^2-8*SB*SC-21*SW^2)*S^4-2*(SA^2-2*SB*SC-9*SW^2)*SW*S^3+SW*(-SW*(SA^2+9*SW^2)-4*(SA+SW)*(2*SA-3*SW)*R^2+8*R^4*SW)*S^2-2*(4*R^2*SW-SA^2-SW^2)*SW^3*S-(2*R^2-SW)*(2*R^2*SW-SA^2)*SW^3) : :

X(39802) lies on these lines: {5028,11985}, {6402,6776}

X(39803) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO 7th BROCARD

Barycentrics    (2*S^4+(2*SA^2+4*SW*R^2-3*SB*SC-SW^2)*S^2+(4*R^2-SA-SW)*SW^2*SA)*(SB+SC) : :
X(39803) = 3*X(11402)-4*X(39805) = 3*X(11402)-2*X(39810)

The reciprocal orthologic center of these triangles is X(3)

X(39803) lies on these lines: {3,76}, {4,13175}, {6,39817}, {22,147}, {25,114}, {115,7395}, {148,7503}, {184,39820}, {378,13172}, {427,39813}, {542,13171}, {690,12168}, {1398,39815}, {1593,23698}, {1597,10723}, {1609,12829}, {1634,19165}, {1993,39807}, {2794,11414}, {2936,39854}, {2967,4558}, {3023,10831}, {3027,10832}, {3425,20794}, {3515,39825}, {3516,10992}, {3964,32458}, {5094,39816}, {5285,24469}, {5410,39823}, {5411,39824}, {5562,39849}, {5984,6636}, {6033,7387}, {6036,7484}, {6054,9909}, {6321,9818}, {6642,15561}, {6721,11284}, {7071,39822}, {7393,38224}, {7509,14651}, {7517,38743}, {7539,39120}, {7592,39808}, {7970,12410}, {8724,14070}, {9715,14981}, {9772,22655}, {9777,39806}, {9798,9864}, {9860,37557}, {9862,10323}, {9915,36776}, {10722,39568}, {10753,37491}, {10768,13222}, {10833,12185}, {11005,12310}, {11245,39804}, {11402,39805}, {11403,39809}, {11405,39819}, {11406,39821}, {11408,39829}, {11409,39830}, {11410,39831}, {11479,14639}, {12083,38744}, {12085,38730}, {12160,39839}, {12177,37488}, {12178,37577}, {12184,18954}, {14118,20094}, {16030,39814}, {17834,39846}, {17974,36790}, {18386,39818}, {18534,22505}, {19118,39811}, {19404,39826}, {19405,39827}, {21312,38738}, {35243,38741}, {37198,38749}, {38552,39193}

X(39803) = reflection of X(i) in X(j) for these (i,j): (12160, 39839), (39810, 39805), (39832, 3)
X(39803) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22, 147, 9861), (114, 39828, 25), (38738, 39841, 21312), (39805, 39810, 11402)

X(39804) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO 7th BROCARD

Barycentrics    (8*R^2-3*SA+SW)*S^4+(2*(SA-2*SW)*R^2+SA^2-SB*SC)*SW*S^2+2*SB*SC*SW^2*R^2 : :

The reciprocal orthologic center of these triangles is X(3)

X(39804) lies on these lines: {2,39810}, {4,39806}, {52,39842}, {69,6036}, {98,6515}, {99,18916}, {114,11433}, {115,11411}, {147,37644}, {542,7714}, {690,18932}, {1899,39813}, {2782,18951}, {3044,3147}, {6321,18917}, {6721,18928}, {6722,11487}, {6776,39828}, {11245,39803}, {12324,39809}, {13175,18914}, {13567,39820}, {14912,39805}, {18909,23698}, {18911,39807}, {18912,39808}, {18913,39812}, {18915,39815}, {18918,39818}, {18919,39819}, {18921,39821}, {18922,39822}, {18923,39823}, {18924,39824}, {18925,39825}, {18926,39826}, {18927,39827}, {18929,39829}, {18930,39830}, {18931,39831}, {19119,39811}, {19166,39814}, {23291,39816}

X(39804) = reflection of X(39833) in X(18951)
X(39804) = {X(1899), X(39817)}-harmonic conjugate of X(39813)

X(39805) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-CONWAY TO 7th BROCARD

Barycentrics
a^2*(a^12-4*(b^2+c^2)*a^10+7*(b^4+b^2*c^2+c^4)*a^8-(b^2+c^2)*(7*b^4-5*b^2*c^2+7*c^4)*a^6+4*(b^6-c^6)*(b^2-c^2)*a^4-(b^6+c^6)*(b^4-4*b^2*c^2+c^4)*a^2-(b^2-c^2)^2*(b^4+c^4)*b^2*c^2) : :
X(39805) = 3*X(11402)+X(39803) = 3*X(11402)-X(39810)

The reciprocal orthologic center of these triangles is X(3)

X(39805) lies on these lines: {2,98}, {3,39839}, {6,39806}, {26,39835}, {49,15561}, {54,99}, {115,569}, {389,39825}, {567,6321}, {578,23698}, {620,1147}, {690,12228}, {1092,38748}, {1176,10753}, {2782,32046}, {5477,19131}, {9666,15452}, {10539,36519}, {10723,15033}, {10984,38749}, {11402,39803}, {11422,39807}, {11423,39808}, {11424,39809}, {11425,39812}, {11426,13175}, {11427,39813}, {11428,39821}, {11429,39822}, {11430,39831}, {11597,15545}, {13336,38737}, {13346,38736}, {13352,38738}, {13353,38224}, {13366,39817}, {13434,14639}, {14912,39804}, {15074,39848}, {17809,39820}, {18388,39818}, {18475,39854}, {19365,39815}, {19408,39826}, {19409,39827}, {21166,34148}, {22115,38750}, {23292,39816}, {37471,38739}, {37472,38730}, {37476,39849}, {37495,38731}

X(39805) = midpoint of X(i) and X(j) for these {i,j}: {3, 39839}, {39803, 39810}
X(39805) = reflection of X(39834) in X(32046)
X(39805) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 39828, 39806), (11402, 39803, 39810)

X(39806) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CONWAY TO 7th BROCARD

Barycentrics    (-2*S^4+(SA^2+2*SW*R^2-3*SB*SC+SW^2)*S^2+(2*R^2-SW)*SW^2*SA)*(SB+SC) : :
X(39806) = 3*X(51)-X(114) = 3*X(51)+X(39817) = X(98)+3*X(3060) = X(99)-5*X(3567) = X(147)-9*X(11002) = 3*X(568)+X(6321) = 3*X(671)+X(39837) = X(5562)-3*X(23514) = 9*X(5640)-X(39807) = X(5889)+3*X(14639) = 3*X(5890)+X(10723) = 3*X(5943)-2*X(6721) = 3*X(5946)-X(33813) = X(6101)-3*X(34127) = X(6243)+3*X(38224) = 3*X(9730)-X(38738) = 7*X(9781)+X(39808) = X(10625)-3*X(38737) = X(11412)-5*X(14061) = X(13188)-9*X(13321)

The reciprocal orthologic center of these triangles is X(3).

X(39806) lies on the nine-point circle of the orthic triangle and on these lines: {4,39804}, {6,39805}, {25,39810}, {26,39834}, {51,114}, {52,115}, {98,3060}, {99,3567}, {143,2782}, {147,11002}, {182,20859}, {185,39809}, {230,511}, {389,23698}, {428,542}, {568,6321}, {576,2967}, {578,39825}, {620,5462}, {671,39837}, {690,12236}, {1216,6722}, {2794,5446}, {3044,3518}, {5186,6746}, {5562,23514}, {5640,39807}, {5889,14639}, {5890,10723}, {5943,6721}, {5946,33813}, {5969,32191}, {6055,21969}, {6101,34127}, {6102,22515}, {6243,38224}, {9729,38736}, {9730,38738}, {9777,39803}, {9781,39808}, {9786,39812}, {9792,39814}, {9880,14831}, {10263,12042}, {10625,38737}, {11412,14061}, {11432,13175}, {11433,39813}, {11435,39821}, {11436,39822}, {11438,39831}, {11591,15092}, {13188,13321}, {13567,39816}, {15043,21166}, {16625,38734}, {17810,39820}, {18390,39818}, {19366,39815}, {19410,39826}, {19411,39827}, {35360,39120}, {37481,38730}, {37484,38739}, {37493,39839}

X(39806) = midpoint of X(i) and X(j) for these {i,j}: {52, 115}, {114, 39817}, {185, 39809}, {6055, 21969}, {6102, 22515}, {9880, 14831}, {10263, 12042}
X(39806) = reflection of X(i) in X(j) for these (i,j): (620, 5462), (1216, 6722), (11591, 15092), (38736, 9729), (39835, 143)
X(39806) = X(118)-of-orthic-triangle if ABC is acute
X(39806) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 39828, 39805), (51, 39817, 114)

X(39807) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO 7th BROCARD

Barycentrics    (S^4+(7*SA^2+8*SW*R^2-6*SB*SC-SW^2)*S^2+(8*R^2-3*SA-2*SW)*SW^2*SA)*(SB+SC) : :
X(39807) = 2*X(98)-3*X(2979) = 4*X(114)-3*X(3060) = 4*X(115)-5*X(11444) = 8*X(620)-7*X(15043) = 4*X(1216)-3*X(14651) = 5*X(3567)-6*X(15561) = 9*X(5640)-8*X(39806) = 3*X(5890)-4*X(33813) = 8*X(6036)-9*X(7998) = 2*X(6321)-3*X(11459) = 16*X(6721)-15*X(11451) = 7*X(7999)-6*X(38224) = 2*X(10263)-3*X(38743) = 5*X(10574)-6*X(21166) = 2*X(10723)-3*X(15305) = 5*X(11439)-4*X(39809) = 4*X(11591)-3*X(38732) = 6*X(14639)-7*X(15056) = 9*X(15045)-10*X(38750) = 5*X(15058)-4*X(22515)

The reciprocal orthologic center of these triangles is X(3)

X(39807) lies on these lines: {2,39817}, {3,39808}, {22,39820}, {98,2979}, {99,5889}, {110,39828}, {114,3060}, {115,11444}, {147,511}, {148,5562}, {542,13201}, {620,15043}, {690,12273}, {1154,13188}, {1216,14651}, {1993,39803}, {2782,11412}, {3567,15561}, {5012,39810}, {5640,39806}, {5876,38733}, {5890,33813}, {6036,7998}, {6101,12188}, {6241,38730}, {6321,11459}, {6721,11451}, {7999,38224}, {9860,31737}, {9862,10625}, {10263,38743}, {10574,21166}, {10723,15305}, {11422,39805}, {11439,39809}, {11440,39812}, {11441,13175}, {11442,39813}, {11443,39819}, {11445,39821}, {11446,39822}, {11447,39823}, {11448,39824}, {11449,39825}, {11452,39829}, {11453,39830}, {11454,39831}, {11591,38732}, {12111,23698}, {13172,13754}, {13391,38744}, {14639,15056}, {15045,38750}, {15058,22515}, {15072,38738}, {15801,39839}, {18392,39818}, {18911,39804}, {19122,39811}, {19167,39814}, {19367,39815}, {19412,39826}, {19413,39827}, {23293,39816}

X(39807) = reflection of X(i) in X(j) for these (i,j): (148, 5562), (5889, 99), (6241, 38730), (9860, 31737), (9862, 10625), (12188, 6101), (38733, 5876), (39808, 3), (39836, 11412), (39837, 13188)
X(39807) = anticomplement of X(39817)

X(39808) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO 7th BROCARD

Barycentrics    (SB+SC)*(S^4+(7*SA^2-4*(3*SA-2*SW)*R^2-6*SB*SC-SW^2)*S^2+(12*R^2-3*SA-2*SW)*SW^2*SA) : :
X(39808) = 2*X(99)-3*X(5890) = 4*X(114)-5*X(3567) = 4*X(115)-3*X(11459) = 4*X(143)-3*X(38743) = 8*X(620)-9*X(15045) = 3*X(2979)-4*X(12042) = 3*X(3060)-2*X(6033) = 2*X(5562)-3*X(14651) = 2*X(5876)-3*X(38732) = 8*X(6036)-7*X(7999) = 3*X(6054)-4*X(39835) = 16*X(6721)-17*X(11465) = 9*X(7998)-10*X(38739) = 7*X(9781)-8*X(39806) = 5*X(10574)-4*X(33813) = 5*X(11444)-6*X(38224) = 3*X(11455)-4*X(39809) = 6*X(14639)-5*X(15058) = 7*X(15043)-6*X(15561) = 3*X(15072)-2*X(38730)

The reciprocal orthologic center of these triangles is X(3)

X(39808) lies on these lines: {3,39807}, {4,39817}, {24,39820}, {52,147}, {54,39810}, {74,39812}, {98,11412}, {99,5890}, {114,3567}, {115,11459}, {143,38743}, {148,13754}, {185,13172}, {511,9862}, {542,7731}, {620,15045}, {690,12284}, {1154,12188}, {1614,39828}, {2782,5889}, {2979,12042}, {3044,11464}, {3060,6033}, {5562,14651}, {5663,38733}, {5876,38732}, {6036,7999}, {6054,39835}, {6102,13188}, {6241,23698}, {6321,12111}, {6721,11465}, {6777,36981}, {6778,36979}, {7592,39803}, {7998,38739}, {9781,39806}, {10263,38744}, {10574,33813}, {10723,12290}, {11423,39805}, {11444,38224}, {11455,39809}, {11456,13175}, {11457,39813}, {11458,39819}, {11460,39821}, {11461,39822}, {11462,39823}, {11463,39824}, {11466,39829}, {11467,39830}, {11468,39831}, {12273,18332}, {13174,31728}, {14639,15058}, {15043,15561}, {15072,38730}, {15305,22515}, {18331,21649}, {18394,39818}, {18912,39804}, {19123,39811}, {19168,39814}, {19368,39815}, {19414,39826}, {19415,39827}, {23294,39816}

X(39808) = reflection of X(i) in X(j) for these (i,j): (4, 39817), (147, 52), (11412, 98), (12111, 6321), (12273, 18332), (12290, 10723), (13172, 185), (13174, 31728), (13188, 6102), (18331, 21649), (38744, 10263), (39807, 3), (39836, 12188), (39837, 5889)
X(39808) = {X(3044), X(39825)}-harmonic conjugate of X(11464)

X(39809) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO 7th BROCARD

Barycentrics    4*a^8-6*(b^2+c^2)*a^6+3*(b^2+c^2)^2*a^4+2*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-(3*b^4-4*b^2*c^2+3*c^4)*(b^2-c^2)^2 : :
X(39809) = 3*X(3)-4*X(6722) = 2*X(3)-3*X(23514) = 3*X(4)-X(99) = 4*X(4)-X(10992) = 5*X(4)-X(13172) = 6*X(5)-5*X(31274) = 4*X(5)-3*X(38748) = 2*X(99)-3*X(114) = X(99)+3*X(10723) = 4*X(99)-3*X(10992) = 5*X(99)-3*X(13172) = X(114)+2*X(10723) = 5*X(114)-2*X(13172) = 8*X(6722)-9*X(23514) = 4*X(10723)+X(10992) = 5*X(10723)+X(13172) = 5*X(10992)-4*X(13172) = 5*X(31274)-3*X(38738) = 10*X(31274)-9*X(38748) = 2*X(38738)-3*X(38748)

The reciprocal orthologic center of these triangles is X(3)

X(39809) lies on these lines: {2,38736}, {3,6722}, {4,99}, {5,31274}, {20,6036}, {24,39831}, {25,39812}, {30,115}, {33,39815}, {34,39822}, {98,3146}, {146,148}, {147,17578}, {183,32152}, {185,39806}, {235,39816}, {376,14061}, {378,39825}, {381,620}, {382,2794}, {543,3830}, {546,33813}, {548,34127}, {549,15092}, {550,38737}, {671,9862}, {690,12295}, {1498,39810}, {1569,14881}, {1593,39828}, {1597,13175}, {1656,38731}, {1657,38224}, {1699,11724}, {2407,8754}, {2482,3845}, {2782,3627}, {2790,38956}, {3044,14157}, {3091,6721}, {3098,33017}, {3529,20398}, {3534,5461}, {3839,12117}, {3843,15561}, {3851,38750}, {3853,14981}, {5059,38735}, {5066,9167}, {5073,11623}, {5076,13188}, {5477,21850}, {6054,20094}, {6564,8997}, {6565,13989}, {6795,35903}, {7517,39854}, {7970,9812}, {8597,19924}, {8703,14971}, {8724,38335}, {9166,11001}, {10113,15357}, {10352,14068}, {10728,10769}, {11361,19130}, {11381,39817}, {11403,39803}, {11424,39805}, {11439,39807}, {11455,39808}, {11470,39819}, {11471,39821}, {11473,39823}, {11474,39824}, {11475,39829}, {11476,39830}, {11632,15684}, {11710,28164}, {11711,18483}, {11725,18481}, {12101,15300}, {12324,39804}, {12943,13183}, {12953,13182}, {14269,35022}, {14645,18440}, {14651,33703}, {14830,36523}, {15359,16111}, {15482,37348}, {15535,34584}, {15690,26614}, {15704,38229}, {15811,39820}, {17800,38742}, {18534,39857}, {19108,23259}, {19109,23249}, {19124,39811}, {19169,39814}, {19416,39826}, {19417,39827}, {19709,22247}, {29012,35377}, {31723,39847}, {32458,32819}, {34981,35345}

X(39809) = midpoint of X(i) and X(j) for these {i,j}: {4, 10723}, {98, 3146}, {148, 10722}, {382, 6321}, {671, 15682}, {5073, 38741}, {6033, 38733}, {10728, 10769}, {11381, 39817}, {11632, 15684}
X(39809) = reflection of X(i) in X(j) for these (i,j): (20, 6036), (98, 38734), (114, 4), (115, 22515), (185, 39806), (1569, 14881), (1657, 38747), (2482, 3845), (3534, 5461), (5477, 21850), (6055, 9880), (10992, 114), (11711, 18483), (13188, 38745), (14830, 36523), (14981, 22505), (15300, 22566), (15357, 10113), (16111, 15359), (18481, 11725), (22505, 3853), (22566, 12101), (33813, 546), (38730, 620), (38738, 5), (38741, 11623), (38749, 115), (39838, 3627)
X(39809) = anticomplement of X(38736)
X(39809) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 38738, 38748), (20, 14639, 6036), (115, 22515, 9880), (115, 38749, 6055), (148, 3543, 10722), (381, 38730, 620), (546, 33813, 36519), (1657, 38224, 38747), (3091, 21166, 6721), (3830, 38733, 6033), (5073, 38732, 38741), (9880, 38749, 115), (33813, 36519, 38751), (38732, 38741, 11623)

X(39810) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-EXTOUCH TO 7th BROCARD

Barycentrics    (2*S^4-(SA^2+2*SW*R^2)*S^2-(2*R^2-SA)*SW^2*SA)*(SB+SC) : :
X(39810) = 3*X(11402)-X(39803) = 3*X(11402)-2*X(39805)

The reciprocal orthologic center of these triangles is X(3)

X(39810) lies on these lines: {2,39804}, {3,39834}, {6,114}, {24,3044}, {25,39806}, {52,39857}, {54,39808}, {98,1993}, {99,7592}, {115,155}, {147,1994}, {184,39817}, {185,39812}, {195,12188}, {394,6036}, {542,5064}, {620,36752}, {690,19456}, {1181,23698}, {1351,9861}, {1498,39809}, {1899,39816}, {2782,12161}, {2794,36747}, {5012,39807}, {5984,11004}, {6033,36749}, {6055,37672}, {6321,18445}, {6721,10601}, {6776,39813}, {9864,16473}, {10602,39819}, {10605,39831}, {10723,11456}, {11402,39803}, {11441,14639}, {12042,16266}, {12160,39832}, {13172,15032}, {13175,19347}, {13188,15087}, {13352,39841}, {14627,38743}, {15561,36753}, {15801,39836}, {15805,31274}, {17814,23514}, {18396,39818}, {19125,39811}, {19170,39814}, {19349,39815}, {19350,39821}, {19354,39822}, {19355,39823}, {19356,39824}, {19357,39825}, {19358,39826}, {19359,39827}, {19363,39829}, {19364,39830}, {22505,39522}, {22515,32139}, {37483,38747}, {37489,39854}, {37493,39835}, {37498,38749}, {37514,38748}

X(39810) = midpoint of X(12160) and X(39832)
X(39810) = reflection of X(i) in X(j) for these (i,j): (3, 39834), (39803, 39805), (39839, 12161)
X(39810) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 39820, 114), (184, 39817, 39828), (11402, 39803, 39805)

X(39811) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER TO 7th BROCARD

Barycentrics    (4*(3*R^2-SW)*S^4-(2*R^2-SW)*SA*SW^3-(SW*(6*R^2-SW)+SA^2-3*SB*SC)*SW*S^2)*(SB+SC) : :
X(39811) = 5*X(3618)-X(39813) = 3*X(5050)+X(13175) = 3*X(5085)-X(39812) = 5*X(19132)-X(39820) = X(39819)+2*X(39828)

The reciprocal orthologic center of these triangles is X(3)

X(39811) lies on these lines: {6,39805}, {68,542}, {98,19121}, {99,19128}, {114,1974}, {115,19131}, {182,2549}, {511,32661}, {690,19138}, {1147,14645}, {1428,39815}, {2330,39822}, {2782,19154}, {2799,8723}, {3589,39816}, {3618,39813}, {4558,34157}, {5050,13175}, {5085,39812}, {5092,39831}, {6036,19126}, {6321,19129}, {6721,19137}, {19118,39803}, {19119,39804}, {19122,39807}, {19123,39808}, {19124,39809}, {19125,39810}, {19130,39818}, {19132,39820}, {19133,39821}, {19134,39826}, {19135,39827}, {19171,39814}, {21637,39817}

X(39811) = midpoint of X(6) and X(39828)
X(39811) = reflection of X(i) in X(j) for these (i,j): (39816, 3589), (39818, 19130), (39819, 6), (39831, 5092), (39840, 19154)

X(39812) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH TO 7th BROCARD

Barycentrics    (SB+SC)*(5*S^4-(2*(SW+12*SA)*R^2-5*SA^2+3*SB*SC)*S^2+(6*R^2-SW)*SA*SW^2) : :
X(39812) = 3*X(3)-X(13175) = 3*X(3)-2*X(39825) = 3*X(5085)-2*X(39811) = 2*X(13175)-3*X(39828) = X(13175)-6*X(39831) = 3*X(37497)-X(39849) = 4*X(39825)-3*X(39828) = X(39825)-3*X(39831) = X(39828)-4*X(39831)

The reciprocal orthologic center of these triangles is X(3)

X(39812) lies on these lines: {3,115}, {4,39816}, {20,39813}, {24,10723}, {25,39809}, {30,39857}, {55,39815}, {56,39822}, {64,39820}, {74,39808}, {98,11413}, {99,264}, {114,1593}, {147,12086}, {148,2071}, {185,39810}, {382,39818}, {542,2935}, {543,39860}, {620,9818}, {690,12302}, {1151,39823}, {1152,39824}, {1204,39817}, {2782,12084}, {2794,12085}, {2936,6033}, {3044,11456}, {3516,10992}, {3520,13172}, {5085,39811}, {5584,39821}, {6644,22515}, {6721,11479}, {7387,39854}, {7395,38748}, {7464,9862}, {7503,21166}, {7526,33813}, {9786,39806}, {11425,39805}, {11440,39807}, {11477,39819}, {11480,39829}, {11481,39830}, {12188,18859}, {13021,39826}, {13022,39827}, {13352,39839}, {14639,17928}, {18913,39804}, {19172,39814}, {21312,38749}, {37497,39849}

X(39812) = midpoint of X(i) and X(j) for these {i,j}: {20, 39813}, {64, 39820}
X(39812) = reflection of X(i) in X(j) for these (i,j): (3, 39831), (4, 39816), (382, 39818), (7387, 39854), (11477, 39819), (13175, 39825), (39828, 3), (39841, 12084)
X(39812) = circumperp conjugate of X(31842)
X(39812) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 13175, 39825), (13175, 39825, 39828), (21312, 39832, 38749)

X(39813) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO 7th BROCARD

Barycentrics    4*R^2*S^4-(2*R^2*SA+3*SB*SC)*SW*S^2-(2*R^2-SW)*SB*SC*SW^2 : :
X(39813) = 3*X(2)-4*X(39816) = 3*X(4)-4*X(39818) = 3*X(376)-4*X(39831) = 5*X(631)-4*X(39825) = 3*X(1992)-4*X(39819) = 5*X(3618)-4*X(39811)

The reciprocal orthologic center of these triangles is X(3)

X(39813) lies on these lines: {2,14656}, {4,99}, {5,13175}, {20,39812}, {30,9861}, {52,39833}, {98,1370}, {115,6643}, {147,7391}, {148,37444}, {376,39831}, {388,39815}, {427,39803}, {497,39822}, {542,13203}, {620,7401}, {631,39825}, {690,12319}, {1352,8783}, {1503,39820}, {1899,39804}, {1992,39819}, {2550,39821}, {2782,14790}, {2794,34938}, {3068,39823}, {3069,39824}, {3618,39811}, {5189,5984}, {6036,7386}, {6321,18531}, {6721,7392}, {6776,39810}, {6803,38748}, {6804,23514}, {6815,21166}, {6816,14639}, {7528,15561}, {10996,38736}, {11427,39805}, {11433,39806}, {11442,39807}, {11457,39808}, {11488,39829}, {11489,39830}, {13188,31723}, {18404,38733}, {18420,33813}, {19174,39814}, {19420,39826}, {19421,39827}, {31305,39857}

X(39813) = reflection of X(i) in X(j) for these (i,j): (20, 39812), (13175, 5), (31305, 39857), (39828, 39816), (39842, 14790)
X(39813) = anticomplement of X(39828)
X(39813) = anticomplementary circle-inverse of-X(3563)
X(39813) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1899, 39817, 39804), (39816, 39828, 2)

X(39814) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN TO 7th BROCARD

Barycentrics    (S^2+SA*SB)*(S^2+SA*SC)*(S^4+(4*SW*R^2-SA^2-3*SB*SC-SW^2)*S^2-(4*R^2-SW)*(SB+SC)*SW^2) : :

The reciprocal orthologic center of these triangles is X(3)

X(39814) lies on these lines: {54,99}, {95,6036}, {97,98}, {114,275}, {115,19179}, {542,19208}, {690,19193}, {2782,19210}, {6321,19176}, {6721,19188}, {8884,23698}, {9792,39806}, {10992,38808}, {13175,19173}, {16030,39803}, {19166,39804}, {19167,39807}, {19168,39808}, {19169,39809}, {19170,39810}, {19171,39811}, {19172,39812}, {19174,39813}, {19175,39815}, {19177,39818}, {19178,39819}, {19180,39820}, {19181,39821}, {19182,39822}, {19183,39823}, {19184,39824}, {19185,39825}, {19186,39826}, {19187,39827}, {19189,39828}, {19190,39829}, {19191,39830}, {19192,39831}, {21638,39817}, {23295,39816}

X(39814) = reflection of X(39843) in X(19210)

X(39815) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC TO 7th BROCARD

Barycentrics
a*(a^10-2*(b^2-b*c+c^2)*a^8+(b^4+c^4-(4*b^2-3*b*c+4*c^2)*b*c)*a^6+(b^6+c^6+(3*b^4+3*c^4-2*(b-c)^2*b*c)*b*c)*a^4-(2*b^8+2*c^8-(3*b^2+4*b*c+3*c^2)*(b-c)^2*b^2*c^2)*a^2+(b^6+c^6-(b^2-c^2)^2*b*c)*(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(3)

X(39815) lies on these lines: {1,23698}, {12,39816}, {30,39851}, {33,39809}, {34,114}, {35,39831}, {36,39825}, {55,39812}, {56,39828}, {65,39821}, {98,4296}, {99,1870}, {115,1060}, {221,39820}, {388,39813}, {542,19505}, {620,37697}, {690,19469}, {999,13175}, {1038,6036}, {1040,38736}, {1062,38738}, {1398,39803}, {1425,39817}, {1428,39811}, {2067,39823}, {2782,32047}, {3585,39818}, {4318,7970}, {4351,10069}, {6198,10723}, {6321,18447}, {6502,39824}, {6721,19372}, {7051,39829}, {11724,34036}, {18455,38730}, {18915,39804}, {19175,39814}, {19349,39810}, {19365,39805}, {19366,39806}, {19367,39807}, {19368,39808}, {19369,39819}, {19370,39826}, {19371,39827}, {19373,39830}, {22515,37729}

X(39815) = reflection of X(i) in X(j) for these (i,j): (39822, 1), (39844, 32047)

X(39816) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA MINOR TO 7th BROCARD

Barycentrics    (16*R^2-3*SW)*S^4-(2*R^2*(SA+2*SW)+3*SB*SC-SW^2)*SW*S^2-(2*R^2-SW)*SB*SC*SW^2 : :
X(39816) = 3*X(2)+X(39813) = 5*X(1656)-X(13175) = 3*X(1853)+X(39820) = X(9861)+3*X(34609)

The reciprocal orthologic center of these triangles is X(3)

X(39816) lies on these lines: {2,14656}, {4,39812}, {5,620}, {11,39822}, {12,39815}, {30,39818}, {98,858}, {99,1594}, {114,136}, {115,11585}, {125,39817}, {140,39825}, {147,31074}, {235,39809}, {403,10723}, {524,39819}, {542,23315}, {590,39823}, {615,39824}, {690,23306}, {1368,6036}, {1656,13175}, {1853,39820}, {1899,39810}, {2072,6321}, {2782,13371}, {2794,23335}, {3044,34224}, {3589,39811}, {3925,39821}, {5094,39803}, {5576,15561}, {5969,20300}, {6823,38736}, {7399,38748}, {7403,36519}, {7405,31274}, {7577,13172}, {9861,34609}, {10024,38730}, {10255,38733}, {13160,21166}, {13567,39806}, {14790,39857}, {15760,38738}, {23291,39804}, {23292,39805}, {23293,39807}, {23294,39808}, {23295,39814}, {23298,39826}, {23299,39827}, {23302,39829}, {23303,39830}, {30771,39120}, {37347,38750}, {37452,38224}

X(39816) = midpoint of X(i) and X(j) for these {i,j}: {4, 39812}, {14790, 39857}, {39813, 39828}, {39818, 39831}
X(39816) = reflection of X(i) in X(j) for these (i,j): (39811, 3589), (39825, 140), (39845, 13371)
X(39816) = complement of X(39828)
X(39816) = {X(2), X(39813)}-harmonic conjugate of X(39828)

X(39817) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO 7th BROCARD

Barycentrics    (S^4-(4*R^2*SW+2*SA^2-3*SB*SC)*S^2-(4*R^2-SA-SW)*SA*SW^2)*(SB+SC) : :
X(39817) = 3*X(51)-2*X(114) = 3*X(51)-4*X(39806) = X(147)-3*X(3060) = 9*X(373)-8*X(6721) = 3*X(568)-X(13188) = 2*X(1216)-3*X(38224) = 2*X(2482)-3*X(16226) = 3*X(3917)-4*X(6036) = 4*X(5447)-5*X(38739) = 4*X(5462)-3*X(15561) = 3*X(5890)-X(13172) = 6*X(5892)-5*X(38750) = 2*X(5907)-3*X(14639) = 4*X(9729)-3*X(21166) = 3*X(9730)-2*X(33813) = X(11412)-3*X(14651) = 2*X(11591)-3*X(38229) = 4*X(11793)-5*X(14061) = 2*X(15644)-3*X(34473) = 3*X(16222)-2*X(33512)

The reciprocal orthologic center of these triangles is X(3)

X(39817) lies on these lines: {2,39807}, {4,39808}, {6,39803}, {25,39820}, {51,114}, {52,2782}, {98,385}, {99,389}, {115,5562}, {125,39816}, {147,3060}, {148,5889}, {184,39810}, {185,23698}, {373,6721}, {542,13417}, {543,14831}, {568,13188}, {690,21649}, {1181,13175}, {1204,39812}, {1216,38224}, {1425,39815}, {1899,39804}, {2482,16226}, {2987,17974}, {3044,10282}, {3270,39822}, {3611,39821}, {3917,6036}, {5392,39120}, {5446,6033}, {5447,38739}, {5462,15561}, {5890,13172}, {5892,38750}, {5907,14639}, {5969,19161}, {6000,10723}, {6054,21849}, {6243,12188}, {6321,13754}, {9729,21166}, {9730,33813}, {9861,33586}, {10625,12042}, {10722,13598}, {11005,11800}, {11381,39809}, {11412,14651}, {11591,38229}, {11793,14061}, {12160,39849}, {12162,22515}, {13366,39805}, {13367,39825}, {13851,39818}, {14981,39835}, {15644,34473}, {16222,33512}, {16625,23235}, {17834,39832}, {18436,38732}, {21636,31757}, {21637,39811}, {21638,39814}, {21639,39819}, {21640,39823}, {21641,39824}, {21642,39826}, {21643,39827}, {21647,39829}, {21648,39830}, {21663,39831}, {34783,38733}, {36987,38747}

X(39817) = midpoint of X(i) and X(j) for these {i,j}: {4, 39808}, {148, 5889}, {6243, 12188}, {34783, 38733}
X(39817) = reflection of X(i) in X(j) for these (i,j): (99, 389), (114, 39806), (5562, 115), (6033, 5446), (6054, 21849), (10625, 12042), (10722, 13598), (11005, 11800), (11381, 39809), (12162, 22515), (14981, 39835), (21636, 31757), (39846, 52)
X(39817) = complement of X(39807)
X(39817) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (114, 39806, 51), (39804, 39813, 1899), (39810, 39828, 184)

X(39818) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO 7th BROCARD

Barycentrics    2*(10*R^2-3*SW)*S^4+((18*SA-19*SW)*R^2+6*(SB+SC)*SW)*SA*S^2-(7*R^2-2*SW)*SB*SC*SW^2 : :
X(39818) = 3*X(4)+X(39813) = 3*X(381)-X(39828) = 5*X(3843)-X(13175) = X(9861)+3*X(34725) = 3*X(18405)+X(39820)

The reciprocal orthologic center of these triangles is X(3)

X(39818) lies on these lines: {4,99}, {5,39825}, {30,39816}, {98,3153}, {115,10316}, {381,39828}, {382,39812}, {542,13248}, {543,18568}, {690,19479}, {2782,18377}, {2794,18569}, {3044,12289}, {3583,39822}, {3585,39815}, {3843,13175}, {6033,31724}, {6036,18531}, {6321,18403}, {6564,39823}, {6565,39824}, {6721,18420}, {7574,38741}, {9861,34725}, {13371,39860}, {13851,39817}, {14791,38747}, {16808,39829}, {16809,39830}, {18386,39803}, {18388,39805}, {18390,39806}, {18392,39807}, {18394,39808}, {18396,39810}, {18405,39820}, {18406,39821}, {18414,39826}, {18415,39827}, {18918,39804}, {19130,39811}, {19177,39814}, {21166,34007}, {22505,22823}, {31723,39838}, {37444,38749}

X(39818) = midpoint of X(382) and X(39812)
X(39818) = reflection of X(i) in X(j) for these (i,j): (39811, 19130), (39825, 5), (39831, 39816), (39847, 18377), (39860, 13371)

X(39819) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EHRMANN TO 7th BROCARD

Barycentrics    (4*(9*R^2-2*SW)*S^4+(2*R^2-SW)*SA*SW^3+(SA^2-10*R^2*SW-3*SB*SC+3*SW^2)*SW*S^2)*(SB+SC) : :
X(39819) = 3*X(6)-X(39828) = 3*X(1992)+X(39813) = 5*X(11482)-X(13175) = 3*X(17813)+X(39820) = 3*X(39811)-2*X(39828)

The reciprocal orthologic center of these triangles is X(3)

X(39819) lies on these lines: {6,39805}, {98,11416}, {99,8537}, {114,8541}, {115,8538}, {511,39831}, {524,39816}, {542,13248}, {575,39825}, {576,7737}, {690,12596}, {1992,39813}, {2782,11255}, {6036,11511}, {6321,18449}, {6721,9813}, {8539,39821}, {8540,39822}, {10602,39810}, {11405,39803}, {11443,39807}, {11458,39808}, {11470,39809}, {11477,39812}, {11482,13175}, {15074,39834}, {17813,39820}, {18919,39804}, {19178,39814}, {19369,39815}, {19426,39826}, {19427,39827}, {21639,39817}

X(39819) = midpoint of X(11477) and X(39812)
X(39819) = reflection of X(i) in X(j) for these (i,j): (39811, 6), (39825, 575), (39848, 11255)

X(39820) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st EXCOSINE TO 7th BROCARD

Barycentrics    (S^4-(2*SA^2+4*SW*R^2-SW^2)*S^2-2*(2*R^2-SA)*SA*SW^2)*(SB+SC) : :
X(39820) = 3*X(154)-2*X(39828) = 3*X(1853)-4*X(39816) = 3*X(10606)-4*X(39831) = 3*X(17813)-4*X(39819) = 5*X(17821)-4*X(39825) = 3*X(18405)-4*X(39818) = 5*X(19132)-4*X(39811) = 3*X(37497)-2*X(39841)

The reciprocal orthologic center of these triangles is X(3)

X(39820) lies on these lines: {6,114}, {22,39807}, {24,39808}, {25,39817}, {64,39812}, {98,394}, {99,1181}, {115,17814}, {147,1993}, {148,11441}, {154,39828}, {155,2782}, {184,39803}, {221,39815}, {323,5984}, {511,9861}, {542,17847}, {620,37514}, {690,17838}, {1498,23698}, {1503,39813}, {1853,39816}, {2192,39822}, {2323,24469}, {2794,37498}, {3044,19357}, {3197,39821}, {5562,39832}, {5969,19149}, {6033,36747}, {6036,17811}, {6321,18451}, {6721,17825}, {6759,13175}, {10606,39831}, {11456,13172}, {12160,39846}, {12164,22552}, {13188,18445}, {13567,39804}, {14981,39839}, {15561,36752}, {15811,39809}, {17809,39805}, {17810,39806}, {17813,39819}, {17819,39823}, {17820,39824}, {17821,39825}, {17826,39829}, {17827,39830}, {17834,39857}, {18405,39818}, {19132,39811}, {19180,39814}, {19430,39826}, {19431,39827}, {36749,38743}, {37476,39834}, {37483,38741}, {37497,39841}

X(39820) = reflection of X(i) in X(j) for these (i,j): (64, 39812), (13175, 6759), (17834, 39857), (39849, 155)
X(39820) = {X(114), X(39810)}-harmonic conjugate of X(6)

X(39821) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXTANGENTS TO 7th BROCARD

Barycentrics
a*(a^13+(b+c)*a^12-3*(b^2+c^2)*a^11-3*(b+c)*(b^2+c^2)*a^10+(3*b^4+3*c^4-(2*b^2-3*b*c+2*c^2)*b*c)*a^9+(b+c)*(3*b^4+3*c^4+(2*b^2+3*b*c+2*c^2)*b*c)*a^8+(5*b^4+5*c^4+2*(b^2+3*b*c+c^2)*b*c)*b*c*a^7-(b+c)*(5*b^4+5*c^4-2*(b^2-3*b*c+c^2)*b*c)*b*c*a^6-(3*b^4+3*c^4+(5*b^2+2*b*c+5*c^2)*b*c)*(b^4+c^4)*a^5-(b+c)*(3*b^4+3*c^4-(5*b^2-2*b*c+5*c^2)*b*c)*(b^4+c^4)*a^4+(3*b^8+3*c^8-(3*b^6+3*c^6-(3*b^2-2*b*c+3*c^2)*b^2*c^2)*b*c)*(b+c)^2*a^3+(b^2-c^2)*(b-c)*(3*b^8+3*c^8+(3*b^6+3*c^6-(3*b^2+2*b*c+3*c^2)*b^2*c^2)*b*c)*a^2-(b^2-c^2)^2*(b+c)^2*(b^6+c^6-(b^2-c^2)^2*b*c)*a+(b^2-c^2)^3*(b-c)*(-b^6-c^6-(b^2-c^2)^2*b*c)) : :

The reciprocal orthologic center of these triangles is X(3)

X(39821) lies on the extangents circle and these lines: {19,114}, {40,13178}, {55,39822}, {65,39815}, {98,3101}, {99,6197}, {115,8251}, {147,9536}, {148,9537}, {542,10119}, {690,12661}, {2550,39813}, {2782,8141}, {3197,39820}, {3611,39817}, {3925,39816}, {5415,39823}, {5416,39824}, {5584,39812}, {6036,10319}, {6321,18453}, {6721,9816}, {7688,39831}, {8539,39819}, {9572,13174}, {9573,9860}, {10306,13175}, {10636,39829}, {10637,39830}, {10902,39825}, {11406,39803}, {11428,39805}, {11435,39806}, {11445,39807}, {11460,39808}, {11471,39809}, {15941,38738}, {18406,39818}, {18921,39804}, {19133,39811}, {19181,39814}, {19350,39810}, {19432,39826}, {19433,39827}

X(39821) = reflection of X(i) in X(j) for these (i,j): (39822, 39828), (39850, 8141)
X(39821) = extangents circle-antipode of-X(39850)

X(39822) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INTANGENTS TO 7th BROCARD

Barycentrics
a*(a^10-2*(b^2+b*c+c^2)*a^8+(b^4+c^4+(4*b^2+3*b*c+4*c^2)*b*c)*a^6+(b^6+c^6-(3*b^4+3*c^4+2*(b+c)^2*b*c)*b*c)*a^4-(2*b^8+2*c^8-(3*b^2-4*b*c+3*c^2)*(b+c)^2*b^2*c^2)*a^2+(b^6+c^6+(b^2-c^2)^2*b*c)*(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(3)

X(39822) lies on the intangents circle and these lines: {1,23698}, {11,39816}, {30,39844}, {33,114}, {34,39809}, {35,39825}, {36,39831}, {55,39821}, {56,39812}, {98,3100}, {99,6198}, {115,1062}, {147,9539}, {148,9538}, {497,39813}, {542,10118}, {620,37696}, {690,12888}, {1038,38736}, {1040,6036}, {1060,38738}, {1250,39830}, {1569,9636}, {1870,10723}, {2023,9594}, {2066,39823}, {2192,39820}, {2330,39811}, {2782,8144}, {3029,9551}, {3044,9638}, {3270,39817}, {3295,13175}, {3583,39818}, {4354,10053}, {5414,39824}, {6321,18455}, {6721,9817}, {7071,39803}, {8540,39819}, {9550,34454}, {9576,9860}, {9577,13174}, {9627,13182}, {9628,12184}, {9629,12185}, {9630,13183}, {9641,12188}, {9642,13188}, {9644,14981}, {9645,39857}, {10638,39829}, {11429,39805}, {11436,39806}, {11446,39807}, {11461,39808}, {18447,38730}, {18922,39804}, {19182,39814}, {19354,39810}, {19434,39826}, {19435,39827}, {33813,37729}

X(39822) = reflection of X(i) in X(j) for these (i,j): (39815, 1), (39821, 39828), (39851, 8144)
X(39822) = intangents circle-antipode of-X(39851)

X(39823) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS TO 7th BROCARD

Barycentrics    (SB+SC)*(S^4+(2*R^2*SW+SA^2-3*SB*SC)*S^2+(2*R^2-SW)*SA*SW^2-S*(3*S^2-SW^2)*(4*R^2-SW)) : :

The reciprocal orthologic center of these triangles is X(3)

X(39823) lies on these lines: {6,39805}, {26,39853}, {98,11417}, {99,10880}, {114,5412}, {115,10897}, {371,23698}, {372,39825}, {542,13287}, {590,39816}, {690,12891}, {1151,39812}, {2066,39822}, {2067,39815}, {2782,11265}, {3068,39813}, {3311,13175}, {5410,39803}, {5415,39821}, {6036,11513}, {6200,39831}, {6230,26922}, {6321,18457}, {6564,39818}, {6721,10961}, {11447,39807}, {11462,39808}, {11473,39809}, {17819,39820}, {18923,39804}, {19183,39814}, {19355,39810}, {19436,39826}, {19438,39827}, {21640,39817}

X(39823) = reflection of X(39852) in X(11265)
X(39823) = {X(6), X(39828)}-harmonic conjugate of X(39824)

X(39824) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS TO 7th BROCARD

Barycentrics    (SB+SC)*(S^4+(2*R^2*SW+SA^2-3*SB*SC)*S^2+(2*R^2-SW)*SA*SW^2+S*(3*S^2-SW^2)*(4*R^2-SW)) : :

The reciprocal orthologic center of these triangles is X(3)

X(39824) lies on these lines: {6,39805}, {26,39852}, {98,11418}, {99,10881}, {114,5413}, {115,10898}, {371,39825}, {372,23698}, {542,13288}, {615,39816}, {690,12892}, {1152,39812}, {2782,11266}, {3069,39813}, {3312,13175}, {5411,39803}, {5414,39822}, {5416,39821}, {6036,11514}, {6321,18459}, {6396,39831}, {6502,39815}, {6565,39818}, {6721,10963}, {11448,39807}, {11463,39808}, {11474,39809}, {17820,39820}, {18924,39804}, {19184,39814}, {19356,39810}, {19437,39827}, {19439,39826}, {21641,39817}

X(39824) = reflection of X(39853) in X(11266)
X(39824) = {X(6), X(39828)}-harmonic conjugate of X(39823)

X(39825) = ORTHOLOGIC CENTER OF THESE TRIANGLES: KOSNITA TO 7th BROCARD

Barycentrics    (SB+SC)*(2*S^4-((6*SA-SW)*R^2-2*SA^2+3*SB*SC)*S^2+(3*R^2-SW)*SA*SW^2) : :
X(39825) = 3*X(3)+X(13175) = 3*X(3)-X(39812) = 5*X(631)-X(39813) = X(9861)-5*X(16195) = X(13175)-3*X(39828) = 2*X(13175)+3*X(39831) = 3*X(14070)-X(39857) = 5*X(17821)-X(39820) = X(39812)+3*X(39828) = 2*X(39812)-3*X(39831) = 2*X(39828)+X(39831)

The reciprocal orthologic center of these triangles is X(3)

X(39825) lies on these lines: {3,115}, {5,39818}, {15,39830}, {16,39829}, {22,38749}, {23,10722}, {24,114}, {26,2794}, {30,39845}, {35,39822}, {36,39815}, {98,7488}, {99,186}, {140,39816}, {148,10298}, {159,542}, {371,39824}, {372,39823}, {378,39809}, {389,39805}, {511,32661}, {543,18324}, {575,39819}, {578,39806}, {620,6644}, {631,39813}, {690,12893}, {1658,2782}, {2070,6033}, {2799,14270}, {2937,38741}, {3044,11464}, {3455,12188}, {3515,39803}, {3520,10723}, {5171,38553}, {5961,7502}, {5969,35228}, {6054,37940}, {6642,6721}, {6722,7514}, {7387,39841}, {7503,23514}, {7506,36519}, {7512,34473}, {7517,39838}, {7556,9862}, {8724,37922}, {8997,9682}, {9590,9864}, {9737,10311}, {9861,16195}, {10902,39821}, {10992,32534}, {11449,39807}, {12117,37941}, {13172,21844}, {13335,32654}, {13367,39817}, {13564,38742}, {14061,35921}, {14118,14639}, {15270,39644}, {17821,39820}, {17928,38748}, {18475,39834}, {18570,22515}, {18925,39804}, {19185,39814}, {19357,39810}, {19440,39826}, {19441,39827}, {21166,22467}, {22505,37440}, {33813,37814}, {34013,37958}, {37489,39839}

X(39825) = midpoint of X(i) and X(j) for these {i,j}: {3, 39828}, {7387, 39841}, {13175, 39812}
X(39825) = reflection of X(i) in X(j) for these (i,j): (39816, 140), (39818, 5), (39819, 575), (39831, 3), (39854, 1658)
X(39825) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 13175, 39812), (11464, 39808, 3044), (39812, 39828, 13175)

X(39826) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS ANTIPODAL TANGENTIAL TO 7th BROCARD

Barycentrics    ((12*R^2+6*SA-SW)*S^4-(2*R^2*SW+SA^2+SB*SC)*SW*S^2+(2*R^2-SW)*SA*SW^3+S*(8*S^4+2*(2*R^2*SW+4*SA^2-3*SB*SC-SW^2)*S^2+2*(2*R^2-SA-SW)*SA*SW^2))*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(3)

X(39826) lies on these lines: {98,19406}, {99,19424}, {114,19446}, {115,19428}, {542,19507}, {690,19482}, {2782,39855}, {6036,19422}, {6321,18462}, {6721,19448}, {8939,39828}, {9723,39827}, {13021,39812}, {13175,19418}, {18414,39818}, {18926,39804}, {18980,23698}, {19134,39811}, {19186,39814}, {19358,39810}, {19370,39815}, {19404,39803}, {19408,39805}, {19410,39806}, {19412,39807}, {19414,39808}, {19416,39809}, {19420,39813}, {19426,39819}, {19430,39820}, {19432,39821}, {19434,39822}, {19436,39823}, {19439,39824}, {19440,39825}, {19450,39829}, {19452,39830}, {19454,39831}, {21642,39817}, {23298,39816}

X(39827) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) ANTIPODAL TANGENTIAL TO 7th BROCARD

Barycentrics    ((12*R^2+6*SA-SW)*S^4-(2*R^2*SW+SA^2+SB*SC)*SW*S^2+(2*R^2-SW)*SA*SW^3-S*(8*S^4+2*(2*R^2*SW+4*SA^2-3*SB*SC-SW^2)*S^2+2*(2*R^2-SA-SW)*SA*SW^2))*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(3)

X(39827) lies on these lines: {98,19407}, {99,19425}, {114,19447}, {115,19429}, {542,19508}, {690,19483}, {2782,39856}, {6036,19423}, {6321,18463}, {6721,19449}, {8943,39828}, {9723,39826}, {13022,39812}, {13175,19419}, {18415,39818}, {18927,39804}, {18981,23698}, {19135,39811}, {19187,39814}, {19359,39810}, {19371,39815}, {19405,39803}, {19409,39805}, {19411,39806}, {19413,39807}, {19415,39808}, {19417,39809}, {19421,39813}, {19427,39819}, {19431,39820}, {19433,39821}, {19435,39822}, {19437,39824}, {19438,39823}, {19441,39825}, {19451,39829}, {19453,39830}, {19455,39831}, {21643,39817}, {23299,39816}

X(39828) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL TO 7th BROCARD

Barycentrics    (S^4+(2*SW*R^2+SA^2-3*SB*SC)*S^2+(2*R^2-SW)*SA*SW^2)*(SB+SC) : :
X(39828) = 3*X(3)-2*X(39831) = 3*X(6)-2*X(39819) = 3*X(154)-X(39820) = 3*X(381)-2*X(39818) = X(9861)-3*X(9909) = X(9876)-3*X(10245) = 2*X(13175)+X(39812) = X(13175)+2*X(39825) = 3*X(13175)+2*X(39831) = 3*X(14070)-2*X(39854) = 3*X(39811)-X(39819) = X(39812)-4*X(39825) = 3*X(39812)-4*X(39831) = 3*X(39825)-X(39831)

The reciprocal orthologic center of these triangles is X(3)

X(39828) lies on the tangential circle and these lines: {2,14656}, {3,115}, {6,39805}, {22,98}, {23,147}, {24,99}, {25,114}, {26,2782}, {30,39841}, {52,39839}, {55,39821}, {56,39815}, {110,39807}, {148,7488}, {154,39820}, {159,39644}, {161,542}, {182,3981}, {184,39810}, {186,13172}, {230,32762}, {378,10723}, {381,39818}, {511,1971}, {543,14070}, {620,6642}, {690,2931}, {1569,9699}, {1593,39809}, {1614,39808}, {2023,9609}, {2070,2936}, {2794,7387}, {2797,14703}, {2799,11616}, {2937,12188}, {2971,14669}, {3029,9570}, {3044,9707}, {3425,36849}, {3515,10992}, {3563,4558}, {5020,6721}, {5899,38744}, {5969,15577}, {5984,37913}, {5989,37123}, {6033,7517}, {6644,33813}, {6676,39120}, {6722,7393}, {6776,39804}, {7395,23514}, {7503,14639}, {7506,15561}, {7509,14061}, {7512,14651}, {7516,34127}, {7526,22515}, {7529,36519}, {7530,22505}, {8185,9864}, {8276,8997}, {8277,13989}, {8591,37940}, {8781,9723}, {8939,39826}, {8943,39827}, {8996,12972}, {9571,34454}, {9590,13174}, {9591,9860}, {9645,39851}, {9658,12184}, {9659,13182}, {9672,13183}, {9673,12185}, {9714,14981}, {9715,39832}, {9862,12088}, {9876,10245}, {10323,34473}, {11365,11724}, {11414,38749}, {12083,38741}, {12085,39860}, {12117,15078}, {12131,15928}, {12178,20872}, {13178,15177}, {14667,16678}, {14790,39845}, {16278,22109}, {17834,39849}, {17928,21166}, {17974,38356}, {18378,38743}, {18534,39838}, {19189,39814}, {31305,39842}, {31953,34131}, {35243,38747}

X(39828) = midpoint of X(i) and X(j) for these {i,j}: {3, 13175}, {17834, 39849}, {31305, 39842}, {39821, 39822}
X(39828) = reflection of X(i) in X(j) for these (i,j): (3, 39825), (6, 39811), (12085, 39860), (14790, 39845), (39812, 3), (39813, 39816), (39857, 26)
X(39828) = anticomplement of X(39816)
X(39828) = complement of X(39813)
X(39828) = tangential circle-antipode of-X(39857)
X(39828) = circumcircle-inverse of-X(31842)
X(39828) = crosssum of X(512) and X(36472)
X(39828) = X(103)-of-tangential-triangle if ABC is acute
X(39828) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 39813, 39816), (25, 39803, 114), (184, 39817, 39810), (13175, 39825, 39812), (39805, 39806, 6), (39823, 39824, 6), (39829, 39830, 6)

X(39829) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL TO 7th BROCARD

Barycentrics    (SB+SC)*(sqrt(3)*(S^4+(2*R^2*SW+SA^2-3*SB*SC)*S^2+(2*R^2-SW)*SA*SW^2)-S*(3*S^2-SW^2)*(4*R^2-SW)) : :

The reciprocal orthologic center of these triangles is X(3)

X(39829) lies on these lines: {6,39805}, {15,23698}, {16,39825}, {26,39859}, {98,11420}, {99,10632}, {114,10641}, {115,10634}, {542,10681}, {690,10663}, {2782,11267}, {6036,11515}, {6321,18468}, {6721,10643}, {7051,39815}, {10636,39821}, {10638,39822}, {10645,39831}, {11408,39803}, {11452,39807}, {11466,39808}, {11475,39809}, {11480,39812}, {11485,13175}, {11488,39813}, {16808,39818}, {17826,39820}, {18929,39804}, {19190,39814}, {19363,39810}, {19450,39826}, {19451,39827}, {21647,39817}, {23302,39816}

X(39829) = reflection of X(39858) in X(11267)
X(39829) = {X(6), X(39828)}-harmonic conjugate of X(39830)

X(39830) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL TO 7th BROCARD

Barycentrics    (SB+SC)*(sqrt(3)*(S^4+(2*R^2*SW+SA^2-3*SB*SC)*S^2+(2*R^2-SW)*SA*SW^2)+S*(3*S^2-SW^2)*(4*R^2-SW)) : :

The reciprocal orthologic center of these triangles is X(3)

X(39830) lies on these lines: {6,39805}, {15,39825}, {16,23698}, {26,39858}, {98,11421}, {99,10633}, {114,10642}, {115,10635}, {542,10682}, {690,10664}, {1250,39822}, {2782,11268}, {6036,11516}, {6321,18470}, {6721,10644}, {10637,39821}, {10646,39831}, {11409,39803}, {11453,39807}, {11467,39808}, {11476,39809}, {11481,39812}, {11486,13175}, {11489,39813}, {16809,39818}, {17827,39820}, {18930,39804}, {19191,39814}, {19364,39810}, {19373,39815}, {19452,39826}, {19453,39827}, {21648,39817}, {23303,39816}

X(39830) = reflection of X(39859) in X(11268)
X(39830) = {X(6), X(39828)}-harmonic conjugate of X(39829)

X(39831) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TRINH TO 7th BROCARD

Barycentrics    (SB+SC)*(4*S^4-((18*SA+SW)*R^2-4*SA^2+3*SB*SC)*S^2+(5*R^2-SW)*SA*SW^2) : :
X(39831) = 5*X(3)-X(13175) = 3*X(3)-X(39828) = 3*X(376)+X(39813) = 3*X(10606)+X(39820) = X(13175)+5*X(39812) = 2*X(13175)-5*X(39825) = 3*X(13175)-5*X(39828) = 2*X(39812)+X(39825) = 3*X(39812)+X(39828) = 3*X(39825)-2*X(39828)

The reciprocal orthologic center of these triangles is X(3)

X(39831) lies on these lines: {3,115}, {24,39809}, {30,39816}, {35,39815}, {36,39822}, {98,2071}, {99,1235}, {114,378}, {186,10723}, {376,39813}, {511,39819}, {542,12302}, {620,7526}, {671,37948}, {690,12901}, {1968,2967}, {2782,11250}, {2794,12084}, {3044,6241}, {3455,18859}, {5092,39811}, {5866,8781}, {5969,15578}, {6200,39823}, {6396,39824}, {6721,9818}, {7503,38748}, {7688,39821}, {9880,15078}, {10605,39810}, {10606,39820}, {10645,39829}, {10646,39830}, {10722,12086}, {10992,35477}, {11410,39803}, {11413,38749}, {11430,39805}, {11438,39806}, {11454,39807}, {11468,39808}, {12085,39857}, {13172,35473}, {13371,39847}, {13496,18570}, {14118,21166}, {14130,15561}, {14639,22467}, {17928,23514}, {18931,39804}, {19192,39814}, {19454,39826}, {19455,39827}, {20094,35494}, {21663,39817}, {22515,37814}, {39653,39832}

X(39831) = midpoint of X(i) and X(j) for these {i,j}: {3, 39812}, {12085, 39857}
X(39831) = reflection of X(i) in X(j) for these (i,j): (39811, 5092), (39818, 39816), (39825, 3), (39847, 13371), (39860, 11250)

X(39832) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO 7th BROCARD

Barycentrics    a^2*(a^8-(b^2+c^2)*a^6+b^2*c^2*a^4+(b^4-c^4)*(b^2-c^2)*a^2-(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(39832) = 3*X(11402)-4*X(39834) = 3*X(11402)-2*X(39839)

The reciprocal parallelogic center of these triangles is X(6).

X(39832) lies on these lines: {2,9876}, {3,76}, {4,9861}, {6,39846}, {13,9915}, {14,9916}, {22,148}, {24,14651}, {25,115}, {112,20975}, {114,7395}, {147,7503}, {159,11646}, {184,39849}, {378,3438}, {427,39842}, {542,12168}, {620,2936}, {671,9909}, {682,3520}, {690,13171}, {1398,39844}, {1593,2794}, {1597,10722}, {1598,14639}, {1916,9917}, {1976,3269}, {1993,39836}, {2353,10828}, {3023,10832}, {3027,10831}, {3044,3167}, {3515,11623}, {3516,10991}, {4611,22143}, {5020,14061}, {5094,39845}, {5410,39852}, {5411,39853}, {5562,39820}, {5938,21177}, {5969,37485}, {5984,14118}, {6033,9818}, {6321,7387}, {6636,20094}, {6642,38224}, {6722,11284}, {7071,39851}, {7393,15561}, {7492,35369}, {7517,38732}, {7592,39837}, {7669,34866}, {7983,12410}, {9715,39828}, {9777,39835}, {9798,13178}, {9918,11606}, {10117,16278}, {10323,13172}, {10723,39568}, {10754,37491}, {10769,13222}, {10833,13183}, {11245,39833}, {11365,38220}, {11402,39834}, {11403,39838}, {11405,39848}, {11406,39850}, {11408,39858}, {11409,39859}, {11410,39860}, {11414,23698}, {11602,22657}, {11603,22656}, {11632,14070}, {12083,38733}, {12085,38741}, {12160,39810}, {12412,18332}, {13173,37577}, {13174,37557}, {13182,18954}, {13861,38229}, {16030,39843}, {17834,39817}, {18386,39847}, {18534,22515}, {19118,39840}, {19404,39855}, {19405,39856}, {20775,35921}, {21312,38749}, {35243,38730}, {37198,38738}, {39653,39831}

X(39832) = reflection of X(i) in X(j) for these (i,j): (12160, 39810), (39803, 3), (39839, 39834)
X(39832) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22, 148, 13175), (115, 3455, 39857), (115, 39857, 25), (2353, 11325, 10828), (38749, 39812, 21312), (39834, 39839, 11402)

X(39833) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO 7th BROCARD

Barycentrics    a^10-4*(b^2+c^2)*a^8+(7*b^4-b^2*c^2+7*c^4)*a^6-(b^2+c^2)*(7*b^4-11*b^2*c^2+7*c^4)*a^4+(4*b^4+3*b^2*c^2+4*c^4)*(b^2-c^2)^2*a^2-(b^6+c^6)*(b^2-c^2)^2 : :

The reciprocal parallelogic center of these triangles is X(6).

X(39833) lies on these lines: {2,39839}, {4,39835}, {52,39813}, {69,620}, {98,18916}, {99,6515}, {114,11411}, {115,11433}, {148,37644}, {542,18932}, {690,18947}, {1899,39842}, {2782,18951}, {2790,18953}, {2794,18909}, {6033,18917}, {6721,11487}, {6722,18928}, {6776,39857}, {9861,18914}, {11245,39832}, {12324,39838}, {13567,39849}, {14912,39834}, {18911,39836}, {18912,39837}, {18913,39841}, {18915,39844}, {18918,39847}, {18919,39848}, {18921,39850}, {18922,39851}, {18923,39852}, {18924,39853}, {18925,39854}, {18926,39855}, {18927,39856}, {18929,39858}, {18930,39859}, {18931,39860}, {19119,39840}, {19166,39843}, {23291,39845}

X(39833) = reflection of X(39804) in X(18951)
X(39833) = {X(1899), X(39846)}-harmonic conjugate of X(39842)

X(39834) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-CONWAY TO 7th BROCARD

Barycentrics    a^2*(a^8-2*(b^2+c^2)*a^6+(2*b^4+b^2*c^2+2*c^4)*a^4-(b^6+c^6)*a^2-(b^2-c^2)^2*b^2*c^2) : :
X(39834) = 3*X(11402)+X(39832) = 3*X(11402)-X(39839)

The reciprocal parallelogic center of these triangles is X(6).

X(39834) lies on these lines: {2,3044}, {3,39810}, {6,14574}, {26,39806}, {39,14601}, {49,38224}, {54,98}, {99,5012}, {110,14061}, {114,569}, {115,184}, {148,11003}, {182,620}, {389,39854}, {542,12228}, {567,6033}, {578,2794}, {690,13198}, {1092,38737}, {1147,6036}, {1176,10754}, {1614,14639}, {2023,3202}, {2782,32046}, {3455,13366}, {6722,9306}, {8723,36955}, {9861,11426}, {10539,23514}, {10722,15033}, {10984,38738}, {11402,39832}, {11422,39836}, {11423,39837}, {11424,39838}, {11425,39841}, {11427,39842}, {11428,39850}, {11429,39851}, {11430,39860}, {13336,38748}, {13346,38747}, {13352,38749}, {13353,15561}, {14645,19126}, {14912,39833}, {15074,39819}, {17809,39849}, {18388,39847}, {18475,39825}, {19365,39844}, {19408,39855}, {19409,39856}, {22115,38739}, {23292,39845}, {34148,34473}, {37471,38750}, {37472,38741}, {37476,39820}, {37495,38742}

X(39834) = midpoint of X(i) and X(j) for these {i,j}: {3, 39810}, {39832, 39839}
X(39834) = reflection of X(39805) in X(32046)
X(39834) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 39857, 39835), (11402, 39832, 39839)

X(39835) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CONWAY TO 7th BROCARD

Barycentrics    a^2*((b^2+c^2)*a^6-2*(b^4+c^4)*a^4+(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^2-(b^4+c^4)*(b^2-c^2)^2) : :
X(39835) = 3*X(51)-X(115) = 3*X(51)+X(39846) = X(98)-5*X(3567) = X(99)+3*X(3060) = X(148)-9*X(11002) = 3*X(568)+X(6033) = 3*X(3917)-5*X(31274) = X(5562)-3*X(36519) = 9*X(5640)-5*X(14061) = 9*X(5640)-X(39836) = 3*X(5890)+X(10722) = 3*X(5943)-2*X(6722) = 3*X(5946)-X(12042) = 3*X(6054)+X(39808) = X(6243)+3*X(15561) = 3*X(9730)-X(38749) = 7*X(9781)-3*X(14639) = 7*X(9781)+X(39837) = 5*X(14061)-X(39836) = 3*X(14639)+X(39837) .

The reciprocal parallelogic center of these triangles is X(6).

X(39835) lies on the nine-point circle of the orthic triangle and on these lines: {4,39833}, {6,14574}, {25,39839}, {26,39805}, {51,115}, {52,114}, {98,3567}, {99,3060}, {143,2782}, {148,11002}, {185,39838}, {389,2794}, {511,620}, {512,12076}, {542,9969}, {543,21849}, {568,6033}, {578,39854}, {690,1112}, {1216,6721}, {1843,5477}, {1994,3044}, {2482,21969}, {3221,38017}, {3455,15004}, {3917,31274}, {5446,23698}, {5462,6036}, {5562,36519}, {5640,14061}, {5890,10722}, {5943,6722}, {5946,12042}, {6054,39808}, {6102,22505}, {6243,15561}, {6746,12131}, {9729,38747}, {9730,38749}, {9777,39832}, {9781,14639}, {9786,39841}, {9792,39843}, {9861,11432}, {10263,33813}, {10625,38748}, {11433,39842}, {11435,39850}, {11436,39851}, {11438,39860}, {11746,15359}, {12188,13321}, {13364,15092}, {13417,15357}, {13567,39845}, {14981,39817}, {15026,34127}, {15043,34473}, {16625,38745}, {17810,39849}, {18390,39847}, {19366,39844}, {19410,39855}, {19411,39856}, {37481,38741}, {37484,38750}, {37493,39810}

X(39835) = midpoint of X(i) and X(j) for these {i,j}: {52, 114}, {115, 39846}, {185, 39838}, {1843, 5477}, {2482, 21969}, {6102, 22505}, {10263, 33813}, {13417, 15357}, {14981, 39817}
X(39835) = reflection of X(i) in X(j) for these (i,j): (1216, 6721), (6036, 5462), (15359, 11746), (38747, 9729), (39806, 143)
X(39835) = X(116)-of-orthic-triangle if ABC is acute
X(39835) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 39857, 39834), (51, 39846, 115), (5640, 39836, 14061), (9781, 39837, 14639)

X(39836) = PARALLELOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO 7th BROCARD

Barycentrics    a^2*((b^2+c^2)*a^6-(2*b^4+b^2*c^2+2*c^4)*a^4+2*(b^6+c^6)*a^2-c^8-b^8+b^4*c^4) : :
X(39836) = 2*X(52)-3*X(14651) = 2*X(99)-3*X(2979) = 4*X(114)-5*X(11444) = 4*X(115)-3*X(3060) = 8*X(620)-9*X(7998) = 5*X(3567)-6*X(38224) = 9*X(5640)-10*X(14061) = 9*X(5640)-8*X(39835) = 3*X(5890)-4*X(12042) = 2*X(6033)-3*X(11459) = 8*X(6036)-7*X(15043) = 16*X(6722)-15*X(11451) = 7*X(7999)-6*X(15561) = 2*X(10263)-3*X(38732) = 5*X(10574)-6*X(34473) = 2*X(10722)-3*X(15305) = 5*X(11439)-4*X(39838) = 4*X(11591)-3*X(38743) = 5*X(14061)-4*X(39835) = 11*X(15024)-12*X(34127)

The reciprocal parallelogic center of these triangles is X(6).

X(39836) lies on these lines: {2,39846}, {3,39837}, {22,39849}, {52,14651}, {98,5889}, {99,2979}, {110,39857}, {114,11444}, {115,3060}, {147,5562}, {148,511}, {542,12219}, {620,7998}, {690,13201}, {1154,12188}, {1993,39832}, {2782,11412}, {2794,12111}, {3044,3455}, {3565,15054}, {3567,38224}, {5012,39839}, {5640,14061}, {5876,38744}, {5890,12042}, {6033,11459}, {6036,15043}, {6101,13188}, {6241,38741}, {6722,11451}, {7731,18332}, {7999,15561}, {9861,11441}, {9862,13754}, {10263,38732}, {10574,34473}, {10625,13172}, {10722,15305}, {11422,39834}, {11439,39838}, {11440,39841}, {11442,39842}, {11443,39848}, {11445,39850}, {11446,39851}, {11447,39852}, {11448,39853}, {11449,39854}, {11452,39858}, {11453,39859}, {11454,39860}, {11591,38743}, {13174,31737}, {13391,38733}, {15024,34127}, {15045,38739}, {15058,22505}, {15072,38749}, {15801,39810}, {18392,39847}, {18911,39833}, {19122,39840}, {19167,39843}, {19367,39844}, {19412,39855}, {19413,39856}, {23293,39845}

X(39836) = reflection of X(i) in X(j) for these (i,j): (147, 5562), (5889, 98), (6241, 38741), (7731, 18332), (13172, 10625), (13174, 31737), (13188, 6101), (38744, 5876), (39807, 11412), (39808, 12188), (39837, 3)
X(39836) = anticomplement of X(39846)
X(39836) = {X(14061), X(39835)}-harmonic conjugate of X(5640)

X(39837) = PARALLELOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO 7th BROCARD

Barycentrics    (SB+SC)*(5*S^4-(4*(9*SA+2*SW)*R^2-11*SA^2+6*SB*SC-3*SW^2)*S^2+(4*R^2+SA-2*SW)*SA*SW^2) : :
X(39837) = 2*X(98)-3*X(5890) = 4*X(114)-3*X(11459) = 4*X(115)-5*X(3567) = 4*X(143)-3*X(38732) = 4*X(389)-3*X(14651) = 8*X(620)-7*X(7999) = 3*X(671)-4*X(39806) = 3*X(2979)-4*X(33813) = 3*X(3060)-2*X(6321) = 2*X(5876)-3*X(38743) = 8*X(6036)-9*X(15045) = 16*X(6722)-17*X(11465) = 9*X(7998)-10*X(38750) = 7*X(9781)-6*X(14639) = 7*X(9781)-8*X(39835) = 5*X(10574)-4*X(12042) = 5*X(11444)-6*X(15561) = 3*X(11455)-4*X(39838) = 10*X(14061)-11*X(15024) = 3*X(14639)-4*X(39835)

The reciprocal parallelogic center of these triangles is X(6).

X(39837) lies on these lines: {3,39836}, {4,39846}, {24,39849}, {52,148}, {54,39839}, {74,39841}, {98,5890}, {99,11412}, {114,11459}, {115,3567}, {143,38732}, {147,11674}, {185,9862}, {389,14651}, {511,13172}, {542,6403}, {620,7999}, {671,39806}, {690,7731}, {1154,13188}, {1614,39857}, {1986,22265}, {2782,5889}, {2794,6241}, {2979,33813}, {3060,6321}, {3563,14094}, {5663,38744}, {5876,38743}, {6033,12111}, {6036,15045}, {6102,12188}, {6722,11465}, {7592,39832}, {7998,38750}, {9781,14639}, {9860,31728}, {9861,11456}, {10263,38733}, {10574,12042}, {10628,18331}, {10722,12290}, {11005,12281}, {11423,39834}, {11444,15561}, {11455,39838}, {11457,39842}, {11458,39848}, {11460,39850}, {11461,39851}, {11462,39852}, {11463,39853}, {11464,39854}, {11466,39858}, {11467,39859}, {11468,39860}, {12243,14831}, {14061,15024}, {15028,34127}, {15043,38224}, {15072,38741}, {15100,15545}, {15305,22505}, {18394,39847}, {18912,39833}, {19123,39840}, {19168,39843}, {19368,39844}, {19414,39855}, {19415,39856}, {23294,39845}

X(39837) = reflection of X(i) in X(j) for these (i,j): (4, 39846), (148, 52), (9860, 31728), (9862, 185), (11412, 99), (12111, 6033), (12188, 6102), (12243, 14831), (12281, 11005), (12290, 10722), (15100, 15545), (22265, 1986), (38733, 10263), (39807, 13188), (39808, 5889), (39836, 3)
X(39837) = {X(14639), X(39835)}-harmonic conjugate of X(9781)

X(39838) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO 7th BROCARD

Barycentrics    4*a^8-4*(b^2+c^2)*a^6+(b^2+c^2)^2*a^4+2*(b^4-c^4)*(b^2-c^2)*a^2-(3*b^4-2*b^2*c^2+3*c^4)*(b^2-c^2)^2 : :
X(39838) = 3*X(3)-4*X(6721) = 4*X(3)-5*X(31274) = 2*X(3)-3*X(36519) = 3*X(4)-X(98) = 5*X(4)-X(9862) = 4*X(4)-X(10991) = 5*X(4)-2*X(11623) = 5*X(4)-3*X(14639) = 7*X(4)-3*X(14651) = 2*X(98)-3*X(115) = 5*X(98)-3*X(9862) = X(98)+3*X(10722) = 4*X(98)-3*X(10991) = 5*X(98)-6*X(11623) = 5*X(98)-9*X(14639) = 7*X(98)-9*X(14651) = 5*X(115)-2*X(9862) = X(115)+2*X(10722) = 5*X(115)-4*X(11623) = 5*X(115)-6*X(14639) = 7*X(115)-6*X(14651) = 16*X(6721)-15*X(31274) = 8*X(6721)-9*X(36519) = 5*X(31274)-6*X(36519)

The reciprocal parallelogic center of these triangles is X(6).

X(39838) lies on these lines: {2,38747}, {3,6721}, {4,32}, {5,38737}, {20,620}, {24,39860}, {25,39841}, {30,114}, {33,39844}, {34,39851}, {99,3146}, {147,543}, {148,17578}, {185,39835}, {235,39845}, {316,32458}, {376,9167}, {378,39854}, {381,6036}, {382,6033}, {542,1351}, {546,12042}, {550,38748}, {574,7694}, {671,5984}, {690,13202}, {868,35282}, {1498,39839}, {1503,1570}, {1513,6781}, {1569,38383}, {1593,39857}, {1597,3455}, {1656,38742}, {1657,15561}, {1699,11725}, {2023,22682}, {2453,10749}, {2456,29012}, {2777,15357}, {2782,3627}, {2797,38956}, {3091,6722}, {3529,20399}, {3534,38750}, {3832,14061}, {3839,5461}, {3843,38224}, {3845,6055}, {3850,34127}, {3851,38739}, {3853,22515}, {3858,15092}, {5059,38746}, {5073,38730}, {5076,12188}, {5182,14927}, {5480,14537}, {6054,13172}, {6564,8980}, {6565,13967}, {6776,11648}, {7517,39825}, {7823,36859}, {7983,9812}, {8724,15684}, {8781,32827}, {9756,18424}, {9880,15687}, {10304,22247}, {10352,33017}, {10353,33019}, {10721,11005}, {10728,10768}, {11001,23234}, {11177,36523}, {11381,39846}, {11403,39832}, {11424,39834}, {11439,39836}, {11455,39837}, {11470,39848}, {11471,39850}, {11473,39852}, {11474,39853}, {11475,39858}, {11476,39859}, {11632,38335}, {11710,18483}, {11711,28164}, {11724,18481}, {11737,26614}, {12007,39593}, {12091,18323}, {12184,12953}, {12185,12943}, {12324,39833}, {12355,35434}, {14269,14830}, {15545,38790}, {15640,36521}, {15704,38751}, {15811,39849}, {17800,38731}, {18534,39828}, {19055,23259}, {19056,23249}, {19124,39840}, {19169,39843}, {19416,39855}, {19417,39856}, {25486,32479}, {29317,35456}, {31723,39818}

X(39838) = midpoint of X(i) and X(j) for these {i,j}: {4, 10722}, {99, 3146}, {147, 10723}, {382, 6033}, {5073, 38730}, {6054, 15682}, {6321, 38744}, {8724, 15684}, {10721, 11005}, {10728, 10768}, {11381, 39846}, {15545, 38790}, {36961, 36962}
X(39838) = reflection of X(i) in X(j) for these (i,j): (20, 620), (99, 38745), (114, 22505), (115, 4), (185, 39835), (1569, 38383), (1657, 38736), (6055, 3845), (6781, 1513), (9862, 11623), (9880, 15687), (10991, 115), (11177, 36523), (11710, 18483), (12042, 546), (12188, 38734), (14981, 6033), (15300, 6054), (18481, 11724), (22515, 3853), (38738, 114), (38741, 6036), (38749, 5), (39809, 3627)
X(39838) = anticomplement of X(38747)
X(39838) = polar-circle-inverse of-X(10735)
X(39838) = X(20) of midpoint triangle of antipedal triangles of X(13) and X(14)
X(39838) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 36519, 31274), (4, 9862, 14639), (5, 38749, 38737), (114, 38738, 2482), (147, 3543, 10723), (381, 38741, 6036), (546, 12042, 23514), (1657, 15561, 38736), (3091, 34473, 6722), (3830, 38744, 6321), (5073, 38743, 38730), (9862, 14639, 11623), (11623, 14639, 115), (12042, 23514, 38740)

X(39839) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-EXTOUCH TO 7th BROCARD

Barycentrics    a^2*(a^8-3*(b^2+c^2)*a^6+(4*b^4+b^2*c^2+4*c^4)*a^4-(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^2+(b^6-c^6)*(b^2-c^2)) : :
X(39839) = 3*X(11402)-X(39832) = 3*X(11402)-2*X(39834)

The reciprocal parallelogic center of these triangles is X(6).

X(39839) lies on these lines: {2,39833}, {3,39805}, {6,13}, {25,39835}, {52,39828}, {54,39837}, {98,7592}, {99,1993}, {114,155}, {148,1994}, {184,39846}, {185,39841}, {195,13188}, {394,620}, {571,2902}, {690,19504}, {1181,2794}, {1199,14651}, {1351,13175}, {1498,39838}, {1899,39845}, {2079,35324}, {2088,9696}, {2482,37672}, {2782,12161}, {2914,18331}, {2936,34986}, {3269,13198}, {3455,17809}, {5012,39836}, {5182,20806}, {5422,14061}, {6036,36752}, {6321,36749}, {6722,10601}, {6776,39842}, {6782,10662}, {6783,10661}, {8745,20774}, {9861,19347}, {9862,15032}, {10602,39848}, {10605,39860}, {10628,32761}, {10677,25236}, {10678,25235}, {10722,11456}, {11004,20094}, {11402,39832}, {12160,39803}, {12829,23128}, {13178,16473}, {13352,39812}, {14627,38732}, {14981,39820}, {15357,17847}, {15801,39807}, {16266,33813}, {16472,38220}, {17811,31274}, {17814,36519}, {18396,39847}, {19125,39840}, {19170,39843}, {19349,39844}, {19350,39850}, {19354,39851}, {19355,39852}, {19356,39853}, {19357,39854}, {19358,39855}, {19359,39856}, {19363,39858}, {19364,39859}, {22505,32139}, {22515,39522}, {23698,36747}, {36753,38224}, {37483,38736}, {37489,39825}, {37493,39806}, {37498,38738}, {37514,38737}

X(39839) = midpoint of X(12160) and X(39803)
X(39839) = reflection of X(i) in X(j) for these (i,j): (3, 39805), (39810, 12161), (39832, 39834)
X(39839) = orthocentroidal circle-inverse of-X(1879)
X(39839) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 39849, 115), (184, 39846, 39857), (11402, 39832, 39834), (31862, 31863, 1879)

X(39840) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER TO 7th BROCARD

Barycentrics    a^2*(a^10-(b^2+c^2)*a^8+b^2*c^2*a^6+(b^4-c^4)*(b^2-c^2)*a^4-(b^2-c^2)^2*(b^4+c^4)*a^2-(b^4-c^4)*(b^2-c^2)*b^2*c^2) : :
X(39840) = 5*X(3618)-X(39842) = 3*X(5050)+X(9861) = 3*X(5085)-X(39841) = 5*X(19132)-X(39849) = X(39848)+2*X(39857)

The reciprocal parallelogic center of these triangles is X(6).

X(39840) lies on these lines: {6,14574}, {98,19128}, {99,19121}, {114,19131}, {115,1974}, {182,2794}, {184,5477}, {193,3044}, {206,542}, {511,39854}, {620,19126}, {690,1177}, {1176,5182}, {1428,39844}, {1576,5661}, {1692,14600}, {1976,2395}, {2030,2386}, {2330,39851}, {2782,19154}, {2790,19156}, {3589,39845}, {3618,39842}, {5026,19127}, {5050,9861}, {5085,39841}, {5092,39860}, {6033,19129}, {6722,19137}, {11646,18374}, {19118,39832}, {19119,39833}, {19122,39836}, {19123,39837}, {19124,39838}, {19125,39839}, {19130,39847}, {19132,39849}, {19133,39850}, {19134,39855}, {19135,39856}, {19171,39843}, {21637,39846}

X(39840) = midpoint of X(6) and X(39857)
X(39840) = reflection of X(i) in X(j) for these (i,j): (39811, 19154), (39845, 3589), (39847, 19130), (39848, 6), (39860, 5092)
X(39840) = crosspoint of X(1177) and X(1976)
X(39840) = crosssum of X(325) and X(858)

X(39841) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH TO 7th BROCARD

Barycentrics    (SB+SC)*(4*S^4-(2*(9*SA-SW)*R^2-4*SA^2+3*SB*SC+SW^2)*S^2+(8*R^2-SA-SW)*SA*SW^2) : :
X(39841) = 3*X(3)-X(9861) = 3*X(3)-2*X(39854) = 3*X(5085)-2*X(39840) = 2*X(9861)-3*X(39857) = X(9861)-6*X(39860) = 3*X(37497)-X(39820) = 4*X(39854)-3*X(39857) = X(39854)-3*X(39860) = X(39857)-4*X(39860)

The reciprocal parallelogic center of these triangles is X(6).

X(39841) lies on these lines: {3,114}, {4,39845}, {20,39842}, {24,10722}, {25,39838}, {30,39828}, {55,39844}, {56,39851}, {64,39849}, {74,39837}, {98,378}, {99,11413}, {115,1593}, {147,2071}, {148,12086}, {185,39839}, {382,39847}, {542,12302}, {690,2935}, {1151,39852}, {1152,39853}, {1204,39846}, {1576,38608}, {2386,18860}, {2782,12084}, {2790,33813}, {3455,11410}, {3516,10991}, {3520,9862}, {5085,39840}, {5584,39850}, {6036,9818}, {6644,22505}, {6722,11479}, {7387,39825}, {7395,38737}, {7464,13172}, {7503,34473}, {7526,12042}, {9786,39835}, {11425,39834}, {11440,39836}, {11477,39848}, {11480,39858}, {11481,39859}, {12085,23698}, {13021,39855}, {13022,39856}, {13188,18859}, {13352,39810}, {14639,35502}, {14651,14865}, {18913,39833}, {19172,39843}, {21312,38738}, {37497,39820}

X(39841) = midpoint of X(i) and X(j) for these {i,j}: {20, 39842}, {64, 39849}
X(39841) = reflection of X(i) in X(j) for these (i,j): (3, 39860), (4, 39845), (382, 39847), (7387, 39825), (9861, 39854), (11477, 39848), (39812, 12084), (39857, 3)
X(39841) = circumperp conjugate of X(127)
X(39841) = circumcircle-inverse of-X(14689)
X(39841) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 9861, 39854), (9861, 39854, 39857), (21312, 39803, 38738)

X(39842) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO 7th BROCARD

Barycentrics    a^10-(b^4+b^2*c^2+c^4)*a^6+(b^6+c^6)*a^4-(b^2-c^2)^2*b^2*c^2*a^2-(b^6+c^6)*(b^2-c^2)^2 : :
X(39842) = 3*X(2)-4*X(39845) = 3*X(4)-4*X(39847) = 3*X(376)-4*X(39860) = 5*X(631)-4*X(39854) = 3*X(1992)-4*X(39848) = 5*X(3618)-4*X(39840)

The reciprocal parallelogic center of these triangles is X(6).

X(39842) lies on these lines: {2,24270}, {4,32}, {5,9861}, {20,39841}, {30,13175}, {52,39804}, {99,1370}, {114,6643}, {127,1632}, {147,28728}, {148,7391}, {315,8783}, {316,2386}, {376,39860}, {388,39844}, {427,39832}, {497,39851}, {542,12319}, {620,7386}, {631,39854}, {690,13203}, {1503,39849}, {1899,39833}, {1992,39848}, {2550,39850}, {2782,14790}, {2790,6033}, {2936,7396}, {3044,37645}, {3068,39852}, {3069,39853}, {3455,8889}, {3618,39840}, {4226,28437}, {5189,20094}, {5477,18935}, {6036,7401}, {6722,7392}, {6776,39839}, {6803,38737}, {6804,36519}, {6815,34473}, {6997,14061}, {7528,38224}, {10996,38747}, {11427,39834}, {11433,39835}, {11442,39836}, {11457,39837}, {11488,39858}, {11489,39859}, {12042,18420}, {12188,31723}, {14064,39644}, {18404,38744}, {19174,39843}, {19420,39855}, {19421,39856}, {23698,34938}, {31305,39828}, {38861,38971}

X(39842) = reflection of X(i) in X(j) for these (i,j): (20, 39841), (9861, 5), (31305, 39828), (39813, 14790), (39857, 39845)
X(39842) = anticomplement of X(39857)
X(39842) = anticomplementary-circle-inverse of-X(112)
X(39842) = X(66)-of-1st-anti-Brocard-triangle
X(39842) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1899, 39846, 39833), (39845, 39857, 2)

X(39843) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN TO 7th BROCARD

Barycentrics    (a^4-(2*b^2+c^2)*a^2+(b^2-c^2)*b^2)*(a^4-(b^2+2*c^2)*a^2-(b^2-c^2)*c^2)*(a^8-(b^2+c^2)*a^6+b^2*c^2*a^4-(b^2-c^2)^2*b^2*c^2) : :

The reciprocal parallelogic center of these triangles is X(6).

X(39843) lies on these lines: {54,98}, {95,620}, {97,99}, {114,19179}, {115,275}, {542,19193}, {690,19208}, {2782,19210}, {2790,19212}, {2794,8884}, {4993,14061}, {4994,14639}, {6033,19176}, {6722,19188}, {9792,39835}, {9861,19173}, {10991,38808}, {16030,39832}, {19166,39833}, {19167,39836}, {19168,39837}, {19169,39838}, {19170,39839}, {19171,39840}, {19172,39841}, {19174,39842}, {19175,39844}, {19177,39847}, {19178,39848}, {19180,39849}, {19181,39850}, {19182,39851}, {19183,39852}, {19184,39853}, {19185,39854}, {19186,39855}, {19187,39856}, {19189,39857}, {19190,39858}, {19191,39859}, {19192,39860}, {21638,39846}, {23295,39845}, {32458,34386}

X(39843) = reflection of X(39814) in X(19210)
X(39843) = barycentric product X(95)*X(9512)
X(39843) = trilinear product X(2167)*X(9512)

X(39844) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC TO 7th BROCARD

Barycentrics    a*(a+b-c)*(a-b+c)*(a^8-(b^2+c^2)*a^6+b^2*c^2*a^4+(b+c)*(b^2-c^2)*(b^3-c^3)*a^2-(b^4+c^4+(b^2+b*c+c^2)*b*c)*(b^2-c^2)^2) : :

The reciprocal parallelogic center of these triangles is X(6).

X(39844) lies on these lines: {1,2794}, {12,39845}, {30,39822}, {33,39838}, {34,115}, {35,39860}, {36,39854}, {55,39841}, {56,39857}, {65,39850}, {98,1870}, {99,4296}, {114,1060}, {221,39849}, {388,39842}, {542,19469}, {620,1038}, {690,19505}, {999,9861}, {1040,38747}, {1062,38749}, {1398,39832}, {1425,39846}, {1428,39840}, {2067,39852}, {2386,5194}, {2782,32047}, {3585,39847}, {4318,7983}, {4351,10089}, {6033,18447}, {6036,37697}, {6198,10722}, {6502,39853}, {6722,19372}, {7051,39858}, {11725,34036}, {18455,38741}, {18915,39833}, {19175,39843}, {19349,39839}, {19365,39834}, {19366,39835}, {19367,39836}, {19368,39837}, {19369,39848}, {19370,39855}, {19371,39856}, {19373,39859}, {22505,37729}

X(39844) = reflection of X(i) in X(j) for these (i,j): (39815, 32047), (39851, 1)

X(39845) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-URSA MINOR TO 7th BROCARD

Barycentrics    (b^2+c^2)*a^8-(b^2+c^2)^2*a^6+(b^2+c^2)*b^2*c^2*a^4+(b^4+c^4)*(b^2-c^2)^2*a^2-(b^6+c^6)*(b^2-c^2)^2 : :
X(39845) = 3*X(2)+X(39842) = 5*X(1656)-X(9861) = 3*X(1853)+X(39849) = X(13175)+3*X(34609)

The reciprocal parallelogic center of these triangles is X(6).

X(39845) lies on these lines: {2,24270}, {4,39841}, {5,2794}, {11,39851}, {12,39844}, {30,39825}, {98,1594}, {99,858}, {114,11585}, {115,427}, {125,39846}, {140,39854}, {148,31074}, {235,39838}, {403,10722}, {524,39848}, {542,23300}, {590,39852}, {615,39853}, {620,1368}, {625,2386}, {690,23315}, {1656,9861}, {1853,39849}, {1899,39839}, {2072,6033}, {2782,13371}, {2790,23333}, {3589,39840}, {3925,39850}, {5094,39832}, {5133,14061}, {5576,38224}, {6823,38747}, {7399,38737}, {7403,23514}, {7577,9862}, {8157,34845}, {8361,39644}, {10024,38741}, {10255,38744}, {13160,34473}, {13175,34609}, {13187,23301}, {13567,39835}, {14639,15559}, {14790,39828}, {15561,37452}, {15760,38749}, {23291,39833}, {23292,39834}, {23293,39836}, {23294,39837}, {23295,39843}, {23298,39855}, {23299,39856}, {23302,39858}, {23303,39859}, {23335,23698}, {30739,31274}, {33332,38229}, {37347,38739}

X(39845) = midpoint of X(i) and X(j) for these {i,j}: {4, 39841}, {14790, 39828}, {39842, 39857}, {39847, 39860}
X(39845) = reflection of X(i) in X(j) for these (i,j): (39816, 13371), (39840, 3589), (39854, 140)
X(39845) = complement of X(39857)
X(39845) = nine-point circle-inverse of-X(6720)
X(39845) = {X(2), X(39842)}-harmonic conjugate of X(39857)

X(39846) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO 7th BROCARD

Barycentrics    a^2*((b^2+c^2)*a^6-2*(b^4+c^4)*a^4+(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^2-(b^6-c^6)*(b^2-c^2)) : :
X(39846) = 3*X(51)-2*X(115) = 3*X(51)-4*X(39835) = X(148)-3*X(3060) = 9*X(373)-8*X(6722) = 3*X(568)-X(12188) = 4*X(620)-3*X(3917) = 2*X(1216)-3*X(15561) = 5*X(3567)-3*X(14651) = 3*X(5182)-2*X(11574) = 4*X(5447)-5*X(38750) = 4*X(5462)-3*X(38224) = 9*X(5650)-10*X(31274) = 3*X(5890)-X(9862) = 6*X(5892)-5*X(38739) = 6*X(5943)-5*X(14061) = 2*X(6055)-3*X(16226) = 4*X(9729)-3*X(34473) = 3*X(9730)-2*X(12042) = 4*X(10095)-3*X(38229) = 4*X(10110)-3*X(14639)

The reciprocal parallelogic center of these triangles is X(6).

X(39846) lies on these lines: {2,39836}, {4,39837}, {6,39832}, {25,39849}, {51,115}, {52,2782}, {98,389}, {99,511}, {114,5562}, {125,39845}, {147,5889}, {148,3060}, {184,39839}, {185,2794}, {373,6722}, {542,1843}, {543,21969}, {568,12188}, {620,3917}, {671,21849}, {690,13417}, {1112,16278}, {1181,9861}, {1204,39841}, {1216,15561}, {1425,39844}, {1899,39833}, {2784,31732}, {2790,6751}, {3044,34986}, {3270,39851}, {3313,5026}, {3455,13366}, {3567,14651}, {3611,39850}, {4173,12830}, {5139,15063}, {5167,6033}, {5182,11574}, {5446,6321}, {5447,38750}, {5462,38224}, {5477,6467}, {5650,31274}, {5890,9862}, {5892,38739}, {5943,14061}, {6000,10722}, {6055,16226}, {6243,13188}, {7731,18331}, {9729,34473}, {9730,12042}, {9969,11646}, {10095,38229}, {10110,14639}, {10625,33813}, {10628,11005}, {10723,13598}, {11381,39838}, {11557,18332}, {11599,31757}, {12160,39820}, {12162,22505}, {13148,13166}, {13175,33586}, {13367,39854}, {13851,39847}, {14531,14981}, {14845,15092}, {15644,21166}, {16222,33511}, {16625,38664}, {16981,35369}, {17834,39803}, {18436,38743}, {21637,39840}, {21638,39843}, {21639,39848}, {21640,39852}, {21641,39853}, {21642,39855}, {21643,39856}, {21647,39858}, {21648,39859}, {21663,39860}, {31739,31839}, {34783,38744}, {36987,38736}

X(39846) = midpoint of X(i) and X(j) for these {i,j}: {4, 39837}, {147, 5889}, {6243, 13188}, {7731, 18331}, {34783, 38744}
X(39846) = reflection of X(i) in X(j) for these (i,j): (98, 389), (115, 39835), (671, 21849), (3313, 5026), (4173, 12830), (5562, 114), (6321, 5446), (6467, 5477), (10625, 33813), (10723, 13598), (11381, 39838), (11599, 31757), (11646, 9969), (12162, 22505), (16278, 1112), (18332, 11557), (31739, 31839), (39817, 52)
X(39846) = complement of X(39836)
X(39846) = crosssum of X(2) and X(9512)
X(39846) = X(150)-of-orthic-triangle if ABC is acute
X(39846) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115, 39835, 51), (39833, 39842, 1899), (39839, 39857, 184)

X(39847) = PARALLELOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO 7th BROCARD

Barycentrics    2*(11*R^2-3*SW)*S^4+(SA-SW)*(R^2*(21*SA+SW)-6*SA*SW)*S^2-2*(3*R^2-SW)*SB*SC*SW^2 : :
X(39847) = 3*X(4)+X(39842) = 3*X(381)-X(39857) = 5*X(3843)-X(9861) = X(13175)+3*X(34725) = 3*X(18405)+X(39849)

The reciprocal parallelogic center of these triangles is X(6).

X(39847) lies on these lines: {4,32}, {5,39854}, {30,39825}, {99,3153}, {114,18404}, {381,39857}, {382,39841}, {542,12596}, {620,18531}, {690,19506}, {2782,18377}, {2790,18380}, {3583,39851}, {3585,39844}, {3843,9861}, {6033,18403}, {6321,31724}, {6564,39852}, {6565,39853}, {6722,18420}, {7574,38730}, {13175,34725}, {13371,39831}, {13851,39846}, {14791,38736}, {16808,39858}, {16809,39859}, {18386,39832}, {18388,39834}, {18390,39835}, {18392,39836}, {18394,39837}, {18396,39839}, {18405,39849}, {18406,39850}, {18414,39855}, {18415,39856}, {18569,23698}, {18918,39833}, {19130,39840}, {19177,39843}, {31723,39809}, {34007,34473}, {37444,38738}

X(39847) = midpoint of X(382) and X(39841)
X(39847) = reflection of X(i) in X(j) for these (i,j): (39818, 18377), (39831, 13371), (39840, 19130), (39854, 5), (39860, 39845)
X(39847) = X(39854)-of-Johnson-triangle

X(39848) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd EHRMANN TO 7th BROCARD

Barycentrics
a^2*(a^10-3*(b^2+c^2)*a^8+(b^2+2*c^2)*(2*b^2+c^2)*a^6+(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^4-(b^2-c^2)^2*(3*b^4+2*b^2*c^2+3*c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(2*b^4-b^2*c^2+2*c^4)) : :
X(39848) = 3*X(6)-X(39857) = 3*X(1992)+X(39842) = X(9861)-5*X(11482) = 3*X(17813)+X(39849) = 3*X(39840)-2*X(39857)

The reciprocal parallelogic center of these triangles is X(6).

X(39848) lies on these lines: {6,14574}, {98,8537}, {99,11416}, {114,8538}, {115,8541}, {511,39860}, {524,39845}, {542,12596}, {575,39854}, {576,2794}, {620,11511}, {690,13248}, {1992,39842}, {2782,11255}, {6033,18449}, {6722,9813}, {8539,39850}, {8540,39851}, {9861,11482}, {10602,39839}, {11405,39832}, {11443,39836}, {11458,39837}, {11470,39838}, {11477,39841}, {15074,39805}, {17813,39849}, {18919,39833}, {19178,39843}, {19369,39844}, {19426,39855}, {19427,39856}, {21639,39846}

X(39848) = midpoint of X(11477) and X(39841)
X(39848) = reflection of X(i) in X(j) for these (i,j): (39819, 11255), (39840, 6), (39854, 575)

X(39849) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st EXCOSINE TO 7th BROCARD

Barycentrics    a^2*(a^8-3*(b^2+c^2)*a^6+(4*b^4+b^2*c^2+4*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(39849) = 3*X(154)-2*X(39857) = 3*X(1853)-4*X(39845) = 3*X(10606)-4*X(39860) = 3*X(17813)-4*X(39848) = 5*X(17821)-4*X(39854) = 3*X(18405)-4*X(39847) = 5*X(19132)-4*X(39840) = 3*X(37497)-2*X(39812)

The reciprocal parallelogic center of these triangles is X(6).

X(39849) lies on these lines: {3,35324}, {6,13}, {22,39836}, {24,39837}, {25,39846}, {64,39841}, {98,1181}, {99,394}, {110,3269}, {112,14094}, {114,17814}, {147,11441}, {148,1993}, {154,39857}, {155,2782}, {184,39832}, {217,5984}, {221,39844}, {323,20094}, {511,13175}, {543,37672}, {574,9876}, {620,17811}, {690,17847}, {1498,2794}, {1503,39842}, {1562,24981}, {1614,22416}, {1853,39845}, {1970,12162}, {1971,13754}, {2192,39851}, {2420,12308}, {2790,17849}, {3197,39850}, {3289,13509}, {5562,39803}, {5663,32661}, {6036,37514}, {6321,36747}, {6722,17825}, {6759,9861}, {7592,14651}, {8571,18356}, {9306,39913}, {9862,11456}, {9915,30402}, {9916,30403}, {10601,14061}, {10606,39860}, {10982,14639}, {10985,14831}, {12111,14585}, {12160,39817}, {13567,39833}, {14591,15102}, {15805,34127}, {15811,39838}, {16278,19504}, {17809,39834}, {17810,39835}, {17813,39848}, {17819,39852}, {17820,39853}, {17821,39854}, {17826,39858}, {17827,39859}, {17834,39828}, {18405,39847}, {19132,39840}, {19180,39843}, {19430,39855}, {19431,39856}, {23128,32139}, {23698,37498}, {33843,34986}, {36749,38732}, {36752,38224}, {37476,39805}, {37483,38730}, {37497,39812}, {38297,38744}

X(39849) = reflection of X(i) in X(j) for these (i,j): (64, 39841), (9861, 6759), (17834, 39828), (39820, 155)
X(39849) = orthocentroidal circle-inverse of-X(36412)
X(39849) = pole of the trilinear polar of X(15351) with respect to MacBeath circumconic
X(39849) = crossdifference of every pair of points on line {X(526), X(11746)}
X(39849) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115, 39839, 6), (399, 22146, 1625), (1625, 22146, 6), (23128, 32139, 32445), (31862, 31863, 36412)

X(39850) = PARALLELOGIC CENTER OF THESE TRIANGLES: EXTANGENTS TO 7th BROCARD

Barycentrics
a*(a^9+b^2*c^2*a^5+(b+c)*a^8+(b+c)*b^2*c^2*a^4-(b^2+c^2)*a^7-(b+c)*(b^2+c^2)*a^6+(b+c)*(b^2-c^2)*(b^3-c^3)*a^3+(b^3+c^3)*(b^2-c^2)^2*a^2-(b^2-c^2)^2*(b^4+c^4+(b^2+b*c+c^2)*b*c)*a-(b^2-c^2)^2*(b+c)*(b^4+c^4-(b^2-b*c+c^2)*b*c)) : :

The reciprocal parallelogic center of these triangles is X(6).

X(39850) lies on the extangents circle and these lines: {19,115}, {40,2794}, {55,39851}, {65,39844}, {98,6197}, {99,3101}, {114,8251}, {147,9537}, {148,9536}, {542,12661}, {620,10319}, {690,10119}, {2550,39842}, {2782,8141}, {3197,39849}, {3611,39846}, {3925,39845}, {5415,39852}, {5416,39853}, {5584,39841}, {6033,18453}, {6722,9816}, {7688,39860}, {8539,39848}, {9572,9860}, {9573,13174}, {9861,10306}, {10636,39858}, {10637,39859}, {10902,39854}, {11406,39832}, {11428,39834}, {11435,39835}, {11445,39836}, {11460,39837}, {11471,39838}, {15941,38749}, {18406,39847}, {18921,39833}, {19133,39840}, {19181,39843}, {19350,39839}, {19432,39855}, {19433,39856}

X(39850) = reflection of X(i) in X(j) for these (i,j): (39821, 8141), (39851, 39857)
X(39850) = extangents circle-antipode of-X(39821)

X(39851) = PARALLELOGIC CENTER OF THESE TRIANGLES: INTANGENTS TO 7th BROCARD

Barycentrics    a*(-a+b+c)*(a^8-(b^2+c^2)*a^6+b^2*c^2*a^4+(b^2-c^2)*(b-c)*(b^3+c^3)*a^2-(b^2-c^2)^2*(b^4+c^4-(b^2-b*c+c^2)*b*c)) : :

The reciprocal parallelogic center of these triangles is X(6).

X(39851) lies on the intangents circle and these lines: {1,2794}, {11,39845}, {30,39815}, {33,115}, {34,39838}, {35,39854}, {36,39860}, {55,39850}, {56,39841}, {98,6198}, {99,3100}, {114,1062}, {147,9538}, {148,9539}, {497,39842}, {542,12888}, {620,1040}, {690,10118}, {1038,38747}, {1060,38749}, {1250,39859}, {1569,9635}, {1870,10722}, {2023,9595}, {2066,39852}, {2192,39849}, {2330,39840}, {2386,5148}, {2782,8144}, {3029,9550}, {3044,9637}, {3270,39846}, {3295,9861}, {3583,39847}, {4354,10086}, {5414,39853}, {6033,18455}, {6036,37696}, {6722,9817}, {7071,39832}, {8540,39848}, {9551,34454}, {9576,13174}, {9577,9860}, {9627,12184}, {9628,13182}, {9629,13183}, {9630,12185}, {9641,13188}, {9642,12188}, {9643,14981}, {9645,39828}, {10638,39858}, {11429,39834}, {11436,39835}, {11446,39836}, {11461,39837}, {12042,37729}, {18447,38741}, {18922,39833}, {19182,39843}, {19354,39839}, {19434,39855}, {19435,39856}

X(39851) = reflection of X(i) in X(j) for these (i,j): (39822, 8144), (39844, 1), (39850, 39857)
X(39851) = intangents circle-antipode of-X(39822)

X(39852) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS TO 7th BROCARD

Barycentrics    (SB+SC)*(2*S^4-(2*(3*SA+SW)*R^2-2*SA^2+3*SB*SC)*S^2+SB*SC*SW^2-S*(3*S^2-SW^2)*(4*R^2-SW)) : :

The reciprocal parallelogic center of these triangles is X(6).

X(39852) lies on these lines: {6,14574}, {26,39824}, {98,10880}, {99,11417}, {114,10897}, {115,5412}, {371,2794}, {372,39854}, {542,12891}, {590,39845}, {620,11513}, {690,13287}, {1151,39841}, {2066,39851}, {2067,39844}, {2782,11265}, {3068,39842}, {3311,9861}, {5410,39832}, {5415,39850}, {6033,18457}, {6200,39860}, {6564,39847}, {6722,10961}, {11447,39836}, {11462,39837}, {11473,39838}, {17819,39849}, {18923,39833}, {19183,39843}, {19355,39839}, {19436,39855}, {19438,39856}, {21640,39846}

X(39852) = reflection of X(39823) in X(11265)
X(39852) = {X(6), X(39857)}-harmonic conjugate of X(39853)

X(39853) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS TO 7th BROCARD

Barycentrics    (SB+SC)*(2*S^4-(2*(3*SA+SW)*R^2-2*SA^2+3*SB*SC)*S^2+SB*SC*SW^2+S*(3*S^2-SW^2)*(4*R^2-SW)) : :

The reciprocal parallelogic center of these triangles is X(6).

X(39853) lies on these lines: {6,14574}, {26,39823}, {98,10881}, {99,11418}, {114,10898}, {115,5413}, {371,39854}, {372,2794}, {542,12892}, {615,39845}, {620,11514}, {690,13288}, {1152,39841}, {2782,11266}, {3069,39842}, {3312,9861}, {5411,39832}, {5414,39851}, {5416,39850}, {6033,18459}, {6396,39860}, {6502,39844}, {6565,39847}, {6722,10963}, {11448,39836}, {11463,39837}, {11474,39838}, {17820,39849}, {18924,39833}, {19184,39843}, {19356,39839}, {19437,39856}, {19439,39855}, {21641,39846}

X(39853) = reflection of X(39824) in X(11266)
X(39853) = {X(6), X(39857)}-harmonic conjugate of X(39852)

X(39854) = PARALLELOGIC CENTER OF THESE TRIANGLES: KOSNITA TO 7th BROCARD

Barycentrics
a^2*(a^12-3*(b^2+c^2)*a^10-4*(b^2+c^2)*b^2*c^2*a^6+(3*b^4+5*b^2*c^2+3*c^4)*a^8-(3*b^8+3*c^8-(3*b^4+2*b^2*c^2+3*c^4)*b^2*c^2)*a^4+3*(b^8-c^8)*(b^2-c^2)*a^2-(b^8+c^8)*(b^2-c^2)^2) : :
X(39854) = 3*X(3)+X(9861) = 3*X(3)-X(39841) = 5*X(631)-X(39842) = X(9861)-3*X(39857) = 2*X(9861)+3*X(39860) = X(13175)-5*X(16195) = 3*X(14070)-X(39828) = 5*X(17821)-X(39849) = X(39841)+3*X(39857) = 2*X(39841)-3*X(39860) = 2*X(39857)+X(39860)

The reciprocal parallelogic center of these triangles is X(6).

X(39854) lies on these lines: {3,114}, {5,39847}, {15,39859}, {16,39858}, {22,38738}, {23,10723}, {24,115}, {26,23698}, {30,39816}, {35,39851}, {36,39844}, {98,186}, {99,7488}, {140,39845}, {147,10298}, {371,39853}, {372,39852}, {378,39838}, {389,39834}, {511,39840}, {542,12893}, {543,14070}, {575,39848}, {578,39835}, {631,39842}, {671,37940}, {690,13289}, {1658,2782}, {2070,6321}, {2790,37813}, {2936,39803}, {2937,38730}, {2967,14676}, {3044,5889}, {3515,11623}, {3518,14639}, {3520,10722}, {5961,14592}, {6036,6644}, {6642,6722}, {6721,7514}, {7387,39812}, {7502,14655}, {7503,36519}, {7506,23514}, {7509,31274}, {7512,21166}, {7517,39809}, {7556,13172}, {7575,13233}, {8980,9682}, {9590,13178}, {9862,21844}, {10902,39850}, {10991,32534}, {11449,39836}, {11464,39837}, {11616,25644}, {11632,37922}, {12042,37814}, {13175,16195}, {13367,39846}, {13564,38731}, {14645,37488}, {14830,37955}, {14981,38444}, {17821,39849}, {17928,38737}, {18475,39805}, {18570,22505}, {18925,39833}, {19185,39843}, {19357,39839}, {19440,39855}, {19441,39856}, {22467,34473}, {22515,37440}, {37489,39810}

X(39854) = midpoint of X(i) and X(j) for these {i,j}: {3, 39857}, {7387, 39812}, {9861, 39841}
X(39854) = reflection of X(i) in X(j) for these (i,j): (39825, 1658), (39845, 140), (39847, 5), (39848, 575), (39860, 3)
X(39854) = circumcircle-inverse of-X(10749)
X(39854) = X(39847)-of-Johnson-triangle
X(39854) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 9861, 39841), (39841, 39857, 9861)

X(39855) = PARALLELOGIC CENTER OF THESE TRIANGLES: LUCAS ANTIPODAL TANGENTIAL TO 7th BROCARD

Barycentrics    (SB+SC)*(6*(2*R^2+SA)*S^4-(6*(SA+SW)*R^2+SB*SC)*SW*S^2+SB*SC*SW^3+S*(10*S^4-2*(2*(3*SA+SW)*R^2-5*SA^2+3*SB*SC)*S^2-2*SA*SW^3)) : :

The reciprocal parallelogic center of these triangles is X(6).

X(39855) lies on these lines: {98,19424}, {99,19406}, {114,19428}, {115,19446}, {542,19482}, {620,19422}, {690,19507}, {2782,39826}, {2794,18980}, {6033,18462}, {6722,19448}, {8939,39857}, {9723,39856}, {9861,19418}, {13021,39841}, {18414,39847}, {18926,39833}, {19134,39840}, {19186,39843}, {19358,39839}, {19370,39844}, {19404,39832}, {19408,39834}, {19410,39835}, {19412,39836}, {19414,39837}, {19416,39838}, {19420,39842}, {19426,39848}, {19430,39849}, {19432,39850}, {19434,39851}, {19436,39852}, {19439,39853}, {19440,39854}, {19450,39858}, {19452,39859}, {19454,39860}, {21642,39846}, {23298,39845}

X(39856) = PARALLELOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) ANTIPODAL TANGENTIAL TO 7th BROCARD

Barycentrics    (SB+SC)*(6*(2*R^2+SA)*S^4-(6*(SA+SW)*R^2+SB*SC)*SW*S^2+SB*SC*SW^3-S*(10*S^4-2*(2*(3*SA+SW)*R^2-5*SA^2+3*SB*SC)*S^2-2*SA*SW^3)) : :

The reciprocal parallelogic center of these triangles is X(6).

X(39856) lies on these lines: {98,19425}, {99,19407}, {114,19429}, {115,19447}, {542,19483}, {620,19423}, {690,19508}, {2782,39827}, {2794,18981}, {6033,18463}, {6722,19449}, {8943,39857}, {9723,39855}, {9861,19419}, {13022,39841}, {18415,39847}, {18927,39833}, {19135,39840}, {19187,39843}, {19359,39839}, {19371,39844}, {19405,39832}, {19409,39834}, {19411,39835}, {19413,39836}, {19415,39837}, {19417,39838}, {19421,39842}, {19427,39848}, {19431,39849}, {19433,39850}, {19435,39851}, {19437,39853}, {19438,39852}, {19441,39854}, {19451,39858}, {19453,39859}, {19455,39860}, {21643,39846}, {23299,39845}

X(39857) = PARALLELOGIC CENTER OF THESE TRIANGLES: TANGENTIAL TO 7th BROCARD

Barycentrics    a^2*(a^8-(b^2+c^2)*a^6+b^2*c^2*a^4+(b^4-c^4)*(b^2-c^2)*a^2-(b^6-c^6)*(b^2-c^2)) : :
X(39857) = 3*X(3)-2*X(39860) = 3*X(6)-2*X(39848) = 3*X(154)-X(39849) = 3*X(381)-2*X(39847) = 2*X(9861)+X(39841) = X(9861)+2*X(39854) = 3*X(9861)+2*X(39860) = 3*X(9876)+X(13175) = 3*X(9909)-X(13175) = 3*X(14070)-2*X(39825) = 3*X(39840)-X(39848) = X(39841)-4*X(39854) = 3*X(39841)-4*X(39860) = 3*X(39854)-X(39860)

The reciprocal parallelogic center of these triangles is X(6).

X(39857) lies on the tangential circle and these lines: {2,24270}, {3,114}, {6,14574}, {22,99}, {23,148}, {24,98}, {25,115}, {26,2782}, {30,39812}, {52,39810}, {55,39850}, {56,39844}, {110,39836}, {147,7488}, {154,39849}, {157,1605}, {159,542}, {160,2934}, {184,39839}, {186,9862}, {187,2386}, {378,10722}, {381,39847}, {543,9876}, {690,10117}, {1569,9700}, {1593,39838}, {1614,39837}, {1658,15270}, {1971,14917}, {1993,3044}, {1995,14061}, {2023,9608}, {2070,5938}, {2079,7669}, {2793,14657}, {2871,32661}, {2937,13188}, {3029,9571}, {3447,18105}, {3515,10991}, {3517,11623}, {3518,14651}, {4226,28438}, {5020,6722}, {5113,20998}, {5152,33802}, {5477,19459}, {5899,38733}, {6036,6642}, {6321,7517}, {6660,8178}, {6721,7393}, {6776,39833}, {7387,23698}, {7395,36519}, {7484,31274}, {7506,38224}, {7526,22505}, {7529,23514}, {7530,22515}, {8185,13178}, {8276,8980}, {8277,13967}, {8939,39855}, {8943,39856}, {9570,34454}, {9590,9860}, {9591,13174}, {9645,39822}, {9658,13182}, {9659,12184}, {9672,12185}, {9673,13183}, {9715,14981}, {9864,15177}, {10323,21166}, {10594,14639}, {11177,37940}, {11365,11725}, {11414,38738}, {11646,20987}, {12083,38730}, {12085,39831}, {12088,13172}, {12243,37939}, {13171,15357}, {13173,20872}, {13558,21525}, {14645,37491}, {14667,23402}, {14729,21006}, {14790,39816}, {17834,39820}, {17928,34473}, {18378,38732}, {18534,39809}, {19189,39843}, {20094,37913}, {20968,39575}, {31305,39813}, {32762,37459}, {35243,38736}, {37123,38654}

X(39857) = midpoint of X(i) and X(j) for these {i,j}: {3, 9861}, {9876, 9909}, {9915, 9916}, {17834, 39820}, {31305, 39813}, {39850, 39851}
X(39857) = reflection of X(i) in X(j) for these (i,j): (3, 39854), (6, 39840), (12085, 39831), (14790, 39816), (39828, 26), (39841, 3), (39842, 39845)
X(39857) = circumperp conjugate of X(14689)
X(39857) = isogonal conjugate of the cyclocevian conjugate of X(11794)
X(39857) = isogonal conjugate of the antigonal conjugate of X(15388)
X(39857) = anticomplement of X(39845)
X(39857) = complement of X(39842)
X(39857) = tangential circle-antipode of-X(39828)
X(39857) = circumcircle-inverse of-X(127)
X(39857) = pole of the trilinear polar of X(850) with respect to circumcircle
X(39857) = crossdifference of every pair of points on line {X(23584), X(34990)}
X(39857) = crosssum of X(511) and X(36471)
X(39857) = X(850)-Ceva conjugate of-X(6)
X(39857) = X(127)-vertex conjugate of-X(2799)
X(39857) = X(101)-of-tangential-triangle if ABC is acute
X(39857) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 39842, 39845), (25, 39832, 115), (115, 3455, 39832), (184, 39846, 39839), (9861, 39854, 39841), (39834, 39835, 6), (39852, 39853, 6), (39858, 39859, 6)

X(39858) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL TO 7th BROCARD

Barycentrics
a^2*(-2*sqrt(3)*(a^8-(b^2+c^2)*a^6+b^2*c^2*a^4+(b^4-c^4)*(b^2-c^2)*a^2-(b^6-c^6)*(b^2-c^2))*S+(a^2-b^2+c^2)*(-a^2+b^2+c^2)*(a^2+b^2-c^2)*(a^4-(b^2+c^2)*a^2+b^4-b^2*c^2+c^4)) : :

The reciprocal parallelogic center of these triangles is X(6).

X(39858) lies on these lines: {6,14574}, {15,2794}, {16,39854}, {26,39830}, {98,10632}, {99,11420}, {114,10634}, {115,10641}, {542,10663}, {620,11515}, {690,10681}, {2782,11267}, {3455,8740}, {6033,18468}, {6722,10643}, {7051,39844}, {9861,11485}, {10636,39850}, {10638,39851}, {10645,39860}, {11408,39832}, {11452,39836}, {11466,39837}, {11475,39838}, {11480,39841}, {11488,39842}, {16808,39847}, {17826,39849}, {18929,39833}, {19190,39843}, {19363,39839}, {19450,39855}, {19451,39856}, {21647,39846}, {23302,39845}

X(39858) = reflection of X(39829) in X(11267)
X(39858) = {X(6), X(39857)}-harmonic conjugate of X(39859)

X(39859) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL TO 7th BROCARD

Barycentrics
a^2*(2*sqrt(3)*(a^8-(b^2+c^2)*a^6+b^2*c^2*a^4+(b^4-c^4)*(b^2-c^2)*a^2-(b^6-c^6)*(b^2-c^2))*S+(a^2-b^2+c^2)*(-a^2+b^2+c^2)*(a^2+b^2-c^2)*(a^4-(b^2+c^2)*a^2+b^4-b^2*c^2+c^4)) : :

The reciprocal parallelogic center of these triangles is X(6).

X(39859) lies on these lines: {6,14574}, {15,39854}, {16,2794}, {26,39829}, {98,10633}, {99,11421}, {114,10635}, {115,10642}, {542,10664}, {620,11516}, {690,10682}, {1250,39851}, {2782,11268}, {3455,8739}, {6033,18470}, {6722,10644}, {9861,11486}, {10637,39850}, {10646,39860}, {11409,39832}, {11453,39836}, {11467,39837}, {11476,39838}, {11481,39841}, {11489,39842}, {16809,39847}, {17827,39849}, {18930,39833}, {19191,39843}, {19364,39839}, {19373,39844}, {19452,39855}, {19453,39856}, {21648,39846}, {23303,39845}

X(39859) = reflection of X(39830) in X(11268)
X(39859) = {X(6), X(39857)}-harmonic conjugate of X(39858)

X(39860) = PARALLELOGIC CENTER OF THESE TRIANGLES: TRINH TO 7th BROCARD

Barycentrics    (SB+SC)*(7*S^4-(2*(15*SA-SW)*R^2-7*SA^2+6*SB*SC+SW^2)*S^2+(12*R^2-SA-2*SW)*SA*SW^2) : :
X(39860) = 5*X(3)-X(9861) = 3*X(3)-X(39857) = 3*X(376)+X(39842) = X(9861)+5*X(39841) = 2*X(9861)-5*X(39854) = 3*X(9861)-5*X(39857) = 3*X(10606)+X(39849) = 2*X(39841)+X(39854) = 3*X(39841)+X(39857) = 3*X(39854)-2*X(39857)

The reciprocal parallelogic center of these triangles is X(6).

X(39860) lies on these lines: {3,114}, {24,39838}, {30,39825}, {35,39844}, {36,39851}, {98,3520}, {99,1236}, {115,378}, {186,10722}, {376,39842}, {511,39848}, {542,12901}, {543,39812}, {690,13293}, {2782,11250}, {3455,9862}, {3516,11623}, {5092,39840}, {5984,35494}, {6036,7526}, {6054,37948}, {6200,39852}, {6396,39853}, {6722,9818}, {7503,38737}, {7527,14061}, {7688,39850}, {10605,39839}, {10606,39849}, {10645,39858}, {10646,39859}, {10723,12086}, {10991,35477}, {11410,39832}, {11413,38738}, {11430,39834}, {11438,39835}, {11454,39836}, {11468,39837}, {12042,18570}, {12084,23698}, {12085,39828}, {13371,39818}, {14118,34473}, {14130,38224}, {14639,14865}, {14651,35475}, {17928,36519}, {18859,38730}, {18931,39833}, {19192,39843}, {19454,39855}, {19455,39856}, {21663,39846}, {22505,37814}

X(39860) = midpoint of X(i) and X(j) for these {i,j}: {3, 39841}, {12085, 39828}
X(39860) = reflection of X(i) in X(j) for these (i,j): (39818, 13371), (39831, 11250), (39840, 5092), (39847, 39845), (39854, 3)
X(39860) = circumperp conjugate of X(10749)

X(39861) = PERSPECTOR OF THESE TRIANGLES: 8th BROCARD AND LUCAS REFLECTION

Barycentrics
SA*(SB+SC)*((4*R^2+3*SW)*S^4-(4*R^2*(6*R^2-SA+SW)+SA^2-3*SW^2)*SW*S^2-(2*R^2-SW)*(2*R^2*SW-SA^2)*SW^2+S*(2*S^4-4*(2*R^2-SW)*(2*R^2+SW)*S^2+2*(2*R^2*(6*R^2*SW-SA^2-3*SW^2)+SW^3)*SW)) : :

X(39861) lies on these lines: {3,6401}, {32,11984}, {1384,22785}, {3053,11986}, {11937,39654}, {11938,39655}, {11959,39649}, {11960,39658}, {11963,39657}, {11964,39664}, {11967,39863}, {11969,39864}, {11971,39865}, {11973,39866}, {11975,39867}, {11977,39868}, {11979,39648}, {11981,39679}, {11983,39862}, {14167,39656}, {19390,39659}, {22499,39652}

X(39862) = PERSPECTOR OF THESE TRIANGLES: 8th BROCARD AND LUCAS(-1) REFLECTION

Barycentrics
SA*(SB+SC)*((4*R^2+3*SW)*S^4-(4*R^2*(6*R^2-SA+SW)+SA^2-3*SW^2)*SW*S^2-(2*R^2-SW)*(2*R^2*SW-SA^2)*SW^2-S*(2*S^4-4*(2*R^2-SW)*(2*R^2+SW)*S^2+2*(2*R^2*(6*R^2*SW-SA^2-3*SW^2)+SW^3)*SW)) : :

X(39862) lies on these lines: {3,6402}, {32,11985}, {1384,22786}, {3053,11987}, {11939,39654}, {11940,39655}, {11961,39649}, {11962,39658}, {11965,39657}, {11966,39664}, {11968,39864}, {11970,39863}, {11972,39866}, {11974,39865}, {11976,39868}, {11978,39867}, {11980,39679}, {11982,39648}, {11983,39861}, {14168,39656}, {19391,39659}, {22500,39652}

X(39863) = PERSPECTOR OF THESE TRIANGLES: 8th BROCARD AND LUCAS SECONDARY CENTRAL

Barycentrics    a^2*(4*(5*a^2-b^2-c^2)*S+(-a^2+b^2+c^2)*(a^2+b^2+c^2)) : :

X(39863) lies on these lines: {3,6}, {8310,39652}, {8326,39650}, {8342,39651}, {10845,39656}, {11291,13993}, {11967,39861}, {11970,39862}, {19392,39659}, {26438,31412}

X(39863) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1384, 39864), (3, 6500, 5024), (187, 3311, 3)

X(39864) = PERSPECTOR OF THESE TRIANGLES: 8th BROCARD AND LUCAS(-1) SECONDARY CENTRAL

Barycentrics    a^2*(-4*(5*a^2-b^2-c^2)*S+(-a^2+b^2+c^2)*(a^2+b^2+c^2)) : :

X(39864) lies on these lines: {3,6}, {8311,39652}, {8327,39650}, {8343,39651}, {10846,39656}, {11292,13925}, {11968,39862}, {11969,39861}, {19393,39659}

X(39864) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1384, 39863), (3, 6501, 5024), (3, 21309, 6417), (187, 3312, 3)

X(39865) = PERSPECTOR OF THESE TRIANGLES: 8th BROCARD AND LUCAS 1st SECONDARY TANGENTS

Barycentrics    a^2*(4*(4*a^2-b^2-c^2)*S+(-a^2+b^2+c^2)*(a^2+b^2+c^2)) : :

X(39865) lies on these lines: {3,6}, {1327,36709}, {8312,39652}, {8328,39650}, {8344,39651}, {10847,39656}, {11971,39861}, {11974,39862}, {19394,39659}, {21736,31414}

X(39865) = {X(3), X(35007)}-harmonic conjugate of X(39866)

X(39866) = PERSPECTOR OF THESE TRIANGLES: 8th BROCARD AND LUCAS(-1) 1st SECONDARY TANGENTS

Barycentrics    a^2*(-4*(4*a^2-b^2-c^2)*S+(-a^2+b^2+c^2)*(a^2+b^2+c^2)) : :

X(39866) lies on these lines: {3,6}, {1328,36714}, {8313,39652}, {8329,39650}, {8345,39651}, {10848,39656}, {11972,39862}, {11973,39861}, {19395,39659}

X(39866) = {X(3), X(35007)}-harmonic conjugate of X(39865)

X(39867) = PERSPECTOR OF THESE TRIANGLES: 8th BROCARD AND LUCAS 2nd SECONDARY TANGENTS

Barycentrics    a^2*((16*a^2-12*b^2-12*c^2)*S-(-a^2+b^2+c^2)*(a^2+b^2+c^2)) : :

X(39867) lies on these lines: {3,6}, {3534,35874}, {8314,39652}, {8330,39650}, {8346,39651}, {8703,39661}, {10849,39656}, {11975,39861}, {11978,39862}, {15696,35830}, {19396,39659}

X(39867) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 3312, 15515), (3, 5023, 39648), (3, 15513, 39868)

X(39868) = PERSPECTOR OF THESE TRIANGLES: 8th BROCARD AND LUCAS(-1) 2nd SECONDARY TANGENTS

Barycentrics    a^2*(-(16*a^2-12*b^2-12*c^2)*S-(-a^2+b^2+c^2)*(a^2+b^2+c^2)) : :

X(39868) lies on these lines: {3,6}, {3534,35873}, {8315,39652}, {8331,39650}, {8347,39651}, {8703,39660}, {10850,39656}, {11976,39862}, {11977,39861}, {15696,35831}, {19397,39659}

X(39868) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 3311, 15515), (3, 5023, 39679), (3, 15513, 39867)

X(39869) = PERSPECTOR OF THESE TRIANGLES: 8th BROCARD AND 1st PAMFILOS-ZHOU

Barycentrics
a*(-a^2+b^2+c^2)*((2*(b+c)*a^9+6*(b+c)*b*c*a^7-2*(2*b^4+2*c^4+b*c*(5*b^2-4*b*c+5*c^2))*a^6-2*(b+c)*(b^4+c^4+8*b*c*(b^2+c^2))*a^5-2*(2*b^6+2*c^6+b*c*(2*b^2+b*c+2*c^2)*(3*b^2+4*b*c+3*c^2))*a^4+2*(b+c)*(b^4+c^4+2*b*c*(b^2-6*b*c+c^2))*b*c*a^3-2*(b^4+c^4-2*b*c*(b^2-5*b*c+c^2))*(b+c)^2*b*c*a^2-4*(b+c)*(b^2+c^2)*b^3*c^3*a+2*(b^4-c^4)*(b^2-c^2)*b^2*c^2)*S-a^12+(2*b^2-b*c+2*c^2)*a^10+(b+c)*(2*b^2-3*b*c+2*c^2)*a^9+(6*b^2+7*b*c+6*c^2)*b*c*a^8-(b+c)*(2*b^4+2*c^4+(b^2-3*b*c+c^2)*b*c)*a^7-(2*b^6+2*c^6+b*c*(2*b^2-b*c+2*c^2)*(5*b^2+11*b*c+5*c^2))*a^6-(b+c)*(2*b^6+2*c^6+(b^4+c^4+b*c*(25*b^2-6*b*c+25*c^2))*b*c)*a^5+(b^8+c^8+(3*b^2+2*b*c+3*c^2)*(2*b^4+2*c^4-b*c*(b^2+5*b*c+c^2))*b*c)*a^4+(b+c)*(2*b^8+2*c^8-(3*b^6+3*c^6-(b^4+c^4-b*c*(b^2+30*b*c+c^2))*b*c)*b*c)*a^3-(b^6+c^6-(b^4+c^4+4*(b^2-3*b*c+c^2)*b*c)*b*c)*(b+c)^2*b*c*a^2+3*(b^4-c^4)*(b^2-c^2)*b^2*c^2*(b+c)*a+(b^4-c^4)*(b^2-c^2)*b^3*c^3) : :

X(39869) lies on the line {3,7594}

X(39870) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO 9th BROCARD

Barycentrics    4*a^6-(b+c)*a^5-(3*b^2-2*b*c+3*c^2)*a^4-2*(b+c)*b*c*a^3+2*(b-c)^2*b*c*a^2+(b^4-c^4)*(b-c)*a-(b^4-c^4)*(b^2-c^2) : :
X(39870) = 3*X(1)+X(39878) = 3*X(1)-X(39898) = X(4)-3*X(16475) = 2*X(5)-3*X(38049) = 2*X(10)-3*X(38118) = X(40)-3*X(25406) = X(69)-3*X(3576) = 2*X(141)-3*X(10165) = 4*X(182)-3*X(38118) = X(193)+3*X(5731) = X(355)-3*X(5050) = X(944)+3*X(14912) = 2*X(1125)-3*X(38029) = X(1352)-3*X(38029) = X(3416)-3*X(5085) = X(3751)-3*X(14912) = 3*X(5085)-2*X(6684) = X(5882)+2*X(8550) = 3*X(6776)-X(39878) = 3*X(6776)+X(39898)

The reciprocal orthologic center of these triangles is X(4).

X(39870) lies on these lines: {1,6776}, {2,39885}, {3,5847}, {4,16475}, {5,38049}, {6,515}, {10,182}, {40,25406}, {69,3576}, {141,10165}, {193,5731}, {355,5050}, {511,4297}, {516,24257}, {518,5882}, {519,11179}, {542,551}, {611,10106}, {613,950}, {730,13354}, {944,3751}, {946,1386}, {999,39883}, {1125,1352}, {1210,1428}, {1319,39873}, {1351,18481}, {1353,34773}, {1385,3564}, {1469,4311}, {2330,31397}, {2646,39897}, {2784,12177}, {2792,3663}, {3056,4304}, {3242,13607}, {3295,39877}, {3416,5085}, {3589,10175}, {3616,5921}, {3618,5587}, {3626,38116}, {3817,3818}, {3827,5884}, {3828,38064}, {4663,12007}, {5092,10164}, {5135,12616}, {5138,6245}, {5182,9864}, {5450,36740}, {5476,34648}, {5480,31673}, {5603,16491}, {5622,13211}, {5691,14853}, {5846,11362}, {5848,11715}, {5886,18440}, {6001,34774}, {6702,38119}, {6796,36741}, {7967,16496}, {7987,10519}, {9955,38040}, {9956,38110}, {10246,39899}, {10755,12119}, {11061,33535}, {11178,19883}, {11180,25055}, {11230,18358}, {11363,39871}, {11364,39872}, {11365,39879}, {11366,39880}, {11367,39881}, {11368,39882}, {11370,39887}, {11371,39888}, {11373,39889}, {11374,39890}, {11375,39891}, {11376,39892}, {11377,39895}, {11378,39896}, {11831,39886}, {12005,24476}, {12017,26446}, {12114,37492}, {12407,25320}, {12588,13411}, {12610,29046}, {12675,34381}, {13464,38315}, {13883,19145}, {13936,19146}, {14561,19925}, {16174,38050}, {16192,33750}, {18357,38167}, {18480,18583}, {18483,36990}, {18991,39875}, {18992,39876}, {19862,24206}, {20986,34822}, {21850,28160}, {28164,31670}, {35762,39893}, {35763,39894}

X(39870) = midpoint of X(i) and X(j) for these {i,j}: {1, 6776}, {944, 3751}, {1351, 18481}, {1353, 34773}, {10755, 12119}, {11061, 33535}, {39878, 39898}
X(39870) = reflection of X(i) in X(j) for these (i,j): (10, 182), (946, 1386), (1352, 1125), (3242, 13607), (3416, 6684), (4663, 12007), (18480, 18583), (24476, 12005), (31673, 5480), (34648, 5476), (36990, 18483), (39884, 9955)
X(39870) = complement of X(39885)
X(39870) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 39878, 39898), (10, 182, 38118), (944, 14912, 3751), (1352, 38029, 1125), (3416, 5085, 6684), (6776, 39898, 39878), (18480, 18583, 38146), (36990, 38035, 18483), (38040, 39884, 9955)

X(39871) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO 9th BROCARD

Barycentrics    (a^2+b^2-c^2)*(a^2-b^2+c^2)*(2*a^8-5*(b^2+c^2)*a^6+5*(b^2+c^2)^2*a^4-3*(b^4-c^4)*(b^2-c^2)*a^2+(b^4-c^4)^2) : :
X(39871) = 2*X(6776)-3*X(11245) = 2*X(9967)-3*X(34664) = 3*X(12022)-X(12283)

The reciprocal orthologic center of these triangles is X(4).

X(39871) lies on these lines: {4,193}, {5,26206}, {6,235}, {25,6776}, {30,6403}, {33,39897}, {34,39873}, {69,1593}, {141,19124}, {182,468}, {185,1503}, {394,427}, {403,18583}, {428,542}, {511,1885}, {524,12294}, {546,18449}, {568,6756}, {1353,1596}, {1594,18358}, {1597,11898}, {1598,13292}, {1902,5847}, {1907,15069}, {1974,8550}, {3089,14912}, {3147,12017}, {3515,25406}, {3516,10519}, {3542,5050}, {3629,11470}, {3818,23047}, {5064,11180}, {5090,39885}, {5410,39876}, {5411,39875}, {5446,12421}, {5480,8541}, {5848,12138}, {6225,12173}, {6467,12241}, {6623,11405}, {6746,9969}, {7487,18914}, {7505,38110}, {7507,13562}, {7713,39878}, {9924,19467}, {9967,34664}, {10019,19130}, {11363,39870}, {11380,39872}, {11382,18909}, {11383,39877}, {11384,39880}, {11385,39881}, {11386,39882}, {11388,39887}, {11389,39888}, {11390,39889}, {11391,39890}, {11392,39891}, {11393,39892}, {11394,39895}, {11395,39896}, {11396,39898}, {11398,39900}, {11399,39901}, {11400,39902}, {11401,39903}, {11442,15809}, {11832,39886}, {12022,12283}, {12220,12362}, {13367,15585}, {13383,19129}, {13488,18436}, {13884,19145}, {13937,19146}, {14853,37197}, {14927,37196}, {16196,26156}, {16252,21637}, {19128,21841}, {19154,37971}, {22479,39883}, {26883,34774}, {35488,38136}, {35764,39893}, {35765,39894}

X(39871) = reflection of X(i) in X(j) for these (i,j): (193, 13142), (3575, 1843), (6467, 12241), (12220, 12362), (39874, 18914), (39899, 13292)
X(39871) = {X(3089), X(14912)}-harmonic conjugate of X(19118)

X(39872) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO 9th BROCARD

Barycentrics    a^10+2*(b^2+c^2)*a^8-(4*b^4+9*b^2*c^2+4*c^4)*a^6+(b^2+c^2)*(2*b^4-7*b^2*c^2+2*c^4)*a^4-(b^6-c^6)*(b^2-c^2)*a^2-(b^4-c^4)*(b^2-c^2)*b^2*c^2 : :

The reciprocal orthologic center of these triangles is X(4).

X(39872) lies on these lines: {3,32451}, {4,5039}, {6,98}, {32,6776}, {54,69}, {83,1352}, {99,35424}, {193,13355}, {194,35387}, {287,34396}, {385,13354}, {511,7470}, {542,12150}, {1351,14880}, {1353,2456}, {1503,12110}, {1691,8550}, {3094,32467}, {3098,7709}, {3398,3564}, {3406,10159}, {5017,11257}, {5038,12007}, {5050,10104}, {5092,21445}, {5171,25406}, {5182,11288}, {5319,14853}, {5847,12197}, {5848,12199}, {5921,7787}, {5999,35439}, {6308,7758}, {7766,35389}, {7856,14561}, {10788,39874}, {10789,39878}, {10790,39879}, {10791,39885}, {10792,39887}, {10793,39888}, {10794,39889}, {10795,39890}, {10796,18440}, {10797,39891}, {10798,39892}, {10799,39897}, {10800,39898}, {10801,39900}, {10802,39901}, {10803,39902}, {10804,39903}, {11364,39870}, {11380,39871}, {11490,39877}, {11837,39880}, {11838,39881}, {11839,39886}, {11840,39895}, {11841,39896}, {11842,39899}, {12215,13335}, {12835,39873}, {13885,19145}, {13938,19146}, {14651,19130}, {18502,39884}, {18993,39875}, {18994,39876}, {22520,39883}, {22712,35423}, {32448,35383}, {33181,39141}, {35766,39893}, {35767,39894}

X(39872) = reflection of X(12110) in X(12212)

X(39873) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO 9th BROCARD

Barycentrics    (-a+b+c)*(2*a^4-(b-c)^2*a^2+(b^2+c^2)*(b-c)^2) : :
X(39873) = 2*X(3056)-3*X(3058) = 3*X(11246)-4*X(24471) = 4*X(17792)-3*X(34612)

The reciprocal orthologic center of these triangles is X(4).

X(39873) lies on these lines: {1,3564}, {3,39900}, {4,39892}, {6,11}, {12,611}, {34,39871}, {55,69}, {56,6776}, {57,39878}, {120,26657}, {141,2330}, {182,5433}, {193,497}, {206,9667}, {388,5921}, {390,20080}, {496,1353}, {499,5050}, {511,6284}, {518,10950}, {524,3056}, {528,25304}, {542,3023}, {599,4995}, {613,37722}, {614,11245}, {895,12904}, {950,34379}, {999,39899}, {1319,39870}, {1350,15338}, {1351,1479}, {1428,8550}, {1469,1503}, {1478,18440}, {1837,3751}, {1858,34381}, {1992,11238}, {2099,39898}, {2175,26932}, {2854,32297}, {2886,15988}, {3057,5847}, {3086,14912}, {3242,5849}, {3295,11898}, {3583,21850}, {3585,39884}, {3614,10516}, {3620,5218}, {3629,8540}, {3763,5326}, {4293,39874}, {4302,33878}, {4459,17365}, {5039,10798}, {5093,9669}, {5204,25406}, {5217,10519}, {5252,39885}, {5298,11179}, {5480,19369}, {5710,10372}, {5845,24840}, {5965,13079}, {7173,14561}, {7741,18583}, {7951,18358}, {8593,12351}, {10385,11160}, {10535,34774}, {10752,12374}, {10753,12185}, {10754,13183}, {10755,13274}, {10759,12764}, {10766,13297}, {10832,19459}, {10833,37491}, {10896,14853}, {10955,12587}, {11180,11237}, {11246,24471}, {11376,16475}, {11393,12167}, {11509,39877}, {11579,32308}, {12428,34382}, {12588,15069}, {12835,39872}, {12836,32451}, {12896,14984}, {14982,32289}, {15171,34380}, {16496,37740}, {16541,19588}, {17635,28849}, {17792,34612}, {18954,39879}, {18955,39880}, {18956,39881}, {18957,39882}, {18958,39886}, {18959,39887}, {18960,39888}, {18961,39889}, {18962,39890}, {18963,39895}, {18964,39896}, {18965,19145}, {18966,19146}, {18967,39903}, {18995,39875}, {18996,39876}, {24837,32029}, {26686,39141}, {35768,39893}, {35769,39894}, {37735,38040}

X(39873) = reflection of X(i) in X(j) for these (i,j): (7354, 1469), (39897, 1)
X(39873) = X(69)-of-Mandart-incircle-triangle
X(39873) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 12589, 11), (141, 2330, 5432), (388, 5921, 39891), (611, 1352, 12), (999, 39899, 39901), (10925, 19029, 11), (10926, 19030, 11)

X(39874) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO 9th BROCARD

Barycentrics    7*a^6-5*(b^2+c^2)*a^4+(b^2-c^2)^2*a^2-3*(b^4-c^4)*(b^2-c^2) : :
X(39874) = 9*X(2)-10*X(12017) = 9*X(2)-8*X(18358) = 6*X(3)-5*X(3620) = 3*X(4)-4*X(6) = 7*X(4)-8*X(5480) = 5*X(4)-8*X(8550) = 11*X(4)-16*X(12007) = 5*X(4)-6*X(14853) = 2*X(4)-3*X(14912) = 5*X(4)-4*X(36990) = 7*X(6)-6*X(5480) = 2*X(6)-3*X(6776) = 5*X(6)-6*X(8550) = 11*X(6)-12*X(12007) = 10*X(6)-9*X(14853) = 8*X(6)-9*X(14912) = 5*X(6)-3*X(36990) = 5*X(3620)-3*X(5921) = 5*X(12017)-4*X(18358) = 5*X(12017)-3*X(18440) = 2*X(16774)-3*X(32603) = 4*X(18358)-3*X(18440)

The reciprocal orthologic center of these triangles is X(4).

X(39874) lies on these lines: {2,8780}, {3,3620}, {4,6}, {20,3564}, {24,39879}, {25,18950}, {30,193}, {54,19124}, {66,3431}, {67,20421}, {69,74}, {98,5033}, {104,39883}, {110,16051}, {125,35260}, {141,3524}, {147,8289}, {154,23291}, {159,186}, {182,1614}, {184,8889}, {185,6403}, {187,8721}, {206,5622}, {323,1370}, {378,19459}, {382,1353}, {388,39900}, {399,18531}, {497,39901}, {511,3529}, {515,39878}, {524,11001}, {550,11898}, {575,19123}, {599,19708}, {631,1352}, {944,16496}, {1205,12281}, {1350,3630}, {1351,3146}, {1428,9638}, {1495,1899}, {1511,12419}, {1513,37689}, {1657,34380}, {1843,5890}, {1974,14157}, {1992,11645}, {2330,8164}, {2777,32234}, {2854,17854}, {3055,9756}, {3085,39891}, {3086,39892}, {3088,31804}, {3091,5050}, {3167,7396}, {3424,13860}, {3426,17040}, {3448,7493}, {3525,5085}, {3528,3631}, {3533,23294}, {3534,11160}, {3543,21850}, {3545,3618}, {3581,18917}, {3589,5071}, {3627,5093}, {3763,15702}, {3830,5032}, {3832,18583}, {3855,14561}, {4232,26869}, {4293,39873}, {4294,39897}, {4869,13635}, {5056,38110}, {5067,10516}, {5095,10721}, {5097,11458}, {5191,18437}, {5207,32823}, {5210,39647}, {5232,13634}, {5477,10722}, {5485,9830}, {5486,11738}, {5603,16491}, {5621,35228}, {5657,39885}, {5663,18438}, {5847,6361}, {5848,12248}, {5984,37182}, {6000,6467}, {6033,16041}, {6054,37690}, {6114,6773}, {6115,6770}, {6193,37483}, {6199,36709}, {6200,12257}, {6221,21736}, {6225,21659}, {6395,36714}, {6396,12256}, {6515,15107}, {6560,39893}, {6561,39894}, {6564,13674}, {6565,13794}, {6622,6759}, {6623,19118}, {6643,15068}, {6803,12134}, {6811,8972}, {6813,13941}, {6816,15052}, {6995,11245}, {6997,15018}, {7378,11402}, {7386,15066}, {7391,11004}, {7408,9777}, {7487,18914}, {7492,13171}, {7494,11442}, {7528,15037}, {7577,23300}, {7687,18918}, {7694,18424}, {7714,11433}, {7728,25321}, {7738,9873}, {7967,39898}, {8588,10991}, {9822,15045}, {9833,11438}, {9924,10605}, {9967,12111}, {10546,18911}, {10575,34382}, {10752,36201}, {10785,39889}, {10786,39890}, {10788,39872}, {10805,39902}, {10806,39903}, {10979,26870}, {10996,14516}, {11177,17008}, {11178,15709}, {11411,37478}, {11427,11550}, {11430,14216}, {11459,11574}, {11468,14810}, {11477,11541}, {11491,39877}, {11511,14094}, {11676,15428}, {11843,39880}, {11844,39881}, {11845,39886}, {11846,39895}, {11847,39896}, {12041,32272}, {12042,33216}, {12082,37491}, {12088,37488}, {12215,32818}, {12220,13754}, {12272,15072}, {12289,29012}, {12290,12294}, {12324,19467}, {13202,34779}, {13596,32621}, {13886,19145}, {13939,19146}, {14001,26316}, {14064,14880}, {14471,15504}, {14644,32250}, {14683,15106}, {14915,15531}, {14982,20125}, {14996,26118}, {15055,32275}, {15073,34146}, {15577,21844}, {15578,23040}, {15698,21356}, {16010,35707}, {16111,32244}, {17855,32260}, {18451,26206}, {18533,26926}, {18913,34782}, {19588,21312}, {21669,37492}, {21735,34507}, {24695,29040}, {25046,37190}, {32001,35474}, {34799,37201}, {35237,35513}, {35424,35951}

X(39874) = midpoint of X(6241) and X(12283)
X(39874) = reflection of X(i) in X(j) for these (i,j): (4, 6776), (193, 39899), (382, 1353), (3146, 1351), (3529, 14927), (5921, 3), (6403, 185), (10721, 5095), (10722, 5477), (11160, 3534), (11898, 550), (12111, 9967), (12272, 37511), (12281, 1205), (12290, 12294), (14227, 39887), (14242, 39888), (15682, 1992), (20080, 33878), (32244, 16111), (32260, 17855), (32272, 12041), (36990, 8550), (39871, 18914)
X(39874) = anticomplement of X(18440)
X(39874) = intersection, other than A,B,C, of conics {{A, B, C, X(69), X(1990)}} and {{A, B, C, X(74), X(2207)}}
X(39874) = crossdifference of every pair of points on line {X(520), X(14398)}
X(39874) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 6776, 14912), (20, 20080, 33878), (154, 23291, 38282), (184, 32064, 8889), (1352, 5092, 3619), (1352, 25406, 631), (1498, 18945, 4), (3087, 16264, 4), (3529, 6241, 32601), (3618, 3818, 3545), (3619, 5092, 631), (5050, 39884, 3091), (5334, 5335, 5254), (5656, 18396, 4), (6146, 34781, 4), (6776, 14853, 8550), (8550, 36990, 14853), (10783, 14227, 4), (10784, 14242, 4), (11206, 37643, 1495), (11433, 31383, 7714), (12017, 18358, 2), (12017, 18440, 18358), (14853, 36990, 4), (31804, 34780, 3088)

X(39875) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO 9th BROCARD

Barycentrics    4*S*(a^2+b^2+c^2)*a^2-(-a^2+b^2+c^2)*(3*a^4+(b^2-c^2)^2) : :
X(39875) = X(10784)-3*X(14912)

The reciprocal orthologic center of these triangles is X(4).

X(39875) lies on these lines: {2,19146}, {4,6}, {32,12256}, {39,12257}, {69,372}, {141,13935}, {182,3068}, {193,11293}, {371,25406}, {485,3618}, {488,14001}, {511,6460}, {524,26620}, {542,19053}, {611,31408}, {1152,10519}, {1352,3069}, {1703,5847}, {1992,32419}, {3299,39900}, {3301,39901}, {3312,3564}, {3619,5420}, {5034,31411}, {5050,7583}, {5085,9540}, {5093,22809}, {5182,33340}, {5207,32435}, {5309,13794}, {5411,39871}, {5848,19081}, {5921,7586}, {6395,11898}, {6409,33750}, {6418,39899}, {6420,39894}, {6424,21736}, {6560,35841}, {6561,14927}, {7584,18440}, {7585,19145}, {7738,26441}, {7753,13674}, {7968,39898}, {8854,18928}, {8976,38110}, {8981,12017}, {10516,13972}, {11179,19054}, {11180,32788}, {12322,39661}, {13354,19090}, {13665,18583}, {13785,39884}, {13893,38118}, {13902,38029}, {13936,39885}, {13951,18358}, {14039,26288}, {14561,31412}, {18991,39870}, {18993,39872}, {18995,39873}, {18999,39877}, {19003,39878}, {19005,39879}, {19007,39880}, {19009,39881}, {19011,39882}, {19013,39883}, {19017,39886}, {19023,39889}, {19025,39890}, {19027,39891}, {19029,39892}, {19031,39895}, {19033,39896}, {19037,39897}, {19047,39902}, {19049,39903}, {24206,32786}, {35770,39893}

X(39875) = reflection of X(1588) in X(6)
X(39875) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 6776, 39887), (6, 3070, 14853), (6, 6776, 39876), (7581, 14912, 6)

X(39876) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO 9th BROCARD

Barycentrics    -4*S*(a^2+b^2+c^2)*a^2-(-a^2+b^2+c^2)*(3*a^4+(b^2-c^2)^2) : :
X(39876) = X(10783)-3*X(14912)

The reciprocal orthologic center of these triangles is X(4).

X(39876) lies on these lines: {2,19145}, {4,6}, {32,12257}, {39,12256}, {69,371}, {141,9540}, {182,3069}, {193,11294}, {372,25406}, {486,3618}, {487,14001}, {511,6459}, {524,26619}, {542,19054}, {1151,10519}, {1350,9541}, {1352,3068}, {1702,5847}, {1992,32421}, {3299,39901}, {3301,39900}, {3311,3564}, {3619,5418}, {3818,31412}, {5050,7584}, {5085,13935}, {5093,22810}, {5182,33341}, {5207,32432}, {5309,13674}, {5410,39871}, {5490,8950}, {5848,19082}, {5921,7585}, {6199,11898}, {6410,33750}, {6417,39899}, {6419,39893}, {6422,21736}, {6560,14927}, {6561,35840}, {6811,31403}, {7583,18440}, {7586,19146}, {7738,8982}, {7753,13794}, {7969,39898}, {8855,18928}, {8976,18358}, {10516,13910}, {11179,19053}, {11180,32787}, {12017,13966}, {12323,39660}, {13354,19089}, {13665,39884}, {13785,18583}, {13883,39885}, {13947,38118}, {13951,38110}, {13959,38029}, {14039,26289}, {18992,39870}, {18994,39872}, {18996,39873}, {19000,39877}, {19004,39878}, {19006,39879}, {19008,39880}, {19010,39881}, {19012,39882}, {19014,39883}, {19018,39886}, {19024,39889}, {19026,39890}, {19028,39891}, {19030,39892}, {19032,39895}, {19034,39896}, {19038,39897}, {19048,39902}, {19050,39903}, {24206,32785}, {35771,39894}

X(39876) = reflection of X(1587) in X(6)
X(39876) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 6776, 39888), (6, 3071, 14853), (6, 6776, 39875), (7582, 14912, 6)

X(39877) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 9th BROCARD

Barycentrics
a*(a^8-(b+c)*a^7-(b^2-6*b*c+c^2)*a^6+(b+c)*(b^2+c^2)*a^5-(b^4+c^4+6*(b^2+b*c+c^2)*b*c)*a^4+(b+c)*(b^4+6*b^2*c^2+c^4)*a^3+(b^2-c^2)^2*(b+c)^2*a^2-(b^4-c^4)*(b^2-c^2)*(b+c)*a-2*(b^4-c^4)*(b^2-c^2)*b*c) : :

The reciprocal orthologic center of these triangles is X(4).

X(39877) lies on these lines: {3,39883}, {6,3073}, {35,39878}, {55,6776}, {56,39898}, {69,10310}, {100,5921}, {182,692}, {197,39879}, {518,3358}, {542,4421}, {1012,3751}, {1158,34381}, {1352,1376}, {1503,11500}, {3295,39870}, {3564,11248}, {4428,11179}, {5687,39885}, {5800,7680}, {5847,10306}, {5848,12332}, {11108,38118}, {11383,39871}, {11490,39872}, {11491,39874}, {11492,39880}, {11493,39881}, {11494,39882}, {11497,39887}, {11498,39888}, {11499,18440}, {11501,39891}, {11502,39892}, {11503,39895}, {11504,39896}, {11507,39900}, {11508,39901}, {11509,39873}, {11510,39903}, {11848,39886}, {11849,39899}, {11898,35000}, {13887,19145}, {13940,19146}, {18491,39884}, {18999,39875}, {19000,39876}, {20872,37488}, {22753,36741}, {35772,39893}, {35773,39894}

X(39877) = reflection of X(i) in X(j) for these (i,j): (11500, 12329), (39883, 3), (39889, 1352)

X(39878) = ORTHOLOGIC CENTER OF THESE TRIANGLES: AQUILA TO 9th BROCARD

Barycentrics    5*a^5-4*(b+c)*a^4-2*(b^2-3*b*c+c^2)*a^3+2*(b^2-c^2)*(b-c)*a^2+(b^2+c^2)*(b-c)^2*a+(b^4-c^4)*(-2*b+2*c) : :
X(39878) = 3*X(1)-4*X(39870) = 3*X(1)-2*X(39898) = 4*X(6)-3*X(1699) = 2*X(69)-3*X(165) = 8*X(182)-7*X(3624) = 2*X(946)-3*X(14912) = 4*X(1352)-5*X(1698) = 10*X(3618)-9*X(7988) = 5*X(3620)-6*X(10164) = 3*X(3679)-2*X(39885) = 6*X(5050)-5*X(8227) = 3*X(5093)-2*X(22793) = 3*X(5587)-2*X(18440) = 3*X(6776)-2*X(39870) = 3*X(6776)-X(39898) = 5*X(7987)-6*X(25406) = 8*X(8550)-5*X(11522) = 4*X(8550)-3*X(16475) = 7*X(9588)-4*X(15069) = 5*X(11522)-6*X(16475)

The reciprocal orthologic center of these triangles is X(4).

X(39878) lies on these lines: {1,6776}, {6,1699}, {10,5921}, {20,34379}, {35,39877}, {36,39883}, {40,3564}, {57,39873}, {69,165}, {159,9590}, {182,3624}, {193,516}, {515,39874}, {517,39899}, {542,2948}, {611,5290}, {946,14912}, {1352,1698}, {1353,12699}, {1503,3751}, {1654,9746}, {1697,39897}, {1768,5848}, {2261,21293}, {2784,3729}, {2807,6467}, {3099,39882}, {3579,11898}, {3618,7988}, {3620,10164}, {4663,36990}, {5050,8227}, {5093,22793}, {5587,18440}, {5588,39888}, {5589,39887}, {5847,7991}, {6403,31728}, {7713,39871}, {7987,25406}, {7996,28849}, {8185,39879}, {8186,39880}, {8187,39881}, {8188,39895}, {8189,39896}, {8550,11522}, {9578,39891}, {9581,39892}, {9588,15069}, {9591,37488}, {9778,20080}, {10519,16192}, {10789,39872}, {10826,39889}, {10827,39890}, {11179,25055}, {11180,19875}, {11852,39886}, {12007,38035}, {13888,19145}, {13942,19146}, {15071,34381}, {18492,39884}, {19003,39875}, {19004,39876}, {19872,24206}, {35774,39893}, {35775,39894}

X(39878) = reflection of X(i) in X(j) for these (i,j): (1, 6776), (5691, 3751), (5921, 10), (6403, 31728), (11898, 3579), (12699, 1353), (36990, 4663), (39898, 39870)
X(39878) = X(8)-Beth conjugate of-X(5921)
X(39878) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6776, 39898, 39870), (39870, 39898, 1)

X(39879) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARA TO 9th BROCARD

Barycentrics    a^2*(a^10-(b^2+c^2)*a^8-2*(b^4+8*b^2*c^2+c^4)*a^6+2*(b^2+c^2)*(b^4+4*b^2*c^2+c^4)*a^4+(b^4-c^4)^2*a^2-(-4*b^2*c^2+(b^2-c^2)^2)*(b^4-c^4)*(b^2-c^2)) : :
X(39879) = 3*X(3)-4*X(15577) = 9*X(3)-8*X(15578) = 11*X(3)-8*X(15579) = 3*X(3)-8*X(15580) = X(3)-4*X(15581) = 5*X(3)-8*X(15582) = 7*X(3)-8*X(35228) = 3*X(159)-2*X(15577) = 9*X(159)-4*X(15578) = 11*X(159)-4*X(15579) = 3*X(159)-4*X(15580) = 5*X(159)-4*X(15582) = 7*X(159)-4*X(35228) = 3*X(15577)-2*X(15578) = 11*X(15577)-6*X(15579) = X(15577)-3*X(15581) = 5*X(15577)-6*X(15582) = 7*X(15577)-6*X(35228) = 11*X(15578)-9*X(15579) = X(15578)-3*X(15580) = 2*X(15578)-9*X(15581) = 5*X(15578)-9*X(15582) = 7*X(15578)-9*X(35228)

The reciprocal orthologic center of these triangles is X(4).

X(39879) lies on these lines: {3,66}, {4,19459}, {5,36851}, {6,1598}, {22,5921}, {24,39874}, {25,6776}, {26,35219}, {64,3098}, {68,10243}, {69,11414}, {154,182}, {161,542}, {197,39877}, {206,5050}, {389,7716}, {394,34750}, {427,38396}, {511,1498}, {575,19132}, {611,26888}, {613,10535}, {1181,1843}, {1350,6000}, {1351,2393}, {1353,7530}, {1482,3827}, {1597,18400}, {1614,19125}, {1656,23300}, {1660,8780}, {1853,16419}, {1971,34809}, {2777,2930}, {2781,12308}, {2854,9934}, {2883,31670}, {2935,12584}, {3089,18935}, {3357,31884}, {3517,20987}, {3519,9920}, {3564,5596}, {3589,11484}, {3763,20299}, {3818,11479}, {3843,18382}, {5055,20300}, {5085,10282}, {5092,17821}, {5093,34117}, {5097,17813}, {5102,34788}, {5198,14853}, {5480,18535}, {5594,39888}, {5595,39887}, {5663,38885}, {5800,7497}, {5847,9911}, {5848,9913}, {5878,29181}, {5895,29317}, {6403,11456}, {6467,26883}, {6617,15595}, {7393,18358}, {7484,32064}, {7517,39899}, {8185,39878}, {8190,39880}, {8191,39881}, {8192,39898}, {8193,39885}, {8194,39895}, {8195,39896}, {9714,22550}, {9818,39884}, {9822,37514}, {9919,32254}, {9967,18451}, {9969,11432}, {10037,39900}, {10046,39901}, {10323,20079}, {10516,18381}, {10519,12324}, {10533,13889}, {10534,13943}, {10594,14912}, {10602,12283}, {10606,14810}, {10752,32240}, {10790,39872}, {10828,39882}, {10829,39889}, {10830,39890}, {10831,39891}, {10832,39892}, {10833,39897}, {10834,39902}, {10835,39903}, {11284,35260}, {11365,39870}, {11441,12220}, {11477,34779}, {11574,17814}, {11576,12167}, {11579,15647}, {11853,39886}, {11898,12083}, {12017,23041}, {12087,20080}, {12121,36201}, {12166,31305}, {12315,18436}, {13093,34778}, {13289,16010}, {13347,17822}, {14561,15583}, {14927,21312}, {15069,37485}, {15141,19377}, {17845,29012}, {18954,39873}, {19005,39875}, {19006,39876}, {19124,19357}, {19457,32250}, {22654,39883}, {35776,39893}, {35777,39894}

X(39879) = midpoint of X(i) and X(j) for these {i,j}: {69, 34781}, {1498, 9924}, {9919, 32254}, {12315, 33878}, {19588, 39568}
X(39879) = reflection of X(i) in X(j) for these (i,j): (3, 159), (6, 6759), (64, 3098), (159, 15581), (1351, 19149), (2935, 12584), (6247, 15585), (8549, 206), (11477, 34779), (11579, 15647), (13093, 34778), (14216, 141), (15577, 15580), (15583, 16252), (16010, 13289), (31670, 2883), (33878, 34787), (34777, 34117), (34780, 66), (36851, 5), (37491, 7387)
X(39879) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (161, 1619, 9909), (206, 8549, 5050), (1351, 32063, 19149), (1614, 39588, 19125), (5050, 14530, 206), (15577, 15580, 159), (15577, 15581, 15580), (15583, 16252, 14561), (34117, 34777, 5093)

X(39880) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO 9th BROCARD

Barycentrics    -(2*a^6+(b+c)*a^5-(3*b^2+2*b*c+3*c^2)*a^4+2*(b+c)*b*c*a^3+2*(b^3-c^3)*(b-c)*a^2-(b^4-c^4)*(b-c)*a-(b^4-c^4)*(b^2-c^2))*sqrt(R*(4*R+r))+S*(-a+b+c)*a^2*(a^2+b^2+c^2) : :

The reciprocal orthologic center of these triangles is X(4).

X(39880) lies on these lines: {6,8196}, {55,39881}, {69,11822}, {518,9835}, {542,11207}, {1352,5599}, {1503,9834}, {3564,11252}, {5597,6776}, {5598,39898}, {5601,5921}, {5847,12458}, {5848,12462}, {8186,39878}, {8190,39879}, {8197,39885}, {8198,39887}, {8199,39888}, {8200,18440}, {8201,39895}, {8202,39896}, {11366,39870}, {11384,39871}, {11492,39877}, {11493,39883}, {11837,39872}, {11843,39874}, {11861,39882}, {11863,39886}, {11865,39889}, {11867,39890}, {11869,39891}, {11871,39892}, {11873,39897}, {11875,39899}, {11877,39900}, {11879,39901}, {11881,39902}, {11883,39903}, {13890,19145}, {13944,19146}, {18495,39884}, {18955,39873}, {19007,39875}, {19008,39876}, {35778,39893}, {35781,39894}

X(39880) = reflection of X(i) in X(j) for these (i,j): (9834, 12452), (39881, 55)

X(39881) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO 9th BROCARD

Barycentrics    -(2*a^6+(b+c)*a^5-(3*b^2+2*b*c+3*c^2)*a^4+2*(b+c)*b*c*a^3+2*(b^3-c^3)*(b-c)*a^2-(b^4-c^4)*(b-c)*a-(b^4-c^4)*(b^2-c^2))*sqrt(R*(4*R+r))-S*(-a+b+c)*a^2*(a^2+b^2+c^2) : :

The reciprocal orthologic center of these triangles is X(4).

X(39881) lies on these lines: {6,8203}, {55,39880}, {69,11823}, {518,9834}, {542,11208}, {1352,5600}, {1503,9835}, {3564,11253}, {5597,39898}, {5598,6776}, {5602,5921}, {5847,12459}, {5848,12463}, {8187,39878}, {8191,39879}, {8204,39885}, {8205,39887}, {8206,39888}, {8207,18440}, {8208,39895}, {8209,39896}, {11367,39870}, {11385,39871}, {11492,39883}, {11493,39877}, {11838,39872}, {11844,39874}, {11862,39882}, {11864,39886}, {11866,39889}, {11868,39890}, {11870,39891}, {11872,39892}, {11874,39897}, {11876,39899}, {11878,39900}, {11880,39901}, {11882,39902}, {11884,39903}, {13891,19145}, {13945,19146}, {18497,39884}, {18956,39873}, {19009,39875}, {19010,39876}, {35779,39894}, {35780,39893}

X(39881) = reflection of X(i) in X(j) for these (i,j): (9835, 12453), (39880, 55)

X(39882) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO 9th BROCARD

Barycentrics    2*a^10+(b^2+c^2)*a^8+(2*b^4+b^2*c^2+2*c^4)*a^6-(b^2+c^2)*(4*b^4+b^2*c^2+4*c^4)*a^4-(b^2-c^2)^2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)*(b^4+b^2*c^2+c^4) : :
X(39882) = 4*X(3098)-3*X(7811) = 3*X(7812)-2*X(31670)

The reciprocal orthologic center of these triangles is X(4).

X(39882) lies on these lines: {6,9993}, {32,6776}, {69,74}, {147,13355}, {182,7846}, {206,15920}, {511,7877}, {1352,3096}, {1503,3094}, {2896,5921}, {3099,39878}, {3564,9821}, {3818,7790}, {5092,7835}, {5847,12497}, {5848,12499}, {7812,31670}, {7865,11180}, {8369,26316}, {9301,39899}, {9857,39885}, {9994,39887}, {9995,39888}, {9996,11287}, {9997,39898}, {10038,39900}, {10047,39901}, {10828,39879}, {10871,39889}, {10872,39890}, {10873,39891}, {10874,39892}, {10875,39895}, {10876,39896}, {10877,39897}, {10878,39902}, {10879,39903}, {11368,39870}, {11386,39871}, {11494,39877}, {11861,39880}, {11862,39881}, {11885,39886}, {13892,19145}, {13946,19146}, {18500,39884}, {18957,39873}, {19011,39875}, {19012,39876}, {22744,39883}, {35782,39893}, {35783,39894}

X(39882) = reflection of X(9873) in X(3094)

X(39883) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 9th BROCARD

Barycentrics
a*(a^8-(b+c)*a^7-(b+c)^2*a^6+(b+c)*(b^2-4*b*c+c^2)*a^5-(b^4+c^4-2*(3*b^2+b*c+3*c^2)*b*c)*a^4+(b^2-c^2)^2*(b+c)*a^3+(b^4+c^4-2*(2*b^2+b*c+2*c^2)*b*c)*(b-c)^2*a^2-(b^4-c^4)*(b-c)^3*a+2*(b^4-c^4)*(b^2-c^2)*b*c) : :

The reciprocal orthologic center of these triangles is X(4).

X(39883) lies on these lines: {3,39877}, {6,22753}, {36,39878}, {55,39898}, {56,6776}, {69,3428}, {104,39874}, {182,25524}, {518,5534}, {542,11194}, {946,37492}, {956,39885}, {958,1352}, {999,39870}, {1503,12114}, {2975,5921}, {3098,11495}, {3149,3751}, {3564,11249}, {5584,10519}, {5847,22770}, {5848,22775}, {6001,7289}, {6261,34381}, {10966,39897}, {11492,39881}, {11493,39880}, {11496,36740}, {16408,38118}, {18440,22758}, {18761,39884}, {19013,39875}, {19014,39876}, {19145,22763}, {19146,22764}, {22479,39871}, {22520,39872}, {22654,39879}, {22744,39882}, {22755,39886}, {22756,39887}, {22757,39888}, {22759,39891}, {22760,39892}, {22761,39895}, {22762,39896}, {22765,39899}, {22766,39900}, {22767,39901}, {22768,39902}, {35784,39893}, {35785,39894}

X(39883) = reflection of X(i) in X(j) for these (i,j): (12114, 22769), (39877, 3), (39890, 1352)

X(39884) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO 9th BROCARD

Barycentrics    4*a^6-(b^2+c^2)*a^4+4*b^2*c^2*a^2-3*(b^4-c^4)*(b^2-c^2) : :
X(39884) = 5*X(3)-7*X(3619) = 3*X(3)-X(14927) = 5*X(4)-X(193) = 3*X(4)-X(1351) = 3*X(4)+X(5921) = 3*X(5)-2*X(182) = 5*X(5)-4*X(3589) = 4*X(5)-3*X(38110) = 7*X(5)-6*X(38317) = 3*X(193)-5*X(1351) = 3*X(193)+5*X(5921) = X(193)+5*X(18440) = 2*X(193)-5*X(21850) = X(1351)+3*X(18440) = 2*X(1351)-3*X(21850) = 7*X(3619)-10*X(18358) = X(5921)-3*X(18440) = 2*X(5921)+3*X(21850) = X(14927)-6*X(18358) = 2*X(18440)+X(21850)

The reciprocal orthologic center of these triangles is X(4).

X(39884) lies on these lines: {3,3619}, {4,193}, {5,182}, {6,546}, {30,599}, {66,31833}, {69,382}, {110,427}, {113,15432}, {140,10516}, {141,550}, {146,32306}, {155,16198}, {159,7526}, {381,6776}, {428,11442}, {511,3627}, {524,15687}, {542,1353}, {549,11645}, {567,7403}, {575,3857}, {597,38071}, {632,5092}, {1368,5651}, {1386,38034}, {1428,10593}, {1478,39892}, {1479,39891}, {1513,17004}, {1595,12134}, {1596,18474}, {1656,25406}, {1657,10519}, {1658,20987}, {1843,12162}, {1853,6677}, {1899,3066}, {1907,14516}, {1992,14269}, {2330,10592}, {3090,12017}, {3091,5050}, {3098,15704}, {3146,33878}, {3167,7378}, {3242,28224}, {3313,11591}, {3410,34603}, {3416,28174}, {3529,3620}, {3530,3763}, {3580,10301}, {3583,39897}, {3585,39873}, {3618,3851}, {3628,5085}, {3796,11548}, {3830,11180}, {3832,14912}, {3843,14853}, {3850,14561}, {3853,15069}, {3854,33748}, {3858,8550}, {3860,38072}, {3867,22660}, {3933,5207}, {5020,32064}, {5066,11179}, {5133,11003}, {5159,35259}, {5476,12007}, {5544,7392}, {5646,7734}, {5663,19161}, {5847,22793}, {5848,22799}, {5893,34779}, {6090,31099}, {6102,9969}, {6287,8362}, {6403,15305}, {6676,31383}, {6723,15647}, {6756,37489}, {6800,37454}, {6823,16655}, {7394,11245}, {7399,16659}, {7401,34780}, {7405,13339}, {7528,18914}, {7530,37488}, {7553,37494}, {7667,21766}, {7715,12359}, {7716,12163}, {7863,18860}, {8703,11178}, {8780,8889}, {9756,10011}, {9818,39879}, {9825,14216}, {9925,11403}, {9955,38040}, {9967,15030}, {10109,38064}, {10154,21243}, {10250,18418}, {10264,32274}, {10272,32233}, {10539,19124}, {10541,12812}, {10620,38321}, {10752,32272}, {10895,39900}, {10896,39901}, {11061,38789}, {11381,37511}, {11477,12102}, {11563,32217}, {11579,11801}, {11646,12212}, {12041,36201}, {12103,31884}, {12220,15058}, {12283,16261}, {12294,12295}, {12315,20079}, {12383,35484}, {12588,15171}, {12589,18990}, {12699,39885}, {13202,32275}, {13474,14913}, {13562,18569}, {13665,39876}, {13785,39875}, {14235,22819}, {14239,22820}, {14826,34609}, {14869,17508}, {14893,20423}, {14929,14994}, {14982,32423}, {15577,18570}, {15578,15646}, {15583,18383}, {15585,34785}, {15681,21356}, {15699,25561}, {15720,33750}, {16774,32601}, {17504,20582}, {17702,23296}, {18357,38165}, {18451,19139}, {18491,39877}, {18492,39878}, {18495,39880}, {18497,39881}, {18500,39882}, {18502,39872}, {18507,39886}, {18509,39887}, {18511,39888}, {18516,39889}, {18517,39890}, {18520,39895}, {18522,39896}, {18525,39898}, {18538,19145}, {18542,39902}, {18544,39903}, {18761,39883}, {18762,19146}, {18952,23411}, {19131,26883}, {19924,35404}, {21358,34200}, {21869,32247}, {23047,39588}, {29181,34507}, {32110,37458}, {33228,39141}, {35786,39893}, {35787,39894}, {37638,37897}

X(39884) = midpoint of X(i) and X(j) for these {i,j}: {4, 18440}, {69, 382}, {113, 32250}, {146, 32306}, {1351, 5921}, {1352, 36990}, {1843, 12162}, {3146, 33878}, {3830, 11180}, {10752, 32272}, {11381, 37511}, {12315, 20079}, {12699, 39885}, {13202, 32275}, {13474, 14913}, {15069, 31670}, {18507, 39886}, {18525, 39898}
X(39884) = reflection of X(i) in X(j) for these (i,j): (3, 18358), (5, 3818), (6, 546), (141, 18553), (550, 141), (1353, 5480), (3313, 11591), (6102, 9969), (6776, 18583), (8550, 19130), (8703, 11178), (10264, 32274), (11179, 5066), (11579, 11801), (15583, 18383), (15704, 3098), (20423, 14893), (21850, 4), (31670, 3853), (32233, 10272), (34776, 16252), (34779, 5893), (34785, 15585), (39870, 9955)
X(39884) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 5921, 1351), (6, 546, 38136), (381, 6776, 18583), (1351, 18440, 5921), (1353, 3845, 5480), (3091, 39874, 5050), (3843, 39899, 14853), (5066, 11179, 38079), (9955, 39870, 38040), (17508, 34573, 14869)

X(39885) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO 9th BROCARD

Barycentrics    3*a^6-(b+c)*a^5-2*(b^2-b*c+c^2)*a^4-2*(b+c)*b*c*a^3+(b^2+c^2)*(b+c)^2*a^2+(b^4-c^4)*(b-c)*a-2*(b^4-c^4)*(b^2-c^2) : :
X(39885) = 4*X(5)-3*X(16475) = 2*X(6)-3*X(5587) = 4*X(141)-3*X(3576) = 4*X(182)-5*X(1698) = 4*X(1386)-5*X(8227) = 2*X(1386)-3*X(10516) = 3*X(1699)-4*X(3818) = 7*X(3090)-6*X(38049) = 5*X(3618)-6*X(10175) = 7*X(3619)-6*X(10165) = 5*X(3620)-3*X(5731) = 7*X(3624)-8*X(24206) = 7*X(3624)-6*X(38029) = 3*X(3679)-X(39878) = 4*X(3844)-3*X(5085) = 8*X(3844)-7*X(31423) = X(5881)+2*X(15069) = 5*X(8227)-6*X(10516) = 3*X(11180)-X(39898) = 4*X(24206)-3*X(38029)

The reciprocal orthologic center of these triangles is X(4).

X(39885) lies on these lines: {1,1352}, {2,39870}, {4,5847}, {5,16475}, {6,5587}, {8,5921}, {10,6776}, {40,1503}, {65,39891}, {67,33535}, {69,515}, {72,39890}, {141,3576}, {159,15177}, {182,1698}, {355,3564}, {511,5691}, {516,4133}, {517,18440}, {518,5735}, {519,11180}, {542,2948}, {611,9578}, {613,9581}, {952,16496}, {956,39883}, {1351,18480}, {1353,18357}, {1386,8227}, {1469,9613}, {1699,3818}, {1737,39901}, {1766,29046}, {1837,39897}, {2330,31434}, {2771,32306}, {2784,4660}, {2792,3729}, {2854,12407}, {3056,3586}, {3057,39892}, {3090,38049}, {3618,10175}, {3619,10165}, {3620,5731}, {3624,24206}, {3827,5693}, {3844,5085}, {4297,10519}, {4663,38144}, {5050,9956}, {5090,39871}, {5252,39873}, {5480,18492}, {5657,39874}, {5663,32261}, {5687,39877}, {5688,39888}, {5689,39887}, {5790,39899}, {5818,14912}, {5846,7982}, {5848,12751}, {5886,16491}, {6210,21061}, {6246,10755}, {6684,25406}, {7989,14561}, {8185,37488}, {8193,39879}, {8197,39880}, {8204,39881}, {8214,39895}, {8215,39896}, {8550,38047}, {9624,38315}, {9625,20987}, {9780,38118}, {9857,39882}, {10039,39900}, {10791,39872}, {10914,39889}, {10915,39902}, {10916,39903}, {11178,25055}, {11179,19875}, {11231,12017}, {11522,18553}, {11898,18525}, {11900,39886}, {12429,14872}, {12587,17857}, {12699,39884}, {12717,29207}, {13883,39876}, {13893,19145}, {13936,39875}, {13947,19146}, {14853,19925}, {14927,31730}, {19876,38064}, {28160,33878}, {28538,31162}, {35788,39893}, {35789,39894}

X(39885) = midpoint of X(i) and X(j) for these {i,j}: {8, 5921}, {11898, 18525}
X(39885) = reflection of X(i) in X(j) for these (i,j): (1, 1352), (40, 3416), (1351, 18480), (1353, 18357), (3751, 355), (6776, 10), (10755, 6246), (12699, 39884), (14927, 31730), (33535, 67)
X(39885) = anticomplement of X(39870)
X(39885) = X(8)-Beth conjugate of-X(6776)
X(39885) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1386, 10516, 8227), (3844, 5085, 31423), (24206, 38029, 3624)

X(39886) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO 9th BROCARD

Barycentrics    (S^2-3*SB*SC)*(2*(6*R^2+SA-2*SW)*S^2-(4*R^2-SW)*(36*R^2*SW-9*SA^2-5*SW^2)) : :
X(39886) = 2*X(6)-3*X(11897) = 4*X(182)-5*X(15183) = 3*X(11831)-2*X(39870) = 3*X(11845)-X(39874) = 3*X(11852)-X(39878) = 3*X(11911)-X(39899)

The reciprocal orthologic center of these triangles is X(4).

X(39886) lies on these lines: {6,11897}, {30,69}, {182,15183}, {402,6776}, {542,1651}, {1352,1650}, {1503,12113}, {3564,11251}, {4240,5921}, {5847,12696}, {5848,12752}, {11831,39870}, {11832,39871}, {11839,39872}, {11845,39874}, {11848,39877}, {11852,39878}, {11853,39879}, {11863,39880}, {11864,39881}, {11885,39882}, {11900,39885}, {11901,39887}, {11902,39888}, {11903,39889}, {11904,39890}, {11905,39891}, {11906,39892}, {11907,39895}, {11908,39896}, {11909,39897}, {11910,39898}, {11911,39899}, {11912,39900}, {11913,39901}, {11914,39902}, {11915,39903}, {13894,19145}, {13948,19146}, {18507,39884}, {18958,39873}, {19017,39875}, {19018,39876}, {22755,39883}, {35790,39893}, {35791,39894}

X(39886) = midpoint of X(4240) and X(5921)
X(39886) = reflection of X(i) in X(j) for these (i,j): (1650, 1352), (6776, 402), (12113, 12583), (18507, 39884)

X(39887) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO 9th BROCARD

Barycentrics    (-a^2+b^2+c^2)*(3*a^4+(b^2-c^2)^2)*S+(a^2+b^2+c^2)*(2*a^6-(b^2+c^2)*a^4-(b^4-c^4)*(b^2-c^2)) : :
X(39887) = 2*X(13749)-3*X(14853) = X(14242)-3*X(14912)

The reciprocal orthologic center of these triangles is X(4).

X(39887) lies on these lines: {4,6}, {69,11824}, {542,5861}, {1161,3564}, {1271,5921}, {1352,5591}, {5491,6281}, {5589,39878}, {5595,39879}, {5605,39898}, {5689,39885}, {5847,12697}, {5848,12753}, {6215,18440}, {7000,13949}, {8198,39880}, {8205,39881}, {8216,39895}, {8217,39896}, {8396,36709}, {8721,12257}, {8974,19145}, {9994,39882}, {10040,39900}, {10048,39901}, {10792,39872}, {10919,39889}, {10921,39890}, {10923,39891}, {10925,39892}, {10927,39897}, {10929,39902}, {10931,39903}, {11291,25406}, {11370,39870}, {11388,39871}, {11497,39877}, {11901,39886}, {11916,39899}, {18509,39884}, {18959,39873}, {22756,39883}, {26341,37342}, {35792,39893}, {35795,39894}

X(39887) = midpoint of X(14227) and X(39874)
X(39887) = reflection of X(i) in X(j) for these (i,j): (5871, 6), (39888, 6776)
X(39887) = {X(4), X(6776)}-harmonic conjugate of X(39875)

X(39888) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO 9th BROCARD

Barycentrics    -(-a^2+b^2+c^2)*(3*a^4+(b^2-c^2)^2)*S+(a^2+b^2+c^2)*(2*a^6-(b^2+c^2)*a^4-(b^4-c^4)*(b^2-c^2)) : :
X(39888) = 2*X(13748)-3*X(14853) = X(14227)-3*X(14912)

The reciprocal orthologic center of these triangles is X(4).

X(39888) lies on these lines: {4,6}, {69,11825}, {542,5860}, {1160,3564}, {1270,5921}, {1352,5590}, {5490,6278}, {5588,39878}, {5594,39879}, {5604,39898}, {5688,39885}, {5847,12698}, {5848,12754}, {6214,18440}, {7374,8975}, {8199,39880}, {8206,39881}, {8218,39895}, {8219,39896}, {8416,36714}, {8721,12256}, {9995,39882}, {10041,39900}, {10049,39901}, {10793,39872}, {10920,39889}, {10922,39890}, {10924,39891}, {10926,39892}, {10928,39897}, {10930,39902}, {10932,39903}, {11292,25406}, {11371,39870}, {11389,39871}, {11498,39877}, {11902,39886}, {11917,39899}, {13950,19146}, {18511,39884}, {18960,39873}, {22757,39883}, {26348,37343}, {35793,39894}, {35794,39893}

X(39888) = midpoint of X(14242) and X(39874)
X(39888) = reflection of X(i) in X(j) for these (i,j): (5870, 6), (39887, 6776)
X(39888) = {X(4), X(6776)}-harmonic conjugate of X(39876)

X(39889) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO 9th BROCARD

Barycentrics
3*a^9-3*(b+c)*a^8-5*(b-c)^2*a^7+(b+c)*(5*b^2-4*b*c+5*c^2)*a^6+(3*b^4+3*c^4-2*(5*b^2-b*c+5*c^2)*b*c)*a^5-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^4-(3*b^4+2*b^2*c^2+3*c^4)*(b-c)^2*a^3+(b^4-c^4)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2+2*(b^4-c^4)*(b^2-c^2)*(b^2-3*b*c+c^2)*a+(b^4-c^4)*(b^2-c^2)^2*(-2*b+2*c) : :

The reciprocal orthologic center of these triangles is X(4).

X(39889) lies on these lines: {6,10893}, {11,6776}, {12,39902}, {69,11826}, {355,18440}, {542,11235}, {1352,1376}, {1503,12114}, {3434,5921}, {3564,10525}, {5820,11496}, {5847,12700}, {5848,12761}, {10523,39900}, {10598,14912}, {10785,39874}, {10794,39872}, {10826,39878}, {10829,39879}, {10871,39882}, {10914,39885}, {10919,39887}, {10920,39888}, {10944,39891}, {10945,39895}, {10946,39896}, {10947,39897}, {10948,39901}, {10949,39903}, {11180,34612}, {11373,39870}, {11390,39871}, {11865,39880}, {11866,39881}, {11903,39886}, {11928,39899}, {13895,19145}, {13952,19146}, {18516,39884}, {18961,39873}, {19023,39875}, {19024,39876}, {35796,39893}, {35797,39894}

X(39889) = reflection of X(i) in X(j) for these (i,j): (12114, 12586), (39877, 1352), (39890, 18440)

X(39890) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO 9th BROCARD

Barycentrics
3*a^9-3*(b+c)*a^8-(5*b^2-2*b*c+5*c^2)*a^7+(b+c)*(5*b^2-8*b*c+5*c^2)*a^6+(3*b^4+3*c^4+2*(b^2+5*b*c+c^2)*b*c)*a^5-(b+c)*(3*b^4+3*c^4-2*(2*b-c)*(b-2*c)*b*c)*a^4-(b^2-c^2)^2*(3*b^2+2*b*c+3*c^2)*a^3+3*(b^4-c^4)*(b^2-c^2)*(b+c)*a^2+2*(b-c)*(b^3+c^3)*(b^4-c^4)*a+(b^4-c^4)*(b^2-c^2)^2*(-2*b+2*c) : :

The reciprocal orthologic center of these triangles is X(4).

X(39890) lies on these lines: {6,10894}, {11,39903}, {12,6776}, {69,11827}, {72,39885}, {355,18440}, {542,11236}, {958,1352}, {1503,11500}, {3436,5921}, {3564,10526}, {5812,5847}, {5848,12762}, {10523,39901}, {10599,14912}, {10786,39874}, {10795,39872}, {10827,39878}, {10830,39879}, {10872,39882}, {10921,39887}, {10922,39888}, {10950,39892}, {10951,39895}, {10952,39896}, {10953,39897}, {10954,39900}, {10955,39902}, {11180,34606}, {11374,39870}, {11391,39871}, {11867,39880}, {11868,39881}, {11904,39886}, {11929,39899}, {13896,19145}, {13953,19146}, {18517,39884}, {18962,39873}, {19025,39875}, {19026,39876}, {35798,39893}, {35799,39894}

X(39890) = reflection of X(i) in X(j) for these (i,j): (11500, 12587), (39883, 1352), (39889, 18440)

X(39891) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO 9th BROCARD

Barycentrics    3*a^6-2*(b^2-b*c+c^2)*a^4+(b^2+c^2)*(b+c)^2*a^2-2*(b^4-c^4)*(b^2-c^2) : :
X(39891) = 2*X(611)-3*X(11237)

The reciprocal orthologic center of these triangles is X(4).

X(39891) lies on these lines: {1,18440}, {4,39897}, {5,39901}, {6,10895}, {12,6776}, {55,1503}, {56,1352}, {65,39885}, {69,7354}, {141,5204}, {159,9659}, {193,5229}, {388,5921}, {495,39900}, {499,18358}, {511,12943}, {518,36971}, {542,611}, {613,3818}, {1351,3585}, {1428,10516}, {1469,9657}, {1478,3564}, {1479,39884}, {1836,5847}, {3056,12953}, {3085,39874}, {3304,12589}, {3416,37567}, {3614,3618}, {5050,7951}, {5432,25406}, {5434,11180}, {5848,12763}, {9578,39878}, {9654,39899}, {9655,11898}, {9658,37488}, {10483,33878}, {10519,15326}, {10590,14912}, {10797,39872}, {10831,39879}, {10873,39882}, {10923,39887}, {10924,39888}, {10944,39889}, {10956,39902}, {10957,39903}, {11375,39870}, {11392,39871}, {11501,39877}, {11670,32261}, {11869,39880}, {11870,39881}, {11905,39886}, {11930,39895}, {11931,39896}, {13897,19145}, {13954,19146}, {14927,15338}, {14982,32243}, {16475,17605}, {19027,39875}, {19028,39876}, {19470,32306}, {22759,39883}, {24248,29235}, {35800,39893}, {35801,39894}

X(39891) = reflection of X(i) in X(j) for these (i,j): (55, 12588), (39900, 495)
X(39891) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18440, 39892), (388, 5921, 39873), (613, 3818, 10896), (3056, 36990, 12953)

X(39892) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO 9th BROCARD

Barycentrics    3*a^6-2*(b^2+b*c+c^2)*a^4+(b^2+c^2)*(b-c)^2*a^2-2*(b^4-c^4)*(b^2-c^2) : :
X(39892) = 2*X(613)-3*X(11238)

The reciprocal orthologic center of these triangles is X(4).

X(39892) lies on these lines: {1,18440}, {4,39873}, {5,39900}, {6,10896}, {11,6776}, {55,1352}, {56,1503}, {69,6284}, {141,5217}, {159,9672}, {193,5225}, {496,39901}, {497,5921}, {498,18358}, {511,12953}, {542,613}, {611,3818}, {1351,3583}, {1469,12943}, {1478,39884}, {1479,3564}, {1898,34381}, {2330,10516}, {3056,9670}, {3057,39885}, {3058,11180}, {3086,39874}, {3303,12588}, {3618,7173}, {5050,7741}, {5433,25406}, {5847,12701}, {5848,12764}, {7727,32306}, {9581,39878}, {9668,11898}, {9669,39899}, {9673,37488}, {10519,15338}, {10591,14912}, {10798,39872}, {10832,39879}, {10874,39882}, {10925,39887}, {10926,39888}, {10950,39890}, {10958,39902}, {10959,39903}, {11376,39870}, {11393,39871}, {11502,39877}, {11871,39880}, {11872,39881}, {11906,39886}, {11932,39895}, {11933,39896}, {13898,19145}, {13955,19146}, {14927,15326}, {14982,32297}, {19029,39875}, {19030,39876}, {22760,39883}, {35802,39893}, {35803,39894}

X(39892) = reflection of X(i) in X(j) for these (i,j): (56, 12589), (39901, 496)
X(39892) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18440, 39891), (497, 5921, 39897), (611, 3818, 10895), (1469, 36990, 12943)

X(39893) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO 9th BROCARD

Barycentrics    2*(a^2+b^2+c^2)*S*a^2+(-a^2+b^2+c^2)*(5*a^4-(b^2+c^2)*a^2+2*(b^2-c^2)^2) : :
X(39893) = 4*X(6)-3*X(35822)

The reciprocal orthologic center of these triangles is X(4).

X(39893) lies on these lines: {2,22616}, {4,22646}, {6,13}, {69,6200}, {193,6561}, {371,3564}, {372,6776}, {485,5921}, {486,14912}, {895,35835}, {1151,11898}, {1328,5032}, {1351,22810}, {1352,10576}, {1353,3071}, {1503,35820}, {5050,10577}, {5093,23261}, {5847,35610}, {5848,35856}, {6419,39876}, {6560,39874}, {8280,11442}, {8281,11402}, {8550,13972}, {8855,11245}, {9541,20080}, {9677,19131}, {9974,36990}, {10783,39661}, {11579,35877}, {14853,35787}, {15069,19145}, {19146,35814}, {35762,39870}, {35764,39871}, {35766,39872}, {35768,39873}, {35769,39901}, {35770,39875}, {35772,39877}, {35774,39878}, {35776,39879}, {35778,39880}, {35780,39881}, {35782,39882}, {35784,39883}, {35786,39884}, {35788,39885}, {35790,39886}, {35792,39887}, {35794,39888}, {35796,39889}, {35798,39890}, {35800,39891}, {35802,39892}, {35804,39895}, {35806,39896}, {35808,39897}, {35809,39900}, {35810,39898}, {35816,39902}, {35818,39903}

X(39893) = reflection of X(35820) in X(35840)
X(39893) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 18440, 6564), (6, 39899, 39894)

X(39894) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO 9th BROCARD

Barycentrics    -2*(a^2+b^2+c^2)*S*a^2+(-a^2+b^2+c^2)*(5*a^4-(b^2+c^2)*a^2+2*(b^2-c^2)^2) : :
X(39894) = 4*X(6)-3*X(35823)

The reciprocal orthologic center of these triangles is X(4).

X(39894) lies on these lines: {2,22645}, {4,22617}, {6,13}, {69,6396}, {193,6560}, {371,6776}, {372,3564}, {485,14912}, {486,5921}, {895,35834}, {1152,11898}, {1327,5032}, {1351,22809}, {1352,10577}, {1353,3070}, {1503,35821}, {5050,10576}, {5093,23251}, {5847,35611}, {5848,35857}, {6420,39875}, {6561,39874}, {8280,11402}, {8281,11442}, {8550,8960}, {8854,11245}, {9975,36990}, {10784,39660}, {11579,35876}, {14853,35786}, {15069,19146}, {19145,35815}, {35763,39870}, {35765,39871}, {35767,39872}, {35768,39901}, {35769,39873}, {35771,39876}, {35773,39877}, {35775,39878}, {35777,39879}, {35779,39881}, {35781,39880}, {35783,39882}, {35785,39883}, {35787,39884}, {35789,39885}, {35791,39886}, {35793,39888}, {35795,39887}, {35797,39889}, {35799,39890}, {35801,39891}, {35803,39892}, {35805,39896}, {35807,39895}, {35808,39900}, {35809,39897}, {35811,39898}, {35817,39902}, {35819,39903}

X(39894) = reflection of X(35821) in X(35841)
X(39894) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 18440, 6565), (6, 39899, 39893)

X(39895) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO 9th BROCARD

Barycentrics    (S^4-(4*R^2*(2*SA-SW)-SA^2+SB*SC+SW^2)*S^2-8*R^2*SB*SC*SW)*S-SW*(S^4-SB*SC*(S^2-SW^2)) : :

The reciprocal orthologic center of these triangles is X(4).

X(39895) lies on these lines: {6,8212}, {69,11828}, {493,6776}, {542,12152}, {1352,8222}, {1503,9838}, {3564,10669}, {5847,22841}, {5848,12765}, {5921,6462}, {6461,39896}, {6465,21736}, {8188,39878}, {8194,39879}, {8201,39880}, {8208,39881}, {8210,39898}, {8214,39885}, {8216,39887}, {8218,39888}, {8220,18440}, {8408,12257}, {10875,39882}, {10945,39889}, {10951,39890}, {11377,39870}, {11394,39871}, {11503,39877}, {11840,39872}, {11846,39874}, {11907,39886}, {11930,39891}, {11932,39892}, {11947,39897}, {11949,39899}, {11951,39900}, {11953,39901}, {11955,39902}, {11957,39903}, {13899,19145}, {13956,19146}, {18520,39884}, {18963,39873}, {19031,39875}, {19032,39876}, {22761,39883}, {35804,39893}, {35807,39894}

X(39895) = reflection of X(9838) in X(12590)

X(39896) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO 9th BROCARD

Barycentrics    -(S^4-(4*R^2*(2*SA-SW)-SA^2+SB*SC+SW^2)*S^2-8*R^2*SB*SC*SW)*S-SW*(S^4-SB*SC*(S^2-SW^2)) : :

The reciprocal orthologic center of these triangles is X(4).

X(39896) lies on these lines: {6,8213}, {69,11829}, {494,6776}, {542,12153}, {1352,8223}, {1503,9839}, {3564,10673}, {5847,22842}, {5848,12766}, {5921,6463}, {6461,39895}, {8189,39878}, {8195,39879}, {8202,39880}, {8209,39881}, {8211,39898}, {8215,39885}, {8217,39887}, {8219,39888}, {8221,18440}, {8420,12256}, {10876,39882}, {10946,39889}, {10952,39890}, {11378,39870}, {11395,39871}, {11504,39877}, {11841,39872}, {11847,39874}, {11908,39886}, {11931,39891}, {11933,39892}, {11948,39897}, {11950,39899}, {11952,39900}, {11954,39901}, {11956,39902}, {11958,39903}, {13900,19145}, {13957,19146}, {18522,39884}, {18964,39873}, {19033,39875}, {19034,39876}, {22762,39883}, {35805,39894}, {35806,39893}

X(39896) = reflection of X(9839) in X(12591)

X(39897) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO 9th BROCARD

Barycentrics    (a+b-c)*(a-b+c)*(2*a^4-(b+c)^2*a^2+(b^2+c^2)*(b+c)^2) : :
X(39897) = 2*X(1469)-3*X(5434)

The reciprocal orthologic center of these triangles is X(4).

X(39897) lies on these lines: {1,3564}, {3,39901}, {4,39891}, {6,12}, {11,613}, {33,39871}, {55,6776}, {56,69}, {57,32861}, {65,5847}, {141,1428}, {182,5432}, {193,388}, {206,9652}, {226,3791}, {495,1353}, {497,5921}, {498,5050}, {511,7354}, {518,10944}, {524,1469}, {542,3024}, {599,5298}, {611,15888}, {612,11245}, {895,12903}, {999,11898}, {1284,15991}, {1317,3242}, {1350,15326}, {1351,1478}, {1386,15950}, {1397,26942}, {1429,33082}, {1479,18440}, {1503,3056}, {1697,39878}, {1837,39885}, {1992,11237}, {2098,39898}, {2330,8550}, {2646,39870}, {2854,32243}, {3085,14912}, {3295,39899}, {3583,39884}, {3585,21850}, {3600,20080}, {3614,14561}, {3620,7288}, {3629,19369}, {3751,5252}, {3763,7294}, {4294,39874}, {4299,33878}, {4995,11179}, {5034,31460}, {5039,10797}, {5093,9654}, {5204,10519}, {5217,25406}, {5443,38040}, {5480,8540}, {5820,10957}, {5845,24816}, {5965,18984}, {7173,10516}, {7235,17362}, {7741,18358}, {7951,18583}, {8593,12350}, {10106,34379}, {10752,12373}, {10753,12184}, {10754,13182}, {10755,13273}, {10759,12763}, {10766,13296}, {10799,39872}, {10831,19459}, {10833,39879}, {10877,39882}, {10895,14853}, {10927,39887}, {10928,39888}, {10947,39889}, {10949,12586}, {10953,39890}, {10965,39902}, {10966,39883}, {11180,11238}, {11375,16475}, {11392,12167}, {11579,32307}, {11873,39880}, {11874,39881}, {11909,39886}, {11947,39895}, {11948,39896}, {12589,15069}, {12837,32451}, {13901,19145}, {13958,19146}, {14982,32290}, {14984,18968}, {15983,28386}, {15988,25466}, {16496,37738}, {16603,33682}, {18954,37491}, {18970,34382}, {18990,34380}, {19037,39875}, {19038,39876}, {24836,32029}, {24953,26543}, {26625,30778}, {26629,39141}, {26888,34774}, {35808,39893}, {35809,39894}

X(39897) = reflection of X(i) in X(j) for these (i,j): (6284, 3056), (39873, 1)
X(39897) = X(69)-of-2nd-anti-circumperp-tangential-triangle
X(39897) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 12588, 12), (141, 1428, 5433), (497, 5921, 39892), (613, 1352, 11), (3295, 39899, 39900), (10923, 19027, 12), (10924, 19028, 12)

X(39898) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR TO 9th BROCARD

Barycentrics    a^6+2*(b+c)*a^5-(3*b^2+4*b*c+3*c^2)*a^4+4*(b+c)*b*c*a^3+(3*b^2+2*b*c+3*c^2)*(b-c)^2*a^2-2*(b^4-c^4)*(b-c)*a-(b^4-c^4)*(b^2-c^2) : :
X(39898) = 3*X(1)-2*X(39870) = 3*X(1)-X(39878) = 2*X(6)-3*X(5603) = 2*X(40)-3*X(10519) = 4*X(141)-3*X(5657) = 4*X(182)-5*X(3616) = 4*X(946)-3*X(14853) = 4*X(1385)-3*X(25406) = 4*X(1386)-5*X(10595) = 4*X(1386)-3*X(14912) = 7*X(3090)-6*X(38047) = 4*X(3098)-3*X(9778) = 5*X(3618)-6*X(5886) = 7*X(3619)-6*X(26446) = 7*X(3622)-6*X(38029) = 2*X(3751)-3*X(14853) = 3*X(6776)-4*X(39870) = 3*X(6776)-2*X(39878) = 5*X(10595)-3*X(14912) = 3*X(11180)-2*X(39885)

The reciprocal orthologic center of these triangles is X(4).

X(39898) lies on these lines: {1,6776}, {4,518}, {6,5603}, {8,1352}, {40,10519}, {55,39883}, {56,39877}, {69,517}, {141,5657}, {145,5921}, {182,3616}, {511,962}, {515,16496}, {519,11180}, {542,3241}, {611,3485}, {944,1503}, {946,3751}, {952,18440}, {1350,6361}, {1351,22791}, {1385,25406}, {1386,10595}, {1469,4295}, {1482,3564}, {1537,10759}, {1992,3656}, {2098,39897}, {2099,39873}, {3056,30305}, {3090,38047}, {3098,9778}, {3416,12245}, {3475,7413}, {3618,5886}, {3619,26446}, {3622,38029}, {3624,38118}, {3654,21356}, {3873,26118}, {4301,34379}, {4430,37456}, {4663,38035}, {5050,5901}, {5182,11724}, {5597,39881}, {5598,39880}, {5604,39888}, {5605,39887}, {5622,11735}, {5693,34378}, {5758,10477}, {5790,18358}, {5800,10532}, {5818,10516}, {5846,15069}, {5847,7982}, {5848,10698}, {5928,17642}, {6905,12329}, {6906,22769}, {7967,39874}, {7968,39875}, {7969,39876}, {7987,33750}, {8148,11898}, {8192,39879}, {8210,39895}, {8211,39896}, {8550,38315}, {9041,34627}, {9624,38049}, {9779,19130}, {9780,24206}, {9812,31670}, {9997,39882}, {10247,39899}, {10800,39872}, {10944,39889}, {10950,39890}, {11179,38314}, {11396,39871}, {11910,39886}, {12017,38028}, {12261,25320}, {12589,18391}, {12672,34381}, {13464,16475}, {13902,19145}, {13959,19146}, {14927,18481}, {17257,31394}, {17316,31395}, {17625,26929}, {18493,18583}, {18525,39884}, {21554,38053}, {28174,33878}, {28538,34631}, {35810,39893}, {35811,39894}

X(39898) = midpoint of X(i) and X(j) for these {i,j}: {145, 5921}, {8148, 11898}
X(39898) = reflection of X(i) in X(j) for these (i,j): (8, 1352), (944, 3242), (1351, 22791), (1992, 3656), (3751, 946), (6361, 1350), (6776, 1), (10759, 1537), (12245, 3416), (14927, 18481), (18525, 39884), (39878, 39870)
X(39898) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 39878, 39870), (946, 3751, 14853), (10595, 14912, 1386), (24206, 38116, 9780), (39870, 39878, 6776), (39902, 39903, 6776)

X(39899) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO 9th BROCARD

Barycentrics    (5*a^4-(b^2+c^2)*a^2+2*(b^2-c^2)^2)*(-a^2+b^2+c^2) : :
X(39899) = 3*X(3)-2*X(69) = 7*X(3)-6*X(10519) = 5*X(3)-6*X(25406) = 17*X(3)-18*X(33750) = 2*X(4)-3*X(5093) = 2*X(5)-3*X(14912) = X(69)-3*X(6776) = 7*X(69)-9*X(10519) = 4*X(69)-3*X(11898) = 5*X(69)-9*X(25406) = 4*X(1353)-3*X(5093) = X(5921)-3*X(14912) = 7*X(6776)-3*X(10519) = 4*X(6776)-X(11898) = 5*X(6776)-3*X(25406) = 17*X(6776)-9*X(33750) = 12*X(10519)-7*X(11898) = 5*X(10519)-7*X(25406) = 5*X(11898)-12*X(25406) = 17*X(25406)-15*X(33750)

The reciprocal orthologic center of these triangles is X(4).

X(39899) lies on these lines: {2,21968}, {3,69}, {4,1353}, {5,5921}, {6,13}, {20,34380}, {22,37779}, {25,37644}, {30,193}, {49,182}, {68,19347}, {110,26869}, {125,32272}, {141,5054}, {146,12165}, {147,7806}, {154,32223}, {155,10116}, {156,19128}, {159,2070}, {184,19129}, {185,34382}, {195,8549}, {302,6770}, {303,6773}, {316,39646}, {323,31152}, {376,20080}, {378,12317}, {382,1351}, {511,1657}, {517,39878}, {518,18526}, {524,3534}, {549,3620}, {568,1843}, {575,5079}, {576,5076}, {599,5092}, {895,12902}, {999,39873}, {1147,26944}, {1154,12220}, {1181,12429}, {1350,5965}, {1352,1656}, {1495,21970}, {1498,10112}, {1531,18396}, {1598,13292}, {1614,19154}, {1853,34986}, {1899,3167}, {1974,10540}, {1992,3830}, {1993,34609}, {1994,5064}, {1995,14683}, {2937,37488}, {3090,33748}, {3098,15688}, {3295,39897}, {3410,7539}, {3448,5094}, {3517,18951}, {3580,26864}, {3618,5055}, {3619,15694}, {3630,14093}, {3631,15700}, {3751,18525}, {3843,14853}, {3845,5032}, {3851,18583}, {4846,6391}, {5020,11245}, {5039,18501}, {5070,38110}, {5072,12007}, {5085,7666}, {5095,7728}, {5181,15040}, {5480,11482}, {5544,35283}, {5596,18534}, {5621,35495}, {5622,9703}, {5644,7392}, {5663,15531}, {5790,39885}, {5800,37230}, {5847,12702}, {5848,12773}, {5889,12283}, {5890,12272}, {5984,13860}, {6090,18911}, {6102,6403}, {6146,12164}, {6417,39876}, {6418,39875}, {6467,10938}, {6515,9909}, {6644,18932}, {6677,18950}, {7387,32358}, {7517,39879}, {7776,14880}, {7777,11177}, {7850,12203}, {8593,11156}, {8681,37511}, {8703,11160}, {8716,38749}, {8780,13567}, {9301,39882}, {9544,37453}, {9654,39891}, {9669,39892}, {9730,14913}, {9744,37688}, {9761,36383}, {9763,36382}, {9862,31859}, {9924,32326}, {9967,18436}, {9969,13321}, {10246,39870}, {10247,39898}, {10297,18919}, {10605,16163}, {10620,34778}, {10752,38790}, {10753,38744}, {10754,38733}, {10758,38768}, {10759,38756}, {10764,38780}, {11004,31133}, {11008,15681}, {11061,12308}, {11165,14830}, {11206,20850}, {11216,18382}, {11225,17810}, {11264,32139}, {11328,25046}, {11402,11442}, {11432,12134}, {11477,29012}, {11574,23039}, {11579,11935}, {11645,15534}, {11806,32260}, {11842,39872}, {11849,39877}, {11875,39880}, {11876,39881}, {11911,39886}, {11916,39887}, {11917,39888}, {11928,39889}, {11929,39890}, {11949,39895}, {11950,39896}, {12000,39902}, {12001,39903}, {12041,32244}, {12083,37491}, {12160,34224}, {12161,34514}, {12167,18494}, {12173,34799}, {12284,32248}, {12294,18439}, {12309,12421}, {12585,37490}, {12601,13749}, {12602,13748}, {13142,34781}, {13188,14928}, {13634,17343}, {13635,17375}, {13690,22484}, {13743,37492}, {13811,22485}, {13861,32165}, {13903,19145}, {13961,19146}, {14645,38730}, {14852,21637}, {14927,17800}, {15061,32275}, {15068,26206}, {15089,25335}, {15317,15321}, {15576,39569}, {15578,35496}, {15701,21356}, {15716,22165}, {15723,34573}, {15760,19119}, {15905,18437}, {15988,17528}, {16475,18493}, {17809,21243}, {18325,32220}, {18481,34379}, {18531,18935}, {18539,23249}, {18553,39561}, {19124,37472}, {19544,37656}, {20423,32455}, {21487,32863}, {21974,26958}, {22728,35439}, {22765,39883}, {23049,39125}, {23259,26438}, {24981,35259}, {31723,36851}, {32002,33971}, {32063,34774}, {32113,37958}, {32257,38728}, {32599,35707}, {32954,39141}, {32972,33684}, {33749,38317}, {34780,36747}, {35424,35464}, {36318,37785}, {36320,37786}

X(39899) = midpoint of X(i) and X(j) for these {i,j}: {193, 39874}, {5889, 12283}, {12284, 32248}
X(39899) = reflection of X(i) in X(j) for these (i,j): (3, 6776), (4, 1353), (382, 1351), (1352, 8550), (3830, 1992), (5921, 5), (6033, 5477), (6403, 6102), (7728, 5095), (9924, 34776), (11160, 8703), (11898, 3), (12294, 32284), (12308, 11061), (12902, 895), (15069, 182), (17800, 14927), (18325, 32220), (18436, 9967), (18438, 6467), (18439, 12294), (18440, 6), (18525, 3751), (31670, 3629), (32244, 12041), (32260, 11806), (32272, 125), (32306, 11579), (36990, 576), (38733, 10754), (38744, 10753), (38756, 10759), (38768, 10758), (38780, 10764), (38790, 10752), (39871, 13292)
X(39899) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(11060)}} and {{A, B, C, X(69), X(1989)}}
X(39899) = Stammler circle-inverse of-X(35463)
X(39899) = crossdifference of every pair of points on line {X(526), X(2489)}
X(39899) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 1353, 5093), (6, 18440, 381), (6, 19130, 14848), (141, 11179, 12017), (141, 12017, 5054), (147, 9755, 37071), (1352, 5050, 1656), (1352, 8550, 5050), (1899, 3167, 30771), (3618, 18358, 5055), (3629, 31670, 1351), (5921, 14912, 5), (6193, 18914, 3), (11411, 31804, 3), (13665, 13785, 115), (14853, 39884, 3843), (18510, 26336, 381), (18512, 26346, 381), (39873, 39901, 999), (39893, 39894, 6), (39897, 39900, 3295)

X(39900) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO 9th BROCARD

Barycentrics    3*a^6-(3*b^2+2*b*c+3*c^2)*a^4+(b^4+c^4-2*(b^2+b*c+c^2)*b*c)*a^2-(b^4-c^4)*(b^2-c^2) : :
X(39900) = 3*X(10056)-2*X(12588)

The reciprocal orthologic center of these triangles is X(4).

X(39900) lies on these lines: {1,6776}, {3,39873}, {5,39892}, {6,1479}, {11,5050}, {12,18440}, {35,69}, {36,25406}, {55,3564}, {182,499}, {193,4294}, {388,39874}, {495,39891}, {497,14912}, {498,1352}, {511,4302}, {542,10053}, {611,1478}, {613,8550}, {895,12896}, {1351,6284}, {1353,15171}, {1428,10072}, {1469,4299}, {1570,9664}, {3056,4309}, {3085,5921}, {3295,39897}, {3299,39875}, {3301,39876}, {3583,14853}, {3584,11180}, {3618,7741}, {3751,10572}, {4304,34379}, {5010,10519}, {5028,9598}, {5034,9599}, {5093,9668}, {5119,5847}, {5135,12586}, {5268,14826}, {5274,33748}, {5310,6515}, {5433,12017}, {5848,10058}, {7727,11061}, {10037,39879}, {10038,39882}, {10039,39885}, {10040,39887}, {10041,39888}, {10079,13354}, {10483,14927}, {10523,39889}, {10801,39872}, {10895,39884}, {10896,18583}, {10954,39890}, {11393,39588}, {11398,39871}, {11507,39877}, {11579,32297}, {11877,39880}, {11878,39881}, {11912,39886}, {11951,39895}, {11952,39896}, {12950,34779}, {12953,21850}, {13904,19145}, {13962,19146}, {15069,31452}, {15338,33878}, {16475,30384}, {19369,31670}, {22766,39883}, {23708,38049}, {24248,29307}, {35808,39894}, {35809,39893}

X(39900) = midpoint of X(6776) and X(39902)
X(39900) = reflection of X(i) in X(j) for these (i,j): (1478, 611), (39891, 495)
X(39900) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6776, 39901), (182, 12589, 499), (1352, 2330, 498), (3295, 39899, 39897)

X(39901) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO 9th BROCARD

Barycentrics    3*a^6-(3*b^2-2*b*c+3*c^2)*a^4+(b^4+c^4+2*(b^2-b*c+c^2)*b*c)*a^2-(b^4-c^4)*(b^2-c^2) : :
X(39901) = 3*X(10072)-2*X(12589)

The reciprocal orthologic center of these triangles is X(4).

X(39901) lies on these lines: {1,6776}, {3,39897}, {5,39891}, {6,1478}, {11,18440}, {12,5050}, {35,25406}, {36,69}, {46,5847}, {56,3564}, {182,498}, {193,4293}, {388,14912}, {496,39892}, {497,39874}, {499,1352}, {511,4299}, {542,10069}, {611,8550}, {613,1479}, {895,18968}, {999,39873}, {1351,7354}, {1353,18990}, {1469,4317}, {1570,9651}, {1737,39885}, {2330,10056}, {3056,4302}, {3086,5921}, {3299,39876}, {3301,39875}, {3582,11180}, {3585,14853}, {3618,7951}, {4311,34379}, {5028,9597}, {5034,9596}, {5093,9655}, {5261,33748}, {5272,14826}, {5322,6515}, {5432,12017}, {5848,10074}, {5849,36741}, {7280,10519}, {8540,31670}, {10046,39879}, {10047,39882}, {10048,39887}, {10049,39888}, {10063,13354}, {10523,39890}, {10802,39872}, {10895,18583}, {10896,39884}, {10948,39889}, {11061,19470}, {11392,39588}, {11399,39871}, {11508,39877}, {11579,32243}, {11879,39880}, {11880,39881}, {11913,39886}, {11953,39895}, {11954,39896}, {12047,16475}, {12940,34779}, {12943,21850}, {13905,19145}, {13963,19146}, {15326,33878}, {22767,39883}, {35768,39894}, {35769,39893}, {37692,38049}

X(39901) = midpoint of X(6776) and X(39903)
X(39901) = reflection of X(i) in X(j) for these (i,j): (1479, 613), (39892, 496)
X(39901) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6776, 39900), (182, 12588, 498), (999, 39899, 39873), (1352, 1428, 499)

X(39902) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO 9th BROCARD

Barycentrics
3*a^9-3*(b+c)*a^8-2*(3*b^2-4*b*c+3*c^2)*a^7+2*(b+c)*(3*b^2-b*c+3*c^2)*a^6+4*(b^4+c^4-(3*b^2+b*c+3*c^2)*b*c)*a^5-2*(b+c)*(2*b^4+2*c^4-(3*b^2+2*b*c+3*c^2)*b*c)*a^4-2*(b^4+c^4-2*(b^2+b*c+c^2)*b*c)*(b-c)^2*a^3+2*(b^2-c^2)*(b-c)^2*(b^3-c^3)*a^2+(b^4-c^4)*(b^2-c^2)*(b^2-4*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(4).

X(39902) lies on these lines: {1,6776}, {6,10531}, {12,39889}, {69,11248}, {542,11239}, {1352,5552}, {1503,12115}, {2077,10519}, {3564,10679}, {5847,12703}, {5848,12775}, {5921,10528}, {6833,12586}, {6938,8679}, {10269,25406}, {10596,14912}, {10803,39872}, {10805,39874}, {10834,39879}, {10878,39882}, {10915,39885}, {10929,39887}, {10930,39888}, {10942,18440}, {10955,39890}, {10956,39891}, {10958,39892}, {10965,39897}, {11400,39871}, {11509,39873}, {11881,39880}, {11882,39881}, {11914,39886}, {11955,39895}, {11956,39896}, {12000,39899}, {13906,19145}, {13964,19146}, {14853,26333}, {18542,39884}, {19047,39875}, {19048,39876}, {22768,39883}, {35816,39893}, {35817,39894}

X(39902) = reflection of X(i) in X(j) for these (i,j): (6776, 39900), (12115, 12594)
X(39902) = {X(6776), X(39898)}-harmonic conjugate of X(39903)

X(39903) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO 9th BROCARD

Barycentrics
3*a^9-3*(b+c)*a^8-2*(3*b^2-2*b*c+3*c^2)*a^7+2*(b+c)*(3*b^2-5*b*c+3*c^2)*a^6+4*(b^4+3*b^2*c^2+c^4)*a^5-2*(b+c)*(2*b^2-b*c+c^2)*(b^2-b*c+2*c^2)*a^4-2*(b^2+c^2)*(b^2+4*b*c+c^2)*(b-c)^2*a^3+2*(b^2-c^2)*(b-c)*(b^4+c^4+3*(b^2+c^2)*b*c)*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(4).

X(39903) lies on these lines: {1,6776}, {6,10532}, {11,39890}, {69,11249}, {542,11240}, {674,6934}, {1352,10527}, {1503,12116}, {3564,10680}, {5847,12704}, {5848,12776}, {5921,10529}, {6834,12587}, {10267,25406}, {10519,11012}, {10597,14912}, {10804,39872}, {10806,39874}, {10835,39879}, {10879,39882}, {10916,39885}, {10931,39887}, {10932,39888}, {10943,18440}, {10949,39889}, {10957,39891}, {10959,39892}, {10966,39883}, {11401,39871}, {11510,39877}, {11883,39880}, {11884,39881}, {11915,39886}, {11957,39895}, {11958,39896}, {12001,39899}, {13907,19145}, {13965,19146}, {14853,26332}, {18544,39884}, {18967,39873}, {19049,39875}, {19050,39876}, {35818,39893}, {35819,39894}

X(39903) = reflection of X(i) in X(j) for these (i,j): (6776, 39901), (12116, 12595)
X(39903) = {X(6776), X(39898)}-harmonic conjugate of X(39902)

X(39904) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO 9th BROCARD

Barycentrics    (5*a^6-3*(b^2+c^2)*a^4+3*(b^4-b^2*c^2+c^4)*a^2-b^6-c^6)*(b^2-c^2) : :
X(39904) = 2*X(3569)-3*X(9185) = 4*X(5027)-3*X(9123) = 2*X(6333)-3*X(9168) = 3*X(9191)-4*X(24284)

The reciprocal parallelogic center of these triangles is X(4).

X(39904) lies on these lines: {99,110}, {351,3566}, {512,9979}, {523,32220}, {525,9131}, {888,15531}, {1499,1513}, {2780,14698}, {2793,6776}, {3309,13251}, {7664,38650}, {9191,24284}, {9208,13318}, {9811,28478}

X(39904) = reflection of X(i) in X(j) for these (i,j): (3268, 5652), (9131, 9135), (14698, 32313), (39905, 351)
X(39904) = crossdifference of every pair of points on line {X(3124), X(5028)}

X(39905) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO 9th BROCARD

Barycentrics    (a^6-5*(b^2+c^2)*a^4+(5*b^4+b^2*c^2+5*c^4)*a^2-b^6-c^6)*(b^2-c^2) : :
X(39905) = 5*X(3569)-2*X(14316) = 3*X(9123)-2*X(9135) = 3*X(9185)-4*X(9208) = 5*X(9979)-4*X(14316)

The reciprocal parallelogic center of these triangles is X(4).

X(39905) lies on these lines: {2,690}, {110,4235}, {297,525}, {351,3566}, {512,9131}, {526,18311}, {1499,8598}, {2780,32312}, {2793,11161}, {3309,13250}, {5027,13315}, {9810,28478}, {14270,35296}, {16230,37174}, {30514,32228}, {36790,36792}

X(39905) = reflection of X(i) in X(j) for these (i,j): (9979, 3569), (14698, 5653), (39904, 351)
X(39905) = barycentric product X(850)*X(35298)
X(39905) = trilinear product X(1577)*X(35298)
X(39905) = crossdifference of every pair of points on line {X(184), X(2502)}
X(39905) = crosssum of X(i) and X(j) for these {i,j}: {32, 9135}, {647, 5107}

X(39906) = X(2)-OF-7th BROCARD TRIANGLE

Barycentrics    (b^2+c^2)*a^6-(b^4+3*b^2*c^2+c^4)*a^4-(b^2-c^2)^2*b^2*c^2 : :
X(39906) = 2*X(182)+X(39910) = X(182)+2*X(39912) = X(39910)-4*X(39912)

X(39906) lies on these lines: {2,2782}, {30,568}, {39,37988}, {76,14096}, {98,37457}, {182,39910}, {184,1316}, {194,37190}, {237,11257}, {401,34396}, {427,1180}, {538,3917}, {566,7668}, {1899,2549}, {2450,5254}, {2790,6776}, {2979,32515}, {3094,20021}, {3095,14957}, {3148,39646}, {3410,9996}, {5063,9512}, {5112,13567}, {5422,35930}, {7739,11205}, {8584,11594}, {9755,35941}, {10169,14836}, {10796,34545}, {11007,26913}, {11442,37242}, {13351,34845}, {14880,37183}, {14912,32428}, {18335,36163}, {21531,32448}, {26881,37906}, {32429,36213}

X(39906) = {X(182), X(39912)}-harmonic conjugate of X(39910)

X(39907) = X(2)-OF-8th BROCARD TRIANGLE

Barycentrics
a^2*(6*a^10-18*(b^2+c^2)*a^8+(29*b^4+20*b^2*c^2+29*c^4)*a^6-9*(b^2+c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^4+(13*b^8+13*c^8-2*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^4+2*b^2*c^2-3*c^4)) : :

X(39907) lies on these lines: {3,2987}, {187,237}, {1976,5023}, {3167,9155}

X(39908) = X(2)-OF-9th BROCARD TRIANGLE

Barycentrics
a^12-8*(b^2+c^2)*a^10+9*(b^2+c^2)^2*a^8-32*(b^2+c^2)*b^2*c^2*a^6-(5*b^8+5*c^8-2*b^2*c^2*(14*b^4-15*b^2*c^2+14*c^4))*a^4+8*(b^4-c^4)*(b^2-c^2)^3*a^2-(b^2-c^2)^2*(5*b^8+5*c^8-2*b^2*c^2*(4*b^4-11*b^2*c^2+4*c^4)) : :
X(39908) = 2*X(39909)+X(39911)

X(39908) lies on these lines: {2,10256}, {4,287}, {297,30227}, {2966,7735}, {39909,39911}

X(39909) = X(3)-OF-9th BROCARD TRIANGLE

Barycentrics    2*S^6-2*(6*R^2*(SA-SW)+SW^2-SA^2+2*SB*SC)*S^4+(4*R^2-SW)*(SA^2-4*SB*SC)*SW*S^2+(4*R^2-SW)*SB*SC*SW^3 : :
X(39909) = 3*X(39908)-X(39911)

X(39909) lies on the line {39908,39911}

X(39910) = X(4)-OF-7th BROCARD TRIANGLE

Barycentrics
(b^2+c^2)*a^10-(3*b^4+5*b^2*c^2+3*c^4)*a^8+(b^2+c^2)*(3*b^4+2*b^2*c^2+3*c^4)*a^6-(b^8+c^8+2*(b^4-b^2*c^2+c^4)*b^2*c^2)*a^4+2*(b^4-c^4)*(b^2-c^2)*b^2*c^2*a^2-(b^2-c^2)^4*b^2*c^2 : :
X(39910) = 2*X(182)-3*X(39906) = 3*X(39906)-4*X(39912)

X(39910) lies on these lines: {5,13351}, {182,39906}, {1352,2549}, {5392,23195}

X(39910) = reflection of X(182) in X(39912)
X(39910) = {X(182), X(39912)}-harmonic conjugate of X(39906)

X(39911) = X(4)-OF-9th BROCARD TRIANGLE

Barycentrics    SC*SB*(S^2-(SB+SC)*SW)*(S^4+SA^2*(3*S^2-SW^2)) : :
X(39911) = 3*X(39908)-2*X(39909)

X(39911) lies on these lines: {5921,33971}, {6620,6776}, {39908,39909}

X(39912) = X(5)-OF-7th BROCARD TRIANGLE

Barycentrics    (b^2+c^2)*a^10-3*(b^2+c^2)^2*a^8+3*(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^6-(b^8+c^8+2*(b^2-c^2)^2*b^2*c^2)*a^4+2*(b^4-c^4)*(b^2-c^2)*b^2*c^2*a^2-(b^2-c^2)^4*b^2*c^2 : :
X(39912) = X(182)-3*X(39906) = 3*X(39906)+X(39910)

X(39912) lies on these lines: {39,39506}, {182,39906}, {2782,4045}, {6776,36181}

X(39912) = midpoint of X(182) and X(39910)
X(39912) = {X(39906), X(39910)}-harmonic conjugate of X(182)

X(39913) = X(6)-OF-7th BROCARD TRIANGLE

Barycentrics    a^2*(-a^2+b^2+c^2)*(a^6-2*(b^2+c^2)*a^4+3*(b^2-c^2)^2*a^2-2*(b^4-c^4)*(b^2-c^2)) : :

X(39913) lies on these lines: {6,1597}, {32,185}, {39,1181}, {115,1899}, {155,22401}, {184,574}, {187,10605}, {217,7772}, {232,11456}, {287,3734}, {577,13754}, {974,14901}, {1015,19354}, {1204,5206}, {1498,3199}, {1500,19349}, {1562,11648}, {1968,6241}, {1970,3357}, {1971,11438}, {2088,35901}, {2207,12174}, {2549,2794}, {3767,18909}, {5013,19347}, {5107,10602}, {5254,18914}, {5309,15341}, {5890,10311}, {6146,7748}, {6751,14917}, {7749,26937}, {7756,19467}, {7781,9289}, {8588,21663}, {9306,39849}, {9730,10314}, {10316,34783}, {10984,22416}, {12112,33885}, {13367,15515}, {13881,26944}, {14216,27371}, {14961,18445}, {18931,21843}, {19357,37512}

X(39913) = isogonal conjugate of the polar conjugate of X(26869)
X(39913) = barycentric product X(3)*X(26869)
X(39913) = trilinear product X(48)*X(26869)
X(39913) = crossdifference of every pair of points on line {X(9007), X(9979)}
X(39913) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (184, 3269, 574), (185, 39643, 32), (1204, 14585, 5206), (5890, 13509, 10311)

X(39914) = X(1)X(87)∩X(2)X(2319)

Barycentrics    (a^2 - b*c)*(a*b - a*c - b*c)*(a*b - a*c + b*c) : :

X(39914) lies on the cubics K739 and K770 and these lines: {1, 87}, {2, 2319}, {6, 17743}, {7, 7153}, {69, 27444}, {75, 24343}, {83, 21759}, {86, 16744}, {238, 385}, {239, 1966}, {335, 20363}, {668, 20669}, {932, 14665}, {940, 2162}, {1001, 2053}, {1016, 5383}, {1019, 4785}, {1429, 4366}, {1914, 27982}, {3403, 17117}, {3500, 3662}, {4383, 27439}, {4513, 27496}, {4687, 24679}, {9263, 23579}, {9452, 10025}, {17149, 23538}, {17277, 25107}, {17291, 26959}, {17300, 27436}, {17349, 27430}, {17375, 27465}, {18139, 27458}, {18194, 23508}, {18830, 35172}, {20142, 27450}, {20179, 34249}, {21226, 22343}, {23417, 31008}, {27438, 32911}

X(39914) = reflection of X(i) in X(j) for these {i,j}: {33680, 20363}, {33681, 24502}
X(39914) = antitomic conjugate of X(10030)
X(39914) = X(34252)-Ceva conjugate of X(239)
X(39914) = X(350)-cross conjugate of X(239)
X(39914) = X(i)-isoconjugate of X(j) for these (i,j): {43, 292}, {192, 1911}, {291, 2176}, {335, 2209}, {660, 20979}, {741, 20691}, {813, 4083}, {875, 4595}, {1403, 4876}, {1423, 7077}, {1922, 6376}, {1967, 17752}, {3835, 34067}, {3971, 18268}, {4562, 8640}, {5378, 6377}, {6382, 14598}, {18265, 30545}, {21760, 33680}
X(39914) = crosspoint of X(18299) and X(32020)
X(39914) = crosssum of X(18758) and X(21760)
X(39914) = trilinear pole of line {659, 4107}
X(39914) = crossdifference of every pair of points on line {20667, 20979}
X(39914) = barycentric product X(i)*X(j) for these {i,j}: {75, 34252}, {87, 350}, {238, 6384}, {239, 330}, {385, 27447}, {659, 18830}, {812, 4598}, {932, 3766}, {1429, 27424}, {1447, 7155}, {1914, 6383}, {1921, 2162}, {2053, 18033}, {2319, 10030}, {3684, 7209}, {3975, 7153}, {5383, 27918}, {7121, 18891}, {8843, 18032}, {16606, 30940}
X(39914) = barycentric quotient X(i)/X(j) for these {i,j}: {87, 291}, {238, 43}, {239, 192}, {330, 335}, {350, 6376}, {385, 17752}, {659, 4083}, {740, 3971}, {812, 3835}, {874, 36863}, {932, 660}, {1428, 1403}, {1429, 1423}, {1447, 3212}, {1914, 2176}, {1921, 6382}, {2053, 7077}, {2162, 292}, {2210, 2209}, {2238, 20691}, {2319, 4876}, {3226, 33680}, {3570, 4595}, {3684, 3208}, {3685, 27538}, {3716, 4147}, {3766, 20906}, {3975, 4110}, {4010, 21051}, {4164, 24533}, {4598, 4562}, {4974, 4970}, {5009, 38832}, {6383, 18895}, {6384, 334}, {7121, 1911}, {7155, 4518}, {7193, 20760}, {8632, 20979}, {8843, 1757}, {8848, 18788}, {10030, 30545}, {14199, 32937}, {15373, 2196}, {18830, 4583}, {20769, 22370}, {21832, 21834}, {22384, 22090}, {23086, 295}, {27447, 1916}, {27846, 3123}, {27918, 21138}, {30940, 31008}, {33295, 33296}, {33891, 33890}, {34071, 813}, {34252, 1}

X(39915) = X(1)X(2668)∩X(2)X(2669)

Barycentrics    (a + b)*(a + c)*(a^2*b^2 + a^2*b*c + a*b^2*c + a^2*c^2 + a*b*c^2 - b^2*c^2) : :

X(39915) lies on the cubic K770 and these lines: {1, 2668}, {2, 2669}, {38, 3571}, {42, 799}, {81, 1977}, {86, 310}, {99, 171}, {238, 7304}, {239, 257}, {274, 9401}, {335, 20362}, {354, 18827}, {385, 38814}, {940, 1975}, {1045, 34021}, {1509, 4038}, {1655, 21779}, {3736, 7168}, {3995, 4576}, {5208, 8822}, {7192, 27804}, {17149, 34022}, {18169, 30940}, {30950, 33779}, {32853, 34016}

X(39915) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 86}, {8033, 333}
X(39915) = X(1655)-cross conjugate of X(34021)
X(39915) = X(1918)-isoconjugate of X(18298)
X(39915) = cevapoint of X(1045) and X(1655)
X(39915) = crosspoint of X(1) and X(1045)
X(39915) = barycentric product X(i)*X(j) for these {i,j}: {1, 34021}, {86, 1655}, {274, 1045}, {310, 21779}, {873, 21883}, {6385, 18756}
X(39915) = barycentric quotient X(i)/X(j) for these {i,j}: {274, 18298}, {1045, 37}, {1655, 10}, {9402, 4079}, {18756, 213}, {21779, 42}, {21883, 756}, {23079, 71}, {27890, 3963}, {34021, 75}
X(39915) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 8033, 2668}, {310, 10458, 86}, {873, 3720, 86}

X(39916) = X(1)X(1655)∩X(2)X(24578)

Barycentrics    (a^2 - b*c)*(a^2*b^2 + a^2*b*c - a*b^2*c + a^2*c^2 - a*b*c^2 - b^2*c^2) : :

X(39916) lies on the cubic K770 and these lines: {1, 1655}, {2, 24578}, {37, 86}, {192, 869}, {194, 19582}, {239, 350}, {257, 3057}, {385, 3747}, {870, 4713}, {1281, 2292}, {1959, 10025}, {2664, 17759}, {2667, 17319}, {3159, 4568}, {3226, 25426}, {3802, 17793}, {4366, 33701}, {4375, 21385}, {5283, 18061}, {9278, 25368}, {9359, 17379}, {12263, 16823}, {16369, 33295}, {16815, 21264}, {16819, 17761}, {16827, 28660}, {17248, 26045}, {17461, 24428}, {18277, 27853}, {20331, 29578}, {27912, 27924}

X(39916) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 239}, {4368, 6651}
X(39916) = X(i)-isoconjugate of X(j) for these (i,j): {292, 2665}, {2107, 37128}
X(39916) = crosspoint of X(i) and X(j) for these (i,j): {1, 2664}, {3570, 4601}
X(39916) = crosssum of X(i) and X(j) for these (i,j): {1, 2665}, {3121, 3572}
X(39916) = crossdifference of every pair of points on line {875, 2107}
X(39916) = barycentric product X(i)*X(j) for these {i,j}: {1, 39028}, {190, 27854}, {239, 17759}, {350, 2664}, {740, 2669}, {1921, 21788}, {2106, 3948}, {21897, 30940}
X(39916) = barycentric quotient X(i)/X(j) for these {i,j}: {238, 2665}, {2106, 37128}, {2664, 291}, {2669, 18827}, {3747, 2107}, {17759, 335}, {20796, 295}, {21788, 292}, {27854, 514}, {38978, 3122}, {39028, 75}
X(39916) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {350, 16514, 239}, {4465, 17475, 39044}, {4713, 16515, 870}, {17475, 39044, 239}

X(39917) = X(1)-CEVA CONJUGATE OF X(257)

Barycentrics    (b^2 + a*c)*(a*b + c^2)*(a^3*b^3 + 2*a^4*b*c + a^2*b^2*c^2 + a^3*c^3 - b^3*c^3) : :

X(39917) lies on the cubic K770 and these lines: {86, 16744}, {256, 26801}, {257, 350}, {904, 4366}, {1432, 39775}, {3903, 17752}, {20691, 27805}

X(39917) = X(1)-Ceva conjugate of X(257)
X(39917) = X(16360)-isoconjugate of X(30657)
X(39917) = crosspoint of X(1) and X(18754)
X(39917) = barycentric product X(i)*X(j) for these {i,j}: {257, 30661}, {7018, 18754}
X(39917) = barycentric quotient X(i)/X(j) for these {i,j}: {16362, 18787}, {18754, 171}, {30661, 894}

X(39918) = X(1)-CEVA CONJUGATE OF X(335)

Barycentrics    (b^2 - a*c)*(a*b - c^2)*(a^3*b^3 - 2*a^4*b*c + a^2*b^2*c^2 + a^3*c^3 - b^3*c^3) : :

X(39918) lies on the cubic K770 and these lines: {1, 18795}, {2, 3252}, {239, 292}, {291, 740}, {335, 350}, {385, 1911}, {1914, 37207}, {2106, 4589}

X(39918) = X(1)-Ceva conjugate of X(335)
X(39918) = X(i)-isoconjugate of X(j) for these (i,j): {893, 16361}, {1914, 18795}
X(39918) = crosspoint of X(1) and X(18794)
X(39918) = crosssum of X(1) and X(18795)
X(39918) = barycentric product X(i)*X(j) for these {i,j}: {334, 18794}, {335, 30667}
X(39918) = barycentric quotient X(i)/X(j) for these {i,j}: {171, 16361}, {291, 18795}, {16363, 18786}, {18794, 238}, {30667, 239}

X(39919) = X(1)-CEVA CONJUGATE OF X(39775)

Barycentrics    (a + b - c)*(a - b + c)*(a^2 - b*c)*(a^2*b^2 - a*b^3 + b^3*c + a^2*c^2 - a*c^3 + b*c^3) : :

X(39919) lies on the cubic K770 and these lines: {1, 18299}, {2, 257}, {85, 30982}, {348, 982}, {350, 18033}, {693, 3907}, {1429, 4366}, {4554, 19567}, {7168, 33674}, {9312, 20917}, {9436, 20358}, {17760, 28391}, {20935, 24524}

X(39919) = X(1)-Ceva conjugate of X(39775)
X(39919) = X(18265)-isoconjugate of X(18299)
X(39919) = barycentric product X(i)*X(j) for these {i,j}: {85, 8844}, {350, 28391}, {1447, 17760}, {10030, 17792}
X(39919) = barycentric quotient X(i)/X(j) for these {i,j}: {8844, 9}, {10030, 18299}, {17760, 4518}, {17792, 4876}, {28391, 291}

X(39920) = X(1)X(18760)∩X(2)X(7061)

Barycentrics    (a^3 + b^3 - a*b*c - c^3)*(a^3 - b^3 - a*b*c + c^3)*(a^4 - a*b^3 - a^2*b*c - 2*b^2*c^2 - a*c^3) : :

X(39920) lies on the cubic K770 and these lines: {1, 18760}, {2, 7061}, {86, 39775}, {257, 3512}, {350, 18036}, {4357, 7261}, {7281, 17793}

X(39920) = X(i)-isoconjugate of X(j) for these (i,j): {3509, 18784}, {18262, 18760}
X(39920) = barycentric product X(i)*X(j) for these {i,j}: {3512, 30660}, {7261, 17739}
X(39920) = barycentric quotient X(i)/X(j) for these {i,j}: {8424, 3509}, {8852, 18784}, {17739, 4645}, {18759, 19554}, {30660, 17789}

X(39921) = X(1)X(148)∩X(2)X(9278)

Barycentrics    (a^2 - a*b - b^2 - a*c - b*c - c^2)*(a^2 + a*b + b^2 - a*c - b*c - c^2)*(a^2 - a*b - b^2 + a*c - b*c + c^2) : :

X(39921) lies on the cubic K770 and these lines: {1, 148}, {2, 9278}, {86, 1086}, {239, 7261}, {257, 17793}, {350, 18032}, {1125, 1929}, {2702, 5195}, {6542, 6543}, {8044, 21221}, {20072, 37135}, {21295, 28604}

X(39921) = X(11599)-Ceva conjugate of X(6650)
X(39921) = X(i)-isoconjugate of X(j) for these (i,j): {1757, 2248}, {6542, 18757}, {6625, 18266}, {13610, 17735}
X(39921) = crossdifference of every pair of points on line {17990, 20668}
X(39921) = barycentric product X(i)*X(j) for these {i,j}: {846, 18032}, {1654, 6650}, {1929, 17762}, {6626, 11599}, {21196, 35148}
X(39921) = barycentric quotient X(i)/X(j) for these {i,j}: {846, 1757}, {1654, 6542}, {1929, 13610}, {2905, 423}, {4213, 17927}, {6626, 17731}, {6650, 6625}, {17762, 20947}, {17962, 2248}, {18755, 17735}, {21085, 6541}, {21196, 2786}, {21879, 20693}, {22139, 17976}, {38814, 1931}

X(39922) = X(1)X(148)∩X(2)X(7061)

Barycentrics    (a^2 - b*c)*(a^2 + a*b - b^2 + a*c - b*c - c^2)*(a^2 + a*b + b^2 + a*c + b*c - c^2)*(a^2 + a*b - b^2 + a*c + b*c + c^2) : :

X(39922) lies on the cubic K770 and these lines: {1, 148}, {2, 7061}, {192, 13174}, {335, 20362}, {385, 740}, {4037, 27982}

X(39922) = X(i)-isoconjugate of X(j) for these (i,j): {846, 9506}, {9505, 18755}, {17762, 18263}
X(39922) = barycentric product X(i)*X(j) for these {i,j}: {2248, 18035}, {6625, 6651}
X(39922) = barycentric quotient X(i)/X(j) for these {i,j}: {2248, 9506}, {6651, 1654}, {8298, 846}, {13610, 9505}, {27929, 21196}

X(39923) = X(2)X(210)∩X(726)X(32041)

Barycentrics    (a^2 - b*c)*(a*b - b^2 + 2*a*c + b*c)*(2*a*b + a*c + b*c - c^2)*(a^2*b - a*b^2 + a^2*c + a*b*c + b^2*c - a*c^2 + b*c^2) : :

X(39923) lies on the cubic K770 and these lines: {2, 210}, {726, 32041}, {740, 10030}, {2279, 3508}, {3226, 33674}, {32922, 37138}

X(39924) = X(1)X(18299)∩X(55)X(192)

Barycentrics    (a - b - c)*(a^3*b + a*b^3 - a^3*c - b^3*c + a^2*c^2 + b^2*c^2)*(a^3*b - a^2*b^2 - a^3*c - b^2*c^2 - a*c^3 + b*c^3) : :

X(39924) lies on the cubic K770 and these lines: {1, 18299}, {33, 24307}, {55, 192}, {200, 4110}, {220, 27538}, {335, 20359}, {740, 7220}, {7281, 17793}, {14942, 19581}

X(39924) = X(350)-cross conjugate of X(14942)
X(39924) = X(i)-isoconjugate of X(j) for these (i,j): {6, 28391}, {7, 18758}, {56, 17792}, {604, 17760}, {18269, 30545}
X(39924) = cevapoint of X(1) and X(24728)
X(39924) = trilinear pole of line {657, 4147}
X(39924) = barycentric product X(9)*X(18299)
X(39924) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 28391}, {8, 17760}, {9, 17792}, {41, 18758}, {3684, 8844}, {7155, 27436}, {18299, 85}

X(39925) = X(1)X(1655)∩X(2)X(3121)

Barycentrics    (a^2*b^2 + a^2*b*c + a*b^2*c - a^2*c^2 - a*b*c^2 - b^2*c^2)*(a^2*b^2 - a^2*b*c + a*b^2*c - a^2*c^2 - a*b*c^2 + b^2*c^2) : :
X(39925) = 2 X[17793] - 3 X[30571], 5 X[27195] - 4 X[36812]

X(39925) lies on the conic {{A,B,C,X(12),X(2)}}, the cubic K770, and these lines: {1, 1655}, {2, 3121}, {57, 17027}, {81, 1977}, {88, 17029}, {105, 385}, {192, 1002}, {274, 1015}, {291, 740}, {330, 4441}, {538, 3227}, {668, 16589}, {1432, 39775}, {1929, 8937}, {3122, 25054}, {3720, 9401}, {8056, 17026}, {8682, 17946}, {13576, 30669}, {17032, 25430}, {17034, 39797}, {17493, 30941}, {17793, 30571}, {21223, 39694}, {27195, 36812}, {31298, 39738}

X(39925) = midpoint of X(1655) and X(9263)
X(39925) = reflection of X(i) in X(j) for these {i,j}: {274, 1015}, {668, 16589}
X(39925) = isogonal conjugate of X(21788)
X(39925) = isotomic conjugate of X(17759)
X(39925) = anticomplement of X(39028)
X(39925) = antitomic conjugate of X(274)
X(39925) = isotomic conjugate of the anticomplement of X(350)
X(39925) = X(2665)-anticomplementary conjugate of X(20554)
X(39925) = X(i)-cross conjugate of X(j) for these (i,j): {350, 2}, {18827, 6650}
X(39925) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21788}, {6, 2664}, {19, 20796}, {31, 17759}, {42, 2106}, {58, 21897}, {71, 15148}, {213, 2669}, {1922, 39028}, {27854, 34067}
X(39925) = cevapoint of X(i) and X(j) for these (i,j): {1, 24727}, {514, 39786}, {740, 16589}, {812, 1015}
X(39925) = trilinear pole of line {513, 3739}
X(39925) = barycentric product X(i)*X(j) for these {i,j}: {75, 2665}, {310, 2107}
X(39925) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2664}, {2, 17759}, {3, 20796}, {6, 21788}, {28, 15148}, {37, 21897}, {81, 2106}, {86, 2669}, {350, 39028}, {812, 27854}, {2107, 42}, {2665, 1}, {8934, 1282}, {8937, 846}, {24727, 39056}

X(39926) = X(1)X(2668)∩X(385)X(3747)

Barycentrics    (b + c)*(-a^2 + b*c)*(a^2*b^2 + a^2*b*c + a*b^2*c - a^2*c^2 + a*b*c^2 + b^2*c^2)*(-(a^2*b^2) + a^2*b*c + a*b^2*c + a^2*c^2 + a*b*c^2 + b^2*c^2) : :
X(39926) = 2 X[3229] - 3 X[10180]

X(39926) lies on the cubic K770 and these lines: {1, 2668}, {385, 3747}, {512, 4369}, {740, 3978}, {1215, 1500}, {3229, 10180}, {5209, 7315}

X(39926) = X(350)-cross conjugate of X(740)
X(39926) = X(i)-isoconjugate of X(j) for these (i,j): {741, 1045}, {1655, 18268}, {1922, 34021}, {9402, 36066}, {18756, 18827}, {21779, 37128}
X(39926) = barycentric product X(2238)*X(18298)
X(39926) = barycentric quotient X(i)/X(j) for these {i,j}: {350, 34021}, {740, 1655}, {2238, 1045}, {3747, 21779}

X(39927) = X(2)X(3504)∩X(6)X(194)

Barycentrics    (a^2 - b*c)*(a^2 + b*c)*(a^2*b^2 - a^2*c^2 - b^2*c^2)*(a^2*b^2 - a^2*c^2 + b^2*c^2) : :

X(39927) lies on the cubics K739, K757, K777, and these lines: {2, 3504}, {6, 194}, {182, 15389}, {239, 14199}, {287, 34238}, {385, 19590}, {699, 35524}, {1691, 16985}, {1916, 8858}, {3222, 5970}, {3407, 24733}, {7793, 32746}, {14041, 18105}, {17752, 20964}

X(39927) = antitomic conjugate of X(17984)
X(39927) = X(3978)-cross conjugate of X(385)
X(39927) = X(i)-isoconjugate of X(j) for these (i,j): {194, 1967}, {694, 1740}, {1581, 1613}, {1927, 6374}, {3221, 37134}, {8789, 18837}, {9468, 17149}, {17938, 20910}, {18829, 23503}
X(39927) = barycentric product X(i)*X(j) for these {i,j}: {385, 2998}, {804, 3222}, {1580, 18832}, {1926, 34248}, {1966, 3223}, {3224, 3978}, {3504, 17984}
X(39927) = barycentric quotient X(i)/X(j) for these {i,j}: {385, 194}, {419, 3186}, {804, 23301}, {1580, 1740}, {1691, 1613}, {1926, 18837}, {1966, 17149}, {2998, 1916}, {3222, 18829}, {3223, 1581}, {3224, 694}, {3504, 36214}, {3978, 6374}, {4039, 21080}, {4107, 21191}, {5027, 3221}, {14296, 23807}, {15389, 17970}, {18832, 1934}, {34248, 1967}

X(39928) = X(2)X(3510)∩X(6)X(19567)

Barycentrics    (a^2 + b*c)*(a^3*b^3 + a^2*b^2*c^2 + a^3*c^3 - b^3*c^3) : :

X(39928) lies on the cubic K739 and these lines: {2, 3510}, {6, 19567}, {83, 21759}, {194, 8866}, {239, 3778}, {789, 7104}, {894, 3978}, {1916, 8857}, {2082, 17033}, {4027, 7122}, {17121, 20457}, {17752, 20964}, {19579, 21751}, {24732, 34251}, {27996, 27998}

X(39928) = X(6)-Ceva conjugate of X(894)
X(39928) = X(893)-isoconjugate of X(7346)
X(39928) = crosspoint of X(6) and X(34251)
X(39928) = barycentric product X(i)*X(j) for these {i,j}: {171, 24732}, {1909, 6196}, {1920, 34251}
X(39928) = barycentric quotient X(i)/X(j) for these {i,j}: {171, 7346}, {6196, 256}, {23192, 7015}, {24732, 7018}, {34251, 893}

X(39929) = X(2)X(3508)∩X(6)X(7033)

Barycentrics    (a^2 + b*c)*(a^2*b^2 + a^2*b*c - a*b^2*c + a^2*c^2 - a*b*c^2 + b^2*c^2) : :

X(39929) lies on the cubic K739 and these lines: {2, 3508}, {6, 7033}, {42, 19579}, {43, 3729}, {83, 213}, {171, 385}, {190, 893}, {194, 869}, {3503, 26752}, {3507, 17499}, {3978, 17752}, {17120, 21757}, {17261, 21838}, {17350, 20665}

X(39929) = X(i)-Ceva conjugate of X(j) for these (i,j): {6, 17752}, {7033, 7081}
X(39929) = X(i)-isoconjugate of X(j) for these (i,j): {904, 39746}, {1431, 3495}
X(39929) = barycentric product X(i)*X(j) for these {i,j}: {894, 26752}, {3503, 17787}
X(39929) = barycentric quotient X(i)/X(j) for these {i,j}: {894, 39746}, {2329, 3495}, {3503, 1432}, {26752, 257}

X(39930) = X(2)X(7)∩X(3)X(30546)

Barycentrics    (a + b - c)*(a - b + c)*(a^2 - b*c)*(a^2*b - a*b^2 + a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2) : :

X(39930) lies on the cubic K739 and these lines: {2, 7}, {3, 30546}, {6, 30545}, {194, 8862}, {241, 337}, {350, 34253}, {1275, 5383}, {1458, 17794}, {2238, 10030}, {2269, 30547}, {3169, 20537}, {3674, 17499}, {3985, 39775}, {6063, 34283}

X(39930) = X(3500)-isoconjugate of X(7077)
X(39930) = crossdifference of every pair of points on line {663, 20460}
X(39930) = barycentric product X(i)*X(j) for these {i,j}: {1429, 17786}, {1447, 32937}, {3212, 14199}, {3501, 10030}, {18033, 34247}
X(39930) = barycentric quotient X(i)/X(j) for these {i,j}: {1429, 3500}, {3501, 4876}, {14199, 7155}, {32937, 4518}, {34247, 7077}

X(39931) = X(2)X(9475)∩X(4)X(147)

Barycentrics    (a^2 - b*c)*(a^2 + b*c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4) : :

X(39931) lies on the Feuerbach circumhyperbola of the orthic triangle, the cubics K718 and K739, and these lines:{2, 9475}, {4, 147}, {6, 264}, {25, 3511}, {52, 7762}, {76, 9419}, {127, 6656}, {155, 7754}, {193, 3186}, {232, 297}, {385, 419}, {732, 17984}, {736, 33874}, {1594, 33548}, {1843, 27377}, {1858, 11988}, {2996, 15751}, {3164, 37190}, {3398, 37124}, {5117, 7777}, {5895, 9289}, {6103, 7792}, {6392, 32605}, {7778, 11331}, {8743, 10349}, {9766, 35142}, {12824, 33294}, {14295, 24284}, {14382, 32542}, {14957, 38368}, {35002, 35474}

X(39931) = polar-circle inverse of X(38947)
X(39931) = isogonal conjugate of X(15391)
X(39931) = polar conjugate of X(36897)
X(39931) = polar conjugate of the isotomic conjugate of X(5976)
X(39931) = polar conjugate of the isogonal conjugate of X(36213)
X(39931) = orthic-isogonal conjugate of X(297)
X(39931) = X(4)-Ceva conjugate of X(297)
X(39931) = X(36213)-cross conjugate of X(5976)
X(39931) = X(i)-isoconjugate of X(j) for these (i,j): {1, 15391}, {48, 36897}, {63, 34238}, {248, 1581}, {287, 1967}, {293, 694}, {336, 9468}, {810, 39291}, {878, 37134}, {1821, 17970}, {1910, 36214}, {1934, 14600}
X(39931) = crosspoint of X(4) and X(419)
X(39931) = crosssum of X(3) and X(36214)
X(39931) = crossdifference of every pair of points on line {878, 17970}
X(39931) = X(9442)-of-orthic-triangle if ABC is acute
X(39931) = barycentric product X(i)*X(j) for these {i,j}: {4, 5976}, {232, 3978}, {240, 1966}, {264, 36213}, {297, 385}, {325, 419}, {511, 17984}, {804, 877}, {880, 17994}, {2211, 14603}, {2967, 14382}, {4230, 14295}, {6530, 12215}, {16230, 17941}
X(39931) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 36897}, {6, 15391}, {25, 34238}, {232, 694}, {237, 17970}, {240, 1581}, {297, 1916}, {385, 287}, {419, 98}, {511, 36214}, {648, 39291}, {685, 18858}, {804, 879}, {877, 18829}, {1580, 293}, {1691, 248}, {1966, 336}, {2211, 9468}, {2679, 20975}, {4230, 805}, {5027, 878}, {5976, 69}, {12215, 6394}, {14602, 14600}, {17941, 17932}, {17984, 290}, {17994, 882}, {34854, 17980}, {36213, 3}
{X(7774),X(39355)}-harmonic conjugate of X(36214)

X(39932) = X(2)X(22252)∩X(6)X(19590)

Barycentrics    a^2*(a^6*b^6 + a^4*b^4*c^4 - a^6*c^6 - b^6*c^6)*(a^6*b^6 - a^4*b^4*c^4 - a^6*c^6 + b^6*c^6) : :

X(39932) lies on the cubic K739 and these lines: {2, 22252}, {6, 19590}, {194, 3499}, {1613, 23173}, {8789, 16985}, {11325, 33786}, {19579, 21751}, {38817, 39080}

X(39932) = isogonal conjugate of X(19566)
X(39932) = isotomic conjugate of X(19573)
X(39932) = antitomic conjugate of X(695)
X(39932) = isogonal conjugate of the anticomplement of X(14603)
X(39932) = X(3978)-cross conjugate of X(6)
X(39932) = X(i)-isoconjugate of X(j) for these (i,j): {1, 19566}, {2, 18272}, {6, 18271}, {31, 19573}, {32, 18276}, {694, 18270}, {1581, 19585}, {1966, 8871}, {1967, 19590}, {9468, 18273}
X(39932) = barycentric product X(695)*X(22252)
X(39932) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 18271}, {2, 19573}, {6, 19566}, {31, 18272}, {75, 18276}, {385, 19590}, {1580, 18270}, {1691, 19585}, {1966, 18273}, {9468, 8871}, {22252, 9230}

X(39933) = X(2)X(7168)∩X(6)X(19579)

Barycentrics    (a^2 + b*c)*(a^3*b^3 + a^2*b^2*c^2 - a^3*c^3 - b^3*c^3)*(a^3*b^3 - a^2*b^2*c^2 - a^3*c^3 + b^3*c^3) : :

X(39933) lies on the cubics K323 and K739 and on these lines: {2, 7168}, {6, 19579}, {194, 3212}, {239, 1967}, {894, 19590}, {1575, 2669}, {1916, 16364}, {4027, 7122}, {17752, 38382}

X(39933) = antitomic conjugate of X(1909)
X(39933) = X(i)-cross conjugate of X(j) for these (i,j): {291, 7061}, {3978, 894}
X(39933) = X(i)-isoconjugate of X(j) for these (i,j): {256, 18278}, {694, 19580}, {893, 3510}, {904, 19565}, {1581, 18274}, {1916, 30634}, {1927, 18277}, {1967, 19579}, {7104, 19567}, {9468, 19581}
X(39933) = barycentric product X(i)*X(j) for these {i,j}: {1580, 30633}, {1909, 7168}, {1966, 24576}
X(39933) = barycentric quotient X(i)/X(j) for these {i,j}: {171, 3510}, {172, 18278}, {385, 19579}, {894, 19565}, {1580, 19580}, {1691, 18274}, {1909, 19567}, {1920, 18275}, {1933, 30634}, {1966, 19581}, {3955, 23186}, {3978, 18277}, {7168, 256}, {24576, 1581}, {30633, 1934}

X(39934) = X(6)X(19567)∩X(194)X(869)

Barycentrics    (a^2 - b*c)*(a^3*b^3 - a^2*b^2*c^2 - a^3*c^3 - b^3*c^3)*(a^3*b^3 + a^2*b^2*c^2 - a^3*c^3 + b^3*c^3) : :

X(39934) lies on the cubic K739 and these lines: {6, 19567}, {194, 869}, {239, 19590}, {894, 9288}, {2210, 16985}, {17752, 38817}

X(39934) = X(3978)-cross conjugate of X(239)
X(39934) = X(i)-isoconjugate of X(j) for these (i,j): {291, 34251}, {292, 6196}, {1922, 24732}
X(39934) = barycentric product X(350)*X(7346)
X(39934) = barycentric quotient X(i)/X(j) for these {i,j}: {238, 6196}, {350, 24732}, {1914, 34251}, {7193, 23192}, {7346, 291}

X(39935) = X(6)X(14251)∩X(194)X(8871)

Barycentrics    (b^2 - a*c)*(b^2 + a*c)*(a*b - c^2)*(a*b + c^2)*(a^6*b^6 - 2*a^8*b^2*c^2 + a^4*b^4*c^4 + a^6*c^6 - b^6*c^6) : :

X(39935) lies on the cubic K739 and these lines: {2, 14251}, {194, 8871}, {239, 1967}, {287, 34238}, {385, 3225}, {694, 732}, {1916, 3978}, {8789, 16985}

X(39935) = X(6)-Ceva conjugate of X(1916)

X(39936) = X(6)X(30545)∩X(63)X(194)

Barycentrics    (a - b - c)*(a^2 + b*c)*(a^2*b - a*b^2 - a^2*c + a*b*c - b^2*c - a*c^2 + b*c^2)*(a^2*b + a*b^2 - a^2*c - a*b*c - b^2*c + a*c^2 + b*c^2) : :

X(39936) lies on the cubic K739 and these lines: {6, 30545}, {63, 194}, {287, 19579}, {1916, 23438}, {1943, 3512}, {16985, 17752}, {23522, 31008}

X(39936) = X(i)-isoconjugate of X(j) for these (i,j): {1431, 3501}, {1432, 34247}, {21348, 29055}, {23655, 37137}
X(39936) = barycentric product X(3500)*X(17787)
X(39936) = barycentric quotient X(i)/X(j) for these {i,j}: {2329, 3501}, {2330, 34247}, {3287, 21348}, {3500, 1432}, {3907, 17072}, {4140, 21958}, {4459, 23772}, {7081, 32937}, {17787, 17786}

X(39937) = X(2)X(3510)∩X(6)X(7033)

Barycentrics    (a^2 - b*c)*(a^2*b^2 - a^2*b*c + a*b^2*c + a^2*c^2 - a*b*c^2 + b^2*c^2)*(a^2*b^2 - a^2*b*c - a*b^2*c + a^2*c^2 + a*b*c^2 + b^2*c^2) : :

X(39937) lies on the cubic K739 and these lines: {2, 3510}, {6, 7033}, {194, 3495}, {239, 2236}, {350, 732}, {894, 3112}, {1107, 39717}, {16696, 32010}, {27807, 28596}

X(39937) = X(i)-isoconjugate of X(j) for these (i,j): {1911, 26752}, {3503, 7077}
X(39937) = barycentric product X(i)*X(j) for these {i,j}: {239, 39746}, {3495, 10030}
X(39937) = barycentric quotient X(i)/X(j) for these {i,j}: {239, 26752}, {1429, 3503}, {3495, 4876}, {39746, 335}

X(39938) = X(6)X(1031)∩X(83)X(115)

Barycentrics    (a^4 - a^2*b^2 - b^4 - a^2*c^2 - b^2*c^2 - c^4)*(a^4 + a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 - c^4)*(a^4 - a^2*b^2 - b^4 + a^2*c^2 - b^2*c^2 + c^4) : :

X(39938) lies on the cubic K739 and these lines: {6, 1031}, {83, 115}, {384, 9484}, {385, 21536}, {7779, 15573}, {9229, 39080}

X(39938) = X(17949)-Ceva conjugate of X(11606)
X(39938) = X(i)-isoconjugate of X(j) for these (i,j): {2076, 39725}, {14370, 17799}, {21355, 34054}
X(39938) = barycentric product X(2896)*X(11606)
X(39938) = barycentric quotient X(i)/X(j) for these {i,j}: {2896, 7779}, {10329, 2076}, {11606, 1031}, {16556, 17799}, {17949, 33665}

X(39939) = X(2)X(19590)∩X(6)X(19585)

Barycentrics    (a^4*b^4 + a^4*b^2*c^2 + a^2*b^4*c^2 - a^4*c^4 - a^2*b^2*c^4 - b^4*c^4)*(a^4*b^4 - a^4*b^2*c^2 + a^2*b^4*c^2 - a^4*c^4 - a^2*b^2*c^4 + b^4*c^4) : :

X(39939) lies on the conic {{A,B,C,X(2),X(6)}}, the cubic K739, and these lines: {2, 19590}, {6, 19585}, {37, 19581}, {42, 19579}, {194, 263}, {251, 4027}, {308, 1084}, {694, 732}, {702, 3228}, {1976, 16985}, {2998, 20023}, {9403, 17319}

X(39939) = reflection of X(308) in X(1084)
X(39939) = antitomic conjugate of X(308)
X(39939) = cyclocevian conjugate of X(34214)
X(39939) = anticomplement of center of hyperbola {{A,B,C,PU(11)}}
X(39939) = anticomplement of X(2)-Ceva conjugate of X(3978)
X(39939) = isotomic conjugate of the anticomplement of X(3978)
X(39939) = X(i)-cross conjugate of X(j) for these (i,j): {3978, 2}, {14970, 11606}
X(39939) = X(i)-isoconjugate of X(j) for these (i,j): {1755, 8870}, {1967, 38382}
X(39939) = cevapoint of X(i) and X(j) for these (i,j): {6, 24729}, {804, 1084}
X(39939) = trilinear pole of line {512, 3934}
X(39939) = barycentric quotient X(i)/X(j) for these {i,j}: {98, 8870}, {385, 38382}

X(39940) = X(194)X(3212)∩X(239)X(385)

Barycentrics    (a + b - c)*(a - b + c)*(a^2 - b*c)*(a^3*b^2 - a^2*b^3 + a^3*b*c - a^2*b^2*c + a*b^3*c + a^3*c^2 - a^2*b*c^2 + a*b^2*c^2 - b^3*c^2 - a^2*c^3 + a*b*c^3 - b^2*c^3) : :

X(39940) lies on the cubic K739 and these lines: {194, 3212}, {239, 385}, {287, 651}, {292, 664}, {732, 10030}, {1423, 3169}, {1432, 1916}, {5255, 28850}, {9312, 17754}, {30667, 36276}

X(39940) = X(7077)-isoconjugate of X(7167)
X(39940) = barycentric product X(3508)*X(10030)
X(39940) = barycentric quotient X(i)/X(j) for these {i,j}: {1429, 7167}, {3508, 4876}

X(39941) = X(6)X(14382)∩X(32)X(30226)

Barycentrics    (a^4 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - a^2*b^2 - b^2*c^2 + c^4)*(a^6*b^4 - a^4*b^6 + a^6*b^2*c^2 - a^4*b^4*c^2 - a^2*b^6*c^2 + a^6*c^4 - a^4*b^2*c^4 + a^2*b^4*c^4 + b^6*c^4 - a^4*c^6 - a^2*b^2*c^6 + b^4*c^6) : :

X(39941) lies on the cubic K739 and these lines: {6, 14382}, {32, 30226}, {98, 385}, {182, 32545}, {194, 8870}, {239, 1821}, {287, 3978}, {290, 732}, {1078, 34359}, {1691, 2966}, {1976, 16985}, {2211, 22456}, {5182, 23878}, {6531, 17984}, {8861, 10350}, {9218, 39652}, {10352, 36899}, {12203, 18338}, {13196, 34369}, {14265, 32451}

X(39941) = X(i)-Ceva conjugate of X(j) for these (i,j): {6, 287}, {14382, 98}
X(39941) = barycentric product X(i)*X(j) for these {i,j}: {98, 25332}, {290, 3511}, {14382, 39092}
X(39941) = barycentric quotient X(i)/X(j) for these {i,j}: {3511, 511}, {25332, 325}

leftri

Points on cubics: X(39942)-X(39984)

rightri

This preamble is contributed by Clark Kimberling, October 7, 2020.

Suppose that P = p(a,b,c) : p(b,c,a) : p(c,a,b) is an polynomial triangle center such that the point (b-c)p(a,b,c) : (c-a)p(b,c,a) : (a-b)p(c,a,b) is on the line at infinity; i.e.,

(b-c)p(a,b,c) + (c-a)p(b,c,a) + (a-b)p(c,a,b) = 0.

Let ζ(P) denote the cubic given by

(b+c)p(a,b,c)(cy-bz)(bx+ay)(cx+az) + (c+a)p(b,c,a)(az-cx)(cy+bz)(ay+bx) + (a+b)p(c,a,b)(bx-ay)(az+cx)(bz+cy) = 0.

Let A'B'C' denote the excentral triangle of ABC. It is easy to check that the following 9 points lie on ζ(P): A, B, C, A', B', C', X(1), X(2), X(6) .

It is also easy to check that if U is a point on ζ(P), then the isotomic conjugate of U also lies on ζ(P). This section consists of isogonal conjugates of points on cubics ζ(P) for selected points P. In column 1 of the following table, the appearance of u(a,b,c) in a row means that the points indicated in column 2 lie on the cubic

u(a,b,c)(cy-bz)(bx+ay)(cx+az) + u(b,c,a)(az-cx)(cy+bz)(a+bx) + u(c,a,b)(bx-ay)(az+cx)(bz+cy) = 0;

so that the point P is given by u(a,b,c) = (b+c)p(a,b,c). This is the cubic pK(X(6),P*), where P* = a*((b + c)*p - (a + c)*q - (a + b)*r) : :

The appearance of {i,j} in column 2 means that {X(i),X(j)} are a pair of isogonal conjugate points that lie on ζ(P).

b+c; (K102) {43,87}, {1944,3224} (and others)
(b+c)^2 {3216,39748}, {32911,39798}
a^2 (b+c)^2 {2350,17277}, {16552,39797}
(b+c)(b+c-a) {978,979}, {1743,8056}, 4383,39956}
a(b+c)(b+c-a)(b^2+c^2-a^2) {1722,39946}, {1723,39947}
(b+c)^2 cos^2 A {579,1751}, 586,39942}, {1708,39943}, {1713,39974}, (1714,39945}
(b+c)(2a-b-c) {44,88}, {37680,39982}
(b+c)(3a+b+c) {940,941},{3731,39987}
(b+c)(-3a+b+c) {3973,36603}, {21214,39969}, 37679,39975}
(b+c)(4a+b+c) {45,89}, {37633, 39974}
(b+c)(-4a+b+c) {16885,26745}, {26745,16885}, {37687,39984}
(b+c)(3a+2b+2c) {3247,39948}, {37685,39983}
(b+c)(2a+3b+3c) {1100,1255}
(b+c)(-2a+3b+3c) {16669,39962}
(b+c)(a+2b+2c) {1449,25430}
(b+c)(-a+2b+2c) {14997,39960}, {16670,39963}
(b+c)(a^2+b^2+c^2+a(a+b+c)) {984,985}, {5276,39957}
(b+c)(a^2+b^2+c^2-a(a+b+c)) {238,291}, {33854,39979}
(b+c)(2(a^2+b^2+c^2)+a(a+b+c)) {7174,39958}
(b+c)(2(a^2+b^2+c^2)-a(a+b+c)) {7290,39959}
(b+c)(bc+ca+ab+a(a+b+c)) {24512,39971}
(b+c)(bc+ca+ab-a(a+b+c)) {1757,1929, 2238,37128, 2664,2965, 9509,9510}
(b+c)(bc+ca+ab-2a(a+b+c)) {37673,39986}
(b+c)(2(bc+ca+ab)-a(a+b+c)) {3751,39954}, {37657,39985}
(b+c)(abc+a(a^2+b^2+c^2)) {3108,3589}, {29821,39973}
(b+c)(abc-a(a^2+b^2+c^2)); (K655) {141,251}, 2896,14370} (and others)
(b+c)(abc-2a(a^2+b^2+c^2)) {3763,39955}
(b+c)(2abc+a(a^2+b^2+c^2)) {1593,15740}, {2297,2299}, {3618,29951}, {31521,39978}
(b+c)(2abc-a(a^2+b^2+c^2)); (K169) {20,64}, {25,69}, {159,13575}, {200,269} (and others)
(b+c)(abc+a(bc+ca+ab)) {2350,17277}, {16552,39797}
(b+c)(abc-a(bc+ca+ab)) {4286,39950}, {8049,8053}
(b+c)(abc+2a(bc+ca+ab)) {15668,39961}
(b+c)(abc-2a(bc+ca+ab)) {17259,39965}, {26102,39972}
(b+c)(2abc+a(bc+ca+ab)) {17379,39967}
(b+c)(2abc-a(bc+ca+ab)) {16569,36598}, {17349,39966}, {21384,39970}
(b+c)(b^2c^2+c^2a^2+a^2b^2+a^2bc) {20965,39968}
(b+c)(b^2c^2+c^2a^2+a^2b^2-a^2bc) {308,3051}, {3499,39953}
(b+c)(2a+b+c) {37,81}, {3293,39949}, {17147,39988}

X(39942) = ISOGONAL CONJUGATE OF X(586)

Barycentrics    a^2*(a*b*c*(a + b - c) + (a*b - a*c - b*c)*S)*(a*b*c*(a - b + c) - (a*b - a*c + b*c)*S) : :

X(39942) lies on these lines: {171, 1124}, {172, 605}, {7029, 17103}

X(39943) = ISOGONAL CONJUGATE OF X(1708)

Barycentrics    a*(a - b - c)*(a^3 + a^2*b + a*b^2 + b^3 - a^2*c - 2*a*b*c - b^2*c - a*c^2 - b*c^2 + c^3)*(a^3 - a^2*b - a*b^2 + b^3 + a^2*c - 2*a*b*c - b^2*c + a*c^2 - b*c^2 + c^3) : :

X(39943) lies on these lines: {1, 2259}, {3, 2164}, {6, 169}, {9, 2478}, {19, 46}, {57, 15474}, {63, 1751}, {142, 39273}, {273, 1708}, {284, 1800}, {610, 909}, {673, 1760}, {1713, 1729}, {1741, 3559}, {1766, 2161}, {1945, 27628}, {2170, 2289}, {2174, 37615}, {2291, 13397}, {2339, 5745}, {2911, 24474}, {3512, 27626}, {7008, 8558}, {7297, 37500}, {7300, 37504}, {21370, 35466}

X(39943) = isogonal conjugate of X(1708)
X(39943) = cevapoint of X(652) and X(2170)
X(39943) = crosssum of X(46) and X(1723)

X(39944) = ISOGONAL CONJUGATE OF X(1713)

Barycentrics    a*(a^6*b + a^5*b^2 - 2*a^4*b^3 - 2*a^3*b^4 + a^2*b^5 + a*b^6 + a^6*c - a^4*b^2*c - a^2*b^4*c + b^6*c - a^5*c^2 - 3*a^4*b*c^2 - 3*a*b^4*c^2 - b^5*c^2 - 2*a^4*c^3 + 4*a^2*b^2*c^3 - 2*b^4*c^3 + 2*a^3*c^4 + 3*a^2*b*c^4 + 3*a*b^2*c^4 + 2*b^3*c^4 + a^2*c^5 + b^2*c^5 - a*c^6 - b*c^6)*(a^6*b - a^5*b^2 - 2*a^4*b^3 + 2*a^3*b^4 + a^2*b^5 - a*b^6 + a^6*c - 3*a^4*b^2*c + 3*a^2*b^4*c - b^6*c + a^5*c^2 - a^4*b*c^2 + 4*a^2*b^3*c^2 + 3*a*b^4*c^2 + b^5*c^2 - 2*a^4*c^3 + 2*b^4*c^3 - 2*a^3*c^4 - a^2*b*c^4 - 3*a*b^2*c^4 - 2*b^3*c^4 + a^2*c^5 - b^2*c^5 + a*c^6 + b*c^6) : :

X(39944) lies on these lines: {28, 580}, {278, 579}, {1713, 37279}

X(39944) = isogonal conjugate of X(1713)
X(39944) = cevapoint of X(244) and X(822)

X(39945) = ISOGONAL CONJUGATE OF X(1714)

Barycentrics    a^2*(a^4 + 2*a*b^3 + b^4 - 2*a^2*b*c + 2*b^3*c - 2*a^2*c^2 - 2*a*b*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*b*c - 2*a*b^2*c + 2*a*c^3 + 2*b*c^3 + c^4) : :

X(39945) lies on these lines: {6, 11517}, {34, 1708}, {56, 3173}, {58, 37300}, {579, 1474}, {1714, 5125}, {1751, 23604}, {2191, 3953}

X(39946) = ISOGONAL CONJUGATE OF X(1722)

Barycentrics    a*(a^3 + a^2*b + a*b^2 + b^3 - 3*a^2*c + b^2*c - 3*a*c^2 + b*c^2 + c^3)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3 + a^2*c + b^2*c + a*c^2 + b*c^2 + c^3) : :

X(39946) lies on these lines: {1, 26065}, {6, 37552}, {34, 1722}, {43, 937}, {56, 4641}, {58, 14868}, {87, 936}, {200, 979}, {269, 978}, {960, 3445}, {970, 9432}, {1474, 1732}, {1490, 9359}, {2191, 30478}, {2297, 5293}, {2334, 3744}, {5529, 36602}, {17962, 39248}

X(39947) = ISOGONAL CONJUGATE OF X(1723)

Barycentrics    a*(a^4 - 2*a^2*b^2 + b^4 - 2*a^3*c - 2*b^3*c + 2*a*b*c^2 + 2*a*c^3 + 2*b*c^3 - c^4)*(a^4 - 2*a^3*b + 2*a*b^3 - b^4 + 2*a*b^2*c + 2*b^3*c - 2*a^2*c^2 - 2*b*c^3 + c^4) : :

X(39947) lies on these lines: {1, 11517}, {2, 24179}, {9, 37887}, {28, 46}, {57, 3173}, {63, 15474}, {81, 6505}, {90, 23604}, {278, 1708}, {997, 1257}, {1054, 2947}, {3928, 34578}, {9445, 24478}

X(39947) = isogonal conjugate of X(1723)
X(39947) = cevapoint of X(244) and X(652)

X(39948) = ISOGONAL CONJUGATE OF X(3247)

Barycentrics    a*(3*a + 3*b + c)*(3*a + b + 3*c) : :

X(39948) lies on these lines: {1, 3683}, {2, 1449}, {6, 25430}, {9, 1255}, {28, 4658}, {57, 1100}, {63, 25417}, {81, 3928}, {88, 5256}, {89, 17011}, {105, 28148}, {239, 39736}, {274, 16834}, {278, 1419}, {279, 553}, {291, 5269}, {330, 29584}, {940, 8056}, {959, 1420}, {961, 3340}, {985, 3677}, {1002, 10389}, {1219, 3241}, {1224, 3679}, {3247, 27789}, {4393, 39740}, {6173, 15474}, {7308, 16667}, {9401, 18194}, {10980, 14969}, {14996, 26745}, {16475, 30571}, {16666, 17022}, {16670, 17019}, {17310, 39729}, {17389, 39722}, {29574, 30701}, {29580, 39738}, {29597, 32009}, {37631, 37887}

X(39948) = isogonal conjugate of X(3247)
X(39948) = cevapoint of X(6) and X(3303)

X(39949) = ISOGONAL CONJUGATE OF X(3293)

Barycentrics    a*(a + b)*(a + c)*(a*b + b^2 - a*c + b*c)*(a*b - a*c - b*c - c^2) : :

X(39949) lies on these lines: {1, 596}, {2, 39748}, {6, 474}, {10, 8050}, {21, 106}, {34, 37168}, {56, 20615}, {58, 5253}, {81, 1126}, {87, 3624}, {274, 3226}, {292, 3294}, {870, 17175}, {979, 1698}, {1010, 1222}, {1019, 16744}, {1220, 25526}, {1411, 37558}, {1724, 19250}, {2163, 21214}, {2334, 5711}, {3248, 6533}, {3286, 34444}, {3737, 23345}, {3953, 16726}, {4674, 18178}, {5192, 18169}, {5506, 9359}, {6532, 8054}, {16709, 39693}, {21173, 24161}, {37129, 37205}

X(39949) = isogonal conjugate of X(3293)
X(39949) = SS(a → b + c) of X(4) (trilinear substitution)

X(39950) = ISOGONAL CONJUGATE OF X(3294)

Barycentrics    a*(a + b)*(a + c)*(a*b - b^2 + a*c + b*c)*(a*b + a*c + b*c - c^2) : :

X(39950) lies on these lines: {1, 3286}, {2, 2350}, {6, 39797}, {56, 20614}, {58, 105}, {65, 18184}, {81, 20367}, {86, 3294}, {213, 16726}, {239, 39747}, {274, 20448}, {278, 31922}, {291, 3293}, {310, 29758}, {959, 4306}, {1014, 1170}, {1280, 35978}, {1412, 2982}, {1434, 29775}, {1475, 17205}, {1929, 3337}, {3227, 33296}, {3970, 16728}, {5563, 18723}, {5902, 18176}, {10471, 29766}, {16549, 30941}, {16704, 39706}, {17207, 39738}, {17753, 26818}, {25264, 39698}, {29750, 29811}

X(39950) = isogonal conjugate of X(3294)
X(39950) = isotomic conjugate of X(4043)
X(39950) = SS(a → ab + ac) of X(4) (trilinear substitution)
X(39950) = X(19)-isoconjugate of X(20925)

X(39951) = ISOGONAL CONJUGATE OF X(3618)

Barycentrics    a^2*(a^2 + 3*b^2 + c^2)*(a^2 + b^2 + 3*c^2) : :

In the plane of a triangle, let
Da = directrix of A-Artzt parabola
Ac = Da∩AC
Ab = Da∩AB
Lab = line through Ab parallel to BC
Lac = line through Ac parallel to BC
A2 = Lab∩BC, and define B3 and C1 cyclically
A3 = Lac&cajp;BC, and define B1 and C2 cyclically
The points A2, A3, B3, B1, C1, C2 lie on a conic, and the perspector of the conic is X(39951). (Angel Montesdeoca, January 27, 2023)

X(39951) lies on these lines: {2, 3933}, {3, 251}, {6, 3917}, {22, 1383}, {25, 39}, {37, 614}, {42, 16502}, {111, 907}, {232, 34818}, {263, 9777}, {305, 308}, {393, 427}, {428, 7738}, {941, 33854}, {967, 4253}, {1169, 5120}, {1184, 7772}, {1194, 8770}, {1196, 21448}, {1239, 3266}, {1384, 15246}, {1427, 15487}, {1627, 34572}, {1976, 11402}, {1993, 30535}, {2056, 6090}, {2165, 3815}, {2395, 11123}, {2548, 5064}, {2549, 15433}, {2963, 31489}, {2987, 5422}, {2998, 3329}, {3051, 11175}, {3108, 5359}, {3516, 17409}, {5050, 34945}, {5094, 13854}, {5275, 39798}, {5286, 37439}, {5644, 39024}, {6421, 8577}, {6422, 8576}, {6997, 15048}, {7386, 37665}, {7391, 15484}, {7485, 30435}, {7499, 31400}, {7786, 37876}, {8882, 16030}, {8891, 34816}, {9178, 10567}, {9300, 31152}, {9766, 21248}, {10542, 15004}, {16949, 38278}, {16950, 31859}, {22246, 39389}, {30734, 34481}

X(39951) = isogonal conjugate of X(3618)
X(39951) = isotomic conjugate of X(40022)
X(39951) = crossdifference of every pair of points on line X(3800)X(3804)
X(39951) = X(92)-isoconjugate of X(3796)

X(39952) = ISOGONAL CONJUGATE OF X(37673)

Barycentrics    a*(-(a*b^2) + 2*a^2*c + a*b*c - b^2*c + 2*a*c^2)*(2*a^2*b + 2*a*b^2 + a*b*c - a*c^2 - b*c^2) : :

X(39952) lies on these lines: {6, 19308}, {37, 3758}, {42, 3097}, {194, 27809}, {941, 37677}, {16606, 37652}, {17349, 39798}, {20158, 37128}

X(39953) = ISOGONAL CONJUGATE OF X(3499)

Barycentrics   (a^4*b^4 - a^4*b^2*c^2 - a^2*b^4*c^2 - a^4*c^4 - a^2*b^2*c^4 - b^4*c^4)*(a^4*b^4 + a^4*b^2*c^2 + a^2*b^4*c^2 - a^4*c^4 + a^2*b^2*c^4 + b^4*c^4) : :

X(39953) lies on these lines: {4, 8874}, {39, 8790}, {384, 3051}, {2896, 8871}, {3491, 3498}, {3492, 8870}, {3503, 8866}, {6196, 8865}, {6655, 37888}, {9495, 9496}, {14946, 14970}, {21814, 26752}

X(39954) = ISOGONAL CONJUGATE OF X(3751)

Barycentrics    a*(a^2 + b^2 - 2*a*c - 2*b*c - c^2)*(a^2 - 2*a*b - b^2 - 2*b*c + c^2) : :

X(39954) lies on these lines: {1, 2271}, {2, 968}, {9, 291}, {10, 30701}, {25, 16778}, {46, 39797}, {57, 238}, {81, 614}, {89, 7292}, {105, 28847}, {242, 278}, {274, 988}, {277, 6857}, {279, 1447}, {330, 16823}, {612, 1255}, {961, 4223}, {985, 7290}, {1002, 3751}, {1219, 39581}, {1280, 16496}, {1390, 26242}, {1402, 5020}, {1432, 18786}, {1929, 15485}, {2006, 36815}, {2362, 30386}, {3333, 16476}, {3679, 34892}, {3749, 26241}, {3757, 39694}, {3920, 27789}, {3931, 19321}, {5268, 25430}, {5573, 36540}, {6714, 17278}, {7081, 39703}, {7191, 25417}, {8056, 17596}, {8297, 35227}, {8769, 11997}, {9507, 16973}, {16232, 30385}, {16830, 39738}, {24175, 24283}, {25055, 34914}, {26227, 39698}, {28600, 36404}, {32009, 39586}, {35577, 39567}

X(39954) = isogonal conjugate of X(3751)
X(39954) = isotomic conjugate of X(30758)

X(39955) = ISOGONAL CONJUGATE OF X(3763)

Barycentrics    a^2*(2*a^2 + 2*b^2 + c^2)*(2*a^2 + b^2 + 2*c^2) : :

X(39955) lies on these lines: {2, 5007}, {6, 6636}, {25, 34482}, {32, 3108}, {37, 5332}, {111, 7954}, {183, 24861}, {251, 14247}, {263, 13366}, {308, 7766}, {385, 34816}, {393, 7408}, {694, 20976}, {1194, 1383}, {1239, 7787}, {1241, 9464}, {1627, 39389}, {1976, 15004}, {1989, 7394}, {2165, 5304}, {2963, 7735}, {2998, 8267}, {5319, 37349}, {5354, 21448}, {5359, 8770}, {7296, 17024}, {7378, 13854}, {7485, 30435}, {7492, 34572}, {7500, 34288}, {7760, 16952}, {8791, 8889}, {9462, 16932}, {10311, 33631}, {11002, 18898}, {12150, 16953}, {13342, 36414}, {13595, 36616}

X(39956) = ISOGONAL CONJUGATE OF X(4383)

Barycentrics    a*(a*b + b^2 - 2*a*c + b*c)*(2*a*b - a*c - b*c - c^2) : :

X(39956) lies on these lines: {2, 34283}, {6, 404}, {9, 23649}, {25, 5120}, {37, 2275}, {39, 941}, {42, 1449}, {111, 8690}, {251, 5019}, {308, 27005}, {346, 1015}, {393, 4200}, {572, 36510}, {966, 39798}, {967, 32911}, {978, 1400}, {1014, 4383}, {1169, 5042}, {1278, 27809}, {1427, 5435}, {2295, 16884}, {2350, 37657}, {3950, 9336}, {4188, 16946}, {4268, 34079}, {5296, 16604}, {6172, 28366}, {14624, 16975}, {16606, 26038}, {16668, 28625}, {16671, 28658}, {17349, 37128}, {20227, 26690}, {20331, 21785}, {26106, 27109}, {28358, 35578}

X(39956) = isogonal conjugate of X(4383)
X(39956) = isotomic conjugate of X(18135)

X(39957) = ISOGONAL CONJUGATE OF X(5276)

Barycentrics    a*(a^2*b + b^3 + a^2*c + a*b*c + a*c^2 + b*c^2)*(a^2*b + a*b^2 + a^2*c + a*b*c + b^2*c + c^3) : :

X(39957) lies on these lines: {2, 16703}, {6, 980}, {25, 940}, {37, 141}, {38, 42}, {39, 16728}, {69, 941}, {75, 14624}, {81, 251}, {85, 26978}, {111, 37633}, {241, 1400}, {263, 37516}, {308, 6385}, {757, 1169}, {1383, 14996}, {1386, 20985}, {1500, 4437}, {1880, 5236}, {1976, 5135}, {3108, 32911}, {3589, 39798}, {3752, 16606}, {4260, 37597}, {4265, 5337}, {8770, 37674}, {13476, 18183}, {14829, 32010}, {16600, 17758}, {21415, 31993}, {22116, 38995}, {25994, 29983}, {26242, 27475}, {37680, 39389}

X(39958) = ISOGONAL CONJUGATE OF X(7174)

Barycentrics    a*(3*a^2 + 2*a*b + 3*b^2 + c^2)*(3*a^2 + b^2 + 2*a*c + 3*c^2) : :

X(39958) lies on these lines: {1, 30435}, {2, 3883}, {9, 1390}, {57, 1386}, {81, 3677}, {171, 8056}, {277, 12436}, {278, 1890}, {291, 16475}, {985, 16491}, {1001, 25430}, {1002, 1449}, {1420, 1432}, {2224, 11529}, {2362, 11371}, {3340, 7132}, {7962, 10800}, {11370, 16232}

X(39959) = ISOGONAL CONJUGATE OF X(7290)

Barycentrics    a*(a^2 + 3*b^2 - 2*a*c + c^2)*(a^2 - 2*a*b + b^2 + 3*c^2) : :

X(39959) lies on these lines: {1, 728}, {2, 3677}, {8, 279}, {9, 105}, {10, 277}, {28, 7719}, {57, 200}, {75, 34018}, {78, 1170}, {81, 3870}, {88, 37223}, {89, 3935}, {278, 1861}, {291, 16496}, {346, 18216}, {522, 4659}, {959, 11523}, {961, 6762}, {982, 8056}, {985, 3751}, {1002, 3243}, {1022, 11525}, {1255, 4666}, {1420, 7132}, {1422, 8270}, {1432, 3340}, {2006, 5231}, {2224, 3576}, {2362, 3641}, {3640, 16232}, {3679, 34578}, {3729, 4907}, {3872, 34056}, {3932, 30813}, {3957, 25417}, {4906, 10582}, {5119, 7313}, {5223, 37658}, {7962, 14839}, {9053, 25355}, {10389, 36483}, {15474, 25006}, {18193, 36603}, {24352, 28582}, {24600, 32922}, {27789, 29817}, {28503, 31146}, {33169, 37887}, {34051, 36819}

X(39959) = isogonal conjugate of X(7290)
X(39959) = cevapoint of X(i) and X(j) for these {i,j}: {1, 5223}, {3729, 4384}
X(39959) = trilinear pole of line X(513)X(4130)

X(39960) = ISOGONAL CONJUGATE OF X(14997)

Barycentrics    a*(2*a*b + 2*b^2 - 3*a*c + 2*b*c)*(3*a*b - 2*a*c - 2*b*c - 2*c^2) : :

X(39960) lies on these lines: {6, 16417}, {37, 25055}, {1383, 33854}, {3444, 5120}, {4740, 27809}, {5275, 39389}, {16667, 28625}, {16670, 28658}

X(39960) = isogonal conjugate of X(14997)
X(39960) = cevapoint of X(1015) and X(4893)

X(39961) = ISOGONAL CONJUGATE OF X(15668)

Barycentrics    a^2*(2*a*b + b^2 + 2*a*c + 2*b*c)*(2*a*b + 2*a*c + 2*b*c + c^2) : :

X(39961) lies on these lines: {2, 20970}, {6, 4184}, {32, 1171}, {37, 3681}, {42, 3730}, {386, 2350}, {967, 2271}, {1400, 4270}, {2276, 28625}, {3240, 16606}, {4277, 23632}, {14624, 20012}, {34816, 37678}, {37128, 37685}

X(39962) = ISOGONAL CONJUGATE OF X(16669)

Barycentrics    a*(a + b - 4*c)*(a - 4*b + c) : :

X(39962) lies on these lines: {1, 3833}, {2, 4398}, {28, 1878}, {57, 37680}, {63, 36603}, {81, 16610}, {88, 3219}, {89, 32911}, {105, 28218}, {321, 36805}, {959, 27625}, {1022, 14838}, {1255, 3752}, {2006, 26724}, {3227, 27002}, {3305, 8056}, {4359, 32017}, {4669, 24858}, {4678, 6553}, {4850, 25430}, {5235, 39747}, {9342, 17598}, {16669, 27003}, {16885, 26745}, {20014, 35577}, {20182, 27789}, {21907, 31204}, {24183, 30831}, {24589, 30710}, {24620, 39703}, {25417, 37633}, {26739, 31231}, {31197, 33761}, {31228, 32933}, {32019, 33113}, {33172, 39696}

X(39962) = isogonal conjugate of X(16669)
X(39962) = cevapoint of X(i) and X(j) for these {i,j}: {1, 16885}, {6, 7280}, {9, 10912}
X(39962) = trilinear product X(514)*X(28218)
X(39962) = trilinear quotient X(28218)/X(101)
X(39962) = X(i)-isoconjugate of X(j) for these {i,j}: {6, 3244}, {101, 28217}

X(39963) = ISOGONAL CONJUGATE OF X(16670)

Barycentrics    a*(a + b - 5*c)*(a - 5*b + c) : :

X(39963) lies on these lines: {1, 3689}, {2, 1266}, {9, 88}, {44, 57}, {45, 31197}, {63, 26745}, {75, 36805}, {81, 5437}, {89, 3306}, {105, 6014}, {278, 8756}, {279, 3911}, {330, 16815}, {650, 1022}, {678, 35227}, {899, 1002}, {959, 27627}, {1219, 9780}, {1224, 19872}, {1419, 34051}, {1422, 36636}, {2006, 17278}, {2401, 7658}, {3227, 4384}, {3617, 6553}, {3621, 35577}, {3676, 21129}, {3679, 24858}, {3752, 16672}, {3929, 36603}, {4346, 5316}, {4792, 7962}, {4859, 37691}, {4887, 31142}, {7308, 8056}, {16816, 38247}, {16832, 36871}, {17012, 25417}, {17021, 27789}, {17022, 39260}, {17028, 39925}, {17284, 34892}, {19804, 32017}, {20196, 24175}, {23511, 37520}, {24457, 35348}, {24627, 26071}, {24857, 25055}, {29578, 39738}, {29596, 30701}, {29598, 34914}, {29608, 39722}, {31183, 34578}, {32019, 32851}

X(39963) = isogonal conjugate of X(16670)
X(39963) = isotomic conjugate of X(30829)
X(39963) = trilinear product X(i)*X(j) for these {i,j}: {6, 36588}, {57, 4900}, {106, 36915}, {514, 6014}
X(39963) = trilinear quotient X(i)/X(j) for these (i,j): (4900, 9), (6014, 101), (36588, 2), (36915, 519)
X(39963) = trilinear pole of line X(4895)X(21343)
X(39963) = cevapoint of X(i) and X(j) for these {i,j}: {2, 24620}, {244, 4893}
X(39963) = X(i)-isoconjugate of X(j) for these {i,j}: {4, 23073}, {6, 3241}, {9, 13462}, {31, 30829}, {58, 4029}, {81, 21870}

X(39964) = ISOGONAL CONJUGATE OF X(16671)

Barycentrics    a^2*(a^2*b + a*b^2 - a^2*c - b^2*c - a*c^2 - b*c^2)*(a^2*b + a*b^2 - a^2*c + b^2*c - a*c^2 + b*c^2) : :

X(39964) lies on these lines: {6, 3293}, {31, 20990}, {81, 4360}, {572, 9456}, {594, 1977}, {595, 1333}, {604, 5301}, {739, 38871}, {1030, 1979}, {1911, 8053}, {2162, 16777}, {2176, 34819}, {4264, 28615}, {14621, 18046}, {16969, 36619}, {17277, 20332}, {21769, 28607}

X(39964) = isogonal conjugate of X(16671)
X(39964) = cevapoint of X(1) and X(15492)
X(39964) = trilinear product X(514)*X(8698)
X(39964) = trilinear quotient X(8698)/X(101)

X(39965) = ISOGONAL CONJUGATE OF X(17259)

Barycentrics    a^2*(2*a*b - b^2 + 2*a*c + 2*b*c)*(2*a*b + 2*a*c + 2*b*c - c^2) : :

X(39965) lies on these lines: {6, 4210}, {37, 3873}, {42, 4253}, {5069, 28625}, {16670, 27635}, {34816, 37686}, {37657, 39798}

X(39966) = ISOGONAL CONJUGATE OF X(17349)

Barycentrics    a^2*(a*b - 2*b^2 + a*c + b*c)*(a*b + a*c + b*c - 2*c^2) : :

X(39966) lies on these lines: {2, 4754}, {6, 16059}, {25, 33863}, {37, 982}, {42, 2275}, {45, 25501}, {393, 4212}, {941, 24512}, {1400, 7248}, {2162, 20459}, {2176, 28360}, {2248, 5021}, {2295, 29814}, {2998, 37686}, {3210, 27809}, {4383, 37128}, {5022, 8770}, {8033, 17259}, {16569, 16606}, {18755, 37309}, {23470, 36614}, {37673, 39798}

X(39967) = ISOGONAL CONJUGATE OF X(17379)

Barycentrics    a^2*(a*b + 2*b^2 + a*c + b*c)*(a*b + a*c + b*c + 2*c^2) : :

X(39967) lies on these lines: {1, 7275}, {2, 21024}, {3, 2248}, {6, 16058}, {25, 18755}, {37, 43}, {42, 2176}, {393, 4213}, {940, 37128}, {941, 2238}, {967, 33863}, {1400, 1403}, {1427, 7146}, {2056, 5114}, {2162, 2309}, {2998, 37678}, {3725, 28637}, {3736, 21001}, {4255, 8770}, {6685, 21071}, {7033, 17319}, {17243, 27264}

X(39968) = ISOGONAL CONJUGATE OF X(20965)

Barycentrics   (2*a^2*b^2 + a^2*c^2 + b^2*c^2)*(a^2*b^2 + 2*a^2*c^2 + b^2*c^2) : :

X(39968) lies on these lines: {2, 31613}, {6, 1078}, {25, 11174}, {37, 26959}, {39, 308}, {42, 18170}, {69, 11175}, {76, 34816}, {99, 18092}, {251, 3329}, {263, 3618}, {385, 3108}, {393, 37337}, {670, 39939}, {694, 3589}, {2165, 7803}, {2350, 37678}, {2963, 7828}, {20148, 37128}

X(39968) = isogonal conjugate of X(20965)
X(39968) = isotomic conjugate of X(3934)
X(39968) = polar conjugate of crosssum of X(3) and X(20775)
X(39968) = trilinear pole of line X(512)X(14712)
X(39968) = X(19)-isoconjugate of X(22062)

X(39969) = ISOGONAL CONJUGATE OF X(21214)

Barycentrics    a*(a^2*b + a*b^2 - a^2*c + 3*a*b*c - b^2*c - a*c^2 - b*c^2)*(a^2*b + a*b^2 - a^2*c - 3*a*b*c + b^2*c - a*c^2 + b*c^2) : :

X(39969) lies on these lines: {1, 4090}, {6, 3208}, {8, 87}, {43, 56}, {58, 3550}, {86, 24524}, {106, 978}, {269, 3212}, {519, 979}, {1027, 5691}, {1431, 3751}, {2136, 9359}, {2163, 3293}, {3445, 12513}, {3632, 39748}

X(39970) = ISOGONAL CONJUGATE OF X(21384)

Barycentrics    a*(a^2*b + a*b^2 + a^2*c - a*b*c + b^2*c - a*c^2 - b*c^2)*(a^2*b - a*b^2 + a^2*c - a*b*c - b^2*c + a*c^2 + b*c^2) : :

X(39970) lies on these lines: {1, 34247}, {2, 1334}, {9, 274}, {28, 2212}, {41, 81}, {46, 1929}, {57, 213}, {105, 3915}, {239, 39694}, {277, 27626}, {278, 2333}, {279, 1400}, {291, 978}, {330, 16827}, {961, 1471}, {1002, 1193}, {1219, 5686}, {2270, 16782}, {3208, 29968}, {3294, 25590}, {3500, 4209}, {4384, 30710}, {6553, 20036}, {8056, 20367}, {9401, 17594}, {15474, 27659}, {16552, 36871}, {16831, 37870}, {17033, 39925}, {17143, 32017}, {18785, 34018}, {19522, 20368}, {19591, 33825}, {29960, 30701}

X(39970) = isogonal conjugate of X(21384)
X(39970) = isotomic conjugate of X(20923)

X(39971) = ISOGONAL CONJUGATE OF X(24512)

Barycentrics    a*(a*b^2 + a^2*c + 2*a*b*c + b^2*c + a*c^2)*(a^2*b + a*b^2 + 2*a*b*c + a*c^2 + b*c^2) : :

X(39971) lies on these lines: {2, 30940}, {6, 16367}, {25, 37502}, {37, 239}, {39, 1509}, {42, 238}, {81, 2350}, {83, 1500}, {86, 39798}, {308, 3948}, {941, 17349}, {1019, 2084}, {1218, 20140}, {1400, 1429}, {1427, 33765}, {2276, 14621}, {3112, 3995}, {4261, 20159}, {14624, 17280}, {16606, 39914}, {17023, 20166}, {20131, 24598}, {20153, 31198}

X(39971) = isogonal conjugate of X(24512)
X(39971) = isotomic conjugate of X(20913)

X(39972) = ISOGONAL CONJUGATE OF X(26102)

Barycentrics    a*(3*a*b + a*c + b*c)*(a*b + 3*a*c + b*c) : :

X(39972) lies on these lines: {1, 4991}, {6, 3750}, {9, 25426}, {42, 87}, {43, 86}, {56, 4649}, {58, 2209}, {106, 29199}, {238, 2334}, {292, 1449}, {870, 2663}, {1126, 16468}, {1220, 32941}, {2191, 29821}, {2279, 16667}, {2309, 36598}, {3226, 17393}, {3240, 25528}, {4489, 5936}, {4685, 20146}, {5145, 36602}, {5256, 7194}, {10013, 17259}, {24766, 29584}

X(39973) = ISOGONAL CONJUGATE OF X(29831)

Barycentrics    a^2*(a^3 + 2*a^2*b + 3*b^3 + 2*b*c^2 + c^3)*(a^3 + b^3 + 2*a^2*c + 2*b^2*c + 3*c^3) : :

X(39973) lies on these lines: {86, 29832}, {1126, 16790}, {2163, 16797}, {2334, 16794}

X(39974) = ISOGONAL CONJUGATE OF X(37633)

Barycentrics    a*(a*b + b^2 + 3*a*c + b*c)*(3*a*b + a*c + b*c + c^2) : :

X(39974) lies on these lines: {1, 20973}, {2, 4277}, {6, 16370}, {25, 37503}, {37, 519}, {42, 44}, {111, 5276}, {251, 33882}, {308, 18145}, {553, 1427}, {941, 37654}, {1100, 1319}, {1169, 2220}, {1171, 2185}, {1500, 4370}, {1839, 1877}, {2092, 39798}, {2245, 2350}, {2276, 36872}, {3666, 4715}, {4251, 34079}, {4274, 16666}, {4646, 37150}, {4664, 27809}, {4850, 39704}, {5035, 17549}, {5275, 21448}, {9331, 36911}, {14020, 37548}, {14584, 20616}, {14624, 17281}, {16669, 28625}, {17196, 25060}, {33854, 39389}

X(39975) = ISOGONAL CONJUGATE OF X(37679)

Barycentrics    a*(a*b + b^2 - 4*a*c + b*c)*(4*a*b - a*c - b*c - c^2) : :

X(39975) lies on these lines: {6, 4188}, {37, 3622}, {42, 16667}, {251, 5042}, {391, 39798}, {1400, 16670}, {3973, 21214}, {4788, 27809}, {8770, 33854}, {20059, 24625}

X(39976) = ISOGONAL CONJUGATE OF X(27680)

Barycentrics    a*(a^4*b + a*b^4 - a^4*c - 3*a^2*b^2*c - b^4*c - a^3*c^2 - b^3*c^2 + 2*a*b*c^3)*(a^4*b + a^3*b^2 - a^4*c - 2*a*b^3*c + 3*a^2*b*c^2 + b^2*c^3 - a*c^4 + b*c^4) : :

X(39976) lies on these lines: {238, 32920}, {1914, 3501}, {5009, 13588}

X(39977) = ISOGONAL CONJUGATE OF X(29821)

Barycentrics    a*(a^2 + a*b + b^2 - a*c + b*c + c^2)*(a^2 - a*b + b^2 + a*c + b*c + c^2) : :

X(39977) lies on these lines: {1, 3790}, {2, 7194}, {6, 3961}, {12, 5018}, {37, 17962}, {42, 9277}, {56, 984}, {58, 1757}, {86, 1215}, {87, 612}, {181, 1431}, {269, 5268}, {870, 3596}, {894, 7166}, {2191, 26102}, {2330, 7281}, {3589, 29821}, {5297, 7184}, {5524, 34916}, {13610, 20715}, {17245, 17719}, {17381, 29820}, {17954, 21035}

X(39977) = isogonal conjugate of X(29821)
X(39977) = isotomic conjugate of X(33944)

X(39978) = ISOGONAL CONJUGATE OF X(31521)

Barycentrics   (a^6 + 7*a^4*b^2 + 7*a^2*b^4 + b^6 + a^4*c^2 + 10*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 - b^2*c^4 - c^6)*(a^6 + a^4*b^2 - a^2*b^4 - b^6 + 7*a^4*c^2 + 10*a^2*b^2*c^2 - b^4*c^2 + 7*a^2*c^4 + b^2*c^4 + c^6) : :

X(39978) lies on these lines: {2, 14259}, {3618, 7485}

X(39979) = ISOGONAL CONJUGATE OF X(33854)

Barycentrics    a*(a^2*b + b^3 - a^2*c - a*b*c - a*c^2 + b*c^2)*(a^2*b + a*b^2 - a^2*c + a*b*c - b^2*c - c^3) : :

X(39979) lies on these lines: {2, 16707}, {6, 3882}, {25, 4383}, {37, 3589}, {39, 16728}, {42, 1386}, {81, 3108}, {111, 37680}, {141, 16726}, {251, 32911}, {263, 4259}, {518, 20456}, {662, 1169}, {941, 3618}, {1015, 4437}, {1383, 14997}, {1400, 27633}, {1880, 5723}, {2350, 37676}, {3797, 27809}, {4033, 14624}, {4145, 9178}, {8770, 37679}, {24625, 32029}, {37633, 39389}

X(39980) = ISOGONAL CONJUGATE OF X(3731)

Barycentrics    a*(3*a + 3*b - c)*(3*a - b + 3*c) : :

Let DEF be the intouch triangle and IaIbIc the excentral-triangle. Let D' be the trisector nearest Ia of segment DIa. Let La be the radical axis of (DEF) and (BCD'), and define Lb and Lc cyclically. The triangle A'B'C' formed by the lines La, Lb, Lc is perspective to ABC, and the perspector is X(39980). See also X(34578). (Angel Montesdeoca, December 31, 2020)

X(39980) lies on these lines: {1, 3052}, {2, 1743}, {6, 8056}, {57, 16667}, {63, 1255}, {81, 18186}, {88, 2999}, {89, 5256}, {105, 10980}, {165, 1002}, {239, 39740}, {274, 16833}, {278, 553}, {330, 16834}, {519, 1219}, {940, 3731}, {959, 3361}, {961, 3339}, {985, 18193}, {1022, 2441}, {1170, 17074}, {1224, 19875}, {1280, 3722}, {1390, 32913}, {2006, 4654}, {3218, 25417}, {3227, 14759}, {3241, 6553}, {3752, 36603}, {3973, 37674}, {4384, 39736}, {4697, 35613}, {4859, 37666}, {4888, 37642}, {5223, 37604}, {6173, 37887}, {7277, 30827}, {8915, 28194}, {9315, 17474}, {10447, 30710}, {14996, 27789}, {17294, 39722}, {20942, 32017}, {23511, 37520}, {25590, 37683}, {26745, 37685}, {29573, 30701}, {29584, 38247}, {29597, 39738}

X(39981) = ISOGONAL CONJUGATE OF X(37657)

Barycentrics    a*(-2*a*b^2 + a^2*c - a*b*c - 2*b^2*c + a*c^2)*(a^2*b + a*b^2 - a*b*c - 2*a*c^2 - 2*b*c^2) : :

X(39981) lies on these lines: {6, 16412}, {25, 3286}, {37, 980}, {39, 20135}, {42, 750}, {86, 941}, {241, 1880}, {393, 15149}, {1400, 6180}, {2350, 4383}, {5737, 16606}, {17259, 39798}, {20131, 24598}, {20154, 37128}, {20156, 31198}

X(39981) = isogonal conjugate of X(37657)
X(39981) = isotomic conjugate of X(30830)

X(39982) = ISOGONAL CONJUGATE OF X(37680)

Barycentrics    a*(a*b + b^2 - 3*a*c + b*c)*(3*a*b - a*c - b*c - c^2) : :

X(39982) lies on these lines: {2, 16723}, {6, 16371}, {37, 537}, {42, 678}, {111, 33854}, {251, 5035}, {291, 16507}, {536, 27809}, {662, 1171}, {903, 24625}, {941, 5069}, {1018, 1100}, {1400, 16669}, {5053, 34079}, {5276, 39389}, {9336, 36911}, {13587, 33882}, {16590, 16604}, {17330, 39798}, {20972, 28658}

X(39982) = isogonal conjugate of X(37680)
X(39982) = isotomic conjugate of X(18145)
X(39982) = complement of X(39996)

X(39983) = ISOGONAL CONJUGATE OF X(37685)

Barycentrics    a*(2*a*b + 2*b^2 + a*c + 2*b*c)*(a*b + 2*a*c + 2*b*c + 2*c^2) : :

X(39983) lies on these lines: {1, 28625}, {3, 3444}, {6, 4658}, {9, 28658}, {25, 1030}, {37, 1698}, {42, 3711}, {45, 1400}, {251, 5275}, {393, 451}, {589, 31473}, {940, 1171}, {941, 1213}, {967, 37674}, {1218, 18140}, {1383, 37675}, {1427, 5219}, {3989, 7241}, {5283, 39798}, {15668, 24944}, {25508, 27269}, {27268, 27809}, {30473, 32009}

X(39983) = isogonal conjugate of X(37685)
X(39983) = isotomic conjugate of complement of X(27269)

X(39984) = ISOGONAL CONJUGATE OF X(37687)

Barycentrics    a*(a*b + b^2 - 5*a*c + b*c)*(5*a*b - a*c - b*c - c^2) : :

X(39984) lies on these lines: {6, 19537}, {37, 3636}, {42, 16668}, {1400, 16671}, {1427, 26741}, {3644, 27809}

X(39985) = X(4)X(526)∩X(30)X(110)

Barycentrics    (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 + a^6*b^2 - 4*a^4*b^4 + a^2*b^6 + b^8 - 3*a^6*c^2 + 2*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - 3*b^6*c^2 + 3*a^4*c^4 - 2*a^2*b^2*c^4 + 3*b^4*c^4 - a^2*c^6 - b^2*c^6)*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6 + a^6*c^2 + 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - b^6*c^2 - 4*a^4*c^4 + 2*a^2*b^2*c^4 + 3*b^4*c^4 + a^2*c^6 - 3*b^2*c^6 + c^8) : :

X(39985) lies on the cubic K025 and these lines: {4, 526}, {5, 14933}, {30, 110}, {74, 3134}, {113, 15329}, {1300, 10152}, {1989, 18320}, {5962, 34170}, {7471, 15469}

X(39985) = reflection of X(i) in X(j) for these {i,j}: {74, 3134}, {14933, 5}, {15329, 113}
X(39985) = isogonal conjugate of X(39986)
X(39985) = isotomic conjugate of X(39988)
X(39985) = antigonal image of X(15329)
X(39985) = symgonal image of X(3134)
X(39985) = X(i)-isoconjugate of X(j) for these (i,j): {5504, 36063}, {5663, 36053}
X(39985) = trilinear product X(i)*X(j) for these {i,j}: {403, 36062}, {477, 1725}, {3003, 36102}, {3580, 36151}, {13754, 36130}
X(39985) = barycentric product X(i)*X(j) for these {i,j}: {477, 3580}, {1725, 36102}, {14220, 16237}
X(39985) = barycentric quotient X(i)/X(j) for these {i,j}: {477, 2986}, {2436, 15470}, {3003, 5663}, {3580, 35520}, {14220, 15421}, {30528, 18878}, {32663, 5504}, {36151, 36053}

X(39986) = X(3)X(523)∩X(74)X(323)

Barycentrics    a^2*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 2*a^4*c^2 + 2*a^2*b^2*c^2 - 2*b^4*c^2 + a^2*c^4 + b^2*c^4)*(a^6 - 2*a^4*b^2 + a^2*b^4 - a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 - 2*b^2*c^4 + c^6)*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 + 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - b^6*c^2 - 3*a^4*c^4 - 2*a^2*b^2*c^4 + 4*b^4*c^4 + 3*a^2*c^6 - b^2*c^6 - c^8) : :

X(39986) lies on the cubic K039 and these lines: {3, 523}, {74, 323}, {186, 476}, {187, 14910}, {1624, 36169}, {2986, 7464}, {3520, 38936}, {7480, 25641}, {11589, 15478}, {15329, 34150}, {31941, 34148}

X(39986) = isogonal conjugate of X(39985)
X(39986) = circumcircle-inverse of X(15328)
X(39986) = X(i)-isoconjugate of X(j) for these (i,j): {403, 36062}, {477, 1725}, {3003, 36102}, {3580, 36151}, {13754, 36130}
X(39986) = trilinear product X(i)*X(j) for these {i,j}: {5504, 36063}, {5663, 36053}
X(39986) = barycentric product X(i)*X(j) for these {i,j}: {2410, 15470}, {2986, 5663}, {6070, 18879}, {7480, 15421}, {14910, 35520}
X(39986) = barycentric quotient X(i)/X(j) for these {i,j}: {5663, 3580}, {7480, 16237}, {10420, 30528}, {14910, 477}, {15470, 2411}, {36053, 36102}
X(39986) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {476, 12028, 39375}, {5504, 10420, 39371}, {10419, 10420, 5504}

X(39987) = X(3)X(526)∩X(30)X(125)

Barycentrics    a^2*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 + 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - b^6*c^2 - 3*a^4*c^4 - 2*a^2*b^2*c^4 + 4*b^4*c^4 + 3*a^2*c^6 - b^2*c^6 - c^8)*(2*a^12 - 5*a^10*b^2 + 10*a^6*b^6 - 10*a^4*b^8 + 3*a^2*b^10 - 5*a^10*c^2 + 18*a^8*b^2*c^2 - 17*a^6*b^4*c^2 + a^4*b^6*c^2 + 2*a^2*b^8*c^2 + b^10*c^2 - 17*a^6*b^2*c^4 + 22*a^4*b^4*c^4 - 5*a^2*b^6*c^4 - 4*b^8*c^4 + 10*a^6*c^6 + a^4*b^2*c^6 - 5*a^2*b^4*c^6 + 6*b^6*c^6 - 10*a^4*c^8 + 2*a^2*b^2*c^8 - 4*b^4*c^8 + 3*a^2*c^10 + b^2*c^10) : :

X(39987) lies on the cubic K038 and these lines: {3, 526}, {30, 125}, {74, 15329}, {131, 3184}, {2420, 18877}, {3134, 6699}, {12095, 12096}

X(39987) = midpoint of X(i) and X(j) for these {i,j}: {3, 14933}, {74, 15329}
X(39987) = reflection of X(3134) in X(6699)
X(39987) = X(36053)-complementary conjugate of X(5663)
X(39987) = X(74)-Ceva conjugate of X(5663)

X(39988) = ISOTOMIC CONJUGATE OF X(39985)

Barycentrics    (a^6 - a^4*b^2 - a^2*b^4 + b^6 - 2*a^4*c^2 + 2*a^2*b^2*c^2 - 2*b^4*c^2 + a^2*c^4 + b^2*c^4)*(a^6 - 2*a^4*b^2 + a^2*b^4 - a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 - 2*b^2*c^4 + c^6)*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 + 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - b^6*c^2 - 3*a^4*c^4 - 2*a^2*b^2*c^4 + 4*b^4*c^4 + 3*a^2*c^6 - b^2*c^6 - c^8) : :

X(39988) lies on these lines: {69, 850}, {340, 18817}, {524, 2986}, {1494, 7799}, {3260, 13754}

X(39988) = X(3003)-isoconjugate of X(36151)
X(39988) = barycentric product X(2986)*X(35520)
X(39988) = barycentric quotient X(i)/X(j) for these {i,j}: {2986, 477}, {5504, 32663}, {5663, 3003}, {15421, 14220}, {15470, 2436}, {18878, 30528}, {35520, 3580}, {36053, 36151}

X(39989) = X(4)X(3096)∩X(30)X(7953)

Barycentrics    (a^2 + 2*b^2 + c^2)*(a^2 + b^2 + 2*c^2)*(3*a^6 + a^4*b^2 - 3*a^2*b^4 - b^6 + a^4*c^2 + a^2*b^2*c^2 + b^4*c^2 - 3*a^2*c^4 + b^2*c^4 - c^6) : :

X(39989) lies on the cubic K025 and these lines: {4, 3096}, {30, 7953}, {316, 35137}, {7927, 31065}, {11703, 34175}

X(39989) = antigonal image of X(20063)
X(39989) = symgonal image of X(5189)
X(39989) = barycentric product X(10159)*X(20063)
X(39989) = barycentric quotient X(20063)/X(3589)

X(39990) = X(4)X(2778)∩X(30)X(1295)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 + a^2*b*c + a*b^2*c - a*b*c^2 - c^4)*(a^4 - b^4 + 2*a^2*b*c - 2*a*b^2*c - 2*a*b*c^2 + 2*b^2*c^2 - c^4)*(a^4 - b^4 + a^2*b*c - a*b^2*c - 2*a^2*c^2 + a*b*c^2 + c^4) : :

X(39990) lies on the cubic K025 and these lines: {4, 2778}, {30, 1295}, {33, 998}, {108, 429}, {123, 16049}, {2829, 7414}, {5203, 34173}, {5962, 38952}

X(39990) = reflection of X(i) in X(j) for these {i,j}: {108, 429}, {16049, 123}
X(39990) = antigonal image of X(16049)
X(39990) = symgonal image of X(429)
X(39990) = barycentric quotient X(i)/X(j) for these {i,j}: {197, 22123}, {6588, 2850}, {14257, 37798}

X(39991) = X(4)X(8674)∩X(30)X(1000)

Barycentrics    (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)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^5*c + 2*a^4*b*c - a^3*b^2*c - a^2*b^3*c + 2*a*b^4*c - b^5*c - 2*a^4*c^2 + 2*a^2*b^2*c^2 - 2*b^4*c^2 + 2*a^3*c^3 - a^2*b*c^3 - a*b^2*c^3 + 2*b^3*c^3 + a^2*c^4 + b^2*c^4 - a*c^5 - b*c^5)*(-a^6 + a^5*b + 2*a^4*b^2 - 2*a^3*b^3 - a^2*b^4 + a*b^5 - 2*a^4*b*c + a^2*b^3*c + b^5*c + a^4*c^2 + a^3*b*c^2 - 2*a^2*b^2*c^2 + a*b^3*c^2 - b^4*c^2 + a^2*b*c^3 - 2*b^3*c^3 + a^2*c^4 - 2*a*b*c^4 + 2*b^2*c^4 + b*c^5 - c^6) : :

X(39991) lies on the cubic K025 and these lines: {4, 8674}, {30, 100}, {104, 3139}, {119, 3658}, {265, 11604}, {2829, 7429}, {5962, 38949}

X(39991) = reflection of X(i) in X(j) for these {i,j}: {104, 3139}, {3658, 119}
X(39991) = antigonal image of X(3658)
X(39991) = symgonal image of X(3139)
X(39991) = X(2771)-isoconjugate of X(36052)
X(39991) = barycentric quotient X(i)/X(j) for these {i,j}: {2687, 2990}, {8609, 2771}

X(39992) = X(4)X(2779)∩X(30)X(102)

Barycentrics    (-(a^2*b) + b^3 - a^2*c + a*b*c + c^3)*(a^5 + b^5 + a^3*b*c + a^2*b^2*c + a*b^3*c - 2*a^3*c^2 - a^2*b*c^2 - a*b^2*c^2 - 2*b^3*c^2 - a*b*c^3 + a*c^4 + b*c^4)*(a^5 - 2*a^3*b^2 + a*b^4 + a^3*b*c - a^2*b^2*c - a*b^3*c + b^4*c + a^2*b*c^2 - a*b^2*c^2 + a*b*c^3 - 2*b^2*c^3 + c^5) : :

X(39992) lies on the cubic K025 and these lines: {4, 2779}, {11, 58}, {30, 102}, {109, 3142}, {124, 4225}, {994, 1836}, {3417, 5450}

X(39992) = reflection of X(i) in X(j) for these {i,j}: {109, 3142}, {4225, 124}
X(39992) = antigonal image of X(4225)
X(39992) = symgonal image of X(3142)
X(39992) = X(2773)-isoconjugate of X(36050)
X(39992) = barycentric quotient X(6589)/X(2773)

X(39993) = X(4)X(27725)∩X(30)X(103)

Barycentrics    (-(a*b) + b^2 - a*c + b*c + c^2)*(a^4 + a^3*b + a^2*b^2 + a*b^3 + b^4 - a^3*c - a^2*b*c - a*b^2*c - b^3*c - a^2*c^2 - a*b*c^2 - b^2*c^2 + a*c^3 + b*c^3)*(a^4 - a^3*b - a^2*b^2 + a*b^3 + a^3*c - a^2*b*c - a*b^2*c + b^3*c + a^2*c^2 - a*b*c^2 - b^2*c^2 + a*c^3 - b*c^3 + c^4) : :

X(39993) lies on the cubic K025 and these lines: {4, 2772}, {30, 103}, {101, 3136}, {116, 4184}, {150, 8049}, {1002, 1478}, {1126, 1834}, {3730, 21045}

X(39993) = reflection of X(i) in X(j) for these {i,j}: {101, 3136}, {4184, 116}
X(39993) = antigonal image of X(4184)
X(39993) = symgonal image of X(3136)
X(39993) = barycentric product X(2690)*X(25259)
X(39993) = barycentric quotient X(6586)/X(2774)

leftri

Points associated with cubics: X(39994)-X(40106)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, October 26, 2020.

In the plane of a triangle ABC, suppose that P = p : q : r and U = u : v : w are distinct points such that p*q*r*u*v*w is not zero, and that the point

(b-c)*p : (c-a)*q : (a-b)*r is on the line at infinity; i.e.,

(b-c)p(a,b,c) + (c-a)p(b,c,a) + (a-b)p(c,a,b) = 0.

Let CC(P,U) denote the cubic given by

(b + c)*p*(y*v - z*w)*(x*u + y*v)*(x*u + z*w) + (c + a)*q*(z*w - x*u)*(y*v + z*w)*(y*v + x*u) + (a + b)*r*(x*u - y*v)*(z*w + x*u)*(z*w + y*v) = 0.

It is easy to verify that the following twelve points lie on CC(P,U):

A, B, C; the vertices of the anticevian triangle of U; 1/u : 1/v : 1/w; a/u : b/v : c/w; b*c/u : c*a/v : a*b/w, and the vertices of the cevian triangle of the point U' shown below.

Note that the first nine points are invariant of P, but that for fixed U, distinct points P yield distinct cubics CC(P,U). Thus, for fixed U, these cubics provide examples of "Cayley's Paradox" (see Cramer's Theorem or When Nine Points Are Worth But Eight: Euler's Resolution of Cramer's Paradox.

The cubic CC(P,U) is the cubic pK(P',U'), where

P' = v^2w^2 : w^2 u^2 : u^2 v^2      and      U' = v*w*(b+c)*p - (a+c)*q - (a+b)*r : :

For a discussion of pK cubics, see Special Isocubics in the Triangle Plane..

See also the preambles just before X(39693) and X(39765).

Example 1. The cubic CC(X(10),X(86)) = Pk(X(594),X(3995)) passes through A, B, C, the vertices of the cevian triangle of X(3995), the vertices of the anticevian triangle of X(10), and X(i) for i = 10, 37, 321, 3159, 3995. The tangents at A, B, C concur in X(40085).

Example 2. The cubic CC(X(10),X(514)) = Pk(X(1086),X(20295)) passes through A, B, C, the vertices of the cevian triangle of X(20295), the vertices of the anticevian triangle of X(514), and X(i) for i = 513, 514, 693, 14078, 20295. The tangents at A, B, C concur in X(40086).

Example 3. The cubic CC(X(2),X(6)) = Pk(X(1502),X(75)) passes through X(i) for i = 75, 76, 561, 6382, 6383, 18832, 18837.

Example 4. The cubic CC(X(10),X(6)) = Pk(X(1502),X(40087)) passes through X(40034).

Example 5. The cubic CC(X(42),X(6)) = Pk(X(1502),X(40088)) passes through X(40005).

Example 6. The cubic CC(X(519),X(6)) = Pk(X(1502),X(40089)) passes through X(3264).

Example 7. The cubic CC(X(524),X(6)) = Pk(X(1502),X(40090)).

Example 8. The cubic CC(X(2),X(76)) = Pk(X(32),X(31)) passes through X(i) for i = 1, 6, 31, 1740, 2162, 2176, 34248.

Example 9. The cubic CC(X(10),X(76)) = Pk(X(32),X(595)) passes through the vertices of the tangential triangle, the vertices of the cevian triangle of X(595), and X(i) for i = 1, 6, 31, 595, 16685, 39964.

Example 10. The cubic CC(X(42),X(76)) = Pk(X(32),X(1621)) passes through the vertices of the tangential triangle, the vertices of the cevian triangle of X(1621), and X(i) for i = 1, 6, 31, 1621, 8053, 34444.

Example 11. The cubic CC(X(519),X(76)) = Pk(X(32),X(40091)).

Example 12. The cubic CC(X(524),X(76)) = Pk(X(32),X(40096)).

Example 13. The cubic CC(X(2),X(894)) = Pk(X(40098),X(256)) passes through A, B, C, the vertices of the cevian triangle of X(256), the vertices of the anticevian triangle of X(257) and X(i) for i = 256, 257, 7018, 27447.

X(39994) = X(2)X(16723)∩X(4)X(5482)

Barycentrics    b*c*(a*b + b^2 - 3*a*c + b*c)*(-3*a*b + a*c + b*c + c^2) : :

X(39994) lies on these lines: {2, 16723}, {4, 5482}, {10, 244}, {37, 26844}, {75, 27797}, {76, 16727}, {83, 37633}, {88, 30866}, {226, 30566}, {321, 1086}, {799, 32014}, {2051, 18139}, {3762, 4049}, {3834, 4080}, {3837, 35353}, {3936, 14554}, {4033, 4359}, {4945, 36805}, {13576, 29824}, {16594, 17245}, {33172, 34258}

X(39994) = isogonal conjugate of X(33882)
X(39994) = isotomic conjugate of X(37680)

X(39995) = X(2)X(39706)∩X(10)X(75)

Barycentrics    b*c*(-(a^2*b) - a*b^2 - a^2*c + 2*a*b*c + b^2*c - a*c^2 + b*c^2) : :

X(39995) lies on these lines: {2, 39706}, {6, 29764}, {7, 18147}, {9, 29484}, {10, 75}, {37, 18143}, {38, 24688}, {86, 39949}, {141, 4043}, {192, 18040}, {312, 17184}, {314, 17273}, {319, 17152}, {320, 350}, {321, 17237}, {333, 19821}, {519, 39996}, {536, 4033}, {668, 17160}, {714, 3123}, {894, 18046}, {903, 18145}, {1086, 3948}, {1111, 35550}, {1278, 30473}, {1909, 17320}, {2234, 17793}, {3294, 29757}, {3644, 17786}, {3662, 18137}, {3666, 27792}, {3729, 18044}, {3760, 17274}, {3765, 17301}, {3770, 17302}, {3834, 4080}, {3912, 18150}, {3963, 17246}, {3975, 37756}, {3992, 25351}, {4110, 4764}, {4359, 4708}, {4364, 20913}, {4440, 17790}, {4479, 17149}, {4664, 20917}, {5249, 18743}, {6384, 39704}, {6549, 20924}, {10452, 17361}, {16696, 26971}, {16705, 16709}, {16726, 27166}, {17116, 29388}, {17143, 17271}, {17144, 17360}, {17231, 22016}, {17235, 20891}, {17252, 20174}, {17260, 29446}, {17261, 29396}, {17262, 29712}, {17277, 29756}, {17336, 28287}, {17349, 29802}, {17380, 34283}, {17495, 26844}, {17791, 18159}, {18073, 24004}, {20336, 21432}, {21858, 26756}, {24487, 30950}, {25269, 29542}, {25660, 26806}, {29437, 29812}, {29558, 29747}, {29760, 29795}

X(39996) = X(2)X(16723)∩X(75)X(537)

Barycentrics    b*c*(4*a^3 - a^2*b - 5*a*b^2 - a^2*c + 2*a*b*c + 3*b^2*c - 5*a*c^2 + 3*b*c^2) : :

X(39996) lies on these lines: {2, 16723}, {75, 537}, {519, 39995}, {545, 4033}, {3248, 26102}, {4479, 17135}, {5224, 17205}, {6376, 39704}, {17195, 17378}, {17330, 18143}, {17333, 18040}, {17361, 20245}, {17487, 24004}, {18073, 20072}, {18137, 27772}, {18145, 30939}, {18743, 31171}, {18750, 20942}, {29484, 37654}

X(39996) = anticomplement of X(39982)

X(39997) = X(2)X(4033)∩X(75)X(27797)

Barycentrics    b*c*(a^3*b - a^2*b^2 - 2*a*b^3 + a^3*c + 6*a^2*b*c - 5*a*b^2*c + 2*b^3*c - a^2*c^2 - 5*a*b*c^2 + 4*b^2*c^2 - 2*a*c^3 + 2*b*c^3) : :

X(39997) lies on these lines: {2, 4033}, {75, 27797}, {76, 1978}, {192, 26844}, {321, 18150}, {1086, 28605}, {3762, 21297}, {4358, 39699}, {4417, 30566}, {6376, 31035}, {14554, 33077}, {16594, 17243}, {18040, 27794}, {24004, 30579}, {27070, 31061}, {29699, 35595}

X(39998) = X(2)X(39)∩X(22)X(157)

Barycentrics    b^2*c^2*(2*a^2 + b^2 + c^2) : :

X(39998) lies on these lines: {2, 39}, {22, 157}, {23, 1799}, {25, 1235}, {32, 16932}, {51, 14994}, {69, 3060}, {75, 29679}, {83, 34482}, {99, 15246}, {111, 1241}, {115, 21248}, {141, 6664}, {187, 35929}, {251, 308}, {264, 6995}, {312, 18045}, {313, 26234}, {315, 7394}, {316, 1369}, {321, 26582}, {325, 1232}, {339, 6676}, {350, 3920}, {384, 1627}, {428, 7767}, {511, 33798}, {612, 3760}, {614, 3761}, {668, 33090}, {698, 8041}, {732, 20965}, {850, 10278}, {1078, 6636}, {1234, 37325}, {1236, 13595}, {1269, 3263}, {1447, 34388}, {1501, 8177}, {1502, 16986}, {1691, 10328}, {1909, 7191}, {1916, 31630}, {1920, 23989}, {1921, 28654}, {1930, 20703}, {1975, 7485}, {2979, 18906}, {3051, 24256}, {3108, 7839}, {3231, 4074}, {3260, 37671}, {3267, 18310}, {3329, 33769}, {3589, 11205}, {3734, 16949}, {3770, 33854}, {3785, 7500}, {3818, 33796}, {3819, 4576}, {3836, 21415}, {3933, 37439}, {3936, 18052}, {3972, 16952}, {4121, 24206}, {4441, 10327}, {4993, 34384}, {5031, 16893}, {5359, 7770}, {5976, 36901}, {6376, 29667}, {7018, 18066}, {7081, 34387}, {7391, 11185}, {7484, 39906}, {7493, 18018}, {7533, 7768}, {7750, 34603}, {7824, 38862}, {7861, 31124}, {8617, 35294}, {8878, 16044}, {10330, 22352}, {14907, 20062}, {16951, 17128}, {17143, 33091}, {18067, 25760}, {18138, 18139}, {18835, 29838}, {21590, 31019}, {21813, 27030}, {22253, 39951}, {29648, 30963}, {29666, 31997}, {31076, 39565}, {36417, 37778}

X(39998) = isotomic conjugate of X(3108)
X(39998) = polar conjugate of isogonal conjugate of X(7767)

X(39999) = X(75)X(24169)∩X(76)X(3763)

Barycentrics    b^2*c^2*(3*a^4 + 5*a^2*b^2 + b^4 + 5*a^2*c^2 + b^2*c^2 + c^4) : :

X(39999) lies on these lines: {75, 24169}, {76, 3763}, {183, 5020}, {6292, 24733}, {6374, 16986}, {7771, 33801}, {9230, 26235}, {18143, 20452}

X(40000) = X(2)X(1031)∩X(6)X(76)

Barycentrics    (a^2 + b^2)*(a^2 + c^2)*(a^4 + a^2*b^2 + b^4 + a^2*c^2 + 3*b^2*c^2 + c^4) : :

X(40000) lies on these lines: {2, 1031}, {5, 8928}, {6, 76}, {1799, 16986}, {3589, 4577}, {5117, 32085}, {5157, 14247}, {7804, 16095}, {8266, 39557}, {9481, 19689}, {10159, 40003}

This is the end of PART 20: Centers X(38001) - X(40000)

PART 1: Introduction and Centers X(1) - X(1000) PART 2: Centers X(1001) - X(3000) PART 3: Centers X(3001) - X(5000)
PART 4: Centers X(5001) - X(7000) PART 5: Centers X(7001) - X(10000) PART 6: Centers X(10001) - X(12000)
PART 7: Centers X(12001) - X(14000) PART 8: Centers X(14001) - X(16000) PART 9: Centers X(16001) - X(18000)
PART 10: Centers X(18001) - X(20000) PART 11: Centers X(20001) - X(22000) PART 12: Centers X(22001) - X(24000)
PART 13: Centers X(24001) - X(26000) PART 14: Centers X(26001) - X(28000) PART 15: Centers X(28001) - X(30000)
PART 16: Centers X(30001) - X(32000) PART 17: Centers X(32001) - X(34000) PART 18: Centers X(34001) - X(36000)
PART 19: Centers X(36001) - X(38000) PART 20: Centers X(38001) - X(40000) PART 21: Centers X(40001) - X(42000)
PART 22: Centers X(42001) - X(44000) PART 23: Centers X(44001) - X(46000) PART 24: Centers X(46001) - X(48000)
PART 25: Centers X(48001) - X(50000) PART 26: Centers X(50001) - X(52000) PART 27: Centers X(52001) - X(54000)
PART 28: Centers X(54001) - X(56000) PART 29: Centers X(56001) - X(58000) PART 30: Centers X(58001) - X(60000)
PART 31: Centers X(60001) - X(62000) PART 32: Centers X(62001) - X(64000) PART 33: Centers X(64001) - X(66000)
PART 34: Centers X(66001) - X(68000) PART 35: Centers X(68001) - X(70000) PART 36: Centers X(70001) - X(72000)