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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)




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

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).

underbar



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)






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

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.

underbar



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)






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

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)

underbar



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)

underbar

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 PjPkP, 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)

underbar

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.

underbar



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 elllipseX(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.

underbar



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)

underbar



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)

underbar



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)

underbar

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)

underbar

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)

underbar



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)

underbar



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)

underbar



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),

underbar



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

underbar



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}}

underbar



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)

underbar



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)

underbar

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}, {5615,2300