## This is PART 13: Centers X(24001) -

 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)

### X(24001) = TRILINEAR POLE OF LINE X(1099)X(1784)

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

Line X(1099)X(1784) is the tangent to the inellipse that is the trilinear square of the Euler line, at X(1099) (the trilinear square of X(30)).

X(4240) is the unique point on the Euler line whose trilinear polar is parallel to the Euler line. X(24001) is the trilinear product X(2)*X(4240).

X(24001) lies on these lines: {19, 27}, {662, 811}, {24000, 24024}

X(24001) = trilinear pole of line X(1099)X(1784)
X(24001) = trilinear product X(i)*X(j) for these {i,j}: {2, 4240}, {4, 2407}, {30, 648}, {99, 1990}, {107, 11064}, {112, 3260}, {162, 14206}, {264, 2420}, {662, 1784}, {670, 14581}, {811, 2173}

### X(24002) = TRILINEAR POLE OF LINE X(11)X(1111)

Barycentrics    b c (b - c)/(b + c - a) : :

Line X(11)X(1111) is the tangent to the inellipse that is the trilinear square of the Gergonne line, at X(1111) (the trilinear square of X(514)). This inellipse has center X(11019) and Brianchon point (perspector) X(1088).

Let P1 and P2 be the two points on the Gergonne line whose trilinear polars are parallel to the Gergonne line. P1 and P2 lie on the circumconic centered at X(1086) (hyperbola {{A, B, C, X(2), X(7)}}), and circle {{X(2), X(109), X(675)}}. The midpoint of P1 and P2 is X(1638). X(24002) is the trilinear product P1*P2.

X(24002) lies on these lines: {7, 513}, {85, 20949}, {109, 2860}, {226, 4776}, {273, 2400}, {279, 2401}, {514, 7216}, {522, 693}, {651, 666}, {655, 658}, {764, 7185}, {918, 3261}, {929, 934}, {1014, 17212}, {1090, 1111}, {1441, 20504}, {1445, 21390}, {1446, 23100}, {2402, 4000}, {2517, 4411}, {3323, 7336}, {3663, 23798}, {3667, 23819}, {3669, 4560}, {4130, 4885}, {4378, 7176}, {4397, 20907}, {4462, 10029}, {4486, 21206}, {4554, 4582}, {4617, 13149}, {4905, 19594}, {5228, 21007}, {7658, 14282}, {20520, 21189}

X(24002) = reflection of X(4130) in X(4885)
X(24002) = isotomic conjugate of X(644)
X(24002) = trilinear pole of line X(11)X(1111)
X(24002) = crossdifference of every pair of points on line X(41)X(1253)
X(24002) = trilinear product X(i)*X(j) for these {i,j}: {2, 3676}, {7, 514}, {10, 17096}, {11, 658}, {34, 15413}, {56, 3261}, {57, 693}, {59, 23100}, {65, 7199}, {75, 3669}, {77, 17924}, {85, 513}, {226, 7192}, {244, 4554}, {273, 905}, {278, 4025}, {279, 522}, {307, 17925}, {314, 7216}, {331, 1459}, {479, 8834}, {552, 4024}, {553, 4608}, {650, 1088}, {651, 1111}, {653, 1565}, {664, 1086}, {738, 4397}, {934, 4858}, {961, 4509}, {1014, 1577}, {2401, 22464}, {20615, 20949}

### X(24003) = CENTER OF TRILINEAR SQUARE OF NAGEL LINE

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

The inellipse that is the trilinear square of the Nagel line passes through X(1), X(75), X(341), X(872), X(1089) and X(4738). The Brianchon point (perspector) of this inellipse is X(7035). The vertex conjugate of the foci of this inellipse is X(765),

X(24003) lies on these lines: {1, 1120}, {2, 38}, {10, 11}, {120, 124}, {900, 3035}, {3740, 3741} et al

X(24003) = midpoint of X(1) and X(4738)
X(24003) = complement of X(244)
X(24003) = perspector of mid- and side-triangles of Gemini triangles 15 and 16

### X(24004) = TRILINEAR POLE OF LINE X(519)X(3992)

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

Line X(519)X(3992) is the tangent to the inellipse that is the trilinear square of the Nagel line, at X(4738) (the trilinear square of X(519)).

X(17780) is the unique point on the Nagel line whose trilinear polar is parallel to the Nagel line. X(24004) is the trilinear product X(2)*X(17780).

X(24004) lies on these lines: {2, 37}, {100, 9059}, {190, 646}, {662, 7258}, {666, 5387}, {889, 3572}, {1016, 4585}, {2325, 3264} et al

X(24004) = isotomic conjugate of X(1022)
X(24004) = trilinear pole of line X(519)X(3992)
X(24004) = crossdifference of every pair of points on line X(667)X(3248)
X(24004) = trilinear product X(i)*X(j) for these {i,j}: {2, 17780}, {44, 668}, {75, 1023}, {86, 4169}, {99, 3943}, {100, 4358}, {101, 3264}, {145, 2415}, {190, 519}, {646, 1319}, {651, 4723}, {662, 3992}, {664, 2325}, {765, 3762}, {900, 1016}, {902, 1978}, {1145, 13136}, {1275, 4528}, {1317, 4582}, {1635, 7035}, {1639, 4998}, {1647, 6632}, {1897, 3977}, {2251, 6386}, {3257, 4738}, {3689, 4554}, {3699, 3911}, {3952, 16704}, {4120, 4600}, {4370, 4555}, {4432, 4562}, {4439, 4586}, {4561, 8756}, {4564, 4768}, {4584, 4783}, {4597, 4908}, {4601, 4730}, {4606, 4742}, {4633, 4819}, {4969, 6540}, {5440, 6335}, {6079, 16594}
X(24004) = barycentric product X(i)*X(j) for these {i,j}: {44, 1978}, {75, 17780}, {76, 1023}, {99, 3992}, {100, 3264}, {101, 9456}, {190, 4358}, {274, 4169}, {519, 668}, {646, 3911}, {664, 4723}, {765, 900}, {789, 4439}, {799, 3943}, {902, 6386}, {1016, 3762}, {2325, 4554}, {2415, 18743}, {3689, 4572}, {3977, 6335}, {4033, 16704}, {4120, 4601}, {4432, 4583}, {4555, 4738}, {4589, 4783}, {4768, 4998}, {4975, 6540}, {6079, 20900}

### X(24005) = CENTER OF TRILINEAR SQUARE OF ORTHIC AXIS

Barycentrics    cos B cot B + cos C cot C : :
Barycentrics    (b + c) (a^4 - 2 a^2 (b - c)^2 + (b^2 - c^2)^2) : :

The inellipse that is the trilinear square of the orthic axis passes through X(1109) and X(2310). The Brianchon point (perspector) of this inellipse is X(158). The vertex conjugate of the foci of this inellipse is X(1096).

X(24005) lies on these lines: {2, 326}, {6, 1210}, {9, 1737}, {10, 37}, {11, 2262}, {198, 1837}, {226, 7363} et al

X(24005) = complement of X(326)

### X(24006) = TRILINEAR POLE OF LINE X(1109)X(3708)

Barycentrics    (sec A) (cot B - cot C) : :
Barycentrics    tan A sin(B - C) : :
Barycentrics    b c (b^2 - c^2)/(b^2 + c^2 - a^2) : :

Line X(1109)X(3708) is the tangent to the inellipse that is the trilinear square of the orthic axis, at X(1109) (the trilinear square of X(523)).

X(24007) and X(24008) are the two points on the orthic axis whose trilinear polars are parallel to the orthic axis. X(24006) is the trilinear product X(24007)*X(24008).

X(24006) is the perspector of the anticevian triangle of X(19) and tangential triangle, wrt the excentral triangle, of the bianticevian conic of X(1) and X(4). This conic is a rectangular hyperbola passing through X(1), X(4), X(19), and the vertices of their anticevian triangles. It has center X(107) and is the excentral isogonal conjugate of line X(40)X(2939), the anticomplementary conjugate of line X(20)X(1330), and the anticomplementary isotomic conjugate of line X(1654)X(3164).

X(24006) lies on these lines: {4, 6003}, {19, 798}, {92, 14207}, {108, 2689}, {136, 15608}, {225, 4017}, {240, 522}, {424, 2501}, {661, 3064}, {810, 2616}, {823, 24000}, {1824, 4132}, {4064, 4086} et al

X(24006) = isogonal conjugate of X(4575)
X(24006) = isotomic conjugate of X(4592)
X(24006) = pole wrt polar circle of trilinear polar of X(662) (line X(1)X(21))
X(24006) = polar conjugate of X(662)
X(24006) = trilinear pole of line X(1109)X(3708)
X(24006) = crossdifference of every pair of points on line X(48)X(255)
X(24006) = trilinear product X(i)*X(j) for these {i,j}: {2, 2501}, {4, 523}, {6, 14618}, {10, 7649}, {19, 1577}, {25, 850}, {27, 4024}, {28, 4036}, {30, 18808}, {34, 4086}, {37, 17924}, {76, 2489}, {92, 661}, {98, 16230}, {99, 8754}, {107, 250}, {110, 2970}, {112, 338}, {162, 1109}, {225, 522}, {226, 3064}, {264, 512}, {393, 525}, {520, 1093}, {526, 6344}, {671, 14273}, {798, 1969}, {811, 2643}, {1096, 14208}, {4064, 8747}, {24007, 24008}
X(24006) = barycentric product X(4)*X(1577)

### X(24007) = POLAR CONJUGATE OF X(2479)

Barycentrics    SB*SC/(3 SB*SC - S^2 - K): : , K as at X(2454)

X(24007) is one of two points on the orthic axis (X(24008) is the other) whose trilinear polar is parallel to the orthic axis.

The trilinear polar of X(24007) meets the line at infinity at X(523).

X(24007) lies on the Kiepert hyperbola, the Dao-Moses-Telv circle, circle {{X(2), X(98), X(112)}} and these lines: {230, 231}, {671, 2480}

X(24007) = reflection of X(24008) in X(1637)
X(24007) = antipode of X(24008) in the Dao-Moses-Telv circle
X(24007) = pole wrt polar circle of trilinear polar of X(2479) (line X(2)X(24008))
X(24007) = polar conjugate of X(2479)
X(24007) = X(107)-Ceva conjugate of X(24008)
X(24007) = X(125)-cross conjugate of X(24008)
X(24007) = PU(4)-harmonic conjugate of X(24008)
X(24007) = {X(i),X(j)}-harmonic conjugate of X(24008) for these {i,j}: {468, 6103}, {647, 6130}, {2501, 16230}, {3018, 11657}

### X(24008) = POLAR CONJUGATE OF X(2480)

Barycentrics    SB*SC/(3 SB*SC - S^2 + K): : , K as at X(2454)

X(24008) is one of two points on the orthic axis (X(24007) is the other) whose trilinear polar is parallel to the orthic axis.

The trilinear polar of X(24008) meets the line at infinity at X(523).

X(24008) lies on the Kiepert hyperbola, the Dao-Moses-Telv circle, circle {{X(2), X(98), X(112)}} and these lines: {230, 231}, {671, 2479}

X(24008) = reflection of X(24007) in X(1637)
X(24008) = antipode of X(24007) in the Dao-Moses-Telv circle
X(24008) = pole wrt polar circle of trilinear polar of X(2480) (line X(2)X(24007))
X(24008) = polar conjugate of X(2480)
X(24008) = X(107)-Ceva conjugate of X(24007)
X(24008) = X(125)-cross conjugate of X(24007)
X(24008) = PU(4)-harmonic conjugate of X(24007)
X(24008) = {X(i),X(j)}-harmonic conjugate of X(24007) for these {i,j}: {468, 6103}, {647, 6130}, {2501, 16230}, {3018, 11657}

### X(24009) = CENTER OF TRILINEAR SQUARE OF SODDY LINE

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

The inellipse that is the trilinear square of the Soddy line passes through X(1), X(1088), X(1097) and X(24014). This inellipse is tangent to the Soddy line at X(1).

X(24009) lies on these lines: {1, 85}, {2, 24010}, {4904, 11019}

X(24009) = midpoint of X(1) and X(24014)
X(24009) = complement of X(24010)

### X(24010) = TRILINEAR SQUARE OF X(3900)

Trilinears    (b - c)^2 (b + c - a)^4 : :
Trilinears    squared distance from A to Soddy line : :

X(24010) lies on the inellipse centered at X(10) and these lines: {2, 24009}, {8, 3177}, {10, 24014}, {200, 644}, {244, 2968}, {756, 7046}, {1146, 2310}, {3022, 3119}, {3679, 24028}, {4642, 23667}, {23529, 24034}

X(24010) = reflection of X(24014) in X(10)
X(24010) = isogonal conjugate of X(24013)
X(24010) = isotomic conjugate of X(24011)
X(24010) = anticomplement of X(24009)
X(24010) = trilinear square of X(3900)
X(24010) = antipode of X(24014) in inellipse centered at X(10)
X(24010) = crossdifference of every pair of points on line X(1461)X(4617)
X(24010) = X(i)-isoconjugate of X(j) for these (i,j): {1, 24013}, {31, 24011}, {934, 934}

### X(24011) = PERSPECTOR OF TRILINEAR SQUARE OF SODDY LINE

Trilinears    1/((1 + cos A) (cos B - cos C))^2 : :
Trilinears    1/(sec^2(B/2) - sec^2(C/2))^2 : :
Trilinears    1/(tan^2(B/2) - tan^2(C/2))^2 : :
Trilinears    1/(a^2 (b - c)^2 (b + c - a)^4) : :

As a point P moves on the Soddy line, the locus of the trilinear pole of the tangent at P to hyperbola {{A, B, C, X(1), P}} is the trilinear polar of X(24011) (line X(658)X(3732)).

X(24011) lies on these lines: {657, 658}, {1275, 10025}, {4635, 15419}

X(24011) = isogonal conjugate of X(24012)
X(24011) = isotomic conjugate of X(24010)
X(24011) = trilinear square of X(658)
X(24011) = trilinear pole of line X(658)X(3732)
X(24011) = X(i)-isoconjugate of X(j) for these (i,j): {1, 24012}, {31, 24010}, {657, 657}

### X(24012) = TRILINEAR SQUARE OF X(657)

Trilinears    (1 + cos A)^2 (cos B - cos C)^2 : :
Trilinears    (sec^2(B/2) - sec^2(C/2))^2 : :
Trilinears    (tan^2(B/2) - tan^2(C/2))^2 : :
Trilinears    a^2 (b - c)^2 (b + c - a)^4 : :

X(24012) lies on these lines: {1, 658}, {244, 14714}, {872, 7071}, {1097, 7038}, {3022, 14936} et al

X(24012) = isogonal conjugate of X(24011)
X(24012) = trilinear square of X(657)
X(24012) = crossdifference of every pair of points on line X(658)X(3732)
X(24012) = trilinear pole, wrt incentral triangle, of Soddy line
X(24012) = X(i)-isoconjugate of X(j) for these (i,j): {1, 24011}, {658, 658}

### X(24013) = TRILINEAR SQUARE OF X(934)

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

Lies on the inellipse centered at X(10) and these lines: {658, 14837}, {1262, 6610}, {1323, 7045}, {3669, 4617}, {4637, 7254}, {7128, 7177} et al

X(24013) = isogonal conjugate of X(24010)
X(24013) = trilinear square of X(934)
X(24013) = trilinear pole of line X(1461)X(4617)
X(24013) = vertex conjugate of foci of inellipse centered at X(24009) (the trilinear square of the Soddy line)

### X(24014) = TRILINEAR SQUARE OF X(516)

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

X(24014) lies on the inellipse centered at X(10), the inellipse centered at X(24009), and these lines: {1, 85}, {2, 24026}, {10, 24010}, {149, 10580}, {200, 1897}, {244, 3914}, {1088, 4312}, {1089, 7952}, {1109, 1962}, {6533, 17102} et al

X(24014) = reflection of X(1) in X(24009)
X(24014) = reflection of X(24010) in X(10)
X(24014) = antipode of X(1) in inellipse centered at X(24009)
X(24014) = antipode of X(24010) in inellipse centered at X(10)
X(24014) = trilinear square of X(516)

### X(24015) = TRILINEAR POLE OF LINE X(516)X(24014)

Trilinears    (a^2 - b^2 cos C - c^2 cos B) / ((1 + cos A)(cos B - cos C)) : :
Barycentrics    b*c*(2*a^3-(b+c)*a^2-(b^2-c^2)*(b-c))*(a-b+c)^2*(c-a)*(a+b-c)^2*(a-b) : :

Line X(516)X(24014) is the tangent to the inellipse that is the trilinear square of the Soddy line, at X(24014) (the trilinear square of X(516)).

Let P1 and P2 be as at X(23973). X(24015) is the trilinear product P1*P2.

X(24015) lies on these lines: {2, 85}, {658, 3732}, {664, 1897}, {934, 9057}, {1275, 4585} et al

X(24015) = trilinear product X(516)*X(658)
X(24015) = trilinear pole of line X(516)X(24014)
X(24015) = crossdifference of every pair of points on line X(8641)X(24012)

### X(24016) = TRILINEAR POLE OF LINE X(6)X(911)

Trilinears    a/((b - c) (2 a^3 - a^2 (b + c) - (b - c)^2 (b + c)) (b + c - a)^2)

Let A', B', C' be the intersections of the Soddy line and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(24016).

X(24016) lies on the circumcircle and these lines: {7, 2723}, {77, 2739}, {100, 677}, {101, 1262}, {103, 1458}, {105, 1456}, {107, 7192}, {108, 4617}, {109, 7339}, {112, 7254}, {269, 2717}, {279, 2724}, {658, 9057}, {663, 6614}, {934, 24012}, {1042, 2700} et al

X(24016) = trilinear pole of line X(6)X(911)
X(24016) = Ψ(X(6), X(911))
X(24016) = trilinear product X(103)*X(934) (circumcircle-X(1) antipodes)

### X(24017) = CENTER OF TRILINEAR SQUARE OF VAN AUBEL LINE

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

X(24017) lies on this line: {2, 24020}

X(24017) = complement of X(24020)

### X(24018) = X(6)-ISOCONJUGATE OF X(107)

Trilinears    (cot A)(tan B - tan C) : :
Trilinears    (b^2 - c^2) (b^2 + c^2 - a^2)^2 : :
Trilinears    directed distance from A to van Aubel line : :
Barycentrics    (cos A)(tan B - tan C) : :

The trilinear polar of X(24018) passes through X(2632).

X(24018) lies on these lines: {69, 15411}, {100, 2727}, {514, 661}, {521, 656}, {525, 8611}, {822, 4131}, {1019, 6003} et al

X(24018) = isogonal conjugate of X(24019)
X(24018) = isotomic conjugate of X(823)
X(24018) = X(6)-isoconjugate of X(107)
X(24018) = trilinear product X(i)*X(j) for these {i,j}: {2, 520}, {6, 3265}, {63, 656}
X(24018) = crossdifference of every pair of points on line X(19)X(31)

### X(24019) = TRILINEAR PRODUCT X(6)*X(107)

Trilinears    (tan A)/(tan B - tan C) : :
Trilinears    1/((b^2 - c^2) (b^2 + c^2 - a^2)^2) : :

X(24019) is the trilinear product of the circumcircle intercepts of the van Aubel line.

X(24019) lies on these lines: {19, 2159}, {27, 21621}, {28, 911}, {101, 107}, {108, 112}, {158, 1910}, {162, 163}, {648, 3732}, {662, 811}, {692, 1783}, {909, 1474}, {923, 1096}, {1172,1905} et al

X(24019) = isogonal conjugate of X(24018)
X(24019) = trilinear product X(i)*X(j) for these {i,j}: {6, 107}, {19, 162}
X(24019) = trilinear pole of line X(19)X(31)
X(24019) = polar conjugate of X(14208)
X(24019) = X(i)-isoconjugate of X(j) for these (i,j): {1, 24018}, {2, 520}, {6, 3265}, {48, 14208}

### X(24020) = TRILINEAR SQUARE OF X(24018)

Trilinears    cot^2 A (tan B - tan C)^2 : :
Trilinears    (b^2 - c^2)^2 (b^2 + c^2 - a^2)^4 : :
Trilinears    squared distance from A to van Aubel line : :

X(24020) lies on these lines: {2, 24017}, {304, 1956}, {326, 4592}

X(24020) = isogonal conjugate of X(24022)
X(24020) = isotomic conjugate of X(24021)
X(24020) = anticomplement of X(24017)

### X(24021) = PERSPECTOR OF TRILINEAR SQUARE OF VAN AUBEL LINE

Trilinears    (sec^2 A)/(tan B - tan C)^2 : :
Trilinears    b^2 c^2/((b^2 - c^2)^2 (b^2 + c^2 - a^2)^4) : :

X(24021) lies on these lines: {1784, 1955}

X(24021) = isotomic conjugate of X(24020)
X(24021) = trilinear square of X(107)

### X(24022) = TRILINEAR SQUARE OF X(24019)

Trilinears    (tan^2 A)/(tan B - tan C)^2 : :
Trilinears    1/((b^2 - c^2)^2 (b^2 + c^2 - a^2)^4) : :

X(24022) lies on these lines: {1784, 1955}

X(24022) = isogonal conjugate of X(24020)
X(24022) = trilinear square of X(24019)
X(24022) = vertex conjugate of foci of inellipse centered at X(24017) (the trilinear square of the van Aubel line)

### X(24023) = TRILINEAR SQUARE OF X(1503)

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

X(24023) lies on the inellipse centered at X(10), the inellipse centered at X(24017), and these lines: {31, 92}, {1111, 4292} {11031, 24031}

X(24023) = trilinear square of X(1503)

### X(24024) = TRILINEAR POLE OF LINE X(2312)X(24023)

Trilinears    b^2 c^2 (b^6 + c^6 - 2 a^6 + a^4 b^2 +a^4 c^2 - b^4 c^2 - c^4 b^2)/((b^2 - c^2) (b^2 + c^2 - a^2)^2) : :

Line X(2312)X(24023) is the tangent to the inellipse that is the trilinear square of the van Aubel line, at X(24023) (the trilinear square of X(1503)).

Let P1 and P2 be as at X(23977). X(24024) is the trilinear product P1*P2.

X(24024) lies on these lines: {1, 29}, {24000, 24001}

X(24024) = trilinear product X(107)*X(1503)
X(24024) = trilinear pole of line X(2312)X(24023)

### X(24025) = CENTER OF TRILINEAR SQUARE OF LINE X(1)X(3)

Trilinears    c ((c - a) (c + a - b))^2 + b ((a - b) (a + b - c))^2 : :

The inellipse that is the trilinear square of line X(1)X(3) passes through X(1), X(255), X(269), X(1079), X(1103), X(1106), X(1253), X(1254) and X(24028). The Brianchon point (perspector) of this inellipse is X(7045).

X(24025) lies on these lines: {1, 88}, {2, 24026}, {3, 2817}, {10, 2968}, {227, 4297}, {498, 15065}, {516, 1465}, {651, 1768}, {676, 2804}, {758, 1735}, {774, 3216}, {899, 1736}, {1062,6796} et al

X(24025) = midpoint of X(1) and X(24028)
X(24025) = complement of X(24026)

### X(24026) = TRILINEAR SQUARE OF X(522)

Trilinears    (b c (b - c) (b + c - a))^2 : :
Trilinears    squared distance from A to line X(1)X(3) : :
Barycentrics    csc A (cos B - cos C)^2 : :
Barycentrics    b c (b - c)^2 (b + c - a)^2 : :

X(24026) lies on the inellipse centered at X(10) and these lines: {1, 318}, {2, 24025}, {8, 80}, {11, 123}, {92, 1699}, {124, 20620}, {321, 4712}, {522, 7004}, {1099, 1125}, {1111, 3120}, {4647, 4736} et al

X(24026) = reflection of X(24028) in X(10)
X(24026) = isogonal conjugate of X(24027)
X(24026) = isotomic conjugate of X(7045)
X(24026) = anticomplement of X(24025)
X(24026) = trilinear square of X(522)
X(24026) = antipode of X(24028) in inellipse centered at X(10)
X(24026) = pole wrt polar circle of trilinear polar of X(7128) (line X(108)X(109))
X(24026) = polar conjugate of X(7128)
X(24026) = X(i)-isoconjugate of X(j) for these (i,j): {1, 24027}, {31, 7045}, {48, 7128}, {109, 109}

### X(24027) = TRILINEAR SQUARE OF X(109)

Trilinears    a^2/(cos B - cos C)^2 : :
Trilinears    a^2/((b - c) (b + c - a))^2 : :

X(24027) lies on these lines: {1, 7012}, {36, 59}, {109, 1459}, {1106, 16944}, {1275, 5384}, {1415, 3063}, {1758, 4570} et al

X(24027) = isogonal conjugate of X(24026)
X(24027) = trilinear square of X(109)
X(24027) = vertex conjugate of foci of inellipse centered at X(24025) (the trilinear square of line X(1)X(3))

### X(24028) = TRILINEAR SQUARE OF X(517)

Trilinears    (-1 + cos B + cos C)^2 : :
Trilinears    (b^3 + c^3 - (a^2 + b c) (b + c) + 2 a b c)^2 : :

X(24028) lies on the inellipse centered at X(10), the inellipse centered at X(24025), and these lines: {1, 88}, {8, 24031}, {10, 24026}, {31, 998}, {38, 12647}, {40, 109}, {46, 1106}, {55, 1411}, {65, 1066}, {80, 2310}, {201, 5690}, {269, 2093}, {347, 4566}, {517, 1457}, {519, 1735}, {656, 6739}, {756, 15065}, {774, 10573}, {952, 7004}, {1074, 1111}, {1103, 7991}, {1109, 2292}, {1110, 2717}, {1145, 1769}, {1210, 3987}, {1253, 1718}, {1254, 5903} et al

X(24028) = reflection of X(1) in X(24025)
X(24028) = reflection of X(24026) in X(10)
X(24028) = antipode of X(1) in inellipse centered at X(24025)
X(24028) = antipode of X(24026) in inellipse centered at X(10)
X(24028) = trilinear square of X(517)

### X(24029) = TRILINEAR POLE OF LINE X(517)X(1457)

Trilinears    (-1 + cos B + cos C)/(cos B - cos C) : :
Trilinears    (b^3 + c^3 - (a^2 + b c) (b + c) + 2 a b c)/((b - c) (b + c - a)) : :

Line X(517)X(1457) is the tangent to the inellipse that is the trilinear square of line X(1)X(3), at X(24028) (the trilinear square of X(517)).

Let P1 and P2 be as at X(23981). X(24029) is the trilinear product P1*P2.

X(24029) lies on the circumconic centered at X(23980) and these lines: {2, 7}, {101, 651}, {108, 1331}, {109, 9058}, {190, 653}, {513, 2283}, {901, 2222}, {1360, 6068} et al

X(24029) = trilinear pole of line X(517)X(1457)
X(24029) = crossdifference of every pair of points on line X(663)X(2310)
X(24029) = trilinear product X(i)*X(j) for these {i,j}: {7, 2427}, {101, 22464}, {109, 908}, {190, 1457}, {517, 651}, {664, 2183}, {2222, 16586}

### X(24030) = CENTER OF TRILINEAR SQUARE OF LINE X(1)X(4)

Barycentrics    (sec C - sec A)^2 (csc B) + (sec A - sec B)^2 (csc C) : :

The inellipse that is the trilinear square of line X(1)X(4) passes through X(1), X(158), X(7138) and X(24034).

X(24030) lies on these lines: {1, 318}, {2, 24030}, {10, 7358}, {11, 118}, {1359, 4297}, {1745, 1895} et al

X(24030) = midpoint of X(1) and X(24034)
X(24030) = complement of X(24031)

### X(24031) = TRILINEAR SQUARE OF X(521)

Trilinears    (sec B - sec C)^2 (csc^2 A) : :
Trilinears    (b - c)^2 (b + c - a)^2 (b^2 + c^2 - a^2)^2 : :
Trilinears    squared distance from A to line X(1)X(4) : :

X(24031) lies on the inellipse centered at X(10) and these lines: {1, 271}, {2, 24030}, {10, 24034}, {36, 7111}, {63, 100}, {78, 255}, {84, 1295}, {1099, 6734}, {2155, 7097} et al

X(24031) = reflection of X(24034) in X(10)
X(24031) = isogonal conjugate of X(24033)
X(24031) = isotomic conjugate of X(24032)
X(24031) = anticomplement of X(24030)
X(24031) = antipode of X(24034) in inellipse centered at X(10)
X(24031) = trilinear square of X(521)
X(24031) = X(i)-isoconjugate of X(j) for these (i,j): {1, 24033}, {31, 24032}, {108, 108}

### X(24032) = PERSPECTOR OF TRILINEAR SQUARE OF LINE X(1)X(4)

Trilinears    1/(sec B - sec C)^2 : :

As a point P moves on line X(1)X(4), the locus of the trilinear pole of the tangent at P to hyperbola {{A, B, C, X(1), P}} is line X(653)X(1020), the trilinear polar of X(24032).

X(24032) lies on these lines: {652, 653}, {693, 13149}, {7045, 15146} et al

X(24032) = isogonal conjugate of X(2638)
X(24032) = isotomic conjugate of X(24031)
X(24032) = trilinear square of X(653)
X(24032) = trilinear pole of line X(653)X(1020)
X(24032) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2638}, {31, 24031}, {652, 652}

### X(24033) = TRILINEAR SQUARE OF X(108)

Trilinears    a^2/(sec B - sec C)^2 : :
Trilinears    1/((b - c) (b + c - a) (b^2 + c^2 - a^2))^2 : :

X(24033) lies on these lines: {108, 6129}, {516, 1785}, {910, 7115}, {1456, 1875} et al

X(24033) = isogonal conjugate of X(24031)
X(24033) = trilinear square of X(108)
X(24033) = vertex conjugate of foci of inellipse centered at X(24030) (the trilinear square of line X(1)X(4))

### X(24034) = TRILINEAR SQUARE OF X(515)

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

X(24034) lies on the inellipse centered at X(10), the inellipse centered at X(24030), and these lines: {1, 318}, {7, 80}, {10, 24031}, {158, 5691}, {244, 1210}, {1109, 2650} et al

X(24034) = reflection of X(1) in X(24030)
X(24034) = reflection of X(24031) in X(10)
X(24034) = antipode of X(1) in inellipse centered at X(24030)
X(24034) = antipode of X(24031) in inellipse centered at X(10)
X(24034) = trilinear square of X(515)

### X(24035) = TRILINEAR POLE OF LINE X(515)X(24034)

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

Line X(515)X(24034) is the tangent to the inellipse that is the trilinear square of line X(1)X(4), at X(24030) (the trilinear square of X(515)).

Let P1 and P2 be as at X(23987). X(24035) is the trilinear product P1*P2.

X(24035) lies on these lines: {2, 92}, {108, 9056}, {653, 1020}, {664, 6335}

X(24035) = trilinear product X(515)*X(653)
X(24035) = trilinear pole of line X(515)X(24034)
X(24035) = crossdifference of every pair of points on line X(1946)X(2638)

### X(24036) = CENTER OF TRILINEAR SQUARE OF LINE X(1)X(6)

Trilinears    a^2 (b + c) - 2 a (b^2 + c^2) + b^3 + c^3 : :
Trilinears    b (a - b)^2 + c (a - c)^2 : :

The inellipse that is the trilinear square of line X(1)X(6) passes through X(1), X(31), X(200), X(678), X(756) and X(4712). The Brianchon point (perspector) of this inellipse is X(765). The vertex conjugate of the foci of this inellipse is X(1110).

X(24036) lies on these lines: {1, 644}, {2, 1111}, {6, 4574}, {9, 48}, {10, 1146}, {37, 537}, {45, 9259}, {100, 5540}, {106, 2297}, {292, 5283}, {519, 3693}, {620, 14838}, {665, 1639}, {672, 758}, {918, 3960}, {1001, 11716}, {1017, 11224}, {1018, 2170}, {1083, 11712}, {1108, 3950} et al

X(24036) = midpoint of X(1) and X(4712)
X(24036) = complement of X(1111)

### X(24037) = PERSPECTOR OF TRILINEAR SQUARE OF LINE X(2)X(6)

Trilinears    b^2 c^2/(b^2 - c^2)^2 : :

The trilinear square of line X(2)X(6) is the inellipse centered at X(21254). This inellipse passes through X(31), X(75), X(757), X(873), X(4094) and X(24038). The vertex conjugate of the foci of this inellipse is X(24041).

X(24037) lies on these lines: {99, 4367}, {512, 9425}, {662, 1924}, {799, 1577}, {1016, 1509}, {1580, 14210} et al

X(24037) = isotomic conjugate of X(2643)
X(24037) = trilinear square of X(99)
X(24037) = trilinear pole of line X(662)X(799)

### X(24038) = TRILINEAR SQUARE OF X(524)

Trilinears    b^2 c^2 (2 a^2 - b^2 - c^2)^2 : :

X(24038) lies on the inellipse centered at X(10), the inellipse centered at X(21254), and these lines: {31, 4592}, {63, 2157}, {75, 799}, {244, 4357}, {350, 1227}, {1111, 4359}, {1366, 7067}, {2642, 4750}, {3218, 17755}, {6629, 14210} et al

X(24038) = trilinear square of X(524)

### X(24039) = TRILINEAR POLE OF LINE X(6629)X(14210)

Trilinears    b^2 c^2 (2a^2 - b^2 - c^2)/(b^2 - c^2) : :

Line X(6629)X(14210) is the tangent to the inellipse that is the trilinear square of line X(2)X(6), at X(24038) (the trilinear square of X(524)).

X(5468) is the unique point on line X(2)X(6) whose trilinear polar is parallel to line X(2)X(6). X(24039) is the trilinear product X(2)*X(5468).

X(24039) lies on these lines: {1, 75}, {99, 8691}, {662, 799}

X(24039) = trilinear pole of line X(6629)X(14210)
X(24039) = trilinear product X(i)*X(j) for these {i,j}: {2, 5468}, {76, 5467}, {99, 524}, {110, 3266}, {187, 670}, {662, 14210}, {690, 4590}, {799, 896}, {892, 2482}, {1992, 2418}, {4601, 14419}

### X(24040) = CENTER OF TRILINEAR SQUARE OF LINE X(115)X(125)

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

The Brianchon point (perspector) of this inellipse is X(1109). The vertex conjugate of the foci of this inellipse is X(2643).

X(24040) lies on these lines: {2, 24041}, {115, 3700}, {2615, 7180}, {4092, 12072}, {4369, 8287}

X(24040) = complement of X(24041)

### X(24041) = TRILINEAR SQUARE OF X(662)

Trilinears    1/(cot B - cot C)^2 : : Trilinears    1/(cos 2B - cos 2C)^2 : :
Trilinears    1/(b^2 - c^2)^2 : :
Trilinears    squared distance from A to line X(115)X(125) : :
Barycentrics    csc A csc^2(B - C) : :

X(24041) is the vertex conjugate of the foci of the inellipse that is the trilinear square of line X(2)X(6). This inellipse is centered at X(21254) and has Brianchon point (perspector) X(24037).

X(24041) lies on these lines: {2, 24040}, {99, 4467}, {249, 1931}, {261, 4998}, {415, 4620}, {593, 1252}, {661, 662}, {799, 4575}, {896, 1101}, {1580, 14210} et al

X(24041) = isogonal conjugate of X(2643)
X(24041) = isotomic conjugate of X(1109)
X(24041) = anticomplement of X(24040)
X(24041) = trilinear square of X(662)
X(24041) = trilinear pole of line X(163)X(662)
X(24041) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2643}, {31, 1109}, {661, 661}

### X(24042) =  X(1)X(4)∩X(3)X(20107)

Barycentrics    2 a^7-2 a^6 b-2 a^5 b^2+2 a^4 b^3-2 a^3 b^4+2 a^2 b^5+2 a b^6-2 b^7-2 a^6 c+4 a^5 b c-a^4 b^2 c+a^3 b^3 c+a^2 b^4 c-5 a b^5 c+2 b^6 c-2 a^5 c^2-a^4 b c^2+2 a^3 b^2 c^2-3 a^2 b^3 c^2-2 a b^4 c^2+6 b^5 c^2+2 a^4 c^3+a^3 b c^3-3 a^2 b^2 c^3+10 a b^3 c^3-6 b^4 c^3-2 a^3 c^4+a^2 b c^4-2 a b^2 c^4-6 b^3 c^4+2 a^2 c^5-5 a b c^5+6 b^2 c^5+2 a c^6+2 b c^6-2 c^7 : :
X(24042) = 3 X[4881] - 5 X[8227], 5 X[3843] - X[18524], 2 X[6681] - 3 X[23513].

See Antreas Hatzipolakis and Peter Moses Hyacinthos 28370.

X(24042) lies on these lines: {1,4}, {3,20107}, {30,6713}, {382,10893}, {517,6246}, {519,10738}, {1319,16174}, {2077,10724}, {3627,7681}, {3667,21179}, {3814,5840}, {3830,22753}, {3843,11496}, {3845,7680}, {4299,10598}, {4302,6968}, {4324,6949}, {4881,8227}, {5046,6684}, {5176,14217}, {5450,10896}, {5806,16125}, {6681,23513}, {6796,12953}, {6923,10165}, {6929,10175}, {6965,10172}, {7704,21842}, {7743,11715}, {7956,15687}, {8666,11928}, {10265,12764}, {10525,11362}

X(24042) = midpoint of X(i) and X(j) for these {i,j}: {4, 3583}, {2077, 10724}, {5176, 14217}
X(24042) = reflection of X(i) in X(j) for these {i,j}: {1319, 16174}, {1519, 18483}, {11715, 7743}
X(24042) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 5225, 6256), (4, 5603, 18513)

### X(24043) =  X(4)X(11538)∩X(30)X(137)

Barycentrics    2 a^16-9 a^14 b^2+17 a^12 b^4-17 a^10 b^6+5 a^8 b^8+13 a^6 b^10-21 a^4 b^12+13 a^2 b^14-3 b^16-9 a^14 c^2+34 a^12 b^2 c^2-41 a^10 b^4 c^2+10 a^8 b^6 c^2-9 a^6 b^8 c^2+56 a^4 b^10 c^2-61 a^2 b^12 c^2+20 b^14 c^2+17 a^12 c^4-41 a^10 b^2 c^4+28 a^8 b^4 c^4-a^6 b^6 c^4-48 a^4 b^8 c^4+105 a^2 b^10 c^4-60 b^12 c^4-17 a^10 c^6+10 a^8 b^2 c^6-a^6 b^4 c^6+26 a^4 b^6 c^6-57 a^2 b^8 c^6+108 b^10 c^6+5 a^8 c^8-9 a^6 b^2 c^8-48 a^4 b^4 c^8-57 a^2 b^6 c^8-130 b^8 c^8+13 a^6 c^10+56 a^4 b^2 c^10+105 a^2 b^4 c^10+108 b^6 c^10-21 a^4 c^12-61 a^2 b^2 c^12-60 b^4 c^12+13 a^2 c^14+20 b^2 c^14-3 c^16 : :

See Antreas Hatzipolakis and Peter Moses Hyacinthos 28370.

X(24043) lies on these lines: {4,11538}, {30,137}

### X(24044) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24044) lies on these lines: {2, 24049}, {8, 4115}, {10, 4037}, {37, 1574}, {321, 1930}, {514, 17762}, {594, 4075}, {1089, 4103}, {1921, 4043}, {2321, 4053}, {3625, 21839}, {3626, 21879}, {3969, 24056}, {4058, 21810}, {4060, 21873}, {4135, 22206}, {4647, 21921}, {5295, 18250}, {17233, 24050}, {21883, 25092}, {24045, 24064}, {24047, 24052}, {24069, 24071}, {24077, 24080}

### X(24045) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24045) lies on these lines: {2, 1796}, {3, 5134}, {4, 101}, {5, 3730}, {6, 9665}, {9, 6990}, {11, 4253}, {41, 3583}, {85, 5074}, {115, 2176}, {116, 17753}, {169, 1699}, {190, 7752}, {218, 10896}, {220, 381}, {312, 4153}, {382, 3207}, {388, 9327}, {430, 3190}, {469, 22000}, {499, 5030}, {573, 15908}, {595, 3767}, {626, 4713}, {672, 7741}, {847, 1826}, {908, 21073}, {910, 22793}, {946, 5179}, {995, 5254}, {1018, 11681}, {1055, 10483}, {1146, 22791}, {1212, 9955}, {1334, 7951}, {1479, 4251}, {1724, 17737}, {1759, 5057}, {1802, 18406}, {2140, 17671}, {2280, 4857}, {2476, 3294}, {3061, 11813}, {3501, 3814}, {3585, 9310}, {3760, 4766}, {3825, 17754}, {3835, 23100}, {3944, 16600}, {4056, 9318}, {4193, 16549}, {4209, 6710}, {4256, 9598}, {4258, 9668}, {4262, 6284}, {4417, 21070}, {5011, 12699}, {5046, 16788}, {5087, 25066}, {5692, 21029}, {5903, 21044}, {6603, 18480}, {6604, 10708}, {7746, 17735}, {7748, 21008}, {7755, 21793}, {8818, 16777}, {9259, 9651}, {9664, 18755}, {9956, 21872}, {11680, 16552}, {13881, 14974}, {16601, 17605}, {17233, 24057}, {17451, 18393}, {17717, 25092}, {21384, 24387}, {24044, 24064}, {24049, 24071}, {24068, 24070}

### X(24046) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24046) lies on these lines: {1, 88}, {2, 3670}, {3, 17054}, {5, 1086}, {6, 5708}, {8, 1739}, {10, 982}, {28, 34}, {31, 3336}, {36, 3924}, {38, 1698}, {39, 20271}, {40, 5573}, {42, 18398}, {43, 3874}, {45, 16853}, {46, 595}, {56, 1324}, {65, 995}, {72, 16610}, {75, 24166}, {85, 17205}, {145, 3987}, {191, 748}, {227, 3660}, {386, 942}, {405, 17595}, {484, 3915}, {496, 3756}, {497, 12534}, {519, 3976}, {573, 20227}, {581, 10202}, {596, 4385}, {602, 5535}, {726, 20923}, {758, 978}, {899, 5904}, {936, 1257}, {975, 5437}, {984, 3634}, {986, 1125}, {991, 9940}, {997, 11512}, {998, 4320}, {999, 15955}, {1015, 3959}, {1064, 15016}, {1068, 4000}, {1089, 17155}, {1104, 4257}, {1149, 5697}, {1193, 5902}, {1201, 5903}, {1210, 1785}, {1254, 3361}, {1279, 3579}, {1465, 4306}, {1468, 3337}, {1471, 15932}, {1616, 12702}, {1722, 18193}, {1724, 3218}, {1735, 3086}, {1736, 5704}, {1737, 23536}, {1738, 10916}, {1837, 6788}, {2140, 24185}, {2275, 3125}, {2292, 3624}, {2650, 5313}, {2901, 3210}, {3120, 7741}, {3159, 18743}, {3214, 17449}, {3216, 3868}, {3242, 9709}, {3290, 3730}, {3293, 3873}, {3454, 3662}, {3555, 3999}, {3616, 4424}, {3632, 4695}, {3663, 9843}, {3666, 5439}, {3673, 21208}, {3678, 16569}, {3735, 16604}, {3742, 3931}, {3772, 24910}, {3778, 3841}, {3782, 4187}, {3821, 17065}, {3825, 3944}, {3831, 24165}, {3833, 17591}, {3836, 4446}, {3878, 21214}, {3970, 17756}, {4253, 16583}, {4255, 15934}, {4283, 5791}, {4347, 9364}, {4359, 10479}, {4392, 9780}, {4414, 5259}, {4415, 17527}, {4419, 17559}, {4646, 5045}, {4657, 24923}, {4658, 5256}, {4859, 5705}, {5011, 16502}, {5030, 16968}, {5044, 16602}, {5121, 21616}, {5221, 16466}, {5247, 18201}, {5248, 17596}, {5264, 7191}, {5272, 12514}, {5400, 12528}, {5687, 17597}, {5709, 13329}, {6682, 19858}, {6997, 15434}, {7004, 9581}, {7204, 10521}, {7226, 19877}, {7759, 24699}, {8572, 10246}, {9534, 24620}, {10039, 23675}, {10306, 15287}, {10449, 17490}, {10571, 18838}, {11227, 15852}, {11263, 14815}, {12609, 24239}, {14964, 16716}, {15048, 21049}, {15839, 16200}, {15854, 25405}, {16600, 17754}, {16611, 21384}, {16700, 18180}, {16706, 20083}, {16817, 24627}, {16818, 24629}, {16823, 24464}, {16975, 21951}, {17734, 24914}, {17861, 24173}, {20320, 24218}, {21077, 24231}, {24789, 24880}

### X(24047) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24047) lies on these lines: {1, 5030}, {2, 1796}, {3, 101}, {5, 5134}, {9, 2173}, {21, 16549}, {35, 672}, {36, 1334}, {39, 595}, {41, 5010}, {54, 71}, {55, 4253}, {56, 9327}, {58, 2276}, {100, 16552}, {106, 16969}, {140, 17747}, {165, 169}, {171, 25092}, {190, 1078}, {213, 4256}, {218, 4262}, {404, 3294}, {484, 17451}, {573, 10310}, {574, 2176}, {596, 9108}, {644, 5303}, {902, 5299}, {993, 3501}, {995, 5013}, {1018, 2975}, {1155, 16601}, {1191, 5024}, {1212, 3579}, {1385, 21872}, {1475, 3746}, {1571, 16968}, {1724, 17756}, {1759, 25082}, {1766, 7549}, {2141, 4184}, {2170, 11010}, {2329, 5267}, {3052, 9605}, {3208, 8666}, {3218, 3970}, {3295, 5022}, {3336, 21808}, {3496, 24036}, {3683, 25068}, {3693, 3916}, {3930, 6763}, {4189, 16788}, {4284, 5301}, {4420, 17746}, {4520, 17614}, {4640, 25066}, {4652, 17742}, {4653, 17750}, {4713, 7815}, {4877, 17303}, {5074, 17095}, {5179, 6684}, {5248, 17754}, {5254, 17734}, {5445, 21044}, {6603, 13624}, {7280, 9310}, {7573, 22000}, {7772, 21793}, {8671, 23851}, {8715, 21384}, {14439, 17744}, {14829, 21070}, {14964, 17524}, {16600, 17596}, {16784, 23649}, {17682, 17729}, {24044, 24052}

### X(24048) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24048) lies on these lines: {1, 6}, {2, 24050}, {144, 22003}, {344, 22047}, {346, 4115}, {594, 8818}, {758, 21033}, {1018, 3949}, {1400, 4067}, {1761, 16553}, {1766, 3984}, {2171, 3678}, {3169, 3899}, {4006, 21871}, {4058, 21060}, {4361, 24076}, {5295, 18492}, {5949, 17757}, {17314, 24049}, {21075, 21675}

### X(24049) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24049) lies on these lines: {1, 4037}, {2, 24044}, {37, 1698}, {145, 4115}, {321, 16831}, {1018, 4099}, {2901, 3294}, {3175, 3970}, {3178, 3947}, {3632, 21879}, {3633, 21839}, {3661, 3995}, {3679, 21816}, {17314, 24048}, {24045, 24071}, {24077, 24092}

### X(24050) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24050) lies on these lines: {2, 24048}, {7, 22003}, {37, 527}, {72, 12563}, {75, 22047}, {142, 4053}, {226, 21069}, {344, 4115}, {514, 18714}, {3674, 3970}, {4043, 20924}, {6666, 21873}, {8818, 24086}, {17233, 24044}, {17243, 24067}, {17732, 24051}, {21879, 25072}, {22008, 22011}, {24077, 24081}, {24185, 24530}

### X(24051) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24051) lies on these lines: {1, 4115}, {2, 24044}, {10, 37}, {519, 21816}, {1125, 4037}, {1509, 24074}, {1959, 3970}, {3244, 21879}, {3635, 21839}, {3995, 16826}, {4664, 22003}, {16777, 24067}, {17732, 24050}, {21820, 25092}

### X(24052) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24052) lies on these lines: {2, 24064}, {37, 1054}, {171, 21883}, {1018, 4551}, {3699, 4115}, {17122, 21820}, {24044, 24047}

### X(24053) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24053) lies on these lines: {1, 6}, {17732, 24064}, {24044, 24047}

### X(24054) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24054) lies on these lines: {2, 1796}, {22, 5134}, {101, 7391}, {220, 5064}, {427, 17747}, {3730, 5133}, {4713, 21248}, {8750, 13854}

### X(24055) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24055) lies on these lines: {2, 1796}, {23, 5134}, {101, 5189}, {858, 17747}, {3730, 5169}, {24085, 24088}

### X(24056) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24056) lies on these lines: {312, 22038}, {3969, 24044}, {4153, 4177}, {17233, 24060}, {21070, 22010}, {22008, 22009}, {24061, 24071}

### X(24057) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24057) lies on these lines: {190, 7917}, {4150, 4174}, {17233, 24045}, {17732, 24074}, {24070, 24080}

### X(24058) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24058) lies on these lines: {2, 24067}, {37, 4472}, {75, 24224}, {226, 306}, {594, 24076}, {4043, 18157}, {4053, 4431}, {4115, 17277}, {4967, 21810}, {17233, 24044}, {17243, 24081}

### X(24059) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24059) lies on these lines: {2, 24092}, {350, 22012}, {873, 4568}, {1621, 4115}, {3969, 24044}, {4359, 22011}, {21817, 22035}

### X(24060) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24060) lies on these lines: {2, 24044}, {306, 1230}, {1920, 4043}, {3741, 4037}, {3969, 24071}, {4115, 17135}, {17233, 24056}, {20500, 22032}

### X(24061) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24061) lies on these lines: {2, 24044}, {3687, 21070}, {3840, 4037}, {4115, 10453}, {24056, 24071}

### X(24062) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24062) lies on these lines: {37, 3631}, {3912, 4053}, {3970, 22008}, {17233, 24044}, {17243, 24090}, {17297, 22003}, {24076, 24081}, {24083, 24084}

### X(24063) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24063) lies on these lines: {2, 24069}, {37, 25269}, {75, 4053}, {304, 3970}, {4687, 21816}, {4751, 21810}, {17233, 24044}, {17243, 24077}

### X(24064) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24064) lies on these lines: {2, 24052}, {37, 17889}, {4043, 22009}, {4417, 22038}, {17233, 24056}, {17717, 21883}, {17732, 24053}, {21090, 21923}, {22015, 22019}, {24044, 24045}

### X(24065) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24065) lies on these lines: {10, 20970}, {306, 1230}, {519, 16886}, {3695, 4115}, {3936, 22011}, {3969, 24044}, {4035, 22017}, {4071, 21081}, {7230, 24070}, {17233, 24045}, {20500, 21092}, {20501, 22008}, {21066, 21076}

### X(24066) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24066) lies on these lines: {2, 24048}, {226, 4053}, {312, 22047}, {1848, 22000}, {1870, 3191}, {3969, 24044}, {4115, 17776}, {5905, 22003}, {21072, 21077}, {24080, 24081}

### X(24067) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24067) lies on these lines: {2, 24058}, {6, 4115}, {37, 39}, {190, 757}, {321, 4967}, {519, 21873}, {594, 4075}, {2171, 7276}, {2321, 3971}, {3175, 17133}, {3686, 21879}, {3949, 4099}, {3950, 4053}, {3995, 17011}, {4065, 4272}, {5105, 19582}, {5257, 21816}, {8818, 24070}, {16777, 24051}, {17233, 24081}, {17242, 22047}, {17243, 24050}, {17262, 22003}, {17314, 24048}, {21101, 21820}

### X(24068) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24068) lies on these lines: {1, 3159}, {2, 596}, {5, 4884}, {10, 75}, {38, 1089}, {145, 14774}, {190, 595}, {192, 4065}, {194, 4568}, {518, 2901}, {519, 3869}, {522, 3913}, {537, 3874}, {714, 3743}, {1046, 24821}, {1107, 22036}, {1125, 3971}, {1126, 4360}, {1330, 21289}, {1698, 17155}, {1724, 3891}, {2176, 4115}, {2276, 21067}, {2292, 4692}, {3157, 23874}, {3175, 3555}, {3216, 3952}, {3293, 17147}, {3295, 17262}, {3454, 3703}, {3634, 24165}, {3670, 3701}, {3741, 4066}, {3831, 4125}, {3953, 4358}, {3987, 4723}, {3992, 24443}, {4013, 11681}, {4424, 4696}, {5267, 8669}, {5283, 22011}, {6693, 17602}, {6763, 17763}, {7226, 10479}, {9053, 15171}, {9369, 15955}, {16777, 24051}, {17114, 24816}, {17314, 17732}, {18135, 21208}, {18140, 24166}, {21101, 25092}, {24045, 24070}

### X(24069) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24069) lies on these lines: {2, 24063}, {244, 19862}, {321, 3452}, {3701, 21081}, {3969, 24086}, {4053, 4358}, {24044, 24071}

### X(24070) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24070) lies on these lines: {2, 24074}, {10, 115}, {190, 14061}, {226, 1358}, {1018, 4013}, {4120, 21090}, {4370, 5257}, {7230, 24065}, {8818, 24067}, {19862, 24956}, {21089, 21092}, {24045, 24068}, {24057, 24080}, {24072, 24076}

### X(24071) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24071) lies on these lines: {2, 24052}, {37, 3120}, {321, 20500}, {3936, 22038}, {3969, 24060}, {4728, 22031}, {21090, 21092}, {24044, 24069}, {24045, 24049}, {24056, 24061}, {24073, 24086}, {24076, 24078}

### X(24072) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24072) lies on these lines: {4568, 18035}, {21091, 21092}, {24070, 24076}

### X(24073) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24073) lies on these lines: {306, 20500}, {1826, 2970}, {21091, 22032}, {24071, 24086}

### X(24074) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24074) lies on these lines: {2, 24070}, {10, 148}, {99, 101}, {543, 21711}, {551, 894}, {1509, 24051}, {2891, 7283}, {3219, 5540}, {4037, 6629}, {6651, 17205}, {6758, 25265}, {17732, 24057}, {24187, 24956}

### X(24075) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24075) lies on these lines: {2, 24044}, {8, 4037}, {37, 19877}, {321, 5308}, {346, 24883}, {3621, 4115}, {17230, 21070}, {20053, 21839}

### X(24076) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24076) lies on these lines: {2, 24063}, {37, 86}, {321, 4033}, {536, 4053}, {545, 22003}, {594, 24058}, {918, 22031}, {3739, 21810}, {3943, 22047}, {4115, 4422}, {4361, 24048}, {4755, 21816}, {6370, 12078}, {17233, 24077}, {17243, 24050}, {17348, 21873}, {24062, 24081}, {24070, 24072}, {24071, 24078}

### X(24077) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24077) lies on these lines: {2, 24058}, {37, 24598}, {321, 3596}, {3159, 19858}, {3247, 3995}, {4037, 17242}, {4115, 17349}, {4135, 21829}, {17233, 24076}, {17243, 24063}, {22016, 22036}, {24044, 24080}, {24049, 24092}, {24050, 24081}

### X(24078) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24078) lies on these lines: {824, 4079}, {24071, 24076}

### X(24079) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24079) lies on these lines: {17233, 24045}, {24082, 24083}

### X(24080) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24080) lies on these lines: {2, 596}, {190, 7751}, {192, 21067}, {712, 20943}, {3661, 4066}, {4568, 20081}, {6376, 22036}, {7230, 17233}, {18832, 22039}, {24044, 24077}, {24057, 24070}, {24066, 24081}

### X(24081) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24081) lies on these lines: {2, 24044}, {37, 25358}, {3912, 4037}, {3969, 4075}, {3995, 17292}, {4115, 6542}, {17233, 24067}, {17243, 24058}, {17776, 24880}, {24050, 24077}, {24062, 24076}, {24066, 24080}

### X(24082) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24082) lies on these lines: {24079, 24083}

### X(24083) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24083) lies on these lines: {10, 8663}, {661, 22043}, {3261, 3835}, {4502, 23803}, {16892, 22037}, {24062, 24084}, {24079, 24082}

### X(24084) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24084) lies on these lines: {2, 24089}, {1577, 8611}, {3676, 7265}, {3835, 22044}, {24062, 24083}

### X(24085) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24085) lies on these lines: {693, 4079}, {24055, 24088}, {24062, 24083}, {24071, 24076}

### X(24086) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24086) lies on these lines: {10, 528}, {142, 24224}, {3912, 18073}, {3969, 24069}, {6666, 21675}, {8287, 22003}, {8818, 24050}, {21090, 21091}, {24070, 24072}, {24071, 24073}

### X(24087) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24087) lies on these lines: {24079, 24082}

### X(24088) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24088) lies on these lines: {850, 4079}, {24055, 24085}, {24079, 24082}

### X(24089) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24089) lies on these lines: {2, 24084}, {37, 4976}, {522, 649}, {4151, 4170}, {4468, 7265}, {4765, 22042}, {21196, 22043}

### X(24090) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24090) lies on these lines: {2, 24048}, {10, 20360}, {37, 524}, {321, 4052}, {594, 24058}, {3912, 21810}, {4053, 4357}, {4115, 17280}, {6646, 22003}, {17233, 24067}, {17243, 24062}, {17353, 21873}, {21879, 25101}, {24044, 24077}

### X(24091) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24091) lies on these lines: {3969, 24044}, {4053, 4071}, {22038, 22047}, {24062, 24083}

### X(24092) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(24092) lies on these lines: {2, 24059}, {75, 2140}, {1001, 4115}, {16777, 24051}, {17233, 24044}, {24049, 24077}

### X(24093) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24093) lies on these lines: {2, 24097}, {8, 900}, {10, 6550}, {191, 21385}, {513, 5836}, {514, 1125}, {523, 2292}, {659, 2975}, {676, 3445}, {891, 3878}, {1089, 3762}, {1220, 4581}, {1329, 3837}, {2826, 5690}, {3766, 20955}, {4448, 21105}, {5330, 21343}, {6366, 10912}, {10015, 24174}, {19947, 21198}, {24095, 24098}, {24101, 24104}, {24106, 24127}, {24111, 24112}, {24118, 24121}, {24122, 24130}, {24123, 24126}

### X(24094) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24094) lies on these lines: {1, 522}, {1265, 14429}, {24098, 24099}, {24117, 24119}

### X(24095) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24095) lies on these lines: {2, 24117}, {514, 894}, {523, 17246}, {4120, 17233}, {4361, 4750}, {16574, 21832}, {16892, 21133}, {20505, 21119}, {24093, 24098}, {24100, 24116}, {24101, 24107}, {24102, 24105}, {24106, 24112}, {24108, 24122}, {24118, 24133}, {24123, 24124}, {24129, 24130}

### X(24096) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24096) lies on these lines: {2, 24098}, {37, 650}, {219, 4435}, {346, 1639}, {900, 16885}, {3287, 23617}, {24097, 24116}, {24112, 24121}

### X(24097) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24097) lies on these lines: {1, 6550}, {2, 24093}, {10, 514}, {56, 659}, {145, 900}, {513, 3057}, {523, 1222}, {891, 5903}, {1482, 2826}, {2804, 15347}, {3336, 21385}, {3837, 11681}, {3904, 19582}, {4977, 5484}, {14421, 21201}, {18220, 23770}, {24096, 24116}, {24130, 24142}

### X(24098) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24098) lies on these lines: {2, 24096}, {192, 918}, {344, 1639}, {513, 11997}, {523, 3728}, {650, 25067}, {812, 4032}, {900, 17365}, {1638, 4000}, {2804, 20507}, {21104, 23757}, {21133, 24457}, {24093, 24095}, {24094, 24099}, {24118, 24129}, {24130, 24133}

### X(24099) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24099) lies on these lines: {1, 900}, {2, 24093}, {12, 3837}, {513, 12579}, {514, 3634}, {659, 5253}, {764, 4977}, {891, 3754}, {1022, 1224}, {1125, 6550}, {2826, 5901}, {4448, 23764}, {4943, 6161}, {18004, 21222}, {24094, 24098}, {24118, 24119}

### X(24100) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24100) lies on these lines: {1459, 2309}, {24094, 24098}, {24095, 24116}, {24103, 24111}, {24115, 24118}

### X(24101) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24101) lies on these lines: {16892, 20511}, {24093, 24104}, {24095, 24107}, {24102, 24106}, {24108, 24126}

### X(24102) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24102) lies on these lines: {24095, 24105}, {24101, 24106}, {24125, 24132}

### X(24103) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24103) lies on these lines: {2, 24118}, {141, 4728}, {812, 1654}, {1635, 17277}, {3762, 3963}, {4977, 21135}, {24093, 24095}, {24100, 24111}, {24121, 24129}

### X(24104) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24104) lies on these lines: {2, 24142}, {764, 24169}, {1621, 4448}, {17494, 24381}, {24093, 24101}

### X(24105) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24105) lies on these lines: {2, 24119}, {24095, 24102}

### X(24106) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24106) lies on these lines: {24093, 24127}, {24095, 24112}, {24101, 24102}

### X(24107) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24107) lies on these lines: {2, 24093}, {514, 6685}, {900, 17135}, {3741, 6550}, {24095, 24101}, {24120, 24126}

### X(24108) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24108) lies on these lines: {2, 24093}, {514, 6686}, {900, 10453}, {3840, 6550}, {24095, 24122}, {24101, 24126}

### X(24109) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24109) lies on these lines: {2, 24124}, {514, 4480}, {523, 2254}, {24093, 24095}, {24129, 24133}, {24134, 24135}

### X(24110) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24110) lies on these lines: {2, 24123}, {45, 6546}, {4389, 6545}, {14475, 17290}, {16892, 24457}, {24093, 24095}

### X(24111) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24111) lies on these lines: {2, 24126}, {55, 21105}, {514, 17596}, {982, 21132}, {16892, 20505}, {21118, 23761}, {24093, 24112}, {24095, 24101}, {24100, 24103}

### X(24112) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24112) lies on these lines: {4642, 17420}, {24093, 24111}, {24095, 24106}, {24096, 24121}, {24126, 24128}

### X(24113) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24113) lies on these lines: {2, 24096}, {345, 1639}, {650, 824}, {918, 17490}, {20505, 20508}, {21120, 21129}, {24093, 24111}

### X(24114) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24114) lies on these lines: {21121, 21134}, {24093, 24101}, {24095, 24102}

### X(24115) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24115) lies on these lines: {2, 24096}, {650, 25091}, {918, 3210}, {1639, 17776}, {24093, 24101}, {24100, 24118}, {24132, 24133}

### X(24116) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24116) lies on these lines: {10459, 17418}, {24093, 24111}, {24095, 24100}, {24096, 24097}

### X(24117) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24117) lies on these lines: {2, 24095}, {514, 3662}, {523, 17340}, {579, 21832}, {661, 17452}, {4120, 17314}, {24094, 24119}, {24096, 24097}

### X(24118) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24118) lies on these lines: {2, 24103}, {6, 1635}, {513, 16814}, {514, 24199}, {649, 2269}, {661, 1213}, {665, 21796}, {812, 17300}, {3572, 4581}, {4728, 17234}, {14284, 21834}, {21106, 21127}, {24093, 24121}, {24095, 24133}, {24096, 24097}, {24098, 24129}, {24099, 24119}, {24100, 24115}

### X(24119) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24119) lies on these lines: {2, 24105}, {661, 23897}, {2176, 14407}, {4079, 21879}, {14825, 17192}, {24094, 24117}, {24099, 24118}

### X(24120) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24120) lies on these lines: {2, 24095}, {63, 21832}, {3969, 4120}, {4988, 6546}, {21124, 21129}, {24093, 24101}, {24107, 24126}, {24123, 24138}

### X(24121) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24121) lies on these lines: {2, 24095}, {9, 21832}, {594, 4120}, {4988, 6544}, {24093, 24118}, {24096, 24112}, {24103, 24129}

### X(24122) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24122) lies on these lines: {24093, 24130}, {24095, 24108}

### X(24123) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24123) lies on these lines: {2, 24110}, {88, 6546}, {4750, 4763}, {6545, 24183}, {10015, 21115}, {14442, 16594}, {24093, 24126}, {24095, 24124}, {24120, 24138}

### X(24124) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24124) lies on these lines: {2, 24109}, {16892, 21129}, {24095, 24123}

### X(24125) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24125) lies on these lines: {3125, 14442}, {24102, 24132}, {24127, 24129}

### X(24126) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24126) lies on these lines: {2, 24111}, {100, 21105}, {244, 21132}, {514, 1054}, {659, 22379}, {2254, 2804}, {3120, 6545}, {24093, 24123}, {24101, 24108}, {24107, 24120}, {24112, 24128}, {24129, 24131}

### X(24127) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24127) lies on these lines: {514, 9259}, {7200, 21132}, {21133, 21138}, {24093, 24106}, {24125, 24129}

### X(24128) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^3 - 2 a^2 b - 3 a b^2 + 2 b^3 - 2 a^2 c + 11 a b c - 3 b^2 c - 3 a c^2 - 3 b c^2 + 2 c^3) : :

X(24128) lies on these lines: {1, 514}, {2, 24093}, {8, 6550}, {513, 14923}, {900, 3621}, {2826, 12245}, {12513, 13266}, {24112, 24126}

### X(24129) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24129) lies on these lines: {2, 24110}, {190, 6546}, {513, 21889}, {514, 4440}, {649, 16560}, {812, 3904}, {900, 4088}, {1086, 6545}, {2786, 21385}, {4422, 6544}, {4473, 10196}, {7202, 21143}, {24095, 24130}, {24098, 24118}, {24103, 24121}, {24109, 24133}, {24125, 24127}, {24126, 24131}

### X(24130) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24130) lies on these lines: {2, 24103}, {69, 812}, {513, 17351}, {514, 3663}, {1635, 17349}, {3596, 3762}, {4444, 21143}, {4728, 17232}, {24093, 24122}, {24095, 24129}, {24097, 24142}, {24098, 24133}

### X(24131) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24131) lies on these lines: {11, 244}, {519, 4645}, {903, 1644}, {1054, 4440}, {1320, 24715}, {6547, 6550}, {21087, 24821}, {24126, 24129}

### X(24132) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24132) lies on these lines: {2, 24105}, {514, 3959}, {24093, 24122}, {24102, 24125}, {24115, 24133}

### X(24133) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24133) lies on these lines: {2, 24093}, {241, 514}, {900, 6542}, {3912, 6550}, {24095, 24118}, {24098, 24130}, {24109, 24129}, {24115, 24132}

### X(24134) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24134) lies on these lines: {1015, 21196}, {1086, 3004}, {16892, 21138}, {24109, 24135}

### X(24135) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24135) lies on these lines: {2, 24140}, {2170, 4369}, {21104, 21139}, {24109, 24134}

### X(24136) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24136) lies on these lines: {244, 23772}, {1647, 4088}, {21140, 24193}, {24109, 24134}, {24126, 24129}

### X(24137) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24137) lies on these lines: {3121, 24900}, {24109, 24134}

### X(24138) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24138) lies on these lines: {1086, 21131}, {10015, 20509}, {24120, 24123}, {24125, 24127}

### X(24139) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24139) lies on these lines: {3801, 21142}

### X(24140) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24140) lies on these lines: {2, 24135}, {115, 661}, {2223, 21805}, {2325, 3930}, {3119, 4521}, {5375, 17439}

### X(24141) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24141) lies on these lines: {2, 24096}, {75, 918}, {523, 25124}, {650, 25099}, {659, 8424}, {900, 17332}, {1639, 17280}, {17069, 21202}, {24093, 24122}, {24095, 24118}, {24103, 24121}

### X(24142) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24142) lies on these lines: {2, 24104}, {764, 3821}, {1001, 4448}, {24093, 24095}, {24097, 24130}, {24099, 24118}

### X(24143) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

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

X(24143) lies on these lines: {4513, 14427}, {24096, 24097}

### X(24144) = CIRCUMCIRCLE-INVERSE OF X(252)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28378.

X(24144) lies on the cubic K466 and these lines: {3,252}, {54,1263}, {12026,19268}, {14072,21975}

X(24144) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (930, 1141, 252), (1263, 6343, 10285)
X(24144) = circumcircle-inverse of X(252)

### X(24145) = X(2)X(2006)∩X(81)X(1086)

Barycentrics    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 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28378.

X(24145) lies on these lines: {2,2006}, {81,1086}, {278,23958}, {651,1994}, {3960,4560}, {5718,17395}, {5723,17484}, {17013,19785}, {17732,24046}

### X(24146) = X(2)X(2006)∩X(343)X(17483)

Barycentrics    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 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28378.

X(24146) lies on these lines: {2,2006}, {343,17483}, {3448,21028}

### X(24147) = MIDPOINT OF X(1157) AND X(19553)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28378.

X(24147) lies on these lines: {140,1493}, {195,252}, {1157,19553}, {5965,10615}, {7159,14102}

X(24147) = midpoint of X(1157) and X(19553)
X(24147) = {X(195),X(252)}-harmonic conjugate of X(13856)

### X(24148) = X(239)X(514)∩X(323)X(4858)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28378.

X(24148) lies on these lines: {239,514}, {323,4858}, {1994,14213}, {3187,17035}, {3219,18662}, {17479,19742}

### X(24149) = X(2)X(7)∩X(37)X(16698)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28378.

X(24149) lies on these lines: {2,7}, {37,16698}, {81,18359}, {323,6358}, {1994,14213}, {17168,22002}

### X(24150) = X(1)X(3161)∩X(2)X(8051)

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

Recalling that triangle centers are functions, at (a,b,c) = (6,9,13), the values of X(4098) and X(24150) are equal.

X(24150) lies on the cubic K1077 and these lines: {1, 3161}, {2, 8051}, {9, 2137}

X(24150) = X(8051)-Ceva conjugate of X(6553)
X(24150) = X(i)-cross conjugate of X(j) for these (i,j): {9, 3161}, {6555, 145}
X(24150) = X(i)-isoconjugate of X(j) for these (i,j): {1616, 8056}, {3445, 23511}, {5382, 17071}, {8055, 16945}
X(24150) = barycentric product X(i)X(j) for these {i,j}: {145, 6553}, {3161, 8051}
X(24150) = barycentric quotient X(i)/X(j) for these {i,j}: {145, 4452}, {1743, 23511}, {2137, 19604}, {3052, 1616}, {3158, 2136}, {3161, 8055}, {4849, 21896}, {6553, 4373}, {6555, 6552}, {20818, 23089}

### X(24151) = X(8)X(3452)∩X(9)X(8056)

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

X(24151) lies on the cubic K1077 and these lines: {2, 8051}, {8, 3452}, {9, 8056}, {3038, 5573}, {4373, 18228}, {15601, 17958}

X(24151) = complement of X(8051)
X(24151) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 21627}, {31, 8056}, {41, 346}, {1616, 142}, {2136, 141}, {4452, 17046}, {6552, 21244}, {8055, 2887}, {21896, 17052}, {23511, 2886} X(24151) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 8056}, {19604, 3680} X(24151) = crosspoint of X(2) and X(8055)
X(24151) = X(i)-isoconjugate of X(j) for these (i,j): {1743, 2137}, {3052, 8051} X(24151) = barycentric product X(i)X(j) for these {i,j}: {2136, 4373}, {3680, 4452}, {6552, 19604}, {6557, 23511}, {8055, 8056} X(24151) = barycentric quotient X(i)/X(j) for these {i,j}: {1616, 1420}, {2136, 145}, {3445, 2137}, {3680, 6553}, {8055, 18743}, {8056, 8051}, {21896, 4848}, {23511, 5435}

### X(24152) (name pending)

Barycentrics    a*(-a + b + c)*(-a + b + c + Sqrt[-a^2 + 2*a*b - b^2 + 2*a*c + 2*b*c - c^2]) : :
X(24152) = (r + 4 R) (Sqrt[r (r + 4 R)] + s) X[9] - 2 (r + R) s X[55]

X(24152) lies on the cubic K1077 and this line: {9,55}

### X(24153) (name pending)

Barycentrics    a*(-a + b + c)*(-a + b + c - Sqrt[-a^2 + 2*a*b - b^2 + 2*a*c + 2*b*c - c^2]) : :
X(24153) = (r + 4 R) (Sqrt[r (r + 4 R)] - s) X[9] + 2 (r + R) s X[55]

X(24153) lies on the cubic K1077 and this line: {9,55}

### X(24154) (name pending)

Barycentrics    a*b - b^2 + a*c + 2*b*c - c^2 + a*Sqrt[-a^2 + 2*a*b - b^2 + 2*a*c + 2*b*c - c^2] : :
X(24154) = s X[1] + 2 Sqrt[r (r + 4 R)] X[142]

X(24154) lies on the conic {{A,B,C,X(2),X(7)}} and the cubic K1077, and on this line: {1,142}

### X(24155) (name pending)

Barycentrics    a*b - b^2 + a*c + 2*b*c - c^2 - a*Sqrt[-a^2 + 2*a*b - b^2 + 2*a*c + 2*b*c - c^2] : :
X(24155) = s X[1] - 2 Sqrt[r (r + 4 R)] X[142]

X(24155) lies on the conic {{A,B,C,X(2),X(7)}} and the cubic K1077, and on this line: {1,142}

### X(24156) (name pending)

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

X(24156) lies on the cubic K1077, and on this line: {277,5745}

### X(24157) (name pending)

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

X(24157) lies on the cubic K1077, and on this line: {277,5745}

### X(24158) X(1)X(236) ∩X(2)X(1488)

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

X(24158) lies on these lines: {1, 236}, {2, 1488}, {188, 3161}

X(24158) = X(9)-cross conjugate of X(236)
X(24158) = X(289)-isoconjugate of X(8078)
X(24158) = barycentric product X(7057)*X(12644)
X(24158) = barycentric quotient X(12644)/X(7048)

### X(24159) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24159) lies on these lines: {1, 224}, {2, 3670}, {3, 1086}, {5, 17054}, {6, 6147}, {7, 58}, {8, 24222}, {38, 19854}, {46, 3011}, {72, 24789}, {88, 17566}, {142, 975}, {221, 15253}, {226, 19372}, {244, 499}, {278, 4306}, {344, 3159}, {348, 17205}, {386, 2140}, {387, 11036}, {405, 3782}, {498, 24443}, {595, 4295}, {614, 12047}, {936, 4859}, {942, 3772}, {946, 990}, {982, 24161}, {986, 10198}, {988, 1125}, {995, 3485}, {997, 24178}, {1056, 15955}, {1279, 12699}, {1330, 19851}, {1387, 3445}, {1453, 4654}, {1478, 3924}, {1479, 3120}, {1616, 22791}, {1698, 24223}, {1714, 3868}, {1722, 21077}, {1724, 5905}, {1738, 3811}, {1739, 5552}, {1788, 17734}, {1834, 15934}, {1930, 4441}, {2345, 22011}, {3616, 4653}, {3649, 16466}, {3682, 24779}, {3752, 11374}, {3767, 20271}, {3953, 10527}, {3987, 10528}, {4252, 24470}, {4255, 5719}, {4256, 5703}, {4310, 19843}, {4346, 17558}, {4363, 17698}, {4364, 16844}, {4389, 11110}, {4415, 11108}, {4419, 16845}, {4642, 10056}, {4694, 10529}, {4962, 21172}, {5044, 17278}, {5230, 5902}, {5248, 24248}, {5266, 5880}, {5272, 21616}, {5439, 17720}, {5573, 8227}, {5687, 17724}, {5711, 17061}, {5758, 13329}, {6700, 24175}, {7040, 7649}, {7483, 17595}, {9579, 16485}, {9895, 24476}, {10200, 17063}, {10916, 17064}, {11269, 18398}, {13411, 24177}, {14023, 24699}, {14450, 17127}, {17597, 24390}, {17719, 24174}, {17871, 23555}, {19755, 20256}, {23581, 24204}, {24166, 24187}

### X(24160) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24160) lies on these lines: {1, 2476}, {2, 3670}, {10, 3699}, {35, 3120}, {37, 24937}, {58, 226}, {65, 17734}, {72, 24880}, {73, 2006}, {106, 23675}, {140, 1086}, {171, 11263}, {386, 3772}, {495, 15955}, {595, 3011}, {894, 6693}, {995, 11375}, {1125, 13161}, {1201, 5443}, {1279, 9955}, {1616, 18493}, {1656, 17054}, {1834, 5719}, {3487, 5292}, {3584, 4642}, {3782, 7483}, {3811, 17064}, {3838, 5266}, {3841, 5293}, {3915, 18393}, {3924, 7951}, {3944, 5248}, {3970, 17737}, {4256, 13411}, {4415, 6675}, {4861, 24222}, {5687, 17783}, {5904, 24892}, {6788, 17606}, {7746, 20271}, {7780, 24699}, {16706, 20108}, {17095, 17205}, {17724, 24390}, {17749, 24789}, {19862, 24171}, {19878, 25072}, {24170, 24187}

### X(24161) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24161) lies on these lines: {1, 442}, {2, 986}, {3, 17889}, {10, 3699}, {21, 3120}, {35, 24715}, {43, 11374}, {58, 11263}, {140, 1054}, {171, 12609}, {226, 5247}, {238, 1780}, {405, 3944}, {498, 24440}, {499, 1725}, {758, 24880}, {846, 6675}, {896, 14450}, {905, 16744}, {946, 7413}, {978, 11375}, {982, 24159}, {984, 19854}, {988, 23681}, {1010, 1125}, {1046, 3649}, {1086, 4999}, {1104, 3838}, {1330, 4892}, {1699, 13442}, {1717, 23708}, {1722, 5219}, {1738, 13411}, {1962, 24936}, {2476, 3924}, {2650, 24883}, {3011, 5255}, {3523, 7613}, {3584, 3987}, {3624, 4657}, {3754, 17734}, {3782, 24953}, {3821, 19270}, {3846, 16817}, {3868, 24892}, {3925, 5293}, {3976, 10527}, {4425, 11110}, {5015, 21241}, {5272, 8227}, {5497, 24387}, {5886, 15973}, {5988, 24181}, {6536, 17557}, {6857, 24248}, {7379, 16020}, {7483, 17596}, {8143, 11230}, {8616, 12699}, {13740, 25385}, {16887, 24187}, {17123, 21616}, {17593, 19862}, {17733, 18134}, {17737, 21808}, {19269, 24788}, {21264, 24774}, {23536, 24541}

X(24161) = perspector of Gemini triangle 7 and cross-triangle of Gemini triangles 3 and 7

### X(24162) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24162) lies on these lines: {7, 614}, {10, 24173}, {56, 2218}, {75, 24178}, {142, 3666}, {226, 20227}, {244, 307}, {982, 4357}, {988, 1125}, {1086, 18589}, {1441, 23675}, {1444, 17205}, {4656, 5750}, {17282, 20106}, {17863, 23536}, {21174, 23808}, {23663, 23677}

### X(24163) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24163) lies on these lines: {2, 3670}, {25, 17054}, {31, 57}, {38, 17306}, {191, 5272}, {427, 1086}, {612, 24443}, {1194, 20271}, {1739, 10327}, {3924, 5322}, {3987, 20020}, {4694, 19993}, {17871, 21210}

### X(24164) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24164) lies on these lines: {2, 3670}, {238, 244}, {614, 3337}, {858, 1086}, {1995, 17054}, {2228, 3006}, {3920, 4642}, {5297, 24443}, {9465, 20271}

### X(24165) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24165) lies on these lines: {1, 3210}, {2, 726}, {10, 38}, {21, 8720}, {42, 17140}, {43, 17490}, {57, 4362}, {63, 16825}, {75, 982}, {76, 20340}, {86, 17600}, {142, 24180}, {190, 17123}, {210, 537}, {244, 321}, {310, 4609}, {312, 4871}, {354, 740}, {404, 8669}, {514, 8027}, {518, 4685}, {519, 3873}, {522, 21204}, {536, 3742}, {551, 1962}, {553, 5847}, {614, 3923}, {714, 4688}, {750, 3891}, {752, 11246}, {756, 24589}, {846, 16823}, {899, 4090}, {968, 24331}, {984, 19804}, {1015, 21345}, {1054, 7081}, {1086, 2887}, {1125, 24598}, {1201, 17164}, {1215, 3752}, {1266, 24210}, {1269, 20598}, {1386, 4697}, {1575, 21101}, {2321, 3726}, {2886, 7263}, {3058, 17764}, {3159, 6532}, {3244, 3896}, {3338, 17733}, {3452, 21093}, {3634, 24068}, {3663, 4425}, {3666, 24325}, {3687, 24231}, {3696, 21342}, {3703, 3836}, {3705, 7897}, {3706, 3999}, {3720, 3993}, {3729, 4011}, {3739, 21080}, {3757, 17596}, {3771, 17740}, {3782, 3846}, {3819, 14839}, {3826, 4884}, {3831, 24046}, {3835, 21197}, {3953, 4647}, {3966, 4655}, {3967, 16602}, {4003, 6682}, {4025, 20518}, {4028, 5542}, {4038, 4360}, {4135, 4358}, {4385, 24174}, {4418, 7191}, {4438, 24789}, {4453, 20525}, {4514, 24715}, {4641, 4974}, {4651, 17154}, {4659, 5573}, {4671, 9335}, {4699, 17157}, {4703, 17276}, {4706, 4946}, {4709, 17135}, {4847, 20882}, {4850, 6685}, {4865, 5880}, {5263, 17598}, {6327, 24692}, {6377, 16606}, {7186, 25048}, {8167, 17262}, {10009, 17149}, {10980, 17151}, {11031, 20880}, {11269, 19789}, {11679, 18193}, {13407, 17748}, {13476, 22316}, {14829, 18201}, {16569, 24620}, {16604, 17459}, {16703, 17205}, {17146, 20011}, {17447, 23944}, {17448, 22184}, {18143, 25140}, {18144, 25121}, {18169, 21352}, {18837, 20888}, {19805, 23537}, {20459, 21369}, {20892, 23677}, {20911, 24215}, {21085, 24235}, {21416, 24221}, {21814, 22011}, {22232, 22343}, {24239, 25385}

### X(24166) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24166) lies on these lines: {2, 22011}, {75, 24046}, {76, 596}, {244, 1930}, {330, 514}, {626, 1086}, {712, 16604}, {982, 16739}, {1574, 9055}, {3216, 17141}, {3760, 17155}, {3953, 20911}, {4025, 21201}, {4441, 24226}, {4446, 24180}, {6534, 18146}, {16600, 24631}, {18140, 24068}, {18837, 20888}, {19805, 24177}, {20530, 22036}, {20893, 24221}, {21443, 24207}, {24159, 24187}, {24185, 24190}

### X(24167) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24167) lies on these lines: {1, 1392}, {10, 982}, {38, 3634}, {57, 4347}, {58, 17513}, {191, 7292}, {244, 1125}, {519, 3953}, {551, 986}, {596, 3831}, {726, 18137}, {942, 4719}, {978, 4067}, {993, 17054}, {995, 4084}, {1104, 4973}, {1393, 4298}, {1698, 4392}, {1739, 3626}, {3159, 4871}, {3244, 3976}, {3293, 17449}, {3315, 3746}, {3336, 7191}, {3337, 4351}, {3579, 4906}, {3624, 9335}, {3625, 24440}, {3635, 4642}, {3636, 4424}, {3678, 16610}, {3741, 24176}, {3742, 3743}, {3752, 3874}, {3782, 3825}, {3881, 3999}, {3892, 4646}, {4003, 5439}, {4066, 17155}, {4134, 17749}, {4695, 4701}, {4850, 18398}, {4868, 5045}, {5248, 17595}, {5573, 12514}, {8715, 17597}, {10916, 24177}, {11263, 24239}, {16887, 24189}, {17050, 24185}, {17063, 19862}, {21208, 24214}

### X(24168) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24168) lies on these lines: {1, 9335}, {10, 982}, {36, 88}, {38, 3828}, {57, 16428}, {244, 519}, {484, 7292}, {551, 17063}, {726, 24196}, {756, 3634}, {758, 16610}, {978, 4084}, {986, 19862}, {995, 3919}, {997, 8056}, {1086, 3814}, {1125, 4424}, {1149, 4674}, {1698, 7226}, {2292, 19878}, {3244, 24440}, {3625, 3976}, {3626, 3953}, {3635, 3987}, {3636, 4642}, {3666, 3833}, {3742, 4868}, {3752, 5883}, {3831, 24176}, {3840, 4717}, {3874, 4849}, {4067, 17749}, {4125, 17155}, {4134, 16569}, {4392, 19875}, {5121, 11813}, {10176, 16602}, {17164, 19847}, {20335, 24185}, {20517, 21186}

### X(24169) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24169) lies on these lines: {2, 846}, {6, 4987}, {10, 38}, {35, 404}, {43, 3662}, {82, 171}, {141, 21085}, {142, 16056}, {244, 4972}, {354, 4085}, {516, 19649}, {614, 4660}, {756, 24988}, {764, 24104}, {899, 17184}, {908, 6686}, {946, 19514}, {982, 4429}, {1051, 20090}, {1086, 1215}, {1155, 6679}, {1376, 17290}, {1738, 3741}, {1961, 17302}, {2292, 17674}, {2887, 3752}, {3125, 23914}, {3263, 3663}, {3336, 8258}, {3589, 4697}, {3666, 3836}, {3740, 17235}, {3782, 21093}, {3791, 17366}, {3816, 25377}, {3831, 19810}, {3840, 3914}, {3846, 16610}, {3912, 4970}, {3925, 6682}, {4000, 4362}, {4028, 21255}, {4383, 4655}, {4415, 24003}, {4434, 17061}, {4438, 17595}, {4640, 17356}, {4672, 11246}, {4682, 17382}, {4734, 17232}, {4871, 24210}, {5047, 12579}, {5249, 6685}, {5268, 17304}, {6377, 18905}, {6541, 17147}, {7191, 17766}, {7262, 17352}, {10180, 17245}, {11263, 20108}, {13741, 24851}, {15523, 17495}, {16062, 24174}, {16707, 17205}, {17023, 24610}, {17122, 19786}, {17123, 24723}, {17234, 17592}, {17282, 17594}, {21241, 24239}, {21242, 21949}, {21817, 22011}, {24170, 24211}, {24318, 25345}, {25349, 25357}

### X(24170) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24170) lies on these lines: {2, 2140}, {10, 274}, {39, 20255}, {75, 24046}, {116, 6656}, {141, 1574}, {171, 17200}, {335, 21067}, {384, 17729}, {514, 7187}, {538, 21025}, {995, 21281}, {1086, 3934}, {1258, 3997}, {1575, 21240}, {1739, 20911}, {1930, 24443}, {3125, 16720}, {3159, 20947}, {3216, 17137}, {3263, 3670}, {3634, 4357}, {3721, 4568}, {3831, 20888}, {4642, 14210}, {4651, 17208}, {5247, 6629}, {5280, 24602}, {6381, 24214}, {7770, 14377}, {8362, 21258}, {9436, 10521}, {16589, 25349}, {17210, 19874}, {17759, 21070}, {19810, 24177}, {22008, 24530}, {22011, 24185}, {24160, 24187}, {24169, 24211}

### X(24171) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24171) lies on these lines: {1, 6904}, {3, 12592}, {4, 5573}, {10, 982}, {36, 1612}, {58, 1414}, {244, 1210}, {386, 5542}, {387, 10980}, {443, 3677}, {515, 17054}, {553, 16466}, {614, 4292}, {936, 4310}, {946, 1086}, {975, 4353}, {978, 24231}, {988, 1125}, {995, 3671}, {1738, 3976}, {1739, 6736}, {3086, 23681}, {3333, 4000}, {3361, 3668}, {3624, 4656}, {3646, 4419}, {3660, 5930}, {3752, 21620}, {3755, 5045}, {3924, 4311}, {3953, 4847}, {4859, 19843}, {5265, 22464}, {5272, 12572}, {6700, 11512}, {7174, 17582}, {9843, 13161}, {10445, 20227}, {10461, 16752}, {10595, 15839}, {11019, 19790}, {11496, 15287}, {16610, 21075}, {16825, 24215}, {17749, 21060}, {19862, 24160}, {23675, 24443}

### X(24172) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24172) lies on these lines: {1, 1447}, {10, 4986}, {11, 4920}, {37, 17048}, {75, 3831}, {85, 3976}, {142, 20861}, {244, 20880}, {307, 1210}, {726, 3760}, {982, 3673}, {1086, 17046}, {1111, 3953}, {1193, 20247}, {1334, 24403}, {1930, 3840}, {3125, 20257}, {3662, 7938}, {3664, 18398}, {3934, 21101}, {3970, 24786}, {3999, 4059}, {16887, 23824}, {17050, 20271}, {17197, 18167}, {17448, 21138}, {18833, 21443}, {18837, 20888}, {20435, 21330}, {21084, 24575}

### X(24173) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24173) lies on these lines: {10, 24162}, {75, 3831}, {142, 24195}, {240, 1210}, {244, 1441}, {273, 1860}, {322, 3976}, {1086, 20305}, {1149, 21271}, {1393, 3668}, {3663, 21208}, {3778, 23677}, {3840, 18697}, {4858, 20259}, {10436, 18906}, {16609, 20227}, {17861, 24046}, {17863, 24443}

### X(24174) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24174) lies on these lines: {1, 474}, {2, 986}, {3, 1054}, {5, 17889}, {8, 244}, {10, 982}, {34, 9364}, {36, 20842}, {38, 9780}, {40, 5272}, {43, 942}, {46, 238}, {55, 16422}, {57, 1722}, {58, 11116}, {65, 978}, {72, 16569}, {75, 3831}, {88, 2975}, {106, 22837}, {142, 5530}, {145, 4695}, {171, 16478}, {277, 291}, {386, 5883}, {404, 3924}, {405, 17596}, {442, 20338}, {499, 1772}, {517, 21214}, {518, 6048}, {614, 5255}, {726, 24182}, {750, 5262}, {756, 19877}, {758, 17749}, {774, 5704}, {846, 11108}, {899, 3868}, {946, 5121}, {960, 16602}, {984, 1698}, {995, 3754}, {1015, 4051}, {1046, 4383}, {1086, 1329}, {1149, 14923}, {1210, 1738}, {1247, 16575}, {1254, 5435}, {1278, 22220}, {1393, 1788}, {1401, 23841}, {1466, 2647}, {1479, 24715}, {1575, 20271}, {1714, 24779}, {1724, 3336}, {1735, 17064}, {2275, 21951}, {2478, 24851}, {2944, 19517}, {3061, 3125}, {3091, 7613}, {3120, 4193}, {3178, 17234}, {3214, 3873}, {3216, 5902}, {3290, 3501}, {3293, 18398}, {3339, 23511}, {3579, 8616}, {3616, 4642}, {3624, 4424}, {3632, 4694}, {3679, 3953}, {3701, 17155}, {3731, 25086}, {3756, 3813}, {3826, 4446}, {3833, 17592}, {3915, 7292}, {3923, 13741}, {3944, 4187}, {3959, 16604}, {3961, 9709}, {3980, 13740}, {4000, 17048}, {4208, 11031}, {4253, 16611}, {4385, 24165}, {4386, 16787}, {4413, 5293}, {4414, 5047}, {4418, 5192}, {4420, 9350}, {4662, 21342}, {4674, 5697}, {4904, 20255}, {4926, 15079}, {5055, 5492}, {5084, 24248}, {5248, 17601}, {5266, 16498}, {5400, 15071}, {5529, 12635}, {6682, 19853}, {6735, 23675}, {8582, 13161}, {8715, 17715}, {9843, 24210}, {10015, 24093}, {10479, 25122}, {11533, 16863}, {12514, 17123}, {12609, 17717}, {12782, 24181}, {14837, 19950}, {16062, 24169}, {16583, 17754}, {16605, 21384}, {16736, 18178}, {16969, 21888}, {17337, 18253}, {17606, 24430}, {17697, 24850}, {17719, 24159}, {17748, 18134}, {17756, 21808}, {18180, 18792}, {21075, 24231}, {21173, 23345}, {23536, 24982}, {24311, 24342}, {24369, 24653}, {24372, 24923}, {24777, 24785}, {24789, 24914}

### X(24175) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    a^2 b + 2 a b^2 + b^3 + a^2 c - 8 a b c - b^2 c + 2 a c^2 - b c^2 + c^3 : :

X(24175) lies on these lines: {1, 11024}, {2, 2415}, {7, 23511}, {10, 982}, {43, 5542}, {57, 169}, {142, 2092}, {165, 16020}, {226, 16610}, {244, 4847}, {321, 24183}, {329, 4887}, {333, 17205}, {516, 5272}, {519, 18141}, {551, 3750}, {553, 4383}, {978, 3671}, {1054, 10164}, {1074, 1210}, {1086, 3452}, {1125, 11512}, {1266, 18743}, {1699, 7613}, {1722, 4298}, {1738, 11019}, {1743, 21454}, {2550, 5573}, {2999, 3664}, {3210, 3950}, {3306, 24779}, {3622, 7963}, {3666, 24181}, {3668, 5435}, {3687, 21255}, {3742, 3755}, {3756, 21949}, {3772, 6692}, {3782, 5316}, {3817, 5121}, {3834, 4035}, {3873, 4924}, {3911, 24789}, {3912, 17490}, {3971, 24200}, {4000, 5437}, {4021, 17022}, {4031, 4641}, {4082, 17155}, {4301, 21214}, {4310, 8580}, {4353, 5268}, {4359, 24219}, {4384, 24215}, {4862, 18228}, {5325, 17337}, {5745, 17278}, {6555, 15590}, {6700, 24159}, {6736, 23675}, {7658, 23801}, {8582, 23536}, {9843, 19802}, {14837, 21198}, {16569, 21060}, {16736, 17197}, {16832, 24214}, {17282, 20106}, {17593, 19862}, {21342, 24393}

X(24175) = complement of X(30568)

### X(24176) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24176) lies on these lines: {1, 17495}, {2, 3159}, {5, 7263}, {10, 38}, {75, 24046}, {244, 4647}, {386, 17490}, {519, 942}, {537, 4015}, {540, 24470}, {726, 3634}, {1070, 10916}, {1086, 3454}, {1125, 3666}, {1574, 21067}, {1575, 22011}, {1698, 17155}, {2292, 6533}, {2901, 5439}, {3293, 17140}, {3626, 24235}, {3720, 4065}, {3741, 24167}, {3752, 20108}, {3828, 6534}, {3831, 24168}, {3836, 24180}, {4000, 20083}, {4066, 20891}, {4142, 8714}, {4257, 19851}, {4361, 5708}, {9843, 17863}, {16825, 20367}, {16853, 17262}, {17205, 20911}, {17591, 19858}, {17749, 24620}, {20888, 21208}, {21187, 21203}, {21240, 24185}

### X(24177) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24177) lies on these lines: {1, 6904}, {2, 2415}, {6, 553}, {7, 2999}, {10, 38}, {19, 57}, {42, 5542}, {43, 24231}, {63, 3008}, {81, 17205}, {142, 3666}, {200, 4310}, {226, 1086}, {239, 24215}, {244, 3914}, {306, 17495}, {312, 1266}, {329, 4862}, {345, 17282}, {354, 3755}, {376, 16485}, {496, 18257}, {497, 5573}, {516, 614}, {527, 4383}, {528, 4906}, {551, 19336}, {612, 4353}, {726, 4082}, {899, 21060}, {940, 3946}, {950, 17054}, {968, 1125}, {980, 17050}, {982, 1738}, {986, 24178}, {1122, 14557}, {1149, 4342}, {1193, 3671}, {1201, 4301}, {1210, 1785}, {1462, 23601}, {1463, 23638}, {1722, 12527}, {1743, 9965}, {1817, 17189}, {2221, 14377}, {2550, 3677}, {2968, 20309}, {3011, 10164}, {3120, 3817}, {3210, 3912}, {3264, 18739}, {3306, 19785}, {3452, 3782}, {3474, 7290}, {3662, 3687}, {3664, 5256}, {3672, 17022}, {3710, 17674}, {3720, 4356}, {3772, 3911}, {3823, 4884}, {3875, 18141}, {3915, 5493}, {3924, 4297}, {3925, 4003}, {3944, 5121}, {3950, 17147}, {4021, 5287}, {4031, 17366}, {4114, 17365}, {4198, 4292}, {4346, 18228}, {4349, 17017}, {4357, 19804}, {4384, 24214}, {4392, 25006}, {4398, 18743}, {4415, 5316}, {4419, 7308}, {4430, 4924}, {4512, 16020}, {4642, 23675}, {4850, 5249}, {4858, 20205}, {4887, 5905}, {4896, 17012}, {5222, 10481}, {5272, 24248}, {5313, 11551}, {5435, 22464}, {5712, 6173}, {5743, 17235}, {5745, 17067}, {5853, 17597}, {6692, 17720}, {6703, 17382}, {6736, 24440}, {7058, 24378}, {7191, 20097}, {7963, 24558}, {8582, 13161}, {10521, 24590}, {10916, 24167}, {13411, 24159}, {14552, 16833}, {14555, 17274}, {16700, 17197}, {16752, 17185}, {17020, 17483}, {17063, 24210}, {17740, 20106}, {17889, 24239}, {19805, 24166}, {19810, 24170}, {20367, 24790}, {24181, 25080}

### X(24178) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24178) lies on these lines: {1, 142}, {2, 988}, {4, 5272}, {5, 5121}, {8, 23675}, {10, 982}, {12, 16610}, {20, 16020}, {28, 36}, {43, 21620}, {56, 24789}, {72, 24231}, {75, 24162}, {171, 12436}, {225, 7288}, {226, 978}, {238, 4292}, {244, 6734}, {274, 4357}, {377, 614}, {388, 1722}, {404, 3011}, {442, 24239}, {475, 499}, {553, 1046}, {631, 1072}, {905, 19947}, {946, 17889}, {958, 17278}, {960, 1086}, {962, 7613}, {986, 24177}, {995, 12609}, {997, 24159}, {1010, 1125}, {1054, 6684}, {1074, 3086}, {1191, 5880}, {1193, 5249}, {1210, 17063}, {1329, 16602}, {1698, 8056}, {1714, 3338}, {1739, 10039}, {1743, 4355}, {1834, 3742}, {2292, 24564}, {2475, 7292}, {3008, 4298}, {3216, 13407}, {3306, 5230}, {3315, 5178}, {3612, 23604}, {3616, 3914}, {3710, 17155}, {3752, 5530}, {3813, 21949}, {3912, 17670}, {3976, 4847}, {4201, 16823}, {4340, 16475}, {4383, 10404}, {4696, 24988}, {4719, 17056}, {4904, 21240}, {4968, 17674}, {5253, 17518}, {5267, 11102}, {5268, 17582}, {5290, 23511}, {5302, 17337}, {5345, 17562}, {5794, 17054}, {6700, 17719}, {7963, 25055}, {8583, 23681}, {10624, 24715}, {10916, 24216}, {11321, 17023}, {12572, 17123}, {13728, 24774}, {16458, 19758}, {16569, 21075}, {16828, 24778}, {17749, 21077}, {20880, 24548}, {21189, 23808}, {24443, 24987}

### X(24179) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24179) lies on these lines: {1, 1441}, {7, 90}, {36, 18655}, {46, 17220}, {57, 24220}, {69, 10916}, {75, 997}, {77, 1111}, {86, 3673}, {273, 4341}, {307, 499}, {614, 17188}, {988, 1125}, {1086, 17073}, {1447, 10446}, {1479, 18650}, {3664, 11019}, {3729, 25078}, {3875, 22836}, {5231, 17272}, {7289, 17197}, {8583, 24547}, {10826, 21270}, {13374, 24471}, {16831, 25065}, {18698, 19861}, {20880, 24540}, {24207, 24227}, {24559, 25252}

### X(24180) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24180) lies on these lines: {37, 39}, {38, 3778}, {142, 24165}, {522, 21194}, {732, 17049}, {742, 13476}, {894, 7191}, {993, 16684}, {1086, 21249}, {3836, 24176}, {3874, 5847}, {3963, 21210}, {4446, 24166}, {6646, 17154}, {10436, 17155}, {21193, 23808}, {21208, 21238}

### X(24181) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24181) lies on these lines: {1, 142}, {2, 24771}, {7, 16572}, {10, 4904}, {170, 17889}, {218, 226}, {220, 17278}, {354, 15493}, {946, 16020}, {1086, 1212}, {1125, 20269}, {1210, 21258}, {3663, 16601}, {3666, 24175}, {4869, 6764}, {5222, 5249}, {5882, 9317}, {5988, 24161}, {9957, 20328}, {12053, 17761}, {12610, 13442}, {12782, 24174}, {16699, 17205}, {17023, 24781}, {17158, 17234}, {17306, 19855}, {17760, 24199}, {19862, 24784}, {20335, 21075}, {24177, 25080}

### X(24182) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    a^2 b^3 - a^2 b^2 c + 2 a b^3 c - a^2 b c^2 - 8 a b^2 c^2 + b^3 c^2 + a^2 c^3 + 2 a b c^3 + b^2 c^3 : :

X(24182) lies on these lines: {10, 3662}, {75, 3840}, {726, 24174}, {1278, 4871}, {3741, 4772}, {4066, 20891}, {4135, 20923}, {5836, 24325}, {7263, 21257}, {10009, 24215}, {17351, 24742}, {20892, 23677}

### X(24183) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    a^2 b + 2 a b^2 + b^3 + a^2 c - 8 a b c + 2 a c^2 + c^3 : :

X(24183) lies on these lines: {2, 45}, {10, 244}, {75, 24184}, {100, 474}, {142, 14554}, {145, 3315}, {169, 3306}, {321, 24175}, {551, 678}, {1054, 3624}, {1266, 4358}, {1500, 4850}, {1647, 25351}, {2254, 23808}, {3008, 24593}, {3636, 3722}, {3762, 4453}, {3834, 3936}, {3943, 17495}, {3994, 24200}, {4033, 19804}, {4359, 24195}, {4442, 4871}, {4450, 5272}, {4706, 9507}, {4728, 25381}, {6545, 24123}, {9324, 25055}, {16729, 17205}, {17230, 20255}

### X(24184) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    a^2 b + 2 a b^2 + b^3 + a^2 c - 11 a b c - 3 b^2 c + 2 a c^2 - 3 b c^2 + c^3 : :

X(24184) lies on these lines: {2, 1266}, {9, 20092}, {10, 4392}, {75, 24183}, {145, 11024}, {169, 3218}, {1022, 17494}, {4025, 21198}, {4389, 24589}, {7263, 16594}, {14475, 25381}, {17244, 17495}, {17250, 19804}, {23809, 24457}

### X(24185) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24185) lies on these lines: {2, 4115}, {115, 116}, {244, 17761}, {335, 4103}, {514, 3125}, {537, 3739}, {596, 20255}, {620, 1125}, {942, 2809}, {2140, 24046}, {3017, 4000}, {4359, 4986}, {4904, 6741}, {14377, 17054}, {17050, 24167}, {17179, 17497}, {17310, 17495}, {17758, 24443}, {20335, 24168}, {21192, 24226}, {21240, 24176}, {22011, 24170}, {24050, 24530}, {24166, 24190}

### X(24186) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24186) lies on these lines: {37, 24938}, {125, 1086}, {244, 1109}, {3035, 3666}, {3120, 24224}, {3670, 24982}, {3752, 24885}, {3937, 21138}, {4025, 23824}, {4357, 20903}, {4359, 24988}, {5205, 17495}, {21208, 24226}, {21209, 23811}

### X(24187) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24187) lies on these lines: {2, 4115}, {99, 1125}, {350, 20893}, {620, 1086}, {903, 17322}, {11263, 17200}, {16887, 24161}, {24074, 24956}, {24159, 24166}, {24160, 24170}

### X(24188) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24188) lies on these lines: {2, 24416}, {11, 244}, {75, 24233}, {106, 14028}, {190, 11814}, {335, 19961}, {519, 1738}, {903, 24228}, {6550, 7336}, {6788, 25436}, {21112, 21140}, {24222, 25351}

### X(24189) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24189) lies on these lines: {75, 24046}, {244, 14210}, {625, 1086}, {6381, 20340}, {6629, 18201}, {16887, 24167}, {24200, 24240}

### X(24190) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24190) lies on these lines: {2, 2140}, {7, 17499}, {10, 3662}, {75, 21240}, {76, 1086}, {86, 2241}, {116, 5025}, {150, 17680}, {330, 17205}, {384, 14377}, {1266, 21071}, {1278, 21070}, {1500, 17234}, {3210, 3912}, {3496, 10436}, {3501, 17282}, {3552, 17729}, {3834, 20691}, {3836, 12782}, {3959, 20924}, {4000, 17034}, {4389, 16589}, {6656, 21258}, {7263, 21024}, {7786, 25350}, {9941, 24325}, {16706, 17750}, {17030, 17050}, {17033, 24790}, {24166, 24185}

### X(24191) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24191) lies on these lines: {244, 693}, {350, 1266}, {1086, 3835}, {3663, 25377}, {3912, 17495}, {4025, 21208}, {6381, 20340}, {6549, 7192}

### X(24192) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24192) lies on these lines: {2, 24198}, {11, 4106}, {244, 4025}, {350, 1266}, {1086, 4885}, {1111, 3676}, {2170, 11068}, {3026, 14027}, {3911, 20367}, {4887, 9436}, {4998, 24203}, {17197, 18197}, {17205, 18155}, {20651, 24196}, {21208, 23799}, {23791, 23823}, {24209, 24236}, {24216, 24232}

### X(24193) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24193) lies on these lines: {2, 24419}, {11, 244}, {334, 24233}, {350, 1266}, {2228, 3006}, {21140, 24136}

### X(24194) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24194) lies on these lines: {244, 4467}, {320, 17731}, {350, 1266}, {693, 17205}, {908, 3008}, {1086, 4369}, {3120, 20295}, {5209, 24234}, {17197, 23803}, {17761, 23554}, {23799, 23824}

### X(24195) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24195) lies on these lines: {115, 116}, {142, 24173}, {244, 1109}, {514, 16726}, {522, 3122}, {942, 2810}, {1125, 21254}, {3125, 17197}, {3666, 6692}, {3739, 9055}, {4000, 24884}, {4359, 24183}, {16732, 17205}, {21200, 23810}

### X(24196) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24196) lies on these lines: {244, 514}, {726, 24168}, {1086, 21260}, {6381, 20340}, {8714, 21208}, {20517, 21210}, {20651, 24192}

### X(24197) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24197) lies on these lines: {244, 661}, {693, 3122}, {1086, 23301}, {2228, 3006}, {6381, 20340}, {18891, 24238}

### X(24198) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24198) lies on these lines: {2, 24192}, {11, 5519}, {244, 7658}, {650, 1086}, {899, 17067}, {908, 3008}, {1111, 4468}, {1936, 3011}, {3835, 17761}, {16751, 17205}

### X(24199) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24199) lies on these lines: {1, 4780}, {2, 2415}, {6, 4795}, {7, 391}, {8, 5542}, {9, 4480}, {10, 3662}, {37, 1266}, {69, 4034}, {75, 142}, {86, 3946}, {141, 4688}, {190, 6666}, {192, 4098}, {193, 4888}, {226, 5233}, {239, 3664}, {274, 1432}, {319, 4545}, {320, 3686}, {333, 553}, {344, 4659}, {514, 24118}, {516, 16823}, {519, 17117}, {527, 7321}, {536, 17245}, {545, 16814}, {551, 17396}, {594, 3834}, {894, 3008}, {903, 17258}, {908, 24589}, {966, 17274}, {1086, 1213}, {1100, 4395}, {1125, 3685}, {1278, 3950}, {1699, 9801}, {1738, 4085}, {1781, 3218}, {2325, 17263}, {2345, 17282}, {3244, 17391}, {3264, 18143}, {3616, 4356}, {3620, 3679}, {3625, 17373}, {3626, 17287}, {3634, 17326}, {3644, 4029}, {3661, 4772}, {3672, 16831}, {3687, 3936}, {3696, 4684}, {3707, 17347}, {3717, 3826}, {3742, 4890}, {3759, 4667}, {3817, 9950}, {3828, 21087}, {3875, 4648}, {3879, 4361}, {3883, 5880}, {3943, 4726}, {3945, 4402}, {4000, 10436}, {4021, 16826}, {4044, 20923}, {4058, 17230}, {4060, 17295}, {4292, 16817}, {4298, 16824}, {4346, 5296}, {4353, 16830}, {4363, 17278}, {4389, 4751}, {4398, 4687}, {4399, 17374}, {4440, 17260}, {4452, 5308}, {4454, 18230}, {4464, 17390}, {4472, 17384}, {4654, 14555}, {4665, 17231}, {4670, 17366}, {4686, 17243}, {4698, 17246}, {4740, 17242}, {4851, 17119}, {4852, 17392}, {4862, 16832}, {4869, 17294}, {4887, 6646}, {4896, 16816}, {4898, 17151}, {5121, 25385}, {5287, 19789}, {5564, 17297}, {5750, 16706}, {6533, 12047}, {7227, 17357}, {7232, 17275}, {7238, 17344}, {9436, 24631}, {9776, 11679}, {10455, 16752}, {10521, 17739}, {15668, 17301}, {16610, 21796}, {16709, 17197}, {16738, 17205}, {16819, 24214}, {17077, 22464}, {17118, 17279}, {17133, 17315}, {17140, 25006}, {17160, 17317}, {17164, 24564}, {17207, 17474}, {17259, 17276}, {17265, 17281}, {17290, 17303}, {17299, 17313}, {17330, 17345}, {17337, 17351}, {17348, 17365}, {17356, 17369}, {17362, 17376}, {17382, 17398}, {17449, 21027}, {17760, 24181}, {17889, 24230}, {20234, 20893}, {20236, 20880}, {20881, 25001}, {20888, 20891}, {20892, 20913}, {20907, 21195}, {21242, 24216}, {23677, 24207}, {23682, 24212}

### X(24200) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24200) lies on these lines: {10, 38}, {244, 4442}, {350, 1266}, {354, 4743}, {897, 18201}, {1086, 4892}, {2796, 7292}, {3616, 3980}, {3971, 24175}, {3994, 24183}, {4062, 17495}, {4831, 4974}, {5212, 24231}, {16610, 21093}, {16741, 17205}, {24189, 24240}

### X(24201) =  MIDPOINT OF X(65) AND X(13756)

Barycentrics    -a (a^7 (b+c)+6 a^5 b c (b+c)-2 a^6 (b+c)^2-(b-c)^6 (b+c)^2+2 a^2 b (b-c)^2 c (3 b^2-2 b c+3 c^2)+a^4 (3 b^4-6 b^3 c-2 b^2 c^2-6 b c^3+3 c^4) +a^3 (-3 b^5+3 b^4 c+b^3 c^2+b^2 c^3+3 b c^4-3 c^5)+a (b-c)^2 (2 b^5-6 b^4 c+3 b^3 c^2+3 b^2 c^3-6 b c^4+2 c^5)) : :
X(24201) = X[65]+X[13756], 3*X[354]-X[3025], X[14115]-2*X[18240]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28380.

X(24201) lies on these lines: {1,953}, {57,901}, {59,1421}, {65,13756}, {226,3259}, {354,3025}, {513,5083}, {517,1387}, {1319,18593}, {2835,15635}, {5570,12016}, {11028,18839}, {14115,18240}

X(24201) = midpoint of X(65) and X(13756)
X(24201) = reflection of X(14115) in X(18240)

### X(24202) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24202) lies on these lines: {1, 1441}, {7, 5563}, {36, 17220}, {86, 99}, {322, 22837}, {1086, 17043}, {1111, 1442}, {1201, 17189}, {1429, 24220}, {2893, 24387}, {3582, 5740}, {3673, 17394}, {4360, 24209}, {4909, 14828}, {5738, 10072}, {7741, 21270}, {16826, 25065}, {17134, 21842}, {17197, 18162}, {17859, 17877}, {21625, 24213}, {24212, 24227}, {24559, 25078}

### X(24203) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24203) lies on these lines: {1, 85}, {2, 644}, {7, 104}, {11, 150}, {56, 17753}, {75, 997}, {81, 1625}, {86, 99}, {101, 673}, {190, 18061}, {404, 20244}, {517, 1447}, {658, 18240}, {938, 5543}, {946, 4911}, {1086, 9259}, {1319, 5088}, {1320, 21272}, {1429, 6996}, {1434, 5563}, {1482, 3212}, {1633, 23392}, {1807, 7269}, {2170, 3732}, {2329, 17681}, {3086, 6604}, {3315, 4453}, {3674, 13464}, {3699, 4986}, {3811, 17158}, {3879, 24213}, {4059, 20323}, {4360, 17861}, {4845, 11019}, {4858, 21602}, {4919, 21232}, {4998, 24192}, {5736, 15956}, {5886, 7179}, {5901, 17084}, {7176, 24928}, {7247, 12047}, {7272, 18393}, {8286, 23674}, {9310, 17682}, {9317, 17439}, {9327, 17758}, {9446, 17626}, {10186, 24248}, {11376, 17181}, {17160, 24209}, {17197, 18723}, {17719, 24228}, {24559, 25083}

### X(24204) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24204) lies on these lines: {1, 1441}, {4, 1111}, {277, 2345}, {321, 17284}, {349, 3673}, {1086, 18636}, {16502, 16732}, {17197, 18727}, {17877, 18693}, {18698, 19836}, {23581, 24159}

### X(24205) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24205) lies on these lines: {1, 1441}, {1086, 18637}, {1111, 3583}, {16732, 16784}, {17197, 18728}, {17877, 18694}, {18208, 24233}

### X(24206) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24206) lies on these lines: {2, 98}, {3, 2916}, {4, 3096}, {5, 141}, {6, 17}, {30, 14810}, {66, 6759}, {67, 14643}, {69, 576}, {76, 3399}, {115, 3094}, {126, 12494}, {140, 1503}, {159, 7393}, {193, 7486}, {262, 3314}, {265, 12584}, {315, 10358}, {323, 7570}, {325, 14994}, {343, 5943}, {373, 3580}, {381, 1350}, {389, 7405}, {394, 7539}, {403, 12294}, {427, 3819}, {498, 12589}, {499, 12588}, {517, 3844}, {518, 6583}, {524, 547}, {549, 11645}, {550, 21167}, {575, 3564}, {578, 14786}, {597, 1353}, {599, 1351}, {620, 4048}, {621, 16002}, {622, 16001}, {631, 17508}, {632, 20190}, {732, 7764}, {754, 10796}, {858, 5650}, {1078, 5207}, {1386, 11230}, {1469, 7951}, {1495, 7495}, {1513, 15819}, {1533, 16261}, {1568, 7699}, {1594, 1843}, {1595, 13348}, {1691, 7749}, {1853, 16419}, {1974, 7505}, {1993, 7571}, {2030, 3054}, {2072, 9967}, {2076, 7747}, {2080, 7810}, {2393, 20300}, {2548, 5039}, {2777, 4550}, {2782, 4045}, {2783, 3821}, {2792, 24295}, {2794, 9996}, {2854, 16511}, {2896, 12110}, {3056, 7741}, {3091, 7938}, {3095, 7794}, {3242, 5790}, {3292, 14389}, {3313, 5576}, {3398, 7889}, {3416, 5886}, {3524, 14927}, {3525, 25406}, {3526, 5085}, {3533, 23294}, {3549, 19126}, {3574, 11444}, {3618, 5067}, {3620, 5056}, {3629, 22330}, {3734, 23698}, {3826, 21252}, {3836, 17047}, {3841, 25144}, {3917, 5133}, {3923, 24250}, {4260, 6881}, {4265, 7489}, {4549, 18420}, {5017, 5475}, {5050, 5070}, {5052, 7603}, {5071, 20423}, {5079, 11477}, {5116, 11646}, {5138, 6861}, {5157, 10539}, {5169, 7998}, {5171, 7800}, {5181, 13162}, {5422, 11225}, {5448, 14128}, {5510, 20551}, {5562, 14788}, {5663, 6698}, {5846, 5901}, {5891, 18388}, {5907, 7399}, {6144, 11482}, {6194, 9993}, {6247, 17704}, {6248, 6656}, {6289, 11314}, {6290, 11313}, {6403, 7577}, {6639, 19131}, {6643, 15435}, {6676, 15448}, {6680, 10104}, {6688, 13567}, {6803, 18931}, {6829, 10477}, {6998, 17307}, {7380, 17234}, {7395, 18396}, {7403, 15644}, {7404, 13346}, {7485, 11550}, {7504, 15988}, {7514, 15577}, {7533, 15107}, {7569, 20806}, {7605, 15019}, {7619, 9830}, {7692, 21736}, {7753, 12212}, {7765, 13108}, {7769, 12215}, {7780, 20576}, {7795, 9737}, {7811, 10788}, {7819, 13335}, {7831, 11676}, {7844, 20398}, {7853, 15980}, {7867, 13355}, {7868, 13860}, {7876, 11257}, {7919, 14651}, {7934, 22677}, {8252, 19146}, {8253, 19145}, {8362, 13334}, {8370, 13449}, {8542, 12596}, {9127, 10162}, {9669, 10387}, {9698, 13331}, {9753, 16990}, {9821, 15821}, {9863, 16895}, {10193, 15578}, {10203, 14940}, {10255, 18438}, {10347, 12176}, {11171, 14981}, {11173, 18584}, {11291, 12975}, {11292, 12974}, {11303, 20429}, {11304, 20428}, {11459, 14789}, {11548, 23292}, {11574, 11585}, {11695, 12359}, {13154, 15581}, {13347, 14216}, {13352, 14787}, {13399, 20791}, {13561, 16239}, {13862, 16986}, {14848, 15533}, {14913, 20303}, {14982, 15061}, {14984, 15088}, {15026, 21230}, {15060, 15101}, {17283, 21554}, {18906, 23514}

X(24206) = complement of X(182)
X(24206) = X(5)-of-1st-Brocard-triangle
X(24206) = X(141)-of-McCay-triangle
X(24206) = centroid of PU(5)PU(11)

### X(24207) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24207) lies on these lines: {75, 21210}, {244, 20236}, {514, 18194}, {982, 17861}, {1086, 17047}, {3551, 3667}, {3670, 23689}, {3953, 23690}, {17065, 21208}, {17197, 18168}, {18208, 24220}, {21185, 24802}, {21443, 24166}, {23677, 24199}, {24179, 24227}, {24224, 24230}

### X(24208) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24208) lies on these lines: {7, 3585}, {10, 75}, {273, 10481}, {516, 23689}, {1099, 1111}, {1441, 4021}, {3244, 20930}, {3672, 10056}, {3946, 16732}, {4353, 23690}, {4464, 17791}, {4698, 25065}, {4887, 23521}, {12649, 21296}, {20171, 21255}, {24214, 24219}

### X(24209) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24209) lies on these lines: {2, 25076}, {7, 5270}, {10, 75}, {44, 4957}, {78, 17151}, {85, 4896}, {91, 3668}, {498, 3672}, {516, 1733}, {519, 3262}, {527, 16732}, {850, 4025}, {1111, 4887}, {1441, 3664}, {1737, 18815}, {3008, 4858}, {3739, 25065}, {3875, 22836}, {3950, 20171}, {4021, 13411}, {4353, 23689}, {4360, 24202}, {4452, 5552}, {4847, 17871}, {5294, 20886}, {6737, 20895}, {7264, 23521}, {17132, 20881}, {17160, 24203}, {17355, 20236}, {18698, 24993}, {24192, 24236}, {25001, 25072}

### X(24210) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24210) lies on these lines: {1, 4}, {2, 968}, {5, 3931}, {6, 24703}, {8, 4104}, {10, 312}, {11, 114}, {36, 4221}, {37, 2886}, {42, 908}, {43, 3452}, {55, 17720}, {57, 12717}, {63, 11269}, {65, 15488}, {81, 5057}, {142, 4335}, {149, 3920}, {171, 516}, {181, 517}, {192, 3705}, {238, 2328}, {256, 314}, {284, 2201}, {329, 3751}, {354, 3782}, {381, 5725}, {386, 21616}, {390, 3749}, {442, 6051}, {496, 20256}, {511, 21334}, {518, 4415}, {519, 4514}, {612, 3434}, {614, 19785}, {693, 23793}, {740, 3687}, {756, 25006}, {846, 5745}, {940, 1836}, {960, 1834}, {982, 3663}, {984, 4656}, {986, 1210}, {988, 3086}, {1001, 3772}, {1010, 1125}, {1045, 21246}, {1054, 6692}, {1076, 4295}, {1086, 3742}, {1211, 3706}, {1266, 24165}, {1279, 17061}, {1329, 4646}, {1362, 12915}, {1402, 4192}, {1500, 20544}, {1621, 3011}, {1707, 5698}, {1722, 5084}, {1725, 3670}, {1737, 4424}, {1756, 22097}, {1839, 2303}, {1961, 20539}, {1999, 4388}, {2292, 6734}, {2310, 11031}, {2550, 5268}, {2887, 3912}, {3006, 3995}, {3008, 17123}, {3058, 3744}, {3120, 3720}, {3175, 3703}, {3616, 23536}, {3622, 23675}, {3664, 4038}, {3672, 5274}, {3673, 6063}, {3679, 3974}, {3696, 5743}, {3702, 5051}, {3717, 3971}, {3727, 23903}, {3736, 17182}, {3748, 17724}, {3750, 13405}, {3752, 3816}, {3814, 4868}, {3817, 4356}, {3821, 3840}, {3826, 21949}, {3838, 15569}, {3883, 4362}, {3896, 5741}, {3911, 17596}, {3961, 5853}, {4000, 5272}, {4001, 4683}, {4011, 17353}, {4021, 17600}, {4028, 4417}, {4123, 22836}, {4138, 18134}, {4292, 24851}, {4307, 9812}, {4310, 10580}, {4343, 21617}, {4353, 17598}, {4358, 4972}, {4359, 4442}, {4383, 4679}, {4416, 4703}, {4419, 24477}, {4423, 16849}, {4429, 18743}, {4432, 6679}, {4642, 24982}, {4689, 5432}, {4743, 5212}, {4780, 5233}, {4862, 10980}, {4871, 24169}, {4891, 4966}, {5046, 17016}, {5080, 17015}, {5230, 5250}, {5247, 12572}, {5255, 10624}, {5266, 15171}, {5269, 9580}, {5292, 12514}, {5316, 16569}, {5710, 12701}, {5711, 12699}, {5718, 17605}, {5739, 17156}, {6358, 23690}, {6381, 18057}, {6686, 11814}, {7174, 24392}, {8167, 17278}, {8582, 24440}, {9284, 21345}, {9711, 21896}, {9843, 24174}, {10164, 17601}, {10448, 24541}, {10458, 17167}, {10582, 23681}, {11238, 17599}, {11375, 19765}, {11599, 14534}, {14008, 25060}, {14009, 25058}, {14829, 24723}, {15280, 25098}, {15507, 20967}, {16826, 23682}, {17063, 24177}, {17122, 24715}, {17197, 18169}, {17595, 17728}, {17715, 17725}, {17723, 20182}, {18905, 20363}, {19815, 20106}, {21935, 24987}, {23812, 23823}, {24386, 25353}

### X(24211) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24211) lies on these lines: {1, 147}, {10, 1930}, {37, 17046}, {39, 24318}, {65, 4920}, {75, 18835}, {116, 16600}, {171, 4911}, {307, 1210}, {626, 4071}, {742, 4109}, {894, 7797}, {976, 21285}, {1086, 17048}, {3664, 13407}, {3767, 24333}, {3831, 4357}, {3944, 17753}, {4056, 5264}, {4434, 7767}, {4847, 23667}, {4865, 7776}, {4872, 5255}, {5209, 16887}, {5283, 25353}, {7278, 24222}, {9436, 13161}, {16788, 20267}, {17790, 25364}, {20255, 25345}, {20880, 21935}, {24169, 24170}

### X(24212) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24212) lies on these lines: {1, 9551}, {38, 20236}, {239, 17153}, {256, 2481}, {514, 18207}, {982, 17861}, {1086, 17049}, {3662, 21278}, {3670, 23690}, {3953, 23689}, {4021, 17600}, {4353, 13161}, {4858, 17446}, {6385, 16887}, {7184, 24237}, {7321, 23823}, {16706, 18082}, {16732, 18183}, {17197, 18170}, {23682, 24199}, {24202, 24227}

### X(24213) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24213) lies on these lines: {7, 1210}, {10, 75}, {11, 1122}, {269, 1256}, {946, 24471}, {1086, 24005}, {1111, 3668}, {1423, 10445}, {1737, 4862}, {3123, 23663}, {3664, 11019}, {3772, 20205}, {3879, 24203}, {3986, 25065}, {4000, 20262}, {4328, 18391}, {4373, 25005}, {4452, 6735}, {4847, 17272}, {6736, 17151}, {7318, 15866}, {9843, 10436}, {10521, 21628}, {21625, 24202}

### X(24214) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24214) lies on these lines: {1, 7}, {10, 3761}, {38, 20880}, {39, 20335}, {76, 3831}, {85, 986}, {142, 5283}, {194, 3662}, {213, 527}, {226, 980}, {239, 4001}, {274, 4357}, {310, 3741}, {519, 17137}, {538, 21240}, {726, 1930}, {982, 3673}, {1086, 1107}, {1111, 3670}, {1125, 4368}, {1193, 18600}, {1266, 17143}, {1975, 24549}, {2176, 17276}, {2887, 3933}, {3008, 16552}, {3665, 3782}, {3666, 4059}, {3720, 17169}, {3727, 7200}, {3760, 3840}, {3912, 25264}, {3944, 17181}, {3946, 20963}, {3953, 7264}, {4000, 21384}, {4384, 24177}, {4398, 17144}, {4656, 16831}, {4872, 24851}, {5254, 17046}, {6381, 24170}, {6646, 16827}, {6706, 25073}, {7195, 22097}, {7754, 24586}, {9436, 13161}, {9534, 17272}, {10521, 20367}, {16819, 24199}, {16832, 24175}, {16975, 20257}, {17197, 18171}, {17758, 25092}, {20018, 21296}, {21208, 24167}, {24208, 24219}

### X(24215) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24215) lies on these lines: {1, 7}, {2, 23649}, {10, 274}, {42, 18600}, {57, 20606}, {76, 3840}, {85, 982}, {142, 1107}, {171, 1434}, {194, 3912}, {239, 24177}, {304, 726}, {330, 3662}, {335, 7187}, {519, 21281}, {527, 2176}, {538, 21071}, {1086, 17448}, {1111, 3953}, {1201, 20347}, {1231, 24218}, {1266, 17144}, {1401, 20358}, {1565, 4920}, {2200, 18723}, {2275, 20335}, {2664, 21060}, {3008, 21384}, {3673, 3976}, {3687, 24621}, {3721, 7200}, {3761, 3831}, {3771, 3926}, {3950, 25264}, {4364, 25130}, {4384, 24175}, {4416, 16827}, {4419, 24654}, {4424, 7278}, {4656, 16826}, {4685, 16711}, {4694, 7264}, {4713, 24652}, {4847, 23682}, {4871, 18135}, {10009, 24182}, {13161, 24241}, {16284, 24440}, {16720, 21101}, {16779, 17691}, {16825, 24171}, {16969, 17276}, {16975, 17050}, {17095, 17719}, {17197, 18172}, {17449, 20247}, {17866, 24219}, {18157, 21080}, {20036, 21296}, {20911, 24165}, {24656, 25349}, {25102, 25350}

### X(24216) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    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 : :

X(24216) lies on these lines: {1, 631}, {2, 16496}, {11, 3999}, {37, 17051}, {106, 519}, {244, 1738}, {354, 5718}, {497, 18193}, {516, 18201}, {517, 6018}, {518, 3756}, {527, 24398}, {614, 24597}, {908, 1647}, {982, 3663}, {1054, 5853}, {1210, 3976}, {1357, 15310}, {1447, 3945}, {1699, 4902}, {1737, 4694}, {2191, 5272}, {3011, 3315}, {3035, 4864}, {3664, 17722}, {3676, 4905}, {3705, 17232}, {3717, 4871}, {3741, 4967}, {3742, 17245}, {3749, 5435}, {3816, 21342}, {3953, 13161}, {3961, 6692}, {4038, 4909}, {4847, 17063}, {4860, 17721}, {4888, 10980}, {4899, 24003}, {5045, 5530}, {5048, 22102}, {5211, 5847}, {5537, 22942}, {5542, 17717}, {10164, 17715}, {10580, 17594}, {10916, 24178}, {17197, 18173}, {17597, 17728}, {17889, 24386}, {21214, 24391}, {21242, 24199}, {24192, 24232}

### X(24217) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24217) lies on these lines: {1, 5}, {2, 2177}, {43, 3816}, {149, 750}, {171, 497}, {238, 11269}, {354, 3944}, {519, 5233}, {940, 11238}, {982, 3663}, {1086, 17051}, {1647, 4850}, {1699, 4888}, {1757, 4679}, {1834, 21214}, {2886, 17245}, {2887, 17232}, {3058, 3550}, {3306, 24715}, {3338, 24851}, {3434, 17122}, {3622, 21935}, {3632, 4023}, {3679, 5241}, {3705, 17242}, {3720, 11680}, {3741, 5224}, {3742, 17889}, {3755, 5121}, {3829, 17056}, {3846, 10453}, {3848, 21949}, {3911, 17601}, {3914, 17063}, {3945, 4038}, {4080, 17146}, {4256, 10199}, {4424, 24223}, {4425, 17249}, {4429, 4871}, {4693, 17740}, {4854, 17591}, {4883, 17605}, {4890, 24230}, {4902, 10980}, {5268, 24392}, {5284, 24892}, {6048, 17527}, {7777, 17319}, {10582, 17064}, {16592, 23903}, {17197, 18174}, {17234, 21241}, {17592, 24239}, {17596, 17728}, {18201, 24248}

### X(24218) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24218) lies on these lines: {57, 7009}, {75, 982}, {225, 1210}, {226, 24235}, {244, 17862}, {345, 726}, {614, 20223}, {1231, 24215}, {1441, 11031}, {1891, 4292}, {3971, 17719}, {4310, 17155}, {11019, 24225}, {17197, 18175}, {20320, 24046}, {23661, 24443}

### X(24219) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24219) lies on these lines: {1, 21271}, {10, 24162}, {75, 24046}, {244, 18698}, {307, 1210}, {511, 942}, {982, 17861}, {1086, 17052}, {1227, 4357}, {1441, 3953}, {3159, 18147}, {3666, 3911}, {3739, 9055}, {4022, 21210}, {4256, 4360}, {4359, 24175}, {4887, 14755}, {17151, 17495}, {17197, 18179}, {17866, 24215}, {24208, 24214}

### X(24220) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24220) lies on these lines: {1, 9551}, {2, 573}, {3, 142}, {4, 991}, {5, 141}, {6, 17197}, {7, 1020}, {10, 3781}, {11, 21746}, {30, 6176}, {57, 24179}, {63, 22000}, {69, 5816}, {81, 1746}, {86, 572}, {116, 117}, {140, 6707}, {222, 226}, {355, 4851}, {381, 17313}, {514, 18161}, {517, 3739}, {908, 1150}, {952, 17390}, {980, 3663}, {995, 4000}, {1086, 17053}, {1429, 24202}, {1482, 4361}, {1656, 17327}, {1699, 1742}, {1766, 10436}, {1777, 12047}, {1953, 4858}, {1959, 20236}, {2876, 17059}, {3091, 4869}, {3136, 3917}, {3452, 5737}, {3589, 19512}, {3616, 10470}, {3667, 21191}, {3729, 22019}, {3741, 20788}, {3814, 21244}, {3817, 3840}, {3834, 5482}, {3878, 21233}, {3912, 21069}, {3946, 13464}, {4224, 17188}, {4301, 17050}, {4399, 5844}, {4402, 5734}, {4445, 5790}, {4852, 10222}, {4859, 11522}, {5249, 19645}, {5587, 17296}, {5733, 6776}, {5736, 21617}, {5786, 21620}, {5788, 21077}, {5792, 20818}, {5799, 8728}, {5806, 6706}, {5901, 17045}, {6210, 8227}, {6831, 18635}, {7146, 17861}, {7377, 17234}, {7381, 18647}, {7384, 17300}, {7406, 18648}, {7522, 17811}, {7749, 20666}, {9956, 17239}, {10444, 16831}, {10476, 19858}, {10888, 17022}, {11376, 20270}, {11679, 22020}, {12053, 19765}, {13329, 21554}, {15488, 15973}, {16435, 19701}, {16552, 16713}, {16574, 17139}, {17077, 17220}, {17262, 22031}, {17355, 20258}, {17792, 20544}, {18208, 24207}, {19549, 20108}, {20245, 21061}, {20274, 21210}, {20883, 21429}

### X(24221) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24221) lies on these lines: {75, 3840}, {85, 982}, {1086, 20528}, {3663, 3817}, {3752, 3774}, {3912, 17490}, {20893, 24166}, {21416, 24165}

### X(24222) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24222) lies on these lines: {1, 5}, {2, 106}, {8, 24159}, {10, 244}, {36, 19335}, {109, 388}, {141, 3679}, {150, 3945}, {519, 3936}, {535, 902}, {542, 6126}, {595, 20060}, {668, 5224}, {982, 24223}, {1056, 24618}, {1074, 1111}, {1086, 1145}, {1149, 3814}, {1320, 24864}, {1647, 6702}, {1737, 4694}, {1739, 6735}, {1769, 3762}, {1772, 3670}, {2093, 4902}, {2789, 5988}, {2802, 3120}, {2948, 24345}, {3216, 12607}, {3244, 21935}, {3315, 6788}, {3944, 6018}, {3963, 4692}, {3976, 18395}, {3987, 10915}, {4013, 25031}, {4389, 20568}, {4737, 18040}, {4861, 24160}, {4967, 4986}, {5255, 5270}, {5541, 24416}, {5790, 17597}, {7208, 24318}, {7278, 24211}, {10246, 17783}, {11236, 16483}, {13259, 14430}, {16486, 17556}, {17249, 18159}, {17460, 21630}, {24188, 25351}

### X(24223) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24223) lies on these lines: {1, 631}, {10, 4392}, {80, 17595}, {982, 24222}, {1086, 17057}, {1647, 17461}, {1698, 24159}, {1734, 4443}, {1737, 3663}, {1739, 4446}, {3245, 17721}, {3585, 15434}, {3670, 18395}, {4414, 6788}, {4424, 24217}, {4859, 19875}, {4967, 10468}, {5718, 5902}, {10260, 14793}, {20516, 21199}

### X(24224) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24224) lies on these lines: {2, 22003}, {11, 1365}, {75, 24058}, {115, 116}, {142, 24086}, {514, 7202}, {522, 2486}, {942, 2801}, {1111, 24237}, {3120, 24186}, {3667, 4459}, {3739, 4422}, {4359, 20881}, {4605, 18815}, {4858, 17761}, {5954, 7336}, {7146, 17861}, {17863, 21617}, {18208, 24229}, {24207, 24230}

### X(24225) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24225) lies on these lines: {1, 4552}, {11, 1365}, {244, 17888}, {514, 3271}, {522, 1086}, {596, 3976}, {942, 2783}, {1111, 2310}, {2293, 24799}, {2643, 4151}, {3122, 20525}, {3667, 4014}, {3675, 4459}, {4081, 4904}, {4516, 17761}, {8299, 22003}, {8714, 23823}, {11019, 24218}, {17197, 17463}, {17205, 17877}, {17885, 24802}, {21185, 24232}, {23798, 24236}

### X(24226) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24226) lies on these lines: {244, 17877}, {1111, 7004}, {3666, 13006}, {4025, 17205}, {4441, 24166}, {21192, 24185}, {21208, 24186}

### X(24227) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24227) lies on these lines: {1, 24224}, {86, 23823}, {1086, 11725}, {3120, 18645}, {24179, 24207}, {24202, 24212}

### X(24228) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24228) lies on these lines: {11, 1111}, {903, 24188}, {1647, 4453}, {4887, 9436}, {17719, 24203}

### X(24229) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24229) lies on these lines: {982, 17861}, {4424, 23689}, {4694, 23690}, {18208, 24224}, {24232, 24236}

### X(24230) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24230) lies on these lines: {1, 9551}, {1086, 17065}, {1111, 20567}, {1210, 24231}, {3551, 23821}, {3663, 3817}, {4890, 24217}, {16732, 18168}, {17197, 18194}, {17861, 18208}, {17889, 24199}, {24207, 24224}

### X(24231) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    -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(24231) lies on these lines: {1, 7}, {2, 18193}, {8, 7613}, {10, 3662}, {11, 3999}, {38, 5249}, {43, 24177}, {44, 5852}, {57, 6211}, {72, 24178}, {142, 984}, {144, 16020}, {145, 4780}, {171, 553}, {226, 262}, {238, 527}, {240, 5236}, {244, 908}, {291, 20335}, {306, 17155}, {320, 5847}, {329, 5272}, {335, 726}, {354, 3782}, {511, 1463}, {518, 1086}, {519, 4645}, {528, 4864}, {536, 4966}, {537, 3717}, {611, 5228}, {613, 6180}, {614, 5905}, {740, 1266}, {748, 17781}, {894, 1125}, {942, 13161}, {946, 3976}, {978, 24171}, {986, 21620}, {988, 3487}, {1001, 17276}, {1054, 6745}, {1111, 23690}, {1155, 17724}, {1210, 24230}, {1278, 4133}, {1279, 17768}, {1386, 17365}, {1707, 9965}, {1737, 24232}, {1757, 3008}, {1785, 1876}, {1836, 17597}, {2285, 3338}, {2298, 5557}, {2550, 16496}, {2886, 21342}, {3006, 17154}, {3011, 3218}, {3120, 17449}, {3210, 4028}, {3242, 5880}, {3315, 5057}, {3416, 7232}, {3452, 17063}, {3474, 3749}, {3475, 17594}, {3616, 17247}, {3624, 3986}, {3632, 15590}, {3634, 17291}, {3667, 4017}, {3670, 5530}, {3677, 4654}, {3685, 4440}, {3687, 24165}, {3696, 7263}, {3705, 4138}, {3742, 4415}, {3744, 11246}, {3751, 4000}, {3756, 5087}, {3775, 4967}, {3790, 17232}, {3834, 3932}, {3868, 23536}, {3869, 23675}, {3873, 3914}, {3874, 23537}, {3883, 4655}, {3911, 17719}, {3944, 11019}, {3946, 4649}, {3953, 12047}, {3982, 17598}, {4003, 5718}, {4014, 15310}, {4026, 17235}, {4031, 17725}, {4078, 17234}, {4104, 19804}, {4114, 17716}, {4357, 24325}, {4416, 16825}, {4432, 17767}, {4644, 16475}, {4663, 17366}, {4733, 4739}, {4847, 17889}, {4854, 4883}, {4859, 5223}, {4860, 17720}, {4871, 21093}, {4890, 5045}, {4941, 12053}, {4974, 17771}, {4989, 16477}, {5212, 24200}, {5220, 17278}, {5263, 7321}, {5266, 24470}, {5268, 9776}, {5279, 6763}, {5293, 12436}, {5772, 19875}, {5843, 15251}, {5846, 7238}, {5853, 24715}, {5988, 9436}, {6173, 7174}, {6646, 16823}, {7191, 17483}, {7290, 24695}, {7292, 17484}, {13405, 17596}, {15254, 17334}, {15481, 17337}, {15569, 17246}, {16569, 21060}, {17064, 24477}, {17140, 17184}, {17156, 19789}, {17205, 18792}, {17368, 19862}, {17595, 17718}, {17766, 24692}, {19868, 24342}, {20348, 21214}, {20470, 24405}, {21075, 24174}, {21077, 24046}

### X(24232) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24232) lies on these lines: {244, 4391}, {1086, 17072}, {1111, 4905}, {1737, 24231}, {4147, 6547}, {6381, 20340}, {6549, 23789}, {20517, 21208}, {21185, 24225}, {24192, 24216}, {24229, 24236}

### X(24233) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24233) lies on these lines: {11, 1111}, {75, 24188}, {334, 24193}, {1647, 3766}, {18208, 24205}, {24192, 24216}

### X(24234) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24234) lies on these lines: {11, 6002}, {36, 516}, {667, 17197}, {4905, 23821}, {5209, 24194}, {8714, 17205}, {24192, 24216}

### X(24235) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24235) lies on these lines: {11, 1365}, {226, 24218}, {244, 1109}, {514, 18191}, {522, 3120}, {542, 553}, {942, 952}, {1086, 2968}, {1111, 7004}, {1125, 16598}, {1357, 2789}, {1772, 3670}, {3626, 24176}, {3666, 4353}, {3935, 17495}, {4025, 16727}, {4359, 25006}, {4904, 6741}, {17197, 18210}, {21085, 24165}

### X(24236) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24236) lies on these lines: {244, 20907}, {1111, 3667}, {17761, 23886}, {21178, 21210}, {23798, 24225}, {24192, 24209}, {24229, 24232}

### X(24237) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24237) lies on these lines: {2, 21362}, {7, 1020}, {10, 2810}, {11, 1357}, {57, 2051}, {116, 5514}, {142, 3589}, {226, 4896}, {244, 13246}, {514, 3942}, {522, 17463}, {545, 22031}, {651, 24618}, {812, 1015}, {908, 24593}, {946, 20418}, {1111, 24224}, {1125, 15507}, {1358, 3026}, {1407, 13478}, {2140, 6173}, {2310, 3667}, {2347, 24778}, {3675, 4459}, {4129, 8287}, {4675, 17750}, {4817, 15634}, {7184, 24212}, {10436, 10469}, {10478, 21454}, {17355, 19593}, {20258, 21255}

### X(24238) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24238) lies on these lines: {1111, 4170}, {18208, 24205}, {18891, 24197}, {24229, 24232}

### X(24239) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    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(24239) lies on these lines: {1, 2}, {4, 988}, {5, 13161}, {7, 18193}, {9, 7736}, {11, 114}, {22, 14793}, {25, 8071}, {35, 19649}, {36, 4220}, {37, 3815}, {38, 908}, {44, 9300}, {55, 16434}, {56, 19544}, {57, 6210}, {140, 5266}, {142, 17063}, {171, 3911}, {183, 3879}, {226, 262}, {230, 1100}, {238, 5745}, {240, 1848}, {244, 5249}, {325, 3846}, {354, 5718}, {427, 1785}, {442, 24178}, {443, 11512}, {496, 3931}, {497, 17594}, {516, 17596}, {553, 18201}, {611, 4383}, {613, 940}, {946, 986}, {950, 7413}, {984, 3452}, {999, 5725}, {1007, 17321}, {1072, 6830}, {1086, 3838}, {1104, 4999}, {1279, 6690}, {1319, 5724}, {1368, 17102}, {1447, 3664}, {1449, 7735}, {1699, 24248}, {1707, 5744}, {1733, 4359}, {1738, 2886}, {1834, 4719}, {1836, 17595}, {2476, 23536}, {3055, 3723}, {3058, 4689}, {3290, 21796}, {3361, 7390}, {3523, 4339}, {3550, 10164}, {3598, 4888}, {3662, 4138}, {3663, 3817}, {3670, 8229}, {3674, 3865}, {3677, 5219}, {3742, 3756}, {3744, 5432}, {3745, 17726}, {3749, 5218}, {3751, 24477}, {3755, 24386}, {3782, 4003}, {3813, 4646}, {3826, 16602}, {3848, 17245}, {3914, 4850}, {3925, 16610}, {3928, 24695}, {3953, 13407}, {3967, 4884}, {3976, 21620}, {4000, 17064}, {4008, 19804}, {4044, 10079}, {4054, 17155}, {4078, 18743}, {4090, 4899}, {4104, 5233}, {4298, 7385}, {4307, 5435}, {4310, 5226}, {4353, 10171}, {4388, 24627}, {4415, 5087}, {4416, 7774}, {4438, 11174}, {4518, 11814}, {4643, 9766}, {4648, 24752}, {4657, 7778}, {4734, 4780}, {4851, 15271}, {5133, 8070}, {5217, 21487}, {5255, 6684}, {5304, 16667}, {5306, 16666}, {5710, 24914}, {5716, 7288}, {5717, 6998}, {5847, 14829}, {6184, 25075}, {6636, 14792}, {6692, 17122}, {7322, 20196}, {7392, 10629}, {7484, 8069}, {7952, 8889}, {9592, 24247}, {11263, 24167}, {12609, 24046}, {15283, 25098}, {15491, 17243}, {17592, 24217}, {17597, 17718}, {17598, 17719}, {17599, 17720}, {17889, 24177}, {18483, 24851}, {21241, 24169}, {21795, 25074}, {24165, 25385}

### X(24240) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24240) lies on these lines: {1, 7616}, {307, 1210}, {3582, 15903}, {3831, 4967}, {17048, 17245}, {24189, 24200}, {24192, 24216}

### X(24241) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

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

X(24241) lies on these lines: {1, 147}, {7, 3944}, {10, 304}, {37, 25353}, {171, 4872}, {222, 226}, {942, 4920}, {982, 3663}, {1125, 24549}, {2276, 24318}, {3314, 3912}, {3840, 4357}, {3997, 5074}, {4021, 17599}, {4799, 4987}, {4860, 4887}, {5275, 24694}, {5276, 24712}, {6063, 17861}, {10436, 17064}, {13161, 24215}, {14828, 17719}, {16783, 20267}

### X(24242) =  ISOGONAL CONJUGATE OF X(8078)

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

X(24242) lies on the cubic K1079 and these lines: {1,289}, {57,2089}, {164,10215}, {173,266}, {188,258}, {505,10231}, {7588,10233}, {8094,10234}

X(24242) = isogonal conjugate of X(8078)
X(24242) = X(9)-cross conjugate of X(258)
X(24242) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8078}, {56, 5430}, {266, 12646}
X(24242) = cevapoint of X(1) and X(18291)
X(24242) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 8078}, {236, 173}
X(24242) = barycentric product X(174)*X(12644)
X(24242) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8078}, {9, 5430}, {259, 12646}, {12644, 556}

### X(24243) =  ISOGONAL CONJUGATE OF X(10133)

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

X(24243) lies on the curves Q124, K070a, K170, and on these lines: {4,487}, {136,13430}, {193,13439}, {254,488}, {264,3128}, {393,494}, {427,1007}, {648,19041}, {1300,1307}, {1322,12221}, {1826,13386}, {8884,8982}, {13441,14593}

X(24243) = isogonal conjugate of X(10133)
X(24243) = isotomic conjugate of X(487)
X(24243) = polar conjugate of X(3069)
X(24243) = X(19217)-anticomplementary conjugate of X(638)
X(24243) = X(i)-cross conjugate of X(j) for these (i,j): {486, 2}, {494, 5491}
X(24243) = X(i)-isoconjugate of X(j) for these (i,j): {1, 10133}, {6, 19216}, {31, 487}, {48, 3069}, {63, 6424}, {163, 17432}, {1973, 8223}
X(24243) = cevapoint of X(i) and X(j) for these (i,j): {2, 12221}, {4, 3536}, {494, 8946}
X(24243) = trilinear pole of line {2501, 14333}
X(24243) = barycentric product X(i)*X(j) for these {i,j}: {4, 5491}, {75, 19217}, {76, 8946}, {264, 494}, {1307, 14618}
X(24243) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 19216}, {2, 487}, {4, 3069}, {6, 10133}, {25, 6424}, {69, 8223}, {494, 3}, {523, 17432}, {1307, 4558}, {5491, 69}, {6406, 19033}, {6421, 19447}, {8946, 6}, {19217, 1}

### X(24244) =  ISOGONAL CONJUGATE OF X(10132)

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

X(24244) lies on the curves Q124, K070b, K170, and on these lines: {4,488}, {136,13441}, {193,13428}, {254,487}, {264,3127}, {393,493}, {427,1007}, {648,19042}, {1300,1306}, {1321,12222}, {1588,8950}, {1826,13387}, {13430,14593}

X(24244) = isogonal conjugate of X(10132)
X(24244) = isotomic conjugate of X(488)
X(24244) = polar conjugate of X(3068)
X(24244) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {8950, 6360}, {19218, 637}
X(24244) = X(i)-cross conjugate of X(j) for these (i,j): {485, 2}, {493, 5490}
X(24244) = cevapoint of X(i) and X(j) for these (i,j): {2, 12222}, {4, 3535}, {493, 8948}
X(24244) = trilinear pole of line {2501, 14334}
X(24244) = X(i)-isoconjugate of X(j) for these (i,j): {1, 10132}, {6, 19215}, {31, 488}, {48, 3068}, {63, 6423}, {163, 17431}, {255, 5200}, {1973, 8222}
X(24244) = barycentric product X(i)*X(j) for these {i,j}: {4, 5490}, {75, 19218}, {76, 8948}, {264, 493}, {1306, 14618}
X(24244) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 19215}, {2, 488}, {4, 3068}, {6, 10132}, {25, 6423}, {69, 8222}, {393, 5200}, {493, 3}, {523, 17431}, {1306, 4558}, {5490, 69}, {6291, 19032}, {6422, 19446}, {8948, 6}, {8950, 8911}, {19218, 1}

### X(24245) =  X(2)-CEVA CONJUGATE OF X(486)

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

X(24245) lies on the cubic K168 and these lines: {3,486}, {6,8940}, {20,1322}, {69,1590}, {131,6560}, {570,18289}, {631,13429}, {1579,10666}, {1609,18290}, {2165,6676}, {5418,8961}, {7586,21464}, {8576,11513}, {10963,11291}

X(24245) = complement of X(24243)
X(24245) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 486}, {487, 2887}, {3069, 20305}, {6424, 226}, {10133, 10}, {17432, 21253}, {19216, 141}
X(24245) = X(2)-Ceva conjugate of X(486)
X(24245) = X(372)-isoconjugate of X(19217)
X(24245) = crosspoint of X(2) and X(487)
X(24245) = crosssum of X(6) and X(8946)
X(24245) = barycentric product X(i)*X(j) for these {i,j}: {486, 487}, {3069, 11091}
X(24245) = barycentric quotient X(i)/X(j) for these {i,j}: {487, 491}, {3069, 1586}, {6414, 494}, {6424, 5412}, {8576, 8946}, {10133, 372}, {11091, 5491}

### X(24246) =  X(2)-CEVA CONJUGATE OF X(485)

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

X(24246) lies on the cubic K168 and these lines: {3,485}, {6,8944}, {20,1321}, {69,1589}, {131,6561}, {570,18290}, {631,13440}, {1609,18289}, {2165,6676}, {7585,21463}, {8577,11514}, {10961,11292}

X(24246) = complement of X(24244)
X(24246) = X(24246) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 485}, {488, 2887}, {3068, 20305}, {6423, 226}, {10132, 10}, {17431, 21253}, {19215, 141}
X(24246) = X(24246) = X(2)-Ceva conjugate of X(485)
X(24246) = X(24246) = X(i)-isoconjugate of X(j) for these (i,j): {92, 8950}, {371, 19218}
X(24246) = X(24246) = crosspoint of X(2) and X(488)
X(24246) = X(24246) = crosssum of X(6) and X(8948)
X(24246) = X(24246) = barycentric product X(i)*X(j) for these {i,j}: {485, 488}, {3068, 11090}
X(24246) = X(24246) = barycentric quotient X(i)/X(j) for these {i,j}: {184, 8950}, {488, 492}, {3068, 1585}, {6413, 493}, {6423, 5413}, {8577, 8948}, {10132, 371}, {11090, 5490}

### X(24247) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24247) lies on these lines: {1, 5286}, {2, 9317}, {4, 3061}, {8, 672}, {9, 515}, {20, 3496}, {169, 17647}, {257, 7791}, {348, 17046}, {355, 25066}, {377, 17451}, {944, 2329}, {986, 7738}, {997, 5179}, {1018, 12647}, {1146, 1376}, {1212, 5794}, {1475, 12649}, {1573, 6184}, {1759, 4299}, {2170, 3434}, {2266, 3189}, {2267, 2345}, {2549, 2795}, {2784, 3923}, {3488, 16503}, {3509, 4293}, {3693, 5252}, {3721, 9597}, {3727, 9598}, {3785, 17739}, {3878, 17732}, {4051, 5082}, {4165, 17740}, {4317, 17736}, {4513, 10944}, {5013, 21965}, {5289, 17747}, {5438, 23058}, {5816, 25078}, {6506, 15842}, {9592, 24239}, {10527, 21029}, {10573, 16549}, {11185, 18061}, {17754, 18391}, {24265, 24275}

X(24247) = X(7)-of-1st-Brocard-triangle

### X(24248) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

Let AaBaCa, AbBbCb, AcBcCc be the A-, B- and C-anti-altimedial triangles. Let A' be the trilinear product Aa*Ab*Ac, and define B' and C' cyclically. A'B'C' is also the anticomplementary triangle of the 1st Brocard triangle. Also, X(24248) = X(1)-of-A'B'C'. (Randy Hutson, October 15, 2018)

X(24248) lies on these lines: {1, 7}, {2, 846}, {3, 1284}, {4, 240}, {6, 17768}, {8, 726}, {9, 1738}, {10, 2996}, {11, 17595}, {30, 7986}, {31, 19785}, {35, 24309}, {37, 5880}, {38, 3434}, {40, 1423}, {42, 5905}, {43, 329}, {45, 3826}, {46, 1400}, {55, 3782}, {57, 12717}, {63, 1711}, {65, 17635}, {69, 740}, {72, 18252}, {75, 23690}, {79, 941}, {86, 11104}, {141, 5695}, {144, 1757}, {145, 17766}, {149, 4392}, {171, 3474}, {182, 5091}, {190, 4429}, {191, 1714}, {192, 4645}, {193, 17770}, {194, 3865}, {226, 17594}, {228, 14455}, {238, 4000}, {291, 24458}, {344, 3836}, {345, 2887}, {354, 12722}, {377, 2292}, {387, 1046}, {497, 982}, {499, 17077}, {517, 1469}, {518, 17276}, {519, 11160}, {527, 3751}, {528, 3242}, {536, 3416}, {573, 1756}, {611, 5762}, {896, 24597}, {938, 9801}, {940, 4854}, {942, 12723}, {946, 988}, {966, 24697}, {968, 5249}, {984, 2550}, {1001, 1086}, {1058, 3976}, {1125, 17304}, {1150, 4442}, {1155, 17720}, {1193, 11415}, {1210, 21629}, {1266, 3883}, {1352, 2783}, {1376, 4415}, {1386, 17301}, {1403, 4192}, {1463, 3057}, {1478, 4424}, {1479, 3670}, {1633, 7295}, {1698, 5296}, {1699, 24239}, {1720, 2961}, {1722, 12572}, {1733, 17861}, {1737, 12618}, {1836, 3666}, {1899, 18202}, {1957, 17903}, {2385, 7289}, {2478, 24443}, {2549, 2795}, {2551, 24440}, {2782, 19637}, {2784, 5921}, {2792, 6776}, {3052, 17061}, {3056, 15310}, {3058, 17597}, {3061, 7738}, {3094, 24289}, {3210, 4388}, {3218, 11269}, {3240, 17484}, {3339, 7996}, {3436, 4642}, {3475, 3750}, {3496, 5286}, {3550, 9778}, {3618, 4672}, {3649, 19765}, {3662, 3685}, {3673, 4008}, {3677, 9580}, {3679, 10005}, {3683, 24789}, {3696, 4643}, {3721, 9598}, {3727, 9597}, {3736, 17139}, {3752, 24703}, {3772, 4640}, {3784, 21334}, {3842, 24693}, {3844, 17281}, {3868, 12530}, {3875, 5847}, {3886, 17274}, {3891, 4450}, {3924, 6872}, {3932, 17262}, {3938, 20075}, {3946, 16475}, {3961, 17784}, {3993, 17316}, {4001, 17156}, {4003, 17721}, {4026, 4363}, {4085, 17767}, {4133, 17294}, {4137, 25254}, {4259, 20718}, {4389, 5263}, {4440, 24349}, {4443, 24316}, {4512, 23681}, {4644, 4649}, {4656, 5268}, {4675, 15569}, {4676, 16706}, {4679, 16610}, {4683, 5739}, {4689, 17718}, {4703, 14555}, {4716, 5839}, {4733, 17251}, {4743, 17771}, {4850, 5057}, {4966, 7232}, {4995, 17783}, {5084, 24174}, {5132, 16382}, {5218, 17601}, {5220, 17334}, {5222, 16468}, {5248, 24159}, {5250, 23536}, {5255, 6361}, {5272, 24177}, {5292, 16566}, {5530, 9612}, {5712, 17592}, {5724, 12943}, {5745, 17064}, {5759, 9441}, {5800, 8680}, {5853, 16496}, {6007, 10477}, {6327, 17147}, {6392, 17739}, {6857, 24161}, {6986, 8238}, {7390, 8245}, {8424, 13723}, {9812, 17591}, {10186, 24203}, {10385, 17715}, {10431, 11031}, {11019, 18193}, {11677, 17446}, {12047, 12610}, {12514, 23537}, {12579, 13736}, {13097, 16056}, {14450, 19767}, {15254, 17278}, {15485, 16020}, {16569, 18228}, {16600, 17732}, {16830, 17247}, {16910, 25248}, {17018, 17483}, {17150, 20064}, {17164, 17676}, {17183, 18792}, {17593, 17717}, {17950, 18391}, {18201, 24217}, {20017, 20290}, {24282, 24293}, {24347, 25372}, {24424, 24445}

X(24248) = X(8)-of-1st-Brocard-triangle

### X(24249) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24249) lies on these lines: {1, 20335}, {2, 9317}, {10, 7795}, {55, 21232}, {56, 17048}, {76, 16822}, {142, 515}, {355, 17062}, {730, 16825}, {999, 6647}, {1111, 16788}, {1352, 2784}, {1837, 17046}, {2329, 3673}, {2795, 3734}, {3061, 17681}, {3212, 3496}, {3663, 9620}, {3735, 24281}, {3754, 14377}, {3816, 17044}, {3980, 4112}, {4011, 24255}, {5088, 17754}, {11343, 16609}, {17050, 19860}, {17619, 24784}, {17683, 21921}, {17738, 24282}, {24263, 24279}, {24264, 24283}

X(24249) = X(9)-of-1st-Brocard-triangle

### X(24250) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24250) lies on these lines: {1, 19890}, {2, 1083}, {5, 10}, {9, 2957}, {11, 14839}, {36, 1009}, {75, 7336}, {141, 513}, {542, 5150}, {760, 908}, {1211, 3140}, {1352, 24265}, {3923, 24206}, {3944, 19950}, {4011, 21243}, {4092, 18151}, {4422, 21252}, {5057, 17292}, {11679, 19987}, {17047, 17279}, {24254, 25385}

X(24250) = X(11)-of-1st-Brocard-triangle

### X(24251) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24251) lies on these lines: {2, 24265}, {141, 20718}, {542, 24253}, {1352, 24264}, {3821, 24269}, {3923, 24206}, {3980, 21243}, {5880, 17047}, {19950, 24274}

X(24251) = X(12)-of-1st-Brocard-triangle

### X(24252) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24252) lies on these lines: {1231, 1973}, {1352, 2784}, {3734, 24269}, {3735, 24268}, {3923, 24263}, {3980, 9306}, {9317, 14826}

X(24252) = X(19)-of-1st-Brocard-triangle

### X(24253) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24253) lies on these lines: {10, 7193}, {182, 2783}, {184, 3980}, {542, 24251}, {692, 24325}, {726, 2330}, {740, 5135}, {1376, 23095}, {1580, 4279}, {2175, 16825}, {3821, 5091}, {4418, 5012}, {16549, 19554}, {19561, 24295}, {24259, 24267}

X(24253) = X(35)-of-1st-Brocard-triangle

### X(24254) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

Barycentrics    a^3 b + a^3 c + a b^2 c + b^3 c + a b c^2 + b c^3 : :

X(24254) lies on these lines: {2, 3125}, {6, 8682}, {10, 626}, {32, 16822}, {37, 712}, {65, 21240}, {141, 758}, {213, 20911}, {274, 2106}, {304, 17750}, {517, 3739}, {742, 3997}, {766, 17792}, {894, 20924}, {1015, 24631}, {1018, 24326}, {1111, 24330}, {1215, 21232}, {1572, 4384}, {1573, 17755}, {1930, 2295}, {2284, 4363}, {2795, 3734}, {3727, 16818}, {3754, 20255}, {3980, 9306}, {4372, 5264}, {4376, 16788}, {4418, 9317}, {5091, 5108}, {7816, 24850}, {9620, 10436}, {10800, 16825}, {14210, 24512}, {14839, 24325}, {16549, 16720}, {16705, 25248}, {17034, 17762}, {17357, 21331}, {17489, 21802}, {17499, 20955}, {24250, 25385}, {24256, 24295}, {24275, 24281}

X(24254) = X(37)-of-1st-Brocard-triangle

### X(24255) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24255) lies on these lines: {1, 17789}, {2, 19933}, {6, 744}, {321, 730}, {1045, 17788}, {2278, 4381}, {2309, 20234}, {2795, 24259}, {3285, 4836}, {3923, 4112}, {3980, 24266}, {4011, 24249}, {4703, 5692}, {4812, 7032}, {6327, 9857}, {14210, 18138}, {16732, 24327}, {20432, 21352}

X(24255) = X(38)-of-1st-Brocard-triangle

### X(24256) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24256) lies on these lines: {2, 694}, {5, 141}, {6, 76}, {13, 25167}, {14, 25157}, {32, 8177}, {39, 698}, {51, 8891}, {69, 7785}, {99, 5116}, {182, 2782}, {183, 5017}, {194, 3618}, {262, 7778}, {384, 1691}, {385, 12212}, {518, 12263}, {524, 5052}, {538, 597}, {575, 13196}, {591, 22723}, {599, 7809}, {611, 10079}, {613, 10063}, {726, 3946}, {730, 1386}, {736, 7804}, {742, 21443}, {1078, 2076}, {1180, 10191}, {1235, 2211}, {1350, 13860}, {1352, 6033}, {1428, 18982}, {1501, 16932}, {1503, 6248}, {1915, 16950}, {1974, 12143}, {1991, 22722}, {2021, 8369}, {2330, 13077}, {3095, 7795}, {3098, 7815}, {3104, 23024}, {3105, 23018}, {3329, 9865}, {3763, 7887}, {3788, 11272}, {5007, 15870}, {5012, 10328}, {5034, 17130}, {5038, 12215}, {5039, 7751}, {5050, 13108}, {5085, 11257}, {5092, 7816}, {5108, 7698}, {5207, 16044}, {5939, 22498}, {6309, 9605}, {6656, 10292}, {7664, 10160}, {7783, 12055}, {7787, 9983}, {7800, 9821}, {7808, 8149}, {8024, 20965}, {8267, 11205}, {9230, 14603}, {9516, 11060}, {9902, 16475}, {10130, 15107}, {10329, 16276}, {12251, 14853}, {13511, 19121}, {16964, 23019}, {16965, 23025}, {17353, 17760}, {24254, 24295}, {25324, 25325}

X(24256) = midpoint of X(6) and X(76)
X(24256) = midpoint of PU(181)
X(24256) = complement of X(3094)
X(24256) = anticomplement of X(10007)
X(24256) = X(39)-of-1st-Brocard-triangle

### X(24257) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24257) lies on these lines: {3, 740}, {40, 3875}, {182, 2783}, {192, 6211}, {239, 6210}, {355, 4085}, {511, 24728}, {515, 990}, {912, 4523}, {946, 3946}, {1352, 2784}, {1429, 4008}, {1999, 20368}, {2187, 17862}, {2321, 6684}, {2791, 24283}, {2792, 6776}, {2796, 11179}, {3564, 4655}, {3576, 3886}, {4133, 10164}, {4297, 4780}, {4672, 5050}, {4743, 18481}, {5085, 5695}, {5278, 11203}, {5853, 5882}, {5988, 9744}, {7413, 17592}, {7609, 17349}, {7613, 9317}, {8235, 9534}, {8550, 17768}, {14912, 24695}

X(24257) = X(40)-of-1st-Brocard-triangle

### X(24258) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24258) lies on these lines: {1083, 3923}, {1233, 9447}, {3734, 24263}, {4048, 4112}, {9306, 24259}

X(24258) = X(41)-of-1st-Brocard-triangle

### X(24259) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24259) lies on these lines: {2, 846}, {31, 17031}, {42, 726}, {43, 3729}, {76, 4039}, {171, 350}, {190, 24463}, {321, 2239}, {384, 1580}, {516, 1764}, {672, 21369}, {986, 1008}, {1009, 24850}, {1086, 24425}, {1215, 24326}, {1269, 1918}, {1376, 4713}, {1707, 17026}, {2223, 12263}, {2245, 24688}, {2795, 24255}, {3734, 4112}, {3747, 20913}, {3769, 4479}, {3961, 17794}, {4286, 25347}, {4362, 4441}, {4436, 24327}, {4640, 21264}, {4697, 24512}, {5091, 24271}, {5108, 24279}, {9306, 24258}, {10453, 20101}, {17135, 17766}, {24253, 24267}

X(24259) = X(42)-of-1st-Brocard-triangle

### X(24260) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24260) lies on these lines: {2, 846}, {43, 726}, {194, 17795}, {239, 9902}, {516, 3840}, {962, 18788}, {1376, 17793}, {2239, 4362}, {3434, 3741}, {3663, 6685}, {3729, 16569}, {3734, 24276}, {3771, 12610}, {4112, 5091}, {4192, 24728}, {10453, 17766}, {16557, 17754}, {17290, 24425}, {24264, 24294}

X(24260) = X(43)-of-1st-Brocard-triangle

### X(24261) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24261) lies on these lines: {2, 24278}, {2787, 4107}, {2795, 3734}, {16822, 17130}, {17738, 24281}

X(24261) = X(44)-of-1st-Brocard-triangle

### X(24262) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24262) lies on these lines: {2, 24277}, {10, 7869}, {2795, 3734}, {7751, 16822}

X(24262) = X(45)-of-1st-Brocard-triangle

### X(24263) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24263) lies on these lines: {349, 9247}, {3734, 24258}, {3735, 24288}, {3923, 24252}, {4112, 4172}, {24249, 24279}

X(24263) = X(48)-of-1st-Brocard-triangle

### X(24264) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24264) lies on these lines: {2, 1083}, {6, 6007}, {31, 10800}, {32, 1580}, {38, 9997}, {55, 14839}, {63, 760}, {75, 2175}, {99, 24282}, {101, 1376}, {171, 2242}, {182, 2783}, {184, 4418}, {517, 993}, {573, 8424}, {692, 4363}, {740, 5138}, {846, 3735}, {1352, 24251}, {1486, 17049}, {1572, 1707}, {1754, 18235}, {1836, 24630}, {2278, 4436}, {2329, 9441}, {2330, 3729}, {2344, 24578}, {2795, 24268}, {3550, 11364}, {3556, 9565}, {3734, 4112}, {3875, 19133}, {3980, 9306}, {4039, 7751}, {4858, 24329}, {5114, 24696}, {5135, 5695}, {9620, 17594}, {13323, 24850}, {16792, 17301}, {17792, 24309}, {24249, 24283}, {24260, 24294}

X(24264) = X(55)-of-1st-Brocard-triangle

### X(24265) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24265) lies on these lines: {2, 24251}, {182, 2783}, {312, 1397}, {613, 14839}, {1352, 24250}, {1428, 3729}, {2175, 4676}, {3501, 12194}, {3734, 24258}, {4011, 9306}, {4672, 5138}, {5091, 24280}, {13478, 20545}, {24247, 24275}, {24276, 24294}

X(24265) = X(56)-of-1st-Brocard-triangle

### X(24266) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24266) lies on these lines: {2, 9317}, {182, 2783}, {239, 3501}, {515, 997}, {604, 1229}, {1083, 4011}, {2795, 24283}, {3008, 16968}, {3061, 6996}, {3980, 24255}, {4858, 24334}, {7803, 17023}, {16609, 21477}

X(24266) = X(57)-of-1st-Brocard-triangle

### X(24267) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24267) lies on these lines: {6, 726}, {182, 2782}, {384, 4279}, {515, 12618}, {1230, 2206}, {3770, 5009}, {3923, 4112}, {4713, 23095}, {24253, 24259}, {24275, 24294}

X(24267) = X(58)-of-1st-Brocard-triangle

### X(24268) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24268) lies on these lines: {1, 4}, {2, 9317}, {3, 16609}, {6, 8680}, {9, 25252}, {57, 5088}, {63, 194}, {92, 2202}, {213, 22001}, {284, 286}, {321, 4390}, {326, 21246}, {355, 16603}, {379, 17451}, {527, 1992}, {572, 17861}, {604, 17863}, {730, 4362}, {870, 1438}, {940, 7223}, {966, 24435}, {980, 13478}, {993, 4124}, {1400, 17134}, {1429, 3673}, {1431, 24728}, {1441, 2268}, {2278, 16732}, {2792, 6776}, {2795, 24264}, {3735, 24252}, {3923, 4112}, {3980, 24293}, {4011, 24294}, {4384, 5745}, {4393, 5905}, {5091, 24283}, {5244, 7354}, {5283, 25080}, {6996, 7146}, {7384, 17084}, {7709, 17596}, {9028, 16973}, {9741, 16833}, {17738, 18906}, {17797, 21299}

X(24268) = X(63)-of-1st-Brocard-triangle

### X(24269) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24269) lies on these lines: {182, 2783}, {740, 4260}, {1009, 1733}, {1284, 20236}, {1352, 2791}, {2549, 2795}, {2886, 16579}, {3734, 24252}, {3821, 24251}, {14453, 17647}

X(24269) = X(65)-of-1st-Brocard-triangle

### X(24270) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24270) lies on these lines: {30, 18382}, {182, 2794}, {1899, 2549}, {2386, 7761}, {2790, 9996}, {3734, 21243}, {7514, 23333}

X(24270) = X(66)-of-1st-Brocard-triangle

### X(24271) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

Barycentrics    a^5 + a^4 b + a^4 c + a^3 b c + a b^3 c + 2 a b^2 c^2 + 2 b^3 c^2 + a b c^3 + 2 b^2 c^3 : :

X(24271) lies on these lines: {2, 99}, {6, 536}, {30, 1211}, {76, 5337}, {81, 538}, {172, 4044}, {187, 16046}, {321, 5291}, {535, 11355}, {754, 2895}, {980, 1975}, {1083, 1316}, {1230, 17587}, {2795, 4418}, {3175, 16785}, {3923, 4112}, {3934, 21495}, {3948, 5277}, {4383, 11286}, {4721, 20769}, {5091, 24259}, {5278, 11352}, {5283, 19281}, {5739, 7737}, {7815, 21537}, {7816, 21511}, {11354, 19701}, {14033, 14555}, {15271, 16431}, {16609, 24850}, {16732, 24335}, {21024, 24632}, {24277, 24288}

X(24271) = X(81)-of-1st-Brocard-triangle

### X(24272) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24272) lies on these lines: {10, 20994}, {3735, 4112}, {3821, 24273}, {3923, 4172}, {4418, 9317}

X(24272) = X(82)-of-1st-Brocard-triangle

### X(24273) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

Barycentrics    a^6 + a^4 b^2 + a^2 b^4 + a^4 c^2 + 3 a^2 b^2 c^2 + 2 b^4 c^2 + a^2 c^4 + 2 b^2 c^4 : :

X(24273) lies on these lines: {2, 4048}, {3, 2916}, {6, 76}, {69, 20088}, {75, 17741}, {99, 10007}, {141, 384}, {182, 7697}, {183, 3407}, {511, 13111}, {518, 9903}, {597, 19570}, {599, 754}, {623, 23019}, {624, 23025}, {698, 17128}, {782, 17997}, {1003, 21358}, {1350, 22677}, {1503, 12252}, {1691, 3934}, {1915, 8891}, {3053, 6308}, {3094, 3734}, {3114, 10010}, {3314, 9866}, {3416, 17766}, {3552, 3619}, {3589, 7797}, {3821, 24272}, {4074, 4563}, {5031, 7832}, {5085, 14880}, {5103, 16044}, {5149, 11646}, {6683, 12055}, {6704, 7834}, {7716, 12144}, {7804, 12212}, {7808, 13331}, {7935, 18500}, {8369, 13086}, {8627, 10130}, {9743, 9751}, {10387, 13078}, {11324, 21001}, {12156, 15533}, {13330, 18548}, {13586, 20582}, {14621, 20917}

X(24273) = X(83)-of-1st-Brocard-triangle

### X(24274) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24274) lies on these lines: {971, 1212}, {984, 6184}, {2549, 2795}, {2550, 17435}, {3125, 7613}, {3501, 9941}, {3721, 4862}, {19950, 24251}

X(24274) = X(85)-of-1st-Brocard-triangle

### X(24275) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24275) lies on these lines: {1, 4037}, {2, 99}, {6, 519}, {10, 17735}, {30, 1213}, {32, 4195}, {39, 13740}, {58, 21024}, {76, 17688}, {86, 538}, {187, 4234}, {754, 1654}, {964, 5283}, {966, 7737}, {1010, 16589}, {1043, 20970}, {1220, 1500}, {1573, 5263}, {1724, 12194}, {2322, 14581}, {2702, 2758}, {2795, 24342}, {3125, 4418}, {3735, 3923}, {3934, 16061}, {4201, 7756}, {5224, 7761}, {5254, 17698}, {5275, 16394}, {5277, 11115}, {5793, 14974}, {7748, 16062}, {7798, 17379}, {7804, 17277}, {7816, 16060}, {7848, 17271}, {7865, 17238}, {8818, 24935}, {9346, 10453}, {9597, 19836}, {9598, 19784}, {9903, 16552}, {11286, 17259}, {11287, 17327}, {15048, 17398}, {16975, 24552}, {17330, 18907}, {23897, 24880}, {24247, 24265}, {24254, 24281}, {24267, 24294}

X(24275) = X(86)-of-1st-Brocard-triangle

### X(24276) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24276) lies on these lines: {10, 3500}, {3734, 24260}, {3923, 18906}, {24265, 24294}

X(24276) = X(87)-of-1st-Brocard-triangle

### X(24277) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24277) lies on these lines: {2, 24262}, {952, 16594}, {3734, 24278}, {3923, 5091}, {24271, 24288}

X(24277) = X(88)-of-1st-Brocard-triangle

### X(24278) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24278) lies on these lines: {2, 24261}, {3734, 24277}

X(24278) = X(89)-of-1st-Brocard-triangle

### X(24279) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24279) lies on the Brocard circle and these lines: {2, 5991}, {3, 142}, {6, 514}, {86, 4237}, {238, 19950}, {384, 24502}, {726, 17976}, {894, 5088}, {1023, 3729}, {1083, 3923}, {5011, 20179}, {5108, 24259}, {5179, 17353}, {24249, 24263}

X(24279) = X(101)-of-1st-Brocard-triangle
X(24279) = inverse-in-circumcircle of X(8618)
X(24279) = isogonal conjugate of antitomic conjugate of X(101)

### X(24280) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24280) lies on these lines: {1, 7240}, {2, 846}, {7, 3685}, {8, 144}, {63, 12717}, {69, 5695}, {72, 12530}, {75, 5698}, {78, 1721}, {145, 726}, {190, 2550}, {192, 4307}, {193, 740}, {312, 3474}, {344, 5880}, {345, 1836}, {346, 4645}, {390, 4454}, {518, 12718}, {527, 3886}, {545, 3242}, {960, 17635}, {962, 20348}, {984, 20073}, {990, 4511}, {1266, 7290}, {2345, 24723}, {2783, 6776}, {2792, 5921}, {2795, 24282}, {3241, 17132}, {3332, 25252}, {3522, 24728}, {3616, 3663}, {3617, 4660}, {3620, 4655}, {3621, 17766}, {3705, 9812}, {3868, 12723}, {3872, 12652}, {3873, 12722}, {3876, 18252}, {3877, 12721}, {3883, 4659}, {3912, 4312}, {4000, 4676}, {4018, 21848}, {4190, 25253}, {4295, 7283}, {4310, 4440}, {4385, 6361}, {4387, 11246}, {4419, 5263}, {4442, 24597}, {4779, 11038}, {5057, 17740}, {5091, 24265}, {5550, 17304}, {6734, 21629}, {6872, 17164}, {6904, 19582}, {7081, 9778}, {9780, 17355}, {9965, 10453}, {11358, 13097}, {11415, 20245}, {15310, 25304}, {17127, 19789}, {17135, 20078}, {17165, 20075}, {17770, 20080}, {17776, 20292}

X(24280) = X(145)-of-1st-Brocard-triangle

### X(24281) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24281) lies on these lines: 1, 16377}, {2, 24262}, {6, 514}, {8, 7794}, {10, 19886}, {32, 3212}, {101, 21138}, {192, 1016}, {239, 980}, {519, 599}, {664, 1015}, {903, 24807}, {952, 1086}, {1506, 17084}, {3125, 9317}, {3699, 13466}, {3734, 24282}, {3735, 24249}, {3752, 5662}, {3912, 17720}, {4000, 6547}, {4237, 5170}, {4360, 6631}, {4393, 4555}, {4482, 9055}, {5091, 24286}, {6633, 17318}, {9259, 21208}, {17119, 24873}, {17395, 24864}, {17738, 24261}, {24254, 24275}

X(24281) = X(190)-of-1st-Brocard-triangle

### X(24282) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24282) lies on these lines: {2, 3125}, {7, 20924}, {8, 315}, {40, 7350}, {65, 304}, {69, 758}, {75, 517}, {76, 3212}, {85, 12709}, {99, 24264}, {145, 17141}, {192, 712}, {193, 8682}, {213, 21216}, {222, 664}, {239, 1572}, {668, 6382}, {766, 25304}, {874, 24351}, {883, 4454}, {894, 9620}, {942, 18156}, {1376, 4561}, {1655, 25270}, {1930, 5903}, {2795, 24280}, {3496, 16822}, {3501, 17760}, {3734, 24281}, {3869, 20911}, {3905, 5255}, {3923, 18906}, {4165, 4766}, {4647, 25307}, {4713, 21138}, {5902, 14210}, {7763, 17084}, {10449, 17762}, {14839, 24349}, {17165, 21272}, {17279, 21331}, {17738, 24249}, {18135, 25253}, {18140, 19582}, {24248, 24293}

X(24282) = X(192)-of-1st-Brocard-triangle

### X(24283) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24283) lies on these lines: {2, 846}, {43, 10025}, {57, 497}, {165, 1447}, {200, 726}, {982, 14942}, {986, 13727}, {1376, 24352}, {1403, 7580}, {1707, 24600}, {2791, 24257}, {2795, 24266}, {3663, 13405}, {3673, 9441}, {3729, 8580}, {3870, 4970}, {4660, 4847}, {5091, 24268}, {14828, 17592}, {24249, 24264}

X(24283) = X(200)-of-1st-Brocard-triangle

### X(24284) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24284) lies on these lines: {2, 3569}, {6, 9035}, {113, 126}, {125, 127}, {141, 526}, {441, 525}, {512, 625}, {620, 690}, {684, 879}, {804, 4107}, {850, 3288}, {888, 6131}, {1499, 16235}, {1640, 3268}, {2492, 3589}, {2799, 14316}, {3049, 3267}, {3050, 23285}, {3618, 14398}, {3800, 9134}, {5108, 18332}, {5652, 6033}, {22159, 23148}

X(24284) = X(647)-of-1st-Brocard-triangle

### X(24285) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24285) lies on these lines: {2, 24290}, {141, 8674}, {512, 4369}, {525, 14838}, {620, 690}, {812, 8659}, {826, 21196}, {2487, 3566}, {2787, 4107}, {3063, 15413}, {3309, 4885}, {3835, 6004}, {8062, 17066}, {14316, 21209}, {17023, 23829}

X(24285) = X(650)-of-1st-Brocard-triangle

### X(24286) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24286) lies on these lines: {1, 512}, {99, 110}, {513, 1386}, {659, 8297}, {663, 1201}, {818, 15413}, {826, 4560}, {840, 14665}, {891, 4435}, {1027, 6372}, {1385, 3309}, {2775, 11699}, {2786, 4164}, {2787, 4107}, {3566, 14529}, {4025, 8633}, {4378, 16971}, {4750, 5040}, {5029, 14419}, {5091, 24281}, {7927, 17166}, {8034, 17017}

X(24286) = X(659)-of-1st-Brocard-triangle

### X(24287) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24287) lies on these lines: {86, 24506}, {141, 900}, {665, 4486}, {798, 18160}, {802, 15413}, {812, 17738}, {816, 21007}, {876, 4010}, {1019, 1577}, {2254, 21261}, {2530, 3835}, {2642, 5224}, {2786, 23596}, {2787, 4107}, {3733, 8060}, {3766, 20913}, {3798, 17072}, {4444, 23829}, {4705, 21196}, {4750, 21053}, {6707, 24959}, {17069, 21051}

X(24287) = X(661)-of-1st-Brocard-triangle

### X(24288) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24288) lies on these lines: {6, 514}, {1316, 5091}, {1724, 19950}, {3735, 24263}, {24271, 24277}

X(24288) = X(662)-of-1st-Brocard-triangle

### X(24289) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24289) lies on these lines: {6, 513}, {9, 1575}, {37, 24338}, {192, 4562}, {292, 3123}, {350, 3662}, {536, 599}, {646, 20532}, {730, 4660}, {1045, 18795}, {1086, 2481}, {3094, 24248}, {3888, 17475}, {3959, 17435}, {4876, 4947}, {6646, 17759}, {9025, 16973}, {11650, 14419}, {17282, 20530}, {17303, 25382}

X(24289) = X(668)-of-1st-Brocard-triangle

### X(24290) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24290) lies on these lines: {2, 24285}, {6, 8674}, {115, 125}, {119, 20623}, {512, 661}, {513, 21389}, {514, 23780}, {521, 20980}, {525, 1577}, {647, 21837}, {649, 6004}, {650, 1734}, {656, 3709}, {665, 1642}, {826, 4024}, {832, 2484}, {918, 4437}, {926, 20455}, {1018, 4551}, {1635, 8659}, {1643, 3887}, {1769, 4526}, {2509, 3063}, {2530, 3250}, {2642, 14407}, {2775, 5540}, {2786, 23596}, {2821, 3030}, {3239, 17072}, {3251, 14438}, {3287, 6003}, {3566, 14321}, {3800, 4841}, {3906, 4931}, {3912, 23829}, {4017, 4171}, {4079, 8061}, {4086, 21958}, {4130, 7655}, {4486, 21261}, {4750, 5098}, {5029, 14419}, {6161, 8632}, {7180, 8611}, {9508, 9509}, {10097, 10693}, {10099, 18785}, {13277, 20331}, {21348, 23800}, {21797, 21828}

X(24290) = X(693)-of-1st-Brocard-triangle

### X(24291) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24291) lies on these lines: {1, 76}, {2, 9317}, {32, 17739}, {34, 9312}, {85, 2647}, {99, 17596}, {141, 25434}, {257, 384}, {286, 447}, {515, 4357}, {535, 17274}, {668, 1965}, {712, 3729}, {846, 4112}, {986, 1975}, {996, 3875}, {1757, 10791}, {3061, 7770}, {3212, 4195}, {3585, 17211}, {3734, 3735}, {3905, 4385}, {3923, 18906}, {3944, 11185}, {5086, 24995}, {5293, 6376}, {7789, 21965}, {10436, 17861}, {17451, 17686}, {24254, 24275}, {24293, 24342}

X(24291) = X(894)-of-1st-Brocard-triangle

### X(24292) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24292) lies on these lines: {2, 11711}, {537, 3751}, {2787, 4107}, {3923, 4112}, {16732, 24350}

X(24292) = X(896)-of-1st-Brocard-triangle

### X(24293) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24293) lies on these lines: {1, 20924}, {2, 19933}, {10, 12933}, {75, 730}, {758, 4259}, {760, 4660}, {894, 10791}, {991, 24325}, {2795, 3734}, {3094, 3735}, {3673, 12263}, {3980, 24268}, {4112, 4418}, {4645, 9857}, {24248, 24282}, {24291, 24342}

X(24293) = X(984)-of-1st-Brocard-triangle

### X(24294) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24294) lies on these lines: {1, 83}, {2, 4112}, {10, 8301}, {99, 2108}, {101, 17793}, {238, 730}, {560, 18044}, {668, 8300}, {993, 19263}, {1083, 4368}, {1125, 9259}, {1281, 19950}, {2329, 12263}, {2787, 4164}, {2795, 3734}, {3821, 24272}, {4011, 24268}, {4154, 17738}, {5291, 17031}, {12194, 17752}, {12782, 17743}, {24260, 24264}, {24265, 24276}, {24267, 24275}

X(24294) = X(238)-of-1st-Brocard-triangle

### X(24295) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

Barycentrics    2 a^3 + a^2 b + a b^2 + b^3 + a^2 c + 2 b^2 c + a c^2 + 2 b c^2 + c^3 : :

X(24295) lies on these lines: {1, 3790}, {2, 846}, {5, 516}, {10, 82}, {37, 39}, {44, 3775}, {141, 4672}, {182, 2784}, {267, 5506}, {319, 16477}, {519, 597}, {551, 17264}, {594, 4974}, {631, 24728}, {740, 3589}, {752, 3844}, {984, 17354}, {1213, 20546}, {1215, 17724}, {1698, 4660}, {2345, 16825}, {2792, 24206}, {3619, 24695}, {3624, 3729}, {3661, 16468}, {3663, 17322}, {3740, 12722}, {3741, 5294}, {3763, 4655}, {3797, 3993}, {3836, 17357}, {3842, 4422}, {3912, 20132}, {4026, 4432}, {4407, 15481}, {4527, 4852}, {4676, 17371}, {5745, 20545}, {6535, 17150}, {10022, 12040}, {12579, 13728}, {15310, 25144}, {16475, 17286}, {17123, 19808}, {17229, 17772}, {17235, 17767}, {17260, 19856}, {17307, 24697}, {18139, 23812}, {19561, 24253}, {24254, 24256}, {24267, 24275}, {24317, 25372}, {24340, 25364}

X(24295) = X(1125)-of-1st-Brocard-triangle

### X(24296) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = 1ST BROCARD TRIANGLE)

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

X(24296) lies on these lines: {2, 99}, {30, 5718}, {36, 17720}, {182, 5091}, {538, 1150}, {980, 6996}, {2276, 24630}, {2795, 4414}, {3936, 7761}, {4653, 11355}, {5114, 17139}, {7739, 24597}, {7798, 16704}, {7804, 11352}, {17595, 24618}

X(24296) = X(1150)-of-1st-Brocard-triangle

### X(24297) =  ISOGONAL CONJUGATE OF X(5126)

Barycentrics    a*(a^3-(4*b+c)*a^2-(b^2-6*b*c+c^2)*a+(b^2-c^2)*(4*b-c))*(a^3-(b+4*c)*a^2-(b^2-6*b*c+c^2)*a+(b^2-c^2)*(b-4*c)) : :
X(24297) = 2*X(6594)-3*X(9623)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28386.

X(24297) lies on the Feuerbach hyperbola and these lines: {1, 6946}, {4, 12762}, {7, 952}, {8, 4767}, {9, 2802}, {11, 1000}, {21, 10914}, {100, 2320}, {104, 1155}, {517, 1156}, {519, 3254}, {1317, 18490}, {1320, 3935}, {1656, 7320}, {2800, 3062}, {2826, 23836}, {2829, 10307}, {3427, 12247}, {3577, 10698}, {3625, 6598}, {3887, 4792}, {4900, 12653}, {5424, 10087}, {5557, 10106}, {5559, 12053}, {5561, 9897}, {5854, 6601}, {6594, 9623}, {7319, 8148}, {7972, 14563}, {10039, 13606}, {11604, 12531}, {12641, 21630}, {12738, 17097}, {13602, 16173}

X(24297) = midpoint of X(4900) and X(12653)
X(24297) = reflection of X(7972) in X(14563)
X(24297) = isogonal conjugate of X(5126)
X(24297) = antigonal conjugate of X(1000)
X(24297) = antipode of X(1000) in the Feuerbach hyperbola
X(24297) = trilinear pole of the line {45, 650}

### X(24298) =  ANTIGONAL CONJUGATE OF X(943)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28386.

X(24298) lies on the Feuerbach hyperbola and these lines: {1, 6901}, {9, 21090}, {11, 943}, {21, 149}, {104, 5842}, {388, 15173}, {497, 15175}, {952, 17097}, {1156, 5762}, {1479, 3467}, {2320, 3434}, {2346, 6881}, {4302, 15446}, {5083, 5557}, {6596, 21630}, {7319, 10526}

X(24298) = antigonal conjugate of X(943)
X(24298) = isotomic conjugate of the anticomplement of X(17796)
X(24298) = antipode of X(943) in the Feuerbach hyperbola

### X(24299) =  MIDPOINT OF X(1) AND X(10902)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28386.

X(24299) lies on these lines: {1, 3}, {21, 912}, {78, 6883}, {140, 5440}, {225, 7510}, {226, 7491}, {284, 8609}, {355, 6861}, {382, 5715}, {443, 10806}, {515, 6841}, {631, 12649}, {938, 6954}, {944, 6824}, {950, 6842}, {952, 6675}, {971, 13743}, {1006, 14054}, {1100, 5755}, {1104, 5396}, {1125, 6881}, {1621, 21740}, {2320, 5768}, {3358, 12687}, {3487, 6868}, {3488, 6825}, {3526, 5705}, {3560, 18446}, {3616, 6826}, {3655, 5787}, {3683, 5694}, {3868, 6875}, {3897, 6857}, {3916, 7508}, {4313, 6850}, {4330, 16155}, {5178, 6989}, {5248, 5887}, {5259, 6326}, {5267, 12005}, {5436, 5720}, {5439, 6924}, {5603, 6869}, {5703, 6827}, {5722, 6863}, {5731, 6851}, {5735, 15696}, {5745, 13607}, {5761, 6987}, {5777, 7489}, {5780, 16857}, {5842, 11281}, {5901, 20420}, {6847, 10587}, {6882, 13411}, {6906, 13369}, {6928, 11374}, {6951, 11015}, {7497, 11363}, {8728, 10943}, {9942, 10179}, {10165, 10916}, {10526, 17718}, {11520, 21165}

X(24299) = midpoint of X(1) and X(10902)
X(24299) = X(5576)-of-2nd circumperp triangle
X(24299) = X(7568)-of-hexyl triangle
X(24299) = X(10902)-of-anti-Aquila triangle
X(24299) = X(14130)-of-Ascella triangle
X(24299) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3576, 11249), (1, 7987, 12704), (1, 16208, 7982), (3, 1385, 13151), (1385, 15178, 1319), (3587, 7987, 3), (3601, 18443, 3), (6906, 18444, 13369), (10246, 16202, 1)

### X(24300) =  ANTIGONAL CONJUGATE OF X(1061)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28386.

X(24300) lies on the Feuerbach hyperbola and this line: {11, 1061}

X(24300) = antigonal conjugate of X(1061)
X(24300) = antipode of X(1061) in the Feuerbach hyperbola

### X(24301) =  MIDPOINT OF X(1) AND X(15177)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28386.

X(24301) lies on these lines: {1, 3}, {5, 11363}, {10, 7542}, {26, 1829}, {184, 912}, {355, 3549}, {515, 15760}, {518, 19131}, {944, 3547}, {946, 12605}, {952, 6676}, {1125, 11585}, {1386, 9967}, {1902, 7526}, {2072, 11230}, {3616, 6643}, {3817, 10297}, {5886, 18531}, {5901, 12362}, {6146, 12259}, {6639, 9956}, {6923, 11393}, {6928, 11392}, {7404, 7718}, {7494, 7967}, {7713, 9714}, {7723, 11699}, {7968, 10897}, {7969, 10898}, {9619, 23115}, {9715, 11396}, {9928, 19357}, {9955, 18404}, {10024, 18480}, {10165, 10257}, {10575, 12262}, {11720, 12358}, {12266, 12363}, {18563, 22793}, {18670, 22054}

X(24301) = midpoint of X(1) and X(15177)
X(24301) = X(15177)-of-anti-Aquila triangle

### X(24302) =  TRILINEAR POLE OF X(650)X(15492)

Barycentrics    a*(2*a^3-(3*b+2*c)*a^2-(2*b^2-7*b*c+2*c^2)*a+(b^2-c^2)*(3*b-2*c))*(2*a^3-(2*b+3*c)*a^2-(2*b^2-7*b*c+2*c^2)*a+(b^2-c^2)*(2*b-3*c)) : :
X(24302) = 2*X(15079)-3*X(16173)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28386.

X(24302) lies on the Feuerbach hyperbola and these lines: {4, 7972}, {8, 6702}, {79, 14217}, {80, 5048}, {84, 13253}, {952, 5560}, {1317, 5561}, {1392, 2802}, {1476, 11009}, {2098, 15446}, {2099, 15180}, {3065, 12737}, {5424, 5919}, {5557, 11011}, {7284, 16200}, {10308, 10698}, {12331, 21398}, {12740, 13143}

X(24302) = trilinear pole of the line {650, 15492}

### X(24303) =  X(6)X(16459)∩X(15)X(1495)

Barycentrics    (SB+SC)*(3*(6*R^2+SA)*S^2-sqrt(3)*(5*S^2-3*(6*R^2-3*SA+2*SW)*SA)*S-9*SB*SC*SW) : :

X(24303) lies on these lines: {6, 16459}, {15, 1495}, {22, 2923}

X(24303) = isogonal conjugate of the cyclocevian conjugate of X(19778)

### X(24304) =  X(6)X(16460)∩X(16)X(1495)

Barycentrics    (SB+SC)*(3*(6*R^2+SA)*S^2-sqrt(3)*(5*S^2+3*(6*R^2-3*SA+2*SW)*SA)*S-9*SB*SC*SW) : :

X(24304) lies on these lines: {6, 16460}, {16, 1495}, {22, 2924}

X(24304) = isogonal conjugate of the cyclocevian conjugate of X(19779)

### X(24305) =  X(5)X(16764)∩X(30)X(11671)

Barycentrics    6 a^16-33 a^14 b^2+75 a^12 b^4-89 a^10 b^6+55 a^8 b^8-11 a^6 b^10-7 a^4 b^12+5 a^2 b^14-b^16-33 a^14 c^2+102 a^12 b^2 c^2-97 a^10 b^4 c^2+4 a^8 b^6 c^2+41 a^6 b^8 c^2-14 a^4 b^10 c^2-7 a^2 b^12 c^2+4 b^14 c^2+75 a^12 c^4-97 a^10 b^2 c^4+8 a^8 b^4 c^4-3 a^6 b^6 c^4+30 a^4 b^8 c^4-9 a^2 b^10 c^4-4 b^12 c^4-89 a^10 c^6+4 a^8 b^2 c^6-3 a^6 b^4 c^6-18 a^4 b^6 c^6+11 a^2 b^8 c^6-4 b^10 c^6+55 a^8 c^8+41 a^6 b^2 c^8+30 a^4 b^4 c^8+11 a^2 b^6 c^8+10 b^8 c^8-11 a^6 c^10-14 a^4 b^2 c^10-9 a^2 b^4 c^10-4 b^6 c^10-7 a^4 c^12-7 a^2 b^2 c^12-4 b^4 c^12+5 a^2 c^14+4 b^2 c^14-c^16 : :
X(24305) = 3 X[1157] - X[14072], X[137] - 3 X[24147]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28391.

X(24305) lies on these lines: {5,16764}, {30,11671}, {137,24147}, {195,10126}, {1157,14072}, {5965,6592}, {14143,22051}

### X(24306) =  33RD HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    2 a^16-11 a^14 b^2+25 a^12 b^4-27 a^10 b^6+5 a^8 b^8+23 a^6 b^10-29 a^4 b^12+15 a^2 b^14-3 b^16-11 a^14 c^2+34 a^12 b^2 c^2-35 a^10 b^4 c^2+12 a^8 b^6 c^2-21 a^6 b^8 c^2+62 a^4 b^10 c^2-61 a^2 b^12 c^2+20 b^14 c^2+25 a^12 c^4-35 a^10 b^2 c^4+8 a^8 b^4 c^4+7 a^6 b^6 c^4-38 a^4 b^8 c^4+93 a^2 b^10 c^4-60 b^12 c^4-27 a^10 c^6+12 a^8 b^2 c^6+7 a^6 b^4 c^6+10 a^4 b^6 c^6-47 a^2 b^8 c^6+108 b^10 c^6+5 a^8 c^8-21 a^6 b^2 c^8-38 a^4 b^4 c^8-47 a^2 b^6 c^8-130 b^8 c^8+23 a^6 c^10+62 a^4 b^2 c^10+93 a^2 b^4 c^10+108 b^6 c^10-29 a^4 c^12-61 a^2 b^2 c^12-60 b^4 c^12+15 a^2 c^14+20 b^2 c^14-3 c^16 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28391.

X(24306) lies on these lines: {2,3}, {137,24147}, {1263,19552}, {3448,11584}, {5663,24043}, {8254,20414}

X(24306) = midpoint of X(1263) and X(19552)
X(24306) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 5, 20030), (5, 10205, 140), (3850, 10289, 5)

Perspectors involving obverse triangles: X(24307) - X(24309)

Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC. The obverse triangle of P is here introduced as the triangle A'B'C', where A' = p : r : q, B' = r : q : p, C' = q : p : r. If P is a triangle center, then the obverse triangle of P is a central triangle. (Clark Kimberling, October 3, 2018)

### X(24307) =  PERSPECTOR OF THESE TRIANGLES: INTANGENTS AND OBVERSE OF X(1)

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

The obverse triangle of X(1) is the central triangle having A-vertex a : c : b = AX(75)∩GAX(9), where GAGBGC = medial. Following is a list of central triangles perspective to the obverse triangle of X(1), provided by Peter Moses, October 3, 2018:

ABC (TCCT 6.1) at X(75)
medial (TCCT 6.2) at X(9)
anticomplementary (TCCT 6.3) at X(1654)
tangential (TCCT 6.5) at X(8424)
excentral (TCCT 6.7) at X(9)
second circumperp (TCCT 6.22) at X(993)
Fuhrmann (TCCT 8.25) at X(10)
first Sharygin (see ETC X(8229)) at X(8424)
second Sharygin (see ETC X(8229)) at X(4363)
Aquilla (see ETC X(5586) / T(1,2) TCCT (6.40) at X(3679)
inner Garcia (see ETC X(5587)) at X(1)
outer Garcia (see ETC X(5587)) at X(3679)
second extouch (see ETC X(5927)) at X(9)
tangential of excentral at X(9)
Garcia reflection, see X(15995) at X(9)
second Zaniah, see X(18214) at X(9)
intangents, at X(24307)
extangents, at X(24308)
first circumperp, at X(24309)

X(24307) lies on these lines: {1,1463}, {9,3063}, {55,23857}, {3938,4336}

### X(24308) =  PERSPECTOR OF THESE TRIANGLES: EXTANGENTS AND OBVERSE OF X(1)

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

X(24308) lies on these lines: {2,71}, {40,15973}, {55,1740}, {65,984}, {75,21231}, {1762,15985}, {6254,9840}, {15830,20258}

### X(24309) =  PERSPECTOR OF THESE TRIANGLES: 1ST CIRCUMPERP AND OBVERSE OF X(1)

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

X(24309) lies on these lines: {1,7225}, {3,142}, {9,1633}, {35,24248}, {40,990}, {55,3663}, {87,572}, {100,3729}, {105,4859}, {106,1292}, {165,846}, {182,15310}, {515,996}, {517,3098}, {527,12329}, {573,9441}, {726,8715}, {993,4660}, {1060,11719}, {1155,12723}, {1293,9082}, {1376,17355}, {1473,4847}, {1621,17304}, {1699,19649}, {1738,7295}, {2182,17668}, {2550,3220}, {2726,2736}, {2796,13173}, {2809,7289}, {2823,3359}, {3008,7083}, {3056,5091}, {3295,4353}, {3434,7293}, {3474,5285}, {3817,16434}, {3923,19548}, {4186,8582}, {4219,10860}, {4292,8193}, {4297,9305}, {4298,12410}, {4421,17132}, {4640,18252}, {5853,22769}, {6011,6015}, {6210,13329}, {6684,12618}, {7411,10434}, {7580,10445}, {7688,19262}, {7987,12652}, {8666,17766}, {10164,19544}, {10267,12517}, {12530,16566}, {16686,17278}, {17792,24264}

Perspectors involving N-obverse triangles: X(24310) - X(24312)

Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC. The N-obverse triangle of P is here introduced as the triangle A'B'C', where A' = - p : r : q, B' = r : - q : p, C' = q : p : - r. If P is a triangle center, then the obverse triangle of P is a central triangle. (Clark Kimberling, October 3, 2018)

### X(24310) =  PERSPECTOR OF THESE TRIANGLES: EXTANGENTS AND N-OBVERSE OF X(1)

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

The N-obverse triangle of X(1) is the central triangle having A-vertex - a : c : b = AX(75)∩GAX(1), where GAGBGC = medial. Following is a list of central triangles perspective to the N-obverse triangle of X(1), provided by Peter Moses, October 3, 2018:

ABC (TCCT 6.1) at X(75)
medial (TCCT 6.2) at X(1)
tangential (TCCT 6.5) at X(8301)
second Sharygin (see ETC X(8229)) at X(8301)
outer Garcia (see ETC X(5587)) at X(1)
first Zaniah, see X(18214) at X(1)
extangents at X(24310)
first Sharygin at X(24311)
fifth extouch at X(24312)

X(24310) lies on these lines: {1,3}, {2,71}, {7,22097}, {9,1730}, {12,9548}, {19,27}, {37,19730}, {43,209}, {48,1817}, {200,22275}, {219,11347}, {226,573}, {238,19728}, {255,1396}, {307,1848}, {312,4019}, {329,2183}, {345,21281}, {380,18163}, {440,16608}, {469,20305}, {518,3198}, {916,2947}, {970,7066}, {1001,19727}, {1423,3782}, {1435,7013}, {1465,22134}, {1708,1766}, {1726,16548}, {1745,5752}, {1763,3684}, {1790,2302}, {1802,11349}, {1812,1958}, {1836,6210}, {1869,6734}, {1943,5088}, {2245,3772}, {2264,4641}, {2269,5712}, {2550,6817}, {2997,19645}, {3101,3164}, {3185,20718}, {3197,15509}, {3306,7573}, {3509,7075}, {3687,8896}, {3752,21866}, {3781,16056}, {3868,18673}, {3955,11428}, {4192,11435}, {4897,18197}, {5219,21363}, {5312,8274}, {5745,15830}, {6197,7554}, {7193,10536}, {7330,7534}, {7513,11471}, {9536,18661}, {10436,17185}, {11679,16574}, {16520,21001}, {16609,20606}, {17135,17784}, {17221,23958}, {18134,22370}, {18161,18607}, {18615,20875}

### X(24311) =  PERSPECTOR OF THESE TRIANGLES: FIRST SHARYGIN AND N-OBVERSE OF X(1)

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

X(24311) lies on these lines: {6,1281}, {45,1213}, {256,4363}, {381,2796}, {536,8245}, {740,3913}, {846,17259}, {4361,4471}, {4425,17327}

### X(24312) =  PERSPECTOR OF THESE TRIANGLES: FIFTH EXTOUCH AND N-OBVERSE OF X(1)

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

X(24312) lies on these lines: {8,1038}, {57,9363}, {388,4000}, {5226,5262}

### X(24313) =  ANTICOMPLEMENT OF X(24155)

Barycentrics    (a-b-c) (2 a-Sqrt[-a^2+2 a b-b^2+2 a c+2 b c-c^2]) : :
X(24313) = s X[8] - 2 Sqrt[r (r + 4 R)] X[9]

X(24313) lies on the cubics K1078 and K1082, and on these lines: {2,24153}, {8,9}, {4373,24154}

X(24313) = reflection of X(4373) in X(24154)
X(24313) = anticomplement X(24155)
X(24313) = isotomic conjugate of the anticomplement X(24157)
X(24313) = X(24153)-anticomplementary conjugate of X(3436)
X(24313) = X(24157)-cross conjugate of X(2)
X(24313) = barycentric quotient X(24157)/X(24155)

### X(24314) =  ANTICOMPLEMENT OF X(24154)

Barycentrics    (a-b-c) (2 a+Sqrt[-a^2+2 a b-b^2+2 a c+2 b c-c^2]) : :
X(24314) = s X[8] + 2 Sqrt[r (r + 4 R)] X[9]

X(24314) lies on the cubics K1078 and K1082, and on these lines: {2,24152}, {8,9}, {4373,24155}

X(24314) = anticomplement X(24154)
X(24314) = reflection of X(4373) in X(24155)
X(24314) = isotomic of the anticomplement X(24156)
X(24314) = X(24152)-anticomplementary conjugate of X(3436)
X(24314) = X(24156)-cross conjugate of X(2)
X(24314) = barycentric quotient X(24156)/X(24154)

### X(24315) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24315) lies on these lines: {2, 1762}, {3, 8680}, {4, 24683}, {5, 24682}, {6, 16609}, {9, 15669}, {10, 9028}, {48, 1441}, {55, 24424}, {69, 21076}, {75, 18162}, {85, 7119}, {86, 423}, {219, 21231}, {284, 17861}, {379, 2173}, {527, 6684}, {534, 946}, {742, 8177}, {1215, 1376}, {1281, 24351}, {1788, 4644}, {1826, 18650}, {2178, 4032}, {2278, 16732}, {2294, 5736}, {2304, 17866}, {2391, 4758}, {2640, 17889}, {3812, 4670}, {3821, 24337}, {3826, 4472}, {3923, 4381}, {4364, 6690}, {4414, 4419}, {4443, 24345}, {5025, 24726}, {5133, 24686}, {5169, 24687}, {7289, 10436}, {8299, 24329}, {17134, 22054}, {18690, 20883}, {21011, 21270}, {22769, 24325}, {25342, 25369}

### X(24316) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24316) lies on these lines: {1, 9028}, {2, 1762}, {3, 24683}, {4, 8680}, {7, 2260}, {9, 18589}, {19, 307}, {38, 497}, {40, 534}, {63, 1848}, {69, 1959}, {71, 4329}, {226, 8557}, {281, 20305}, {524, 12635}, {527, 946}, {545, 11235}, {631, 24684}, {960, 4643}, {1001, 4364}, {1370, 24686}, {1468, 1935}, {1839, 18655}, {1944, 17181}, {2173, 24580}, {2183, 5813}, {2294, 5738}, {2886, 4363}, {2893, 25252}, {4357, 7289}, {4381, 24348}, {4443, 24248}, {4454, 24400}, {5327, 24695}, {7791, 24726}, {12589, 17447}, {16063, 24687}, {17043, 20818}, {17221, 20074}, {17272, 18725}, {17321, 18162}, {21061, 22006}, {24318, 24334}, {24319, 24333}, {24336, 24694}

### X(24317) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24317) lies on these lines: {2, 1762}, {3, 24682}, {5, 8680}, {11, 24424}, {140, 24684}, {226, 19720}, {534, 6684}, {631, 24683}, {1125, 9028}, {2173, 24581}, {2294, 5740}, {3816, 4364}, {3821, 25346}, {4363, 24318}, {4419, 10589}, {4443, 5988}, {5224, 18161}, {5845, 25358}, {7485, 24686}, {7496, 24687}, {7824, 24726}, {8167, 24328}, {8609, 16603}, {17052, 25078}, {17322, 18162}, {21012, 21271}, {24295, 25372}, {24327, 24348}

### X(24318) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24318) lies on these lines: {1, 19887}, {2, 7}, {10, 514}, {39, 24211}, {100, 24712}, {116, 24036}, {214, 544}, {291, 5988}, {325, 4071}, {524, 4434}, {545, 25367}, {1083, 1125}, {1376, 24694}, {1565, 21232}, {1647, 24403}, {2246, 24582}, {2276, 24241}, {2329, 4564}, {2786, 13277}, {2886, 25355}, {3035, 5845}, {3257, 17256}, {3501, 17181}, {3570, 4416}, {4292, 16377}, {4363, 24317}, {4364, 25377}, {4419, 24398}, {4425, 25349}, {4438, 25343}, {4465, 11814}, {4472, 25359}, {4865, 9766}, {6647, 7181}, {7208, 24222}, {17046, 25066}, {21013, 21272}, {21029, 25244}, {21616, 24455}, {22279, 22325}, {24169, 25345}, {24316, 24334}, {24323, 25341}, {24348, 25382}

### X(24319) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24319) lies on these lines: {2, 24334}, {11, 25371}, {527, 6763}, {545, 25366}, {946, 24705}, {958, 24702}, {2486, 2886}, {3035, 25358}, {3944, 4419}, {4363, 24317}, {4472, 25367}, {4708, 5836}, {4999, 24700}, {5988, 24463}, {6856, 7961}, {21014, 21273}, {24316, 24333}, {24324, 25341}, {24328, 24694}, {24341, 24456}

### X(24320) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24320) lies on these lines: {1, 7083}, {2, 1473}, {3, 9}, {6, 7193}, {7, 4223}, {10, 24309}, {21, 17257}, {22, 3219}, {25, 63}, {28, 8822}, {45, 4265}, {55, 846}, {56, 87}, {57, 5020}, {72, 13730}, {75, 242}, {101, 991}, {105, 4310}, {154, 3955}, {169, 990}, {170, 5584}, {191, 8185}, {197, 4640}, {218, 4260}, {219, 511}, {220, 1350}, {222, 9306}, {228, 20835}, {240, 6059}, {255, 2212}, {329, 4224}, {405, 4357}, {474, 17353}, {518, 1486}, {611, 2175}, {613, 3271}, {894, 19310}, {920, 10037}, {956, 3883}, {958, 1503}, {960, 20876}, {999, 7290}, {1001, 4364}, {1158, 9910}, {1259, 3145}, {1351, 2323}, {1370, 21015}, {1460, 1707}, {1466, 23085}, {1597, 18540}, {1598, 5709}, {1633, 2550}, {1716, 5247}, {1736, 16551}, {1957, 7337}, {1995, 3218}, {2003, 3167}, {2178, 3286}, {2195, 2810}, {2267, 22390}, {2911, 4259}, {3173, 20122}, {3211, 5751}, {3242, 16686}, {3295, 7174}, {3303, 7301}, {3305, 7293}, {3306, 11284}, {3428, 6210}, {3452, 16434}, {3487, 17560}, {3662, 16048}, {3717, 5687}, {3718, 7283}, {3784, 17811}, {3786, 16876}, {3923, 24336}, {3927, 20831}, {3929, 5285}, {3937, 5651}, {4186, 6734}, {4192, 15509}, {4220, 5273}, {4228, 5905}, {4254, 16517}, {4363, 24332}, {5017, 16514}, {5096, 16885}, {5220, 12329}, {5250, 8192}, {5257, 16849}, {5329, 7262}, {5745, 19544}, {5750, 16852}, {6090, 22128}, {6211, 10310}, {6511, 23181}, {6642, 24467}, {6646, 17522}, {6677, 20266}, {7009, 18750}, {7082, 10832}, {7308, 16419}, {8817, 15344}, {9798, 12514}, {10436, 19309}, {10477, 23151}, {11108, 17306}, {11344, 22345}, {11350, 22060}, {14520, 22153}, {15494, 16678}, {15650, 20833}, {17260, 19314}, {18228, 19649}, {18589, 24701}, {24325, 24333}

### X(24321) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24321) lies on these lines: {2, 1762}, {22, 8680}, {427, 24686}, {1370, 24683}, {2173, 24584}, {5133, 24682}, {7485, 24684}, {17492, 21016}

### X(24322) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24322) lies on these lines: {2, 1762}, {23, 8680}, {858, 24687}, {1281, 2786}, {2173, 24585}, {5169, 24682}, {7496, 24684}, {16063, 24683}, {21017, 21274}

### X(24323) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24323) lies on these lines: {2, 25366}, {545, 4995}, {1215, 1376}, {3454, 4643}, {4364, 9318}, {4400, 9022}, {4670, 5439}, {21018, 21276}, {24318, 25341}, {24330, 24335}, {24345, 24463}

### X(24324) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24324) lies on these lines: {2, 2161}, {291, 24345}, {536, 5440}, {545, 6174}, {812, 14288}, {1215, 1376}, {3262, 7113}, {3739, 24618}, {3753, 4670}, {3925, 4472}, {4364, 5432}, {8299, 24346}, {21019, 21277}, {24319, 25341}, {24350, 25382}

### X(24325) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

Barycentrics    a^2 b + a^2 c + 2 a b c + b^2 c + b c^2 : :

X(24325) lies on these lines: {1, 75}, {2, 38}, {6, 4974}, {7, 4655}, {8, 4699}, {10, 141}, {11, 25385}, {31, 4697}, {37, 39}, {42, 4359}, {43, 19804}, {55, 3980}, {56, 4032}, {81, 3791}, {85, 4334}, {105, 24339}, {145, 4772}, {171, 3757}, {192, 3616}, {226, 3846}, {229, 409}, {238, 894}, {239, 4649}, {240, 17913}, {321, 3720}, {354, 3741}, {511, 17049}, {519, 3696}, {536, 551}, {594, 4966}, {689, 741}, {692, 24253}, {730, 21443}, {742, 1386}, {750, 4434}, {752, 3883}, {758, 18805}, {872, 3216}, {899, 24589}, {940, 4362}, {946, 21629}, {960, 20258}, {964, 4812}, {976, 16454}, {985, 16998}, {991, 24293}, {993, 8680}, {1001, 3923}, {1086, 3821}, {1089, 18137}, {1220, 7194}, {1278, 3622}, {1441, 1458}, {1469, 16609}, {1574, 21897}, {1575, 3774}, {1582, 5009}, {1621, 4418}, {1698, 4751}, {1738, 4085}, {1757, 17277}, {1909, 1921}, {1962, 17147}, {1965, 8033}, {1999, 4038}, {2235, 23660}, {2550, 24693}, {2805, 21630}, {2886, 20256}, {2887, 5249}, {3210, 17592}, {3244, 4709}, {3264, 4710}, {3286, 16684}, {3416, 4675}, {3485, 7201}, {3624, 4687}, {3636, 4686}, {3664, 5847}, {3666, 24165}, {3679, 17297}, {3685, 16484}, {3706, 4883}, {3716, 4809}, {3740, 4090}, {3742, 3840}, {3751, 4384}, {3752, 6685}, {3758, 16468}, {3761, 21615}, {3773, 3912}, {3782, 4425}, {3790, 17244}, {3797, 16826}, {3831, 5439}, {3878, 20718}, {3879, 17772}, {3891, 5311}, {3932, 17245}, {3953, 4022}, {3979, 3996}, {4011, 4423}, {4078, 4439}, {4357, 24231}, {4364, 25354}, {4365, 4980}, {4368, 24330}, {4385, 20923}, {4407, 24603}, {4416, 17771}, {4429, 25351}, {4431, 4527}, {4440, 9791}, {4458, 24353}, {4472, 9055}, {4523, 17050}, {4535, 17233}, {4645, 17153}, {4660, 5880}, {4663, 17348}, {4664, 25055}, {4676, 15485}, {4681, 15808}, {4683, 17483}, {4684, 4967}, {4698, 19862}, {4703, 5905}, {4716, 17117}, {4722, 19742}, {4755, 19883}, {4968, 20891}, {4991, 16666}, {5211, 17722}, {5220, 17259}, {5223, 16832}, {5247, 16817}, {5248, 24850}, {5695, 17118}, {5698, 7222}, {5749, 16020}, {5836, 24182}, {5852, 17332}, {5886, 20430}, {5949, 20546}, {6211, 21554}, {6532, 20108}, {6541, 17243}, {6646, 24697}, {6686, 16602}, {6703, 17061}, {7081, 17122}, {7321, 24723}, {7336, 19890}, {8025, 17150}, {8299, 24326}, {9284, 16592}, {9505, 24505}, {9507, 25368}, {9941, 24190}, {10180, 17155}, {10459, 20892}, {10479, 18398}, {10582, 20173}, {10980, 18229}, {11997, 12053}, {14839, 24254}, {15254, 17351}, {15523, 18139}, {16477, 17120}, {17017, 19684}, {17031, 24512}, {17135, 21020}, {17365, 17770}, {17599, 19701}, {18201, 24627}, {19623, 24345}, {19856, 25120}, {21197, 24675}, {21926, 24390}, {22769, 24315}, {24320, 24333}, {24377, 24440}, {25345, 25373}

### X(24326) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

Barycentrics    a^2 b^2 + 2 a^2 b c + b^3 c + a^2 c^2 + b c^3 : :

X(24326) lies on these lines: {1, 16720}, {2, 37}, {38, 9055}, {42, 742}, {55, 4363}, {76, 21021}, {141, 3930}, {244, 25350}, {518, 24690}, {524, 4030}, {538, 4692}, {712, 4424}, {756, 3123}, {894, 1914}, {902, 4797}, {984, 4493}, {1018, 24254}, {1045, 3961}, {1215, 24259}, {1403, 7211}, {1500, 1930}, {1573, 4986}, {1621, 24358}, {2242, 7267}, {3571, 24448}, {3663, 21101}, {3681, 4643}, {3703, 4665}, {3727, 17760}, {3744, 4670}, {3758, 5332}, {3873, 24691}, {3934, 7264}, {3938, 9022}, {3970, 21240}, {4396, 7081}, {4644, 20101}, {4659, 17594}, {4660, 4799}, {4894, 7759}, {4972, 25345}, {6664, 21035}, {7187, 25303}, {7235, 12837}, {7272, 7761}, {8299, 24325}, {17318, 17599}, {18059, 21217}, {20255, 21808}, {20331, 24631}, {20691, 20911}, {24309, 24333}, {24431, 24454}

### X(24327) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

Barycentrics    a^3 b^2 + a^2 b^2 c + a^3 c^2 + a^2 b c^2 + b^3 c^2 + b^2 c^3 : :

X(24327) lies on these lines: {1, 4494}, {2, 3122}, {6, 4039}, {10, 674}, {37, 700}, {39, 714}, {76, 24696}, {313, 2309}, {536, 12263}, {696, 17760}, {730, 4377}, {1213, 25120}, {1215, 17369}, {1964, 3963}, {2234, 20913}, {2278, 4112}, {3264, 21352}, {3821, 25374}, {3923, 4381}, {3934, 6007}, {4363, 24405}, {4364, 17793}, {4422, 6690}, {4436, 24259}, {4472, 9055}, {5988, 25346}, {8299, 24425}, {9025, 25102}, {16732, 24255}, {17337, 25106}, {17398, 25124}, {18048, 18209}, {21022, 21278}, {21238, 21746}, {24317, 24348}, {24340, 24345}

### X(24328) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24328) lies on these lines: {3, 527}, {6, 1423}, {7, 198}, {37, 7289}, {45, 16560}, {48, 6180}, {55, 1633}, {56, 4644}, {100, 4454}, {222, 1020}, {226, 15509}, {513, 14205}, {524, 9840}, {536, 3913}, {545, 4421}, {954, 3220}, {958, 4643}, {999, 4667}, {1001, 4364}, {1215, 1376}, {1282, 24341}, {1473, 21319}, {1696, 4648}, {1732, 4641}, {2097, 7146}, {2178, 17365}, {2182, 8545}, {2183, 5228}, {2262, 7190}, {2270, 4328}, {2347, 7225}, {2391, 6767}, {3247, 18725}, {3553, 24471}, {3663, 4254}, {3929, 21483}, {4413, 4470}, {4428, 20834}, {4471, 24405}, {4654, 11347}, {4659, 5687}, {4715, 11194}, {4747, 5253}, {5085, 23693}, {5745, 23089}, {6600, 24309}, {7175, 20991}, {8167, 24317}, {8715, 17132}, {11343, 17274}, {11495, 15624}, {16367, 17333}, {16672, 18735}, {16777, 18161}, {23085, 24700}, {24319, 24694}, {24352, 24424}

### X(24329) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24329) lies on these lines: {2, 25348}, {9, 20544}, {744, 4372}, {2175, 20236}, {2330, 20927}, {3923, 15507}, {4363, 24346}, {4376, 4381}, {4858, 24264}, {5698, 7379}, {8299, 24315}, {17046, 24689}, {19133, 20171}, {21023, 21280}, {24428, 24456}

### X(24330) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

Barycentrics    a^3 b + a^3 c + a b^2 c + a b c^2 + 2 b^2 c^2 : :

X(24330) lies on these lines: {1, 538}, {2, 45}, {6, 4441}, {10, 4721}, {38, 25368}, {42, 536}, {43, 4659}, {44, 24592}, {75, 2238}, {76, 2295}, {141, 20347}, {171, 4396}, {213, 20888}, {291, 24448}, {321, 742}, {350, 894}, {524, 17135}, {527, 3741}, {672, 17351}, {756, 25384}, {870, 17475}, {1111, 24254}, {1215, 24259}, {1836, 4643}, {1909, 10027}, {2276, 3729}, {3120, 25345}, {3720, 4670}, {3734, 16788}, {3758, 4479}, {3760, 17750}, {3780, 17143}, {3923, 4376}, {3934, 16549}, {4368, 24325}, {4400, 5255}, {4418, 9318}, {4644, 10453}, {4651, 4665}, {4664, 17032}, {4672, 17031}, {4686, 21904}, {4971, 20011}, {5263, 17794}, {5264, 7751}, {6685, 17132}, {9055, 17165}, {15985, 17220}, {17002, 21793}, {17018, 17318}, {17137, 21024}, {17140, 24403}, {17355, 20335}, {17790, 18152}, {20292, 24699}, {24323, 24335}, {25353, 25385}

### X(24331) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24331) lies on these lines: {1, 2}, {36, 23407}, {45, 537}, {75, 4693}, {106, 789}, {142, 4660}, {238, 3758}, {244, 24464}, {274, 4653}, {392, 20358}, {528, 24693}, {740, 17119}, {752, 4675}, {894, 15485}, {968, 24165}, {984, 24841}, {993, 24333}, {1001, 3923}, {1150, 17450}, {1215, 4423}, {1621, 3980}, {2177, 24589}, {3242, 3842}, {3246, 4670}, {3750, 19804}, {3976, 11110}, {3986, 16517}, {4011, 5284}, {4085, 17278}, {4090, 7308}, {4378, 4448}, {4655, 7238}, {4688, 4702}, {5257, 16973}, {6173, 24692}, {6706, 25130}, {7176, 13462}, {10180, 17599}, {11365, 20834}, {14621, 16801}, {15569, 25384}, {21027, 21283}, {24349, 24821}

### X(24332) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24332) lies on these lines: {75, 7193}, {92, 3955}, {182, 4858}, {184, 14213}, {242, 894}, {984, 2607}, {1001, 24346}, {1733, 2175}, {1944, 3781}, {1985, 21368}, {3060, 24149}, {3410, 24146}, {3923, 15507}, {4363, 24320}, {4459, 7295}, {6358, 9306}, {11422, 24148}, {11442, 21028}

### X(24333) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24333) lies on these lines: {2, 7}, {41, 20880}, {69, 4071}, {75, 3684}, {85, 2329}, {87, 2191}, {169, 17050}, {524, 4865}, {742, 4362}, {846, 4419}, {870, 1438}, {993, 24331}, {1001, 24352}, {1111, 16788}, {1215, 1376}, {1281, 9451}, {1282, 2550}, {1478, 9903}, {1486, 8424}, {2082, 20257}, {2246, 24596}, {2886, 5845}, {2887, 4643}, {3035, 25355}, {3158, 4659}, {3496, 17753}, {3729, 21101}, {3742, 4670}, {3767, 24211}, {3923, 4376}, {4011, 4713}, {4364, 25363}, {4434, 8667}, {4444, 8774}, {4454, 5281}, {4644, 24477}, {4754, 9840}, {6180, 22163}, {6647, 7223}, {7032, 24403}, {7201, 11683}, {7264, 16783}, {11680, 24712}, {17739, 21281}, {21029, 21285}, {24309, 24326}, {24316, 24319}, {24320, 24325}

### X(24334) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24334) lies on these lines: {2, 24319}, {19, 20258}, {40, 24705}, {46, 527}, {55, 25371}, {75, 1429}, {322, 7175}, {524, 8256}, {604, 20895}, {958, 24700}, {1215, 1376}, {1281, 24451}, {1329, 24702}, {1766, 21246}, {1944, 3501}, {2268, 24993}, {2550, 4470}, {2886, 4472}, {3035, 4364}, {3507, 16571}, {3923, 15507}, {4381, 25382}, {4419, 17596}, {4454, 9318}, {4659, 5438}, {4670, 5836}, {4858, 24266}, {5783, 21233}, {7961, 17567}, {17452, 24540}, {21030, 21286}, {24316, 24318}, {24343, 24358}

### X(24335) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

Barycentrics    a^5 + a^4 b + a^4 c + a^3 b c + a b^3 c + b^4 c + 2 a b^2 c^2 + b^3 c^2 + a b c^3 + b^2 c^3 + b c^4 : :

X(24335) lies on these lines: {2, 25370}, {55, 4363}, {75, 5301}, {141, 1761}, {191, 4643}, {524, 1046}, {536, 5266}, {846, 4364}, {1333, 18697}, {1781, 17279}, {2108, 24340}, {3454, 24704}, {3694, 17351}, {3923, 4381}, {3931, 4670}, {3980, 25363}, {4026, 4472}, {4419, 4427}, {5255, 9022}, {5695, 19761}, {8680, 24850}, {16732, 24271}, {17587, 20896}, {17762, 19623}, {20654, 21287}, {24323, 24330}, {24346, 24425}

### X(24336) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24336) lies on these lines: {2, 25371}, {3, 24700}, {4, 24702}, {9, 6996}, {11, 25369}, {69, 4032}, {72, 527}, {75, 1423}, {536, 5836}, {573, 21233}, {742, 17792}, {894, 20769}, {960, 12545}, {984, 2550}, {1215, 1376}, {1706, 4385}, {2269, 24547}, {2486, 2886}, {3035, 4472}, {3718, 3729}, {3923, 24320}, {4454, 6555}, {4643, 5794}, {4665, 8256}, {5082, 7961}, {8299, 25375}, {20245, 21033}, {24316, 24694}

### X(24337) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

Barycentrics    a^9 - a^8 b + 2 a^4 b^5 - a b^8 - b^9 - a^8 c + b^8 c + 2 a b^4 c^4 + 2 a^4 c^5 - a c^8 + b c^8 - c^9 : :

X(24337) lies on these lines: {2, 25372}, {206, 24706}, {3821, 24315}, {3923, 25343}, {21034, 21288}, {24424, 24444}

### X(24338) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(2), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24338) lies on these lines: {1, 513}, {2, 19945}, {6, 4941}, {37, 24289}, {43, 23343}, {45, 1575}, {75, 24517}, {190, 3123}, {192, 23354}, {244, 903}, {256, 17246}, {291, 545}, {350, 4389}, {536, 984}, {751, 18822}, {894, 24503}, {900, 24427}, {986, 24433}, {1015, 24722}, {1054, 16561}, {1086, 4947}, {1655, 9295}, {1769, 24409}, {2254, 24408}, {3122, 4440}, {3242, 9025}, {3248, 4499}, {3571, 9318}, {3821, 17790}, {4363, 24456}, {4364, 24450}, {4419, 4443}, {4562, 4664}, {4713, 24458}, {6646, 24437}, {7321, 22172}, {9296, 18140}, {16495, 24482}, {17273, 22167}, {17290, 20530}, {17334, 24575}, {24346, 24428}, {24396, 24398}, {24403, 24413}, {24404, 24414}

### X(24339) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24339) lies on these lines: {2, 25373}, {105, 24325}, {141, 17744}, {251, 20898}, {984, 4376}, {1369, 21037}, {4363, 7295}, {4364, 24340}

### X(24340) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

Barycentrics    a^5 + a^3 b^2 + a^2 b^3 + a^2 b^2 c + b^4 c + a^3 c^2 + a^2 b c^2 + a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 + b c^4 : :

X(24340) lies on these lines: {2, 4381}, {38, 190}, {58, 4672}, {83, 744}, {141, 17799}, {1078, 4836}, {1281, 25347}, {2108, 24335}, {2161, 2886}, {2244, 24614}, {3096, 4837}, {3923, 4283}, {4364, 24339}, {6292, 24707}, {21038, 21289}, {24295, 25364}, {24327, 24345}

### X(24341) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24341) lies on these lines: {1, 5696}, {2, 2310}, {9, 1721}, {37, 4335}, {38, 4346}, {45, 846}, {75, 4073}, {144, 21039}, {170, 1212}, {171, 1711}, {198, 8245}, {377, 4331}, {774, 4208}, {982, 1086}, {984, 2550}, {1282, 24328}, {1458, 10861}, {1734, 17057}, {1742, 17668}, {2293, 24554}, {2345, 12725}, {3000, 24635}, {3779, 7201}, {3925, 24430}, {4051, 9025}, {4363, 24411}, {4454, 4712}, {4518, 24451}, {5296, 10868}, {17868, 25304}, {24319, 24456}

### X(24342) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

Barycentrics    a^3 + a^2 b + a b^2 + a^2 c + 3 a b c + 2 b^2 c + a c^2 + 2 b c^2 : :

X(24342) lies on these lines: {1, 75}, {2, 846}, {3, 10892}, {8, 20090}, {9, 46}, {10, 894}, {40, 15973}, {81, 21020}, {165, 7413}, {190, 3842}, {238, 3739}, {239, 4991}, {320, 3775}, {321, 1961}, {333, 4697}, {516, 2938}, {524, 3416}, {726, 16830}, {756, 4756}, {758, 3786}, {870, 10009}, {873, 18059}, {896, 5235}, {966, 24695}, {984, 4363}, {986, 2049}, {996, 7312}, {1001, 4436}, {1051, 19717}, {1100, 4716}, {1125, 3685}, {1215, 3699}, {1386, 4688}, {1441, 5018}, {1699, 2941}, {1738, 5750}, {1762, 3925}, {1770, 19857}, {1909, 4710}, {1962, 5333}, {2292, 14005}, {2550, 4470}, {2640, 4429}, {2663, 3293}, {2795, 24275}, {2887, 19808}, {2895, 8013}, {2959, 4660}, {3122, 24923}, {3336, 16574}, {3624, 4657}, {3634, 17260}, {3696, 4649}, {3706, 4038}, {3836, 17289}, {3932, 7227}, {3993, 16826}, {4026, 4472}, {4205, 24851}, {4357, 19856}, {4365, 17019}, {4384, 14621}, {4527, 17315}, {4650, 5737}, {4655, 5224}, {4672, 17277}, {4676, 4751}, {4693, 15569}, {4699, 16825}, {4967, 5847}, {5564, 17772}, {5695, 15668}, {5902, 10477}, {5999, 9746}, {6998, 8245}, {7262, 19732}, {8025, 17163}, {8424, 19329}, {9780, 17350}, {11110, 24850}, {11263, 24931}, {11533, 17164}, {12047, 21246}, {16477, 17348}, {17258, 17767}, {17301, 25055}, {17592, 19701}, {17778, 21085}, {17790, 25382}, {18140, 24731}, {19868, 24231}, {19879, 20258}, {20880, 23689}, {23690, 24993}, {24174, 24311}, {24291, 24293}

### X(24343) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24343) lies on these lines: {2, 25376}, {9, 25107}, {894, 4110}, {2162, 6382}, {3242, 4363}, {3739, 24758}, {3923, 4713}, {10009, 24502}, {24334, 24358}

### X(24344) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24344) lies on these lines: {2, 4432}, {89, 519}, {100, 4363}, {239, 17126}, {750, 4693}, {2163, 4793}, {3240, 3758}, {3679, 24616}, {3923, 9458}, {3952, 24451}, {4392, 24464}, {4427, 9330}, {5561, 21251}, {21042, 21291}

### X(24345) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24345) lies on these lines: {1, 523}, {2, 24348}, {6, 24365}, {99, 24714}, {115, 24711}, {230, 8557}, {291, 24324}, {524, 3416}, {545, 13174}, {645, 21254}, {894, 9421}, {1046, 12607}, {1086, 2640}, {1109, 14616}, {1281, 24350}, {2161, 17719}, {2948, 24222}, {2959, 17768}, {3035, 16575}, {3110, 8672}, {3571, 9318}, {4381, 24351}, {4427, 17487}, {4436, 24820}, {4443, 24315}, {4590, 17103}, {5539, 24722}, {10026, 17369}, {13610, 17365}, {14985, 17724}, {19623, 24325}, {21043, 21221}, {24323, 24463}, {24327, 24340}, {24405, 24416}

### X(24346) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24346) lies on these lines: {1, 523}, {2, 25379}, {6, 4124}, {11, 2161}, {75, 3573}, {92, 23353}, {110, 14616}, {116, 24713}, {242, 8679}, {281, 12586}, {674, 1944}, {692, 4858}, {894, 24482}, {900, 24411}, {993, 8680}, {1001, 24332}, {1146, 5848}, {3826, 5805}, {3877, 5263}, {3923, 25382}, {4363, 24329}, {4448, 9318}, {4459, 16686}, {4579, 18151}, {4679, 17369}, {8299, 24324}, {21045, 21293}, {24335, 24425}, {24338, 24428}, {24347, 24442}, {24401, 24455}, {24406, 24416}

### X(24347) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24347) lies on these lines: {1, 23879}, {116, 16562}, {163, 17886}, {525, 3109}, {544, 2948}, {894, 24504}, {2247, 24619}, {2607, 9318}, {5845, 14985}, {21046, 21294}, {21381, 24712}, {24248, 25372}, {24346, 24442}

### X(24348) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24348) lies on these lines: {2, 24345}, {10, 523}, {99, 24711}, {524, 551}, {545, 11599}, {620, 24714}, {903, 3120}, {2615, 16598}, {3923, 25362}, {4381, 24316}, {4443, 25346}, {4590, 6626}, {4934, 21254}, {17793, 25367}, {21047, 21295}, {24317, 24327}, {24318, 25382}, {25347, 25364}

### X(24349) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24349) lies on these lines: {1, 87}, {2, 38}, {4, 4812}, {7, 8}, {9, 16823}, {10, 3662}, {37, 2275}, {42, 3210}, {43, 17490}, {57, 7081}, {63, 3757}, {78, 4327}, {79, 4894}, {81, 3891}, {142, 3717}, {145, 740}, {182, 4579}, {190, 1001}, {210, 19804}, {226, 3705}, {238, 17350}, {239, 3751}, {274, 3786}, {312, 354}, {318, 1876}, {321, 3873}, {341, 3812}, {345, 3475}, {346, 11038}, {386, 596}, {390, 4454}, {404, 18048}, {519, 4740}, {527, 3883}, {536, 3241}, {668, 10009}, {698, 4754}, {742, 4644}, {869, 24621}, {870, 18906}, {871, 20023}, {899, 24620}, {938, 20171}, {942, 4385}, {962, 9801}, {985, 7766}, {1002, 4441}, {1086, 4429}, {1089, 18398}, {1125, 17368}, {1219, 1432}, {1266, 3755}, {1279, 4676}, {1281, 9451}, {1362, 6063}, {1376, 8850}, {1386, 3758}, {1698, 17291}, {1757, 16825}, {1836, 4514}, {1892, 5081}, {2263, 3872}, {2292, 25124}, {2321, 4684}, {2805, 9802}, {2975, 13733}, {3061, 20706}, {3175, 4883}, {3240, 17495}, {3242, 4363}, {3243, 3886}, {3306, 5205}, {3600, 4032}, {3617, 4772}, {3621, 4821}, {3622, 4704}, {3623, 4788}, {3632, 4709}, {3644, 20057}, {3681, 4359}, {3699, 4413}, {3701, 20923}, {3702, 3889}, {3703, 18134}, {3739, 5772}, {3742, 3967}, {3753, 4737}, {3759, 4663}, {3761, 21443}, {3773, 17230}, {3774, 17756}, {3790, 3912}, {3797, 17316}, {3813, 21927}, {3844, 17227}, {3869, 20348}, {3874, 10449}, {3932, 17234}, {3938, 4418}, {3961, 3980}, {3974, 18141}, {3976, 21330}, {3994, 17450}, {4026, 4389}, {4030, 11246}, {4078, 17244}, {4090, 16569}, {4253, 22011}, {4260, 20913}, {4301, 9950}, {4318, 4861}, {4353, 17023}, {4384, 5223}, {4388, 5905}, {4393, 4649}, {4416, 5850}, {4419, 9791}, {4430, 17135}, {4440, 24248}, {4470, 25384}, {4651, 4661}, {4664, 15569}, {4671, 17146}, {4678, 4732}, {4686, 20050}, {4687, 5550}, {4692, 5902}, {4696, 20892}, {4697, 17716}, {4702, 15570}, {4706, 21870}, {4726, 20053}, {4751, 19877}, {4767, 24594}, {4884, 17056}, {4891, 22034}, {4901, 6173}, {4966, 17233}, {5014, 20292}, {5045, 22016}, {5220, 17277}, {5247, 19851}, {5603, 20430}, {5846, 17365}, {5847, 17364}, {5852, 17347}, {5904, 9534}, {6194, 8924}, {6327, 17483}, {6685, 17591}, {7172, 21454}, {7174, 10436}, {9369, 19860}, {9785, 11997}, {10580, 20173}, {11688, 23853}, {14839, 24282}, {15254, 17336}, {15481, 17335}, {16475, 17120}, {16484, 25269}, {16496, 17116}, {16606, 20284}, {16710, 18792}, {17018, 17147}, {17126, 20045}, {17169, 18157}, {17276, 24723}, {17754, 19587}, {18144, 22279}, {18194, 23579}, {18412, 20236}, {19586, 20917}, {20909, 25301}, {21020, 25294}, {21197, 24749}, {24331, 24821}

### X(24350) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24350) lies on these lines: {625, 24716}, {659, 24353}, {922, 20912}, {1281, 24345}, {2795, 24714}, {3923, 4381}, {16732, 24292}, {21048, 21298}, {24324, 25382}

### X(24351) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24351) lies on these lines: {2, 3122}, {6, 7155}, {8, 674}, {55, 190}, {75, 20358}, {76, 6007}, {192, 700}, {194, 714}, {313, 21299}, {335, 24445}, {523, 7985}, {874, 24282}, {1043, 5327}, {1278, 17142}, {1281, 24315}, {1654, 25291}, {1918, 17350}, {2234, 24621}, {3242, 4363}, {3596, 21746}, {3779, 17787}, {4381, 24345}, {4419, 17794}, {9025, 24524}, {17349, 25277}, {17379, 25295}, {17755, 25375}, {17786, 17792}, {17793, 24456}, {19582, 19765}

### X(24352) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24352) lies on these lines: {2, 45}, {6, 10025}, {7, 17747}, {44, 24600}, {55, 9318}, {200, 536}, {218, 7264}, {220, 3673}, {497, 5845}, {527, 11019}, {742, 20173}, {982, 24398}, {1001, 24333}, {1376, 24283}, {2391, 4342}, {3242, 14942}, {3693, 17262}, {3870, 17318}, {3967, 4659}, {4421, 24685}, {4643, 4847}, {4644, 10580}, {4670, 10582}, {4971, 20015}, {6604, 21049}, {9436, 17276}, {11235, 24694}, {11238, 24712}, {14548, 17365}, {14828, 16777}, {17132, 20103}, {17251, 25006}, {17597, 24403}, {17790, 18153}, {24328, 24424}

### X(24353) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24353) lies on these lines: {10, 2774}, {659, 24350}, {810, 17899}, {2786, 3716}, {3700, 8062}, {4458, 24325}, {21050, 21300}

### X(24354) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24354) lies on these lines: {2, 25356}, {75, 8632}, {190, 14408}, {514, 21112}, {659, 24350}, {786, 3737}, {812, 14288}, {816, 4086}, {894, 3768}, {900, 4375}, {1919, 20906}, {3758, 23650}, {4363, 4491}, {4472, 24721}, {4508, 21606}, {4509, 8060}, {9318, 24416}, {17303, 21261}, {20949, 20981}, {21055, 21304}, {21260, 24698}

### X(24355) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24355) lies on these lines: {1, 17046}, {2, 21687}, {3, 4361}, {3187, 7146}, {4372, 24367}, {24432, 24435}

### X(24356) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24356) lies on these lines: {659, 24350}, {1281, 2786}, {8631, 23807}, {21056, 21305}

### X(24357) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

Barycentrics    a^2 b^2 + 3 a^2 b c + a b^2 c + b^3 c + a^2 c^2 + a b c^2 + b c^3 : :

X(24357) lies on these lines: {1, 742}, {2, 37}, {45, 17755}, {85, 24656}, {335, 4389}, {354, 24691}, {518, 4643}, {537, 24441}, {740, 17318}, {982, 25349}, {984, 4026}, {1001, 3923}, {1215, 4713}, {1279, 4670}, {1486, 8424}, {1621, 4376}, {2667, 3938}, {3616, 16720}, {3873, 24690}, {3932, 4665}, {4078, 25352}, {4377, 21615}, {4419, 9791}, {4429, 25357}, {4660, 24699}, {4709, 17133}, {4797, 8616}, {12618, 20430}, {17063, 25350}, {18135, 21021}, {20331, 24629}

### X(24358) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24358) lies on these lines: {1, 9055}, {2, 4376}, {6, 17755}, {9, 141}, {21, 16720}, {37, 86}, {63, 24691}, {75, 4366}, {100, 4760}, {171, 4797}, {238, 742}, {304, 4426}, {344, 4473}, {385, 20947}, {513, 9470}, {524, 1757}, {536, 1279}, {545, 24416}, {673, 3739}, {812, 14288}, {846, 25349}, {1001, 3923}, {1086, 4657}, {1447, 20530}, {1486, 24822}, {1621, 24326}, {1914, 3263}, {1966, 17790}, {2082, 24735}, {2243, 24602}, {2345, 20533}, {3219, 24690}, {3496, 20255}, {3512, 15985}, {3701, 4400}, {3758, 20132}, {3836, 24699}, {4011, 4713}, {4026, 4472}, {4078, 4667}, {4154, 21254}, {4358, 4396}, {4364, 24339}, {4370, 4795}, {4437, 4851}, {4440, 7222}, {4465, 9318}, {4553, 7077}, {4567, 4590}, {4708, 17260}, {4715, 17297}, {4872, 20541}, {5263, 25384}, {5291, 14210}, {8424, 24700}, {11814, 25342}, {17596, 25350}, {17739, 21025}, {18589, 20258}, {20483, 20553}, {24003, 24685}, {24334, 24343}

### X(24359) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

Barycentrics    a^5 - 3 a^4 b + 2 a^3 b^2 - 3 a^4 c + 5 a^3 b c - 2 a^2 b^2 c - 5 a b^3 c + b^4 c + 2 a^3 c^2 - 2 a^2 b c^2 + 10 a b^2 c^2 - b^3 c^2 - 5 a b c^3 - b^2 c^3 + b c^4 : :

See X(24310).

X(24359) lies on these lines: {1, 900}, {1086, 17718}, {4448, 9318}

### X(24360) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24360) lies on these lines: {2, 23898}, {24362, 24368}

### X(24361) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24361) lies on these lines: {2, 23898}

### X(24362) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24362) lies on these lines: {2, 23901}, {4039, 24435}, {4362, 4412}, {24360, 24368}, {24365, 24369}, {24372, 24378}, {24376, 24377}

### X(24363) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24363) lies on these lines: {2, 23902}, {7, 8}, {4039, 17314}, {4371, 24435}, {6857, 24640}, {20539, 21271}, {21085, 21675}, {24364, 24372}

### X(24364) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24364) lies on these lines: {1, 142}, {2, 23897}, {194, 966}, {390, 16527}, {3212, 9278}, {3618, 16913}, {4419, 21879}, {7735, 9509}, {7738, 17691}, {9607, 17277}, {24363, 24372}

### X(24365) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24365) lies on these lines: {2, 23902}, {6, 24345}, {8, 3826}, {474, 24640}, {1001, 7235}, {1376, 24435}, {4039, 4361}, {4733, 11281}, {8424, 16609}, {24362, 24369}, {24372, 24376}, {24377, 24378}

### X(24366) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24366) lies on these lines: {2, 23897}, {6, 17680}, {8, 141}, {1213, 1655}, {3589, 17686}, {4085, 24656}, {4657, 24439}, {6707, 14005}, {7864, 17352}, {16926, 17398}, {17045, 24435}, {24372, 24373}

### X(24367) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24367) lies on these lines: {2, 23909}, {55, 4361}, {4372, 24355}

### X(24368) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24368) lies on these lines: {2, 23910}, {24360, 24362}

### X(24369) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24369) lies on these lines: {2, 23913}, {1376, 4361}, {2640, 17277}, {24174, 24653}, {24362, 24365}, {24370, 24384}, {24374, 24376}

### X(24370) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24370) lies on these lines: {2, 23915}, {24360, 24362}, {24369, 24384}

### X(24371) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24371) lies on these lines: {2, 23902}, {3, 24640}, {8, 3475}, {55, 7235}, {56, 24432}, {999, 24641}, {17718, 21085}

### X(24372) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24372) lies on these lines: {2, 23913}, {6, 2640}, {10, 75}, {982, 4361}, {1045, 3125}, {1376, 1961}, {1740, 3959}, {2277, 17891}, {2643, 24530}, {17063, 17380}, {17872, 24478}, {24174, 24923}, {24362, 24378}, {24363, 24364}, {24365, 24376}, {24366, 24373}, {24443, 24575}

### X(24373) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24373) lies on these lines: {2, 23915}, {8, 4446}, {24366, 24372}

### X(24374) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24374) lies on these lines: {1, 75}, {2, 23901}, {261, 11711}, {7170, 24575}, {24369, 24376}, {24383, 24384}

### X(24375) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

Barycentrics    3 a^4 - 5 a^3 b - 4 a^2 b^2 - a b^3 - b^4 - 5 a^3 c + 3 a^2 b c + 7 a b^2 c - b^3 c - 4 a^2 c^2 + 7 a b c^2 + 8 b^2 c^2 - a c^3 - b c^3 - c^4 : :

X(24375) lies on these lines: {2, 23897}, {3622, 4000}

### X(24376) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24376) lies on these lines: {2, 23936}, {1054, 2606}, {24362, 24377}, {24365, 24372}, {24369, 24374}

### X(24377) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24377) lies on these lines: {2, 23913}, {1376, 4360}, {2640, 17349}, {2669, 3212}, {24325, 24440}, {24362, 24376}, {24365, 24378}

### X(24378) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24378) lies on these lines: {2, 23897}, {75, 24439}, {86, 4000}, {99, 3008}, {239, 9278}, {333, 16722}, {645, 1266}, {662, 1429}, {673, 2669}, {1434, 16711}, {4360, 24435}, {4384, 6626}, {4395, 19623}, {4417, 17680}, {4560, 7178}, {5222, 17103}, {5550, 14005}, {7058, 24177}, {14829, 24610}, {24362, 24372}, {24365, 24377}

### X(24379) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24379) lies on these lines: {2, 23949}, {17061, 17069}, {24380, 24381}

### X(24380) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24380) lies on these lines: {2, 23950}, {24379, 24381}

### X(24381) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24381) lies on these lines: {2, 23951}, {523, 661}, {693, 18059}, {2517, 24674}, {2533, 3907}, {17494, 24104}, {21085, 21196}, {23948, 23954}, {24379, 24380}

### X(24382) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24382) lies on these lines: {2, 23949}, {1734, 21212}, {2254, 3004}, {4041, 4369}

### X(24383) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24383) lies on these lines: {2, 23909}, {740, 1001}, {24362, 24365}, {24374, 24384}

### X(24384) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24384) lies on these lines: {2, 23897}, {10, 620}, {141, 404}, {1211, 19308}, {2134, 6626}, {2482, 6537}, {4000, 5550}, {5224, 7891}, {5743, 21511}, {7267, 20653}, {10026, 17103}, {17292, 24636}, {17694, 21024}, {24369, 24370}, {24374, 24383}, {24435, 24953}

### X(24385) = X(2)X(16762)∩X(3)X(16766)

Barycentrics    2 a^16-11 a^14 b^2+25 a^12 b^4-31 a^10 b^6+25 a^8 b^8-17 a^6 b^10+11 a^4 b^12-5 a^2 b^14+b^16-11 a^14 c^2+34 a^12 b^2 c^2-31 a^10 b^4 c^2-4 a^8 b^6 c^2+31 a^6 b^8 c^2-38 a^4 b^10 c^2+27 a^2 b^12 c^2-8 b^14 c^2+25 a^12 c^4-31 a^10 b^2 c^4-5 a^6 b^6 c^4+34 a^4 b^8 c^4-51 a^2 b^10 c^4+28 b^12 c^4-31 a^10 c^6-4 a^8 b^2 c^6-5 a^6 b^4 c^6-14 a^4 b^6 c^6+29 a^2 b^8 c^6-56 b^10 c^6+25 a^8 c^8+31 a^6 b^2 c^8+34 a^4 b^4 c^8+29 a^2 b^6 c^8+70 b^8 c^8-17 a^6 c^10-38 a^4 b^2 c^10-51 a^2 b^4 c^10-56 b^6 c^10+11 a^4 c^12+27 a^2 b^2 c^12+28 b^4 c^12-5 a^2 c^14-8 b^2 c^14+c^16 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28392.

X(24385) lies on these lines: {2,16762}, {3,16766}, {5,195}, {30,1141}, {137,24147}, {140,15345}, {252,10126}, {523,10096}, {546,23337}, {1154,12026}, {3850,15307}, {6592,10615}, {7604,13469}, {11016,16239}, {13856,23280}, {14051,20414}

X(24385) = midpoint of X(i) and X(j) for these {i,j}: {5, 19553}, {137, 24147}, {1157, 1263}
X(24385) = reflection of X(6592) in X(10614)
X(24385) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (195, 3459, 5), (16762, 16764, 2), (16766, 16768, 3)

### X(24386) = COMPLEMENT OF X(3158)

Barycentrics    (a-b-c) (a b-3 b^2+a c+6 b c-3 c^2) : :
X(24386) = X[3189] - 7 X[3624], 5 X[3617] + X[3680], X[10] + 2 X[3813], 4 X[3634] - X[3913], X[6601] + 2 X[6666], 5 X[3091] + X[6762], 11 X[5056] + X[6764], 7 X[3090] - X[6765], X[2136] - 7 X[9780], 2 X[3626] + X[10912], X[946] + 2 X[10916], 4 X[1125] - X[12437], 5 X[3616] + X[12625], 5 X[5818] + X[12629], 4 X[10] - X[12640], 8 X[3813] + X[12640], X[12632] - 13 X[19877], X[12513] + 2 X[19925], 4 X[3813] - X[21627], 2 X[10] + X[21627], X[12640] + 2 X[21627]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.

X(24386) lies on these lines: {1,6856}, {2,3158}, {8,18220}, {9,5274}, {10,496}, {11,210}, {142,2886}, {200,10589}, {226,3873}, {312,17774}, {321,4939}, {497,4512}, {516,11235}, {518,3817}, {519,5055}, {522,21204}, {527,1699}, {528,10164}, {758,946}, {908,4661}, {950,10527}, {1058,5705}, {1125,12437}, {1210,3753}, {1329,4711}, {1420,5175}, {1706,5704}, {2136,9780}, {2321,3705}, {2550,6692}, {2887,17059}, {2900,4666}, {3090,6765}, {3091,6762}, {3189,3624}, {3243,5226}, {3434,3911}, {3616,12625}, {3617,3680}, {3626,10912}, {3634,3913}, {3687,4923}, {3717,4903}, {3755,24239}, {3756,21949}, {3794,17197}, {3825,3956}, {3838,5542}, {3847,4662}, {3877,5837}, {3892,21620}, {3894,12047}, {3921,4187}, {3928,9812}, {3939,17123}, {4031,20292}, {4035,10453}, {4134,21616}, {4314,4999}, {4525,11813}, {4669,5854}, {4731,8582}, {4859,21267}, {4863,6745}, {4906,17070}, {5056,6764}, {5087,21060}, {5325,11238}, {5573,17067}, {5744,9580}, {5784,17626}, {5795,9581}, {5818,12629}, {6557,10005}, {6600,8167}, {6601,6666}, {6736,17606}, {6737,11376}, {7681,9842}, {7741,21075}, {8666,18519}, {9669,12572}, {10106,10529}, {10896,12527}, {12513,19925}, {12632,19877}, {17889,24216}, {18483,18540}

X(24386) = midpoint of X(3928) and X(9812)
X(24386) = reflection of X(3817) in X(3829)
X(24386) = complement of X(3158)
X(24386) = X(i)-complementary conjugate of X(j) for these (i,j): {7, 2885}, {56, 3161}, {269, 12640}, {1293, 4521}, {3445, 9}, {3676, 5510}, {4373, 1329}, {8056, 3452}, {10029, 20540}, {16079, 8}, {16945, 37}, {19604, 10}
X(24386) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 3813, 21627), (10, 21627, 12640), (11, 4847, 3452), (497, 5231, 5745), (2886, 11019, 142), (3756, 21949, 24175), (6734, 12053, 5837)

### X(24387) = COMPLEMENT OF X(8715)

Barycentrics    a^2 b^2-b^4-2 a^2 b c+a b^2 c+a^2 c^2+a b c^2+2 b^2 c^2-c^4 : :
X(24387) = X[5] - 3 X[3829], X[3813] + 3 X[3829], 5 X[1656] - X[3913], 7 X[3526] - 3 X[4421], X[6765] - 9 X[7988], X[3811] - 5 X[8227], 3 X[5790] + X[10912], 3 X[10175] - X[10915], X[382] + 3 X[11194], X[3] + 3 X[11235], 7 X[3851] - 3 X[11236], 3 X[381] + X[12513], 3 X[5] - X[12607], 9 X[3829] - X[12607], 3 X[3813] + X[12607], 7 X[9624] + X[12625], 7 X[7989] + X[12629], 17 X[7486] - X[12632], X[12635] - 5 X[18493], 3 X[3817] - X[21077], 3 X[10175] + X[21627], 3 X[5886] - X[22836]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.

X(24387) lies on these lines: {1,2476}, {2,3746}, {3,11235}, {4,535}, {5,519}, {8,3814}, {10,11}, {12,3244}, {21,4857}, {35,149}, {72,11813}, {78,23708}, {80,4861}, {115,17448}, {116,20257}, {140,528}, {145,7951}, {214,5533}, {226,3881}, {354,11263}, {355,22837}, {377,10072}, {381,12513}, {382,11194}, {404,3582}, {405,11238}, {442,551}, {474,10199}, {495,3635}, {496,1125}, {497,5248}, {499,3434}, {501,19642}, {515,10943}, {516,6705}, {518,9955}, {529,546}, {536,7764}, {758,946}, {956,10896}, {958,9669}, {960,7743}, {993,1479}, {1058,10198}, {1210,3754}, {1212,21090}, {1329,3626}, {1385,1484}, {1478,10529}, {1506,20691}, {1656,3913}, {1770,4973}, {2475,5563}, {2550,10200}, {2801,12608}, {2893,24202}, {2975,3583}, {3035,20107}, {3058,7483}, {3086,6904}, {3120,3953}, {3241,5141}, {3303,10197}, {3304,17532}, {3337,20292}, {3419,11376}, {3452,4015}, {3526,4421}, {3555,17605}, {3584,7504}, {3625,7173}, {3632,11681}, {3634,3816}, {3678,4847}, {3679,4193}, {3680,6975}, {3811,8227}, {3817,18908}, {3820,3847}, {3826,19878}, {3828,9710}, {3838,5045}, {3851,11236}, {3868,18393}, {3872,10826}, {3874,12047}, {3878,6734}, {3880,9956}, {3889,10129}, {3892,13407}, {3925,19862}, {3968,8582}, {4085,20108}, {4297,15908}, {4301,6831}, {4309,6910}, {4324,5303}, {4330,17549}, {4426,9665}, {4479,7796}, {4658,14009}, {4669,17533}, {4745,9711}, {4972,19864}, {4999,15171}, {5046,5258}, {5057,6763}, {5080,5288}, {5082,10589}, {5129,5274}, {5223,7678}, {5231,9614}, {5267,6284}, {5270,17577}, {5299,17737}, {5450,10525}, {5497,24161}, {5537,6972}, {5603,6873}, {5690,13463}, {5777,22835}, {5790,10912}, {5794,11373}, {5881,6941}, {5882,6842}, {5886,22836}, {6154,7294}, {6690,15172}, {6765,7988}, {6828,11522}, {6829,9624}, {6830,7982}, {6871,11240}, {6881,12437}, {6882,11362}, {6933,10056}, {6943,7991}, {6990,11523}, {7486,12632}, {7681,19925}, {7752,17144}, {7956,12571}, {7989,12629}, {9670,16370}, {10175,10915}, {10948,17647}, {11019,12446}, {11260,18480}, {11928,22758}, {12635,18493}, {14872,21635}, {14923,18395}, {15888,17530}, {16174,20117}, {16829,17669}, {17046,17761}, {17529,19883}, {21073,24036}, {21384,24045}

X(24387) = midpoint of X(i) and X(j) for these {i,j}: {4, 8666}, {5, 3813}, {355, 22837}, {946, 10916}, {5450, 10525}, {5690, 13463}, {10915, 21627}, {11260, 18480}
X(24387) = complement of X(8715)
X(24387) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 7741, 3814), (10, 11, 3825), (10, 12053, 3884), (10, 21630, 3057), (21, 10707, 4857), (496, 2886, 1125), (1125, 2886, 3841), (1479, 10527, 993), (3813, 3829, 5), (4847, 21616, 3678), (5231, 9614, 12514), (9710, 17527, 3828), (10175, 21627, 10915), (10914, 17606, 10)

### X(24388) = X(1)X(475)∩X(10)X(1001)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.

X(24388) lies on these lines: {1,475}, {10,1001}, {141,15733}, {142,17059}, {281,497}, {515,1597}, {518,12618}, {522,3663}, {527,21629}, {740,10916}, {946,1871}, {1210,3755}, {2321,3693}, {2345,6601}, {3174,17284}, {3886,6734}, {3939,17353}, {3946,11019}, {4523,21616}, {4858,23529}, {4953,17246}, {5572,16608}, {5856,17351}, {10915,17765}, {18216,18634}

X(24388) = X(i)-complementary conjugate of X(j) for these (i,j): {56, 5452}, {3433, 9}, {13577, 1329}

### X(24389) = COMPLEMENT OF X(3174)

Barycentrics    (a-b-c) (a^3 b-3 a^2 b^2+3 a b^3-b^4+a^3 c+2 a^2 b c-3 a b^2 c+4 b^3 c-3 a^2 c^2-3 a b c^2-6 b^2 c^2+3 a c^3+4 b c^3-c^4) : :
X(24389) = X[7674] - 5 X[18230]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.

X(24389) lies on these lines: {2,3174}, {9,497}, {10,1001}, {11,3059}, {142,2886}, {226,15185}, {390,6734}, {480,4863}, {516,1158}, {518,946}, {527,11235}, {528,10265}, {908,7678}, {960,4342}, {1210,2550}, {1445,3434}, {3243,3485}, {3826,9843}, {4134,18254}, {4326,5231}, {4662,18255}, {4686,4953}, {5223,9614}, {5249,11025}, {5542,11263}, {5745,8730}, {6067,14100}, {7674,18230}, {7675,10527}, {11680,21617}, {12609,20116}, {12731,21631}, {17059,21255}

X(24389) = midpoint of X(9) and X(6601)
X(24389) = reflection of X(6600) in X(6666)
X(24389) = complement of X(3174)
X(24389) = X(4578)-Ceva conjugate of X(522)
X(24389) = crosssum of X(56) and X(21059)
X(24389) = barycentric product X(8)*X(24181)
X(24389) = barycentric quotient X(24181)/X(7)
X(24389) = {X(2886),X(5572)}-harmonic conjugate of X(142)

### X(24390) = COMPLEMENT OF X(3871)

Barycentrics    a^2 b^2-b^4-2 a^2 b c+2 a b^2 c+a^2 c^2+2 a b c^2+2 b^2 c^2-c^4 : :
X(24390) = 3 X[3584] - 4 X[6668], 2 X[12] - 3 X[17530], 3 X[17577] - X[20060], 3 X[17549] - X[20066] : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.

X(24390) lies on these lines: {1,442}, {2,496}, {3,3434}, {4,956}, {5,8}, {9,9614}, {10,11}, {12,519}, {21,149}, {30,2975}, {35,528}, {39,21956}, {40,5231}, {55,7483}, {56,11112}, {63,3650}, {65,10916}, {72,946}, {75,17181}, {78,5886}, {92,15763}, {100,140}, {145,495}, {200,8227}, {210,21616}, {226,3555}, {325,17143}, {354,12609}, {355,1532}, {377,999}, {381,3436}, {388,17532}, {390,6857}, {404,15325}, {405,497}, {443,14986}, {474,2550}, {498,3913}, {499,1376}, {515,15908}, {516,3916}, {517,6734}, {518,12047}, {529,3585}, {546,5080}, {548,5303}, {551,3841}, {594,17444}, {631,17784}, {726,21927}, {867,17164}, {908,9955}, {942,20612}, {944,5175}, {952,4861}, {954,6601}, {958,1479}, {962,5789}, {993,6284}, {997,11376}, {1001,11517}, {1056,5177}, {1086,3953}, {1125,3925}, {1191,1714}, {1210,3753}, {1212,21073}, {1260,6832}, {1278,7906}, {1319,10949}, {1329,3679}, {1385,10609}, {1478,12513}, {1519,5777}, {1537,5887}, {1538,9947}, {1565,20880}, {1621,6675}, {1656,5552}, {1697,5705}, {1698,3816}, {1737,5836}, {1738,17055}, {1883,4968}, {2170,21029}, {2475,18990}, {2478,9669}, {2551,10591}, {2646,10959}, {2932,6940}, {2968,23528}, {3006,3695}, {3035,5533}, {3058,5248}, {3090,7080}, {3091,3421}, {3136,17135}, {3244,3822}, {3303,10198}, {3338,5880}, {3340,15844}, {3452,3697}, {3582,6691}, {3583,5258}, {3584,6668}, {3614,3625}, {3616,8728}, {3617,3820}, {3621,5141}, {3622,4197}, {3624,3826}, {3626,3814}, {3632,7951}, {3649,3874}, {3656,11682}, {3678,11813}, {3746,6690}, {3811,4863}, {3817,21075}, {3824,5049}, {3838,13407}, {3847,9711}, {3869,6841}, {3870,11374}, {3873,6147}, {3878,21677}, {3880,10039}, {3881,11263}, {3893,10915}, {3927,11415}, {3933,4441}, {3976,17889}, {4002,8582}, {4026,19863}, {4294,16370}, {4299,11194}, {4353,21955}, {4413,10200}, {4423,17590}, {4511,5178}, {4662,5087}, {4678,5154}, {4853,5587}, {4857,5251}, {4865,17733}, {4875,5179}, {4882,7988}, {4904,17046}, {4915,7989}, {5044,7743}, {5045,5249}, {5084,5274}, {5100,7081}, {5173,14054}, {5176,18357}, {5180,11684}, {5219,6765}, {5226,6764}, {5250,5791}, {5254,16975}, {5260,10707}, {5267,15338}, {5291,7745}, {5292,5710}, {5305,17737}, {5432,8715}, {5439,11019}, {5450,11826}, {5484,17677}, {5499,6224}, {5559,12653}, {5657,6922}, {5686,7678}, {5690,6882}, {5697,13463}, {5711,11269}, {5722,19860}, {5724,15955}, {5726,11519}, {5744,6361}, {5745,10624}, {5815,9779}, {5833,10384}, {5842,11012}, {5853,13411}, {5854,8068}, {5855,11009}, {5881,18242}, {5904,18393}, {6174,7294}, {6244,6890}, {6653,7824}, {6705,17613}, {6735,9956}, {6736,10175}, {6737,13464}, {6743,7958}, {6762,9612}, {6763,17768}, {6829,10595}, {6830,12245}, {6833,10306}, {6862,10679}, {6871,9654}, {6872,9668}, {6889,10806}, {6910,20075}, {6913,10531}, {6917,10680}, {6929,10522}, {6933,10528}, {6937,7967}, {6980,10942}, {6984,10597}, {6991,20007}, {7288,16371}, {7354,8666}, {7373,11240}, {7680,7982}, {7767,20553}, {8256,18395}, {9578,12629}, {9581,9623}, {9655,20076}, {9780,17527}, {9874,18228}, {10058,13271}, {10523,10912}, {10525,22758}, {10572,12690}, {10585,11239}, {10944,22837}, {11256,12749}, {11373,19861}, {11544,17483}, {11698,12531}, {12514,12701}, {12527,18483}, {12608,13257}, {12670,21631}, {14019,16020}, {14740,16174}, {15733,16193}, {15868,22767}, {15950,22836}, {16552,17747}, {16842,19855}, {17514,19858}, {17549,20066}, {17577,20060}, {17597,24159}, {17605,21077}, {17724,24160}

X(24390) = complement of X(3871)
X(24390) = midpoint of X(i) and X(j) for these {i,j}: {3585, 5288}, {4861, 5086}
X(24390) = reflection of X(i) in X(j) for these {i,j}: {35, 4999}, {15338, 5267}, {20612, 942}
X(24390) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2886, 442), (2, 5082, 5687), (5, 8, 17757), (8, 5603, 5730), (8, 11680, 5), (10, 11, 4187), (10, 10914, 1145), (10, 12053, 392), (10, 21630, 3884), (21, 149, 15171), (145, 2476, 495), (377, 10529, 999), (497, 19843, 405), (499, 1376, 13747), (946, 4847, 72), (958, 1479, 11113), (958, 11235, 1479), (1125, 3925, 17529), (1329, 3829, 7741), (1329, 7741, 17533), (1698, 3816, 17575), (2550, 3086, 474), (2551, 10591, 17556), (2886, 3813, 1), (3006, 3702, 3695), (3244, 3822, 15888), (3434, 10527, 3), (3617, 4193, 3820), (3626, 3814, 21031), (3632, 7951, 12607), (3679, 3829, 17533), (3679, 7741, 1329), (3816, 9710, 1698), (3820, 10593, 4193), (4863, 11375, 3811), (6675, 15172, 1621), (6980, 12645, 10942), (7173, 21031, 3814), (9669, 9708, 2478), (12608, 14872, 13257), (17046, 20257, 4904)
X(24390) = complement of the isogonal conjugate of X(20615)
X(24390) = X(i)-complementary conjugate of X(j) for these (i,j): {56, 4075}, {596, 1329}, {20615, 10}

### X(24391) = COMPLEMENT OF X(11523)

Barycentrics    3 a^3 b-a^2 b^2-3 a b^3+b^4+3 a^3 c+2 a^2 b c-a b^2 c-a^2 c^2-a b c^2-2 b^2 c^2-3 a c^3+c^4 : :
X(24391) = 3 X[165] - X[3189], X[20] - 3 X[3928], 3 X[5770] - 2 X[6705], 3 X[5657] - X[6765], 3 X[5709] - X[6869], 5 X[3522] - X[12536], 3 X[10] - 2 X[12607], 3 X[3928] + X[12625], X[6769] - 3 X[14647], 3 X[10175] - 2 X[21077], 3 X[10165] - 2 X[22836]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.

X(24391) lies on these lines: {1,5745}, {2,3984}, {3,519}, {4,527}, {8,57}, {9,938}, {10,141}, {11,3962}, {20,3928}, {40,5768}, {56,6737}, {63,950}, {65,4847}, {72,1210}, {78,3911}, {144,10392}, {145,3601}, {165,3189}, {200,1467}, {210,8582}, {226,2476}, {329,9581}, {341,4899}, {354,21677}, {355,2095}, {377,553}, {387,3946}, {405,5325}, {443,3679}, {452,3929}, {497,12526}, {515,5709}, {516,5787}, {517,6245}, {528,5493}, {529,20420}, {551,6675}, {579,2321}, {610,5839}, {758,946}, {908,5154}, {912,6260}, {936,6692}, {958,6738}, {960,11019}, {986,3755}, {1001,6744}, {1040,15954}, {1125,5791}, {1329,21060}, {1376,6743}, {1385,5771}, {1737,5904}, {1770,4880}, {1834,3663}, {1837,12527}, {1998,10393}, {2093,5082}, {2136,6764}, {2323,3562}, {2478,3951}, {2550,3339}, {2551,5223}, {2886,3671}, {2900,10884}, {3008,17054}, {3243,5775}, {3244,5855}, {3419,4292}, {3485,5231}, {3487,5705}, {3522,12536}, {3617,9776}, {3626,5708}, {3632,15803}, {3730,21096}, {3811,6684}, {3813,4301}, {3825,4127}, {3869,12053}, {3876,5316}, {3894,13407}, {3901,12047}, {3916,4304}, {3927,5722}, {3940,6700}, {4001,5016}, {4067,21616}, {4208,6173}, {4298,5794}, {4313,20008}, {4314,4640}, {4654,5177}, {4667,5717}, {4685,16056}, {4711,10855}, {4855,20013}, {5044,9843}, {5046,17781}, {5128,17784}, {5175,9579}, {5220,18250}, {5273,5436}, {5435,5438}, {5657,6765}, {5690,9940}, {5770,6705}, {5777,7682}, {5795,18391}, {5805,5850}, {5843,22792}, {5854,13226}, {6282,12245}, {6763,10572}, {6769,14647}, {6824,13464}, {6847,7982}, {9943,15733}, {10165,22836}, {10175,21077}, {10202,10915}, {10477,20258}, {10529,11682}, {10578,18231}, {11529,19843}, {12832,14740}, {14054,18389}, {14986,15829}, {15933,17558}, {17449,23675}, {21214,24216}

X(24391) = midpoint of X(i) and X(j) for these {i,j}: {8, 6762}, {20, 12625}, {2136, 6764}, {12245, 12629}
X(24391) = reflection of X(i) in X(j) for these {i,j}: {946, 10916}, {3244, 11260}, {3811, 6684}, {4301, 3813}, {5882, 8666}, {12437, 3}, {12635, 1125}, {12640, 11362}
X(24391) = complement of X(11523)
X(24391) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 942, 142), (10, 3874, 21620), (63, 12649, 950), (72, 1210, 3452), (145, 5744, 3601), (1737, 5904, 21075), (3868, 6734, 226), (3927, 5722, 12572), (3928, 12625, 20), (5175, 9965, 9579), (5435, 20007, 5438), (5777, 7682, 9842), (5791, 15934, 1125), (6744, 18249, 1001)

### X(24392) = X(1)X(442)∩X(2)X(3158)

Barycentrics    (a-b-c) (a^2-a b+2 b^2-a c-4 b c+2 c^2) : :
X(24392) = 4 X[10] - X[2136], 4 X[2886] - X[2900], 4 X[1125] - X[3189], 2 X[8] + X[3680], X[1] - 4 X[3813], 5 X[1698] - 2 X[3913], X[9] + 2 X[6601], 2 X[4] + X[6762], 5 X[3091] + X[6764], 4 X[5] - X[6765], 4 X[6666] - X[7674], 4 X[3829] - 3 X[7988], 2 X[3811] - 5 X[8227], X[3632] + 2 X[10912], X[40] - 4 X[10916], 4 X[946] - X[11523], 5 X[3616] - 2 X[12437], X[5691] + 2 X[12513], 7 X[3622] - X[12536], 5 X[3617] + X[12541], 7 X[7989] - 4 X[12607], 2 X[1] + X[12625], 8 X[3813] + X[12625], 2 X[355] + X[12629], 7 X[9780] - X[12632], 5 X[11522] - 2 X[12635], 5 X[3617] - 2 X[12640], X[12541] + 2 X[12640], 4 X[3036] - X[12641], X[1768] + 2 X[13271], X[11531] - 4 X[13463], 2 X[3174] - 5 X[20195], X[3680] - 4 X[21627], X[8] + 2 X[21627], 7 X[9624] - 4 X[22836]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.

X(24392) lies on these lines: {1,442}, {2,3158}, {4,6762}, {5,6765}, {8,3452}, {9,497}, {10,1058}, {11,200}, {40,6899}, {55,5231}, {57,3434}, {63,149}, {72,9614}, {120,5272}, {142,10580}, {145,5226}, {165,528}, {210,11238}, {226,3243}, {312,4901}, {354,6173}, {355,7956}, {390,5745}, {392,3679}, {496,936}, {516,3928}, {518,1699}, {519,3545}, {521,11193}, {522,6545}, {527,9812}, {674,10439}, {748,3939}, {946,11523}, {956,3586}, {1024,2319}, {1125,3189}, {1146,5574}, {1210,1706}, {1329,4882}, {1420,10529}, {1697,6734}, {1698,3913}, {1738,5573}, {1768,13271}, {1837,4853}, {2481,20935}, {2550,5437}, {2898,9436}, {2999,17721}, {3036,4900}, {3058,4512}, {3086,5438}, {3091,6764}, {3174,3925}, {3295,5705}, {3340,12649}, {3555,9612}, {3601,10527}, {3616,12437}, {3617,12541}, {3622,12536}, {3632,5087}, {3633,10827}, {3677,3914}, {3681,10707}, {3703,4873}, {3705,3886}, {3706,4007}, {3756,8056}, {3811,8227}, {3816,8580}, {3829,7988}, {3870,5219}, {3872,5727}, {3873,4654}, {3875,7179}, {3911,17784}, {3940,7743}, {3944,16496}, {3966,4034}, {4323,20008}, {4326,6067}, {4373,16078}, {4423,6600}, {4514,11679}, {4677,5854}, {4859,21949}, {4862,21342}, {5049,17528}, {5123,8168}, {5175,10106}, {5178,19861}, {5225,12527}, {5261,9797}, {5268,24217}, {5269,11269}, {5436,19843}, {5691,12513}, {5722,9623}, {5748,20015}, {5784,12915}, {5791,15172}, {5837,9785}, {5855,11224}, {5880,10980}, {6666,7674}, {6745,10589}, {7082,13274}, {7174,24210}, {7989,12607}, {8055,10005}, {9624,22836}, {9708,18527}, {9780,12632}, {10453,17296}, {10591,21075}, {11522,12635}, {11531,13463}, {12526,12701}, {16572,21073}, {17284,19589}, {17597,23681}, {21740,22837}

X(24392) = reflection of X(i) in X(j) for these {i,j}: {1699, 11235}, {3158, 2}
X(24392) = X(8)-waw conjugate of X(9)
X(24392) = X(6558)-Ceva conjugate of X(522)
X(24392) = crosspoint of X(8) and X(4373)
X(24392) = crosssum of X(i) and X(j) for these (i,j): {56, 3052}, {604, 21059}
X(24392) = barycentric product X(i)*X(j) for these {i,j}: {8, 4859}, {333, 21949}
X(24392) = barycentric quotient X(i)/X(j) for these {i,j}: {4859, 7}, {21949, 226}
X(24392) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 5274, 3452), (8, 6557, 6555), (8, 12053, 15829), (8, 21627, 3680), (11, 4863, 200), (63, 149, 9580), (497, 4847, 9), (1210, 5082, 1706), (2550, 11019, 5437), (3617, 12541, 12640), (3870, 11680, 5219), (3925, 10582, 20195), (8056, 21267, 3756)

### X(24393) = COMPLEMENT OF X(3243)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.

X(24393) lies on these lines: {1,4878}, {2,3243}, {7,3617}, {8,9}, {10,141}, {11,210}, {55,5325}, {75,4899}, {144,4678}, {145,18230}, {200,5218}, {226,3681}, {333,7256}, {355,382}, {480,6737}, {515,3358}, {519,1001}, {527,1478}, {528,4669}, {551,15570}, {944,21153}, {946,3678}, {952,6594}, {958,6600}, {960,4342}, {971,5690}, {984,3755}, {1210,3697}, {1445,10106}, {1788,4321}, {2346,5260}, {2551,4866}, {2886,21060}, {3008,3242}, {3036,5856}, {3057,7064}, {3059,4111}, {3158,5273}, {3174,4882}, {3189,5234}, {3219,20095}, {3244,4974}, {3340,8232}, {3621,8236}, {3625,6541}, {3671,9710}, {3703,4061}, {3706,4082}, {3711,6745}, {3715,4863}, {3740,11019}, {3751,4667}, {3786,17197}, {3876,12053}, {3913,18249}, {3929,17784}, {3946,7174}, {3983,8582}, {3984,11526}, {4015,10916}, {4133,4439}, {4310,17067}, {4314,5302}, {4344,16670}, {4345,15829}, {4349,4663}, {4661,5249}, {4668,5698}, {4690,5845}, {4691,5850}, {4710,4738}, {4711,15733}, {4864,17337}, {4915,8275}, {4929,16833}, {5252,12573}, {5534,5770}, {5657,5732}, {5790,5805}, {5806,12599}, {5817,12245}, {5904,11551}, {6067,21031}, {6692,8580}, {6734,7705}, {7951,21075}, {8256,15587}, {9053,17348}, {9623,11041}, {9668,12572}, {9780,11038}, {9948,11495}, {9956,20330}, {10039,18412}, {10164,13226}, {10389,20015}, {10573,15298}, {10624,15650}, {10950,15837}, {12447,12513}, {12527,12943}, {12564,12855}, {12647,15299}, {18357,18482}, {21342,24175}

X(24393) = midpoint of X(i) and X(j) for these {i,j}: {8, 9}, {2550, 5223}, {7674, 12625}
X(24393) = reflection of X(i) in X(j) for these {i,j}: {1, 6666}, {142, 10}, {5542, 3826}, {12437, 6600}, {18482, 18357}, {20330, 9956}
X(24393) = complement of X(3243)
X(24393) = X(8)-beth conjugate of X(142)
X(24393) = crosssum of X(56) and X(1471)
X(24393) = barycentric product X(4924)*X(6557)
X(24393) = barycentric quotient X(4924)/X(5435)
X(24393) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 2321, 4923), (8, 3717, 2321), (8, 5686, 9), (8, 5837, 12640), (8, 10005, 4901), (10, 5542, 3826), (10, 21255, 3823), (210, 4847, 3452), (958, 6743, 12437), (1697, 5809, 15006), (3679, 5223, 2550), (3703, 4113, 4061), (3706, 4126, 4082), (3826, 5542, 142), (9780, 11038, 20195)

### X(24394) = X(1)X(17959)∩X(30)X(511)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.

X(24394) lies on these lines: {1,17959}, {30,511}, {37,4097}, {210,2321}, {612,1962}, {2136,2292}, {2294,3174}, {2643,19589}, {2650,3680}, {3169,3728}, {3678,4527}, {3686,11997}, {3742,3946}, {3743,3913}, {3747,3939}, {3753,3755}, {3754,4743}, {3773,3956}, {3786,3877}, {3794,4483}, {3873,3875}, {3919,4780}, {3943,21889}, {3950,21867}, {3968,4085}, {4015,4535}, {4068,6600}, {4133,4134}, {4433,4516}, {4512,4877}, {6601,18698}, {10158,10180}, {12541,17164}, {17163,18697}, {21871,22312}

X(24394) = barycentric product X(i)*X(j) for these {i,j}: {8, 16611}, {9, 4442}, {2321, 7292}, {3701, 16784}, {7017, 23230}
X(24394) = barycentric quotient X(i)/X(j) for these {i,j}: {2832, 17096}, {4442, 85}, {6019, 16611}, {7292, 1434}, {16611, 7}, {16784, 1014}, {23230, 222}

### X(24395) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24395) lies on these lines: {1, 522}, {988, 1125}, {1331, 17888}, {2550, 10039}, {2607, 9318}, {4011, 25377}, {6604, 24695}, {24397, 24416}

### X(24396) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24396) lies on these lines: {1, 905}, {2, 24449}, {6, 43}, {2254, 9318}, {2426, 3550}, {3571, 24410}, {4859, 17063}, {4925, 24420}, {24338, 24398}, {24397, 24412}

### X(24397) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24397) lies on these lines: {1, 513}, {8, 19945}, {45, 16604}, {87, 1001}, {244, 4997}, {528, 4947}, {978, 23343}, {982, 24433}, {984, 1698}, {1909, 4389}, {3038, 17071}, {3123, 24841}, {3242, 10912}, {3616, 24485}, {24395, 24416}, {24396, 24412}, {24420, 24427}

### X(24398) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24398) lies on these lines: {1, 514}, {2, 2415}, {87, 2191}, {527, 24216}, {982, 24352}, {3570, 3875}, {3961, 23669}, {4384, 19965}, {4419, 24318}, {4454, 25377}, {4919, 21138}, {24311, 25368}, {24338, 24396}, {24399, 24400}, {24409, 24422}, {24413, 24419}, {24420, 24423}

### X(24399) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24399) lies on these lines: {1, 513}, {10, 75}, {244, 4080}, {537, 3123}, {982, 1647}, {1125, 24487}, {1739, 25025}, {3216, 23343}, {3670, 24433}, {3976, 23869}, {4444, 24403}, {4947, 24715}, {17063, 25377}, {24398, 24400}, {24413, 24414}

### X(24400) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24400) lies on these lines: {1, 525}, {19, 27}, {1025, 3729}, {4454, 24316}, {24398, 24399}, {24403, 24409}

### X(24401) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24401) lies on these lines: {1, 824}, {3, 4363}, {812, 24410}, {894, 24484}, {2607, 9318}, {17354, 17671}, {24346, 24455}, {24442, 24453}

### X(24402) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24402) lies on these lines: {1, 522}, {2, 846}, {109, 14628}, {740, 4585}

### X(24403) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24403) lies on these lines: {1, 514}, {2, 37}, {524, 17145}, {527, 17449}, {1018, 21208}, {1107, 25261}, {1334, 24172}, {1642, 5222}, {1647, 24318}, {2176, 20247}, {2275, 25237}, {3570, 4360}, {3571, 24419}, {3722, 24685}, {3726, 20347}, {3952, 4465}, {4392, 4419}, {4396, 20045}, {4444, 24399}, {4643, 20042}, {4713, 17165}, {5276, 6654}, {6545, 23810}, {7032, 24333}, {9259, 17136}, {9309, 12915}, {16604, 25244}, {17140, 24330}, {17318, 17780}, {17597, 24352}, {20244, 20271}, {20248, 21785}, {21138, 21272}, {21282, 24699}, {24338, 24413}, {24400, 24409}, {24404, 24429}

### X(24404) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24404) lies on these lines: {1, 812}, {7, 24484}, {39, 1086}, {190, 16549}, {1655, 4440}, {2607, 9318}, {3821, 25373}, {6646, 24505}, {17483, 24504}, {24338, 24414}, {24403, 24429}

### X(24405) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24405) lies on these lines: {1, 513}, {2, 23343}, {7, 2283}, {37, 3675}, {42, 19945}, {45, 672}, {100, 903}, {527, 551}, {545, 8299}, {674, 1469}, {894, 24487}, {942, 24433}, {984, 5902}, {1086, 4557}, {1644, 4413}, {3110, 4840}, {3720, 24494}, {4068, 17246}, {4363, 24327}, {4389, 20347}, {4436, 4440}, {4447, 7238}, {4448, 9318}, {4471, 24328}, {4862, 15624}, {5091, 23344}, {6646, 16684}, {8053, 17276}, {16494, 24482}, {16679, 17365}, {16687, 17483}, {17290, 20335}, {17724, 23845}, {20470, 24231}, {24345, 24416}

### X(24406) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24406) lies on these lines: {1, 513}, {43, 19945}, {192, 21140}, {545, 2108}, {903, 1054}, {984, 3753}, {4912, 24441}, {4947, 21320}, {16569, 23343}, {24346, 24416}, {24409, 24428}

### X(24407) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24407) lies on these lines: {1, 514}, {2, 45}, {524, 20042}, {527, 1647}, {536, 17780}, {678, 24685}, {3570, 17160}, {4393, 18822}, {4659, 9458}, {4971, 20058}, {24419, 24423}

### X(24408) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

Barycentrics    a^4 - 3 a^3 b - 2 a b^3 - 3 a^3 c + 9 a^2 b c + 3 b^3 c - 6 b^2 c^2 - 2 a c^3 + 3 b c^3 : :

X(24408) lies on these lines: {1, 514}, {2, 1266}, {536, 9458}, {545, 24709}, {676, 24409}, {1647, 4419}, {2254, 24338}, {17132, 25377}, {24441, 25378}

### X(24409) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24409) lies on these lines: {1, 522}, {551, 3923}, {676, 24408}, {894, 24485}, {1769, 24338}, {2191, 2414}, {3756, 4370}, {4000, 24980}, {4448, 9318}, {4904, 5856}, {11125, 24427}, {24398, 24422}, {24400, 24403}, {24406, 24428}

### X(24410) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24410) lies on these lines: {1, 522}, {4, 9}, {34, 2405}, {109, 24026}, {318, 7012}, {348, 22464}, {812, 24401}, {867, 2886}, {993, 14127}, {1376, 9458}, {1457, 10538}, {1818, 3685}, {1935, 23528}, {2284, 4513}, {2792, 21293}, {3035, 16594}, {3073, 17869}, {3100, 24307}, {3571, 24396}, {4318, 4861}, {5263, 9414}, {5438, 6789}, {24416, 24418}

### X(24411) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24411) lies on these lines: {1, 522}, {85, 10001}, {527, 1478}, {900, 24346}, {990, 997}, {1026, 3729}, {1146, 5851}, {1275, 9312}, {1376, 23343}, {1768, 2250}, {1996, 5231}, {2263, 3872}, {3035, 4370}, {4363, 24341}, {4858, 9355}, {18816, 24026}, {24338, 24396}

### X(24412) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24412) lies on these lines: {1, 8678}, {3571, 9318}, {4561, 23686}, {24396, 24397}

### X(24413) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24413) lies on these lines: {1, 512}, {38, 75}, {87, 24492}, {982, 4493}, {1739, 20006}, {2276, 18795}, {3670, 19950}, {4392, 4443}, {9318, 24423}, {19951, 24443}, {24338, 24403}, {24396, 24397}, {24398, 24419}, {24399, 24414}

### X(24414) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24414) lies on these lines: {1, 788}, {10, 4443}, {3122, 19974}, {24338, 24404}, {24399, 24413}

### X(24415) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24415) lies on these lines: {1, 525}, {81, 4360}, {918, 9318}

### X(24416) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24416) lies on these lines: {1, 900}, {2, 24188}, {106, 4432}, {190, 244}, {545, 24358}, {1054, 1086}, {1411, 24816}, {1769, 24338}, {1772, 24845}, {4427, 4440}, {5541, 24222}, {9318, 24354}, {9457, 24841}, {24345, 24405}, {24346, 24406}, {24395, 24397}, {24410, 24418}, {24428, 24461}

### X(24417) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24417) lies on these lines: {31, 9811}, {812, 1769}, {824, 21189}, {918, 4364}, {1638, 3782}, {3960, 9259}

### X(24418) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24418) lies on these lines: {1, 513}, {145, 19945}, {984, 3624}, {3242, 17792}, {3622, 24507}, {3976, 24433}, {4389, 25303}, {17290, 25102}, {21214, 23343}, {24410, 24416}

### X(24419) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24419) lies on these lines: {1, 659}, {2, 24193}, {190, 244}, {291, 4465}, {2254, 24338}, {3248, 3315}, {3571, 24403}, {4809, 24427}, {9318, 24420}, {16676, 20331}, {24398, 24413}, {24407, 24423}, {24421, 24460}

### X(24420) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24420) lies on these lines: {1, 667}, {37, 982}, {4925, 24396}, {9318, 24419}, {16495, 17449}, {17597, 18170}, {24338, 24403}, {24397, 24427}, {24398, 24423}

### X(24421) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24421) lies on these lines: {1, 667}, {982, 1107}, {24338, 24404}, {24419, 24460}

### X(24422) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

Barycentrics    a (a^4 - 3 a^3 b + 4 a^2 b^2 + a b^3 + b^4 - 3 a^3 c + 5 a^2 b c - 7 a b^2 c - 7 b^3 c + 4 a^2 c^2 - 7 a b c^2 + 16 b^2 c^2 + a c^3 - 7 b c^3 + c^4) : :

X(24422) lies on these lines: {1, 513}, {200, 19945}, {23343, 23511}, {24398, 24409}

### X(24423) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24423) lies on these lines: {1, 513}, {45, 25427}, {239, 19945}, {292, 24722}, {870, 4389}, {984, 4363}, {2664, 23343}, {9318, 24413}, {16826, 24510}, {24398, 24420}, {24407, 24419}

### X(24424) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24424) lies on these lines: {1, 5327}, {11, 24317}, {35, 24684}, {38, 497}, {55, 24315}, {71, 17863}, {527, 3874}, {534, 10624}, {536, 960}, {596, 17132}, {950, 9028}, {1001, 3923}, {1479, 24682}, {1953, 25255}, {2294, 17220}, {2486, 2886}, {3663, 18589}, {3782, 25361}, {4019, 18147}, {4032, 16777}, {4294, 24683}, {4454, 17140}, {5432, 25341}, {5572, 13476}, {6508, 18662}, {7289, 10889}, {11235, 24441}, {11238, 25362}, {12635, 17318}, {17861, 21231}, {20531, 25348}, {24248, 24445}, {24328, 24352}, {24337, 24444}

### X(24425) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24425) lies on these lines: {1, 714}, {38, 190}, {45, 4011}, {75, 3747}, {670, 1965}, {744, 1930}, {1001, 3923}, {1086, 24259}, {1191, 5695}, {1918, 20891}, {4056, 4837}, {4364, 4368}, {4376, 4381}, {5263, 24450}, {8299, 24327}, {13476, 17351}, {17153, 17280}, {17290, 24260}, {18137, 20964}, {23383, 24850}, {24335, 24346}

### X(24426) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24426) lies on these lines: {1, 667}, {2, 24448}, {6, 982}, {38, 3570}, {3571, 24403}, {9318, 24413}, {24338, 24396}

### X(24427) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24427) lies on these lines: {1, 514}, {2, 726}, {900, 24338}, {1647, 4392}, {3807, 17793}, {4809, 24419}, {9458, 17495}, {11125, 24409}, {17449, 17484}, {24397, 24420}, {24399, 24413}

### X(24428) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24428) lies on these lines: {1, 514}, {2, 846}, {740, 3570}, {1083, 19950}, {1647, 3218}, {2607, 3571}, {3724, 13589}, {6548, 17960}, {6654, 8300}, {17194, 18646}, {24329, 24456}, {24338, 24346}, {24406, 24409}, {24416, 24461}

### X(24429) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24429) lies on these lines: {1, 513}, {8, 192}, {10, 24004}, {38, 20042}, {986, 6788}, {1001, 18792}, {1125, 19945}, {2607, 3571}, {2796, 3122}, {3123, 4432}, {3293, 23343}, {3821, 18046}, {3909, 14752}, {4389, 16887}, {4424, 24433}, {24403, 24404}

### X(24430) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24430) lies on these lines: {1, 90}, {2, 7004}, {3, 3460}, {9, 1040}, {11, 982}, {20, 201}, {22, 21368}, {31, 1776}, {33, 63}, {37, 10391}, {38, 497}, {55, 846}, {57, 1736}, {73, 12528}, {84, 1038}, {210, 9371}, {212, 3100}, {238, 7082}, {240, 1857}, {244, 10589}, {255, 6198}, {388, 774}, {390, 7226}, {522, 17860}, {651, 20277}, {756, 5218}, {916, 20122}, {920, 3072}, {958, 1854}, {971, 1214}, {986, 1837}, {990, 1708}, {991, 16577}, {1001, 24434}, {1062, 3074}, {1254, 5229}, {1376, 24433}, {1393, 3091}, {1465, 5927}, {1473, 16560}, {1478, 1725}, {1709, 8270}, {1710, 8185}, {1726, 3220}, {1735, 5587}, {1777, 7701}, {1807, 6914}, {1859, 24310}, {1864, 3666}, {1870, 18477}, {2201, 16567}, {2292, 3486}, {2293, 3989}, {2361, 7262}, {2635, 17080}, {2654, 3868}, {3056, 9017}, {3075, 24467}, {3468, 8757}, {3661, 22418}, {3670, 9581}, {3703, 4073}, {3731, 10383}, {3876, 22072}, {3925, 24341}, {3929, 7070}, {3955, 10535}, {3976, 11376}, {4003, 17604}, {4376, 9414}, {4392, 5274}, {4424, 5727}, {4459, 4493}, {4650, 5348}, {5220, 7074}, {5285, 21375}, {5745, 16870}, {5777, 17102}, {5784, 25091}, {6001, 24806}, {7008, 23052}, {7046, 24031}, {7193, 11429}, {7299, 9630}, {7416, 23067}, {10394, 14547}, {10395, 23537}, {11220, 22053}, {11502, 17596}, {13243, 17074}, {15064, 24025}, {16452, 22347}, {17606, 24174}, {18161, 21318}, {20305, 22069}, {22410, 23585}

### X(24431) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24431) lies on these lines: {1, 195}, {2, 24433}, {11, 38}, {37, 20752}, {55, 846}, {63, 5348}, {201, 7354}, {343, 21920}, {354, 1736}, {495, 1725}, {497, 7226}, {756, 5432}, {774, 15888}, {1393, 3614}, {1776, 3920}, {2167, 2602}, {2292, 10950}, {2310, 3058}, {2361, 3219}, {2801, 16577}, {3074, 9630}, {3670, 17606}, {3989, 14547}, {4392, 10589}, {5492, 5903}, {8679, 21318}, {9414, 24448}, {11031, 17602}, {24326, 24454}

### X(24432) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24432) lies on these lines: {1, 2}, {3, 7235}, {56, 24371}, {58, 1733}, {740, 1792}, {23922, 24953}, {24355, 24435}

### X(24433) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24433) lies on these lines: {1, 6}, {2, 24431}, {8, 2397}, {10, 522}, {21, 4570}, {38, 1647}, {65, 24029}, {212, 9639}, {244, 6667}, {528, 2310}, {756, 6690}, {774, 12607}, {942, 24405}, {982, 24397}, {986, 24338}, {1376, 24430}, {1725, 17757}, {1735, 5123}, {2801, 16578}, {2810, 17463}, {2836, 21362}, {2886, 7069}, {3035, 7004}, {3036, 24028}, {3580, 21920}, {3670, 24399}, {3754, 5492}, {3976, 24418}, {4026, 25024}, {4424, 24429}, {4429, 25005}, {4640, 23703}, {4694, 24871}, {5794, 18340}, {6211, 20872}, {8758, 17615}, {17256, 24490}, {19945, 24443}, {20962, 21326}, {20989, 21368}

### X(24434) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24434) lies on these lines: {1, 6597}, {2, 24431}, {38, 2886}, {756, 3035}, {984, 1376}, {1001, 24430}, {1040, 15296}, {1736, 3742}, {1818, 3989}, {2550, 7226}, {2801, 16579}, {3816, 7069}, {3878, 5492}, {6690, 7004}, {9959, 23846}, {11031, 17061}

### X(24435) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)<

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

X(24435) lies on these lines: {1, 142}, {2, 21675}, {9, 22003}, {48, 75}, {56, 4361}, {63, 18653}, {100, 21231}, {141, 9317}, {144, 18661}, {239, 2260}, {284, 18698}, {354, 4852}, {966, 24268}, {1376, 24365}, {1474, 18697}, {1761, 14953}, {2173, 11683}, {2194, 4418}, {2287, 8680}, {2294, 16054}, {2360, 4647}, {2646, 3739}, {2893, 4466}, {3187, 17490}, {3419, 6739}, {3958, 8822}, {4039, 24362}, {4360, 24378}, {4371, 24363}, {4483, 14210}, {5086, 20305}, {5271, 6350}, {6741, 23902}, {16713, 17136}, {17043, 24390}, {17045, 24366}, {17052, 24780}, {17119, 20991}, {18162, 20880}, {24355, 24432}, {24384, 24953}

### X(24436) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24436) lies on these lines: {1, 2836}, {3, 7609}, {9, 1030}, {21, 24697}, {22, 7262}, {23, 896}, {25, 4650}, {35, 4557}, {36, 238}, {55, 846}, {56, 7312}, {58, 3122}, {999, 5429}, {1046, 20831}, {1473, 17063}, {1633, 24715}, {1757, 20872}, {2112, 24484}, {2607, 4459}, {2792, 7427}, {3751, 20468}, {3929, 7298}, {4068, 23861}, {4655, 17522}, {5150, 24482}, {5284, 6536}, {5358, 24851}, {6186, 13486}, {7193, 8540}, {7280, 15601}, {7293, 17123}

### X(24437) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24437) lies on these lines: {1, 6}, {2, 21699}, {8, 4446}, {10, 24520}, {38, 3571}, {69, 4443}, {75, 670}, {86, 3728}, {141, 24575}, {190, 22167}, {239, 4022}, {256, 524}, {291, 594}, {314, 714}, {319, 3778}, {668, 21238}, {700, 17157}, {846, 4068}, {982, 4361}, {1030, 8298}, {1045, 16696}, {1654, 3122}, {2228, 17287}, {2664, 22271}, {3123, 17273}, {3670, 4716}, {3764, 17363}, {3961, 20990}, {4392, 17162}, {4445, 4484}, {4735, 17372}, {4941, 17255}, {6542, 21035}, {6646, 24338}, {7170, 9457}, {12329, 16877}, {12782, 17299}, {16556, 20474}, {16738, 25295}, {17065, 17275}, {17135, 17148}, {17253, 24456}, {17256, 22172}, {17277, 21330}, {17289, 20456}, {17362, 24478}, {17365, 24463}

### X(24438) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24438) lies on these lines: {1, 474}, {8, 4446}, {2292, 24450}, {2664, 22299}, {8256, 24643}

### X(24439) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24439) lies on these lines: {1, 142}, {75, 24378}, {982, 4361}, {4657, 24366}, {5271, 17148}

### X(24440) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24440) lies on these lines: {1, 474}, {2, 4642}, {4, 9365}, {8, 982}, {9, 16605}, {10, 75}, {12, 17889}, {21, 17601}, {36, 2933}, {38, 3617}, {40, 238}, {43, 65}, {46, 4650}, {56, 1054}, {57, 9363}, {72, 6048}, {100, 3924}, {145, 244}, {279, 291}, {386, 3754}, {484, 1724}, {498, 24161}, {517, 978}, {518, 21896}, {519, 3976}, {750, 17016}, {899, 3869}, {958, 17596}, {960, 16569}, {988, 9623}, {1046, 2640}, {1086, 12607}, {1104, 3550}, {1149, 3885}, {1201, 14923}, {1220, 3980}, {1329, 3944}, {1574, 3735}, {1575, 3061}, {1697, 5272}, {1698, 4424}, {1707, 5128}, {1772, 10573}, {2176, 21888}, {2275, 4051}, {2276, 21951}, {2292, 4903}, {2550, 24478}, {2551, 24248}, {2650, 3240}, {2810, 17114}, {2899, 8421}, {3057, 16610}, {3086, 24028}, {3120, 11681}, {3169, 20227}, {3208, 3290}, {3214, 3868}, {3216, 4674}, {3244, 24168}, {3293, 5902}, {3339, 3751}, {3501, 16583}, {3623, 9335}, {3625, 24167}, {3632, 3953}, {3633, 4694}, {3666, 3698}, {3670, 3679}, {3730, 16611}, {3755, 17065}, {3782, 21031}, {3840, 4673}, {3871, 17715}, {3878, 17749}, {3914, 24982}, {4073, 17872}, {4392, 4678}, {4414, 5260}, {4415, 9711}, {4696, 17155}, {4714, 10479}, {4850, 10459}, {4859, 6706}, {4882, 16496}, {5121, 12053}, {5143, 13738}, {5250, 17123}, {5261, 7613}, {5262, 17716}, {5264, 16478}, {5293, 9709}, {5529, 5730}, {5552, 17719}, {6735, 23536}, {6736, 24177}, {6762, 18193}, {7991, 23511}, {8582, 24210}, {9364, 21147}, {16284, 24215}, {17145, 20047}, {17451, 17756}, {18178, 18792}, {19582, 24003}, {20271, 20691}, {21935, 24991}, {24325, 24377}

### X(24441) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24441) lies on these lines: {1, 4715}, {2, 45}, {6, 17247}, {9, 17323}, {37, 7232}, {144, 17045}, {192, 4445}, {320, 16672}, {514, 23352}, {519, 4643}, {524, 3241}, {527, 551}, {536, 984}, {537, 24357}, {597, 6172}, {599, 742}, {3247, 17345}, {3644, 17252}, {3662, 16675}, {3663, 17259}, {3672, 17332}, {3729, 17327}, {3731, 17235}, {3763, 17249}, {3821, 3828}, {3834, 16676}, {4357, 17262}, {4361, 17246}, {4384, 16590}, {4393, 17488}, {4428, 20834}, {4484, 24456}, {4659, 4708}, {4665, 4748}, {4670, 25055}, {4677, 4690}, {4681, 17272}, {4698, 4862}, {4704, 17273}, {4718, 17270}, {4755, 6173}, {4762, 24457}, {4798, 19883}, {4908, 17237}, {4912, 24406}, {5296, 7263}, {5308, 7238}, {5851, 10186}, {6144, 17393}, {6646, 16777}, {6707, 7222}, {11235, 24424}, {15533, 17389}, {15668, 17276}, {16673, 17376}, {16674, 17300}, {16677, 17234}, {16814, 17304}, {16884, 17347}, {16885, 17302}, {17118, 17248}, {17119, 17256}, {17236, 17267}, {17264, 21358}, {17307, 25269}, {17319, 17329}, {17321, 17334}, {17324, 17336}, {17790, 18146}, {24408, 25378}

### X(24442) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24442) lies on these lines: {1, 826}, {894, 24488}, {3185, 4418}, {24346, 24347}, {24401, 24453}

### X(24443) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24443) lies on these lines: {1, 88}, {2, 986}, {3, 3924}, {5, 3120}, {6, 1406}, {8, 982}, {10, 38}, {12, 1086}, {21, 17596}, {31, 46}, {34, 9316}, {35, 4218}, {39, 3125}, {40, 614}, {42, 942}, {43, 3868}, {55, 17054}, {56, 2933}, {57, 961}, {58, 1325}, {63, 1722}, {65, 1193}, {72, 899}, {73, 18838}, {75, 1237}, {141, 20653}, {145, 3976}, {171, 5262}, {192, 22220}, {201, 3772}, {227, 1458}, {239, 24610}, {240, 5125}, {321, 3831}, {354, 4646}, {377, 11031}, {386, 2650}, {405, 4414}, {442, 20966}, {484, 595}, {496, 1647}, {498, 24159}, {517, 1201}, {518, 3214}, {519, 3953}, {580, 5535}, {581, 15016}, {612, 24163}, {672, 16583}, {726, 3701}, {740, 21330}, {748, 12514}, {756, 1698}, {758, 3216}, {774, 1210}, {846, 5047}, {896, 1724}, {902, 3579}, {958, 17595}, {960, 16610}, {964, 3980}, {975, 17124}, {976, 1376}, {978, 3869}, {984, 4699}, {988, 19860}, {995, 5903}, {1042, 1465}, {1104, 1155}, {1107, 21951}, {1125, 4424}, {1149, 3057}, {1329, 3782}, {1334, 3290}, {1388, 8572}, {1401, 16980}, {1403, 13738}, {1451, 1454}, {1574, 3954}, {1575, 3721}, {1697, 5573}, {1706, 3677}, {1737, 21935}, {1738, 3778}, {1788, 4000}, {1837, 7004}, {1838, 2181}, {1930, 24170}, {1959, 24598}, {2140, 24786}, {2170, 2275}, {2171, 2277}, {2269, 20227}, {2276, 20271}, {2294, 4261}, {2310, 9581}, {2476, 17889}, {2478, 24248}, {2999, 3339}, {3008, 24633}, {3011, 6684}, {3178, 18139}, {3218, 5247}, {3244, 4694}, {3293, 3874}, {3555, 17449}, {3616, 17063}, {3617, 4392}, {3622, 9335}, {3660, 4322}, {3663, 8582}, {3666, 3812}, {3681, 6048}, {3691, 16605}, {3696, 4022}, {3698, 4003}, {3702, 3840}, {3720, 3931}, {3724, 16453}, {3726, 20691}, {3727, 16604}, {3743, 3833}, {3753, 10459}, {3755, 21346}, {3764, 5880}, {3821, 5051}, {3826, 21035}, {3876, 16569}, {3877, 21214}, {3913, 17597}, {3923, 5192}, {3936, 17748}, {3938, 5687}, {3944, 4193}, {3992, 24068}, {4284, 16547}, {4296, 9364}, {4310, 7080}, {4346, 8165}, {4385, 17155}, {4418, 13740}, {4429, 4446}, {4645, 24478}, {5011, 5299}, {5046, 24851}, {5128, 7290}, {5177, 7613}, {5249, 5530}, {5250, 5272}, {5254, 21044}, {5255, 7191}, {5264, 17469}, {5283, 21921}, {5297, 24164}, {5396, 5885}, {5445, 17734}, {5563, 15955}, {5692, 17749}, {5711, 17017}, {5904, 21805}, {6187, 23850}, {6650, 16044}, {7069, 17606}, {7264, 21208}, {8056, 8583}, {8728, 21674}, {9310, 9620}, {9317, 9413}, {10479, 21020}, {11319, 24850}, {11512, 19861}, {11533, 17535}, {12526, 23511}, {12579, 14020}, {12702, 16483}, {13161, 24982}, {15048, 21950}, {16478, 17126}, {16549, 16600}, {16552, 16611}, {16700, 18178}, {16720, 25350}, {16824, 24627}, {16825, 24464}, {17053, 17452}, {17114, 23154}, {17145, 20051}, {17187, 18180}, {17495, 17751}, {17541, 17738}, {17750, 21840}, {17758, 24185}, {17863, 24173}, {19945, 24433}, {19951, 24413}, {20284, 20707}, {20360, 24530}, {20366, 25141}, {21342, 21896}, {21725, 23929}, {23661, 24218}, {23675, 24171}, {23928, 24923}, {24178, 24987}, {24372, 24575}

### X(24444) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24444) lies on these lines: {24337, 24424}

### X(24445) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24445) lies on these lines: {1, 15076}, {7, 17446}, {55, 24822}, {69, 17872}, {75, 23682}, {335, 24351}, {982, 1086}, {984, 4363}, {986, 5880}, {3959, 9025}, {4073, 4446}, {4459, 4493}, {7184, 18161}, {17063, 17290}, {18252, 20227}, {20930, 23688}, {24248, 24424}

### X(24446) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24446) lies on these lines: {1, 442}, {2, 21692}, {199, 8301}, {1376, 24365}, {2160, 2319}, {2606, 9451}, {3187, 3873}, {4362, 4412}

### X(24447) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24447) lies on these lines: {1, 512}, {38, 1755}, {55, 16378}, {86, 1621}, {659, 9318}, {894, 24492}, {7225, 9316}, {20347, 20475}, {24346, 24347}

### X(24448) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24448) lies on these lines: {2, 24426}, {38, 190}, {291, 24330}, {984, 4376}, {3571, 24326}, {4363, 4493}, {9414, 24431}

### X(24449) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24449) lies on these lines: {1, 2389}, {2, 24396}, {982, 1086}, {8255, 17056}, {15726, 21856}

### X(24450) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24450) lies on these lines: {2, 3122}, {10, 17790}, {37, 1045}, {45, 846}, {86, 3728}, {190, 756}, {256, 1213}, {274, 714}, {291, 4472}, {984, 4363}, {1046, 5220}, {1654, 21699}, {1740, 17038}, {2292, 24438}, {2663, 22271}, {3571, 9458}, {4364, 24338}, {4427, 9330}, {4492, 24456}, {5263, 24425}, {5283, 24696}, {5988, 25359}, {6007, 16589}, {17398, 24575}, {18140, 24688}

### X(24451) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24451) lies on these lines: {2, 3123}, {8, 9025}, {75, 7155}, {190, 1376}, {192, 1740}, {1281, 24334}, {3242, 4363}, {3952, 24344}, {4110, 25311}, {4414, 9458}, {4443, 25382}, {4454, 17794}, {4518, 24341}, {6382, 18830}, {17790, 24717}

### X(24452) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

Barycentrics    4 a^3 + 3 a b^2 - 2 b^3 + 9 a b c + 6 b^2 c + 3 a c^2 + 6 b c^2 - 2 c^3 : :

X(24452) lies on these lines: {2, 4432}, {9, 484}, {75, 519}, {545, 984}, {1654, 7229}, {3679, 4363}, {3828, 17354}, {4859, 15668}

### X(24453) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24453) lies on these lines: {1, 23877}, {894, 24498}, {3716, 9318}, {24401, 24442}

### X(24454) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24454) lies on these lines: {543, 4736}, {620, 7266}, {984, 3571}, {2786, 13277}, {4376, 9414}, {4971, 23928}, {24326, 24431}

### X(24455) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24455) lies on these lines: {1, 514}, {2, 2140}, {101, 2481}, {517, 25368}, {894, 24491}, {993, 24331}, {3570, 17143}, {4251, 6654}, {6684, 25342}, {17761, 18785}, {19974, 24514}, {21616, 24318}, {24346, 24401}

### X(24456) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24456) lies on these lines: {1, 4503}, {2, 3123}, {37, 19584}, {75, 4941}, {244, 25378}, {256, 17321}, {291, 4419}, {982, 3122}, {984, 4026}, {1086, 3816}, {2228, 4664}, {3662, 22172}, {3663, 17065}, {3672, 24478}, {3764, 17320}, {3778, 17247}, {4022, 17249}, {4110, 25121}, {4363, 24338}, {4446, 17246}, {4484, 24441}, {4492, 24450}, {4699, 22174}, {5988, 25369}, {17236, 21330}, {17238, 22167}, {17253, 24437}, {17257, 24575}, {17333, 20456}, {17396, 23659}, {17793, 24351}, {19945, 25377}, {24319, 24341}, {24329, 24428}

### X(24457) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24457) lies on these lines: {1, 513}, {10, 522}, {11, 244}, {44, 1643}, {45, 650}, {521, 10912}, {523, 2292}, {661, 14437}, {693, 4389}, {946, 3667}, {984, 4777}, {1283, 4491}, {1491, 14434}, {1854, 15313}, {2787, 14286}, {2804, 21112}, {2827, 11715}, {3242, 9001}, {3309, 7986}, {3445, 6129}, {3649, 4017}, {3663, 23810}, {3669, 15306}, {3900, 11525}, {4762, 24441}, {4885, 17290}, {4926, 15079}, {4977, 21105}, {5542, 6006}, {6163, 9268}, {8641, 23855}, {14205, 15726}, {16892, 24110}, {21133, 24098}, {23809, 24184}

X(24457) = intouch-to-excentral similarity image of X(5)

### X(24458) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24458) lies on these lines: 2, 3123}, {75, 20598}, {192, 2227}, {291, 24248}, {350, 4941}, {984, 4493}, {3571, 4376}, {3662, 23462}, {4443, 25349}, {4713, 24338}, {7148, 20081}

### X(24459) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24459) lies on these lines: {525, 656}, {647, 14208}, {659, 812}, {905, 15411}, {1565, 3942}, {1577, 3709}, {3265, 21107}, {4086, 23878}, {4391, 21225}, {4643, 9034}, {6563, 21117}, {7212, 16591}, {7650, 23882}

### X(24460) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24460) lies on these lines: {1, 514}, {2, 1334}, {41, 6654}, {894, 24509}, {2481, 9317}, {3570, 17144}, {24419, 24421}

### X(24461) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24461) lies on these lines: {1, 4777}, {894, 16494}, {900, 14191}, {1001, 3923}, {4448, 9318}, {24416, 24428}

### X(24462) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24462) lies on these lines: {1, 926}, {10, 20525}, {38, 4453}, {43, 3310}, {171, 654}, {240, 522}, {659, 17990}, {876, 6373}, {918, 984}, {982, 1638}, {2254, 2786}, {2488, 25084}, {2530, 3805}, {3123, 21208}, {3550, 6139}, {3709, 4040}, {4443, 14315}, {4502, 6005}, {9511, 17596}, {21010, 23188}

### X(24463) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24463) lies on these lines: {1, 6007}, {2, 3123}, {38, 4440}, {75, 256}, {190, 24259}, {291, 4363}, {560, 8296}, {894, 20456}, {984, 2550}, {987, 19844}, {1582, 8424}, {2277, 16571}, {3728, 6646}, {3778, 17116}, {3961, 21320}, {4022, 7321}, {4364, 24338}, {4389, 4947}, {4446, 17118}, {4471, 8297}, {4659, 12782}, {4941, 17321}, {5988, 24319}, {17117, 23659}, {17300, 22167}, {17305, 19945}, {17365, 24437}, {17790, 24688}, {24323, 24345}, {25347, 25382}

### X(24464) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = OBVERSE TRIANGLE OF X(1)

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

X(24464) lies on these lines: {1, 3}, {10, 3765}, {38, 3980}, {213, 1759}, {244, 24331}, {292, 3735}, {752, 3764}, {758, 869}, {984, 4363}, {1107, 3753}, {1739, 4384}, {2664, 5692}, {3778, 4660}, {4283, 4429}, {4392, 24344}, {4443, 24715}, {4674, 16975}, {4703, 21936}, {5283, 16549}, {6327, 20966}, {16823, 24046}, {16825, 24443}

### X(24465) =  X(7)X(100)∩X(11)X(57)

Barycentrics    (2*a^4-2*(b+c)*a^3-(b^2-4*b*c+c^2)*a^2+(b^2-c^2)^2)*(a+b-c)*(a-b+c) : :
X(24465) = X(11)+3*X(11246), 3*X(65)+X(18976), 3*X(553)-X(5083)

See Antreas Hatzipolakis, César Lozada, and Ercole Suppa, Hyacinthos 28398 and Hyacinthos 28399.

X(24465) lies on these lines: {5, 7702}, {7, 100}, {11, 57}, {30, 18838}, {56, 1387}, {65, 952}, {80, 3339}, {109, 1086}, {149, 21454}, {214, 3671}, {226, 3035}, {388, 1145}, {516, 3660}, {528, 553}, {938, 10724}, {942, 5840}, {1155, 5762}, {1317, 3340}, {1320, 3600}, {1466, 10090}, {1470, 11729}, {1537, 4295}, {1617, 3474}, {2099, 11046}, {2802, 4298}, {2829, 4292}, {2834, 3937}, {3036, 4848}, {3218, 5857}, {3333, 14217}, {3336, 8068}, {3337, 5533}, {3361, 16173}, {3911, 5087}, {4355, 5541}, {4440, 14594}, {4654, 6174}, {4860, 13274}, {5221, 12019}, {5708, 10738}, {5854, 10106}, {6147, 10044}, {7972, 18421}, {9945, 12739}, {10404, 10956}, {10742, 18541}, {11529, 12119}, {14151, 20095}, {15803, 21154}, {17579, 18419}, {17724, 23703}

X(24465) = midpoint of X(4292) and X(12736)
X(24465) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5221, 13273, 12832), (12832, 13273, 12019)

### X(24466) =  MIDPOINT OF X(20) AND X(100)

Barycentrics    -4 a^7+4 a^6 (b+c)-a (b-c)^4 (b+c)^2+(b-c)^4 (b+c)^3+a^5 (7 b^2-10 b c+7 c^2)-a^4 (7 b^3+b^2 c+b c^2+7 c^3)-2 a^3 (b^4-4 b^3 c+2 b^2 c^2-4 b c^3+c^4) +2 a^2 (b^5-b^4 c-b c^4+c^5) : :
X(24466) = 3*X[2]-X[10724], 2*X[3]-X[11], X[4]-2*X[3035], X[20]+X[100], X[80]-3*X[165], X[104]-3*X[376], 2*X[140]-X[22938], X[149]-5*X[3522], 5*X[631]-4*X[6667], X[1320]-3*X[5731], 2*X[1387]-3*X[3576], X[1484]-3*X[8703], X[1657]+X[10742], 2*X[3036]-3*X[5657], X[3146]-4*X[20400], X[3529]+X[10728], X[4316]+X[5537], X[5538]+X[15228]

See Ercole Suppa, Hyacinthos 28400.

X(24466) lies on these lines: {2,10724}, {3,11}, {4,3035}, {20,100}, {30,119}, {40,550}, {55,6948}, {80,165}, {104,376}, {120,7427}, {140,22938}, {149,3522}, {214,516}, {411,17100}, {515,1145}, {517,1317}, {548,11012}, {631,6667}, {944,5854}, {971,6068}, {1001,6955}, {1155,12743}, {1320,5731}, {1350,5848}, {1376,6938}, {1387,3576}, {1484,8703}, {1657,10742}, {1753,1862}, {2771,16111}, {2800,10609}, {2802,4297}, {2803,3184}, {2806,14689}, {2886,6950}, {2932,7580}, {2950,10860}, {2951,6282}, {3036,5657}, {3058,10269}, {3146,20400}, {3529,10728}, {3534,6244}, {3925,6914}, {4188,7681}, {4190,11496}, {4299,10087}, {4304,12736}, {4314,18240}, {4316,5537}, {4333,5812}, {4413,6930}, {4421,12115}, {4880,5844}, {4996,5842}, {5010,6907}, {5204,13274}, {5217,6850}, {5326,6980}, {5432,6923}, {5434,10679}, {5533,7280}, {5538,15228}, {5732,5856}, {5759,5851}, {6200,13913}, {6224,9778}, {6246,6684}, {6361,10698}, {6690,6951}, {6702,10164}, {6827,12764}, {6891,12953}, {6921,10893}, {6925,12761}, {6934,12775}, {6961,10896}, {7354,10956}, {7680,17579}, {7957,17660}, {7972,7991}, {7982,12735}, {7987,16173}, {8227,17563}, {8674,16163}, {8735,22055}, {9541,19082}, {9803,9963}, {10074,22770}, {10165,16174}, {10167,15528}, {10265,12690}, {10304,10707}, {10711,11001}, {10767,15035}, {10768,21166}, {10778,15055}, {11698,15704}, {11729,12699}, {11849,15888}, {12116,13271}, {12248,17538}, {12773,15696}, {13268,16190}, {14740,14872}

X(24466) = midpoint of X(i) and X(j) for these {i,j}: {20,100}, {40,12119}, {104,13199}, {1657,10742}, {3529,10728}, {4316,5537}, {5538,15228}, {6361,10698}, {7957,17660}, {7972,7991}, {9803,9963}, {10711,11001}, {11698,15704}
X(24466) = reflection of X(i) in X(j) for these {i,j}: {4,3035}, {11,3}, {149,20418}, {1537,214}, {6154,10993}, {6246,6684}, {6326,9945}, {7982,12735}, {10738,6713}, {12690,10265}, {12699,11729}, {14217,1387}, {14872,14740}, {22938,140}
X(24466) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,11,21154}, {3,10525,5433}, {3,10738,6713}, {3,11826,15908}, {20,10310,11827}, {376,13199,104}, {1155,12743,12832}, {3576,14217,1387}, {5010,6907,21155}, {6713,10738,11}

### X(24467) =  X(3)X(63)∩X(5)X(57)

Barycentrics    a*(-a^2+b^2+c^2)*(a^4-2*(b^2-b*c+c^2)*a^2+(b^2-c^2)^2) : :
X(24467) = X(4)-3*X(5770), 2*R*X(5)-(2*R-r)*X(57), X(84)+3*X(3928), 3*X(165)-X(5534), 3*X(3928)-X(5709)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28401.

X(24467) lies on these lines: {1, 1399}, {3, 63}, {4, 3218}, {5, 57}, {7, 6824}, {8, 6948}, {9, 140}, {11, 90}, {12, 17700}, {26, 3220}, {30, 84}, {36, 5693}, {38, 601}, {40, 550}, {46, 355}, {56, 920}, {65, 22758}, {104, 3869}, {144, 6926}, {155, 222}, {165, 5534}, {191, 3576}, {226, 6862}, {255, 1062}, {329, 6891}, {392, 16203}, {405, 10202}, {411, 13243}, {484, 5881}, {499, 7082}, {513, 8279}, {517, 1158}, {518, 11248}, {527, 6705}, {546, 18540}, {548, 3587}, {549, 3929}, {602, 896}, {603, 1060}, {631, 3219}, {632, 7308}, {758, 5450}, {908, 6958}, {938, 6930}, {942, 3560}, {960, 10269}, {971, 6985}, {982, 3073}, {993, 5884}, {997, 5694}, {999, 12709}, {1001, 13373}, {1147, 7193}, {1155, 11499}, {1181, 22129}, {1210, 6929}, {1216, 3784}, {1364, 6238}, {1385, 5289}, {1407, 17814}, {1445, 5779}, {1454, 1478}, {1465, 8757}, {1482, 4018}, {1483, 1697}, {1484, 9614}, {1565, 7183}, {1656, 3306}, {1699, 7701}, {1708, 5777}, {1709, 10943}, {1737, 18961}, {1776, 3086}, {1858, 8071}, {2003, 12161}, {2077, 5904}, {2096, 6850}, {2323, 16266}, {2771, 6261}, {2800, 8666}, {2801, 6796}, {3072, 4650}, {3091, 23958}, {3157, 17102}, {3305, 3526}, {3333, 5901}, {3336, 5587}, {3337, 8227}, {3338, 3649}, {3358, 5762}, {3359, 5690}, {3487, 6892}, {3555, 10679}, {3564, 7289}, {3628, 5437}, {3651, 11220}, {3781, 5447}, {3811, 12341}, {3868, 6906}, {3870, 11849}, {3876, 6940}, {3911, 6959}, {3937, 5562}, {4292, 6917}, {4293, 7098}, {4640, 10267}, {4995, 7162}, {5223, 10270}, {5248, 12005}, {5249, 6861}, {5250, 10246}, {5251, 15016}, {5273, 6989}, {5435, 5811}, {5442, 5660}, {5535, 5691}, {5704, 6973}, {5708, 6913}, {5714, 6859}, {5720, 6924}, {5744, 6825}, {5761, 6935}, {5768, 6868}, {5780, 16417}, {5789, 18541}, {5844, 6762}, {5851, 18243}, {5905, 6833}, {6001, 11249}, {6326, 7280}, {6734, 6923}, {6846, 21454}, {6847, 9965}, {6869, 9799}, {6875, 18444}, {6883, 9940}, {6887, 9776}, {6888, 17483}, {6890, 20078}, {6905, 12528}, {6922, 13226}, {6938, 12649}, {6972, 17484}, {7686, 18761}, {10200, 15297}, {10461, 15952}, {10525, 10916}, {10526, 12616}, {10680, 12672}, {10785, 11415}, {10942, 21031}, {11012, 15071}, {12532, 18861}, {12705, 22791}, {13465, 19861}, {15888, 17699}, {16574, 19543}, {18732, 22659}, {19549, 21371}

X(24467) = midpoint of X(i) and X(j) for these {i,j}: {84, 5709}, {6869, 9799}
X(24467) = reflection of X(i) in X(j) for these (i,j): (10525, 10916), (10526, 12616), (12699, 10943)
X(24467) = X(156)-of-excentral triangle
X(24467) = X(13561)-of-6th-mixtilinear-triangle
X(24467) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (40, 7171, 550), (40, 10085, 18481), (57, 7330, 5), (84, 3928, 5709), (90, 17437, 11), (255, 7004, 1062), (1155, 14872, 11499), (1709, 12704, 12699), (1768, 6763, 40), (3587, 9841, 548), (4640, 12675, 10267), (4652, 18446, 3), (5435, 5811, 6944), (5720, 15803, 6924), (10884, 21165, 3)

### X(24468) =  X(1)X(3)∩X(9)X(6990)

Barycentrics    a*(a^6-(b+c)*a^3*b*c-3*(b^2+b*c+c^2)*a^4+(b^2-c^2)*(b-c)*a*b*c+(b^2+b*c+c^2)*(3*b^2-2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b-c)^2) : :
X(24468) = 3*X(40)-2*X(11010), 3*X(165)-2*X(11849)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28401.

X(24468) lies on these lines: {1, 3}, {9, 6990}, {10, 6900}, {30, 6763}, {48, 16553}, {63, 2894}, {191, 12699}, {920, 9580}, {946, 6884}, {1732, 1766}, {3218, 20066}, {3219, 18483}, {3651, 3874}, {3868, 16132}, {4654, 14526}, {5127, 5358}, {5250, 15674}, {5587, 6894}, {5762, 15908}, {5904, 6985}, {6284, 16113}, {7098, 10624}, {7289, 9047}, {7411, 12005}, {10021, 16139}, {11499, 15104}, {12248, 12625}, {16127, 20078}

X(24468) = reflection of X(7982) in X(11014)
X(24468) = X(35)-of-tangential-of-excentral-triangle
X(24468) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (40, 5709, 5535), (40, 6766, 12703), (40, 12704, 3576), (484, 12702, 40), (2095, 5584, 15016), (11009, 11010, 1697)

### X(24469) =  X(9)X(114)∩X(57)X(98)

Barycentrics    a*(a^10-2*(2*b^2+b*c+2*c^2)*a^8+(7*b^4+7*c^4-(2*b^2-7*b*c+2*c^2)*b*c)*a^6-(7*b^6+7*c^6-2*(2*b^4+2*c^4-(2*b^2-3*b*c+2*c^2)*b*c)*b*c)*a^4+(4*b^6+4*c^6+(6*b^4+6*c^4+(7*b^2+6*b*c+7*c^2)*b*c)*b*c)*(b-c)^2*a^2-(b^2-c^2)^2*(b^6+c^6-2*(b^4+b^2*c^2+c^4)*b*c)) : :
X(24469) = 3*X(165)-2*X(12178)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28401.

X(24469) lies on these lines: {1, 22504}, {9, 114}, {40, 99}, {46, 9860}, {57, 98}, {63, 147}, {84, 2794}, {165, 12178}, {542, 3928}, {1697, 7970}, {1709, 12182}, {2782, 5709}, {3218, 5984}, {3220, 9861}, {3333, 11710}, {3929, 6054}, {5437, 6036}, {6033, 7330}, {6769, 13173}, {7061, 7350}, {12514, 21636}, {18540, 22505}

X(24469) = reflection of X(i) in X(j) for these (i,j): (1, 22504), (6769, 13173)
X(24469) = X(22504)-of-Aquila-triangle

### X(24470) =  X(3)X(7)∩X(5)X(57)

Barycentrics    2*a^4+2*(b+c)*a^3-(b^2-4*b*c+c^2)*a^2-2*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(24470) = X(1)+3*X(11246), 3*X(354)+X(1770), 3*X(354)-X(15171), 3*X(553)-X(942), 9*X(553)-X(950), 3*X(553)+X(4292), 6*X(553)-X(12433), 3*X(942)-X(950), X(950)+3*X(4292), 2*X(950)-3*X(12433), X(3868)+3*X(11112), 2*X(4292)+X(12433), 3*X(5049)-X(10624), 3*X(5434)+X(5903), 3*X(5902)+X(7354)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28401.

X(24470) lies on these lines: {1, 550}, {3, 7}, {4, 5708}, {5, 57}, {11, 79}, {12, 3336}, {20, 15934}, {30, 553}, {36, 3649}, {40, 4355}, {46, 495}, {56, 5901}, {58, 1086}, {63, 8728}, {65, 952}, {78, 17563}, {84, 5805}, {140, 226}, {144, 17582}, {273, 7546}, {329, 16408}, {354, 1770}, {355, 3339}, {376, 11036}, {382, 938}, {386, 17365}, {388, 5690}, {390, 3296}, {404, 17483}, {442, 3218}, {443, 3927}, {474, 5905}, {484, 15888}, {496, 1836}, {516, 5045}, {517, 4298}, {527, 5044}, {528, 3881}, {529, 3754}, {540, 24176}, {545, 3159}, {546, 1210}, {548, 4114}, {549, 4654}, {596, 5846}, {632, 5219}, {944, 1159}, {946, 20418}, {962, 7373}, {975, 17276}, {990, 8144}, {999, 4295}, {1056, 12702}, {1125, 17235}, {1155, 13407}, {1385, 3671}, {1387, 5563}, {1399, 15253}, {1407, 5707}, {1420, 10283}, {1434, 1565}, {1466, 6924}, {1478, 5221}, {1479, 4860}, {1482, 3600}, {1483, 3340}, {1595, 1892}, {1656, 5435}, {1657, 3488}, {1788, 9654}, {1876, 6756}, {2094, 17528}, {2095, 6850}, {2099, 4317}, {2476, 23958}, {2646, 11551}, {3073, 15251}, {3075, 15252}, {3219, 17529}, {3295, 3474}, {3306, 17527}, {3333, 4312}, {3361, 5886}, {3526, 5226}, {3530, 3982}, {3534, 4313}, {3579, 21620}, {3585, 12019}, {3601, 8703}, {3616, 17571}, {3627, 5722}, {3628, 3911}, {3648, 5284}, {3662, 17698}, {3678, 5852}, {3824, 5745}, {3845, 9581}, {3851, 5704}, {3868, 11112}, {3916, 5249}, {3925, 6763}, {3928, 5791}, {3937, 18180}, {3940, 6904}, {3947, 11231}, {4252, 24159}, {4303, 5453}, {4304, 12103}, {4308, 10247}, {4315, 23339}, {4316, 10543}, {4325, 5425}, {4338, 12701}, {4757, 5855}, {4880, 21677}, {4973, 4999}, {5049, 10624}, {5253, 14450}, {5260, 9782}, {5266, 24231}, {5267, 11281}, {5298, 5443}, {5326, 5442}, {5434, 5903}, {5586, 11529}, {5735, 9841}, {5777, 5843}, {5779, 6864}, {5789, 6843}, {5840, 6583}, {5842, 12005}, {5844, 10106}, {5857, 12609}, {5902, 7354}, {6284, 18398}, {6361, 6767}, {6831, 13226}, {6887, 8732}, {6894, 13243}, {6915, 13257}, {7091, 12700}, {7682, 22792}, {9655, 18391}, {9657, 10573}, {9776, 11108}, {10593, 17728}, {11009, 12735}, {11019, 22793}, {11041, 18526}, {11496, 20330}, {11518, 15704}, {11544, 12047}, {11827, 15016}, {13374, 18260}, {14054, 17616}, {15170, 17609}, {15681, 15933}, {16056, 22458}, {16415, 20805}, {16863, 18228}, {17484, 17531}, {17580, 20059}

X(24470) = midpoint of X(i) and X(j) for these {i,j}: {65, 18990}, {942, 4292}, {1770, 15171}
X(24470) = reflection of X(i) in X(j) for these (i,j): (5044, 12436), (12433, 942), (15172, 5045)
X(24470) = X(143)-of-intouch-triangle
X(24470) = X(5901)-of-2nd-anti-circumperp-tangential-triangle
X(24470) = X(6101)-of-inverse-in-incircle-triangle
X(24470) = X(10263)-of-incircle-circles-triangle
X(24470) = excentral-to-intouch similarity image of X(5)
X(24470) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 7, 6147), (3, 6147, 5719), (4, 21454, 5708), (46, 10404, 495), (79, 3337, 11), (354, 1770, 15171), (443, 9965, 3927), (553, 4292, 942), (999, 4295, 22791), (1836, 3338, 496), (4654, 15803, 11374), (5435, 5714, 1656), (5557, 15228, 1), (5708, 18541, 4), (11374, 15803, 549)

### X(24471) =  X(6)X(57)∩X(7)X(8)

Barycentrics    a*((b+c)*a+b^2+c^2)*(a+b-c)*(a-b+c) : :
X(24471) = 3*X(354)-X(3056)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28401.

X(24471) lies on these lines: {1, 1350}, {6, 57}, {7, 8}, {12, 3844}, {37, 1423}, {46, 611}, {56, 77}, {60, 757}, {72, 17272}, {86, 1431}, {141, 226}, {181, 9436}, {193, 17490}, {210, 5232}, {241, 1400}, {273, 1875}, {279, 959}, {307, 3665}, {354, 3056}, {511, 942}, {513, 21202}, {517, 3663}, {524, 553}, {599, 4654}, {613, 3338}, {674, 5173}, {742, 4032}, {760, 18252}, {946, 24213}, {960, 3674}, {1100, 1429}, {1108, 18161}, {1155, 2330}, {1210, 5480}, {1284, 15569}, {1319, 1442}, {1351, 5708}, {1357, 1366}, {1358, 2836}, {1401, 9025}, {1432, 3863}, {1466, 7013}, {1470, 1804}, {1503, 4292}, {1834, 5929}, {1836, 12589}, {1837, 21279}, {1843, 1876}, {1887, 7282}, {2099, 7190}, {2257, 18725}, {2260, 3942}, {2262, 4000}, {2264, 7291}, {2269, 3666}, {2285, 6180}, {3057, 3672}, {3218, 15988}, {3242, 3340}, {3339, 3751}, {3361, 16475}, {3435, 7053}, {3487, 10519}, {3589, 3911}, {3618, 5435}, {3619, 5226}, {3629, 4031}, {3630, 4114}, {3631, 3982}, {3668, 3827}, {3673, 10446}, {3676, 9002}, {3739, 15985}, {3740, 5224}, {3763, 5219}, {3812, 10436}, {3875, 3880}, {4021, 9957}, {4260, 10481}, {4267, 4719}, {4269, 16696}, {4298, 5847}, {4341, 22769}, {4359, 15983}, {4419, 21871}, {4452, 14923}, {4662, 17270}, {4663, 5221}, {4731, 5936}, {4862, 5903}, {4888, 5902}, {4909, 5049}, {5083, 9024}, {5085, 15803}, {5092, 5122}, {5572, 21746}, {5846, 10106}, {6385, 18033}, {7023, 7177}, {7197, 14256}, {7269, 11011}, {7274, 18421}, {7686, 17861}, {8679, 20617}, {9004, 22277}, {9021, 15556}, {9037, 18838}, {9612, 10516}, {10360, 18935}, {10391, 18650}, {10456, 10477}, {10520, 12915}, {10914, 17151}, {12722, 15310}, {12723, 15726}, {13374, 24179}, {16603, 17239}, {17084, 17322}, {17189, 18180}, {17276, 21853}, {17705, 22464}, {18440, 18541}, {21239, 24005}

X(24471) = midpoint of X(i) and X(j) for these {i,j}: {65, 1469}, {17276, 21853}
X(24471) = X(53)-of-intouch triangle
X(24471) = X(1386)-of-2nd anti-circumperp-tangential triangle
X(24471) = X(20477)-of-inverse-in-incircle triangle
X(24471) = intouch-isogonal conjugate of X(4854)
X(24471) = excentral-to-intouch similarity image of X(6)
X(24471) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 2097, 7289), (7, 3212, 75), (7, 5933, 6604), (65, 1122, 7), (1423, 7146, 37)

### X(24472) =  X(1)X(99)∩X(57)X(98)

Barycentrics    (b+c)*a^7-(b-c)^2*a^6-2*(b^3+c^3)*a^5+(b^2+c^2)*(b-c)^2*a^4+(b+c)*(2*b^4+2*c^4-(2*b^2-b*c+2*c^2)*b*c)*a^3+(2*b^2+3*b*c+2*c^2)*(b-c)^2*b*c*a^2-(b^2-c^2)*(b-c)*(b^4-b^2*c^2+c^4)*a+(b^2-c^2)^2*b^2*c^2 : :
X(24472) = 3*X(354)-X(3027)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28401.

X(24472) lies on these lines: {1, 99}, {7, 147}, {46, 10053}, {56, 11710}, {57, 98}, {65, 1355}, {114, 226}, {115, 1210}, {148, 938}, {291, 23996}, {354, 3027}, {388, 9864}, {542, 553}, {620, 13411}, {942, 2782}, {950, 23698}, {1281, 11031}, {1836, 12185}, {1837, 13182}, {1876, 12131}, {2783, 5083}, {2784, 4298}, {2785, 12016}, {2786, 11028}, {2787, 12736}, {2794, 4292}, {2795, 10122}, {3218, 5985}, {3338, 10069}, {3339, 9860}, {3340, 7970}, {3488, 13172}, {3586, 10723}, {3601, 21166}, {3671, 21636}, {3911, 6036}, {4654, 6054}, {4697, 16598}, {5708, 12188}, {5722, 6321}, {5984, 21454}, {5988, 9436}, {8591, 15933}, {9579, 10722}, {9581, 14639}, {10072, 12258}, {10404, 12184}, {11019, 11599}, {11374, 15561}, {11518, 23235}, {13178, 18391}, {13188, 15934}

X(24472) = midpoint of X(65) and X(3023)
X(24472) = incircle-inverse of X(741)
X(24472) = center of the circle {{X(65), X(1356), X(3023)}}
X(24472) = X(129)-of-intouch-triangle
X(24472) = X(1298)-of-inverse-in-incircle-triangle
X(24472) = X(11710)-of-2nd anti-circumperp-tangential-triangle
X(24472) = {X(1), X(10089)}-harmonic conjugate of X(11711)

### X(24473) =  X(1)X(3052)∩X(2)X(72)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28407.

X(24473) lies on these lines: {1,3052}, {2,72}, {3,11520}, {7,3419}, {8,4004}, {9,17542}, {30,1071}, {46,4421}, {56,12559}, {57,5440}, {63,15934}, {65,519}, {78,5708}, {210,3828}, {226,17530}, {354,392}, {376,517}, {381,912}, {405,3929}, {474,11523}, {518,599}, {527,5728}, {528,11570}, {537,21080}, {549,10202}, {597,9021}, {956,11529}, {960,3901}, {971,3543}, {995,3999}, {997,4860}, {999,4930}, {1125,3962}, {1159,3872}, {1210,17533}, {1385,16139}, {1698,4533}, {1739,4849}, {1858,11238}, {2093,3243}, {2095,18446}, {2771,10706}, {2796,12723}, {2802,4744}, {2886,11551}, {3057,3881}, {3058,12711}, {3218,17549}, {3219,16861}, {3244,4757}, {3306,3940}, {3333,5730}, {3338,12635}, {3339,5687}, {3488,9965}, {3524,9940}, {3534,13369}, {3545,5777}, {3601,19704}, {3626,3922}, {3632,10107}, {3634,4005}, {3649,10916}, {3653,13373}, {3654,11239}, {3656,11240}, {3681,3921}, {3697,3812}, {3698,4745}, {3740,4539}, {3742,5692}, {3754,4669}, {3811,5221}, {3829,12047}, {3833,4134}, {3839,5806}, {3869,5045}, {3877,5049}, {3878,17609}, {3889,9957}, {3892,5919}, {3899,10179}, {3902,17145}, {3927,16857}, {3951,11108}, {3984,16408}, {4067,19883}, {4127,19862}, {4234,5208}, {4654,14054}, {4677,5836}, {4737,20892}, {4906,5315}, {4980,5295}, {5057,18527}, {5122,23958}, {5563,16126}, {5570,10072}, {5693,13374}, {5722,5905}, {5844,10273}, {5887,6583}, {6147,6734}, {6875,15178}, {6906,10222}, {7373,11682}, {7672,18419}, {7682,13257}, {9841,11531}, {10056,13750}, {10122,17525}, {10156,15721}, {10246,21165}, {11111,15933}, {11220,15683}, {11227,15692}, {12005,14110}, {12109,21849}, {12527,17706}, {14450,22793}, {15650,19536}, {15803,19705}, {16465,17579}, {17619,21077}, {19251,22345}, {21969,23154}

### X(24474) =  X(1)X(3)∩X(2)X(5761)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28407.

Let A'B'C' be the orthic triangle. Let LA be the reflection of line B'C' in the internal angle bisector of A, and define LB and LC cyclically. Let A" = LB∩LC and define B" and C" cyclically. X(24474) = X(3)-of-A"B"C". (Randy Hutson, November 30, 2018)

X(24474) lies on these lines: {1,3}, {2,5761}, {4,912}, {5,72}, {7,6850}, {8,6826}, {10,6881}, {19,3211}, {20,13369}, {28,110}, {30,1071}, {34,3157}, {37,5755}, {52,1866}, {63,3560}, {68,5130}, {78,6911}, {119,21077}, {140,5439}, {142,3754}, {155,1829}, {210,9956}, {225,1830}, {226,6842}, {227,5399}, {283,18180}, {329,5804}, {355,518}, {381,5715}, {382,971}, {389,12109}, {392,5771}, {429,12259}, {443,10597}, {500,15852}, {511,13408}, {515,3874}, {516,5884}, {546,5927}, {550,10167}, {579,8609}, {758,946}, {916,12162}, {938,5758}, {944,3873}, {950,7491}, {952,3555}, {960,5791}, {962,5768}, {1046,3073}, {1064,2650}, {1066,1254}, {1104,5398}, {1210,6882}, {1320,12776}, {1389,4861}, {1393,22350}, {1439,22464}, {1479,1858}, {1512,10942}, {1656,5044}, {1699,3901}, {1770,5840}, {1828,5446}, {1836,10525}, {1837,10526}, {1870,3562}, {1871,15762}, {1891,12134}, {1898,3583}, {1902,11472}, {1998,19541}, {2262,2323}, {2771,7728}, {2778,12262}, {2800,4084}, {2802,9946}, {2808,13474}, {2836,9970}, {3090,3876}, {3218,6906}, {3219,6920}, {3419,6917}, {3487,6825}, {3488,6868}, {3529,11220}, {3577,6762}, {3649,15908}, {3651,18444}, {3678,10175}, {3681,5818}, {3753,5690}, {3811,11499}, {3812,10198}, {3817,4067}, {3827,19149}, {3851,10157}, {3869,5603}, {3877,6857}, {3878,5745}, {3881,5882}, {3889,7967}, {3892,13607}, {3894,5691}, {3916,6914}, {3927,6913}, {3940,6918}, {3962,5694}, {4018,8727}, {4185,9928}, {4219,15062}, {4297,12005}, {4311,5083}, {4325,12119}, {4463,5797}, {4848,12736}, {4857,16155}, {5231,18493}, {5295,20237}, {5435,6961}, {5440,6924}, {5480,9021}, {5497,6011}, {5534,18518}, {5587,5904}, {5657,6989}, {5692,8227}, {5703,6954}, {5704,6978}, {5713,5752}, {5714,6982}, {5720,11523}, {5722,5812}, {5728,5762}, {5734,5744}, {5748,6981}, {5763,6922}, {5770,6847}, {5787,5878}, {5841,10572}, {5844,10914}, {5883,6684}, {6147,6907}, {6261,12559}, {6288,14872}, {6797,19914}, {6863,11374}, {6885,14923}, {6908,11036}, {6985,11520}, {7970,13190}, {7978,13218}, {7983,12190}, {7984,12382}, {8261,16139}, {9581,18397}, {9856,18544}, {10156,15720}, {10449,20928}, {10531,11415}, {10625,11573}, {10693,12261}, {10698,13279}, {10705,13119}, {11246,11826}, {11571,12750}, {12635,22753}, {12650,12687}, {12675,18481}, {12695,13743}, {12711,15171}, {13099,13314}, {13226,17654}, {13729,17484}, {17563,17612}, {17625,18990}, {17661,22799}, {17857,18491}, {18239,22792}, {18254,23513}, {18357,18908}

X(24474) = reflection of X(3) in X(942)

### X(24475) =  X(1)X(1399)∩X(3)X(3218)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28407.

X(24475) lies on these lines: {1,1399}, {3,3218}, {4,17483}, {5,226}, {7,6917}, {10,5885}, {30,1071}, {40,3894}, {52,23154}, {57,6924}, {65,952}, {72,140}, {145,6948}, {182,9021}, {354,5887}, {355,5270}, {381,12528}, {495,13750}, {496,1858}, {517,550}, {518,5690}, {548,10167}, {549,9940}, {632,5044}, {758,1385}, {916,5876}, {938,6929}, {946,1484}, {960,13373}, {971,3627}, {1125,5694}, {1482,3873}, {1657,11220}, {1735,5399}, {1807,3075}, {1870,23070}, {2095,6985}, {2167,19210}, {2800,3881}, {2801,18480}, {3157,12161}, {3337,6326}, {3339,5534}, {3487,5770}, {3526,3876}, {3555,5844}, {3560,15934}, {3562,18455}, {3576,3901}, {3628,5439}, {3635,10284}, {3670,5396}, {3678,11231}, {3845,5806}, {3850,5927}, {3869,10246}, {3877,17571}, {3878,15178}, {3889,10247}, {3916,7508}, {3927,6883}, {4067,10165}, {4325,5903}, {4430,12245}, {4880,10902}, {5045,10283}, {5083,19907}, {5208,15952}, {5221,11499}, {5446,12109}, {5536,16132}, {5563,6265}, {5693,5886}, {5708,6911}, {5728,5843}, {5730,16203}, {5883,9956}, {5904,15016}, {5905,6928}, {6001,22791}, {6101,11573}, {6824,11036}, {6868,9965}, {6885,21454}, {6902,17484}, {6907,14054}, {6923,12649}, {6936,20078}, {6942,23958}, {6955,20013}, {7171,7982}, {7330,11518}, {10269,12635}, {10273,10914}, {10942,18838}, {11009,11571}, {11227,15712}, {11230,20117}, {11362,13145}, {11567,11715}, {12009,19862}, {12558,22798}, {12699,15071}, {12711,15172}, {13243,21669}, {14872,18357}, {15096,18406}, {15931,16139}, {21740,22765}

### X(24476) =  X(1)X(159)∩X(6)X(169)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28407.

X(24476) lies on these lines: {1,159}, {6,169}, {7,8}, {38,22097}, {46,12329}, {72,141}, {81,105}, {182,10202}, {210,3844}, {511,13408}, {517,990}, {611,13750}, {613,5570}, {912,1352}, {1071,1503}, {1172,18178}, {1284,17447}, {1738,21867}, {1824,3782}, {1843,23154}, {1858,12589}, {1876,6180}, {2002,18838}, {2352,18607}, {2771,14982}, {2809,3755}, {2879,23770}, {3057,3100}, {3313,11573}, {3555,5846}, {3589,5439}, {3619,3876}, {3666,17441}, {3744,20999}, {3751,5902}, {3763,5044}, {3821,4523}, {3873,19993}, {3874,5847}, {4267,18183}, {4463,17184}, {5085,9940}, {5728,5845}, {5777,10516}, {5848,11570}, {5903,16496}, {9028,18389}, {9053,10914}, {9895,24159}, {11220,14927}, {12723,17768}, {16475,18398}

Centers associated with the obverse and N-obverse triangles of X(1): X(24477) - X(24481))

These centers were contributed by Randy Hutson, October 5, 2018. Obverse and N-obverse triangles are introduced in preambles just before X(24307) and X(24310).

### X(24477) =  CENTROID OF N-OBVERSE TRIANGLE OF X(1)

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

X(24477) lies on these lines: {1, 5745}, {2, 210}, {3, 3189}, {4, 10916}, {7, 2886}, {8, 56}, {9, 11019}, {10, 1056}, {11, 329}, {38, 11269}, {46, 5082}, {48, 3684}, {55, 5744}, {57, 2550}, {63, 497}, {69, 1447}, {72, 3086}, {75, 7196}, {78, 7288}, {142, 10980}, {144, 5274}, {145, 2646}, {165, 5853}, {200, 3911}, {226, 5231} et al

### X(24478) =  PERSPECTOR OF THESE TRIANGLES: N-OBVERSE OF X(1) AND (CROSS-TRIANGLE OF OBVERSE OF X(1) AND N-OBVERSE OF X(1))

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

X(24478) lies on these lines: {1,2092}, {6, 291}, {9, 12782}, {38, 1654}, {43, 3779}, {69, 982}, {71, 238}, {75, 256}, {239, 3778}, {319, 4022}, {350, 21257}, {614, 22370}, {894, 23659}, {966, 984}, {978, 3781}, {1045, 21746} et al

### X(24479) =  TRILINEAR POLE OF PERSPECTRIX OF THESE TRIANGLES: OBVERSE OF X(1) AND N-OBVERSE OF X(1)

Trilinears    1/((a^2 - b c) (a^3 - b^3 - c^3 + a b c)) : :
Barycentrics    a/((a^2 - b c) (a^3 - b^3 - c^3 + a b c)) : :

X(24479) lies on these lines: {39, 8868}, {239, 1916}, {291, 1580}, {511, 1757}, {672, 1931}, {1959, 3930} et al

X(24479) = isogonal conjugate of X(19557)
X(24479) = isotomic conjugate of X(18037)

### X(24480) =  EIGENCENTER OF OBVERSE TRIANGLE OF X(1)

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

X(24480) lies on these lines: {101, 3923}, {292, 4493}, {870, 1438}, {1001, 9259}

### X(24481) =  EIGENCENTER OF N-OBVERSE TRIANGLE OF X(1)

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

X(24481) lies on these lines: {274, 284}, {893, 7242}, {1376, 18755} : :

Collineation mappings involving trilinear obverse triangles: X(24482) - X(24518)

Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC. Trilinears for P are p/a : q/b : r/c. The trilinear obverse triangle of P is here introduced as the triangle A'B'C', where A' = p/a : r/c : q/b, B' = r/c : q/b : p/a, C' = q/b : p/a : r/c (trilinears). The same triangle is given by barycentrics A' = b c p : b^2 r : c^2 q, B' = a^2 r : c a q : p c^2 p, C' = a^2 q : b^2 p : a b r. For example, the trilinear obverse triangle of X(2) has first barycentric b c : b^2 : c^2. (Clark Kimberling, October 5, 2018)

### X(24482) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24482) lies on these lines: {2, 513}, {6, 3573}, {8, 24004}, {9, 1026}, {44, 751}, {45, 660}, {55, 765}, {75, 4124}, {190, 3271}, {238, 993}, {244, 24722}, {320, 4679}, {344, 9309}, {518, 1992}, {644, 10755}, {668, 25333}, {846, 9282}, {894, 24346}, {1001, 3257}, {1052, 17596}, {1083, 4585}, {1086, 4499}, {1621, 6163}, {2876, 25316}, {3056, 17336}, {3097, 3764}, {3122, 9359}, {3768, 24493}, {3792, 4759}, {3799, 4370}, {3888, 4422}, {4459, 18151}, {4473, 4553}, {4579, 16686}, {4965, 20881}, {5150, 24436}, {6373, 24508}, {9025, 17264}, {16494, 24405}, {16495, 24338}, {17777, 18191}, {24483, 24490}, {24484, 24486}, {24488, 24492}, {24502, 24505}, {24509, 24515}

### X(24483) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24483) lies on these lines: {2, 656}, {24482, 24490}, {24484, 24489}, {24485, 24518}

### X(24484) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24484) lies on these lines: {2, 661}, {3, 6}, {7, 24404}, {100, 20974}, {171, 23636}, {291, 9471}, {513, 5701}, {537, 4876}, {672, 3792}, {894, 24401}, {984, 2113}, {1655, 17300}, {2110, 20670}, {2112, 24436}, {2170, 24715}, {2284, 3730}, {2810, 6184}, {3061, 4655}, {3684, 20729}, {3937, 23988}, {3939, 10756}, {8298, 20465}, {14825, 21143}, {17234, 17671}, {17265, 17675}, {23466, 24508}, {24482, 24486}, {24483, 24489}, {24488, 24494}, {24502, 24510}

### X(24485) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

Barycentrics    3 a^3 b^2 - a^2 b^3 - 7 a^3 b c + 3 a^2 b^2 c - a b^3 c + 3 a^3 c^2 + 3 a^2 b c^2 - 3 a b^2 c^2 + b^3 c^2 - a^2 c^3 - a b c^3 + b^2 c^3 : :

X(24485) lies on these lines: {2, 513}, {145, 24004}, {894, 24409}, {3616, 24397}, {16495, 25382}, {24483, 24518}

### X(24486) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24486) lies on these lines: {2, 650}, {37, 2275}, {1001, 22116}, {1015, 24841}, {6376, 17263}, {12782, 16484}, {16706, 20269}, {24482, 24484}, {24502, 24508}, {24509, 24510}

### X(24487) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24487) lies on these lines: {1, 24004}, {2, 513}, {75, 25382}, {87, 18040}, {551, 726}, {751, 2228}, {894, 24405}, {1125, 24399}, {3248, 4033}, {24502, 24503}

### X(24488) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24488) lies on these lines: {2, 1491}, {42, 2183}, {101, 23646}, {659, 24499}, {876, 14947}, {894, 24442}, {2503, 20975}, {17463, 24712}, {24482, 24492}, {24484, 24494}

### X(24489) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24489) lies on these lines: {2, 8060}, {6, 20839}, {21339, 24713}, {24483, 24484}

### X(24490) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24490) lies on these lines: {1, 3122}, {17256, 24433}, {24482, 24483}

### X(24491) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24491) lies on these lines: {1, 6}, {2, 649}, {672, 4759}, {748, 21369}, {874, 17277}, {875, 24494}, {894, 24455}, {1018, 4432}, {2112, 5150}, {4090, 23415}, {4676, 16549}, {9360, 16569}, {17335, 23891}, {24482, 24484}, {24493, 24517}, {24505, 24508}

### X(24492) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24492) lies on these lines: {2, 659}, {38, 3248}, {87, 24413}, {190, 1621}, {846, 9359}, {894, 24447}, {17601, 20331}, {24482, 24488}

### X(24493) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24493) lies on these lines: {2, 798}, {2245, 24512}, {3768, 24482}, {16482, 20979}, {24483, 24484}, {24491, 24517}

### X(24494) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24494) lies on these lines: {2, 513}, {6, 31}, {875, 24491}, {993, 1083}, {2810, 8299}, {3240, 16467}, {3251, 9320}, {3271, 20331}, {3720, 24405}, {3873, 4664}, {9016, 14439}, {17135, 24004}, {24484, 24488}

### X(24495) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24495) lies on these lines: {2, 513}, {7, 24801}, {354, 4664}, {749, 1100}, {4595, 23456}, {10453, 24004}, {20331, 25048}, {24498, 24516}

### X(24496) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24496) lies on these lines: {1, 39}, {2, 514}, {672, 1023}, {1575, 2087}, {1698, 19934}, {3573, 5030}, {4089, 20335}, {24482, 24484}, {24508, 24510}

### X(24497) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24497) lies on these lines: {2, 812}, {45, 1015}, {1001, 4752}, {8671, 23855}, {24482, 24484}

### X(24498) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24498) lies on these lines: {2, 2254}, {55, 2316}, {894, 24453}, {982, 4419}, {24482, 24483}, {24484, 24488}, {24495, 24516}

### X(24499) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24499) lies on these lines: {2, 650}, {57, 9319}, {81, 3658}, {100, 1814}, {222, 7045}, {241, 2275}, {659, 24488}, {664, 7117}, {673, 14936}, {1027, 14947}, {1155, 3240}, {1376, 9503}, {1447, 8608}, {3732, 13006}, {6376, 18751}, {7493, 8758}, {9358, 17074}, {9441, 12782}, {20096, 21859}, {24482, 24483}

### X(24500) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24500) lies on these lines: {662, 20975}, {895, 5546}, {2611, 24711}, {3448, 24713}, {4552, 25051}, {4565, 14060}, {17463, 21221}, {24482, 24488}, {24483, 24484}, {24505, 24516}

### X(24501) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24501) lies on these lines: {2, 650}, {55, 22116}, {2275, 3666}, {2991, 4571}, {3750, 12782}, {24482, 24488}

### X(24502) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24502) lies on these lines: {1, 87}, {2, 649}, {6, 874}, {101, 24815}, {384, 24279}, {889, 4363}, {1977, 1978}, {2108, 19579}, {3249, 21211}, {3758, 24004}, {4116, 24696}, {4568, 24825}, {4672, 17499}, {6382, 23538}, {8054, 21224}, {10009, 24343}, {18046, 24731}, {24482, 24505}, {24484, 24510}, {24486, 24508}, {24487, 24503}

### X(24503) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24503) lies on these lines: {2, 798}, {894, 24338}, {6386, 21762}, {24487, 24502}

### X(24504) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24504) lies on these lines: {2, 661}, {39, 1931}, {99, 20982}, {100, 20670}, {148, 2170}, {329, 1655}, {335, 21220}, {662, 2503}, {756, 9510}, {894, 24347}, {896, 3240}, {1005, 2651}, {2185, 2644}, {2608, 3219}, {2641, 4641}, {3930, 20536}, {4366, 20974}, {6625, 17451}, {17483, 24404}, {18139, 18140}, {21221, 24712}, {24482, 24488}

### X(24505) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24505) lies on these lines: {2, 661}, {8, 192}, {39, 2669}, {141, 18140}, {148, 24715}, {244, 21220}, {645, 10754}, {799, 3124}, {894, 2607}, {3952, 25047}, {6646, 24404}, {9505, 24325}, {24482, 24502}, {24491, 24508}, {24500, 24516}

### X(24506) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24506) lies on these lines: {2, 2642}, {39, 4526}, {86, 24287}, {798, 8062}, {900, 17398}, {992, 4435}, {1213, 24959}, {3716, 3768}, {4800, 24512}, {7253, 8061}, {8060, 17217}, {8632, 14288}, {8674, 21053}, {17362, 21714}

### X(24507) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

Barycentrics    5 a^3 b^2 - 3 a^2 b^3 - 13 a^3 b c + 7 a^2 b^2 c - a b^3 c + 5 a^3 c^2 + 7 a^2 b c^2 - 5 a b^2 c^2 + b^3 c^2 - 3 a^2 c^3 - a b c^3 + b^2 c^3 : :

X(24507) lies on these lines: {2, 513}, {3621, 24004}, {3622, 24418}

### X(24508) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24508) lies on these lines: {2, 812}, {190, 1015}, {291, 4432}, {668, 4422}, {3227, 4370}, {4473, 9263}, {6373, 24482}, {23466, 24484}, {24486, 24502}, {24491, 24505}, {24496, 24510}

### X(24509) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24509) lies on these lines: {1, 4704}, {2, 649}, {874, 17349}, {894, 24460}, {23466, 24484}, {24482, 24515}, {24486, 24510}

### X(24510) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24510) lies on these lines: {2, 513}, {335, 24722}, {2276, 3758}, {4664, 4795}, {6542, 24004}, {16826, 24423}, {24484, 24502}, {24486, 24509}, {24496, 24508}

### X(24511) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24511) lies on these lines: {2, 1755}, {6, 3725}, {9, 1215}, {10, 2179}, {39, 14815}, {71, 3683}, {169, 2083}, {354, 20785}, {375, 8608}, {579, 7262}, {1125, 4020}, {2260, 4641}, {2276, 3764}, {4672, 17754}, {5275, 20665}, {10176, 14964}, {14963, 15049}, {20470, 22099}, {20610, 24631}, {22061, 23383}

### X(24512) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

Barycentrics    a (a^2 b + a^2 c + 2 a b c + b^2 c + b c^2) : :
Trilinears    (sin A + sin B + sin C) csc A + (csc A + csc B + csc C) sin A : :

X(24512) lies on these lines: {1, 39}, {2, 6}, {10, 3780}, {32, 16783}, {37, 38}, {42, 1100}, {43, 1051}, {55, 16693}, {58, 16850}, {65, 3727}, {75, 17027}, {76, 4754}, {83, 1509}, {171, 1914}, {213, 1125}, {244, 21840}, {274, 17034}, {284, 16056}, {350, 894}, {384, 17103}, {404, 18755}, {405, 5021}, {474, 2271}, {519, 16971}, {551, 3230}, {572, 4192}, {579, 8731}, {594, 17135}, {612, 16973}, {614, 16972}, {750, 2280}, {851, 2278}, {899, 16666}, {942, 3721}, {999, 19263}, {1002, 1390}, {1030, 4210}, {1107, 1475}, {1172, 4212}, {1193, 16604}, {1468, 4426}, {1574, 3293}, {1621, 17735}, {1931, 19237}, {1985, 3330}, {2176, 3616}, {2229, 5109}, {2235, 4670}, {2239, 21764}, {2241, 5264}, {2242, 16788}, {2245, 24493}, {2269, 20359}, {2345, 10453}, {2549, 11355}, {3122, 24513}, {3125, 5883}, {3216, 20970}, {3290, 3742}, {3550, 10987}, {3571, 9506}, {3622, 16969}, {3664, 20335}, {3678, 25089}, {3684, 17122}, {3693, 4883}, {3735, 5902}, {3739, 24592}, {3741, 5750}, {3751, 3789}, {3758, 4465}, {3770, 18152}, {3783, 4649}, {3812, 21951}, {3833, 16611}, {3874, 3954}, {4161, 23493}, {4184, 5124}, {4199, 5019}, {4204, 5115}, {4251, 5277}, {4253, 5283}, {4254, 16059}, {4283, 4476}, {4357, 24690}, {4360, 17759}, {4363, 4441}, {4368, 4672}, {4435, 14438}, {4503, 4667}, {4559, 15950}, {4651, 17362}, {4653, 5030}, {4657, 24691}, {4675, 24712}, {4697, 24259}, {4800, 24506}, {5013, 19765}, {5110, 13588}, {5120, 16058}, {5253, 21008}, {5254, 23903}, {5332, 16779}, {5439, 16583}, {5710, 16781}, {5711, 16502}, {5819, 20995}, {6625, 16044}, {6626, 16927}, {6685, 20228}, {7109, 16685}, {10176, 21839}, {10436, 17026}, {10459, 17448}, {10582, 16970}, {14210, 24254}, {14968, 18907}, {16514, 16826}, {16517, 17022}, {16521, 17021}, {16522, 17012}, {16523, 17316}, {16525, 17032}, {16552, 16589}, {16569, 16667}, {16777, 17597}, {16782, 17023}, {16818, 21240}, {16884, 17018}, {17031, 24325}, {17045, 25349}, {17126, 21793}, {17141, 25263}, {17365, 20347}, {17499, 18140}, {17752, 25303}, {20464, 23532}

X(24512) = {X(2),X(6)}-harmonic conjugate of X(2238)

### X(24513) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24513) lies on these lines: {2, 256}, {6, 23398}, {42, 51}, {291, 4672}, {672, 3778}, {1009, 14815}, {1193, 23414}, {1201, 23473}, {2085, 16600}, {2276, 3764}, {3122, 24512}, {3720, 22172}, {4022, 24690}, {17135, 22167}, {17759, 24478}, {21753, 23444}, {22343, 23632}

### X(24514) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

Barycentrics    a^3 b + a^3 c + b^2 c^2 : :

X(24514) lies on these lines: {1, 1655}, {2, 7}, {6, 350}, {31, 385}, {37, 17032}, {41, 384}, {42, 192}, {43, 3729}, {44, 17028}, {72, 1008}, {75, 2238}, {76, 213}, {171, 16997}, {190, 2276}, {193, 10453}, {194, 1193}, {218, 7770}, {238, 16998}, {239, 4441}, {257, 3869}, {316, 4805}, {330, 1201}, {386, 25264}, {536, 21904}, {748, 17000}, {750, 16999}, {870, 16514}, {1575, 17351}, {1654, 4388}, {1743, 17026}, {1909, 2176}, {1958, 16956}, {2106, 8033}, {2251, 3972}, {2280, 4366}, {2295, 6376}, {2887, 16991}, {3195, 9308}, {3314, 4766}, {3570, 4386}, {3720, 17379}, {3741, 4416}, {3758, 4465}, {3759, 4479}, {3760, 17034}, {3780, 17144}, {3783, 3923}, {3789, 5263}, {3963, 7109}, {3997, 6381}, {4110, 25287}, {4431, 4685}, {4672, 17793}, {4676, 8299}, {5025, 24995}, {5276, 14621}, {6180, 7196}, {6327, 17007}, {6645, 9310}, {6679, 17003}, {7754, 16466}, {7766, 21764}, {12263, 16476}, {16468, 17031}, {16552, 17030}, {16670, 17029}, {16969, 25303}, {17001, 17126}, {17002, 17127}, {17018, 17319}, {17135, 17363}, {17334, 25349}, {17347, 24690}, {17349, 24592}, {17743, 18031}, {17750, 18140}, {17751, 20109}, {19974, 24455}

### X(24515) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24515) lies on these lines: {2, 650}, {39, 874}, {75, 2275}, {5263, 22116}, {6376, 17279}, {17280, 21226}, {24482, 24509}, {24484, 24502}, {24491, 24505}

### X(24516) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24516) lies on these lines: {1, 3122}, {2, 1491}, {190, 22116}, {320, 3675}, {2245, 3573}, {14947, 23838}, {24482, 24484}, {24495, 24498}, {24500, 24505}

### X(24517) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24517) lies on these lines: {2, 513}, {10, 24004}, {37, 23354}, {75, 24338}, {740, 3679}, {751, 17250}, {16709, 22174}, {20006, 25036}, {24491, 24493}, {24500, 24505}

### X(24518) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24518) lies on these lines: {2, 661}, {24483, 24485}

Collineation mappings involving trilinear N-obverse triangles: X(24519) - X(24536)

Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC. Trilinears for P are p/a : q/b : r/c. The trilinear N-obverse triangle of P is here introduced as the triangle A'B'C', where A' = - p/a : r/c : q/b, B' = r/c : - q/b : p/a, C' = q/b : p/a : - r/c (trilinears). The same triangle is given by barycentrics A' = - b c p : b^2 r : c^2 q, B' = a^2 r : - c a q : p c^2 p, C' = a^2 q : b^2 p : - a b r. For example, the trilinear obverse triangle of X(2) has first barycentric - b c : b^2 : c^2. (Clark Kimberling, October 5, 2018)

### X(24519) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24519) lies on these lines: {2, 17448}, {1386, 3779}, {1654, 16744}, {2275, 3759}, {3616, 24944}, {18194, 24520}, {24880, 24962}

### X(24520) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24520) lies on these lines: {2, 3728}, {3, 6}, {10, 24437}, {43, 24530}, {244, 24919}, {17038, 24944}, {18194, 24519}, {22167, 24958}, {24525, 24528}, {24526, 24527}

### X(24521) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24521) lies on these lines: {2, 3056}, {24522, 24529}

### X(24522) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24522) lies on these lines: {2, 17448}, {194, 1992}, {330, 5839}, {3189, 3779}, {3227, 24897}, {3780, 9263}, {16722, 20090}, {24521, 24529}

### X(24523) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24523) lies on these lines: {2, 3056}, {1100, 2275}, {1246, 6384}, {1964, 24478}, {3769, 3779}, {3794, 15985}, {16574, 18788}, {18194, 24519}

### X(24524) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24524) lies on these lines: {1, 668}, {2, 17448}, {6, 17743}, {7, 8}, {41, 18047}, {42, 17149}, {43, 6384}, {76, 519}, {99, 8715}, {145, 350}, {183, 12513}, {190, 3208}, {192, 21219}, {193, 17787}, {194, 20691}, {239, 20917}, {274, 3679}, {304, 4737}, {312, 3765}, {313, 17377}, {321, 20055}, {325, 12607}, {330, 1575}, {335, 3959}, {341, 18156}, {386, 18148}, {529, 7750}, {535, 7802}, {536, 20081}, {740, 9902}, {799, 2177}, {869, 19567}, {894, 4110}, {1078, 8666}, {1334, 17336}, {1655, 4664}, {1965, 3870}, {1966, 3751}, {1975, 3913}, {2170, 18055}, {2176, 10027}, {2275, 9263}, {2276, 21226}, {2295, 3758}, {3187, 19803}, {3241, 18135}, {3244, 6381}, {3264, 17378}, {3501, 4595}, {3570, 9310}, {3596, 3879}, {3621, 4441}, {3625, 20888}, {3632, 3761}, {3633, 3760}, {3673, 18159}, {3681, 18059}, {3691, 17335}, {3729, 4050}, {3759, 3780}, {3769, 4039}, {3770, 17299}, {3873, 20352}, {3948, 17389}, {3963, 17363}, {3975, 17316}, {4251, 4482}, {4377, 4725}, {4385, 17762}, {4386, 6645}, {4474, 20954}, {4505, 4672}, {4651, 22293}, {4738, 7278}, {4774, 7199}, {4851, 20923}, {4852, 18144}, {4865, 21590}, {4876, 9311}, {7257, 17103}, {7773, 11236}, {8026, 20537}, {9025, 24351}, {10987, 17692}, {14584, 20566}, {16549, 23891}, {17373, 20891}, {17375, 20892}, {17386, 18137}, {17393, 18133}, {19581, 19586}, {20060, 20553}, {21223, 21877}, {24661, 25120}, {25274, 25290}, {25276, 25281}, {25277, 25279}, {25282, 25294}, {25292, 25295}

### X(24525) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24525) lies on these lines: {2, 24657}, {42, 2260}, {24520, 24528}

### X(24526) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24526) lies on these lines: {2, 24658}, {24520, 24527}

### X(24527) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24527) lies on these lines: {2, 24660}, {24520, 24526}

### X(24528) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24528) lies on these lines: {2, 17448}, {6, 31}, {8, 16606}, {43, 2275}, {200, 292}, {291, 2319}, {312, 21893}, {518, 20284}, {893, 3751}, {1015, 16569}, {1575, 5839}, {2162, 23579}, {2277, 21904}, {3121, 3681}, {3240, 23632}, {3961, 23543}, {4651, 25293}, {4661, 8620}, {6384, 9263}, {6685, 16975}, {20012, 21877}, {20271, 22173}, {24520, 24525}

### X(24529) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24529) lies on these lines: {1, 4531}, {517, 3500}, {3212, 20460}, {3759, 6647}, {24521, 24522}

### X(24530) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24530) lies on these lines: {2, 37}, {6, 662}, {39, 2669}, {43, 24520}, {81, 4272}, {82, 8301}, {86, 2092}, {190, 21796}, {256, 2234}, {291, 872}, {319, 21857}, {386, 2274}, {749, 1469}, {869, 4446}, {897, 5968}, {980, 5224}, {986, 6042}, {1045, 3122}, {1054, 13610}, {1333, 19308}, {1654, 16696}, {1698, 21730}, {1740, 3764}, {1757, 3216}, {1964, 24478}, {2275, 3759}, {2352, 19345}, {2643, 24372}, {2664, 21035}, {2895, 18601}, {3125, 18714}, {3783, 4022}, {3959, 18041}, {4277, 17379}, {4360, 17053}, {5069, 17349}, {6542, 21858}, {6703, 25059}, {7235, 24883}, {8610, 17319}, {16700, 17778}, {16726, 20090}, {17256, 21892}, {17315, 20691}, {18042, 21008}, {20360, 24443}, {22008, 24170}, {24050, 24185}, {24880, 25065}, {24890, 24899}

### X(24531) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24531) lies on these lines: {2, 25138}

### X(24532) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24532) lies on these lines: {650, 1459}, {5256, 23654}, {23655, 25142}

### X(24533) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24533) lies on these lines: {2, 669}, {43, 18197}, {44, 513}, {512, 24782}, {667, 17072}, {876, 7180}, {890, 20295}, {1499, 19543}, {2517, 4874}, {2533, 3907}, {3004, 25300}, {3835, 8640}, {4083, 22090}, {4467, 17989}, {5518, 22386}, {8630, 24601}, {8637, 21260}, {8651, 25084}, {16695, 21051}, {18093, 18106}, {19308, 21005}, {21191, 24674}, {21351, 22322}, {24755, 25128}

### X(24534) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24534) lies on these lines: {42, 23658}, {661, 3907}, {669, 2451}, {8640, 23466}, {22199, 23574}

### X(24535) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24535) lies on these lines: {2, 23655}, {7234, 17033}

### X(24536) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24536) lies on these lines: {2, 3056}, {2275, 3759}

Collineation mappings involving Gemini triangle 1: X(24537) - X(24571)

Following is a list of central triangles, by barycentric coordinates of A-vertex. The full names are Gemini triangle 1, Gemini triangle 2, Gemini triangle 3, etc. (Gemini is a constellation of stars - and the Latin noun for twin.) These triangles are cited in the sequel. (Clark Kimberling, October 6, 2018)

Gemini 1      a : b + c : b + c
Gemini 2      - a : b + c : b + c

Gemini 3      a : a + b : a + c
Gemini 4      - a : a + b : a + c

Gemini 5      a : a - b : a - c
Gemini 6      a : b - a : c - a

Gemini 7      a : a - c : a - b
Gemini 8      a : c - a : b - a (Garcia reflection triangle)

Gemini 9      a : b + c - a : b + c - a
Gemini 10      - a : b + c - a : b + c - a

Gemini 11      a : a + b + c : a + b + c
Gemini 12      - a : a + b + c : a + b + c

Gemini 13      b + c : a : a
Gemini 14      b + c : 2a : 2a

Gemini 15      b + c : b : c (Gergonne line extraversion triangle; see X(10180). See note below.)
Gemini 16      b + c : c : b (See note below.)

Gemini 17      b + c : b - c : c - b (See note below.)
Gemini 18      b + c : c - b : b - c (See note below.)

Gemini 19      b + c : a + b : a + c (See note below.)
Gemini 20      2b + 2c : a : a (See note below.)

Gemini 21      a + b + c : a : a
Gemini 22      a + b + c : - a : - a

Gemini 23      a + b + c : b + c : b + c
Gemini 24      a + b + c : - b - c : - b - c

Gemini 25      a + b + c : a + b : a + c
Gemini 26      a + b + c : a + c : a + b

Gemini 27      a - b - c : a : a
Gemini 28      a - b - c : b + c : b + c (See note below.)

Gemini 29      a : b - c : c - b (See note below.)
Gemini 30      a : c - b : b - c (Inner Conway triangle; see note below.)

Gemini 31      b c : a^2 : a^2
Gemini 32      - b c : a^2 : a^2

Gemini 33      a^2 : b c : b c
Gemini 34      - a^2 : b c : b c

Gemini 35      cos A : 1 : 1
Gemini 36      - cos A : 1 : 1

Gemini 37      sec A : 1 : 1
Gemini 38      - sec A : 1 : 1

Gemini 39      -a + b + c : a + b + c : a + b + c
Gemini 40      a + b + c : - a + b + c : - a + b + c

Notes: (Randy Hutson, beginning November 9, 2018)
Gemini triangle 15 = complement of cevian triangle of X(75)
Gemini triangle 16 = complement of incentral triangle
Gemini triangle 17 = anticomplement of cevian triangle of X(75)
Gemini triangle 17 = anticomplement of anticomplement of Gemini triangle 15
Gemini triangle 18 = anticomplement of incentral triangle
Gemini triangle 19 = medial triangle of obverse triangle of X(1)
Gemini triangle 19 = obverse triangle of X(10)
Gemini triangle 20 = complement of Gemini triangle 28
Gemini triangle 23 = complement of Gemini triangle 39
Gemini triangle 28 = anticomplement of Gemini triangle 20
Gemini triangle 29 = anticomplement of extouch triangle
Gemini triangle 29 = anticomplement of anticomplement of 1st Zaniah triangle
Gemini triangle 30 = anticomplement of intouch triangle
Gemini triangle 30 = anticomplement of anticomplement of 2nd Zaniah triangle
Gemini triangle 33 = unary cofactor triangle of 2nd Sharygin triangle
Gemini triangle 34 = unary cofactor triangle of 1st Sharygin triangle
Gemini triangle 39 = anticomplement of Gemini triangle 23

For more Gemini triangles, see the preambles just before X(26153) and X(27378).

If T is a central triangle A'B'C' with A' of the form f(a,b,c) : g(a,b,c) : g(a,b,c), then the (A,B,C,X(2); A',B',C',X(2)) collineation image of the Euler line is the Euler line. Examples include Gemini triangles 1,2,9,10,11,12,13,14,21,22,23,24,27,28, and 31-40.

Let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 1, as in centers X(24537)-X(24571). Then

m(X) = a (a + b - c)(a - b + c) x + (b + c - a) (a + b - c) (a + c) y + (b + c - a) (a - b + c) (a + b) z : : ,

and m(X) is on the Euler line if and only if X is on the Euler line.

### X(24537) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24537) lies on these lines: {1, 17860}, {2, 3}, {8, 394}, {10, 255}, {77, 1441}, {92, 4296}, {189, 1220}, {1074, 1125}, {1437, 5767}, {1944, 3869}, {2199, 20262}, {5930, 18652}, {19861, 24552}, {24540, 24546}

### X(24538) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24538) lies on these lines: {2, 3}, {4294, 23541}, {4296, 6350}, {8270, 19860}, {10436, 22464}, {11427, 19767}, {19765, 23292}, {24553, 24555}

### X(24539) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24539) lies on these lines: {2, 3}, {5248, 23541}, {10436, 24999}, {19861, 24542}

### X(24540) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24540) lies on these lines: {1, 20895}, {2, 6}, {3, 17183}, {7, 5253}, {56, 20245}, {105, 24669}, {319, 25005}, {326, 17863}, {404, 10446}, {604, 21246}, {1329, 21286}, {1388, 18654}, {3672, 24558}, {3879, 24982}, {4855, 10889}, {5289, 21273}, {7190, 10436}, {9310, 20258}, {10455, 16454}, {14868, 19752}, {17182, 19645}, {17201, 19520}, {17272, 24998}, {17321, 24999}, {17452, 24334}, {20880, 24179}, {24537, 24546}, {24545, 24550}, {24554, 24559}

### X(24541) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24541) lies on these lines: {1, 2}, {21, 946}, {36, 12609}, {37, 5830}, {40, 6910}, {56, 5249}, {63, 3485}, {65, 4999}, {140, 3753}, {142, 5253}, {149, 4314}, {214, 3841}, {225, 11109}, {226, 2975}, {283, 11110}, {354, 11281}, {377, 3576}, {392, 5771}, {404, 10165}, {405, 5812}, {442, 1385}, {443, 12116}, {452, 5715}, {474, 10267}, {515, 2476}, {516, 4189}, {517, 7483}, {908, 958}, {944, 6856}, {950, 11680}, {956, 11374}, {960, 15950}, {993, 12047}, {1001, 5832}, {1004, 8273}, {1072, 13740}, {1512, 6863}, {1519, 3560}, {1621, 12053}, {1699, 6872}, {1770, 5267}, {1953, 22447}, {2323, 5257}, {2475, 4297}, {2478, 8227}, {2550, 4855}, {2646, 2886}, {3035, 3698}, {3057, 6690}, {3072, 19270}, {3193, 17557}, {3218, 3671}, {3306, 7288}, {3359, 6977}, {3434, 3601}, {3436, 5219}, {3452, 5260}, {3523, 10268}, {3555, 5719}, {3628, 17619}, {3812, 5433}, {3817, 5046}, {3824, 5126}, {3838, 7354}, {3869, 5745}, {3877, 13464}, {3925, 10959}, {3947, 20060}, {4187, 11230}, {4190, 7987}, {4295, 4652}, {4423, 10966}, {4512, 11522}, {4857, 5426}, {4870, 17781}, {4972, 24985}, {5045, 14054}, {5084, 10532}, {5141, 19925}, {5154, 10171}, {5177, 5731}, {5178, 12437}, {5187, 7988}, {5204, 5880}, {5250, 5603}, {5251, 5443}, {5258, 21077}, {5265, 9776}, {5290, 20076}, {5303, 20292}, {5432, 5836}, {5439, 15325}, {5536, 15674}, {5587, 6933}, {5691, 6871}, {5730, 5791}, {5795, 11681}, {5985, 21636}, {6598, 24386}, {6974, 12705}, {7504, 10175}, {7705, 10172}, {8609, 17398}, {8666, 13407}, {8728, 10943}, {9578, 10585}, {9624, 12704}, {9812, 17576}, {9955, 11113}, {10404, 11194}, {10436, 22464}, {10448, 24210}, {10597, 17559}, {10680, 11108}, {10785, 18443}, {10806, 17582}, {11112, 13624}, {11114, 18483}, {11260, 15888}, {11344, 22753}, {11520, 24477}, {12001, 16853}, {12512, 17548}, {12513, 17718}, {12699, 16370}, {13408, 13745}, {15670, 22937}, {16137, 24473}, {16202, 16408}, {16295, 20470}, {16418, 18493}, {17202, 17588}, {17353, 25024}, {17528, 18544}, {17529, 22935}, {17530, 18480}, {17532, 18481}, {23536, 24161}, {24390, 24929}, {24554, 24555}

### X(24542) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24542) lies on these lines: {2, 11}, {10, 3722}, {31, 18139}, {35, 17674}, {88, 5550}, {238, 3936}, {244, 1125}, {405, 20999}, {678, 3634}, {748, 3771}, {902, 3836}, {1015, 24956}, {1054, 3624}, {1086, 4427}, {1279, 3006}, {1283, 5051}, {1478, 11346}, {1623, 16865}, {2246, 5257}, {3011, 4358}, {3120, 4432}, {3240, 17352}, {3271, 3909}, {3315, 3616}, {3683, 17184}, {3685, 4442}, {3712, 17495}, {3720, 6679}, {3891, 17776}, {3932, 20045}, {3952, 4422}, {3995, 17061}, {4062, 4974}, {4202, 5248}, {4302, 17679}, {4450, 8616}, {4453, 14432}, {4473, 4756}, {4552, 15253}, {4689, 17356}, {4712, 5294}, {4760, 25357}, {4966, 16704}, {5192, 10198}, {5297, 17263}, {5750, 14439}, {7664, 16597}, {9347, 17244}, {11731, 24871}, {15338, 17690}, {16020, 17740}, {16342, 19836}, {16842, 23858}, {17126, 17234}, {17127, 18134}, {17398, 20331}, {17766, 21026}, {19861, 24539}

### X(24543) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24543) lies on these lines: {2, 3}, {3100, 24552}

### X(24544) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24544) lies on these lines: {2,3}

### X(24545) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24545) lies on these lines: {1, 20237}, {2, 31}, {3, 1519}, {6, 24996}, {394, 24997}, {908, 1460}, {1010, 19861}, {2049, 24987}, {5711, 10601}, {5783, 25006}, {24540, 24550}, {24546, 24549}

### X(24546) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24546) lies on these lines: {1, 21420}, {2, 32}, {24537, 24540}, {24545, 24549}, {24555, 24571}

### X(24547) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24547) lies on these lines: {2, 37}, {7, 3869}, {9, 24612}, {10, 20895}, {65, 21273}, {142, 2171}, {322, 5232}, {392, 17183}, {960, 20245}, {1108, 16713}, {1266, 24548}, {1400, 21233}, {1441, 4357}, {2269, 24336}, {2292, 3663}, {3100, 5263}, {3262, 5224}, {3702, 10447}, {3718, 4696}, {3766, 25020}, {3875, 19860}, {3877, 10446}, {3925, 21333}, {4858, 5257}, {4967, 24982}, {5250, 10444}, {5252, 21286}, {5831, 20270}, {7190, 10436}, {8583, 24179}, {17869, 19858}, {19859, 20320}, {20235, 25017}, {24552, 24566}, {25000, 25023}

### X(24548) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24548) lies on these lines: {2, 39}, {142, 17451}, {978, 21405}, {1266, 24547}, {20880, 24178}, {24537, 24540}

### X(24549) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 1

Barycentrics    a^4 + a b^3 + b^3 c + a c^3 + b c^3 : :

X(24549) lies on these lines: {1, 75}, {2, 41}, {7, 4195}, {10, 16992}, {31, 17137}, {32, 21240}, {56, 9436}, {63, 16050}, {69, 5247}, {85, 2647}, {141, 4426}, {142, 11321}, {183, 3831}, {218, 17353}, {239, 16787}, {315, 2887}, {384, 3662}, {405, 4357}, {441, 18639}, {742, 16974}, {894, 17688}, {940, 3912}, {976, 3263}, {993, 16887}, {1125, 24241}, {1220, 14828}, {1944, 3061}, {1973, 15149}, {1975, 24214}, {3494, 6384}, {3721, 4376}, {3915, 17152}, {3924, 20911}, {4056, 17211}, {4384, 16780}, {4386, 20255}, {4950, 16886}, {5249, 19281}, {5255, 21281}, {7770, 20335}, {9259, 24652}, {10448, 16705}, {11319, 20347}, {11320, 17184}, {13735, 17274}, {14829, 17284}, {16061, 17754}, {16783, 16818}, {16912, 17248}, {16914, 17236}, {17023, 19701}, {17050, 20172}, {17282, 17682}, {24545, 24546}, {24550, 24566}

### X(24550) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24550) lies on these lines: {1, 2}, {141, 24991}, {392, 8731}, {1064, 18465}, {1385, 4199}, {4322, 5484}, {16056, 17614}, {22053, 24723}, {24540, 24545}, {24549, 24566}

### X(24551) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24551) lies on these lines: {1, 2}, {63, 1284}, {497, 24669}, {2171, 17754}, {3794, 6210}, {3925, 24655}, {5250, 8731}, {17194, 17202}, {20258, 20557}, {21334, 22370}, {24552, 24563}

### X(24552) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 1

Barycentrics    a^3 + a b^2 + b^2 c + a c^2 + b c^2 : :

X(24552) lies on these lines: {1, 321}, {2, 11}, {6, 17135}, {8, 13740}, {9, 4981}, {10, 748}, {21, 15654}, {31, 1150}, {36, 16393}, {38, 3923}, {56, 11115}, {75, 7191}, {76, 18064}, {81, 10453}, {86, 4441}, {141, 6327}, {145, 1220}, {190, 7226}, {238, 5278}, {244, 3980}, {312, 3920}, {333, 17127}, {392, 1824}, {595, 10479}, {599, 20290}, {612, 4358}, {614, 4359}, {740, 17017}, {750, 3840}, {756, 4011}, {870, 18059}, {894, 3873}, {956, 11354}, {958, 11319}, {960, 4463}, {982, 4418}, {999, 16394}, {1008, 4388}, {1010, 3616}, {1125, 3914}, {1215, 3938}, {1386, 3187}, {1479, 5051}, {1836, 17184}, {2176, 18091}, {2177, 6685}, {2887, 24943}, {2975, 4195}, {3100, 24543}, {3112, 6382}, {3219, 4676}, {3240, 3996}, {3242, 17165}, {3589, 21283}, {3662, 20292}, {3763, 21282}, {3886, 3896}, {3974, 20020}, {3995, 4387}, {4042, 19742}, {4202, 19836}, {4361, 17163}, {4362, 17469}, {4363, 17140}, {4376, 25368}, {4383, 4651}, {4414, 6682}, {4514, 17289}, {4649, 19738}, {4666, 10436}, {4670, 4883}, {4673, 17016}, {4693, 17600}, {4847, 5294}, {4865, 15523}, {4871, 17124}, {5047, 19853}, {5224, 20553}, {5248, 16342}, {5251, 11346}, {5259, 19858}, {5260, 17697}, {5271, 7290}, {5272, 24589}, {5297, 18743}, {5695, 17147}, {5710, 17751}, {6284, 17676}, {6679, 21242}, {6686, 9350}, {7292, 19804}, {7770, 20556}, {9580, 17306}, {9780, 13741}, {9812, 13727}, {11105, 11390}, {11108, 19874}, {11322, 16678}, {11393, 24989}, {13728, 15171}, {14829, 17126}, {15569, 19791}, {16405, 23853}, {16475, 17156}, {16738, 20992}, {16825, 21020}, {16975, 24275}, {17155, 17598}, {17293, 18082}, {17353, 25006}, {17398, 21956}, {17526, 19843}, {17698, 24390}, {17716, 17763}, {17871, 20886}, {17884, 20236}, {17887, 21407}, {19861, 24537}, {24540, 24545}, {24547, 24566}, {24551, 24563}

### X(24553) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24553) lies on these lines: {1, 23529}, {2, 6}, {7, 17073}, {48, 21279}, {281, 1442}, {377, 1800}, {1014, 5746}, {1125, 15299}, {1958, 3434}, {3616, 24554}, {10431, 17188}, {10446, 24580}, {17183, 24609}, {24538, 24555}

### X(24554) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24554) lies on these lines: {1, 5785}, {2, 37}, {7, 24635}, {9, 7190}, {21, 990}, {63, 4328}, {81, 2257}, {86, 16743}, {142, 22464}, {144, 1212}, {219, 7269}, {672, 7201}, {991, 10861}, {1001, 3100}, {1108, 3945}, {1423, 17451}, {1621, 4319}, {2293, 24341}, {2975, 4327}, {3616, 24553}, {3668, 5249}, {3673, 25255}, {3681, 21039}, {3729, 25082}, {3755, 24987}, {4648, 8609}, {4666, 18216}, {4859, 25065}, {5333, 24567}, {6173, 17092}, {7174, 19860}, {7229, 25066}, {11220, 17194}, {12530, 23407}, {14923, 17868}, {15988, 16973}, {16578, 20195}, {16579, 17080}, {16589, 25004}, {17019, 17811}, {17248, 25000}, {19861, 24557}, {20880, 25252}, {23681, 25080}, {24540, 24559}, {24541, 24555}

### X(24555) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24555) lies on these lines: {2, 39}, {7, 22070}, {1424, 23636}, {3177, 16588}, {24538, 24553}, {24541, 24554}, {24546, 24571}

### X(24556) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24556) lies on these lines: {2, 6}, {21, 10309}, {27, 17182}, {329, 1014}, {1010, 19861}, {1043, 17862}, {1408, 24954}, {1412, 3452}, {1434, 5905}, {1817, 17183}, {4658, 8582}, {14007, 24987}, {16054, 17167}, {16700, 25091}

### X(24557) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24557) lies on these lines: {2, 6}, {9, 1014}, {21, 3062}, {144, 1434}, {274, 25001}, {314, 20905}, {651, 5257}, {1412, 7308}, {1621, 1958}, {2297, 3305}, {3193, 17551}, {3562, 19859}, {5308, 5783}, {15601, 16948}, {16054, 17183}, {16696, 25067}, {16887, 25019}, {17260, 23617}, {19861, 24554}, {24563, 24564}

### X(24558) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24558) lies on these lines: {1, 2}, {21, 10305}, {56, 9965}, {144, 7677}, {153, 2478}, {214, 4294}, {329, 1420}, {392, 9940}, {404, 10306}, {443, 5901}, {452, 1385}, {474, 10595}, {631, 10698}, {908, 4308}, {944, 6919}, {1387, 5082}, {1388, 2551}, {1466, 5253}, {1482, 17567}, {3306, 4323}, {3434, 18220}, {3522, 4881}, {3523, 3877}, {3576, 17576}, {3672, 24540}, {3869, 5265}, {3890, 5281}, {3897, 5129}, {4187, 7967}, {4188, 10310}, {4189, 22775}, {4855, 9785}, {5046, 12667}, {5084, 10246}, {5177, 5886}, {5289, 7288}, {5328, 6049}, {5330, 6921}, {5435, 11682}, {5603, 6904}, {5731, 6260}, {5744, 15829}, {5748, 10106}, {6850, 11729}, {7963, 24177}, {8148, 17564}, {10283, 16408}, {12245, 13747}, {13462, 20214}

### X(24559) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24559) lies on these lines: {1, 2}, {297, 17923}, {1447, 1959}, {3685, 24563}, {4881, 14953}, {5249, 17084}, {16054, 17614}, {24179, 25252}, {24202, 25078}, {24203, 25083}, {24540, 24554}

### X(24560) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24560) lies on these lines: {2, 661}, {394, 23092}, {441, 525}, {693, 17215}, {4077, 10436}, {4529, 25258}, {8062, 18155}

### X(24561) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24561) lies on these lines: {2, 667}, {1010, 3309}, {1220, 20317}, {3737, 24563}, {4162, 5263}

### X(24562) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24562) lies on these lines: {2, 650}, {394, 22383}, {405, 8760}, {441, 525}, {652, 4131}, {1001, 11934}, {3126, 8641}, {3309, 8642}, {4130, 25259}, {4190, 8142}, {4765, 14838}, {6129, 16757}, {7253, 16751}, {14077, 19860}

### X(24563) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24563) lies on these lines: {2, 31}, {81, 24997}, {142, 8543}, {516, 21495}, {2323, 5750}, {2975, 20258}, {3685, 24559}, {3737, 24561}, {5847, 25007}, {7190, 10436}, {15310, 16377}, {16830, 25024}, {24551, 24552}, {24557, 24564}

### X(24564) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24564) lies on these lines: {1, 2}, {21, 18653}, {142, 3869}, {355, 16842}, {388, 3305}, {392, 8728}, {404, 7688}, {405, 18481}, {442, 9955}, {443, 3587}, {515, 5047}, {517, 17529}, {553, 11684}, {944, 17552}, {946, 4197}, {950, 5284}, {960, 3649}, {984, 23675}, {1213, 4875}, {1266, 24547}, {1512, 1656}, {1837, 8167}, {2292, 24178}, {2478, 3646}, {3057, 3826}, {3177, 17248}, {3218, 18249}, {3219, 4298}, {3340, 20195}, {3436, 7308}, {3475, 3984}, {3522, 21628}, {3615, 14005}, {3648, 4292}, {3740, 15888}, {3742, 21677}, {3876, 21620}, {4054, 19582}, {4190, 4512}, {4205, 6739}, {4297, 16865}, {4355, 20078}, {4423, 5794}, {4513, 17303}, {4881, 15674}, {5234, 20076}, {5253, 5745}, {5259, 5441}, {5260, 6666}, {5270, 5506}, {5302, 5434}, {5316, 11681}, {5484, 17260}, {5586, 12526}, {5790, 16855}, {5919, 9710}, {6175, 18483}, {6675, 13151}, {6684, 17531}, {6701, 12047}, {7354, 15254}, {7987, 12617}, {9956, 17575}, {10164, 17572}, {10176, 13407}, {10395, 13384}, {10404, 17781}, {11025, 24393}, {11108, 18525}, {12536, 24389}, {13624, 15670}, {17164, 24199}, {18250, 20060}, {24557, 24563}

### X(24565) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24565) lies on these lines: {1, 280}, {2, 3}, {86, 6527}, {189, 1394}, {253, 3945}, {285, 1498}, {347, 10436}, {388, 1359}, {1295, 11496}, {1804, 2975}, {3100, 19861}, {3616, 10538}, {4296, 19860}

### X(24566) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24566) lies on these lines: {2, 3}, {24547, 24552}, {24549, 24550}

### X(24567) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24567) lies on these lines: {2, 3}, {1778, 13567}, {5333, 24554}, {10436, 20223}

### X(24568) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24568) lies on these lines: {2,3}

### X(24569) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 1

Barycentrics    7 a^7 - a^6 b - 9 a^5 b^2 + 3 a^4 b^3 - 3 a^3 b^4 - 3 a^2 b^5 + 5 a b^6 + b^7 - a^6 c + 10 a^5 b c - 5 a^4 b^2 c - 8 a^3 b^3 c + a^2 b^4 c - 2 a b^5 c + 5 b^6 c - 9 a^5 c^2 - 5 a^4 b c^2 + 14 a^3 b^2 c^2 + 2 a^2 b^3 c^2 - 5 a b^4 c^2 + 3 b^5 c^2 + 3 a^4 c^3 - 8 a^3 b c^3 + 2 a^2 b^2 c^3 + 4 a b^3 c^3 - 9 b^4 c^3 - 3 a^3 c^4 + a^2 b c^4 - 5 a b^2 c^4 - 9 b^3 c^4 - 3 a^2 c^5 - 2 a b c^5 + 3 b^2 c^5 + 5 a c^6 + 5 b c^6 + c^7 : :

X(24569) lies on these lines: {2,3}

### X(24570) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 1

Barycentrics    3 a^7 - a^6 b - 5 a^5 b^2 + 3 a^4 b^3 + a^3 b^4 - 3 a^2 b^5 + a b^6 + b^7 - a^6 c - a^4 b^2 c + 4 a^3 b^3 c + a^2 b^4 c - 4 a b^5 c + b^6 c - 5 a^5 c^2 - a^4 b c^2 + 14 a^3 b^2 c^2 + 10 a^2 b^3 c^2 - a b^4 c^2 - b^5 c^2 + 3 a^4 c^3 + 4 a^3 b c^3 + 10 a^2 b^2 c^3 + 8 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 - 3 a^2 c^5 - 4 a b c^5 - b^2 c^5 + a c^6 + b c^6 + c^7 : :

X(24570) lies on these lines: {2, 3}, {348, 3668}, {1724, 18928}, {1935, 18915}, {1944, 3485}, {3616, 24553}, {19766, 23292}

### X(24571) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 1

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

X(24571) lies on these lines: {2, 3}, {24546, 24555}

### X(24572) =  X(3)X(12278)∩X(5)X(1986)

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

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28414.

X(24572) lies on these lines: {3,12278}, {5,1986}, {54,15059}, {68,6640}, {125,12038}, {1352,6639}, {2072,5449}, {4549,18404}, {5889,10255}, {8548,11898}, {10024,12162}, {10254,15058}, {10721,15062}, {11585,12606}

### X(24573) =  REFLECTION OF X(252) IN X(5)

Barycentrics    -a^16+5 a^14 (b^2+c^2)+(b^2-c^2)^6 (b^4-b^2 c^2+c^4)-3 a^12 (3 b^4+4 b^2 c^2+3 c^4)+4 a^10 (b^6+b^4 c^2+b^2 c^4+c^6)-a^2 (b^2-c^2)^4 (6 b^6-b^4 c^2-b^2 c^4+6 c^6)+a^8 (10 b^8+6 b^6 c^2+7 b^4 c^4+6 b^2 c^6+10 c^8)+a^4 (b^2-c^2)^2 (15 b^8-b^6 c^2-2 b^4 c^4-b^2 c^6+15 c^8)+a^6 (-19 b^10+10 b^8 c^2+10 b^2 c^8-19 c^10) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28418.

X(24573) lies on these lines: {4,11671}, {5,252}, {137,195}, {546,6288}, {5501,21230}, {10285,14140}, {14072,20120}, {16336,20414}

X(24573) = reflection of X(252) in X(5)
X(24573) = (X(i),X(j))-harmonic conjugate of X(k) for these {i,j,k}: {15307,19552,5}

Centers associated with the trilinear obverse and trilinear N-obverse triangles of X(2): X(24574) - X(24579))

These centers were contributed by Randy Hutson, October 6, 2018. Obverse and N-obverse triangles are introduced in preambles just before X(24482) and X(24519).

### X(24574) =  CENTROID OF TRILINEAR OBVERSE TRIANGLE OF X(2)

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

X(24574) lies on these lines: {2, 742}, {190, 2276}, {291, 751}, {3060, 9018}, {3809, 4562}

### X(24575) =  X(6)X(256)∩X(75)X(291)

Trilinears    a^2 b c - a b^3 - a c^3 - b^2 c^2 : :
Barycentrics    a (a^2 b c - a b^3 - a c^3 - b^2 c^2) : :

X(24575) is the perspector of the trilinear N-obverse triangle of X(2) and (cross-triangle of trilinear obverse triangle of X(2) and trilinear N-obverse triangle of X(2)).

X(24575) lies on these lines: {1, 3688}, {2, 3728}, {6, 256}, {39, 1045}, {43, 2277}, {75, 291}, {171, 2260}, {239, 3778}, {319, 2228}, {387, 986}, {740, 4283}, {894, 20456}, {982, 4000}, {983, 12329}, {1582, 17798} et al

### X(24576) =  TRILINEAR POLE OF PERSPECTRIX OF THESE TRIANGLES: TRILINEAR OBVERSE OF X(2) AND TRILINEAR N-OBVERSE OF X(2)

Trilinears    1/((a^2 - b c) (a^3 b^3 + a^3 c^3 - a^2 b^2 c^2 - b^3 c^3)) : :
Barycentrics    a/((a^2 - b c) (a^3 b^3 + a^3 c^3 - a^2 b^2 c^2 - b^3 c^3)) : :

X(24576) lies on these lines: {239, 1967}, {291, 3978}, {1580, 1922} et al

X(24576) = isogonal conjugate of X(19580)
X(24576) = isotomic conjugate of X(19581)

### X(24577) =  CENTROID OF SIDE-TRIANGLE OF THESE TRIANGLES: TRILINEAR OBVERSE OF X(2) AND TRILINEAR N-OBVERSE OF X(2)

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

X(24577) lies on these lines: {43, 798}, {661, 2276}, {1022, 9460} et al

### X(24578) =  X(1)X(39)∩X(9)X(75)

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

X(24578) is the perspector of the trilinear N-obverse triangle of X(2) and the unary cofactor triangle of the trilinear obverse triangle of X(2).

X(24578) lies on these lines: {1, 39}, {6, 1045}, {9, 75}, {40, 21384}, {57, 7075}, {99, 2311}, {165, 2319}, {191, 2795}, {194, 3212}, {239, 672}, {659, 8659}, {726, 3508}, {869, 17756}, {970, 1490}, {986, 3125}, {1050, 2176}, {1107, 5836} et al

### X(24579) =  EIGENCENTER OF TRILINEAR N-OBVERSE TRIANGLE OF X(2)

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

X(24579) lies on these lines: {1, 3511}, {6, 3503}, {9, 1966}, {40, 511}, {57, 893}, {1697, 8844} et al

Collineation mappings involving Gemini triangle 2: X(24580) - X(24638)

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 2, as in centers X(24580)-X(24638). Then

m(X) = a x - (a + c) y - (a + b) z : : , and m(X) is on the Euler line if and only if X is on the Euler line.

Fixed points: m(X(2)) = X(2), m(X(239))= X(239), m(X(649))= X(649)
Cycles: m(X(57)) = X(63), m(X(63)) = X(57), and m(X(88)) = X(190), m(X(190)) = X(88)
(Clark Kimberling, October 7, 2018)

### X(24580) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24580) lies on these lines: {2, 3}, {69, 662}, {241, 2275}, {281, 17134}, {284, 5738}, {307, 610}, {347, 653}, {348, 7291}, {673, 2164}, {910, 5813}, {911, 5773}, {1631, 11677}, {2173, 24316}, {3002, 24597}, {3101, 6349}, {3306, 5011}, {3912, 4855}, {4329, 17073}, {4466, 24683}, {5256, 8555}, {5739, 24632}, {5740, 5802}, {10446, 24553}, {18634, 18650}, {20110, 20818}, {24582, 24612}, {24583, 24596}, {24587, 24636}

### X(24581) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24581) lies on these lines: {2, 3}, {216, 23585}, {284, 5740}, {572, 25000}, {1055, 16603}, {2173, 24317}, {4359, 17864}, {4384, 24582}, {4466, 24684}, {5222, 24737}, {5741, 24632}, {7291, 17095}, {17043, 21271}, {19804, 20926}, {20305, 22054}, {20913, 24636}

### X(24582) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24582) lies on these lines: {2, 11}, {498, 17683}, {662, 24619}, {908, 3234}, {1015, 24918}, {1018, 6710}, {1146, 17136}, {1958, 25000}, {2246, 24318}, {3119, 5745}, {3936, 24602}, {4209, 11681}, {4359, 20901}, {4384, 24581}, {4437, 17780}, {4604, 17346}, {4750, 4763}, {5741, 24586}, {17044, 21272}, {19804, 20940}, {24580, 24612}

### X(24583) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24583) lies on these lines: {2, 12}, {594, 18654}, {1959, 17023}, {3869, 17397}, {4384, 24581}, {5303, 6999}, {17045, 21273}, {17327, 21286}, {17398, 20245}, {24580, 24596}

### X(24584) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24584) lies on these lines: {2, 3}, {2173, 24321}, {3088, 3547}, {4466, 24686}, {4872, 16706}, {17023, 18651}, {17492, 18636}, {18750, 24612}, {19804, 24596}

### X(24585) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24585) lies on these lines: {2, 3}, {2173, 24322}, {4466, 24687}, {6084, 24623}, {18637, 21274}

### X(24586) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 2

Barycentrics    a^4 - a b^3 - b^3 c - a c^3 - b c^3 : :

X(24586) lies on these lines: {1, 16060}, {2, 31}, {6, 24691}, {8, 4209}, {10, 11321}, {32, 21240}, {41, 17137}, {55, 3912}, {57, 85}, {63, 15487}, {65, 16822}, {69, 3684}, {75, 3509}, {81, 17208}, {141, 4386}, {142, 16992}, {183, 20335}, {239, 982}, {304, 3496}, {312, 17738}, {315, 17046}, {385, 3662}, {660, 3681}, {673, 2319}, {940, 16502}, {1150, 24592}, {1395, 15149}, {1460, 16412}, {1759, 1930}, {2243, 4376}, {2329, 21281}, {3263, 5282}, {3416, 8301}, {3500, 19806}, {3550, 17284}, {3677, 16834}, {3687, 11347}, {3721, 3905}, {3741, 20172}, {3750, 17316}, {3759, 17731}, {3831, 7770}, {3996, 17294}, {4357, 5275}, {4393, 17598}, {4416, 6180}, {4426, 20255}, {4850, 24613}, {5269, 16831}, {5329, 11329}, {5741, 24582}, {6996, 20368}, {7754, 24214}, {9310, 17152}, {14377, 20888}, {14828, 17298}, {16816, 18201}, {16826, 17716}, {16833, 18193}, {16910, 21935}, {16993, 17248}, {16995, 17236}, {17211, 20267}, {17277, 17754}, {17279, 17735}, {19591, 21371}, {19732, 24603}, {24587, 24591}, {24595, 24637}

### X(24587) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 2

Barycentrics    a^5 - a b^4 - b^4 c - a c^4 - b c^4 : :

X(24587) lies on these lines: {2, 32}, {10, 19271}, {19, 21442}, {35, 5300}, {63, 20432}, {75, 5341}, {239, 3959}, {379, 1150}, {980, 21997}, {1760, 20234}, {2175, 17138}, {2244, 4381}, {3765, 24630}, {4118, 4412}, {4202, 20083}, {4645, 5138}, {5737, 19281}, {17047, 21275}, {24580, 24636}, {24586, 24591}, {24598, 24610}, {24617, 24621}

### X(24588) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24588) lies on these lines: {2, 35}, {32, 24789}, {63, 169}, {81, 24790}, {86, 142}, {239, 3874}, {2140, 20769}, {3647, 16815}, {3912, 17682}, {4251, 5249}, {5299, 16752}, {16831, 17683}, {19808, 24603}, {24592, 24632}

### X(24589) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 2

Barycentrics    b c (4 a + b + c) : :

X(24589) lies on these lines: {2, 37}, {8, 4002}, {10, 244}, {57, 5278}, {81, 17121}, {86, 17012}, {88, 274}, {100, 16823}, {142, 3936}, {210, 17140}, {239, 16971}, {354, 4651}, {404, 16817}, {443, 5016}, {518, 17146}, {519, 17450}, {551, 3902}, {693, 14475}, {748, 3980}, {750, 16825}, {756, 24165}, {899, 24325}, {908, 24199}, {982, 4981}, {1086, 5241}, {1104, 19284}, {1150, 3306}, {1441, 3911}, {1465, 17077}, {1647, 21027}, {1698, 4968}, {2087, 22192}, {2177, 24331}, {2999, 19684}, {3006, 3826}, {3121, 16604}, {3218, 17277}, {3624, 3702}, {3626, 4935}, {3634, 3701}, {3687, 18139}, {3706, 3848}, {3707, 4031}, {3720, 3896}, {3740, 17165}, {3742, 17135}, {3758, 14997}, {3759, 14996}, {3762, 25038}, {3828, 4692}, {3840, 21020}, {3891, 5268}, {3923, 17125}, {3977, 6666}, {4054, 5316}, {4360, 17021}, {4362, 17124}, {4385, 19877}, {4389, 24184}, {4418, 17123}, {4427, 15254}, {4647, 19862}, {4682, 17150}, {4696, 9780}, {4706, 15569}, {4723, 19875}, {4742, 25055}, {4883, 20011}, {4975, 19883}, {5249, 5741}, {5253, 16824}, {5263, 7292}, {5271, 5437}, {5272, 24552}, {5739, 9776}, {5743, 17184}, {6377, 20899}, {6381, 20913}, {7081, 9342}, {7283, 17536}, {7321, 17484}, {8041, 21954}, {14210, 17023}, {16704, 16709}, {16738, 16753}, {17013, 17394}, {17259, 17595}, {17367, 20911}, {21418, 23521}, {24596, 24605}

### X(24590) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24590) lies on these lines: {1, 11349}, {2, 40}, {7, 2270}, {9, 7229}, {19, 273}, {46, 3008}, {57, 279}, {63, 169}, {77, 2262}, {198, 7190}, {239, 4209}, {910, 5228}, {942, 5256}, {1697, 5308}, {1817, 8726}, {2183, 8545}, {2999, 3339}, {3218, 24599}, {3295, 5287}, {3305, 3730}, {3306, 5011}, {3333, 17014}, {3359, 7397}, {3753, 21514}, {3895, 17316}, {5813, 9436}, {8257, 16548}, {10444, 20905}, {10521, 24177}, {11343, 19860}, {12514, 16832}, {16412, 19861}, {17092, 18725}, {21370, 24600}

### X(24591) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24591) lies on these lines: {2, 41}, {3, 3419}, {10, 19314}, {63, 6996}, {75, 16560}, {78, 21554}, {239, 4051}, {908, 23151}, {1150, 3765}, {1210, 19310}, {1726, 20882}, {1958, 17077}, {3306, 16054}, {3434, 9441}, {3911, 11329}, {4384, 24618}, {5745, 16367}, {9843, 19316}, {14829, 20917}, {16551, 20236}, {17206, 20925}, {19545, 22449}, {24586, 24587}, {24592, 24605}

### X(24592) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24592) lies on these lines: {1, 2}, {9, 4441}, {31, 20172}, {44, 24330}, {63, 20605}, {71, 20174}, {75, 672}, {76, 3691}, {274, 1475}, {310, 333}, {321, 17755}, {350, 17277}, {748, 20154}, {940, 1206}, {1086, 24690}, {1150, 24586}, {1334, 17143}, {1468, 11321}, {2238, 17348}, {2276, 4361}, {2280, 16992}, {2308, 14621}, {2350, 4359}, {2886, 4766}, {3112, 18087}, {3219, 17738}, {3686, 20335}, {3696, 8299}, {3739, 24512}, {3797, 4365}, {3883, 13576}, {3975, 18152}, {4395, 25349}, {4416, 20347}, {4423, 20156}, {4479, 17335}, {5247, 17686}, {14829, 24602}, {16552, 20888}, {16748, 18206}, {16752, 17208}, {16887, 24790}, {16994, 20180}, {17050, 17137}, {17117, 17759}, {17149, 18739}, {17152, 20257}, {17349, 24514}, {19804, 24629}, {20179, 21764}, {24588, 24632}, {24591, 24605}

### X(24593) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24593) lies on these lines: {2, 44}, {57, 321}, {88, 239}, {190, 3218}, {312, 23958}, {484, 4742}, {518, 17780}, {649, 693}, {752, 1647}, {896, 4759}, {899, 4753}, {908, 24237}, {1150, 3306}, {1155, 4702}, {1997, 20078}, {3008, 24183}, {3230, 16826}, {3262, 4359}, {3336, 3702}, {3337, 4968}, {3689, 17145}, {3891, 18193}, {3911, 3936}, {3999, 20045}, {4001, 6692}, {4031, 4054}, {4393, 4850}, {4434, 17449}, {4450, 11019}, {4706, 17162}, {4722, 6686}, {4767, 5205}, {4981, 17122}, {5087, 17491}, {5278, 5437}, {5372, 17791}, {5744, 14154}, {6327, 17728}, {9352, 10453}, {16602, 19742}, {16610, 16704}, {16729, 18206}, {17318, 17595}, {17763, 18201}

### X(24594) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 2

Barycentrics    a^3 - a b^2 + 8 a b c + b^2 c - a c^2 + b c^2 : :

X(24594) lies on these lines: {2, 45}, {1001, 4781}, {1150, 3306}, {1647, 24693}, {3218, 17335}, {3711, 17146}, {3891, 17124}, {4358, 4659}, {4359, 5437}, {4393, 24620}, {4413, 17780}, {4670, 16610}, {4759, 17125}, {4767, 24349}, {4850, 16826}, {5741, 9776}, {17020, 19738}, {17051, 21283}, {17277, 24616}, {17318, 17495}

### X(24595) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24595) lies on these lines: {2, 48}, {63, 1746}, {75, 21368}, {92, 21367}, {212, 20242}, {355, 20999}, {1150, 3765}, {1726, 14213}, {1985, 7193}, {3306, 24618}, {4384, 24606}, {4850, 24624}, {5303, 15486}, {11064, 21239}, {14206, 16551}, {20879, 21375}, {24586, 24637}

### X(24596) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24596) lies on these lines: {1, 17683}, {2, 11}, {8, 17682}, {31, 3008}, {41, 17050}, {63, 169}, {81, 277}, {142, 2280}, {220, 20244}, {239, 3873}, {333, 16748}, {1150, 24586}, {1914, 17278}, {2246, 24333}, {2975, 4209}, {3306, 24600}, {3739, 4376}, {4000, 5276}, {4512, 16832}, {5260, 17691}, {6996, 9778}, {7195, 21454}, {9310, 20257}, {9780, 17681}, {16454, 16818}, {17026, 24602}, {19804, 24584}, {21258, 21285}, {24580, 24583}, {24589, 24605}

### X(24597) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24597) lies on these lines: {2, 6}, {4, 162}, {8, 5266}, {20, 5721}, {21, 387}, {22, 5324}, {31, 3434}, {44, 17720}, {57, 15474}, {58, 377}, {63, 1723}, {77, 3911}, {88, 279}, {238, 11269}, {239, 17740}, {283, 1724}, {345, 3187}, {386, 6910}, {464, 2193}, {497, 2361}, {580, 6836}, {582, 6899}, {614, 24216}, {631, 5396}, {896, 24248}, {908, 1743}, {1104, 12649}, {1191, 10529}, {1269, 19818}, {1453, 6734}, {1465, 5435}, {1707, 3914}, {1748, 17903}, {1754, 10431}, {1760, 3218}, {1788, 4296}, {1824, 21848}, {1834, 6872}, {1865, 6994}, {1999, 5336}, {2082, 24611}, {2308, 24892}, {2550, 17126}, {3002, 24580}, {3008, 3306}, {3011, 3751}, {3015, 24608}, {3052, 20075}, {3086, 3562}, {3120, 24695}, {3193, 16471}, {3216, 6921}, {3240, 5218}, {3332, 10883}, {3436, 5230}, {3617, 5724}, {3718, 4358}, {3752, 18607}, {3769, 10327}, {3772, 4641}, {3782, 20078}, {3791, 4438}, {3875, 3977}, {4031, 17067}, {4190, 4252}, {4197, 4340}, {4259, 18191}, {4339, 5178}, {4389, 19823}, {4398, 19824}, {4442, 24280}, {4663, 17718}, {4850, 5222}, {4888, 5249}, {4902, 23681}, {4967, 5271}, {5219, 16670}, {5220, 17602}, {5231, 16469}, {5256, 5745}, {5269, 25006}, {5294, 11679}, {5706, 6837}, {5707, 6832}, {5725, 9780}, {5729, 15252}, {6776, 8229}, {6857, 19767}, {7191, 24477}, {7739, 24296}, {10449, 17526}, {10527, 16466}, {11240, 16483}, {14206, 17861}, {16342, 19766}, {16477, 17717}, {16729, 18135}, {17249, 19786}, {17250, 19832}, {17366, 17595}, {17367, 24627}, {18750, 19788}

### X(24598) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24598) lies on these lines: {2, 39}, {3, 1244}, {6, 662}, {32, 19308}, {37, 24077}, {43, 20456}, {63, 978}, {81, 386}, {86, 4261}, {87, 23659}, {99, 11320}, {192, 17053}, {239, 2275}, {291, 869}, {379, 24617}, {894, 2277}, {1015, 4393}, {1054, 17795}, {1125, 24165}, {1150, 24614}, {1575, 3661}, {1724, 1931}, {1740, 3778}, {1959, 24443}, {1964, 4446}, {2092, 17379}, {2234, 4443}, {2276, 16826}, {2309, 17065}, {3002, 24580}, {3008, 24635}, {3009, 12782}, {3095, 19522}, {3122, 24696}, {3216, 18206}, {3596, 17148}, {3666, 16604}, {3752, 17367}, {3766, 22092}, {4253, 11349}, {4359, 17030}, {4664, 8610}, {4850, 17023}, {5013, 16367}, {5069, 17277}, {5222, 24625}, {5224, 16696}, {7032, 20862}, {7976, 20352}, {9605, 16412}, {11337, 19762}, {12263, 17155}, {16700, 18134}, {16726, 17378}, {16728, 17352}, {16887, 18601}, {17027, 20868}, {17316, 17756}, {17331, 21892}, {17350, 21796}, {17363, 21857}, {17377, 21858}, {17389, 20691}, {19804, 21608}, {24587, 24610}

### X(24599) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 2

Barycentrics    5 a^2 - 2 a b + b^2 - 2 a c - 6 b c + c^2 : :

X(24599) lies on these lines: {1, 2}, {6, 4747}, {9, 4402}, {44, 4454}, {144, 673}, {333, 18600}, {346, 4361}, {379, 9965}, {391, 4000}, {597, 4470}, {966, 17325}, {1266, 6172}, {1278, 17755}, {3161, 17151}, {3177, 17490}, {3210, 25237}, {3218, 24590}, {3672, 17277}, {3752, 4875}, {3759, 3945}, {3875, 18230}, {3946, 5296}, {3973, 4488}, {4307, 4974}, {4371, 17279}, {4395, 4419}, {4405, 17269}, {4461, 17117}, {4700, 6173}, {4748, 17382}, {4859, 21296}, {4869, 5839}, {4936, 8055}, {5232, 16706}, {5435, 9312}, {5437, 17474}, {6996, 20070}, {7195, 21454}, {7222, 16669}, {7613, 24692}, {12245, 19512}, {16054, 16704}, {17314, 17337}, {20078, 24630}

### X(24600) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24600) lies on these lines: {1, 2}, {44, 24352}, {57, 658}, {142, 14548}, {728, 17158}, {1449, 14828}, {1453, 13727}, {1707, 24283}, {1743, 10025}, {2402, 3676}, {3306, 24596}, {3333, 17682}, {3361, 4209}, {3598, 5838}, {3673, 16572}, {3693, 3875}, {3975, 18153}, {4000, 9436}, {4395, 25355}, {5435, 14189}, {6173, 24694}, {7290, 14942}, {9776, 15493}, {15299, 24014}, {16750, 18206}, {17755, 20173}, {21370, 24590}

### X(24601) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24601) lies on these lines: {2, 667}, {27, 18344}, {239, 4083}, {514, 4508}, {649, 3766}, {673, 6008}, {1019, 4391}, {2483, 3261}, {2484, 4374}, {2515, 4411}, {3063, 17217}, {3250, 4107}, {3309, 6996}, {3835, 8632}, {4040, 4776}, {4063, 4384}, {4408, 10566}, {4462, 24612}, {4782, 16815}, {8630, 24533}, {8646, 25299}, {17367, 24719}, {20906, 21389}, {20981, 21191}

### X(24602) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24602) lies on these lines: {2, 31}, {100, 2725}, {172, 20255}, {239, 244}, {320, 3570}, {333, 873}, {673, 4607}, {799, 3975}, {1019, 4391}, {1086, 4396}, {1150, 3306}, {2177, 17316}, {2243, 24358}, {3218, 17755}, {3263, 3509}, {3662, 16997}, {3691, 17206}, {3760, 14377}, {3831, 17686}, {3936, 24582}, {4358, 17738}, {4372, 20271}, {4495, 4817}, {5011, 14210}, {5235, 6629}, {5277, 21240}, {5280, 24170}, {9310, 21281}, {10448, 22267}, {14829, 24592}, {16784, 17023}, {17026, 24596}, {17680, 21935}, {24612, 24615}

### X(24603) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24603) lies on these lines: {1, 2}, {6, 4798}, {37, 4431}, {44, 4472}, {75, 4044}, {76, 19804}, {86, 3686}, {142, 4751}, {165, 7406}, {192, 3986}, {274, 3975}, {333, 1509}, {346, 5936}, {379, 5179}, {516, 7384}, {527, 17256}, {594, 4698}, {673, 1268}, {908, 24633}, {950, 16053}, {958, 16412}, {966, 4416}, {993, 11329}, {1086, 1213}, {1100, 6707}, {1211, 21240}, {1266, 4364}, {1654, 3664}, {2321, 4687}, {2345, 25101}, {3305, 3730}, {3501, 7308}, {3619, 20195}, {3663, 4699}, {3666, 16589}, {3674, 16609}, {3691, 18206}, {3707, 3758}, {3723, 4399}, {3729, 5296}, {3740, 20683}, {3797, 25354}, {3842, 4439}, {3879, 15668}, {3883, 20156}, {3925, 20544}, {3943, 4755}, {3946, 17322}, {3948, 4359}, {4021, 17117}, {4022, 21699}, {4058, 17242}, {4060, 17315}, {4363, 4480}, {4383, 17750}, {4395, 25358}, {4407, 24325}, {4413, 21477}, {4473, 17260}, {4648, 17270}, {4667, 17346}, {4670, 17330}, {4675, 17251}, {4690, 17392}, {4700, 4758}, {4733, 15569}, {4739, 17246}, {4741, 4896}, {4748, 17274}, {4772, 17247}, {4887, 17254}, {5232, 17298}, {5235, 6629}, {5249, 17746}, {5251, 21511}, {5260, 11349}, {5267, 19308}, {5717, 14007}, {5743, 20255}, {5745, 6626}, {5750, 17277}, {5814, 16456}, {6376, 18739}, {6381, 20913}, {6537, 20337}, {6684, 6996}, {6999, 19925}, {7227, 16814}, {7377, 10175}, {8185, 11340}, {10022, 16590}, {11353, 20172}, {16752, 17210}, {17067, 17305}, {17238, 21255}, {17239, 17245}, {17259, 17303}, {17278, 17327}, {17280, 25072}, {17337, 17385}, {17348, 17398}, {19732, 24586}, {19808, 24588}

X(24603) = complement of X(16826)

### X(24604) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24604) lies on these lines: {2, 3}, {7, 610}, {19, 347}, {57, 279}, {198, 8232}, {284, 3945}, {673, 1436}, {910, 948}, {942, 17014}, {1448, 2999}, {1730, 4253}, {2182, 10402}, {2270, 7013}, {3008, 15803}, {3188, 5435}, {3601, 5308}, {3668, 18594}, {4258, 5712}, {4384, 5088}, {5745, 23058}, {9776, 17023}, {14552, 24632}

### X(24605) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24605) lies on these lines: {2, 3}, {305, 799}, {607, 18659}, {3162, 17903}, {4384, 21370}, {5089, 17134}, {14189, 17080}, {18596, 20235}, {24589, 24596}, {24591, 24592}

### X(24606) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24606) lies on these lines: {2, 3}, {7, 2303}, {63, 1930}, {81, 17170}, {190, 5279}, {272, 673}, {284, 18650}, {333, 7291}, {1172, 4329}, {1474, 18589}, {3188, 17080}, {4280, 16580}, {4384, 24595}

### X(24607) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24607) lies on these lines: {2, 3}, {81, 18652}, {284, 18634}, {307, 651}, {333, 348}, {347, 2322}, {1172, 17073}, {1440, 16713}, {2897, 18643}, {4384, 17080}, {5235, 24635}, {7177, 18206}

### X(24608) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24608) lies on these lines: {2, 3}, {3015, 24597}, {7291, 17079}

### X(24609) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24609) lies on these lines: {2, 3}, {32, 5712}, {57, 348}, {69, 284}, {333, 3926}, {387, 19758}, {579, 3618}, {610, 4357}, {1119, 17086}, {3053, 17056}, {3601, 3912}, {3785, 18134}, {3933, 14552}, {4384, 5745}, {4648, 5337}, {4850, 5222}, {5273, 25083}, {5737, 7789}, {5746, 8822}, {12437, 17294}, {16834, 24391}, {17183, 24553}, {17316, 24929}

### X(24610) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24610) lies on these lines: {2, 3}, {239, 24443}, {1054, 4384}, {1150, 24621}, {1958, 3662}, {4466, 24726}, {7187, 7291}, {14829, 24378}, {17023, 24169}, {17797, 17872}, {20913, 24614}, {24587, 24598}

### X(24611) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24611) lies on these lines: {1, 11337}, {2, 19}, {3, 1829}, {9, 21368}, {21, 7713}, {22, 1040}, {30, 5155}, {33, 20243}, {34, 16049}, {40, 78}, {46, 386}, {57, 77}, {63, 573}, {92, 23512}, {165, 5314}, {193, 3218}, {224, 3430}, {608, 1465}, {908, 1766}, {1062, 2915}, {1150, 24633}, {1214, 11350}, {1452, 4225}, {1726, 21363}, {1753, 6838}, {1824, 19544}, {2082, 24597}, {2172, 16586}, {3149, 8251}, {3306, 5011}, {4228, 5338}, {4384, 24595}, {5057, 12717}, {5119, 17461}, {5219, 16548}, {5744, 7291}, {6197, 6988}, {10444, 18690}, {18651, 20266}, {21361, 21375}, {24589, 24596}

### X(24612) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24612) lies on these lines: {2, 12}, {6, 20245}, {8, 6996}, {9, 24547}, {63, 169}, {141, 21286}, {144, 673}, {218, 329}, {239, 3869}, {527, 1405}, {604, 21246}, {908, 16788}, {1150, 3765}, {1229, 5227}, {1766, 20895}, {2285, 24993}, {2995, 20891}, {3008, 12527}, {3421, 7397}, {3434, 7406}, {3452, 9310}, {3661, 5176}, {3687, 5016}, {4361, 21273}, {4462, 24601}, {4696, 11679}, {5080, 7377}, {5228, 20347}, {5235, 5744}, {5773, 23151}, {7384, 11680}, {11682, 16834}, {11684, 16816}, {12526, 16833}, {12531, 20055}, {16777, 18654}, {16832, 17683}, {18750, 24584}, {19798, 20921}, {24580, 24582}, {24602, 24615}

### X(24613) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24613) lies on these lines: {2, 82}, {306, 4097}, {4384, 24637}, {4850, 24586}, {16555, 20898}, {16784, 17023}

### X(24614) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 2

Barycentrics    a^5 - a^3 b^2 - a^2 b^3 - a b^4 - a^2 b^2 c - b^4 c - a^3 c^2 - a^2 b c^2 - a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 - a c^4 - b c^4 : :

X(24614) lies on these lines: {2, 32}, {10, 36}, {141, 662}, {239, 3752}, {1150, 24598}, {2244, 24340}, {3661, 18047}, {3662, 18048}, {16784, 17023}, {16830, 19270}, {17055, 21289}, {20913, 24610}

### X(24615) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24615) lies on these lines: {2, 87}, {63, 20370}, {1150, 17026}, {2319, 17149}, {4384, 24620}, {4598, 6376}, {5209, 24632}, {6382, 16557}, {24602, 24612}

### X(24616) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24616) lies on these lines: {2, 44}, {8, 4781}, {63, 4659}, {190, 1150}, {3218, 4384}, {3219, 5372}, {3240, 4753}, {3679, 24344}, {4393, 16704}, {4689, 20048}, {4767, 5220}, {4793, 16558}, {5278, 23958}, {14996, 16826}, {14997, 24627}, {17057, 21291}, {17277, 24594}

### X(24617) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24617) lies on these lines: {2, 99}, {36, 759}, {81, 17205}, {86, 17382}, {88, 4049}, {105, 4236}, {110, 5091}, {162, 14119}, {239, 9278}, {244, 19642}, {274, 21997}, {333, 16759}, {379, 24598}, {448, 24781}, {523, 897}, {645, 4440}, {662, 1086}, {673, 24625}, {757, 17366}, {1414, 5723}, {1931, 3008}, {1963, 3946}, {2226, 4615}, {3015, 24597}, {5196, 7292}, {6185, 14953}, {6626, 16815}, {17058, 21221}, {17103, 17367}, {20913, 24610}, {24587, 24621}

### X(24618) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24618) lies on these lines: {1, 21554}, {2, 101}, {11, 5091}, {36, 80}, {57, 1111}, {88, 4049}, {294, 3008}, {572, 17077}, {651, 24237}, {668, 14829}, {673, 812}, {911, 1375}, {1056, 24222}, {1156, 3667}, {1565, 5723}, {2398, 10695}, {2481, 6996}, {3306, 24595}, {3739, 24324}, {4384, 24591}, {4657, 25367}, {4858, 16560}, {4986, 11679}, {5435, 13478}, {5662, 11998}, {9708, 19261}, {16832, 24685}, {17059, 21293}, {17181, 17367}, {17595, 24296}

### X(24619) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24619) lies on these lines: {2, 98}, {81, 17198}, {116, 163}, {150, 5546}, {333, 16703}, {662, 24582}, {673, 24624}, {850, 1821}, {2247, 24347}, {2249, 2864}, {4237, 6740}, {16562, 17886}, {21253, 24917}

### X(24620) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 2

Barycentrics    a^2 b + a b^2 + a^2 c - 5 a b c - b^2 c + a c^2 - b c^2 : :

X(24620) lies on these lines: {2, 37}, {8, 244}, {88, 330}, {145, 3812}, {194, 16815}, {236, 16018}, {239, 3306}, {354, 20012}, {404, 19851}, {899, 24349}, {1054, 16825}, {1086, 5233}, {1122, 21454}, {1266, 5316}, {1999, 5437}, {2550, 5211}, {3177, 5744}, {3218, 17349}, {3241, 17450}, {3617, 17480}, {3622, 4719}, {3687, 21255}, {3711, 24841}, {3720, 4734}, {3753, 20037}, {3891, 9342}, {3977, 17338}, {4384, 24615}, {4389, 5241}, {4393, 24594}, {5212, 5542}, {5222, 17497}, {8056, 11679}, {9335, 17135}, {9534, 24046}, {9776, 17778}, {10453, 17063}, {16569, 24165}, {16704, 16710}, {16753, 17178}, {16817, 19278}, {17012, 17379}, {17028, 19565}, {17230, 20255}, {17277, 17595}, {17367, 21216}, {17722, 24693}, {17749, 24176}, {19742, 23958}, {19998, 22295}

### X(24621) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24621) lies on these lines: {1, 3210}, {2, 39}, {57, 239}, {63, 16827}, {69, 18827}, {81, 16915}, {192, 3247}, {347, 3164}, {385, 11329}, {869, 24349}, {988, 16823}, {1107, 19804}, {1150, 24610}, {1575, 20917}, {1730, 4209}, {2234, 24351}, {3009, 17155}, {3122, 24717}, {3687, 24215}, {3705, 23682}, {4189, 23407}, {4384, 24615}, {4393, 14996}, {4699, 17148}, {5361, 16816}, {6904, 20018}, {7520, 19851}, {7754, 16412}, {7783, 16367}, {7793, 19308}, {9902, 20340}, {12251, 19522}, {16349, 16994}, {16574, 17349}, {16710, 17379}, {16831, 25264}, {16998, 21511}, {17065, 21299}, {17114, 20036}, {17316, 17759}, {17394, 20170}, {21218, 24635}, {24587, 24617}

### X(24622) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24622) lies on these lines: {2, 647}, {75, 3700}, {512, 25299}, {523, 18154}, {649, 3766}, {650, 3261}, {661, 4374}, {1021, 4384}, {1577, 17899}, {1639, 21611}, {3239, 20907}, {3661, 21719}, {3757, 4477}, {4025, 4391}, {4086, 4458}, {4359, 4467}, {4394, 4408}, {4522, 23684}, {4841, 7199}, {4976, 20954}, {6590, 20906}, {7180, 17066}, {9404, 17277}, {17069, 18155}, {20891, 21437}, {20909, 21433}, {20923, 21610}

### X(24623) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24623) lies on these lines: {2, 659}, {88, 673}, {239, 891}, {335, 875}, {514, 4508}, {649, 693}, {890, 4366}, {900, 4784}, {918, 7192}, {1960, 16826}, {2483, 7199}, {2786, 4932}, {2826, 6996}, {4374, 21389}, {4375, 4728}, {4384, 21385}, {4393, 21343}, {4724, 4776}, {6084, 17494}, {16729, 17755}, {18197, 21391}, {24593, 24628}

### X(24624) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = GEMINI TRIANGLE 2

Barycentrics    (a + b) (a + c) (a^2 - a b + b^2 - c^2) (a^2 - b^2 - a c + c^2) : :
Trilinears    1/(sin(A - B) + sin(A - C)) : :

X(24624) lies on these lines: {2, 662}, {4, 162}, {5, 60}, {10, 21}, {11, 110}, {27, 653}, {58, 3585}, {76, 799}, {81, 226}, {83, 4599}, {86, 4604}, {88, 4049}, {98, 8229}, {125, 24916}, {149, 643}, {190, 321}, {229, 1210}, {239, 11611}, {265, 1175}, {267, 2166}, {655, 16548}, {658, 1434}, {660, 17763}, {673, 24619}, {823, 2052}, {897, 5466}, {1090, 2625}, {1098, 5046}, {1109, 21381}, {1155, 5196}, {1171, 1989}, {1325, 1737}, {1411, 5331}, {1465, 18609}, {1492, 14009}, {1746, 2051}, {1837, 11101}, {1931, 11608}, {2194, 14008}, {2349, 2394}, {2580, 2593}, {2581, 2592}, {2606, 2608}, {2617, 5400}, {2651, 5057}, {3109, 12019}, {3257, 4080}, {3448, 8286}, {3583, 5127}, {3615, 7741}, {3911, 18653}, {4052, 4921}, {4359, 8052}, {4467, 14223}, {4606, 5325}, {4850, 24595}, {5297, 7474}, {5333, 18645}, {5741, 7058}, {5972, 24955}, {7951, 9275}, {10826, 13746}, {13576, 14956}, {14204, 15343}, {14628, 18163}, {19810, 20566}

X(24624) = isogonal conjugate of X(2245)
X(24624) = isotomic conjugate of X(3936)
X(24624) = polar conjugate of X(860)
X(24624) = trilinear pole of line X(1)X(523) (the perspectrix of Gemini triangles 10 and 27)

### X(24625) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24625) lies on these lines: {1, 3799}, {2, 668}, {88, 1022}, {214, 5313}, {239, 292}, {673, 24617}, {2229, 17029}, {2275, 17367}, {3123, 4499}, {3248, 3888}, {3779, 7189}, {4383, 9259}, {4585, 9456}, {5069, 17380}, {5222, 24598}, {6184, 17014}, {8632, 20332}, {8649, 14997}, {9460, 16610}, {16604, 16815}, {17292, 17448}, {17795, 20985}, {19945, 24722}

### X(24626) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24626) lies on these lines: {2, 669}, {649, 3766}, {3004, 6084}, {4384, 18197}

### X(24627) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24627) lies on these lines: {1, 19278}, {2, 7}, {8, 988}, {10, 1054}, {11, 24723}, {21, 22344}, {38, 7081}, {75, 17595}, {81, 20228}, {88, 274}, {171, 6682}, {191, 19864}, {239, 980}, {244, 16823}, {312, 17262}, {320, 5718}, {333, 3752}, {405, 23085}, {740, 17593}, {750, 16830}, {752, 17722}, {846, 3840}, {940, 17394}, {942, 19270}, {982, 3757}, {984, 5205}, {1155, 5263}, {1401, 9564}, {1999, 3666}, {2329, 23622}, {3187, 5372}, {3210, 11679}, {3230, 16826}, {3336, 19863}, {3661, 17740}, {3685, 4414}, {3741, 17596}, {3769, 17599}, {3883, 5211}, {3916, 13740}, {3936, 17288}, {3977, 17280}, {4054, 4440}, {4195, 4652}, {4201, 6734}, {4203, 22060}, {4358, 17261}, {4362, 17591}, {4384, 24615}, {4388, 24239}, {4389, 17720}, {4643, 5233}, {4655, 17717}, {4734, 17156}, {5241, 17256}, {5271, 17148}, {5439, 11110}, {5708, 19273}, {5737, 19804}, {6292, 17292}, {10453, 17594}, {14997, 24616}, {15934, 19279}, {16610, 17277}, {16704, 17012}, {16817, 24046}, {16824, 24443}, {17020, 19742}, {17117, 17495}, {17367, 24597}, {18201, 24325}, {19243, 23169}, {19260, 23206}, {21242, 24715}

### X(24628) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24628) lies on these lines: {1, 21937}, {2, 896}, {57, 85}, {171, 3747}, {190, 3509}, {239, 18201}, {291, 4753}, {320, 24685}, {325, 4987}, {335, 4434}, {649, 693}, {982, 4393}, {1155, 3712}, {3218, 17755}, {4771, 17731}, {5221, 16822}, {6629, 16611}, {17316, 17601}, {17335, 17754}

### X(24629) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24629) lies on these lines: {2, 38}, {239, 750}, {1150, 3306}, {4359, 17026}, {4850, 17023}, {5222, 17497}, {5253, 16822}, {16818, 24046}, {17316, 17450}, {19804, 24592}, {20331, 24357}

### X(24630) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24630) lies on these lines: {2, 36}, {27, 3687}, {63, 169}, {83, 226}, {101, 908}, {239, 758}, {515, 3912}, {527, 666}, {1019, 4391}, {1111, 3218}, {1150, 3761}, {1441, 16566}, {1836, 24264}, {2242, 17720}, {2276, 24296}, {2323, 17139}, {3262, 16548}, {3573, 5057}, {3765, 24587}, {5222, 5905}, {5279, 20236}, {5745, 6626}, {5773, 20347}, {5792, 23151}, {7397, 12115}, {8680, 17755}, {11364, 17719}, {17209, 24632}, {17798, 20544}, {20078, 24599}

### X(24631) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24631) lies on these lines: {1, 16061}, {2, 38}, {10, 17670}, {39, 712}, {56, 16822}, {57, 85}, {75, 17026}, {76, 17048}, {171, 239}, {354, 3703}, {740, 17027}, {894, 3329}, {985, 4974}, {1015, 24254}, {1086, 25345}, {1475, 20911}, {2350, 4359}, {3290, 17353}, {3509, 17277}, {3670, 16818}, {3677, 16831}, {3687, 4260}, {3736, 5256}, {3752, 16606}, {3759, 4771}, {3980, 20172}, {4393, 17716}, {4697, 14621}, {4734, 17014}, {5222, 17490}, {5269, 16834}, {9436, 24199}, {10180, 17397}, {16600, 24166}, {16815, 18201}, {16826, 17598}, {16832, 18193}, {17369, 25368}, {20331, 24326}, {20610, 24511}

### X(24632) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24632) lies on these lines: {2, 58}, {8, 14953}, {21, 3912}, {27, 318}, {35, 306}, {57, 85}, {63, 1930}, {69, 284}, {81, 5299}, {86, 17227}, {141, 1333}, {239, 3670}, {286, 20236}, {307, 651}, {314, 17738}, {334, 2311}, {344, 4877}, {379, 1150}, {524, 4273}, {599, 3285}, {1010, 17308}, {1043, 17294}, {1211, 5277}, {1761, 18697}, {1778, 17353}, {1817, 3687}, {2303, 4357}, {3286, 21477}, {3662, 17189}, {3729, 8822}, {4253, 5278}, {4267, 11343}, {4269, 16574}, {4276, 21511}, {4278, 21495}, {4281, 16060}, {4388, 17188}, {4653, 17316}, {5021, 19732}, {5209, 24615}, {5222, 16704}, {5235, 6629}, {5308, 17588}, {5333, 17200}, {5739, 24580}, {5741, 24581}, {10479, 19281}, {11110, 16831}, {14552, 24604}, {14829, 18148}, {16047, 17266}, {16050, 17284}, {17052, 21287}, {17209, 24630}, {21024, 24271}, {21245, 24890}, {24588, 24592}

### X(24633) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24633) lies on these lines: {2, 65}, {9, 24993}, {37, 21273}, {63, 169}, {71, 1229}, {144, 4699}, {239, 2975}, {673, 11683}, {908, 24603}, {1150, 24611}, {1400, 21233}, {1441, 16574}, {1959, 17023}, {2262, 16713}, {3008, 24443}, {3177, 17490}, {3666, 20247}, {3687, 9568}, {3739, 20245}, {4850, 5222}, {4852, 18654}, {5057, 7384}, {5086, 6999}, {5745, 17451}, {11682, 16831}, {11684, 16815}, {12526, 16832}, {17275, 21286}, {17755, 20892}, {20895, 21061}, {21371, 25001}

### X(24634) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 2

Barycentrics    3 a^9 + a^8 b - 2 a^5 b^4 - 2 a^4 b^5 - a b^8 + b^9 + a^8 c - b^8 c - 2 a^5 c^4 + 2 a b^4 c^4 - 2 a^4 c^5 - a c^8 - b c^8 + c^9 : :

X(24634) lies on these lines: {2, 66}, {315, 4599}, {17068, 21288}

### X(24635) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24635) lies on these lines: {1, 21}, {2, 85}, {3, 7291}, {7, 24554}, {8, 25083}, {9, 77}, {37, 144}, {57, 17451}, {71, 18161}, {75, 16713}, {92, 15149}, {142, 17092}, {169, 11349}, {219, 1442}, {220, 394}, {227, 18231}, {277, 15474}, {294, 21511}, {321, 25242}, {326, 2287}, {329, 5308}, {379, 5088}, {572, 16551}, {579, 18726}, {672, 7146}, {756, 11678}, {857, 17181}, {958, 4296}, {966, 5942}, {1005, 2000}, {1108, 3672}, {1214, 3160}, {1444, 1760}, {1743, 25065}, {2267, 16560}, {2310, 24708}, {2340, 3681}, {2346, 8271}, {3000, 24341}, {3008, 24598}, {3101, 5584}, {3218, 5228}, {3661, 6184}, {3666, 17014}, {3912, 25082}, {3928, 16579}, {3929, 16577}, {3998, 14552}, {4419, 8609}, {4511, 23151}, {4517, 23612}, {4558, 7054}, {4850, 5222}, {5235, 24607}, {5249, 10481}, {5745, 17080}, {10025, 16826}, {13726, 18732}, {14021, 17170}, {14953, 16699}, {15853, 25091}, {17147, 17158}, {17272, 25078}, {17284, 24036}, {17447, 20992}, {20173, 25237}, {21218, 24621}

### X(24636) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24636) lies on these lines: {2, 99}, {4359, 17886}, {17292, 24384}, {17397, 20538}, {19804, 20951}, {20913, 24581}, {24580, 24587}

### X(24637) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(560), WHERE A'B'C' = GEMINI TRIANGLE 2

Barycentrics    a^6-a b^5-b^5 c-a c^5-b c^5 : :

X(24637) lies on these lines: {2, 560}, {4384, 24613}, {20627, 21366}, {24586, 24595}

### X(24638) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 2

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

X(24638) lies on these lines: {2, 187}, {379, 1150}, {649, 3766}, {2245, 17277}, {3285, 15668}, {3734, 11351}, {4257, 17679}, {8626, 21241}, {16568, 20912}

### X(24639) = (name pending)

Barycentrics    -(a^4 (b^2+c^2)-2 a^3 b c (b+c)-2 a^2 (b^4-b^3 c-b^2 c^2-b c^3+c^4)+2 a b c (b-c)^2 (b+c)+(b-c)^4 (b+c)^2) (2 a^9-4 a^8 (b+c)+12 a^7 b c+a^6 (5 b^3-9 b^2 c-9 b c^2+5 c^3)-a^5 (5 b^4+2 b^3 c-16 b^2 c^2+2 b c^3+5 c^4)+a^4 (b-c)^2 (2 b^3+11 b^2 c+11 b c^2+2 c^3)+2 a^3 (b-c)^2 (b^4-3 b^3 c-4 b^2 c^2-3 b c^3+c^4)-a^2 (b-c)^2 (3 b^5-b^4 c-4 b^3 c^2-4 b^2 c^3-b c^4+3 c^5)+a (b^2-c^2)^4-b c (b-c)^4 (b+c)^3) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28423.

X(24639) lies on this line: {1830,7649}

### X(24640) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24640) lies on these lines:

### X(24641) = (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24641) lies on these lines:

### X(24642) = (name pending)

Barycentrics    -a (a+b-c) (a-b+c) (a^7 (b-c)^2+2 a^6 b c (b+c)+b (b-c)^2 c (b+c)^3 (b^2-3 b c+c^2)-a^5 (b+c)^2 (3 b^2-2 b c+3 c^2)+3 a^4 b c (3 b^3+b^2 c+b c^2+3 c^3)-a (b^2-c^2)^2 (b^4-4 b^3 c-3 b^2 c^2-4 b c^3+c^4)+a^2 b c (-12 b^5+9 b^4 c+5 b^3 c^2+5 b^2 c^3+9 b c^4-12 c^5)+a^3 (3 b^6+2 b^5 c-16 b^4 c^2+6 b^3 c^3-16 b^2 c^4+2 b c^5+3 c^6)) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28423.

X(24642) lies on this line: {8758,22465}

### X(24643) = (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = N-OBVERSE TRIANGLE OF X(1)

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

X(24643) lies on these lines:

### X(24644) = X(1)X(971)∩X(2)X(165)

Barycentrics    a (3 a^5-5 a^4 b-2 a^3 b^2+6 a^2 b^3-a b^4-b^5-5 a^4 c+20 a^3 b c-6 a^2 b^2 c+4 a b^3 c-13 b^4 c-2 a^3 c^2-6 a^2 b c^2-6 a b^2 c^2+14 b^3 c^2+6 a^2 c^3+4 a b c^3+14 b^2 c^3-a c^4-13 b c^4-c^5) : :
X(24644) = 4 X[1001] - X[2951], 2 X[1] + X[3062], X[144] + 2 X[4301], 4 X[946] - X[4312], 2 X[390] + X[5691], 2 X[5779] + X[7982], 8 X[1001] - 5 X[7987], 2 X[2951] - 5 X[7987], 4 X[2550] - 7 X[7989], 4 X[9] - X[7991], 2 X[5759] + X[9589], X[3062] - 4 X[11372], X[1] + 2 X[11372], 2 X[7] - 5 X[11522], 2 X[5223] + X[11531], X[7995] + 2 X[12560], 2 X[1156] + X[13253], 2 X[9856] + X[14100], 4 X[5572] - X[15071], X[3243] + 2 X[16112], 7 X[9588] - 10 X[18230], 2 X[12672] + X[18412], 3 X[165] - 4 X[21153]

X(24644) lies on the cubic K1083 and these lines: {1,971}, {2,165}, {4,5726}, {7,11522}, {9,5836}, {88,24637}, {144,4301}, {390,5691}, {518,11224}, {946,3361}, {954,1750}, {962,5234}, {990,16487}, {1001,2951}, {1012,13462}, {1156,11526}, {1709,10980}, {2550,7989}, {2800,10398}, {3243,16112}, {3339,12705}, {3656,5843}, {3679,5817}, {3731,12652}, {3869,4853}, {4321,9814}, {4326,8543}, {5175,6736}, {5572,15071}, {5759,9589}, {5779,7982}, {5805,15803}, {5833,6837}, {5880,16209}, {7965,9580}, {7995,12560}, {9588,18230}, {9819,15298}, {9955,10270}, {10268,22793}, {10679,18529}, {12672,18412}, {12699,15909}

X(24644) = reflection of X(3679) in X(5817)
X(24644) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 11372, 3062), (1001, 2951, 7987)

### X(24645) = X(7)X(1699)∩X(63)X(5303)

Barycentrics    a (3 a^5+3 a^4 b-18 a^3 b^2+6 a^2 b^3+15 a b^4-9 b^5+3 a^4 c+20 a^3 b c-6 a^2 b^2 c-28 a b^3 c+11 b^4 c-18 a^3 c^2-6 a^2 b c^2+26 a b^2 c^2-2 b^3 c^2+6 a^2 c^3-28 a b c^3-2 b^2 c^3+15 a c^4+11 b c^4-9 c^5) : :
X(24645) = 2 X[3339] + X[9851]

X(24645) lies on the cubic K1083 and these lines: {7,1699}, {63,5303}, {88,24637}, {145,4297}, {165,518}, {515,3339}, {1071,3361}, {1768,14151}, {3873,11224}, {5223,10164}, {5603,7992}, {5658,10398}, {5691,21454}, {5745,11407}, {6667,11219}, {7967,9819}, {7989,9776}, {7995,12563}, {10582,13243}, {10860,11531}

### X(24646) = X(1)X(3308)∩X(2)X(11)

Barycentrics    (a-b-c) (a (a^4-a^3 b-a^2 b^2+a b^3-a^3 c+3 a^2 b c-a b^2 c+b^3 c-a^2 c^2-a b c^2-2 b^2 c^2+a c^3+b c^3)+2 (a^2+b^2-2 b c+c^2) Sqrt[R (-2 r+R)] S) : :
X(24646) = 3 (1 - R / Sqrt[R (R - 2 r)]) X[2] - 2 X[11]

X(24646) lies on the cubics K040 and K1083, and on and these lines: {1,3308}, {2,11}, {3,23477}, {9,2590}, {516,1381}, {518,2446}, {1382,3911}, {3295,23517}

X(24646) = X(i)-Hirst inverse of X(j) for these (i,j): {1, 3308}, {9, 3307}
X(24646) = X(9)-line conjugate of X(2590)
X(24646) = crossdifference of every pair of points on line {665, 2590}

### X(24647) = X(1)X(3302)∩X(2)X(11)

Barycentrics    (a-b-c) (a (a^4-a^3 b-a^2 b^2+a b^3-a^3 c+3 a^2 b c-a b^2 c+b^3 c-a^2 c^2-a b c^2-2 b^2 c^2+a c^3+b c^3)-2 (a^2+b^2-2 b c+c^2) Sqrt[R (-2 r+R)] S) : : : :
X(24647) = 3 (1 + R / Sqrt[R (R - 2 r)]) X[2] - 2 X[11]

X(24647) lies on the cubics K040 and K1083, and on and these lines: {1,3307}, {2,11}, {3,23517}, {9,2591}, {516,1382}, {518,2447}, {1381,3911}, {3295,23477}

X(24647) = X(i)-Hirst inverse of X(j) for these (i,j): {1, 3307}, {9, 3308}
X(24647) = X(9)-line conjugate of X(2591)
X(24647) = crossdifference of every pair of points on line {665, 2591}

### X(24648) = X(7)X(3308)∩X(57)X(100)

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

X(24648) lies on the cubic K1083 and these lines: {7,3308}, {57,100}, {145,3307}, {518,2447}, {1381,3218}

### X(24649) = X(7)X(3307)∩X(57)X(100)

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

X(24649) lies on the cubic K1083 and these lines: {7,3307}, {57,100}, {145,3308}, {518,2446}, {1382,3218}

### X(24650) = X(5)X(2575)∩X(6)X(15461)

Barycentrics    a^2 ((a^2-b^2) (a^2-c^2)+(a^2 b^2-b^4+a^2 c^2+b^2 c^2-c^4) J) : : : :
X(24650) = (3 + J^2) X[25] - J (J - 1) X[110]

X(24650) lies on the cubics K289 and K907, and on these lines: {4,2575}, {6,15461}, {24,15460}, {25,110}, {51,13415}, {193,2574}, {511,1113}, {1312,3580}, {1345,5640}, {1347,19130}, {2104,2393}

### X(24651) = X(5)X(2575)∩X(6)X(15460)

Barycentrics    a^2 ((a^2-b^2) (a^2-c^2)-(a^2 b^2-b^4+a^2 c^2+b^2 c^2-c^4) J) : : : :
X(24651) = (3 + J^2) X[25] - J (J + 1) X[110]

X(24651) lies on the cubics K289 and K907, and on these lines: {4,2574}, {6,15460}, {24,15461}, {25,110}, {51,13414}, {193,2575}, {511,1114}, {1313,3580}, {1344,5640}, {1346,19130}, {2105,2393}

Collineation mappings involving Gemini triangle 3: X(24652) - X(24679)

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 3, as in centers X(24652)-X(24679). Then

m(X) = a (a b + a c - b c) x + (a^2 b - a^2 c + a b ^2 + b^2 c) y + (a^2 c - a^2 b + a c ^2 + c^2 b) z : :

(Clark Kimberling, October 8, 2018)

### X(24652) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24652) lies on these lines: {1, 20255}, {2, 17448}, {8, 24770}, {141, 21214}, {1125, 16846}, {1193, 4851}, {1279, 3616}, {2176, 24691}, {2275, 17279}, {3208, 25350}, {4713, 24215}, {9259, 24549}, {24653, 24655}, {24657, 24670}

### X(24653) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24653) lies on these lines: {1, 3836}, {2, 3728}, {1125, 19518}, {3741, 17278}, {3757, 16986}, {4657, 20335}, {24174, 24369}, {24652, 24655}, {24658, 24660}, {24661, 24663}, {24671, 24672}

### X(24654) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 3

Barycentrics    a^3 b + a b^3 + a^3 c - 7 a^2 b c - a b^2 c + b^3 c - a b c^2 - 2 b^2 c^2 + a c^3 + b c^3 : :

X(24654) lies on these lines: {1, 142}, {2, 17448}, {7, 16969}, {330, 344}, {2176, 4644}, {3009, 3475}, {4419, 24215}, {4851, 20036}, {4916, 20018}, {5819, 16524}, {5839, 16827}, {24661, 24669}, {24662, 24677}, {24672, 24679}, {25136, 25295}

### X(24655) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24655) lies on these lines: {2, 3056}, {1284, 5880}, {1376, 15668}, {1818, 5794}, {2550, 3616}, {2886, 17245}, {3035, 17398}, {3925, 24551}, {4039, 4851}, {4645, 11375}, {4648, 4682}, {4660, 5886}, {4871, 17265}, {5432, 22370}, {17300, 17718}, {24652, 24653}, {24661, 24671}, {24663, 24672}, {24735, 25106}

### X(24656) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24656) lies on these lines: {1, 21264}, {2, 17448}, {8, 3739}, {37, 1655}, {85, 24357}, {86, 17752}, {274, 20691}, {551, 3934}, {1100, 17033}, {1334, 4754}, {2295, 4670}, {3208, 4363}, {3616, 20530}, {3780, 17348}, {3788, 10197}, {4039, 4682}, {4085, 24366}, {4664, 20081}, {4675, 21281}, {10987, 16919}, {16827, 21904}, {17137, 17376}, {17792, 25124}, {24215, 25349}, {24661, 24662}

### X(24657) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24657) lies on these lines: {2, 24525}, {2352, 4657}, {24652, 24670}

### X(24658) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24658) lies on these lines: {2, 24526}, {24653, 24660}

### X(24659) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24659) lies on these lines: {2, 18194}, {1125, 3883}, {1964, 3840}, {3759, 6686}, {3971, 16696}, {4434, 4682}, {6685, 17394}, {17448, 25106}, {24652, 24653}, {24660, 24668}, {24671, 24678}

### X(24660) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24660) lies on these lines: {2, 24527}, {17792, 24739}, {24653, 24658}, {24659, 24668}

### X(24661) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24661) lies on these lines: {1, 75}, {2, 18194}, {37, 87}, {43, 1100}, {171, 21769}, {330, 21080}, {1125, 14823}, {1385, 1742}, {1386, 21214}, {1449, 2664}, {3009, 17379}, {3248, 4687}, {3510, 20530}, {3723, 24696}, {3731, 9359}, {3759, 16569}, {3941, 8616}, {4419, 7240}, {4991, 17749}, {7032, 16826}, {7184, 17321}, {9902, 18147}, {15668, 18170}, {16525, 17754}, {16679, 16690}, {16710, 17155}, {16831, 20140}, {17260, 23524}, {17349, 23532}, {17398, 20532}, {24524, 25120}, {24653, 24663}, {24654, 24669}, {24655, 24671}, {24656, 24662}

### X(24662) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24662) lies on these lines: {1, 1575}, {2, 24527}, {24654, 24677}, {24656, 24661}

### X(24663) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24663) lies on these lines: {2, 17448}, {86, 1429}, {388, 1319}, {4417, 21214}, {4648, 16706}, {16705, 17078}, {24653, 24661}, {24655, 24672}

### X(24664) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24664) lies on these lines: {2, 25103}, {24653, 24658}

### X(24665) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24665) lies on these lines: {2, 24748}, {21191, 24667}

### X(24666) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24666) lies on these lines: {2, 23655}, {238, 23568}, {649, 2666}, {650, 4449}, {659, 24768}, {663, 4369}, {676, 20508}, {1459, 4874}, {7662, 17418}, {14296, 17215}, {24747, 25142}

### X(24667) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24667) lies on these lines: {2, 24525}, {3, 142}, {21191, 24665}, {24652, 24653}

### X(24668) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24668) lies on these lines: {2, 17448}, {1125, 21240}, {3624, 16476}, {4648, 5550}, {24659, 24660}

### X(24669) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24669) lies on these lines: {1, 20258}, {2, 3056}, {7, 21}, {71, 17754}, {105, 24540}, {497, 24551}, {1125, 6210}, {1740, 4000}, {2345, 4876}, {3475, 17379}, {3816, 17234}, {4039, 5839}, {4648, 20335}, {6690, 17381}, {24575, 24737}, {24654, 24661}

### X(24670) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24670) lies on these lines: {2, 24679}, {1964, 3720}, {24652, 24657}

### X(24671) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24671) lies on these lines: {2, 25118}, {3837, 17719}, {24653, 24672}, {24655, 24661}, {24659, 24678}, {24673, 24676}

### X(24672) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24672) lies on these lines: {2, 18194}, {1279, 3616}, {4682, 17394}, {24653, 24671}, {24654, 24679}, {24655, 24663}

### X(24673) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24673) lies on these lines: {2, 24754}, {17392, 25356}, {21191, 25423}, {24671, 24676}

### X(24674) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24674) lies on these lines: {2, 20983}, {661, 4874}, {1459, 3837}, {1980, 19684}, {2517, 24381}, {3835, 4367}, {4010, 6129}, {8640, 23803}, {19701, 21005}, {21191, 24533}, {23301, 23655}, {24749, 25126}

### X(24675) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24675) lies on these lines: {2, 23655}, {521, 25380}, {522, 6589}, {788, 6685}, {810, 17072}, {3221, 25142}, {3835, 7234}, {3907, 4885}, {21191, 24533}, {21197, 24325}

### X(24676) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24676) lies on these lines: {2, 23656}, {3837, 4107}, {21191, 24533}, {24671, 24673}

### X(24677) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24677) lies on these lines: {2, 25312}, {24654, 24662}

### X(24678) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24678) lies on these lines: {2, 3056}, {69, 3769}, {86, 171}, {319, 4039}, {1279, 3616}, {1284, 24723}, {1964, 16706}, {2887, 17234}, {3662, 21010}, {3745, 17300}, {4871, 17283}, {7081, 15985}, {10436, 18788}, {17720, 21299}, {22279, 22281}, {24653, 24661}, {24659, 24671}

### X(24679) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 3

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

X(24679) lies on these lines: {1, 3696}, {2, 24670}, {1966, 16777}, {3403, 17318}, {17394, 17752}, {18194, 25102}, {24652, 24653}, {24654, 24672}, {24656, 24661}

### X(24680) = (name pending)

Barycentrics    a^2 (a^8-4 a^7 (b+c) -8 a^6 (b^2-3 b c+c^2) +4 a^5 (3 b^3+b^2 c+b c^2+3 c^3) +2 a^4 (9 b^4-32 b^3 c+54 b^2 c^2-32 b c^3+9 c^4) -4 a^3 (b-c)^2 (3 b^3+5 b^2 c+5 b c^2+3 c^3) -8 a^2 (b-c)^2 (2 b^4-3 b^3 c-2 b^2 c^2-3 b c^3+2 c^4) +4 a (b-c)^4 (b+c)^3 +(b-c)^2 (b+c)^4 (5 b^2-26 b c+5 c^2)) : :

See Angel Montesdeoca, HG130918.

X(24680) lies on this line: {1709,4915}

### X(24681) = X(502)X(8702)∩X(7984)X(11009)

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

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28430.

X(24681) lies on these lines: {502,8702}, {7984,11009}

Collineation mappings involving Gemini triangle 4: X(24682) - X(24726)

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 4, as in centers X(24682)-X(24726). Then m(X) = a x - (a + b) y - (a + c) y : : . (Clark Kimberling, October 8, 2018)

### X(24682) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24682) lies on these lines: {2, 2173}, {3, 24317}, {4, 8680}, {5, 24315}, {10, 534}, {19, 20305}, {25, 25343}, {30, 25362}, {307, 1839}, {379, 4466}, {405, 25359}, {442, 25363}, {469, 1762}, {524, 17733}, {527, 10916}, {946, 9028}, {1479, 24424}, {1656, 25341}, {1953, 21270}, {1995, 25344}, {3741, 4643}, {3838, 4670}, {4329, 21231}, {4419, 5225}, {4644, 11269}, {4655, 4837}, {4672, 24706}, {4708, 15254}, {5020, 25360}, {5133, 24321}, {5169, 24322}, {7377, 16560}, {7522, 25361}, {7770, 25364}, {17438, 20074}, {24696, 24711}

### X(24683) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24683) lies on these lines: {1, 534}, {2, 2173}, {3, 24316}, {4, 24315}, {19, 18650}, {20, 8680}, {40, 9028}, {42, 3000}, {48, 4329}, {69, 1761}, {144, 3949}, {443, 25363}, {464, 1762}, {527, 3811}, {610, 18589}, {631, 24317}, {976, 4419}, {1370, 24321}, {1953, 20061}, {1973, 17170}, {2267, 5813}, {3090, 25341}, {3524, 25362}, {4294, 24424}, {4466, 24580}, {4640, 4643}, {4670, 5880}, {4837, 24714}, {5845, 11495}, {6353, 25360}, {6857, 25359}, {7490, 25361}, {7493, 25344}, {7494, 25343}, {14001, 25364}, {16063, 24322}, {17251, 18253}, {20074, 21271}, {24685, 24702}, {24694, 24700}, {24695, 24696}

### X(24684) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24684) lies on these lines: {2, 2173}, {3, 8680}, {5, 25341}, {35, 24424}, {48, 21231}, {140, 24317}, {474, 25363}, {524, 17748}, {534, 1125}, {536, 8669}, {631, 24316}, {1441, 22054}, {1762, 7573}, {2278, 16609}, {4466, 24581}, {4643, 24685}, {4670, 6685}, {4672, 4836}, {5054, 25362}, {6676, 25360}, {6684, 9028}, {7483, 25359}, {7485, 24321}, {7495, 25344}, {7496, 24322}, {7499, 25343}, {7536, 25361}, {7807, 25364}, {17438, 21271}, {18650, 20305}, {24688, 24714}

### X(24685) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24685) lies on these lines: {2, 2246}, {10, 544}, {11, 25342}, {100, 9318}, {101, 21232}, {214, 514}, {239, 385}, {320, 24628}, {527, 1155}, {659, 812}, {678, 24407}, {740, 4396}, {742, 4434}, {910, 20335}, {1055, 6647}, {1211, 5745}, {1215, 1376}, {3035, 5845}, {3570, 17755}, {3722, 24403}, {3758, 17754}, {3912, 4070}, {4109, 7767}, {4251, 17048}, {4362, 8667}, {4419, 17601}, {4421, 24352}, {4643, 24684}, {4721, 24850}, {4763, 22108}, {4892, 24699}, {5432, 25353}, {9451, 24841}, {16831, 18785}, {16832, 24618}, {17136, 21139}, {17439, 21272}, {24003, 24358}, {24683, 24702}, {25358, 25359}

### X(24686) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24686) lies on these lines: {2, 2173}, {22, 25343}, {25, 25344}, {427, 24321}, {1370, 24316}, {1995, 25360}, {4466, 24584}, {5133, 24315}, {7391, 8680}, {7485, 24317}, {7571, 25341}, {16545, 16607}, {17442, 17492}

### X(24687) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24687) lies on these lines: {2, 2173}, {23, 25344}, {858, 24322}, {4466, 24585}, {5169, 24315}, {5189, 8680}, {6636, 25343}, {7496, 24317}, {7570, 25341}, {13595, 25360}, {16063, 24316}, {16546, 21234}, {18669, 21274}

### X(24688) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 4

Barycentrics    a^2 b^3 + 2 a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 : :

X(24688) lies on these lines: {2, 2234}, {10, 536}, {39, 25347}, {75, 3123}, {76, 714}, {141, 2486}, {321, 2228}, {674, 12263}, {700, 21443}, {726, 4735}, {1269, 3778}, {2245, 24259}, {3122, 20913}, {3728, 18133}, {3739, 21257}, {3741, 17237}, {3770, 24575}, {3834, 3838}, {3934, 6007}, {4479, 17250}, {4655, 4837}, {4688, 20340}, {4749, 17031}, {6385, 24732}, {17229, 25140}, {17445, 21278}, {17790, 24463}, {18040, 22167}, {18140, 24450}, {18143, 21330}, {21024, 25141}, {24684, 24714}, {24707, 24711}

### X(24689) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24689) lies on these lines: {41, 25348}, {744, 4950}, {4643, 24713}, {4655, 24702}, {4799, 4837}, {16551, 17047}, {17046, 24329}, {17447, 21280}

### X(24690) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24690) lies on these lines: {2, 44}, {10, 4754}, {38, 742}, {42, 524}, {63, 4376}, {69, 2276}, {72, 16720}, {141, 672}, {172, 17206}, {213, 16887}, {319, 17759}, {350, 6646}, {518, 24326}, {527, 3741}, {536, 17135}, {896, 4797}, {899, 25350}, {1086, 24592}, {1107, 17137}, {1575, 17344}, {2238, 4416}, {3219, 24358}, {3691, 20255}, {3720, 4364}, {3840, 4465}, {3873, 24357}, {4022, 24513}, {4357, 24512}, {4389, 17027}, {4396, 14829}, {4419, 10453}, {4424, 8682}, {4441, 17276}, {4640, 4760}, {4651, 4690}, {4655, 4799}, {4683, 24712}, {4725, 20011}, {4981, 25384}, {7196, 17950}, {16552, 21240}, {17026, 17274}, {17032, 17378}, {17152, 17448}, {17184, 25345}, {17272, 17754}, {17345, 20347}, {17347, 24514}

### X(24691) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24691) lies on these lines: {2, 44}, {6, 24586}, {7, 21264}, {43, 524}, {63, 24358}, {69, 1575}, {141, 17754}, {193, 21904}, {291, 3416}, {350, 17276}, {354, 24357}, {527, 3840}, {536, 10453}, {672, 17279}, {742, 982}, {1086, 17026}, {2176, 24652}, {2275, 17137}, {2276, 4851}, {3218, 4376}, {3662, 25345}, {3741, 4363}, {3868, 16720}, {3873, 24326}, {4253, 21240}, {4426, 17206}, {4440, 4479}, {4650, 4797}, {4657, 24512}, {4667, 6685}, {4725, 20012}, {4799, 24712}, {7232, 20335}, {16887, 17750}, {17027, 17301}, {17149, 17790}, {17299, 17759}, {17345, 20530}, {17448, 21281}, {20255, 21384}, {24694, 24699}

### X(24692) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24692) lies on these lines: {2, 4759}, {7, 4660}, {10, 527}, {44, 25351}, {69, 4709}, {79, 3831}, {320, 519}, {528, 7238}, {545, 4439}, {551, 4675}, {726, 4440}, {740, 17374}, {752, 1086}, {899, 17491}, {908, 23831}, {1125, 17305}, {1155, 4892}, {1738, 17770}, {1836, 3840}, {2796, 3912}, {2887, 11246}, {3218, 21241}, {3474, 3771}, {3634, 24697}, {3679, 4741}, {3741, 20292}, {3821, 14621}, {3828, 17256}, {3834, 4432}, {3836, 4422}, {3923, 4312}, {3932, 17767}, {3993, 17316}, {4084, 4259}, {4085, 17365}, {4090, 5905}, {4693, 17297}, {4715, 4753}, {4732, 17344}, {4784, 4785}, {4871, 5057}, {4966, 17764}, {6173, 24331}, {6327, 24165}, {7613, 24599}, {15310, 24827}, {17449, 20042}, {17766, 24231}

### X(24693) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24693) lies on these lines: {1, 4743}, {2, 4432}, {10, 527}, {45, 2796}, {69, 4732}, {142, 20181}, {320, 3679}, {519, 4675}, {528, 24331}, {551, 17067}, {740, 17316}, {752, 4384}, {984, 4440}, {1150, 21027}, {1647, 24594}, {1698, 17354}, {1738, 17023}, {1739, 3764}, {2550, 24325}, {2640, 4429}, {3306, 21242}, {3696, 17374}, {3739, 4660}, {3754, 4259}, {3821, 17325}, {3826, 3923}, {3836, 17284}, {3842, 24248}, {3925, 3980}, {4085, 10436}, {4307, 4974}, {4359, 4865}, {4407, 17274}, {4413, 25385}, {4439, 4659}, {4644, 4753}, {4663, 4796}, {4693, 17244}, {4703, 20292}, {4709, 4851}, {9780, 24697}, {17256, 19875}, {17450, 21283}, {17722, 24620}

### X(24694) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24694) lies on these lines: {2, 2246}, {9, 4872}, {10, 24701}, {55, 25353}, {169, 17046}, {239, 7837}, {320, 17026}, {524, 4362}, {527, 1836}, {742, 4865}, {984, 20539}, {1376, 24318}, {2082, 17062}, {2886, 5845}, {3684, 7179}, {3729, 4119}, {3741, 4643}, {4056, 16552}, {4109, 7776}, {4376, 4438}, {4655, 4799}, {4911, 21384}, {5275, 24241}, {6173, 24600}, {9318, 11680}, {11235, 24352}, {17451, 21285}, {24316, 24336}, {24319, 24328}, {24683, 24700}, {24691, 24699}

### X(24695) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24695) lies on these lines: {1, 527}, {2, 896}, {4, 1046}, {6, 17768}, {7, 238}, {8, 752}, {31, 5905}, {38, 20078}, {43, 2635}, {44, 5880}, {46, 2183}, {56, 15507}, {57, 5121}, {58, 17139}, {69, 3923}, {79, 1714}, {144, 984}, {145, 537}, {171, 329}, {193, 740}, {226, 1707}, {320, 4676}, {387, 24851}, {513, 4259}, {516, 3751}, {524, 5695}, {553, 5272}, {579, 1756}, {583, 24744}, {611, 5857}, {612, 17781}, {651, 4331}, {846, 5712}, {966, 24342}, {982, 9965}, {1001, 17365}, {1125, 4896}, {1281, 7774}, {1386, 17276}, {1468, 11415}, {1738, 1743}, {1742, 5759}, {1757, 2550}, {1836, 4641}, {1992, 2796}, {2094, 18201}, {2308, 19785}, {2650, 6872}, {2795, 7737}, {2835, 5903}, {3120, 24597}, {3242, 5852}, {3416, 17351}, {3475, 8616}, {3618, 3821}, {3619, 24295}, {3648, 19767}, {3663, 16475}, {3685, 17364}, {3729, 5847}, {3758, 24723}, {3779, 15310}, {3826, 16885}, {3928, 24239}, {3979, 10385}, {3980, 14555}, {4000, 16468}, {4011, 18141}, {4295, 5247}, {4310, 20059}, {4353, 16491}, {4383, 11246}, {4418, 5739}, {4523, 12530}, {4645, 17350}, {4675, 15254}, {4748, 19856}, {4862, 16469}, {5057, 11269}, {5222, 16477}, {5263, 17347}, {5327, 24316}, {5744, 17717}, {5745, 16570}, {5850, 16496}, {5988, 7735}, {5992, 7766}, {6173, 15601}, {6604, 24395}, {6650, 20158}, {7290, 24231}, {9776, 17123}, {9791, 17379}, {10186, 11364}, {14912, 24257}, {16476, 17753}, {16830, 17333}, {17063, 21454}, {17122, 18228}, {17126, 17484}, {17127, 17483}, {17165, 20064}, {17716, 20214}, {24683, 24696}, {24728, 25406}

### X(24696) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24696) lies on these lines: {1, 536}, {2, 2234}, {6, 1045}, {37, 1740}, {39, 4443}, {42, 3758}, {43, 44}, {45, 2664}, {75, 2309}, {76, 24327}, {87, 1100}, {190, 869}, {192, 1964}, {194, 714}, {256, 4261}, {386, 4672}, {513, 3795}, {674, 12782}, {740, 5145}, {872, 17350}, {899, 17335}, {978, 15254}, {995, 4432}, {1193, 4676}, {1278, 17445}, {1580, 2278}, {1613, 21883}, {2235, 2276}, {2274, 3685}, {2667, 17379}, {3009, 4664}, {3097, 4735}, {3122, 24598}, {3240, 4781}, {3248, 4393}, {3720, 4479}, {3723, 24661}, {3736, 3923}, {3739, 16571}, {3759, 22343}, {3783, 4643}, {3875, 18170}, {4116, 24502}, {4360, 7032}, {4377, 9902}, {4446, 21746}, {4493, 21814}, {4644, 24722}, {4655, 24707}, {4836, 24714}, {4837, 24726}, {4851, 7184}, {4852, 18194}, {5069, 24575}, {5110, 8424}, {5114, 24264}, {5283, 24450}, {7786, 25347}, {9020, 9941}, {9025, 20691}, {17443, 17891}, {21238, 21299}, {24682, 24711}, {24683, 24695}

### X(24697) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 4

Barycentrics    a^3 - a^2 b - 2 a b^2 - b^3 - a^2 c - 3 a b c - b^2 c - 2 a c^2 - b c^2 - c^3 : :

X(24697) lies on these lines: {1, 524}, {2, 896}, {5, 7609}, {8, 192}, {9, 46}, {10, 190}, {21, 24436}, {58, 86}, {81, 6536}, {319, 3993}, {333, 4425}, {726, 17258}, {752, 16830}, {846, 1211}, {960, 3792}, {966, 24248}, {1001, 17253}, {1043, 12579}, {1046, 4205}, {1054, 5241}, {1423, 10404}, {1503, 8245}, {1757, 4026}, {1962, 2895}, {2792, 6998}, {3120, 5235}, {3244, 3883}, {3616, 4741}, {3622, 5625}, {3624, 4675}, {3633, 7174}, {3634, 24692}, {3647, 24931}, {3685, 3775}, {3686, 4716}, {3717, 4691}, {3773, 17261}, {3821, 17277}, {3836, 17260}, {3842, 4645}, {3844, 16814}, {3923, 5224}, {3944, 5737}, {4001, 4038}, {4259, 5692}, {4389, 16825}, {4416, 4649}, {4505, 25280}, {4657, 16468}, {4676, 17250}, {4708, 19856}, {4748, 5698}, {4886, 4970}, {4974, 17302}, {5057, 14009}, {5248, 7301}, {5302, 19879}, {5327, 25359}, {5333, 8040}, {5695, 17251}, {5739, 17592}, {5743, 17596}, {6210, 12699}, {6646, 24325}, {6650, 17755}, {6651, 17292}, {6707, 17306}, {9534, 24717}, {9780, 24693}, {10180, 17778}, {15254, 17237}, {15569, 17344}, {16477, 17023}, {16823, 17254}, {17116, 17767}, {17307, 24295}, {17319, 17772}, {17367, 20142}, {17889, 19732}, {22174, 24923}

### X(24698) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24698) lies on these lines: {513, 4486}, {667, 25356}, {1919, 17397}, {4675, 21191}, {4785, 24699}, {6542, 17458}, {17256, 20979}, {17308, 21262}, {21260, 24354}, {24716, 24718}

### X(24699) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24699) lies on these lines: {2, 2243}, {10, 527}, {79, 4721}, {171, 25345}, {238, 25357}, {239, 320}, {315, 20271}, {536, 24715}, {742, 4645}, {1836, 4713}, {3120, 4396}, {3509, 20541}, {3662, 20179}, {3726, 20553}, {3836, 24358}, {4364, 16830}, {4465, 5057}, {4660, 24357}, {4670, 24707}, {4675, 17023}, {4708, 19856}, {4785, 24698}, {4805, 5902}, {4892, 24685}, {4911, 20255}, {5277, 17211}, {7759, 24046}, {7780, 24160}, {14023, 24159}, {17256, 17369}, {20292, 24330}, {21282, 24403}, {24691, 24694}

### X(24700) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24700) lies on these lines: {2, 24702}, {3, 24336}, {21, 25371}, {527, 3916}, {572, 21233}, {958, 24334}, {1444, 4032}, {2182, 5745}, {4640, 4670}, {4643, 24684}, {4644, 4650}, {4798, 24703}, {4999, 24319}, {5433, 25369}, {8424, 24358}, {17440, 21273}, {23085, 24328}, {24683, 24694}

### X(24701) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24701) lies on these lines: {1, 30}, {4, 85}, {10, 24694}, {19, 25365}, {25, 18651}, {77, 1892}, {304, 1330}, {1368, 1763}, {1370, 17441}, {1930, 5814}, {4655, 24705}, {4670, 5880}, {5777, 5928}, {6998, 17181}, {16502, 23537}, {18589, 24320}, {18596, 21530}, {19544, 21621}

### X(24702) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24702) lies on these lines: {2, 24700}, {4, 24336}, {9, 7377}, {56, 25369}, {527, 1210}, {958, 24319}, {1329, 24334}, {1766, 21244}, {2478, 25371}, {3452, 5782}, {3741, 4643}, {4640, 4708}, {4655, 24689}, {4670, 5087}, {4748, 5698}, {5816, 21233}, {17452, 21286}, {24683, 24685}

### X(24703) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24703) lies on these lines: {1, 529}, {2, 1155}, {3, 12608}, {4, 960}, {5, 12514}, {6, 24210}, {7, 3742}, {8, 3967}, {9, 1699}, {10, 381}, {11, 63}, {12, 5250}, {20, 12679}, {21, 11375}, {30, 997}, {31, 17720}, {38, 17721}, {40, 1329}, {46, 4187}, {55, 908}, {57, 3816}, {65, 2478}, {72, 1479}, {78, 6284}, {79, 3624}, {80, 3899}, {142, 8167}, {144, 5274}, {149, 3681}, {165, 3035}, {190, 3705}, {191, 7741}, {200, 528}, {210, 3434}, {214, 16128}, {226, 1001}, {238, 3772}, {283, 7299}, {306, 4387}, {312, 3416}, {321, 3966}, {329, 497}, {354, 5905}, {355, 3878}, {382, 17647}, {392, 1478}, {404, 24954}, {405, 12047}, {430, 1211}, {452, 3485}, {474, 1770}, {499, 3916}, {513, 3784}, {515, 5289}, {516, 1376}, {517, 6929}, {527, 11019}, {614, 3782}, {748, 3120}, {758, 5722}, {846, 17717}, {946, 958}, {950, 12635}, {962, 2551}, {965, 1839}, {968, 5718}, {978, 24851}, {982, 17276}, {993, 5886}, {1056, 10179}, {1086, 5272}, {1125, 16418}, {1158, 6922}, {1191, 13161}, {1519, 3428}, {1621, 17718}, {1633, 19649}, {1697, 12607}, {1698, 17530}, {1706, 9589}, {1708, 14022}, {1709, 15842}, {1737, 17556}, {1788, 6919}, {1837, 3869}, {1848, 11323}, {2550, 3740}, {2646, 6872}, {2887, 4011}, {2975, 11376}, {3036, 14217}, {3057, 3436}, {3058, 3870}, {3185, 4192}, {3218, 17728}, {3219, 11680}, {3244, 4930}, {3286, 17182}, {3295, 21077}, {3305, 3925}, {3306, 11246}, {3419, 3583}, {3421, 3880}, {3612, 16154}, {3616, 4870}, {3647, 16159}, {3650, 16153}, {3685, 4417}, {3687, 5695}, {3689, 20075}, {3696, 14555}, {3702, 10371}, {3706, 5739}, {3715, 25006}, {3720, 24725}, {3730, 20544}, {3741, 4643}, {3752, 24248}, {3756, 18193}, {3771, 4432}, {3811, 15171}, {3812, 4295}, {3813, 9614}, {3814, 6980}, {3817, 5745}, {3826, 7308}, {3829, 3929}, {3840, 4655}, {3846, 3923}, {3848, 9776}, {3873, 17484}, {3877, 5080}, {3890, 20060}, {3913, 10624}, {3914, 4383}, {3927, 9669}, {3940, 9668}, {3952, 5014}, {3962, 12649}, {3971, 4865}, {4009, 10327}, {4193, 24914}, {4297, 6259}, {4299, 17614}, {4301, 5795}, {4302, 5440}, {4307, 4682}, {4310, 4906}, {4312, 5437}, {4338, 17575}, {4358, 6327}, {4421, 6745}, {4423, 5249}, {4425, 4657}, {4428, 13405}, {4465, 4799}, {4511, 11114}, {4512, 5219}, {4517, 20556}, {4641, 11269}, {4645, 18743}, {4652, 5433}, {4654, 10582}, {4662, 5082}, {4675, 24708}, {4798, 24700}, {4847, 5220}, {4853, 13463}, {4854, 5256}, {4855, 15338}, {4857, 5904}, {4862, 5573}, {4952, 17765}, {4957, 17890}, {4973, 10199}, {4999, 8227}, {5016, 25253}, {5044, 22793}, {5068, 18231}, {5119, 17757}, {5123, 5657}, {5187, 17606}, {5218, 5748}, {5223, 24392}, {5248, 11374}, {5251, 18393}, {5273, 5832}, {5302, 19843}, {5328, 9778}, {5436, 11281}, {5537, 15813}, {5603, 6976}, {5691, 15829}, {5705, 18253}, {5709, 7681}, {5712, 15569}, {5720, 5842}, {5727, 5855}, {5730, 10572}, {5731, 12678}, {5744, 10589}, {5762, 7956}, {5784, 10431}, {5837, 19925}, {5853, 21060}, {5887, 6928}, {5909, 11254}, {6001, 6827}, {6210, 20545}, {6691, 15803}, {6734, 10896}, {6836, 12688}, {6837, 15823}, {6865, 9943}, {6893, 7686}, {7270, 19582}, {7290, 17061}, {7354, 19861}, {7991, 8256}, {8165, 20070}, {8583, 9579}, {8616, 17719}, {8666, 11373}, {9581, 12526}, {9809, 11220}, {10401, 17183}, {10738, 18254}, {10895, 24987}, {10947, 17615}, {10950, 11682}, {10980, 17051}, {11108, 12609}, {11238, 17781}, {11362, 12700}, {12053, 12513}, {12116, 14872}, {12433, 12559}, {12571, 18249}, {13271, 14740}, {15320, 17259}, {15601, 17070}, {16152, 17525}, {16569, 24715}, {17123, 17278}, {17139, 18165}, {17732, 25066}, {20076, 20323}

### X(24704) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24704) lies on these lines: {58, 25370}, {79, 4363}, {536, 24851}, {1761, 21245}, {1836, 4643}, {3454, 24335}, {4016, 21287}, {4644, 17491}, {4655, 4837}, {4708, 19856}

### X(24705) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24705) lies on these lines: {1, 527}, {9, 10446}, {40, 24334}, {65, 25371}, {71, 20258}, {226, 17139}, {314, 4044}, {573, 21246}, {946, 24319}, {960, 12545}, {966, 10456}, {1400, 17183}, {1764, 2183}, {1944, 3496}, {1999, 17781}, {2269, 20245}, {2300, 3663}, {3686, 10447}, {3741, 4643}, {4640, 4670}, {4655, 24701}, {4708, 5087}, {5257, 10455}, {5745, 10478}, {5795, 12435}, {5819, 10442}, {7175, 8822}, {10441, 12572}, {10465, 15829}, {10480, 12527}, {17276, 21769}, {21061, 22031}

### X(24706) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 4

Barycentrics    3 a^9 + a^8 b - 2 a^5 b^4 - a b^8 - b^9 + a^8 c - 2 a^4 b^4 c + b^8 c - 2 a^5 c^4 - 2 a^4 b c^4 + 2 a b^4 c^4 - a c^8 + b c^8 - c^9 : :

X(24706) lies on these lines: {66, 25372}, {206, 24337}, {4672, 24682}, {16544, 21247}, {17453, 21288}

### X(24707) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 4

Barycentrics    a^5 - a^3 b^2 - a^2 b^3 - a b^4 - b^5 - a^2 b^2 c - a^3 c^2 - a^2 b c^2 - a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 - a c^4 - c^5 : :

X(24707) lies on these lines: {2, 2244}, {42, 320}, {83, 25374}, {744, 2896}, {1078, 25346}, {3096, 4381}, {4640, 17237}, {4655, 24696}, {4670, 24699}, {6292, 24340}, {16556, 21249}, {17457, 21289}, {24688, 24711}

### X(24708) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24708) lies on these lines: {1, 527}, {2, 3000}, {6, 4335}, {9, 1742}, {40, 13598}, {43, 44}, {85, 25375}, {87, 8424}, {144, 2293}, {165, 2183}, {170, 1212}, {405, 1044}, {978, 15601}, {1001, 4334}, {1042, 11106}, {1423, 20992}, {1633, 2267}, {2269, 9309}, {2310, 24635}, {2340, 6172}, {2951, 5819}, {3208, 9025}, {3576, 15507}, {3672, 20978}, {3870, 20072}, {4499, 23407}, {4675, 24703}, {6007, 21384}, {8572, 21214}, {17026, 24717}, {17139, 17194}

### X(24709) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24709) lies on these lines: {2, 4432}, {10, 10707}, {11, 4422}, {88, 2796}, {100, 11814}, {121, 5541}, {149, 24003}, {190, 1647}, {244, 4440}, {519, 4767}, {528, 9458}, {545, 24408}, {551, 4945}, {3315, 21093}, {3716, 4728}, {3840, 4683}, {3952, 20042}, {3994, 5211}, {4465, 4643}, {4655, 24710}, {4800, 14286}, {4871, 5057}, {17460, 21290}

### X(24710) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24710) lies on these lines: {2, 4759}, {89, 25378}, {4643, 5057}, {4655, 24709}, {16558, 21251}, {17461, 21291}

### X(24711) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24711) lies on these lines: {2, 24714}, {99, 24348}, {115, 24345}, {523, 13178}, {2611, 24500}, {2640, 8287}, {2643, 21221}, {4837, 24717}, {24682, 24696}, {24688, 24707}, {24712, 24722}, {24713, 24715}

### X(24712) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24712) lies on these lines: {1, 544}, {2, 2246}, {4, 7960}, {10, 10712}, {11, 5845}, {38, 497}, {41, 17181}, {80, 514}, {81, 226}, {100, 24318}, {116, 5540}, {150, 2170}, {210, 4553}, {291, 812}, {294, 3008}, {319, 3807}, {524, 17763}, {527, 1156}, {672, 4872}, {1475, 4911}, {1565, 9317}, {1621, 25353}, {1699, 9355}, {1959, 3930}, {2112, 17397}, {2280, 7179}, {2503, 8287}, {3061, 21285}, {3732, 21044}, {3925, 4472}, {4056, 4253}, {4423, 25367}, {4465, 4643}, {4644, 11269}, {4670, 25383}, {4675, 24512}, {4683, 24690}, {4712, 20539}, {4799, 24691}, {5074, 5526}, {5276, 24241}, {5308, 9502}, {7247, 17474}, {11238, 24352}, {11680, 24333}, {14439, 20533}, {17308, 20540}, {17439, 20096}, {17463, 24488}, {21221, 24504}, {21381, 24347}, {24711, 24722}

### X(24713) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24713) lies on these lines: {101, 25379}, {116, 24346}, {692, 4466}, {1565, 5848}, {3448, 24500}, {4643, 24689}, {16560, 21252}, {17463, 21293}, {21339, 24489}, {24711, 24715}

### X(24714) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24714) lies on these lines: {2, 24711}, {99, 24345}, {523, 11711}, {620, 24348}, {662, 21254}, {2795, 24350}, {4836, 24696}, {4837, 24683}, {17467, 21295}, {24684, 24688}

### X(24715) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24715) lies on these lines: {1, 528}, {2, 4432}, {4, 9365}, {8, 537}, {10, 190}, {11, 1054}, {30, 8481}, {35, 24161}, {42, 20292}, {43, 1836}, {46, 16560}, {55, 17889}, {65, 24836}, {72, 24835}, {75, 4660}, {79, 3293}, {80, 900}, {88, 1647}, {100, 3120}, {106, 14028}, {142, 16484}, {148, 24505}, {149, 244}, {165, 17064}, {171, 3914}, {238, 516}, {239, 752}, {291, 812}, {319, 4709}, {320, 519}, {334, 874}, {335, 740}, {390, 7613}, {484, 2161}, {497, 17063}, {515, 24813}, {517, 3792}, {536, 24699}, {545, 3679}, {668, 4783}, {846, 3925}, {851, 5143}, {894, 4085}, {899, 5057}, {956, 24826}, {978, 12699}, {982, 3434}, {984, 2550}, {1320, 24131}, {1376, 3944}, {1479, 24174}, {1581, 19637}, {1633, 24436}, {1644, 4945}, {1654, 4732}, {1698, 4422}, {1737, 24846}, {1739, 3583}, {1757, 17768}, {1770, 5247}, {1837, 24840}, {1929, 20531}, {1961, 4854}, {2006, 23703}, {2170, 24484}, {2475, 4642}, {2607, 2640}, {2786, 13178}, {2802, 24864}, {2810, 4014}, {2836, 21889}, {2886, 17596}, {3057, 24837}, {3210, 4865}, {3240, 24725}, {3306, 24217}, {3416, 9055}, {3474, 4650}, {3509, 21956}, {3550, 3772}, {3585, 3987}, {3632, 9041}, {3685, 3836}, {3696, 4690}, {3699, 21093}, {3722, 20095}, {3749, 23681}, {3750, 5249}, {3751, 4312}, {3755, 4649}, {3782, 3961}, {3791, 20101}, {3821, 5263}, {3834, 4702}, {3842, 9791}, {3879, 4780}, {3912, 4693}, {3923, 4429}, {4026, 4472}, {4051, 9597}, {4080, 17780}, {4259, 5903}, {4307, 17014}, {4366, 17397}, {4370, 19875}, {4407, 17254}, {4409, 4668}, {4418, 4972}, {4442, 17763}, {4443, 24464}, {4473, 9780}, {4514, 24165}, {4518, 5992}, {4640, 21949}, {4651, 4683}, {4695, 5080}, {4716, 5847}, {4753, 20072}, {4850, 17722}, {4862, 16496}, {4947, 24399}, {5014, 17155}, {5090, 24814}, {5134, 16611}, {5235, 21027}, {5252, 24816}, {5255, 23537}, {5272, 9580}, {5308, 20533}, {5541, 24222}, {5587, 24828}, {5657, 24817}, {5687, 24820}, {5688, 24832}, {5689, 24831}, {5695, 17269}, {5790, 24844}, {5853, 24231}, {6154, 17724}, {6174, 9324}, {6187, 13589}, {8193, 24822}, {8197, 24823}, {8204, 24824}, {8214, 24838}, {8215, 24839}, {8616, 24789}, {9614, 11512}, {9802, 17460}, {9857, 24825}, {10039, 24845}, {10624, 24178}, {10791, 24815}, {10914, 24834}, {10915, 24847}, {10916, 24848}, {11900, 24830}, {12701, 21214}, {13883, 24819}, {13893, 24842}, {13936, 24818}, {13947, 24843}, {15485, 17278}, {16173, 24871}, {16569, 24703}, {17122, 24210}, {17308, 17738}, {17484, 21805}, {17491, 19998}, {17495, 21282}, {17715, 20075}, {17716, 19785}, {17777, 24003}, {18193, 24392}, {21098, 21944}, {21242, 24627}, {24711, 24713}

### X(24716) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24716) lies on these lines: {625, 24350}, {4655, 4837}, {16568, 21256}, {17472, 21298}, {24698, 24718}

### X(24717) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24717) lies on these lines: {2, 2234}, {8, 536}, {75, 21299}, {76, 6007}, {194, 4443}, {320, 1836}, {714, 20081}, {1278, 21278}, {3122, 24621}, {3210, 3764}, {4277, 4734}, {4655, 10449}, {4740, 20352}, {4741, 17135}, {4837, 24711}, {9025, 17144}, {9534, 24697}, {16571, 21257}, {17026, 24708}, {17155, 23633}, {17299, 25311}, {17790, 24451}, {22016, 25279}

### X(24718) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24718) lies on these lines: {10, 4913}, {513, 23301}, {650, 20316}, {656, 18155}, {1577, 10479}, {3716, 6003}, {4874, 9013}, {6002, 17072}, {7662, 25128}, {17478, 21300}, {23783, 23874}, {24698, 24716}

### X(24719) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24719) lies on these lines: {2, 4782}, {8, 4083}, {79, 4905}, {320, 350}, {514, 4122}, {522, 4810}, {523, 4382}, {649, 3837}, {659, 3835}, {663, 4992}, {667, 1125}, {812, 1491}, {1019, 23815}, {1698, 4063}, {3309, 12699}, {3667, 4458}, {3669, 10404}, {3700, 4813}, {3777, 6002}, {4170, 6004}, {4380, 9508}, {4448, 4679}, {4498, 21051}, {4724, 4806}, {4728, 4874}, {4762, 4824}, {4784, 4785}, {5880, 6008}, {17367, 24601}, {17458, 23656}, {24698, 24716}

### X(24720) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24720) lies on these lines: {2, 4724}, {10, 514}, {145, 4449}, {512, 23815}, {513, 3716}, {522, 693}, {523, 3776}, {650, 25380}, {661, 4521}, {663, 3616}, {918, 4522}, {1125, 4794}, {1577, 4905}, {1734, 4978}, {2523, 16612}, {2526, 25128}, {3624, 4040}, {3667, 4010}, {3669, 3907}, {3810, 7178}, {4041, 4801}, {4106, 7659}, {4142, 21188}, {4389, 4406}, {4394, 4830}, {4462, 21052}, {4474, 21222}, {4728, 6006}, {4762, 4913}, {4784, 4785}, {4791, 23796}, {4823, 8714}, {4977, 20316}, {6372, 21260}, {8672, 23301}, {8689, 24924}, {16737, 18299}, {20521, 21189}

### X(24721) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24721) lies on these lines: {291, 812}, {513, 4486}, {514, 4774}, {659, 25381}, {3768, 17256}, {3776, 4382}, {3837, 4375}, {4369, 4728}, {4458, 4810}, {4472, 24354}, {4491, 25356}, {4784, 4785}, {6542, 21303}, {8632, 17397}, {17308, 21261}

### X(24722) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24722) lies on these lines: {1, 545}, {37, 7240}, {87, 17276}, {244, 24482}, {291, 513}, {292, 24423}, {335, 24510}, {668, 25382}, {1015, 24338}, {1045, 7277}, {1086, 9359}, {2234, 20072}, {3122, 4499}, {3248, 4440}, {3783, 4715}, {4398, 23524}, {4644, 24696}, {5539, 24345}, {7184, 17351}, {7321, 22343}, {19945, 24625}, {24711, 24712}

### X(24723) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 4

Barycentrics    a^3 - a^2 b - a b^2 - b^3 - a^2 c - a b c - a c^2 - c^3 : :

X(24723) lies on these lines: {1, 320}, {2, 1155}, {8, 536}, {9, 4429}, {10, 190}, {11, 24627}, {21, 1633}, {31, 19786}, {37, 4645}, {38, 4514}, {42, 4683}, {43, 4703}, {65, 17950}, {75, 23690}, {85, 4331}, {141, 3685}, {145, 4741}, {171, 4425}, {192, 3416}, {238, 3821}, {257, 19637}, {274, 19643}, {319, 740}, {333, 3914}, {516, 4357}, {518, 6646}, {528, 17254}, {744, 2292}, {846, 2887}, {894, 4026}, {960, 4201}, {968, 18134}, {984, 4660}, {986, 3764}, {1001, 3662}, {1010, 1770}, {1086, 16823}, {1125, 17305}, {1279, 17235}, {1284, 24678}, {1330, 3931}, {1386, 17302}, {1456, 17086}, {1621, 17184}, {1654, 3696}, {1698, 17354}, {1738, 17277}, {1757, 4085}, {1999, 4854}, {2245, 5051}, {2345, 24280}, {2550, 17257}, {2895, 3896}, {3057, 5484}, {3210, 3966}, {3219, 4972}, {3242, 17255}, {3286, 17202}, {3616, 4675}, {3617, 20073}, {3661, 5695}, {3663, 3883}, {3666, 4388}, {3745, 20101}, {3751, 17347}, {3755, 4416}, {3757, 3782}, {3758, 24695}, {3775, 17764}, {3790, 17262}, {3792, 3878}, {3823, 16814}, {3826, 17260}, {3836, 17263}, {3844, 17280}, {3846, 17596}, {3869, 4259}, {3879, 4356}, {3886, 17272}, {3920, 4450}, {3923, 17289}, {3932, 17261}, {3993, 17315}, {4003, 5211}, {4295, 13725}, {4307, 17321}, {4312, 10436}, {4337, 18465}, {4360, 5847}, {4364, 16830}, {4415, 7081}, {4417, 17594}, {4418, 19808}, {4488, 5772}, {4649, 17770}, {4663, 20072}, {4749, 5262}, {4966, 17288}, {5014, 7226}, {5220, 17333}, {5846, 17246}, {5992, 20716}, {6284, 24726}, {7290, 17304}, {7321, 24325}, {11110, 12609}, {11375, 19278}, {12047, 19270}, {12514, 16062}, {13735, 19869}, {14829, 24210}, {15569, 17300}, {16475, 17380}, {17123, 24169}, {17276, 24349}, {22053, 24550}

### X(24724) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24724) lies on these lines: {1, 3849}, {79, 4754}, {524, 4442}, {896, 25383}, {1836, 4643}, {4440, 7779}, {4465, 5057}, {4760, 4892}, {4784, 4785}, {9055, 21282}, {16702, 17204}

### X(24725) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24725) lies on these lines: {1, 535}, {2, 896}, {4, 2650}, {6, 3120}, {7, 244}, {11, 17365}, {31, 226}, {38, 5905}, {42, 1836}, {43, 20292}, {79, 386}, {81, 3944}, {142, 17125}, {329, 756}, {516, 2177}, {614, 4654}, {748, 5249}, {750, 908}, {899, 5880}, {902, 17718}, {982, 17483}, {984, 17484}, {986, 14450}, {1046, 2476}, {1150, 17770}, {1201, 10404}, {1215, 6327}, {1468, 12047}, {1647, 4860}, {1724, 11263}, {1962, 5712}, {2308, 3772}, {2550, 21805}, {3218, 17717}, {3240, 24715}, {3452, 17124}, {3649, 3924}, {3664, 9345}, {3720, 24703}, {3782, 17017}, {3838, 4641}, {3915, 13407}, {3923, 3936}, {3980, 5741}, {4011, 18139}, {4054, 5847}, {4062, 5695}, {4138, 5294}, {4295, 4642}, {4300, 5812}, {4392, 17722}, {4414, 5718}, {4415, 5311}, {4417, 4418}, {4425, 19684}, {4519, 17374}, {4644, 11269}, {4675, 4679}, {4865, 17165}, {5226, 9340}, {5230, 5714}, {5739, 21020}, {5902, 6788}, {6536, 19701}, {6690, 17775}, {9355, 10883}, {9612, 21935}, {16272, 20277}, {17126, 17719}, {17276, 17723}, {17300, 17777}, {17449, 17721}, {17771, 21242}, {19767, 24851}, {20430, 21326}

### X(24726) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 4

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

X(24725) lies on these lines: {2, 2173}, {190, 3661}, {320, 1999}, {384, 25364}, {4466, 24610}, {4655, 10449}, {4837, 24696}, {5025, 24315}, {6284, 24723}, {6655, 8680}, {7791, 24316}, {7824, 24317}, {24688, 24707}

### X(24727) =  X(1)X(6)∩X(39)X(1045)

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

Let U = obverse triangle of X(1), and let V(X) = N-obverse triangle of X. The locus of X such that V(X) is perspective to U is the cubic given by

b^3 x^3-c^3 x^3+a b^2 x^2 y-b c^2 x^2 y-a^2 b x y^2+a c^2 x y^2-a^3 y^3+c^3 y^3+b^2 c x^2 z-a c^2 x^2 z-a^2 c y^2 z+b c^2 y^2 z-a b^2 x z^2+a^2 c x z^2+a^2 b y z^2-b^2 c y z^2+a^3 z^3-b^3 z^3 = 0.

This cubic passes through X(i) for i = 1,2,9,239,516,3496,3509,13174,16557,17738, 24727,24728. (Peter Moses, October 8, 2018)

X(24727) lies on these lines: {1,6}, {39,1045}, {57,6063}, {63,17027}, {239,19565}, {672,3685}, {740,24578}, {894,1475}, {2082,17033}, {2111,17738}, {2344,5138}, {2350,4418}, {3218,17029}, {3305,17032}, {3306,17028}, {3496,16574}, {3501,3886}, {3509,17031}, {3691,17260}, {3923,4253}, {3928,16557}, {4063,4762}, {4876,9052}, {5364,10453}, {10436,17030}

### X(24728) =  X(1)X(7)∩X(2)X(8245)

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

See X(24727).

X(24728) lies on the cubic introduced at X(24727) and on these lines: {1,7}, {2,8245}, {3,3923}, {4,3821}, {40,726}, {69,2784}, {103,789}, {165,3729}, {192,18788}, {376,2796}, {511,24257}, {515,4660}, {572,2344}, {573,24578}, {631,24295}, {740,1350}, {944,17766}, {971,18252}, {1403,7580}, {1431,24268}, {1503,4655}, {1699,17304}, {1766,3509}, {1768,16566}, {2783,3098}, {2938,6194}, {3522,24280}, {3831,12618}, {3840,21629}, {3980,4220}, {4011,19649}, {4192,24260}, {4672,5085}, {5918,17635}, {6210,16825}, {6685,10445}, {6776,17770}, {7155,17738}, {7411,11688}, {8669,12512}, {8931,16822}, {9740,17132}, {9841,12717}, {10164,17355}, {10167,12723}, {11220,12530}, {12725,19645}, {24309,24326}

### X(24729) =  X(3)X(6)∩X(25)X(8840)

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

Let U = obverse triangle of X(6), and let V(X) = N-obverse triangle of X. The locus of X such that V(X) is perspective to U is the cubic given by

b^6 x^3-c^6 x^3+a^2 b^4 x^2 y-b^2 c^4 x^2 y-a^4 b^2 x y^2+a^2 c^4 x y^2-a^6 y^3+c^6 y^3+b^4 c^2 x^2 z-a^2 c^4 x^2 z-a^4 c^2 y^2 z+b^2 c^4 y^2 z-a^2 b^4 x z^2+a^4 c^2 x z^2+a^4 b^2 y z^2-b^4 c^2 y z^2+a^6 z^3-b^6 z^3 = 0.

This cubic passes through X(i) for i = 2,3,6,385,1503,3511,5989,6660,8301,8424,15588, 24729,24730. (Peter Moses, October 8, 2018)

X(24729) lies on these lines: {3,6}, {25,8840}, {237,12215}, {385,19566}, {732,3511}, {3499,8265}, {7824,22062}, {8424,23851}, {15588,20854}, {21006,23878}

### X(24730) =  X(4)X(6)∩X(154)X(1975)

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

See X(24729).

X(24730) lies on the cubic introduced at X(24729) and on these lines: {4,6}, {154,1975}, {159,698}, {206,3492}, {1853,7851}, {2782,6759}, {3511,15270}, {3556,8301}, {6000,14880}, {7789,10192}

### X(24731) =  X(10)X(75)∩X(141)X(17762)

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

Let U = obverse triangle of X(75), and let V(X) = N-obverse triangle of X. The locus of X such that V(X) is perspective to U is the cubic given by

a^3 b^3 x^3-a^3 c^3 x^3+a^3 b^2 c x^2 y-a^2 b c^3 x^2 y-a^2 b^3 c x y^2+a b^2 c^3 x y^2-a^3 b^3 y^3+b^3 c^3 y^3+a^2 b^3 c x^2 z-a^3 b c^2 x^2 z-a^3 b^2 c y^2 z+a b^3 c^2 y^2 z-a b^3 c^2 x z^2+a^2 b c^3 x z^2+a^3 b c^2 y z^2-a b^2 c^3 y z^2+a^3 c^3 z^3-b^3 c^3 z^3 = 0.

This cubic passes through X(i) for i = 2,75,350,6376,19567,20935,24731,24732. (Peter Moses, October 8, 2018)

X(24731) lies on these lines: {10,75}, {141,17762}, {350,4645}, {940,2162}, {1909,17302}, {1966,17738}, {2533,17159}, {3416,17144}, {3502,3905}, {3836,20947}, {4785,4823}, {18046,24502}, {18140,24342}

### X(24732) =  X(69)X(350)∩X(75)X(700)

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

See X(24731).

X(24732) lies on the cubic introduced at X(24731) and on these lines: {6,19579}, {37,2998}, {69,350}, {75,700}, {141,3863}, {256,9230}, {789,1582}, {1921,21257}, {3978,24575}, {4485,5224}, {6374,18277}, {6384,21264}, {6385,24688}, {9229,17788}, {17234,18149}

### X(24733) =  X(76)X(141)∩X(3978)X(5207)

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

Let U = obverse triangle of X(76), and let V(X) = N-obverse triangle of X. The locus of X such that V(X) is perspective to U is the cubic given by

a^6 b^6 x^3-a^6 c^6 x^3+a^6 b^4 c^2 x^2 y-a^4 b^2 c^6 x^2 y-a^4 b^6 c^2 x y^2+a^2 b^4 c^6 x y^2-a^6 b^6 y^3+b^6 c^6 y^3+a^4 b^6 c^2 x^2 z-a^6 b^2 c^4 x^2 z-a^6 b^4 c^2 y^2 z+a^2 b^6 c^4 y^2 z-a^2 b^6 c^4 x z^2+a^4 b^2 c^6 x z^2+a^6 b^2 c^4 y z^2-a^2 b^4 c^6 y z^2+a^6 c^6 z^3-b^6 c^6 z^3 = 0.

This cubic passes through X(i) for i = 2,76,3978,6374,19573,24733,24734. (Peter Moses, October 8, 2018)

X(24733) lies on these lines: {76,141}, {3978,5207}, {7797,9230}, {8920,14880}

### X(24734) =  X(32)X(8790)∩X(39)X(6374)

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

See X(24733).

X(24734) lies on the cubic introduced at X(24733) and on these lines: {32,8790}, {39,6374}, {76,14820}, {315,1899}

Collineation mappings involving Gemini triangle 5: X(24735) - X(24758)

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 5, as in centers X(24735)-X(24758). Then

m(X) = a(a b + a c - b c) x + (a^2 c - a^2 b + a b^2 + b^2 c - 2 a b c) y + (a^2 b - a^2 c + a c^2 + b c^2 - 2 a b c) z : :

(Clark Kimberling, October 9, 2018)

### X(24735) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24735) lies on these lines: {2, 17448}, {8, 1279}, {44, 21281}, {85, 1418}, {1655, 17301}, {1909, 17278}, {2082, 24358}, {3208, 4422}, {3691, 4643}, {3765, 24789}, {3772, 3975}, {3780, 4851}, {3959, 17755}, {4713, 20257}, {4754, 24796}, {17026, 21025}, {17277, 17743}, {17352, 17752}, {18194, 24746}, {20255, 21384}, {24655, 25106}, {24736, 24738}

### X(24736) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24736) lies on these lines: {2, 3728}, {1125, 3242}, {3840, 4438}, {18194, 24757}, {24735, 24738}, {24741, 24743}

### X(24737) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24737) lies on these lines: {2, 17448}, {348, 2275}, {1475, 4644}, {3242, 3589}, {5222, 24581}, {6714, 16020}, {16713, 16744}, {18194, 24744}, {24575, 24669}, {24753, 24758}

### X(24738) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24738) lies on these lines: {2, 3056}, {6714, 17337}, {24735, 24736}

### X(24739) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24739) lies on these lines: {2, 17448}, {274, 4602}, {551, 6683}, {1575, 17144}, {2275, 18135}, {4465, 23649}, {4657, 17095}, {4698, 5550}, {7834, 10199}, {17235, 24798}, {17792, 24660}, {18194, 24745}

### X(24740) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24740) lies on these lines: {2, 24525}

### X(24741) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24741) lies on these lines: {2, 24526}, {24736, 24743}

### X(24742) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24742) lies on these lines: {2, 18194}, {10, 1279}, {3008, 21257}, {3759, 4871}, {3840, 17348}, {17351, 24182}, {17352, 20340}, {24735, 24736}, {24743, 24751}, {24746, 24757}, {25117, 25279}

### X(24743) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24743) lies on these lines: {2, 24527}, {24736, 24741}, {24742, 24751}

### X(24744) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24744) lies on these lines: {2, 3728}, {583, 24695}, {18194, 24737}

### X(24745) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24745) lies on these lines: {2, 24527}, {43, 16969}, {1575, 14823}, {3216, 3795}, {18194, 24739}

### X(24746) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24746) lies on these lines: {2, 3728}, {9, 46}, {16569, 17279}, {18194, 24735}, {21020, 24958}, {24742, 24757}, {24750, 24751}

### X(24747) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24747) lies on these lines: {2, 669}, {659, 4885}, {24666, 25142}, {24755, 24756}

### X(24748) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24748) lies on these lines: {2, 24665}

### X(24749) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24749) lies on these lines: {2, 23655}, {43, 3835}, {44, 513}, {171, 23568}, {4449, 4885}, {8640, 14426}, {21051, 22090}, {21197, 24349}, {24674, 25126}

### X(24750) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24750) lies on these lines: {2, 24525}, {24735, 24736}, {24746, 24751}

### X(24751) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24751) lies on these lines: {2, 17448}, {17385, 19877}, {24742, 24743}, {24746, 24750}

### X(24752) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24752) lies on these lines: {2, 3056}, {4, 9}, {8, 1284}, {69, 1447}, {329, 20487}, {345, 8844}, {497, 22370}, {978, 1818}, {1376, 4203}, {1788, 4645}, {2886, 5224}, {3035, 17352}, {4000, 24478}, {4073, 16609}, {4648, 24239}, {10446, 20544}, {18194, 24737}

### X(24753) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24753) lies on these lines: {2, 18194}, {8, 1279}, {24737, 24758}

### X(24754) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24754) lies on these lines: {2, 24673}, {14408, 16816}, {16815, 23650}

### X(24755) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24755) lies on these lines: {2, 20983}, {1150, 1980}, {1491, 4369}, {3741, 8640}, {4147, 4367}, {4397, 9508}, {4874, 17420}, {10453, 23506}, {24533, 25128}, {24747, 24756}

### X(24756) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24756) lies on these lines: {2, 23655}, {513, 3716}, {17063, 21197}, {24747, 24755}

### X(24757) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24757) lies on these lines: {2, 3056}, {8, 1279}, {43, 17352}, {256, 16706}, {3741, 17123}, {18194, 24736}, {24742, 24746}

### X(24758) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 5

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

X(24758) lies on these lines: {2, 24670}, {3242, 16823}, {3739, 24343}, {17348, 21214}, {18194, 24739}, {24735, 24736}, {24737, 24753}

Collineation mappings involving Gemini triangle 6: X(24759) - X(24770)

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 6, as in centers X(24759)-X(24770). Then

m(X) = a(a b + a c - 3 b c) x + (a^2 b - 3 a^2 c - a b^2 - b^2 c + 4 a b c) y + (a^2 c - 3 a^2 b - a c^2 -b c^2 + 4 a b c) z : :

(Clark Kimberling, October 9, 2018)

### X(24759) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 6

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

X(24759) lies on these lines: {2, 24761}, {8, 20530}, {145, 3834}, {536, 9311}, {24760, 24765}

### X(24760) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 6

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

X(24760) lies on these lines: {4851, 4865}, {24759, 24765}

### X(24761) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 6

Barycentrics    3 a^3 b + 4 a^2 b^2 - a b^3 + 3 a^3 c - 21 a^2 b c + 5 a b^2 c - b^3 c + 4 a^2 c^2 + 5 a b c^2 + 2 b^2 c^2 - a c^3 - b c^3 : :

X(24761) lies on these lines: {1, 142}, {2, 24759}, {145, 20530}, {1997, 4393}

### X(24762) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 6

Barycentrics    2 a^3 b + 3 a^2 b^2 + 2 a^3 c - 12 a^2 b c + a b^2 c + 3 a^2 c^2 + a b c^2 + 2 b^2 c^2 : :

X(24762) lies on these lines: {1, 20530}, {2, 24759}, {1201, 4852}, {3622, 4000}, {24766, 24767}

### X(24763) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 6

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

X(24763) lies on these lines: {4865, 4950}

### X(24764) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 6

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

X(24764) lies on these lines: (none)

### X(24765) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 6

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

X(24765) lies on these lines: {2, 24766}, {519, 17232}, {4852, 4906}, {24759, 24760}

### X(24766) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 6

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

X(24766) lies on these lines: {1, 87}, {2, 24765}, {43, 4852}, {3723, 4906}, {17336, 18194}, {24762, 24767}

### X(24767) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 6

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

X(24767) lies on these lines: {24762, 24766}

### X(24768) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 6

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

X(24768) lies on these lines: {659, 24666}, {4369, 4879}

### X(24769) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 6

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

X(24769) lies on these lines: {42, 4382}, {4369, 4449}, {4380, 23655}

### X(24770) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 6

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

X(24770) lies on these lines: {2, 24759}, {8, 24652}, {10, 20530}, {3617, 4000}, {17050, 25102}, {17235, 25280}

### X(24771) =  X(9)X(497)∩X(200)X(220)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28432.

X(24771) lies on these lines: {2, 24181}, {9, 497}, {37, 3677}, {40, 15487}, {57, 16593}, {190, 1088}, {200, 220}, {440, 17742}, {518, 15490}, {644, 3870}, {1040, 2324}, {2325, 15286}, {3161, 10580}, {4370, 8557}, {4853, 15853}, {16572, 21096}, {17755, 20173}

X(24771) = complement of the isogonal conjugate of X(21002)
X(24771) = barycentric product X(i)*X(j) for these {i,j}: {8, 3174}, {220, 20946}, {341, 21002}, {346, 16572} , {728, 8732}
X(24771) = trilinear product X(i)*X(j) for these {i,j}: {9, 3174}, {200, 16572}, {346, 21002}, {480, 8732}
X(24771) = medial-isotomic conjugate of X(200)

### X(24772) =  X(4)X(14980)∩X(1154)X(14072)

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

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28437.

X(24772) lies on these lines: {4,14980}, {1154,14072}, {5663,15907}

Collineation mappings involving Gemini triangle 7: X(24773) - X(24795)

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 7, as in centers X(24773)-X(24795). Then

m(X) = a^2 (a - b - c) x + b ( -a b + b^2 + a c - 2 b c + c^2) y + c (-a c + c^2 + a b - 2 b c + b^2) z : :

(Clark Kimberling, October 9, 2018)

### X(24773) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24773) lies on these lines: {2, 17861}, {307, 3008}, {326, 17282}, {3772, 20268}, {4056, 16580}, {4859, 10436}, {4904, 17366}, {16551, 16888}, {17278, 20269}, {17384, 24774}, {24777, 24788}

### X(24774) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24774) lies on these lines: {1, 6706}, {2, 277}, {10, 4904}, {37, 7264}, {65, 2140}, {72, 20335}, {116, 17606}, {142, 442}, {354, 17758}, {496, 20328}, {650, 23100}, {905, 16604}, {984, 1698}, {1111, 1212}, {1155, 14377}, {1385, 9317}, {1447, 17682}, {1737, 21258}, {2348, 24805}, {3057, 17761}, {3085, 4000}, {3663, 25073}, {3752, 24786}, {3753, 17050}, {4059, 4253}, {4657, 24779}, {10914, 20257}, {13728, 24178}, {16593, 21073}, {17384, 24773}, {21264, 24161}, {22011, 24787}

### X(24775) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24775) lies on these lines: {2, 21436}, {244, 7658}, {905, 7208}, {3756, 4904}, {4885, 4939}, {20269, 24786}

### X(24776) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24776) lies on these lines: {2, 20890}, {20269, 24787}

### X(24777) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24777) lies on these lines: {2, 21414}, {24174, 24785}, {24773, 24788}

### X(24778) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24778) lies on these lines: {2, 2415}, {2347, 24237}, {3008, 17077}, {4904, 17245}, {11512, 19863}, {16706, 24780}, {16828, 24178}, {17278, 20269}, {17384, 24784}, {20905, 25065}, {21173, 22343}, {25076, 25243}

### X(24779) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24779) lies on these lines: {1, 142}, {2, 17861}, {63, 15474}, {75, 24781}, {219, 1086}, {278, 3911}, {281, 17885}, {387, 5883}, {499, 17073}, {885, 21189}, {948, 1743}, {1214, 24789}, {1445, 1723}, {1479, 18589}, {1714, 24174}, {1737, 18634}, {3306, 24175}, {3332, 7613}, {3682, 24159}, {3739, 19854}, {4021, 19785}, {4361, 4904}, {4657, 24774}, {6737, 21255}, {10573, 16608}, {16752, 17189}

### X(24780) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24780) lies on these lines: {1, 16608}, {2, 17861}, {35, 18589}, {57, 1744}, {80, 20305}, {86, 4612}, {142, 10090}, {256, 21189}, {284, 4466}, {499, 1733}, {905, 16696}, {1030, 16581}, {1375, 1781}, {2294, 24884}, {3624, 4657}, {3739, 24784}, {4904, 17045}, {7110, 16732}, {16706, 24778}, {17052, 24435}, {17322, 24781}

### X(24781) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24781) lies on these lines: {2, 277}, {75, 24779}, {92, 15474}, {226, 17682}, {239, 4904}, {379, 4911}, {443, 5440}, {448, 24617}, {475, 1895}, {666, 3008}, {693, 905}, {1010, 1125}, {1086, 1944}, {1375, 1447}, {3811, 18134}, {4000, 20227}, {4859, 10436}, {5249, 16054}, {17023, 24181}, {17322, 24780}

### X(24782) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24782) lies on these lines: {1, 4477}, {2, 647}, {43, 4524}, {241, 514}, {512, 24533}, {659, 6363}, {693, 3310}, {1021, 2999}, {1491, 21189}, {3239, 25098}, {3666, 3700}, {3752, 17069}, {4383, 9404}, {4467, 4850}, {4885, 6589}, {4897, 21894}, {21828, 24924}, {24793, 24794}

### X(24783) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24783) lies on these lines: {905, 24794}

### X(24784) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24784) lies on these lines: {2, 277}, {116, 2646}, {142, 13747}, {905, 1107}, {1125, 4904}, {3739, 24780}, {3752, 20267}, {5440, 17046}, {9317, 9956}, {10039, 17044}, {14377, 17605}, {17062, 17614}, {17384, 24778}, {17619, 24249}, {19862, 24181}

### X(24785) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24785) lies on these lines: {2, 23581}, {24174, 24777}

### X(24786) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24786) lies on these lines: {1, 17048}, {2, 1930}, {39, 1111}, {41, 2224}, {244, 17758}, {325, 17192}, {350, 4099}, {498, 4000}, {1015, 7278}, {1089, 3934}, {1447, 5280}, {1739, 17050}, {2140, 24443}, {2275, 7208}, {2276, 7264}, {2548, 4056}, {3294, 25073}, {3670, 20335}, {3752, 24774}, {3970, 24172}, {3987, 20257}, {4372, 7815}, {4376, 7808}, {4642, 17761}, {4647, 21264}, {4680, 7800}, {6292, 16886}, {6683, 16720}, {7272, 9596}, {20269, 24775}, {21208, 21808}

### X(24787) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24787) lies on these lines: {2, 21404}, {3925, 4904}, {20269, 24776}, {22011, 24774}

### X(24788) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24788) lies on these lines: {2, 25090}, {905, 16716}, {3739, 24780}, {3772, 17068}, {19269, 24161}, {20267, 20268}, {20269, 24776}, {24773, 24777}

### X(24789) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24789) lies on these lines: {1, 3925}, {2, 37}, {3, 2218}, {6, 5249}, {7, 4641}, {9, 3782}, {11, 1040}, {12, 1722}, {31, 5880}, {32, 24588}, {43, 17718}, {44, 5905}, {55, 1738}, {56, 24178}, {57, 1723}, {63, 1086}, {72, 24159}, {81, 4675}, {141, 5271}, {142, 940}, {200, 17724}, {218, 226}, {223, 5723}, {238, 1836}, {239, 18134}, {241, 277}, {244, 24892}, {273, 18679}, {306, 4361}, {333, 3662}, {377, 1104}, {405, 23537}, {497, 16020}, {499, 17102}, {612, 3826}, {614, 2886}, {650, 24793}, {748, 3120}, {942, 1714}, {958, 23536}, {975, 17529}, {978, 11375}, {980, 24790}, {988, 24953}, {990, 8226}, {1001, 3914}, {1125, 16458}, {1211, 4384}, {1214, 24779}, {1279, 3434}, {1376, 3011}, {1445, 6354}, {1698, 7322}, {1707, 11246}, {1717, 7741}, {1721, 7965}, {1743, 4654}, {1754, 5805}, {1999, 17234}, {2006, 8056}, {2178, 16438}, {2275, 16699}, {2352, 16056}, {2550, 3744}, {2887, 3966}, {2999, 5718}, {3187, 4851}, {3216, 11374}, {3219, 17276}, {3242, 25006}, {3305, 4415}, {3474, 7613}, {3654, 4674}, {3670, 5791}, {3683, 24248}, {3742, 11269}, {3757, 4429}, {3759, 17778}, {3765, 24735}, {3784, 18191}, {3812, 5230}, {3816, 17070}, {3823, 10327}, {3836, 4362}, {3911, 24175}, {3924, 5794}, {3929, 4862}, {3944, 4679}, {3946, 20182}, {3980, 6679}, {3999, 24477}, {4001, 7232}, {4197, 5262}, {4208, 5716}, {4357, 19732}, {4363, 5294}, {4416, 19723}, {4423, 16849}, {4438, 24165}, {4643, 5278}, {4656, 6666}, {4847, 17597}, {4886, 16816}, {4966, 17156}, {5137, 9306}, {5219, 23511}, {5222, 5712}, {5231, 5573}, {5247, 10404}, {5256, 17056}, {5268, 17602}, {5287, 17245}, {5292, 5439}, {5433, 11512}, {5721, 18443}, {5737, 17290}, {5739, 17348}, {5745, 17067}, {6734, 17054}, {6745, 17783}, {7270, 19851}, {7292, 11680}, {7299, 7702}, {7365, 8732}, {7522, 19728}, {8270, 15253}, {8616, 24715}, {10528, 21896}, {11376, 21214}, {11679, 17282}, {12513, 23675}, {12609, 16466}, {15474, 18607}, {16062, 16817}, {16569, 17719}, {16580, 19724}, {16736, 18603}, {16885, 17781}, {17022, 20195}, {17023, 19701}, {17063, 17728}, {17073, 20268}, {17092, 18625}, {17127, 20292}, {17299, 20483}, {17325, 19744}, {17749, 24160}, {20269, 24776}, {20470, 23339}, {21020, 24943}, {24046, 24880}, {24174, 24914}, {25048, 25308}

### X(24790) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24790) lies on these lines: {1, 142}, {2, 3760}, {31, 14377}, {42, 17758}, {81, 24588}, {116, 21935}, {213, 1086}, {274, 16706}, {609, 4209}, {673, 5299}, {980, 24789}, {1111, 16583}, {1193, 2140}, {1201, 17761}, {1475, 17205}, {3008, 16552}, {3216, 20335}, {3290, 7264}, {3294, 3663}, {3739, 16828}, {3752, 24774}, {3772, 20267}, {3987, 21232}, {5280, 17682}, {5283, 17278}, {7795, 19846}, {16600, 20880}, {16752, 17023}, {16831, 19785}, {16887, 24592}, {17033, 24190}, {17326, 24919}, {17366, 20963}, {17489, 20893}, {19864, 21264}, {20367, 24177}

### X(24791) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24791) lies on these lines: {2, 25095}

### X(24792) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24792) lies on these lines: {905, 4762}, {1769, 17067}

### X(24793) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24793) lies on these lines: {2, 21438}, {650, 24789}, {693, 905}, {3772, 4885}, {5249, 22383}, {5272, 15283}, {15280, 17064}, {23733, 23806}, {24782, 24794}

### X(24794) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24794) lies on these lines: {905, 24783}, {24782, 24793}

### X(24795) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 7

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

X(24795) lies on these lines: {2, 25272}, {141, 3679}, {277, 291}, {333, 17205}, {3008, 3732}, {4000, 21232}, {17278, 21138}, {21264, 24161}

Collineation mappings involving the Garcia reflection triangle (Gemini triangle 8): X(24796) - X(24805)

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 8, as in centers X(24796)-X(24805). Then

m(X) = a (a^2 - a b + 2 b^2 - a c - 4 c b + 2c^2) x - (2a^b + a b^2 - b^3 + 2 a^2 c + 3 a b c + 2 b^2 c - 4 a c^2 - 3 b c^2 + 2 c^3) y - (2a^c + a c^2 - c^3 + 2 a^2 b + 3 a b c + 2 b c^2 - 4 a b^2 - 3 b^2 c + 2 b^3) z : :

(Clark Kimberling, October 9, 2018)

### X(24796) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GARCIA REFLECTION TRIANGLE

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

X(24796) lies on these lines: {1, 1358}, {2, 24797}, {7, 8}, {56, 10481}, {277, 2348}, {279, 1319}, {354, 14268}, {553, 19723}, {1086, 2082}, {1111, 1837}, {1323, 1388}, {1334, 17276}, {1420, 21314}, {1467, 7271}, {1565, 11376}, {1697, 4862}, {3303, 3663}, {3361, 20121}, {3665, 11375}, {4373, 12632}, {4675, 21808}, {4754, 24735}, {4888, 11518}, {4902, 7991}, {5221, 10521}, {9436, 24914}, {12701, 17170}

### X(24797) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GARCIA REFLECTION TRIANGLE

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

X(24797) lies on these lines: {2, 24796}, {7, 21}, {8, 1358}, {9, 17107}, {85, 25280}, {279, 3476}, {388, 10481}, {1788, 7195}, {4862, 12053}, {4902, 11522}, {5290, 20121}, {6604, 7185}, {10106, 21314}

### X(24798) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GARCIA REFLECTION TRIANGLE

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

X(24798) lies on these lines: {2, 24796}, {7, 5550}, {10, 1358}, {12, 10481}, {65, 760}, {279, 5252}, {307, 1122}, {348, 1319}, {1071, 1367}, {1111, 17606}, {1323, 10944}, {1565, 3057}, {2348, 20269}, {3669, 17448}, {4059, 7179}, {6604, 11011}, {7195, 24914}, {9578, 21314}, {17235, 24739}

### X(24799) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GARCIA REFLECTION TRIANGLE

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

X(24799) lies on these lines: {1, 87}, {2, 24802}, {1111, 14100}, {2293, 24225}, {7671, 17861}

### X(24800) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GARCIA REFLECTION TRIANGLE

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

X(24800) lies on these lines: {2, 24796}, {7, 350}, {42, 1358}, {85, 25287}

### X(24801) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GARCIA REFLECTION TRIANGLE

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

X(24801) lies on these lines: {2, 24796}, {7, 24495}, {43, 1358}, {65, 1088}, {1836, 4106}

### X(24802) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GARCIA REFLECTION TRIANGLE

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

X(24802) lies on these lines: {1, 6}, {2, 24799}, {982, 21267}, {2310, 4862}, {4888, 21346}, {7671, 25065}, {17885, 24225}, {21185, 24207}

### X(24803) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GARCIA REFLECTION TRIANGLE

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

X(24803) lies on these lines: {2, 24796}, {7, 37}, {57, 20602}, {63, 17107}, {239, 1358}, {552, 553}, {3669, 4380}, {5228, 7185}

### X(24804) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GARCIA REFLECTION TRIANGLE

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

X(24804) lies on these lines: (none)

### X(24805) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GARCIA REFLECTION TRIANGLE

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

X(24805) lies on these lines: {2, 24796}, {7, 10588}, {85, 1319}, {1111, 2646}, {1125, 1358}, {1434, 1447}, {2348, 24774}, {3598, 10404}, {3649, 10521}, {3674, 4870}, {3748, 7264}, {5183, 17753}, {5433, 10481}, {6706, 9318}, {7195, 11375}

Centers related to obverse triangles: X(24806) - X(24854)

This preamble and centers X(24806)-X(24854) were contributed by César Eliud Lozada, October 10, 2018.

• Obverse triangle of X(1) (defined just before of X24307):

• Perspective triangles and perspectors:
• (ABC, 75), (anti-tangential-midarc, 24806), (anticomplementary, 1654), (Aquila, 3679), (1st circumperp, 24309), (2nd circumperp, 993), (excentral, 9), (extangents, 24308), (2nd extouch, 9), (Fuhrmann, 10), (inner-Garcia, 1), (outer-Garcia, 3679), (Garcia-reflection, 9), (intangents, 24307), (medial, 9), (1st Sharygin, 8424), (2nd Sharygin, 4363), (tangential, 8424), (2nd Zaniah, 9), (N-obverse of X(1), 75), (trilinear obverse of X(2), 894),

• Orthologic triangles and orthologic centers:
• (anti-Artzt, 3679, 24807), (Artzt, 3679, 24808), (1st Parry, 1, 24809), (2nd Parry, 1, 24810), (1st tri-squares, 3679, 24811), (2nd tri-squares, 3679, 24812)

• Parallelogic triangles and parallelogic centers:
• (ABC, 1, 190), (ABC-X3 reflections, 1, 24813), (anti-Aquila, 1, 4432), (anti-Ara, 1, 24814), (5th anti-Brocard, 1, 24815), (2nd anti-circumperp-tangential, 1, 24816), (anti-Euler, 1, 24817), (anti-inner-Grebe, 1, 24818), (anti-outer-Grebe, 1, 24819), (anti-Mandart-incircle, 1, 24820), (anticomplementary, 1, 4440), (Aquila, 1, 24821), (Ara, 1, 24822), (1st Auriga, 1, 24823), (2nd Auriga, 1, 24824), (5th Brocard, 1, 24825), (2nd circumperp tangential, 1, 24826), (Ehrmann-mid, 1, 24827), (Euler, 1, 24828), (outer-Garcia, 1, 24715), (Gossard, 1, 24830), (inner-Grebe, 1, 24831), (outer-Grebe, 1, 24832), (Johnson, 1, 24833), (inner-Johnson, 1, 24834), (outer-Johnson, 1, 24835), (1st Johnson-Yff, 1, 24836), (2nd Johnson-Yff, 1, 24837), (Lucas homothetic, 1, 24838), (Lucas(-1) homothetic, 1, 24839), (Mandart-incircle, 1, 24840), (medial, 1, 1086), (5th mixtilinear, 1, 24841), (3rd tri-squares-central, 1, 24842), (4th tri-squares-central, 1, 24843), (X3-ABC reflections, 1, 24844), (Yff contact, 10, 190), (inner-Yff, 1, 24845), (outer-Yff, 1, 24846), (inner-Yff tangents, 1, 24847), (outer-Yff tangents, 1, 24848)

• N-obverse triangle of X(1) (defined just before of X24310)

• Perspective triangles and perspectors:
• (ABC, 75), (anti-tangential-midarc, 24849), (extangents, 24310), (5th extouch, 24312), (outer-Garcia, 1), (medial, 1), (1st Sharygin, 24311), (2nd Sharygin, 8301), (tangential, 8301), (1st Zaniah, 1), (obverse of X(1), 75), (trilinear N-obverse of X(2), 239)

• Orthologic triangles and orthologic centers:
• (excenters-midpoints, 1, 24850), (Garcia-reflection, 1, 24851), (2nd Schiffler, 1, 24852)

• Trilinear obverse triangle of X(2) (defined just before of X24482)

• Perspective triangles and perspectors:
• (ABC, 6), (anti-Conway, 6), (2nd anti-Conway, 6), (anti-inner-Grebe, 6), (anti-outer-Grebe, 6), (anti-Honsberger, 6), (anticomplementary, 192), (2nd Brocard, 6), (circumsymmedial, 6), (2nd Ehrmann, 6), (excentral, 1045), (9th Fermat-Dao, 6), (10th Fermat-Dao, 6), (13th Fermat-Dao, 6), (14th Fermat-Dao, 6), (inner-Grebe, 6), (outer-Grebe, 6), (incentral, 192), (1st Kenmotu diagonals, 6), (2nd Kenmotu diagonals, 6), (2nd mixtilinear, 24853), (2nd orthosymmedial, 6), (symmedial, 6), (tangential, 6), (inner tri-equilateral, 6), (outer tri-equilateral, 6), (obverse of X(1), 894), (trilinear N-obverse of X(2), 6)

• Orthologic triangles and orthologic centers:
• (1st Ehrmann,6,24854)

• Trilinear N-obverse triangle of X(2) (defined just before of X24519)

• Perspective triangles and perspectors:
• (ABC, 6), (anti-Conway, 6), (2nd anti-Conway, 6), (anti-inner-Grebe, 6), (anti-outer-Grebe, 6), (anti-Honsberger, 6), (2nd Brocard, 6), (circumsymmedial, 6), (2nd Ehrmann, 6), (9th Fermat-Dao, 6), (10th Fermat-Dao, 6), (13th Fermat-Dao, 6), (14th Fermat-Dao, 6), (inner-Grebe, 6), (outer-Grebe, 6), (incentral, 2), (1st Kenmotu diagonals, 6), (2nd Kenmotu diagonals, 6), (2nd orthosymmedial, 6), (symmedial, 6), (tangential, 6), (inner tri-equilateral, 6), (outer tri-equilateral, 6), (N-obverse of X(1), 239), trilinear obverse of X(2), 6)

• Orthologic triangles and orthologic centers:
• (1st Ehrmann,6,24829)

### X(24806) = PERSPECTOR OF THESE TRIANGLES: OBVERSE OF X(1) AND ANTI-TANGENTIAL-MIDARC

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

X(24806) lies on these lines: {1,3}, {2,1457}, {8,73}, {9,4559}, {10,10571}, {34,92}, {75,77}, {87,1411}, {109,993}, {201,3869}, {221,958}, {222,956}, {223,9623}, {227,5836}, {269,7223}, {348,21281}, {355,1745}, {386,4848}, {388,1042}, {603,2975}, {944,4303}, {962,2654}, {995,3911}, {1044,7354}, {1064,18391}, {1193,1788}, {1201,7288}, {1406,22759}, {1442,17015}, {1450,5435}, {1458,3476}, {1464,5252}, {1465,3753}, {1943,16821}, {2263,8680}, {3486,4300}, {3679,4551}, {4296,6360}, {4306,10106}, {4334,5434}, {4386,17966}, {4392,18419}, {5433,21214}, {5657,22350}, {5731,22053}, {6001,24430}, {6708,19372}, {15656,21767}, {21147,23555}

X(24806) = X(23853)-of-2nd anti-circumperp-tangential triangle
X(24806) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (65, 1319, 1403), (221, 958, 1935), (1042, 10459, 388)

### X(24807) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARTZT TO OBVERSE OF X(1)

Barycentrics    7*a^4-7*(b+c)*a^3-(b^2-11*b*c+c^2)*a^2+(b+c)*(5*b^2-12*b*c+5*c^2)*a-2*b^4-b^3*c+8*b^2*c^2-b*c^3-2*c^4 : :

The reciprocal orthologic center of these triangles is X(3679)

X(24807) lies on these lines: {2,952}, {519,4482}, {903,24281}, {2784,11161}, {2789,8593}, {13637,24811}, {13757,24812}

X(24807) = reflection of X(903) in X(24281)

### X(24808) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARTZT TO OBVERSE OF X(1)

Barycentrics    a^6-(b+c)*a^5+(b+2*c)*(2*b+c)*a^4-2*(b+c)^3*a^3-(b^2-4*b*c+c^2)*(b+c)^2*a^2+(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a-(2*b^2-b*c+2*c^2)*(b^2-c^2)^2 : :
X(24808) = 7*X(3090)-4*X(15251)

The reciprocal orthologic center of these triangles is X(3679).

Let P be a point on the Steiner circumellipse. Let A' be the orthocenter of BCP, and define B' and C' cyclically. Let Q be the centroid of A'B'C'. The locus of Q as P varies is an ellipse similar and orthogonal to the Steiner circumellipse, and also centered at X(2). When P = X(190), Q = X(24808). See also X(98) and X(6054). (Randy Hutson, October 15, 2018)

X(24808) lies on these lines: {2,952}, {4,10743}, {10,98}, {100,7427}, {355,21554}, {515,13635}, {517,3799}, {953,3006}, {2789,6054}, {2826,10711}, {3090,15251}, {5690,7385}, {5818,7380}, {9956,16830}, {13638,24811}, {13758,24812}

### X(24809) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO OBVERSE OF X(1)

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

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

X(24809) lies on these lines: {2,2784}, {10,5168}, {110,1010}, {351,24810}, {952,9978}, {2789,9485}, {2826,13250}

X(24809) = reflection of X(24810) in X(351)
X(24809) = X(190)-of-1st Parry triangle
X(24809) = X(24813)-of-2nd Parry triangle

### X(24810) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO OBVERSE OF X(1)

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

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

X(24810) lies on these lines: {10,5029}, {351,24809}, {952,9980}, {2784,9147}, {2789,5466}, {2826,13251}

X(24810) = reflection of X(24809) in X(351)
X(24810) = X(190)-of-2nd Parry triangle
X(24810) = X(24813)-of-1st Parry triangle

### X(24811) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO OBVERSE OF X(1)

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

The reciprocal orthologic center of these triangles is X(3679)

X(24811) lies on these lines: {2,24812}, {952,3068}, {2784,13653}, {2789,13640}, {13637,24807}, {13638,24808}

### X(24812) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO OBVERSE OF X(1)

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

The reciprocal orthologic center of these triangles is X(3679)

X(24812) lies on these lines: {2,24811}, {952,3069}, {2784,13773}, {2789,13760}, {13757,24807}, {13758,24808}

### X(24813) = PARALLELOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO OBVERSE OF X(1)

Barycentrics    a^6-(b+c)*a^5+(2*b^2-3*b*c+2*c^2)*a^4-(3*b^2+2*b*c+3*c^2)*(b-c)^2*a^2+(b^2-c^2)^2*(b+c)*a-(b^2-c^2)^2*b*c : :
X(24813) = 3*X(3)-X(24844) = 3*X(165)-X(24821) = 3*X(190)-2*X(24844) = 3*X(376)-X(24817) = 5*X(631)-4*X(4422) = 3*X(903)-2*X(24833) = 7*X(3523)-5*X(4473) = 3*X(3524)-2*X(4370) = 3*X(3576)-2*X(4432) = 2*X(4409)+5*X(17538) = 2*X(4437)-3*X(10519) = 3*X(10304)-X(17487) = 2*X(16593)-3*X(21151)

The reciprocal parallelogic center of these triangles is X(1)

X(24813) lies on these lines: {2,24828}, {3,190}, {4,1086}, {20,4440}, {30,903}, {35,24845}, {36,24846}, {40,537}, {55,24816}, {56,24840}, {98,103}, {104,900}, {165,24821}, {182,24815}, {371,24819}, {372,24818}, {376,545}, {382,24827}, {515,24715}, {517,24841}, {528,944}, {631,4422}, {673,971}, {726,19589}, {1350,9055}, {1593,24814}, {1897,3937}, {2796,4297}, {3098,24825}, {3428,24826}, {3523,4473}, {3524,4370}, {3576,4432}, {4409,17538}, {4437,10519}, {4997,19515}, {5845,6776}, {6284,24837}, {7354,24836}, {9041,12245}, {9540,24842}, {10304,17487}, {10305,18283}, {10310,24820}, {11248,24847}, {11249,24848}, {11414,24822}, {11822,24823}, {11823,24824}, {11824,24831}, {11825,24832}, {11827,24835}, {11828,24838}, {11829,24839}, {13935,24843}, {16593,21151}

X(24813) = midpoint of X(20) and X(4440)
X(24813) = reflection of X(i) in X(j) for these (i,j): (4, 1086), (190, 3), (382, 24827)
X(24813) = anticomplement of X(24828)
X(24813) = X(190)-of-ABC-X3 reflections triangle
X(24813) = X(1086)-of-anti-Euler triangle
X(24813) = X(24816)-of-anti-Mandart-incircle triangle
X(24813) = X(24840)-of-2nd circumperp tangential triangle

### X(24814) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24814) lies on these lines: {4,4440}, {25,190}, {33,24840}, {34,24816}, {235,24828}, {427,1086}, {428,545}, {468,4422}, {528,12135}, {537,1829}, {900,1830}, {903,5064}, {1593,24813}, {1598,24844}, {1843,9055}, {2786,5185}, {2796,12132}, {4432,11363}, {4473,6353}, {5090,24715}, {5411,24818}, {7487,24817}, {7713,24821}, {7714,17487}, {11380,24815}, {11383,24820}, {11384,24823}, {11385,24824}, {11386,24825}, {11388,24831}, {11389,24832}, {11390,24834}, {11391,24835}, {11392,24836}, {11393,24837}, {11394,24838}, {11395,24839}, {11396,24841}, {11398,24845}, {11399,24846}, {11400,24847}, {11401,24848}, {11832,24830}, {13884,24842}, {13937,24843}, {22479,24826}

X(24814) = X(190)-of-anti-Ara triangle

### X(24815) = PARALLELOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24815) lies on these lines: {32,190}, {83,1086}, {98,24828}, {101,24502}, {182,24813}, {528,12195}, {537,12194}, {545,12150}, {900,13194}, {1078,4422}, {2786,4027}, {2796,12191}, {4432,11364}, {4440,7787}, {4473,7793}, {9055,12212}, {10788,24817}, {10789,24821}, {10790,24822}, {10791,24715}, {10792,24831}, {10793,24832}, {10794,24834}, {10795,24835}, {10796,24833}, {10797,24836}, {10798,24837}, {10799,24840}, {10800,24841}, {10801,24845}, {10802,24846}, {10803,24847}, {10804,24848}, {11380,24814}, {11490,24820}, {11837,24823}, {11838,24824}, {11839,24830}, {11840,24838}, {11841,24839}, {11842,24844}, {12835,24816}, {13885,24842}, {13938,24843}, {18502,24827}, {18993,24818}, {18994,24819}, {22520,24826}

X(24815) = X(190)-of-5th anti-Brocard triangle

### X(24816) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24816) lies on these lines: {1,24840}, {3,24845}, {4,24837}, {11,24828}, {12,1086}, {34,24814}, {56,190}, {57,24821}, {65,537}, {388,4440}, {528,3885}, {545,5434}, {726,1463}, {900,1317}, {903,11237}, {999,24844}, {1227,3264}, {1319,4432}, {1362,2786}, {1411,24416}, {1469,9055}, {1478,24833}, {2099,24841}, {2796,10106}, {3585,24827}, {3837,20366}, {4293,24817}, {4370,5298}, {4422,5433}, {4473,7288}, {5252,24715}, {11509,24820}, {12835,24815}, {17114,24068}, {18954,24822}, {18955,24823}, {18956,24824}, {18957,24825}, {18958,24830}, {18959,24831}, {18960,24832}, {18961,24834}, {18962,24835}, {18963,24838}, {18964,24839}, {18965,24842}, {18966,24843}, {18967,24848}, {18995,24818}, {18996,24819}

X(24816) = reflection of X(24840) in X(1)
X(24816) = X(190)-of-2nd anti-circumperp-tangential triangle
X(24816) = X(24813)-of-Mandart-incircle triangle
X(24816) = X(24840)-of-5th mixtilinear triangle
X(24816) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 4440, 24836), (999, 24844, 24846)

### X(24817) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO OBVERSE OF X(1)

Barycentrics    3*a^6-3*(b+c)*a^5-(3*b^2-b*c+3*c^2)*a^4+4*(b+c)*(b^2+c^2)*a^3+(b^4+c^4-2*(2*b^2+b*c+2*c^2)*b*c)*a^2-(b^2-c^2)^2*(b+c)*a-(b^2-3*b*c+c^2)*(b^2-c^2)^2 : :
X(24817) = 3*X(4)-4*X(24828) = 4*X(5)-5*X(4473) = 3*X(190)-2*X(24828) = 3*X(376)-2*X(24813) = 5*X(631)-4*X(1086) = 2*X(673)-3*X(21168) = 2*X(903)-3*X(3524) = 7*X(3090)-8*X(4422) = 5*X(3091)-4*X(24827) = 3*X(3545)-4*X(4370) = 4*X(4409)-11*X(21735) = 4*X(4432)-3*X(5603) = 3*X(5657)-2*X(24715) = 3*X(7967)-2*X(24841) = 3*X(17487)-2*X(24844)

The reciprocal parallelogic center of these triangles is X(1)

X(24817) lies on these lines: {2,24833}, {3,4440}, {4,190}, {5,4473}, {24,24822}, {30,17487}, {40,2796}, {104,24826}, {376,545}, {388,24845}, {497,24846}, {515,24821}, {528,11827}, {537,944}, {631,1086}, {673,21168}, {900,13199}, {903,3524}, {2786,13172}, {3085,24836}, {3086,24837}, {3090,4422}, {3091,24827}, {3545,4370}, {4293,24816}, {4294,24840}, {4409,21735}, {4432,5603}, {5657,24715}, {5762,20533}, {6776,9055}, {7487,24814}, {7581,24819}, {7582,24818}, {7967,24841}, {9862,24825}, {10783,24831}, {10784,24832}, {10785,24834}, {10786,24835}, {10788,24815}, {10805,24847}, {10806,24848}, {11491,24820}, {11843,24823}, {11844,24824}, {11845,24830}, {11846,24838}, {11847,24839}, {13886,24842}, {13939,24843}

X(24817) = reflection of X(i) in X(j) for these (i,j): (4, 190), (4440, 3)
X(24817) = anticomplement of X(24833)
X(24817) = X(190)-of-anti-Euler triangle
X(24817) = X(4440)-of-ABC-X3 reflections triangle

### X(24818) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24818) lies on these lines: {2,24843}, {6,190}, {372,24813}, {528,19065}, {537,18992}, {545,19053}, {900,19112}, {1086,3069}, {1587,24828}, {2786,19108}, {2796,19057}, {3068,4422}, {3299,24845}, {3301,24846}, {4370,19054}, {4432,18991}, {4440,7586}, {4473,7585}, {5411,24814}, {6418,24844}, {7582,24817}, {7584,24833}, {7968,24841}, {13785,24827}, {13936,24715}, {18993,24815}, {18995,24816}, {18999,24820}, {19003,24821}, {19005,24822}, {19007,24823}, {19009,24824}, {19011,24825}, {19013,24826}, {19017,24830}, {19023,24834}, {19025,24835}, {19027,24836}, {19029,24837}, {19031,24838}, {19033,24839}, {19037,24840}, {19047,24847}, {19049,24848}

X(24818) = X(190)-of-anti-inner-Grebe triangle
X(24818) = {X(4473), X(7585)}-harmonic conjugate of X(24842)

### X(24819) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24819) lies on these lines: {2,24842}, {6,190}, {371,24813}, {528,19066}, {537,18991}, {545,19054}, {900,19113}, {1086,3068}, {1588,24828}, {2786,19109}, {2796,19058}, {3069,4422}, {3299,24846}, {3301,24845}, {4370,19053}, {4432,18992}, {4440,7585}, {4473,7586}, {6417,24844}, {7581,24817}, {7583,24833}, {7969,24841}, {13665,24827}, {13883,24715}, {18994,24815}, {18996,24816}, {19000,24820}, {19004,24821}, {19006,24822}, {19008,24823}, {19010,24824}, {19012,24825}, {19014,24826}, {19018,24830}, {19024,24834}, {19026,24835}, {19028,24836}, {19030,24837}, {19032,24838}, {19034,24839}, {19038,24840}, {19048,24847}, {19050,24848}

X(24819) = X(190)-of-anti-outer-Grebe triangle
X(24819) = {X(4473), X(7586)}-harmonic conjugate of X(24843)

### X(24820) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24820) lies on these lines: {3,537}, {4,528}, {6,983}, {35,24821}, {55,190}, {56,24841}, {100,4440}, {105,4578}, {197,24822}, {335,11329}, {480,673}, {518,6211}, {545,4421}, {900,13205}, {1001,4422}, {1083,4069}, {1086,1376}, {1621,4473}, {2786,13173}, {2796,8715}, {3295,4432}, {3303,17697}, {3939,14839}, {4370,4428}, {4436,24345}, {4557,8301}, {5687,24715}, {7202,16504}, {8177,9055}, {8865,18755}, {9025,19589}, {9041,12513}, {10310,24813}, {11383,24814}, {11490,24815}, {11491,24817}, {11492,24823}, {11493,24824}, {11494,24825}, {11496,24828}, {11497,24831}, {11498,24832}, {11499,24833}, {11500,24835}, {11501,24836}, {11502,24837}, {11503,24838}, {11504,24839}, {11507,24845}, {11508,24846}, {11509,24816}, {11510,24848}, {11848,24830}, {11849,24844}, {13887,24842}, {13940,24843}, {16686,23343}, {18491,24827}, {18999,24818}, {19000,24819}

X(24820) = reflection of X(i) in X(j) for these (i,j): (24826, 3), (24834, 1086)
X(24820) = X(190)-of-anti-Mandart-incircle triangle
X(24820) = X(24841)-of-2nd circumperp tangential triangle

### X(24821) = PARALLELOGIC CENTER OF THESE TRIANGLES: AQUILA TO OBVERSE OF X(1)

Barycentrics    a^3+(b+c)*a^2-(3*b^2+b*c+3*c^2)*a+2*(b+c)*b*c : :
X(24821) = 3*X(1)-4*X(4432) = 3*X(1)-2*X(24841) = 3*X(165)-2*X(24813) = 3*X(190)-2*X(4432) = 3*X(190)-X(24841) = 2*X(239)-3*X(1757) = 2*X(903)-3*X(19875) = 4*X(1086)-5*X(1698) = 4*X(1125)-5*X(4473) = 3*X(1699)-4*X(24828) = 7*X(3624)-8*X(4422) = 3*X(3679)-2*X(24715) = 3*X(3932)-2*X(7238) = 3*X(5587)-2*X(24833) = 3*X(11852)-2*X(24830)

The reciprocal parallelogic center of these triangles is X(1)

X(24821) lies on these lines: {1,190}, {8,2796}, {10,4440}, {35,24820}, {36,24826}, {57,24816}, {165,24813}, {239,726}, {244,4756}, {291,4465}, {320,4439}, {335,16831}, {515,24817}, {517,24844}, {518,4693}, {519,4480}, {528,3632}, {545,3679}, {846,17165}, {900,5541}, {903,17250}, {984,4363}, {1046,24068}, {1054,3952}, {1086,1698}, {1125,4473}, {1282,2786}, {1697,24840}, {1699,24828}, {2108,17794}, {3099,24825}, {3624,4422}, {3633,9041}, {3751,9055}, {3932,7238}, {4009,18201}, {4370,16676}, {4753,17160}, {5220,17119}, {5587,24833}, {5588,24832}, {5589,24831}, {5850,20533}, {6763,16560}, {7713,24814}, {8185,24822}, {8186,24823}, {8187,24824}, {8188,24838}, {8189,24839}, {9324,17780}, {9457,13541}, {9578,24836}, {9581,24837}, {10789,24815}, {10826,24834}, {10827,24835}, {11852,24830}, {13888,24842}, {13942,24843}, {16832,17755}, {18492,24827}, {19003,24818}, {19004,24819}, {20375,21391}, {21087,24131}, {24331,24349}

X(24821) = reflection of X(i) in X(j) for these (i,j): (1, 190), (320, 4439), (4440, 10), (17160, 4753), (24841, 4432)
X(24821) = X(190)-of-Aquila triangle
X(24821) = X(4440)-of-outer-Garcia triangle
X(24821) = {X(4432), X(24841)}-harmonic conjugate of X(1)

### X(24822) = PARALLELOGIC CENTER OF THESE TRIANGLES: ARA TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24822) lies on these lines: {3,1086}, {22,4440}, {24,24817}, {25,190}, {55,24445}, {159,9055}, {197,24820}, {528,12410}, {537,9798}, {545,9909}, {900,13222}, {1486,24358}, {1598,24828}, {1995,4473}, {2786,13175}, {2796,9876}, {4363,20834}, {4422,5020}, {4432,11365}, {4446,7083}, {5594,24832}, {5595,24831}, {7517,24844}, {8185,24821}, {8190,24823}, {8191,24824}, {8192,24841}, {8193,24715}, {8194,24838}, {8195,24839}, {9818,24827}, {10037,24845}, {10046,24846}, {10790,24815}, {10828,24825}, {10829,24834}, {10830,24835}, {10831,24836}, {10832,24837}, {10833,24840}, {10834,24847}, {10835,24848}, {11414,24813}, {11853,24830}, {13889,24842}, {13943,24843}, {18954,24816}, {19005,24818}, {19006,24819}, {22654,24826}

X(24822) = X(190)-of-Ara triangle

### X(24823) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO OBVERSE OF X(1)

Barycentrics    ((b+c)*a^2-2*(b^2+c^2)*a+b*c*(b+c))*D+a^2*(a+b+c)*(-a+b+c)*(a^2-(b+c)*a-b*c+c^2+b^2) : : , where D=4*S*sqrt(R*(4*R+r))

The reciprocal parallelogic center of these triangles is X(1)

X(24823) lies on these lines: {55,537}, {190,5597}, {528,12454}, {545,11207}, {900,13228}, {1086,5599}, {2786,13176}, {2796,12345}, {4432,11366}, {4440,5601}, {5598,24841}, {8186,24821}, {8190,24822}, {8196,24828}, {8197,24715}, {8198,24831}, {8199,24832}, {8200,24833}, {8201,24838}, {8202,24839}, {9041,12455}, {9055,12452}, {11384,24814}, {11492,24820}, {11493,24826}, {11822,24813}, {11837,24815}, {11843,24817}, {11861,24825}, {11865,24834}, {11867,24835}, {11869,24836}, {11871,24837}, {11873,24840}, {11875,24844}, {11877,24845}, {11879,24846}, {11881,24847}, {11883,24848}, {13890,24842}, {13944,24843}, {18495,24827}, {18955,24816}, {19007,24818}, {19008,24819}

X(24823) = reflection of X(24824) in X(55)
X(24823) = X(190)-of-1st Auriga triangle
X(24823) = X(24841)-of-2nd Auriga triangle

### X(24824) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO OBVERSE OF X(1)

Barycentrics    -((b+c)*a^2-2*(b^2+c^2)*a+b*c*(b+c))*D+a^2*(a+b+c)*(-a+b+c)*(a^2-(b+c)*a-b*c+c^2+b^2) : : , where D=4*S*sqrt(R*(4*R+r))

The reciprocal parallelogic center of these triangles is X(1)

X(24824) lies on these lines: {55,537}, {190,5598}, {528,12455}, {545,11208}, {900,13230}, {1086,5600}, {2786,13177}, {2796,12346}, {4432,11367}, {4440,5602}, {5597,24841}, {8187,24821}, {8191,24822}, {8203,24828}, {8204,24715}, {8205,24831}, {8206,24832}, {8207,24833}, {8208,24838}, {8209,24839}, {9041,12454}, {9055,12453}, {11385,24814}, {11492,24826}, {11493,24820}, {11823,24813}, {11838,24815}, {11844,24817}, {11862,24825}, {11866,24834}, {11868,24835}, {11870,24836}, {11872,24837}, {11874,24840}, {11876,24844}, {11878,24845}, {11880,24846}, {11882,24847}, {11884,24848}, {13891,24842}, {13945,24843}, {18497,24827}, {18956,24816}, {19009,24818}, {19010,24819}

X(24824) = reflection of X(24823) in X(55)
X(24824) = X(190)-of-2nd Auriga triangle
X(24824) = X(24841)-of-1st Auriga triangle

### X(24825) = PARALLELOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24825) lies on these lines: {32,190}, {528,12495}, {537,9941}, {545,7811}, {900,13235}, {903,7865}, {1086,3096}, {2786,8782}, {2796,9878}, {2896,4440}, {3094,9055}, {3098,24813}, {3099,24821}, {4422,7846}, {4432,11368}, {4473,10583}, {4568,24502}, {9301,24844}, {9857,24715}, {9862,24817}, {9993,24828}, {9994,24831}, {9995,24832}, {9996,24833}, {9997,24841}, {10038,24845}, {10047,24846}, {10828,24822}, {10871,24834}, {10872,24835}, {10873,24836}, {10874,24837}, {10875,24838}, {10876,24839}, {10877,24840}, {10878,24847}, {10879,24848}, {11386,24814}, {11494,24820}, {11861,24823}, {11862,24824}, {11885,24830}, {13892,24842}, {13946,24843}, {18500,24827}, {18957,24816}, {19011,24818}, {19012,24819}, {22744,24826}

X(24825) = X(190)-of-5th Brocard triangle

### X(24826) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24826) lies on these lines: {3,537}, {20,528}, {36,24821}, {55,24841}, {56,190}, {104,24817}, {335,16367}, {518,9441}, {545,11194}, {659,918}, {900,22560}, {956,24715}, {958,1086}, {999,4432}, {1001,4310}, {2786,22514}, {2796,8666}, {2975,4440}, {3428,24813}, {3913,9041}, {4473,5253}, {9055,22769}, {10966,24840}, {11492,24824}, {11493,24823}, {12114,24834}, {16412,17755}, {18761,24827}, {19013,24818}, {19014,24819}, {22479,24814}, {22520,24815}, {22654,24822}, {22744,24825}, {22753,24828}, {22755,24830}, {22756,24831}, {22757,24832}, {22758,24833}, {22759,24836}, {22760,24837}, {22761,24838}, {22762,24839}, {22763,24842}, {22764,24843}, {22765,24844}, {22766,24845}, {22767,24846}, {22768,24847}

X(24826) = reflection of X(i) in X(j) for these (i,j): (24820, 3), (24835, 1086)
X(24826) = X(190)-of-2nd circumperp tangential triangle
X(24826) = X(24841)-of-anti-Mandart-incircle triangle

### X(24827) = PARALLELOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO OBVERSE OF X(1)

Barycentrics    2*a^6-2*(b+c)*a^5+(b+c)*(b^2+c^2)*a^3-(3*b^2-4*b*c+3*c^2)*b*c*a^2+(b^2-c^2)^2*(b+c)*a-(2*b^2-3*b*c+2*c^2)*(b^2-c^2)^2 : :
X(24827) = 3*X(4)+X(4440) = 3*X(5)-2*X(4422) = X(190)-3*X(381) = 5*X(3091)-X(24817) = 9*X(3545)-5*X(4473) = 5*X(3843)-X(24844) = X(4409)+6*X(14893) = X(4440)-3*X(24833) = 5*X(18492)-X(24821)

The reciprocal parallelogic center of these triangles is X(1)

X(24827) lies on these lines: {4,4440}, {5,4422}, {30,1086}, {190,381}, {382,24813}, {528,22791}, {545,3845}, {546,24828}, {900,22938}, {903,3830}, {978,12699}, {1478,24837}, {1479,24836}, {2786,22515}, {2796,18483}, {3091,24817}, {3545,4473}, {3583,24840}, {3585,24816}, {3818,9055}, {3843,24844}, {4370,5066}, {4409,14893}, {4432,9955}, {4437,18358}, {5845,21850}, {9818,24822}, {10895,24845}, {10896,24846}, {13785,24818}, {18491,24820}, {18492,24821}, {18495,24823}, {18497,24824}, {18500,24825}, {18502,24815}, {18507,24830}, {18509,24831}, {18511,24832}, {18516,24834}, {18517,24835}, {18520,24838}, {18522,24839}, {18525,24841}, {18542,24847}, {18544,24848}, {18761,24826}, {18762,24843}

X(24827) = midpoint of X(i) and X(j) for these {i,j}: {4, 24833}, {382, 24813}, {903, 3830}, {12699, 24715}, {18507, 24830}, {18525, 24841}
X(24827) = reflection of X(i) in X(j) for these (i,j): (4370, 5066), (4432, 9955), (4437, 18358), (24828, 546)
X(24827) = X(190)-of-Ehrmann-mid triangle
X(24827) = X(24833)-of-Euler triangle

### X(24828) = PARALLELOGIC CENTER OF THESE TRIANGLES: EULER TO OBVERSE OF X(1)

Barycentrics    (3*b^2-2*b*c+3*c^2)*a^4-2*(b+c)*(b^2+c^2)*a^3-2*(b^4+c^4-(b+c)^2*b*c)*a^2+2*(b^2-c^2)^2*(b+c)*a-(b^4-c^4)*(b^2-c^2) : :
X(24828) = 3*X(4)+X(24817) = X(20)-5*X(4473) = 3*X(190)-X(24817) = 3*X(381)-X(24833) = 3*X(381)+X(24844) = X(673)-3*X(5817) = X(903)-3*X(3545) = 3*X(1699)+X(24821) = 5*X(3091)-X(4440) = 3*X(3839)+X(17487) = 8*X(3850)-X(4409) = 3*X(5587)-X(24715) = 3*X(5603)-X(24841) = 3*X(11897)-X(24830)

The reciprocal parallelogic center of these triangles is X(1)

X(24828) lies on these lines: {2,24813}, {3,4422}, {4,190}, {5,1086}, {11,24816}, {12,24840}, {20,4473}, {30,4370}, {98,24815}, {114,118}, {119,900}, {235,24814}, {355,528}, {371,24842}, {372,24843}, {381,545}, {511,4437}, {515,4432}, {537,946}, {546,24827}, {673,5817}, {903,3545}, {971,16593}, {1331,1862}, {1352,5779}, {1478,24846}, {1479,24845}, {1482,9041}, {1587,24818}, {1588,24819}, {1598,24822}, {1699,24821}, {2796,9880}, {3091,4440}, {3839,17487}, {3850,4409}, {3932,15310}, {5480,9055}, {5587,24715}, {5603,24841}, {6201,24832}, {6202,24831}, {7330,16560}, {8196,24823}, {8203,24824}, {8212,24838}, {8213,24839}, {9993,24825}, {10531,24847}, {10532,24848}, {10893,24834}, {10894,24835}, {10895,24836}, {10896,24837}, {11496,24820}, {11897,24830}, {16561,18540}, {16594,19515}, {22753,24826}

X(24828) = midpoint of X(i) and X(j) for these {i,j}: {4, 190}, {24833, 24844}
X(24828) = reflection of X(i) in X(j) for these (i,j): (3, 4422), (1086, 5), (24827, 546)
X(24828) = complement of X(24813)
X(24828) = X(190)-of-Euler triangle
X(24828) = X(1086)-of-Johnson triangle
X(24828) = X(4422)-of-X3-ABC reflections triangle
X(24828) = X(24833)-of-Ehrmann-mid triangle
X(24828) = {X(381), X(24844)}-harmonic conjugate of X(24833)

### X(24829) = ORTHOLOGIC CENTER THESE TRIANGLES: 1st EHRMANN TO TRILINEAR N-OBVERSE OF X(2)

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

The reciprocal orthologic center of these triangles is X(6)

X(24829) lies on the line {3,3923}

### X(24830) = PARALLELOGIC CENTER OF THESE TRIANGLES: GOSSARD TO OBVERSE OF X(1)

Barycentrics
(b-c)^2*(a^8+2*b*c*a^6-(b+c)^3*a^5-(4*b^4+4*c^4+3*(b^2-b*c+c^2)*b*c)*a^4+2*(b^2+c^2)*(b+c)^3*a^3+4*(b^4-c^4)*(b^2-c^2)*a^2-(b^2+c^2)^2*(b+c)^3*a-(b^6+c^6-(3*b^4+3*c^4-4*(b-c)^2*b*c)*b*c)*(b+c)^2)*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(24830) = 4*X(4422)-5*X(15183) = 2*X(4432)-3*X(11831) = 3*X(11845)-X(24817) = 3*X(11852)-X(24821) = 3*X(11897)-2*X(24828) = 3*X(11911)-X(24844)

The reciprocal parallelogic center of these triangles is X(1)

X(24830) lies on these lines: {30,903}, {190,402}, {528,12626}, {537,12438}, {545,1651}, {900,13268}, {1086,1650}, {2786,13179}, {2796,12347}, {4240,4440}, {4422,15183}, {4432,11831}, {9055,12583}, {11832,24814}, {11839,24815}, {11845,24817}, {11848,24820}, {11852,24821}, {11853,24822}, {11885,24825}, {11897,24828}, {11900,24715}, {11901,24831}, {11902,24832}, {11903,24834}, {11904,24835}, {11905,24836}, {11906,24837}, {11907,24838}, {11908,24839}, {11909,24840}, {11910,24841}, {11911,24844}, {11912,24845}, {11913,24846}, {11914,24847}, {11915,24848}, {13894,24842}, {13948,24843}, {18507,24827}, {18958,24816}, {19017,24818}, {19018,24819}, {22755,24826}

X(24830) = midpoint of X(4240) and X(4440)
X(24830) = reflection of X(i) in X(j) for these (i,j): (190, 402), (1650, 1086), (18507, 24827)
X(24830) = X(190)-of-Gossard triangle

### X(24831) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24831) lies on these lines: {6,190}, {528,12627}, {537,3641}, {545,5861}, {900,13269}, {1086,5591}, {1271,4440}, {2786,6319}, {2796,9882}, {4432,11370}, {5589,24821}, {5595,24822}, {5605,24841}, {5689,24715}, {6202,24828}, {6215,24833}, {8198,24823}, {8205,24824}, {8216,24838}, {8217,24839}, {8974,24842}, {9994,24825}, {10040,24845}, {10048,24846}, {10783,24817}, {10792,24815}, {10919,24834}, {10921,24835}, {10923,24836}, {10925,24837}, {10927,24840}, {10929,24847}, {10931,24848}, {11388,24814}, {11497,24820}, {11824,24813}, {11901,24830}, {11916,24844}, {13949,24843}, {18509,24827}, {18959,24816}, {22756,24826}

X(24831) = reflection of X(24832) in X(190)
X(24831) = X(190)-of-inner-Grebe triangle

### X(24832) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24832) lies on these lines: {6,190}, {528,12628}, {537,3640}, {545,5860}, {900,13270}, {1086,5590}, {1270,4440}, {2786,6320}, {2796,9883}, {4432,11371}, {5588,24821}, {5594,24822}, {5604,24841}, {5688,24715}, {6201,24828}, {6214,24833}, {8199,24823}, {8206,24824}, {8218,24838}, {8219,24839}, {8975,24842}, {9995,24825}, {10041,24845}, {10049,24846}, {10784,24817}, {10793,24815}, {10920,24834}, {10922,24835}, {10924,24836}, {10926,24837}, {10928,24840}, {10930,24847}, {10932,24848}, {11389,24814}, {11498,24820}, {11825,24813}, {11902,24830}, {11917,24844}, {13950,24843}, {18511,24827}, {18960,24816}, {22757,24826}

X(24832) = reflection of X(24831) in X(190)
X(24832) = X(190)-of-outer-Grebe triangle

### X(24833) = PARALLELOGIC CENTER OF THESE TRIANGLES: JOHNSON TO OBVERSE OF X(1)

Barycentrics    a^6-(b+c)*a^5-(b^2-b*c+c^2)*a^4+(b+c)*(b^2+c^2)*a^3+(b^2+c^2)*(b^2-3*b*c+c^2)*a^2-(b^2-c^2)^2*(b-c)^2 : :
X(24833) = 3*X(381)-2*X(24828) = 3*X(381)-X(24844) = 3*X(903)-X(24813) = 5*X(1656)-4*X(4422) = 7*X(3090)-5*X(4473) = 3*X(3545)-X(17487) = 5*X(3843)+2*X(4409) = 2*X(4370)-3*X(5055) = 2*X(4432)-3*X(5886) = X(4440)+2*X(24827) = 3*X(5587)-X(24821)

The reciprocal parallelogic center of these triangles is X(1)

X(24833) lies on these lines: {1,24836}, {2,24817}, {3,1086}, {4,4440}, {5,190}, {12,24845}, {30,903}, {88,19515}, {355,537}, {381,545}, {517,3792}, {528,1482}, {673,5762}, {900,10738}, {946,2796}, {952,24841}, {1351,5845}, {1352,9055}, {1478,24816}, {1479,24840}, {1656,4422}, {2786,6321}, {2969,22148}, {3090,4473}, {3545,17487}, {3843,4409}, {4370,5055}, {4432,5886}, {5587,24821}, {5805,20430}, {6214,24832}, {6215,24831}, {7583,24819}, {7584,24818}, {8200,24823}, {8207,24824}, {8220,24838}, {8221,24839}, {8976,24842}, {9041,12645}, {9996,24825}, {10796,24815}, {10942,24847}, {10943,24848}, {11499,24820}, {13951,24843}, {15310,20358}, {22758,24826}

X(24833) = midpoint of X(i) and X(j) for these {i,j}: {4, 4440}, {24834, 24835}
X(24833) = reflection of X(i) in X(j) for these (i,j): (3, 1086), (4, 24827), (190, 5), (24844, 24828)
X(24833) = complement of X(24817)
X(24833) = X(190)-of-Johnson triangle
X(24833) = X(1086)-of-X3-ABC reflections triangle
X(24833) = X(4440)-of-Euler triangle
X(24833) = X(24827)-of-anti-Euler triangle
X(24833) = X(24844)-of-Ehrmann-mid triangle
X(24833) = X(24845)-of-outer-Johnson triangle
X(24833) = X(24846)-of-inner-Johnson triangle
X(24833) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 24844, 24828), (24836, 24837, 1)

### X(24834) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24834) lies on these lines: {11,190}, {12,24847}, {355,537}, {528,944}, {545,11235}, {900,13271}, {1086,1376}, {2786,13180}, {2796,12348}, {3434,4440}, {4432,11373}, {4473,10584}, {9055,12586}, {10523,24845}, {10785,24817}, {10794,24815}, {10826,24821}, {10829,24822}, {10871,24825}, {10893,24828}, {10914,24715}, {10919,24831}, {10920,24832}, {10944,24836}, {10945,24838}, {10946,24839}, {10947,24840}, {10948,24846}, {10949,24848}, {11390,24814}, {11865,24823}, {11866,24824}, {11903,24830}, {11928,24844}, {12114,24826}, {13895,24842}, {13952,24843}, {18516,24827}, {18961,24816}, {19023,24818}, {19024,24819}

X(24834) = reflection of X(i) in X(j) for these (i,j): (24820, 1086), (24835, 24833)
X(24834) = X(190)-of-inner-Johnson triangle
X(24834) = X(24847)-of-outer-Johnson triangle

### X(24835) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24835) lies on these lines: {11,24848}, {12,190}, {72,24715}, {355,537}, {528,962}, {545,11236}, {900,13272}, {958,1086}, {2786,13181}, {2796,12349}, {3436,4440}, {4432,11374}, {4473,10585}, {9055,12587}, {10523,24846}, {10786,24817}, {10795,24815}, {10827,24821}, {10830,24822}, {10872,24825}, {10894,24828}, {10921,24831}, {10922,24832}, {10950,24837}, {10951,24838}, {10952,24839}, {10953,24840}, {10954,24845}, {10955,24847}, {11391,24814}, {11500,24820}, {11827,24813}, {11867,24823}, {11868,24824}, {11904,24830}, {11929,24844}, {13896,24842}, {13953,24843}, {18517,24827}, {18962,24816}, {19025,24818}, {19026,24819}

X(24835) = reflection of X(i) in X(j) for these (i,j): (24826, 1086), (24834, 24833)
X(24835) = X(190)-of-outer-Johnson triangle
X(24835) = X(24848)-of-inner-Johnson triangle

### X(24836) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24836) lies on these lines: {1,24833}, {4,24840}, {5,24846}, {7,528}, {12,190}, {55,3782}, {56,1086}, {65,24715}, {109,1365}, {226,2796}, {388,4440}, {495,24845}, {517,1463}, {537,5252}, {545,11237}, {900,13273}, {1284,5172}, {1400,1989}, {1479,24827}, {2786,13182}, {2807,4014}, {3085,24817}, {4432,11375}, {4473,10588}, {6690,9791}, {7354,24813}, {9055,12588}, {9578,24821}, {9654,24844}, {10797,24815}, {10831,24822}, {10873,24825}, {10895,24828}, {10923,24831}, {10924,24832}, {10944,24834}, {10956,24847}, {10957,24848}, {11392,24814}, {11501,24820}, {11869,24823}, {11870,24824}, {11905,24830}, {11930,24838}, {11931,24839}, {13897,24842}, {13954,24843}, {19027,24818}, {19028,24819}, {22759,24826}

X(24836) = reflection of X(24845) in X(495)
X(24836) = X(190)-of-1st Johnson-Yff triangle
X(24836) = X(24833)-of-inner-Yff triangle
X(24836) = X(24837)-of-inner-Yff tangents triangle
X(24836) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 24833, 24837), (388, 4440, 24816)

### X(24837) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24837) lies on these lines: {1,24833}, {4,24816}, {5,24845}, {11,190}, {55,1086}, {496,24846}, {497,4440}, {528,2098}, {537,1837}, {545,11238}, {900,13274}, {903,3058}, {1478,24827}, {2786,13183}, {2796,12053}, {3057,24715}, {3086,24817}, {3254,4516}, {3939,7336}, {4432,11376}, {4473,10589}, {6284,24813}, {9055,12589}, {9581,24821}, {9669,24844}, {10798,24815}, {10832,24822}, {10874,24825}, {10896,24828}, {10925,24831}, {10926,24832}, {10950,24835}, {10958,24847}, {10959,24848}, {11393,24814}, {11502,24820}, {11871,24823}, {11872,24824}, {11906,24830}, {11932,24838}, {11933,24839}, {13898,24842}, {13955,24843}, {19029,24818}, {19030,24819}, {22760,24826}

X(24837) = reflection of X(24846) in X(496)
X(24837) = X(190)-of-2nd Johnson-Yff triangle
X(24837) = X(24833)-of-outer-Yff triangle
X(24837) = X(24836)-of-outer-Yff tangents triangle
X(24837) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 24833, 24836), (497, 4440, 24840)

### X(24838) = PARALLELOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO OBVERSE OF X(1)

Barycentrics
(a^6-(b+c)*a^5-(b^2+b*c+c^2)*a^4+2*(b+c)*(b^2+c^2)*a^3-(b^2+c^2)*(b-c)^2*a^2-(b+c)*(b^2+c^2)^2*a+b^6+c^6-(b^4+c^4+b*c*(5*b^2-14*b*c+5*c^2))*b*c)*a^2+(a^6-(b+c)*a^5-(2*b^2-b*c+2*c^2)*a^4+2*(b+c)*(b^2+c^2)*a^3-3*(b^2+c^2)*(b-c)^2*a^2-(b+c)*(b^2+c^2)^2*a+(b^2+c^2)^2*b*c)*S : :

The reciprocal parallelogic center of these triangles is X(1)

X(24838) lies on these lines: {190,493}, {528,12636}, {537,12440}, {545,12152}, {900,13275}, {1086,8222}, {2786,13184}, {2796,12352}, {4432,11377}, {4440,6462}, {6461,24839}, {8188,24821}, {8194,24822}, {8201,24823}, {8208,24824}, {8210,24841}, {8212,24828}, {8214,24715}, {8216,24831}, {8218,24832}, {8220,24833}, {9055,12590}, {10875,24825}, {10945,24834}, {10951,24835}, {11394,24814}, {11503,24820}, {11828,24813}, {11840,24815}, {11846,24817}, {11907,24830}, {11930,24836}, {11932,24837}, {11947,24840}, {11949,24844}, {11951,24845}, {11953,24846}, {11955,24847}, {11957,24848}, {13899,24842}, {13956,24843}, {18520,24827}, {18963,24816}, {19031,24818}, {19032,24819}, {22761,24826}

X(24838) = X(190)-of-Lucas homothetic triangle

### X(24839) = PARALLELOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO OBVERSE OF X(1)

Barycentrics
(a^6-(b+c)*a^5-(b^2+b*c+c^2)*a^4+2*(b+c)*(b^2+c^2)*a^3-(b^2+c^2)*(b-c)^2*a^2-(b+c)*(b^2+c^2)^2*a+b^6+c^6-(b^4+c^4+b*c*(5*b^2-14*b*c+5*c^2))*b*c)*a^2-(a^6-(b+c)*a^5-(2*b^2-b*c+2*c^2)*a^4+2*(b+c)*(b^2+c^2)*a^3-3*(b^2+c^2)*(b-c)^2*a^2-(b+c)*(b^2+c^2)^2*a+(b^2+c^2)^2*b*c)*S : :

The reciprocal parallelogic center of these triangles is X(1)

X(24839) lies on these lines: {190,494}, {528,12637}, {537,12441}, {545,12153}, {900,13276}, {1086,8223}, {2786,13185}, {2796,12353}, {4432,11378}, {4440,6463}, {6461,24838}, {8189,24821}, {8195,24822}, {8202,24823}, {8209,24824}, {8211,24841}, {8213,24828}, {8215,24715}, {8217,24831}, {8219,24832}, {8221,24833}, {9055,12591}, {10876,24825}, {10946,24834}, {10952,24835}, {11395,24814}, {11504,24820}, {11829,24813}, {11841,24815}, {11847,24817}, {11908,24830}, {11931,24836}, {11933,24837}, {11948,24840}, {11950,24844}, {11952,24845}, {11954,24846}, {11956,24847}, {11958,24848}, {13900,24842}, {13957,24843}, {18522,24827}, {18964,24816}, {19033,24818}, {19034,24819}, {22762,24826}

X(24839) = X(190)-of-Lucas(-1) homothetic triangle

### X(24840) = PARALLELOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24840) lies on these lines: {1,24816}, {3,24846}, {4,24836}, {11,244}, {12,24828}, {33,24814}, {55,190}, {56,24813}, {87,24307}, {497,4440}, {513,23772}, {522,3271}, {528,10394}, {537,3057}, {545,3058}, {903,11238}, {950,2796}, {1479,24833}, {1697,24821}, {1837,24715}, {2098,24841}, {2646,4432}, {2786,3022}, {3056,9055}, {3295,24844}, {3583,24827}, {3667,4014}, {4294,24817}, {4370,4995}, {4422,5432}, {4473,5218}, {10385,17487}, {10799,24815}, {10833,24822}, {10877,24825}, {10927,24831}, {10928,24832}, {10947,24834}, {10953,24835}, {10965,24847}, {10966,24826}, {11873,24823}, {11874,24824}, {11909,24830}, {11947,24838}, {11948,24839}, {13901,24842}, {13958,24843}, {19037,24818}, {19038,24819}

X(24840) = reflection of X(i) in X(j) for these (i,j): (4014, 24225), (24816, 1)
X(24840) = X(190)-of-Mandart-incircle triangle
X(24840) = X(24813)-of-2nd anti-circumperp-tangential triangle
X(24840) = X(24816)-of-5th mixtilinear triangle
X(24840) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (497, 4440, 24837), (2310, 4459, 11), (3295, 24844, 24845)

### X(24841) = PARALLELOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR TO OBVERSE OF X(1)

Barycentrics    a^3-2*(b+c)*a^2+(3*b^2-b*c+3*c^2)*a-(b+c)*b*c : :
X(24841) = 3*X(1)-2*X(4432) = 3*X(1)-X(24821) = 3*X(190)-4*X(4432) = 3*X(190)-2*X(24821) = 3*X(903)-2*X(24715) = 5*X(3616)-4*X(4422) = 7*X(3622)-5*X(4473) = 3*X(4645)-4*X(7238) = 3*X(5603)-2*X(24828) = 3*X(7967)-X(24817) = 3*X(10247)-X(24844) = 3*X(11038)-2*X(16593)

The reciprocal parallelogic center of these triangles is X(1)

X(24841) lies on these lines: {1,190}, {2,4126}, {8,599}, {55,24826}, {56,24820}, {88,17780}, {100,17154}, {145,528}, {239,335}, {244,3699}, {320,519}, {517,24813}, {545,3241}, {726,4693}, {900,1120}, {918,1280}, {952,24833}, {984,24331}, {1015,24486}, {1317,20098}, {1621,20068}, {1647,4997}, {1830,1897}, {2098,24840}, {2099,24816}, {2786,7983}, {2796,3244}, {3123,24397}, {3242,4363}, {3315,3952}, {3616,4422}, {3622,4473}, {3679,17227}, {3685,4864}, {3891,4430}, {3996,17155}, {3999,5205}, {4080,10707}, {4310,4429}, {4366,20145}, {4370,16672}, {4465,17794}, {4552,14151}, {4645,7238}, {5083,14594}, {5597,24824}, {5598,24823}, {5603,24828}, {5604,24832}, {5605,24831}, {6559,17435}, {7081,21342}, {7174,16831}, {7967,24817}, {7968,24818}, {7969,24819}, {8192,24822}, {8210,24838}, {8211,24839}, {9457,24416}, {9997,24825}, {10247,24844}, {10800,24815}, {10944,24834}, {10950,24835}, {11038,16593}, {11396,24814}, {11910,24830}, {12635,17480}, {13902,24842}, {13959,24843}, {18525,24827}

X(24841) = midpoint of X(145) and X(4440)
X(24841) = reflection of X(i) in X(j) for these (i,j): (8, 1086), (190, 1), (3685, 4864), (18525, 24827), (24821, 4432)
X(24841) = X(190)-of-5th mixtilinear triangle
X(24841) = X(24820)-of-2nd anti-circumperp-tangential triangle
X(24841) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 24821, 4432), (3242, 24349, 5263), (4080, 20042, 10707)

### X(24842) = PARALLELOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24842) lies on these lines: {2,24819}, {6,344}, {190,3068}, {371,24828}, {528,13911}, {537,8983}, {545,13846}, {590,1086}, {900,13922}, {2161,7347}, {2786,8997}, {2796,13908}, {4432,13883}, {4440,8972}, {4473,7585}, {8974,24831}, {8975,24832}, {8976,24833}, {9055,13910}, {9540,24813}, {13884,24814}, {13885,24815}, {13886,24817}, {13887,24820}, {13888,24821}, {13889,24822}, {13890,24823}, {13891,24824}, {13892,24825}, {13893,24715}, {13894,24830}, {13895,24834}, {13896,24835}, {13897,24836}, {13899,24838}, {13900,24839}, {13901,24840}, {13902,24841}, {13903,24844}, {13904,24845}, {13905,24846}, {13906,24847}, {13907,24848}, {18538,24827}, {18965,24816}, {22763,24826}

X(24842) = X(190)-of-3rd tri-squares-central triangle
X(24842) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 4422, 24843), (4473, 7585, 24818)

### X(24843) = PARALLELOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24843) lies on these lines: {2,24818}, {6,344}, {190,3069}, {372,24828}, {528,13973}, {537,13971}, {545,13847}, {615,1086}, {900,13991}, {2161,7348}, {2786,13989}, {2796,13968}, {4432,13936}, {4440,13941}, {4473,7586}, {9055,13972}, {13935,24813}, {13937,24814}, {13938,24815}, {13939,24817}, {13940,24820}, {13942,24821}, {13943,24822}, {13944,24823}, {13945,24824}, {13946,24825}, {13947,24715}, {13948,24830}, {13949,24831}, {13950,24832}, {13951,24833}, {13952,24834}, {13953,24835}, {13954,24836}, {13955,24837}, {13956,24838}, {13957,24839}, {13958,24840}, {13959,24841}, {13961,24844}, {13962,24845}, {13963,24846}, {13964,24847}, {13965,24848}, {18762,24827}, {18966,24816}, {22764,24826}

X(24843) = X(190)-of-4th tri-squares-central triangle
X(24843) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 4422, 24842), (4473, 7586, 24819)

### X(24844) = PARALLELOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO OBVERSE OF X(1)

Barycentrics    a^6-(b+c)*a^5-(4*b^2-3*b*c+4*c^2)*a^4+3*(b+c)*(b^2+c^2)*a^3+(b^2+b*c+c^2)*(3*b^2-8*b*c+3*c^2)*a^2-2*(b^2-c^2)^2*(b+c)*a+2*(b^2-c^2)^2*b*c : :
X(24844) = 3*X(3)-2*X(24813) = 4*X(140)-5*X(4473) = 3*X(190)-X(24813) = 3*X(381)-4*X(24828) = 3*X(381)-2*X(24833) = 2*X(903)-3*X(5055) = 4*X(1086)-5*X(1656) = 7*X(3526)-8*X(4422) = 5*X(3843)-4*X(24827) = 4*X(4370)-3*X(5054) = 4*X(4409)-11*X(5072) = 4*X(4432)-3*X(10246) = 3*X(5790)-2*X(24715) = 3*X(10247)-2*X(24841) = 3*X(17487)-X(24817)

The reciprocal parallelogic center of these triangles is X(1)

X(24844) lies on these lines: {3,190}, {5,4440}, {30,17487}, {140,4473}, {355,2796}, {381,545}, {517,24821}, {528,12645}, {537,1482}, {900,12331}, {903,5055}, {999,24816}, {1086,1656}, {1351,9055}, {1598,24814}, {2786,13188}, {3295,24840}, {3526,4422}, {3843,24827}, {4370,5054}, {4409,5072}, {4432,10246}, {5790,24715}, {5843,20533}, {5845,11898}, {6417,24819}, {6418,24818}, {7517,24822}, {9301,24825}, {9654,24836}, {9669,24837}, {10247,24841}, {11842,24815}, {11849,24820}, {11875,24823}, {11876,24824}, {11911,24830}, {11916,24831}, {11917,24832}, {11928,24834}, {11929,24835}, {11949,24838}, {11950,24839}, {12000,24847}, {12001,24848}, {13903,24842}, {13961,24843}, {22765,24826}

X(24844) = reflection of X(i) in X(j) for these (i,j): (3, 190), (4440, 5), (24833, 24828)
X(24844) = X(190)-of-X3-ABC reflections triangle
X(24844) = X(4440)-of-Johnson triangle
X(24844) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (24816, 24846, 999), (24828, 24833, 381), (24840, 24845, 3295)

### X(24845) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24845) lies on these lines: {1,190}, {3,24816}, {5,24837}, {12,24833}, {35,24813}, {388,24817}, {495,24836}, {498,1086}, {499,4422}, {528,12647}, {545,10056}, {611,9055}, {900,10087}, {903,3584}, {1479,24828}, {1772,24416}, {2786,10086}, {2796,10054}, {3085,4440}, {3086,4473}, {3295,24840}, {3299,24818}, {3301,24819}, {4370,10072}, {10037,24822}, {10038,24825}, {10039,24715}, {10040,24831}, {10041,24832}, {10523,24834}, {10801,24815}, {10895,24827}, {10954,24835}, {11398,24814}, {11507,24820}, {11877,24823}, {11878,24824}, {11912,24830}, {11951,24838}, {11952,24839}, {13904,24842}, {13962,24843}, {22766,24826}

X(24845) = midpoint of X(190) and X(24847)
X(24845) = reflection of X(24836) in X(495)
X(24845) = X(190)-of-inner-Yff triangle
X(24845) = X(24833)-of-1st Johnson-Yff triangle
X(24845) = X(24846)-of-inner-Yff tangents triangle
X(24845) = X(24847)-of-outer-Yff triangle
X(24845) = {X(3295), X(24844)}-harmonic conjugate of X(24840)

### X(24846) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24846) lies on these lines: {1,190}, {3,24840}, {5,24836}, {36,24813}, {496,24837}, {497,24817}, {498,4422}, {499,1086}, {528,5729}, {545,10072}, {613,9055}, {673,1733}, {900,10090}, {903,3582}, {920,16560}, {999,24816}, {1210,2796}, {1478,24828}, {1737,24715}, {2786,10089}, {3085,4473}, {3086,4440}, {3299,24819}, {3301,24818}, {4370,10056}, {10046,24822}, {10047,24825}, {10048,24831}, {10049,24832}, {10523,24835}, {10802,24815}, {10896,24827}, {10948,24834}, {11399,24814}, {11508,24820}, {11879,24823}, {11880,24824}, {11913,24830}, {11953,24838}, {11954,24839}, {13905,24842}, {13963,24843}, {22767,24826}

X(24846) = midpoint of X(190) and X(24848)
X(24846) = reflection of X(24837) in X(496)
X(24846) = X(190)-of-outer-Yff triangle
X(24846) = X(24833)-of-2nd Johnson-Yff triangle
X(24846) = X(24845)-of-outer-Yff tangents triangle
X(24846) = X(24848)-of-inner-Yff triangle
X(24846) = {X(999), X(24844)}-harmonic conjugate of X(24816)

### X(24847) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24847) lies on these lines: {1,190}, {12,24834}, {528,12648}, {545,11239}, {900,13278}, {1086,5552}, {2786,13189}, {2796,12356}, {4440,10528}, {4473,10586}, {9055,12594}, {10531,24828}, {10803,24815}, {10805,24817}, {10834,24822}, {10878,24825}, {10915,24715}, {10929,24831}, {10930,24832}, {10942,24833}, {10955,24835}, {10956,24836}, {10958,24837}, {10965,24840}, {11248,24813}, {11400,24814}, {11509,24816}, {11881,24823}, {11882,24824}, {11914,24830}, {11955,24838}, {11956,24839}, {12000,24844}, {13906,24842}, {13964,24843}, {18542,24827}, {19047,24818}, {19048,24819}, {22768,24826}

X(24847) = reflection of X(190) in X(24845)
X(24847) = X(190)-of-inner-Yff tangents triangle
X(24847) = X(24834)-of-1st Johnson-Yff triangle

### X(24848) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO OBVERSE OF X(1)

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

The reciprocal parallelogic center of these triangles is X(1)

X(24848) lies on these lines: {1,190}, {11,24835}, {528,12649}, {545,11240}, {900,13279}, {1086,10527}, {2786,13190}, {2796,12357}, {4440,10529}, {4473,10587}, {9055,12595}, {10532,24828}, {10804,24815}, {10806,24817}, {10835,24822}, {10879,24825}, {10916,24715}, {10931,24831}, {10932,24832}, {10943,24833}, {10949,24834}, {10957,24836}, {10959,24837}, {10966,24826}, {11249,24813}, {11401,24814}, {11510,24820}, {11883,24823}, {11884,24824}, {11915,24830}, {11957,24838}, {11958,24839}, {12001,24844}, {13907,24842}, {13965,24843}, {18544,24827}, {18967,24816}, {19049,24818}, {19050,24819}

X(24848) = reflection of X(190) in X(24846)
X(24848) = X(190)-of-outer-Yff tangents triangle
X(24848) = X(24835)-of-2nd Johnson-Yff triangle

### X(24849) = PERSPECTOR OF THESE TRIANGLES: N-OBVERSE OF X(1) AND ANTI-TANGENTIAL-MIDARC

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

X(24849) lies on these lines: {1,15973}, {8,73}, {43,65}, {56,1740}, {978,1403}, {986,5887}, {1193,3485}, {1457,20036}, {7355,24310}

### X(24850) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-MIDPOINTS TO N-OBVERSE OF X(1)

Barycentrics    2*a^4+(b+c)*a^3-(b^2+c^2)*a^2+(b+c)*b*c*a+(b+c)^2*b*c : :
X(24850) = X(1)-3*X(4234) = 4*X(3634)-3*X(16052)

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

X(24850) lies on these lines: {1,4234}, {2,24851}, {3,3923}, {8,896}, {10,30}, {21,4418}, {35,1215}, {58,740}, {79,4892}, {100,24852}, {171,7283}, {190,5293}, {386,4672}, {405,3980}, {474,4011}, {540,21081}, {726,5266}, {846,1010}, {902,4968}, {964,4414}, {978,2234}, {986,4195}, {993,15952}, {1003,16822}, {1009,24259}, {1043,1046}, {1045,4281}, {1054,13741}, {1086,1125}, {1089,4434}, {1155,3831}, {1580,11104}, {1755,3501}, {1770,2887}, {1834,8258}, {2292,4427}, {3178,3712}, {3210,16478}, {3550,4385}, {3634,16052}, {3648,4683}, {3741,3916}, {3821,17698}, {3985,5277}, {4205,12579}, {4252,5695}, {4267,4436}, {4650,10449}, {4754,4760}, {5248,24325}, {6651,16917}, {6679,23537}, {7262,9534}, {7295,19845}, {7816,24254}, {8680,24335}, {11110,24342}, {11319,24443}, {12512,17355}, {12567,19259}, {13323,24264}, {13740,17596}, {16061,17738}, {16609,24271}, {17164,17539}, {17697,24174}, {23383,24425}

X(24850) = midpoint of X(i) and X(j) for these {i,j}: {100, 24852}, {1043, 1046}
X(24850) = reflection of X(1834) in X(8258)
X(24850) = complement of X(24851)
X(24850) = {X(4252), X(5695)}-harmonic conjugate of X(17733)

### X(24851) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GARCIA-REFLECTION TO N-OBVERSE OF X(1)

Barycentrics    a^4-(b^2+b*c+c^2)*a^2-(b^3+c^3)*a-(b^2-c^2)^2 : :
X(24851) = 2*X(10)-3*X(17677) = 4*X(1125)-3*X(4234) = 5*X(1698)-6*X(16052)

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

X(24851) lies on these lines: {1,30}, {2,24850}, {3,3944}, {4,240}, {5,17596}, {8,4683}, {10,190}, {11,24852}, {21,3120}, {35,17719}, {36,15952}, {40,1756}, {145,17491}, {148,257}, {171,1770}, {238,23537}, {405,17889}, {442,846}, {497,3976}, {498,17601}, {513,18178}, {516,5255}, {522,21134}, {535,15955}, {726,5015}, {740,1330}, {896,3648}, {982,1479}, {987,19645}, {988,1699}, {1010,4425}, {1046,1834}, {1054,4187}, {1125,4234}, {1193,5057}, {1698,16052}, {1714,7262}, {1738,12572}, {2292,2475}, {2476,4414}, {2478,24174}, {2549,3061}, {2650,14450}, {2782,3865}, {2887,7283}, {3338,24217}, {3496,5254}, {3583,3670}, {3585,4424}, {3663,4911}, {3735,7748}, {3750,13407}, {3821,13740}, {3914,5247}, {3923,16062}, {3924,11114}, {3953,4857}, {3980,17533}, {4205,24342}, {4292,24210}, {4309,17715}, {4385,4660}, {4415,5293}, {4418,5051}, {4642,5080}, {4650,5292}, {4653,11263}, {4655,10449}, {4703,9534}, {4872,24214}, {4887,6744}, {5046,24443}, {5129,7613}, {5134,16600}, {5302,21949}, {5358,24436}, {5988,6998}, {6175,21674}, {6536,14005}, {6650,17685}, {6651,17673}, {6656,17738}, {7847,18061}, {8040,17551}, {9612,17594}, {9670,17597}, {10896,17595}, {11110,12579}, {13741,24169}, {16478,19785}, {18483,24239}

X(24851) = reflection of X(i) in X(j) for these (i,j): (1046, 1834), (5255, 13161), (24852, 11)
X(24851) = anticomplement of X(24850)
X(24851) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 24248, 986), (21, 3120, 24161), (3782, 6284, 1)

### X(24852) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd SCHIFFLER TO N-OBVERSE OF X(1)

Barycentrics    a^7-(b+c)*a^6-(3*b^2-4*b*c+3*c^2)*a^5+(b+c)*(b^2+c^2)*a^4+(2*b^4+2*c^4-(2*b^2-b*c+2*c^2)*b*c)*a^3-(b+c)*b^2*c^2*a^2-(b^2-c^2)^2*b*c*a+(b^2-c^2)^2*(b+c)*b*c : :
X(24852) = 2*X(214)-3*X(4234) = 4*X(6702)-3*X(17677)

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

X(24852) lies on these lines: {11,24851}, {30,80}, {100,24850}, {214,4234}, {1222,2802}, {2796,21630}, {4418,12746}, {6702,17677}

X(24852) = reflection of X(i) in X(j) for these (i,j): (100, 24850), (24851, 11)

### X(24853) = PERSPECTOR OF THESE TRIANGLES: TRILINEAR OBVERSE OF X(2) AND 2nd MIXTILINEAR

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

X(24853) lies on the line {220,2276}

### X(24854) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st EHRMANN TO TRILINEAR OBVERSE OF X(2)

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

The reciprocal orthologic center of these triangles is X(6)

X(24854) lies on the line {3,8301}

### X(24855) = X(2)X(6)∩X(5)X(8585)

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

See Angel Montesdeoca, HG140918.

X(24855) lies on these lines: {2,6}, {5,8585}, {30,10418}, {111,858}, {113,9193}, {115,16317}, {126,6390}, {187,468}, {427,18424}, {625,6719}, {690,9209}, {1503,2502}, {1513,1514}, {2030,5972}, {3291,5159}, {5107,6388}, {5210,7493}, {5915,10564}, {6791,13857}, {9759,14982}

### X(24856) = X(7)X(10405)∩X(281)X(1886)

Barycentrics    a^8 - a^7(b+c) - 5a^6(b-c)^2 + 11a^5(b-c)^2(b+c) - 5a^4(b-c)^2(3b^2+2b c+3c^2) + a^3(b-c)^2(29b^3-13b^2c-13b c^2+29c^3) - a^2(b-c)^4(39b^2+82b c+39c^2) + a(b-c)^4(25b^3+87b^2c+87b c^2+25c^3) - 2(b-c)^4(b+c)^2(3b^2+10b c+3c^2) : :

See Angel Montesdeoca, HG190918.

X(24856) lies on these lines: {7,10405}, {281,1886}

### X(24857) = X(81)X(16489)∩X(88)X(551)

Barycentrics    1/(a^2+2a(b+c)+b^2+11b c+c^2) : :

See Alexandr Skutin and Angel Montesdeoca, HG220918.

X(24857) lies on these lines: {81,16489}, {88,551}

### X(24858) = REFLECTION OF X(4767) IN X(551)

Barycentrics    a^4-5a^3(b+c)- 3a^2(4b^2-19b c+4c^2)-5a(b+c)^3+(b+c)^4 : :

See Alexandr Skutin and Angel Montesdeoca, HG220918.

X(24858) lies on these lines: {1,4370}, {2,4738}, {57,1317}, {81,16490}, {88,519}, {89,3241}, {106,8028}, {551,4767}, {900,1022}, {4677,8056}

X(24858) = reflection of X(4767) in X(551)

### X(24859) = (name pending)

Barycentrics    a (a^3 + a^2 b - a b^2 - b^3 + a^2 c - 3 a b c + b^2 c - a c^2 + b c^2 - c^3) (a^12 - 2 a^11 b - 4 a^10 b^2 + 10 a^9 b^3 + 5 a^8 b^4 - 20 a^7 b^5 + 20 a^5 b^7 - 5 a^4 b^8 - 10 a^3 b^9 + 4 a^2 b^10 + 2 a b^11 - b^12 - 2 a^11 c + 4 a^10 b c + 6 a^9 b^2 c - 26 a^8 b^3 c + 12 a^7 b^4 c + 48 a^6 b^5 c - 52 a^5 b^6 c - 28 a^4 b^7 c + 54 a^3 b^8 c - 4 a^2 b^9 c - 18 a b^10 c + 6 b^11 c - 4 a^10 c^2 + 6 a^9 b c^2 - 4 a^8 b^2 c^2 + 40 a^7 b^3 c^2 - 52 a^6 b^4 c^2 - 58 a^5 b^5 c^2 + 126 a^4 b^6 c^2 - 28 a^3 b^7 c^2 - 60 a^2 b^8 c^2 + 40 a b^9 c^2 - 6 b^10 c^2 + 10 a^9 c^3 - 26 a^8 b c^3 + 40 a^7 b^2 c^3 - 68 a^6 b^3 c^3 + 98 a^5 b^4 c^3 - 2 a^4 b^5 c^3 - 156 a^3 b^6 c^3 + 114 a^2 b^7 c^3 + 8 a b^8 c^3 - 18 b^9 c^3 + 5 a^8 c^4 + 12 a^7 b c^4 - 52 a^6 b^2 c^4 + 98 a^5 b^3 c^4 - 217 a^4 b^4 c^4 + 134 a^3 b^5 c^4 + 56 a^2 b^6 c^4 - 114 a b^7 c^4 + 33 b^8 c^4 - 20 a^7 c^5 + 48 a^6 b c^5 - 58 a^5 b^2 c^5 - 2 a^4 b^3 c^5 + 134 a^3 b^4 c^5 - 220 a^2 b^5 c^5 + 82 a b^6 c^5 + 12 b^7 c^5 - 52 a^5 b c^6 + 126 a^4 b^2 c^6 - 156 a^3 b^3 c^6 + 56 a^2 b^4 c^6 + 82 a b^5 c^6 - 52 b^6 c^6 + 20 a^5 c^7 - 28 a^4 b c^7 - 28 a^3 b^2 c^7 + 114 a^2 b^3 c^7 - 114 a b^4 c^7 + 12 b^5 c^7 - 5 a^4 c^8 + 54 a^3 b c^8 - 60 a^2 b^2 c^8 + 8 a b^3 c^8 + 33 b^4 c^8 - 10 a^3 c^9 - 4 a^2 b c^9 + 40 a b^2 c^9 - 18 b^3 c^9 + 4 a^2 c^10 - 18 a b c^10 - 6 b^2 c^10 + 2 a c^11 + 6 b c^11 - c^12) : :

See Antreas Hatzipolakis and Angel Montesdeoca, HG280918.

X(24859) lies on this line: {5,11014}

### X(24860) = X(10095)X(10620)∩X(12006)X(16835)

Barycentrics    a^2(a^14-6a^12(b^2+c^2)+ a^10(13b^4+4b^2c^2+13c^4)+(b^2-c^2)^4(2b^6- 21b^4c^2-21b^2c^4+2c^6)- 2a^8(5b^6-3b^4c^2-3b^2c^4+5c^6)+ a^6(-5b^8+47b^6c^2-243b^4c^4+47b^2c^6-5c^8)- 3a^2(b^2-c^2)^2(3b^8-29b^6c^2-91b^4c^4- 29b^2c^6+3c^8)+ a^4(14b^10-127b^8c^2+59b^6c^4+59b^4c^6- 127b^2c^8+14c^10)) : :

See Antreas Hatzipolakis and Angel Montesdeoca, HG280918.

X(24860) lies on these lines: {10095,10620}, {12006,16835}

### X(24861) = X(251)X(15271)∩X(263)X(3763)

Barycentrics    1/(4a^2(b^2+c^2)+3b^2c^2) : :

See Angel Montesdeoca, HG290918.

X(24861) lies on these lines: {251,15271}, {263,3763}

### X(24862) = X(4)X(16813)∩X(115)X(130)

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

See Angel Montesdeoca, HG031018.

X(24862) lies on these lines: {4, 16813}, {115, 130}, {125, 137}, {184, 11082}, {338, 23290}, {1879, 6752}, {3574, 5095}, {7687, 18402}, {13851, 16337}, {15451, 20975}

X(24862) = trilinear pole, wrt orthic triangle, of line X(4)X(54)

### X(24863) = (name pending)

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

See Angel Montesdeoca, HG041018.

X(24863) lies on this line: {37, 109}

Collineation mappings involving Gemini triangle 9: X(24864) - X(24879)

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 9, as in centers X(24864)-X(24879). Then

m(X) = (a^3 - a^2 b - 2 a b^2 - a^2 c + 5 a b c - 2 a c^2) x + (- 2 a^3 + a^2 b + 2 a b^2 - b^3 + 3 a^2 c - 7 a b c + 2 b^2 c + 3 a c^2 + b c^2 - 2 c^3) y + (- 2 a^3 + a^2 c + 2 a c^2 - c^3 + 3 a^2 b - 7 a b c + 2 b c^2 + 3 a b^2 + b^2 c - 2 b^3) z : : ,

and m(X) is on the Euler line if and only if X is on the Euler line.

(Clark Kimberling, October 10, 2018)

### X(24864) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 9

Barycentrics    a^4 - 3 a^3 b - a^2 b^2 + 2 a b^3 - b^4 - 3 a^3 c + 11 a^2 b c - 4 a b^2 c - a^2 c^2 - 4 a b c^2 + 2 b^2 c^2 + 2 a c^3 - c^4 : :

X(24864) lies on these lines: {1, 2}, {11, 10700}, {80, 17460}, {106, 1145}, {514, 1000}, {952, 24828}, {996, 1016}, {1086, 4792}, {1168, 25026}, {1317, 24867}, {1320, 24222}, {2802, 24715}, {3057, 18340}, {3120, 12653}, {3476, 23703}, {3722, 7972}, {4346, 6549}, {4389, 4555}, {4675, 24868}, {5252, 24865}, {6631, 17305}, {10713, 16594}, {13541, 21630}, {17119, 25035}, {17395, 24281}, {17724, 25416}

### X(24865) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 9

Barycentrics    a^7 - a^6 b - a^5 b^2 - 2 a^3 b^4 + 2 a^2 b^5 + 2 a b^6 - b^7 - a^6 c + 5 a^5 b c - 2 a^4 b^2 c + 2 a^3 b^3 c + a^2 b^4 c - 7 a b^5 c + 2 b^6 c - a^5 c^2 - 2 a^4 b c^2 + 2 a^3 b^2 c^2 - 3 a^2 b^3 c^2 - 2 a b^4 c^2 + 4 b^5 c^2 + 2 a^3 b c^3 - 3 a^2 b^2 c^3 + 14 a b^3 c^3 - 5 b^4 c^3 - 2 a^3 c^4 + a^2 b c^4 - 2 a b^2 c^4 - 5 b^3 c^4 + 2 a^2 c^5 - 7 a b c^5 + 4 b^2 c^5 + 2 a c^6 + 2 b c^6 - c^7 : :

X(24865) lies on these lines: {2, 3}, {1387, 24869}, {5252, 24864}, {11376, 18340}, {23708, 24871}

### X(24866) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 9

Barycentrics    5 a^7 - 5 a^6 b - 11 a^5 b^2 + 9 a^4 b^3 + 5 a^3 b^4 - 5 a^2 b^5 + a b^6 + b^7 - 5 a^6 c + 19 a^5 b c - a^4 b^2 c - 14 a^3 b^3 c + 5 a^2 b^4 c - 5 a b^5 c + b^6 c - 11 a^5 c^2 - a^4 b c^2 + 10 a^3 b^2 c^2 - a b^4 c^2 - b^5 c^2 + 9 a^4 c^3 - 14 a^3 b c^3 + 10 a b^3 c^3 - b^4 c^3 + 5 a^3 c^4 + 5 a^2 b c^4 - a b^2 c^4 - b^3 c^4 - 5 a^2 c^5 - 5 a b c^5 - b^2 c^5 + a c^6 + b c^6 + c^7 : :

X(24866) lies on these lines: {2, 3}, {24870, 24871}

### X(24867) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 9

Barycentrics    4 a^7 - 4 a^6 b - 10 a^5 b^2 + 9 a^4 b^3 + 7 a^3 b^4 - 7 a^2 b^5 - a b^6 + 2 b^7 - 4 a^6 c + 14 a^5 b c + a^4 b^2 c - 16 a^3 b^3 c + 4 a^2 b^4 c + 2 a b^5 c - b^6 c - 10 a^5 c^2 + a^4 b c^2 + 8 a^3 b^2 c^2 + 3 a^2 b^3 c^2 + a b^4 c^2 - 5 b^5 c^2 + 9 a^4 c^3 - 16 a^3 b c^3 + 3 a^2 b^2 c^3 - 4 a b^3 c^3 + 4 b^4 c^3 + 7 a^3 c^4 + 4 a^2 b c^4 + a b^2 c^4 + 4 b^3 c^4 - 7 a^2 c^5 + 2 a b c^5 - 5 b^2 c^5 - a c^6 - b c^6 + 2 c^7 : :

X(24867) lies on these lines: {2, 3}, {1317, 24864}

### X(24868) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 9

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

X(24868) lies on these lines: {2, 6}, {1086, 1320}, {4675, 24864}, {17374, 25034}

### X(24869) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 9

Barycentrics    5 a^4 - 6 a^3 b - 8 a^2 b^2 + 4 a b^3 + b^4 - 6 a^3 c + 25 a^2 b c - 5 a b^2 c - 8 a^2 c^2 - 5 a b c^2 - 2 b^2 c^2 + 4 a c^3 + c^4 : :

X(24869) lies on these lines: {1, 2}, {1387, 24865}, {17056, 25040}, {18220, 18340}, {24874, 24878}

### X(24870) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 9

Barycentrics    a^5 - 4 a^3 b^2 + a^2 b^3 + 3 a b^4 - b^5 + 5 a^3 b c - 12 a b^3 c + 5 b^4 c - 4 a^3 c^2 + 18 a b^2 c^2 - 4 b^3 c^2 + a^2 c^3 - 12 a b c^3 - 4 b^2 c^3 + 3 a c^4 + 5 b c^4 - c^5 : :

X(24870) lies on these lines: {2, 7}, {106, 1086}, {4675, 24864}, {14475, 23838}, {24866, 24871}

### X(24871) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 9

Barycentrics    4 a^4 - 3 a^3 b - 7 a^2 b^2 + 2 a b^3 + 2 b^4 - 3 a^3 c + 14 a^2 b c - a b^2 c - 7 a^2 c^2 - a b c^2 - 4 b^2 c^2 + 2 a c^3 + 2 c^4 : :

X(24871) lies on these lines: {1, 2}, {1319, 25026}, {1387, 4674}, {4694, 24433}, {11731, 24542}, {16173, 24715}, {17395, 24873}, {23708, 24865}, {24866, 24870}

### X(24872) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 9

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

X(24872) lies on this line: {2, 3}

### X(24873) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 9

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

X(24873) lies on these lines: {2, 37}, {10, 25029}, {1086, 1145}, {4675, 24864}, {14475, 23757}, {17119, 24281}, {17237, 25030}, {17395, 24871}, {25031, 25035}

### X(24874) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 9

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

X(24874) lies on these lines: {2, 37}, {16589, 25037}, {24869, 24878}

### X(24875) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 9

Barycentrics    7 a^7 - 7 a^6 b - 13 a^5 b^2 + 9 a^4 b^3 + a^3 b^4 - a^2 b^5 + 5 a b^6 - b^7 - 7 a^6 c + 29 a^5 b c - 5 a^4 b^2 c - 10 a^3 b^3 c + 7 a^2 b^4 c - 19 a b^5 c + 5 b^6 c - 13 a^5 c^2 - 5 a^4 b c^2 + 14 a^3 b^2 c^2 - 6 a^2 b^3 c^2 - 5 a b^4 c^2 + 7 b^5 c^2 + 9 a^4 c^3 - 10 a^3 b c^3 - 6 a^2 b^2 c^3 + 38 a b^3 c^3 - 11 b^4 c^3 + a^3 c^4 + 7 a^2 b c^4 - 5 a b^2 c^4 - 11 b^3 c^4 - a^2 c^5 - 19 a b c^5 + 7 b^2 c^5 + 5 a c^6 + 5 b c^6 - c^7 : :

X(24875) lies on this line: {2, 3}

### X(24876) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 9

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

X(24876) lies on this line: {2, 3}

### X(24877) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 9

Barycentrics    3 a^7 - 3 a^6 b - 5 a^5 b^2 + 3 a^4 b^3 - a^3 b^4 + a^2 b^5 + 3 a b^6 - b^7 - 3 a^6 c + 13 a^5 b c - 3 a^4 b^2 c - 2 a^3 b^3 c + 3 a^2 b^4 c - 11 a b^5 c + 3 b^6 c - 5 a^5 c^2 - 3 a^4 b c^2 + 6 a^3 b^2 c^2 - 4 a^2 b^3 c^2 - 3 a b^4 c^2 + 5 b^5 c^2 + 3 a^4 c^3 - 2 a^3 b c^3 - 4 a^2 b^2 c^3 + 22 a b^3 c^3 - 7 b^4 c^3 - a^3 c^4 + 3 a^2 b c^4 - 3 a b^2 c^4 - 7 b^3 c^4 + a^2 c^5 - 11 a b c^5 + 5 b^2 c^5 + 3 a c^6 + 3 b c^6 - c^7 : :

X(24877) lies on this line: {2, 3}

### X(24878) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 9

Barycentrics    5 a^7 - 5 a^6 b - 11 a^5 b^2 + 9 a^4 b^3 + 5 a^3 b^4 - 5 a^2 b^5 + a b^6 + b^7 - 5 a^6 c + 13 a^5 b c - a^4 b^2 c - 2 a^3 b^3 c + 5 a^2 b^4 c - 11 a b^5 c + b^6 c - 11 a^5 c^2 - a^4 b c^2 + 4 a^3 b^2 c^2 - 6 a^2 b^3 c^2 - a b^4 c^2 - b^5 c^2 + 9 a^4 c^3 - 2 a^3 b c^3 - 6 a^2 b^2 c^3 + 22 a b^3 c^3 - b^4 c^3 + 5 a^3 c^4 + 5 a^2 b c^4 - a b^2 c^4 - b^3 c^4 - 5 a^2 c^5 - 11 a b c^5 - b^2 c^5 + a c^6 + b c^6 + c^7 : :

X(24878) lies on these lines: {2, 3}, {24869, 24874}

### X(24879) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 9

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

X(24879) lies on this line: {2, 3}

Collineation mappings involving Gemini triangle 10: X(24880) - X(24925)

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 10, as in centers X(24880)-X(24925). Then

m(X) = a (a^2 + a b + a c + b c) x + (-a^2 b + b^2 - a^2 c - a b c - a c^2 - b c^2) y + (-a^2 c + c^2 - a^2 b - a b c - a b^2 - b^2 c) z : : ,

and m(X) is on the Euler line if and only if X is on the Euler line.

(Clark Kimberling, October 10, 2018)

### X(24880) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24880) lies on these lines: {1, 2}, {3, 759}, {5, 580}, {6, 24937}, {58, 442}, {72, 24160}, {79, 896}, {171, 3841}, {191, 3120}, {201, 2006}, {333, 3454}, {377, 4257}, {595, 2886}, {748, 7741}, {758, 24161}, {851, 4278}, {991, 6889}, {1010, 6693}, {1046, 11263}, {1054, 1247}, {1324, 16453}, {1724, 2476}, {1754, 6828}, {1834, 4653}, {3019, 3090}, {3294, 17737}, {3579, 21949}, {3647, 24851}, {3678, 17719}, {3772, 5791}, {3822, 5247}, {3825, 17123}, {4220, 5358}, {4252, 17528}, {4256, 7483}, {4658, 17056}, {5251, 21935}, {5400, 6960}, {5432, 24934}, {5433, 24885}, {5445, 24916}, {5713, 6858}, {5745, 23537}, {5956, 11231}, {6175, 16948}, {6211, 24468}, {6831, 13329}, {7413, 7683}, {7828, 17277}, {7887, 20154}, {9581, 24933}, {12514, 17064}, {17070, 18253}, {17275, 24935}, {17278, 24884}, {17337, 17527}, {17776, 24081}, {23897, 24275}, {24046, 24789}, {24519, 24962}, {24530, 25065}, {24881, 24910}, {24888, 24904}, {24903, 24909}, {24919, 24922}

### X(24881) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24881) lies on these lines: {2, 3}, {1718, 3216}, {24880, 24910}, {24883, 24916}, {24889, 24890}

### X(24882) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24882) lies on these lines: {2, 3}, {386, 20277}, {1698, 21686}, {2939, 4466}, {3216, 3468}, {24889, 24918}, {24895, 24897}

### X(24883) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24883) lies on these lines: {1, 2}, {4, 162}, {6, 2476}, {21, 1834}, {23, 5358}, {30, 16948}, {58, 2475}, {81, 442}, {149, 595}, {213, 17737}, {267, 3336}, {333, 5051}, {346, 24075}, {377, 2363}, {411, 5721}, {579, 16548}, {580, 6840}, {582, 6903}, {896, 3648}, {897, 9214}, {940, 4197}, {1046, 3120}, {1150, 16062}, {1330, 16704}, {1449, 24937}, {1479, 17127}, {1724, 5046}, {1751, 7557}, {1754, 6895}, {2006, 15556}, {2650, 24161}, {2895, 3454}, {3019, 5068}, {3218, 23537}, {3332, 6870}, {3649, 17070}, {3769, 5300}, {3772, 3868}, {3876, 17720}, {4193, 4383}, {4202, 14829}, {4205, 5235}, {4252, 17579}, {4418, 8258}, {4653, 15674}, {4854, 18253}, {4921, 16052}, {5080, 5247}, {5154, 14997}, {5178, 5266}, {5315, 24387}, {5396, 6853}, {5706, 6828}, {5707, 6829}, {5722, 24933}, {6703, 14005}, {6941, 7592}, {7235, 24530}, {11680, 16466}, {15325, 24925}, {17337, 17534}, {17362, 24935}, {17366, 24890}, {17669, 20142}, {24881, 24916}, {24903, 24913}, {24919, 24923}

### X(24884) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24884) lies on these lines: {2, 7}, {6, 17058}, {140, 3019}, {284, 1375}, {1781, 4466}, {2160, 16581}, {2294, 24780}, {3017, 17366}, {3763, 16408}, {4000, 24195}, {15803, 24933}, {16054, 17052}, {17278, 24880}, {24899, 24919}

### X(24885) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24885) lies on these lines: {2, 11}, {140, 24902}, {1054, 2607}, {3752, 24186}, {5433, 24880}, {6739, 13747}, {14310, 21180}, {15059, 19629}, {15368, 18191}

### X(24886) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24886) lies on this line: {2, 3}

### X(24887) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24887) lies on this line: {2, 3}

### X(24888) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24888) lies on these lines: {2, 31}, {17728, 24893}, {24880, 24904}, {24890, 24892}, {24896, 24916}

### X(24889) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24889) lies on these lines: {2, 32}, {24881, 24890}, {24882, 24918}, {24897, 24908}

### X(24890) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24890) lies on these lines: {2, 6}, {1333, 17052}, {2893, 3285}, {4261, 25065}, {4675, 24937}, {17278, 24880}, {17366, 24883}, {20195, 24902}, {21245, 24632}, {24530, 24899}, {24881, 24889}, {24888, 24892}

### X(24891) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24891) lies on these lines: {2, 39}, {17278, 24902}, {24881, 24889}, {24918, 24925}

### X(24892) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24892) lies on these lines: {1, 2}, {9, 17737}, {11, 212}, {31, 2886}, {38, 3772}, {44, 17605}, {63, 3120}, {149, 8616}, {238, 11680}, {244, 24789}, {321, 4438}, {345, 4365}, {442, 1468}, {672, 16567}, {750, 3925}, {756, 17720}, {851, 2217}, {896, 1836}, {902, 3434}, {958, 21935}, {1150, 2887}, {1155, 21949}, {1834, 10448}, {2177, 6690}, {2292, 5791}, {2308, 24597}, {2476, 5247}, {2611, 20254}, {2933, 4191}, {3035, 9350}, {3218, 17889}, {3219, 3944}, {3681, 17719}, {3691, 3767}, {3722, 4863}, {3782, 17070}, {3816, 17125}, {3826, 17124}, {3838, 4641}, {3846, 5278}, {3868, 24161}, {3914, 4414}, {3915, 24390}, {4001, 4138}, {4650, 20292}, {4799, 25383}, {5057, 7262}, {5127, 14008}, {5284, 24217}, {5904, 24160}, {6327, 21241}, {6679, 21242}, {16968, 21029}, {17278, 17728}, {17449, 24477}, {24888, 24890}

### X(24893) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24893) lies on these lines: {2, 44}, {656, 24921}, {17278, 24880}, {17728, 24888}

### X(24894) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24894) lies on these lines: {2, 45}, {17245, 17531}, {17278, 24880}, {17282, 24935}

### X(24895) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24895) lies on these lines: {2, 6}, {5292, 25065}, {7235, 24530}, {24882, 24897}

### X(24896) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24896) lies on these lines: {1, 2}, {17728, 24923}, {17737, 21061}, {24888, 24916}

### X(24897) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24897) lies on these lines: {1, 24530}, {2, 39}, {1654, 18171}, {2223, 19345}, {3227, 24522}, {17053, 17143}, {24882, 24895}, {24889, 24908}

### X(24898) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24898) lies on these lines: {1, 2}, {896, 20084}, {1834, 15674}, {2475, 16948}, {6933, 14997}, {11684, 17070}

### X(24899) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24899) lies on these lines: {1, 2}, {5741, 7828}, {21124, 24921}, {24530, 24890}, {24884, 24919}

### X(24900) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24900) lies on these lines: {2, 650}, {43, 21727}, {647, 4789}, {661, 18197}, {3121, 24137}, {3666, 17161}, {4369, 16751}, {4850, 21196}, {14077, 19767}, {14838, 16754}

### X(24901) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24901) lies on these lines: {2, 31}, {17278, 24880}

### X(24902) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24902) lies on these lines: {1, 2}, {140, 24885}, {896, 6701}, {1743, 24937}, {5127, 7504}, {5400, 6853}, {17278, 24891}, {20195, 24890}, {23897, 24956}, {24904, 24910}

### X(24903) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24903) lies on these lines: {2, 3}, {24880, 24909}, {24883, 24913}

### X(24904) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24904) lies on these lines: {2, 3}, {3615, 5972}, {7741, 24955}, {24880, 24888}, {24902, 24910}

### X(24905) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24905) lies on this line: {2, 3}

### X(24906) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 10

Barycentrics    5 a^7 + 7 a^6 b - 3 a^5 b^2 - 3 a^4 b^3 - 3 a^3 b^4 - 9 a^2 b^5 + a b^6 + 5 b^7 + 7 a^6 c + 7 a^5 b c - a^4 b^2 c - 2 a^3 b^3 c - 7 a^2 b^4 c - 5 a b^5 c + b^6 c - 3 a^5 c^2 - a^4 b c^2 + 10 a^3 b^2 c^2 + 16 a^2 b^3 c^2 - a b^4 c^2 - 9 b^5 c^2 - 3 a^4 c^3 - 2 a^3 b c^3 + 16 a^2 b^2 c^3 + 10 a b^3 c^3 + 3 b^4 c^3 - 3 a^3 c^4 - 7 a^2 b c^4 - a b^2 c^4 + 3 b^3 c^4 - 9 a^2 c^5 - 5 a b c^5 - 9 b^2 c^5 + a c^6 + b c^6 + 5 c^7 : :

X(24906) lies on this line: {2, 3}

### X(24907) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24907) lies on these lines: {2, 3}, {7235, 24530}

### X(24908) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24908) lies on these lines: {2, 3}, {24889, 24897}

### X(24909) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24909) lies on these lines: {2, 7}, {1014, 1213}, {1170, 24934}, {5433, 24936}, {7235, 24530}, {24880, 24903}

### X(24910) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24910) lies on these lines: {2, 36}, {3772, 24046}, {24880, 24881}, {24902, 24904}

### X(24911) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24911) lies on these lines: {2, 38}, {3017, 4695}, {3214, 5956}, {3216, 5253}, {6703, 20966}, {24880, 24888}

### X(24912) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24912) lies on these lines: {2, 7}, {56, 24931}, {227, 5956}, {1211, 1412}, {5433, 24934}, {24880, 24881}

### X(24913) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24913) lies on these lines: {1, 24907}, {2, 7}, {1412, 3580}, {24880, 24888}, {24883, 24903}

### X(24914) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24914) lies on these lines: {1, 140}, {2, 65}, {3, 1737}, {4, 1155}, {5, 46}, {7, 10588}, {8, 1319}, {10, 56}, {11, 40}, {12, 57}, {30, 10826}, {35, 5722}, {36, 355}, {43, 2594}, {55, 1210}, {63, 1329}, {72, 18838}, {73, 899}, {78, 3035}, {79, 19919}, {80, 5442}, {100, 20118}, {109, 7299}, {119, 24467}, {165, 6284}, {201, 3772}, {226, 3634}, {333, 1408}, {354, 3085}, {381, 1770}, {388, 5435}, {392, 10200}, {404, 5794}, {405, 11509}, {406, 1887}, {427, 1452}, {442, 1454}, {475, 1875}, {484, 7741}, {495, 3338}, {496, 5119}, {497, 5704}, {498, 942}, {499, 517}, {515, 5204}, {516, 10896}, {518, 5552}, {519, 1388}, {549, 3612}, {550, 12019}, {553, 3947}, {580, 5348}, {590, 2362}, {604, 17275}, {615, 16232}, {631, 2646}, {750, 1451}, {920, 6842}, {936, 21677}, {938, 5218}, {940, 5530}, {950, 5217}, {958, 1470}, {962, 5183}, {978, 24806}, {986, 17720}, {993, 19524}, {997, 13747}, {999, 10039}, {1038, 1722}, {1125, 2099}, {1145, 20586}, {1158, 1532}, {1213, 2285}, {1214, 1714}, {1284, 17279}, {1317, 3632}, {1376, 6734}, {1385, 10573}, {1393, 17278}, {1399, 1724}, {1400, 17303}, {1402, 5955}, {1403, 3831}, {1415, 4426}, {1420, 3679}, {1428, 3416}, {1445, 3826}, {1450, 10459}, {1464, 16569}, {1466, 5745}, {1467, 8580}, {1469, 17077}, {1478, 9956}, {1479, 3579}, {1512, 12114}, {1617, 9709}, {1656, 12047}, {1697, 9588}, {1699, 5128}, {1702, 19029}, {1703, 19030}, {1706, 5231}, {1728, 6907}, {1768, 6259}, {1771, 2361}, {1776, 6932}, {1858, 6825}, {1864, 6908}, {1882, 14018}, {1898, 6838}, {1905, 3541}, {1935, 9364}, {1940, 5125}, {2067, 13973}, {2093, 8227}, {2098, 11362}, {2245, 3142}, {2263, 17337}, {2475, 9352}, {2476, 5880}, {2478, 4640}, {2551, 5744}, {2975, 25005}, {3011, 17054}, {3057, 3086}, {3090, 4295}, {3091, 3474}, {3149, 12616}, {3212, 17095}, {3218, 11681}, {3256, 5259}, {3303, 11019}, {3305, 16140}, {3333, 15888}, {3336, 7951}, {3339, 3649}, {3340, 3624}, {3359, 15908}, {3361, 5434}, {3419, 5172}, {3436, 5123}, {3476, 3617}, {3486, 3523}, {3524, 4305}, {3576, 10950}, {3582, 3654}, {3583, 15079}, {3584, 18398}, {3586, 15338}, {3614, 9612}, {3616, 11011}, {3627, 4333}, {3653, 24926}, {3680, 13996}, {3683, 5084}, {3697, 17625}, {3698, 19843}, {3714, 17740}, {3752, 5230}, {3814, 7702}, {3816, 5250}, {3828, 4298}, {3833, 12432}, {3838, 6933}, {3844, 12588}, {3872, 8256}, {3873, 13751}, {3880, 10529}, {3884, 10199}, {3916, 17619}, {3925, 5705}, {4187, 4679}, {4188, 5086}, {4193, 24703}, {4292, 10175}, {4293, 5818}, {4299, 5122}, {4309, 18527}, {4423, 9843}, {4511, 17566}, {4646, 11269}, {4654, 19876}, {4860, 21620}, {4863, 5687}, {4999, 19860}, {5045, 10056}, {5048, 12245}, {5057, 5154}, {5061, 5814}, {5080, 7705}, {5087, 6931}, {5131, 10483}, {5141, 20292}, {5193, 5288}, {5224, 10401}, {5225, 9778}, {5235, 5323}, {5248, 14882}, {5255, 17721}, {5316, 18249}, {5326, 11529}, {5439, 10198}, {5535, 5812}, {5587, 7354}, {5691, 15326}, {5698, 6919}, {5708, 13407}, {5710, 24239}, {5711, 17723}, {5727, 7987}, {5770, 14872}, {5836, 10527}, {5883, 15556}, {5886, 5903}, {5887, 6959}, {5901, 25415}, {5902, 11374}, {5919, 14986}, {6001, 6834}, {6175, 18977}, {6361, 10591}, {6502, 13911}, {6691, 19861}, {6702, 13273}, {6735, 12513}, {6745, 24391}, {6796, 10265}, {6833, 7686}, {6848, 12688}, {6883, 11507}, {6891, 14110}, {6964, 17634}, {7080, 24477}, {7181, 9312}, {7195, 24798}, {7235, 24530}, {7580, 10395}, {7742, 11499}, {7989, 9579}, {8162, 21625}, {8543, 17534}, {8581, 8732}, {9436, 24796}, {9708, 22759}, {9947, 11575}, {9957, 10072}, {10085, 13226}, {10090, 12619}, {10179, 10586}, {10366, 11347}, {10371, 14829}, {10571, 17749}, {10592, 24470}, {10624, 11238}, {10786, 12675}, {10827, 18990}, {12647, 24928}, {12678, 18242}, {12831, 20400}, {12943, 19925}, {13161, 17595}, {13373, 15867}, {13883, 18995}, {13893, 19028}, {13912, 19038}, {13936, 18996}, {13947, 19027}, {13975, 19037}, {15016, 18397}, {15228, 18514}, {16371, 17647}, {17078, 17090}, {17124, 21674}, {17734, 24046}, {18525, 21578}, {18965, 18991}, {18966, 18992}, {22791, 23708}, {24174, 24789}, {24836, 25351}, {24880, 24881}

### X(24915) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24915) lies on these lines: {2, 85}, {7, 24944}, {1434, 16589}, {7235, 24530}

### X(24916) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24916) lies on these lines: {1, 24904}, {2, 11}, {81, 22156}, {110, 8286}, {125, 24624}, {2607, 3120}, {5445, 24880}, {8287, 15059}, {24881, 24883}, {24888, 24896}

### X(24917) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24917) lies on these lines: {2, 101}, {21253, 24619}

### X(24918) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24918) lies on these lines: {2, 99}, {1015, 24582}, {3125, 15903}, {6710, 20331}, {6714, 7483}, {7874, 17674}, {24882, 24889}, {24891, 24925}

### X(24919) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24919) lies on these lines: {2, 37}, {244, 24520}, {1654, 16752}, {17326, 24790}, {24880, 24922}, {24883, 24923}, {24884, 24899}

### X(24920) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24920) lies on these lines: {2, 900}, {1698, 21714}, {2642, 17398}, {13277, 24938}

### X(24921) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24921) lies on these lines: {2, 649}, {656, 24893}, {4728, 24961}, {21124, 24899}

### X(24922) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24922) lies on these lines: {2, 7}, {17052, 19308}, {17276, 24957}, {24530, 24890}, {24880, 24919}

### X(24923) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24923) lies on these lines: {1, 24530}, {2, 38}, {6, 20746}, {10, 24437}, {1500, 4261}, {1698, 21699}, {3122, 24342}, {3670, 17322}, {4022, 19856}, {4272, 4658}, {4657, 24046}, {5009, 19329}, {5045, 5956}, {5333, 20966}, {9050, 17749}, {17278, 24880}, {17728, 24896}, {22174, 24697}, {22220, 24958}, {23928, 24443}, {24174, 24372}, {24883, 24919}

### X(24924) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24924) lies on these lines: {2, 661}, {649, 4106}, {650, 4379}, {656, 24893}, {693, 1635}, {1125, 4761}, {1638, 16892}, {1698, 4160}, {2254, 4874}, {2487, 3700}, {2490, 6546}, {2527, 23729}, {3676, 21115}, {3763, 9013}, {3835, 4979}, {3911, 4077}, {4024, 17069}, {4120, 4897}, {4191, 23864}, {4367, 21052}, {4378, 14430}, {4382, 4394}, {4383, 18199}, {4406, 14408}, {4467, 4931}, {4763, 17494}, {4776, 4932}, {4789, 4838}, {4804, 9508}, {4928, 20295}, {6590, 7658}, {6608, 15584}, {8689, 24720}, {11068, 21183}, {11123, 12072}, {14425, 21116}, {14838, 16754}, {21828, 24782}

### X(24925) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 10

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

X(24925) lies on these lines: {2, 3}, {5433, 24880}, {15325, 24883}, {24891, 24918}

### X(24926) =  X(1)X(3)∩X(10)X(5444)

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

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28442.

X(24926) lies on these lines: {1,3}, {10,5444}, {79,21578}, {80,1125}, {214,3918}, {498,7967}, {515,5443}, {551,10572}, {944,7951}, {950,16173}, {1387,10543}, {1476,5424}, {1479,3622}, {1483,5432}, {2320,5248}, {2975,4067}, {3583,5901}, {3584,10944}, {3585,15950}, {3616,3822}, {3636,5441}, {3655,11375}, {3678,4511}, {3723,5356}, {3754,4881}, {3897,5251}, {3901,11194}, {4188,21398}, {4302,10595}, {4324,22791}, {4848,5442}, {5288,22836}, {5445,10165}, {5494,6126}, {5557,12563}, {5727,15079}, {5731,10483}, {5836,15015}, {6284,10283}, {7294,11545}, {7972,10039}, {9897,9956}, {11715,12691}, {18393,18481}, {18493,18514}

X(24926) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3,11009}, {1,56,5425}, {1,484,11011}, {1,1385,36}, {1,1420,18398}, {1,2646,3746}, {1,3576,5903}, {1,3612,5697}, {1,5010,1482}, {1,7280,2099}, {1,11010,10222}, {1,21842,5563}, {3,11009,3245}, {1385,11567,3}, {1420,18398,5563}, {2646,15178,1}, {3612,5697,35}, {3746,14803,35}, {10039,13607,7972}, {11011,13624,484}, {18398,21842,1420}

### X(24927) =  X(1)X(3)∩X(119)X(1125)

Barycentrics    a (2 a^6-3 a^5 b-3 a^4 b^2+6 a^3 b^3-3 a b^5+b^6-3 a^5 c+14 a^4 b c-8 a^3 b^2 c-12 a^2 b^3 c+11 a b^4 c-2 b^5 c-3 a^4 c^2-8 a^3 b c^2+20 a^2 b^2 c^2-8 a b^3 c^2-b^4 c^2+6 a^3 c^3-12 a^2 b c^3-8 a b^2 c^3+4 b^3 c^3+11 a b c^4-b^2 c^4-3 a c^5-2 b c^5+c^6) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28442.

X(24927) lies on these lines: {1,3}, {119,1125}, {140,6735}, {214,13607}, {355,10200}, {551,12608}, {631,12648}, {944,6944}, {952,17614}, {1483,5440}, {1519,5901}, {2320,5553}, {3616,6893}, {3897,5084}, {4308,6827}, {4311,7491}, {5554,7967}, {5731,10531}, {5777,12773}, {5836,12737}, {5886,6256}, {6265,12675}, {6713,10039}, {6848,10586}, {6882,10106}, {6923,11373}, {9956,12751}, {10165,10915}, {10942,17527}

X(24927) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3359,1482}, {1,3576,11248}, {1,5193,942}, {1,7987,12703}, {1,16203,10202}, {1,16209,7982}, {1385,15178,2646}, {1388,22768,1}, {10246,16203,1}

### X(24928) =  MIDPOINT OF X(1) AND X(56)

Barycentrics    a (2 a^3-a^2 b-2 a b^2+b^3-a^2 c+6 a b c-b^2 c-2 a c^2-b c^2+c^3) : :
X(24928) = X[10]-2*X[6691], X[1837]-3*X[10072], X[4299]+X[12701], X[10090]+X[20586], X[10573]-3*X[17728]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28442.

X(24928) lies on these lines: {1,3}, {4,4308}, {5,10106}, {7,10595}, {8,17567}, {10,6691}, {11,18480}, {12,11230}, {21,15179}, {30,4311}, {37,5053}, {58,18211}, {72,17624}, {104,1476}, {145,5440}, {210,5288}, {214,3244}, {219,1732}, {226,5901}, {355,3086}, {376,9785}, {381,9613}, {382,9614}, {388,5886}, {392,2975}, {404,10914}, {405,11035}, {452,3487}, {474,3872}, {495,1125}, {496,515}, {497,18481}, {499,5252}, {519,8256}, {529,551}, {550,10624}, {938,6049}, {944,5722}, {946,1387}, {952,1210}, {956,5044}, {958,12128}, {960,8666}, {997,12513}, {998,3445}, {1056,3436}, {1058,5731}, {1064,4322}, {1066,1201}, {1100,4266}, {1108,1731}, {1191,3157}, {1193,5399}, {1317,22935}, {1386,8679}, {1412,15952}, {1475,17439}, {1478,9955}, {1519,22792}, {1538,6256}, {1621,10569}, {1656,9578}, {1657,9580}, {1699,9655}, {1706,16417}, {1737,10944}, {1743,22147}, {1776,5887}, {1828,1870}, {1836,4317}, {1837,10072}, {1872,15500}, {2257,20818}, {2320,3296}, {2649,9444}, {2771,10074}, {2841,11700}, {3028,11699}, {3035,10915}, {3073,9363}, {3218,5330}, {3419,10529}, {3485,6930}, {3486,3655}, {3555,4511}, {3582,17606}, {3585,12764}, {3600,5603}, {3623,11041}, {3633,3689}, {3636,12577}, {3656,4295}, {3680,17573}, {3752,15854}, {3753,4861}, {3813,17647}, {3871,4881}, {3877,3916}, {3884,4640}, {3885,4188}, {3898,5267}, {3911,5690}, {3927,15829}, {3940,6762}, {4002,17531}, {4253,6603}, {4292,22791}, {4293,12699}, {4297,15171}, {4298,13464}, {4299,12701}, {4304,15172}, {4314,15170}, {4848,5844}, {4853,9709}, {5030,21872}, {5083,19907}, {5176,17619}, {5248,10179}, {5265,5657}, {5270,17605}, {5284,14150}, {5290,9624}, {5427,22937}, {5433,10039}, {5434,12047}, {5435,12245}, {5438,12629}, {5533,18976}, {5542,5625}, {5550,8164}, {5691,9669}, {5727,18526}, {5836,22837}, {5882,11019}, {6147,10283}, {6284,21578}, {6377,9434}, {6735,13747}, {6738,13607}, {6797,12737}, {6921,12648}, {6948,12700}, {6970,18391}, {7320,10299}, {7354,22793}, {8227,9654}, {8583,9708}, {9259,16583}, {9581,18525}, {9612,18493}, {9623,16408}, {9856,12114}, {10090,20586}, {10572,18527}, {10573,17728}, {10593,19925}, {11036,17576}, {11037,11111}, {11194,12514}, {12573,20330}, {12616,20418}, {12650,19541}, {13407,15950}, {14100,16132}, {15180,15446}, {17100,17652}, {17644,18908}

X(24928) = midpoint of X(i) and X(j) for these {i,j}: {1,56}, {46,2098}, {4299,12701}, {4311,12053}, {10074,12740}, {10090,20586}
X(24928) = reflection of X(i) in X(j) for these {i,j}: {10,6691}, {1329,1125}, {9957,20789}
X(24928) = complement of isogonal conjugate of X15617
X(24928) = X(24)-of-incircle-circles-triangle
X(24928) = inner-Johnson-to-ABC similarity image of X(18480)
X(24928) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3,9957}, {1,35,5919}, {1,36,3057}, {1,46,2098}, {1,57,1482}, {1,65,10222}, {1,999,942}, {1,1319,1385}, {1, 1385, 24929}, {1,1388,15178}, {1,1420,3}, {1,3304,5045}, {1,3337,11009}, {1,3338,2099}, {1,3339,16200}, {1,3340,10247}, {1,3361,7982}, {1,3576,3295}, {1,3601,6767}, {1,3612,3303}, {1,5563,65}, {1,5902,11011}, {1,5903,5048}, {1,7373,5049}, {1,13462,40}, {1,15803,7962}, {1,21842,2646}, {3,1420,5126}, {4,11373,7743}, {36,3057,3579}, {56,2098,46}, {499,5252,9956}, {938,6049,7967}, {944,14986,5722}, {946,4315,18990}, {956,19861,5044}, {1056,3616,11374}, {1125,5795,17527}, {1319,2646,21842}, {1319,20323,1}, {1387,18990,946}, {1388,3304,1}, {1478,11376,9955}, {2646,21842,1385}, {3086,3476,355}, {3337,11009,65}, {4861,5253,3753}, {5045,9940,16218}, {5045,15178,1}, {5048,5903,11278}, {5126,9957,3}, {5433,10039,11231}, {5563,11009,3337}, {5708,10247,3340}, {6583,11567,10222}, {6583,15178,11567}, {7373,10246,1}, {7962,15803,12702}

### X(24929) =  COMPLEMENT OF X(3419)

Barycentrics    a (2 a^3-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(24929) = 3*X[2]-X[3419], X[63]-3*X[16370], X[956]+X[3870], X[1478]-3*X[17718], X[1836]+X[4302], X[5252]-3*X[10056]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28477.

X(24929) lies on these lines: {1,3}, {2,3419}, {4,4313}, {5,950}, {7,376}, {8,5791}, {9,3940}, {10,6675}, {11,6881}, {12,6841}, {20,3487}, {21,72}, {28,1255}, {30,226}, {33,7497}, {37,101}, {41,16601}, {42,8731}, {63,16370}, {73,500}, {74,934}, {78,405}, {79,4324}, {80,3584}, {100,3753}, {104,2346}, {105,20219}, {140,1210}, {142,214}, {145,3897}, {169,4258}, {191,3962}, {200,9708}, {210,5251}, {212,5398}, {228,859}, {329,11111}, {355,3085}, {377,3824}, {378,1876}, {381,3586}, {382,9612}, {386,1104}, {388,4305}, {390,5603}, {392,1621}, {404,5439}, {443,1058}, {474,4855}, {495,515}, {496,1125}, {497,5886}, {498,1837}, {516,8255}, {518,993}, {519,5745}, {548,4114}, {549,3911}, {550,3982}, {553,8703}, {579,1100}, {582,1451}, {610,3247}, {631,938}, {674,1386}, {758,4640}, {855,21319}, {894,4234}, {908,11113}, {910,4262}, {912,6914}, {936,2900}, {944,5787}, {946,4314}, {951,7100}, {954,971}, {956,3870}, {958,3811}, {960,5248}, {975,7535}, {976,10448}, {991,6610}, {995,1279}, {997,1001}, {1000,2320}, {1006,5728}, {1026,16443}, {1043,5295}, {1056,5731}, {1064,2293}, {1071,6906}, {1212,4251}, {1400,14636}, {1426,7414}, {1447,13634}, {1464,4337}, {1478,17718}, {1479,9955}, {1656,9581}, {1657,9579}, {1699,9668}, {1737,5432}, {1766,21848}, {1770,3649}, {1785,7510}, {1817,17019}, {1829,14017}, {1836,4302}, {1858,5694}, {1864,7489}, {1870,4219}, {1892,18533}, {1895,7531}, {2003,23071}, {2256,3211}, {2269,5755}, {2271,16968}, {2329,3991}, {2475,11015}, {2649,9367}, {2650,9340}, {2687,14733}, {2771,10058}, {2807,11700}, {2808,15730}, {2975,3555}, {2999,16485}, {3024,11699}, {3058,15950}, {3086,6989}, {3146,5714}, {3149,5806}, {3158,9623}, {3189,19843}, {3218,17549}, {3244,5855}, {3306,16371}, {3475,4293}, {3476,3655}, {3485,4294}, {3522,11036}, {3524,5435}, {3525,5704}, {3534,4654}, {3560,5777}, {3582,5444}, {3583,17605}, {3585,5441}, {3622,6904}, {3636,12436}, {3647,4067}, {3656,10385}, {3671,16137}, {3678,5302}, {3679,3689}, {3683,5692}, {3693,16788}, {3697,4420}, {3720,16056}, {3752,4256}, {3820,6745}, {3868,3916}, {3871,10914}, {3874,5267}, {3876,16865}, {3892,15570}, {3920,4224}, {3927,11523}, {3984,15650}, {3996,16821}, {4134,15481}, {4276,18165}, {4297,18990}, {4299,10404}, {4309,12701}, {4428,5289}, {4649,5429}, {4652,11520}, {4857,5443}, {4870,18393}, {4999,10916}, {5088,14828}, {5110,20227}, {5175,6856}, {5218,18391}, {5249,11112}, {5250,5730}, {5252,10056}, {5256,21483}, {5262,7523}, {5281,5657}, {5287,11347}, {5290,9655}, {5325,15673}, {5341,16777}, {5396,14547}, {5428,10122}, {5434,21578}, {5437,16417}, {5438,16408}, {5450,12675}, {5453,13754}, {5529,17123}, {5687,19860}, {5691,9654}, {5705,12625}, {5720,6913}, {5727,5790}, {5761,5812}, {5768,6935}, {5771,5844}, {5794,10198}, {5804,6927}, {5836,8715}, {5882,6245}, {5887,12711}, {5901,12053}, {5927,6912}, {6175,9963}, {6261,9856}, {6284,12047}, {6326,14100}, {6684,6738}, {6692,17564}, {6700,17527}, {6705,13607}, {6734,7483}, {6762,7160}, {6796,7686}, {6909,10167}, {6910,12649}, {6920,9844}, {7098,16139}, {7308,16857}, {7354,13407}, {7428,22345}, {7508,17010}, {7520,9538}, {8068,12743}, {8227,9669}, {9578,18525}, {9605,16780}, {9614,18493}, {9619,16781}, {9785,10595}, {9946,15558}, {9947,17857}, {10039,10950}, {10165,11019}, {10175,12019}, {10176,15254}, {10304,21454}, {10360,18931}, {10386,10624}, {10436,19276}, {10592,19925}, {11246,11551}, {11281,12609}, {11552,15228}, {11712,15746}, {11715,12735}, {12109,15489}, {12114,12260}, {12512,12563}, {12514,12635}, {12575,13464}, {12672,21740}, {12690,17530}, {13726,19767}, {15008,15299}, {15677,17484}, {15837,18412}, {16054,16826}, {16457,19859}, {16583,18755}, {16843,19753}, {17525,17781}, {17558,20007}, {17576,20214}, {17637,22936}, {18482,21617}, {19520,19861}

X(24929) = complement of X3419
X(24929) = midpoint of X(i) and X/j) for these {i,j}: {1,55}, {72,16465}, {226,4304}, {954,7675}, {956,3870}, {1012,18446}, {1836,4302}, {2099,5119}, {10058,12739}
X(24929) = reflection of X(i) in X(j) for these {i,j}: {10,6690}, {226,5719}, {495,13405}, {942,11018}, {2886,1125}, {5173,5045}, {18407,9955}
X(24929) = X(22)-of-incircle-circles-triangle
X(24929) = outer-Johnson-to-ABC similarity image of X(18480)
X(24929) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3,942}, {1,35,65}, {1,36,354}, {1,56,5045}, {1,57,15934}, {1,165,11529}, {1,484,5425}, {1,999,5049}, {1, 1385, 24928}, {1,1420,7373}, {1,1697,1482}, {1,2646,1385}, {1,3057,10222}, {1,3295,9957}, {1,3576,999}, {1,3601,3}, {1,3612,56}, {1,3746,3057}, {1,5010,5902}, {1,5119,2099}, {1,5563,17609}, {1,5697,11011}, {1,7280,18398}, {1,7962,10247}, {1,7987,3333}, {1,9819,16200}, {1,10383,18443}, {1,10389,6767}, {1,13384,10246}, {1,14793,5570}, {1,14800,13751}, {1,15803,11518}, {1,21842,20323}, {1,22766,16193}, {1,22768,13373}, {2,3488,5722}, {3,57,5122}, {3,1482,5709}, {3,5708,15803}, {3,10246,18443}, {3,10247,2095}, {3,15934,57}, {3,18443,11227}, {4,5703,11374}, {8,6857,5791}, {12,10543,10572}, {12,10572,18480}, {35,65,3579}, {35,5425,484}, {55,2099,5119}, {55,5172,35}, {56,3612,13624}, {57,15934,942}, {78,405,5044}, {140,12433,1210}, {388,4305,18481}, {484,5425,65}, {497,5886,7743}, {498,1837,9956}, {550,6147,4292}, {936,5436,11108}, {942,5122,57}, {944,6847,5787}, {946,4314,15171}, {950,13411,5}, {999,3576,5126}, {1058,3616,11373}, {1319,3748,1}, {1479,11375,9955}, {1621,4511,392}, {1737,5432,11231}, {2646,3748,1319}, {3085,3486,355}, {3485,4294,12699}, {3576,6282,3}, {3586,5219,381}, {3601,13384,10383}, {3649,15338,1770}, {3868,4189,3916}, {3940,16418,9}, {4297,21620,18990}, {4313,5703,4}, {4420,5260,3697}, {5010,5902,1155}, {5045,13624,56}, {5049,5126,999}, {5119,13462,3428}, {5248,22836,960}, {5697,11011,11278}, {5708,11518,942}, {5720,6913,10157}, {5731,10578,1056}, {5761,6868,5812}, {5901,15172,12053}, {6261,11496,9856}, {6284,12047,22793}, {6767,10246,1}, {6767,10383,942}, {6909,18444,10167}, {6935,7967,5768}, {10165,11019,15325}, {10386,22791,10624}, {10389,13384,1}, {11230,18527,11}, {11518,15803,5708}, {14547,22350,5396}

### X(24930) =  MIDPOINT OF X(107) AND X(125)

Barycentrics    2 a^18-4 a^16 b^2-5 a^14 b^4+14 a^12 b^6+4 a^10 b^8-27 a^8 b^10+19 a^6 b^12-4 a^2 b^16+b^18-4 a^16 c^2+22 a^14 b^2 c^2-18 a^12 b^4 c^2-55 a^10 b^6 c^2+92 a^8 b^8 c^2-12 a^6 b^10 c^2-46 a^4 b^12 c^2+21 a^2 b^14 c^2-5 a^14 c^4-18 a^12 b^2 c^4+104 a^10 b^4 c^4-65 a^8 b^6 c^4-127 a^6 b^8 c^4+138 a^4 b^10 c^4-12 a^2 b^12 c^4-15 b^14 c^4+14 a^12 c^6-55 a^10 b^2 c^6-65 a^8 b^4 c^6+240 a^6 b^6 c^6-92 a^4 b^8 c^6-77 a^2 b^10 c^6+35 b^12 c^6+4 a^10 c^8+92 a^8 b^2 c^8-127 a^6 b^4 c^8-92 a^4 b^6 c^8+144 a^2 b^8 c^8-21 b^10 c^8-27 a^8 c^10-12 a^6 b^2 c^10+138 a^4 b^4 c^10-77 a^2 b^6 c^10-21 b^8 c^10+19 a^6 c^12-46 a^4 b^2 c^12-12 a^2 b^4 c^12+35 b^6 c^12+21 a^2 b^2 c^14-15 b^4 c^14-4 a^2 c^16+c^18 : :
X(24930) = X[5667] + 3 X[14644], X[107] - 3 X[14847], X[125] + 3 X[14847], X[16163] - 3 X[23239], X[10745] - 3 X[23515].

See Antreas Hatzipolakis, Angel Montesdeoca, and Peter Moses, Hyacinthos 28446 and Hyacinthos 28449.

X(24930) lies on cubic K818 and these lines: {4,74}, {122,6723}, {402,5972}, {10745,23515}, {12295,23240}, {16111,22337}, {16163,23239}

X(24930) = midpoint of X(i) and X(j) for these {i,j}: {107, 125}, {12295, 23240}, {16111, 22337}
X(24930) = reflection of X(i) in X(j) for these {i,j}: {122, 6723}, {5972, 6716}}
X(24930) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (125, 14847, 107)

Collineation mappings involving Gemini triangle 11: X(24931) - X(24962)

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 11, as in centers X(24931)-X(24962). Then

m(X) = a (a^2 + a b + a c + b c) x + (a^2 b + 2 a b^2 + b^3 + a^2 c + 3 a b c + 2 b^2 c + a c^2 + b c^2) y + (a^2 c + 2 a c^2 + c^3 + a^2 b + 3 a b c + 2 b c^2 + a b^2 + b^2 c) z : : ,

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, October 11, 2018)

### X(24931) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24931) lies on these lines: {1, 2}, {56, 24912}, {58, 1211}, {261, 1078}, {284, 1213}, {333, 6693}, {405, 759}, {1010, 3454}, {1324, 16287}, {3647, 24697}, {3752, 5956}, {3763, 16408}, {4053, 11374}, {4205, 4653}, {4256, 13728}, {4658, 6703}, {5743, 17698}, {7308, 24933}, {7832, 17307}, {11263, 24342}, {17327, 19273}, {19933, 19936}, {24932, 24954}, {24934, 24938}, {24941, 24950}, {24949, 24952}

### X(24932) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24932) lies on these lines: {1, 21670}, {2, 3}, {24931, 24954}, {24935, 24942}, {24936, 24955}

### X(24933) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 11

Barycentrics    a^7 - a^6 b - 3 a^5 b^2 + a^4 b^3 + a^3 b^4 - a^2 b^5 + a b^6 + b^7 - a^6 c - 5 a^5 b c - a^4 b^2 c + 6 a^3 b^3 c + a^2 b^4 c - a b^5 c + b^6 c - 3 a^5 c^2 - a^4 b c^2 + 10 a^3 b^2 c^2 + 8 a^2 b^3 c^2 - a b^4 c^2 - b^5 c^2 + a^4 c^3 + 6 a^3 b c^3 + 8 a^2 b^2 c^3 + 2 a b^3 c^3 - b^4 c^3 + a^3 c^4 + a^2 b c^4 - a b^2 c^4 - b^3 c^4 - a^2 c^5 - a b c^5 - b^2 c^5 + a c^6 + b c^6 + c^7 : :

X(24933) lies on these lines: {1, 21671}, {2, 3}, {943, 21015}, {3624, 24937}, {5506, 18598}, {5722, 24883}, {5747, 25063}, {7308, 24931}, {9581, 24880}, {11374, 24936}, {15803, 24884}, {24942, 24956}

### X(24934) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 11

Barycentrics    2 a^6 b + 2 a^5 b^2 - 3 a^4 b^3 - 3 a^3 b^4 + a^2 b^5 + a b^6 + 2 a^6 c + 6 a^5 b c - a^4 b^2 c - 10 a^3 b^3 c - 2 a^2 b^4 c + 4 a b^5 c + b^6 c + 2 a^5 c^2 - a^4 b c^2 - 12 a^3 b^2 c^2 - 11 a^2 b^3 c^2 - a b^4 c^2 + b^5 c^2 - 3 a^4 c^3 - 10 a^3 b c^3 - 11 a^2 b^2 c^3 - 8 a b^3 c^3 - 2 b^4 c^3 - 3 a^3 c^4 - 2 a^2 b c^4 - a b^2 c^4 - 2 b^3 c^4 + a^2 c^5 + 4 a b c^5 + b^2 c^5 + a c^6 + b c^6 : :

X(24934) lies on these lines: {1, 21672}, {2, 3}, {1170, 24909}, {5432, 24880}, {5433, 24912}, {15325, 24936}, {24931, 24938}

### X(24935) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 11

Barycentrics    a^5 + a^4 b + a^2 b^3 + 2 a b^4 + b^5 + a^4 c + a^3 b c + a^2 b^2 c + 3 a b^3 c + 2 b^4 c + a^2 b c^2 + 2 a b^2 c^2 + b^3 c^2 + a^2 c^3 + 3 a b c^3 + b^2 c^3 + 2 a c^4 + 2 b c^4 + c^5 : :

X(24935) lies on these lines: {1, 20654}, {2, 6}, {1125, 21076}, {1333, 3454}, {3061, 7110}, {3943, 24946}, {4053, 11374}, {8818, 24275}, {17275, 24880}, {17282, 24894}, {17362, 24883}, {24932, 24942}, {24941, 24943}, {24944, 24947}, {24957, 24958}

### X(24936) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24936) lies on these lines: {1, 2}, {3, 229}, {21, 17056}, {58, 15674}, {81, 6675}, {581, 6884}, {846, 14450}, {1211, 17557}, {1330, 17588}, {1962, 24161}, {2475, 4653}, {3247, 24937}, {3936, 11110}, {4197, 19765}, {4648, 6910}, {5047, 5718}, {5224, 19334}, {5433, 24909}, {9782, 17596}, {11374, 24933}, {11375, 17080}, {13408, 21161}, {15325, 24934}, {15670, 16948}, {16342, 18134}, {17245, 17531}, {18139, 19270}, {24932, 24955}, {24944, 24952}, {24945, 24962}

### X(24937) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24937) lies on these lines: {1, 21675}, {2, 7}, {5, 572}, {6, 24880}, {37, 24160}, {86, 17052}, {284, 442}, {594, 5719}, {1100, 3017}, {1449, 24883}, {1743, 24902}, {1761, 11263}, {1765, 6852}, {1901, 4877}, {3247, 24936}, {3624, 24933}, {4053, 11374}, {4675, 24890}, {5044, 21873}, {5816, 6858}, {7110, 21921}, {8818, 11108}, {24944, 24957}, {24947, 24958}

### X(24938) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24938) lies on these lines: {1, 21676}, {2, 11}, {37, 24186}, {4010, 24959}, {13277, 24920}, {24931, 24934}

### X(24939) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24939) lies on these lines: {1, 21679}, {2, 3}

### X(24940) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24940) lies on these lines: {1, 21680}, {2, 3}

### X(24941) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24941) lies on these lines: {1, 20655}, {2, 31}, {2206, 21245}, {24931, 24950}, {24935, 24943}

### X(24942) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24942) lies on these lines: {1, 21681}, {2, 32}, {24932, 24935}, {24933, 24956}, {24945, 24951}

### X(24943) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 11

Barycentrics    a^3 + a b^2 + b^3 + b^2 c + a c^2 + b c^2 + c^3 : :

X(24943) lies on these lines: {1, 2}, {31, 141}, {55, 3763}, {69, 2308}, {210, 17357}, {748, 1211}, {756, 17279}, {902, 3619}, {968, 17306}, {1011, 1626}, {1150, 6679}, {1213, 2280}, {1468, 17698}, {1962, 4657}, {2187, 7499}, {2223, 7822}, {2887, 24552}, {2895, 16468}, {3052, 21358}, {3416, 17469}, {3620, 21747}, {3662, 4418}, {3683, 17237}, {3744, 3844}, {3745, 17231}, {3773, 3891}, {3775, 5278}, {3923, 17184}, {3930, 17303}, {3989, 17776}, {4046, 17366}, {4365, 19785}, {4676, 4683}, {5743, 17125}, {6535, 17286}, {6703, 9345}, {21020, 24789}, {24935, 24941}

### X(24944) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24944) lies on these lines: {1, 21699}, {2, 37}, {7, 24915}, {86, 16589}, {1268, 1500}, {1655, 16709}, {3616, 24519}, {3624, 24945}, {17038, 24520}, {24935, 24947}, {24936, 24952}, {24937, 24957}

### X(24945) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24945) lies on these lines: {1, 21700}, {2, 39}, {1045, 3216}, {3624, 24944}, {24936, 24962}, {24942, 24951}

### X(24946) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24946) lies on these lines: {1, 2}, {1211, 15674}, {3454, 15680}, {3943, 24935}, {6740, 6919}, {20084, 24850}

### X(24947) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24947) lies on these lines: {1, 2}, {1931, 24956}, {24935, 24944}, {24937, 24958}

### X(24948) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24948) lies on these lines: {1, 21727}, {2, 650}, {37, 17161}, {661, 1019}, {1491, 4057}, {3709, 4467}, {4010, 8043}, {4789, 25084}

### X(24949) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24949) lies on these lines: {2, 3}, {24931, 24952}

### X(24950) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 11

Barycentrics    a^7 + a^6 b - a^5 b^2 - 2 a^4 b^3 - 2 a^3 b^4 + 2 a b^6 + b^7 + a^6 c - 6 a^4 b^2 c - 10 a^3 b^3 c - 5 a^2 b^4 c + 2 a b^5 c + 2 b^6 c - a^5 c^2 - 6 a^4 b c^2 - 15 a^3 b^2 c^2 - 16 a^2 b^3 c^2 - 6 a b^4 c^2 - 2 a^4 c^3 - 10 a^3 b c^3 - 16 a^2 b^2 c^3 - 12 a b^3 c^3 - 3 b^4 c^3 - 2 a^3 c^4 - 5 a^2 b c^4 - 6 a b^2 c^4 - 3 b^3 c^4 + 2 a b c^5 + 2 a c^6 + 2 b c^6 + c^7 : :

X(24950) lies on these lines: {2, 3}, {3624, 24955}, {24931, 24941}

### X(24951) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24951) lies on these lines: {2, 3}, {24942, 24945}

### X(24952) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24952) lies on these lines: {1, 21673}, {2, 7}, {24931, 24949}, {24936, 24944}

### X(24953) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24953) lies on these lines: {1, 5791}, {2, 12}, {3, 3925}, {5, 5251}, {8, 6690}, {9, 11375}, {10, 2646}, {11, 405}, {21, 2886}, {36, 8728}, {48, 1213}, {55, 5082}, {63, 3649}, {65, 5745}, {72, 354}, {100, 9710}, {120, 19314}, {140, 355}, {149, 15676}, {210, 13411}, {377, 15326}, {404, 3826}, {442, 993}, {443, 5204}, {452, 10896}, {468, 5130}, {495, 5258}, {496, 5259}, {497, 2894}, {498, 9708}, {499, 11108}, {590, 19026}, {615, 19025}, {631, 4413}, {632, 10942}, {908, 5302}, {946, 3683}, {956, 10198}, {960, 15950}, {988, 24789}, {1001, 1259}, {1212, 22070}, {1376, 6910}, {1377, 13958}, {1378, 13901}, {1453, 17723}, {1468, 17056}, {1470, 19520}, {1479, 16418}, {1621, 3813}, {1656, 10526}, {1834, 10448}, {1837, 5705}, {2260, 17398}, {2352, 19858}, {2478, 7173}, {2550, 5217}, {2829, 6937}, {3035, 6224}, {3058, 5248}, {3086, 4423}, {3090, 10894}, {3333, 3624}, {3419, 10543}, {3428, 6824}, {3485, 5273}, {3525, 10786}, {3526, 6713}, {3560, 15908}, {3616, 12635}, {3634, 13747}, {3698, 6684}, {3763, 12587}, {3782, 24161}, {3816, 5047}, {3820, 10954}, {3829, 16858}, {3841, 5267}, {3868, 11281}, {3869, 18253}, {3901, 16137}, {3916, 11246}, {3983, 6745}, {4026, 16342}, {4299, 17528}, {4302, 17571}, {4429, 19278}, {4512, 12701}, {4652, 5880}, {4679, 5812}, {4699, 7891}, {4870, 5325}, {4972, 16347}, {4995, 5687}, {5044, 16193}, {5054, 18518}, {5067, 10599}, {5070, 11929}, {5084, 10953}, {5094, 11391}, {5129, 10589}, {5177, 12943}, {5219, 5234}, {5220, 5550}, {5221, 5744}, {5225, 11106}, {5231, 5436}, {5247, 5718}, {5316, 17590}, {5326, 10955}, {5584, 6847}, {5719, 5904}, {5837, 11011}, {5842, 6875}, {5886, 12704}, {6147, 6763}, {6174, 9709}, {6832, 7958}, {6837, 7965}, {6852, 7680}, {6853, 18242}, {6856, 10895}, {6861, 11249}, {6878, 10785}, {6889, 12114}, {6892, 10310}, {6914, 11826}, {6920, 7681}, {6976, 10893}, {7484, 10830}, {7515, 23207}, {7561, 19857}, {7786, 12933}, {7807, 16819}, {7808, 10795}, {7914, 10872}, {9669, 16866}, {10021, 16139}, {10200, 16842}, {10523, 17527}, {10944, 24987}, {11111, 12953}, {11238, 17561}, {11499, 21155}, {11680, 16865}, {11904, 15184}, {12572, 17605}, {12699, 16617}, {13181, 14061}, {13214, 15059}, {15338, 16370}, {16478, 17726}, {17057, 18357}, {17560, 20988}, {17566, 19877}, {17698, 19863}, {19273, 19784}, {23922, 24432}, {24384, 24435}, {24931, 24934}

### X(24954) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24954) lies on these lines: {1, 3820}, {2, 65}, {3, 4679}, {9, 5433}, {10, 2098}, {11, 936}, {12, 8583}, {55, 6700}, {56, 3452}, {63, 6691}, {72, 10200}, {78, 3816}, {210, 3086}, {377, 5087}, {388, 5328}, {404, 24703}, {405, 22768}, {443, 17605}, {474, 1836}, {496, 4863}, {499, 5044}, {631, 3683}, {908, 10404}, {946, 4413}, {956, 1125}, {975, 17723}, {978, 17720}, {997, 1837}, {1058, 3689}, {1155, 17567}, {1319, 2551}, {1329, 5252}, {1376, 12701}, {1388, 5795}, {1408, 24556}, {1698, 3628}, {1770, 16417}, {2099, 8582}, {2646, 5084}, {3035, 5250}, {3303, 6745}, {3305, 4999}, {3333, 3624}, {3419, 3825}, {3421, 20323}, {3476, 8165}, {3616, 11260}, {3649, 5437}, {3678, 10199}, {3679, 11373}, {3698, 5603}, {3740, 10527}, {3782, 11512}, {3872, 9711}, {3925, 6769}, {4005, 24477}, {4193, 5794}, {4423, 13411}, {4640, 6921}, {4662, 10529}, {4860, 15650}, {5057, 17572}, {5204, 12572}, {5221, 6692}, {5233, 10371}, {5289, 24982}, {5293, 17721}, {5438, 6284}, {5880, 17531}, {5919, 7080}, {6001, 6967}, {6910, 15254}, {6926, 12688}, {6944, 14110}, {6983, 7686}, {7288, 18228}, {10179, 10528}, {11230, 19854}, {12047, 16408}, {12053, 20103}, {12514, 13747}, {12609, 16862}, {15481, 18230}, {17303, 21801}, {17556, 17647}, {24931, 24932}, {24936, 24944}

### X(24955) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24955) lies on these lines: {1, 21054}, {2, 11}, {110, 8287}, {1125, 21098}, {1290, 5954}, {3624, 24950}, {4092, 6742}, {5972, 24624}, {7741, 24904}, {8286, 15059}, {18004, 24957}, {24932, 24936}

### X(24956) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24956) lies on these lines: {1, 21711}, {2, 99}, {39, 16614}, {1015, 24542}, {1125, 4115}, {1213, 15670}, {1931, 24947}, {3731, 4370}, {5283, 22398}, {5475, 11346}, {14148, 16711}, {16430, 21004}, {19862, 24070}, {23897, 24902}, {24074, 24187}, {24933, 24942}

### X(24957) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24957) lies on these lines: {1, 21043}, {2, 45}, {115, 662}, {645, 14061}, {1125, 21089}, {1698, 21047}, {4092, 11725}, {6543, 17398}, {13178, 17467}, {17276, 24922}, {18004, 24955}, {24935, 24958}, {24937, 24944}

### X(24958) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24958) lies on these lines: {1, 21713}, {2, 37}, {3923, 20369}, {7760, 17379}, {21020, 24746}, {22167, 24520}, {22220, 24923}, {24935, 24957}, {24937, 24947}

### X(24959) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24959) lies on these lines: {1, 21714}, {2, 900}, {1213, 24506}, {4010, 24938}, {6707, 24287}, {8062, 9013}, {14321, 24961}, {18004, 24955}

### X(24960) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24960) lies on these lines: {1, 21719}, {2, 647}, {523, 24961}, {649, 21260}, {1021, 1698}, {1213, 9404}, {2523, 4391}, {2605, 21721}, {3700, 17303}, {20316, 22383}

### X(24961) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24961) lies on these lines: {1, 21721}, {2, 650}, {523, 24960}, {4024, 17303}, {4728, 24921}, {14321, 24959}, {18147, 21438}

### X(24962) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 11

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

X(24962) lies on these lines: {1, 21725}, {2, 668}, {99, 16592}, {14061, 16613}, {24519, 24880}, {24936, 24945}

### X(24963) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 12

Barycentrics    a^4 + a^3 b - 3 a^2 b^2 - 4 a b^3 - b^4 + a^3 c - 9 a^2 b c - 18 a b^2 c - 6 b^3 c - 3 a^2 c^2 - 18 a b c^2 - 10 b^2 c^2 - 4 a c^3 - 6 b c^3 - c^4 : :

X(24963) lies on these lines: {1, 2}, {17275, 24967}

### X(24964) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 12

Barycentrics    a^7 + 3 a^6 b + 3 a^5 b^2 + 2 a^4 b^3 - 4 a^2 b^5 - 4 a b^6 - b^7 + 3 a^6 c + 5 a^5 b c + 4 a^4 b^2 c + 6 a^3 b^3 c - 3 a^2 b^4 c - 11 a b^5 c - 4 b^6 c + 3 a^5 c^2 + 4 a^4 b c^2 + 14 a^3 b^2 c^2 + 19 a^2 b^3 c^2 + 4 a b^4 c^2 - 2 b^5 c^2 + 2 a^4 c^3 + 6 a^3 b c^3 + 19 a^2 b^2 c^3 + 22 a b^3 c^3 + 7 b^4 c^3 - 3 a^2 b c^4 + 4 a b^2 c^4 + 7 b^3 c^4 - 4 a^2 c^5 - 11 a b c^5 - 2 b^2 c^5 - 4 a c^6 - 4 b c^6 - c^7 : :

X(24964) lies on this line: {2, 3}

### X(24965) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 12

Barycentrics    5 a^7 + 15 a^6 b + 9 a^5 b^2 - 11 a^4 b^3 - 15 a^3 b^4 - 5 a^2 b^5 + a b^6 + b^7 + 15 a^6 c + 27 a^5 b c - a^4 b^2 c - 22 a^3 b^3 c - 15 a^2 b^4 c - 5 a b^5 c + b^6 c + 9 a^5 c^2 - a^4 b c^2 - 14 a^3 b^2 c^2 - 4 a^2 b^3 c^2 - a b^4 c^2 - b^5 c^2 - 11 a^4 c^3 - 22 a^3 b c^3 - 4 a^2 b^2 c^3 + 10 a b^3 c^3 - b^4 c^3 - 15 a^3 c^4 - 15 a^2 b c^4 - a b^2 c^4 - b^3 c^4 - 5 a^2 c^5 - 5 a b c^5 - b^2 c^5 + a c^6 + b c^6 + c^7 : :

X(24965) lies on this line: {2, 3}

### X(24966) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 12

Barycentrics    4 a^7 + 12 a^6 b + 6 a^5 b^2 - 13 a^4 b^3 - 15 a^3 b^4 - a^2 b^5 + 5 a b^6 + 2 b^7 + 12 a^6 c + 22 a^5 b c - 5 a^4 b^2 c - 28 a^3 b^3 c - 12 a^2 b^4 c + 6 a b^5 c + 5 b^6 c + 6 a^5 c^2 - 5 a^4 b c^2 - 28 a^3 b^2 c^2 - 23 a^2 b^3 c^2 - 5 a b^4 c^2 + b^5 c^2 - 13 a^4 c^3 - 28 a^3 b c^3 - 23 a^2 b^2 c^3 - 12 a b^3 c^3 - 8 b^4 c^3 - 15 a^3 c^4 - 12 a^2 b c^4 - 5 a b^2 c^4 - 8 b^3 c^4 - a^2 c^5 + 6 a b c^5 + b^2 c^5 + 5 a c^6 + 5 b c^6 + 2 c^7 : :

X(24966) lies on this line: {2, 3}

### X(24967) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 12

Barycentrics    a^5 + 3 a^4 b - 5 a^2 b^3 - 4 a b^4 - b^5 + 3 a^4 c + 5 a^3 b c - 7 a^2 b^2 c - 11 a b^3 c - 4 b^4 c - 7 a^2 b c^2 - 14 a b^2 c^2 - 7 b^3 c^2 - 5 a^2 c^3 - 11 a b c^3 - 7 b^2 c^3 - 4 a c^4 - 4 b c^4 - c^5 : :

X(24967) lies on these lines: {2, 6}, {4969, 24968}, {17275, 24963}

### X(24968) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 12

Barycentrics    5 a^4 + 14 a^3 b + 12 a^2 b^2 + 4 a b^3 + b^4 + 14 a^3 c + 17 a^2 b c - a b^2 c + 12 a^2 c^2 - a b c^2 - 2 b^2 c^2 + 4 a c^3 + c^4 : :

X(24968) lies on these lines: {1, 2}, {1051, 9782}, {4969, 24967}

### X(24969) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 12

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

X(24969) lies on these lines: {1, 2}

### X(24970) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 12

Barycentrics    7 a^7 + 21 a^6 b + 15 a^5 b^2 - 7 a^4 b^3 - 15 a^3 b^4 - 13 a^2 b^5 - 7 a b^6 - b^7 + 21 a^6 c + 37 a^5 b c + 7 a^4 b^2 c - 10 a^3 b^3 c - 21 a^2 b^4 c - 27 a b^5 c - 7 b^6 c + 15 a^5 c^2 + 7 a^4 b c^2 + 14 a^3 b^2 c^2 + 34 a^2 b^3 c^2 + 7 a b^4 c^2 - 5 b^5 c^2 - 7 a^4 c^3 - 10 a^3 b c^3 + 34 a^2 b^2 c^3 + 54 a b^3 c^3 + 13 b^4 c^3 - 15 a^3 c^4 - 21 a^2 b c^4 + 7 a b^2 c^4 + 13 b^3 c^4 - 13 a^2 c^5 - 27 a b c^5 - 5 b^2 c^5 - 7 a c^6 - 7 b c^6 - c^7 : :

X(24970) lies on this line: {2, 3}

### X(24971) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 12

Barycentrics    a^7 + 3 a^6 b + 3 a^5 b^2 + 2 a^4 b^3 - 4 a^2 b^5 - 4 a b^6 - b^7 + 3 a^6 c + 8 a^5 b c + 16 a^4 b^2 c + 24 a^3 b^3 c + 9 a^2 b^4 c - 8 a b^5 c - 4 b^6 c + 3 a^5 c^2 + 16 a^4 b c^2 + 49 a^3 b^2 c^2 + 54 a^2 b^3 c^2 + 16 a b^4 c^2 - 2 b^5 c^2 + 2 a^4 c^3 + 24 a^3 b c^3 + 54 a^2 b^2 c^3 + 40 a b^3 c^3 + 7 b^4 c^3 + 9 a^2 b c^4 + 16 a b^2 c^4 + 7 b^3 c^4 - 4 a^2 c^5 - 8 a b c^5 - 2 b^2 c^5 - 4 a c^6 - 4 b c^6 - c^7 : :

X(24971) lies on this line: {2, 3}

### X(24972) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 12

Barycentrics    a^7 + 3 a^6 b - 7 a^4 b^3 - 9 a^3 b^4 - 7 a^2 b^5 - 4 a b^6 - b^7 + 3 a^6 c + 5 a^5 b c - 5 a^4 b^2 c - 11 a^3 b^3 c - 12 a^2 b^4 c - 11 a b^5 c - 4 b^6 c - 5 a^4 b c^2 - 7 a^3 b^2 c^2 - 2 a^2 b^3 c^2 - 5 a b^4 c^2 - 5 b^5 c^2 - 7 a^4 c^3 - 11 a^3 b c^3 - 2 a^2 b^2 c^3 + 5 a b^3 c^3 - 2 b^4 c^3 - 9 a^3 c^4 - 12 a^2 b c^4 - 5 a b^2 c^4 - 2 b^3 c^4 - 7 a^2 c^5 - 11 a b c^5 - 5 b^2 c^5 - 4 a c^6 - 4 b c^6 - c^7 : :

X(24972) lies on this line: {2, 3}

4th intersections of inconics and the circumcircles of their polar triangles: X(24973) - X(24980)

Let E be an inconic (other than the incircle) with Brianchon point (perspector) P. Let A'B'C' be the polar triangle of E, which is the cevian triangle of P. Let O* be the circumcircle of A'B'C' (i.e., the cevian circle of P). E and O* intersect in 4 points: A', B', C' and a 4th intersection which is a triangle center. The appearance of (E,i,O*,j) in the following list means that inconic E with perspector X(i) intersects circle O* (other than at A', B', C') at X(j):

(Brocard inellipse, 6, symmedial circle, 24973)
(incentral inellipse, 1, incentral circle, 23063)
(Kiepert parabola, 99, 2nd Steiner circle, 24974)
(Lemoine inellipse, 598, 3rd Lemoine circle, 20383)
(MacBeath inconic, 264, MacBeath circle, 24977)
(Mandart inellipse, 8, Mandart circle, 11)
(orthic inconic, 4, nine-point circle, 125)
(Steiner inellipse, 2, nine-point circle, 115)
(Yff parabola, 190, cevian circle of X(190), 24979)

Contributed by Randy Hutson, October 11, 2018.

### X(24973) = 4th INTERSECTION OF BROCARD INELLIPSE AND SYMMEDIAL CIRCLE

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

X(24973) lies on the Brocard inellipse, the symmedial circle, and these lines: {511, 9482}, {323, 5104}, {805, 8041}, {2679, 3005}

### X(24974) = 4th INTERSECTION OF KIEPERT PARABOLA AND 2nd STEINER CIRCLE

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

X(24974) lies on the Kiepert parabola, the 2nd Steiner circle, and these lines: {476, 8029}, {523, 14683}, {669, 11641}, {868, 1649}, {11123, 14731}

### X(24975) = TRILINEAR POLE OF TANGENT TO KIEPERT PARABOLA AT X(24974)

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

X(24975) lies on these lines: {2, 6}, {3, 18121}, {30, 14356}, {114, 6593}, {140, 18114}, {297, 14590}, {620, 2492}, {868, 5467}, {1576, 6033} et al

X(24975) = complement of isotomic conjugate of isogonal conjugate of X(2088)

### X(24976) = TRILINEAR POLE OF TANGENT TO LEMOINE INELLIPSE AT X(20383)

Barycentrics    (b^2 - c^2) (13 a^6 + 9 a^4 (b^2 + c^2) - 3 a^2 (b^4 + 11 b^2 c^2 + c^4) + (b^2 + c^2)^3) : :

X(24976) lies on these lines: {99, 12074}, {351, 523}, {620, 2793}, {1499, 11616}, {6088, 9125}

### X(24977) = 4th INTERSECTION OF MacBEATH INCONIC AND MacBEATH CIRCLE

Barycentrics    a^2 (b^2 - c^2)^2 (a^2 - b^2 - c^2) (a^8 - 2 a^6 (b^2 + c^2) + a^4 b^2 c^2 + a^2 (b^2 + c^2) (2 b^4 - 3 b^2 c^2 + 2 c^4) - (b^2 - c^2)^2 (b^4 + c^4))^2 : :

X(24977) lies on the MacBeath inconic, the MacBeath circle, and these lines: {3, 1291}, {23, 2967}, {2972, 3258}, {12091, 18403}, {13409, 14731}

### X(24978) = TRILINEAR POLE OF TANGENT TO MacBEATH INCONIC AT X(24977)

Barycentrics    (b^2 - c^2) (a^8 - 2 a^6 (b^2 + c^2) + a^4 b^2 c^2 + a^2 (b^2 + c^2) (2 b^4 - 3 b^2 c^2 + 2 c^4) - (b^2 - c^2)^2 (b^4 + c^4)) : :

X(24978) lies on these lines: {83, 14223}, {297, 525}, {523, 5926}, {526, 12236}, {620, 2492}, {648, 16813}, {1263, 10264}, {1637, 6334}, {2394, 18366}, {6132, 8151}, {9517, 16230}, {10214, 11557}, {15412, 20577}

### X(24979) = 4th INTERSECTION OF YFF PARABOLA AND CEVIAN CIRCLE OF X(190)

Barycentrics    (b - c) (2 a^5 - 3 a^4 (b + c) + a^3 (b^2 + 4 b c + c^2) - a^2 b c (b + c) - a (b - c)^2 (b^2 + c^2) + (b - c)^2 (b^3 + c^3))^2 : :

X(24979) lies on the Yff parabola, the cevian circle of X(190), and these lines: {514, 20096}, {927, 6545}, {1566, 3259}, {3239, 17777}

### X(24980) = TRILINEAR POLE OF TANGENT TO YFF PARABOLA AT X(24979)

Barycentrics    2 a^5 - 3 a^4 (b + c) + a^3 (b^2 + 4 b c + c^2) - a^2 b c (b + c) - a (b - c)^2 (b^2 + c^2) + (b - c)^2 (b^3 + c^3) : :

X(24980) lies on these lines: {1, 2}, {226, 5091}, {597, 8255}, {908, 3573}, {1083, 3452}, {3035, 6366}, {3310, 23988}, {3675, 3911}, {15252, 24003}

X(24980) = complement of crossdifference of X(109) and X(649)

### X(24981) =  MIDPOINT OF X(12121) AND X(12308)

Barycentrics    4 a^6 - 4 a^4 (b^2 + c^2) - (b^2 - c^2)^2 (b^2 + c^2) + a^2 (b^2 + c^2)^2 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28452.

X(24981) lies on these lines: {2,98}, {4,6053}, {23,5965}, {51,1353}, {74,3528}, {113,137}, {115,20976}, {155,382}, {159,2930}, {265,3851}, {373,8550}, {539,10540}, {541,11820}, {550,5562}, {1112,1843}, {1495,3564}, {1503,3292}, {1511,3530}, {2502,6388}, {2777,3529}, {2931,9908}, {2948,3632}, {3124,5477}, {3167,11550}, {3233,6070}, {3544,14644}, {3631,5181}, {3636,11720}, {3855,7687}, {4576,14928}, {5079,14643}, {5191,14981}, {5448,18430}, {5655,12902}, {5907,10619}, {6030,15108}, {6154,8674}, {6329,6593}, {6699,15720}, {6791,20998}, {7706,18445}, {7984,20057}, {9155,10991}, {10116,18350}, {10264,14869}, {10299,12317}, {10620,15688}, {11008,11061}, {11225,13595}, {11422,19130}, {11441,21659}, {11693,22250}, {11737,11801}, {11800,12824}, {12295,15687}, {12310,20850}, {12412,22109}, {13366,18583}, {13417,14984}, {13431,14449}, {13605,15808}, {14389,18553}, {15036,15715}, {15039,20397}, {15040,15700}, {15303,20583}

X(24981) = midpoint of X(i) and X(j) for these {i,j}: {12121,12308}, {12383,14094}
X(24981) = reflection of X(i) in X(j) for these {i,j}: {4,6053}, {113,5609}, {125,110}, {265,16534}, {3448,5972}, {6070,3233}, {10990,16163}, {12317,20417}, {15063,399}, {16003,1511}
X(24981) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {110,125,5642}, {110,3448,5972}, {3448,5972,125}, {6723,9140,125}, {9143,14683,110}, {12317,15035,20417}

Collineation mappings involving Gemini triangle 13: X(24982) - X(25024)

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 13, as in centers X(24982)-X(25024). Then

m(X) = (-a^2 b + b^3 - a^2 c - b^2 c - b c^2 + c^3) x + b(a^2 - b^2 - 2 a c + c ^2) y + c(a^2 - c^2 -2a b + b^2) z : : ,

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, October 12, 2018)

### X(24982) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c + 3 a b^2 c - a^2 c^2 + 3 a b c^2 - 2 b^2 c^2 - a c^3 + c^4 : :

X(24982) lies on these lines: {1, 2}, {3, 1512}, {4, 3359}, {5, 1519}, {9, 5555}, {11, 5836}, {12, 3812}, {20, 10270}, {21, 2077}, {40, 2478}, {57, 3436}, {63, 1788}, {65, 908}, {72, 3820}, {80, 14803}, {84, 377}, {100, 950}, {119, 125}, {142, 7679}, {165, 6872}, {210, 9711}, {226, 11681}, {329, 8165}, {354, 12607}, {355, 474}, {388, 3306}, {392, 5690}, {404, 515}, {405, 11248}, {443, 5818}, {495, 5439}, {496, 10914}, {516, 5046}, {517, 4187}, {518, 21031}, {942, 17757}, {944, 17567}, {946, 4193}, {952, 17614}, {958, 1470}, {962, 6919}, {1058, 3895}, {1145, 9957}, {1266, 23521}, {1319, 6691}, {1376, 1837}, {1385, 13747}, {1476, 5176}, {1699, 5187}, {1706, 3434}, {1738, 21935}, {1739, 23537}, {1834, 25091}, {1861, 4429}, {1877, 5294}, {2093, 11415}, {2345, 24005}, {2475, 19925}, {2476, 7705}, {2646, 3035}, {2886, 3698}, {2975, 3911}, {3036, 20118}, {3057, 3816}, {3218, 12527}, {3219, 18250}, {3339, 5905}, {3361, 20076}, {3419, 9709}, {3452, 3869}, {3486, 4855}, {3576, 6921}, {3579, 11113}, {3614, 3838}, {3628, 11729}, {3670, 24186}, {3681, 24391}, {3693, 21049}, {3701, 17862}, {3717, 20905}, {3740, 21677}, {3742, 15888}, {3754, 3814}, {3817, 5154}, {3826, 5784}, {3832, 21628}, {3868, 21075}, {3877, 11362}, {3879, 24540}, {3897, 10165}, {3914, 24440}, {3918, 6702}, {3922, 17605}, {3925, 10958}, {4026, 25099}, {4188, 4297}, {4189, 10164}, {4190, 5691}, {4197, 15016}, {4292, 5080}, {4298, 20060}, {4357, 24986}, {4413, 5794}, {4423, 10965}, {4512, 9588}, {4642, 24210}, {4731, 9710}, {4967, 24547}, {4972, 24983}, {5051, 24991}, {5084, 5250}, {5087, 10107}, {5177, 11024}, {5179, 16549}, {5251, 5445}, {5260, 5745}, {5261, 9776}, {5289, 24954}, {5316, 5837}, {5437, 9578}, {5438, 5727}, {5687, 5722}, {5711, 10601}, {5749, 20262}, {5782, 17303}, {5790, 16203}, {5791, 19520}, {5828, 11037}, {5880, 10895}, {5883, 13407}, {5902, 21077}, {5903, 21616}, {6174, 10543}, {6245, 6904}, {6871, 7989}, {6920, 12775}, {6931, 8227}, {6957, 12705}, {6992, 10268}, {7483, 11231}, {7504, 10172}, {7951, 12609}, {8728, 10202}, {9955, 17533}, {10265, 12751}, {10404, 11236}, {10596, 17559}, {10679, 11108}, {10786, 18443}, {10805, 17582}, {11112, 18480}, {12000, 16853}, {12053, 14923}, {12512, 15680}, {12513, 17728}, {12699, 17556}, {12709, 18236}, {13161, 24443}, {13373, 17529}, {13906, 19065}, {13911, 19047}, {13964, 19066}, {13973, 19048}, {14011, 17167}, {15829, 20196}, {16371, 18481}, {16417, 18525}, {17063, 23675}, {17528, 18542}, {18239, 18242}, {23536, 24174}, {24985, 24988}, {25001, 25004}

### X(24983) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^6 b - a^4 b^3 - a^2 b^5 + b^7 + a^6 c - 2 a^3 b^3 c - a^2 b^4 c + 2 a b^5 c + 2 a^2 b^3 c^2 - 2 b^5 c^2 - a^4 c^3 - 2 a^3 b c^3 + 2 a^2 b^2 c^3 - 4 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 + 2 a b c^5 - 2 b^2 c^5 + c^7 : :

X(24983) lies on these lines: {2, 3}, {10, 23528}, {1210, 23542}, {1737, 23518}, {1785, 23661}, {4972, 24982}, {19861, 21147}, {24986, 24992}

### X(24984) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^6 b - a^4 b^3 - a^2 b^5 + b^7 + a^6 c - 2 a^5 b c + 2 a^3 b^3 c - a^2 b^4 c + 2 a^2 b^3 c^2 - 2 b^5 c^2 - a^4 c^3 + 2 a^3 b c^3 + 2 a^2 b^2 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - 2 b^2 c^5 + c^7 : :

X(24984) lies on these lines: {1, 23518}, {2, 3}, {8, 343}, {10, 1074}, {1441, 4357}, {3100, 5174}, {4296, 17923}, {4972, 19860}, {5797, 18180}, {5906, 7078}, {17862, 23537}, {25000, 25002}

### X(24985) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^6 b - a^4 b^3 - a^2 b^5 + b^7 + a^6 c - 4 a^5 b c + 6 a^3 b^3 c - a^2 b^4 c - 2 a b^5 c + 2 a^2 b^3 c^2 - 2 b^5 c^2 - a^4 c^3 + 6 a^3 b c^3 + 2 a^2 b^2 c^3 + 4 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - 2 a b c^5 - 2 b^2 c^5 + c^7 : :

X(24985) lies on these lines: {2, 3}, {1125, 23541}, {4972, 24541}, {13411, 23542}, {17282, 19861}, {24982, 24988}

### X(24986) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^4 b - 2 a^2 b^3 + b^5 + a^4 c + a^2 b^2 c + 2 a b^3 c + a^2 b c^2 - b^3 c^2 - 2 a^2 c^3 + 2 a b c^3 - b^2 c^3 + c^5 : :

X(24986) lies on these lines: {2, 6}, {7, 11681}, {10, 20895}, {56, 21286}, {75, 25005}, {1229, 24005}, {1329, 20245}, {1400, 21244}, {4187, 17183}, {4193, 10446}, {4357, 24982}, {4972, 24991}, {5554, 17321}, {17270, 19861}, {17452, 25369}, {24983, 24992}, {25001, 25007}, {25018, 25021}

### X(24987) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c - 4 a^2 b c - a b^2 c - a^2 c^2 - a b c^2 - 2 b^2 c^2 - a c^3 + c^4 : :

X(24987) lies on these lines: {1, 2}, {4, 5250}, {5, 392}, {9, 3436}, {12, 908}, {20, 10268}, {21, 515}, {35, 17647}, {40, 377}, {55, 5794}, {63, 388}, {65, 5249}, {72, 495}, {80, 5259}, {92, 225}, {140, 17614}, {142, 4848}, {149, 12575}, {165, 4190}, {191, 5270}, {210, 12607}, {226, 3869}, {283, 1010}, {322, 5224}, {329, 5261}, {355, 405}, {390, 5175}, {394, 5711}, {404, 6684}, {442, 517}, {443, 5657}, {474, 11249}, {516, 2475}, {518, 8261}, {527, 11684}, {529, 18253}, {535, 3647}, {536, 4918}, {758, 13407}, {944, 6857}, {946, 2476}, {950, 1621}, {952, 6675}, {956, 5791}, {958, 5252}, {962, 5177}, {966, 8557}, {982, 23675}, {986, 23536}, {1001, 1837}, {1004, 5584}, {1072, 16062}, {1104, 5724}, {1108, 5742}, {1145, 6797}, {1213, 8609}, {1220, 5294}, {1224, 2990}, {1266, 24999}, {1319, 4999}, {1385, 7483}, {1441, 4357}, {1478, 12514}, {1519, 6842}, {1616, 17721}, {1697, 3434}, {1699, 6871}, {1738, 4642}, {1758, 5484}, {1788, 3306}, {1861, 5174}, {2049, 24545}, {2078, 5176}, {2292, 13161}, {2322, 8755}, {2323, 5750}, {2345, 3692}, {2346, 5178}, {2478, 5587}, {2550, 22370}, {2551, 3305}, {2646, 6690}, {2784, 5985}, {2886, 3057}, {2975, 5745}, {3146, 21628}, {3193, 14005}, {3218, 4298}, {3219, 12527}, {3290, 21965}, {3294, 5179}, {3295, 3419}, {3359, 6897}, {3452, 11681}, {3475, 11520}, {3485, 11682}, {3576, 6910}, {3577, 6933}, {3579, 11112}, {3600, 5744}, {3614, 5087}, {3698, 3826}, {3710, 4385}, {3717, 3963}, {3740, 21031}, {3753, 5690}, {3755, 24554}, {3813, 5919}, {3816, 17606}, {3817, 5141}, {3822, 3878}, {3868, 21620}, {3873, 24391}, {3876, 21075}, {3883, 5016}, {3890, 11680}, {3895, 5082}, {3897, 5882}, {3898, 24387}, {3911, 5253}, {3916, 18990}, {3925, 5836}, {3932, 25099}, {4018, 6147}, {4084, 11551}, {4187, 9956}, {4188, 10164}, {4189, 4297}, {4193, 10175}, {4197, 11362}, {4293, 4652}, {4331, 17257}, {4413, 10966}, {4424, 23537}, {4512, 5691}, {4520, 17747}, {4640, 7354}, {4855, 5218}, {4967, 24993}, {5044, 17757}, {5046, 19925}, {5051, 23541}, {5080, 12572}, {5084, 5818}, {5123, 10957}, {5187, 7989}, {5219, 10585}, {5229, 5698}, {5248, 10572}, {5251, 14798}, {5267, 21578}, {5289, 11375}, {5290, 5905}, {5296, 20262}, {5330, 13464}, {5436, 5727}, {5603, 6856}, {5687, 19520}, {5692, 21077}, {5707, 16458}, {5720, 10786}, {5725, 16466}, {5730, 11374}, {5731, 6245}, {5775, 11037}, {5790, 11108}, {5847, 15988}, {5855, 11281}, {5903, 12609}, {6907, 12672}, {6925, 12705}, {7330, 12115}, {7680, 14110}, {7741, 17057}, {7951, 21616}, {8236, 24389}, {8608, 16699}, {9612, 11415}, {9943, 17616}, {9948, 11220}, {9955, 17530}, {9957, 24390}, {10389, 12625}, {10597, 17582}, {10680, 16408}, {10806, 17559}, {10895, 24703}, {10943, 17527}, {10944, 24953}, {11113, 18480}, {11231, 13747}, {11237, 17781}, {11344, 11500}, {12001, 16863}, {12635, 17718}, {12699, 17532}, {12702, 17528}, {13907, 19065}, {13911, 19049}, {13965, 19066}, {13973, 19050}, {14007, 24556}, {16370, 18481}, {16418, 18525}, {17072, 19948}, {17525, 22798}, {17577, 18483}, {21935, 24210}, {24178, 24443}, {25255, 25260}

### X(24988) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^2 b + b^3 + a^2 c - 4 a b c + c^3 : :

X(24988) lies on these lines: {2, 11}, {8, 3315}, {10, 244}, {43, 18139}, {88, 9780}, {121, 1054}, {321, 17888}, {404, 23850}, {474, 20999}, {678, 19862}, {748, 4450}, {756, 24169}, {899, 3836}, {1086, 3952}, {1125, 3722}, {1213, 20331}, {1623, 17572}, {1738, 4358}, {2246, 5750}, {3006, 3823}, {3120, 24003}, {3240, 17234}, {3634, 5051}, {3740, 17184}, {3771, 9350}, {3890, 6018}, {3932, 17495}, {4126, 20068}, {4359, 24186}, {4389, 9330}, {4422, 4427}, {4440, 4756}, {4453, 14430}, {4660, 17125}, {4696, 24178}, {4768, 21180}, {4966, 19998}, {5014, 5272}, {5257, 14439}, {5297, 16706}, {5741, 16569}, {6187, 19846}, {9324, 19872}, {9347, 17367}, {9458, 17719}, {16062, 19877}, {16862, 23858}, {17126, 17352}, {17724, 17780}, {24982, 24985}

### X(24989) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    (a^2 + b^2 - c^2) (a^2 - b^2 + c^2) (a^4 b - b^5 + a^4 c + a^2 b^2 c - 2 a b^3 c + a^2 b c^2 - b^3 c^2 - 2 a b c^3 - b^2 c^3 - c^5) : :

X(24989) lies on these lines: {2, 3}, {321, 23541}, {1861, 4972}, {1876, 17184}, {3192, 5741}, {3920, 5081}, {5014, 23050}, {11393, 24552}

### X(24990) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^8 b - 2 a^4 b^5 + b^9 + a^8 c + a^6 b^2 c - 2 a^5 b^3 c - a^4 b^4 c - a^2 b^6 c + 2 a b^7 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 - 2 a^5 b c^3 + a^4 b^2 c^3 + 4 a^3 b^3 c^3 + 2 a^2 b^4 c^3 - 2 a b^5 c^3 + b^6 c^3 - a^4 b c^4 + 2 a^2 b^3 c^4 - b^5 c^4 - 2 a^4 c^5 - a^2 b^2 c^5 - 2 a b^3 c^5 - b^4 c^5 - a^2 b c^6 + b^3 c^6 + 2 a b c^7 - b^2 c^7 + c^9 : :

X(24990) lies on these lines: {2, 3}, {4442, 23541}

### X(24991) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^5 b - a^3 b^3 - a^2 b^4 + b^6 + a^5 c + a^3 b^2 c + 2 a b^4 c + a^3 b c^2 - b^4 c^2 - a^3 c^3 - a^2 c^4 + 2 a b c^4 - b^2 c^4 + c^6 : :

X(24991) lies on these lines: {2, 31}, {10, 20237}, {141, 24550}, {4429, 21912}, {4972, 24986}, {4981, 24998}, {5051, 24982}, {21935, 24440}, {23541, 24997}, {24992, 24995}

### X(24992) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^6 b - a^4 b^3 - a^2 b^5 + b^7 + a^6 c + a^4 b^2 c + 2 a b^5 c + a^4 b c^2 - b^5 c^2 - a^4 c^3 - a^2 c^5 + 2 a b c^5 - b^2 c^5 + c^7 : :

X(24992) lies on these lines: {2, 32}, {10, 21420}, {24983, 24986}, {24991, 24995}, {25002, 25018}

### X(24993) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    b c (2 a^3 + a^2 b + b^3 + a^2 c + 4 a b c - b^2 c - b c^2 + c^3) : :

X(24993) lies on these lines: {1, 20895}, {2, 37}, {7, 3436}, {9, 24633}, {38, 23677}, {65, 20245}, {69, 5554}, {77, 1441}, {86, 3262}, {322, 3945}, {517, 17183}, {960, 21273}, {1266, 24994}, {1319, 18654}, {2171, 21246}, {2268, 24334}, {2269, 25371}, {2285, 24612}, {3596, 4696}, {3663, 8582}, {3875, 19861}, {4357, 24982}, {4467, 25022}, {4858, 5750}, {4967, 24987}, {4972, 25012}, {4981, 24996}, {5224, 25005}, {8583, 17151}, {10401, 21286}, {17451, 20258}, {17861, 20880}, {18698, 24209}, {23690, 24342}, {25004, 25023}, {25083, 25255}

### X(24994) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    b c (a b + b^2 + a c - 2 b c + c^2) (a^3 + a^2 b + a^2 c - a b c + b^2 c + b c^2) : :

X(24994) lies on these lines: {2, 39}, {1266, 24993}, {3831, 21405}, {24983, 24986}

### X(24995) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^3 b + b^4 + a^3 c + c^4 : :

X(24995) lies on these lines: {2, 41}, {8, 3905}, {9, 17550}, {10, 1930}, {31, 315}, {32, 4805}, {69, 5230}, {76, 21935}, {115, 4721}, {213, 626}, {218, 7866}, {325, 1193}, {672, 6656}, {742, 16886}, {758, 17211}, {2251, 6680}, {2295, 20541}, {2887, 17137}, {3314, 17033}, {3662, 16906}, {3721, 25345}, {4136, 17489}, {4165, 21216}, {4167, 17497}, {4197, 10436}, {4357, 5051}, {4429, 10030}, {4950, 16974}, {5025, 24514}, {5086, 24291}, {5255, 20553}, {7762, 21764}, {7776, 16466}, {7933, 17350}, {24991, 24992}, {24996, 25012}

### X(24996) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    (b+c)^2*a^4-(b^3+c^3)*a^3-(b^4+c^4-b*c*(b+c)^2)*a^2+(b^3+c^3)*(b^2+c^2)*a+(b^2-c^2)^2*b*c : :

X(24996) lies on these lines: {1, 2}, {6, 24545}, {181, 908}, {355, 11358}, {515, 13588}, {1512, 4192}, {2478, 9548}, {4972, 24986}, {4981, 24993}, {5087, 10406}, {23541, 25003}, {24995, 25012}

### X(24997) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^4 b^2 - a^3 b^3 - a^2 b^4 + a b^5 + 2 a^4 b c + a^2 b^3 c + b^5 c + a^4 c^2 - 2 a^2 b^2 c^2 + a b^3 c^2 - a^3 c^3 + a^2 b c^3 + a b^2 c^3 - 2 b^3 c^3 - a^2 c^4 + a c^5 + b c^5 : :

X(24997) lies on these lines: {1, 2}, {11, 25135}, {81, 24563}, {394, 24545}, {497, 22370}, {515, 4203}, {518, 20487}, {908, 20545}, {1423, 20557}, {1512, 19540}, {3925, 25144}, {3989, 25245}, {4972, 25010}, {4981, 25024}, {23541, 24991}

### X(24998) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    2 a^4 b - 4 a^2 b^3 + 2 b^5 + 2 a^4 c - 2 a^3 b c + a^2 b^2 c + 4 a b^3 c - b^4 c + a^2 b c^2 - 4 a b^2 c^2 - b^3 c^2 - 4 a^2 c^3 + 4 a b c^3 - b^2 c^3 - b c^4 + 2 c^5 : :

X(24998) lies on these lines: {2, 44}, {693, 17420}, {3262, 5224}, {4357, 24982}, {4389, 25005}, {4981, 24991}, {17272, 24540}

### X(24999) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^4 b - 2 a^2 b^3 + b^5 + a^4 c - 4 a^3 b c - a^2 b^2 c + 2 a b^3 c - 2 b^4 c - a^2 b c^2 - 8 a b^2 c^2 + b^3 c^2 - 2 a^2 c^3 + 2 a b c^3 + b^2 c^3 - 2 b c^4 + c^5 : :

X(24999) lies on these lines: {2, 45}, {7, 2975}, {1266, 24987}, {2292, 3663}, {4357, 24982}, {10436, 24539}, {10915, 20895}, {17183, 19531}, {17250, 25005}, {17274, 19860}, {17321, 24540}

### X(25000) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^4 b - 2 a^2 b^3 + b^5 + a^4 c - 2 a^3 b c + a^2 b^2 c + a^2 b c^2 - b^3 c^2 - 2 a^2 c^3 - b^2 c^3 + c^5 : :

X(25000) lies on these lines: {2, 6}, {10, 774}, {75, 23978}, {198, 21270}, {307, 20262}, {311, 18740}, {348, 5942}, {379, 5816}, {442, 5796}, {572, 24581}, {573, 857}, {594, 25243}, {1958, 24582}, {2183, 20305}, {3207, 20074}, {4271, 8287}, {4357, 20905}, {5729, 9780}, {5753, 8728}, {17239, 25067}, {17248, 24554}, {17863, 24005}, {24547, 25023}, {24984, 25002}

### X(25001) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    b c (2 a^3 - 3 a^2 b + b^3 - 3 a^2 c - b^2 c - b c^2 + c^3) : :

X(25001) lies on these lines: {2, 37}, {8, 5728}, {9, 1441}, {10, 774}, {45, 17895}, {85, 144}, {274, 24557}, {318, 9780}, {322, 391}, {338, 1213}, {377, 9962}, {379, 1766}, {672, 4032}, {1713, 5278}, {2183, 21231}, {2262, 21271}, {2293, 25375}, {3262, 17277}, {3294, 17866}, {3553, 5736}, {3710, 4968}, {3717, 3963}, {3729, 20880}, {3731, 17861}, {3886, 19860}, {4384, 20895}, {4858, 6666}, {5819, 20061}, {6605, 10025}, {11997, 13576}, {16601, 25255}, {16676, 23521}, {16732, 16814}, {17220, 21871}, {17355, 18698}, {20236, 25101}, {20881, 24199}, {21371, 24633}, {24209, 25072}, {24982, 25004}, {24986, 25007}, {25082, 25252}

### X(25002) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    b c (a^4 b - 2 a^3 b^2 + a^2 b^3 + a^4 c - 2 a^2 b^2 c + b^4 c - 2 a^3 c^2 - 2 a^2 b c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3 + b c^4) : :

X(25002) lies on these lines: {2, 39}, {349, 1212}, {3177, 6063}, {3717, 3963}, {5179, 17866}, {17760, 21404}, {20905, 21418}, {24984, 25000}, {24992, 25018}

### X(25003) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    (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(25003) lies on these lines: {2, 6}, {321, 24005}, {442, 5811}, {3948, 17862}, {3995, 21049}, {4972, 21912}, {5051, 24982}, {23541, 24996}, {25013, 25019}

### X(25004) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    (b + c) (a^4 - 2 a^2 b^2 + b^4 + 7 a^2 b c + 2 a b^2 c - b^3 c - 2 a^2 c^2 + 2 a b c^2 - b c^3 + c^4) : :

X(25004) lies on these lines: {2, 6}, {442, 5817}, {3948, 20905}, {5051, 8582}, {16589, 24554}, {24982, 25001}, {24993, 25023}, {25010, 25011}

### X(25005) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c - a^2 b c + 2 a b^2 c - a^2 c^2 + 2 a b c^2 - 2 b^2 c^2 - a c^3 + c^4 : :

X(25005) lies on these lines: {1, 2}, {5, 1537}, {11, 8256}, {20, 1512}, {40, 5046}, {46, 5080}, {56, 5176}, {57, 20060}, {65, 5123}, {75, 24986}, {80, 17100}, {100, 1837}, {104, 355}, {140, 3897}, {149, 9581}, {165, 15680}, {319, 24540}, {322, 5740}, {346, 24005}, {377, 2096}, {392, 10284}, {405, 11849}, {443, 5770}, {474, 5790}, {496, 1145}, {515, 4188}, {517, 4193}, {908, 4848}, {944, 4881}, {946, 5154}, {952, 13747}, {962, 5187}, {984, 25010}, {993, 5445}, {1158, 2475}, {1311, 8685}, {1320, 11373}, {1329, 3869}, {1376, 5086}, {1385, 17566}, {1788, 3218}, {2476, 3753}, {2478, 5657}, {2550, 15297}, {2551, 3219}, {2975, 24914}, {3035, 10950}, {3036, 6691}, {3208, 21013}, {3306, 9578}, {3339, 17483}, {3501, 21044}, {3579, 11114}, {3586, 20066}, {3614, 10129}, {3681, 21031}, {3754, 7951}, {3814, 5903}, {3816, 3890}, {3820, 3876}, {3825, 5697}, {3838, 3922}, {3841, 17057}, {3868, 17757}, {3871, 5722}, {3873, 12607}, {3877, 4187}, {4147, 21105}, {4189, 6684}, {4190, 12616}, {4197, 5885}, {4373, 24213}, {4389, 24998}, {4429, 24433}, {4661, 24391}, {4855, 5727}, {5141, 10175}, {5175, 10395}, {5177, 5811}, {5178, 8668}, {5221, 11236}, {5224, 24993}, {5252, 5253}, {5422, 5711}, {5435, 20076}, {5603, 6931}, {5836, 11680}, {5881, 10265}, {5884, 12665}, {6681, 21842}, {6702, 7741}, {6945, 12672}, {7354, 9352}, {7704, 23513}, {9335, 23675}, {9711, 21677}, {9803, 17857}, {9947, 17616}, {10107, 17605}, {10164, 17548}, {10786, 18444}, {10895, 20292}, {11112, 18357}, {12531, 20118}, {12666, 18242}, {12702, 17556}, {13587, 18481}, {14011, 17174}, {15079, 24387}, {15679, 22798}, {15803, 20067}, {16371, 18525}, {17072, 21132}, {17250, 24999}, {17533, 22791}, {17579, 18480}, {21935, 24440}

### X(25006) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    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(25006) lies on these lines: {1, 2}, {5, 3697}, {9, 3434}, {11, 3740}, {12, 4662}, {38, 1738}, {40, 10431}, {63, 2550}, {65, 9710}, {75, 1233}, {92, 1861}, {100, 5745}, {142, 3873}, {210, 908}, {226, 3681}, {319, 14828}, {321, 3717}, {329, 5686}, {354, 3826}, {355, 7580}, {497, 3305}, {515, 7411}, {516, 3219}, {517, 3690}, {518, 3925}, {527, 20292}, {528, 3683}, {594, 3693}, {756, 24210}, {946, 3876}, {950, 5178}, {952, 13151}, {958, 20835}, {966, 3692}, {984, 3914}, {1001, 4863}, {1004, 5794}, {1005, 5086}, {1109, 4712}, {1211, 8286}, {1329, 3983}, {1441, 4967}, {1512, 5790}, {1621, 5853}, {1654, 10025}, {1697, 10395}, {1734, 23811}, {1836, 5220}, {2346, 5284}, {2475, 12527}, {2476, 21075}, {2911, 17275}, {3058, 15254}, {3059, 8255}, {3242, 24789}, {3306, 24477}, {3416, 4042}, {3419, 9708}, {3452, 11680}, {3485, 3984}, {3555, 8728}, {3587, 5657}, {3663, 7226}, {3678, 12047}, {3686, 4071}, {3689, 6690}, {3691, 20556}, {3696, 3703}, {3706, 3932}, {3715, 24703}, {3782, 21949}, {3786, 17167}, {3814, 3956}, {3820, 3921}, {3841, 13407}, {3883, 5014}, {3886, 17776}, {3890, 21627}, {3951, 4295}, {3967, 4126}, {4001, 4645}, {4082, 4671}, {4090, 25385}, {4197, 21620}, {4312, 20078}, {4314, 16865}, {4357, 4972}, {4359, 24235}, {4392, 24177}, {4416, 6327}, {4430, 5542}, {4431, 17163}, {4512, 20075}, {4514, 17277}, {4679, 11235}, {4684, 18139}, {4733, 6741}, {4849, 5718}, {4866, 6871}, {4883, 17245}, {4899, 17165}, {5044, 7743}, {5045, 17529}, {5046, 18250}, {5082, 5250}, {5173, 21617}, {5174, 14004}, {5177, 5815}, {5223, 5905}, {5234, 6872}, {5258, 17647}, {5263, 5294}, {5269, 24597}, {5273, 17784}, {5302, 6284}, {5316, 24386}, {5534, 6889}, {5687, 5791}, {5690, 8727}, {5783, 24545}, {5836, 21677}, {5837, 14923}, {5850, 17483}, {5904, 12609}, {6692, 9342}, {6769, 6837}, {6907, 18908}, {7160, 12620}, {7174, 19785}, {7308, 24392}, {7674, 18230}, {7991, 12617}, {9711, 17606}, {10584, 20196}, {10883, 11362}, {10914, 14022}, {11108, 18530}, {12699, 15650}, {15587, 17616}, {17072, 21183}, {17140, 24199}, {17251, 24352}, {17278, 17597}, {17286, 24388}, {17353, 24552}, {20070, 21628}, {21283, 25101}

### X(25007) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^4 b - 2 a^2 b^3 + b^5 + a^4 c + 2 a^3 b c - 2 a^2 b^2 c + 2 a b^3 c + b^4 c - 2 a^2 b c^2 - 2 b^3 c^2 - 2 a^2 c^3 + 2 a b c^3 - 2 b^2 c^3 + b c^4 + c^5 : :

X(25007) lies on these lines: {1, 2}, {297, 1861}, {344, 24005}, {355, 21477}, {515, 21495}, {908, 16609}, {1213, 25099}, {1512, 6996}, {1738, 25010}, {4297, 21537}, {5249, 16603}, {5750, 15988}, {5790, 21526}, {5847, 24563}, {6684, 21511}, {10164, 21508}, {10521, 17483}, {16431, 18481}, {18525, 21539}, {24986, 25001}

### X(25008) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    (b - c) (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c - a^2 b^2 c + a^3 c^2 - a^2 b c^2 + 2 b^3 c^2 - a^2 c^3 + 2 b^2 c^3 - a c^4) : :

X(25008) lies on these lines: {2, 661}, {656, 18155}, {693, 17420}, {4025, 4391}, {4077, 4357}, {4131, 17215}, {4374, 25022}, {14208, 23788}

### X(25009) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    b c (b - c) (-3 a^3 + 3 a^2 b - a b^2 + b^3 + 3 a^2 c - 2 a b c - b^2 c - a c^2 - b c^2 + c^3) : :

X(25009) lies on these lines: {2, 650}, {443, 8760}, {656, 4811}, {850, 6587}, {1577, 4765}, {2550, 11934}, {3239, 17896}, {4025, 4391}, {4077, 4521}, {4380, 23819}, {4397, 7649}, {4462, 7178}, {4468, 24002}, {8142, 17576}, {17862, 21438}

### X(25010) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^5 b - a^3 b^3 - a^2 b^4 + b^6 + a^5 c + a^3 b^2 c + 2 a b^4 c + a^3 b c^2 - 4 a^2 b^2 c^2 - b^4 c^2 - a^3 c^3 - a^2 c^4 + 2 a b c^4 - b^2 c^4 + c^6 : :

X(25010) lies on these lines: {2, 31}, {10, 20633}, {984, 25005}, {1423, 11681}, {1738, 25007}, {1756, 3814}, {4193, 6210}, {4357, 24982}, {4972, 24997}, {17306, 19860}, {25004, 25011}

### X(25011) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^3 b-a^2 b^2-a b^3+b^4+a^3 c+4 a^2 b c+7 a b^2 c-a^2 c^2+7 a b c^2-2 b^2 c^2-a c^3+c^4 : :

X(25011) lies on these lines: {1, 2}, {140, 1512}, {142, 11681}, {354, 9711}, {355, 16862}, {377, 18492}, {442, 22798}, {474, 18481}, {496, 4002}, {515, 17531}, {517, 17575}, {908, 3649}, {1266, 24993}, {1329, 5249}, {1738, 22174}, {1788, 3305}, {2551, 3306}, {2975, 6692}, {3218, 18250}, {3359, 6898}, {3436, 5437}, {3648, 12572}, {3698, 3816}, {3742, 21031}, {3753, 17527}, {3813, 4731}, {3814, 6701}, {3820, 5439}, {3826, 14100}, {3833, 13407}, {3848, 15888}, {3869, 5316}, {3911, 5260}, {4187, 9955}, {4197, 10175}, {4297, 17572}, {5047, 6684}, {5068, 21628}, {5080, 12436}, {5084, 6361}, {5221, 17781}, {5250, 17559}, {5251, 5442}, {5253, 5795}, {5440, 15174}, {5586, 5905}, {5750, 7110}, {6666, 8543}, {6919, 11024}, {8165, 9776}, {8728, 17612}, {9842, 9961}, {9956, 17529}, {10164, 16865}, {16408, 18525}, {25004, 25010}

### X(25012) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^8 b - 2 a^4 b^5 + b^9 + a^8 c + a^6 b^2 c - 2 a^5 b^3 c - a^4 b^4 c - a^2 b^6 c + 2 a b^7 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 - 2 a^5 b c^3 + a^4 b^2 c^3 + 8 a^3 b^3 c^3 + 2 a^2 b^4 c^3 - 2 a b^5 c^3 + b^6 c^3 - a^4 b c^4 + 2 a^2 b^3 c^4 - b^5 c^4 - 2 a^4 c^5 - a^2 b^2 c^5 - 2 a b^3 c^5 - b^4 c^5 - a^2 b c^6 + b^3 c^6 + 2 a b c^7 - b^2 c^7 + c^9 : :

X(25012) lies on these lines: {2, 3}, {3914, 17862}, {4972, 24993}, {24995, 24996}

### X(25013) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    (b + c) (a^8 + 2 a^7 b - 2 a^5 b^3 - 2 a^4 b^4 - 2 a^3 b^5 + 2 a b^7 + b^8 + 2 a^7 c - a^6 b c - 2 a^5 b^2 c + 3 a^4 b^3 c - 2 a^3 b^4 c - 3 a^2 b^5 c + 2 a b^6 c + b^7 c - 2 a^5 b c^2 + 2 a^4 b^2 c^2 + 8 a^3 b^3 c^2 - 6 a b^5 c^2 - 2 b^6 c^2 - 2 a^5 c^3 + 3 a^4 b c^3 + 8 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 - 2 a^3 b c^4 + 2 a b^3 c^4 + 2 b^4 c^4 - 2 a^3 c^5 - 3 a^2 b c^5 - 6 a b^2 c^5 - b^3 c^5 + 2 a b c^6 - 2 b^2 c^6 + 2 a c^7 + b c^7 + c^8) : :

X(25013) lies on these lines: {2, 3}, {25003, 25019}

### X(25014) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    (b + c) (a^9 + a^8 b - 2 a^5 b^4 - 2 a^4 b^5 + a b^8 + b^9 + a^8 c + 3 a^7 b c - a^6 b^2 c - 3 a^5 b^3 c - a^4 b^4 c - 3 a^3 b^5 c + a^2 b^6 c + 3 a b^7 c - a^6 b c^2 + 2 a^5 b^2 c^2 + 5 a^4 b^3 c^2 - 3 a^2 b^5 c^2 - 2 a b^6 c^2 - b^7 c^2 - 3 a^5 b c^3 + 5 a^4 b^2 c^3 + 14 a^3 b^3 c^3 + 2 a^2 b^4 c^3 - 3 a b^5 c^3 + b^6 c^3 - 2 a^5 c^4 - a^4 b c^4 + 2 a^2 b^3 c^4 + 2 a b^4 c^4 - b^5 c^4 - 2 a^4 c^5 - 3 a^3 b c^5 - 3 a^2 b^2 c^5 - 3 a b^3 c^5 - b^4 c^5 + a^2 b c^6 - 2 a b^2 c^6 + b^3 c^6 + 3 a b c^7 - b^2 c^7 + a c^8 + c^9) : :

X(25014) lies on this line: {2, 3}

### X(25015) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    (b + c) (a^6 - a^4 b^2 - a^2 b^4 + b^6 + a^4 b c + 2 a^3 b^2 c - 2 a b^4 c - b^5 c - a^4 c^2 + 2 a^3 b c^2 + 6 a^2 b^2 c^2 + 2 a b^3 c^2 - b^4 c^2 + 2 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - 2 a b c^4 - b^2 c^4 - b c^5 + c^6) : :

X(25015) lies on these lines: {2, 3}, {10, 307}, {78, 3936}, {387, 5738}, {936, 3454}, {938, 1834}, {948, 1211}, {1074, 1210}, {1213, 6554}, {1257, 16086}, {1330, 2287}, {1869, 18589}, {4359, 6734}, {4429, 11024}, {4658, 15936}, {5249, 23542}, {5703, 17056}, {8680, 21671}, {12609, 23518}, {17052, 18634}, {17863, 23537}, {18673, 25361}

### X(25016) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    3 a^6 b - 3 a^4 b^3 - 3 a^2 b^5 + 3 b^7 + 3 a^6 c - 2 a^5 b c - 2 a^3 b^3 c - 3 a^2 b^4 c + 4 a b^5 c + 6 a^2 b^3 c^2 - 6 b^5 c^2 - 3 a^4 c^3 - 2 a^3 b c^3 + 6 a^2 b^2 c^3 - 8 a b^3 c^3 + 3 b^4 c^3 - 3 a^2 b c^4 + 3 b^3 c^4 - 3 a^2 c^5 + 4 a b c^5 - 6 b^2 c^5 + 3 c^7 : :

X(25016) lies on these lines: {2, 3}, {3679, 23541}

### X(25017) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^6 b - a^4 b^3 - a^2 b^5 + b^7 + a^6 c - 2 a^5 b c + 2 a^3 b^3 c - a^2 b^4 c + 4 a^3 b^2 c^2 + 6 a^2 b^3 c^2 - 2 b^5 c^2 - a^4 c^3 + 2 a^3 b c^3 + 6 a^2 b^2 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - 2 b^2 c^5 + c^7 : :

X(25017) lies on these lines: {2, 3}, {8, 13567}, {10, 774}, {1446, 4357}, {3695, 25243}, {5296, 6554}, {19860, 23541}, {20235, 24547}, {20905, 23537}

### X(25018) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^6 b - a^4 b^3 - a^2 b^5 + b^7 + a^6 c + 2 a^3 b^3 c - a^2 b^4 c + 2 a b^5 c + 2 a^2 b^3 c^2 - 2 b^5 c^2 - a^4 c^3 + 2 a^3 b c^3 + 2 a^2 b^2 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 + 2 a b c^5 - 2 b^2 c^5 + c^7 : :

X(25018) lies on these lines: {2, 3}, {24986, 25021}, {24992, 25002}

### X(25019) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^4 b - 2 a^2 b^3 + b^5 + a^4 c - 4 a^3 b c + 4 a^2 b^2 c - b^4 c + 4 a^2 b c^2 - 2 a^2 c^3 - b c^4 + c^5 : :

X(25019) lies on these lines: {2, 7}, {10, 774}, {141, 25067}, {198, 18650}, {306, 3965}, {857, 10445}, {1125, 1496}, {1441, 20262}, {1730, 21062}, {1861, 4429}, {2183, 18589}, {2270, 4329}, {2293, 21914}, {2321, 25243}, {2654, 19860}, {3663, 20905}, {4001, 17811}, {5051, 8582}, {5256, 18928}, {5942, 9312}, {16887, 24557}, {17220, 21068}, {25003, 25013}

### X(25020) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    (b - c) (a^4 - 4 a^3 b + 4 a b^3 - b^4 - 4 a^3 c + 7 a^2 b c - 2 a b^2 c - b^3 c - 2 a b c^2 + 4 a c^3 - b c^3 - c^4) : :

X(25020) lies on these lines: {2, 900}, {522, 21052}, {1769, 23541}, {2815, 3877}, {3766, 24547}

### X(25021) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    b c (a^4 b + a^2 b^3 + a^4 c - 2 a^2 b^2 c + b^4 c - 2 a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3 + b c^4) : :

X(25021) lies on these lines: {2, 39}, {1738, 21935}, {24986, 25018}

### X(25022) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    b c (b - c) (-a^3 - a^2 b - a^2 c - a b c + b^2 c + b c^2) (-a^3 - a^2 b + a b^2 + b^3 - a^2 c + 2 a b c - b^2 c + a c^2 - b c^2 + c^3) : :

X(25022) lies on these lines: {2, 647}, {4374, 25008}, {4467, 24993}, {14837, 17896}

### X(25023) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a^4 b - 2 a^2 b^3 + b^5 + a^4 c - 2 a^3 b c + 4 a^2 b^2 c + 2 a b^3 c - b^4 c + 4 a^2 b c^2 - 2 a^2 c^3 + 2 a b c^3 - b c^4 + c^5 : :

X(25023) lies on these lines: {2, 7}, {75, 24005}, {1738, 21935}, {3755, 5554}, {3821, 8582}, {24547, 25000}, {24986, 25001}, {24993, 25004}

### X(25024) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 13

Barycentrics    a (a^3 b^2-a^2 b^3-a b^4+b^5-3 a^2 b^2 c-b^4 c+a^3 c^2-3 a^2 b c^2-2 b^3 c^2-a^2 c^3-2 b^2 c^3-a c^4-b c^4+c^5) : :

X(25024) lies on these lines: {2, 38}, {9, 1405}, {10, 1733}, {21, 6211}, {201, 1220}, {256, 4642}, {518, 25099}, {740, 25245}, {1240, 3718}, {1756, 3754}, {1757, 15988}, {2292, 17261}, {3508, 21808}, {3617, 4073}, {3717, 3963}, {4026, 24433}, {4357, 24982}, {4981, 24997}, {6187, 7295}, {7081, 11031}, {7174, 15839}, {16830, 24563}, {17353, 24541}

Collineation mappings involving Gemini triangle 14: X(25025) - X(25041)

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 13, as in centers X(25025)-X(25041). Then

m(X) = : : (a^2 b + a b^2 + 2 b^3 - a^2 c + 2 a b c - 3 b^2 c + a c^2 - 3 b c^2 + 2 c^3) x + (4 a^2 b + 2 a b^2 - 2 b^3 - 10 a b c + 2 b^2 c + 4 b c^2) y + (4 a^2 c - 10 a b c + 4 b^2 c + 2 a c^2 + 2 b c^2 - 2 c^3) z : :

and m(X) is on the Euler line if and only if X is on the Euler line.

(Clark Kimberling, October 12, 2018)

### X(25025) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 14

Barycentrics    a^3 b - 5 a^2 b^2 - 4 a b^3 + 2 b^4 + a^3 c - 2 a^2 b c + 13 a b^2 c - 2 b^3 c - 5 a^2 c^2 + 13 a b c^2 - 8 b^2 c^2 - 4 a c^3 - 2 b c^3 + 2 c^4 : :

X(25025) lies on these lines: {1, 2}, {88, 10713}, {121, 4674}, {1739, 24399}, {17237, 25029}, {25036, 25037}

### X(25026) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 14

Barycentrics    a^6 b - a^5 b^2 + a^4 b^3 + 3 a^3 b^4 - 4 a^2 b^5 - 2 a b^6 + 2 b^7 + a^6 c - 2 a^5 b c + 2 a^4 b^2 c - 8 a^3 b^3 c - a^2 b^4 c + 10 a b^5 c - 2 b^6 c - a^5 c^2 + 2 a^4 b c^2 + 2 a^3 b^2 c^2 + 7 a^2 b^3 c^2 + 2 a b^4 c^2 - 6 b^5 c^2 + a^4 c^3 - 8 a^3 b c^3 + 7 a^2 b^2 c^3 - 20 a b^3 c^3 + 6 b^4 c^3 + 3 a^3 c^4 - a^2 b c^4 + 2 a b^2 c^4 + 6 b^3 c^4 - 4 a^2 c^5 + 10 a b c^5 - 6 b^2 c^5 - 2 a c^6 - 2 b c^6 + 2 c^7 : :

X(25026) lies on these lines: {2, 3}, {1145, 25030}, {1168, 24864}, {1319, 24871}

### X(25027) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 14

Barycentrics    5 a^6 b + a^5 b^2 - 4 a^4 b^3 - 5 a^2 b^5 - a b^6 + 4 b^7 + 5 a^6 c - 22 a^5 b c + a^4 b^2 c + 20 a^3 b^3 c - 5 a^2 b^4 c + 2 a b^5 c - b^6 c + a^5 c^2 + a^4 b c^2 - 8 a^3 b^2 c^2 + 2 a^2 b^3 c^2 + a b^4 c^2 - 9 b^5 c^2 - 4 a^4 c^3 + 20 a^3 b c^3 + 2 a^2 b^2 c^3 - 4 a b^3 c^3 + 6 b^4 c^3 - 5 a^2 b c^4 + a b^2 c^4 + 6 b^3 c^4 - 5 a^2 c^5 + 2 a b c^5 - 9 b^2 c^5 - a c^6 - b c^6 + 4 c^7 : :

X(25027) lies on this line: {2, 3}

### X(25028) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 14

Barycentrics    4 a^6 b + 2 a^5 b^2 - 5 a^4 b^3 - 3 a^3 b^4 - a^2 b^5 + a b^6 + 2 b^7 + 4 a^6 c - 20 a^5 b c - a^4 b^2 c + 28 a^3 b^3 c - 4 a^2 b^4 c - 8 a b^5 c + b^6 c + 2 a^5 c^2 - a^4 b c^2 - 10 a^3 b^2 c^2 - 5 a^2 b^3 c^2 - a b^4 c^2 - 3 b^5 c^2 - 5 a^4 c^3 + 28 a^3 b c^3 - 5 a^2 b^2 c^3 + 16 a b^3 c^3 - 3 a^3 c^4 - 4 a^2 b c^4 - a b^2 c^4 - a^2 c^5 - 8 a b c^5 - 3 b^2 c^5 + a c^6 + b c^6 + 2 c^7 : :

X(25028) lies on this line: {2, 3}

### X(25029) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 14

Barycentrics    a^4 b - a^3 b^2 - 6 a^2 b^3 - 2 a b^4 + 2 b^5 + a^4 c - 2 a^3 b c + 3 a^2 b^2 c + 10 a b^3 c - 2 b^4 c - a^3 c^2 + 3 a^2 b c^2 - 4 b^3 c^2 - 6 a^2 c^3 + 10 a b c^3 - 4 b^2 c^3 - 2 a c^4 - 2 b c^4 + 2 c^5 : :

X(25029) lies on these lines: {2, 6}, {10, 24873}, {17237, 25025}

### X(25030) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 14

Barycentrics    5 a^3 b - 4 a^2 b^2 - 5 a b^3 + 4 b^4 + 5 a^3 c - 16 a^2 b c + 14 a b^2 c - b^3 c - 4 a^2 c^2 + 14 a b c^2 - 10 b^2 c^2 - 5 a c^3 - b c^3 + 4 c^4 : :

X(25030) lies on these lines: {1, 2}, {1145, 25026}, {17237, 24873}

### X(25031) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 14

Barycentrics    4 a^3 b+a^2 b^2-a b^3+2 b^4+4 a^3 c-14 a^2 b c+a b^2 c+b^3 c+a^2 c^2+a b c^2-2 b^2 c^2-a c^3+b c^3+2 c^4 : :

X(25031) lies on these lines: {1, 2}, {514, 4364}, {2802, 25351}, {4013, 24222}, {4389, 6549}, {6667, 11731}, {21630, 25040}, {24873, 25035}

### X(25032) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 14

Barycentrics    a^8 b - a^7 b^2 + 2 a^6 b^3 + 2 a^5 b^4 - 3 a^4 b^5 + a^3 b^6 - 2 a^2 b^7 - 2 a b^8 + 2 b^9 + a^8 c - 2 a^7 b c + 3 a^6 b^2 c - 10 a^5 b^3 c + a^4 b^4 c + 2 a^3 b^5 c - 3 a^2 b^6 c + 10 a b^7 c - 2 b^8 c - a^7 c^2 + 3 a^6 b c^2 + 4 a^4 b^3 c^2 + a^3 b^4 c^2 - 3 a^2 b^5 c^2 - 4 b^7 c^2 + 2 a^6 c^3 - 10 a^5 b c^3 + 4 a^4 b^2 c^3 + 6 a^2 b^4 c^3 - 10 a b^5 c^3 + 4 b^6 c^3 + 2 a^5 c^4 + a^4 b c^4 + a^3 b^2 c^4 + 6 a^2 b^3 c^4 + 4 a b^4 c^4 - 3 a^4 c^5 + 2 a^3 b c^5 - 3 a^2 b^2 c^5 - 10 a b^3 c^5 + a^3 c^6 - 3 a^2 b c^6 + 4 b^3 c^6 - 2 a^2 c^7 + 10 a b c^7 - 4 b^2 c^7 - 2 a c^8 - 2 b c^8 + 2 c^9 : :

X(25032) lies on this line: {2, 3}

### X(25033) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 14

Barycentrics    a^8 b - a^7 b^2 + 2 a^6 b^3 + 2 a^5 b^4 - 3 a^4 b^5 + a^3 b^6 - 2 a^2 b^7 - 2 a b^8 + 2 b^9 + a^8 c - 2 a^7 b c + 3 a^6 b^2 c - 10 a^5 b^3 c + a^4 b^4 c + 2 a^3 b^5 c - 3 a^2 b^6 c + 10 a b^7 c - 2 b^8 c - a^7 c^2 + 3 a^6 b c^2 + a^4 b^3 c^2 - 2 a^3 b^4 c^2 - 3 a^2 b^5 c^2 - 4 b^7 c^2 + 2 a^6 c^3 - 10 a^5 b c^3 + a^4 b^2 c^3 + 18 a^3 b^3 c^3 + 3 a^2 b^4 c^3 - 10 a b^5 c^3 + 4 b^6 c^3 + 2 a^5 c^4 + a^4 b c^4 - 2 a^3 b^2 c^4 + 3 a^2 b^3 c^4 + 4 a b^4 c^4 - 3 a^4 c^5 + 2 a^3 b c^5 - 3 a^2 b^2 c^5 - 10 a b^3 c^5 + a^3 c^6 - 3 a^2 b c^6 + 4 b^3 c^6 - 2 a^2 c^7 + 10 a b c^7 - 4 b^2 c^7 - 2 a c^8 - 2 b c^8 + 2 c^9 : :

X(25033) lies on this line: {2, 3}

### X(25034) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 14

Barycentrics    3 a^3 b^2 + 3 a^2 b^3 - 2 a^3 b c - 3 a^2 b^2 c + 3 a b^3 c - 2 b^4 c + 3 a^3 c^2 - 3 a^2 b c^2 - 18 a b^2 c^2 + 6 b^3 c^2 + 3 a^2 c^3 + 3 a b c^3 + 6 b^2 c^3 - 2 b c^4 : :

X(25034) lies on these lines: {2, 37}, {121, 1086}, {17237, 25025}, {17374, 24868}

### X(25035) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 14

Barycentrics    a^4 b + 5 a^3 b^2 - 2 a b^4 + 2 b^5 + a^4 c - 6 a^3 b c - 3 a^2 b^2 c + 16 a b^3 c - 6 b^4 c + 5 a^3 c^2 - 3 a^2 b c^2 - 36 a b^2 c^2 + 8 b^3 c^2 + 16 a b c^3 + 8 b^2 c^3 - 2 a c^4 - 6 b c^4 + 2 c^5 : :

X(25035) lies on these lines: {2, 45}, {17119, 24864}, {17237, 25025}, {24873, 25031}

### X(25036) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 14

Barycentrics    b c (8 a^3 - 9 a^2 b + 3 a b^2 + 2 b^3 - 9 a^2 c + 6 a b c - 3 b^2 c + 3 a c^2 - 3 b c^2 + 2 c^3) : :

X(25036) lies on these lines: {2, 37}, {190, 20568}, {4033, 4738}, {20006, 24517}, {25025, 25037}

### X(25037) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 14

Barycentrics    (b + c) (a^4 - 4 a^3 b - 9 a^2 b^2 - 2 a b^3 + 2 b^4 - 4 a^3 c + 24 a^2 b c + 6 a b^2 c - 4 b^3 c - 9 a^2 c^2 + 6 a b c^2 - 3 b^2 c^2 - 2 a c^3 - 4 b c^3 + 2 c^4) : :

X(25037) lies on these lines: {2, 6}, {16589, 24874}, {25025, 25036}

### X(25038) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 14

Barycentrics    b c (b - c) (-16 a^3 + 15 a^2 b - 3 a b^2 + 2 b^3 + 15 a^2 c - 6 a b c - 3 b^2 c - 3 a c^2 - 3 b c^2 + 2 c^3) : :

X(25038) lies on these lines: {2, 650}, {3762, 24589}

### X(25039) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 14

Barycentrics    7 a^6 b - a^5 b^2 - 2 a^4 b^3 + 6 a^3 b^4 - 13 a^2 b^5 - 5 a b^6 + 8 b^7 + 7 a^6 c - 26 a^5 b c + 5 a^4 b^2 c + 4 a^3 b^3 c - 7 a^2 b^4 c + 22 a b^5 c - 5 b^6 c - a^5 c^2 + 5 a^4 b c^2 - 4 a^3 b^2 c^2 + 16 a^2 b^3 c^2 + 5 a b^4 c^2 - 21 b^5 c^2 - 2 a^4 c^3 + 4 a^3 b c^3 + 16 a^2 b^2 c^3 - 44 a b^3 c^3 + 18 b^4 c^3 + 6 a^3 c^4 - 7 a^2 b c^4 + 5 a b^2 c^4 + 18 b^3 c^4 - 13 a^2 c^5 + 22 a b c^5 - 21 b^2 c^5 - 5 a c^6 - 5 b c^6 + 8 c^7 : :

X(25039) lies on this line: {2, 3}

### X(25040) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 14

Barycentrics    (b + c) (a^6 - a^5 b + a^4 b^2 + 3 a^3 b^3 - 4 a^2 b^4 - 2 a b^5 + 2 b^6 - a^5 c + 4 a^4 b c - 5 a^3 b^2 c + 6 a^2 b^3 c + 12 a b^4 c - 4 b^5 c + a^4 c^2 - 5 a^3 b c^2 - 11 a^2 b^2 c^2 - 7 a b^3 c^2 - 2 b^4 c^2 + 3 a^3 c^3 + 6 a^2 b c^3 - 7 a b^2 c^3 + 8 b^3 c^3 - 4 a^2 c^4 + 12 a b c^4 - 2 b^2 c^4 - 2 a c^5 - 4 b c^5 + 2 c^6) : :

X(25040) lies on these lines: {2, 3}, {17056, 24869}, {21630, 25031}

### X(25041) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 14

Barycentrics    a^6 b - a^5 b^2 - 2 a^4 b^3 - 4 a^2 b^5 - 2 a b^6 + 2 b^7 + a^6 c - 2 a^5 b c - a^4 b^2 c + 10 a^3 b^3 c - 4 a^2 b^4 c + 10 a b^5 c - 2 b^6 c - a^5 c^2 - a^4 b c^2 - 4 a^3 b^2 c^2 + a^2 b^3 c^2 - a b^4 c^2 - 6 b^5 c^2 - 2 a^4 c^3 + 10 a^3 b c^3 + a^2 b^2 c^3 - 2 a b^3 c^3 + 3 b^4 c^3 - 4 a^2 b c^4 - a b^2 c^4 + 3 b^3 c^4 - 4 a^2 c^5 + 10 a b c^5 - 6 b^2 c^5 - 2 a c^6 - 2 b c^6 + 2 c^7 : :

X(25041) lies on this line: {2, 3}

### X(25042) =  ISOGONAL CONJUGATE OF X(22335)

Barycentrics    a^2 (a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4) (a^6+a^4 b^2-5 a^2 b^4+3 b^6+a^4 c^2-a^2 b^2 c^2-3 b^4 c^2-5 a^2 c^4-3 b^2 c^4+3 c^6) : :

See Peter Moses, Hyacinthos 28457.

X(25042) lies on these lines: {3,54}, {140,1141}, {252,549}, {546,7604}, {4994,18559}, {12026,19268}, {13367,20574}

X(25042) = isogonal conjugate of X(22335)
X(25042) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 54, 1157), (3, 16762, 12307), (10610, 18016, 3)

### X(25043) =  ANTICOMPLEMENT OF X(15345)

Barycentrics    b^2 c^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^4-a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) : :

See Peter Moses, Hyacinthos 28457.

X(25043) lies on the conic {{A,B,C,X(4),X(5)}}, the cubics K028, K115, K526, and on these lines: {2,3459}, {3,252}, {4,93}, {17,11082}, {18,11087}, {26,1601}, {30,15619}, {343,565}, {566,2165}, {631,18016}, {1263,21230}, {1487,1656}, {1658,21394}, {2937,11816}, {2962,18395}, {2980,17714}, {5449,6368}, {5965,20414}, {10095,17500}, {12044,23181}, {14072,19268}, {14247,14265}, {20424,22335}

X(25043) = anticomplement X(15345)
X(25043) = cevapoint of X(3519) and X(21975)
X(25043) = crosspoint of X(11140) and X(20572)
X(25043) = X(i)-cross conjugate of X(j) for these (i,j): {233, 324}, {1209, 5}, {6368, 930}
X(25043) = X(i)-isoconjugate of X(j) for these (i,j): {49, 2190}, {54, 2964}, {1994, 2148}, {2167, 2965}, {2169, 3518}
X(25043) = barycentric product X(i)*X(j) for these {i,j}: {5, 11140}, {93, 343}, {216, 20572}, {311, 2963}, {324, 3519}, {930, 18314}, {2962, 14213}
X(25043) = Kirikami-Euler image of X(5)
X(25043) = pedal antipodal perspector of X(54)
X(25043) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 1994}, {51, 2965}, {53, 3518}, {93, 275}, {216, 49}, {233, 1493}, {311, 7769}, {930, 18315}, {1487, 288}, {1953, 2964}, {2962, 2167}, {2963, 54}, {3519, 97}, {10412, 2413}, {11140, 95}, {12077, 1510}, {20572, 276}
X(25043) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (93, 11140, 3519), (252, 930, 3), (3519, 18370, 4)

### X(25044) =  X(2)X(252)∩X(3)X(54)

Barycentrics    a^4 (a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4-2 a^2 c^2-b^2 c^2+c^4) : :

See Peter Moses, Hyacinthos 28457.

X(25044) lies on the cubic K009 and these lines: {2,252}, {3,54}, {4,137}, {5,23338}, {24,3432}, {32,14586}, {49,15787}, {96,7578}, {140,24305}, {184,20574}, {186,15620}, {288,13472}, {1147,15958}, {2120,3567}, {3090,7604}, {3484,6241}, {3574,8154}, {8254,14143}, {8884,18559}, {10285,24385}, {11671,24144}, {11815,15454}

X(25044) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (54, 1157, 3), (1493, 6150, 15345), (6150, 15345, 3)
X(25044) = X(i)-Ceva conjugate of X(j) for these (i,j): {288, 14533}, {933, 1510}, {1166, 54}
X(25044) = isogonal conjugate of the anticomplement X(15345)
X(25044) = crosssum of X(3519) and X(21975)
X(25044) = X(i)-isoconjugate of X(j) for these (i,j): {5, 2962}, {252, 1087}, {564, 12044}, {930, 2618}, {1953, 11140}, {2963, 14213}
X(25044) = barycentric product X(i)*X(j) for these {i,j}: {49, 275}, {54, 1994}, {95, 2965}, {97, 3518}, {288, 1493}, {1510, 18315}, {2167, 2964}, {15721, 16306}
X(25044) = barycentric quotient X(i)/X(j) for these {i,j}: {49, 343}, {54, 11140}, {275, 20572}, {1510, 18314}, {1994, 311}, {2148, 2962}, {2964, 14213}, {2965, 5}, {3518, 324}, {8882, 93}, {14533, 3519}, {14586, 930}

Points associated with a family of ellipses: X(25045) - X(25050)

Let E be the circumellipse with center X(141), given by

a^2 (b^2 + c^2) y z + b^2 (c^2 + a^2) x z + c^2 (a^2 + b^2) x y = 0.

This ellipse is the isogonal conjugate of the anticomplement of the de Longchamps line, and it passes through X(i) for i = 67, 110, 660, 670, 694, 1634, 4553, 4576, 8050, 20021.

For arbitrary fixed nonzero constant k, let E(k) be the ellipse obtained from E by dilation from X(141) with ratio k; then E(k) is given by

(b^2 + c^2) ((a^2 + b^2) (a^2 + c^2) k^2 - b^2 c^2) x^2 + 2 (a^2 b^2 c^2 + (b^2 + c^2) (c^2 + a^2) (a^2 + b^2) k^2) y z + (cyclic) = 0.

If you have GeoGebra, you can view Incenter.

The major axis of every E(k) passes through X(i) for i = 6, 1344, 2574, 8105, 9173, 13414, and the minor axis, through X(i) for i = 6, 1345, 2575, 8106, 9174, 13415, 14899. The two axes are parallel to the Simson lines of X(1113) and X(1114).

These ellipses are associated with a problem in navigation posed by William Lionheart (see the link at X(6)), in connection with the fact that if a point P moves on E(k), the sum of squared distances from P to the sidelines BC, CA, AB stays constant., The value of the constant is

(a^2 b^2 c^2 + (b^2 + c^2) (c^2 + a^2) (a^2 + b^2) k^2) / (4 (a^2 + b^2 + c^2) R^2).

Suppose that P = p : q : r is a point on the circumcircle. Then the point

LM(P,k) = (b^2 + c^2) k ((2 a^2 + b^2 + c^2) p + (-a^2 - c^2) q + (-a^2 - b^2) r) - a^2 ((b^2 + c^2) p + (c^2 + a^2) q + (a^2 + b^2) r) : :

is on the ellipse E(k). The point LM(P,k) is here named the Lionheart-Moses k-image of P. Centers X(25045)-X(250540) are examples of such images; see also X(25314)-X(25336). (Peter Moses, October 10, 2018)

The elliipse E(1) is the X(82)-anticomplementary conjugate of the infinity line; specifically, if a point (b - c)(a + t) is on the infinity line, then its image on E(1) is the point

- a^2 (b^2 + c^2) / ((a - c) (b + t) - (a - b) (c + t))
+ b^2 (c^2 + a^2) / ((b - a) (c + t) - (b - c) (a + t))
+ c^2 (a^2 + b^2) / ((c - b) (a + t) - (c - a) (b + t)) : :

### X(25045) =  LIONHEART-MOSES 1-IMAGE OF X(74)

Barycentrics    3*a^10*b^2 - 6*a^8*b^4 + 6*a^4*b^8 - 3*a^2*b^10 + 3*a^10*c^2 + 4*a^8*b^2*c^2 - a^6*b^4*c^2 - 9*a^4*b^6*c^2 + 2*a^2*b^8*c^2 + b^10*c^2 - 6*a^8*c^4 - a^6*b^2*c^4 + 10*a^4*b^4*c^4 + a^2*b^6*c^4 - 4*b^8*c^4 - 9*a^4*b^2*c^6 + a^2*b^4*c^6 + 6*b^6*c^6 + 6*a^4*c^8 + 2*a^2*b^2*c^8 - 4*b^4*c^8 - 3*a^2*c^10 + b^2*c^10 : :

X(25045) lies on these lines: {4, 94}, {23, 14611}, {1302, 14919}, {2986, 10752}, {6194, 7493}, {9003, 11061}

X(25045) = {30, 21289}, {82, 30}, {251, 18668}, {1495, 21217}, {2173, 2896}, {4599, 3268}, {14206, 1369}

### X(25046) =  LIONHEART-MOSES 1-IMAGE OF X(98)

Barycentrics    3*a^8*b^2 - 3*a^6*b^4 + a^4*b^6 - a^2*b^8 + 3*a^8*c^2 - 2*a^6*b^2*c^2 - 2*a^4*b^4*c^2 + 2*a^2*b^6*c^2 - b^8*c^2 - 3*a^6*c^4 - 2*a^4*b^2*c^4 + 2*a^2*b^4*c^4 + b^6*c^4 + a^4*c^6 + 2*a^2*b^2*c^6 + b^4*c^6 - a^2*c^8 - b^2*c^8 : :
X(25046) = 5 X[3618] - 4 X[7668], 2 X[6] - 3 X[25314], X[25051] - 3 X[25314], 4 X[6] - 3 X[25317], 2 X[25051] - 3 X[25317], 9 X[25317] - 8 X[25324], 3 X[25051] - 4 X[25324], 9 X[25314] - 4 X[25324], 3 X[6] - 2 X[25324]

X(25046) lies on these lines: {2, 98}, {6, 17500}, {22, 10330}, {69, 160}, {193, 8264}, {237, 3564}, {325, 6038}, {526, 11061}, {1503, 14957}, {2871, 14570}, {2987, 10753}, {3060, 8267}, {3618, 7668}, {14958, 15588}, {16684, 18612}, {23360, 23371}

X(25046) = reflection of X(i) in X(j) for these {i,j}: {69, 1634}, {25051, 6}, {25317, 25314}
X(25046) = anticomplement X(20021)
X(25046) = X(20022)-Ceva conjugate of X(2)
X(25046) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 511}, {237, 21217}, {511, 21289}, {1755, 2896}, {1959, 1369}, {3112, 14957}, {3405, 69}, {4593, 14295}, {20022, 6327}
X(25046) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25051, 25317), (25051, 25314, 6)

### X(25047) =  LIONHEART-MOSES 1-IMAGE OF X(99)

Barycentrics    a^4*b^2 - 3*a^2*b^4 + a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 - 3*a^2*c^4 + b^2*c^4 : :
X(25047) =

3 X[2] - 4 X[3124], 2 X[6] - 3 X[25315], X[25052] - 3 X[25315], X[25052] - 4 X[25322], 3 X[25315] - 4 X[25322], 3 X[25052] - 4 X[25325], 9 X[25315] - 4 X[25325], 3 X[6] - 2 X[25325], 3 X[25322] - X[25325], 2 X[25322] + X[25334], X[25052] + 2 X[25334], 3 X[25315] + 2 X[25334], 2 X[25325] + 3 X[25334]

X(25047) lies on these lines: {2, 694}, {6, 10330}, {69, 6664}, {94, 2996}, {110, 10754}, {111, 4563}, {148, 690}, {193, 2854}, {194, 11002}, {698, 20977}, {3060, 8267}, {3952, 24505}, {5468, 20998}, {7783, 15018}, {17154, 21220}

X(25047) = X(25047) = midpoint of X(6) and X(25334)
X(25047) = reflection of X(i) in X(j) for these {i,j}: {6, 25322}, {4576, 3124}, {25052, 6}
X(25047) = anticomplement X(4576)
X(25047) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 512}, {83, 17217}, {251, 7192}, {308, 21305}, {512, 21289}, {661, 1369}, {669, 21217}, {798, 2896}, {827, 21295}, {4593, 670}, {4599, 4576}, {4630, 6758}, {10566, 17137}, {18070, 315}, {18082, 21301}, {18098, 20295}, {18105, 8}, {18108, 17135}
X(25047) = X(6)-line conjugate of X(14778)
X(25047) = crossdifference of every pair of points on line {5027, 14778}
X(25047) = barycentric product X(1)*X(18060)
X(25047) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25322, 25315), (3124, 4576, 2), (25052, 25315, 6), (25322, 25334, 25052)

### X(25048) =  LIONHEART-MOSES 1-IMAGE OF X(100)

Barycentrics    a*(a^2*b^2 - a*b^3 - a^2*b*c + a^2*c^2 + b^2*c^2 - a*c^3) : :
X(25048) = 3 X[903] - 2 X[4014], 3 X[3799] - 4 X[4422], 5 X[4473] - 6 X[16482], 2 X[190] - 3 X[24482], 4 X[3271] - 3 X[24482], 4 X[6] - 3 X[25316], 2 X[25050] - 3 X[25316], 9 X[25316] - 8 X[25323], 3 X[25050] - 4 X[25323], 3 X[6] - 2 X[25323]

X(25048) lies on these lines: {1, 3122}, {2, 4553}, {6, 82}, {8, 4033}, {75, 3056}, {86, 17049}, {105, 1332}, {149, 3448}, {190, 3271}, {239, 674}, {256, 17445}, {291, 3248}, {320, 9025}, {335, 9016}, {350, 20863}, {513, 4440}, {519, 21100}, {545, 4499}, {651, 10755}, {662, 8301}, {903, 4014}, {1086, 3888}, {1964, 24478}, {2836, 3868}, {3060, 3891}, {3270, 14942}, {3573, 16686}, {3688, 17277}, {3759, 3779}, {3770, 17142}, {3778, 18170}, {3799, 4422}, {3885, 7671}, {4124, 18151}, {4360, 21746}, {4443, 4787}, {4446, 7032}, {4473, 16482}, {4517, 17335}, {4922, 9263}, {4969, 9054}, {6007, 17160}, {7186, 24165}, {13476, 20090}, {16507, 22323}, {16706, 17792}, {17121, 22277}, {17790, 20352}, {18042, 23868}, {20331, 24495}
X(25048) = anticomplement X(4553)
X(25048) = reflection of X(i) in X(j) for these {i,j}: {190, 3271}, {320, 20358}, {3888, 1086}, {25050, 6}
X(25048) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 513}, {83, 20295}, {251, 514}, {308, 21304}, {513, 21289}, {514, 1369}, {649, 2896}, {667, 21217}, {827, 4427}, {1176, 20294}, {3112, 21301}, {4628, 190}, {10566, 69}, {18070, 21287}, {18105, 1654}, {18108, 8}
X(25048) = barycentric product X(1)*X(18061)
X(25048) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25050, 25316), (190, 3271, 24482), (1964, 24478, 24530)

### X(25049) =  LIONHEART-MOSES 1-IMAGE OF X(101)

Barycentrics    a^4*b^2 - a^2*b^4 - 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 - a^2*c^4 - b^2*c^4 : :

X(25049) lies on these lines: {7, 3240}, {69, 8050}, {75, 3909}, {149, 20295}, {150, 2774}, {675, 1331}, {2810, 23989}, {4440, 21220}, {6650, 8049}, {11061, 17220}, {17029, 17484}

X(25049) = anticomplement X(1634)
X(25049) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 514}, {83, 513}, {251, 17494}, {308, 21301}, {513, 2896}, {514, 21289}, {649, 21217}, {693, 1369}, {3112, 20295}, {4599, 4427}, {6591, 8878}, {10566, 8}, {18070, 1330}, {18105, 1655}, {18108, 2}, {18833, 21304}

### X(25050) =  LIONHEART-MOSES 1-IMAGE OF X(105)

Barycentrics    a*(a^4*b^2 - a^3*b^3 + a^2*b^4 - a*b^5 + a^4*b*c - 2*a^3*b^2*c + a^2*b^3*c + a^4*c^2 - 2*a^3*b*c^2 + a^2*b^2*c^2 - a*b^3*c^2 + b^4*c^2 - a^3*c^3 + a^2*b*c^3 - a*b^2*c^3 + a^2*c^4 + b^2*c^4 - a*c^5) : :
X(25050) = 3 X[3799] - 2 X[4437], 2 X[6] - 3 X[25316], X[25048] - 3 X[25316], X[25048] - 4 X[25323], 3 X[25316] - 4 X[25323]

X(25050) lies on these lines: {6, 82}, {69, 4553}, {100, 1814}, {190, 2876}, {319, 20552}, {518, 2113}, {2110, 22116}, {2836, 2895}, {2991, 10760}, {3573, 20468}, {3799, 4437}, {3827, 17789}, {3888, 5845}, {8674, 11061}, {10327, 18138}, {14839, 20455}, {20248, 21278} : :

X(25050) = reflection of X(i) in X(j) for these {i,j}: {6, 25323}, {69, 4553}, {25048, 6}
X(25050) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25323, 25316), (25048, 25316, 6)
X(25050) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 518}, {83, 20347}, {251, 239}, {518, 21289}, {672, 2896}, {1176, 3100}, {2223, 21217}, {2356, 8878}, {3112, 20556}, {3912, 1369}, {4628, 918}

### X(25051) =  LIONHEART-MOSES 1-IMAGE OF X(110)

Barycentrics    a^6*b^2 - a^2*b^6 + a^6*c^2 - 4*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - b^6*c^2 + 2*a^2*b^2*c^4 + 2*b^4*c^4 - a^2*c^6 - b^2*c^6 : :
X(25051) = 3 X[2] - 4 X[7668], 4 X[6] - 3 X[25314], 2 X[25046] - 3 X[25314], 2 X[6] - 3 X[25317], X[25046] - 3 X[25317], 3 X[25314] - 8 X[25324], X[25046] - 4 X[25324], 3 X[25317] - 4 X[25324]

X(25051) lies on these lines: {2, 1634}, {4, 542}, {6, 17500}, {69, 290}, {98, 4558}, {99, 14060}, {148, 804}, {311, 6467}, {338, 2854}, {524, 14957}, {526, 3448}, {1232, 11574}, {2407, 9512}, {2782, 14570}, {3260, 8681}, {4226, 7669}, {4552, 24500}, {6128, 6997}, {6148, 7386}, {12042, 22085}

X(25051) = reflection of X(i) in X(j) for these {i,j}: {6, 25324}, {69, 20021}, {1634, 7668}, {14570, 20975}, {25046, 6}, {25314, 25317}
X(25051) = anticomplement X(1634)
X(25051) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25046, 25314), (6, 25324, 25317), (1634, 7668, 2), (9512, 22143, 2407), (12188, 22143, 9512), (25046, 25317, 6)
X(25051) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 523}, {83, 7192}, {251, 4560}, {308, 17217}, {512, 21217}, {523, 21289}, {661, 2896}, {827, 6758}, {1577, 1369}, {3112, 512}, {4577, 21295}, {4580, 4329}, {4593, 4576}, {4599, 99}, {10566, 75}, {18070, 69}, {18082, 513}, {18097, 693}, {18098, 514}, {18105, 192}, {18107, 17149}, {18108, 1}

### X(25052) =  LIONHEART-MOSES 1-IMAGE OF X(111)

Barycentrics    3*a^6*b^2 - 3*a^2*b^6 + 3*a^6*c^2 - 8*a^4*b^2*c^2 + 2*a^2*b^4*c^2 + b^6*c^2 + 2*a^2*b^2*c^4 + 2*b^4*c^4 - 3*a^2*c^6 + b^2*c^6 : :
X(25052) = 4 X[3124] - 5 X[3618], 4 X[6] - 3 X[25315], 2 X[25047] - 3 X[25315], 9 X[25315] - 8 X[25322], 3 X[25047] - 4 X[25322], 3 X[6] - 2 X[25322], 3 X[25315] - 8 X[25325], X[25047] - 4 X[25325], X[25322] - 3 X[25325], 9 X[25315] - 4 X[25334], 3 X[25047] - 2 X[25334], 3 X[6] - X[25334], 6 X[25325] - X[25334]

X(25052) lies on these lines: {2, 17413}, {6, 10330}, {67, 69}, {99, 895}, {194, 1992}, {690, 11061}, {2393, 3266}, {2407, 5989}, {2930, 5468}, {3124, 3618}, {5095, 14928}, {7664, 15118}, {11059, 11188} X(25052) = reflection of X(i) in X(j) for these {i,j}: {6, 25325}, {69, 4576}, {25047, 6}, {25334, 25322}
X(25052) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25047, 25315), (6, 25334, 25322), (25322, 25334, 25047)
X(25052) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 524}, {83, 17491}, {187, 21217}, {251, 17497}, {308, 21298}, {524, 21289}, {896, 2896}, {3112, 316}, {4599, 690}, {14210, 1369}, {22105, 21221}

### X(25053) =  LIONHEART-MOSES 1-IMAGE OF X(112)

Barycentrics    a^10*b^2 - 2*a^8*b^4 + 2*a^4*b^8 - a^2*b^10 + a^10*c^2 + a^6*b^4*c^2 - 3*a^4*b^6*c^2 + 2*a^2*b^8*c^2 - b^10*c^2 - 2*a^8*c^4 + a^6*b^2*c^4 + 2*a^4*b^4*c^4 - a^2*b^6*c^4 - 3*a^4*b^2*c^6 - a^2*b^4*c^6 + 2*b^6*c^6 + 2*a^4*c^8 + 2*a^2*b^2*c^8 - a^2*c^10 - b^2*c^10 : :

X(25053) lies on these lines: {4, 23962}, {22, 10330}, {69, 110}, {287, 6515}, {339, 1112}, {1370, 2882}, {3060, 18018}, {3448, 9517}

X(25053) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 525}, {83, 7253}, {251, 17498}, {308, 21300}, {525, 21289}, {647, 21217}, {656, 2896}, {661, 8878}, {798, 10340}, {1176, 4560}, {1799, 7192}, {3112, 850}, {4580, 8}, {4599, 110}, {10566, 3868}, {14208, 1369}, {18070, 4}, {18082, 4391}, {18097, 521}, {18098, 25259}, {18105, 21216}, {18108, 3187}

### X(25054) =  LIONHEART-MOSES 1-IMAGE OF X(689)

Barycentrics    -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(25054) = 3 X[2] - 4 X[1084], X[670] - 3 X[3228], 2 X[1084] - 3 X[3228], 2 X[6] - 3 X[25318], 4 X[6] - 3 X[25319], 3 X[25319] - 8 X[25326], 3 X[25318] - 4 X[25326], 9 X[25319] - 8 X[25327], 9 X[25318] - 4 X[25327], 3 X[6] - 2 X[25327], 3 X[25326] - X[25327], 4 X[25327] - 3 X[25332], 3 X[25319] - 2 X[25332], 3 X[25318] - X[25332], 4 X[25326] - X[25332]

X(25054) lies on these lines: {2, 670}, {6, 19585}, {69, 694}, {148, 804}, {193, 8264}, {194, 1992}, {385, 9512}, {511, 19566}, {1370, 16098}, {1655, 4664}, {1916, 20975}, {2882, 20065}, {3978, 6379}, {4440, 21220}, {4922, 9263}, {6339, 6527}, {10340, 11061}, {10754, 11596}, {14360, 14948}

X(25054) = reflection of X(i) in X(j) for these {i,j}: {2, 3228}, {6, 25326}, {69, 694}, {670, 1084}, {1370, 16098}, {14360, 14948}, {25319, 25318}, {25332, 6}
X(25054) = anticomplement X(670)
X(25054) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2, 21305}, {6, 17217}, {25, 21300}, {31, 512}, {32, 7192}, {37, 21304}, {42, 21301}, {82, 688}, {163, 4576}, {213, 20295}, {512, 6327}, {513, 17138}, {523, 21275}, {560, 523}, {604, 4374}, {649, 17137}, {661, 315}, {662, 670}, {667, 17135}, {669, 8}, {688, 21289}, {798, 69}, {810, 1370}, {1084, 21221}, {1333, 17159}, {1402, 21302}, {1501, 4560}, {1576, 21295}, {1910, 14295}, {1918, 513}, {1919, 75}, {1924, 2}, {1927, 804}, {1973, 850}, {1974, 7253}, {1980, 1}, {2084, 1369}, {2205, 514}, {2206, 17166}, {2489, 21270}, {3049, 4329}, {3063, 20245}, {3113, 9006}, {3121, 150}, {3122, 21293}, {3124, 21294}, {3709, 21286}, {4017, 21280}, {4079, 21287}, {4117, 148}, {4455, 20554}, {5029, 20560}, {7180, 21285}, {9178, 21298}, {9236, 3005}, {9247, 6563}, {9426, 192}, {9427, 21220}, {9494, 21217}, {14574, 6758}, {18105, 21278}, {18267, 876}, {22383, 18659}
X(25054) = X(512)-Ceva conjugate of X(2)
X(25054) = X(9428)-cross conjugate of X(2)
X(25054) = X(2)-Hirst inverse of X(1084)
X(25054) = cevapoint of X(9431) and X(23180)
X(25054) = barycentric product X(i)*X(j) for these {i,j}: {76, 9431}, {264, 23180}, {512, 9428}
X(25054) = barycentric quotient X(i)/X(j) for these {i,j}: {9428, 670}, {9431, 6}, {23180, 3}, {23868, 15593}
X(25054) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25326, 25318), (6, 25332, 25319), (670, 1084, 2), (670, 3228, 1084), (25318, 25332, 6)

### X(25055) = X(1) + 2 X(2)

Trilinears    3 r + 4 R sin B sin C : :
Barycentrics    5 a + 2 b + 2 c : :

X(25055) lies on these lines: {1, 2}, {3, 9589}, {5, 3655}, {11, 13384}, {21, 25056}, {30, 1699}, {35, 4428}, {36, 1001}, {40, 549}, {56, 4355}, {57, 5298}, {86, 2163}, {99, 12258}, {115, 9875}, {140, 3654}, {165, 3524}, {191, 3338}, {210, 5049}, {214, 9963}, {226, 13462}, {238, 4715}, {274, 4479}, {354, 3894}, {355, 547}, {376, 946}, {377, 4857}, {381, 1385}, {392, 3742}, {405, 5563}, {442, 3829}, {452, 4317}, {474, 3746}, {484, 3306}, {515, 3545}, {516, 10304}, {517, 5054}, {518, 4539}, {524, 16475}, {528, 15015}, {540, 5429}, {553, 3361}, {597, 3751}, {599, 1386}, {620, 9881}, {631, 7991}, {671, 11711}, {726, 9334}, {758, 15671}, {903, 4432}, {940, 5315}, {944, 5071}, {952, 15699}, {956, 8167}, {960, 3901}, {962, 15692}, {984, 4694}, {993, 5284}, {999, 4423}, {1022, 4448}, {1054, 11731} et al

X(25055) = midpoint of X(1) and X(19875)
X(25055) = reflection of X(i) in X(j) for these (i,j): (2, 19883), (19875, 2)
X(25055) = {X(1),X(2)}-harmonic conjugate of X(3679)
X(25055) = homothetic center of anti-Aquila triangle and (cross-triangle of Aquila and anti-Aquila triangles)

### X(25056) = CENTROID OF GEMINI TRIANGLE 1

Barycentrics    8 a^3 + 19 a^2 (b + c) + a (13 b^2 + 29 b c + 13 c^2) + 2 (b + c) (b^2 + 4 b c + c^2) : :

X(25056) lies on these lines: {2, 319}, {21, 25055}

### X(25057) = CENTROID OF GEMINI TRIANGLE 2

Barycentrics    8 a^3 - 3 a^2 (b + c) - 3 a (3 b^2 + b c + 3 c^2) + 2 (b + c) (b^2 - 4 b c + c^2) : :

X(25057) lies on these lines: {2, 44}, {100, 993}

### X(25058) = PERSPECTOR OF GEMINI TRIANGLE 1 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 1 AND 2

Trilinears    (a b^2 + a c^2 + 3 a b c + b^3 + 2 b^2 c + 2 b c^2 + c^3)/(b + c) : :
Barycentrics    a (a b^2 + a c^2 + 3 a b c + b^3 + 2 b^2 c + 2 b c^2 + c^3)/(b + c) : :

X(25058) lies on these lines: {1, 21}, {2, 314}, {37, 333}, {42, 3786}, {86, 3666}, {274, 3210}, {940, 2305}, {960, 5331}, {1010, 3931}, {1402, 4184} et al

X(25058) = {X(2),X(25060)}-harmonic conjugate of X(25059)

### X(25059) = PERSPECTOR OF GEMINI TRIANGLE 2 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 1 AND 2

Trilinears    (a b^2 + a c^2 + b^3 + 3 a b c + c^3)/(b + c) : :
Barycentrics    a (a b^2 + a c^2 + b^3 + 3 a b c + c^3)/(b + c) : :

X(25059) lies on these lines: {2, 314}, {27, 1880}, {42, 5208}, {43, 3786}, {57, 77}, {65, 5331}, {86, 3752}, {239, 257}, {940, 11329}, {986, 4281}, {1043, 4646}, {1402, 4225}, {1403, 4267}, {2258, 3869} et al

X(25059) = {X(2),X(25060)}-harmonic conjugate of X(25058)

### X(25060) = {X(25058),X(25059)}-HARMONIC CONJUGATE OF X(2)

Trilinears    (a b^2 + a c^2 + 3 a b c + b^3 + b^2 c + b c^2 + c^3)/(b + c) : :
Barycentrics    a (a b^2 + a c^2 + 3 a b c + b^3 + b^2 c + b c^2 + c^3)/(b + c) : :

X(25060) lies on these lines: {1, 994}, {2, 314}, {21, 3931}, {37, 5235}, {58, 4414}, {81, 593}, {86, 4850}, {187, 980}, {286, 3672}, {333, 3187}, {940, 1030}, {2292, 4281} et al

X(25060) = {X(25058),X(25059)}-harmonic conjugate of X(2)

Collineation mappings involving Gemini triangle 15: X(25061) - X(25101)

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 15 (the Gergonne line extraversion triangle; see X(10180)), as in centers X(25061)-X(25100). Then

m(X) = a [(b + c)(a - b - c) x + (b^2 - a b - b c) y + (c^2 - a c - b c) z]) : :

(Clark Kimberling, October 13, 2018)

### X(25061) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a^2 (a^2 b^2-2 a b^3+b^4+2 b^3 c+a^2 c^2+4 b^2 c^2-2 a c^3+2 b c^3+c^4) : :

X(25061) lies on these lines: {2, 21404}, {37, 17061}, {756, 23988}, {846, 6184}, {5943, 20684}, {16601, 25070}

### X(25062) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^5 b - a^4 b^2 - a b^5 + b^6 + a^5 c - 2 a^4 b c - a^3 b^2 c + 3 a^2 b^3 c - b^5 c - a^4 c^2 - a^3 b c^2 + 2 a^2 b^2 c^2 + a b^3 c^2 - b^4 c^2 + 3 a^2 b c^3 + a b^2 c^3 + 2 b^3 c^3 - b^2 c^4 - a c^5 - b c^5 + c^6) : :

X(25062) lies on these lines: {1, 1802}, {2, 23581}, {6, 10122}, {37, 1210}, {169, 1040}, {212, 1729}, {1212, 17102}, {1698, 17916}, {3008, 3666}, {3074, 8558}, {4000, 7264}, {5089, 6684}, {12005, 20752}, {25065, 25071}, {25066, 25096}

### X(25063) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^5 b - a^4 b^2 - a b^5 + b^6 + a^5 c - a^3 b^2 c - a^2 b^3 c + b^5 c - a^4 c^2 - a^3 b c^2 + 2 a^2 b^2 c^2 + a b^3 c^2 - b^4 c^2 - a^2 b c^3 + a b^2 c^3 - 2 b^3 c^3 - b^2 c^4 - a c^5 + b c^5 + c^6) : :

X(25063) lies on these lines: {1, 1783}, {2, 23581}, {9, 255}, {37, 216}, {169, 9817}, {226, 241}, {284, 3469}, {946, 5089}, {3075, 8558}, {3731, 7523}, {5747, 24933}, {12059, 22164}, {16600, 17720}, {17911, 24026}, {25067, 25068}

### X(25064) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^5 b - a^4 b^2 - a b^5 + b^6 + a^5 c + 2 a^4 b c - a^3 b^2 c - 5 a^2 b^3 c + 3 b^5 c - a^4 c^2 - a^3 b c^2 + 2 a^2 b^2 c^2 + a b^3 c^2 - b^4 c^2 - 5 a^2 b c^3 + a b^2 c^3 - 6 b^3 c^3 - b^2 c^4 - a c^5 + 3 b c^5 + c^6) : :

X(25064) lies on these lines: {2, 23581}, {3624, 17916}, {16601, 25069}

### X(25065) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c - 2 a^2 b c - b^3 c - a^2 c^2 - a c^3 - b c^3 + c^4) : :

X(25065) lies on these lines: {1, 1170}, {2, 17861}, {37, 142}, {41, 16551}, {214, 17457}, {241, 3664}, {256, 14947}, {284, 16560}, {573, 7146}, {758, 16574}, {975, 988}, {986, 3743}, {1418, 4896}, {1743, 24635}, {1781, 11349}, {2171, 20367}, {2183, 18726}, {2809, 17447}, {3306, 16673}, {3666, 3911}, {3673, 4687}, {3739, 24209}, {3752, 16579}, {3912, 4019}, {3946, 8609}, {3986, 24213}, {4261, 24890}, {4266, 18161}, {4424, 12736}, {4698, 24208}, {4859, 24554}, {4888, 17092}, {5292, 24895}, {5662, 16696}, {5718, 18593}, {6586, 21182}, {7671, 24802}, {7832, 17289}, {16566, 21511}, {16601, 25067}, {16826, 24202}, {16831, 24179}, {17046, 21069}, {17047, 25348}, {17353, 24036}, {17355, 25083}, {20881, 25245}, {20905, 24778}, {24530, 24880}, {25062, 25071}, {25070, 25074}, {25097, 25101}

### X(25066) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^2 b - 2 a b^2 + b^3 + a^2 c + b^2 c - 2 a c^2 + b c^2 + c^3) : :

X(25066) lies on these lines: {1, 728}, {2, 277}, {3, 9}, {6, 3694}, {10, 1146}, {11, 21073}, {32, 44}, {36, 17744}, {37, 39}, {41, 5440}, {45, 5013}, {56, 17742}, {57, 21526}, {63, 21477}, {65, 16549}, {72, 672}, {78, 218}, {169, 1376}, {187, 15492}, {200, 16572}, {210, 2223}, {220, 997}, {241, 17284}, {344, 3926}, {346, 14986}, {354, 3970}, {355, 24247}, {392, 1334}, {475, 5089}, {500, 15984}, {517, 3061}, {518, 4253}, {519, 4515}, {574, 16814}, {579, 16298}, {894, 16061}, {939, 2297}, {942, 17754}, {960, 3730}, {988, 3731}, {1018, 3057}, {1100, 7772}, {1108, 2321}, {1155, 1759}, {1210, 8568}, {1214, 14376}, {1222, 6558}, {1329, 5179}, {1384, 3973}, {1385, 2329}, {1418, 21255}, {1427, 20106}, {1475, 3555}, {1574, 16605}, {1575, 16583}, {1723, 3713}, {1743, 3965}, {2049, 5283}, {2082, 5687}, {2170, 10914}, {2276, 3931}, {2345, 5831}, {2646, 16788}, {3053, 16885}, {3208, 9957}, {3218, 21540}, {3219, 21495}, {3305, 11343}, {3306, 21519}, {3496, 3579}, {3679, 4875}, {3691, 3697}, {3721, 20331}, {3744, 5299}, {3752, 16600}, {3753, 17451}, {3825, 21090}, {3846, 21856}, {3878, 21872}, {3912, 3933}, {3916, 5282}, {3929, 21539}, {3950, 21625}, {3998, 5294}, {4005, 17746}, {4357, 8362}, {4416, 7767}, {4422, 7789}, {4640, 24047}, {4641, 5337}, {4643, 7800}, {4851, 7758}, {4885, 23100}, {5007, 16669}, {5041, 16666}, {5087, 24045}, {5120, 5227}, {5437, 21542}, {5439, 21808}, {5525, 5563}, {5703, 5749}, {6292, 17237}, {6390, 25100}, {7123, 22131}, {7229, 24554}, {7308, 21514}, {7794, 17231}, {7795, 17279}, {7819, 17353}, {7822, 17357}, {7854, 17344}, {7855, 17374}, {8609, 17281}, {9310, 17614}, {11019, 21096}, {12447, 15853}, {13006, 17102}, {13161, 15048}, {16043, 17257}, {16060, 17260}, {16675, 22332}, {16728, 16887}, {17046, 24318}, {17619, 21044}, {17732, 24703}, {17747, 21616}, {18758, 20372}, {21258, 25355}, {25062, 25096}, {25087, 25091}

### X(25067) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c - 4 a^2 b c + 5 a b^2 c - 2 b^3 c - a^2 c^2 + 5 a b c^2 + 2 b^2 c^2 - a c^3 - 2 b c^3 + c^4) : :

X(25067) lies on these lines: {1, 480}, {2, 37}, {9, 241}, {57, 1696}, {141, 25019}, {144, 1418}, {200, 18216}, {219, 8257}, {220, 1445}, {374, 18161}, {474, 990}, {517, 19250}, {518, 21346}, {650, 24098}, {1212, 3160}, {1214, 7308}, {1376, 4319}, {1427, 18228}, {1465, 5316}, {1738, 21955}, {2257, 4383}, {2310, 15587}, {2324, 5228}, {2340, 5572}, {3242, 3445}, {3452, 3668}, {3740, 21039}, {3755, 8582}, {3912, 3965}, {4328, 5437}, {4413, 9371}, {4641, 17811}, {4875, 17277}, {5129, 15852}, {5723, 6666}, {6172, 17092}, {7174, 8583}, {8609, 17337}, {16601, 25065}, {16696, 24557}, {17239, 25000}, {17355, 25097}, {25063, 25068}, {25083, 25101}

### X(25068) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^2 b - 2 a b^2 + b^3 + a^2 c + 2 a b c + 3 b^2 c - 2 a c^2 + 3 b c^2 + c^3) : :

X(25068) lies on these lines: {1, 4515}, {2, 277}, {9, 474}, {10, 4875}, {37, 3624}, {44, 5277}, {72, 17754}, {169, 4413}, {210, 4253}, {218, 936}, {392, 3501}, {612, 9605}, {672, 5044}, {960, 16549}, {1125, 3693}, {1212, 1698}, {2082, 9709}, {2276, 6051}, {2329, 17614}, {2345, 3086}, {3061, 3753}, {3305, 16412}, {3616, 3991}, {3634, 24036}, {3683, 24047}, {3697, 21384}, {3731, 11512}, {3740, 16552}, {3742, 3970}, {3816, 21073}, {3930, 5045}, {3931, 17756}, {4390, 24928}, {4679, 17732}, {4936, 8583}, {5750, 5830}, {8580, 16572}, {16600, 16610}, {16728, 17210}, {17303, 19854}, {17355, 25100}, {25063, 25067}, {25085, 25093}, {25088, 25096}, {25089, 25092}

### X(25069) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^4 b-2 a^3 b^2+2 a^2 b^3-2 a b^4+b^5+a^4 c-a^2 b^2 c-2 a b^3 c+2 b^4 c-2 a^3 c^2-a^2 b c^2+8 a b^2 c^2-3 b^3 c^2+2 a^2 c^3-2 a b c^3-3 b^2 c^3-2 a c^4+2 b c^4+c^5) : :

X(25069) lies on these lines: {1, 3119}, {2, 21436}, {37, 24025}, {1054, 3731}, {1639, 17724}, {5400, 9502}, {11019, 20310}, {11814, 24036}, {16600, 17720}, {16601, 25064}, {17435, 18240}

### X(25070) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^4 b - a^3 b^2 - a b^4 + b^5 + a^4 c - 2 a^3 b c - b^4 c - a^3 c^2 - a c^4 - b c^4 + c^5) : :

X(25070) lies on these lines: {1, 14827}, {2, 20890}, {37, 2886}, {9502, 11031}, {16601, 25061}, {25065, 25074}, {25075, 25096}

### X(25071) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^5 b - a^4 b^2 - a b^5 + b^6 + a^5 c - 2 a^4 b c - b^5 c - a^4 c^2 - a c^5 - b c^5 + c^6) : :

X(25071) lies on these lines: {2, 21414}, {37, 17046}, {16599, 25104}, {25062, 25065}

### X(25072) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    2 a^2 - 5 a b + b^2 - 5 a c - 2 b c + c^2 : :

X(25072) lies on these lines: {1, 4924}, {2, 2415}, {6, 4909}, {9, 3664}, {10, 344}, {37, 3008}, {45, 142}, {144, 4896}, {346, 16832}, {516, 7385}, {519, 16484}, {527, 16814}, {551, 3618}, {749, 984}, {1654, 3912}, {1721, 10164}, {1723, 3305}, {1736, 13411}, {1738, 3634}, {1743, 5308}, {2321, 17259}, {2325, 3739}, {3589, 4755}, {3624, 4310}, {3626, 17233}, {3635, 3759}, {3686, 17243}, {3707, 4851}, {3828, 17342}, {3842, 19868}, {3875, 4098}, {3879, 17335}, {3945, 3973}, {3950, 4384}, {4000, 16676}, {4029, 4361}, {4357, 17263}, {4416, 17244}, {4419, 20195}, {4422, 4698}, {4431, 16815}, {4667, 16885}, {4700, 17390}, {4856, 17349}, {4888, 6172}, {4907, 5218}, {4967, 17264}, {5222, 16673}, {5257, 17279}, {5296, 17284}, {5435, 7274}, {6687, 17045}, {7961, 20196}, {8232, 10481}, {15492, 17392}, {16601, 25065}, {16675, 17278}, {16677, 17301}, {17023, 17338}, {17050, 22019}, {17067, 17246}, {17234, 17329}, {17257, 21255}, {17280, 24603}, {19878, 24160}, {21879, 24050}, {24209, 25001}, {25073, 25086}, {25081, 25097}

### X(25073) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a^3 b - 3 a^2 b^2 + a^3 c - 2 a^2 b c - a b^2 c + b^3 c - 3 a^2 c^2 - a b c^2 - 2 b^2 c^2 + b c^3 : :

X(25073) lies on these lines: {2, 986}, {37, 17048}, {846, 17681}, {1107, 21232}, {3294, 24786}, {3663, 24774}, {3934, 3985}, {6706, 24214}, {17596, 17682}, {17672, 21674}, {17686, 24850}, {25062, 25065}, {25072, 25086}, {25093, 25095}

### X(25074) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^3 b^2 - 2 a^2 b^3 + a b^4 + 2 a^3 b c - 3 a^2 b^2 c - 2 a b^3 c + b^4 c + a^3 c^2 - 3 a^2 b c^2 - b^3 c^2 - 2 a^2 c^3 - 2 a b c^3 - b^2 c^3 + a c^4 + b c^4) : :

X(25074) lies on these lines: {2, 277}, {37, 982}, {43, 1212}, {672, 8731}, {1107, 3961}, {3693, 3741}, {3991, 10453}, {3999, 14746}, {4212, 5089}, {4685, 4875}, {5268, 5283}, {6685, 24036}, {17717, 21856}, {21795, 24239}, {25065, 25070}, {25080, 25096}

### X(25075) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^3 b^2 - 2 a^2 b^3 + a b^4 + 2 a^3 b c - 3 a^2 b^2 c - 2 a b^3 c + b^4 c + a^3 c^2 - 3 a^2 b c^2 + 4 a b^2 c^2 - b^3 c^2 - 2 a^2 c^3 - 2 a b c^3 - b^2 c^3 + a c^4 + b c^4) : :

X(25075) lies on these lines: {2, 277}, {37, 3848}, {672, 11227}, {1212, 16569}, {3693, 3840}, {5121, 21795}, {6184, 24239}, {6686, 24036}, {25070, 25096}

### X(25076) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (2 a^3 b - 2 a^2 b^2 - 2 a b^3 + 2 b^4 + 2 a^3 c - 6 a^2 b c + 5 a b^2 c - 3 b^3 c - 2 a^2 c^2 + 5 a b c^2 + 2 b^2 c^2 - 2 a c^3 - 3 b c^3 + 2 c^4) : :

X(25076) lies on these lines: {2, 24209}, {37, 17067}, {3008, 8609}, {7658, 25084}, {16601, 25065}, {24778, 25243}, {25083, 25097}

### X(25077) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c - 6 a^2 b c + 10 a b^2 c - 3 b^3 c - a^2 c^2 + 10 a b c^2 + 4 b^2 c^2 - a c^3 - 3 b c^3 + c^4) : :

X(25077) lies on these lines: {2, 20881}, {6666, 8609}, {16601, 25065}

### X(25078) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c + b^3 c - a^2 c^2 - a c^3 + b c^3 + c^4) : :

X(25078) lies on these lines: {1, 1257}, {2, 17861}, {6, 22836}, {9, 48}, {10, 5721}, {36, 5279}, {37, 39}, {71, 3878}, {78, 1723}, {241, 21255}, {346, 3086}, {404, 1781}, {519, 1108}, {573, 3061}, {579, 758}, {583, 4053}, {672, 21078}, {1018, 17452}, {1212, 12447}, {1214, 20106}, {1826, 3814}, {2171, 16549}, {2257, 3811}, {2260, 3874}, {2264, 5440}, {2277, 16600}, {2294, 5883}, {2321, 8609}, {3454, 18591}, {3663, 25083}, {3693, 3950}, {3729, 24179}, {3731, 7963}, {3986, 16601}, {3991, 6744}, {4016, 4286}, {4149, 21059}, {4261, 20083}, {5227, 8666}, {5816, 24247}, {8804, 21616}, {10582, 16673}, {14439, 21809}, {16552, 21033}, {16566, 21495}, {16578, 17073}, {17052, 24317}, {17272, 24635}, {24202, 24559}

### X(25079) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    (b+c)*a^3+(b-c)^2*a^2-b*c*(b+c)*a+b*c*(b+c)^2 : :

X(25079) lies on these lines: {1, 341}, {2, 986}, {3, 4011}, {9, 4020}, {10, 11}, {35, 4432}, {39, 3985}, {72, 3840}, {140, 9959}, {312, 978}, {386, 25123}, {404, 24850}, {474, 3923}, {499, 4438}, {537, 3953}, {595, 4434}, {740, 3216}, {899, 3702}, {942, 4871}, {960, 3831}, {975, 25496}, {982, 25492}, {1043, 5529}, {1125, 1215}, {1149, 4696}, {1193, 4358}, {1201, 3701}, {1437, 5150}, {2887, 21616}, {3109, 6789}, {3120, 17674}, {3230, 4095}, {3293, 4975}, {3555, 4090}, {3624, 4687}, {3670, 19847}, {3729, 11512}, {3741, 5044}, {3743, 20108}, {3836, 12047}, {3842, 19863}, {3980, 16408}, {4003, 19862}, {4383, 17733}, {4385, 21214}, {4418, 17531}, {4422, 4999}, {4465, 16720}, {4673, 6048}, {5205, 5255}, {6051, 6685}, {6651, 7824}, {6682, 19864}, {6700, 9371}, {8728, 25385}, {9955, 21241}, {12567, 16302}, {17048, 20530}, {17355, 25068}, {17777, 24851}, {17793, 21336}, {21232, 25103}, {24945, 25124}, {25062, 25095}, {25063, 25078}, {25065, 25093}

### X(25080) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (b + c) (a^4 - 2 a^2 b^2 + b^4 - a^2 b c - 2 a b^2 c - b^3 c - 2 a^2 c^2 - 2 a b c^2 - b c^3 + c^4) : :

X(25080) lies on these lines: {1, 21}, {2, 17861}, {10, 3998}, {35, 3101}, {37, 226}, {57, 2294}, {284, 1762}, {323, 5483}, {345, 18697}, {347, 18625}, {515, 13442}, {740, 4847}, {912, 5453}, {984, 3190}, {1376, 11221}, {1419, 18675}, {1723, 5256}, {1726, 2268}, {1730, 17451}, {1754, 4137}, {1781, 1817}, {2256, 3173}, {2335, 15314}, {2801, 3989}, {2954, 13329}, {3601, 18673}, {3664, 18607}, {3666, 3946}, {3724, 5322}, {3841, 23604}, {3929, 3958}, {4413, 10158}, {4647, 19843}, {5173, 20718}, {5283, 24268}, {5294, 24036}, {10180, 16598}, {14838, 21209}, {16601, 25061}, {16603, 21072}, {18726, 22097}, {19822, 19854}, {23681, 24554}, {24177, 24181}, {25074, 25096}

### X(25081) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (b + c) (a^3 - a^2 b - a b^2 + b^3 - a^2 c - 3 a b c - 2 b^2 c - a c^2 - 2 b c^2 + c^3) : :

X(25081) lies on these lines: {1, 2287}, {2, 17861}, {9, 758}, {10, 37}, {19, 5248}, {21, 1781}, {45, 4016}, {55, 11221}, {71, 3754}, {142, 8680}, {200, 1962}, {281, 451}, {344, 18697}, {346, 4647}, {551, 1108}, {579, 5883}, {965, 22836}, {1125, 3002}, {1376, 10158}, {1743, 2650}, {1761, 3647}, {1766, 12567}, {1826, 3822}, {1901, 11263}, {1953, 3884}, {2171, 3294}, {2292, 3731}, {3161, 17164}, {3247, 3811}, {3678, 22021}, {3919, 21866}, {3949, 4015}, {4047, 4084}, {4068, 6600}, {4364, 16608}, {4698, 16578}, {4856, 4875}, {5251, 5279}, {5296, 18391}, {5333, 16585}, {5436, 18673}, {5750, 24036}, {6536, 21717}, {6708, 16579}, {8804, 12609}, {10180, 13405}, {11683, 16053}, {16601, 17355}, {17052, 25359}, {20228, 21332}, {21061, 21808}, {21811, 21921}, {25072, 25097}, {25085, 25086}

### X(25082) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a - b - c) (a b - b^2 + a c + b c - c^2) : :

X(25082) lies on these lines: {1, 644}, {2, 277}, {8, 1212}, {9, 21}, {37, 2275}, {39, 16614}, {45, 5782}, {100, 169}, {145, 3991}, {190, 18055}, {220, 4511}, {329, 14021}, {344, 348}, {346, 3702}, {391, 3694}, {672, 3868}, {728, 3872}, {964, 5283}, {993, 17744}, {1018, 14923}, {1334, 3061}, {1475, 3889}, {1759, 24047}, {2082, 3871}, {2141, 3681}, {2170, 3208}, {2329, 3897}, {2975, 17742}, {3219, 16367}, {3305, 15490}, {3501, 14439}, {3665, 16593}, {3705, 21795}, {3729, 24554}, {3730, 3869}, {3731, 7963}, {3870, 16572}, {3873, 3970}, {3912, 24635}, {3930, 21384}, {3976, 23649}, {4200, 5089}, {4251, 4587}, {4391, 6376}, {4513, 4861}, {4664, 25268}, {4666, 24771}, {4676, 5764}, {4687, 16705}, {4850, 16600}, {5030, 17736}, {5057, 17732}, {5179, 11681}, {5294, 17023}, {5333, 16831}, {5525, 8666}, {5526, 22836}, {5540, 8715}, {5552, 6554}, {5830, 17340}, {7191, 9605}, {7705, 21044}, {7741, 21090}, {8609, 11240}, {11109, 17916}, {11680, 21073}, {16583, 17756}, {16720, 17279}, {16728, 17169}, {17077, 20946}, {17092, 17234}, {17350, 17696}, {17754, 21808}, {20271, 20331}, {25001, 25252}

### X(25083) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^2 - b^2 - c^2) (a b - b^2 + a c - c^2) : :

X(25083) lies on these lines: {2, 277}, {3, 63}, {8, 24635}, {9, 11343}, {32, 4641}, {37, 980}, {39, 712}, {57, 21477}, {69, 3694}, {75, 25252}, {77, 3692}, {81, 5266}, {99, 448}, {100, 7291}, {110, 2754}, {190, 1944}, {219, 326}, {222, 3719}, {241, 3693}, {304, 345}, {306, 3933}, {441, 525}, {517, 1959}, {518, 2223}, {536, 8609}, {910, 20602}, {942, 1009}, {988, 8583}, {1010, 16735}, {1026, 23102}, {1108, 3875}, {1155, 20715}, {1212, 4384}, {1229, 17077}, {1332, 6510}, {1418, 17298}, {1465, 17789}, {1742, 4073}, {2108, 18208}, {2325, 16578}, {2340, 4712}, {2795, 20544}, {3008, 24036}, {3218, 21495}, {3219, 21511}, {3263, 5089}, {3305, 21514}, {3306, 21526}, {3501, 7146}, {3509, 21989}, {3616, 6051}, {3663, 25078}, {3712, 8758}, {3739, 20236}, {3928, 16431}, {3929, 16436}, {3965, 4416}, {3990, 23124}, {3991, 17316}, {4001, 7767}, {4515, 17294}, {4574, 20744}, {4869, 17092}, {5256, 9605}, {5273, 24609}, {5294, 7819}, {5437, 21519}, {7308, 21496}, {17355, 25065}, {17754, 21985}, {20740, 20808}, {22134, 23125}, {22363, 23167}, {24203, 24559}, {24993, 25255}, {25067, 25101}, {25076, 25097}

### X(25084) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (b - c) (a^3 b - a b^3 + a^3 c + 2 a^2 b c - 3 a b^2 c - 3 a b c^2 + 2 b^2 c^2 - a c^3) : :

X(25084) lies on these lines: {1, 4524}, {2, 647}, {37, 17069}, {665, 3835}, {804, 25126}, {905, 3239}, {940, 9404}, {1021, 17022}, {2488, 24462}, {3709, 4369}, {4477, 5268}, {4789, 24948}, {4885, 6586}, {7658, 25076}, {8651, 24533}

### X(25085) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^4 b - a^3 b^2 - a b^4 + b^5 + a^4 c - 2 a^3 b c - b^4 c - a^3 c^2 + 4 a b^2 c^2 - a c^4 - b c^4 + c^5) : :

X(25085) lies on these lines: {2, 20890}, {37, 1738}, {16601, 25065}, {25068, 25093}, {25081, 25086}

### X(25086) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^2 b - 2 a b^2 + b^3 + a^2 c - 6 a b c - 5 b^2 c - 2 a c^2 - 5 b c^2 + c^3) : :

X(25086) lies on these lines: {2, 277}, {9, 5439}, {37, 1698}, {169, 4423}, {451, 5089}, {517, 21921}, {551, 4875}, {910, 5259}, {1210, 5257}, {1212, 3624}, {1759, 15254}, {3208, 4002}, {3290, 16589}, {3294, 3812}, {3634, 3693}, {3691, 5045}, {3731, 24174}, {3739, 7264}, {3740, 3970}, {3742, 16552}, {3754, 4520}, {3826, 21073}, {3991, 9780}, {4009, 22011}, {4515, 19875}, {5044, 21808}, {5302, 17736}, {6051, 16583}, {6666, 17048}, {9331, 21896}, {16610, 25092}, {19878, 24036}, {25072, 25073}, {25081, 25085}

### X(25087) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^5 b - a^4 b^2 - a b^5 + b^6 + a^5 c - a^4 b c - a^3 b^2 c + a^2 b^3 c - a^4 c^2 - a^3 b c^2 + 2 a^2 b^2 c^2 + a b^3 c^2 - b^4 c^2 + a^2 b c^3 + a b^2 c^3 - b^2 c^4 - a c^5 + c^6) : :

X(25087) lies on these lines: {1, 6}, {2, 23581}, {10, 17916}, {33, 169}, {40, 5089}, {201, 3730}, {225, 5179}, {255, 8558}, {1729, 1936}, {3220, 18596}, {6554, 7952}, {17860, 17911}, {25066, 25091}

### X(25088) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (b + c) (a^5 - a^4 b - a b^4 + b^5 - a^4 c - a^3 b c + 3 a^2 b^2 c + a b^3 c - 2 b^4 c + 3 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 - 2 b c^4 + c^5) : :

X(25088) lies on these lines: {2, 23581}, {10, 37}, {33, 11221}, {201, 3294}, {220, 758}, {2294, 2324}, {2318, 3970}, {3191, 21808}, {16601, 25061}, {25068, 25096}

### X(25089) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^2 b + b^3 + a^2 c + 4 a b c + 4 b^2 c + 4 b c^2 + c^3) : :

X(25089) lies on these lines: {2, 1930}, {37, 19862}, {1574, 4868}, {1575, 3743}, {1759, 17124}, {3290, 19878}, {3634, 16611}, {3678, 24512}, {3721, 3833}, {3727, 3918}, {3780, 3956}, {4015, 20963}, {5297, 5299}, {7832, 17263}, {10176, 17750}, {16589, 24036}, {16601, 25064}, {20331, 21816}, {25068, 25092}

### X(25090) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (b + c) (a^5 - a^3 b^2 - a^2 b^3 + b^5 - a^3 b c - a^2 b^2 c - a b^3 c - b^4 c - a^3 c^2 - a^2 b c^2 - a^2 c^3 - a b c^3 - b c^4 + c^5) : :

X(25090) lies on these lines: {2, 24788}, {37, 442}, {38, 22126}, {1375, 3666}, {4245, 17451}, {4254, 18674}, {4267, 18669}, {16577, 16600}, {16601, 25061}, {25062, 25065}, {25081, 25085}

### X(25091) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^4 b - 2 a^2 b^3 + b^5 + a^4 c - 2 a^3 b c + 2 a b^3 c - b^4 c + 4 a b^2 c^2 - 2 a^2 c^3 + 2 a b c^3 - b c^4 + c^5) : :

X(25091) lies on these lines: {1, 939}, {2, 37}, {3, 16389}, {9, 223}, {33, 1376}, {40, 13737}, {57, 2324}, {63, 220}, {198, 10319}, {201, 960}, {210, 8758}, {219, 1708}, {225, 1329}, {227, 2551}, {306, 3965}, {329, 1427}, {394, 2911}, {452, 15852}, {517, 1730}, {518, 2318}, {650, 24115}, {908, 6354}, {910, 3101}, {936, 17102}, {940, 3553}, {942, 3191}, {1211, 15526}, {1418, 9965}, {1465, 3452}, {1766, 11347}, {1818, 10391}, {1834, 24982}, {2003, 6510}, {2123, 7952}, {2287, 18603}, {2654, 5836}, {3190, 5728}, {3219, 6610}, {3753, 19257}, {4383, 8557}, {4875, 5278}, {5256, 17825}, {5294, 23292}, {5745, 16578}, {5784, 24430}, {6358, 6708}, {6666, 16579}, {15853, 24635}, {16601, 25061}, {16700, 24556}, {17044, 18652}, {17080, 18228}, {19542, 21062}, {20103, 24025}, {21871, 24310}, {25066, 25087}

### X(25092) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (2 a b^2 + 2 a b c + b^2 c + 2 a c^2 + b c^2) : :

X(25092) lies on these lines: {1, 672}, {2, 3760}, {6, 5248}, {9, 386}, {10, 2276}, {21, 5280}, {35, 5276}, {37, 39}, {38, 3970}, {42, 16552}, {44, 20970}, {145, 9331}, {169, 17594}, {171, 24047}, {172, 5267}, {192, 17030}, {218, 19765}, {519, 1107}, {551, 2275}, {609, 4189}, {941, 1743}, {978, 3731}, {1001, 9605}, {1015, 3636}, {1018, 10459}, {1193, 3294}, {1212, 3931}, {1573, 3626}, {1574, 3828}, {1575, 3634}, {1621, 5299}, {1655, 6381}, {1698, 17756}, {1739, 21921}, {1759, 4414}, {1962, 20456}, {2140, 3663}, {2277, 3986}, {2303, 4278}, {2345, 19858}, {2975, 16785}, {3008, 3666}, {3061, 15953}, {3244, 16975}, {3293, 3691}, {3635, 17448}, {3670, 21808}, {3693, 19868}, {3702, 4099}, {3707, 4277}, {3741, 21070}, {3743, 24036}, {3811, 16517}, {3815, 3825}, {3821, 22380}, {3822, 5254}, {3912, 16887}, {4261, 5257}, {4424, 17451}, {4745, 21868}, {5179, 5530}, {5286, 10198}, {5294, 17023}, {5305, 6690}, {6683, 20530}, {6685, 21838}, {7746, 20104}, {10448, 16788}, {13006, 13411}, {16610, 25086}, {16819, 17759}, {17717, 24045}, {17758, 24214}, {21101, 24068}, {21240, 25349}, {21820, 24051}, {21883, 24044}, {25068, 25089}

### X(25093) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^5 b - a^4 b^2 - a b^5 + b^6 + a^5 c - 2 a^4 b c - 4 a^2 b^3 c - b^5 c - a^4 c^2 - 4 a^2 b c^3 - 4 b^3 c^3 - a c^5 - b c^5 + c^6) : :

X(25093) lies on these lines: {2, 21414}, {25068, 25085}, {25073, 25095}

### X(25094) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (2 a^4 b - 4 a^2 b^3 + 2 b^5 + 2 a^4 c - 2 a^3 b c - 3 a^2 b^2 c - b^4 c - 3 a^2 b c^2 - 6 a b^2 c^2 - b^3 c^2 - 4 a^2 c^3 - b^2 c^3 - b c^4 + 2 c^5) : :

X(25094) lies on these lines: {1, 89}, {2, 24209}, {37, 16586}, {226, 17080}, {3666, 8609}, {3977, 17760}

### X(25095) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (b + c) (a^5 - a^4 b - a b^4 + b^5 - a^4 c - 2 a^3 b c + 2 a^2 b^2 c + a b^3 c - 2 b^4 c + 2 a^2 b c^2 + a b^2 c^2 + a b c^3 - a c^4 - 2 b c^4 + c^5) : :

X(25095) lies on these lines: {1, 7259}, {2, 24791}, {9, 3708}, {37, 16613}, {523, 3039}, {4422, 16599}, {16598, 24036}, {25073, 25093}

### X(25096) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a - b - c) (a^3 b - a^2 b^2 + a b^3 - b^4 + a^3 c - 2 a^2 b c + 3 b^3 c - a^2 c^2 - 4 b^2 c^2 + a c^3 + 3 b c^3 - c^4) : :

X(25096) lies on these lines: {2, 21436}, {9, 7004}, {37, 3756}, {644, 3315}, {1212, 14936}, {1638, 16578}, {2284, 5083}, {3239, 24003}, {3310, 13006}, {4422, 23587}, {4850, 16600}, {5573, 24771}, {14740, 17435}, {15490, 23511}, {25062, 25066}, {25068, 25088}, {25070, 25075}, {25074, 25080}

### X(25097) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c - 6 a^2 b c + 6 a b^2 c - 3 b^3 c - a^2 c^2 + 6 a b c^2 - a c^3 - 3 b c^3 + c^4) : :

X(25097) lies on these lines: {1, 4578}, {2, 20881}, {9, 3942}, {918, 3960}, {11716, 24820}, {17355, 25067}, {24003, 24025}, {25065, 25101}, {25072, 25081}, {25076, 25083}

### X(25098) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (b - c) (a b + a c - b c) (a^2 - b^2 - c^2) : :

X(25098) lies on these lines: {1, 1938}, {2, 21438}, {37, 4411}, {63, 22383}, {441, 525}, {650, 824}, {665, 3798}, {918, 6589}, {1946, 23093}, {3239, 24782}, {3310, 4468}, {3663, 23806}, {3931, 14077}, {4063, 8662}, {4083, 8640}, {4369, 21348}, {4458, 6129}, {6586, 17069}, {7658, 25076}, {9443, 17592}, {15280, 24210}, {15283, 24239}, {17921, 20906}

### X(25099) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a (a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c - 4 a^2 b c + a b^2 c - 2 b^3 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 - a c^3 - 2 b c^3 + c^4) : :

X(25099) lies on these lines: {2, 37}, {9, 7146}, {44, 15988}, {65, 21371}, {241, 894}, {257, 17260}, {264, 14571}, {326, 5782}, {458, 1841}, {517, 19243}, {518, 25024}, {650, 24141}, {1001, 3057}, {1108, 3618}, {1155, 8424}, {1213, 25007}, {1284, 3812}, {1766, 11343}, {3589, 8609}, {3661, 3965}, {3694, 17286}, {3932, 24987}, {4026, 24982}, {4875, 17349}, {5750, 16578}, {16601, 17760}, {17355, 25065}, {25072, 25081}

### X(25100) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    a^2 (a^2 b^2 - 2 a b^3 + b^4 + 2 b^3 c + a^2 c^2 + 8 b^2 c^2 - 2 a c^3 + 2 b c^3 + c^4) : :

X(25100) lies on these lines:

### X(25101) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 15

Barycentrics    2 a^2 - 3 a b + b^2 - 3 a c + c^2 : :

X(25101) lies on these lines: {1, 4899}, {2, 2415}, {7, 4480}, {9, 69}, {10, 3685}, {37, 3589}, {44, 3629}, {45, 3763}, {75, 2325}, {141, 16814}, {142, 190}, {144, 17298}, {165, 9801}, {192, 3008}, {193, 3973}, {238, 4078}, {239, 3950}, {319, 3707}, {345, 7308}, {346, 4384}, {391, 17294}, {519, 15485}, {524, 15492}, {527, 17234}, {536, 17337}, {597, 3723}, {894, 4473}, {966, 17286}, {1001, 3717}, {1026, 2293}, {1083, 2330}, {1125, 17368}, {1213, 17359}, {1265, 5436}, {1266, 17262}, {1654, 17268}, {1696, 21477}, {1743, 17316}, {2321, 5564}, {2345, 24603}, {2346, 4578}, {3241, 16487}, {3244, 17121}, {3247, 3618}, {3305, 3687}, {3664, 17244}, {3686, 17233}, {3710, 5047}, {3730, 21371}, {3739, 17340}, {3748, 4126}, {3757, 4082}, {3834, 17334}, {3883, 3932}, {3929, 18141}, {3943, 17348}, {3946, 4664}, {4021, 4704}, {4029, 4360}, {4098, 17319}, {4364, 17357}, {4370, 17245}, {4389, 17341}, {4395, 4718}, {4398, 17067}, {4419, 17282}, {4643, 17267}, {4657, 16675}, {4667, 17317}, {4681, 6687}, {4684, 5220}, {4687, 5750}, {4698, 17369}, {4700, 17377}, {4755, 17398}, {4851, 6144}, {4856, 17389}, {4862, 20073}, {4869, 6172}, {4966, 15481}, {4967, 17259}, {5224, 17342}, {5257, 17289}, {5296, 17308}, {5325, 14829}, {5745, 18743}, {5749, 16831}, {6390, 25066}, {6646, 17266}, {8236, 10005}, {8245, 9950}, {10030, 18140}, {15828, 17300}, {16601, 17760}, {16669, 17390}, {16676, 17321}, {17229, 17330}, {17230, 17331}, {17231, 17332}, {17232, 17333}, {17240, 17346}, {17241, 17347}, {17246, 17356}, {17248, 17358}, {17256, 17285}, {17257, 17284}, {17258, 17283}, {17265, 17276}, {17269, 17275}, {17312, 20072}, {17781, 18139}, {20236, 25001}, {20881, 20905}, {21283, 25006}, {21873, 22047}, {21879, 24090}, {22214, 24575}, {25065, 25097}, {25067, 25083}

Collineation mappings involving Gemini triangle 16: X(25102) - X(25146)

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 16, as in centers X(25102)-X(25146). Then

m(X) = (b + c)(a b + a c - b c) + (a b c - a c^2 + b^2 c) y + (a b c - a b^2 + b c^2) z : :

(Clark Kimberling, October 13, 2018)

### X(25102) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^2 b^2 + 2 a^2 b c - a b^2 c + a^2 c^2 - a b c^2 + 2 b^2 c^2 : :

X(25102) lies on these lines: {1, 20530}, {2, 17448}, {8, 21264}, {10, 141}, {12, 20541}, {37, 2998}, {75, 21868}, {76, 536}, {192, 20943}, {519, 3934}, {535, 7830}, {668, 1107}, {712, 21067}, {742, 4095}, {1018, 4721}, {1084, 4755}, {1500, 4681}, {1575, 1909}, {2228, 7148}, {3208, 4713}, {3501, 17351}, {3661, 4417}, {3666, 18136}, {3734, 8715}, {3742, 20340}, {3752, 20917}, {3795, 9902}, {3912, 21025}, {4437, 21965}, {4726, 20888}, {7784, 11236}, {7815, 8666}, {9025, 24327}, {10449, 17372}, {10987, 16920}, {12513, 15271}, {17033, 21904}, {17050, 24770}, {17229, 21024}, {17290, 24418}, {17792, 25131}, {18194, 24679}, {20911, 21021}, {24215, 25350}, {25103, 25119}, {25105, 25110}, {25106, 25108}, {25111, 25123}, {25121, 25124}

### X(25103) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^5 b^2 - a^3 b^4 + 2 a^5 b c - a^4 b^2 c - 3 a^3 b^3 c + a^2 b^4 c + a b^5 c + a^5 c^2 - a^4 b c^2 + b^5 c^2 - 3 a^3 b c^3 - 2 a b^3 c^3 - b^4 c^3 - a^3 c^4 + a^2 b c^4 - b^3 c^4 + a b c^5 + b^2 c^5 : :

X(25103) lies on these lines: {2, 24664}, {960, 21231}, {25102, 25119}, {25106, 25112}, {25107, 25139}, {25122, 25138}

### X(25104) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^5 b^2 - a^3 b^4 - a^4 b^2 c + a^3 b^3 c + a^2 b^4 c - a b^5 c + a^5 c^2 - a^4 b c^2 + b^5 c^2 + a^3 b c^3 + 2 a b^3 c^3 - b^4 c^3 - a^3 c^4 + a^2 b c^4 - b^3 c^4 - a b c^5 + b^2 c^5 : :

X(25104) lies on these lines: {2, 24664}, {141, 1329}, {1714, 24174}, {1755, 18738}, {2083, 16560}, {14829, 20917}, {16599, 25071}, {25108, 25109}, {25120, 25122}

### X(25105) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^5 b^2 - a^3 b^4 - 2 a^5 b c - a^4 b^2 c + 5 a^3 b^3 c + a^2 b^4 c - 3 a b^5 c + a^5 c^2 - a^4 b c^2 + b^5 c^2 + 5 a^3 b c^3 + 6 a b^3 c^3 - b^4 c^3 - a^3 c^4 + a^2 b c^4 - b^3 c^4 - 3 a b c^5 + b^2 c^5 : :

X(25105) lies on these lines: {2, 24664}, {25102, 25110}

### X(25106) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^3 b^2 + 2 a^3 b c - a^2 b^2 c + a b^3 c + a^3 c^2 - a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 + a b c^3 + b^2 c^3 : :

X(25106) lies on these lines: {2, 3728}, {37, 6377}, {43, 20923}, {75, 16569}, {740, 3216}, {899, 20891}, {1215, 3739}, {1698, 4751}, {3752, 21080}, {16706, 17793}, {17337, 24327}, {17448, 24659}, {24655, 24735}, {25102, 25108}, {25103, 25112}, {25111, 25115}, {25121, 25125}, {25133, 25141}, {25140, 25146}

### X(25107) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^2 b^2 - 3 a b^2 c + a^2 c^2 - 3 a b c^2 + 2 b^2 c^2 : :

X(25107) lies on these lines: {1, 24745}, {2, 17448}, {8, 20530}, {9, 24343}, {10, 3934}, {141, 2885}, {194, 1575}, {350, 21868}, {536, 18135}, {668, 16604}, {1107, 7786}, {1329, 20541}, {1574, 6381}, {3214, 4852}, {3739, 5772}, {17792, 25120}, {18140, 20691}, {25103, 25139}, {25119, 25132}, {25131, 25137}, {25140, 25145}

### X(25108) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a (a^2 b^2 - a b^3 + 2 a^2 b c - 4 a b^2 c + 3 b^3 c + a^2 c^2 - 4 a b c^2 - 4 b^2 c^2 - a c^3 + 3 b c^3) : :

X(25108) lies on these lines: {2, 3056}, {37, 4941}, {141, 3740}, {210, 17232}, {518, 17234}, {960, 3836}, {3742, 17245}, {3826, 5836}, {4085, 10179}, {9025, 17337}, {15587, 16593}, {25102, 25106}, {25104, 25109}, {25121, 25146}, {25125, 25140}

### X(25109) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^2 b^2 - 2 a^2 b c - 5 a b^2 c + a^2 c^2 - 5 a b c^2 + 2 b^2 c^2 : :

X(25109) lies on these lines: {2, 17448}, {10, 20530}, {536, 18140}, {984, 1698}, {1575, 1655}, {3760, 4726}, {3828, 3934}, {3831, 17239}, {4698, 20363}, {4852, 6048}, {5257, 25376}, {9780, 21264}, {25104, 25108}, {25121, 25122}, {25132, 25139}

### X(25110) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^4 b^2 - a^3 b^3 - 2 a^4 b c + 4 a^2 b^3 c - 3 a b^4 c + a^4 c^2 - 4 a^2 b^2 c^2 + 2 a b^3 c^2 + b^4 c^2 - a^3 c^3 + 4 a^2 b c^3 + 2 a b^2 c^3 - 2 b^3 c^3 - 3 a b c^4 + b^2 c^4 : :

X(25110) lies on these lines: {2, 25119}, {25102, 25105}

### X(25111) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^4 b^2 + 2 a^4 b c - a^3 b^2 c + a b^4 c + a^4 c^2 - a^3 b c^2 - a b^3 c^2 + b^4 c^2 - a b^2 c^3 + a b c^4 + b^2 c^4 : :

X(25111) lies on these lines: {2, 24525}, {141, 210}, {742, 18138}, {2176, 21590}, {3011, 4074}, {4437, 17056}, {25102, 25123}, {25106, 25115}, {25116, 25139}

### X(25112) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^5 b^2 + 2 a^5 b c - a^4 b^2 c + a b^5 c + a^5 c^2 - a^4 b c^2 - a b^4 c^2 + b^5 c^2 - a b^2 c^4 + a b c^5 + b^2 c^5 : :

X(25112) lies on these lines: {2, 24526}, {25103, 25106}, {25122, 25133}, {25138, 25141}

### X(25113) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^2 b^3 + 3 a^2 b^2 c - 2 a b^3 c + 3 a^2 b c^2 + 2 a b^2 c^2 + b^3 c^2 + a^2 c^3 - 2 a b c^3 + b^2 c^3 : :

X(25113) lies on these lines: {2, 18194}, {10, 4684}, {726, 18143}, {3741, 17231}, {3840, 17241}, {3971, 18144}, {4871, 21238}, {17244, 21257}, {17245, 20340}, {20917, 21080}, {25102, 25106}, {25114, 25130}, {25124, 25144}

### X(25114) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^3 b^4 + 3 a^3 b^3 c - a^2 b^4 c - a b^4 c^2 + 3 a^3 b 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 : :

X(25114) lies on these lines: {2, 24527}, {17448, 25140}, {25103, 25106}, {25113, 25130}, {25133, 25138}

### X(25115) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^3 b^3 + 3 a^3 b^2 c + 3 a^3 b c^2 - 4 a^2 b^2 c^2 + a b^3 c^2 + a^3 c^3 + a b^2 c^3 + 2 b^3 c^3 : :

X(25115) lies on these lines: {2, 17448}, {42, 20530}, {210, 3739}, {536, 18152}, {1575, 3770}, {25106, 25111}, {25123, 25139}

### X(25116) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^3 b^3 + 3 a^3 b^2 c + 3 a^3 b c^2 - 8 a^2 b^2 c^2 + a b^3 c^2 + a^3 c^3 + a b^2 c^3 + 2 b^3 c^3 : :

X(25116) lies on these lines: {2, 17448}, {43, 4852}, {1215, 3739}, {25111, 25139}, {25119, 25129}

### X(25117) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    2 a^3 b^2 - a^2 b^3 + 4 a^3 b c - 5 a^2 b^2 c + 4 a b^3 c + 2 a^3 c^2 - 5 a^2 b c^2 - 6 a b^2 c^2 + b^3 c^2 - a^2 c^3 + 4 a b c^3 + b^2 c^3 : :

X(25117) lies on these lines: {2, 23633}, {21191, 25126}, {24742, 25279}, {25102, 25106}, {25125, 25146}

### X(25118) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^3 b^2 - 2 a^2 b^3 + 2 a^3 b c - 7 a^2 b^2 c + 5 a b^3 c + a^3 c^2 - 7 a^2 b c^2 - 6 a b^2 c^2 - b^3 c^2 - 2 a^2 c^3 + 5 a b c^3 - b^2 c^3 : :

X(25118) lies on these lines: {2, 24671}, {4096, 18150}, {25102, 25106}

### X(25119) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^4 b^2 - a^3 b^3 + 2 a^4 b c - 4 a^3 b^2 c + a b^4 c + a^4 c^2 - 4 a^3 b c^2 + 4 a^2 b^2 c^2 - 2 a b^3 c^2 + b^4 c^2 - a^3 c^3 - 2 a b^2 c^3 - 2 b^3 c^3 + a b c^4 + b^2 c^4 : :

X(25119) lies on these lines: {2, 25110}, {25102, 25103}, {25106, 25111}, {25107, 25132}, {25116, 25129}

### X(25120) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^3 b^2 - a^2 b^2 c - a b^3 c + a^3 c^2 - a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 - a b c^3 + b^2 c^3 : :

X(25120) lies on these lines: {2, 3728}, {10, 3688}, {75, 17793}, {141, 20339}, {740, 18147}, {992, 4039}, {1045, 18140}, {1213, 24327}, {1215, 17303}, {1575, 21080}, {1740, 6376}, {2234, 18133}, {2887, 23305}, {4967, 12263}, {17279, 24003}, {17334, 25382}, {17792, 25107}, {19856, 24325}, {24524, 24661}, {25104, 25122}

### X(25121) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^2 b^3 - a^2 b^2 c - 2 a b^3 c - a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 + a^2 c^3 - 2 a b c^3 + b^2 c^3 : :

X(25121) lies on these lines: {2, 18194}, {10, 75}, {37, 23643}, {141, 20340}, {1084, 1213}, {3661, 21257}, {3741, 17239}, {3840, 17228}, {3844, 15985}, {4110, 24456}, {4871, 17231}, {4970, 21858}, {17792, 25107}, {18144, 24165}, {24575, 25280}, {25102, 25124}, {25106, 25125}, {25108, 25146}, {25109, 25122}

### X(25122) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^3 b^4 - a^3 b^3 c - a^2 b^4 c - a b^4 c^2 - a^3 b 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 : :

X(25122) lies on these lines: {2, 24527}, {10, 1015}, {141, 20338}, {714, 18148}, {1698, 18194}, {3831, 3844}, {10479, 24174}, {25103, 25138}, {25104, 25120}, {25109, 25121}, {25112, 25133}

### X(25123) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    (b + c) (a^4 b + a^3 b^2 + a^4 c + a^3 b c + a b^3 c + a^3 c^2 - a b^2 c^2 + b^3 c^2 + a b c^3 + b^2 c^3) : :

X(25123) lies on these lines: {2, 3728}, {10, 12}, {43, 312}, {714, 3752}, {2092, 3985}, {2667, 18743}, {3936, 21688}, {4647, 6048}, {6685, 10180}, {17793, 19786}, {25102, 25111}, {25115, 25139}

### X(25124) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    (b + c) (a^3 b + a^3 c + 3 a^2 b c + a b^2 c + a b c^2 + b^2 c^2) : :

X(25124) lies on these lines: {1, 75}, {2, 3728}, {10, 4111}, {37, 714}, {192, 1962}, {385, 3757}, {512, 21211}, {518, 15985}, {523, 24141}, {726, 3743}, {894, 3747}, {1221, 3510}, {1402, 4032}, {1486, 8424}, {2292, 24349}, {3720, 20891}, {3739, 3741}, {4022, 6682}, {4068, 4363}, {4698, 24003}, {4699, 10453}, {4772, 17163}, {7155, 20146}, {17322, 17793}, {17398, 24327}, {17792, 24656}, {23668, 23944}, {25102, 25121}, {25113, 25144}, {25129, 25130}

### X(25125) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^3 b^2 + a^2 b^3 + 2 a^3 b c - 2 a^2 b^2 c - a b^3 c + a^3 c^2 - 2 a^2 b c^2 - 4 a b^2 c^2 + 2 b^3 c^2 + a^2 c^3 - a b c^3 + 2 b^2 c^3 : :

X(25125) lies on these lines: {2, 17448}, {5, 10}, {37, 4110}, {239, 20530}, {312, 21868}, {518, 20340}, {536, 3264}, {1575, 3975}, {3596, 21892}, {3662, 3739}, {3687, 21025}, {3752, 6376}, {3823, 20486}, {4394, 6002}, {16602, 20917}, {17490, 20943}, {23301, 25142}, {25106, 25121}, {25108, 25140}, {25117, 25146}, {25137, 25141}

### X(25126) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    (b - c) (-a^2 b^3 + 2 a^3 b c - 3 a^2 b^2 c - 3 a^2 b c^2 + 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(25126) lies on these lines: {2, 669}, {523, 18154}, {650, 3837}, {804, 25084}, {4369, 21051}, {4507, 4992}, {14426, 23803}, {21128, 21349}, {21191, 25117}, {24674, 24749}

### X(25127) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    (b - c) (a^4 b^2 + 2 a^4 b c - a^3 b^2 c - a^2 b^3 c + a^4 c^2 - a^3 b c^2 - a^2 b c^3 + b^3 c^3) : :

X(25127) lies on these lines: {2, 24665}, {514, 6586}, {21191, 25129}

### X(25128) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    (b - c) (a - b - c) (a^2 b^2 + a^2 c^2 - b^2 c^2) : :

X(25128) lies on these lines: {2, 23655}, {10, 7234}, {38, 20909}, {514, 23768}, {521, 3716}, {522, 4087}, {650, 3907}, {663, 21300}, {788, 3741}, {982, 21197}, {2526, 24720}, {3221, 21191}, {4369, 8678}, {4874, 20316}, {5737, 23865}, {7662, 24718}, {21056, 23572}, {24533, 24755}

### X(25129) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^4 b^2 + 2 a^4 b c - a^3 b^2 c + a b^4 c + a^4 c^2 - a^3 b c^2 - 4 a^2 b^2 c^2 - a b^3 c^2 + b^4 c^2 - a b^2 c^3 + a b c^4 + b^2 c^4 : :

X(25129) lies on these lines: {2, 24525}, {75, 16969}, {698, 3290}, {742, 18157}, {3242, 15668}, {3739, 5836}, {21191, 25127}, {25102, 25106}, {25116, 25119}, {25124, 25130}

### X(25130) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^2 b^2 + 6 a^2 b c + 3 a b^2 c + a^2 c^2 + 3 a b c^2 + 2 b^2 c^2 : :

X(25130) lies on these lines: {1, 3696}, {2, 17448}, {37, 194}, {274, 536}, {330, 4687}, {730, 1125}, {1100, 16827}, {1107, 4698}, {2176, 4670}, {3230, 17175}, {3616, 21264}, {4364, 24215}, {4688, 17144}, {4739, 17143}, {4755, 5283}, {6706, 24331}, {7786, 16604}, {9534, 17372}, {10436, 16969}, {25113, 25114}, {25124, 25129}

### X(25131) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^5 b^2 - a^3 b^4 + a^5 b c - a^4 b^2 c - a^3 b^3 c + a^2 b^4 c + a^5 c^2 - a^4 b c^2 + b^5 c^2 - a^3 b c^3 - b^4 c^3 - a^3 c^4 + a^2 b c^4 - b^3 c^4 + b^2 c^5 : :

X(25131) lies on these lines: {2, 24664}, {17792, 25102}, {25107, 25137}

### X(25132) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    (b + c) (a^5 b - a^3 b^3 + a^5 c - 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 - 5 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 - 2 b^3 c^3 + a b c^4 + b^2 c^4) : :

X(25132) lies on these lines: {2, 24664}, {65, 21231}, {85, 8680}, {2294, 3212}, {16609, 17056}, {17090, 25255}, {17792, 24656}, {25102, 25111}, {25107, 25119}, {25109, 25139}

### X(25133) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^5 b^2 - a^3 b^4 + 2 a^5 b c - a^4 b^2 c + a^3 b^3 c + a^2 b^4 c + a b^5 c + a^5 c^2 - a^4 b c^2 + b^5 c^2 + a^3 b c^3 + 2 a b^3 c^3 - b^4 c^3 - a^3 c^4 + a^2 b c^4 - b^3 c^4 + a b c^5 + b^2 c^5 : :

X(25133) lies on these lines: {2, 24664}, {1107, 21232}, {25106, 25141}, {25112, 25122}, {25114, 25138}

### X(25134) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^2 b^4 + 2 a^2 b^3 c - 2 a b^4 c + 4 a^2 b^2 c^2 + b^4 c^2 + 2 a^2 b c^3 + a^2 c^4 - 2 a b c^4 + b^2 c^4 : :

X(25134) lies on these lines: {2, 24670}, {9055, 18052}, {25102, 25111}

### X(25135) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a (a^3 b^2 - a b^4 + 2 a^3 b c - a^2 b^2 c - 2 a b^3 c + 3 b^4 c + a^3 c^2 - a^2 b c^2 + 4 a b^2 c^2 - b^3 c^2 - 2 a b c^3 - b^2 c^3 - a c^4 + 3 b c^4) : :

X(25135) lies on these lines: {2, 3056}, {5, 10}, {11, 24997}, {238, 1376}, {325, 24471}, {518, 3705}, {908, 20487}, {978, 1104}, {3035, 6679}, {3061, 20697}, {3123, 3666}, {3742, 3756}, {4640, 6210}, {16298, 17749}, {25102, 25103}

### X(25136) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    (b + c) (a^5 b + a^4 b^2 + a^5 c + a^4 b c + a^2 b^3 c + a b^4 c + a^4 c^2 - 3 a^2 b^2 c^2 + b^4 c^2 + a^2 b c^3 + a b c^4 + b^2 c^4) : :

X(25136) lies on these lines: {2, 25307}, {758, 20255}, {960, 3739}, {24654, 25295}, {25102, 25111}, {25103, 25106}, {25124, 25129}

### X(25137) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a (a^3 b^2 - a b^4 + 2 a^3 b c - a^2 b^2 c - 2 a b^3 c + 3 b^4 c + a^3 c^2 - a^2 b c^2 - 4 a b^2 c^2 - b^3 c^2 - 2 a b c^3 - b^2 c^3 - a c^4 + 3 b c^4) : :

X(25137) lies on these lines: {2, 3056}, {10, 20256}, {120, 125}, {518, 18134}, {960, 2887}, {3925, 5836}, {25102, 25111}, {25107, 25131}, {25125, 25141}

### X(25138) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    (b + c) (a^5 b - 2 a^3 b^3 + a^5 c - a^4 b c + 2 a^2 b^3 c + a b^4 c - 2 a^2 b^2 c^2 + b^4 c^2 - 2 a^3 c^3 + 2 a^2 b c^3 - 3 b^3 c^3 + a b c^4 + b^2 c^4) : :

X(25138) lies on these lines: {2, 24531}, {21232, 21254}, {25103, 25122}, {25112, 25141}, {25114, 25133}

### X(25139) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^4 b^2 - a^3 b^3 + 2 a^4 b c - 4 a^3 b^2 c + a b^4 c + a^4 c^2 - 4 a^3 b c^2 + 8 a^2 b^2 c^2 - 2 a b^3 c^2 + b^4 c^2 - a^3 c^3 - 2 a b^2 c^3 - 2 b^3 c^3 + a b c^4 + b^2 c^4 : :

X(25139) lies on these lines: {2, 25110}, {891, 4928}, {25103, 25107}, {25109, 25132}, {25111, 25116}, {25115, 25123}

### X(25140) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^2 b^3 + a^2 b^2 c - 2 a b^3 c + a^2 b c^2 + b^3 c^2 + a^2 c^3 - 2 a b c^3 + b^2 c^3 : :

X(25140) lies on these lines: {2, 18194}, {10, 141}, {37, 20532}, {726, 18144}, {1502, 6381}, {2228, 18040}, {3741, 17228}, {3840, 17231}, {3912, 21257}, {3971, 18133}, {4518, 18207}, {4871, 17241}, {6374, 6376}, {17229, 24688}, {17234, 20340}, {17308, 20139}, {17448, 25114}, {18143, 24165}, {25106, 25146}, {25107, 25145}, {25108, 25125}

### X(25141) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^3 b^4 + a^3 b^3 c - a^2 b^4 c - a b^4 c^2 + a^3 b c^3 + b^4 c^3 + a^3 c^4 - a^2 b c^4 - a b^2 c^4 + b^3 c^4 : :

X(25141) lies on these lines: {2, 24527}, {10, 141}, {6376, 6386}, {7148, 18040}, {20366, 24443}, {21024, 24688}, {25106, 25133}, {25112, 25138}, {25125, 25137}

### X(25142) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a (b - c) (a b + a c - b c)^2 : :

X(25142) lies on these lines: {2, 20983}, {42, 23506}, {43, 8640}, {44, 513}, {693, 14404}, {756, 21350}, {3221, 24675}, {3808, 11068}, {3835, 4083}, {3837, 20316}, {4010, 4397}, {4383, 20473}, {4940, 9400}, {21191, 25117}, {23301, 25125}, {23655, 24532}, {24666, 24747}

### X(25143) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    (b - c) (a^3 b^2 - a^2 b^3 + 4 a^3 b c - 5 a^2 b^2 c + a^3 c^2 - 5 a^2 b c^2 + 3 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(25143) lies on these lines: {2, 23655}, {984, 21197}, {3835, 4507}, {4147, 4885}, {21191, 25117}

### X(25144) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a (a^2 b^2 - a b^3 + 2 a^2 b c + 3 b^3 c + a^2 c^2 - a c^3 + 3 b c^3) : :

X(25144) lies on these lines: {2, 3056}, {6, 16569}, {10, 141}, {37, 19584}, {210, 17238}, {354, 17232}, {511, 3634}, {613, 16862}, {978, 1386}, {1100, 17795}, {1428, 17531}, {1469, 9780}, {3589, 6686}, {3619, 3779}, {3705, 3742}, {3740, 5224}, {3799, 17324}, {3838, 20545}, {3841, 24206}, {3848, 17245}, {3880, 4085}, {3912, 4890}, {3925, 24997}, {4357, 7064}, {4429, 5836}, {4553, 17384}, {4699, 19586}, {5249, 20487}, {6210, 15254}, {8167, 10387}, {9342, 15988}, {15310, 24295}, {17235, 21865}, {17291, 20358}, {25106, 25121}, {25113, 25124}

### X(25145) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^2 b^4 + 2 a^2 b^3 c - 2 a b^4 c + 8 a^2 b^2 c^2 + b^4 c^2 + 2 a^2 b c^3 + a^2 c^4 - 2 a b c^4 + b^2 c^4 : :

X(25145) lies on these lines: {2, 24670}, {25102, 25106}, {25107, 25140}, {25109, 25121}

### X(25146) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 16

Barycentrics    a^3 b^2 - 2 a^2 b^3 + 2 a^3 b c - 3 a^2 b^2 c + 5 a b^3 c + a^3 c^2 - 3 a^2 b c^2 - 2 a b^2 c^2 - b^3 c^2 - 2 a^2 c^3 + 5 a b c^3 - b^2 c^3 : :

X(25146) lies on these lines: {2, 24671}, {537, 18150}, {3837, 14426}, {25106, 25140}, {25108, 25121}, {25113, 25124}, {25117, 25125}

### X(25147) =  X(4)X(12026)∩X(5)X(128)

Barycentrics    (-(b^2-c^2)^2+a^2 (b^2+c^2)) (a^12-4 a^10 (b^2+c^2)+(b^2-c^2)^4 (3 b^4-2 b^2 c^2+3 c^4)+a^8 (9 b^4+4 b^2 c^2+9 c^4)-4 a^2 (b^2-c^2)^2 (3 b^6-b^4 c^2-b^2 c^4+3 c^6)-4 a^6 (4 b^6-b^4 c^2-b^2 c^4+4 c^6)+a^4 (19 b^8-18 b^6 c^2+7 b^4 c^4-18 b^2 c^6+19 c^8)) : :
X(25147) = X[4]+2*X[12026], 5*X[5]-2*X[128], 2*X[546]+X[1141], X[930]-4*X[3628], 5*X[1656]-2*X[6592], 7*X[3090]-X[13512], X[6343]-4*X[8254]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28460.

X(25147) lies on these lines: {4,12026}, {5,128}, {381,9143}, {546,1141}, {930,3628}, {1656,6592}, {3090,13512}, {3327,10593}, {3459,14143}, {6343,8254}, {7159,10592}, {7741,14101}, {10126,20413}, {13406,15367}, {14652,18378}, {15307,22051}

X(25147) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {5,137,1263}, {5,1263,14072}, {1656,11671,6592}

### X(25148) =  ISOGONAL CONJUGATE OF X(6150)

Barycentrics    (a^4-a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (a^10-3 a^8 b^2+2 a^6 b^4+2 a^4 b^6-3 a^2 b^8+b^10-4 a^8 c^2+5 a^6 b^2 c^2-2 a^4 b^4 c^2+5 a^2 b^6 c^2-4 b^8 c^2+8 a^6 c^4+a^4 b^2 c^4+a^2 b^4 c^4+8 b^6 c^4-10 a^4 c^6-10 a^2 b^2 c^6-10 b^4 c^6+7 a^2 c^8+7 b^2 c^8-2 c^10) (a^10-4 a^8 b^2+8 a^6 b^4-10 a^4 b^6+7 a^2 b^8-2 b^10-3 a^8 c^2+5 a^6 b^2 c^2+a^4 b^4 c^2-10 a^2 b^6 c^2+7 b^8 c^2+2 a^6 c^4-2 a^4 b^2 c^4+a^2 b^4 c^4-10 b^6 c^4+2 a^4 c^6+5 a^2 b^2 c^6+8 b^4 c^6-3 a^2 c^8-4 b^2 c^8+c^10) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28460.

X(25148) lies on the hyperbola {{A, B, C, X(4), X(5)}} and these lines: {5,930}, {137,252}, {1263,19552}, {10126,20413}, {15643,21394}, {20030,22335}

X(25148) = isogonal conjugate of X(6150)
X(25148) = reflection of X(i) in X(j) for these {i,j}: {930,21975}, {3459,137}
X(25148) = antigonal conjugate of X(3459)
X(25148) = antipode in hyperbola {{A, B, C, X(4), X(5)}} of X(3459)

### X(25149) =  X(30)X(511)∩X(128)X(8562)

Barycentrics    (b-c) (b+c) (-a^6+2 a^4 b^2-a^2 b^4-a^2 b^3 c+b^5 c+2 a^4 c^2+a^2 b^2 c^2-a^2 b c^3-2 b^3 c^3-a^2 c^4+b c^5) (a^6-2 a^4 b^2+a^2 b^4-a^2 b^3 c+b^5 c-2 a^4 c^2-a^2 b^2 c^2-a^2 b c^3-2 b^3 c^3+a^2 c^4+b c^5) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28460.

X(25149) lies on these lines: {30,511}, {128,8562}, {137,8901}, {1141,10412}, {6132,14769}, {14225,15619}

### X(25150) =  ISOGONAL CONJUGATE OF X(15907)

Barycentrics    (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (a^12-4 a^10 b^2+6 a^8 b^4-4 a^6 b^6+a^4 b^8-4 a^10 c^2+10 a^8 b^2 c^2-8 a^6 b^4 c^2+3 a^4 b^6 c^2-2 a^2 b^8 c^2+b^10 c^2+6 a^8 c^4-8 a^6 b^2 c^4+a^4 b^4 c^4+2 a^2 b^6 c^4-4 b^8 c^4-4 a^6 c^6+3 a^4 b^2 c^6+2 a^2 b^4 c^6+6 b^6 c^6+a^4 c^8-2 a^2 b^2 c^8-4 b^4 c^8+b^2 c^10) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28460.

X(25150) lies on these lines: {1,3327}, {3,252}, {4,11671}, {5,128}, {12,14101}, {26,15959}, {30,511}, {54,14071}, {140,6592}, {156,6069}, {1157,19553}, {5576,14769}, {5889,13505}, {7387,15960}, {7512,14652}, {7568,13467}, {10205,14143}, {11412,13504}, {13160,15367}, {14118,14674}, {14865,14978}, {15425,15957}

X(25150) = isogonal conjugate of X(15907)

Related centers to Fermat-Dao equilateral triangles: X(25151) - X(25236)

This preamble and centers X(25151)-X(25236) were contributed by César Eliud Lozada, October 13, 2018.

For definitions and coordinates of all Fermat-Dao equilateral triangles, see Index of triangles referenced in ETC.

A complete list of orthologic and parallelogic triangles to Fermat-Dao triangles and centers can be downloaded from here.

### X(25151) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd FERMAT-DAO TO 6th FERMAT-DAO

Barycentrics    (SB+SC)*(3*S^4+(3*R^2*(-8*SW+3*SA)+SA^2+2*SB*SC+5*SW^2)*S^2+sqrt(3)*((3*R^2-4*SW)*S^2+(3*(4*SA+SW)*R^2-4*SA^2)*SW)*S-(3*R^2-3*SA+2*SW)*SA*SW^2) : :
X(25151) = 2*X(396)+X(23007) = 4*X(11542)-X(22999) = 5*X(16960)+X(25182) = 5*X(16960)-2*X(25220) = X(25182)+2*X(25220)

The reciprocal orthologic center of these triangles is X(25152)

X(25151) lies on these lines: {17,14182}, {61,25180}, {396,23007}, {511,16529}, {512,5470}, {6787,22846}, {11542,22999}, {16267,25217}, {16808,23017}, {16960,25182}

X(25151) = reflection of X(25217) in X(16267)
X(25151) = {X(16960), X(25182)}-harmonic conjugate of X(25220)

### X(25152) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th FERMAT-DAO TO 3rd FERMAT-DAO

Barycentrics    (sqrt(3)*SB-S)*(sqrt(3)*SC-S)*(3*(3*R^2+SB+SC)*S^2-sqrt(3)*(S^2-(6*R^2-SW)*SW)*S-3*(3*R^2*SW-SA^2-SB*SC)*SW) : :

The reciprocal orthologic center of these triangles is X(25151)

X(25152) lies on these lines: {13,11085}, {14,1916}, {531,25215}, {2782,16462}, {6033,11582}, {8015,25155}, {14181,16464}, {16639,25207}

### X(25153) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 10th FERMAT-DAO TO 3rd FERMAT-DAO

Barycentrics    3*(6*R^2+SB+SC)*S^2-sqrt(3)*(2*S^2-3*(3*SA-SW)*R^2+3*SA^2-SW^2)*S-3*(3*R^2+SA)*(SB+SC)*SW : :

The reciprocal orthologic center of these triangles is X(25151)

X(25153) lies on these lines: {6,23896}, {13,11085}, {39,395}, {523,11624}, {531,25221}, {2782,11626}, {2854,25232}, {11083,12205}, {11141,14181}, {16641,25207}

X(25153) = reflection of X(25231) in X(11626)

### X(25154) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 12th FERMAT-DAO TO 3rd FERMAT-DAO

Barycentrics    (3*SA-2*SW)*S^2-6*sqrt(3)*SB*SC*S-3*SB*SC*SW : :
X(25154) = X(4)+2*X(16001) = X(20)-4*X(20415) = 2*X(549)-3*X(22489) = X(616)-3*X(3545) = 2*X(618)-3*X(5055) = 3*X(3839)-2*X(22796) = 3*X(5054)-4*X(6669) = X(5473)-4*X(20252) = X(5473)-3*X(22489) = 4*X(5478)-X(5617) = 2*X(5478)+X(13103) = X(5617)+2*X(13103) = 2*X(6321)+X(22509) = 2*X(6774)-3*X(9166) = 4*X(20252)-3*X(22489)

The reciprocal orthologic center of these triangles is X(5470)

X(25154) lies on these lines: {2,9736}, {3,5459}, {4,542}, {5,5463}, {13,15}, {20,20415}, {62,5469}, {115,10653}, {376,6771}, {381,530}, {395,23006}, {511,25223}, {512,25175}, {524,20428}, {531,6321}, {543,5613}, {549,5473}, {616,3545}, {618,5055}, {622,11178}, {2782,22694}, {3058,10062}, {3543,6770}, {3655,11705}, {3839,22796}, {5054,6669}, {5335,11179}, {5434,10078}, {5470,16631}, {5472,10654}, {5969,25195}, {6054,22797}, {6774,9166}, {6778,22515}, {8703,21156}, {8724,16627}, {9115,18581}, {9140,16770}, {9762,9771}, {9830,22575}, {10056,13076}, {10072,18974}, {12355,16629}, {16808,25156}, {16809,22998}

X(25154) = midpoint of X(i) and X(j) for these {i,j}: {381, 13103}, {3543, 6770}
X(25154) = reflection of X(i) in X(j) for these (i,j): (3, 5459), (376, 6771), (381, 5478), (549, 20252), (3655, 11705), (5463, 5), (5473, 549), (5617, 381), (6054, 22797), (25164, 9880)
X(25154) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 20423, 25164), (13, 23005, 6772), (549, 20252, 22489), (5473, 22489, 549), (5478, 13103, 5617)

### X(25155) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 14th FERMAT-DAO TO 3rd FERMAT-DAO

Barycentrics    3*(6*R^2-SA)*S^2-sqrt(3)*(2*S^2-6*R^2*SW+3*SA^2-SW^2)*S+3*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(25151)

X(25155) lies on these lines: {2,3107}, {6,23896}, {13,94}, {2088,6772}, {2782,5640}, {3129,14181}, {6582,11146}, {8015,25152}, {15018,22687}, {16637,25207}, {16770,23005}

X(25155) = reflection of X(25233) in X(5640)

### X(25156) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO TO 3rd FERMAT-DAO

Barycentrics    (9*SA-2*SW)*S^2-sqrt(3)*(12*S^2-(15*SA+SW)*(SB+SC))*S-3*SB*SC*SW : :
X(25156) = 5*X(14)-4*X(6782) = 3*X(14)-2*X(22998) = 2*X(16)-3*X(5470) = 2*X(99)-3*X(21360) = 4*X(115)-3*X(16963) = 3*X(5469)-2*X(9115) = 6*X(5469)-5*X(16961) = 2*X(6779)-3*X(16963) = 6*X(6782)-5*X(22998) = 2*X(6782)-5*X(23005) = 8*X(6782)-5*X(25235) = 4*X(9115)-5*X(16961) = X(22998)-3*X(23005) = 4*X(22998)-3*X(25235) = 4*X(23005)-X(25235)

The reciprocal orthologic center of these triangles is X(5470)

X(25156) lies on these lines: {3,13}, {14,530}, {16,5470}, {30,6778}, {99,21360}, {115,6779}, {148,532}, {511,25227}, {512,25177}, {524,22578}, {542,19107}, {543,22494}, {622,22850}, {2782,22696}, {5463,16967}, {5469,9115}, {5969,25203}, {6781,16962}, {8595,22490}, {9116,9886}, {10654,20088}, {12355,25166}, {16808,25154}, {19106,19924}

X(25156) = reflection of X(i) in X(j) for these (i,j): (14, 23005), (6779, 115), (25235, 14)
X(25156) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115, 6779, 16963), (5469, 9115, 16961)

### X(25157) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd FERMAT-DAO TO 1st NEUBERG

Barycentrics    sqrt(3)*SA*S^2-(4*S^2+3*SA^2+SW^2)*S-sqrt(3)*SB*SC*SW : :
X(25157) = X(13)+2*X(25183) = 2*X(76)+X(3105) = 2*X(141)+X(23024) = X(3104)-4*X(3934) = 4*X(5459)-3*X(22688) = 4*X(11542)-X(23000) = 5*X(16960)+X(25199) = 3*X(22489)-2*X(22691)

The reciprocal orthologic center of these triangles is X(6582)

X(25157) lies on these lines: {2,3106}, {13,76}, {14,24256}, {15,3734}, {16,183}, {17,6581}, {39,16644}, {61,7770}, {62,7751}, {141,23024}, {381,511}, {524,22701}, {538,3107}, {624,3314}, {633,16044}, {635,5025}, {2782,5464}, {3104,3934}, {5459,22688}, {5463,22712}, {5470,5969}, {5981,8289}, {6671,7835}, {6672,17004}, {6694,16895}, {6695,7856}, {7761,23005}, {11185,23004}, {11305,22846}, {11542,23000}, {16634,18906}, {16808,23018}, {16960,25199}, {22486,22493}, {22489,22691}

X(25157) = reflection of X(i) in X(j) for these (i,j): (3106, 2), (25167, 9466)
X(25157) = {X(76), X(11303)}-harmonic conjugate of X(25195)

### X(25158) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd FERMAT-DAO TO 2nd NEUBERG

Barycentrics    3*S^4+3*(2*SA^2-5*SB*SC-9*SW^2)*S^2+sqrt(3)*((3*SA+4*SW)*S^2+(9*SA^2-3*SB*SC-13*SW^2)*SW)*S-15*SB*SC*SW^2 : :
X(25158) = X(13)+2*X(25184) = 4*X(11542)-X(23001) = 5*X(16960)+X(25200)

The reciprocal orthologic center of these triangles is X(6298)

X(25158) lies on these lines: {2,22688}, {13,83}, {17,6296}, {61,25192}, {754,16267}, {5459,22510}, {11542,23001}, {11632,25168}, {16808,23019}, {16960,25200}

### X(25159) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd FERMAT-DAO TO 1st TRI-SQUARES-CENTRAL

Barycentrics    (3*(SA-15*SW)*S^2-(370*S^2-3*(SB+SC)*(89*SA+9*SW))*S-48*SB*SC*SW)*sqrt(3)+(213*SA-218*SW)*S^2+3*S*(98*S^2-3*(SB+SC)*(31*SA+16*SW))+27*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(13705)

X(25159) lies on these lines: {13,1327}, {17,13706}, {61,25193}, {530,22631}, {11542,23002}, {16808,23020}, {16960,25201}

### X(25160) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd FERMAT-DAO TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    -(3*(SA-15*SW)*S^2+(370*S^2-3*(SB+SC)*(89*SA+9*SW))*S-48*SB*SC*SW)*sqrt(3)+(213*SA-218*SW)*S^2-3*S*(98*S^2-3*(SB+SC)*(31*SA+16*SW))+27*SB*SC*SW : :
X(25160) = X(13)+2*X(25186)

The reciprocal orthologic center of these triangles is X(13825)

X(25160) lies on these lines: {13,1328}, {17,13826}, {61,25194}, {530,22602}, {11542,23003}, {16808,23021}, {16960,25202}

### X(25161) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th FERMAT-DAO TO 5th FERMAT-DAO

Barycentrics    (SB+SC)*(3*S^4+(3*R^2*(-8*SW+3*SA)+SA^2+2*SB*SC+5*SW^2)*S^2-sqrt(3)*((3*R^2-4*SW)*S^2+(3*(4*SA+SW)*R^2-4*SA^2)*SW)*S-(3*R^2-3*SA+2*SW)*SA*SW^2) : :
X(25161) = X(14)+2*X(25173) = 2*X(395)+X(23014) = 4*X(11543)-X(23008) = 5*X(16961)+X(25177)

The reciprocal orthologic center of these triangles is X(25162)

X(25161) lies on these lines: {14,25173}, {18,14178}, {62,25175}, {395,23014}, {511,16530}, {512,5469}, {6787,22891}, {11543,23008}, {16268,25214}, {16809,23023}, {16961,25177}

X(25161) = reflection of X(25214) in X(16268)
X(25161) = {X(16961), X(25177)}-harmonic conjugate of X(25219)

### X(25162) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th FERMAT-DAO TO 4th FERMAT-DAO

Barycentrics    (sqrt(3)*SB+S)*(sqrt(3)*SC+S)*(3*(3*R^2+SB+SC)*S^2+sqrt(3)*(S^2-(6*R^2-SW)*SW)*S-3*(3*R^2*SW-SA^2-SB*SC)*SW) : :

The reciprocal orthologic center of these triangles is X(25161)

X(25162) lies on these lines: {14,11080}, {530,25218}, {2782,16461}, {6033,11581}, {8014,25165}, {14177,16463}, {16638,25208}

### X(25163) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 9th FERMAT-DAO TO 4th FERMAT-DAO

Barycentrics    3*(6*R^2+SB+SC)*S^2+sqrt(3)*(2*S^2-3*(3*SA-SW)*R^2+3*SA^2-SW^2)*S-3*(3*R^2+SA)*(SB+SC)*SW : :

The reciprocal orthologic center of these triangles is X(25161)

X(25163) lies on these lines: {6,23895}, {14,11080}, {39,396}, {523,11626}, {530,25222}, {2782,11624}, {2854,25231}, {11088,12204}, {11142,14177}, {16640,25208}

X(25163) = reflection of X(25232) in X(11624)

### X(25164) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 11th FERMAT-DAO TO 4th FERMAT-DAO

Barycentrics    (3*SA-2*SW)*S^2+6*sqrt(3)*SB*SC*S-3*SB*SC*SW : :
X(25164) = X(4)+2*X(16002) = X(20)-4*X(20416) = 2*X(549)-3*X(22490) = X(617)-3*X(3545) = 2*X(619)-3*X(5055) = 3*X(3839)-2*X(22797) = 3*X(5054)-4*X(6670) = X(5474)-4*X(20253) = X(5474)-3*X(22490) = 4*X(5479)-X(5613) = 2*X(5479)+X(13102) = X(5613)+2*X(13102) = 2*X(6321)+X(22507) = 2*X(6771)-3*X(9166) = 4*X(20253)-3*X(22490)

The reciprocal orthologic center of these triangles is X(5469)

X(25164) lies on these lines: {2,9735}, {3,5460}, {4,542}, {5,5464}, {14,16}, {20,20416}, {61,5470}, {115,10654}, {376,6774}, {381,531}, {396,23013}, {511,25224}, {512,25180}, {524,20429}, {530,6321}, {543,5617}, {549,5474}, {617,3545}, {619,5055}, {621,11178}, {2782,22693}, {3058,10061}, {3543,6773}, {3655,11706}, {3839,22797}, {5054,6670}, {5334,11179}, {5434,10077}, {5469,16630}, {5471,10653}, {5969,25191}, {6054,22796}, {6771,9166}, {6777,22515}, {8703,21157}, {8724,16626}, {9117,18582}, {9140,16771}, {9760,9771}, {9830,22576}, {10056,13075}, {10072,18975}, {12355,16628}, {16808,22997}, {16809,25166}

X(25164) = midpoint of X(i) and X(j) for these {i,j}: {381, 13102}, {3543, 6773}
X(25164) = reflection of X(i) in X(j) for these (i,j): (3, 5460), (376, 6774), (381, 5479), (549, 20253), (3655, 11706), (5464, 5), (5474, 549), (5613, 381), (6054, 22796), (25154, 9880)
X(25164) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 20423, 25154), (14, 23004, 6775), (549, 20253, 22490), (5474, 22490, 549), (5479, 13102, 5613)

### X(25165) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 13th FERMAT-DAO TO 4th FERMAT-DAO

Barycentrics    3*(6*R^2-SA)*S^2+sqrt(3)*(2*S^2-6*R^2*SW+3*SA^2-SW^2)*S+3*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(25161)

X(25165) lies on these lines: {2,3106}, {6,23895}, {14,94}, {530,25226}, {2088,6775}, {2782,5640}, {3130,14177}, {6295,11145}, {8014,25162}, {15018,22689}, {16636,25208}, {16771,23004}

X(25165) = reflection of X(25234) in X(5640)

### X(25166) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO TO 4th FERMAT-DAO

Barycentrics    (9*SA-2*SW)*S^2+sqrt(3)*(12*S^2-(15*SA+SW)*(SB+SC))*S-3*SB*SC*SW : :
X(25166) = 5*X(13)-4*X(6783) = 3*X(13)-2*X(22997) = 2*X(15)-3*X(5469) = 2*X(99)-3*X(21359) = 4*X(115)-3*X(16962) = 3*X(5470)-2*X(9117) = 6*X(5470)-5*X(16960) = 2*X(6780)-3*X(16962) = 6*X(6783)-5*X(22997) = 2*X(6783)-5*X(23004) = 8*X(6783)-5*X(25236) = 4*X(9117)-5*X(16960) = X(22997)-3*X(23004) = 4*X(22997)-3*X(25236) = 4*X(23004)-X(25236)

The reciprocal orthologic center of these triangles is X(5469)

X(25166) lies on these lines: {3,14}, {13,531}, {15,5469}, {30,6777}, {99,21359}, {115,6780}, {148,533}, {511,25228}, {512,25182}, {524,22577}, {542,19106}, {543,22493}, {621,22894}, {2782,22695}, {5464,16966}, {5470,9117}, {5969,25199}, {6781,16963}, {8594,22489}, {9114,9885}, {10653,20088}, {12355,25156}, {16809,25164}, {19107,19924}

X(25166) = reflection of X(i) in X(j) for these (i,j): (13, 23004), (6780, 115), (25236, 13)
X(25166) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115, 6780, 16962), (5470, 9117, 16960)

### X(25167) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th FERMAT-DAO TO 1st NEUBERG

Barycentrics    sqrt(3)*SA*S^2+(4*S^2+3*SA^2+SW^2)*S-sqrt(3)*SB*SC*SW : :
X(25167) = X(14)+2*X(25187) = 2*X(76)+X(3104) = 2*X(141)+X(23018) = X(3105)-4*X(3934) = 4*X(5460)-3*X(22690) = 4*X(11543)-X(23009) = 5*X(16961)+X(25203) = 3*X(22490)-2*X(22692)

The reciprocal orthologic center of these triangles is X(6295)

X(25167) lies on these lines: {2,3107}, {13,24256}, {14,76}, {15,183}, {16,3734}, {18,6294}, {39,16645}, {61,7751}, {62,7770}, {141,23018}, {381,511}, {524,22702}, {538,3106}, {623,3314}, {634,16044}, {636,5025}, {2782,5463}, {3105,3934}, {5460,22690}, {5464,22712}, {5469,5969}, {5980,8289}, {6671,17004}, {6672,7835}, {6694,7856}, {6695,16895}, {7761,23004}, {11185,23005}, {11306,22891}, {11543,23009}, {16635,18906}, {16809,23024}, {16961,25203}, {22486,22494}, {22490,22692}

X(25167) = reflection of X(i) in X(j) for these (i,j): (3107, 2), (25157, 9466)
X(25167) = {X(76), X(11304)}-harmonic conjugate of X(25191)

### X(25168) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th FERMAT-DAO TO 2nd NEUBERG

Barycentrics    3*S^4+3*(2*SA^2-5*SB*SC-9*SW^2)*S^2-sqrt(3)*((3*SA+4*SW)*S^2+(9*SA^2-3*SB*SC-13*SW^2)*SW)*S-15*SB*SC*SW^2 : :
X(25168) = X(14)+2*X(25188) = 4*X(11543)-X(23010) = 5*X(16961)+X(25204)

The reciprocal orthologic center of these triangles is X(6299)

X(25168) lies on these lines: {2,22690}, {14,83}, {18,6297}, {62,25196}, {754,16268}, {5460,22511}, {11543,23010}, {11632,25158}, {16809,23025}, {16961,25204}

### X(25169) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th FERMAT-DAO TO 1st TRI-SQUARES-CENTRAL

Barycentrics    -(3*(SA-15*SW)*S^2-(370*S^2-3*(SB+SC)*(89*SA+9*SW))*S-48*SB*SC*SW)*sqrt(3)+(213*SA-218*SW)*S^2+3*S*(98*S^2-3*(SB+SC)*(31*SA+16*SW))+27*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(13703)

X(25169) lies on these lines: {14,1327}, {18,13704}, {62,25197}, {531,22633}, {11543,23011}, {16809,23026}, {16961,25205}

### X(25170) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th FERMAT-DAO TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    (3*(SA-15*SW)*S^2+(370*S^2-3*(SB+SC)*(89*SA+9*SW))*S-48*SB*SC*SW)*sqrt(3)+(213*SA-218*SW)*S^2-3*S*(98*S^2-3*(SB+SC)*(31*SA+16*SW))+27*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(13823)

X(25170) lies on these lines: {14,1328}, {18,13824}, {62,25198}, {531,22604}, {11543,23012}, {16809,23027}, {16961,25206}

### X(25171) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th FERMAT-DAO TO 6th FERMAT-DAO

Barycentrics
(SB+SC)*(S+sqrt(3)*SB)*(S+sqrt(3)*SC)*(sqrt(3)*(9*S^4-(72*R^2*SW+3*SA^2-12*SB*SC-13*SW^2)*S^2+(9*SA-4*SW)*SA*SW^2)*S+9*(12*R^2+4*SA-7*SW)*S^4+3*(108*R^2*SA^2-SW*(33*SA^2-4*SW*SA-3*SW^2))*S^2-3*SA^2*SW^3) : :

The reciprocal orthologic center of these triangles is X(25172)

X(25171) lies on these lines: {530,25230}, {8014,25181}, {11080,25179}, {14175,16463}, {16461,25218}, {16638,25209}

### X(25172) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th FERMAT-DAO TO 5th FERMAT-DAO

Barycentrics
(SB+SC)*(sqrt(3)*SB-S)*(sqrt(3)*SC-S)*(sqrt(3)*(-9*S^4+(72*R^2*SW+3*SA^2-12*SB*SC-13*SW^2)*S^2-(9*SA-4*SW)*SA*SW^2)*S+9*(12*R^2+4*SA-7*SW)*S^4+3*(3*SA^2*(-11*SW+36*R^2)+SW^2*(3*SW+4*SA))*S^2-3*SA^2*SW^3) : :

The reciprocal orthologic center of these triangles is X(25171)

X(25172) lies on these lines: {531,25229}, {8015,25176}, {11085,25174}, {14176,16464}, {16462,25215}, {16639,25210}

### X(25173) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 8th FERMAT-DAO TO 5th FERMAT-DAO

Barycentrics    (SB+SC)*(sqrt(3)*SA+S)*(2*S^2+sqrt(3)*(SB+SC)*S+SA^2-SB*SC)*(4*S^2-(3*SA+SW)*(SB+SC)) : :
X(25173) = X(14)-3*X(25161) = 3*X(16268)-X(23008) = 5*X(16961)-3*X(25214) = 5*X(16961)-X(25227) = 3*X(16963)+X(25177) = 3*X(25214)-X(25227)

The reciprocal orthologic center of these triangles is X(25162)

X(25173) lies on these lines: {6,14186}, {14,25161}, {16,23014}, {115,512}, {381,23023}, {395,25219}, {511,9115}, {6785,22512}, {10653,25175}, {11486,23028}, {14178,16645}, {16268,23008}, {16961,25214}, {16963,25177}, {18581,25223}

X(25173) = midpoint of X(16) and X(23014)
X(25173) = reflection of X(i) in X(j) for these (i,j): (25178, 14113), (25219, 395)
X(25173) = {X(16961), X(25227)}-harmonic conjugate of X(25214)

### X(25174) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 10th FERMAT-DAO TO 5th FERMAT-DAO

Barycentrics    (SB+SC)*(-6*sqrt(3)*S*R^2*(3*S^2-SW^2)+9*S^4-(36*R^2*(3*SA-SW)-21*SA^2+12*SB*SC+7*SW^2)*S^2-(3*SA-4*SW)*SA*SW^2) : :

The reciprocal orthologic center of these triangles is X(25171)

X(25174) lies on these lines: {6,691}, {512,11624}, {531,25231}, {11085,25172}, {11141,14176}, {11626,25221}, {16641,25210}

X(25174) = reflection of X(25221) in X(11626)

### X(25175) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 12th FERMAT-DAO TO 5th FERMAT-DAO

Barycentrics    (SB+SC)*(2*sqrt(3)*S*(-SA*SW*(9*R^2-2*SA)+(3*R^2+2*SW)*S^2)+3*S^4-(6*R^2*SW-SA^2-2*SB*SC-5*SW^2)*S^2-(6*R^2-3*SA+2*SW)*SA*SW^2) : :

The reciprocal orthologic center of these triangles is X(25162)

X(25175) lies on these lines: {5,14178}, {13,23014}, {30,14186}, {62,25161}, {298,511}, {381,25223}, {395,23028}, {512,25154}, {5318,23023}, {6785,25180}, {10653,25173}, {11080,11624}, {16808,25177}, {16809,23008}, {18581,25219}

X(25175) = reflection of X(i) in X(j) for these (i,j): (14178, 5), (25223, 381)

### X(25176) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 14th FERMAT-DAO TO 5th FERMAT-DAO

Barycentrics    (SB+SC)*(-36*sqrt(3)*S^3*R^2+9*S^4-(3*SA*(36*R^2-7*SA)+12*SB*SC+7*SW^2)*S^2-(3*SA-4*SW)*SA*SW^2) : :

The reciprocal orthologic center of these triangles is X(25171)

X(25176) lies on these lines: {6,691}, {511,8595}, {531,25233}, {3129,14176}, {5640,25225}, {8015,25172}, {10654,11002}, {14174,15018}, {16637,25210}

X(25176) = reflection of X(25225) in X(5640)

### X(25177) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO TO 5th FERMAT-DAO

Barycentrics    (SB+SC)*(sqrt(3)*((-9*R^2-2*SW)*S^2+(R^2*(18*SA+SW)-2*SA^2)*SW)*S+3*S^4-(9*R^2*(3*SA-2*SW)-7*SA^2+4*SB*SC+7*SW^2)*S^2+(3*R^2-3*SA+4*SW)*SA*SW^2) : :
X(25177) = 3*X(14)-2*X(23008) = 5*X(16961)-6*X(25161) = 3*X(16963)-4*X(25173)

The reciprocal orthologic center of these triangles is X(25162)

X(25177) lies on these lines: {14,23008}, {511,25235}, {512,25156}, {14178,16967}, {14186,16241}, {16808,25175}, {16961,25161}, {16963,25173}, {16965,23023}

X(25177) = reflection of X(i) in X(j) for these (i,j): (14, 23014), (25227, 14)
X(25177) = {X(25161), X(25219)}-harmonic conjugate of X(16961)

### X(25178) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 7th FERMAT-DAO TO 6th FERMAT-DAO

Barycentrics    (SB+SC)*(sqrt(3)*SA-S)*(2*S^2-sqrt(3)*(SB+SC)*S+SA^2-SB*SC)*(4*S^2-(SW+3*SA)*(SB+SC)) : :
X(25178) = 3*X(16267)-X(22999) = 5*X(16960)-3*X(25217) = 5*X(16960)-X(25228) = 3*X(16962)+X(25182) = 3*X(25217)-X(25228)

The reciprocal orthologic center of these triangles is X(25152)

X(25178) lies on these lines: {6,14188}, {15,23007}, {115,512}, {381,23017}, {396,25220}, {511,9117}, {6785,22513}, {10654,25180}, {11081,16248}, {11485,23022}, {14182,16644}, {16267,22999}, {16960,25217}, {16962,25182}, {18582,25224}

X(25178) = midpoint of X(15) and X(23007)
X(25178) = reflection of X(i) in X(j) for these (i,j): (25173, 14113), (25220, 396)
X(25178) = {X(16960), X(25228)}-harmonic conjugate of X(25217)

### X(25179) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 9th FERMAT-DAO TO 6th FERMAT-DAO

Barycentrics    (SB+SC)*(6*sqrt(3)*S*R^2*(3*S^2-SW^2)+9*S^4-(36*R^2*(3*SA-SW)-21*SA^2+12*SB*SC+7*SW^2)*S^2-(3*SA-4*SW)*SA*SW^2) : :

The reciprocal orthologic center of these triangles is X(25172)

X(25179) lies on these lines: {6,691}, {512,11626}, {530,25232}, {11080,25171}, {11142,14175}, {11624,25222}, {16640,25209}

X(25179) = reflection of X(25222) in X(11624)

### X(25180) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 11th FERMAT-DAO TO 6th FERMAT-DAO

Barycentrics    (SB+SC)*(-2*sqrt(3)*S*(-SA*SW*(9*R^2-2*SA)+(3*R^2+2*SW)*S^2)+3*S^4-(6*R^2*SW-SA^2-2*SB*SC-5*SW^2)*S^2-(6*R^2-3*SA+2*SW)*SA*SW^2) : :

The reciprocal orthologic center of these triangles is X(25152)

X(25180) lies on these lines: {5,14182}, {14,23007}, {30,14188}, {61,25151}, {299,383}, {381,25224}, {396,23022}, {512,25164}, {5321,23017}, {6785,25175}, {10654,25178}, {11085,11626}, {16808,22999}, {16809,25182}, {18582,25220}

X(25180) = reflection of X(i) in X(j) for these (i,j): (14182, 5), (25224, 381)

### X(25181) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 13th FERMAT-DAO TO 6th FERMAT-DAO

Barycentrics    (SB+SC)*(36*sqrt(3)*S^3*R^2+9*S^4-(3*SA*(36*R^2-7*SA)+12*SB*SC+7*SW^2)*S^2-(3*SA-4*SW)*SA*SW^2) : :

The reciprocal orthologic center of these triangles is X(25172)

X(25181) lies on these lines: {6,691}, {511,8594}, {530,25234}, {3130,14175}, {5640,25226}, {8014,25171}, {10653,11002}, {14180,15018}, {16636,25209}

X(25181) = reflection of X(25226) in X(5640)

### X(25182) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO TO 6th FERMAT-DAO

Barycentrics    (SB+SC)*(-sqrt(3)*((-9*R^2-2*SW)*S^2+(R^2*(18*SA+SW)-2*SA^2)*SW)*S+3*S^4-(9*R^2*(3*SA-2*SW)-7*SA^2+4*SB*SC+7*SW^2)*S^2+(3*R^2-3*SA+4*SW)*SA*SW^2) : :
X(25182) = 3*X(13)-2*X(22999) = 5*X(16960)-6*X(25151) = 5*X(16960)-4*X(25220) = 3*X(16962)-4*X(25178)

The reciprocal orthologic center of these triangles is X(25152)

X(25182) lies on these lines: {13,22999}, {511,25236}, {512,25166}, {14182,16966}, {14188,16242}, {16809,25180}, {16960,25151}, {16962,25178}, {16964,23017}

X(25182) = reflection of X(i) in X(j) for these (i,j): (13, 23007), (25228, 13)
X(25182) = {X(25151), X(25220)}-harmonic conjugate of X(16960)

### X(25183) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 7th FERMAT-DAO TO 1st NEUBERG

Barycentrics    2*sqrt(3)*((b^2+c^2)*a^2+4*b^2*c^2)*S+3*(b^2+c^2)*a^4-3*(b^2-c^2)^2*a^2+6*(b^2+c^2)*b^2*c^2 : :
X(25183) = X(13)-3*X(25157) = 4*X(6669)-3*X(22691) = 3*X(16267)-X(23000) = 3*X(16962)+X(25199)

The reciprocal orthologic center of these triangles is X(6582)

X(25183) lies on these lines: {13,76}, {39,22892}, {99,6295}, {115,141}, {183,6582}, {381,23018}, {395,18806}, {396,538}, {511,22796}, {616,6194}, {618,5976}, {3934,22847}, {6581,16644}, {6669,22691}, {10654,25191}, {13876,13877}, {13929,13930}, {16267,23000}, {16635,18906}, {16962,25199}

### X(25184) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 7th FERMAT-DAO TO 2nd NEUBERG

Barycentrics    2*(4*a^4+6*(b^2+c^2)*a^2+b^4+6*b^2*c^2+c^4)*sqrt(3)*S+3*(4*a^4+3*(b^2+c^2)*a^2+b^4+c^4)*(b^2+c^2) : :
X(25184) = X(13)-3*X(25158) = 3*X(16267)-X(23001) = 3*X(16962)+X(25200)

The reciprocal orthologic center of these triangles is X(6298)

X(25184) lies on these lines: {13,83}, {115,3589}, {381,23019}, {396,754}, {618,22691}, {3329,14904}, {6109,6669}, {6292,22892}, {6296,16644}, {6298,11174}, {6704,22847}, {10654,25192}, {13876,13878}, {13929,13931}, {16267,23001}, {16962,25200}

### X(25185) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 7th FERMAT-DAO TO 1st TRI-SQUARES-CENTRAL

Barycentrics    2*(2*a^4-7*b^4+16*b^2*c^2-7*c^4-2*(b^2+c^2)*a^2+6*(b^2-c^2)^2*sqrt(3))*S+3*(b^2-c^2)^2*(a^2+b^2+c^2)*sqrt(3)+6*a^6-6*(b^2+c^2)*a^4+6*b^2*c^2*a^2-6*(b^4-c^4)*(b^2-c^2) : :
X(25185) = X(13)-3*X(25159) = 3*X(16267)-X(23002) = 3*X(16962)+X(25201)

The reciprocal orthologic center of these triangles is X(13705)

X(25185) lies on these lines: {13,1327}, {115,13703}, {381,23020}, {530,10668}, {5459,13929}, {6302,13701}, {6306,22847}, {9112,22541}, {10654,25193}, {13706,16644}, {16267,23002}, {16962,25201}

### X(25186) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 7th FERMAT-DAO TO 2nd TRI-SQUARES-CENTRAL

Barycentrics
-2*(2*a^4-7*b^4+16*b^2*c^2-7*c^4-2*(b^2+c^2)*a^2-6*(b^2-c^2)^2*sqrt(3))*S-3*(b^2-c^2)^2*(a^2+b^2+c^2)*sqrt(3)+6*a^6-6*(b^2+c^2)*a^4+6*b^2*c^2*a^2-6*(b^4-c^4)*(b^2-c^2) : :
X(25186) = X(13)-3*X(25160) = 3*X(16267)-X(23003) = 3*X(16962)+X(25202)

The reciprocal orthologic center of these triangles is X(13825)

X(25186) lies on these lines: {13,1328}, {115,13823}, {381,23021}, {530,10672}, {5459,13850}, {6302,22847}, {6306,13821}, {9112,19101}, {10654,25194}, {13826,16644}, {16267,23003}, {16962,25202}

### X(25187) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 8th FERMAT-DAO TO 1st NEUBERG

Barycentrics    -2*sqrt(3)*((b^2+c^2)*a^2+4*b^2*c^2)*S+3*(b^2+c^2)*a^4-3*(b^2-c^2)^2*a^2+6*(b^2+c^2)*b^2*c^2 : :
X(25187) = X(14)-3*X(25167) = 4*X(6670)-3*X(22692) = 3*X(16268)-X(23009) = 3*X(16963)+X(25203)

The reciprocal orthologic center of these triangles is X(6295)

X(25187) lies on these lines: {14,76}, {39,22848}, {99,6582}, {115,141}, {183,6295}, {381,23024}, {395,538}, {396,18806}, {511,22797}, {617,6194}, {619,5976}, {3934,22893}, {6294,16645}, {6670,22692}, {10653,25195}, {13875,13877}, {13928,13930}, {16268,23009}, {16634,18906}, {16963,25203}

### X(25188) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 8th FERMAT-DAO TO 2nd NEUBERG

Barycentrics    -2*(4*a^4+6*(b^2+c^2)*a^2+b^4+6*b^2*c^2+c^4)*sqrt(3)*S+3*(4*a^4+3*(b^2+c^2)*a^2+b^4+c^4)*(b^2+c^2) : :
X(25188) = X(14)-3*X(25168) = 3*X(16268)-X(23010) = 3*X(16963)+X(25204)

The reciprocal orthologic center of these triangles is X(6299)

X(25188) lies on these lines: {14,83}, {115,3589}, {381,23025}, {395,754}, {619,22692}, {3329,14905}, {6108,6670}, {6292,22848}, {6297,16645}, {6299,11174}, {6704,22893}, {10653,25196}, {13875,13878}, {13928,13931}, {16268,23010}, {16963,25204}

### X(25189) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 8th FERMAT-DAO TO 1st TRI-SQUARES-CENTRAL

Barycentrics    2*(2*a^4-7*b^4+16*b^2*c^2-7*c^4-2*(b^2+c^2)*a^2-6*(b^2-c^2)^2*sqrt(3))*S-3*(b^2-c^2)^2*(a^2+b^2+c^2)*sqrt(3)+6*a^6-6*(b^2+c^2)*a^4+6*b^2*c^2*a^2-6*(b^4-c^4)*(b^2-c^2) : :
X(25189) = X(14)-3*X(25169) = 3*X(16268)-X(23011) = 3*X(16963)+X(25205)

The reciprocal orthologic center of these triangles is X(13703)

X(25189) lies on these lines: {14,1327}, {115,13703}, {381,23026}, {531,10667}, {5460,13928}, {6303,13701}, {6307,22893}, {9113,22541}, {10653,25197}, {13704,16645}, {16268,23011}, {16963,25205}

### X(25190) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 8th FERMAT-DAO TO 2nd TRI-SQUARES-CENTRAL

Barycentrics
-2*(2*a^4-7*b^4+16*b^2*c^2-7*c^4-2*(b^2+c^2)*a^2+6*(b^2-c^2)^2*sqrt(3))*S+3*(b^2-c^2)^2*(a^2+b^2+c^2)*sqrt(3)+6*a^6-6*(b^2+c^2)*a^4+6*b^2*c^2*a^2-6*(b^4-c^4)*(b^2-c^2) : :
X(25190) = X(14)-3*X(25170) = 3*X(16268)-X(23012) = 3*X(16963)+X(25206)

The reciprocal orthologic center of these triangles is X(13823)

X(25190) lies on these lines: {14,1328}, {115,13823}, {381,23027}, {531,10671}, {5460,13850}, {6303,22893}, {6307,13821}, {9113,19101}, {10653,25198}, {13824,16645}, {16268,23012}, {16963,25206}

### X(25191) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 11th FERMAT-DAO TO 1st NEUBERG

Barycentrics    -2*sqrt(3)*(a^2+b^2+c^2)*((b^2+c^2)*a^2-2*b^2*c^2)*S+3*(b^2+c^2)*a^6+9*(b^2+c^2)*a^2*b^2*c^2-6*(b^2-c^2)^2*b^2*c^2-3*(b^6+c^6)*a^2 : :

The reciprocal orthologic center of these triangles is X(6582)

X(25191) lies on these lines: {5,6581}, {14,76}, {39,11305}, {61,7770}, {194,16634}, {381,538}, {533,7812}, {732,25196}, {2782,6294}, {3107,11303}, {5321,23018}, {5969,25164}, {9466,11297}, {9760,13085}, {10654,25183}, {16808,23000}, {16809,25199}

X(25191) = reflection of X(i) in X(j) for these (i,j): (6581, 5), (25195, 6248)
X(25191) = {X(76), X(11304)}-harmonic conjugate of X(25167)

### X(25192) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 11th FERMAT-DAO TO 2nd NEUBERG

Barycentrics    -2*(a^2+b^2+c^2)*(2*a^4-b^4-c^4)*S+(4*(b^2+c^2)*a^6+(b^4+10*b^2*c^2+c^4)*a^4-2*(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^2-(b^2-c^2)^2*(b^4+4*b^2*c^2+c^4))*sqrt(3) : :

The reciprocal orthologic center of these triangles is X(6298)

X(25192) lies on these lines: {5,6296}, {14,83}, {61,25158}, {316,25204}, {381,754}, {732,25195}, {5321,23019}, {6292,11306}, {7770,16964}, {10654,25184}, {16634,20088}, {16808,23001}, {16809,25200}

X(25192) = reflection of X(i) in X(j) for these (i,j): (6296, 5), (25196, 6249)

### X(25193) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 11th FERMAT-DAO TO 1st TRI-SQUARES-CENTRAL

Barycentrics    -3*(S^2+3*(-5+4*sqrt(3))*SB*SC)*S+(2*sqrt(3)-3)*(3*SA-sqrt(3)*SW-4*SW)*S^2-9*(sqrt(3)-1)*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(13705)

X(25193) lies on these lines: {5,13706}, {14,1327}, {61,25159}, {381,13687}, {530,22634}, {5321,23020}, {10654,25185}, {16808,23002}, {16809,25201}

X(25193) = reflection of X(i) in X(j) for these (i,j): (13706, 5), (25197, 13687)

### X(25194) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 11th FERMAT-DAO TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    3*(S^2+3*(-5-4*sqrt(3))*SB*SC)*S+(-2*sqrt(3)-3)*(3*SA+sqrt(3)*SW-4*SW)*S^2-9*(-sqrt(3)-1)*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(13825)

X(25194) lies on these lines: {5,13826}, {14,1328}, {61,25160}, {381,13807}, {530,22605}, {5321,23021}, {10654,25186}, {16808,23003}, {16809,25202}

X(25194) = reflection of X(i) in X(j) for these (i,j): (13826, 5), (25198, 13807)

### X(25195) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 12th FERMAT-DAO TO 1st NEUBERG

Barycentrics    2*sqrt(3)*(a^2+b^2+c^2)*((b^2+c^2)*a^2-2*b^2*c^2)*S+3*(b^2+c^2)*a^6+9*(b^2+c^2)*a^2*b^2*c^2-6*(b^2-c^2)^2*b^2*c^2-3*(b^6+c^6)*a^2 : :

The reciprocal orthologic center of these triangles is X(6295)

X(25195) lies on these lines: {5,6294}, {13,76}, {39,11306}, {62,7770}, {194,16635}, {381,538}, {532,7812}, {732,25192}, {2782,6581}, {3106,11304}, {5318,23024}, {5969,25154}, {9466,11298}, {9762,13085}, {10653,25187}, {16808,25203}, {16809,23009}

X(25195) = reflection of X(i) in X(j) for these (i,j): (6294, 5), (25191, 6248)
X(25195) = {X(76), X(11303)}-harmonic conjugate of X(25157)

### X(25196) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 12th FERMAT-DAO TO 2nd NEUBERG

Barycentrics    2*(a^2+b^2+c^2)*(2*a^4-b^4-c^4)*S+(4*(b^2+c^2)*a^6+(b^4+10*b^2*c^2+c^4)*a^4-2*(-c^2+2*b^2)*(b^2-2*c^2)*(b^2+c^2)*a^2-(b^2-c^2)^2*(b^4+4*b^2*c^2+c^4))*sqrt(3) : :

The reciprocal orthologic center of these triangles is X(6299)

X(25196) lies on these lines: {5,6297}, {13,83}, {62,25168}, {316,25200}, {381,754}, {732,25191}, {5318,23025}, {6292,11305}, {7770,16965}, {10653,25188}, {16635,20088}, {16808,25204}, {16809,23010}

X(25196) = reflection of X(i) in X(j) for these (i,j): (6297, 5), (25192, 6249)

### X(25197) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 12th FERMAT-DAO TO 1st TRI-SQUARES-CENTRAL

Barycentrics    -3*(S^2+3*(-5-4*sqrt(3))*SB*SC)*S+(-2*sqrt(3)-3)*(3*SA+sqrt(3)*SW-4*SW)*S^2-9*(-sqrt(3)-1)*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(13703)

X(25197) lies on these lines: {5,13704}, {13,1327}, {62,25169}, {381,13687}, {531,22635}, {5318,23026}, {10653,25189}, {16808,25205}, {16809,23011}

X(25197) = reflection of X(i) in X(j) for these (i,j): (13704, 5), (25193, 13687)

### X(25198) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 12th FERMAT-DAO TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    3*(S^2+3*(-5+4*sqrt(3))*SB*SC)*S+(2*sqrt(3)-3)*(3*SA-sqrt(3)*SW-4*SW)*S^2-9*(sqrt(3)-1)*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(13823)

X(25198) lies on these lines: {5,13824}, {13,1328}, {62,25170}, {381,13807}, {531,22606}, {5318,23027}, {10653,25190}, {16808,25206}, {16809,23012}

X(25198) = reflection of X(i) in X(j) for these (i,j): (13824, 5), (25194, 13807)

### X(25199) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO TO 1st NEUBERG

Barycentrics    -sqrt(3)*((SA-8*SW)*S^2-(11*SA^2+SB*SC-3*SW^2)*SW)*S+12*S^4+3*(5*SA^2-SA*SW-SW^2)*S^2+3*SB*SC*SW^2 : :
X(25199) = 3*X(13)-2*X(23000) = 4*X(621)-3*X(22695) = 5*X(16960)-6*X(25157) = 3*X(16962)-4*X(25183)

The reciprocal orthologic center of these triangles is X(6582)

X(25199) lies on these lines: {13,538}, {18,76}, {621,22695}, {622,20081}, {732,23024}, {3818,13108}, {5969,25166}, {6581,16966}, {16809,25191}, {16960,25157}, {16962,25183}, {16964,23018}

### X(25200) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO TO 2nd NEUBERG

Barycentrics    sqrt(3)*((SA-12*SW)*S^2-(7*SA^2-19*SB*SC+SW^2)*SW)*S+15*S^4+3*(-9*SB*SC-3*SW^2+4*SA^2)*S^2-15*SB*SC*SW^2 : :
X(25200) = 3*X(13)-2*X(23001) = 5*X(16960)-6*X(25158) = 3*X(16962)-4*X(25184)

The reciprocal orthologic center of these triangles is X(6298)

X(25200) lies on these lines: {13,754}, {18,83}, {316,25196}, {621,6777}, {624,2896}, {6296,16966}, {13111,19130}, {16809,25192}, {16960,25158}, {16962,25184}, {16964,23019}

### X(25201) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO TO 1st TRI-SQUARES-CENTRAL

Barycentrics
2*((55*a^4-35*(b^2+c^2)*a^2-38*(b^2-c^2)^2)*S+3*(a^2-b^2-c^2)*(3*(b^2+c^2)*a^2-(b^2-c^2)^2))*sqrt(3)-12*(24*a^4-9*(b^2+c^2)*a^2-17*(b^2-c^2)^2)*S+23*a^6-104*(b^2+c^2)*a^4+5*(17*b^4-6*b^2*c^2+17*c^4)*a^2-4*(b^4-c^4)*(b^2-c^2) : :
X(25201) = 5*X(16960)-6*X(25159) = 3*X(16962)-4*X(25185)

The reciprocal orthologic center of these triangles is X(13705)

X(25201) lies on these lines: {13,23002}, {18,1327}, {530,22636}, {13706,16966}, {13713,25205}, {16809,25193}, {16960,25159}, {16962,25185}, {16964,23020}

### X(25202) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO TO 2nd TRI-SQUARES-CENTRAL

Barycentrics
-2*(-(55*a^4-35*(b^2+c^2)*a^2-38*(b^2-c^2)^2)*S+3*(a^2-b^2-c^2)*(3*(b^2+c^2)*a^2-(b^2-c^2)^2))*sqrt(3)+12*(24*a^4-9*(b^2+c^2)*a^2-17*(b^2-c^2)^2)*S+23*a^6-104*(b^2+c^2)*a^4+5*(17*b^4-6*b^2*c^2+17*c^4)*a^2-4*(b^4-c^4)*(b^2-c^2) : :
X(25202) = 3*X(13)-2*X(23003) = 5*X(16960)-6*X(25160) = 3*X(16962)-4*X(25186)

The reciprocal orthologic center of these triangles is X(13825)

X(25202) lies on these lines: {13,23003}, {18,1328}, {530,22607}, {13826,16966}, {13836,25206}, {16809,25194}, {16960,25160}, {16962,25186}, {16964,23021}

### X(25203) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO TO 1st NEUBERG

Barycentrics    sqrt(3)*((SA-8*SW)*S^2-(11*SA^2+SB*SC-3*SW^2)*SW)*S+12*S^4+3*(5*SA^2-SW*SA-SW^2)*S^2+3*SB*SC*SW^2 : :
X(25203) = 3*X(14)-2*X(23009) = 4*X(622)-3*X(22696) = 5*X(16961)-6*X(25167) = 3*X(16963)-4*X(25187)

The reciprocal orthologic center of these triangles is X(6295)

X(25203) lies on these lines: {14,538}, {17,76}, {621,20081}, {622,22696}, {732,23018}, {3818,13108}, {5969,25156}, {6294,16967}, {16808,25195}, {16961,25167}, {16963,25187}, {16965,23024}

### X(25204) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO TO 2nd NEUBERG

Barycentrics    -sqrt(3)*((SA-12*SW)*S^2-(7*SA^2-19*SB*SC+SW^2)*SW)*S+15*S^4+3*(-9*SB*SC-3*SW^2+4*SA^2)*S^2-15*SB*SC*SW^2 : :
X(25204) = 3*X(14)-2*X(23010) = 5*X(16961)-6*X(25168) = 3*X(16963)-4*X(25188)

The reciprocal orthologic center of these triangles is X(6299)

X(25204) lies on these lines: {14,754}, {17,83}, {316,25192}, {622,6778}, {623,2896}, {6297,16967}, {13111,19130}, {16808,25196}, {16961,25168}, {16963,25188}, {16965,23025}

### X(25205) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO TO 1st TRI-SQUARES-CENTRAL

Barycentrics
-2*((55*a^4-35*(b^2+c^2)*a^2-38*(b^2-c^2)^2)*S+3*(a^2-b^2-c^2)*(3*(b^2+c^2)*a^2-(b^2-c^2)^2))*sqrt(3)-12*(24*a^4-9*(b^2+c^2)*a^2-17*(b^2-c^2)^2)*S+23*a^6-104*(b^2+c^2)*a^4+5*(17*b^4-6*b^2*c^2+17*c^4)*a^2-4*(b^4-c^4)*(b^2-c^2) : :
X(25205) = 3*X(14)-2*X(23011) = 5*X(16961)-6*X(25169) = 3*X(16963)-4*X(25189)

The reciprocal orthologic center of these triangles is X(13703)

X(25205) lies on these lines: {14,23011}, {17,1327}, {531,22637}, {13704,16967}, {13713,25201}, {16808,25197}, {16961,25169}, {16963,25189}, {16965,23026}

### X(25206) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO TO 2nd TRI-SQUARES-CENTRAL

Barycentrics
2*(-(55*a^4-35*(b^2+c^2)*a^2-38*(b^2-c^2)^2)*S+3*(a^2-b^2-c^2)*(3*(b^2+c^2)*a^2-(b^2-c^2)^2))*sqrt(3)+12*(24*a^4-9*(b^2+c^2)*a^2-17*(b^2-c^2)^2)*S+23*a^6-104*(b^2+c^2)*a^4+5*(17*b^4-6*b^2*c^2+17*c^4)*a^2-4*(b^4-c^4)*(b^2-c^2) : :
X(25206) = 5*X(16961)-6*X(25170) = 3*X(16963)-4*X(25190)

The reciprocal orthologic center of these triangles is X(13823)

X(25206) lies on these lines: {14,23012}, {17,1328}, {531,22608}, {13824,16967}, {13836,25202}, {16808,25198}, {16961,25170}, {16963,25190}, {16965,23027}

### X(25207) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd FERMAT-DAO TO 3rd FERMAT-DAO

Barycentrics
-sqrt(3)*((3*SA*(6*R^2+SA)+6*SB*SC-SW^2)*S^2-(3*SA*(2*R^2-SA)-6*SB*SC+SW^2)*SW^2)*S+(SW+3*SA+90*R^2)*S^4+(18*R^2*(-SW^2-4*SA*SW+6*SA^2)+SW*(-2*SW^2+3*SA^2))*S^2+3*SB*SC*SW^3 : :

The reciprocal orthologic center of these triangles is X(25151)

X(25207) lies on these lines: {13,99}, {530,16248}, {531,25212}, {2782,16260}, {14181,16643}, {16262,25208}, {16637,25155}, {16639,25152}, {16641,25153}

X(25207) = reflection of X(25216) in X(16260)

### X(25208) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st FERMAT-DAO TO 4th FERMAT-DAO

Barycentrics
sqrt(3)*((3*SA*(6*R^2+SA)+6*SB*SC-SW^2)*S^2-(3*SA*(2*R^2-SA)-6*SB*SC+SW^2)*SW^2)*S+(90*R^2+3*SA+SW)*S^4+(18*R^2*(-SW^2-4*SA*SW+6*SA^2)+SW*(-2*SW^2+3*SA^2))*S^2+3*SB*SC*SW^3 : :

The reciprocal orthologic center of these triangles is X(25161)

X(25208) lies on these lines: {14,99}, {530,25211}, {531,16247}, {2782,16259}, {14177,16642}, {16262,25207}, {16636,25165}, {16638,25162}, {16640,25163}

X(25208) = reflection of X(25213) in X(16259)

### X(25209) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st FERMAT-DAO TO 2nd FERMAT-DAO

Barycentrics
(SB+SC)*(-2*sqrt(3)*(-SA*SW^3*(36*R^2-9*SA+4*SW)+(18*R^2*(54*R^2*SA-9*SA^2+6*SW*SA-SW^2)+SW*(9*SA^2-36*SW*SA+13*SW^2))*S^2+9*(5*SW-36*R^2)*S^4)*S+81*S^6+(216*R^2*(9*R^2-3*SA-2*SW)+117*SA^2-36*SB*SC+24*SW^2)*S^4-3*(12*R^2*(9*SA^2+6*SW*SA-SW^2)-SW*(24*SA^2+16*SW*SA-7*SW^2))*SW*S^2-3*(3*SA-4*SW)*SA*SW^4) : :

The reciprocal orthologic center of these triangles is X(25210)

X(25209) lies on these lines: {530,25213}, {14175,16642}, {16259,25211}, {16636,25181}, {16638,25171}, {16640,25179}

X(25209) = reflection of X(25211) in X(16259)

### X(25210) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd FERMAT-DAO TO 1st FERMAT-DAO

Barycentrics
(SB+SC)*(2*sqrt(3)*(-SA*SW^3*(36*R^2-9*SA+4*SW)+(18*R^2*(54*R^2*SA-9*SA^2+6*SW*SA-SW^2)+SW*(9*SA^2-36*SW*SA+13*SW^2))*S^2+9*(5*SW-36*R^2)*S^4)*S+81*S^6+(216*R^2*(9*R^2-3*SA-2*SW)+117*SA^2-36*SB*SC+24*SW^2)*S^4-3*(12*R^2*(9*SA^2+6*SW*SA-SW^2)-SW*(24*SA^2+16*SW*SA-7*SW^2))*SW*S^2-3*(3*SA-4*SW)*SA*SW^4) : :

The reciprocal orthologic center of these triangles is X(25209)

X(25210) lies on these lines: {531,25216}, {14176,16643}, {16260,25212}, {16637,25176}, {16639,25172}, {16641,25174}

X(25210) = reflection of X(25212) in X(16260)

### X(25211) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st FERMAT-DAO TO 2nd FERMAT-DAO

Barycentrics
(SB+SC)*(81*S^6+(648*R^4-72*(6*SA+SW)*R^2+105*SA^2-24*SB*SC-36*SW^2)*S^4-3*SW*(4*(SW^2+9*SA^2)*R^2+SW*(4*SA^2-8*SA*SW-SW^2))*S^2+3*SA^2*SW^4-2*sqrt(3)*((-72*R^2+7*SW)*S^4+(18*R^2*(18*R^2*SA-3*SA^2-SW^2)+SA*SW*(3*SA-4*SW)+7*SW^3)*S^2+SA*SW^3*(3*SA-4*SW))*S) : :

The reciprocal parallelogic center of these triangles is X(25212)

X(25211) lies on these lines: {530,25208}, {14183,16642}, {16259,25209}, {16636,25226}, {16638,25218}, {16640,25222}

X(25211) = reflection of X(25209) in X(16259)

### X(25212) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd FERMAT-DAO TO 1st FERMAT-DAO

Barycentrics
(SB+SC)*(81*S^6+(648*R^4-72*(6*SA+SW)*R^2+105*SA^2-24*SB*SC-36*SW^2)*S^4-3*SW*(4*(SW^2+9*SA^2)*R^2+SW*(4*SA^2-8*SA*SW-SW^2))*S^2+3*SA^2*SW^4+2*sqrt(3)*((-72*R^2+7*SW)*S^4+(18*R^2*(18*R^2*SA-3*SA^2-SW^2)+SA*SW*(3*SA-4*SW)+7*SW^3)*S^2+SA*SW^3*(3*SA-4*SW))*S) : :

The reciprocal parallelogic center of these triangles is X(25211)

X(25212) lies on these lines: {531,25207}, {14184,16643}, {16260,25210}, {16637,25225}, {16639,25215}, {16641,25221}

X(25212) = reflection of X(25210) in X(16260)

### X(25213) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st FERMAT-DAO TO 4th FERMAT-DAO

Barycentrics
sqrt(3)*(-3*(SA-SW+6*R^2)*S^4+(18*(-SW+2*SA)*R^2+5*SA^2+4*SB*SC-2*SW^2)*SW*S^2+(36*R^2+SW)*SB*SC*SW^2)*S+18*S^6-9*(6*SA*R^2-3*SA^2+SW^2)*S^4+3*(6*SA*R^2+SA^2+2*SB*SC-SW^2)*SW^2*S^2+6*SB*SC*SW^4 : :

The reciprocal parallelogic center of these triangles is X(25214)

X(25213) lies on these lines: {530,25209}, {2782,16259}, {14185,16642}, {16262,25216}, {16636,25234}, {16638,25230}, {16640,25232}

X(25213) = reflection of X(25208) in X(16259)

### X(25214) = PARALLELOGIC CENTER OF THESE TRIANGLES: 4th FERMAT-DAO TO 1st FERMAT-DAO

Barycentrics    (SB+SC)*(-sqrt(3)*(SW*(3*SW*R^2-4*SA^2)+(15*R^2-4*SW)*S^2)*S+9*S^4+(-3*(4*SW+9*SA)*R^2+11*SA^2-2*SB*SC-SW^2)*S^2-(3*R^2-2*SW-SA)*SA*SW^2) : :
X(25214) = X(14)+2*X(25219) = 2*X(395)+X(23008) = 4*X(11543)-X(23014) = 5*X(16961)-2*X(25173) = 5*X(16961)+X(25227)

The reciprocal parallelogic center of these triangles is X(25213)

X(25214) lies on these lines: {14,25219}, {18,14186}, {62,25223}, {395,23008}, {511,5469}, {512,16530}, {11543,23014}, {16268,25161}, {16809,23028}, {16961,25173}

X(25214) = reflection of X(25161) in X(16268)
X(25214) = {X(16961), X(25227)}-harmonic conjugate of X(25173)

### X(25215) = PARALLELOGIC CENTER OF THESE TRIANGLES: 6th FERMAT-DAO TO 1st FERMAT-DAO

Barycentrics
(SB+SC)*(sqrt(3)*SB-S)*(sqrt(3)*SC-S)*(sqrt(3)*(7*S^4+(24*SW*R^2+7*SA^2+8*SB*SC-9*SW^2)*S^2-SW^2*SA^2)*S+(36*R^2+5*SW)*S^4+(108*R^2*SA^2-(3*SA^2+8*SA*SW+3*SW^2)*SW)*S^2-3*SA^2*SW^3) : :

The reciprocal parallelogic center of these triangles is X(25211)

X(25215) lies on these lines: {531,25152}, {8015,25225}, {11085,25221}, {14184,16464}, {16462,25172}, {16639,25212}

### X(25216) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd FERMAT-DAO TO 3rd FERMAT-DAO

Barycentrics
-sqrt(3)*(-3*(SA-SW+6*R^2)*S^4+(18*(-SW+2*SA)*R^2+5*SA^2+4*SB*SC-2*SW^2)*SW*S^2+(36*R^2+SW)*SB*SC*SW^2)*S+18*S^6-9*(6*R^2*SA-3*SA^2+SW^2)*S^4+3*(6*R^2*SA+SA^2+2*SB*SC-SW^2)*SW^2*S^2+6*SB*SC*SW^4 : :

The reciprocal parallelogic center of these triangles is X(25217)

X(25216) lies on these lines: {531,25210}, {2782,16260}, {14187,16643}, {16262,25213}, {16639,25229}, {16641,25231}

X(25216) = reflection of X(25207) in X(16260)

### X(25217) = PARALLELOGIC CENTER OF THESE TRIANGLES: 3rd FERMAT-DAO TO 2nd FERMAT-DAO

Barycentrics    (SB+SC)*(sqrt(3)*(SW*(3*SW*R^2-4*SA^2)+(15*R^2-4*SW)*S^2)*S+9*S^4+(-3*(4*SW+9*SA)*R^2+11*SA^2-2*SB*SC-SW^2)*S^2-(3*R^2-2*SW-SA)*SA*SW^2) : :
X(25217) = X(13)+2*X(25220) = 2*X(396)+X(22999) = 4*X(11542)-X(23007) = 5*X(16960)-2*X(25178) = 5*X(16960)+X(25228) = 2*X(25178)+X(25228)

The reciprocal parallelogic center of these triangles is X(25216)

X(25217) lies on these lines: {13,25220}, {17,14188}, {61,25224}, {396,22999}, {511,5470}, {512,16529}, {11542,23007}, {16267,25151}, {16808,23022}, {16960,25178}

X(25217) = reflection of X(25151) in X(16267)
X(25217) = {X(16960), X(25228)}-harmonic conjugate of X(25178)

### X(25218) = PARALLELOGIC CENTER OF THESE TRIANGLES: 5th FERMAT-DAO TO 2nd FERMAT-DAO

Barycentrics
(SB+SC)*(sqrt(3)*SB+S)*(sqrt(3)*SC+S)*(-sqrt(3)*(7*S^4+(24*SW*R^2+7*SA^2+8*SB*SC-9*SW^2)*S^2-SW^2*SA^2)*S+(36*R^2+5*SW)*S^4+(108*R^2*SA^2-(3*SA^2+8*SA*SW+3*SW^2)*SW)*S^2-3*SA^2*SW^3) : :

The reciprocal parallelogic center of these triangles is X(25212)

X(25218) lies on these lines: {530,25162}, {16461,25171}, {16638,25211}

### X(25219) = PARALLELOGIC CENTER OF THESE TRIANGLES: 8th FERMAT-DAO TO 1st FERMAT-DAO

Barycentrics    (SB+SC)*(sqrt(3)*(SW*((-SW+3*SA)*R^2+SA^2)-(6*R^2-SW)*S^2)*S+3*S^4-(3*(SW+3*SA)*R^2-4*SA^2+SB*SC+SW^2)*S^2+SW^3*SA) : :
X(25219) = X(14)-3*X(25214) = 3*X(16268)-X(23014) = 5*X(16961)-3*X(25161) = 5*X(16961)-X(25177) = 3*X(16963)+X(25227) = 3*X(25161)-X(25177)

The reciprocal parallelogic center of these triangles is X(25213)

X(25219) lies on these lines: {6,14178}, {14,25214}, {16,23008}, {115,511}, {381,23028}, {395,25173}, {512,9115}, {10653,25223}, {11486,23023}, {14186,16645}, {16268,23014}, {16961,25161}, {16963,25227}, {18581,25175}

X(25219) = midpoint of X(16) and X(23008)
X(25219) = reflection of X(25173) in X(395)
X(25219) = {X(16961), X(25177)}-harmonic conjugate of X(25161)

### X(25220) = PARALLELOGIC CENTER OF THESE TRIANGLES: 7th FERMAT-DAO TO 2nd FERMAT-DAO

Barycentrics    (SB+SC)*(-sqrt(3)*(SW*((-SW+3*SA)*R^2+SA^2)-(6*R^2-SW)*S^2)*S+3*S^4-(3*(SW+3*SA)*R^2-4*SA^2+SB*SC+SW^2)*S^2+SW^3*SA) : :
X(25220) = X(13)-3*X(25217) = 3*X(16267)-X(23007) = 5*X(16960)-3*X(25151) = 5*X(16960)-X(25182) = 3*X(16962)+X(25228) = 3*X(25151)-X(25182)

The reciprocal parallelogic center of these triangles is X(25216)

X(25220) lies on these lines: {6,14182}, {13,25217}, {15,22999}, {115,511}, {381,23022}, {396,25178}, {512,9117}, {10654,25224}, {11485,23017}, {14188,16644}, {16267,23007}, {16960,25151}, {16962,25228}, {18582,25180}

X(25220) = midpoint of X(15) and X(22999)
X(25220) = reflection of X(25178) in X(396)
X(25220) = {X(16960), X(25182)}-harmonic conjugate of X(25151)

### X(25221) = PARALLELOGIC CENTER OF THESE TRIANGLES: 10th FERMAT-DAO TO 1st FERMAT-DAO

Barycentrics    (SB+SC)*(-2*sqrt(3)*(SW*(3*SW*R^2-2*SA^2-4*SB*SC)+3*(9*R^2-2*SW)*S^2)*S+27*S^4-3*(12*(3*SA+SW)*R^2-9*SA^2-SW^2)*S^2+3*SA^2*SW^2) : :

The reciprocal parallelogic center of these triangles is X(25211)

X(25221) lies on these lines: {6,25225}, {531,25153}, {671,25222}, {11085,25215}, {11141,14184}, {11626,25174}, {16641,25212}

X(25221) = reflection of X(25174) in X(11626)

### X(25222) = PARALLELOGIC CENTER OF THESE TRIANGLES: 9th FERMAT-DAO TO 2nd FERMAT-DAO

Barycentrics    (SB+SC)*(2*sqrt(3)*(SW*(3*SW*R^2-2*SA^2-4*SB*SC)+3*(9*R^2-2*SW)*S^2)*S+27*S^4-3*(12*(3*SA+SW)*R^2-9*SA^2-SW^2)*S^2+3*SA^2*SW^2) : :

The reciprocal parallelogic center of these triangles is X(25212)

X(25222) lies on these lines: {6,25226}, {530,25163}, {671,25221}, {11080,25218}, {11142,14183}, {11624,25179}, {16640,25211}

X(25222) = reflection of X(25179) in X(11624)

### X(25223) = PARALLELOGIC CENTER OF THESE TRIANGLES: 12th FERMAT-DAO TO 1st FERMAT-DAO

Barycentrics    (SB+SC)*(-2*sqrt(3)*(SA*SW*(3*R^2-2*SA)+(3*R^2-2*SW)*S^2)*S+9*S^4-(6*(6*SA-SW)*R^2-11*SA^2+2*SB*SC+SW^2)*S^2-(6*R^2-2*SW-SA)*SA*SW^2) : :

The reciprocal parallelogic center of these triangles is X(25213)

X(25223) lies on these lines: {5,14186}, {13,23008}, {30,14178}, {62,25214}, {381,25175}, {395,23023}, {511,25154}, {512,5617}, {5318,23028}, {6787,11178}, {10653,25219}, {16808,25227}, {16809,23014}, {18581,25173}

X(25223) = reflection of X(i) in X(j) for these (i,j): (14186, 5), (25175, 381)

### X(25224) = PARALLELOGIC CENTER OF THESE TRIANGLES: 11th FERMAT-DAO TO 2nd FERMAT-DAO

Barycentrics    (SB+SC)*(2*sqrt(3)*(SA*SW*(3*R^2-2*SA)+(3*R^2-2*SW)*S^2)*S+9*S^4-(6*(6*SA-SW)*R^2-11*SA^2+2*SB*SC+SW^2)*S^2-(6*R^2-2*SW-SA)*SA*SW^2) : :

The reciprocal parallelogic center of these triangles is X(25216)

X(25224) lies on these lines: {5,14188}, {14,22999}, {30,14182}, {61,25217}, {381,25180}, {396,23017}, {511,25164}, {512,5613}, {5321,23022}, {6787,11178}, {10654,25220}, {16808,23007}, {16809,25228}, {18582,25178}

X(25224) = reflection of X(i) in X(j) for these (i,j): (14188, 5), (25180, 381)

### X(25225) = PARALLELOGIC CENTER OF THESE TRIANGLES: 14th FERMAT-DAO TO 1st FERMAT-DAO

Barycentrics    (SB+SC)*(-4*sqrt(3)*((9*R^2-5*SW)*S^2-(3*SA-2*SW)*SA*SW)*S+27*S^4-3*(36*R^2*SA-9*SA^2-SW^2)*S^2+3*SA^2*SW^2) : :

The reciprocal parallelogic center of these triangles is X(25211)

X(25225) lies on these lines: {6,25221}, {23,14174}, {511,25154}, {621,11581}, {3129,14184}, {5640,25176}, {8015,25215}, {11317,25226}, {16637,25212}

X(25225) = reflection of X(25176) in X(5640)

### X(25226) = PARALLELOGIC CENTER OF THESE TRIANGLES: 13th FERMAT-DAO TO 2nd FERMAT-DAO

Barycentrics    (SB+SC)*(4*sqrt(3)*((9*R^2-5*SW)*S^2-(3*SA-2*SW)*SA*SW)*S+27*S^4-3*(36*R^2*SA-9*SA^2-SW^2)*S^2+3*SA^2*SW^2) : :

The reciprocal parallelogic center of these triangles is X(25212)

X(25226) lies on these lines: {6,25222}, {23,14180}, {511,25164}, {530,25165}, {622,11582}, {3130,14183}, {5640,25181}, {8014,25218}, {11317,25225}, {16636,25211}

X(25226) = reflection of X(25181) in X(5640)

### X(25227) = PARALLELOGIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO TO 1st FERMAT-DAO

Barycentrics    (SB+SC)*(-3*sqrt(3)*(-SW*((-SW+6*SA)*R^2+2*SA^2)+(15*R^2-2*SW)*S^2)*S+27*S^4-3*(3*(15*SA-2*SW)*R^2-13*SA^2+4*SB*SC+5*SW^2)*S^2-3*(3*R^2-4*SW+SA)*SA*SW^2) : :
X(25227) = 5*X(16961)-4*X(25173) = 3*X(16963)-4*X(25219)

The reciprocal parallelogic center of these triangles is X(25213)

X(25227) lies on these lines: {14,23008}, {511,25156}, {512,25235}, {14178,16241}, {14186,16967}, {16808,25223}, {16961,25173}, {16963,25219}, {16965,23028}

X(25227) = reflection of X(i) in X(j) for these (i,j): (14, 23008), (25177, 14)
X(25227) = {X(25173), X(25214)}-harmonic conjugate of X(16961)

### X(25228) = PARALLELOGIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO TO 2nd FERMAT-DAO

Barycentrics    (SB+SC)*(3*sqrt(3)*(-SW*((-SW+6*SA)*R^2+2*SA^2)+(15*R^2-2*SW)*S^2)*S+27*S^4-3*(3*(15*SA-2*SW)*R^2-13*SA^2+4*SB*SC+5*SW^2)*S^2-3*(3*R^2-4*SW+SA)*SA*SW^2) : :
X(25228) = 5*X(16960)-4*X(25178) = 5*X(16960)-6*X(25217) = 3*X(16962)-4*X(25220)

The reciprocal parallelogic center of these triangles is X(25216)

X(25228) lies on these lines: {13,22999}, {511,25166}, {512,25236}, {14182,16242}, {14188,16966}, {16809,25224}, {16960,25178}, {16962,25220}, {16964,23022}

X(25228) = reflection of X(i) in X(j) for these (i,j): (13, 22999), (25182, 13)
X(25228) = {X(25178), X(25217)}-harmonic conjugate of X(16960)

### X(25229) = PARALLELOGIC CENTER OF THESE TRIANGLES: 6th FERMAT-DAO TO 3rd FERMAT-DAO

Barycentrics    (sqrt(3)*SB-S)*(sqrt(3)*SC-S)*(sqrt(3)*((R^2-SW)*SW^2+(15*R^2-SW)*S^2)*S+(6*R^2*SW+2*SA^2+SB*SC-2*SW^2)*S^2+SW^2*SB*SC) : :

The reciprocal parallelogic center of these triangles is X(25217)

X(25229) lies on these lines: {531,25172}, {2782,16462}, {6321,11582}, {8015,25233}, {11085,25231}, {14187,16464}, {16639,25216}

### X(25230) = PARALLELOGIC CENTER OF THESE TRIANGLES: 5th FERMAT-DAO TO 4th FERMAT-DAO

Barycentrics    (sqrt(3)*SB+S)*(sqrt(3)*SC+S)*(-sqrt(3)*((R^2-SW)*SW^2+(15*R^2-SW)*S^2)*S+(6*R^2*SW+2*SA^2+SB*SC-2*SW^2)*S^2+SW^2*SB*SC) : :

The reciprocal parallelogic center of these triangles is X(25214)

X(25230) lies on these lines: {530,25171}, {2782,16461}, {6321,11581}, {8014,25234}, {11080,25232}, {14185,16463}, {16638,25213}

### X(25231) = PARALLELOGIC CENTER OF THESE TRIANGLES: 10th FERMAT-DAO TO 3rd FERMAT-DAO

Barycentrics    sqrt(3)*(-(3*R^2+SA)*(SB+SC)*SW+(6*R^2-SA+SW)*S^2)*S+(3*(3*SA-SW)*R^2+SA^2+2*SB*SC-SW^2)*S^2+2*SW^2*SB*SC : :

The reciprocal parallelogic center of these triangles is X(25217)

X(25231) lies on these lines: {6,25233}, {531,25174}, {2782,11626}, {2854,25163}, {11085,25229}, {11141,14187}, {16641,25216}

X(25231) = reflection of X(25153) in X(11626)

### X(25232) = PARALLELOGIC CENTER OF THESE TRIANGLES: 9th FERMAT-DAO TO 4th FERMAT-DAO

Barycentrics    -sqrt(3)*(-(3*R^2+SA)*(SB+SC)*SW+(6*R^2-SA+SW)*S^2)*S+(3*(3*SA-SW)*R^2+SA^2+2*SB*SC-SW^2)*S^2+2*SW^2*SB*SC : :

The reciprocal parallelogic center of these triangles is X(25214)

X(25232) lies on these lines: {6,25234}, {530,25179}, {2782,11624}, {2854,25153}, {11080,25230}, {11142,14185}, {16640,25213}

X(25232) = reflection of X(25163) in X(11624)

### X(25233) = PARALLELOGIC CENTER OF THESE TRIANGLES: 3rd FERMAT-DAO TO 14th FERMAT-DAO

Barycentrics    sqrt(3)*(SW*SB*SC+(6*R^2-SA)*S^2)*S+(6*R^2*SW+SA^2+2*SB*SC-SW^2)*S^2+2*SW^2*SB*SC : :

The reciprocal parallelogic center of these triangles is X(25234)

X(25233) lies on these lines: {6,25231}, {23,22687}, {531,25176}, {2782,5640}, {3129,14187}, {5617,11078}, {8015,25229}, {16637,25216}, {21466,22998}

X(25233) = reflection of X(25155) in X(5640)

### X(25234) = PARALLELOGIC CENTER OF THESE TRIANGLES: 13th FERMAT-DAO TO 4th FERMAT-DAO

Barycentrics    -sqrt(3)*(SW*SB*SC+(6*R^2-SA)*S^2)*S+(6*R^2*SW+SA^2+2*SB*SC-SW^2)*S^2+2*SW^2*SB*SC : :

The reciprocal parallelogic center of these triangles is X(25214)

X(25234) lies on these lines: {6,25232}, {23,22689}, {530,25181}, {2782,5640}, {3130,14185}, {5613,11092}, {8014,25230}, {16636,25213}, {21467,22997}

X(25234) = reflection of X(25165) in X(5640)

### X(25235) = PARALLELOGIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO TO 3rd FERMAT-DAO

Barycentrics    -sqrt(3)*(4*S^2-(9*SA-SW)*(SB+SC))*S+3*(5*SA-2*SW)*S^2+3*SW*SB*SC : :
X(25235) = 2*X(13)-3*X(16963) = 3*X(14)-4*X(6782) = 3*X(14)-2*X(23005) = 2*X(115)-3*X(16530) = 4*X(115)-5*X(16961) = 4*X(187)-3*X(16529) = 4*X(395)-3*X(5470) = 4*X(618)-3*X(21360) = 2*X(6782)-3*X(22998) = 8*X(6782)-3*X(25156) = 4*X(9115)-3*X(16963) = 6*X(16530)-5*X(16961) = 3*X(22998)-X(23005) = 4*X(22998)-X(25156) = 4*X(23005)-3*X(25156)

The reciprocal parallelogic center of these triangles is X(16529)

X(25235) lies on these lines: {13,5055}, {14,530}, {15,532}, {16,5613}, {17,5472}, {115,16530}, {187,16529}, {395,5470}, {511,25177}, {512,25227}, {524,6780}, {542,6779}, {618,21360}, {5463,5569}, {5464,8595}, {5617,16808}, {5965,6781}, {5980,7777}, {6108,23234}, {6770,10646}, {6777,13102}, {8860,9762}, {9114,10488}, {11489,22846}, {13103,16809}, {16965,23006}, {22496,22578}, {22507,23004}

X(25235) = reflection of X(i) in X(j) for these (i,j): (13, 9115), (14, 22998), (5464, 8595), (6778, 16), (19107, 6777), (22494, 5463), (22578, 22496), (23005, 6782), (25156, 14), (25236, 6781)
X(25235) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 9115, 16963), (115, 16530, 16961), (6782, 23005, 14), (22998, 23005, 6782)

### X(25236) = PARALLELOGIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO TO 4th FERMAT-DAO

Barycentrics    sqrt(3)*(4*S^2-(9*SA-SW)*(SB+SC))*S+3*(5*SA-2*SW)*S^2+3*SW*SB*SC : :
X(25236) = 3*X(13)-4*X(6783) = 3*X(13)-2*X(23004) = 2*X(14)-3*X(16962) = 2*X(115)-3*X(16529) = 4*X(115)-5*X(16960) = 4*X(187)-3*X(16530) = 4*X(396)-3*X(5469) = 4*X(619)-3*X(21359) = 2*X(6783)-3*X(22997) = 8*X(6783)-3*X(25166) = 4*X(9117)-3*X(16962) = 6*X(16529)-5*X(16960) = 3*X(22997)-X(23004) = 4*X(22997)-X(25166) = 4*X(23004)-3*X(25166)

The reciprocal parallelogic center of these triangles is X(16530)

X(25236) lies on these lines: {13,531}, {14,5055}, {15,5617}, {16,533}, {18,5471}, {115,16529}, {187,16530}, {396,5469}, {511,25182}, {512,25228}, {524,6779}, {542,6780}, {619,21359}, {5463,8594}, {5464,5569}, {5613,16809}, {5965,6781}, {5981,7777}, {6109,23234}, {6773,10645}, {6778,13103}, {8860,9760}, {9116,10488}, {11488,22891}, {13102,16808}, {16964,23013}, {22495,22577}, {22509,23005}

X(25236) = reflection of X(i) in X(j) for these (i,j): (13, 22997), (14, 9117), (5463, 8594), (6777, 15), (19106, 6778), (22493, 5464), (22577, 22495), (23004, 6783), (25166, 13), (25235, 6781)
X(25236) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 9117, 16962), (115, 16529, 16960), (6783, 23004, 13), (22997, 23004, 6783)

Collineation mappings involving Gemini triangle 17: X(25237) - X(25272)

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 17, as in centers X(25237)-X(25272). Then

m(X) = a (b + c) (b + c - a) x + b (c - a)(c + a - b) y + c (b - a) (b + a - c) z : :

(Clark Kimberling, October 14, 2018)

### X(25237) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^3 b - 2 a^2 b^2 + a b^3 + a^3 c - 2 a^2 b c - b^3 c - 2 a^2 c^2 + 2 b^2 c^2 + a c^3 - b c^3 : :

X(25237) lies on these lines: {2, 277}, {144, 145}, {148, 1655}, {194, 16865}, {239, 3219}, {536, 4875}, {672, 20247}, {894, 11319}, {2269, 20248}, {2275, 24403}, {3208, 21272}, {3210, 24599}, {3729, 17164}, {3732, 3871}, {3970, 3995}, {4080, 17244}, {4513, 17262}, {4552, 8545}, {4936, 25268}, {7264, 24036}, {7754, 20045}, {9310, 17136}, {16920, 17350}, {17257, 17676}, {17261, 25241}, {17451, 20244}, {17760, 25253}, {20173, 24635}, {21029, 25353}, {25245, 25269}, {25246, 25254}

X(25237) = anticomplement of X(20880)

### X(25238) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^6 b - a^5 b^2 - a^2 b^5 + a b^6 + a^6 c - 2 a^5 b c - a^4 b^2 c + 2 a^3 b^3 c + a^2 b^4 c - b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 - a b^4 c^2 + b^5 c^2 + 2 a^3 b c^3 + a^2 b c^4 - a b^2 c^4 - a^2 c^5 + b^2 c^5 + a c^6 - b c^6 : :

X(25238) lies on these lines: {2, 23581}, {192, 12649}, {239, 3219}, {17247, 25261}, {25241, 25247}, {25242, 25266}, {25253, 25265}

### X(25239) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^6 b - a^5 b^2 - a^2 b^5 + a b^6 + a^6 c - a^5 b c - a^4 b^2 c + a^2 b^4 c + a b^5 c - b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 - a b^4 c^2 + b^5 c^2 - 2 a b^3 c^3 + a^2 b c^4 - a b^2 c^4 - a^2 c^5 + a b c^5 + b^2 c^5 + a c^6 - b c^6 : :

X(25239) lies on these lines: {2, 23581}, {192, 3100}, {3970, 3995}, {17489, 18359}, {25243, 25244}, {25252, 25253}

### X(25240) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^6 b - a^5 b^2 - a^2 b^5 + a b^6 + a^6 c - a^4 b^2 c - 2 a^3 b^3 c + a^2 b^4 c + 2 a b^5 c - b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 - a b^4 c^2 + b^5 c^2 - 2 a^3 b c^3 - 4 a b^3 c^3 + a^2 b c^4 - a b^2 c^4 - a^2 c^5 + 2 a b c^5 + b^2 c^5 + a c^6 - b c^6 : :

X(25240) lies on these lines: {2, 23581}, {4080, 17244}

### X(25241) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - 2 a^3 b c - b^4 c - a^3 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 + a c^4 - b c^4 : :

X(25241) lies on these lines: {2, 17861}, {7, 192}, {1444, 19623}, {1764, 1999}, {2397, 17262}, {3210, 5744}, {3729, 25245}, {3995, 22019}, {4293, 20009}, {5088, 25264}, {11683, 21511}, {17033, 17489}, {17261, 25237}, {21023, 25348}, {25238, 25247}, {25246, 25249}, {25262, 25270}, {25268, 25269}

### X(25242) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^3 b - 2 a^2 b^2 + a b^3 + a^3 c - a^2 b c + a b^2 c - b^3 c - 2 a^2 c^2 + a b c^2 + 2 b^2 c^2 + a c^3 - b c^3 : :

X(25242) lies on these lines: {1, 87}, {2, 277}, {8, 3177}, {9, 17691}, {20, 72}, {37, 4352}, {75, 1212}, {76, 6184}, {78, 10025}, {85, 3693}, {190, 220}, {218, 384}, {239, 16572}, {241, 312}, {279, 304}, {306, 18663}, {321, 24635}, {329, 6999}, {345, 948}, {480, 9305}, {517, 20535}, {536, 17158}, {664, 4513}, {728, 9312}, {2896, 4741}, {3008, 17490}, {3160, 4552}, {3210, 5222}, {3212, 3501}, {3474, 20715}, {3732, 5687}, {3880, 9311}, {3912, 10481}, {4201, 17257}, {4393, 7839}, {4416, 6743}, {4419, 7738}, {4515, 16284}, {4699, 20236}, {4740, 20881}, {5088, 17742}, {5308, 25263}, {5584, 11683}, {6542, 7960}, {6646, 7791}, {7229, 25255}, {7750, 17347}, {7787, 17745}, {7876, 17236}, {9607, 17246}, {17014, 17147}, {17090, 21232}, {17170, 20533}, {17181, 21073}, {17759, 21216}, {20065, 20072}, {21049, 25355}, {25238, 25266}, {25253, 25272}

### X(25243) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - 4 a^3 b c + 4 a^2 b^2 c - b^4 c - a^3 c^2 + 4 a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 + a c^4 - b c^4 : :

X(25243) lies on these lines: {2, 37}, {8, 1736}, {9, 4552}, {145, 5728}, {594, 25000}, {883, 17350}, {1713, 19742}, {1766, 14953}, {2183, 21271}, {2267, 17221}, {2321, 25019}, {3161, 25252}, {3305, 18662}, {3695, 25017}, {3729, 25268}, {3731, 25255}, {3876, 20222}, {3969, 13567}, {11433, 20017}, {12848, 20110}, {16578, 17077}, {17220, 21801}, {17261, 25237}, {21273, 21371}, {24778, 25076}, {25239, 25244}, {25257, 25269}

### X(25244) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^3 b - 2 a^2 b^2 + a b^3 + a^3 c + 2 a b^2 c - b^3 c - 2 a^2 c^2 + 2 a b c^2 + 2 b^2 c^2 + a c^3 - b c^3 : :

X(25244) lies on these lines: {2, 277}, {37, 18600}, {144, 4190}, {192, 3622}, {894, 5764}, {2329, 17136}, {3061, 20244}, {3177, 3617}, {3219, 14953}, {3729, 19861}, {3970, 17169}, {3995, 16826}, {7229, 25252}, {16604, 24403}, {16919, 17350}, {17164, 17760}, {17355, 25260}, {17367, 17489}, {17496, 17741}, {17754, 20247}, {21029, 24318}, {25239, 25243}

### X(25245) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - 4 a^3 b c + 2 a^2 b^2 c - b^4 c - a^3 c^2 + 2 a^2 b c^2 - 4 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 + a c^4 - b c^4 : :

X(25245) lies on these lines: {2, 37}, {190, 15988}, {740, 25024}, {894, 1442}, {3729, 25241}, {3731, 25251}, {3989, 24997}, {4451, 4651}, {11997, 20863}, {17116, 25250}, {17261, 25255}, {20881, 25065}, {25237, 25269}

### X(25246) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^5 b - a^4 b^2 - a^2 b^4 + a b^5 + a^5 c - 2 a^4 b c - b^5 c - a^4 c^2 + b^4 c^2 - a^2 c^4 + b^2 c^4 + a c^5 - b c^5 : :

X(25246) lies on these lines: {2, 20890}, {192, 3434}, {3995, 22015}, {25237, 25254}, {25241, 25249}

### X(25247) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^6 b - a^5 b^2 - a^2 b^5 + a b^6 + a^6 c - 2 a^5 b c - b^6 c - a^5 c^2 + b^5 c^2 - a^2 c^5 + b^2 c^5 + a c^6 - b c^6 : :

X(25247) lies on these lines: {2, 21414}, {192, 21285}, {25238, 25241}, {25253, 25262}, {25265, 25270}

### X(25248) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^3 b - 2 a^2 b^2 - a b^3 + a^3 c - 2 a^2 b c + b^3 c - 2 a^2 c^2 - a c^3 + b c^3 : :

X(25248) lies on these lines: {2, 986}, {192, 20247}, {384, 4427}, {846, 16931}, {1500, 17141}, {3730, 17489}, {3797, 17751}, {4414, 16822}, {4418, 16930}, {4642, 17755}, {7783, 17136}, {16549, 25263}, {16552, 17497}, {16705, 24254}, {16827, 17495}, {16910, 24248}, {17033, 17147}, {17261, 25261}, {21226, 21272}, {25238, 25241}, {25262, 25265}

### X(25249) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^4 b^2 - 2 a^3 b^3 + a^2 b^4 + 2 a^4 b c - 3 a^3 b^2 c - a^2 b^3 c + a^4 c^2 - 3 a^3 b c^2 + a b^3 c^2 - b^4 c^2 - 2 a^3 c^3 - a^2 b c^3 + a b^2 c^3 + 2 b^3 c^3 + a^2 c^4 - b^2 c^4 : :

X(25249) lies on these lines: {2, 277}, {37, 16708}, {192, 3873}, {194, 23407}, {335, 3995}, {1999, 18206}, {3177, 20012}, {17177, 22015}, {25241, 25246}, {25254, 25266}

### X(25250) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - 3 a^3 b c + 2 a^2 b^2 c - b^4 c - a^3 c^2 + 2 a^2 b c^2 - a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 + a c^4 - b c^4 : :

X(25250) lies on these lines: {2, 24209}, {142, 192}, {239, 4552}, {4025, 25258}, {17116, 25245}, {17261, 25237}, {17339, 25252}, {25257, 25268}

### X(25251) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - 6 a^3 b c + 8 a^2 b^2 c - b^4 c - a^3 c^2 + 8 a^2 b c^2 - 4 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 + a c^4 - b c^4 : :

X(25251) lies on these lines: {2, 20881}, {45, 2397}, {192, 4402}, {3731, 25245}, {17261, 25237}

### X(25252) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - a^3 b c + a b^3 c - b^4 c - a^3 c^2 + b^3 c^2 - a^2 c^3 + a b c^3 + b^2 c^3 + a c^4 - b c^4 : :

X(25252) lies on these lines: {1, 87}, {2, 17861}, {3, 11683}, {9, 24268}, {75, 25083}, {92, 3998}, {144, 2801}, {190, 219}, {200, 21084}, {239, 1723}, {278, 345}, {306, 6360}, {312, 1214}, {322, 3694}, {326, 1944}, {329, 21078}, {346, 347}, {536, 1108}, {573, 22003}, {1278, 20881}, {1330, 18666}, {1943, 3719}, {1959, 10446}, {2256, 17262}, {2257, 3875}, {2414, 6601}, {2550, 21804}, {2893, 24316}, {3161, 25243}, {3177, 4416}, {3332, 24280}, {3668, 3912}, {4431, 4847}, {5279, 17134}, {7229, 25244}, {7560, 21376}, {17339, 25250}, {17740, 21442}, {20880, 24554}, {24179, 24559}, {25001, 25082}, {25239, 25253}

### X(25253) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^3 b - a b^3 + a^3 c - 2 a^2 b c + b^3 c + 2 b^2 c^2 - a c^3 + b c^3 : :

X(25253) lies on these lines: {1, 3159}, {2, 986}, {3, 4427}, {8, 80}, {65, 4358}, {72, 3702}, {100, 23844}, {145, 3994}, {190, 2975}, {304, 20347}, {312, 3869}, {313, 21271}, {321, 960}, {341, 14923}, {346, 21801}, {350, 20247}, {392, 4968}, {517, 3701}, {518, 22016}, {726, 1201}, {740, 25277}, {846, 16347}, {902, 8669}, {946, 3006}, {962, 10327}, {978, 17495}, {1001, 17142}, {1043, 7424}, {1125, 3977}, {1191, 3891}, {1193, 17147}, {1230, 22076}, {1265, 3434}, {1975, 17136}, {2415, 3445}, {2899, 5554}, {3057, 3967}, {3230, 22036}, {3263, 20244}, {3485, 17776}, {3555, 4742}, {3616, 17140}, {3622, 4704}, {3649, 18139}, {3681, 4673}, {3704, 5741}, {3729, 19861}, {3874, 4975}, {3876, 4651}, {3877, 4385}, {3884, 4692}, {3885, 4737}, {3886, 3984}, {3893, 4487}, {3915, 20045}, {3923, 11115}, {3924, 4011}, {3971, 10459}, {3976, 17154}, {3985, 17451}, {4009, 5836}, {4082, 4301}, {4115, 16552}, {4190, 24280}, {4387, 12635}, {4418, 19284}, {4485, 21272}, {4511, 7283}, {4552, 10571}, {4647, 10176}, {4723, 10914}, {5014, 12701}, {5016, 24703}, {5045, 17146}, {5057, 7270}, {5300, 12699}, {5687, 17780}, {6327, 11415}, {9534, 17163}, {11684, 14829}, {16466, 17150}, {16704, 17733}, {17155, 21214}, {17220, 20336}, {17760, 25237}, {18135, 24282}, {18137, 20718}, {20018, 25294}, {21674, 25385}, {25238, 25265}, {25239, 25252}, {25242, 25272}, {25247, 25262}

### X(25254) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    (b + c) (-a^5 + 2 a^3 b^2 - a b^4 + a^3 b c + a^2 b^2 c + a b^3 c + b^4 c + 2 a^3 c^2 + a^2 b c^2 - 2 a b^2 c^2 - b^3 c^2 + a b c^3 - b^2 c^3 - a c^4 + b c^4) : :

X(25254) lies on these lines: {2, 17861}, {63, 3187}, {145, 758}, {192, 3151}, {226, 3995}, {345, 20896}, {3743, 10587}, {3957, 6758}, {4137, 24248}, {5745, 17495}, {17164, 25257}, {25237, 25246}, {25249, 25266}

### X(25255) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    (b + c) (-a^4 + a^3 b + a^2 b^2 - a b^3 + a^3 c + a^2 b c + a b^2 c + b^3 c + a^2 c^2 + a b c^2 - 2 b^2 c^2 - a c^3 + b c^3) : :

X(25255) lies on these lines: {2, 17861}, {7, 2294}, {8, 192}, {21, 11683}, {37, 1441}, {42, 21084}, {92, 18662}, {144, 758}, {284, 14543}, {306, 3950}, {322, 4664}, {346, 18697}, {523, 885}, {1781, 14953}, {1953, 24424}, {1959, 17183}, {1962, 10578}, {2650, 3177}, {3673, 24554}, {3729, 17164}, {3731, 25243}, {3958, 6172}, {4016, 4419}, {4032, 21808}, {4065, 6765}, {4294, 20061}, {4313, 18673}, {4431, 17163}, {4461, 4647}, {5271, 17147}, {7229, 25242}, {7672, 20718}, {7674, 24394}, {13576, 21804}, {16601, 25001}, {16609, 21811}, {16824, 25264}, {17090, 25132}, {17116, 25257}, {17139, 18714}, {17261, 25245}, {17776, 20896}, {21675, 25359}, {24987, 25260}, {24993, 25083}

### X(25256) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    2 a^3 b - 4 a^2 b^2 + 2 a b^3 + 2 a^3 c - 3 a^2 b c + a b^2 c - 2 b^3 c - 4 a^2 c^2 + a b c^2 + 4 b^2 c^2 + 2 a c^3 - 2 b c^3 : :

X(25256) lies on these lines: {2, 277}, {144, 15680}, {192, 3623}, {3177, 3621}, {4393, 25264}, {17489, 25266}, {19582, 25272}

### X(25257) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - 2 a^2 b^2 c - b^4 c - a^3 c^2 - 2 a^2 b c^2 + 4 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 + a c^4 - b c^4 : :

X(25257) lies on these lines: {2, 277}, {20, 912}, {192, 3945}, {194, 712}, {323, 401}, {1443, 4552}, {2795, 20556}, {3729, 25241}, {5440, 16380}, {6999, 17484}, {17116, 25255}, {17164, 25254}, {25243, 25269}, {25250, 25268}

### X(25258) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    (b - c) (-a^4 b + a^2 b^3 - a^4 c - a^3 b c + 2 a^2 b^2 c + 2 a^2 b c^2 - a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3) : :

X(25258) lies on these lines: {2, 647}, {37, 18155}, {192, 4467}, {321, 16751}, {693, 21225}, {804, 25299}, {905, 21438}, {1021, 1999}, {3210, 17069}, {3250, 20295}, {4024, 4560}, {4025, 25250}, {4524, 20012}, {4529, 24560}, {6332, 17496}, {6590, 11068}

### X(25259) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    (b - c) (a b - b^2 + a c - b c - c^2) : :

X(25259) lies on these lines: {2, 2400}, {190, 1252}, {192, 23757}, {193, 9031}, {297, 525}, {513, 4122}, {514, 4024}, {522, 3935}, {523, 8663}, {649, 2786}, {650, 4467}, {652, 3219}, {661, 824}, {677, 1897}, {693, 918}, {768, 4079}, {900, 4380}, {1011, 23093}, {1491, 18004}, {1577, 23875}, {1639, 17069}, {2254, 4522}, {2340, 8714}, {2509, 16757}, {2785, 4474}, {3004, 4776}, {3151, 20298}, {3762, 23876}, {3776, 4728}, {3810, 23745}, {3835, 4120}, {3910, 4462}, {3952, 25266}, {4064, 20294}, {4130, 24562}, {4184, 22388}, {4453, 4885}, {4500, 4931}, {4656, 23801}, {4707, 4791}, {4762, 4820}, {4893, 21196}, {6332, 17496}, {6370, 23758}, {6586, 16751}, {6590, 7192}, {7253, 14954}, {17233, 21133}, {17280, 21202}, {17375, 23730}, {18155, 21611}

X(25259) = anticomplement of X(4025)
X(25259) = isotomic conjugate of trilinear pole of line X(3)X(142)
X(25259) = pole wrt polar circle of trilinear polar of X(26705) (line X(6)X(1836))
X(25259) = polar conjugate of X(26705)

### X(25260) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^5 b - a^4 b^2 - a^2 b^4 + a b^5 + a^5 c - 2 a^4 b c - b^5 c - a^4 c^2 + 2 a^2 b^2 c^2 + b^4 c^2 - a^2 c^4 + b^2 c^4 + a c^5 - b c^5 : :

X(25260) lies on these lines: {2, 20890}, {192, 2550}, {17261, 25237}, {17355, 25244}, {17489, 17752}, {24987, 25255}

### X(25261) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^3 b - 2 a^2 b^2 + a b^3 + a^3 c - 4 a^2 b c - 2 a b^2 c - b^3 c - 2 a^2 c^2 - 2 a b c^2 + 2 b^2 c^2 + a c^3 - b c^3 : :

X(25261) lies on these lines: {2, 277}, {9, 20247}, {37, 20448}, {192, 3617}, {1107, 24403}, {3177, 3622}, {3290, 18600}, {3488, 20071}, {3661, 3995}, {12649, 17257}, {14949, 17496}, {16815, 17495}, {17247, 25238}, {17261, 25248}, {17266, 25263}, {17494, 21201}, {20347, 21808}, {24987, 25255}

### X(25262) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^6 b - a^5 b^2 - a^2 b^5 + a b^6 + a^6 c - 2 a^5 b c - a^4 b^2 c + a^2 b^4 c - b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 - a b^4 c^2 + b^5 c^2 - 2 a b^3 c^3 + a^2 b c^4 - a b^2 c^4 - a^2 c^5 + b^2 c^5 + a c^6 - b c^6 : :

X(25262) lies on these lines: {2, 23581}, {25241, 25270}, {25247, 25253}, {25248, 25265}

### X(25263) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^3 b + a b^3 + a^3 c + 2 a^2 b c + 2 a b^2 c - b^3 c + 2 a b c^2 + a c^3 - b c^3 : :

X(25263) lies on these lines: {1, 4568}, {2, 1930}, {10, 17497}, {37, 16705}, {190, 17103}, {192, 3616}, {335, 17169}, {762, 8682}, {1655, 6625}, {3952, 17499}, {3995, 16826}, {4080, 17244}, {4552, 17084}, {5308, 25242}, {5692, 20109}, {9780, 21216}, {16549, 25248}, {17141, 24512}, {17147, 17397}, {17266, 25261}

### X(25264) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^2 b^2 + a^2 b c + a^2 c^2 - b^2 c^2 : :

X(25264) lies on these lines: {1, 87}, {2, 3760}, {10, 1655}, {35, 385}, {36, 7783}, {37, 274}, {39, 350}, {42, 17499}, {55, 7754}, {75, 5283}, {76, 2276}, {99, 172}, {144, 20018}, {148, 3585}, {190, 213}, {193, 4294}, {239, 3219}, {257, 4424}, {312, 980}, {315, 9598}, {321, 10471}, {335, 3970}, {346, 4352}, {384, 5280}, {386, 24514}, {519, 21226}, {536, 1107}, {538, 1500}, {609, 3552}, {668, 20691}, {672, 17034}, {1018, 17752}, {1045, 21080}, {1078, 4396}, {1266, 17050}, {1449, 20170}, {1479, 7774}, {1575, 18140}, {1914, 7760}, {1920, 21883}, {1930, 3797}, {1999, 18206}, {2176, 17262}, {2241, 7798}, {2242, 7781}, {2275, 7757}, {2397, 16820}, {2664, 3971}, {2996, 10590}, {3085, 6392}, {3159, 4568}, {3177, 12526}, {3210, 4384}, {3244, 9263}, {3294, 16827}, {3295, 22253}, {3583, 7785}, {3584, 19570}, {3644, 16975}, {3730, 17033}, {3761, 20081}, {3875, 21384}, {3891, 23407}, {3912, 24214}, {3920, 8267}, {3950, 24215}, {3972, 7296}, {3995, 16826}, {4037, 16720}, {4099, 7187}, {4253, 17027}, {4302, 20065}, {4324, 14712}, {4360, 20963}, {4366, 5299}, {4393, 25256}, {4441, 17030}, {4493, 14620}, {4552, 7176}, {4718, 17448}, {4857, 13571}, {5010, 7793}, {5088, 25241}, {5248, 16998}, {5259, 17000}, {5332, 7894}, {6284, 7762}, {6645, 16785}, {7031, 7766}, {7741, 7777}, {7759, 9664}, {9331, 20105}, {9534, 17257}, {9596, 11185}, {10754, 19369}, {16667, 20168}, {16815, 17495}, {16818, 17302}, {16824, 25255}, {16831, 24621}, {16832, 17490}, {16887, 21070}, {17756, 18135}, {21024, 25349}

### X(25265) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    (a - b) (a - c) (b + c) (a^4 - b^4 - 3 a^2 b c - a b^2 c + 2 b^3 c - a b c^2 - 2 b^2 c^2 + 2 b c^3 - c^4) : :

X(25265) lies on these lines: {2, 24791}, {525, 644}, {6758, 24074}, {25238, 25253}, {25247, 25270}, {25248, 25262}, {25266, 25272}, {25267, 25268}

### X(25266) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    (a - b) (a - c) (a^3 b - a^2 b^2 + a b^3 - b^4 + a^3 c - 2 a^2 b c - a b^2 c + 2 b^3 c - a^2 c^2 - a b c^2 - 2 b^2 c^2 + a c^3 + 2 b c^3 - c^4) : :

X(25266) lies on these lines: {2, 21436}, {643, 17498}, {644, 2414}, {1025, 4552}, {1331, 2398}, {3952, 25259}, {17489, 25256}, {25238, 25242}, {25249, 25254}, {25265, 25272}

### X(25267) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    (a - b) (a - c) (a^4 b - a^3 b^2 + a b^4 - b^5 + a^4 c - 2 a^3 b c - a b^3 c + 2 b^4 c - a^3 c^2 - b^3 c^2 - a b c^3 - b^2 c^3 + a c^4 + 2 b c^4 - c^5) : :

X(25267) lies on these lines: {3882, 4552}, {25265, 25268}

### X(25268) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    (a - b) (a - c) (a - b - c) (a b + b^2 + a c - 2 b c + c^2) : :

X(25268) lies on these lines: {2, 20881}, {8, 2310}, {190, 644}, {192, 3161}, {522, 4069}, {645, 4560}, {646, 4526}, {3210, 8055}, {3729, 25243}, {4033, 4391}, {4587, 14543}, {4664, 25082}, {4750, 22003}, {4936, 25237}, {6557, 17490}, {8834, 17480}, {17183, 21809}, {17261, 25245}, {21272, 21362}, {25241, 25269}, {25250, 25257}, {25265, 25267}

### X(25269) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    2 a^2 - 3 a b - 3 a c + 3 b c : :

X(25269) lies on these lines: {2, 2415}, {6, 190}, {9, 1278}, {37, 24063}, {44, 3644}, {45, 4699}, {75, 16814}, {86, 16674}, {144, 6542}, {194, 3230}, {239, 3973}, {344, 4440}, {346, 3620}, {527, 17242}, {536, 15492}, {545, 17234}, {726, 15485}, {894, 3247}, {903, 17265}, {1266, 17338}, {1334, 20081}, {2321, 17333}, {2325, 3662}, {3159, 4257}, {3208, 21219}, {3619, 4419}, {3621, 5223}, {3630, 3943}, {3631, 4741}, {3685, 16496}, {3723, 4664}, {3758, 4681}, {3759, 4718}, {3950, 4480}, {3995, 14996}, {4000, 4473}, {4029, 17391}, {4363, 16677}, {4370, 17352}, {4384, 4821}, {4389, 17340}, {4398, 4422}, {4416, 20055}, {4431, 17331}, {4488, 17316}, {4659, 4772}, {4686, 17335}, {4715, 17386}, {4740, 17277}, {4764, 17348}, {4862, 17266}, {4873, 17287}, {4903, 17596}, {4908, 17240}, {11008, 17314}, {14997, 17147}, {16484, 24349}, {16486, 17480}, {16885, 17160}, {17229, 17329}, {17232, 17264}, {17235, 17342}, {17238, 17258}, {17246, 17354}, {17249, 17359}, {17254, 17286}, {17255, 17285}, {17268, 17274}, {17269, 17273}, {17300, 17487}, {17307, 24441}, {25237, 25245}, {25241, 25268}, {25243, 25257}

### X(25270) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    a^3 b - a^2 b^2 - a b^3 + a^3 c - 2 a^2 b c + b^3 c - a^2 c^2 + b^2 c^2 - a c^3 + b c^3 : :

X(25270) lies on these lines: {2, 986}, {1655, 24282}, {3552, 4427}, {3797, 3869}, {3924, 6651}, {21219, 21272}, {21295, 25283}, {25237, 25245}, {25241, 25262}, {25247, 25265}

### X(25271) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    (b - c) (-a^4 b + a^2 b^3 - a^4 c + a^3 b c + 2 a^2 b^2 c - 2 a b^3 c + 2 a^2 b c^2 - a b^2 c^2 + b^3 c^2 + a^2 c^3 - 2 a b c^3 + b^2 c^3) : :

X(25271) lies on these lines: {2, 21438}, {145, 1938}, {192, 693}, {323, 401}, {650, 3210}, {824, 17147}, {3666, 21611}, {4025, 25250}, {4467, 21225}, {17132, 23733}

### X(25272) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 17

Barycentrics    (a - b) (a - c) (a b - 3 b^2 + a c + 2 b c - 3 c^2) : :

X(25272) lies on these lines: {2, 24795}, {37, 16811}, {145, 14759}, {190, 17136}, {1026, 3952}, {3732, 17780}, {4595, 21272}, {17164, 17760}, {19582, 25256}, {25242, 25253}, {25265, 25266}

Collineation mappings involving Gemini triangle 18: X(25273) - X(25313)

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 18, as in centers X(25273)-X(25313). Then

m(X) = (b + c) (b c + c a + a b) x + (c - a)(b c + c a + a b) y + ( (b - a) (b c + c a + a b) z : :

(Clark Kimberling, October 14, 2018)

### X(25273) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a^4 b^3 - a^3 b^4 - a^4 b^2 c - a^3 b^3 c + a^2 b^4 c - a^4 b c^2 + a b^4 c^2 + a^4 c^3 - a^3 b c^3 + a b^3 c^3 - b^4 c^3 - a^3 c^4 + a^2 b c^4 + a b^2 c^4 - b^3 c^4 : :

X(25273) lies on these lines: {2, 24527}, {7, 8}, {4598, 20996}, {25277, 25313}, {25283, 25309}, {25298, 25308}

### X(25274) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a^5 b^2 + a^4 b^3 - a^3 b^4 - a^2 b^5 + 2 a^5 b c - 2 a^4 b^2 c - 2 a^3 b^3 c + 2 a^2 b^4 c + a^5 c^2 - 2 a^4 b c^2 + b^5 c^2 + a^4 c^3 - 2 a^3 b c^3 - b^4 c^3 - a^3 c^4 + 2 a^2 b c^4 - b^3 c^4 - a^2 c^5 + b^2 c^5 : :

X(25274) lies on these lines: {2, 24664}, {322, 3869}, {4485, 21272}, {4566, 7138}, {24524, 25290}, {25277, 25283}, {25278, 25310}, {25293, 25309}

### X(25275) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a^5 b^2 + a^4 b^3 - a^3 b^4 - a^2 b^5 + a^5 b c - 2 a^4 b^2 c + 2 a^2 b^4 c - a b^5 c + a^5 c^2 - 2 a^4 b c^2 + b^5 c^2 + a^4 c^3 + 2 a b^3 c^3 - b^4 c^3 - a^3 c^4 + 2 a^2 b c^4 - b^3 c^4 - a^2 c^5 - a b c^5 + b^2 c^5 : :

X(25275) lies on these lines: {2, 24664}, {69, 313}, {279, 291}, {25279, 25280}, {25291, 25293}

### X(25276) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a^5 b^2 + a^4 b^3 - a^3 b^4 - a^2 b^5 - 2 a^4 b^2 c + 2 a^3 b^3 c + 2 a^2 b^4 c - 2 a b^5 c + a^5 c^2 - 2 a^4 b c^2 + b^5 c^2 + a^4 c^3 + 2 a^3 b c^3 + 4 a b^3 c^3 - b^4 c^3 - a^3 c^4 + 2 a^2 b c^4 - b^3 c^4 - a^2 c^5 - 2 a b c^5 + b^2 c^5 : :

X(25276) lies on these lines: {2, 24664}, {24524, 25281}

### X(25277) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a^3 b^2 - a^2 b^3 + 2 a^3 b c + a^3 c^2 - 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 : :

X(25277) lies on these lines: {2, 3728}, {8, 17153}, {43, 22024}, {75, 3681}, {192, 872}, {518, 20892}, {726, 3214}, {740, 25253}, {1278, 21805}, {1921, 20247}, {3264, 22277}, {3617, 4772}, {3739, 3999}, {3779, 20352}, {4361, 17142}, {4699, 17140}, {4704, 17592}, {4738, 5902}, {10009, 17141}, {17135, 20891}, {17147, 21080}, {17157, 17495}, {17349, 24351}, {17792, 25292}, {24524, 25279}, {25273, 25313}, {25274, 25283}, {25282, 25286}

### X(25278) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    b c (3 a^2 - 3 a b - 3 a c + 2 b c) : :

X(25278) lies on these lines: {2, 17448}, {8, 76}, {69, 3264}, {75, 4678}, {144, 4110}, {145, 6376}, {304, 4723}, {350, 3621}, {391, 17786}, {519, 18135}, {646, 6172}, {1909, 3617}, {3241, 18140}, {3263, 10513}, {3266, 10327}, {3436, 20553}, {3625, 3760}, {3626, 3761}, {3632, 6381}, {3730, 23891}, {4033, 25333}, {4371, 18144}, {4668, 20888}, {4737, 20911}, {6555, 20935}, {6683, 16975}, {17756, 21226}, {17759, 20105}, {18133, 20047}, {20052, 20943}, {25274, 25310}, {25291, 25292}, {25293, 25312}

### X(25279) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a (a^2 b^2 - a b^3 - a b^2 c + b^3 c + a^2 c^2 - a b c^2 - b^2 c^2 - a c^3 + b c^3) : :

X(25279) lies on these lines: {2, 3056}, {8, 20892}, {9, 3888}, {69, 3263}, {75, 4553}, {100, 1253}, {210, 17343}, {513, 17336}, {518, 17375}, {674, 17234}, {1654, 3909}, {2550, 14923}, {3006, 7998}, {3271, 17338}, {3573, 24309}, {3662, 3688}, {3705, 3819}, {3729, 3799}, {3779, 3873}, {3781, 3869}, {3877, 4660}, {4110, 23354}, {4358, 21299}, {4517, 6646}, {6007, 17242}, {7064, 17333}, {9024, 17337}, {9025, 17349}, {17244, 21746}, {17278, 25048}, {17360, 22271}, {17364, 20683}, {17378, 22277}, {20923, 21278}, {22016, 24717}, {22294, 22301}, {24524, 25277}, {24742, 25117}, {25275, 25280}, {25290, 25297}, {25298, 25311}

### X(25280) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    b c (a^2 - 2 a b - 2 a c + b c) : :

X(25280) lies on these lines: {2, 17448}, {8, 350}, {10, 274}, {75, 3617}, {76, 3679}, {85, 24797}, {319, 17751}, {325, 21031}, {335, 21951}, {519, 18140}, {966, 17786}, {1078, 5258}, {1334, 4595}, {1575, 21226}, {1654, 17787}, {1655, 20691}, {2238, 17752}, {2329, 3570}, {3263, 20955}, {3264, 17271}, {3294, 23891}, {3596, 17270}, {3626, 6381}, {3661, 3975}, {3701, 17762}, {3760, 4668}, {3807, 17760}, {3934, 13466}, {4033, 17256}, {4104, 7018}, {4110, 17257}, {4441, 4678}, {4505, 24697}, {4669, 18145}, {4677, 18146}, {4690, 17790}, {4691, 20888}, {4723, 20911}, {5564, 18133}, {6626, 7257}, {9263, 16604}, {10987, 16914}, {17235, 24770}, {17759, 21868}, {18159, 20880}, {24575, 25121}, {25275, 25279}, {25292, 25293}

### X(25281) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a^4 b^2 - a^2 b^4 - 3 a^3 b^2 c + 5 a^2 b^3 c - 2 a b^4 c + a^4 c^2 - 3 a^3 b c^2 - 2 a^2 b^2 c^2 + a b^3 c^2 + b^4 c^2 + 5 a^2 b c^3 + a b^2 c^3 - 2 b^3 c^3 - a^2 c^4 - 2 a b c^4 + b^2 c^4 : :

X(25281) lies on these lines: {2, 25110}, {24524, 25276}

### X(25282) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a^4 b^2 - a^2 b^4 + 2 a^4 b c - a^3 b^2 c + a^2 b^3 c + a^4 c^2 - a^3 b c^2 - a b^3 c^2 + b^4 c^2 + a^2 b c^3 - a b^2 c^3 - a^2 c^4 + b^2 c^4 : :

X(25282) lies on these lines: {2, 24525}, {69, 4661}, {518, 18138}, {1909, 17165}, {3681, 16703}, {3952, 17149}, {24524, 25294}, {25277, 25286}, {25287, 25310}

### X(25283) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a^5 b^2 - a^2 b^5 + 2 a^5 b c - a^4 b^2 c + a^2 b^4 c + a^5 c^2 - a^4 b c^2 - a b^4 c^2 + b^5 c^2 + a^2 b c^4 - a b^2 c^4 - a^2 c^5 + b^2 c^5 : :

X(25283) lies on these lines: {2, 24526}, {3681, 20955}, {21295, 25270}, {25273, 25309}, {25274, 25277}, {25293, 25313}

### X(25284) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a^3 b^2 - a^2 b^3 - 2 a^3 b c - 2 a^2 b^2 c + 2 a b^3 c + a^3 c^2 - 2 a^2 b c^2 - b^3 c^2 - a^2 c^3 + 2 a b c^3 - b^2 c^3 : :

X(25284) lies on these lines: {2, 18194}, {8, 4772}, {10, 17178}, {319, 4651}, {894, 23354}, {4851, 17721}, {17135, 17373}, {17300, 20352}, {17792, 25295}, {24524, 25277}, {25285, 25303}

### X(25285) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a^4 b^3 - a^3 b^4 - a^4 b^2 c - 2 a^3 b^3 c + a^2 b^4 c - a^4 b c^2 + a b^4 c^2 + a^4 c^3 - 2 a^3 b c^3 - b^4 c^3 - a^3 c^4 + a^2 b c^4 + a b^2 c^4 - b^3 c^4 : :

X(25285) lies on these lines: {2, 24527}, {24575, 25311}, {25274, 25277}, {25284, 25303}, {25309, 25313}

### X(25286) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    b c (2 a^3 b - a^2 b^2 + 2 a^3 c - a^2 b c - a^2 c^2 + b^2 c^2) : :

X(25286) lies on these lines: {2, 17448}, {42, 668}, {75, 4661}, {210, 18059}, {310, 4685}, {350, 20011}, {519, 18152}, {1909, 4651}, {1920, 21805}, {1965, 3935}, {1978, 4090}, {3240, 17149}, {6376, 17018}, {17752, 21753}, {20048, 20943}, {25277, 25282}, {25294, 25310}

### X(25287) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    b c (2 a^3 b - a^2 b^2 + 2 a^3 c - 2 a^2 b c - a^2 c^2 + b^2 c^2) : :

X(25287) lies on these lines: {2, 17448}, {42, 6376}, {43, 668}, {75, 3681}, {76, 4685}, {85, 24800}, {200, 1965}, {350, 20012}, {561, 21805}, {899, 6384}, {1575, 21223}, {2238, 17786}, {3952, 8026}, {4033, 7109}, {4090, 6382}, {4110, 24514}, {17144, 18152}, {19998, 20943}, {25282, 25310}, {25290, 25302}

### X(25288) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    3 a^3 b^2 - 3 a^2 b^3 + 2 a^3 b c - 2 a^2 b^2 c + 2 a b^3 c + 3 a^3 c^2 - 2 a^2 b c^2 - 4 a b^2 c^2 + b^3 c^2 - 3 a^2 c^3 + 2 a b c^3 + b^2 c^3 : :

X(25288) lies on these lines: {2, 23633}, {20983, 25299}, {24524, 25277}

### X(25289) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    3 a^3 b^2 - 3 a^2 b^3 - 2 a^3 b c - 4 a^2 b^2 c + 4 a b^3 c + 3 a^3 c^2 - 4 a^2 b c^2 - 2 a b^2 c^2 - b^3 c^2 - 3 a^2 c^3 + 4 a b c^3 - b^2 c^3 : :

X(25289) lies on these lines: {2, 24671}, {4553, 17165}, {24524, 25277}

### X(25290) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a^4 b^2 - a^2 b^4 + 2 a^4 b c - 5 a^3 b^2 c + 3 a^2 b^3 c + a^4 c^2 - 5 a^3 b c^2 + 2 a^2 b^2 c^2 - a b^3 c^2 + b^4 c^2 + 3 a^2 b c^3 - a b^2 c^3 - 2 b^3 c^3 - a^2 c^4 + b^2 c^4 : :

X(25290) lies on these lines: {2, 25110}, {8, 21416}, {3681, 16284}, {3930, 20248}, {24524, 25274}, {25277, 25282}, {25279, 25297}, {25287, 25302}

### X(25291) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a^3 b^2 - a^2 b^3 + a^3 b c - a b^3 c + a^3 c^2 - 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 - a b c^3 + b^2 c^3 : :

X(25291) lies on these lines: {2, 3728}, {8, 4710}, {69, 20351}, {346, 3949}, {1278, 17794}, {1654, 24351}, {3056, 25298}, {3681, 17787}, {6327, 11677}, {25275, 25293}, {25278, 25292}

### X(25292) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a^3 b^2 - a^2 b^3 - 2 a^3 b c + 2 a b^3 c + a^3 c^2 + 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 + 2 a b c^3 - b^2 c^3 : :

X(25292) lies on these lines: {2, 18194}, {8, 726}, {9, 23354}, {10, 23579}, {69, 20352}, {319, 350}, {1654, 25054}, {3416, 15983}, {4651, 21223}, {8050, 17275}, {17792, 25277}, {20055, 21299}, {24524, 25295}, {25278, 25291}, {25280, 25293}

### X(25293) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a^4 b^3 - a^3 b^4 - a^4 b^2 c + a^2 b^4 c - a^4 b c^2 + a b^4 c^2 + a^4 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 : :

X(25293) lies on these lines: {2, 24527}, {8, 291}, {10, 20464}, {69, 20350}, {1837, 3056}, {4651, 24528}, {25274, 25309}, {25275, 25291}, {25278, 25312}, {25280, 25292}, {25283, 25313}

### X(25294) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    (b + c) (a^4 b - a^2 b^3 + a^4 c + 3 a^3 b c - a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(25294) lies on these lines: {2, 3728}, {8, 79}, {42, 3952}, {714, 3210}, {740, 20012}, {3293, 22024}, {20018, 25253}, {21020, 24349}, {24524, 25282}, {25286, 25310}

### X(25295) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    (b + c) (a^3 b - a^2 b^2 + a^3 c + 3 a^2 b c - a^2 c^2 + b^2 c^2) : :

X(25295) lies on these lines: {2, 3728}, {37, 3121}, {75, 3873}, {145, 740}, {190, 4068}, {192, 714}, {354, 20892}, {518, 15983}, {1107, 23432}, {1215, 22167}, {1962, 4704}, {2234, 16710}, {3696, 3922}, {3747, 17350}, {3948, 4890}, {3995, 21080}, {4033, 22279}, {4360, 17142}, {4772, 21020}, {16738, 24437}, {17147, 17157}, {17379, 24351}, {17792, 25284}, {21100, 21803}, {24524, 25292}, {24654, 25136}, {25302, 25303}

### X(25296) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    b c (7 a^2 - 5 a b - 5 a c + 4 b c) : :

X(25296) lies on these lines: {2, 17448}, {8, 3761}, {85, 4487}, {145, 668}, {274, 3617}, {350, 20014}, {1909, 4678}, {3621, 17144}, {3623, 6376}, {3760, 20053}, {4441, 20052}, {4935, 17158}, {6381, 20050}

### X(25297) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    b c (2 a^4 - 5 a^3 b + 4 a^2 b^2 - a b^3 - 5 a^3 c + 5 a^2 b c - a b^2 c + b^3 c + 4 a^2 c^2 - a b c^2 - 2 b^2 c^2 - a c^3 + b c^3) : :

X(25297) lies on these lines: {2, 17448}, {200, 668}, {350, 20015}, {519, 18153}, {3699, 20935}, {3870, 6376}, {4110, 10025}, {10327, 16284}, {25279, 25290}, {25308, 25310}

### X(25298) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    b c (2 a^3 - a b^2 - 2 a b c + b^2 c - a c^2 + b c^2) : :

X(25298) lies on these lines: {2, 17448}, {4, 8}, {10, 20456}, {44, 4033}, {69, 20892}, {75, 4410}, {100, 20777}, {239, 668}, {312, 20055}, {313, 17362}, {319, 20891}, {350, 20016}, {518, 20352}, {519, 3948}, {524, 3264}, {752, 4783}, {1107, 20868}, {1269, 4399}, {1909, 4359}, {1931, 7257}, {3056, 25291}, {3210, 21219}, {3263, 7779}, {3596, 17363}, {3625, 4044}, {3686, 3963}, {3770, 5564}, {3975, 4358}, {4110, 17350}, {4380, 4462}, {4393, 6376}, {4852, 18133}, {4986, 20432}, {5904, 21435}, {16816, 20917}, {17148, 21857}, {17299, 22016}, {17348, 18040}, {17349, 17786}, {17372, 18137}, {17373, 20923}, {17792, 25277}, {25273, 25308}, {25279, 25311}

### X(25299) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    (b - c) (-a^4 b - a^2 b^3 - a^4 c + a^3 b c - 2 a^2 b^2 c - 2 a^2 b c^2 + a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(25299) lies on these lines: {2, 669}, {10, 18197}, {512, 24622}, {649, 17072}, {804, 25258}, {1491, 17494}, {2533, 7192}, {2978, 21302}, {8640, 21260}, {8646, 24601}, {20983, 25288}, {21191, 23655}

### X(25300) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    (b - c) (-a b - a c + b c) (-a^3 b - a^2 b^2 - a^3 c + a^2 b c - a^2 c^2 + b^2 c^2) : :

X(25300) lies on these lines: {2, 24665}, {330, 23572}, {514, 19565}, {693, 14404}, {3004, 24533}, {4083, 17217}, {21272, 25312}, {25302, 25305}

### X(25301) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    (b - c) (-a^4 b + 2 a^3 b^2 - a^2 b^3 - a^4 c + a^3 b c - 2 a^2 b^2 c + 2 a^3 c^2 - 2 a^2 b c^2 - a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(25301) lies on these lines: {2, 23655}, {8, 649}, {333, 23865}, {518, 20952}, {788, 17135}, {3221, 9493}, {3835, 10453}, {3907, 17494}, {4651, 7234}, {7192, 8678}, {20909, 24349}

### X(25302) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    (a b - b^2 + a c - c^2) (a^3 b + a^2 b^2 + a^3 c - a^2 b c + a^2 c^2 - b^2 c^2) : :

X(25302) lies on these lines: {2, 24525}, {75, 14923}, {518, 3263}, {883, 1458}, {4576, 20045}, {24524, 25277}, {25287, 25290}, {25295, 25303}, {25300, 25305}

### X(25303) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    b c (3 a^2 + b c) : :

X(25303) lies on these lines: {1, 76}, {2, 17448}, {37, 21226}, {75, 145}, {83, 16784}, {85, 3476}, {86, 1222}, {99, 3746}, {183, 3304}, {192, 20105}, {257, 3726}, {274, 519}, {316, 5270}, {320, 17152}, {325, 15888}, {330, 2276}, {335, 3727}, {551, 18140}, {612, 11059}, {668, 1125}, {1015, 6683}, {1078, 5563}, {1107, 9263}, {1914, 6645}, {1975, 3303}, {2295, 10027}, {3230, 17499}, {3241, 17144}, {3244, 17143}, {3266, 3920}, {3552, 10987}, {3584, 7769}, {3616, 6376}, {3623, 4441}, {3635, 20888}, {3636, 6381}, {3723, 3770}, {3750, 8033}, {3780, 16827}, {3975, 16826}, {4317, 14907}, {4359, 20016}, {4389, 24418}, {4696, 20947}, {4968, 17762}, {5255, 17103}, {5434, 7750}, {5564, 16709}, {7187, 24326}, {7278, 20924}, {7321, 20244}, {7760, 16785}, {7763, 10056}, {7773, 11237}, {8025, 19811}, {12513, 16992}, {15589, 16284}, {16969, 24514}, {16971, 17034}, {17379, 17787}, {17752, 24512}, {25284, 25285}, {25295, 25302}

### X(25304) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a (2 a^2 b^2 - 2 a b^3 - a^2 b c + b^3 c + 2 a^2 c^2 - 2 a c^3 + b c^3) : :

X(25304) lies on these lines: {2, 3056}, {6, 100}, {7, 3888}, {8, 511}, {55, 15988}, {69, 674}, {141, 11680}, {145, 1469}, {149, 12589}, {193, 3779}, {263, 7155}, {344, 4553}, {404, 613}, {517, 12530}, {518, 1278}, {611, 3871}, {766, 24282}, {1002, 20090}, {1332, 1486}, {1350, 2975}, {1351, 5687}, {1428, 4188}, {1621, 10387}, {1959, 4073}, {1992, 22277}, {2293, 22370}, {2475, 12588}, {2876, 12272}, {3060, 10327}, {3094, 23423}, {3161, 3799}, {3416, 5086}, {3688, 17257}, {3873, 24471}, {4000, 25048}, {4259, 20040}, {4488, 4499}, {4704, 19586}, {4855, 16475}, {5175, 10477}, {5369, 21216}, {5480, 11681}, {5552, 14853}, {7077, 9309}, {10446, 20556}, {10519, 10527}, {15310, 24280}, {17316, 21746}, {17757, 21850}, {17868, 24341}, {25278, 25291}

### X(25305) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    (a^3 b + a^2 b^2 + a^3 c - a^2 b c + a^2 c^2 - b^2 c^2) (a^2 b - b^3 + a^2 c - 2 a b c + b^2 c + b c^2 - c^3) : :

X(25305) lies on these lines: {350, 21272}, {517, 3262}, {17448, 25313}, {24524, 25274}, {25300, 25302}

### X(25306) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a (a^3 b^2 - a b^4 - a b^3 c + b^4 c + a^3 c^2 + a b^2 c^2 - a b c^3 - a c^4 + b c^4) : :

X(25306) lies on these lines: {2, 3056}, {4, 8}, {31, 43}, {57, 3888}, {312, 21278}, {511, 3705}, {674, 4417}, {1193, 16478}, {1469, 3873}, {3006, 3060}, {3212, 20537}, {3741, 11680}, {3772, 25048}, {4553, 18743}, {6210, 11688}, {24524, 25274}

### X(25307) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    (b + c) (a^5 b + a^4 b^2 - a^3 b^3 - a^2 b^4 + a^5 c + a^4 b c + a^3 b^2 c + a^2 b^3 c + a^4 c^2 + a^3 b c^2 - 3 a^2 b^2 c^2 + b^4 c^2 - a^3 c^3 + a^2 b c^3 - a^2 c^4 + b^2 c^4) : :

X(25307) lies on these lines: {2, 25136}, {75, 3869}, {758, 21281}, {4647, 24282}, {24524, 25282}, {25274, 25277}, {25295, 25302}

### X(25308) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a (a^3 b^2 - a b^4 - a b^3 c + b^4 c + a^3 c^2 - a b^2 c^2 - a b c^3 - a c^4 + b c^4) : :

X(25308) lies on these lines: {2, 3056}, {63, 3888}, {100, 212}, {312, 4553}, {674, 18134}, {1026, 21361}, {2836, 2895}, {2979, 3006}, {3705, 3917}, {3779, 17778}, {3781, 4388}, {3792, 4865}, {3869, 6327}, {3873, 4259}, {4438, 7186}, {5794, 14923}, {24524, 25282}, {24789, 25048}, {25273, 25298}, {25297, 25310}

### X(25309) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    (a - b) (a - c) (b + c) (a^3 b + 3 a^2 b^2 + a b^3 + a^3 c - 3 a^2 b c - a b^2 c + b^3 c + 3 a^2 c^2 - a b c^2 - 3 b^2 c^2 + a c^3 + b c^3) : :

X(25309) lies on these lines: {2, 24531}, {6631, 23861}, {21272, 21295}, {25273, 25283}, {25274, 25293}, {25285, 25313}, {25310, 25312}

### X(25310) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    (a - b) (a - c) (a^2 b^2 + a b^3 + 2 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 + a c^3 + b c^3) : :

X(25310) lies on these lines: {2, 25110}, {668, 891}, {3676, 4551}, {25274, 25278}, {25282, 25287}, {25286, 25294}, {25297, 25308}, {25309, 25312}

### X(25311) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a^3 b^2 - a^2 b^3 - 2 a^3 b c - a^2 b^2 c + 2 a b^3 c + a^3 c^2 - a^2 b c^2 + a b^2 c^2 - b^3 c^2 - a^2 c^3 + 2 a b c^3 - b^2 c^3 : :

X(25311) lies on these lines: {2, 18194}, {6, 20532}, {7, 8}, {100, 20794}, {932, 23857}, {1716, 10027}, {3240, 8050}, {4110, 24451}, {4851, 20530}, {5687, 22152}, {6542, 21299}, {7155, 9025}, {7779, 10327}, {10453, 17373}, {17299, 24717}, {17350, 23354}, {17375, 20352}, {21080, 21219}, {24575, 25285}, {25279, 25298}

### X(25312) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    (a - b) (a - c) (a b + a c - b c) (a b^2 - 2 a b c + b^2 c + a c^2 + b c^2) : :

X(25312) lies on these lines: {2, 24677}, {8, 17793}, {668, 3888}, {1018, 20979}, {3699, 23354}, {21272, 25300}, {25278, 25293}, {25309, 25310}

### X(25313) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 18

Barycentrics    a^5 b^2 + a^4 b^3 - a^3 b^4 - a^2 b^5 + 2 a^5 b c - 2 a^4 b^2 c + 2 a^2 b^4 c + a^5 c^2 - 2 a^4 b c^2 + b^5 c^2 + a^4 c^3 + 2 a b^3 c^3 - b^4 c^3 - a^3 c^4 + 2 a^2 b c^4 - b^3 c^4 - a^2 c^5 + b^2 c^5 : :

X(25313) lies on these lines: {2, 24664}, {193, 3212}, {15983, 20955}, {17448, 25305}, {21226, 21272}, {25273, 25277}, {25283, 25293}, {25285, 25309}

Lionheart-Moses images: X(25314) - X(25336)

Lionheart-Moses images are defined in the preamble just before X(25045).

### X(25314) =  LIONHEART-MOSES (1/3)-IMAGE OF X(98)

Barycentrics    5*a^8*b^2 - 7*a^6*b^4 + 3*a^4*b^6 - a^2*b^8 + 5*a^8*c^2 - 2*a^6*b^2*c^2 - 2*a^4*b^4*c^2 + 4*a^2*b^6*c^2 - b^8*c^2 - 7*a^6*c^4 - 2*a^4*b^2*c^4 - 2*a^2*b^4*c^4 + b^6*c^4 + 3*a^4*c^6 + 4*a^2*b^2*c^6 + b^4*c^6 - a^2*c^8 - b^2*c^8 : :
X(25314) = X[193] + 2 X[1634], 5 X[3618] - 2 X[20021], 2 X[6] + X[25046], 4 X[6] - X[25051], 2 X[25046] + X[25051], 5 X[25051] - 8 X[25324], 5 X[25317] - 4 X[25324], 5 X[6] - 2 X[25324], 5 X[25046] + 4 X[25324]

X(25314) lies on these lines: {6, 17500}, {193, 1634}, {263, 1992}, {542, 3545}, {3618, 20021}

X(25314) = midpoint of X(25046) and X(25317)
X(25314) = reflection of X(i) in X(j) for these {i,j}: {25051, 25317}, {25317, 6}
X(25314) = {X(6),X(25046)}-harmonic conjugate of X(25051)

### X(25315) =  LIONHEART-MOSES (1/3)-IMAGE OF X(99)

Barycentrics    a^6*b^2 + 4*a^4*b^4 + 3*a^2*b^6 + a^6*c^2 - 16*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - b^6*c^2 + 4*a^4*c^4 + 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 - b^2*c^6 : :
X(25315) = X[69] - 4 X[3124], 5 X[3618] - 2 X[4576], 2 X[6] + X[25047], 4 X[6] - X[25052], 2 X[25047] + X[25052], X[25047] - 4 X[25322], X[6] + 2 X[25322], X[25052] + 8 X[25322], 5 X[25052] - 8 X[25325], 5 X[6] - 2 X[25325], 5 X[25322] + X[25325], 5 X[25047] + 4 X[25325], 5 X[25047] - 2 X[25334], 10 X[25322] - X[25334], 5 X[6] + X[25334], 2 X[25325] + X[25334], 5 X[25052] + 4 X[25334]

X(25315) lies on these lines: {6, 10330}, {69, 3124}, {1992, 2854}, {3618, 4576}, {5969, 13331}

X(25315) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25047, 25052), (6, 25322, 25047), (6, 25334, 25325)

### X(25316) =  LIONHEART-MOSES (1/3)-IMAGE OF X(105)

Barycentrics    a*(a^4*b^2 - a^3*b^3 + a^2*b^4 - a*b^5 + 3*a^4*b*c - 4*a^3*b^2*c + 3*a^2*b^3*c + a^4*c^2 - 4*a^3*b*c^2 - a^2*b^2*c^2 - a*b^3*c^2 + b^4*c^2 - a^3*c^3 + 3*a^2*b*c^3 - a*b^2*c^3 + a^2*c^4 + b^2*c^4 - a*c^5) : :
X(25316) = X[193] + 2 X[4553], X[190] + 2 X[20455], 4 X[6] - X[25048], 2 X[6] + X[25050], X[25048] + 2 X[25050], X[25050] - 4 X[25323], X[6] + 2 X[25323], X[25048] + 8 X[25323]

X(25316) lies on these lines: {6, 82}, {190, 20455}, {193, 4553}, {518, 17264}, {2876, 24482}

X(25316) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25050, 25048), (6, 25323, 25050)

### X(25317) =  LIONHEART-MOSES (1/3)-IMAGE OF X(110)

Barycentrics    a^8*b^2 - 5*a^6*b^4 + 3*a^4*b^6 + a^2*b^8 + a^8*c^2 + 2*a^6*b^2*c^2 + 2*a^4*b^4*c^2 + 2*a^2*b^6*c^2 + b^8*c^2 - 5*a^6*c^4 + 2*a^4*b^2*c^4 - 10*a^2*b^4*c^4 - b^6*c^4 + 3*a^4*c^6 + 2*a^2*b^2*c^6 - b^4*c^6 + a^2*c^8 + b^2*c^8 : :
X(25317) = 2 X[1634] - 5 X[3618], X[69] - 4 X[7668], X[193] + 2 X[20021], 4 X[6] - X[25046], 2 X[6] + X[25051], X[25046] + 2 X[25051], X[25051] - 4 X[25324], X[6] + 2 X[25324], X[25314] + 4 X[25324], X[25046] + 8 X[25324]

X(25317) lies on these lines: {2, 6784}, {6, 17500}, {69, 7668}, {193, 20021}, {542, 3839}, {1634, 3618}

X(25317) = midpoint of X(25051) and X(25314)
X(25317) = reflection of X(i) in X(j) for these {i,j}: {25046, 25314}, {25314, 6}
X(25317) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25051, 25046), (6, 25324, 25051)

### X(25318) =  LIONHEART-MOSES (1/3)-IMAGE OF X(689)

Barycentrics    3*a^6*b^4 + a^4*b^6 - 5*a^6*b^2*c^2 - 3*a^4*b^4*c^2 + a^2*b^6*c^2 + 3*a^6*c^4 - 3*a^4*b^2*c^4 + 3*a^2*b^4*c^4 - b^6*c^4 + a^4*c^6 + a^2*b^2*c^6 - b^4*c^6 : :
X(25318) = X[193] + 2 X[694], X[69] - 4 X[1084], X[1992] + 2 X[3228], 2 X[670] - 5 X[3618], 2 X[6] + X[25054], X[25054] - 4 X[25326], X[6] + 2 X[25326], X[25319] + 4 X[25326], 5 X[25319] - 4 X[25327], 5 X[6] - 2 X[25327], 5 X[25326] + X[25327], 5 X[25054] + 4 X[25327], 8 X[25327] - 5 X[25332], 4 X[6] - X[25332], 2 X[25054] + X[25332], 8 X[25326] + X[25332]

X(25318) lies on these lines: {6, 19585}, {69, 1084}, {193, 694}, {263, 1992}, {670, 3618}, {5032, 5969}

X(25318) = midpoint of X(25054) and X(25319)
X(25318) = reflection of X(i) in X(j) for these {i,j}: {25319, 6}, {25332, 25319}
X(25318) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25054, 25332), (6, 25326, 25054)

### X(25319) =  LIONHEART-MOSES (1/3)-IMAGE OF X(733)

Barycentrics    3*a^6*b^4 - a^4*b^6 - a^6*b^2*c^2 - 3*a^4*b^4*c^2 - a^2*b^6*c^2 + 3*a^6*c^4 - 3*a^4*b^2*c^4 + 3*a^2*b^4*c^4 + b^6*c^4 - a^4*c^6 - a^2*b^2*c^6 + b^4*c^6 : :
X(25319) = X[193] + 2 X[670], 2 X[694] - 5 X[3618], 4 X[6] - X[25054], 5 X[25054] - 8 X[25326], 5 X[25318] - 4 X[25326], 5 X[6] - 2 X[25326], X[6] + 2 X[25327], X[25318] + 4 X[25327], X[25326] + 5 X[25327], X[25054] + 8 X[25327], 4 X[25327] - X[25332], 2 X[6] + X[25332], X[25054] + 2 X[25332], 4 X[25326] + 5 X[25332]

X(25319) lies on these lines: {2, 6784}, {6, 19585}, {193, 670}, {694, 3618}, {5969, 13331}

X(25319) = midpoint of X(25318) and X(25332)
X(25319) = reflection of X(i) in X(j) for these {i,j}: {25054, 25318}, {25318, 6}
X(25319) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25327, 25332), (6, 25332, 25054)

### X(25320) =  LIONHEART-MOSES (1/3)-IMAGE OF X(827)

Barycentrics    (a^2 - b^2 - c^2)*(a^6 - a^4*b^2 - 3*a^2*b^4 - b^6 - a^4*c^2 + 7*a^2*b^2*c^2 + b^4*c^2 - 3*a^2*c^4 + b^2*c^4 - c^6) : :
X(25320) = X[69] - 4 X[125], 2 X[67] + X[193], 2 X[125] + X[895], X[69] + 2 X[895], 2 X[6] + X[3448], X[2930] - 4 X[3589], 2 X[110] - 5 X[3618], 7 X[3619] - 4 X[5181], X[146] - 4 X[5480], 5 X[2] - 2 X[5648], 5 X[3620] - 8 X[6698], 2 X[265] + X[6776], X[1992] + 2 X[9140], 4 X[597] - X[9143], X[1352] + 2 X[9976], X[1351] + 2 X[10264], 4 X[6] - X[11061], 2 X[3448] + X[11061], X[4] + 2 X[11579], X[1205] + 2 X[11800], 2 X[9970] + X[12317], 4 X[182] - X[12383], X[3751] + 2 X[13605], 4 X[6593] - X[14683], 2 X[10733] + X[14927], 5 X[3091] - 2 X[14982], 7 X[3619] - 10 X[15059], 2 X[5181] - 5 X[15059], 2 X[1352] - 5 X[15081], 4 X[9976] + 5 X[15081], 5 X[3618] - 8 X[15118], X[110] - 4 X[15118], X[10754] + 2 X[15357], X[10117] + 2 X[15583], X[10752] + 2 X[16003], 2 X[5480] + X[16010], X[146] + 2 X[16010], 4 X[11801] - X[18440], X[399] - 4 X[18583], 5 X[15081] - 8 X[20301], X[1352] - 4 X[20301], X[9976] + 2 X[20301], X[10620] + 2 X[21850], X[6391] + 2 X[23296], X[2892] - 4 X[23300], X[3448] - 4 X[25328], X[6] + 2 X[25328], X[25321] + 4 X[25328], X[11061] + 8 X[25328], 5 X[11061] - 8 X[25329], 5 X[25321] - 4 X[25329], 5 X[6] - 2 X[25329], 5 X[25328] + X[25329], 5 X[3448] + 4 X[25329], X[25321] + 2 X[25330], X[11061] + 4 X[25330], 2 X[25329] + 5 X[25330], 6 X[25329] - 5 X[25331], 3 X[11061] - 4 X[25331], 3 X[25321] - 2 X[25331], 3 X[6] - X[25331], 6 X[25328] + X[25331], 3 X[25330] + X[25331], 3 X[3448] + 2 X[25331], 5 X[3448] - 2 X[25335], 10 X[25328] - X[25335], 5 X[25330] - X[25335], 5 X[6] + X[25335], 2 X[25329] + X[25335], 5 X[25321] + 2 X[25335], 5 X[25331] + 3 X[25335], 5 X[11061] + 4 X[25335], 14 X[25329] - 5 X[25336], 7 X[11061] - 4 X[25336], 7 X[25331] - 3 X[25336], 7 X[25321] - 2 X[25336], 7 X[6] - X[25336], 14 X[25328] + X[25336], 7 X[25330] + X[25336], 7 X[3448] + 2 X[25336], 7 X[25335] + 5 X[25336]

X(25320) lies on these lines: {2, 2854}, {4, 11579}, {6, 3448}, {67, 193}, {69, 125}, {110, 3618}, {146, 5480}, {182, 12383}, {265, 6776}, {399, 18583}, {542, 3545}, {597, 7605}, {1205, 11800}, {1351, 10264}, {1352, 9976}, {1992, 9140}, {2072, 3564}, {2892, 23300}, {2930, 3589}, {3091, 14982}, {3619, 5181}, {3620, 6698}, {3751, 13605}, {5476, 16261}, {5622, 17702}, {5663, 14853}, {5987, 16989}, {6391, 23296}, {6593, 14683}, {9970, 12317}, {10117, 15583}, {10519, 14984}, {10620, 21850}, {10733, 14927}, {10752, 16003}, {10754, 15357}, {11801, 18440}, {13198, 18935}

X(25320) = midpoint of X(i) and X(j) for these {i,j}: {6, 25330}, {3448, 25321}
X(25320) = reflection of X(i) in X(j) for these {i,j}: {3448, 25330}, {10519, 15061}, {11061, 25321}, {25321, 6}, {25330, 25328}
X(25320) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 3448, 11061), (6, 25328, 3448), (6, 25335, 25329), (110, 15118, 3618), (125, 895, 69), (1352, 20301, 15081), (5181, 15059, 3619), (5480, 16010, 146), (9976, 20301, 1352)

### X(25321) =  LIONHEART-MOSES (1/3)-IMAGE OF X(9076)

Barycentrics    5*a^8 - 4*a^6*b^2 - 4*a^4*b^4 + 4*a^2*b^6 - b^8 - 4*a^6*c^2 + 9*a^4*b^2*c^2 - 3*a^2*b^4*c^2 - 4*a^4*c^4 - 3*a^2*b^2*c^4 + 2*b^4*c^4 + 4*a^2*c^6 - c^8 : :
X(25321) = 2 X[110] + X[193], X[399] + 2 X[1353], 4 X[6] - X[3448], 2 X[67] - 5 X[3618], X[2930] + 2 X[3629], X[193] - 4 X[5095], X[110] + 2 X[5095], 4 X[113] - X[5921], 5 X[3620] - 8 X[5972], X[69] - 4 X[6593], X[146] + 2 X[6776], 2 X[1992] + X[9143], X[7731] + 2 X[9967], X[146] - 4 X[9970], X[6776] + 2 X[9970], X[20] + 2 X[10752], 2 X[6] + X[11061], X[3448] + 2 X[11061], 4 X[5642] - X[11160], X[6403] - 4 X[11557], 4 X[10272] - X[11898], X[12270] + 2 X[12294], 2 X[1351] + X[12383], 5 X[2] - 2 X[13169], 4 X[11574] - X[13201], X[5596] + 2 X[13248], X[12220] + 2 X[13417], 2 X[895] + X[14683], X[13169] - 10 X[15303], X[2] - 4 X[15303], 2 X[5477] + X[15342], X[3580] - 4 X[15471], 4 X[12007] - X[16010], 2 X[141] + X[16176], 5 X[15081] - 8 X[18583], 4 X[5181] - X[20080], 5 X[3448] - 8 X[25328], 5 X[25320] - 4 X[25328], 5 X[6] - 2 X[25328], 5 X[11061] + 4 X[25328], X[11061] - 4 X[25329], X[6] + 2 X[25329], X[25320] + 4 X[25329], X[25328] + 5 X[25329], X[3448] + 8 X[25329], 6 X[25328] - 5 X[25330], 3 X[3448] - 4 X[25330], 3 X[25320] - 2 X[25330], 3 X[6] - X[25330], 6 X[25329] + X[25330], 3 X[11061] + 2 X[25330], X[25320] + 2 X[25331], X[25330] + 3 X[25331], X[3448] + 4 X[25331], 2 X[25328] + 5 X[25331], 14 X[25328] - 5 X[25335], 7 X[3448] - 4 X[25335], 7 X[25330] - 3 X[25335], 7 X[25320] - 2 X[25335], 7 X[6] - X[25335], 14 X[25329] + X[25335], 7 X[25331] + X[25335], 7 X[11061] + 2 X[25335], 5 X[11061] - 2 X[25336], 10 X[25329] - X[25336], 5 X[25331] - X[25336], 5 X[6] + X[25336], 2 X[25328] + X[25336], 5 X[25320] + 2 X[25336], 5 X[25330] + 3 X[25336], 5 X[3448] + 4 X[25336], 5 X[25335] + 7 X[25336]

X(25321) lies on these lines: {2, 9769}, {6, 3448}, {20, 10752}, {67, 3618}, {69, 6593}, {110, 193}, {113, 5921}, {141, 16176}, {146, 6776}, {399, 1353}, {403, 3564}, {542, 3839}, {895, 14683}, {1351, 12383}, {1992, 2854}, {2914, 19139}, {2930, 3629}, {3580, 15471}, {3620, 5972}, {5181, 20080}, {547