ETC

X(72413) = X(1296)X(66615)∩X(8705)X(10098)

Barycentrics    a^2*(2*a^10+2*b^10-b^8*c^2-11*b^6*c^4+5*b^4*c^6+9*b^2*c^8-4*c^10-a^8*(8*b^2+c^2)+a^6*(6*b^4+8*b^2*c^2-11*c^4)+a^4*(6*b^6-14*b^4*c^2+7*b^2*c^4+5*c^6)+a^2*(-8*b^8+8*b^6*c^2+7*b^4*c^4-16*b^2*c^6+9*c^8))*(2*a^10-4*b^10+9*b^8*c^2+5*b^6*c^4-11*b^4*c^6-b^2*c^8+2*c^10-a^8*(b^2+8*c^2)+a^6*(-11*b^4+8*b^2*c^2+6*c^4)+a^4*(5*b^6+7*b^4*c^2-14*b^2*c^4+6*c^6)+a^2*(9*b^8-16*b^6*c^2+7*b^4*c^4+8*b^2*c^6-8*c^8)) : :

Antreas Hatzipolakis and Ivan Pavlov, euclid 9521.

X(72413) lies on the circumcircle and these lines: {1296, 66615}, {8705, 10098}, {11568, 55135}, {23699, 67731}

X(72413) = intersection, other than A, B, C, of circumconics {{A, B, C, X(64), X(30488)}}, {{A, B, C, X(74), X(98)}}, {{A, B, C, X(8705), X(43720)}}, {{A, B, C, X(54998), X(70363)}}

---

X(72414) = X(4)X(575)∩X(381)X(13378)

Barycentrics    4*a^10-3*a^8*(b^2+c^2)+a^6*(-25*b^4+14*b^2*c^2-25*c^4)+a^2*(b^2-c^2)^2*(9*b^4-2*b^2*c^2+9*c^4)-2*(b^2-c^2)^2*(3*b^6-5*b^4*c^2-5*b^2*c^4+3*c^6)+a^4*(21*b^6-29*b^4*c^2-29*b^2*c^4+21*c^6) : :
X(72414) = 7*X[3832]+2*X[47590], -5*X[3843]+2*X[46673], -11*X[3855]+2*X[47592]

Antreas Hatzipolakis and Ivan Pavlov, euclid 9521.

X(72414) lies on these lines: {4, 575}, {381, 13378}, {524, 14866}, {1531, 13860}, {3832, 47590}, {3843, 46673}, {3845, 14856}, {3855, 47592}, {13168, 48895}, {40261, 58883}

X(72414) = midpoint of X(i) and X(j) for these {i,j}: {13378, 50730}
X(72414) = reflection of X(i) in X(j) for these {i,j}: {13378, 381}
X(72414) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 47589, 31748}

---

X(72415) = X(4)X(263)∩X(115)X(512)

Barycentrics    a^2*(b-c)^2*(b+c)^2*(a^4+4*b^2*c^2-a^2*(b^2+c^2)) : :
X(72415) = X[5167]+2*X[53419], 2*X[6321]+X[65748], -X[6785]+3*X[14639], -3*X[9166]+X[67630], X[9879]+3*X[14041], X[31848]+2*X[38734], -3*X[41135]+X[46303], -4*X[43291]+X[67540], -X[47287]+4*X[59571]

Antreas Hatzipolakis and Ivan Pavlov, euclid 9521.

X(72415) lies on these lines: {4, 263}, {30, 47638}, {51, 3845}, {115, 512}, {125, 2780}, {230, 32442}, {373, 3363}, {511, 8352}, {526, 16278}, {543, 6786}, {671, 6787}, {1084, 52625}, {1514, IsogConj(X54976)}, {2387, 39563}, {2869, 8754}, {3111, 5461}, {3124, 58754}, {3819, 66349}, {3917, 66392}, {4173, 44518}, {5077, 12525}, {5167, 53419}, {5650, 20326}, {6310, 33229}, {6321, 65748}, {6785, 14639}, {7615, 61689}, {7833, 67151}, {7841, 52658}, {8370, 34236}, {8597, 11673}, {9044, 64258}, {9166, 67630}, {9879, 14041}, {14135, 33249}, {22112, 45722}, {31848, 38734}, {32967, 58211}, {32984, 35687}, {33017, 34095}, {34417, 45723}, {34980, 41221}, {39691, 42068}, {40951, 69141}, {41135, 46303}, {43291, 67540}, {47287, 59571}, {56957, 69100}

X(72415) = midpoint of X(i) and X(j) for these {i,j}: {671, 6787}, {8597, 11673}, {9879, 33873}
X(72415) = reflection of X(i) in X(j) for these {i,j}: {3111, 5461}, {6784, 115}, {6786, 67215}, {32442, 230}, {65751, 6784}
X(72415) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(6784)}}, {{A, B, C, X(263), X(65751)}}, {{A, B, C, X(52038), X(58754)}} X(72415) = perspector of circumconic {{A, B, C, X(2395), X(31174)}}
X(72415) = pole of line {877, IsogConj(X6037)} with respect to the polar circle
X(72415) = pole of line {1499, 68786} with respect to the Jerabek hyperbola
X(72415) = pole of line {804, 68778} with respect to the Kiepert hyperbola
X(72415) = lies on inconics with perspector: X(5651)
X(72415) = barycentric product X(i)*X(j) for these (i, j): {115, 5651}, {3124, 69380}, {4079, 69390}, {31174, 512}
X(72415) = barycentric quotient X(i)/X(j) for these (i, j): {5651, 4590}, {31174, 670}, {69380, 34537}, {69390, 52612}
X(72415) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {115, 512, 6784}, {512, 6784, 65751}, {543, 67215, 6786}, {671, 6787, 34383}, {9879, 14041, 33873}, {9879, 33873, 55005}

---

X(72416) = X(5)X(113)∩X(114)X(325)

Barycentrics    a^2*(-b^4-c^4+a^2*(b^2+c^2))*(4*b^2*c^2*(b^2-c^2)^2+a^6*(b^2+c^2)-2*a^4*(b^4-3*b^2*c^2+c^4)+a^2*(b^6-3*b^4*c^2-3*b^2*c^4+c^6)) : :
X(72416) = -X[9862]+4*X[64490], -4*X[20399]+X[67352], -3*X[23234]+X[67639], X[31850]+2*X[38745]

Antreas Hatzipolakis and Ivan Pavlov, euclid 9521.

X(72416) lies on these lines: {4, 6331}, {5, 113}, {114, 325}, {147, 46303}, {446, 9155}, {542, 6784}, {568, 27374}, {1352, 61689}, {2682, 14915}, {2794, 3111}, {5025, 15072}, {5640, 13862}, {5650, 37451}, {6000, 33228}, {6033, 41330}, {6054, 6785}, {6656, 16836}, {7418, 36213}, {9862, 64490}, {11459, 37446}, {13240, 47353}, {23234, 67639}, {31850, 38745}, {33184, 64100}

X(72416) = midpoint of X(i) and X(j) for these {i,j}: {147, 46303}, {6033, 41330}, {6054, 6785}
X(72416) = reflection of X(i) in X(j) for these {i,j}: {6784, 67220}, {6786, 114}, {65748, 6786}, {65751, 41330}
X(72416) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(6786)}}, {{A, B, C, X(325), X(43917)}}, {{A, B, C, X(6393), X(34087)}}
X(72416) = pole of line {888, 53149} with respect to the polar circle
X(72416) = pole of line {2023, 3003} with respect to the Kiepert hyperbola
X(72416) = pole of line {1976, 43574} with respect to the Stammler hyperbola
X(72416) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {114, 511, 6786}, {511, 6786, 65748}, {6054, 6785, 34383}

---

X(72417) = X(4)X(6335)∩X(119)X(517)

Barycentrics    a*(-2*a*b*c+a^2*(b+c)-(b-c)^2*(b+c))*(-2*a^4*b*c+a^5*(b+c)+4*b*c*(b^2-c^2)^2-2*a^2*b*c*(b^2+c^2)+a*(b-c)^2*(b^3-3*b^2*c-3*b*c^2+c^3)-2*a^3*(b^3-2*b^2*c-2*b*c^2+c^3)) : :
X(72417) = X[3937]+2*X[10742], -X[12248]+4*X[64489], -4*X[20400]+X[67420], X[31849]+2*X[38757], -X[38389]+4*X[67864]

Antreas Hatzipolakis and Ivan Pavlov, euclid 9521.

X(72417) lies on these lines: {4, 6335}, {119, 517}, {125, 30444}, {2801, 25436}, {2807, 5587}, {2810, 10711}, {2818, 61729}, {2821, 14431}, {2829, 34583}, {3937, 10742}, {12248, 64489}, {18542, 23154}, {20400, 67420}, {29353, 68548}, {31849, 38757}, {38389, 67864}, {42448, 45631}, {53548, 56416}, {61674, 67216}

X(72417) = midpoint of X(i) and X(j) for these {i,j}: {10711, 61731}
X(72417) = reflection of X(i) in X(j) for these {i,j}: {61672, 119}, {61674, 67216}, {65743, 61672}
X(72417) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(61672)}}, {{A, B, C, X(36798), X(51379)}}, {{A, B, C, X(51367), X(60288)}}
X(72417) = pole of line {891, 43933} with respect to the polar circle
X(72417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {119, 517, 61672}, {517, 61672, 65743}, {10711, 61731, 2810}

---

X(72418) = X(4)X(145)∩X(118)X(516)

Barycentrics    (2*a^3-a^2*(b+c)-(b-c)^2*(b+c))*(2*a^5-a^4*(b+c)-4*a*(b^2-c^2)^2+2*a^3*(b^2+c^2)+a^2*(-4*b^3+2*b^2*c+2*b*c^2-4*c^3)+(b-c)^2*(5*b^3+9*b^2*c+9*b*c^2+5*c^3)) : :
X(72418) = X[152]+2*X[68552], X[1565]+2*X[10741], X[10727]+2*X[17044], X[31851]+2*X[38769]

Antreas Hatzipolakis and Ivan Pavlov, euclid 9521.

X(72418) lies on these lines: {4, 145}, {118, 516}, {152, 68552}, {1565, 10741}, {5845, 10710}, {10727, 17044}, {28182, 36028}, {31851, 38769}

X(72418) = reflection of X(i) in X(j) for these {i,j}: {51406, 118}, {65745, 51406}
X(72418) = perspector of circumconic {{A, B, C, X(2398), X(65336)}}
X(72418) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(51406)}}, {{A, B, C, X(516), X(6336)}}, {{A, B, C, X(910), X(36125)}}, {{A, B, C, X(1320), X(51376)}}, {{A, B, C, X(4080), X(51366)}}, {{A, B, C, X(63851), X(65745)}}
X(72418) = pole of line {900, 53150} with respect to the polar circle
X(72418) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {118, 516, 51406}, {516, 51406, 65745}, {1530, 1541, 1536}

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X(72419) = X(2)X(11166)∩X(599)X(5094)

Barycentrics    (a^2-2*(b^2+c^2))*(4*a^4-5*b^4+14*b^2*c^2-5*c^4+5*a^2*(b^2+c^2)) : :
X(72419) = -4*X[31606]+X[31748], X[34795]+2*X[47589]

Antreas Hatzipolakis and Ivan Pavlov, euclid 9521.

X(72419) lies on these lines: {2, 11166}, {125, 11168}, {141, 12036}, {524, 10162}, {599, 5094}, {1992, 11056}, {3849, 13378}, {3906, 8371}, {5650, 17430}, {6791, 16509}, {8288, 15810}, {9829, 11645}, {9830, 10163}, {20582, 30749}, {31606, 31748}, {34795, 47589}, {44569, 59197}, {51389, 59780}

X(72419) = reflection of X(i) in X(j) for these {i,j}: {13378, 34512}, {30516, 2}, {50729, 30516}
X(72419) = intersection, other than A, B, C, of circumconics {{A, B, C, X(3363), X(5094)}}, {{A, B, C, X(8541), X(11166)}}
X(72419) = pole of line {47352, 47587} with respect to the orthocentroidal circle
X(72419) = pole of line {43697, IsogConj(X3363)} with respect to the Stammler hyperbola
X(72419) = pole of line {8704, 31173} with respect to the Steiner inellipse X(72419) = barycentric quotient X(i)/X(j) for these (i, j): {3363, 598}
X(72419) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3849, 34512, 13378}

---

ETC

X(72409) = (name pending)

Barycentrics    a^2*(2*a^10 - 7*a^8*b^2 + 5*a^6*b^4 + 5*a^4*b^6 - 7*a^2*b^8 + 2*b^10 + 8*a^8*c^2 + 8*a^6*b^2*c^2 - 72*a^4*b^4*c^2 + 8*a^2*b^6*c^2 + 8*b^8*c^2 - 55*a^6*c^4 + 36*a^4*b^2*c^4 + 36*a^2*b^4*c^4 - 55*b^6*c^4 + 50*a^4*c^6 - 16*a^2*b^2*c^6 + 50*b^4*c^6 - a^2*c^8 - b^2*c^8 - 4*c^10)*(2*a^10 + 8*a^8*b^2 - 55*a^6*b^4 + 50*a^4*b^6 - a^2*b^8 - 4*b^10 - 7*a^8*c^2 + 8*a^6*b^2*c^2 + 36*a^4*b^4*c^2 - 16*a^2*b^6*c^2 - b^8*c^2 + 5*a^6*c^4 - 72*a^4*b^2*c^4 + 36*a^2*b^4*c^4 + 50*b^6*c^4 + 5*a^4*c^6 + 8*a^2*b^2*c^6 - 55*b^4*c^6 - 7*a^2*c^8 + 8*b^2*c^8 + 2*c^10) : :

Antreas Hatzipolakis and Ercole Suppa, euclid 9480.

X(72409) lies on the circumcircle and these lines: { }

X(72409) = intersection, other than A, B, C, of the circumconics : {{A, B, C, X (6), X (47588)}, {A, B, C, X (74), X (98)}, {A, B, C, X (13377), X (14490)}}

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X(72410) = X(2)X(47589)∩X(381)X(31748)

Barycentrics    -4*a^10-37*a^8*b^2+185*a^6*b^4-109*a^4*b^6-73*a^2*b^8+38*b^10-37*a^8*c^2-70*a^6*b^2*c^2+225*a^4*b^4*c^2+152*a^2*b^6*c^2-142*b^8*c^2+185*a^6*c^4+225*a^4*b^2*c^4-126*a^2*b^4*c^4+104*b^6*c^4-109*a^4*c^6+152*a^2*b^2*c^6+104*b^4*c^6-73*a^2*c^8-142*b^2*c^8+38*c^10 : :
X(72410) = X[2]+2*X[47589], 2*X[2]+X[50730], 4*X[47589]-X[50730], 2*X[381]+X[31748], 5*X[381]-2*X[46673], 5*X[31748]+4*X[46673], 5*X[1656]+4*X[47591], 7*X[3090]+2*X[47590], 2*X[3545]-X[13378], 2*X[14866]+X[50729], 2*X[46732]-5*X[61985]

Antreas Hatzipolakis and Ercole Suppa, euclid 9480.

X(72410) lies on these lines: {2, 47589}, {381, 31748}, {1656, 47591}, {3090, 47590}, {3545, 13378}, {3839, 11645}, {5056, 47592}, {14866, 50729}, {46732, 61985}

X(72410) = reflection of X(i) in X(j) for these {i,j}: {13378, 3545}
X(72410) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 47589, 50730}

---

X(72411) = X(115)X(1499)∩X(187)X(524)

Barycentrics    (b^2-c^2)^2*(-2*a^2+b^2+c^2)*(7*a^4-4*a^2*b^2+b^4-4*a^2*c^2-b^2*c^2+c^4) : :
X(72411) = X[115] - 2*X[67397], X[843] + X[6792], 2*X[620] - X[56429], 3*X[1691] - X[70601], 2*X[2030] - X[62373], 3*X[5215] - 2*X[37745], 2*X[32525] - X[44956], 3*X[26613] - X[62295], X[31173] - 2*X[37746]

Antreas Hatzipolakis and Ercole Suppa, euclid 9499.

X(72411) lies on these lines: {32, 67227}, {115, 1499}, {187, 524}, {476, 843}, {512, 6791}, {542, 67371}, {620, 56429}, {1648, 5099}, {1691, 70601}, {2030, 15303}, {2549, 67717}, {3815, 53798}, {3849, 62293}, {5104, 43913}, {5215, 37745}, {5475, 9169}, {5913, 67557}, {6077, 45672}, {6388, 14858}, {7603, 9151}, {7746, 15098}, {8288, 35605}, {9181, 41672}, {10418, 50566}, {14916, 21843}, {15357, 44398}, {17964, 34806}, {26613, 62295}, {31173, 37746}, {32761, 48945}, {37637, 57312}

X(72411) = midpoint of X(843) and X(6792)
X(72411) = reflection of X(i) in X(j) for these {i,j} : {115, 67397}, {31173, 37746}, {44956, 32525}, {56429, 620}, {62373, 2030}
X(72411) = reflection of X(i) in the line X(j)X(k) for these {i,j,k} : {2482, 2, 1499}, {5477, 6, 1499}, {18800, 597, 1499}, {50567, 141, 1499}
X(72411) = foot of the perpendicular from X(i) to the line X(j)X(k) for these {i, j, k} : {115, 187, 2482}, {187, 115, 38395}, {1691, 8288, 35605}, {8288, 1691, 70601}
X(72411) = perspector of the circumconic through X(5468) and X(9123)
X(72411) = intersection, other than A, B, C, of the circumconics : {{A, B, C, X(476), X(9123)}, {A,B,C,X(524),X(26613)}, {A,B,C,X(1648),X(39785)},{A,B,C,X(2482),X(20382)}}
X(72411) = center of circle {X(843), X(5912), X(6792)}
X(72411) = pole of the line {2793, 5461} with respect to Kiepert hyperbola
X(72411) = pole of the line {111, 52239} with respect to Stammler hyperbola
X(72411) = pole of the line {125, 62293} with respect to orthoptic circle of Jerabek hyperbola
X(72411) = pole of tripolar of X(62672) with respect to orthoptic circle of Kiepert hyperbola
X(72411) = pole of the line {22110, 42008} with respect to dual conic of Wallace hyperbola
X(72411) = crossdifference of every pair of points on the line X(9124)X(9178)
X(72411) = barycentric product X(i)*X(j) for these (i,j): {690, 9123}, {1648, 26613}
X(72411) = barycentric quotient X(i)/X(j) for these (i,j) : {51, 9124}, {9123, 892}, {21906, 52239}, {26613, 52940}
X(72411) = trilinear product X(2642)*X(9123)
X(72411) = trilinear quotient X(i)/X(j) for these (i,j) : {2642, 9124}, {9123, 36085}

---

X(72412) = X(125)X(1499)∩X(468)X(524)

Barycentrics    (b^2-c^2)^2*(-2*a^2+b^2+c^2)*(3*a^6-a^4*b^2-3*a^2*b^4+b^6-a^4*c^2+7*a^2*b^2*c^2-2*b^4*c^2-3*a^2*c^4-2*b^2*c^4+c^6) : :
X(72412) = X[125]-2*X[67398], X[31655]-2*X[32525], X[2770]+X[6792], 2*X[5972]-X[67394], X[7426]+X[62293], 5*X[47453]-X[70601], 3*X[47455]-X[62373]

Antreas Hatzipolakis and Ercole Suppa, euclid 9499.

X(72412) lies on these lines: {30, 37746}, {115, 47587}, {125, 1499}, {373, 2679}, {468, 524}, {511, 47349}, {512, 57425}, {523, 6791}, {542, 32222}, {1316, 9169}, {1648, 5099}, {2696, 54012}, {2715, 2770}, {3258, 46659}, {5108, 61644}, {5912, 47200}, {5972, 67394}, {6077, 50567}, {7426, 18800}, {9125, 23992}, {10989, 43910}, {13857, 47574}, {15638, 51258}, {34806, 68315}, {37638, 57355}, {37648, 53805}, {43964, 45303}, {44114, 55148}, {44569, 64966}, {47004, 52038}, {47453, 70601}, {47455, 62373}, {48317, 63758}, {57345, 63128}, {61645, 67392}

X(72412) = midpoint of X(i) and X(j) for these {i,j}: {2770, 6792}, {7426, 62293}
X(72412) = reflection of X(i) in X(j) for these {i,j}: {125, 67398}, {31655, 32525}, {67394, 5972}
X(72412) = reflection of X(i) in the line X(j)X(k) for these {i,j,k}: {5095, 6, 1499}, {5181, 141, 1499}, {5642, 2, 1499}, {15303, 597, 1499}
X(72412) = foot of the perpendicular from X(i) to the line X(j)X(k) for these {i,j,k}: {37746, 6791, 67397}, {115, 7426, 18800}, {7426, 115, 47587}, {125, 468, 3292}, {468, 125, 2682}, {3258, 1316, 9169}, {1316, 3258, 46659}
X(72412) = perspector of the circumconic through X(4235) and X(23287)
X(72412) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(468),X(57604)}, {A,B,C,X(5095),X(20382)}, {A,B,C,X(44102),X(52238)}}
X(72412) = center of circle {X(2770), X(6792), X(67393)}
X(72412) = pole of the line {30786, 52236} with respect to Wallace hyperbola
X(72412) = pole of the line {125, 524} with respect to orthoptic circle of Jerabek hyperbola
X(72412) = pole of the line {115, 62373} with respect to orthoptic circle of Kiepert hyperbola
X(72412) = pole of the line {42008, 47097} with respect to dual conic of Wallace hyperbola
X(72412) = cross-difference of every pair of points on the line X(10097)X(32583)
X(72412) = barycentric product X(i)*X(j) for these (i, j): {524, 57604}, {52238, 52628}
X(72412) = barycentric quotient X(i)/X(j) for these (i, j): {1648, 52236}, {52238, 66929}, {57604, 671}
X(72412) = trilinear product X(896)*X(57604)
X(72412) = trilinear quotient X(897)/X(57504)
X(72412) = {X(468),X(47550)}-harmonic conjugate of X(5642)

---

ORTHOLOGIC TRIANGLES

Let ABC be a triangle, P, Q two points, A'B'C' the padal triangle of Q and S a point on the Euler line such that OS/OH= t: number.

Denote:

Ab, Ac = the orthogonal projections of A on BP, CP, resp.
Sa = same to S point of the triangle A'AbAc.
Similary Sb,Sc.

Cases the triangle ABC, SaSbSc are orthologic.

1. Let ABC be a triangle, P a point and A'B'C' the pedal triangle of a point Q.

Denote:

Ab, Ac = the orthogonal projections of A on BP, CP, resp.

Ga = the centroid of A'AbAc.
Similarly Gb, Gc

ABC, GaGbGc are circumorthologic.
ie The Orthologic center (ABC, GaGbGc) = X1 lies on the circumcircle of ABC.
The Orthologic center (GaGbGc, ABC) = X2 lies on the circumcircle of GaGbGc.

2. Let ABC be a triangle, P = I = X(1), A'B'C' the pedal triangle of a point Q and S a point on the Euler line such that OS/OH= t: number.

Denote:

Ab, Ac = the orthogonal projections of A on BI, CI, resp.

Sa = same to S point of the triangle A'AbAc.
Similarly Sb, Sc

ABC, SaSbSc are Orthologic.

3. Let ABC be a triangle, P, Q two isogonal conjugate points, A'B'C' the pedal triangle of Q and S a point on the Euler line such that OS/OH= t: number.

Denote:

Ab, Ac = the orthogonal projections of A on BP, CP, resp.

Sa = same to S point of the triangle A'AbAc.
Similarly Sb, Sc

ABC, SaSbSc are circumoerthologic. ie The Orthologic center (ABC, SaSbSc) = X1 lies on the circumcircle of ABC.
The Orthologic center (SaSbSc, ABC) = X2 lies on the circumcircle of SaSbSc.

EULER

X(72398) = 105TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics 6*a^10 - 11*a^8*b^2 - 2*a^6*b^4 + 12*a^4*b^6 - 4*a^2*b^8 - b^10 - 11*a^8*c^2 + 38*a^6*b^2*c^2 - 21*a^4*b^4*c^2 - 9*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 21*a^4*b^2*c^4 + 26*a^2*b^4*c^4 - 2*b^6*c^4 + 12*a^4*c^6 - 9*a^2*b^2*c^6 - 2*b^4*c^6 - 4*a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(72398) = 5 X[3] - 3 X[16532], 3 X[3] - X[43893], 7 X[3] - 3 X[46451], X[23] - 4 X[62087], 3 X[140] - 2 X[403], 5 X[140] - 4 X[15350], 5 X[140] - 2 X[44267], 3 X[376] + X[35452], 4 X[403] - 3 X[11558], 5 X[403] - 6 X[15350], X[403] - 3 X[34152], 5 X[403] - 3 X[44267], 4 X[468] - 7 X[61784], 5 X[546] - 8 X[5159], 7 X[546] - 8 X[63838], 3 X[547] - 4 X[10257], 7 X[547] - 4 X[47310], 25 X[548] - 4 X[37899], 9 X[548] - 4 X[37931], 7 X[548] - 4 X[47335], 19 X[548] - 4 X[47342], 3 X[549] - X[52403], 5 X[550] + X[5189], 3 X[550] - X[13619], 2 X[858] + X[62151], X[1657] + 3 X[44450], X[2070] - 3 X[8703], 5 X[2071] - X[18403], 3 X[2071] - X[37938], 9 X[2071] - X[64890], 7 X[2072] - 3 X[65087], 5 X[3522] - X[5899], 7 X[3528] - 3 X[37922], 2 X[3530] - 3 X[37948], 3 X[3534] + X[46450], X[3627] - 3 X[65085], 7 X[3853] - 6 X[65087], 3 X[5066] - 2 X[44283], 7 X[5159] - 5 X[63838], 3 X[5189] + 5 X[13619], X[5189] - 5 X[18859], 3 X[7426] - 5 X[15646], 2 X[7426] - 5 X[34200], X[7464] + 2 X[44245], X[7574] + 2 X[62136], 5 X[10096] - 6 X[16532], 3 X[10096] - 2 X[43893], 7 X[10096] - 6 X[46451], 4 X[10151] - 5 X[61940], 7 X[10257] - 3 X[47310], 5 X[11558] - 8 X[15350], X[11558] - 4 X[34152], 5 X[11558] - 4 X[44267], X[11563] - 3 X[37948], 2 X[11799] - 5 X[61790], 3 X[12100] - 2 X[44234], 3 X[12101] - 4 X[23323], X[12103] + 2 X[37950], 5 X[12812] - 2 X[62288], X[13473] - 3 X[15122], 4 X[13473] - 3 X[62026], X[13619] + 3 X[18859], 3 X[14893] - 2 X[64891], 4 X[15122] - X[62026], 2 X[15350] - 5 X[34152], 2 X[15646] - 3 X[34200], 9 X[15688] - X[37949], 3 X[15690] - 2 X[44246], X[15690] + 2 X[54995], 5 X[15712] - 3 X[37943], 9 X[16532] - 5 X[43893], 7 X[16532] - 5 X[46451], 8 X[16976] - 7 X[61821], X[18325] - 4 X[61792], 3 X[18403] - 5 X[37938], 9 X[18403] - 5 X[64890], 2 X[18572] + X[62156], X[20063] - 13 X[62105], 5 X[22248] - 2 X[62344], 3 X[25338] - 4 X[37935], 7 X[25338] - 4 X[47338], 2 X[25338] - 5 X[62064], 5 X[30745] - 2 X[62013], 5 X[34152] - X[44267], X[35001] + 5 X[62104], 3 X[35489] - 7 X[62100], 5 X[37760] - 11 X[62062], 9 X[37899] - 25 X[37931], 7 X[37899] - 25 X[47335], 19 X[37899] - 25 X[47342], 5 X[37923] - 17 X[62084], X[37924] - 7 X[62091], X[37925] - 6 X[62089], 7 X[37931] - 9 X[47335], 19 X[37931] - 9 X[47342], 2 X[37935] - 3 X[37968], 7 X[37935] - 3 X[47338], 8 X[37935] - 15 X[62064], 3 X[37938] - X[64890], 3 X[37941] - 4 X[58190], X[37944] + 4 X[41981], X[37947] - 3 X[44280], 3 X[37955] - 2 X[44264], 3 X[37955] - 5 X[46853], 3 X[37956] - 11 X[62085], 5 X[37958] - 11 X[62079], 7 X[37968] - 2 X[47338], 4 X[37968] - 5 X[62064], 9 X[41982] - 8 X[47114], 3 X[41983] - 2 X[44282], 7 X[43893] - 9 X[46451], 2 X[44214] - 3 X[61782], X[44246] + 3 X[54995], 2 X[44264] - 5 X[46853], 4 X[44452] - 5 X[61810], 8 X[44911] - 9 X[47598], 4 X[44961] - 7 X[61821], 4 X[46031] - 5 X[48154], 4 X[47090] + X[58203], 4 X[47311] + 5 X[62138], 19 X[47335] - 7 X[47342], 4 X[47336] - 7 X[55862], 8 X[47338] - 35 X[62064], 7 X[50693] + X[60466], 5 X[60455] + 7 X[62134], 3 X[60462] + 5 X[62131], 13 X[62092] - X[62290], 3 X[13363] - 2 X[13446]

See Antreas Hatzipolakis and Peter Moses, euclid 9446.

X(72398) lies on these lines: {2, 3}, {74, 50708}, {477, 33639}, {930, 67735}, {1154, 17855}, {1291, 67797}, {1294, 13863}, {2693, 30248}, {2777, 46114}, {6799, 53934}, {13363, 13446}, {13391, 37853}, {13399, 32423}, {13445, 34153}, {14677, 43574}, {22115, 43391}, {29011, 67784}, {40111, 50434}, {53884, 67727}

X(72398) = midpoint of X(i) and X(j) for these {i,j}: {550, 18859}, {3153, 15704}, {13445, 34153}, {14677, 43574}, {16386, 37950}, {40111, 50434}
X(72398) = reflection of X(i) in X(j) for these {i,j}: {140, 34152}, {186, 33923}, {3853, 2072}, {10096, 3}, {11558, 140}, {11563, 3530}, {12103, 16386}, {25338, 37968}, {31726, 3628}, {44267, 15350}, {44961, 16976}, {47096, 22249}
X(72398) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {403, 13473, 44226}, {427, 21284, 65154}, {3520, 13619, 403}, {3530, 50143, 140}, {5159, 7426, 6677}, {5189, 6636, 7426}, {11563, 37948, 3530}, {15690, 66718, 548}, {16387, 47311, 5159}

---

X(72399) = 106TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics 4*a^10 - 12*a^8*b^2 + 8*a^6*b^4 + 8*a^4*b^6 - 12*a^2*b^8 + 4*b^10 - 12*a^8*c^2 + 2*a^6*b^2*c^2 - 7*a^4*b^4*c^2 + 29*a^2*b^6*c^2 - 12*b^8*c^2 + 8*a^6*c^4 - 7*a^4*b^2*c^4 - 34*a^2*b^4*c^4 + 8*b^6*c^4 + 8*a^4*c^6 + 29*a^2*b^2*c^6 + 8*b^4*c^6 - 12*a^2*c^8 - 12*b^2*c^8 + 4*c^10 : :
X(72399) = X[2] - 9 X[37943], X[2] - 3 X[44282], 25 X[2] - 9 X[44450], 7 X[2] + 9 X[46451], 17 X[2] - 9 X[65085], 2 X[5] + X[12105], X[23] + 3 X[5055], 7 X[140] + 2 X[47338], 3 X[186] + X[3830], 5 X[381] - X[10296], X[381] + 3 X[37907], 3 X[403] - X[3845], 3 X[403] + X[44265], 4 X[468] - X[18571], 3 X[468] - X[18579], 5 X[468] - 2 X[22249], 2 X[468] + X[44961], 9 X[468] - X[47031], 13 X[468] - X[47308], 11 X[468] + X[47309], 7 X[468] + X[47310], 3 X[468] + X[47332], 5 X[468] - X[47333], 7 X[468] - X[47335], 5 X[468] + X[47336], 3 X[549] - X[54995], 5 X[632] + X[62344], X[858] - 3 X[15699], 5 X[1656] - X[10989], 5 X[1656] + X[37967], 3 X[2070] + 5 X[19709], 3 X[2071] - 7 X[15701], 3 X[2072] + X[47313], 3 X[2072] - 5 X[61910], 7 X[3090] + X[37901], 3 X[3153] - 11 X[61932], 3 X[3524] + X[18325], X[3534] - 3 X[15646], 3 X[3545] - X[18572], 3 X[3545] + 5 X[37760], 2 X[3628] + X[16619], 5 X[3843] + 7 X[37957], 7 X[3851] + 5 X[37953], X[3853] + 2 X[37934], 3 X[5054] - X[37950], X[5066] + 3 X[10096], X[5066] - 6 X[37942], 2 X[5066] - 3 X[46031], 7 X[5066] - 6 X[63838], 11 X[5070] + X[37946], 5 X[5071] - X[7574], 5 X[5071] + 3 X[37909], 2 X[5159] - 3 X[47599], X[5189] - 9 X[61899], 3 X[5899] + 13 X[61901], X[7464] - 5 X[15694], X[7574] + 3 X[37909], 5 X[7575] + X[10296], X[7575] - 3 X[37907], X[8703] + 3 X[11563], 5 X[8703] - 3 X[16386], 2 X[8703] - 3 X[37968], X[8703] - 3 X[44214], X[8703] - 6 X[44900], X[10096] + 2 X[37942], 2 X[10096] + X[46031], 7 X[10096] + 2 X[63838], 2 X[10109] + X[37904], 3 X[10151] - 2 X[61997], 3 X[10257] - 4 X[11540], X[10296] + 15 X[37907], X[10297] + 2 X[44264], X[11001] + 3 X[31726], X[11558] + 2 X[16531], 3 X[11558] + X[62138], 5 X[11563] + X[16386], 2 X[11563] + X[37968], X[11563] + 2 X[44900], 3 X[11799] + X[54995], 2 X[11812] - 3 X[44452], X[12100] - 3 X[44234], 4 X[12811] - X[47339], 3 X[13619] + 5 X[62007], 3 X[14269] + 5 X[37958], 3 X[14892] + 4 X[47316], 2 X[15350] + X[37971], 6 X[15350] - X[47311], 3 X[15350] - 2 X[61896], X[15681] - 5 X[37952], X[15682] - 3 X[44283], X[15685] - 9 X[37955], 5 X[15693] - 3 X[34152], 5 X[15695] - 9 X[37941], 7 X[15703] + X[37924], 5 X[15713] + 3 X[43893], 2 X[16386] - 5 X[37968], X[16386] - 5 X[44214], X[16386] - 10 X[44900], 6 X[16531] - X[62138], 9 X[16532] - X[19710], 3 X[16532] - X[44280], X[18323] - 3 X[23046], 3 X[18403] - 7 X[41106], 3 X[18571] - 4 X[18579], 5 X[18571] - 8 X[22249], X[18571] + 2 X[44961], 9 X[18571] - 4 X[47031], 13 X[18571] - 4 X[47308], 11 X[18571] + 4 X[47309], 7 X[18571] + 4 X[47310], 3 X[18571] + 4 X[47332], 5 X[18571] - 4 X[47333], X[18571] + 4 X[47334], 7 X[18571] - 4 X[47335], 5 X[18571] + 4 X[47336], X[18572] + 5 X[37760], 5 X[18579] - 6 X[22249], 2 X[18579] + 3 X[44961], 3 X[18579] - X[47031], 13 X[18579] - 3 X[47308], 11 X[18579] + 3 X[47309], 7 X[18579] + 3 X[47310], 5 X[18579] - 3 X[47333], X[18579] + 3 X[47334], 7 X[18579] - 3 X[47335], 5 X[18579] + 3 X[47336], 3 X[18859] - 11 X[61843], 5 X[19708] + 3 X[52403], X[19710] - 3 X[44280], X[20063] + 15 X[61906], 5 X[22248] + 3 X[41987], 4 X[22249] + 5 X[44961], 18 X[22249] - 5 X[47031], 26 X[22249] - 5 X[47308], 22 X[22249] + 5 X[47309], 14 X[22249] + 5 X[47310], 6 X[22249] + 5 X[47332], 2 X[22249] + 5 X[47334], 14 X[22249] - 5 X[47335], 2 X[22249] + X[47336], 3 X[23323] - 4 X[61960], X[25338] + 2 X[68319], 5 X[30745] - 9 X[61887], 7 X[33699] - 9 X[65087], X[35001] - 9 X[61864], 4 X[35018] + X[47312], 4 X[35018] - X[47341], 3 X[35452] - 19 X[61857], 9 X[35489] + 7 X[62009], 2 X[37897] + 3 X[47478], X[37899] + 6 X[45757], X[37900] + 9 X[61909], 4 X[37911] - 3 X[47598], 9 X[37922] + 7 X[61974], 5 X[37923] + 11 X[61925], 3 X[37925] + 17 X[61893], 3 X[37931] + 2 X[62010], 6 X[37935] + X[62022], 3 X[37936] + 7 X[61920], 3 X[37938] - X[47314], 3 X[37938] - 7 X[61898], 9 X[37940] + 11 X[61950], 4 X[37942] - X[46031], 7 X[37942] - X[63838], 9 X[37943] + X[44266], 3 X[37943] - X[44282], 25 X[37943] - X[44450], 7 X[37943] + X[46451], 17 X[37943] - X[65085], 3 X[37944] - 23 X[61862], 3 X[37947] + 11 X[61908], 9 X[37948] - 13 X[61797], X[37968] - 4 X[44900], 3 X[37971] + X[47311], 3 X[37971] + 4 X[61896], 3 X[38335] + X[56369], 3 X[44246] - X[62154], X[44266] + 3 X[44282], 25 X[44266] + 9 X[44450], 7 X[44266] - 9 X[46451], 17 X[44266] + 9 X[65085], 25 X[44282] - 3 X[44450], 7 X[44282] + 3 X[46451], 17 X[44282] - 3 X[65085], 7 X[44450] + 25 X[46451], 17 X[44450] - 25 X[65085], 9 X[44961] + 2 X[47031], 13 X[44961] + 2 X[47308], 11 X[44961] - 2 X[47309], 7 X[44961] - 2 X[47310], 3 X[44961] - 2 X[47332], 5 X[44961] + 2 X[47333], 7 X[44961] + 2 X[47335], 5 X[44961] - 2 X[47336], 7 X[46031] - 4 X[63838], 3 X[46450] - 19 X[61913], 17 X[46451] + 7 X[65085], 13 X[47031] - 9 X[47308], 11 X[47031] + 9 X[47309], 7 X[47031] + 9 X[47310], X[47031] + 3 X[47332], 5 X[47031] - 9 X[47333], X[47031] + 9 X[47334], 7 X[47031] - 9 X[47335], 5 X[47031] + 9 X[47336], 3 X[47096] + 7 X[61851], 11 X[47308] + 13 X[47309], 7 X[47308] + 13 X[47310], 3 X[47308] + 13 X[47332], 5 X[47308] - 13 X[47333], X[47308] + 13 X[47334], 7 X[47308] - 13 X[47335], 5 X[47308] + 13 X[47336], 7 X[47309] - 11 X[47310], 3 X[47309] - 11 X[47332], 5 X[47309] + 11 X[47333], X[47309] - 11 X[47334], 7 X[47309] + 11 X[47335], 5 X[47309] - 11 X[47336], 3 X[47310] - 7 X[47332], 5 X[47310] + 7 X[47333], X[47310] - 7 X[47334], 5 X[47310] - 7 X[47336], X[47311] - 4 X[61896], X[47313] + 5 X[61910], X[47314] - 7 X[61898], 5 X[47332] + 3 X[47333], X[47332] - 3 X[47334], 7 X[47332] + 3 X[47335], 5 X[47332] - 3 X[47336], X[47333] + 5 X[47334], 7 X[47333] - 5 X[47335], 7 X[47334] + X[47335], 5 X[47334] - X[47336], 5 X[47335] + 7 X[47336], X[47340] + 4 X[67236], X[47342] + 4 X[61922], 7 X[55856] - X[62332], 3 X[57584] - 5 X[61998], 5 X[60455] - 21 X[61897], 15 X[61882] + X[62290], 7 X[62000] - 3 X[64890], X[62043] - 3 X[64891], X[110] + 3 X[15362], 3 X[5215] - X[38611], X[9158] + 3 X[57305], X[11179] - 5 X[47453], X[11801] + 2 X[15448], 3 X[14643] + X[15360], 2 X[15088] + X[32237], X[20423] + 3 X[47450], X[21850] + 5 X[47452], X[34315] + 3 X[59403], X[34316] + 3 X[59404], 3 X[47455] - X[50979], X[47471] + 3 X[47562], X[50955] + 3 X[52238]

See Antreas Hatzipolakis and Peter Moses, euclid 9446.

X(72399) lies on these lines: {2, 3}, {110, 15362}, {113, 15361}, {524, 10272}, {952, 47495}, {3564, 47544}, {5215, 38611}, {5844, 47488}, {9158, 57305}, {11178, 32217}, {11179, 47453}, {11645, 20304}, {11649, 13364}, {11801, 15448}, {12900, 19924}, {14643, 15360}, {15088, 32237}, {16328, 18487}, {20423, 47450}, {21850, 47452}, {32423, 35266}, {32515, 46986}, {34315, 59403}, {34316, 59404}, {34380, 47473}, {43291, 47169}, {43656, 53950}, {44204, 47219}, {44569, 46817}, {45969, 61606}, {47455, 50979}, {47471, 47562}, {47556, 47581}, {50955, 52238}, {61572, 62508}, {61619, 63124}

X(72399) = midpoint of X(i) and X(j) for these {i,j}: {2, 44266}, {5, 7426}, {113, 15361}, {376, 44267}, {381, 7575}, {468, 47334}, {547, 25338}, {549, 11799}, {3845, 44265}, {10295, 15687}, {10989, 37967}, {11178, 32217}, {11563, 44214}, {11737, 44264}, {15686, 62288}, {16619, 47097}, {18579, 47332}, {44204, 47219}, {44569, 46817}, {47310, 47335}, {47312, 47341}, {47333, 47336}, {47556, 47581}
X(72399) = reflection of X(i) in X(j) for these {i,j}: {547, 68319}, {10297, 11737}, {12105, 7426}, {14893, 37984}, {15122, 10124}, {37968, 44214}, {44214, 44900}, {44961, 47334}, {47097, 3628}, {47333, 22249}, {62139, 66595}
X(72399) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {381, 37907, 7575}, {403, 44265, 3845}, {403, 66725, 37984}, {468, 44961, 18571}, {468, 47332, 18579}, {468, 47336, 22249}, {5071, 37909, 7574}, {10096, 37942, 46031}, {10096, 44233, 25338}, {10109, 66529, 5066}, {10296, 10298, 16386}, {11563, 44900, 37968}, {13626, 13627, 381}, {14002, 37907, 7426}, {18579, 47334, 47332}, {25338, 44234, 25337}, {34330, 62961, 14893}, {44233, 68319, 46031}, {44266, 44282, 2}, {57322, 57323, 61924}

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TWO PROBLEMS CONNECTED

Francisco Javier García Capitán, Two problems [by Antreas Hatzipolakis] connected.

FGJC
geogebra file

z

X(72407) = X(13)X(42788)∩X(14)X(1506)

Barycentrics    8 a^16 - 84 a^14 b^2 + 375 a^12 b^4 - 908 a^10 b^6 + 1288 a^8 b^8 - 1086 a^6 b^10 + 528 a^4 b^12 - 136 a^2 b^14 + 15 b^16 - 84 a^14 c^2 + 594 a^12 b^2 c^2 - 1492 a^10 b^4 c^2 + 1258 a^8 b^6 c^2 + 552 a^6 b^8 c^2 - 1476 a^4 b^10 c^2 + 766 a^2 b^12 c^2 - 130 b^14 c^2 + 375 a^12 c^4 - 1492 a^10 b^2 c^4 + 1076 a^8 b^4 c^4 + 1002 a^6 b^6 c^4 + 391 a^4 b^8 c^4 - 1492 a^2 b^10 c^4 + 487 b^12 c^4 - 908 a^10 c^6 + 1258 a^8 b^2 c^6 + 1002 a^6 b^4 c^6 + 1082 a^4 b^6 c^6 + 862 a^2 b^8 c^6 - 1018 b^10 c^6 + 1288 a^8 c^8 + 552 a^6 b^2 c^8 + 391 a^4 b^4 c^8 + 862 a^2 b^6 c^8 + 1292 b^8 c^8 - 1086 a^6 c^10 - 1476 a^4 b^2 c^10 - 1492 a^2 b^4 c^10 - 1018 b^6 c^10 + 528 a^4 c^12 + 766 a^2 b^2 c^12 + 487 b^4 c^12 - 136 a^2 c^14 - 130 b^2 c^14 + 15 c^16 - 4 a^14 T + 28 a^12 b^2 T - 86 a^10 b^4 T + 138 a^8 b^6 T - 134 a^6 b^8 T + 78 a^4 b^10 T - 22 a^2 b^12 T + 2 b^14 T + 28 a^12 c^2 T - 112 a^10 b^2 c^2 T + 142 a^8 b^4 c^2 T - 48 a^6 b^6 c^2 T - 58 a^4 b^8 c^2 T + 56 a^2 b^10 c^2 T - 6 b^12 c^2 T - 86 a^10 c^4 T + 142 a^8 b^2 c^4 T - 112 a^6 b^4 c^4 T + 56 a^4 b^6 c^4 T - 84 a^2 b^8 c^4 T + 8 b^10 c^4 T + 138 a^8 c^6 T - 48 a^6 b^2 c^6 T + 56 a^4 b^4 c^6 T + 96 a^2 b^6 c^6 T - 4 b^8 c^6 T - 134 a^6 c^8 T - 58 a^4 b^2 c^8 T - 84 a^2 b^4 c^8 T - 4 b^6 c^8 T + 78 a^4 c^10 T + 56 a^2 b^2 c^10 T + 8 b^4 c^10 T - 22 a^2 c^12 T - 6 b^2 c^12 T + 2 c^14 T : : where T = Sqrt[3] S

Benjamin Lee Warren and Francisco Javier García Capitán, euclid 9479.

X(72407) lies on these lines: {5, 11602}, {13, 42788}, {14, 1506}, {17, 5615}, {18, 59403}, {76, 16966}, {83, 6671}, {5487, 35689}, {6115, 11606}, {10187, 25555}, {11122, 37832}, {11272, 43539}, {12817, 52649}, {16242, 62877}, {16964, 54861}

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X(72408) = X(13)X(1506)∩X(14)X(42788)

Barycentrics    8 a^16-84 a^14 b^2+375 a^12 b^4-908 a^10 b^6+1288 a^8 b^8-1086 a^6 b^10+528 a^4 b^12-136 a^2 b^14+15 b^16-84 a^14 c^2+594 a^12 b^2 c^2-1492 a^10 b^4 c^2+1258 a^8 b^6 c^2+552 a^6 b^8 c^2-1476 a^4 b^10 c^2+766 a^2 b^12 c^2-130 b^14 c^2+375 a^12 c^4-1492 a^10 b^2 c^4+1076 a^8 b^4 c^4+1002 a^6 b^6 c^4+391 a^4 b^8 c^4-1492 a^2 b^10 c^4+487 b^12 c^4-908 a^10 c^6+1258 a^8 b^2 c^6+1002 a^6 b^4 c^6+1082 a^4 b^6 c^6+862 a^2 b^8 c^6-1018 b^10 c^6+1288 a^8 c^8+552 a^6 b^2 c^8+391 a^4 b^4 c^8+862 a^2 b^6 c^8+1292 b^8 c^8-1086 a^6 c^10-1476 a^4 b^2 c^10-1492 a^2 b^4 c^10-1018 b^6 c^10+528 a^4 c^12+766 a^2 b^2 c^12+487 b^4 c^12-136 a^2 c^14-130 b^2 c^14+15 c^16+4 a^14 T-28 a^12 b^2 T+86 a^10 b^4 T-138 a^8 b^6 T+134 a^6 b^8 T-78 a^4 b^10 T+22 a^2 b^12 T-2 b^14 T-28 a^12 c^2 T+112 a^10 b^2 c^2 T-142 a^8 b^4 c^2 T+48 a^6 b^6 c^2 T+58 a^4 b^8 c^2 T-56 a^2 b^10 c^2 T+6 b^12 c^2 T+86 a^10 c^4 T-142 a^8 b^2 c^4 T+112 a^6 b^4 c^4 T-56 a^4 b^6 c^4 T+84 a^2 b^8 c^4 T-8 b^10 c^4 T-138 a^8 c^6 T+48 a^6 b^2 c^6 T-56 a^4 b^4 c^6 T-96 a^2 b^6 c^6 T+4 b^8 c^6 T+134 a^6 c^8 T+58 a^4 b^2 c^8 T+84 a^2 b^4 c^8 T+4 b^6 c^8 T-78 a^4 c^10 T-56 a^2 b^2 c^10 T-8 b^4 c^10 T+22 a^2 c^12 T+6 b^2 c^12 T-2 c^14 T : : where T = Sqrt[3] S

Benjamin Lee Warren and Francisco Javier García Capitán, euclid 9479.

X(72408) lies on these lines: {5, 11603}, {13, 1506}, {14, 42788}, {17, 59404}, {18, 5611}, {76, 16967}, {83, 6672}, {5488, 35688}, {6114, 11606}, {10188, 25555}, {11121, 37835}, {11272, 43538}, {12816, 44289}, {16241, 62876}, {16965, 54860}

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ETC

X(5459)
Let ABC be a triangle, let A', B', C' be the midpoints of BC, CA, AB. Let L_a be the perpendicular through A' to the line X(618)A'. Define L_b, L_c cyclically. Then X(5459) is the center of the equilateral triangle A''B''C'' bounded by L_a, L_b, L_c. The circumcircle of A''B''C'' passes through X(14081) and X(32552) and has squared radius (3 sqrt(3) S^3 + 9 S^2 SW + 3 sqrt(3) S SW^2 + SW^3)/(9 (3 S^2 + 2 sqrt(3) S SW + SW^2)). This circle (A''B''C'') is here named 1st Suppa circle. The 2nd Suppa circle is defined at X(5460)
(Euclid 8675, August 28, 2025)

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X(5960)
Let ABC be a triangle, let A', B', C' be the midpoints of BC, CA, AB. Let L_a be the perpendicular through A' to the line X(619)A'. Define L_b, L_c cyclically. Then X(5460) is the center of the equilateral triangle A''B''C'' bounded by L_a, L_b, L_c. The circumcircle of A''B''C'' passes through X(14082) and X(32553)and has squared radius (-3 sqrt(3) S^3 + 9 S^2 SW - 3 sqrt(3) S SW^2 + SW^3)/(9 (3 S^2 - 2 sqrt(3) S SW + SW^2)). The circle (A''B''C'') is here named 2nd Suppa circle. The 1st Suppa circle is defined at X(5459)
(Euclid 8675, August 28, 2025)