X(15) = 1st ISODYNAMIC POINT

Trilinears

\(sin(A + π/3) : sin(B + π/3) : sin(C + π/3)\)

\(cos(A - π/6) : cos(B - π/6) : cos(C - π/6)\)

\(3 cos A + sqrt(3) sin A : :\)

Barycentrics

\(a sin(A + π/3) : b sin(B + π/3) : c sin(C + π/3)\)

\((SB + SC)/((SA + Sqrt[3] S) (SB + SC) + 4 SB SC) : :\)

\((S + Sqrt[3] SA) (SB + SC) : :\)

\(a^2*(Sqrt[3]*(a^2 - b^2 - c^2) - 2*S) : :\)

Notes

Let U and V be the points on sideline BC met by the interior and exterior bisectors of angle A. The circle having diameter UV is the A-Apollonian circle. The B- and C- Apollonian circles are similarly constructed. Each circle passes through one vertex and both isodynamic points. (This configuration, based on the cevians of X(1), generalizes to arbitrary cevians; see TCCT, p. 98, problem 8.)

Let A’B’C’ be the 4th Brocard triangle and A″B″C″ be the 4th anti-Brocard triangle. The circumcircles of AA’A″, BB’B″, CC’C″ concur in two points, X(15) and X(16). (Randy Hutson, July 20, 2016)

The line X(13)X(15) is parallel to the Euler line, and the distance between the two lines is (SB - SC)(SC - SA)(SA - SB)/(cot(ω) + 31/2)|*((E - 8F)S2)1/2. (Kiminari Shinagawa, February 20, 2018)

The pedal triangle of X(15) is equilateral. If you have The Geometer’s Sketchpad, you can view 1st isodynamic point and X(15), with Brocard axis and Lemoine axis.

Video showing circular porisim-orbits of X(13), X(14), X(15), and X(16): 3-Periodics in a Concentric Homothetic Poncelet Pair: Circular Loci of Four Triangle Centers. (Dan Reznik, August 9, 2020) See also Loci of Centers of Ellipse-Mounted Triangles. –> Loci of Centers of Ellipse-Mounted Triangles. (Dan Reznik, August 26, 2020)

If you have GeoGebra, you can view 1st isodynamic point.

Several notable circles pass through X(15) and X(16). For each circle listed here, the appearance of (i; [name], [list]) means that the center is X(i), and the points with listed indices lie on the circle:

(187; Schoute circle, 15,16)

(351; Parry circle, 2, 15, 16, 23, 110, 111, 352, 353, 5638, 5639, 6141, 6142, 7598, 7599, 7601, 7602, 7711, 9138, 9147, 9153, 9156, 9157, 9158, 9162, 9163, 9212, 9213, 9978, 9980, 9998, 9999, 11199, 11673, 13114, 13242, 14660, 14704, 14705, 32072, 32073, 32074, 32526, 33502, 33503)

(647; Moses radical circle, 15, 16, 5000, 5001, 6112, 6113, 6114, 6115, 6116, 6117)

(649; Gheorghe circle, 15, 16, 1276, 1277, 32622, 32623)

(663; 1, 15, 16, 36, 3465, 4040, 5526, 5529)

(665; 15, 16, 32753, 32754)

(669; 15, 16, 5004, 5005, 5980, 5981)

(887; 15, 16, 99, 729, 13210, 14691)

(890; 15, 16, 100, 739))

(1960; Moses isodynamic circle, 15, 16, 101, 106, 214, 9321, 11716, 38013, 38014, 41183, 41184, 41185, 41186, 41187, 41188, 41189, 41190, 41191, 41192, 41193)

(2488; 15, 16, 3513, 3514)

(2502; Parry isodynamic circle, 15, 16)

(3005; 15, 16, 5002, 5003)

(3250; 15, 16, 32763, 32764)

(3569; 15, 16, 32618, 32619, 39665, 39666)

(5027; 15, 16, 18773, 18774, 22687, 22689)

(5075; 15, 16, 846, 1054, 1283, 5197)

(5638; 15, 16, 39164, 39165)

(5639; 15, 16, 39162, 39163)

(6137; 13, 15, 16, 3165, 5616, 5669, 6104, 10658)

(6138; 14, 15, 16, 3166, 5612, 5668, 6105, 10657)

(6139; Terzic circle, 15, 16, 55, 109, 654, 1155, 2291, 41155, 41156, 41157, 41158, 41159, 41160, 41161, 41162, 41163, 41164, 41165, 41166)

(6140; 15, 16, 115, 128, 399, 1263, 1511, 2079, 10277, 14367)

(8644; 15, 16, 38001, 38002)

(9409; 15, 16, 74, 112, 5667, 9862, 11587, 40894, 40895)

(9420; 15, 16, 98, 13236, 15920, 26714)

(15451; 4, 15, 16, 186, 3484, 11674, 13509, 15412)

(17414; 15, 16, 11629, 11630)

(9527; 15, 16, 351, 2502, 9129, 10166)

(42649; 15, 16, 35, 484, 3483, 14102)

(42650; 15, 16, 54, 1157, 3482, 18335)

(42651; 15, 16, 125, 184, 2081, 13558)

(42652; 15, 16, 385, 805, 5970, 32531)

(42653; 15, 16, 501, 3743, 5127, 14838, 14873, 39149)

(42654; 15, 16, 647, 1495, 14685, 16319, 35901)

(42655; 15, 16, 667, 1083, 3230, 11650, 11651, 11652)

(42656; 15, 16, 1138, 2132, 6794, 12112, 14254)

(42657; 15, 16, 3065, 3464, 5540, 6126)

This list of circles was contributed by Peter Moses, April 17, 2021, with the following notes. Starting with a circle X(15), X(16), and a point P = p : q : r, the center of the circle is given by

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

and the power of vertex A with repect to the circle is

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

If P lies on the Lemoine axis, then the power of A with respect to to the circle is

-b^2*c^2*(c^2*q + b^2*r)/(c^2*(a^2 - b^2)*q + b^2*(a^2 - c^2)*r)).

Consider this experiment, in which 3 regular hexagons, HA, Hb, HC are erected on the sides of T = ABC. Let the hexagon vertices be labeled as Ha = {B,A1,A2,A3,A4,C), Hb = {C,B1,B2,B3,B4,A}, Hc={A,C1,C2,C3,C4,B}, and the vertices of the 3 “flank-triangles”, by Fa = {A,C1,B4}, Fb = “{B,A1,C4}, Fc = {C,B1,A4}. Between each pair of consecutive hexagons, define 3 “flank” triangles Fa = {A,C1,B4}, Fb = {B,A1,C4}, Fc={C,Ba,A4}. Let T’ be the triangle with vertices X(15)-of-Fa, X(15)-of-Fb, X(15)-of-Fc. Claim (1): T’ is perspective to T, and the perspector is X(6). Claim (2): X(16)-of-Fa = X(16)-of-Fb = X(15)-of-Fc . (Dan Reznick, November 11, 2021)

X(15) lies on the Evans conic, Parry circle, Moses radical circle, Schoutte circle, Parry isodynamic circle, the cubics K001, K018, K048, K050, K073, K114, K129b, K148, K193, K206, K261a, K261b, K262a, K262b, K263, K290, K291, K292, K303a, K304, K341a, K390, K435, K438a, K438b, K439, K440, K441, K458, K463, K468, K469, K471, K505, K508, K513, K514, K523, K524, K639, K640, K641, K730, K802, K803, K881, K882, K883, K884, K885, K894, K900, K909, K912, K940, K942, K944, K946, K1052, K1064, K1099, K1105, K1132b, K1133a and the curves Q002, Q016, Q037, Q039, Q043, Q049, Q054, Q067, Q075, Q076, Q090, Q092, Q097, Q123, Q136, Q137, Q138, Q139, Q140, Q142, Q143, as well as these lines: {1, 1251}, {2, 14}, {3, 6}, {4, 17}, {5, 2913}, {11, 11755}, {13, 30}, {18, 140}, {20, 3412}, {21, 5362}, {23, 11629}, {24, 10642}, {35, 1250}, {36, 202}, {40, 10636}, {44, 11790}, {45, 11791}, {51, 3132}, {54, 10678}, {55, 203}, {56, 7005}, {57, 11760}, {74, 5668}, {86, 21898}, {98, 33388}, {99, 22687}, {110, 2378}, {111, 9202}, {115, 6771}, {128, 11600}, {183, 25167}, {184, 2903}, {185, 21647}, {186, 3165}, {214, 5240}, {230, 21156}, {237, 14186}, {298, 533}, {299, 3643}, {302, 34508}, {303, 316}, {323, 3170}, {351, 9162}, {376, 10653}, {378, 8740}, {381, 16644}, {383, 9993}, {385, 5980}, {395, 549}, {397, 550}, {399, 5612}, {404, 5367}, {465, 13567}, {466, 23292}, {470, 6110}, {484, 8444}, {485, 2041}, {486, 2042}, {512, 9163}, {523, 16181}, {524, 5463}, {530, 22495}, {532, 616}, {542, 9117}, {590, 18585}, {597, 35303}, {615, 15765}, {622, 9989}, {625, 11306}, {627, 22901}, {628, 636}, {630, 31706}, {631, 11489}, {633, 7836}, {635, 7832}, {691, 2379}, {740, 5699}, {842, 5994}, {843, 9203}, {846, 2946}, {940, 21476}, {1080, 6115}, {1082, 16577}, {1138, 5624}, {1147, 3205}, {1154, 2902}, {1157, 8447}, {1181, 19363}, {1263, 8173}, {1277, 8482}, {1337, 2981}, {1338, 2381}, {1495, 3129}, {1498, 17826}, {1511, 6105}, {1513, 9749}, {1593, 11408}, {1656, 5339}, {1657, 5340}, {1658, 11268}, {1682, 11758}, {1724, 11098}, {2043, 6560}, {2044, 6561}, {2045, 5420}, {2046, 5418}, {2058, 13391}, {2070, 2923}, {2132, 8445}, {2133, 8448}, {2380, 10409}, {2549, 5474}, {2777, 10681}, {2854, 13859}, {2926, 10329}, {2927, 2937}, {2952, 2959}, {3065, 5673}, {3070, 14814}, {3071, 14813}, {3096, 11290}, {3124, 14705}, {3130, 34417}, {3200, 11137}, {3231, 14178}, {3334, 14146}, {3411, 3530}, {3441, 8478}, {3464, 7326}, {3465, 7059}, {3479, 8451}, {3480, 8175}, {3483, 16883}, {3484, 8479}, {3515, 11409}, {3524, 16963}, {3631, 22845}, {3734, 25157}, {3849, 9763}, {3850, 5349}, {3923, 5700}, {3972, 35918}, {4383, 21475}, {5010, 7127}, {5054, 16268}, {5056, 5343}, {5059, 5344}, {5066, 12817}, {5068, 5365}, {5366, 22235}, {5459, 31710}, {5469, 25166}, {5470, 31709}, {5471, 6774}, {5472, 6781}, {5473, 9112}, {5529, 11789}, {5613, 9981}, {5614, 17403}, {5617, 6777}, {5623, 8491}, {5663, 10657}, {5667, 6111}, {5672, 8501}, {5675, 8456}, {5679, 8455}, {5681, 8462}, {5873, 22746}, {5916, 23895}, {5999, 22691}, {6000, 10675}, {6137, 9213}, {6138, 9138}, {6151, 21462}, {6241, 11466}, {6294, 23009}, {6296, 23019}, {6297, 23010}, {6300, 22611}, {6301, 22610}, {6304, 22640}, {6305, 22639}, {6564, 18587}, {6565, 18586}, {6581, 8177}, {6642, 10644}, {6694, 7859}, {6695, 10583}, {6759, 10676}, {6770, 6778}, {7060, 7089}, {7164, 8449}, {7325, 8508}, {7327, 8476}, {7329, 8472}, {7426, 34315}, {7488, 11421}, {7502, 11135}, {7622, 9761}, {7709, 32466}, {7751, 33466}, {7790, 11303}, {7844, 11305}, {7846, 11308}, {7865, 11297}, {7877, 35689}, {7880, 11301}, {7914, 11312}, {8172, 8495}, {8291, 9865}, {8431, 8453}, {8434, 8454}, {8438, 8457}, {8441, 8471}, {8458, 8535}, {8463, 8490}, {8473, 8486}, {8474, 8487}, {8475, 8494}, {8477, 8496}, {8483, 16882}, {8598, 12155}, {8839, 13367}, {8884, 19190}, {8919, 23721}, {9113, 21157}, {9147, 14447}, {9744, 9750}, {9754, 16652}, {9886, 22579}, {9932, 10660}, {10187, 22237}, {10188, 35018}, {10282, 30403}, {10546, 16259}, {10637, 10902}, {10661, 13754}, {10663, 17702}, {10664, 12893}, {10682, 13289}, {10788, 22696}, {11003, 14169}, {11004, 11126}, {11008, 22844}, {11145, 15018}, {11202, 11244}, {11449, 11453}, {11452, 12111}, {11464, 11467}, {11540, 33606}, {11676, 22701}, {11761, 11770}, {12367, 14173}, {12584, 32302}, {12816, 15682}, {12972, 12981}, {12973, 12983}, {12980, 13058}, {12982, 13057}, {13049, 13059}, {13050, 13060}, {13102, 22891}, {13704, 23011}, {13706, 23020}, {13824, 23012}, {13826, 23021}, {13860, 22693}, {13881, 16630}, {14137, 16940}, {14182, 23017}, {14188, 23022}, {14369, 14972}, {14704, 20998}, {15080, 34009}, {15412, 23872}, {15640, 33607}, {15743, 18776}, {15764, 32788}, {16319, 32460}, {16460, 16639}, {16807, 32628}, {17277, 21869}, {17821, 17827}, {18400, 32397}, {18762, 35738}, {18909, 18929}, {18925, 18930}, {18980, 19450}, {18981, 19451}, {19185, 19191}, {19357, 19364}, {19440, 19452}, {19441, 19453}, {22113, 22895}, {22489, 31693}, {22571, 31695}, {22602, 31697}, {22631, 31699}, {22688, 31701}, {22702, 22714}, {22707, 22715}, {22796, 22892}, {22843, 22862}, {22962, 22975}, {22999, 25220}, {23007, 25178}, {23259, 35732}, {23358, 32398}, {25151, 31707}, {25158, 31713}, {25159, 31715}, {25160, 31717}, {25217, 31719}, {30461, 30467}, {30468, 36185}, {31378, 36210}, {31694, 33475}, {31696, 33477}, {31698, 33447}, {31700, 33446}, {31702, 33479}, {31708, 33480}, {31712, 33483}, {31714, 33485}, {31716, 33489}, {31718, 33487}, {31720, 33490}, {32171, 32208}, {32465, 32515}, {35725, 35727}, {35730, 35740}

X(15) = midpoint of X(i) and X(j) for these {i,j}: {3, 5611}, {14, 6780}, {616, 3180}, {622, 14712}, {2378, 5610}, {6777, 25236}

X(15) = reflection of X(i) in X(j) for these {i,j}: {1, 11707}, {4, 7684}, {13, 396}, {14, 6109}, {16, 187}, {17, 14138}, {298, 618}, {316, 624}, {621, 623}, {623, 6671}, {2902, 11136}, {5318, 11542}, {5463, 35304}, {5668, 5995}, {5978, 619}, {6778, 6783}, {9162, 351}, {10409, 33526}, {11600, 15609}, {16267, 16962}, {19106, 5318}, {20428, 5}, {22997, 9117}, {22999, 25220}, {23004, 115}, {23007, 25178}, {34315, 7426}, {34317, 14170}

X(15) = reflection of X(i) in X(j) for these (i,j): (13,396), (16,187), (298,618), (316,624), (621,623)

X(15) = isogonal conjugate of X(13)

X(15) = isotomic conjugate of X(300)

X(15) = complement of X(621)

X(15) = anticomplement of X(623)

X(15) = circumcircle-inverse of X(16)

X(15) = nine-point-circle-inverse of X(6112)

X(15) = Brocard-circle-inverse of X(16)

X(15) = polar-circle-inverse of X(6116)

X(15) = orthoptic-circle-of-Steiner-inellipse-inverse of X(6114)

X(15) = 2nd-Brocard-circle-inverse of X(3105)

X(15) = circumcircle-of-inner-Napoleon-triangle-inverse of X(14)

X(15) = Lucas-inner-circle-inverse of X(16)

X(15) = Lucas-circles-radical-circle inverse of X(16)

X(15) = outer-Montesdeoca-Lemoine circle-inverse of X(16)

X(15) = inner-Montesdeoca-Lemoine-circle-inverse of X(16)

X(15) = antigonal image of X(11600)

X(15) = symgonal image of X(33526)

X(15) = complement of the isogonal conjugate of X(3438)

X(15) = complement of the isotomic conjugate of X(2992)

X(15) = isogonal conjugate of the anticomplement of X(618)

X(15) = isogonal conjugate of the complement of X(616)

X(15) = isotomic conjugate of the isogonal conjugate of X(34394)

X(15) = isogonal conjugate of the isotomic conjugate of X(298)

X(15) = isotomic conjugate of the polar conjugate of X(8739)

X(15) = isogonal conjugate of the polar conjugate of X(470)

X(15) = Thomson-isogonal conjugate of X(5463)

X(15) = excentral-isogonal conjugate of X(2945)

X(15) = tangential-isogonal conjugate of X(2925)

X(15) = orthic-isogonal conjugate of X(2902)

X(15) = psi-transform of X(16)

X(15) = X(i)-complementary conjugate of X(j) for these (i,j): {2992, 2887}, {3438, 10}

X(15) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 202}, {3, 3165}, {4, 2902}, {6, 3170}, {13, 62}, {14, 5616}, {30, 5668}, {54, 3200}, {74, 16}, {470, 8739}, {2981, 6}, {11117, 11126}, {17402, 6137}, {32036, 35443}

X(15) = X(i)-cross conjugate of X(j) for these (i,j): {74, 8445}, {1094, 7006}, {1154, 11600}, {1511, 16}, {3200, 62}, {6137, 17402}, {14816, 13}, {19295, 323}, {34327, 11146}, {34394, 8739}

X(15) = X(i)-isoconjugate of X(j) for these (i,j): {1, 13}, {2, 2153}, {16, 2166}, {18, 3383}, {31, 300}, {63, 8737}, {75, 3457}, {94, 2152}, {661, 23895}, {662, 20578}, {1081, 19551}, {1577, 5995}, {2154, 11078}, {2306, 7026}, {3179, 14358}, {6138, 32680}, {23871, 32678}, {24041, 30452}

X(15) = crosspoint of X(i) and X(j) for these (i,j): {2, 2992}, {13, 18}, {249, 10409}, {298, 470}, {2380, 16460}, {11600, 36210}

X(15) = crosssum of X(i) and X(j) for these (i,j): {1, 3179}, {2, 3180}, {3, 10661}, {6, 3129}, {15, 62}, {16, 5612}, {395, 30459}, {396, 8014}, {523, 30465}, {532, 619}, {6104, 36208}, {6111, 6116}, {9200, 30467}, {11542, 11555}, {18777, 30466}, {20578, 30452}, {23283, 30460}

X(15) = X(i)-line conjugate of X(j) for these (i,j): {13, 11537}, {549, 395}, {9138, 6138}, {16181, 523}

X(15) = X(i)-vertex conjugate of X(j) for these (i,j): {4, 16257}, {13, 3457}, {16, 512}, {3458, 32906}

X(15) = trilinear pole of line {526, 6137}

X(15) = crossdifference of every pair of points on line {395, 523}

X(15) = X(6)-Hirst inverse of X(16)

X(15) = X(15)-of-2nd-Brocard-triangle

X(15) = X(15)-of-circumsymmedial-triangle

X(15) = {X(371),:ref:X(372) <X(372)>}-harmonic conjugate of X(61)

X(15) = X(75)-isoconjugate of X(3457)

X(15) = X(1577)-isoconjugate of X(5995)

X(15) = outer-Napoleon-to-inner-Napoleon similarity image of X(13)

X(15) = orthocentroidal-to-ABC similarity image of X(13)

X(15) = 4th-Brocard-to-circumsymmedial similarity image of X(13)

X(15) = X(2378)-of-2nd-Parry triangle

X(15) = radical center of Lucas(2/sqrt(3)) circles

X(15) = homothetic center of (equilateral) 1st isogonal triangle of X(13) and pedal triangle of X(15)

X(15) = homothetic center of (equilateral) 1st isogonal triangle of X(14) and triangle formed by circumcenters of BC:ref:X(14) <X(14)>, CA:ref:X(14) <X(14)>, AB:ref:X(14) <X(14)>

X(15) = eigencenter of inner Napoleon triangle

X(15) = X(13)-of-4th-anti-Brocard-triangle

X(15) = X(15)-of-:ref:`X(3) <X(3)>`PU(1)

X(15) = Thomson-isogonal conjugate of X(5463)

X(15) = X(10657)-of-orthocentroidal-triangle

X(15) = {X(16),:ref:X(61) <X(61)>}-harmonic conjugate of X(6)

X(15) = Cundy-Parry Phi transform of X(61)

X(15) = Cundy-Parry Psi transform of X(17)

X(15) = X(1277)-of-orthic-triangle if ABC is acute

X(15) = barycentric product X(i)*X(j) for these {i,j}: {3, 470}, {6, 298}, {13, 11131}, {14, 323}, {16, 11092}, {17, 11146}, {18, 11127}, {50, 301}, {54, 33529}, {62, 19778}, {69, 8739}, {75, 2151}, {76, 34394}, {97, 6117}, {99, 6137}, {110, 23870}, {249, 30465}, {299, 11086}, {302, 8603}, {523, 17402}, {526, 23896}, {533, 6151}, {618, 2981}, {691, 9204}, {1082, 5240}, {2380, 14922}, {2987, 6782}, {3165, 19774}, {3170, 11121}, {3268, 5994}, {3457, 11129}, {3458, 7799}, {5616, 13582}, {6110, 14919}, {10409, 35443}, {10410, 14447}, {10411, 20579}, {10677, 11143}, {11078, 36209}, {11117, 19294}, {11120, 19295}, {11126, 11600}, {11130, 36210}, {11133, 21462}, {11136, 34390}, {11137, 34389}, {17403, 23284}

X(15) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 300}, {6, 13}, {14, 94}, {16, 11078}, {25, 8737}, {31, 2153}, {32, 3457}, {50, 16}, {61, 8838}, {62, 16770}, {110, 23895}, {186, 471}, {298, 76}, {301, 20573}, {323, 299}, {470, 264}, {512, 20578}, {526, 23871}, {1154, 33530}, {1250, 7026}, {1576, 5995}, {2088, 30468}, {2151, 1}, {2154, 2166}, {2981, 11119}, {3124, 30452}, {3165, 19772}, {3170, 3180}, {3200, 11127}, {3457, 11080}, {3458, 1989}, {5994, 476}, {6105, 8836}, {6117, 324}, {6137, 523}, {6138, 23283}, {6151, 11118}, {8603, 17}, {8604, 11601}, {8738, 6344}, {8739, 4}, {9204, 35522}, {10633, 472}, {10677, 11144}, {11062, 6116}, {11081, 36211}, {11083, 11581}, {11086, 14}, {11092, 301}, {11127, 303}, {11131, 298}, {11135, 6104}, {11136, 62}, {11137, 61}, {11146, 302}, {11243, 8919}, {14270, 6138}, {17402, 99}, {19294, 532}, {19295, 619}, {19373, 1081}, {19627, 34395}, {19778, 34390}, {20579, 10412}, {21461, 11139}, {21462, 11082}, {23870, 850}, {23896, 35139}, {30465, 338}, {32729, 9206}, {33529, 311}, {34327, 629}, {34394, 6}, {34395, 11081}, {34397, 8740}, {36209, 11092}

X(15) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 11754, 11753}, {1, 11763, 11762}, {2, 617, 3642}, {2, 621, 623}, {2, 5334, 18581}, {2, 10654, 14}, {2, 18581, 16967}, {3, 6, 16}, {3, 16, 10646}, {3, 61, 62}, {3, 62, 5237}, {3, 371, 3364}, {3, 372, 3365}, {3, 3390, 35739}, {3, 5238, 5352}, {3, 5615, 9736}, {3, 5865, 14540}, {3, 10634, 11515}, {3, 11480, 10645}, {3, 11485, 6}, {3, 11486, 11481}, {3, 13350, 21158}, {3, 15793, 15784}, {3, 18468, 10634}, {3, 22236, 61}, {3, 22238, 5351}, {4, 10632, 10641}, {4, 11488, 18582}, {4, 18582, 16808}, {5, 5321, 16809}, {5, 23302, 16966}, {6, 16, 62}, {6, 10645, 10646}, {6, 10646, 34755}, {6, 11480, 3}, {6, 11481, 11486}, {6, 11485, 61}, {6, 19781, 32}, {6, 22236, 11485}, {13, 396, 16267}, {13, 15441, 11581}, {13, 16960, 11542}, {13, 16962, 396}, {13, 19106, 5318}, {14, 16241, 2}, {14, 16967, 18581}, {14, 33417, 16967}, {16, 61, 6}, {16, 62, 34755}, {16, 5238, 10645}, {16, 6396, 35739}, {16, 10645, 3}, {16, 10646, 5237}, {16, 34754, 61}, {17, 16808, 18582}, {17, 19107, 16808}, {17, 22906, 31704}, {18, 33416, 23303}, {32, 3098, 16}, {32, 3105, 62}, {35, 2307, 7006}, {35, 5353, 1250}, {36, 5357, 19373}, {39, 5092, 16}, {50, 3581, 16}, {61, 3389, 3365}, {61, 3390, 3364}, {61, 5238, 3}, {61, 5352, 5237}, {61, 10645, 16}, {61, 11480, 10646}, {61, 14539, 3107}, {61, 34754, 11485}, {62, 5352, 3}, {62, 10646, 16}, {140, 398, 18}, {140, 11543, 23303}, {140, 23303, 33416}, {182, 574, 16}, {182, 3106, 62}, {182, 9735, 3}, {187, 11480, 21158}, {216, 11430, 16}, {298, 30471, 7799}, {299, 11299, 3643}, {303, 11304, 624}, {323, 11146, 11131}, {323, 34394, 3170}, {371, 372, 61}, {371, 3389, 62}, {371, 5238, 35739}, {371, 6200, 16}, {372, 3390, 62}, {372, 6396, 16}, {389, 22052, 16}, {395, 549, 16242}, {396, 5318, 11542}, {396, 11542, 16960}, {398, 23303, 11543}, {485, 2041, 3391}, {486, 2042, 3392}, {500, 17454, 16}, {566, 14805, 16}, {569, 14806, 16}, {572, 4256, 16}, {573, 4257, 16}, {575, 8589, 16}, {576, 8588, 16}, {576, 9736, 5615}, {577, 11438, 16}, {578, 10979, 16}, {621, 5334, 33518}, {623, 6671, 2}, {628, 22861, 22850}, {991, 4262, 16}, {1151, 6221, 16}, {1152, 6398, 16}, {1250, 2307, 5353}, {1250, 5353, 7006}, {1340, 1341, 16}, {1350, 1384, 16}, {1351, 5210, 16}, {1379, 1380, 16}, {1620, 33636, 16}, {1670, 1671, 3105}, {1689, 1690, 3106}, {1691, 35002, 16}, {2030, 18860, 16}, {2076, 5611, 3105}, {2076, 9301, 16}, {2076, 19781, 187}, {2080, 5104, 16}, {2903, 3166, 3201}, {3003, 10564, 16}, {3053, 33878, 16}, {3094, 26316, 16}, {3105, 3106, 3094}, {3311, 6411, 16}, {3312, 6412, 16}, {3364, 3365, 62}, {3365, 35739, 5237}, {3366, 3367, 5}, {3371, 3372, 3390}, {3385, 3386, 3389}, {3389, 3390, 3}, {3430, 33628, 16}, {3592, 6451, 16}, {3594, 6452, 16}, {5008, 14810, 16}, {5013, 12017, 16}, {5024, 5085, 16}, {5030, 13329, 16}, {5033, 9737, 16}, {5093, 5585, 16}, {5237, 34755, 16}, {5238, 10645, 11480}, {5238, 11485, 10646}, {5238, 22236, 62}, {5238, 34754, 16}, {5318, 11542, 13}, {5321, 16772, 23302}, {5321, 23302, 5}, {5334, 18581, 14}, {5352, 11485, 34755}, {5352, 30560, 21158}, {5357, 19373, 202}, {5473, 9112, 23006}, {5611, 13350, 14538}, {5611, 21401, 21158}, {6105, 36209, 11086}, {6199, 6409, 16}, {6200, 6396, 10645}, {6200, 11485, 3365}, {6295, 22689, 5981}, {6303, 6307, 14905}, {6395, 6410, 16}, {6396, 11485, 3364}, {6407, 6468, 16}, {6408, 6469, 16}, {6425, 6445, 16}, {6426, 6446, 16}, {6429, 9690, 16}, {6437, 6449, 16}, {6438, 6450, 16}, {6439, 9691, 16}, {6453, 6480, 16}, {6454, 6481, 16}, {6671, 33518, 16967}, {6778, 16529, 6783}, {7051, 10638, 1}, {9675, 9738, 16}, {10632, 32585, 8837}, {10641, 11475, 4}, {10645, 11480, 5352}, {10645, 11485, 62}, {10645, 34754, 6}, {10654, 18581, 5334}, {10667, 10671, 6}, {10676, 30402, 6759}, {11127, 11131, 323}, {11137, 22115, 3200}, {11477, 15655, 16}, {11480, 11485, 16}, {11480, 22236, 6}, {11480, 34754, 62}, {11481, 11486, 16}, {11485, 22236, 34754}, {11488, 18582, 17}, {11542, 16960, 16267}, {11543, 23303, 18}, {11581, 11586, 15441}, {11753, 11762, 1}, {11754, 11763, 1}, {11755, 11764, 11}, {11756, 11765, 55}, {11757, 11766, 1}, {11758, 11767, 1682}, {11759, 11768, 56}, {11760, 11769, 57}, {11761, 11770, 11993}, {12054, 12055, 16}, {14538, 21158, 3}, {15037, 15109, 16}, {16241, 16967, 33417}, {16808, 19107, 4}, {16809, 16964, 5321}, {16809, 16966, 5}, {16960, 19106, 13}, {16962, 19106, 16960}, {16964, 16966, 16809}, {16967, 33417, 2}, {17851, 17852, 16}, {21309, 31884, 16}, {22510, 23004, 115}, {33442, 33443, 6299}, {35207, 35208, 36}