X₁₅

1st ISODYNAMIC POINT

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 X₁, generalizes to arbitrary cevians; see TCCT, p. 98, problem 8.)

5, 6, 7 3, 4, 5 Kepler Kimberling

X₁₅ (1st ISODYNAMIC POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Auxiliary Points: O_Ap,A, O_Ap,B, O_Ap,C

• Circle: Circumcircle (circumcircle) — Passes through vertices A, B, C; centered at X₃ CIRCUMCENTER

• Circle: A-Apollonian Circle — Circle through A whose diameter on BC is bounded by the internal and external angle bisectors of ∠A

• Circle: B-Apollonian Circle — Circle through B whose diameter on CA is bounded by the internal and external angle bisectors of ∠B

• Circle: C-Apollonian Circle — Circle through C whose diameter on AB is bounded by the internal and external angle bisectors of ∠C

• Central Lines: {3, 6}

• Related Centers: X₃ (CIRCUMCENTER), X₁₆ (2nd ISODYNAMIC POINT), X₆ (SYMMEDIAN POINT)

Trilinears: \(\sin(A + \pi /3) : \sin(B + \pi /3) : \sin(C + \pi /3)\)

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

Tripolars: \(b c : :\)

Search Value (6, 9, 13): \(3.102444020655\)

▸ Alternate Coordinates5 forms

• Trilinears: \(\cos(A - \pi /6) : \cos(B - \pi /6) : \cos(C - \pi /6)\)

• Trilinears: \(3 \cos A + \sqrt{3} \sin A : :\)

• Barycentrics: \((S_{B} + S_{C})/((S_{A} + \sqrt{3} S) (S_{B} + S_{C}) + 4 S_{B} S_{C}) : :\)

• Barycentrics: \((S + \sqrt{3} S_{A}) (S_{B} + S_{C}) : :\)

• Barycentrics: \(a^{2} (\sqrt{3} (a^{2} - b^{2} - c^{2}) - 2 S) : :\)

▸ Linear Combinations2 combinations

• X₁₅ = \((r^{2} + 2rR + s^{2})\) X₁ - \(6rR\) X₂ - \(2r(r - 3^{1/2}s)\) X₃ = \(\sqrt{3}\) X₃ + \((\cot \omega )\) X₆ (Peter Moses, April 2, 2013)

• X₁₅ = \(3\) X₂ - \(4\) X₆₆₇₁, \(3\) X₁₃ - \(2\) X₅₃₁₈, \(3\) X₁₃ - \(4\) X₁₁₅₄₂, \(2\) X₁₃ - \(3\) X₁₆₂₆₇, \(3\) X₁₃ - \(5\) X₁₆₉₆₀, X₁₃ - \(3\) X₁₆₉₆₂, \(3\) X₁₃ - X₁₉₁₀₆, \(3\) X₁₄ - \(2\) X₃₃₅₁₈, \(3\) X₁₇ - \(2\) X₃₁₇₀₅, \(3\) X₁₈ - X₂₂₈₄₉, \(2\) X₁₁₅ - \(3\) X₂₂₅₁₀, \(4\) X₂₃₀ - \(3\) X₂₂₅₁₁, \(3\) X₃₉₆ - X₅₃₁₈, \(3\) X₃₉₆ - \(2\) X₁₁₅₄₂, \(4\) X₃₉₆ - \(3\) X₁₆₂₆₇, \(6\) X₃₉₆ - \(5\) X₁₆₉₆₀, \(2\) X₃₉₆ - \(3\) X₁₆₉₆₂, \(6\) X₃₉₆ - X₁₉₁₀₆, X₆₂₁ - \(4\) X₆₆₇₁, \(3\) X₃₁₂₉ - X₃₄₄₀, \(4\) X₅₃₁₈ - \(9\) X₁₆₂₆₇, \(2\) X₅₃₁₈ - \(5\) X₁₆₉₆₀, \(2\) X₅₃₁₈ - \(9\) X₁₆₉₆₂, \(3\) X₅₄₆₉ - X₂₅₁₆₆, \(3\) X₅₄₇₀ - \(2\) X₃₁₇₀₉, \(2\) X₆₁₀₉ + X₆₇₈₀, \(3\) X₆₁₀₉ - X₃₃₅₁₈, X₆₇₇₈ - \(3\) X₁₆₅₂₉, \(3\) X₆₇₈₀ + \(2\) X₃₃₅₁₈, \(2\) X₆₇₈₃ - \(3\) X₁₆₅₂₉, X₁₀₆₇₅ - \(3\) X₁₁₂₄₃, \(8\) X₁₁₅₄₂ - \(9\) X₁₆₂₆₇, \(4\) X₁₁₅₄₂ - \(5\) X₁₆₉₆₀, \(4\) X₁₁₅₄₂ - \(9\) X₁₆₉₆₂, \(4\) X₁₁₅₄₂ - X₁₉₁₀₆, \(3\) X₁₄₁₃₈ - X₃₁₇₀₅, \(9\) X₁₆₂₆₇ - \(10\) X₁₆₉₆₀, \(9\) X₁₆₂₆₇ - \(2\) X₁₉₁₀₆, \(5\) X₁₆₉₆₀ - \(9\) X₁₆₉₆₂, \(5\) X₁₆₉₆₀ - X₁₉₁₀₆, \(9\) X₁₆₉₆₂ - X₁₉₁₀₆, \(3\) X₂₂₄₈₉ - \(2\) X₃₁₆₉₃, X₂₂₄₉₅ + \(2\) X₃₅₉₃₁, \(3\) X₂₂₅₁₀ - X₂₃₀₀₄, \(3\) X₂₂₅₇₁ - \(2\) X₃₁₆₉₅, \(3\) X₂₂₆₀₂ - \(2\) X₃₁₆₉₇, \(3\) X₂₂₆₃₁ - \(2\) X₃₁₆₉₉, \(3\) X₂₂₆₈₈ - \(2\) X₃₁₇₀₁, \(3\) X₂₂₈₄₆ - \(2\) X₃₁₇₀₃, \(3\) X₂₅₁₅₁ - \(2\) X₃₁₇₀₇, \(3\) X₂₅₁₅₇ - \(2\) X₃₁₇₁₁, \(3\) X₂₅₁₅₈ - \(2\) X₃₁₇₁₃, \(3\) X₂₅₁₅₉ - \(2\) X₃₁₇₁₅, \(3\) X₂₅₁₆₀ - \(2\) X₃₁₇₁₇, \(3\) X₂₅₂₁₇ - \(2\) X₃₁₇₁₉

X₁₅ 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.

Related Collection 1 Topic Guides

T_1: Isogonal Conjugate • T_5: Circle Inversion • T_8: The Brocard Axis • T_13: The Isogonic & Napoleon Central Lines • T_15: The Circumcircle • T_18: The Apollonian Circles • T_24: Equilateral, Isogonic & Napoleon Triangles

▸ Incident Central Lines331 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}

▸ Functional Relations68 relations

• X₁₅ = midpoint of X_i and X_j for these {i,j}: {3, 5611}, {14, 6780}, {616, 3180}, {622, 14712}, {2378, 5610}, {6777, 25236}

• X₁₅ = 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}

• X₁₅ = reflection of X_i in X_j for these (i,j): (13, 396), (16, 187), (298, 618), (316, 624), (621, 623)

• X₁₅ = isogonal conjugate of X₁₃

• X₁₅ = isotomic conjugate of X₃₀₀

• X₁₅ = complement of X₆₂₁

• X₁₅ = anticomplement of X₆₂₃

• X₁₅ = circumcircle-inverse of X₁₆

• X₁₅ = nine-point-circle-inverse of X₆₁₁₂

• X₁₅ = Brocard-circle-inverse of X₁₆

• X₁₅ = polar-circle-inverse of X₆₁₁₆

• X₁₅ = orthoptic-circle-of-Steiner-inellipse-inverse of X₆₁₁₄

• X₁₅ = 2nd-Brocard-circle-inverse of X₃₁₀₅

• X₁₅ = circumcircle-of-inner-Napoleon-triangle-inverse of X₁₄

• X₁₅ = Lucas-inner-circle-inverse of X₁₆

• X₁₅ = Lucas-circles-radical-circle inverse of X₁₆

• X₁₅ = outer-Montesdeoca-Lemoine circle-inverse of X₁₆

• X₁₅ = inner-Montesdeoca-Lemoine-circle-inverse of X₁₆

• X₁₅ = antigonal image of X₁₁₆₀₀

• X₁₅ = symgonal image of X₃₃₅₂₆

• X₁₅ = complement of the isogonal conjugate of X₃₄₃₈

• X₁₅ = complement of the isotomic conjugate of X₂₉₉₂

• X₁₅ = isogonal conjugate of the anticomplement of X₆₁₈

• X₁₅ = isogonal conjugate of the complement of X₆₁₆

• X₁₅ = isotomic conjugate of the isogonal conjugate of X₃₄₃₉₄

• X₁₅ = isogonal conjugate of the isotomic conjugate of X₂₉₈

• X₁₅ = isotomic conjugate of the polar conjugate of X₈₇₃₉

• X₁₅ = isogonal conjugate of the polar conjugate of X₄₇₀

• X₁₅ = Thomson-isogonal conjugate of X₅₄₆₃

• X₁₅ = excentral-isogonal conjugate of X₂₉₄₅

• X₁₅ = tangential-isogonal conjugate of X₂₉₂₅

• X₁₅ = orthic-isogonal conjugate of X₂₉₀₂

• X₁₅ = psi-transform of X₁₆

• X₁₅ = X_i-complementary conjugate of X_j for these (i,j): {2992, 2887}, {3438, 10}

• X₁₅ = 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}

• X₁₅ = 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}

• X₁₅ = 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}

• X₁₅ = crosspoint of X_i and X_j for these (i,j): {2, 2992}, {13, 18}, {249, 10409}, {298, 470}, {2380, 16460}, {11600, 36210}

• X₁₅ = 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}

• X₁₅ = X_i-line conjugate of X_j for these (i,j): {13, 11537}, {549, 395}, {9138, 6138}, {16181, 523}

• X₁₅ = X_i-vertex conjugate of X_j for these (i,j): {4, 16257}, {13, 3457}, {16, 512}, {3458, 32906}

• X₁₅ = trilinear pole of line {526, 6137}

• X₁₅ = crossdifference of every pair of points on line {395, 523}

• X₁₅ = X₆-Hirst inverse of X₁₆

• X₁₅ = X₁₅-of-2nd-Brocard-triangle

• X₁₅ = X₁₅-of-circumsymmedial-triangle

• X₁₅ = {X₃₇₁,X₃₇₂}-harmonic conjugate of X₆₁

• X₁₅ = X₇₅-isoconjugate of X₃₄₅₇

• X₁₅ = X₁₅₇₇-isoconjugate of X₅₉₉₅

• X₁₅ = outer-Napoleon-to-inner-Napoleon similarity image of X₁₃

• X₁₅ = orthocentroidal-to-ABC similarity image of X₁₃

• X₁₅ = 4th-Brocard-to-circumsymmedial similarity image of X₁₃

• X₁₅ = X₂₃₇₈-of-2nd-Parry triangle

• X₁₅ = radical center of Lucas(2/\(\sqrt{3}\)) circles

• X₁₅ = homothetic center of (equilateral) 1st isogonal triangle of X₁₃ and pedal triangle of X₁₅

• X₁₅ = homothetic center of (equilateral) 1st isogonal triangle of X₁₄ and triangle formed by circumcenters of BCX₁₄, CAX₁₄, ABX₁₄

• X₁₅ = eigencenter of inner Napoleon triangle

• X₁₅ = X₁₃-of-4th-anti-Brocard-triangle

• X₁₅ = X₁₅-of-X₃PU(1)

• X₁₅ = Thomson-isogonal conjugate of X₅₄₆₃

• X₁₅ = X₁₀₆₅₇-of-orthocentroidal-triangle

• X₁₅ = {X₁₆,X₆₁}-harmonic conjugate of X₆

• X₁₅ = Cundy-Parry Phi transform of X₆₁

• X₁₅ = Cundy-Parry Psi transform of X₁₇

• X₁₅ = X₁₂₇₇-of-orthic-triangle if ABC is acute

• X₁₅ = 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}

• X₁₅ = 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}

• X₁₅ = {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}

Notes

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, X₁₅ and X₁₆. (Randy Hutson, July 20, 2016)

The line X₁₃X₁₅ is parallel to the Euler line, and the distance between the two lines is \((S_{B} - S_{C})(S_{C} - S_{A})(S_{A} - S_{B})/(\cot(\omega ) + 3^{1/2})|*((E - 8F)S^{2})^{1/2}\). (Kiminari Shinagawa, February 20, 2018)

The pedal triangle of X₁₅ is equilateral.

Video showing circular porisim-orbits of X₁₃, X₁₄, X₁₅, and X₁₆: 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. (Dan Reznik, August 26, 2020)

Several notable circles pass through X₁₅ and X₁₆. 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:

• (X₁₈₇; Schoute circle, 15,16)

• (X₃₅₁; 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)

• (X₆₄₇; Moses radical circle, 15, 16, 5000, 5001, 6112, 6113, 6114, 6115, 6116, 6117)

• (X₆₄₉; Gheorghe circle, 15, 16, 1276, 1277, 32622, 32623)

• (X₆₆₃; 1, 15, 16, 36, 3465, 4040, 5526, 5529)

• (X₆₆₅; 15, 16, 32753, 32754)

• (X₆₆₉; 15, 16, 5004, 5005, 5980, 5981)

• (X₈₈₇; 15, 16, 99, 729, 13210, 14691)

• (X₈₉₀; 15, 16, 100, 739)

• (X₁₉₆₀; Moses isodynamic circle, 15, 16, 101, 106, 214, 9321, 11716, 38013, 38014, 41183, 41184, 41185, 41186, 41187, 41188, 41189, 41190, 41191, 41192, 41193)

• (X₂₄₈₈; 15, 16, 3513, 3514)

• (X₂₅₀₂; Parry isodynamic circle, 15, 16)

• (X₃₀₀₅; 15, 16, 5002, 5003)

• (X₃₂₅₀; 15, 16, 32763, 32764)

• (X₃₅₆₉; 15, 16, 32618, 32619, 39665, 39666)

• (X₅₀₂₇; 15, 16, 18773, 18774, 22687, 22689)

• (X₅₀₇₅; 15, 16, 846, 1054, 1283, 5197)

• (X₅₆₃₈; 15, 16, 39164, 39165)

• (X₅₆₃₉; 15, 16, 39162, 39163)

• (X₆₁₃₇; 13, 15, 16, 3165, 5616, 5669, 6104, 10658)

• (X₆₁₃₈; 14, 15, 16, 3166, 5612, 5668, 6105, 10657)

• (X₆₁₃₉; Terzic circle, 15, 16, 55, 109, 654, 1155, 2291, 41155, 41156, 41157, 41158, 41159, 41160, 41161, 41162, 41163, 41164, 41165, 41166)

• (X₆₁₄₀; 15, 16, 115, 128, 399, 1263, 1511, 2079, 10277, 14367)

• (X₈₆₄₄; 15, 16, 38001, 38002)

• (X₉₄₀₉; 15, 16, 74, 112, 5667, 9862, 11587, 40894, 40895)

• (X₉₄₂₀; 15, 16, 98, 13236, 15920, 26714)

• (X₁₅₄₅₁; 4, 15, 16, 186, 3484, 11674, 13509, 15412)

• (X₁₇₄₁₄; 15, 16, 11629, 11630)

• (X₉₅₂₇; 15, 16, 351, 2502, 9129, 10166)

• (X₄₂₆₄₉; 15, 16, 35, 484, 3483, 14102)

• (X₄₂₆₅₀; 15, 16, 54, 1157, 3482, 18335)

• (X₄₂₆₅₁; 15, 16, 125, 184, 2081, 13558)

• (X₄₂₆₅₂; 15, 16, 385, 805, 5970, 32531)

• (X₄₂₆₅₃; 15, 16, 501, 3743, 5127, 14838, 14873, 39149)

• (X₄₂₆₅₄; 15, 16, 647, 1495, 14685, 16319, 35901)

• (X₄₂₆₅₅; 15, 16, 667, 1083, 3230, 11650, 11651, 11652)

• (X₄₂₆₅₆; 15, 16, 1138, 2132, 6794, 12112, 14254)

• (X₄₂₆₅₇; 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 X₁₅, X₁₆, 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, H_b, HC are erected on the sides of T = ABC. Let the hexagon vertices be labeled as

• H_a = {B,A₁,A₂,A₃,A₄,C), H_b = {C,B₁,B₂,B₃,B₄,A}, H_c={A,C₁,C₂,C₃,C₄,B},

• and the vertices of the 3 “flank-triangles”, by F_a = {A,C₁,B₄}, F_b = “{B,A₁,C₄}, F_c = {C,B₁,A₄}.

Between each pair of consecutive hexagons, define 3 “flank” triangles F_a = {A,C₁,B₄}, F_b = {B,A₁,C₄}, F_c={C,B_a,A₄}. Let T' be the triangle with vertices X₁₅-of-F_a, X₁₅-of-F_b, X₁₅-of-F_c. Claim (1): T' is perspective to T, and the perspector is X₆. Claim (2): X₁₆-of-F_a = X₁₆-of-F_b = X₁₅-of-F_c . (Dan Reznick, November 11, 2021)