X₁₆

2nd 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 a vertex and both isodynamic points. The pedal triangle of X₁₆ is equilateral.

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

X₁₆ (2nd 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), X₁₅ (1st 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): \(-6.784211495839\)

▸ 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

• X₁₆ = \(-(r^{2} + 2rR + s^{2})\) X₁ + \(6rR\) X₂ + \(2r(r + 3^{1/2}s)\) X₃ = \(\sqrt{3}\) X₃ - \((\cot \omega )\) X₆ (Peter Moses, April 2, 2013)

• X₁₆ = \(3\) X₂ - \(4\) X₆₆₇₂, \(3\) X₁₃ - \(2\) X₃₃₅₁₇, \(3\) X₁₄ - \(2\) X₅₃₂₁, \(3\) X₁₄ - \(4\) X₁₁₅₄₃, \(2\) X₁₄ - \(3\) X₁₆₂₆₈, \(3\) X₁₄ - \(5\) X₁₆₉₆₁, X₁₄ - \(3\) X₁₆₉₆₃, \(3\) X₁₄ - X₁₉₁₀₇, \(3\) X₁₇ - X₂₂₈₉₅, \(3\) X₁₈ - \(2\) X₃₁₇₀₆, \(2\) X₁₁₅ - \(3\) X₂₂₅₁₁, \(4\) X₂₃₀ - \(3\) X₂₂₅₁₀, \(3\) X₃₉₅ - X₅₃₂₁, \(3\) X₃₉₅ - \(2\) X₁₁₅₄₃, \(4\) X₃₉₅ - \(3\) X₁₆₂₆₈, \(6\) X₃₉₅ - \(5\) X₁₆₉₆₁, \(2\) X₃₉₅ - \(3\) X₁₆₉₆₃, \(6\) X₃₉₅ - X₁₉₁₀₇, X₆₂₂ - \(4\) X₆₆₇₂, \(3\) X₃₁₃₀ - X₃₄₄₁, \(4\) X₅₃₂₁ - \(9\) X₁₆₂₆₈, \(2\) X₅₃₂₁ - \(5\) X₁₆₉₆₁, \(2\) X₅₃₂₁ - \(9\) X₁₆₉₆₃, \(3\) X₅₄₆₉ - \(2\) X₃₁₇₁₀, \(3\) X₅₄₇₀ - X₂₅₁₅₆, \(2\) X₆₁₀₈ + X₆₇₇₉, \(3\) X₆₁₀₈ - X₃₃₅₁₇, X₆₇₇₇ - \(3\) X₁₆₅₃₀, \(3\) X₆₇₇₉ + \(2\) X₃₃₅₁₇, \(2\) X₆₇₈₂ - \(3\) X₁₆₅₃₀, X₁₀₆₇₆ - \(3\) X₁₁₂₄₄, \(8\) X₁₁₅₄₃ - \(9\) X₁₆₂₆₈, \(4\) X₁₁₅₄₃ - \(5\) X₁₆₉₆₁, \(4\) X₁₁₅₄₃ - \(9\) X₁₆₉₆₃, \(4\) X₁₁₅₄₃ - X₁₉₁₀₇, \(3\) X₁₄₁₃₉ - X₃₁₇₀₆, \(9\) X₁₆₂₆₈ - \(10\) X₁₆₉₆₁, \(9\) X₁₆₂₆₈ - \(2\) X₁₉₁₀₇, \(5\) X₁₆₉₆₁ - \(9\) X₁₆₉₆₃, \(5\) X₁₆₉₆₁ - X₁₉₁₀₇, \(9\) X₁₆₉₆₃ - X₁₉₁₀₇, \(3\) X₂₂₄₉₀ - \(2\) X₃₁₆₉₄, X₂₂₄₉₆ + \(2\) X₃₅₉₃₂, \(3\) X₂₂₅₁₁ - X₂₃₀₀₅, \(3\) X₂₂₅₇₂ - \(2\) X₃₁₆₉₆, \(3\) X₂₂₆₀₄ - \(2\) X₃₁₆₉₈, \(3\) X₂₂₆₃₃ - \(2\) X₃₁₇₀₀, \(3\) X₂₂₆₉₀ - \(2\) X₃₁₇₀₂, \(3\) X₂₂₈₉₁ - \(2\) X₃₁₇₀₄, \(3\) X₂₅₁₆₁ - \(2\) X₃₁₇₀₈, \(3\) X₂₅₁₆₇ - \(2\) X₃₁₇₁₂, \(3\) X₂₅₁₆₈ - \(2\) X₃₁₇₁₄, \(3\) X₂₅₁₆₉ - \(2\) X₃₁₇₁₈, \(3\) X₂₅₁₇₀ - \(2\) X₃₁₇₁₆, \(3\) X₂₅₂₁₄ - \(2\) X₃₁₇₂₀

X₁₆ lies on the Evans conic, Parry circle, Moses Radical circle, Schoutte circle, Parry isodynamic circle, the cubics K001, K018, K048, K050, K073, K114, K129a, K148, K193, K206, K261a, K261b, K262a, K262b, K263, K290, K291, K292, K303b, K304, K341b, 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, K1132a, K1133b, and the curves on 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_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 Lines332 lines

{1, 1250}, {2, 13}, {3, 6}, {4, 18}, {5, 2912}, {11, 11773}, {14, 30}, {17, 140}, {20, 3411}, {21, 5367}, {23, 11630}, {24, 10641}, {35, 5357}, {36, 203}, {40, 10637}, {44, 11791}, {45, 11790}, {51, 3131}, {54, 10677}, {55, 202}, {56, 7006}, {57, 11778}, {74, 5669}, {86, 21869}, {98, 33389}, {99, 22689}, {110, 2379}, {111, 9203}, {115, 6774}, {128, 11601}, {183, 25157}, {184, 2902}, {185, 21648}, {186, 3166}, {214, 5239}, {230, 21157}, {237, 14188}, {298, 3642}, {299, 532}, {302, 316}, {303, 34509}, {323, 3171}, {351, 9163}, {358, 1135}, {376, 10654}, {378, 8739}, {381, 16645}, {383, 6114}, {385, 5981}, {396, 549}, {398, 550}, {399, 5616}, {404, 5362}, {465, 23292}, {466, 13567}, {471, 6111}, {484, 7052}, {485, 2042}, {486, 2041}, {512, 9162}, {523, 16182}, {524, 5464}, {531, 22496}, {533, 617}, {542, 9115}, {559, 16577}, {590, 15765}, {597, 35304}, {615, 18585}, {621, 9988}, {625, 11305}, {627, 635}, {628, 22855}, {629, 31705}, {631, 11488}, {634, 7836}, {636, 7832}, {691, 2378}, {740, 5700}, {842, 5995}, {843, 9202}, {846, 2945}, {940, 21475}, {1080, 9993}, {1138, 5623}, {1147, 3206}, {1154, 2903}, {1157, 8457}, {1181, 19364}, {1263, 8172}, {1276, 8481}, {1337, 2380}, {1338, 3458}, {1495, 3130}, {1498, 17827}, {1511, 6104}, {1513, 9750}, {1593, 11409}, {1656, 5340}, {1657, 5339}, {1658, 11267}, {1682, 11776}, {1724, 11097}, {2043, 6561}, {2044, 6560}, {2045, 5418}, {2046, 5420}, {2059, 13391}, {2070, 2924}, {2132, 8455}, {2133, 8458}, {2307, 7280}, {2381, 10410}, {2549, 5473}, {2777, 10682}, {2854, 13858}, {2925, 10329}, {2928, 2937}, {2953, 2959}, {2981, 21461}, {3065, 5672}, {3070, 14813}, {3071, 14814}, {3096, 11289}, {3124, 14704}, {3129, 34417}, {3201, 11134}, {3231, 14182}, {3412, 3530}, {3440, 8470}, {3464, 7325}, {3465, 7060}, {3479, 8174}, {3480, 8461}, {3483, 16882}, {3484, 8471}, {3515, 11408}, {3524, 16962}, {3631, 22844}, {3734, 25167}, {3849, 9761}, {3850, 5350}, {3923, 5699}, {3972, 35917}, {4383, 21476}, {5054, 16267}, {5056, 5344}, {5059, 5343}, {5066, 12816}, {5068, 5366}, {5365, 22237}, {5460, 31709}, {5469, 31710}, {5470, 25156}, {5471, 6781}, {5472, 6771}, {5474, 9113}, {5529, 11752}, {5610, 17402}, {5613, 6778}, {5617, 9982}, {5624, 8492}, {5663, 10658}, {5667, 6110}, {5673, 8502}, {5674, 8446}, {5678, 8445}, {5682, 8452}, {5872, 22745}, {5917, 23896}, {5999, 22692}, {6000, 10676}, {6137, 9138}, {6138, 9213}, {6241, 11467}, {6294, 8177}, {6296, 23001}, {6297, 23025}, {6300, 22609}, {6301, 22612}, {6304, 22638}, {6305, 22641}, {6564, 18586}, {6565, 18587}, {6581, 23000}, {6642, 10643}, {6694, 10583}, {6695, 7859}, {6759, 10675}, {6773, 6777}, {7059, 7088}, {7164, 8459}, {7326, 8509}, {7327, 8468}, {7329, 8464}, {7426, 34316}, {7488, 11420}, {7502, 11136}, {7622, 9763}, {7709, 32465}, {7751, 33467}, {7790, 11304}, {7844, 11306}, {7846, 11307}, {7865, 11298}, {7877, 35688}, {7880, 11302}, {7914, 11311}, {8173, 8496}, {8292, 9865}, {8431, 8463}, {8433, 8444}, {8437, 8447}, {8442, 8479}, {8448, 8536}, {8453, 8489}, {8465, 8486}, {8466, 8487}, {8467, 8494}, {8469, 8495}, {8484, 16883}, {8598, 12154}, {8837, 13367}, {8884, 19191}, {8918, 23722}, {9112, 21156}, {9147, 14446}, {9744, 9749}, {9754, 16653}, {9885, 22580}, {9932, 10659}, {10187, 35018}, {10188, 22235}, {10282, 30402}, {10546, 16260}, {10636, 10902}, {10662, 13754}, {10663, 12893}, {10664, 17702}, {10681, 13289}, {10788, 22695}, {11003, 14170}, {11004, 11127}, {11008, 22845}, {11146, 15018}, {11202, 11243}, {11449, 11452}, {11453, 12111}, {11464, 11466}, {11540, 33607}, {11586, 18777}, {11676, 22702}, {11779, 11788}, {12367, 14179}, {12584, 32301}, {12817, 15682}, {12972, 12980}, {12973, 12982}, {12981, 13060}, {12983, 13059}, {13049, 13057}, {13050, 13058}, {13103, 22846}, {13704, 23026}, {13706, 23002}, {13824, 23027}, {13826, 23003}, {13860, 22694}, {13881, 16631}, {14136, 16941}, {14178, 23023}, {14186, 23028}, {14368, 14972}, {14705, 20998}, {15080, 34008}, {15412, 23873}, {15640, 33606}, {15764, 32787}, {16319, 32461}, {16459, 16638}, {16806, 32627}, {17277, 21898}, {17821, 17826}, {18400, 32398}, {18538, 35738}, {18909, 18930}, {18925, 18929}, {18980, 19452}, {18981, 19453}, {19185, 19190}, {19357, 19363}, {19440, 19450}, {19441, 19451}, {22114, 22849}, {22490, 31694}, {22572, 31696}, {22604, 31698}, {22633, 31700}, {22690, 31702}, {22701, 22715}, {22708, 22714}, {22797, 22848}, {22890, 22906}, {22962, 22974}, {23008, 25219}, {23014, 25173}, {23249, 35732}, {23267, 35733}, {23358, 32397}, {25161, 31708}, {25168, 31714}, {25169, 31718}, {25170, 31716}, {25214, 31720}, {30464, 30470}, {30465, 36186}, {31378, 36211}, {31693, 33474}, {31695, 33476}, {31697, 33445}, {31699, 33444}, {31701, 33478}, {31707, 33481}, {31711, 33482}, {31713, 33484}, {31715, 33486}, {31717, 33488}, {31719, 33491}, {32171, 32207}, {32466, 32515}, {32785, 35730}, {35726, 35727}

▸ Functional Relations67 relations

• X₁₆ = midpoint of X_i and X_j for these {i,j}: {3, 5615}, {13, 6779}, {617, 3181}, {621, 14712}, {2379, 5614}, {6778, 25235}

• X₁₆ = reflection of X_i in X_j for these (i,j): (14, 395), (15, 187), (299, 619), (316, 623), (622, 624)

• X₁₆ = isogonal conjugate of X₁₄

• X₁₆ = isotomic conjugate of X₃₀₁

• X₁₆ = complement of X₆₂₂

• X₁₆ = anticomplement of X₆₂₄

• X₁₆ = circumcircle-inverse of X₁₅

• X₁₆ = nine-point-circle-inverse of X₆₁₁₃

• X₁₆ = Brocard-circle-inverse of X₁₅

• X₁₆ = polar-circle-inverse of X₆₁₁₇

• X₁₆ = orthoptic-circle-of-Steiner-inellipse-inverse of X₆₁₁₅

• X₁₆ = 2nd-Brocard-circle-inverse of X₃₁₀₄

• X₁₆ = circumcircle-of-outer-Napoleon-triangle-inverse of X₁₃

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

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

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

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

• X₁₆ = antigonal image of X₁₁₆₀₁

• X₁₆ = symgonal image of X₃₃₅₂₇

• X₁₆ = complement of the isogonal conjugate of X₃₄₃₉

• X₁₆ = complement of the isotomic conjugate of X₂₉₉₃

• X₁₆ = isogonal conjugate of the anticomplement of X₆₁₉

• X₁₆ = isogonal conjugate of the complement of X₆₁₇

• X₁₆ = isotomic conjugate of the isogonal conjugate of X₃₄₃₉₅

• X₁₆ = isogonal conjugate of the isotomic conjugate of X₂₉₉

• X₁₆ = isotomic conjugate of the polar conjugate of X₈₇₄₀

• X₁₆ = isogonal conjugate of the polar conjugate of X₄₇₁

• X₁₆ = Thomson-isogonal conjugate of X₅₄₆₄

• X₁₆ = excentral-isogonal conjugate of X₂₉₄₆

• X₁₆ = tangential-isogonal conjugate of X₂₉₂₆

• X₁₆ = orthic-isogonal conjugate of X₂₉₀₃

• X₁₆ = psi-transform of X₁₅

• X₁₆ = X_i-complementary conjugate of X_j for these (i,j): {2993, 2887}, {3439, 10}

• X₁₆ = X_i-Ceva conjugate of X_j for these (i,j): {1, 203}, {3, 3166}, {4, 2903}, {6, 3171}, {13, 5612}, {14, 61}, {30, 5669}, {54, 3201}, {74, 15}, {471, 8740}, {6151, 6}, {7150, 7005}, {11118, 11127}, {17403, 6138}, {32037, 35444}

• X₁₆ = X_i-cross conjugate of X_j for these (i,j): {74, 8455}, {1095, 7005}, {1154, 11601}, {1511, 15}, {3201, 61}, {6138, 17403}, {14817, 14}, {19294, 323}, {34328, 11145}, {34395, 8740}

• X₁₆ = X_i-isoconjugate of X_j for these (i,j): {1, 14}, {2, 2154}, {15, 2166}, {17, 3376}, {31, 301}, {63, 8738}, {75, 3458}, {94, 2151}, {554, 7126}, {661, 23896}, {662, 20579}, {1577, 5994}, {2153, 11092}, {6137, 32680}, {7043, 33654}, {23870, 32678}, {24041, 30453}

• X₁₆ = crosspoint of X_i and X_j for these (i,j): {1, 7150}, {2, 2993}, {14, 17}, {249, 10410}, {299, 471}, {2381, 16459}, {11601, 36211}

• X₁₆ = crosssum of X_i and X_j for these (i,j): {2, 3181}, {3, 10662}, {6, 3130}, {15, 5616}, {16, 61}, {395, 8015}, {396, 30462}, {523, 30468}, {533, 618}, {6105, 36209}, {6110, 6117}, {9201, 30470}, {11543, 11556}, {18776, 30469}, {20579, 30453}, {23284, 30463}

• X₁₆ = X_i-line conjugate of X_j for these (i,j): {14, 11549}, {549, 396}, {9138, 6137}, {16182, 523}

• X₁₆ = X_i-vertex conjugate of X_j for these (i,j): {4, 16258}, {14, 3458}, {15, 512}, {3457, 32908}

• X₁₆ = trilinear pole of line {526, 6138}

• X₁₆ = crossdifference of every pair of points on line {396, 523}

• X₁₆ = X₆-Hirst inverse of X₁₅

• X₁₆ = X₁₆-of-2nd-Brocard triangle

• X₁₆ = X₁₆-of-circumsymmedial-triangle

• X₁₆ = {X₃₇₁,X₃₇₂}-harmonic conjugate of X₆₂

• X₁₆ = X₇₅-isoconjugate of X₃₄₅₈

• X₁₆ = X₁₅₇₇-isoconjugate of X₅₉₉₄

• X₁₆ = inner-Napoleon-to-outer-Napoleon similarity image of X₁₄

• X₁₆ = orthocentroidal-to-ABC similarity image of X₁₄

• X₁₆ = 4th-Brocard-to-circumsymmedial similarity image of X₁₄

• X₁₆ = X₂₃₇₉-of-2nd-Parry-triangle

• X₁₆ = homothetic center of (equilateral) 1st isogonal triangle of X₁₄ and pedal triangle of X₁₆

• X₁₆ = homothetic center of (equilateral) 1st isogonal triangle of X₁₃ and triangle formed by circumcenters of BCX₁₃, CAX₁₃, ABX₁₃

• X₁₆ = radical center of Lucas(-2/\(\sqrt{3}\)) circles

• X₁₆ = eigencenter of outer Napoleon triangle

• X₁₆ = X₁₄ of 4th anti-Brocard triangle

• X₁₆ = X₁₆-of-X₃PU(1)

• X₁₆ = Thomson-isogonal conjugate of X₅₄₆₄

• X₁₆ = X₁₀₆₅₈-of-orthocentroidal-triangle

• X₁₆ = {X₁₅,X₆₂}-harmonic conjugate of X₆

• X₁₆ = Cundy-Parry Phi transform of X₆₂

• X₁₆ = Cundy-Parry Psi transform of X₁₈

• X₁₆ = X₁₂₇₆-of-orthic-triangle if ABC is acute

• X₁₆ = barycentric product X_i*X_j for these {i,j}: {3, 471}, {6, 299}, {13, 323}, {14, 11130}, {15, 11078}, {17, 11126}, {18, 11145}, {50, 300}, {54, 33530}, {61, 19779}, {69, 8740}, {75, 2152}, {76, 34395}, {97, 6116}, {99, 6138}, {110, 23871}, {249, 30468}, {298, 11081}, {303, 8604}, {523, 17403}, {526, 23895}, {532, 2981}, {559, 5239}, {619, 6151}, {691, 9205}, {2987, 6783}, {3166, 19775}, {3171, 11122}, {3268, 5995}, {3457, 7799}, {3458, 11128}, {5612, 13582}, {6111, 14919}, {10409, 14446}, {10410, 35444}, {10411, 20578}, {10678, 11144}, {11092, 36208}, {11118, 19295}, {11119, 19294}, {11127, 11601}, {11131, 36211}, {11132, 21461}, {11134, 34390}, {11135, 34389}, {14922, 16459}, {17402, 23283}

• X₁₆ = barycentric quotient X_i/X_j for these {i,j}: {2, 301}, {6, 14}, {13, 94}, {15, 11092}, {25, 8738}, {31, 2154}, {32, 3458}, {50, 15}, {61, 16771}, {62, 8836}, {110, 23896}, {186, 470}, {299, 76}, {300, 20573}, {323, 298}, {471, 264}, {512, 20579}, {526, 23870}, {1154, 33529}, {1576, 5994}, {2088, 30465}, {2152, 1}, {2153, 2166}, {2981, 11117}, {3124, 30453}, {3166, 19773}, {3171, 3181}, {3201, 11126}, {3457, 1989}, {3458, 11085}, {5995, 476}, {6104, 8838}, {6116, 324}, {6137, 23284}, {6138, 523}, {6151, 11120}, {7051, 554}, {8603, 11600}, {8604, 18}, {8737, 6344}, {8740, 4}, {9205, 35522}, {10632, 473}, {10638, 7043}, {10678, 11143}, {11062, 6117}, {11078, 300}, {11081, 13}, {11086, 36210}, {11088, 11582}, {11126, 302}, {11130, 299}, {11134, 62}, {11135, 61}, {11136, 6105}, {11145, 303}, {11244, 8918}, {14270, 6137}, {17403, 99}, {19294, 618}, {19295, 533}, {19627, 34394}, {19779, 34389}, {20578, 10412}, {21461, 11087}, {21462, 11138}, {23871, 850}, {23895, 35139}, {30468, 338}, {32729, 9207}, {33530, 311}, {34328, 630}, {34394, 11086}, {34395, 6}, {34397, 8739}, {36208, 11078}

• X₁₆ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 11772, 11771}, {1, 11781, 11780}, {2, 616, 3643}, {2, 622, 624}, {2, 5335, 18582}, {2, 10653, 13}, {2, 18582, 16966}, {3, 6, 15}, {3, 15, 10645}, {3, 61, 5238}, {3, 62, 61}, {3, 371, 3389}, {3, 372, 3390}, {3, 1152, 35739}, {3, 5237, 5351}, {3, 5611, 9735}, {3, 5864, 14541}, {3, 10635, 11516}, {3, 11481, 10646}, {3, 11485, 11480}, {3, 11486, 6}, {3, 13349, 21159}, {3, 15794, 15785}, {3, 18470, 10635}, {3, 22236, 5352}, {3, 22238, 62}, {4, 10633, 10642}, {4, 11489, 18581}, {4, 18581, 16809}, {5, 5318, 16808}, {5, 23303, 16967}, {6, 15, 61}, {6, 10645, 34754}, {6, 10646, 10645}, {6, 11480, 11485}, {6, 11481, 3}, {6, 11486, 62}, {6, 19780, 32}, {6, 22238, 11486}, {13, 16242, 2}, {13, 16966, 18582}, {13, 33416, 16966}, {14, 395, 16268}, {14, 15442, 11582}, {14, 16961, 11543}, {14, 16963, 395}, {14, 19107, 5321}, {15, 61, 34754}, {15, 62, 6}, {15, 5237, 10646}, {15, 10645, 5238}, {15, 10646, 3}, {15, 34755, 62}, {15, 35739, 6396}, {17, 33417, 23302}, {18, 16809, 18581}, {18, 19106, 16809}, {18, 22862, 31703}, {32, 3098, 15}, {32, 3104, 61}, {35, 5357, 10638}, {36, 5353, 7051}, {36, 7127, 203}, {39, 5092, 15}, {50, 3581, 15}, {61, 5351, 3}, {61, 10645, 15}, {62, 3364, 3390}, {62, 3365, 3389}, {62, 5237, 3}, {62, 5351, 5238}, {62, 10646, 15}, {62, 11481, 10645}, {62, 14538, 3106}, {62, 34755, 11486}, {62, 35739, 3365}, {140, 397, 17}, {140, 11542, 23302}, {140, 23302, 33417}, {182, 574, 15}, {182, 3107, 61}, {182, 9736, 3}, {187, 11481, 21159}, {216, 11430, 15}, {298, 11300, 3642}, {299, 30472, 7799}, {302, 11303, 623}, {323, 11145, 11130}, {323, 34395, 3171}, {371, 372, 62}, {371, 3364, 61}, {371, 6200, 15}, {372, 3365, 61}, {372, 6396, 15}, {389, 22052, 15}, {395, 5321, 11543}, {395, 11543, 16961}, {396, 549, 16241}, {397, 23302, 11542}, {485, 2042, 3366}, {486, 2041, 3367}, {500, 17454, 15}, {566, 14805, 15}, {569, 14806, 15}, {572, 4256, 15}, {573, 4257, 15}, {575, 8589, 15}, {576, 8588, 15}, {576, 9735, 5611}, {577, 11438, 15}, {578, 10979, 15}, {622, 5335, 33517}, {624, 6672, 2}, {627, 22907, 22894}, {991, 4262, 15}, {1151, 6221, 15}, {1152, 6398, 15}, {1250, 19373, 1}, {1340, 1341, 15}, {1350, 1384, 15}, {1351, 5210, 15}, {1379, 1380, 15}, {1620, 33636, 15}, {1670, 1671, 3104}, {1689, 1690, 3107}, {1691, 35002, 15}, {2030, 18860, 15}, {2076, 5615, 3104}, {2076, 9301, 15}, {2076, 19780, 187}, {2080, 5104, 15}, {2902, 3165, 3200}, {3003, 10564, 15}, {3053, 33878, 15}, {3094, 26316, 15}, {3104, 3107, 3094}, {3311, 6411, 15}, {3312, 6412, 15}, {3364, 3365, 3}, {3371, 3372, 3365}, {3385, 3386, 3364}, {3389, 3390, 61}, {3391, 3392, 5}, {3430, 33628, 15}, {3592, 6451, 15}, {3594, 6452, 15}, {5008, 14810, 15}, {5013, 12017, 15}, {5024, 5085, 15}, {5030, 13329, 15}, {5033, 9737, 15}, {5093, 5585, 15}, {5237, 10646, 11481}, {5237, 11486, 10645}, {5237, 22238, 61}, {5237, 34755, 15}, {5238, 34754, 15}, {5318, 16773, 23303}, {5318, 23303, 5}, {5321, 11543, 14}, {5335, 18582, 13}, {5351, 11486, 34754}, {5351, 30559, 21159}, {5353, 7051, 203}, {5357, 10638, 7005}, {5474, 9113, 23013}, {5615, 13349, 14539}, {5615, 21402, 21159}, {6104, 36208, 11081}, {6199, 6409, 15}, {6200, 6396, 10646}, {6200, 11486, 3390}, {6302, 6306, 14904}, {6395, 6410, 15}, {6396, 11486, 3389}, {6407, 6468, 15}, {6408, 6469, 15}, {6425, 6445, 15}, {6426, 6446, 15}, {6429, 9690, 15}, {6437, 6449, 15}, {6438, 6450, 15}, {6439, 9691, 15}, {6453, 6480, 15}, {6454, 6481, 15}, {6582, 22687, 5980}, {6672, 33517, 16966}, {6777, 16530, 6782}, {7051, 7127, 5353}, {9675, 9738, 15}, {10633, 32586, 8839}, {10642, 11476, 4}, {10646, 11481, 5351}, {10646, 11486, 61}, {10646, 34755, 6}, {10653, 18582, 5335}, {10668, 10672, 6}, {10675, 30403, 6759}, {11126, 11130, 323}, {11134, 22115, 3201}, {11477, 15655, 15}, {11480, 11485, 15}, {11481, 11486, 15}, {11481, 22238, 6}, {11481, 34755, 61}, {11486, 22238, 34755}, {11489, 18581, 18}, {11542, 23302, 17}, {11543, 16961, 16268}, {11582, 15743, 15442}, {11771, 11780, 1}, {11772, 11781, 1}, {11773, 11782, 11}, {11774, 11783, 55}, {11775, 11784, 1}, {11776, 11785, 1682}, {11777, 11786, 56}, {11778, 11787, 57}, {11779, 11788, 11993}, {12054, 12055, 15}, {14539, 21159, 3}, {15037, 15109, 15}, {16242, 16966, 33416}, {16808, 16965, 5318}, {16808, 16967, 5}, {16809, 19106, 4}, {16961, 19107, 14}, {16963, 19107, 16961}, {16965, 16967, 16808}, {16966, 33416, 2}, {17851, 17852, 15}, {21309, 31884, 15}, {22511, 23005, 115}, {33440, 33441, 6298}, {35209, 35210, 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 X₁₄X₁₆ 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)

Consider this picture, in which (purple) hexagons are erected on the sides of ABC, with (green) flank-triangles:

• 1. X₁₆ of the 3 flank-triangles coincide with X₁₆ of ABC.

• 2. the (red) triangle of the centroids of the hexagons is perspective to ABC, and the perspector is X₁₃.

• 3. the circumcircle of the centroids (apices of equilaterals erect4d on the sides of ABC) has center X₆₂₇.

• 4. the (green) triangle with vertices on X₁₅-of-flank-triangles is perspective to ABC, and the perspector is X₆. (Dan Reznick, October 18, 2021)