X₉

MITTENPUNKT

X₉ is the symmedian point of the excentral triangle.

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

X₉ (MITTENPUNKT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Excentral Triangle △J_aJ_bJ_c — vertices J_a, J_b, J_c (Anticevian triangle of Incenter X₁ (the three excenters))

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Concurrence Lines: J_a–M_a, J_b–M_b, J_c–M_c concurring at X₉

Trilinears: \(b + c - a : c + a - b : a + b - c\)

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

Tripolars: \(\sqrt{b c (a^{4}-2 a^{2} b^{2}+b^{4}-4 b^{3} c-2 a^{2} c^{2}+6 b^{2} c^{2}-4 b c^{3}+c^{4})} : :\)

Search Value (6, 9, 13): \(3.571972699230\)

▸ Alternate Coordinates8 forms

• Trilinears: \(\cot(A/2) : \cot(B/2) : \cot(C/2)\)

• Trilinears: \(a - s : b - s : c - s\)

• Trilinears: \(\csc A + \cot A : :\)

• Trilinears: \(\csc A (1 + \cos A) : :\)

• Trilinears: \(\tan A' : :\), where J_aJ_bJ_c = excentral triangle

• Trilinears: \(d(a,b,c) : :\), where d(a,b,c) = distance from A to the Gergonne line

• Trilinears: square of semi-minor axis of A-Soddy ellipse : :

• Barycentrics: \(1 + \cos A : 1 + \cos B : 1 + \cos C\)

▸ Linear Combinations1 combination

• X₉ = \((r + 2R)\) X₁ - \(6R\) X₂ - \(2r\) X₃ = \(3\) X₂ - X₇

X₉ lies on the these conics: Feuerbach circumhyperbola, Feuerbach circumhyperbola of the medial triangle, Jerabek circumhyperbola of the excentral triangle, Mandart hyperbola

X₉ lies on these curves: K002, K132, K202, K207, K220, K251, K294, K332, K343, K345, K351, K352, K363, K384, K387, K453, K637, K696, K697, K710, K716, K717, K760, K761, K817, K880, K950, K970, K977, K980, K982, K984, K1025, K1038, K1044, K1055, K1059, K1060, K1077, K1079, K1080, K1081, K1082, K1083, K1084, K1090, Q151

X₉ lies on the curve Q151 and on conics and cubics listed below.

Related Collection 1 Topic Guides

T_12: The Fuhrmann & Spieker Central Lines • T_14: The Incircle & Excircles • T_17: The Spieker Circle • T_22: Intouch (Contact) & Extouch Triangles • T_23: Incentral & Excentral Triangles

▸ Incident Central Lines1141 lines

{1, 6}, {2, 7}, {3, 84}, {4, 10}, {5, 1729}, {8, 346}, {11, 3254}, {12, 5857}, {20, 10429}, {21, 41}, {22, 5314}, {25, 5285}, {30, 3587}, {31, 612}, {32, 987}, {33, 212}, {34, 201}, {35, 90}, {36, 2178}, {38, 614}, {39, 978}, {42, 941}, {43, 256}, {46, 79}, {48, 101}, {51, 3690}, {55, 200}, {56, 1696}, {58, 975}, {65, 4047}, {69, 344}, {75, 190}, {77, 651}, {80, 528}, {81, 5287}, {85, 10509}, {86, 2279}, {87, 292}, {92, 6358}, {99, 35106}, {100, 1005}, {105, 4712}, {108, 3213}, {109, 2199}, {114, 24469}, {119, 2950}, {123, 20623}, {124, 5513}, {140, 5843}, {141, 4422}, {145, 4029}, {164, 168}, {165, 910}, {171, 1707}, {172, 5019}, {173, 177}, {182, 7193}, {184, 26885}, {192, 239}, {193, 3879}, {194, 16827}, {205, 2359}, {216, 13006}, {222, 17811}, {223, 1073}, {228, 1011}, {241, 269}, {244, 17125}, {255, 25063}, {257, 17743}, {258, 7028}, {259, 15997}, {261, 645}, {264, 1948}, {273, 26003}, {275, 26941}, {291, 17065}, {294, 1253}, {306, 5739}, {312, 314}, {318, 1896}, {319, 17233}, {320, 17234}, {321, 1751}, {326, 6518}, {335, 17000}, {341, 4095}, {342, 653}, {345, 2339}, {348, 738}, {350, 17026}, {354, 4423}, {355, 31789}, {362, 2090}, {363, 5934}, {364, 366}, {374, 517}, {375, 22276}, {377, 9579}, {379, 18655}, {381, 18482}, {386, 10467}, {388, 12527}, {393, 1785}, {394, 2003}, {404, 4652}, {418, 26901}, {427, 21015}, {440, 1211}, {443, 4292}, {474, 3916}, {478, 1038}, {484, 5036}, {497, 4847}, {498, 920}, {499, 25522}, {511, 3781}, {513, 3126}, {514, 23760}, {515, 3427}, {519, 1000}, {521, 4130}, {522, 657}, {524, 4851}, {536, 4361}, {545, 7263}, {551, 18490}, {581, 3682}, {583, 3338}, {584, 15175}, {595, 28594}, {597, 17045}, {599, 17231}, {604, 1420}, {607, 1039}, {608, 1041}, {609, 1333}, {631, 6700}, {644, 1320}, {646, 4110}, {649, 4521}, {650, 15737}, {652, 3239}, {654, 1639}, {660, 3252}, {661, 35354}, {664, 31169}, {668, 17786}, {726, 16825}, {750, 896}, {758, 2294}, {759, 29127}, {765, 5377}, {813, 2726}, {857, 17052}, {869, 2309}, {899, 4414}, {900, 22108}, {905, 20318}, {912, 3211}, {919, 2751}, {934, 2371}, {937, 2336}, {938, 5129}, {940, 4641}, {942, 3927}, {943, 1802}, {946, 5758}, {948, 3668}, {952, 7966}, {964, 2198}, {970, 15877}, {976, 5037}, {980, 27623}, {981, 1918}, {982, 3290}, {983, 1914}, {986, 1722}, {989, 5255}, {990, 13329}, {991, 1818}, {995, 19246}, {1012, 6282}, {1014, 24557}, {1015, 21826}, {1016, 35957}, {1020, 6356}, {1021, 3700}, {1026, 9319}, {1030, 2932}, {1040, 5452}, {1046, 10381}, {1050, 2275}, {1054, 3038}, {1055, 18450}, {1062, 35194}, {1071, 8726}, {1084, 35095}, {1086, 4859}, {1088, 1223}, {1096, 7076}, {1098, 2150}, {1125, 1732}, {1150, 4358}, {1155, 4413}, {1158, 5811}, {1167, 7129}, {1174, 1621}, {1181, 2910}, {1193, 5105}, {1195, 5552}, {1200, 5281}, {1202, 10578}, {1210, 5084}, {1215, 24511}, {1220, 31359}, {1247, 9560}, {1249, 1712}, {1250, 7126}, {1251, 7127}, {1259, 11344}, {1266, 20073}, {1278, 16816}, {1282, 3041}, {1331, 2000}, {1332, 28978}, {1377, 1703}, {1378, 1702}, {1384, 37589}, 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{1959, 15988}, {1966, 6376}, {1974, 26924}, {1992, 29574}, {2006, 26611}, {2013, 2018}, {2014, 2017}, {2049, 19859}, {2066, 7133}, {2078, 17615}, {2093, 3753}, {2099, 31165}, {2108, 20603}, {2111, 33701}, {2112, 4579}, {2124, 2125}, {2137, 24150}, {2173, 3647}, {2174, 2278}, {2175, 2330}, {2177, 21805}, {2209, 3728}, {2214, 28615}, {2220, 5301}, {2223, 16688}, {2225, 29828}, {2242, 5042}, {2252, 5553}, {2271, 37573}, {2272, 5658}, {2293, 2340}, {2295, 4274}, {2305, 5277}, {2308, 5311}, {2312, 4220}, {2317, 22356}, {2318, 2335}, {2319, 7081}, {2320, 2364}, {2326, 11107}, {2355, 37385}, {2432, 14298}, {2478, 6734}, {2503, 21381}, {2509, 21189}, {2568, 2573}, {2569, 2572}, {2590, 3307}, {2591, 3308}, {2629, 2634}, {2640, 2645}, {2648, 17963}, {2802, 4752}, {2887, 4703}, {2895, 32858}, {2947, 2954}, {2957, 24250}, {2959, 20666}, {2999, 3666}, {3008, 3663}, {3009, 7032}, {3013, 35069}, {3037, 5539}, {3056, 3271}, {3057, 3680}, {3058, 4863}, {3060, 26911}, {3063, 24307}, {3068, 5393}, {3069, 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{17156, 32864}, {17175, 25508}, {17227, 17273}, {17228, 17271}, {17230, 17268}, {17238, 17252}, {17239, 17251}, {17240, 17295}, {17241, 17297}, {17244, 17300}, {17246, 17301}, {17247, 17302}, {17249, 17305}, {17250, 17307}, {17309, 17372}, {17310, 17373}, {17311, 17374}, {17312, 17375}, {17315, 17377}, {17317, 17378}, {17320, 17380}, {17322, 17381}, {17323, 17382}, {17324, 17383}, {17325, 17384}, {17391, 20090}, {17394, 29597}, {17438, 37518}, {17444, 21398}, {17499, 26110}, {17542, 24473}, {17550, 24995}, {17605, 31245}, {17614, 37519}, {17616, 37309}, {17625, 25893}, {17687, 25500}, {17691, 25242}, {17718, 34917}, {17720, 35466}, {17728, 31249}, {17737, 24892}, {17757, 31434}, {17792, 18788}, {17793, 30546}, {17889, 33099}, {18040, 29396}, {18044, 18133}, {18046, 29561}, {18134, 33066}, {18139, 32859}, {18398, 25542}, {18483, 31418}, {18589, 24316}, {18596, 34823}, {18725, 34371}, {18755, 37574}, {18758, 23863}, {19261, 23169}, {19372, 37591}, {19523, 37522}, {19555, 31090}, {19582, 23640}, {19604, 25731}, {19785, 26723}, {19804, 32939}, {19872, 32632}, {20080, 29583}, {20092, 24184}, {20106, 26934}, {20174, 29773}, {20205, 21370}, {20247, 25261}, {20248, 26653}, {20257, 27304}, {20305, 34176}, {20370, 34832}, {20430, 37510}, {20444, 35550}, {20456, 22172}, {20486, 25613}, {20544, 24329}, {20608, 21250}, {20662, 21320}, {20667, 26069}, {20678, 23868}, {20719, 31785}, {20930, 28980}, {20973, 21858}, {20979, 21389}, {20980, 21348}, {20984, 22174}, {21010, 36635}, {21026, 31134}, {21066, 26793}, {21090, 24298}, {21281, 30030}, {21286, 26581}, {21319, 37319}, {21367, 32782}, {21368, 24611}, {21379, 33681}, {21405, 27390}, {21514, 37597}, {21759, 23566}, {21827, 23543}, {21832, 24121}, {21956, 32865}, {22003, 24435}, {22076, 37324}, {22758, 37611}, {23062, 23618}, {23073, 30389}, {23354, 25292}, {23529, 28118}, {23649, 28352}, {24067, 27368}, {24154, 24156}, {24155, 24157}, {24158, 24242}, {24172, 28090}, {24210, 33137}, {24268, 25252}, {24325, 32935}, {24343, 25107}, {24346, 36008}, {24398, 27918}, {24411, 34361}, {24512, 26102}, {24542, 33122}, {24547, 24612}, {24549, 33821}, {24586, 30758}, {24603, 28827}, {24633, 24993}, {24690, 30822}, {24935, 25669}, {25010, 30312}, {25057, 31171}, {25343, 30779}, {25457, 25682}, {25570, 25571}, {25660, 29456}, {25690, 25693}, {25716, 36628}, {25842, 25856}, {25850, 25858}, {25875, 34489}, {25885, 34036}, {25958, 29873}, {25960, 33119}, {25961, 33067}, {26006, 26668}, {26035, 31339}, {26068, 27020}, {26107, 26959}, {26592, 34388}, {26658, 34497}, {26724, 33146}, {26730, 26731}, {26756, 27073}, {26919, 26940}, {26933, 30739}, {27036, 27102}, {27044, 27136}, {27108, 27514}, {27253, 36854}, {27398, 37265}, {27507, 28789}, {28365, 37596}, {28604, 29576}, {29085, 36661}, {29381, 29536}, {29388, 29504}, {29395, 29423}, {29431, 29514}, {29570, 31313}, {29616, 32099}, {29632, 33065}, {29642, 33064}, {29643, 32843}, {29653, 32946}, {29664, 33107}, {29667, 33166}, {29674, 33082}, {29679, 33083}, {29681, 33153}, {29687, 33080}, {29711, 29716}, {29767, 30939}, {29826, 32944}, {29850, 32776}, {29851, 33069}, {29854, 32949}, {29855, 32775}, {29856, 34997}, {30295, 35990}, {30416, 30429}, {30417, 30430}, {30676, 30701}, {30695, 31994}, {30942, 34589}, {31316, 31343}, {31408, 31533}, {31534, 34495}, {31535, 34494}, {31547, 31565}, {31548, 31566}, {31937, 35239}, {32577, 33589}, {32771, 32938}, {32773, 33118}, {32778, 33164}, {32849, 33077}, {32860, 32936}, {32861, 33092}, {32862, 33075}, {32914, 32925}, {32947, 33117}, {33084, 33158}, {33096, 33111}, {33100, 33131}, {33101, 33130}, {33129, 33151}, {33132, 33154}, {33134, 33139}, {33700, 36906}, {33863, 37608}, {33934, 33951}, {34256, 34293}, {34926, 34932}, {35091, 35113}, {35128, 35129}, {35614, 35617}, {35659, 35668}, {36531, 36554}, {37234, 37585}, {37248, 37583}, {37507, 37609}

X₉ lies on the following circumconics: Feuerbach circumhyperbola, Feuerbach circumhyperbola of the medial triangle, Jerabek circumhyperbola of the excentral triangle, Mandart hyperbola, and these: {{A, B, C,2,9,200,281,282,346,2184,2287,2297,4183,6605,7097,7110,14943,15889,15890,15891,15892,16016,19605,21446,23617,28071,28132,30705,31618,34525,36101,36627,36629,36796,36910,36916}} {{A, B, C,3,9,40,271,1167,1819,3342,3347,7013,7078,34902}} {{A, B, C,9,10,72,78,307,318,1793,3694,3710,3718,8806}} {{A, B, C,9,33,37,210,226,312,1826,1903,2250,2321,2341,8818,13455,25430,27475,35144,35354,36800}} {{A, B, C,9,41,213,1334,1400,2212,2279,2311,2333,18784,18785,35106}} {{A, B, C,9,63,219,268,1073,1260,1815,2322,2327,2328,2983,3692,7123,15629,36631}} {{A, B, C,9,75,518,522,765,1861,2751,3693,3717,9436,33676,36819}} {{A, B, C,9,87,238,242,261,1447,2726,3684,3685,3737,4076,7220,9282,9499,18786,36815}}

X₉ lies on the inellipse through X_i for i = 9,57,604,2171,2289,3083,3084,6602,34544; the perspector of this ellipse is X₄₅₆₄.

X₉ lies on the following cubics: K002, K132, K202, K207, K220, K251, K294, K332, K343, K345, K351, K352, K363, K384, K387, K453, K637, K696, K697, K710, K716, K717, K760, K761, K817, K880, K950, K970, K977, K980, K982, K984, K1025, K1038, K1044, K1055, K1059, K1060, K1077, K1079, K1080, K1081, K1082, K1083, K1084, K1090

▸ Functional Relations81 relations

• X₉ = perspector of ABC and the medial triangle of the extouch triangle of ABC. (Randy Hutson, 9/23/2011)

• X₉ = midpoint of X_i and X_j for these (i,j): midpoint of X_i and X_j for these {i,j}: {1, 5223}, {2, 6172}, {3, 5779}, {4, 5759}, {7, 144}, {8, 390}, {11, 6068}, {20, 36991}, {40, 11372}, {57, 36973}, {63, 8545}, {72, 5728}, {100, 1156}, {190, 673}, {329, 12848}, {346, 5838}, {651, 36101}, {1001, 5220}, {2262, 21871}, {2294, 3958}, {2550, 5698}, {2951, 3062}, {3059, 14100}, {3257, 37131}, {3434, 36976}, {3587, 18540}, {3621, 12630}, {3681, 7671}, {3869, 7672}, {4361, 17262}, {4915, 9819}, {5817, 21168}, {5839, 17314}, {7670, 11691}, {7673, 14923}, {9846, 12125}, {11495, 16112}, {11684, 16133}, {12389, 12399}, {12526, 12560}, {12527, 12573}, {12528, 12669}, {12529, 12706}, {12530, 12718}, {12531, 12730}, {12532, 12755}, {12533, 12846}, {12534, 12847}, {12535, 12850}, {13359, 13360}, {15254, 15481}, {30628, 34784}, {31547, 31565}, {31548, 31566}, {34926, 34932}, {35659, 35668}

• X₉ = reflection of X_i in X_j for these {i,j}: {1, 1001}, {3, 31658}, {7, 142}, {8, 24393}, {57, 8257}, {84, 3358}, {100, 6594}, {101, 28345}, {142, 6666}, {1001, 15254}, {2294, 25081}, {2550, 10}, {2951, 11495}, {3062, 16112}, {3174, 6600}, {3243, 1}, {3254, 11}, {3874, 20116}, {4312, 5880}, {4361, 17348}, {4851, 17243}, {5220, 15481}, {5223, 5220}, {5528, 100}, {5542, 1125}, {5732, 3}, {5735, 5805}, {5805, 5}, {5833, 5791}, {5880, 3826}, {6173, 2}, {6601, 24389}, {8255, 6690}, {9623, 9708}, {10427, 3035}, {13159, 6701}, {14943, 35508}, {15185, 5572}, {15298, 15296}, {15299, 15297}, {16593, 4422}, {17151, 4361}, {17314, 3950}, {17668, 15587}, {18443, 6883}, {20195, 18230}, {31657, 140}, {31671, 18482}, {36867, 15935}

• X₉ = isogonal conjugate of X₅₇

• X₉ = isotomic conjugate of X₈₅

• X₉ = complement of X₇

• X₉ = anticomplement of X₁₄₂

• X₉ = polar conjugate of X₂₇₃

• X₉ = antigonal image of X₃₂₅₄

• X₉ = antitomic image of X₁₄₉₄₃

• X₉ = symgonal image of X₆₅₉₄

• X₉ = circumcircle-inverse of X₃₂₆₂₅

• X₉ = Spieker-circle-inverse of X₅₁₉₉

• X₉ = Bevan-circle-inverse of X₅₀₁₁

• X₉ = Stevanovic-circle inverse of X₁₅₇₃₇

• X₉ = Thomson-isogonal conjugate of X₃₅₇₆

• X₉ = medial-isogonal conjugate of X₂₈₈₆

• X₉ = anticomplementary-isogonal conjugate of X₂₈₉₀

• X₉ = excentral-isogonal conjugate of X₁₆₅

• X₉ = tangential-isogonal conjugate of X₂₉₂₁

• X₉ = orthic-isogonal conjugate of X₂₉₀₀

• X₉ = X_i-Ceva conjugate of X_j for these (i,j): {1, 3158}, {2, 1}, {4, 2900}, {6, 3169}, {7, 3174}, {8, 200}, {21, 55}, {27, 3189}, {29, 3190}, {31, 32468}, {41, 7075}, {55, 3208}, {57, 2136}, {63, 40}, {78, 7070}, {81, 3913}, {85, 3870}, {88, 3880}, {92, 3811}, {100, 3900}, {105, 19589}, {144, 2951}, {189, 6765}, {190, 522}, {241, 9451}, {257, 3961}, {294, 3684}, {312, 78}, {318, 33}, {329, 1490}, {333, 8}, {346, 2324}, {348, 8270}, {527, 5528}, {643, 4041}, {644, 650}, {645, 3737}, {650, 4919}, {651, 521}, {653, 8058}, {655, 2804}, {660, 926}, {664, 4105}, {672, 24578}, {673, 5853}, {765, 3939}, {799, 3907}, {894, 1045}, {897, 24394}, {908, 6326}, {1121, 3935}, {1156, 15733}, {1220, 42}, {1223, 142}, {1252, 1018}, {1261, 4513}, {1320, 3689}, {1751, 12625}, {1791, 5285}, {1821, 740}, {2053, 2319}, {2167, 8715}, {2184, 11523}, {2185, 3871}, {2287, 219}, {2297, 1449}, {2319, 4050}, {2322, 281}, {2339, 1697}, {2346, 6600}, {2349, 758}, {2975, 15621}, {3218, 5541}, {3219, 191}, {3257, 3738}, {3305, 3646}, {3596, 4149}, {3699, 663}, {3903, 4477}, {4076, 4069}, {4102, 4420}, {4564, 100}, {4997, 4511}, {5279, 18598}, {5546, 1021}, {6558, 4521}, {6605, 220}, {7123, 3501}, {7131, 57}, {8056, 3680}, {9776, 12658}, {10509, 7674}, {14942, 2340}, {15889, 30557}, {15890, 30556}, {17484, 13146}, {17743, 43}, {21446, 3243}, {23617, 6}, {23618, 7}, {27065, 5506}, {27834, 513}, {28659, 4123}, {30608, 3872}, {30705, 8271}, {30711, 4882}, {31343, 4162}, {31359, 612}, {32008, 2}, {32015, 3957}, {32635, 210}

• X₉ = cevapoint of X_i and X_j for these (i,j): (1, 165), (6, 198), (31, 205), (37, 71), (41, 212), (55, 220). (2066.5414)

• X₉ = X_i-cross conjugate of X_j for these (i,j): {1, 19605}, {6, 282}, {8, 2319}, {37, 281}, {41, 33}, {42, 10570}, {55, 1}, {57, 2125}, {71, 219}, {198, 2324}, {200, 3680}, {210, 8}, {212, 78}, {220, 200}, {518, 14943}, {650, 644}, {652, 101}, {654, 5548}, {657, 3939}, {663, 3699}, {672, 2338}, {1146, 1021}, {1212, 2}, {1334, 55}, {1400, 30457}, {1864, 4}, {1903, 8805}, {2082, 57}, {2170, 650}, {2183, 15629}, {2238, 15628}, {2245, 15627}, {2264, 1172}, {2269, 284}, {2310, 522}, {2340, 14942}, {2347, 6}, {2348, 294}, {3059, 6601}, {3119, 3900}, {3271, 3737}, {3287, 645}, {3683, 21}, {3689, 1320}, {3691, 333}, {3693, 4876}, {3700, 1018}, {3709, 4069}, {3711, 4900}, {3715, 4866}, {3900, 100}, {4041, 643}, {4105, 664}, {4162, 31343}, {4266, 2364}, {4326, 10390}, {4477, 3903}, {4517, 7220}, {4936, 3158}, {7008, 3347}, {7069, 318}, {7082, 90}, {7156, 7070}, {8012, 220}, {9404, 5546}, {10382, 5665}, {11429, 3469}, {11436, 3362}, {14100, 7}, {14298, 1783}, {14547, 29}, {15733, 3254}, {15837, 2346}, {18235, 2329}, {20665, 9439}, {21033, 2321}, {21811, 37}, {23544, 893}, {27538, 3208}, {28070, 728}, {30223, 84}, {30456, 27382}, {33299, 312}

• X₉ = crosspoint of X_i and X_j for these (i,j): {1, 8056}, {2, 8}, {21, 333}, {55, 2053}, {57, 2137}, {63, 271}, {100, 4564}, {188, 7028}, {190, 765}, {236, 24158}, {258, 24242}, {312, 318}, {645, 4076}, {651, 7012}, {1016, 8706}, {1252, 5546}, {1275, 6606}, {2287, 2322}, {3161, 24150}, {6605, 32008}, {24154, 24155}

• X₉ = crosssum of X_i and X_j for these (i,j): {1, 1743}, {2, 3210}, {3, 3211}, {6, 56}, {7, 3212}, {9, 2136}, {19, 208}, {25, 21058}, {34, 3213}, {37, 3214}, {41, 21059}, {44, 17460}, {48, 3215}, {63, 8897}, {65, 1400}, {173, 8078}, {244, 649}, {269, 17106}, {294, 9453}, {513, 2170}, {514, 24237}, {518, 19593}, {603, 604}, {650, 7004}, {661, 2611}, {798, 4128}, {1015, 6363}, {1086, 7178}, {1357, 7180}, {1407, 6611}, {1418, 1475}, {1575, 20366}, {2082, 28017}, {2446, 2590}, {2447, 2591}, {2488, 14936}, {4000, 28110}, {4466, 21124}, {10490, 18888}

• X₉ = crossdifference of every pair of points on line X₅₁₃X₆₆₃

• X₉ = X_i-Hirst inverse of X_j for these (i,j): {1, 518}, {2, 10025}, {8, 3685}, {43, 8844}, {55, 3684}, {57, 6168}, {192, 239}, {200, 3693}, {294, 28071}, {1282, 8299}, {1575, 16557}, {2170, 4919}, {2348, 3158}, {3307, 24646}, {3308, 24647}, {4876, 7077}, {5239, 5240}, {17792, 18788}

• X₉ = X_i-aleph conjugate of X_j for these (i,j): {1, 43}, {2, 9}, {4, 1711}, {7, 16572}, {8, 10860}, {9, 170}, {75, 1759}, {76, 21366}, {99, 21383}, {174, 1743}, {188, 165}, {190, 1018}, {259, 32462}, {291, 18787}, {365, 1740}, {366, 1}, {507, 361}, {508, 57}, {509, 978}, {522, 2958}, {556, 1766}, {4146, 169}, {4182, 2951}, {5374, 1745}, {6728, 1053}, {7025, 503}, {14087, 1018}, {14089, 21383}, {18297, 63}, {20034, 1716}

• X₉ = X_i-beth conjugate of X_j for these (i,j): (9, 6), (190, 6), (346, 346), (644, 9), (645, 75)

• X₉ = X₁-line conjugate of X₁₂₇₉

• X₉ = X_i-vertex conjugate of X_j for these (i,j): {1, 3420}, {9, 1436}, {269, 1037}, {3900, 32625}

• X₉ = perspector of ABC and extraversion triangle of X₅₇

• X₉ = X₆-of-excentral-triangle

• X₉ = X₁₅₉-of-intouch-triangle

• X₉ = X₆-of-2nd-extouch-triangle

• X₉ = X_i-complementary conjugate of X_j for these (i,j): {1, 2886}, {2, 17046}, {3, 34822}, {6, 142}, {8, 2887}, {9, 141}, {19, 16608}, {21, 3741}, {25, 1210}, {31, 1}, {32, 3752}, {33, 5}, {37, 17052}, {40, 20307}, {41, 2}, {42, 442}, {43, 20338}, {48, 17073}, {55, 10}, {56, 11019}, {57, 21258}, {58, 3742}, {63, 18639}, {71, 18642}, {75, 17047}, {78, 1368}, {81, 17050}, {82, 17049}, {100, 17072}, {101, 4885}, {109, 3900}, {163, 17069}, {184, 17102}, {192, 20547}, {198, 20206}, {200, 1329}, {210, 3454}, {212, 3}, {213, 17056}, {219, 18589}, {220, 3452}, {228, 18641}, {251, 17048}, {281, 20305}, {282, 21239}, {284, 3739}, {294, 20335}, {312, 626}, {318, 21243}, {333, 21240}, {346, 21244}, {512, 8286}, {513, 17059}, {522, 21252}, {560, 17053}, {604, 4000}, {607, 226}, {643, 512}, {644, 3835}, {646, 21262}, {649, 4904}, {650, 116}, {657, 26932}, {662, 17066}, {663, 11}, {667, 3756}, {669, 16613}, {672, 17060}, {692, 522}, {756, 34829}, {798, 17058}, {893, 17062}, {902, 1145}, {904, 24239}, {923, 17070}, {983, 17792}, {1106, 5573}, {1110, 3035}, {1172, 34830}, {1174, 6706}, {1252, 21232}, {1253, 9}, {1260, 34823}, {1320, 21241}, {1333, 3946}, {1334, 1211}, {1395, 17054}, {1400, 18635}, {1402, 1834}, {1409, 18643}, {1415, 7658}, {1820, 18638}, {1911, 1738}, {1918, 2092}, {1946, 2968}, {1964, 17055}, {1973, 3772}, {1974, 20227}, {1980, 16614}, {2053, 3840}, {2148, 17043}, {2149, 17044}, {2150, 17045}, {2155, 18634}, {2156, 18636}, {2157, 18637}, {2158, 18640}, {2159, 18644}, {2162, 20257}, {2163, 17051}, {2172, 17068}, {2175, 37}, {2176, 20528}, {2177, 17057}, {2187, 7952}, {2192, 946}, {2194, 1125}, {2195, 518}, {2200, 18592}, {2208, 3086}, {2212, 6}, {2258, 25466}, {2268, 10472}, {2287, 21246}, {2289, 6389}, {2299, 942}, {2316, 3834}, {2318, 21530}, {2319, 20255}, {2320, 21242}, {2321, 21245}, {2328, 960}, {2332, 6708}, {2340, 120}, {2342, 517}, {2344, 21264}, {2361, 214}, {2364, 34824}, {3052, 12640}, {3063, 1086}, {3158, 2885}, {3195, 20264}, {3208, 21250}, {3433, 24388}, {3445, 24386}, {3596, 21235}, {3684, 20333}, {3685, 20542}, {3688, 21249}, {3689, 121}, {3693, 20540}, {3699, 21260}, {3700, 21253}, {3709, 8287}, {3711, 21251}, {3724, 6739}, {3900, 124}, {3939, 513}, {4041, 125}, {4069, 31946}, {4105, 5514}, {4162, 5510}, {4166, 20334}, {4182, 20543}, {4183, 34831}, {4548, 16582}, {4814, 15614}, {4845, 5087}, {4876, 20541}, {4895, 3259}, {5546, 4369}, {5547, 4892}, {5548, 4928}, {6059, 24005}, {6065, 24003}, {6066, 24036}, {6187, 1737}, {6602, 6554}, {6603, 31844}, {6614, 17427}, {7037, 6245}, {7054, 21233}, {7069, 1209}, {7070, 2883}, {7071, 20262}, {7072, 21616}, {7073, 25639}, {7074, 6260}, {7077, 3836}, {7084, 1376}, {7104, 28358}, {7110, 21236}, {7118, 57}, {7121, 17063}, {7139, 20268}, {7156, 20207}, {7252, 17761}, {7257, 23301}, {7339, 24009}, {7367, 20205}, {8611, 127}, {8641, 1146}, {8750, 521}, {8851, 20340}, {9439, 3816}, {9447, 39}, {9448, 16584}, {9456, 17067}, {10482, 3740}, {13455, 639}, {14827, 1212}, {14942, 20544}, {15374, 21629}, {18265, 1575}, {18757, 33135}, {18889, 527}, {19624, 6594}, {21059, 6600}, {21789, 34589}, {23990, 16578}, {32652, 8058}, {32666, 676}, {32739, 905}, {33299, 21248}, {33635, 17239}, {34067, 25380}, {34248, 17065}, {34446, 31397}, {36086, 926}, {36797, 21259}, {36799, 20549}, {36910, 21237}

• X₉ = X_i-anticomplementary conjugate of X_j for these (i,j): {1, 2890}, {1170, 3434}, {1174, 8}, {2346, 69}, {6605, 3436}, {10482, 329}, {21453, 21285}, {31618, 21280}, {32008, 6327}

• X₉ = Thomson-isogonal conjugate of X₃₅₇₆

• X₉ = medial-isogonal conjugate of X₂₈₈₆

• X₉ = anticomplementary-isogonal conjugate of X₂₈₉₀

• X₉ = excentral-isogonal conjugate of X₁₆₅

• X₉ = tangential-isogonal conjugate of X₂₉₂₁

• X₉ = orthic-isogonal conjugate of X₂₉₀₀

• X₉ = trilinear square root of X₂₀₀

• X₉ = trilinear product of extraversions of X₅₇

• X₉ = trilinear product of PU(112)

• X₉ = inverse-in-circumconic-centered-at-X₁ of X₆₆₀₃

• X₉ = orthocenter of X₁X₄X₈₈₅

• X₉ = bicentric sum of PU(56)

• X₉ = midpoint of PU(56)

• X₉ = crossdifference of PU(96)

• X₉ = perspector of circumconic centered at X₁

• X₉ = the point in which the extended legs P(6)P(33) and U(6)U(33) of the trapezoid PU(6)PU(33) meet

• X₉ = trilinear pole of line X₆₅₀X₆₆₃

• X₉ = pole wrt polar circle of trilinear polar of X₂₇₃ (line X₅₁₄X₃₀₆₄)

• X₉ = X₄₈-isoconjugate (polar conjugate) of X₂₇₃

• X₉ = perspector of ABC and unary cofactor triangle of 1st mixtilinear triangle

• X₉ = perspector of ABC and unary cofactor triangle of 3rd mixtilinear triangle

• X₉ = homothetic center of excentral triangle and 2nd extouch triangle

• X₉ = perspector of ABC and medial triangle of extouch triangle

• X₉ = perspector of pedal and anticevian triangles of X₁₄₉₀

• X₉ = SS(A→J_a) of X₁₉, where J_aJ_bJ_c is the excentral triangle

• X₉ = homothetic center of medial triangle and tangential triangle of excentral triangle

• X₉ = homothetic center of excentral triangle and complement of the intouch triangle

• X₉ = Cundy-Parry Phi transform of X₈₄

• X₉ = Cundy-Parry Psi transform of X₄₀

• X₉ = trilinear product of circumcircle intercepts of excircles radical circle

• X₉ = perspector of Gemini triangle 4 and unary cofactor triangle of Gemini triangle 3

• X₉ = eigencenter of Gemini triangle 5

• X₉ = perspector of Gemini triangle 35 and cross-triangle of ABC and Gemini triangle 35

• X₉ = perspector of ABC and unary cofactor triangle of Gemini triangle 35

• X₉ = perspector of ABC and unary cofactor triangle of Gemini triangle 40

• X₉ = internal center of similitude of the Bevan circle and Spieker circle; the external center is X₁₇₀₆

• X₉ = excentral-to-ABC barycentric image of X₉

• X₉ = intouch-to-ABC barycentric image of X₇

• X₉ = intouch-to-excentral similarity image of X₇

• X₉ = ABC-to-excentral barycentric image of X₉

• X₉ = barycentric product X_i*X_j for these {i,j}: {1, 8}, {3, 318}, {4, 78}, {6, 312}, {7, 200}, {10, 21}, {11, 765}, {12, 1098}, {19, 345}, {25, 3718}, {27, 3694}, {28, 3710}, {29, 72}, {31, 3596}, {32, 28659}, {33, 69}, {34, 1265}, {37, 333}, {40, 280}, {41, 76}, {42, 314}, {43, 7155}, {44, 4997}, {45, 30608}, {48, 7017}, {55, 75}, {56, 341}, {57, 346}, {58, 3701}, {59, 24026}, {60, 1089}, {63, 281}, {65, 1043}, {66, 4123}, {71, 31623}, {77, 7046}, {79, 4420}, {80, 4511}, {81, 2321}, {82, 3703}, {83, 33299}, {84, 7080}, {85, 220}, {86, 210}, {87, 27538}, {88, 2325}, {89, 4873}, {90, 5552}, {92, 219}, {95, 7069}, {99, 4041}, {100, 522}, {101, 4391}, {104, 6735}, {105, 3717}, {106, 4723}, {109, 4397}, {110, 4086}, {142, 6605}, {144, 19605}, {145, 3680}, {158, 1259}, {171, 4451}, {174, 6731}, {188, 188}, {189, 2324}, {190, 650}, {192, 2319}, {212, 264}, {213, 28660}, {222, 7101}, {225, 1792}, {226, 2287}, {236, 7028}, {238, 4518}, {239, 4876}, {241, 6559}, {244, 4076}, {253, 7070}, {256, 7081}, {257, 2329}, {259, 556}, {261, 756}, {266, 7027}, {269, 5423}, {270, 3695}, {271, 7952}, {273, 1260}, {274, 1334}, {278, 3692}, {279, 728}, {282, 329}, {284, 321}, {285, 21075}, {286, 2318}, {291, 3685}, {292, 3975}, {294, 3912}, {304, 607}, {305, 2212}, {306, 1172}, {307, 4183}, {309, 7074}, {313, 2194}, {319, 7073}, {322, 2192}, {326, 1857}, {330, 3208}, {331, 1802}, {332, 1824}, {335, 3684}, {348, 7079}, {350, 7077}, {366, 4182}, {391, 25430}, {393, 3719}, {480, 1088}, {483, 3082}, {492, 13455}, {512, 7257}, {513, 3699}, {514, 644}, {518, 14942}, {519, 1320}, {521, 1897}, {523, 643}, {561, 2175}, {594, 2185}, {596, 3871}, {612, 30479}, {645, 661}, {646, 649}, {648, 8611}, {651, 3239}, {652, 6335}, {657, 4554}, {658, 4130}, {660, 3716}, {662, 3700}, {663, 668}, {664, 3900}, {673, 3693}, {679, 4152}, {693, 3939}, {726, 8851}, {749, 4387}, {757, 6057}, {758, 6740}, {799, 3709}, {860, 1793}, {870, 4517}, {872, 18021}, {873, 7064}, {885, 1026}, {893, 17787}, {897, 3712}, {898, 14430}, {901, 4768}, {903, 3689}, {932, 4147}, {934, 4163}, {941, 11679}, {943, 6734}, {947, 23528}, {950, 1257}, {958, 31359}, {960, 1220}, {979, 19582}, {983, 3705}, {985, 3790}, {996, 3877}, {997, 30513}, {1000, 3872}, {1002, 3886}, {1014, 4082}, {1016, 2170}, {1018, 4560}, {1019, 30730}, {1021, 4552}, {1022, 30731}, {1025, 28132}, {1034, 1490}, {1036, 4385}, {1096, 1264}, {1100, 4102}, {1111, 6065}, {1120, 3880}, {1121, 6603}, {1125, 32635}, {1126, 3702}, {1146, 4564}, {1156, 6745}, {1174, 1229}, {1212, 32008}, {1214, 2322}, {1219, 1697}, {1222, 3057}, {1231, 2332}, {1240, 20967}, {1252, 4858}, {1253, 6063}, {1255, 3686}, {1261, 3663}, {1267, 13456}, {1268, 3683}, {1275, 3119}, {1280, 5853}, {1318, 4738}, {1332, 3064}, {1333, 30713}, {1390, 3883}, {1392, 3632}, {1407, 30693}, {1420, 6556}, {1434, 4515}, {1435, 30681}, {1441, 2328}, {1476, 6736}, {1492, 4522}, {1502, 9447}, {1577, 5546}, {1635, 4582}, {1639, 3257}, {1743, 6557}, {1751, 27396}, {1783, 6332}, {1785, 1809}, {1807, 5081}, {1812, 1826}, {1896, 3682}, {1903, 27398}, {1911, 4087}, {1928, 9448}, {1937, 7360}, {1959, 15628}, {1978, 3063}, {2052, 2289}, {2053, 6376}, {2057, 10309}, {2082, 30701}, {2125, 30695}, {2136, 6553}, {2137, 6552}, {2149, 23978}, {2150, 28654}, {2161, 32851}, {2162, 4110}, {2171, 7058}, {2176, 27424}, {2184, 27382}, {2195, 3263}, {2269, 30710}, {2279, 28809}, {2297, 18228}, {2298, 3687}, {2299, 20336}, {2310, 4998}, {2311, 3948}, {2316, 4358}, {2320, 3679}, {2323, 18359}, {2326, 26942}, {2330, 7018}, {2334, 4673}, {2335, 5271}, {2339, 2345}, {2340, 2481}, {2341, 3936}, {2342, 3262}, {2344, 3661}, {2346, 4847}, {2347, 32017}, {2349, 7359}, {2361, 20566}, {2363, 3704}, {2364, 4671}, {2643, 6064}, {2968, 7012}, {2985, 17452}, {2997, 3190}, {2998, 7075}, {3056, 7033}, {3059, 21453}, {3061, 17743}, {3083, 13454}, {3084, 13426}, {3112, 3688}, {3158, 4373}, {3161, 8056}, {3198, 5931}, {3219, 7110}, {3241, 4900}, {3247, 30711}, {3254, 3935}, {3271, 7035}, {3287, 27805}, {3452, 23617}, {3467, 27529}, {3478, 4737}, {3495, 26752}, {3615, 3678}, {3616, 4866}, {3623, 31509}, {3667, 31343}, {3669, 6558}, {3676, 4578}, {3691, 32009}, {3714, 5331}, {3715, 30598}, {3737, 3952}, {3762, 5548}, {3869, 10570}, {3870, 6601}, {3903, 3907}, {3996, 13476}, {3998, 8748}, {4007, 25417}, {4017, 7256}, {4024, 4612}, {4033, 7252}, {4034, 27789}, {4036, 4636}, {4069, 7192}, {4079, 4631}, {4081, 7045}, {4092, 24041}, {4105, 4569}, {4124, 5378}, {4140, 4603}, {4146, 6726}, {4149, 7357}, {4166, 18297}, {4171, 4573}, {4319, 8817}, {4433, 18827}, {4435, 4562}, {4494, 30650}, {4512, 5936}, {4513, 9311}, {4516, 4600}, {4521, 27834}, {4524, 4625}, {4526, 4607}, {4530, 5376}, {4534, 5382}, {4543, 4618}, {4551, 7253}, {4555, 4895}, {4557, 18155}, {4561, 18344}, {4567, 21044}, {4571, 7649}, {4572, 8641}, {4587, 17924}, {4597, 4814}, {4604, 4944}, {4606, 4765}, {4614, 4843}, {4624, 4827}, {4645, 7281}, {4791, 5549}, {4811, 8694}, {4845, 30806}, {4853, 7320}, {4861, 5559}, {4882, 5558}, {4919, 6630}, {4936, 27818}, {4985, 8701}, {5205, 9365}, {5239, 7043}, {5240, 7026}, {5380, 14432}, {5391, 13427}, {5430, 24242}, {5547, 14210}, {5665, 20007}, {6358, 7054}, {6554, 7131}, {6555, 19604}, {6606, 6608}, {6615, 8706}, {6737, 17097}, {7004, 15742}, {7020, 7078}, {7068, 24000}, {7071, 7182}, {7072, 20930}, {7090, 30556}, {7097, 27540}, {7105, 7283}, {7162, 10527}, {7178, 7259}, {7180, 7258}, {7220, 24349}, {7285, 27525}, {8012, 31618}, {8058, 13138}, {8707, 17420}, {8806, 13614}, {9368, 9369}, {9404, 15455}, {9436, 28071}, {9442, 28058}, {10025, 14943}, {10482, 20880}, {11609, 17763}, {12644, 12646}, {14121, 30557}, {14206, 15627}, {14534, 21033}, {14624, 17185}, {14827, 20567}, {15416, 32674}, {15891, 30413}, {15892, 30412}, {17780, 23838}, {18265, 18891}, {18750, 30457}, {19607, 21078}, {23705, 23836}, {24150, 24151}, {24152, 24154}, {24153, 24155}, {28070, 30705}

• X₉ = barycentric quotient X_i/X_j for these {i,j}: {1, 7}, {2, 85}, {3, 77}, {4, 273}, {6, 57}, {7, 1088}, {8, 75}, {10, 1441}, {11, 1111}, {19, 278}, {21, 86}, {22, 7210}, {25, 34}, {29, 286}, {31, 56}, {32, 604}, {33, 4}, {34, 1119}, {35, 1442}, {36, 1443}, {37, 226}, {38, 3665}, {40, 347}, {41, 6}, {42, 65}, {43, 3212}, {44, 3911}, {45, 5219}, {48, 222}, {51, 1393}, {55, 1}, {56, 269}, {57, 279}, {58, 1014}, {59, 7045}, {60, 757}, {63, 348}, {64, 8809}, {65, 3668}, {69, 7182}, {71, 1214}, {72, 307}, {73, 1439}, {75, 6063}, {76, 20567}, {77, 7056}, {78, 69}, {80, 18815}, {81, 1434}, {84, 1440}, {90, 7318}, {92, 331}, {99, 4625}, {100, 664}, {101, 651}, {109, 934}, {110, 1414}, {144, 31627}, {154, 1394}, {163, 4565}, {165, 3160}, {171, 7176}, {172, 7175}, {173, 18886}, {174, 555}, {181, 1254}, {184, 603}, {188, 4146}, {190, 4554}, {192, 30545}, {197, 21147}, {198, 223}, {200, 8}, {201, 6356}, {205, 478}, {210, 10}, {212, 3}, {213, 1400}, {218, 1445}, {219, 63}, {222, 7177}, {223, 14256}, {226, 1446}, {228, 73}, {238, 1447}, {239, 10030}, {244, 1358}, {255, 1804}, {256, 7249}, {258, 21456}, {259, 174}, {261, 873}, {266, 7371}, {269, 479}, {278, 1847}, {279, 23062}, {280, 309}, {281, 92}, {282, 189}, {283, 1444}, {284, 81}, {291, 7233}, {294, 673}, {306, 1231}, {312, 76}, {314, 310}, {318, 264}, {321, 349}, {326, 7055}, {330, 7209}, {333, 274}, {341, 3596}, {344, 21609}, {345, 304}, {346, 312}, {350, 18033}, {354, 10481}, {391, 19804}, {394, 7183}, {461, 5342}, {480, 200}, {497, 3673}, {512, 4017}, {513, 3676}, {514, 24002}, {517, 22464}, {518, 9436}, {521, 4025}, {522, 693}, {523, 4077}, {560, 1397}, {572, 17074}, {573, 17080}, {577, 7125}, {594, 6358}, {603, 7053}, {604, 1407}, {607, 19}, {608, 1435}, {610, 18623}, {612, 388}, {614, 7195}, {643, 99}, {644, 190}, {645, 799}, {646, 1978}, {649, 3669}, {650, 514}, {651, 658}, {652, 905}, {653, 13149}, {654, 3960}, {656, 17094}, {657, 650}, {661, 7178}, {662, 4573}, {663, 513}, {664, 4569}, {668, 4572}, {672, 241}, {678, 1317}, {692, 109}, {728, 346}, {750, 7223}, {756, 12}, {757, 552}, {765, 4998}, {798, 7180}, {846, 17084}, {849, 7341}, {862, 1874}, {869, 1469}, {872, 181}, {884, 1027}, {893, 1432}, {894, 7196}, {896, 7181}, {902, 1319}, {904, 1431}, {906, 1813}, {923, 7316}, {926, 2254}, {934, 4626}, {950, 17863}, {958, 10436}, {960, 4357}, {968, 3485}, {982, 7185}, {984, 7179}, {1018, 4552}, {1019, 17096}, {1021, 4560}, {1026, 883}, {1040, 17170}, {1043, 314}, {1054, 17089}, {1055, 6610}, {1096, 1118}, {1098, 261}, {1100, 553}, {1106, 7023}, {1107, 30097}, {1110, 59}, {1146, 4858}, {1155, 1323}, {1158, 31600}, {1170, 10509}, {1172, 27}, {1174, 1170}, {1193, 24471}, {1201, 1122}, {1212, 142}, {1220, 31643}, {1229, 1233}, {1250, 1082}, {1251, 1081}, {1252, 4564}, {1253, 55}, {1254, 6046}, {1259, 326}, {1260, 78}, {1261, 1222}, {1265, 3718}, {1318, 679}, {1320, 903}, {1331, 6516}, {1333, 1412}, {1334, 37}, {1376, 9312}, {1395, 1398}, {1397, 1106}, {1400, 1427}, {1402, 1042}, {1407, 738}, {1414, 4616}, {1415, 1461}, {1419, 9533}, {1436, 1422}, {1438, 1462}, {1445, 17093}, {1449, 21454}, {1460, 4320}, {1461, 4617}, {1469, 7204}, {1474, 1396}, {1475, 1418}, {1490, 5932}, {1500, 2171}, {1613, 1424}, {1615, 2124}, {1617, 4350}, {1635, 30725}, {1639, 3762}, {1697, 3672}, {1707, 17081}, {1721, 2898}, {1722, 31598}, {1731, 33129}, {1740, 17082}, {1742, 31526}, {1743, 5435}, {1754, 3188}, {1759, 17075}, {1760, 17076}, {1781, 18625}, {1783, 653}, {1792, 332}, {1802, 219}, {1812, 17206}, {1824, 225}, {1837, 17861}, {1857, 158}, {1859, 1838}, {1864, 1210}, {1897, 18026}, {1903, 8808}, {1909, 7205}, {1914, 1429}, {1918, 1402}, {1935, 6359}, {1936, 5088}, {1946, 1459}, {1962, 3649}, {1964, 1401}, {1973, 608}, {1974, 1395}, {2053, 87}, {2066, 13389}, {2082, 4000}, {2098, 4862}, {2115, 9499}, {2124, 17113}, {2136, 4452}, {2149, 1262}, {2150, 593}, {2161, 2006}, {2162, 7153}, {2170, 1086}, {2171, 6354}, {2173, 6357}, {2174, 2003}, {2175, 31}, {2176, 1423}, {2177, 2099}, {2183, 1465}, {2185, 1509}, {2187, 221}, {2188, 1433}, {2192, 84}, {2193, 1790}, {2194, 58}, {2195, 105}, {2199, 6611}, {2200, 1409}, {2204, 1474}, {2206, 1408}, {2208, 1413}, {2209, 1403}, {2210, 1428}, {2212, 25}, {2223, 1458}, {2238, 16609}, {2241, 7225}, {2244, 7214}, {2245, 18593}, {2246, 5723}, {2251, 1404}, {2258, 959}, {2259, 2982}, {2268, 940}, {2269, 3666}, {2276, 7146}, {2280, 5228}, {2284, 1025}, {2285, 7365}, {2287, 333}, {2289, 394}, {2293, 354}, {2295, 4032}, {2299, 28}, {2308, 32636}, {2310, 11}, {2316, 88}, {2318, 72}, {2319, 330}, {2321, 321}, {2322, 31623}, {2323, 3218}, {2324, 329}, {2325, 4358}, {2327, 1812}, {2328, 21}, {2329, 894}, {2330, 171}, {2331, 196}, {2332, 1172}, {2333, 1880}, {2340, 518}, {2341, 24624}, {2342, 104}, {2344, 14621}, {2346, 21453}, {2347, 3752}, {2348, 3008}, {2352, 4306}, {2356, 1876}, {2361, 36}, {2364, 89}, {2427, 24029}, {2632, 1367}, {2638, 1364}, {2640, 17085}, {2643, 1365}, {2646, 3664}, {2900, 12649}, {2911, 1708}, {2939, 18631}, {2951, 31527}, {2968, 17880}, {2997, 15467}, {3009, 1463}, {3022, 2310}, {3024, 7266}, {3052, 1420}, {3056, 982}, {3057, 3663}, {3058, 7264}, {3059, 4847}, {3061, 3662}, {3063, 649}, {3064, 17924}, {3083, 13453}, {3084, 13436}, {3100, 4872}, {3119, 1146}, {3158, 145}, {3161, 18743}, {3169, 3210}, {3172, 3213}, {3185, 10571}, {3190, 3868}, {3195, 208}, {3198, 5930}, {3207, 1419}, {3208, 192}, {3217, 4383}, {3218, 17078}, {3219, 17095}, {3239, 4391}, {3248, 1357}, {3270, 7004}, {3271, 244}, {3287, 4369}, {3295, 7190}, {3303, 4328}, {3304, 7271}, {3306, 17079}, {3309, 31605}, {3445, 19604}, {3452, 26563}, {3496, 17086}, {3596, 561}, {3601, 3945}, {3666, 3674}, {3680, 4373}, {3681, 33298}, {3683, 1125}, {3684, 239}, {3685, 350}, {3686, 4359}, {3687, 20911}, {3688, 38}, {3689, 519}, {3690, 201}, {3691, 3739}, {3692, 345}, {3693, 3912}, {3694, 306}, {3699, 668}, {3700, 1577}, {3701, 313}, {3702, 1269}, {3703, 1930}, {3704, 18697}, {3706, 20888}, {3707, 24589}, {3709, 661}, {3710, 20336}, {3711, 3679}, {3712, 14210}, {3713, 11679}, {3715, 1698}, {3716, 3766}, {3717, 3263}, {3718, 305}, {3719, 3926}, {3720, 4059}, {3721, 16888}, {3723, 3982}, {3724, 1464}, {3731, 5226}, {3733, 7203}, {3737, 7192}, {3738, 4453}, {3745, 4298}, {3746, 7269}, {3747, 1284}, {3786, 30966}, {3870, 6604}, {3871, 4360}, {3876, 5224}, {3877, 4389}, {3880, 1266}, {3883, 26234}, {3885, 4398}, {3886, 4441}, {3900, 522}, {3907, 4374}, {3910, 4509}, {3913, 3875}, {3920, 7247}, {3938, 30617}, {3939, 100}, {3949, 26942}, {3957, 32007}, {3965, 3687}, {3974, 4385}, {3975, 1921}, {3985, 3948}, {3996, 17143}, {4007, 28605}, {4009, 6381}, {4041, 523}, {4042, 32092}, {4046, 4647}, {4050, 1278}, {4055, 22341}, {4060, 4980}, {4069, 3952}, {4073, 3705}, {4076, 7035}, {4081, 24026}, {4082, 3701}, {4086, 850}, {4087, 18891}, {4092, 1109}, {4094, 3027}, {4095, 3963}, {4097, 3896}, {4102, 32018}, {4105, 3900}, {4110, 6382}, {4111, 21020}, {4117, 1356}, {4118, 7217}, {4119, 20432}, {4123, 315}, {4130, 3239}, {4136, 20234}, {4147, 20906}, {4149, 6327}, {4152, 4738}, {4162, 3667}, {4163, 4397}, {4166, 366}, {4167, 21442}, {4171, 3700}, {4178, 20627}, {4182, 18297}, {4183, 29}, {4253, 17092}, {4254, 5256}, {4258, 1449}, {4266, 4850}, {4319, 497}, {4320, 7197}, {4326, 10580}, {4336, 1836}, {4361, 7243}, {4387, 3760}, {4390, 4363}, {4391, 3261}, {4394, 30719}, {4420, 319}, {4421, 25716}, {4433, 740}, {4435, 812}, {4451, 7018}, {4474, 4411}, {4477, 3907}, {4501, 4382}, {4511, 320}, {4512, 3616}, {4513, 3729}, {4515, 2321}, {4516, 3120}, {4517, 984}, {4518, 334}, {4521, 4462}, {4524, 4041}, {4526, 4728}, {4528, 4768}, {4531, 3778}, {4548, 2172}, {4551, 4566}, {4557, 4551}, {4559, 1020}, {4560, 7199}, {4564, 1275}, {4565, 4637}, {4567, 4620}, {4571, 4561}, {4573, 4635}, {4578, 3699}, {4579, 6649}, {4587, 1332}, {4606, 4624}, {4612, 4610}, {4662, 4967}, {4723, 3264}, {4730, 30572}, {4765, 4801}, {4790, 30723}, {4814, 4777}, {4820, 4823}, {4827, 4765}, {4843, 4815}, {4845, 1156}, {4847, 20880}, {4849, 4848}, {4853, 31995}, {4858, 23989}, {4860, 21314}, {4861, 7321}, {4866, 5936}, {4873, 4671}, {4875, 24199}, {4876, 335}, {4877, 5333}, {4882, 32087}, {4895, 900}, {4901, 31130}, {4903, 20943}, {4907, 5274}, {4919, 4440}, {4936, 3161}, {4944, 4791}, {4953, 4939}, {4959, 4926}, {4976, 4978}, {4979, 30724}, {4990, 4985}, {4995, 7278}, {4997, 20568}, {5048, 4887}, {5089, 5236}, {5250, 17321}, {5269, 3600}, {5285, 4296}, {5289, 17274}, {5311, 10404}, {5320, 1451}, {5414, 13388}, {5423, 341}, {5452, 169}, {5532, 1090}, {5546, 662}, {5547, 897}, {5548, 3257}, {5549, 4604}, {5552, 20930}, {5795, 24993}, {5802, 19788}, {5837, 24547}, {6003, 31603}, {6056, 255}, {6057, 1089}, {6058, 1091}, {6059, 1096}, {6060, 1097}, {6061, 1098}, {6062, 1099}, {6064, 24037}, {6065, 765}, {6066, 1110}, {6139, 14413}, {6187, 1411}, {6198, 7282}, {6332, 15413}, {6558, 646}, {6600, 3870}, {6602, 220}, {6603, 527}, {6605, 32008}, {6607, 6608}, {6608, 6362}, {6726, 188}, {6731, 556}, {6735, 3262}, {6736, 20895}, {6740, 14616}, {6741, 17886}, {6745, 30806}, {7004, 1565}, {7007, 7149}, {7014, 558}, {7017, 1969}, {7032, 7248}, {7037, 3345}, {7046, 318}, {7050, 7091}, {7054, 2185}, {7062, 23996}, {7064, 756}, {7067, 24038}, {7068, 17879}, {7069, 5}, {7070, 20}, {7071, 33}, {7072, 90}, {7073, 79}, {7074, 40}, {7075, 194}, {7076, 1940}, {7077, 291}, {7078, 7013}, {7079, 281}, {7080, 322}, {7081, 1909}, {7082, 499}, {7083, 614}, {7084, 1037}, {7085, 1038}, {7087, 7213}, {7101, 7017}, {7110, 30690}, {7115, 7128}, {7117, 3942}, {7118, 1436}, {7123, 7131}, {7124, 7289}, {7131, 30705}, {7133, 1659}, {7154, 7129}, {7155, 6384}, {7156, 1249}, {7177, 30682}, {7180, 7216}, {7252, 1019}, {7253, 18155}, {7256, 7257}, {7257, 670}, {7259, 645}, {7281, 7261}, {7290, 3598}, {7322, 5261}, {7339, 24013}, {7359, 14206}, {7367, 282}, {7368, 2324}, {7675, 14548}, {7707, 234}, {7952, 342}, {7962, 4346}, {8012, 1212}, {8056, 27818}, {8058, 17896}, {8540, 18201}, {8545, 1996}, {8551, 8012}, {8580, 31994}, {8606, 7100}, {8611, 525}, {8641, 663}, {8647, 1279}, {8653, 4822}, {8676, 23800}, {8750, 108}, {8835, 15913}, {8851, 3226}, {9310, 6180}, {9404, 14838}, {9439, 9309}, {9440, 9446}, {9441, 14189}, {9447, 32}, {9448, 560}, {9629, 3583}, {10382, 938}, {10387, 3677}, {10393, 5738}, {10482, 2346}, {10501, 10491}, {10502, 10489}, {10570, 2995}, {10581, 21127}, {10582, 32086}, {10638, 559}, {11075, 26743}, {11124, 21105}, {11193, 21201}, {11429, 3075}, {11934, 21185}, {11997, 24210}, {11998, 24237}, {12329, 8270}, {13427, 1336}, {13455, 485}, {13456, 1123}, {14077, 30181}, {14100, 11019}, {14298, 14837}, {14308, 17898}, {14392, 6366}, {14427, 1639}, {14547, 942}, {14827, 41}, {14936, 2170}, {14942, 2481}, {15627, 2349}, {15628, 1821}, {15733, 26015}, {15837, 13405}, {16011, 2091}, {16012, 177}, {16283, 9310}, {16502, 28017}, {16545, 18626}, {16546, 18627}, {16552, 17077}, {16556, 17083}, {16566, 17087}, {16568, 17088}, {16569, 17090}, {16571, 17091}, {16572, 8732}, {16588, 17451}, {16601, 21617}, {16666, 4031}, {16686, 1421}, {16713, 16708}, {16721, 18176}, {16777, 4654}, {16780, 28079}, {16885, 31231}, {17185, 16705}, {17194, 17169}, {17197, 16727}, {17412, 13401}, {17420, 3004}, {17452, 3782}, {17453, 7251}, {17469, 7198}, {17742, 28739}, {17744, 28780}, {17787, 1920}, {17798, 5018}, {18098, 18097}, {18163, 18600}, {18191, 17205}, {18265, 1911}, {18344, 7649}, {18594, 18624}, {18595, 18628}, {18596, 18629}, {18597, 18630}, {18598, 18632}, {18599, 18633}, {18887, 21624}, {18888, 14596}, {18889, 2291}, {19605, 10405}, {19624, 2078}, {20229, 1475}, {20359, 24215}, {20663, 8850}, {20665, 2275}, {20672, 2114}, {20684, 3721}, {20753, 3784}, {20967, 1193}, {21010, 4334}, {21033, 1211}, {21039, 3925}, {21044, 16732}, {21059, 1617}, {21104, 23599}, {21127, 21104}, {21333, 4920}, {21334, 24214}, {21677, 18698}, {21789, 3737}, {21795, 21808}, {21803, 7211}, {21809, 4415}, {21811, 17056}, {21832, 7212}, {21840, 5244}, {21859, 4605}, {21879, 27691}, {22074, 22097}, {22079, 22053}, {23207, 4303}, {23344, 23703}, {23544, 28358}, {23638, 24443}, {23838, 6548}, {23990, 2149}, {24010, 4081}, {24012, 3022}, {24027, 7339}, {24041, 7340}, {24151, 27828}, {24394, 4442}, {24430, 17181}, {25082, 17234}, {25128, 23807}, {25268, 21580}, {26885, 1935}, {27382, 18750}, {27396, 18134}, {27424, 6383}, {27508, 20921}, {27523, 20923}, {27538, 6376}, {27540, 20914}, {27549, 30758}, {27958, 8033}, {28043, 2550}, {28070, 6554}, {28071, 14942}, {28125, 5880}, {28659, 1502}, {28660, 6385}, {28808, 20925}, {28809, 21615}, {30223, 3086}, {30457, 2184}, {30568, 18135}, {30608, 20569}, {30618, 17353}, {30706, 2082}, {30713, 27801}, {30730, 4033}, {30731, 24004}, {32008, 31618}, {32462, 31604}, {32635, 1268}, {32652, 8059}, {32666, 32735}, {32674, 32714}, {32739, 1415}, {32851, 20924}, {33299, 141}

• X₉ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 6, 1449}, {1, 37, 3247}, {1, 44, 16670}, {1, 45, 16676}, {1, 72, 11523}, {1, 238, 7290}, {1, 405, 5436}, {1, 960, 15829}, {1, 984, 7174}, {1, 1723, 2257}, {1, 1724, 1453}, {1, 1728, 10396}, {1, 1743, 6}, {1, 1757, 3751}, {1, 3731, 37}, {1, 3973, 1743}, {1, 5234, 958}, {1, 10398, 5728}, {1, 16468, 16475}, {1, 16469, 1386}, {1, 16552, 21384}, {1, 16667, 1100}, {1, 16673, 16777}, {1, 17744, 17742}, {1, 30330, 5572}, {2, 7, 142}, {2, 57, 5437}, {2, 63, 57}, {2, 142, 20195}, {2, 144, 7}, {2, 226, 25525}, {2, 307, 18634}, {2, 329, 226}, {2, 672, 17754}, {2, 894, 10436}, {2, 908, 5219}, {2, 3218, 3306}, {2, 3219, 63}, {2, 3305, 7308}, {2, 3452, 30827}, {2, 3662, 17282}, {2, 3911, 31190}, {2, 3929, 3928}, {2, 4357, 17306}, {2, 5273, 5745}, {2, 5282, 3509}, {2, 5296, 5257}, {2, 5435, 6692}, {2, 5744, 3911}, {2, 5749, 5750}, {2, 5905, 5249}, {2, 6646, 3662}, {2, 9965, 9776}, {2, 17236, 17291}, {2, 17257, 4357}, {2, 17333, 17274}, {2, 17350, 894}, {2, 17483, 27186}, {2, 17484, 31019}, {2, 17781, 4654}, {2, 18228, 3452}, {2, 18230, 6666}, {2, 20347, 30949}, {2, 20348, 30097}, {2, 26125, 25521}, {2, 26685, 17353}, {2, 26792, 31053}, {2, 26806, 27147}, {2, 26836, 26997}, {2, 27065, 3305}, {2, 27131, 30852}, {2, 27184, 25527}, {2, 27420, 27384}, {2, 28287, 27626}, {2, 29696, 29740}, {2, 30414, 5242}, {2, 30415, 5243}, {2, 30946, 20335}, {2, 31018, 908}, {2, 31053, 31266}, {2, 31300, 26806}, {3, 84, 9841}, {3, 936, 5438}, {3, 5044, 936}, {3, 5777, 1490}, {3, 7330, 84}, {3, 12684, 31805}, {3, 24320, 3220}, {3, 31445, 31424}, {3, 31658, 21153}, {4, 21168, 5759}, {5, 5791, 5705}, {5, 5812, 5715}, {5, 26921, 5709}, {6, 37, 1}, {6, 44, 1743}, {6, 45, 37}, {6, 219, 2323}, {6, 220, 219}, {6, 1001, 16503}, {6, 1100, 16667}, {6, 1743, 16670}, {6, 2176, 2300}, {6, 3731, 3247}, {6, 8557, 2257}, {6, 8609, 3554}, {6, 15492, 3973}, {6, 16672, 16884}, {6, 16675, 16777}, {6, 16677, 3723}, {6, 16777, 1100}, {6, 16814, 3731}, {6, 16884, 16666}, {6, 16885, 44}, {6, 16969, 21785}, {6, 16970, 7290}, {6, 16972, 16475}, {6, 21769, 20228}, {7, 142, 6173}, {7, 1445, 57}, {7, 6172, 144}, {7, 6666, 20195}, {7, 8232, 226}, {7, 18230, 2}, {7, 29007, 8545}, {8, 346, 2321}, {8, 391, 3686}, {8, 452, 950}, {8, 950, 12625}, {8, 1334, 3208}, {8, 1697, 2136}, {8, 2269, 3169}, {8, 2321, 4007}, {8, 2325, 4873}, {8, 3161, 346}, {8, 3208, 4050}, {8, 3685, 3886}, {8, 3686, 4034}, {8, 3717, 4901}, {8, 5250, 1697}, {8, 5686, 24393}, {8, 27549, 3717}, {10, 40, 1706}, {10, 3730, 3501}, {10, 6554, 23058}, {10, 12514, 40}, {10, 12572, 4}, {10, 12618, 1861}, {10, 17355, 2345}, {10, 18250, 2551}, {10, 31594, 14121}, {10, 31595, 7090}, {19, 71, 40}, {19, 169, 16547}, {19, 1766, 16548}, {19, 2183, 2270}, {19, 7079, 281}, {21, 78, 3601}, {21, 2287, 284}, {21, 3876, 78}, {25, 7085, 5285}, {25, 26867, 7085}, {31, 612, 5269}, {31, 756, 612}, {33, 212, 7070}, {37, 44, 6}, {37, 45, 3731}, {37, 72, 22021}, {37, 220, 2324}, {37, 1100, 16777}, {37, 1743, 1449}, {37, 2911, 3553}, {37, 3723, 16672}, {37, 3731, 16676}, {37, 3973, 16670}, {37, 4426, 5336}, {37, 15479, 15829}, {37, 15492, 44}, {37, 16517, 7174}, {37, 16666, 3723}, {37, 16669, 1100}, {37, 16671, 16884}, {37, 16777, 16673}, {37, 16814, 45}, {37, 16885, 1743}, {37, 21873, 4053}, {37, 21879, 21810}, {38, 614, 3677}, {38, 748, 614}, {39, 21796, 2277}, {39, 31442, 31429}, {41, 2268, 284}, {41, 33299, 78}, {43, 846, 17594}, {43, 21369, 21387}, {44, 45, 1}, {44, 1100, 16669}, {44, 3723, 16671}, {44, 3731, 1449}, {44, 8557, 16572}, {44, 15492, 16885}, {44, 16675, 16667}, {44, 16814, 37}, {44, 16885, 3973}, {45, 1743, 3247}, {45, 3973, 1449}, {45, 15492, 1743}, {45, 16669, 16673}, {45, 16777, 16675}, {45, 16884, 16677}, {45, 16885, 6}, {48, 2265, 2261}, {48, 2267, 572}, {51, 3690, 26893}, {55, 200, 3158}, {55, 210, 200}, {55, 480, 6600}, {55, 1864, 10382}, {55, 2264, 380}, {55, 3059, 3174}, {55, 3683, 4512}, {55, 3711, 3689}, {55, 3715, 210}, {55, 7082, 30223}, {55, 14100, 4326}, {56, 8581, 4321}, {56, 25917, 8583}, {57, 63, 3928}, {57, 3929, 63}, {57, 7308, 2}, {63, 3219, 3929}, {63, 3305, 2}, {63, 3306, 3218}, {63, 7308, 5437}, {63, 21371, 16574}, {63, 25880, 28017}, {63, 25894, 28039}, {63, 27065, 7308}, {69, 344, 3912}, {69, 3912, 17296}, {71, 2183, 573}, {71, 2345, 3501}, {72, 405, 1}, {72, 10396, 6762}, {72, 14054, 5904}, {75, 190, 3729}, {75, 3729, 4659}, {75, 17277, 4384}, {75, 17335, 17277}, {75, 17336, 190}, {75, 20927, 20236}, {77, 651, 1419}, {85, 32024, 30625}, {85, 32088, 32008}, {85, 32100, 32024}, {86, 4687, 16831}, {86, 18206, 18164}, {101, 572, 48}, {101, 2265, 16554}, {101, 12034, 2265}, {105, 4712, 9451}, {141, 4422, 17279}, {141, 4643, 17272}, {141, 17279, 17284}, {141, 17332, 4643}, {142, 6666, 2}, {144, 6666, 6173}, {144, 18230, 142}, {165, 1709, 10860}, {165, 1750, 7580}, {165, 2951, 11495}, {165, 3062, 2951}, {165, 8580, 1376}, {165, 30326, 1750}, {165, 30393, 8580}, {169, 573, 2270}, {169, 1766, 19}, {169, 3730, 40}, {169, 4456, 7713}, {169, 7079, 23058}, {169, 12514, 3496}, {171, 7262, 1707}, {173, 8078, 18888}, {188, 236, 16016}, {190, 4384, 4659}, {190, 17277, 75}, {190, 17335, 4384}, {190, 17336, 25728}, {191, 1698, 46}, {191, 1781, 1761}, {192, 239, 3875}, {192, 17349, 239}, {193, 17316, 3879}, {198, 1436, 1604}, {198, 1903, 1490}, {198, 2182, 610}, {200, 4326, 3174}, {200, 4512, 55}, {200, 10382, 2900}, {210, 3683, 55}, {210, 3689, 3711}, {210, 4512, 3158}, {210, 13615, 2900}, {210, 14100, 3059}, {210, 15837, 480}, {212, 7069, 33}, {213, 5283, 1}, {218, 16601, 1}, {220, 958, 2329}, {220, 1212, 1}, {220, 8557, 15286}, {226, 329, 28609}, {226, 1708, 57}, {238, 984, 1}, {238, 16517, 1449}, {239, 17261, 192}, {241, 6180, 269}, {261, 645, 27958}, {281, 20262, 23058}, {284, 4877, 21}, {312, 333, 11679}, {319, 17233, 17294}, {319, 17264, 17233}, {320, 17234, 17298}, {320, 17263, 17234}, {321, 5278, 5271}, {329, 5749, 5746}, {333, 17185, 18163}, {344, 4416, 17296}, {345, 14555, 3687}, {346, 391, 8}, {346, 2269, 3208}, {346, 2321, 4873}, {346, 2347, 3169}, {346, 3161, 2325}, {346, 3686, 4007}, {346, 3692, 728}, {346, 3707, 4034}, {354, 4423, 10582}, {374, 21871, 2262}, {380, 3694, 3158}, {381, 31671, 18482}, {390, 5686, 8}, {390, 5809, 950}, {390, 5825, 10392}, {391, 452, 5802}, {391, 1334, 3169}, {391, 2321, 4034}, {391, 2325, 4007}, {391, 3161, 2321}, {392, 956, 1}, {405, 954, 1001}, {405, 15650, 72}, {474, 3916, 15803}, {480, 3059, 200}, {480, 4326, 3158}, {480, 14100, 3174}, {497, 4847, 24392}, {573, 1766, 40}, {573, 3730, 71}, {573, 17355, 3501}, {579, 5750, 17754}, {579, 8558, 1741}, {594, 4370, 17340}, {594, 17275, 3679}, {594, 17330, 17275}, {594, 17340, 17281}, {599, 17267, 17231}, {612, 756, 7322}, {644, 2170, 4919}, {672, 1400, 579}, {672, 5282, 63}, {728, 1697, 3208}, {728, 2082, 2136}, {894, 17260, 2}, {894, 21371, 57}, {894, 27420, 1944}, {908, 31018, 31142}, {936, 31424, 3}, {950, 10392, 5809}, {954, 5223, 11523}, {954, 5728, 1}, {954, 5729, 5728}, {958, 960, 1}, {958, 5302, 5234}, {958, 15479, 1449}, {960, 1212, 3061}, {960, 5302, 958}, {960, 30618, 220}, {965, 5782, 5783}, {966, 2345, 10}, {966, 6554, 20262}, {982, 5272, 5573}, {982, 17123, 5272}, {984, 16970, 3247}, {992, 2277, 978}, {993, 997, 3576}, {993, 10176, 997}, {1001, 5223, 3243}, {1001, 15481, 5223}, {1006, 18446, 3576}, {1038, 1935, 1394}, {1086, 17276, 4862}, {1086, 17278, 4859}, {1086, 17334, 17276}, {1086, 17337, 17278}, {1100, 16667, 1449}, {1100, 16669, 6}, {1100, 16675, 16673}, {1100, 16777, 1}, {1107, 2176, 1}, {1108, 2256, 1}, {1124, 8965, 1}, {1156, 6594, 5528}, {1212, 30618, 2329}, {1213, 1901, 442}, {1213, 7359, 7110}, {1213, 17303, 1698}, {1213, 17369, 17303}, {1253, 2310, 4319}, {1253, 21039, 28043}, {1260, 1864, 2900}, {1260, 10382, 3158}, {1260, 13615, 55}, {1276, 1277, 40}, {1276, 6192, 1277}, {1277, 6191, 1276}, {1278, 16816, 17117}, {1279, 3242, 1}, {1329, 18253, 26066}, {1329, 26066, 1698}, {1334, 2082, 1697}, {1334, 2347, 2269}, {1334, 3691, 8}, {1376, 3740, 8580}, {1376, 4640, 165}, {1376, 5574, 19605}, {1376, 16112, 17668}, {1376, 30624, 5574}, {1400, 2285, 57}, {1400, 5279, 3509}, {1400, 5749, 17754}, {1436, 1604, 32625}, {1445, 3305, 6666}, {1445, 8545, 7}, {1449, 3247, 1}, {1449, 16670, 6}, {1449, 16676, 3247}, {1621, 3681, 3870}, {1621, 3870, 10389}, {1621, 6605, 6602}, {1652, 1653, 57}, {1654, 3661, 17270}, {1654, 4473, 17280}, {1654, 17280, 3661}, {1654, 17339, 17286}, {1697, 10384, 390}, {1698, 9612, 442}, {1707, 5268, 171}, {1723, 1728, 1713}, {1723, 1743, 16572}, {1726, 21361, 1763}, {1728, 15298, 5728}, {1731, 4266, 2082}, {1743, 2324, 2323}, {1743, 3731, 1}, {1743, 3973, 44}, {1743, 16673, 16667}, {1743, 16814, 16676}, {1743, 17744, 5227}, {1743, 21061, 21384}, {1750, 30326, 5927}, {1757, 20372, 24727}, {1759, 16549, 46}, {1778, 2303, 58}, {1826, 26063, 5587}, {1864, 3683, 13615}, {1903, 2182, 5776}, {1903, 7367, 282}, {2082, 3692, 3169}, {2161, 4370, 16561}, {2170, 4390, 3872}, {2170, 21809, 17452}, {2220, 5301, 7031}, {2223, 20992, 16688}, {2238, 2276, 43}, {2245, 17369, 16549}, {2246, 14439, 100}, {2257, 5227, 6762}, {2264, 3965, 3684}, {2268, 3217, 41}, {2268, 21033, 78}, {2269, 2347, 4266}, {2269, 3691, 3686}, {2270, 2345, 1706}, {2278, 3204, 2174}, {2280, 3930, 3870}, {2284, 5701, 1}, {2285, 28070, 27382}, {2287, 27396, 78}, {2310, 4319, 4907}, {2318, 14547, 3190}, {2321, 2325, 346}, {2321, 3169, 4050}, {2321, 3686, 8}, {2321, 3707, 3686}, {2321, 4266, 3169}, {2321, 5802, 12625}, {2324, 9119, 11523}, {2325, 3686, 2321}, {2325, 3707, 8}, {2325, 4266, 3208}, {2329, 3061, 1}, {2345, 5819, 2550}, {2345, 6554, 281}, {2347, 3161, 3208}, {2347, 3691, 391}, {2348, 3693, 3684}, {2478, 6734, 9581}, {2886, 24703, 1699}, {2949, 5715, 5709}, {2975, 19861, 1420}, {3008, 3663, 4000}, {3009, 22343, 7032}, {3056, 4517, 3688}, {3057, 4853, 3680}, {3057, 4875, 4051}, {3059, 15837, 6600}, {3062, 8580, 15587}, {3068, 6351, 5393}, {3069, 6352, 5405}, {3161, 3686, 4873}, {3161, 3707, 4007}, {3174, 6600, 3158}, {3218, 3306, 57}, {3219, 3305, 57}, {3219, 7308, 3928}, {3219, 17260, 16574}, {3219, 27065, 2}, {3230, 16975, 1}, {3230, 20228, 21769}, {3247, 16670, 1449}, {3247, 16676, 37}, {3271, 3688, 3056}, {3271, 7064, 3688}, {3287, 3709, 3737}, {3294, 16552, 1}, {3294, 21061, 37}, {3299, 3301, 16473}, {3305, 3929, 5437}, {3333, 3646, 1125}, {3419, 11113, 3586}, {3436, 24987, 9578}, {3452, 5325, 5745}, {3452, 5745, 2}, {3452, 20258, 21246}, {3487, 16845, 1125}, {3496, 3501, 40}, {3509, 17754, 57}, {3586, 3679, 3419}, {3589, 4364, 4657}, {3589, 4657, 29598}, {3596, 17787, 4494}, {3618, 17321, 17023}, {3624, 6763, 3338}, {3633, 4898, 17388}, {3661, 17280, 17286}, {3661, 17331, 1654}, {3661, 17339, 17280}, {3662, 6646, 17274}, {3662, 17333, 6646}, {3662, 17338, 2}, {3664, 25072, 29571}, {3664, 29571, 4648}, {3666, 4383, 2999}, {3672, 5222, 3946}, {3678, 5248, 3811}, {3683, 3715, 200}, {3686, 3707, 391}, {3688, 7064, 4517}, {3689, 3711, 200}, {3693, 3965, 3694}, {3700, 9404, 1021}, {3713, 3965, 200}, {3717, 3883, 8}, {3723, 16666, 16884}, {3723, 16671, 16666}, {3723, 16884, 1}, {3729, 4384, 75}, {3729, 25728, 190}, {3731, 3973, 6}, {3731, 16667, 16673}, {3731, 16885, 16670}, {3739, 4363, 25590}, {3739, 17259, 16832}, {3739, 17351, 4363}, {3740, 4640, 1376}, {3740, 18227, 18236}, {3758, 4687, 86}, {3759, 4360, 16834}, {3759, 4664, 4360}, {3763, 17253, 17237}, {3782, 24789, 23681}, {3834, 17345, 7232}, {3869, 5260, 19860}, {3869, 19860, 3340}, {3872, 3877, 7962}, {3873, 5284, 4666}, {3876, 25082, 33299}, {3877, 4390, 4919}, {3883, 27549, 4901}, {3911, 5316, 2}, {3912, 4416, 69}, {3912, 25101, 344}, {3913, 4662, 4882}, {3927, 11108, 942}, {3928, 5437, 57}, {3929, 7308, 57}, {3940, 16418, 24929}, {3943, 17362, 17299}, {3944, 33138, 17064}, {3951, 5047, 11518}, {3961, 8616, 3749}, {3970, 16783, 1}, {3970, 17746, 5904}, {3973, 16673, 16669}, {3973, 16814, 3247}, {3975, 17787, 3596}, {3995, 19742, 3187}, {4000, 4419, 3663}, {4007, 4034, 8}, {4007, 4873, 2321}, {4029, 4700, 145}, {4030, 4126, 30615}, {4034, 4873, 4007}, {4042, 4387, 3706}, {4053, 21873, 24048}, {4067, 30143, 12559}, {4271, 17340, 1018}, {4313, 20007, 12437}, {4314, 6743, 3189}, {4357, 17353, 2}, {4361, 17348, 16833}, {4363, 17259, 3739}, {4370, 17330, 17281}, {4384, 25728, 3729}, {4386, 17735, 3550}, {4389, 16706, 17304}, {4389, 17352, 16706}, {4393, 4704, 17319}, {4416, 25101, 3912}, {4422, 4643, 17284}, {4422, 17332, 141}, {4445, 17269, 17229}, {4452, 24599, 4402}, {4473, 17280, 17339}, {4473, 17331, 17286}, {4488, 31995, 4454}, {4513, 4875, 4853}, {4520, 4875, 3057}, {4557, 8053, 15624}, {4640, 15587, 11495}, {4643, 17279, 141}, {4644, 4648, 3664}, {4670, 4698, 15668}, {4675, 17365, 4888}, {4681, 4852, 17318}, {4690, 17229, 4445}, {4708, 17385, 17327}, {4741, 17232, 17288}, {4851, 17243, 29573}, {4853, 4936, 4513}, {4859, 4862, 1086}, {4859, 31183, 17278}, {4862, 31183, 4859}, {4866, 4882, 4662}, {4999, 25681, 3624}, {5044, 5779, 5785}, {5044, 31424, 5438}, {5044, 31445, 3}, {5219, 31142, 908}, {5220, 15254, 1}, {5224, 17289, 17308}, {5224, 17354, 17289}, {5239, 5240, 1}, {5242, 5243, 2}, {5245, 5246, 8}, {5249, 5905, 4654}, {5249, 17781, 5905}, {5251, 5526, 16788}, {5251, 5692, 1}, {5257, 5746, 25525}, {5257, 5750, 2}, {5259, 5904, 1}, {5259, 17745, 16783}, {5269, 7322, 612}, {5273, 18228, 2}, {5279, 5749, 2285}, {5296, 5749, 2}, {5316, 5744, 31190}, {5325, 5745, 5273}, {5436, 11523, 1}, {5440, 16370, 30282}, {5506, 6763, 3624}, {5584, 12688, 12565}, {5686, 5838, 3686}, {5692, 18397, 72}, {5705, 31446, 5791}, {5728, 5729, 10398}, {5729, 15650, 5220}, {5732, 5785, 5784}, {5732, 21153, 3}, {5735, 5833, 5832}, {5739, 17776, 306}, {5742, 5830, 5831}, {5745, 18228, 30827}, {5758, 6846, 946}, {5759, 5817, 4}, {5766, 5809, 390}, {5766, 5825, 5809}, {5779, 31658, 5732}, {5785, 31424, 5732}, {5785, 31658, 5438}, {5795, 5837, 8}, {5809, 5838, 5802}, {5811, 6908, 6260}, {5904, 10399, 14054}, {5927, 7580, 1750}, {6172, 18230, 7}, {6173, 20195, 142}, {6191, 6192, 40}, {6203, 6204, 57}, {6210, 6211, 40}, {6211, 7609, 6210}, {6212, 6213, 40}, {6260, 6684, 6908}, {6601, 24389, 24392}, {6646, 17338, 17282}, {6687, 17235, 17356}, {6986, 12528, 10884}, {7001, 7010, 188}, {7026, 7043, 3679}, {7081, 20665, 2319}, {7090, 14121, 10}, {7174, 7290, 1}, {7174, 15601, 7290}, {7232, 17265, 3834}, {7290, 15601, 238}, {8165, 18231, 9780}, {8232, 12848, 7}, {8257, 8545, 6173}, {8580, 30393, 3740}, {9330, 17126, 5297}, {9776, 9965, 553}, {10177, 15185, 5572}, {10396, 31435, 5436}, {10398, 15299, 10396}, {10436, 16574, 57}, {10456, 18229, 10472}, {10857, 30304, 10167}, {11106, 20007, 4313}, {11679, 30568, 312}, {11683, 26671, 16609}, {11752, 11789, 3576}, {12572, 17355, 8804}, {13405, 21060, 25568}, {14100, 15837, 55}, {14829, 18743, 30567}, {15296, 15297, 1001}, {15298, 15299, 1}, {15492, 16814, 6}, {16482, 23343, 1}, {16517, 16970, 1}, {16519, 16974, 1}, {16547, 16548, 19}, {16566, 20602, 1760}, {16572, 17742, 6762}, {16666, 16671, 6}, {16666, 16672, 1}, {16667, 16673, 1}, {16667, 16675, 3247}, {16668, 16674, 1}, {16669, 16673, 1449}, {16669, 16675, 1}, {16669, 16777, 16667}, {16669, 16814, 16675}, {16670, 16676, 1}, {16671, 16677, 1}, {16672, 16677, 37}, {16672, 16884, 3723}, {16673, 16777, 3247}, {16675, 16777, 37}, {16675, 16885, 16669}, {16677, 16884, 16672}, {16706, 17258, 4389}, {16713, 17183, 17197}, {16779, 16973, 1449}, {16788, 21078, 3553}, {16814, 16885, 1}, {16815, 17116, 4699}, {16816, 25269, 1278}, {16826, 17120, 17379}, {16832, 25590, 3739}, {16833, 17151, 4361}, {16969, 17448, 1}, {17121, 17319, 4393}, {17227, 17329, 17273}, {17227, 17341, 17283}, {17228, 17328, 17271}, {17228, 17342, 17285}, {17230, 17343, 17287}, {17231, 17344, 599}, {17233, 17346, 319}, {17234, 17347, 320}, {17235, 17356, 17290}, {17237, 17357, 3763}, {17238, 17358, 17292}, {17239, 17359, 17293}, {17240, 17360, 17295}, {17241, 17361, 17297}, {17242, 17363, 6542}, {17244, 17364, 17300}, {17245, 17365, 4675}, {17246, 17366, 17301}, {17247, 17367, 17302}, {17248, 17368, 2}, {17249, 17370, 17305}, {17250, 17371, 17307}, {17251, 17293, 17239}, {17252, 17292, 17238}, {17254, 17291, 17236}, {17255, 17290, 17235}, {17256, 17289, 5224}, {17256, 17354, 17308}, {17257, 17353, 17306}, {17257, 26685, 2}, {17257, 28287, 1423}, {17257, 29497, 29740}, {17258, 17352, 17304}, {17259, 17351, 25590}, {17260, 17350, 10436}, {17261, 17349, 3875}, {17262, 17348, 17151}, {17263, 17347, 17298}, {17264, 17346, 17294}, {17266, 17288, 17232}, {17268, 17287, 17230}, {17270, 17286, 3661}, {17271, 17285, 17228}, {17272, 17284, 141}, {17273, 17283, 17227}, {17274, 17282, 3662}, {17275, 17281, 594}, {17276, 17278, 1086}, {17276, 17337, 4859}, {17277, 17336, 3729}, {17277, 25728, 4659}, {17278, 17334, 4862}, {17278, 17337, 31183}, {17279, 17332, 17272}, {17280, 17331, 17270}, {17281, 17330, 3679}, {17299, 17362, 3632}, {17300, 20072, 17364}, {17315, 17377, 29605}, {17322, 17381, 29603}, {17324, 29630, 17383}, {17328, 17342, 17228}, {17329, 17341, 17227}, {17330, 17340, 594}, {17331, 17339, 3661}, {17333, 17338, 3662}, {17334, 17337, 1086}, {17335, 17336, 75}, {17353, 29497, 1423}, {17375, 29572, 17312}, {17379, 27268, 16826}, {17484, 31019, 31164}, {18228, 26059, 21246}, {18249, 18250, 10}, {20090, 29569, 17391}, {20196, 31231, 2}, {20683, 21746, 3779}, {21296, 29627, 4869}, {21616, 26363, 8227}, {21811, 33299, 27396}, {24152, 24153, 200}, {24313, 24314, 8}, {24477, 26105, 11019}, {24697, 33159, 32784}, {24909, 24952, 2}, {25019, 28739, 18634}, {25447, 25651, 2}, {25525, 28609, 226}, {25760, 33115, 29857}, {25842, 25856, 25860}, {26059, 27420, 10436}, {26885, 26890, 184}, {27254, 29967, 5219}, {27509, 28731, 63}, {27819, 27834, 19604}, {28606, 32911, 5256}, {29382, 29492, 3662}, {29395, 29423, 29511}, {29492, 29698, 17274}, {29509, 29541, 29396}, {30324, 30325, 226}, {30327, 30328, 226}, {30412, 30413, 2}, {30414, 30415, 2}, {30556, 30557, 1}, {30557, 31438, 1449}, {30557, 31453, 18991}, {31561, 31562, 4}, {31594, 31595, 10}, {32008, 32024, 85}, {32008, 32100, 30625}, {32088, 32100, 85}, {32555, 32556, 3}, {32784, 33159, 1698}, {32864, 32915, 17156}, {32917, 32931, 29828}, {33076, 33165, 3679}

Notes

Let A' be the orthocorrespondent of the A-excenter, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₉. (Randy Hutson, November 18, 2015)

Let H_aH_bH_c be the orthic triangle. Let L_a be the antiorthic axis of AH_bH_c, and define L_b and L_c cyclically. Let A″ = L_b∩L_c. B″ = L_c∩L_a, C″ = L_a∩L_b. Triangle A″B″C″ is inversely similar to ABC, with similitude center X₉. (Randy Hutson, November 18, 2015)

Let E be the locus of the trilinear pole of a line that passes through X₁. The center of E is X₉. Moreover, E passes through the points X₁₀₀, X₆₅₈, X₆₆₂, X₇₉₉, X₁₈₂₁, X₂₅₈₀, X₂₅₈₁ and the bicentric pairs PU(34), PU(75), PU(77), PU(79). Also, E is a circumellipse of ABC and an inellipse of the excentral triangle. (Randy Hutson, February 10, 2016)

The locus E also passes through the vertices of Gemini triangle 2. (Randy Hutson, November 30, 2018)

Let A' be the intersection of the tangents to the A-excircle at the intercepts with the circumcircle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₉. (Randy Hutson, December 2, 2017)

Let A' be the perspector of the A-mixtilinear excircle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₉. (Randy Hutson, December 2, 2017)

Let E be the circumellipse of T = ABC with center X₉. Then ABC is a billiard orbit of E (3-periodic). If we fix E in the plane, all its triangular orbits (a set of “rotating” triangles T) have the same X₉. Note that X₉ is the point of concurrence of lines drawn from each excenter to the midpoint of the corresponding side of T. (Dan Reznik, June 30, 2019) See [1] Triangular Orbits in Elliptic Billiards: the Mittenpunkt X₉ is stationary at the origin and [2] Triangular Orbits in Elliptic Billiards: the Mittenpunkt X₉ is stationary at the origin.

NEW in 2020: A particularly fine article is recommended: ‘ Can the Ellliptic Billiard Still Surprise Us? ‘, by Dan Reznik, Ronaldo Garcia, and Jair Koiller , in Mathematical Intelligencer 42 (2020) 6-17. An online pdf is available: Click here.

Notes from Peter Moses (July 1, 2019) about the circumellipse E with center X₉:

• E passes through the vertices of these triangles: ABC; Honsberger (see X₇₆₇₀); Inner Conway (see X₁₁₆₇₇); Gemini 2, Gemini 30.

• E passes through the point X_i for these i:

• 88, 100, 162, 190, 651, 653, 655, 658, 660, 662, 673, 771, 799,

• 823, 897, 1156, 1492, 1821, 2349, 2580, 2581, 3257, 4598, 4599

• 4604, 4606, 4607, 8052, 20332, 23707, 24624, 27834, 32680.

• The perspector of E is X₁; the major axis of E passes through X_i for these i: 9, 2590, 3307, 24646, and the minor axis, for these i: 9, 2591, 3309, 24647. The ellipse E is the isogonal conjugate of the antiorthic axis.

• Let g = length of semi-major axis of E; then g = Sqrt[(2 R (r + R + \(\sqrt{R(R - 2 r)}\)) s^2)/(r + 4 R)^2].

• Let h = length of semi-minor axis of E: then h = Sqrt[(2 R (r + R - \(\sqrt{R(R - 2 r)}\)) s^2)/(r + 4 R)^2].

• h/g = \(\sqrt{(OI - R) (OI + 3 R) / ((OI - 3 R) (OI + R))}\) = Sqrt[(r + R - \(\sqrt{R (R - 2 r)}\)) / (r + R + \(\sqrt{R (R - 2 r)}\))].

• eccentricity of E: 2 \(\sqrt{OI R / ((3 R - OI) (OI + R))}\).

• If F is a focus of E, then |FX₉|^2 = 4 s^2 R \(\sqrt{R (R - 2 r)}\) / (r + 4 R)^2/.

• The axes of E are the asymptotes of the Feuerbach hyperbola.

• The area of E is π R S /(r (r + 4 R))^(3/2) = 4 π a b c / ( 2 b c + 2 c a + 2 a b - a^2 - b^2 - c^2 )^(3/2) * area(ABC).

The triangle tangent to the vertices of E is the excentral triangle.

Let A'B'C' be the intouch triangle of the anticomplementary triangle of ABC. The ellipse E passes through A', B', C'. See Three orbits in elliptic billiard. (Dan Reznik, July 1, 2019)

The circumellipse with center X₉ meets the circumhyperbola with center X₁₁ (i.e., the Feuerbach hyperbola) in X₁₁₅₆. (Dan Reznik, January 20, 2020.)

Each of the following cubics passes through the four foci (two real and two imaginary) of the ellipse E: K040, K351, K352, K710, K716, K1060. The two real foci are a pair of isogonal conjugates, and likewise for the two imaginary foci. Moreover, if \(p : q : r\) is on the circumcircle, then \(p/a : q/b : r/c\) is on E. (Peter Moses, July 2 and 3, 2019)

The ellipse E is the isogonal conjugate of the antiorthic axis, X₄₄X₅₁₃, with barycentric equation \(x + y + z = 0\), and E is the isotomic conjugate of the X₅₁₄X₆₆₁, with barycentric equation a x + b y + c z = 0. This line, denoted by L(31), is the perspectrix of the anticomplementary triangle and the inner Conway triangle (which is the intouch triangle of the anticomplementary triangle). Points lying on X₅₁₄X₆₆₁ include X_i for i = 514, 661, 693, 857, 908, 914, 1577, 1959, 2084, 2582, 2583, 3239, 3250, 3762, 3766, 3835, 3904, 3912, 3936, 3948, 4129, 4358, 4391, 4462, 4468, 4486, 4728, 4766, 4776, 4789, 4791, 4801, 4823, 4978, 5074, 5179, 6332, 6381, 6590, 8045, 14206, 14207, 14208, 14209, 14210, 14281, 14349, 14350, 14963, 18669, 18715, 24018, 30565, 30566, 30804, 30806, 32679. (Peter Moses, July 2 and 3, 2019)

For a dynamic graphic with many options, based on a triangle inscribed in the circumellipse with center X₉, see Elliptic Billiard: Loci of Triangular Centers. (Dan Reznik, February 4, 2020)