X₃₉₁₂

X₃₇₄₇com(INTOUCH TRIANGLE)

X₃₉₁₂ = complement of X₂₃₉

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

X₃₉₁₂ (X₃₇₄₇com(INTOUCH TRIANGLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Circle: Incircle (incircle) — Inscribed circle tangent to sidelines at T_a, T_b, T_c; centered at X₁ INCENTER

• Central Lines: {1, 2}, {9, 69}

• Related Centers: X₁ (INCENTER), X₂ (CENTROID), X₉ (MITTENPUNKT), X₆₉ (SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE)

Barycentrics: \(b^{2} + c^{2} - ab - ac : :\)

Search Value (6, 9, 13): \(12.578331791095\)

▸ Linear Combinations1 combination

• X₃₉₁₂ = \(3\) X₂ + X₆₅₄₂, \(3\) X₂ - \(5\) X₁₇₂₆₆, \(9\) X₂ - X₂₀₀₁₆, X₇₅ - \(3\) X₂₇₄₈₇, X₁₉₀ - \(3\) X₁₇₂₆₄, X₁₉₀ + \(3\) X₁₇₂₉₇, X₂₃₉ - \(5\) X₁₇₂₆₆, X₂₃₉ + \(3\) X₁₇₃₁₀, \(3\) X₂₃₉ - X₂₀₀₁₆, X₃₂₀ + \(2\) X₂₃₂₅, \(2\) X₃₂₀ + X₄₄₈₀, X₃₂₀ + \(3\) X₁₇₂₆₄, X₃₂₀ - \(3\) X₁₇₂₉₇, X₁₂₆₆ - \(4\) X₃₈₃₄, X₁₂₆₆ + \(2\) X₃₉₄₃, \(3\) X₁₇₃₈ - \(4\) X₂₅₃₅₁, \(4\) X₂₃₂₅ - X₄₄₈₀, \(2\) X₂₃₂₅ - \(3\) X₁₇₂₆₄, \(2\) X₂₃₂₅ + \(3\) X₁₇₂₉₇, \(2\) X₃₀₀₈ + X₆₅₄₂, \(2\) X₃₀₀₈ - \(5\) X₁₇₂₆₆, \(2\) X₃₀₀₈ + \(3\) X₁₇₃₁₀, \(6\) X₃₀₀₈ - X₂₀₀₁₆, X₃₇₁₇ + \(2\) X₄₉₆₆, \(2\) X₃₈₃₄ + X₃₉₄₃, \(3\) X₃₈₃₆ - \(2\) X₂₅₃₅₁, \(2\) X₃₉₃₂ + X₄₆₈₄, \(4\) X₃₉₃₂ - X₄₈₉₉, \(2\) X₄₃₉₅ + X₄₇₂₇, \(2\) X₄₃₉₅ - \(5\) X₃₁₂₄₃, \(2\) X₄₄₂₂ + X₁₇₃₇₄, \(5\) X₄₄₇₃ - X₂₀₀₇₂, X₄₄₈₀ - \(6\) X₁₇₂₆₄, X₄₄₈₀ + \(6\) X₁₇₂₉₇, \(2\) X₄₆₈₄ + X₄₈₉₉, X₄₆₉₃ + \(3\) X₃₁₁₅₁, X₄₇₁₆ - \(5\) X₃₁₂₅₂, X₄₇₂₇ + \(5\) X₃₁₂₄₃, X₄₈₉₉ + \(4\) X₄₉₆₆, \(3\) X₄₉₀₈ + \(2\) X₇₂₃₈, X₄₉₆₉ - \(4\) X₆₆₈₇, \(2\) X₆₅₄₁ + X₂₄₂₃₁, X₆₅₄₂ + \(5\) X₁₇₂₆₆, X₆₅₄₂ - \(3\) X₁₇₃₁₀, \(3\) X₆₅₄₂ + X₂₀₀₁₆, \(2\) X₇₂₃₈ - \(3\) X₃₁₁₃₈, \(4\) X₁₇₀₆₇ - X₁₇₁₆₀, \(4\) X₁₇₀₆₇ - \(5\) X₂₇₁₉₁, \(4\) X₁₇₀₆₇ - \(3\) X₃₇₇₅₆, X₁₇₁₆₀ - \(5\) X₂₇₁₉₁, X₁₇₁₆₀ - \(3\) X₃₇₇₅₆, \(5\) X₁₇₂₆₆ + \(3\) X₁₇₃₁₀, \(15\) X₁₇₂₆₆ - X₂₀₀₁₆, \(9\) X₁₇₃₁₀ + X₂₀₀₁₆, X₂₄₇₁₅ - \(3\) X₃₁₁₅₁, \(5\) X₂₇₁₉₁ - \(3\) X₃₇₇₅₆

X₃₉₁₂ lies on the cubics K766 and K868.

▸ Incident Central Lines622 lines

{1, 2}, {5, 29331}, {6, 3879}, {7, 346}, {9, 69}, {11, 20486}, {21, 24632}, {35, 21511}, {36, 21495}, {37, 141}, {39, 37596}, {40, 36698}, {44, 524}, {45, 599}, {55, 11343}, {56, 21477}, {57, 345}, {58, 16050}, {63, 3730}, {71, 16574}, {72, 30810}, {73, 28777}, {75, 142}, {76, 85}, {77, 28739}, {81, 5280}, {86, 4758}, {88, 34892}, {99, 35163}, {100, 2725}, {101, 2862}, {115, 20337}, {144, 3161}, {171, 33158}, {172, 5337}, {190, 320}, {192, 3662}, {193, 1743}, {194, 24215}, {213, 37676}, {220, 23151}, {238, 5847}, {241, 3693}, {244, 32848}, {281, 27384}, {297, 1785}, {302, 5242}, {303, 5243}, {307, 27396}, {313, 18040}, {318, 37448}, {319, 3686}, {321, 1930}, {322, 20946}, {325, 26012}, {326, 28753}, {329, 15490}, {333, 1174}, {334, 350}, {335, 726}, {354, 3703}, {391, 18230}, {440, 10381}, {469, 5342}, {480, 30620}, {487, 32556}, {488, 32555}, {514, 661}, {515, 6996}, {516, 3685}, {518, 3717}, {522, 30188}, {528, 4702}, {536, 1086}, {537, 4439}, {545, 4908}, {553, 32007}, {573, 21371}, {594, 3739}, {597, 16666}, {637, 31562}, {638, 31561}, {664, 35158}, {666, 2338}, {668, 3975}, {672, 18206}, {673, 5853}, {674, 4553}, {740, 1738}, {748, 32852}, {750, 33156}, {752, 4432}, {756, 33081}, {760, 20715}, {846, 33085}, {894, 3664}, {903, 28301}, {940, 17750}, {942, 3695}, {944, 7397}, {946, 7377}, {948, 9312}, {950, 7270}, {952, 19512}, {960, 30847}, {966, 17270}, {968, 26034}, {980, 2276}, {982, 12782}, {984, 4078}, {988, 16043}, {991, 12618}, {993, 16367}, {999, 21526}, {1001, 3416}, {1009, 37609}, {1015, 35126}, {1016, 1429}, {1018, 20367}, {1043, 16054}, {1089, 13407}, {1100, 3589}, {1121, 4555}, {1146, 34852}, {1150, 16788}, {1155, 3712}, {1211, 6537}, {1213, 4698}, {1214, 18639}, {1229, 1441}, {1265, 11523}, {1269, 4043}, {1270, 30412}, {1271, 30413}, {1278, 4072}, {1279, 5846}, {1329, 30812}, {1330, 12572}, {1332, 2323}, {1334, 17137}, {1375, 5440}, {1376, 16412}, {1388, 31230}, {1432, 32017}, {1438, 3684}, {1442, 28780}, {1445, 3692}, {1449, 3618}, {1453, 13742}, {1468, 33819}, {1475, 27109}, {1500, 3666}, {1574, 31198}, {1621, 33078}, {1654, 17260}, {1708, 3719}, {1724, 11342}, {1742, 21629}, {1757, 34379}, {1763, 8897}, {1781, 27059}, {1818, 1861}, {1826, 18065}, {1836, 4387}, {1848, 17442}, {1921, 20496}, {1931, 6629}, {1948, 6335}, {1962, 32781}, {1978, 18891}, {1992, 16670}, {1997, 30827}, {2183, 3882}, {2223, 4447}, {2227, 23462}, {2228, 3122}, {2238, 16782}, {2239, 3747}, {2243, 4760}, {2294, 18697}, {2324, 27509}, {2329, 5745}, {2345, 4470}, {2550, 3886}, {2796, 24692}, {2887, 24210}, {2895, 27065}, {2901, 19791}, {2951, 9801}, {3061, 3452}, {3120, 31041}, {3175, 3782}, {3208, 21281}, {3210, 24177}, {3218, 3977}, {3219, 4001}, {3243, 4901}, {3246, 28538}, {3247, 3619}, {3262, 4858}, {3263, 3930}, {3264, 4033}, {3271, 9025}, {3295, 21514}, {3303, 21496}, {3304, 21519}, {3305, 5739}, {3306, 17740}, {3314, 24241}, {3454, 26601}, {3475, 3974}, {3596, 17786}, {3601, 24609}, {3620, 3731}, {3629, 16669}, {3630, 15492}, {3631, 16814}, {3644, 4398}, {3668, 25252}, {3672, 17304}, {3673, 20173}, {3675, 22116}, {3689, 26007}, {3694, 16608}, {3696, 3826}, {3701, 33839}, {3704, 3812}, {3706, 3925}, {3707, 17335}, {3710, 3868}, {3713, 25878}, {3714, 25466}, {3723, 17045}, {3746, 21516}, {3748, 4030}, {3750, 33079}, {3752, 20255}, {3755, 4429}, {3758, 4667}, {3759, 17341}, {3760, 21073}, {3763, 4657}, {3772, 3905}, {3773, 24325}, {3775, 3842}, {3778, 21330}, {3779, 35892}, {3781, 10477}, {3790, 5542}, {3814, 26019}, {3821, 3993}, {3823, 28581}, {3844, 4026}, {3873, 32862}, {3875, 4000}, {3876, 4101}, {3888, 29353}, {3914, 25957}, {3933, 25066}, {3944, 4138}, {3945, 5749}, {3946, 4360}, {3963, 20891}, {3965, 25067}, {3966, 4423}, {3969, 4359}, {3971, 33064}, {3973, 20080}, {3986, 17238}, {3991, 37597}, {3994, 32856}, {3995, 17184}, {3998, 18636}, {4007, 20195}, {4011, 32946}, {4019, 25065}, {4021, 17291}, {4022, 21035}, {4029, 4389}, {4037, 24081}, {4038, 32780}, {4053, 22047}, {4054, 4671}, {4058, 4699}, {4060, 5564}, {4070, 24685}, {4080, 6549}, {4082, 32937}, {4085, 31342}, {4095, 17062}, {4098, 4704}, {4102, 32015}, {4104, 33084}, {4109, 18140}, {4110, 30090}, {4144, 4465}, {4150, 18044}, {4273, 30906}, {4292, 7283}, {4297, 37416}, {4312, 24280}, {4314, 17691}, {4339, 33198}, {4361, 17265}, {4363, 4675}, {4366, 17766}, {4370, 4715}, {4383, 16502}, {4385, 21620}, {4395, 4727}, {4409, 28322}, {4415, 35652}, {4419, 17274}, {4434, 26629}, {4440, 4887}, {4441, 30949}, {4445, 17259}, {4461, 31995}, {4464, 4852}, {4473, 20072}, {4475, 20590}, {4488, 20059}, {4513, 5228}, {4515, 21258}, {4552, 22464}, {4642, 26562}, {4649, 33159}, {4656, 27184}, {4659, 4873}, {4665, 4688}, {4670, 17359}, {4681, 17235}, {4686, 7263}, {4687, 5224}, {4690, 17330}, {4693, 24715}, {4700, 16786}, {4708, 4755}, {4716, 31252}, {4719, 25914}, {4725, 4969}, {4741, 17333}, {4767, 36240}, {4779, 30332}, {4780, 27480}, {4855, 24580}, {4856, 17121}, {4859, 17151}, {4864, 9053}, {4884, 21342}, {4890, 25144}, {4936, 20111}, {4968, 17672}, {4970, 24169}, {4974, 17772}, {4975, 30384}, {5010, 21508}, {5015, 17681}, {5081, 26003}, {5174, 37389}, {5204, 16431}, {5217, 16436}, {5219, 28808}, {5223, 27549}, {5226, 31994}, {5227, 26130}, {5232, 5296}, {5233, 5316}, {5247, 33821}, {5266, 7819}, {5273, 37655}, {5278, 16783}, {5279, 16566}, {5284, 33075}, {5295, 8728}, {5299, 32911}, {5435, 32003}, {5461, 30823}, {5563, 21540}, {5587, 36662}, {5590, 6351}, {5591, 6352}, {5603, 7402}, {5687, 37272}, {5691, 7406}, {5695, 5880}, {5717, 13740}, {5718, 30818}, {5814, 11108}, {5839, 37650}, {5977, 16609}, {6174, 32043}, {6185, 14942}, {6329, 16668}, {6510, 36949}, {6547, 16610}, {6550, 24133}, {6557, 10405}, {6631, 37758}, {6646, 17261}, {6651, 17770}, {6656, 13161}, {6767, 21529}, {7155, 27429}, {7232, 17262}, {7237, 17470}, {7280, 21537}, {7291, 16550}, {7308, 14555}, {7373, 21542}, {7384, 19925}, {7490, 7718}, {8055, 30695}, {8069, 37344}, {8162, 21528}, {8193, 11350}, {8287, 21237}, {8359, 37599}, {8362, 37592}, {8369, 37589}, {9028, 16560}, {9294, 21191}, {9310, 25940}, {9441, 28849}, {10026, 20529}, {10164, 28909}, {10445, 10446}, {10452, 21061}, {10481, 25242}, {11329, 25440}, {11340, 37557}, {11353, 26687}, {11712, 26231}, {11997, 18252}, {12263, 30982}, {12410, 37269}, {12699, 36731}, {13006, 36212}, {13466, 16594}, {13476, 21865}, {14001, 37552}, {14839, 20358}, {14943, 33675}, {15488, 19542}, {15507, 33845}, {15668, 17293}, {16060, 37573}, {16061, 37607}, {16484, 33076}, {16524, 37673}, {16578, 37796}, {16602, 21868}, {16603, 17046}, {16671, 32455}, {16672, 17325}, {16675, 17253}, {16676, 21356}, {16704, 36954}, {16738, 27032}, {16779, 17338}, {16781, 37679}, {16784, 37680}, {16785, 32779}, {16887, 25092}, {17050, 17143}, {17053, 27633}, {17056, 21024}, {17063, 32855}, {17066, 21958}, {17067, 17133}, {17077, 27514}, {17116, 26806}, {17120, 20090}, {17122, 33160}, {17123, 32861}, {17126, 35263}, {17144, 20257}, {17178, 27073}, {17197, 30939}, {17256, 17271}, {17258, 17273}, {17290, 17301}, {17305, 17320}, {17307, 17322}, {17331, 17343}, {17334, 17345}, {17336, 17347}, {17337, 17348}, {17339, 17350}, {17340, 17351}, {17358, 17368}, {17370, 17380}, {17371, 17381}, {17382, 17395}, {17383, 17396}, {17385, 17398}, {17463, 35552}, {17484, 32106}, {17490, 24175}, {17495, 24191}, {17499, 17778}, {17592, 33174}, {17647, 37233}, {17670, 24178}, {17671, 21075}, {17682, 24588}, {17698, 37594}, {17717, 30869}, {17719, 30837}, {17720, 24281}, {17754, 30962}, {17781, 32859}, {17787, 30097}, {17790, 21091}, {17791, 18151}, {17792, 21746}, {17798, 21989}, {17861, 20171}, {17862, 20237}, {17977, 26884}, {18073, 21094}, {18135, 30961}, {18149, 19567}, {18150, 22031}, {18480, 36728}, {18692, 26163}, {18714, 20932}, {18904, 20363}, {19582, 27129}, {19623, 25536}, {19786, 34064}, {20078, 25734}, {20132, 24295}, {20172, 32941}, {20262, 28778}, {20333, 20530}, {20336, 22021}, {20343, 20688}, {20541, 25383}, {20549, 21830}, {20610, 20785}, {20881, 30379}, {20895, 20905}, {20911, 21808}, {20912, 21417}, {20927, 20930}, {20942, 20943}, {20956, 21591}, {21010, 21985}, {21025, 25102}, {21026, 33136}, {21060, 27538}, {21069, 24220}, {21072, 30007}, {21074, 29965}, {21076, 25660}, {21101, 33931}, {21257, 25140}, {21331, 35101}, {21454, 32098}, {21810, 24090}, {22016, 22019}, {22020, 22028}, {22267, 37574}, {23058, 28827}, {23536, 33840}, {23681, 30699}, {23682, 30969}, {23829, 24290}, {24050, 27569}, {24214, 25264}, {24479, 36800}, {24629, 33089}, {24635, 25082}, {24803, 35160}, {25019, 26540}, {25099, 26543}, {25245, 26573}, {25378, 25760}, {25681, 30826}, {25772, 25856}, {25959, 33134}, {25961, 32860}, {26580, 31017}, {26738, 30629}, {26764, 27017}, {27003, 33168}, {27186, 28605}, {27254, 27267}, {27269, 27288}, {27431, 27497}, {27495, 31323}, {27523, 36854}, {27751, 35121}, {28526, 32857}, {28603, 33920}, {28606, 33172}, {30808, 36205}, {30824, 34362}, {30856, 35141}, {30857, 36236}, {31053, 31060}, {31225, 33298}, {32024, 33066}, {32913, 33164}, {32919, 33115}, {32925, 33069}, {32926, 33124}, {32928, 33123}, {32930, 32949}, {32936, 33067}, {32942, 33073}, {32943, 33072}, {33942, 34847}, {34234, 34932}, {35075, 35122}, {35085, 35134}, {35153, 35177}, {35168, 36915}, {35595, 37656}, {36605, 38255}, {37176, 37554}

▸ Functional Relations27 relations

• X₃₉₁₂ = midpoint of X_i and X_j for these {i,j}: {2, 17310}, {44, 17374}, {190, 320}, {238, 32846}, {239, 6542}, {335, 3797}, {350, 4562}, {1086, 3943}, {3685, 4645}, {3717, 4684}, {3932, 4966}, {4693, 24715}, {4908, 31138}, {10026, 35080}, {17264, 17297}

• X₃₉₁₂ = reflection of X_i in X_j for these {i,j}: {44, 4422}, {190, 2325}, {239, 3008}, {1086, 3834}, {1266, 1086}, {1738, 3836}, {3717, 3932}, {4440, 4887}, {4480, 190}, {4684, 4966}, {4899, 3717}, {4974, 31289}, {9436, 35094}, {35119, 20530}

• X₃₉₁₂ = isogonal conjugate of X₁₄₃₈

• X₃₉₁₂ = isotomic conjugate of X₆₇₃

• X₃₉₁₂ = complement of X₂₃₉

• X₃₉₁₂ = anticomplement of X₃₀₀₈

• X₃₉₁₂ = complementary conjugate of X₂₀₃₃₃

• X₃₉₁₂ = antitomic image of X₉₄₃₆

• X₃₉₁₂ = inverse-in-Steiner-circumellipse of X₈

• X₃₉₁₂ = inverse-in-Steiner-inellipse of X₁₀

• X₃₉₁₂ = medial isogonal conjugate of X₂₀₃₃₃

• X₃₉₁₂ = X_i-anticomplementary conjugate of X_j for these (i,j): {1280, 69}, {1477, 7}, {1810, 4329}, {6078, 513}, {35160, 21285}, {35355, 150}, {36807, 6327}

• X₃₉₁₂ = X_i-complementary conjugate of X_j for these (i,j): {1, 20333}, {2, 20542}, {6, 17793}, {31, 17755}, {100, 27854}, {171, 39080}, {172, 19563}, {213, 35068}, {291, 141}, {292, 10}, {295, 18589}, {334, 626}, {335, 2887}, {649, 38989}, {660, 3835}, {667, 35119}, {692, 27929}, {694, 3846}, {741, 3739}, {813, 513}, {875, 1086}, {876, 116}, {904, 39044}, {1911, 2}, {1922, 37}, {1967, 4357}, {2196, 3}, {2311, 960}, {3252, 120}, {3572, 11}, {4444, 21252}, {4518, 21244}, {4562, 21260}, {4583, 21262}, {4584, 512}, {4639, 23301}, {4876, 1329}, {5378, 27076}, {7077, 3452}, {7122, 5976}, {7233, 17046}, {9468, 1107}, {14598, 39}, {18265, 1212}, {18268, 1125}, {18827, 21240}, {18893, 8265}, {18895, 21235}, {18897, 16584}, {22116, 20540}, {30663, 20541}, {34067, 514}, {35352, 21253}, {37128, 3741}, {37207, 788}

• X₃₉₁₂ = X_i-Ceva conjugate of X_j for these (i,j): {2, 17755}, {75, 4712}, {92, 20431}, {334, 10}, {350, 726}, {3263, 3717}, {3570, 2786}, {4562, 514}, {4876, 17760}, {5378, 4568}, {9436, 4899}, {14942, 3729}, {18025, 8}, {18031, 75}, {18157, 3263}, {30941, 518}, {36807, 2}

• X₃₉₁₂ = X_i-cross conjugate of X_j for these (i,j): {518, 9436}, {672, 1861}, {3675, 23829}, {3693, 3717}, {3930, 518}, {3932, 3263}, {4712, 75}, {4966, 30941}, {16593, 2}, {17464, 1}, {20860, 6}

• X₃₉₁₂ = X_i-isoconjugate of X_j for these (i,j): {1, 1438}, {3, 8751}, {4, 32658}, {6, 105}, {9, 1416}, {19, 36057}, {25, 1814}, {31, 673}, {32, 2481}, {48, 36124}, {55, 1462}, {56, 294}, {57, 2195}, {58, 18785}, {101, 1027}, {109, 1024}, {112, 10099}, {513, 919}, {514, 32666}, {560, 18031}, {604, 14942}, {649, 36086}, {650, 32735}, {651, 884}, {663, 36146}, {666, 667}, {885, 1415}, {927, 3063}, {1015, 5377}, {1106, 6559}, {1292, 2440}, {1333, 13576}, {1397, 36796}, {1407, 28071}, {1492, 29956}, {1911, 6654}, {1973, 31637}, {1980, 36803}, {2175, 34018}, {2223, 6185}, {3290, 15382}, {3309, 32644}, {4724, 36138}, {4762, 32724}, {6169, 9316}, {23696, 32674}

• X₃₉₁₂ = cevapoint of X_i and X_j for these (i,j): {1, 16550}, {2, 20533}, {6, 20871}, {350, 33677}, {518, 3693}, {672, 1818}, {3675, 24290}, {3930, 3932}, {6084, 27918}

• X₃₉₁₂ = crosspoint of X_i and X_j for these (i,j): {2, 335}, {75, 18031}, {85, 35160}, {4600, 4639}, {9436, 10029}, {18157, 30941}

• X₃₉₁₂ = crosssum of X_i and X_j for these (i,j): {6, 1914}, {31, 9454}, {41, 8647}, {1015, 8659}, {1458, 9316}

• X₃₉₁₂ = trilinear pole of line X₉₁₈X₂₂₅₄

• X₃₉₁₂ = crossdifference of PU(48)

• X₃₉₁₂ = barycentric square root of X₄₄₃₇

• X₃₉₁₂ = polar conjugate of X₃₆₁₂₄

• X₃₉₁₂ = pole wrt polar circle of trilinear polar of X₃₆₁₂₄ (line X₁₉X₁₀₂₄)

• barycentric product X_i*X_j for these {i,j}: {1, 3263}, {7, 3717}, {8, 9436}, {10, 30941}, {37, 18157}, {69, 1861}, {75, 518}, {76, 672}, {85, 3693}, {86, 3932}, {92, 25083}, {99, 4088}, {190, 918}, {241, 312}, {264, 1818}, {274, 3930}, {304, 5089}, {305, 2356}, {306, 15149}, {310, 20683}, {313, 3286}, {321, 18206}, {334, 8299}, {335, 17755}, {341, 34855}, {345, 5236}, {350, 22116}, {522, 883}, {561, 2223}, {665, 1978}, {668, 2254}, {673, 4437}, {693, 1026}, {799, 24290}, {926, 4572}, {1025, 4391}, {1268, 4966}, {1458, 3596}, {1502, 9454}, {1876, 3718}, {1921, 3252}, {1928, 9455}, {1969, 20752}, {2283, 35519}, {2284, 3261}, {2340, 6063}, {2414, 4468}, {2481, 4712}, {2991, 20431}, {3161, 10029}, {3262, 36819}, {3264, 34230}, {3675, 7035}, {3952, 23829}, {4238, 14208}, {4373, 4899}, {4447, 7018}, {4684, 5936}, {5383, 23773}, {6184, 18031}, {6385, 39258}, {14439, 20568}, {16593, 36807}, {20504, 35574}, {31637, 34337}

• X₃₉₁₂ = barycentric quotient X_i/X_j for these {i,j}: {1, 105}, {2, 673}, {3, 36057}, {4, 36124}, {6, 1438}, {8, 14942}, {9, 294}, {10, 13576}, {19, 8751}, {37, 18785}, {48, 32658}, {55, 2195}, {56, 1416}, {57, 1462}, {63, 1814}, {69, 31637}, {75, 2481}, {76, 18031}, {85, 34018}, {100, 36086}, {101, 919}, {109, 32735}, {120, 1738}, {190, 666}, {200, 28071}, {239, 6654}, {241, 57}, {312, 36796}, {344, 31638}, {346, 6559}, {513, 1027}, {518, 1}, {521, 23696}, {522, 885}, {536, 36816}, {650, 1024}, {651, 36146}, {656, 10099}, {663, 884}, {664, 927}, {665, 649}, {672, 6}, {673, 6185}, {692, 32666}, {765, 5377}, {883, 664}, {918, 514}, {926, 663}, {1025, 651}, {1026, 100}, {1292, 36041}, {1362, 1458}, {1458, 56}, {1642, 2246}, {1738, 14267}, {1818, 3}, {1861, 4}, {1876, 34}, {1978, 36803}, {2223, 31}, {2254, 513}, {2283, 109}, {2284, 101}, {2340, 55}, {2356, 25}, {2414, 37206}, {2808, 23694}, {3126, 2254}, {3239, 28132}, {3250, 29956}, {3252, 292}, {3263, 75}, {3286, 58}, {3675, 244}, {3693, 9}, {3699, 36802}, {3717, 8}, {3930, 37}, {3932, 10}, {4088, 523}, {4238, 162}, {4447, 171}, {4468, 2402}, {4518, 33676}, {4553, 35333}, {4554, 34085}, {4684, 3616}, {4712, 518}, {4899, 145}, {4925, 3667}, {4966, 1125}, {5089, 19}, {5236, 278}, {5257, 14625}, {6168, 6180}, {6184, 672}, {8299, 238}, {8693, 36138}, {9436, 7}, {9451, 9453}, {9454, 32}, {9455, 560}, {9502, 910}, {10029, 27818}, {14439, 44}, {14626, 2334}, {15149, 27}, {16593, 3008}, {16728, 18206}, {17435, 2170}, {17464, 3290}, {17755, 239}, {17794, 33674}, {18157, 274}, {18206, 81}, {20482, 21956}, {20504, 23770}, {20683, 42}, {20749, 20780}, {20752, 48}, {20778, 7193}, {21959, 21051}, {22116, 291}, {23102, 4712}, {23773, 21138}, {23829, 7192}, {24290, 661}, {25083, 63}, {25302, 34063}, {26706, 36111}, {27919, 4366}, {28850, 14197}, {30941, 86}, {34230, 106}, {34253, 1429}, {34337, 1861}, {34855, 269}, {35293, 1155}, {36101, 9503}, {36819, 104}, {36905, 14189}, {37908, 2299}, {38989, 27846}, {39046, 20470}, {39258, 213}

• X₃₉₁₂ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 2, 17023}, {1, 3679, 36479}, {1, 17284, 2}, {1, 17316, 29574}, {1, 29573, 17316}, {1, 29579, 29596}, {1, 29583, 29601}, {1, 29598, 26626}, {1, 29602, 29585}, {1, 29637, 1125}, {1, 29660, 551}, {1, 29674, 10}, {1, 29960, 30038}, {2, 8, 4384}, {2, 10, 24603}, {2, 145, 5222}, {2, 239, 3008}, {2, 306, 3687}, {2, 3187, 26723}, {2, 3616, 29603}, {2, 3621, 24599}, {2, 3661, 10}, {2, 4393, 17367}, {2, 5308, 16831}, {2, 6542, 239}, {2, 10453, 17026}, {2, 16815, 31211}, {2, 16816, 29628}, {2, 16826, 1125}, {2, 17135, 24592}, {2, 17230, 3661}, {2, 17244, 29571}, {2, 17284, 29596}, {2, 17292, 29604}, {2, 17316, 1}, {2, 17367, 31191}, {2, 20016, 29590}, {2, 20055, 16816}, {2, 26113, 25510}, {2, 26247, 3011}, {2, 26575, 24982}, {2, 26581, 24987}, {2, 26593, 25006}, {2, 26594, 25007}, {2, 26595, 24996}, {2, 26597, 24997}, {2, 26599, 25011}, {2, 26610, 25005}, {2, 26626, 29598}, {2, 26774, 27044}, {2, 26821, 26982}, {2, 29569, 16826}, {2, 29570, 17397}, {2, 29572, 17244}, {2, 29573, 29574}, {2, 29577, 29594}, {2, 29579, 17284}, {2, 29582, 29600}, {2, 29583, 17316}, {2, 29585, 26626}, {2, 29586, 29614}, {2, 29587, 17292}, {2, 29589, 29569}, {2, 29590, 29607}, {2, 29591, 29610}, {2, 29592, 29609}, {2, 29593, 29576}, {2, 29595, 29612}, {2, 29599, 29581}, {2, 29610, 3634}, {2, 29611, 17308}, {2, 29612, 19862}, {2, 29616, 8}, {2, 29618, 3244}, {2, 29621, 5308}, {2, 29624, 3616}, {2, 29625, 3636}, {2, 30967, 4871}, {2, 31027, 3741}, {2, 31028, 3840}, {2, 31038, 11019}, {2, 31093, 3006}, {2, 32858, 306}, {2, 34255, 11679}, {2, 36845, 24600}, {6, 4851, 3879}, {6, 17267, 17279}, {6, 17279, 17353}, {6, 17311, 4851}, {7, 346, 3729}, {7, 4869, 17298}, {8, 29616, 17294}, {8, 29627, 2}, {8, 29966, 30030}, {9, 69, 4416}, {9, 344, 25101}, {9, 17296, 69}, {10, 29571, 2}, {10, 29594, 3661}, {10, 29600, 29571}, {10, 29968, 30063}, {11, 20486, 20544}, {37, 141, 4357}, {37, 17231, 141}, {37, 17237, 4364}, {42, 30821, 2}, {43, 30822, 2}, {45, 599, 4643}, {69, 344, 9}, {75, 142, 24199}, {75, 2321, 4431}, {75, 17233, 2321}, {75, 17234, 142}, {75, 17240, 17233}, {75, 17241, 17234}, {76, 312, 4044}, {81, 33157, 5294}, {86, 17285, 17289}, {86, 17289, 5750}, {141, 4364, 17237}, {141, 17243, 37}, {142, 2321, 75}, {142, 17233, 4431}, {144, 3161, 25728}, {145, 5222, 16834}, {145, 29986, 30036}, {145, 30833, 2}, {190, 17264, 2325}, {190, 17297, 320}, {192, 3662, 3663}, {192, 17232, 3662}, {192, 17242, 3950}, {193, 26685, 1743}, {200, 30813, 2}, {226, 7146, 3674}, {226, 21071, 4044}, {239, 17266, 2}, {239, 17310, 6542}, {239, 29607, 29590}, {241, 3693, 25083}, {306, 25935, 8}, {312, 7146, 17760}, {312, 18134, 226}, {312, 20917, 76}, {319, 17263, 17277}, {319, 17277, 3686}, {320, 2325, 4480}, {320, 17264, 190}, {321, 3970, 22048}, {321, 18139, 5249}, {321, 20913, 20888}, {344, 17296, 4416}, {345, 18141, 57}, {346, 4869, 7}, {594, 3739, 4967}, {594, 17245, 3739}, {894, 17268, 17280}, {894, 17280, 17355}, {894, 17300, 3664}, {894, 17312, 17300}, {1001, 3416, 3883}, {1100, 17357, 3589}, {1125, 29604, 2}, {1229, 1441, 20236}, {1999, 16826, 30168}, {2321, 17234, 24199}, {2345, 4648, 10436}, {3006, 29824, 26015}, {3161, 21296, 144}, {3178, 3831, 5530}, {3218, 32849, 3977}, {3219, 32863, 4001}, {3247, 17306, 17321}, {3262, 37788, 4858}, {3452, 4035, 4417}, {3589, 17390, 1100}, {3616, 29624, 29597}, {3619, 17321, 17306}, {3620, 17257, 17272}, {3631, 17332, 17344}, {3632, 31183, 16833}, {3661, 17230, 29594}, {3661, 17244, 2}, {3661, 29571, 24603}, {3661, 29572, 29571}, {3661, 29576, 29593}, {3661, 29577, 17230}, {3661, 29581, 29576}, {3661, 29582, 17244}, {3662, 17232, 21255}, {3662, 17242, 192}, {3663, 3950, 192}, {3663, 21255, 3662}, {3664, 17355, 894}, {3686, 6666, 17277}, {3687, 29968, 24603}, {3723, 17384, 17045}, {3729, 17298, 7}, {3731, 17272, 17257}, {3739, 17229, 594}, {3758, 17342, 17354}, {3758, 17378, 4667}, {3758, 17387, 17378}, {3759, 17341, 17352}, {3759, 17386, 17377}, {3763, 16777, 4657}, {3834, 3943, 1266}, {3840, 29671, 24239}, {3844, 15569, 4026}, {3875, 17282, 4000}, {3879, 17353, 6}, {3932, 4684, 4899}, {3936, 4358, 908}, {3950, 21255, 3663}, {4000, 17314, 3875}, {4043, 18143, 1269}, {4360, 16706, 3946}, {4360, 17283, 16706}, {4361, 17265, 17278}, {4361, 17309, 17299}, {4363, 17269, 17281}, {4363, 17313, 4675}, {4364, 17237, 4357}, {4384, 17294, 8}, {4393, 17389, 3244}, {4393, 29629, 31191}, {4416, 25101, 9}, {4417, 18743, 3452}, {4431, 24199, 75}, {4437, 16593, 17755}, {4445, 17259, 17275}, {4447, 8299, 2223}, {4475, 20703, 20590}, {4511, 28757, 26006}, {4664, 17227, 4389}, {4665, 34824, 4688}, {4670, 17359, 17369}, {4671, 31019, 4054}, {4675, 17281, 4363}, {4681, 17235, 17246}, {4687, 5224, 5257}, {4687, 17228, 5224}, {4693, 31151, 24715}, {4698, 17239, 1213}, {4704, 17236, 17247}, {4727, 31243, 4395}, {4851, 17267, 17353}, {4851, 17279, 6}, {4852, 17356, 17366}, {4852, 17388, 4464}, {4873, 6173, 4659}, {5233, 30829, 5316}, {5308, 29611, 2}, {6376, 18743, 30830}, {6542, 17266, 3008}, {6542, 29590, 20016}, {7081, 29839, 13405}, {7232, 17262, 17276}, {7270, 19815, 37086}, {10324, 11019, 8}, {10436, 17286, 2345}, {10453, 29641, 4847}, {14829, 33116, 5745}, {15668, 17293, 17303}, {16284, 18743, 30854}, {16672, 21358, 17325}, {16706, 17315, 4360}, {16814, 17344, 17332}, {16815, 29626, 2}, {16816, 20055, 29617}, {16826, 17292, 2}, {16826, 29575, 29569}, {16826, 29587, 29604}, {16826, 29589, 29606}, {16826, 29609, 29592}, {16831, 17308, 2}, {16834, 29605, 145}, {17023, 29574, 1}, {17023, 29596, 2}, {17023, 29601, 29574}, {17034, 30173, 10}, {17045, 34573, 17384}, {17160, 27191, 37756}, {17229, 17245, 4967}, {17230, 17244, 10}, {17230, 29572, 2}, {17230, 29579, 29674}, {17230, 29582, 29571}, {17230, 29599, 29593}, {17230, 29600, 24603}, {17231, 17243, 4357}, {17232, 17242, 3663}, {17233, 17234, 75}, {17233, 17241, 142}, {17234, 17240, 2321}, {17238, 27268, 17248}, {17240, 17241, 75}, {17244, 29572, 29600}, {17244, 29576, 29581}, {17244, 29577, 3661}, {17244, 29581, 29599}, {17244, 29582, 29572}, {17244, 29594, 24603}, {17244, 29674, 29596}, {17248, 27268, 3986}, {17260, 17287, 1654}, {17261, 17288, 6646}, {17263, 17277, 6666}, {17263, 17295, 3686}, {17265, 17309, 4361}, {17266, 17310, 239}, {17267, 17311, 6}, {17268, 17300, 17355}, {17268, 17312, 894}, {17269, 17313, 4363}, {17277, 17295, 319}, {17278, 17299, 4361}, {17279, 17311, 3879}, {17280, 17300, 894}, {17280, 17312, 3664}, {17283, 17315, 3946}, {17284, 17316, 17023}, {17284, 29573, 1}, {17284, 29583, 29574}, {17284, 29602, 29598}, {17285, 17317, 5750}, {17289, 17317, 86}, {17290, 17318, 17301}, {17291, 17319, 17302}, {17292, 29569, 1125}, {17292, 29575, 16826}, {17292, 29637, 29596}, {17302, 17319, 4021}, {17310, 27757, 6633}, {17316, 26626, 29585}, {17316, 29573, 29601}, {17316, 29579, 2}, {17316, 29583, 29573}, {17316, 29585, 29602}, {17335, 17346, 3707}, {17335, 17360, 17346}, {17336, 17361, 17347}, {17337, 17362, 17348}, {17338, 17363, 17349}, {17339, 17364, 17350}, {17340, 17365, 17351}, {17341, 17386, 3759}, {17342, 17387, 3758}, {17348, 17372, 17362}, {17349, 17373, 17363}, {17350, 17375, 17364}, {17351, 17376, 17365}, {17352, 17377, 3759}, {17354, 17378, 3758}, {17354, 17387, 4667}, {17358, 17379, 17368}, {17366, 17388, 4852}, {17367, 17389, 4393}, {17367, 29618, 17389}, {17367, 29629, 2}, {17368, 17391, 17379}, {17369, 17392, 4670}, {17370, 17393, 17380}, {17371, 17394, 17381}, {17385, 28639, 17398}, {17389, 29629, 17367}, {17397, 29570, 551}, {17397, 29613, 2}, {17758, 21070, 20888}, {17763, 29632, 3011}, {17786, 20923, 3596}, {18040, 18137, 313}, {18139, 20913, 17758}, {18230, 32099, 391}, {19791, 37096, 23537}, {20016, 29590, 239}, {20055, 29617, 3625}, {20337, 23947, 115}, {21371, 22370, 573}, {24899, 24947, 2}, {25957, 32915, 3914}, {26540, 26669, 25019}, {26626, 29585, 1}, {26626, 29593, 29659}, {26626, 29598, 17023}, {27113, 27166, 2}, {27191, 37756, 17067}, {27399, 27526, 6745}, {27474, 27475, 27478}, {28422, 28713, 26006}, {28734, 28789, 2}, {28740, 28795, 2}, {28742, 28797, 2}, {28751, 28805, 2}, {28756, 28812, 2}, {28757, 28813, 2}, {28761, 28816, 2}, {28808, 30828, 5219}, {28813, 28982, 26006}, {29569, 29575, 29606}, {29569, 29579, 29637}, {29569, 29587, 2}, {29569, 29589, 29575}, {29571, 29594, 10}, {29571, 29600, 17244}, {29572, 29577, 10}, {29572, 29593, 29599}, {29573, 29579, 17023}, {29573, 29598, 29602}, {29574, 29596, 17023}, {29574, 29601, 17316}, {29575, 29587, 1125}, {29576, 29581, 2}, {29576, 29593, 10}, {29577, 29582, 2}, {29578, 29591, 3634}, {29578, 29610, 2}, {29579, 29583, 1}, {29580, 29586, 3636}, {29580, 29614, 29586}, {29581, 29599, 29571}, {29584, 29588, 3635}, {29584, 29619, 29588}, {29585, 29602, 29574}, {29587, 29589, 16826}, {29590, 29607, 3008}, {29592, 29609, 1125}, {29593, 29599, 2}, {29594, 29600, 2}, {29595, 29608, 19862}, {29596, 29601, 1}, {29597, 29603, 3616}, {29598, 29602, 1}, {29604, 29606, 16826}, {29608, 29612, 2}, {29608, 29622, 29612}, {29610, 29620, 29578}, {29611, 29621, 16831}, {29612, 29622, 29595}, {29613, 29623, 29570}, {29614, 29625, 29580}, {29615, 29626, 16815}, {29616, 29627, 4384}, {29617, 29628, 16816}, {29618, 29629, 4393}, {29619, 29630, 29584}, {29643, 30942, 29639}, {29988, 30059, 30109}, {31017, 31035, 26580}

Notes

X₃₉₁₂ is the perspector of the circumconic passing through the isogonal conjugates of PU(48). (Randy Hutson, August 29, 2018)