X₉₉

STEINER POINT

X₉₉ = X₆-of-1st-anti-Brocard-triangle

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

X₉₉ (STEINER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

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

• Central Lines: {3, 98}

• Related Centers: X₃ (CIRCUMCENTER), X₉₈ (TARRY POINT)

Trilinears: \(bc/(b^{2} - c^{2}) : ca/(c^{2} - a^{2}) : ab/(a^{2} - b^{2})\)

Barycentrics: \(1/(b^{2} - c^{2}) : 1/(c^{2} - a^{2}) : 1/(a^{2} - b^{2})\)

Tripolars: \(|a (b^{2} - c^{2})| : :\)

Search Value (6, 9, 13): \(3.438729453648\)

▸ Alternate Coordinates2 forms

• Trilinears: \(b^{2}c^{2} \csc(B - C) : c^{2}a^{2} \csc(C - A) : a^{2}b^{2} \csc(A - B)\)

• Barycentrics: \(d(A,L) : :\), where d = directed distance and L = X₁₁₅X₁₂₅

▸ Linear Combinations1 combination

• X₉₉ = \(2(r^{2} + 4rR - s^{2})(r^{2} + 2rR + s^{2})\) X₁ - \(3(r^{4} + 4Rr^{3} + 2r^{2}s^{2} - 4rRs^{2} + s^{4})\) X₂ - \(4r^{2}(r^{2} + 4rR + 5s^{2})\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines931 lines

{1, 741}, {2, 111}, {3, 76}, {4, 114}, {5, 5966}, {6, 729}, {8, 28471}, {9, 35106}, {10, 6626}, {11, 10769}, {12, 13181}, {13, 303}, {14, 302}, {15, 22687}, {16, 22689}, {20, 147}, {21, 105}, {22, 305}, {23, 2770}, {25, 2374}, {27, 9085}, {28, 15344}, {30, 316}, {31, 715}, {32, 194}, {35, 1909}, {36, 350}, {37, 2375}, {38, 745}, {39, 83}, {40, 24469}, {51, 35919}, {53, 35920}, {55, 3023}, {56, 3027}, {57, 35176}, {58, 727}, {61, 22685}, {62, 22683}, {63, 2249}, {69, 74}, {75, 261}, {81, 739}, {86, 106}, {95, 311}, {100, 668}, {101, 190}, {102, 332}, {103, 1043}, {104, 314}, {107, 2797}, {108, 811}, {109, 643}, {110, 690}, {112, 648}, {113, 33512}, {116, 31001}, {125, 35922}, {127, 35923}, {140, 38224}, {141, 755}, {145, 28531}, {159, 2366}, {160, 1502}, {162, 32691}, {163, 825}, {165, 9860}, {172, 25264}, {182, 7709}, {184, 3044}, {186, 5866}, {187, 385}, {192, 2242}, {193, 5477}, {216, 35928}, {230, 14568}, {232, 15014}, {237, 3978}, {238, 2669}, {239, 1931}, {249, 525}, {250, 10423}, {251, 35929}, {253, 5896}, {262, 35930}, {264, 378}, {265, 14850}, {286, 915}, {287, 3269}, {298, 531}, {299, 530}, {304, 26702}, {310, 675}, {313, 2372}, {317, 1299}, {323, 32730}, {326, 26701}, {327, 35934}, {330, 2241}, {333, 1121}, {338, 34866}, {340, 10295}, {343, 35937}, {371, 19056}, {372, 19055}, {381, 22515}, {382, 7773}, {386, 3029}, {393, 35940}, {394, 26717}, {395, 35942}, {396, 35943}, {401, 36212}, {402, 13179}, {404, 18140}, {419, 19599}, {439, 6392}, {448, 25083}, {468, 37803}, {476, 850}, {477, 3260}, {487, 8982}, {488, 26441}, {489, 12123}, {490, 12124}, {491, 6560}, {492, 6561}, {493, 13184}, {494, 13185}, {511, 2698}, {512, 805}, {513, 2703}, {514, 2702}, {515, 2708}, {516, 2700}, {517, 2699}, {518, 2711}, {519, 2712}, {520, 2713}, {521, 2714}, {522, 2701}, {523, 691}, {524, 843}, {526, 9160}, {532, 6779}, {533, 6780}, {536, 16702}, {548, 7767}, {549, 11632}, {550, 3933}, {573, 34454}, {575, 35950}, {576, 10788}, {577, 35952}, {590, 35953}, {593, 17147}, {595, 34063}, {597, 35954}, {598, 5503}, {599, 6323}, {621, 5617}, {622, 5613}, {623, 23004}, {624, 23005}, {625, 7925}, {626, 6655}, {627, 25560}, {628, 25559}, {629, 11602}, {630, 11603}, {631, 6036}, {633, 22507}, {634, 22509}, {637, 6231}, {638, 6230}, {644, 8693}, {647, 9091}, {651, 32038}, {666, 919}, {669, 886}, {670, 804}, {689, 4609}, {692, 785}, {693, 1290}, {694, 11229}, {695, 711}, {697, 2206}, {698, 1691}, {703, 3051}, {707, 3117}, {712, 5006}, {713, 1333}, {719, 1964}, {723, 3116}, {726, 1326}, {731, 3736}, {732, 2076}, {736, 5162}, {737, 3094}, {740, 12031}, {753, 30966}, {754, 6781}, {757, 4360}, {767, 6385}, {769, 32739}, {783, 2531}, {789, 4602}, {801, 6509}, {813, 1016}, {815, 3888}, {826, 9218}, {827, 1576}, {835, 1978}, {839, 6386}, {840, 18821}, {858, 37804}, {859, 2726}, {873, 1621}, {883, 6606}, {889, 898}, {894, 38814}, {895, 13479}, {901, 4555}, {917, 7431}, {931, 4631}, {932, 18830}, {933, 18831}, {934, 4569}, {935, 3267}, {953, 17139}, {972, 1792}, {1010, 5988}, {1014, 8686}, {1015, 4366}, {1030, 3770}, {1046, 25607}, {1086, 25536}, {1113, 15164}, {1114, 15165}, {1125, 11599}, {1180, 9101}, {1194, 16951}, {1232, 13597}, {1235, 3520}, {1236, 2071}, {1270, 26617}, {1271, 26618}, {1273, 14979}, {1285, 1992}, {1287, 23285}, {1292, 2788}, {1293, 2789}, {1294, 2790}, {1295, 2791}, {1296, 2418}, {1298, 5562}, {1301, 2409}, {1302, 15329}, {1304, 3233}, {1305, 4572}, {1308, 7199}, {1310, 1633}, {1311, 4225}, {1316, 9155}, {1325, 2752}, {1331, 28624}, {1332, 36080}, {1350, 8719}, {1370, 34254}, {1379, 3413}, {1380, 3414}, {1384, 11055}, {1415, 14612}, {1434, 1477}, {1499, 2709}, {1500, 6645}, {1503, 2710}, {1506, 16044}, {1511, 18332}, {1562, 30227}, {1593, 12131}, {1605, 2926}, {1606, 2925}, {1613, 6380}, {1625, 2421}, {1627, 8267}, {1649, 9170}, {1655, 5277}, {1656, 38732}, {1657, 7776}, {1789, 21586}, {1799, 3455}, {1812, 32726}, {1921, 14195}, {1963, 17319}, {1995, 9084}, {2023, 5013}, {2052, 6503}, {2080, 32469}, {2106, 3230}, {2108, 24294}, {2134, 27954}, {2142, 14824}, {2185, 32939}, {2207, 6461}, {2222, 4998}, {2311, 24578}, {2368, 8053}, {2382, 18822}, {2383, 7576}, {2384, 16704}, {2502, 5108}, {2525, 9514}, {2548, 14035}, {2644, 21196}, {2668, 3750}, {2679, 31513}, {2687, 3262}, {2688, 35517}, {2689, 35519}, {2690, 3261}, {2695, 35516}, {2704, 3309}, {2705, 3667}, {2706, 6000}, {2707, 6001}, {2717, 35164}, {2718, 30939}, {2719, 4131}, {2721, 16741}, {2722, 15413}, {2725, 18157}, {2727, 30805}, {2748, 4076}, {2758, 3264}, {2764, 4143}, {2766, 7476}, {2857, 37183}, {2862, 16876}, {2868, 16084}, {2896, 7794}, {2930, 36792}, {2971, 36898}, {2975, 17143}, {2986, 34834}, {2996, 32989}, {3006, 5196}, {3008, 24378}, {3053, 6179}, {3058, 12351}, {3068, 8997}, {3069, 13989}, {3090, 6721}, {3091, 20399}, {3095, 12110}, {3096, 7791}, {3097, 10791}, {3108, 38278}, {3110, 14839}, {3146, 32816}, {3180, 9117}, {3181, 9115}, {3244, 32004}, {3286, 14665}, {3290, 16756}, {3314, 7761}, {3360, 33786}, {3363, 12040}, {3398, 32448}, {3407, 10290}, {3428, 22504}, {3448, 15357}, {3491, 14135}, {3522, 3785}, {3523, 11623}, {3524, 6055}, {3525, 20398}, {3526, 34127}, {3529, 32006}, {3533, 38735}, {3534, 7788}, {3543, 32827}, {3545, 9880}, {3564, 23700}, {3565, 35136}, {3566, 10425}, {3569, 14960}, {3570, 28841}, {3571, 7170}, {3576, 11710}, {3579, 29300}, {3582, 10070}, {3584, 10054}, {3589, 6034}, {3616, 11725}, {3618, 14039}, {3628, 38229}, {3632, 28547}, {3658, 9058}, {3666, 11611}, {3679, 28559}, {3699, 4614}, {3701, 37294}, {3729, 27958}, {3732, 28847}, {3746, 25303}, {3760, 7280}, {3761, 5010}, {3767, 7857}, {3788, 5025}, {3799, 29363}, {3815, 8370}, {3830, 22566}, {3832, 32835}, {3849, 7840}, {3854, 32873}, {3882, 6010}, {3886, 28842}, {3906, 9181}, {3912, 35163}, {3934, 7824}, {3948, 19308}, {3952, 4596}, {3977, 18653}, {4025, 35169}, {4037, 7267}, {4074, 10329}, {4093, 34996}, {4094, 7207}, {4108, 9080}, {4176, 11206}, {4188, 18135}, {4189, 5985}, {4203, 34020}, {4210, 18152}, {4215, 9074}, {4218, 30893}, {4228, 9061}, {4240, 9064}, {4243, 9057}, {4246, 9107}, {4249, 26705}, {4256, 37678}, {4357, 38453}, {4367, 24037}, {4391, 9090}, {4393, 9346}, {4467, 24041}, {4482, 4595}, {4499, 29325}, {4552, 4565}, {4553, 29071}, {4557, 8708}, {4570, 32682}, {4575, 15440}, {4582, 4591}, {4583, 4593}, {4588, 4597}, {4594, 29055}, {4613, 4639}, {4620, 7253}, {4658, 28523}, {4720, 28535}, {4736, 7266}, {4756, 28196}, {4760, 7200}, {4840, 29341}, {4897, 6064}, {4921, 8696}, {4996, 19628}, {5007, 7839}, {5008, 33694}, {5017, 32451}, {5024, 11174}, {5030, 37686}, {5055, 12355}, {5056, 32839}, {5059, 32825}, {5064, 12132}, {5088, 20924}, {5097, 22521}, {5099, 36174}, {5111, 13196}, {5113, 14959}, {5116, 24256}, {5171, 12251}, {5189, 30718}, {5201, 34000}, {5206, 7751}, {5207, 29012}, {5210, 8667}, {5235, 28317}, {5248, 31997}, {5254, 7807}, {5267, 20888}, {5283, 16915}, {5284, 33779}, {5286, 7856}, {5291, 17759}, {5309, 7806}, {5333, 28326}, {5434, 12350}, {5467, 11636}, {5469, 6670}, {5470, 6669}, {5473, 36776}, {5475, 7777}, {5479, 36765}, {5485, 11147}, {5490, 6568}, {5491, 6569}, {5597, 13176}, {5598, 13177}, {5603, 11724}, {5642, 9144}, {5648, 36883}, {5649, 5664}, {5663, 9161}, {5745, 11608}, {5840, 10768}, {5860, 33342}, {5861, 33343}, {5897, 6527}, {5921, 10008}, {5972, 16278}, {5987, 6031}, {5990, 9081}, {5992, 11115}, {5994, 17402}, {5995, 17403}, {5999, 9772}, {6071, 22103}, {6093, 9966}, {6114, 11304}, {6115, 11303}, {6194, 8722}, {6195, 9463}, {6200, 22716}, {6226, 11825}, {6227, 11824}, {6248, 37334}, {6284, 12185}, {6292, 10159}, {6295, 10645}, {6340, 38282}, {6353, 19583}, {6376, 25440}, {6396, 22718}, {6551, 6635}, {6563, 10420}, {6574, 7258}, {6582, 10646}, {6628, 27804}, {6646, 25435}, {6656, 7789}, {6658, 7747}, {6671, 22510}, {6672, 22511}, {6680, 7765}, {6683, 31652}, {6776, 35387}, {6782, 23013}, {6783, 23006}, {6786, 6787}, {7061, 33941}, {7255, 8684}, {7260, 30670}, {7279, 34388}, {7283, 15168}, {7354, 12184}, {7417, 9775}, {7419, 9083}, {7424, 33864}, {7426, 10102}, {7436, 32706}, {7450, 9056}, {7463, 26704}, {7471, 9060}, {7488, 28706}, {7495, 11056}, {7496, 13233}, {7502, 21395}, {7603, 17005}, {7610, 11151}, {7735, 32985}, {7736, 14033}, {7737, 7774}, {7738, 7803}, {7739, 16989}, {7745, 7858}, {7746, 7907}, {7749, 33259}, {7753, 19686}, {7758, 7877}, {7759, 7823}, {7762, 7905}, {7772, 7787}, {7778, 7841}, {7780, 15513}, {7784, 7881}, {7792, 7827}, {7800, 32965}, {7805, 35007}, {7815, 15515}, {7817, 16984}, {7818, 7897}, {7819, 7859}, {7821, 7842}, {7822, 7876}, {7825, 7888}, {7829, 10583}, {7834, 7864}, {7838, 13571}, {7843, 7941}, {7849, 7928}, {7851, 7942}, {7852, 7923}, {7853, 7880}, {7854, 7904}, {7855, 7893}, {7861, 7874}, {7862, 32966}, {7865, 9878}, {7866, 7918}, {7867, 7872}, {7868, 7937}, {7869, 7935}, {7873, 7895}, {7875, 14036}, {7878, 9605}, {7879, 7936}, {7884, 33220}, {7886, 33245}, {7887, 7940}, {7889, 19689}, {7894, 30435}, {7896, 7929}, {7900, 7903}, {7902, 7932}, {7915, 7948}, {7916, 7946}, {7926, 9766}, {7943, 33217}, {7953, 35137}, {8025, 8700}, {8029, 14728}, {8041, 10328}, {8149, 34999}, {8266, 33769}, {8287, 25469}, {8295, 10998}, {8299, 19635}, {8352, 22110}, {8362, 9478}, {8586, 12151}, {8588, 17131}, {8589, 9466}, {8597, 31173}, {8652, 32042}, {8666, 17144}, {8691, 24039}, {8703, 14830}, {8704, 13241}, {8707, 27808}, {8709, 16695}, {8715, 24524}, {8783, 23208}, {8787, 15534}, {8856, 28677}, {8860, 11149}, {8884, 23233}, {8980, 9540}, {9070, 13589}, {9078, 17521}, {9079, 17587}, {9082, 13588}, {9097, 16711}, {9108, 17524}, {9109, 17539}, {9110, 16748}, {9111, 35172}, {9112, 12155}, {9113, 12154}, {9136, 9487}, {9143, 9184}, {9185, 9216}, {9202, 27550}, {9203, 27551}, {9230, 20775}, {9293, 10190}, {9308, 10607}, {9428, 9494}, {9509, 35068}, {9512, 22085}, {9861, 11414}, {9870, 11580}, {9875, 19875}, {10007, 24273}, {10130, 31078}, {10278, 37879}, {10299, 35021}, {10302, 15810}, {10303, 32838}, {10304, 11177}, {10310, 12178}, {10350, 32452}, {10461, 29053}, {10488, 15533}, {10684, 11672}, {10796, 32447}, {11005, 17702}, {11060, 14972}, {11101, 33933}, {11112, 37664}, {11116, 30758}, {11121, 22848}, {11122, 22892}, {11123, 31614}, {11145, 34376}, {11146, 34374}, {11155, 11168}, {11176, 23356}, {11184, 11317}, {11237, 18969}, {11238, 12354}, {11248, 12189}, {11249, 12190}, {11285, 15815}, {11307, 36252}, {11308, 36251}, {11319, 27162}, {11320, 24598}, {11322, 30964}, {11328, 32524}, {11329, 30830}, {11353, 31234}, {11413, 34168}, {11593, 27779}, {11596, 22143}, {11612, 13858}, {11613, 13859}, {11822, 12179}, {11823, 12180}, {11826, 12182}, {11827, 12183}, {11828, 12186}, {11829, 12187}, {11842, 32519}, {12030, 35550}, {12054, 32516}, {12074, 24976}, {12093, 33962}, {12156, 18907}, {12195, 12782}, {12212, 32449}, {12258, 25055}, {12348, 34612}, {12349, 34606}, {12941, 18974}, {12942, 18975}, {12951, 13076}, {12952, 13075}, {13587, 18145}, {13637, 13642}, {13638, 35306}, {13712, 32808}, {13738, 30022}, {13757, 13761}, {13758, 35305}, {13835, 32809}, {13881, 33233}, {13935, 13967}, {14023, 33254}, {14060, 25051}, {14096, 22735}, {14118, 26166}, {14558, 18570}, {14574, 33514}, {14658, 19570}, {14720, 18436}, {14727, 21789}, {14734, 23105}, {14810, 14994}, {14849, 38728}, {14867, 31839}, {14927, 15428}, {14953, 35158}, {14961, 15013}, {15022, 32871}, {15034, 31854}, {15035, 22265}, {15059, 15359}, {15061, 15535}, {15075, 28405}, {15545, 32423}, {15574, 22241}, {15588, 35965}, {15696, 29322}, {15697, 32896}, {15701, 26614}, {15705, 32874}, {15708, 32885}, {15717, 32834}, {15814, 17503}, {16041, 37690}, {16049, 26703}, {16370, 16992}, {16371, 18146}, {16589, 16917}, {16592, 24962}, {16678, 18021}, {16806, 32036}, {16807, 32037}, {16887, 28485}, {16912, 36812}, {16914, 27318}, {16921, 31455}, {16924, 31401}, {16948, 28583}, {16956, 21838}, {16990, 33008}, {17004, 33274}, {17008, 21843}, {17034, 33863}, {17058, 26147}, {17132, 37792}, {17150, 33774}, {17183, 38452}, {17210, 28499}, {17280, 25433}, {17475, 18268}, {17499, 18755}, {17549, 37670}, {17596, 24291}, {17677, 30761}, {17729, 30109}, {17738, 18061}, {17777, 25533}, {17780, 28210}, {17944, 29313}, {18153, 37309}, {18206, 35167}, {18315, 32692}, {18481, 29096}, {19057, 32788}, {19058, 32787}, {19598, 36955}, {20023, 37184}, {20172, 31449}, {20580, 23582}, {20624, 31623}, {20982, 24504}, {21004, 21431}, {21163, 37455}, {21220, 21341}, {21359, 25166}, {21360, 25156}, {21937, 26244}, {22089, 22456}, {22239, 37937}, {22401, 28723}, {22407, 36511}, {22489, 22577}, {22490, 22578}, {22561, 22564}, {22691, 33409}, {22692, 33408}, {23096, 37943}, {23334, 25486}, {23610, 38017}, {23889, 35180}, {23999, 36068}, {24264, 24282}, {24345, 24714}, {24348, 24711}, {24630, 32851}, {25332, 30226}, {25526, 28491}, {25532, 30997}, {26243, 35276}, {26257, 30749}, {26615, 32810}, {26616, 32811}, {26619, 32805}, {26620, 32806}, {26716, 32040}, {26860, 28310}, {27082, 34403}, {27269, 33062}, {27656, 29983}, {27705, 34053}, {28348, 30092}, {28574, 33947}, {29039, 31730}, {29052, 35338}, {30528, 32690}, {30736, 37927}, {30785, 31107}, {31400, 32971}, {31404, 32979}, {31415, 33016}, {31450, 33269}, {31644, 36953}, {32465, 36759}, {32466, 36760}, {32483, 32485}, {32522, 37479}, {32823, 33703}, {33376, 33458}, {33377, 33459}, {33511, 38793}, {33705, 37465}, {33801, 35226}, {33932, 37405}, {33943, 37816}, {33972, 35298}, {34283, 36744}, {34389, 37848}, {34390, 37850}, {34505, 37637}, {35178, 35324}, {35287, 37667}, {35540, 37896}, {35549, 37903}, {36036, 36065}, {36070, 36085}, {36166, 38704}, {36329, 36769}, {36330, 36768}, {37205, 37212}, {37459, 38227}

▸ Functional Relations78 relations

• X₉₉ = midpoint of X_i and X_j for these (i,j): (20, 147), (616, 617)

• X₉₉ = reflection of X_i in X_j for these (i,j): (4, 114), (13, 619), (14, 618), (98, 3), (115, 620), (148, 115), (316, 325), (385, 187), (671, 2)

• X₉₉ = isogonal conjugate of X₅₁₂

• X₉₉ = isotomic conjugate of X₅₂₃

• X₉₉ = complement of X₁₄₈

• X₉₉ = anticomplement of X₁₁₅

• X₉₉ = cevapoint of X_i and X_j for these (i,j): (2, 523), (3, 525), (39, 512), (100, 190)

• X₉₉ = X₁₀₁₉-cross conjugate of X₁₅₀₉

• X₉₉ = crossdifference of every pair of points on line X₃₅₁X₈₆₅

• X₉₉ = X_i-cross conjugate of X_j for these (i,j): (3, 249), (22, 250), (512, 83), (523, 2), (525, 76)

• X₉₉ = X₂₁-beth conjugate of X₇₄₁

• X₉₉ = X₆-of-1st-anti-Brocard-triangle

• X₉₉ = X₃₈₁-of-anti-McCay-triangle

• X₉₉ = circumcircle-antipode of X₉₈

• X₉₉ = point of intersection, other than A, B, and C, of the circumcircle and Steiner ellipse

• X₉₉ = Ψ(X_i, X_j for these (i,j): (1, 75), (2, 39), (3, 69), (4, 69), (37, 2), (51, 5), (351, 690)

• X₉₉ = point of intersection, other than A, B, C, of the circumcircle and hyperbola {A, B, C,PU(1)}}

• X₉₉ = point of intersection, other than A, B, C, of the circumcircle and hyperbola {A, B, C,PU(37)}}

• X₉₉ = trilinear product of PU(90)

• X₉₉ = similitude center of (equilateral) antipedal triangles of X₁₃ and X₁₄

• X₉₉ = Steiner-circumellipse-antipode of X₆₇₁

• X₉₉ = projection from Steiner inellipse to Steiner circumellipse of X₂₄₈₂

• X₉₉ = trilinear pole of line X₂X₆

• X₉₉ = pole wrt polar circle of trilinear polar of X₂₅₀₁ (line X₁₁₅X₂₉₇₁)

• X₉₉ = X₄₈-isoconjugate (polar conjugate) of X₂₅₀₁

• X₉₉ = X₆-isoconjugate of X₆₆₁

• X₉₉ = X₁₅₇₇-isoconjugate of X₃₂

• X₉₉ = concurrence of reflections in sides of ABC of line X₄X₆₉

• X₉₉ = Λ(X₁, X₅₁₂)

• X₉₉ = isotomic conjugate wrt 1st Brocard triangle of X₇₆

• X₉₉ = perspector of ABC and the tangential triangle, wrt the anticomplementary triangle, of the bianticevian conic of X₁ and X₂

• X₉₉ = perspector of ABC and the tangential triangle, wrt the tangential triangle, of the Stammler hyperbola

• X₉₉ = reflection of X₆₉₁ in the Euler line

• X₉₉ = reflection of X₈₀₅ in the Brocard axis

• X₉₉ = reflection of X₂₇₀₃ in line X₁X₃

• X₉₉ = reflection of X₃₁₆ in the de Longchamps line

• X₉₉ = X₁₃₀-of-excentral-triangle

• X₉₉ = X₁₂₉-of-hexyl-triangle

• X₉₉ = inverse-in-polar-circle of X₅₁₃₉

• X₉₉ = inverse-in-{circumcircle, nine-point circle}-inverter of X₁₂₆

• X₉₉ = inverse-in-2nd-Brocard-circle of X₇₆

• X₉₉ = trilinear product of vertices of circumcircle antipode of circumorthic triangle

• X₉₉ = 1st-Parry-to-ABC similarity image of X₂

• X₉₉ = crossdifference of PU(105)

• X₉₉ = X₁₆₉₁ of 6th Brocard triangle

• X₉₉ = eigencenter of circummedial triangle

• X₉₉ = eigencenter of circumsymmedial triangle

• X₉₉ = perspector of ABC and cross-triangle of circumcevian triangles of PU(1)

• X₉₉ = X₉₈-of-anti-Artzt-triangle

• X₉₉ = X₂-of-1st-anti-Parry-triangle

• X₉₉ = Thomson-isogonal conjugate of X₅₁₁

• X₉₉ = Lucas-isogonal conjugate of X₅₁₁

• X₉₉ = X₁₃₇₉-of-circummedial-triangle

• X₉₉ = X₆₃₂₃-of-circumsymmedial-triangle

• X₉₉ = Kiepert image of X₂

• X₉₉ = Cundy-Parry Phi transform of X₁₄₂₆₅

• X₉₉ = intersection of antipedal lines of X₁₃₇₉ and X₁₃₈₀

• X₉₉ = homothetic center of anticomplementary triangle and mid-triangle of antipedal triangles of X₁₃ and X₁₄

• X₉₉ = barycentric square root of X₄₅₉₀

• X₉₉ = barycentric product of circumcircle intercepts of line X₂X₃₉

• X₉₉ = perspector of ABC and vertex triangle of 1st and 2nd isodynamic-Dao triangles

• X₉₉ = orthic-to-ABC functional image of X₁₃₀

• X₉₉ = tangential-isogonal conjugate of X₃₃₇₀₄

• X₉₉ = trilinear pole, wrt circumtangential triangle, of Brocard axis

• X₉₉ = Vu circlecevian point of PU(1)

• X₉₉ = Vu circlecevian point of PU(37)

• X₉₉ = areal center of pedal triangles of X₁₅ and X₁₆

• X₉₉ = areal center of pedal triangles of PU(2)

• X₉₉ = areal center of cevian triangles of PU(40)

• X₉₉ = X₂-Ceva conjugate of X₃₁₉₉₈

• X₉₉ = perspector of circumconic centered at X₃₁₉₉₈

• X₉₉ = Cundy-Parry Psi transform of X₃₄₁₅₇

• X₉₉ = X₃₉₁₅₆-of-orthic-triangle if ABC is acute

• X₉₉ = Conway-circle-inverse of X₃₈₄₇₇

• X₉₉ = Steiner-circumellipse-X₁-antipode of X₁₈₈₂₇

• X₉₉ = Steiner-circumellipse-X₃-antipode of X₂₉₀

• X₉₉ = Steiner-circumellipse-X₄-antipode of X₃₅₁₄₂

• X₉₉ = Steiner-circumellipse-X₆-antipode of X₃₂₂₈

Notes

Let L_A be the reflection of the line X₃X₆ in line BC, and define L_B and L_C cyclically. Let A' = L_B∩L_C, B' = L_C∩L_A, C' = L_A∩L_B. The lines AA', BB', CC' concur in X₉₉. (Randy Hutson, 9/23/2011)

X₉₉ is the only point on the circumcircle whose isotomic conjugate lies on the line at infinity. (Randy Hutson, 9/23/2011)

X₉₉ is the center of the bianticevian conic of X₁ and X₂, which is the rectangular hyperbola H that passes through these points: X₁, X₂, X₂₀, X₆₃, X₁₄₇, X₁₉₄, X₄₈₇, X₄₈₈, X₆₁₆, X₆₁₇, X₆₂₇, X₆₂₈, X₁₇₆₄, X₂₈₉₆, the excenters, the vertices of the anticomplementary triangle, and the extraversions of X₆₃. Also, H is the anticomplementary conjugate of line X₄X₆₉, the anticomplementary isotomic conjugate of line X₂X₆, the excentral isogonal conjugate of line X₄₀X₅₁₁, and the excentral isotomic conjugate of line X₁₀₄₅X₂₉₅₁; also, H is tangent to line X₁X₇₅ at X₁, to line X₂X₆ at X₂, and meets the line at infinity (and the Kiepert hyperbola, other than at X₂) at X₃₄₁₃ and X₃₄₁₄. (Randy Hutson, December 26, 2015)

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let A″B″C″ be the tangential triangle of A'B'C'. Let A* be the cevapoint of B″ and C″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₉₉. (Randy Hutson, February 10, 2016)

Let L_a, L_b, L_c be the lines through A, B, C, resp. parallel to the Brocard axis. Let M_a, M_b, M_c be the reflections of BC, CA, AB in L_a, L_b, L_c, resp. Let A' = M_b∩M_c, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to ABC with similarity ratio 3. Let A″B″C″ be the reflection of A'B'C' in the Brocard axis. The triangle A″B″C″ is homothetic to ABC, with center of homothety X₁₁₅ and centroid X₉₉. See Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, February 10, 2016)

Let A', B' and C' be the intersections of the de Longchamps line and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X₉₉. (Randy Hutson, February 10, 2016)

Let A', B', C' be the intersections of the Brocard axis and lines BC, CA, AB, resp. Let O_a, O_b, O_c be the circumcenters of AB'C', BC'A', CA'B', resp. The lines AO_a, BO_b, CO_c concur in X₉₉. (Randy Hutson, February 10, 2016)

Let A'B'C' be the 1st Brocard triangle. Let L_a be the line through A parallel to B'C', and define L_b and L_c cyclically. L_a, L_b, L_c concur in X₉₉. (Randy Hutson, February 10, 2016)

Let A' be the trilinear pole of the perpendicular bisector of BC, and define B' and C' cyclically. A'B'C' is also the anticomplement of the anticomplement of the midheight triangle. X₉₉ is the trilinear product A'*B'*C'. (Randy Hutson, January 29, 2018)

Let A'B'C' be the 1st Brocard triangle. Let A″ be the reflection of A in line B'C', and define B″, C″ cyclically. Let A″’ be the reflection of A' in line BC, and define B″’, C″’ cyclically. Let A* = B″B″’∩C″C″’, and define B*, C* cyclically. Triangle A*B*C* is homothetic to A'B'C' at X₉₉. (Randy Hutson, June 27, 2018)

See a construction: Ercole Suppa, Hyacinthos 29064.

Let N_AN_BN_C and N'_AN'_BN'_C be the inner and outer Napoleon triangles, resp. Let A' be the reflection of N_A in line N'_BN'_C, and define B' and C' cyclically. Let A″ be the reflection of N'_A in line N_BN_C, and define B″ and C″ cyclically. Triangles A'B'C' and A″B″C″ are equilateral and inversely similar, with similitude center X₉₉. (Randy Hutson, January 17, 2020)

For more about the Steiner circumellipse, visit MathWorld.

Let O_A be the circle centered at the A-vertex of the Moses-Steiner osculatory triangle and passing through A; define O_B and O_C cyclically. X₉₉ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

Let L_A be the tangent at A to circle {A,PU(1)}}, and define L_B and L_C cyclically. The lines L_A, L_B, L_C concur in X₉₉. (Randy Hutson, August 30, 2020)

The osculating circle of the Steiner circumellipse at X₉₉ intersects the Steiner circumellipse in two points: X₉₉ and X₈₉₂. See Osculating circle of Steiner circumellipse at X₉₉. (Randy Hutson, November 5, 2021)

Let O be a point (not necessarily X₃. Let Q₁,Q₂,…Qn be a regular n-gon with center O. Let P be a point, and define n triangles T₁=Q1Q2P, T₂=Q2Q3P, …, Tn=QnQ1P.

Claim 1: for all n, the points X₂-of-Ti are concyclic on a first circle whose center O₂ is collinear with O,P.

Claim 2: for all n, the lpoints X₉₉-of-Ti are concyclic with P on a second circle whose center O99 is also collinear with O,P. Figure. (Dan Reznik, December 10, 2021)

X₉₉ is the {X₃₉,X₃₈₄}-harmonic conjugate of X₈₃.