X(99) = STEINER POINT¶
Trilinears
\(bc/(b2 - c2) : ca/(c2 - a2) : ab/(a2 - b2)\)
\(b2c2 csc(B - C) : c2a2 csc(C - A) : a2b2 csc(A - B)\)
Barycentrics
\(1/(b2 - c2) : 1/(c2 - a2) : 1/(a2 - b2)\)
\(d(A,L) : : , where d = directed distance and L = :ref:`X(115) <X(115)>\)X(125) <X(125)>`
Notes
Let LA be the reflection of the line X(3)X(6) in line BC, and define LB and LC cyclically. Let A’ = LB∩LC, B’ = LC∩LA, C’ = LA∩LB. The lines AA’, BB’, CC’ concur in X(99). (Randy Hutson, 9/23/2011)
X(99) is the only point on the circumcircle whose isotomic conjugate lies on the line at infinity. (Randy Hutson, 9/23/2011)
X(99) is the center of the bianticevian conic of X(1) and X(2), which is the rectangular hyperbola H that passes through these points: X(1), X(2), X(20), X(63), X(147), X(194), X(487), X(488), X(616), X(617), X(627), X(628), X(1764), X(2896), the excenters, the vertices of the anticomplementary triangle, and the extraversions of X(63). Also, H is the anticomplementary conjugate of line X(4)X(69), the anticomplementary isotomic conjugate of line X(2)X(6), the excentral isogonal conjugate of line X(40)X(511), and the excentral isotomic conjugate of line X(1045)X(2951); also, H is tangent to line X(1)X(75) at X(1), to line X(2)X(6) at X(2), and meets the line at infinity (and the Kiepert hyperbola, other than at X(2)) at X(3413) and X(3414). (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 X(99). (Randy Hutson, February 10, 2016)
Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Brocard axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A’ = Mb∩Mc, 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 X(115) and centroid X(99). 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 X(99). (Randy Hutson, February 10, 2016)
Let A’, B’, C’ be the intersections of the Brocard axis and lines BC, CA, AB, resp. Let Oa, Ob, Oc be the circumcenters of AB’C’, BC’A’, CA’B’, resp. The lines AOa, BOb, COc concur in X(99). (Randy Hutson, February 10, 2016)
Let A’B’C’ be the 1st Brocard triangle. Let La be the line through A parallel to B’C’, and define Lb and Lc cyclically. La, Lb, Lc concur in X(99). (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. X(99) 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 X(99). (Randy Hutson, June 27, 2018)
See a construction: Ercole Suppa, Hyacinthos 29064.
If you have The Geometer’s Sketchpad, you can view the following dynamic sketches: X(99) and Steiner Circum-ellipse (showing X(99) and an area-ratio property)
Let NANBNC and N’AN’BN’C be the inner and outer Napoleon triangles, resp. Let A’ be the reflection of NA in line N’BN’C, and define B’ and C’ cyclically. Let A″ be the reflection of N’A in line NBNC, and define B″ and C″ cyclically. Triangles A’B’C’ and A″B″C″ are equilateral and inversely similar, with similitude center X(99). (Randy Hutson, January 17, 2020)
For more about the Steiner circumellipse, visit MathWorld.
Let OA be the circle centered at the A-vertex of the Moses-Steiner osculatory triangle and passing through A; define OB and OC cyclically. X(99) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)
Let LA be the tangent at A to circle {A,PU(1)}}, and define LB and LC cyclically. The lines LA, LB, LC concur in X(99). (Randy Hutson, August 30, 2020)
The osculating circle of the Steiner circumellipse at X(99) intersects the Steiner circumellipse in two points: X(99) and X(892). See Osculating circle of Steiner circumellipse at X(99). (Randy Hutson, November 5, 2021)
Let O be a point (not necessarily X(3). 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 X(2)-of-Ti are concyclic on a first circle whose center O2 is collinear with O,P. Claim 2: for all n, the lpoints X(99)-of-Ti are concyclic with P on a second circle whose center O99 is also collinear with O,P. Figure. –> Figure. (Dan Reznik, December 10, 2021)
X(99) lies on these 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}
X(99) is the {X(39),:ref:X(384) <X(384)>}-harmonic conjugate of X(83). For a list of other harmonic conjugates of X(99), click Tables at the top of this page.
X(99) = midpoint of X(i) and X(j) for these (i,j): (20,147), (616,617)
X(99) = 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)
X(99) = isogonal conjugate of X(512)
X(99) = isotomic conjugate of X(523)
X(99) = complement of X(148)
X(99) = anticomplement of X(115)
X(99) = cevapoint of X(i) and X(j) for these (i,j): (2,523), (3,525), (39,512), (100,190)
X(99) = X(1019)-cross conjugate of X(1509)
X(99) = crossdifference of every pair of points on line X(351)X(865)
X(99) = X(i)-cross conjugate of X(j) for these (i,j): (3,249), (22,250), (512,83), (523,2), (525,76)
X(99) = X(21)-beth conjugate of X(741)
X(99) = X(6)-of-1st-anti-Brocard-triangle
X(99) = X(381)-of-anti-McCay-triangle
X(99) = circumcircle-antipode of X(98)
X(99) = point of intersection, other than A, B, and C, of the circumcircle and Steiner ellipse
X(99) = Ψ(X(i), X(j) for these (i,j): (1,75), (2,39), (3,69), (4,69), (37,2), (51,5), (351,690)
X(99) = point of intersection, other than A, B, C, of the circumcircle and hyperbola {A,B,C,PU(1)}}
X(99) = point of intersection, other than A, B, C, of the circumcircle and hyperbola {A,B,C,PU(37)}}
X(99) = trilinear product of PU(90)
X(99) = similitude center of (equilateral) antipedal triangles of X(13) and X(14)
X(99) = Steiner-circumellipse-antipode of X(671)
X(99) = projection from Steiner inellipse to Steiner circumellipse of X(2482)
X(99) = trilinear pole of line X(2)X(6)
X(99) = pole wrt polar circle of trilinear polar of X(2501) (line X(115)X(2971))
X(99) = X(48)-isoconjugate (polar conjugate) of X(2501)
X(99) = X(6)-isoconjugate of X(661)
X(99) = X(1577)-isoconjugate of X(32)
X(99) = concurrence of reflections in sides of ABC of line X(4)X(69)
X(99) = Λ(X(1), X(512))
X(99) = isotomic conjugate wrt 1st Brocard triangle of X(76)
X(99) = perspector of ABC and the tangential triangle, wrt the anticomplementary triangle, of the bianticevian conic of X(1) and X(2)
X(99) = perspector of ABC and the tangential triangle, wrt the tangential triangle, of the Stammler hyperbola
X(99) = reflection of X(691) in the Euler line
X(99) = reflection of X(805) in the Brocard axis
X(99) = reflection of X(2703) in line X(1)X(3)
X(99) = reflection of X(316) in the de Longchamps line
X(99) = X(130)-of-excentral-triangle
X(99) = X(129)-of-hexyl-triangle
X(99) = inverse-in-polar-circle of X(5139)
X(99) = inverse-in-{circumcircle, nine-point circle}-inverter of X(126)
X(99) = inverse-in-2nd-Brocard-circle of X(76)
X(99) = trilinear product of vertices of circumcircle antipode of circumorthic triangle
X(99) = 1st-Parry-to-ABC similarity image of X(2)
X(99) = crossdifference of PU(105)
X(99) = X(1691) of 6th Brocard triangle
X(99) = eigencenter of circummedial triangle
X(99) = eigencenter of circumsymmedial triangle
X(99) = perspector of ABC and cross-triangle of circumcevian triangles of PU(1)
X(99) = X(98)-of-anti-Artzt-triangle
X(99) = X(2)-of-1st-anti-Parry-triangle
X(99) = Thomson-isogonal conjugate of X(511)
X(99) = Lucas-isogonal conjugate of X(511)
X(99) = X(1379)-of-circummedial-triangle
X(99) = X(6323)-of-circumsymmedial-triangle
X(99) = Cundy-Parry Phi transform of X(14265)
X(99) = intersection of antipedal lines of X(1379) and X(1380)
X(99) = homothetic center of anticomplementary triangle and mid-triangle of antipedal triangles of X(13) and X(14)
X(99) = barycentric square root of X(4590)
X(99) = barycentric product of circumcircle intercepts of line X(2)X(39)
X(99) = perspector of ABC and vertex triangle of 1st and 2nd isodynamic-Dao triangles
X(99) = orthic-to-ABC functional image of X(130)
X(99) = tangential-isogonal conjugate of X(33704)
X(99) = trilinear pole, wrt circumtangential triangle, of Brocard axis
X(99) = Vu circlecevian point of PU(1)
X(99) = Vu circlecevian point of PU(37)
X(99) = areal center of pedal triangles of X(15) and X(16)
X(99) = areal center of pedal triangles of PU(2)
X(99) = areal center of cevian triangles of PU(40)
X(99) = X(2)-Ceva conjugate of X(31998)
X(99) = perspector of circumconic centered at X(31998)
X(99) = Cundy-Parry Psi transform of X(34157)
X(99) = X(39156)-of-orthic-triangle if ABC is acute
X(99) = Conway-circle-inverse of X(38477)
X(99) = Steiner-circumellipse-X(1)-antipode of X(18827)
X(99) = Steiner-circumellipse-X(3)-antipode of X(290)