X(23) = FAR-OUT POINT

Trilinears

\(a[b4 + c4 - a4 - b2c2] : :\)

Barycentrics

\(2 sin 2A - 3 tan ω : 2 sin 2B - 3 tan ω : 2 sin 2C - 3 tan ω (M. Iliev, 5/13/07)\)

Notes

As a point on the Euler line, X(23) has Shinagawa coefficients (E + 4F, -4E - 4F).

Let A’B’C’ be the antipedal triangle of X(3) (the tangential triangle). The circumcircles of AA’X(3), BB’X(3), CC’X(3) concur in two points: X(3) and X(23). (Randy Hutson, Octobe3r 13, 2015)

Let A’B’C’ be the anti-orthocentroidal triangle. Let A″ be the reflection of A’ in line BC, and define B″ and C″ cyclically. Then X(23) is the centroid of A″B″C″; see X(9140), X(11002). (Randy Hutson, December 10, 2016)

If you have The Geometer’s Sketchpad, you can view Far-out point. If you have GeoGebra, you can view Far-out point.

X(23) lies on the Parry circle, anti-Brocard circle, anti-McCay circumcircle, and these lines: {2, 3}, {6, 353}, {8, 8185}, {10, 9591}, {15, 11629}, {16, 11630}, {31, 5363}, {32, 5354}, {35, 5297}, {36, 5370}, {49, 10263}, {50, 2493}, {51, 575}, {52, 1614}, {54, 5446}, {55, 5160}, {56, 7286}, {61, 34395}, {62, 34394}, {67, 8262}, {69, 20987}, {74, 9060}, {75, 21408}, {76, 26233}, {94, 98}, {99, 2770}, {100, 2752}, {105, 1290}, {107, 2697}, {108, 37798}, {110, 323}, {111, 187}, {112, 14580}, {114, 40604}, {125, 29012}, {137, 14652}, {141, 32218}, {143, 1199}, {145, 9798}, {146, 2931}, {147, 39828}, {148, 19577}, {154, 1993}, {156, 6243}, {157, 17008}, {159, 193}, {160, 7777}, {161, 6515}, {182, 5640}, {183, 2453}, {184, 576}, {194, 15652}, {195, 14449}, {206, 18882}, {216, 10985}, {230, 11063}, {232, 250}, {238, 27679}, {251, 1194}, {262, 7578}, {316, 5099}, {324, 1629}, {325, 3447}, {343, 3410}, {351, 9213}, {352, 2502}, {373, 5092}, {385, 523}, {388, 9658}, {390, 10833}, {394, 35264}, {477, 1302}, {497, 9673}, {512, 9138}, {515, 9625}, {516, 9590}, {524, 2930}, {530, 13859}, {531, 13858}, {542, 15360}, {568, 15032}, {569, 9781}, {574, 15302}, {577, 15355}, {598, 11628}, {612, 7298}, {614, 5345}, {647, 13114}, {667, 9980}, {671, 3455}, {675, 2690}, {689, 14603}, {895, 1177}, {896, 24436}, {925, 40118}, {935, 2373}, {946, 9626}, {1030, 37675}, {1078, 26235}, {1112, 18449}, {1147, 26882}, {1151, 33502}, {1152, 33503}, {1154, 5609}, {1173, 11692}, {1176, 9969}, {1180, 7772}, {1184, 22331}, {1196, 1627}, {1204, 12279}, {1216, 43598}, {1283, 3724}, {1287, 9076}, {1291, 5966}, {1296, 10102}, {1297, 1304}, {1300, 16167}, {1311, 2689}, {1324, 37764}, {1350, 15066}, {1351, 11004}, {1379, 6141}, {1380, 6142}, {1384, 40126}, {1460, 30652}, {1473, 23958}, {1486, 20061}, {1493, 16982}, {1501, 3981}, {1503, 3448}, {1511, 37477}, {1531, 22109}, {1533, 2777}, {1587, 35776}, {1588, 35777}, {1609, 37689}, {1621, 20988}, {1634, 7840}, {1691, 3124}, {1692, 39024}, {1799, 3456}, {1843, 19121}, {1853, 15579}, {1915, 20859}, {1916, 17938}, {1974, 11511}, {1975, 9464}, {1976, 13137}, {2076, 3231}, {2079, 5913}, {2080, 5191}, {2353, 40232}, {2421, 33928}, {2452, 7766}, {2459, 7599}, {2460, 7598}, {2492, 10561}, {2548, 9700}, {2549, 9699}, {2550, 9713}, {2551, 9712}, {2687, 9058}, {2688, 9057}, {2691, 9061}, {2692, 9083}, {2693, 9064}, {2694, 9107}, {2695, 9056}, {2696, 9084}, {2758, 9059}, {2766, 26703}, {2780, 32222}, {2781, 15139}, {2782, 5987}, {2854, 12367}, {2888, 12134}, {2916, 3589}, {2917, 16252}, {2936, 8591}, {2967, 16978}, {2979, 9306}, {2981, 10613}, {3043, 20773}, {3047, 15647}, {3055, 15109}, {3066, 5085}, {3098, 5651}, {3218, 3220}, {3219, 5285}, {3240, 37576}, {3303, 10149}, {3304, 17024}, {3314, 16335}, {3457, 36759}, {3458, 36760}, {3563, 10420}, {3564, 12310}, {3565, 40119}, {3581, 5663}, {3600, 18954}, {3617, 8193}, {3620, 37485}, {3621, 12410}, {3622, 11365}, {3623, 8192}, {3704, 33091}, {3743, 3746}, {3767, 33802}, {3796, 5422}, {3849, 14682}, {3934, 10130}, {3935, 40910}, {4265, 37633}, {4442, 23848}, {4550, 16261}, {5017, 9463}, {5028, 44116}, {5029, 41185}, {5032, 32621}, {5038, 13410}, {5078, 16686}, {5096, 37680}, {5097, 44109}, {5106, 5162}, {5111, 20976}, {5134, 24055}, {5143, 6187}, {5158, 10311}, {5166, 32740}, {5168, 41183}, {5171, 38528}, {5205, 26262}, {5206, 39576}, {5210, 20481}, {5225, 9672}, {5229, 9659}, {5261, 10831}, {5274, 10832}, {5304, 16303}, {5314, 27065}, {5322, 5563}, {5329, 17127}, {5347, 32911}, {5358, 24883}, {5412, 11418}, {5413, 11417}, {5462, 38848}, {5467, 5968}, {5480, 13394}, {5520, 26231}, {5523, 8428}, {5607, 9163}, {5608, 9162}, {5642, 19924}, {5643, 5943}, {5650, 14810}, {5866, 14360}, {5888, 15082}, {5889, 6759}, {5907, 7691}, {5921, 37488}, {5938, 20099}, {5944, 37472}, {5965, 24981}, {5972, 29317}, {5978, 14368}, {5979, 14369}, {5984, 9861}, {5986, 38664}, {5990, 26249}, {5991, 26277}, {6000, 15054}, {6038, 33997}, {6054, 36829}, {6090, 33878}, {6101, 18350}, {6104, 6109}, {6105, 6108}, {6151, 10614}, {6153, 10203}, {6193, 32048}, {6403, 44080}, {6453, 8854}, {6454, 8855}, {6530, 37766}, {6566, 7602}, {6567, 7601}, {6593, 9019}, {6776, 37644}, {6781, 10418}, {6795, 16311}, {7083, 30653}, {7291, 37782}, {7293, 27003}, {7295, 17126}, {7665, 14712}, {7669, 22329}, {7684, 8838}, {7685, 8836}, {7689, 12290}, {7711, 9155}, {7728, 32227}, {7735, 16306}, {7736, 9609}, {7738, 9608}, {7767, 34992}, {7779, 16316}, {7782, 11059}, {7783, 31088}, {7785, 23208}, {7806, 33801}, {7816, 30749}, {7823, 15270}, {7825, 30747}, {7885, 31076}, {7911, 30785}, {7928, 31124}, {8024, 16276}, {8290, 38998}, {8538, 44077}, {8542, 11188}, {8585, 8588}, {8586, 39689}, {8593, 9966}, {8596, 9876}, {8644, 9137}, {8680, 24322}, {8717, 37470}, {8718, 40647}, {8744, 10317}, {8859, 16092}, {8996, 43133}, {9070, 12030}, {9123, 11616}, {9135, 9212}, {9140, 11645}, {9157, 15562}, {9185, 14270}, {9189, 39477}, {9301, 9999}, {9420, 22734}, {9534, 9571}, {9535, 9570}, {9538, 9645}, {9540, 9683}, {9541, 9682}, {9542, 9695}, {9543, 9694}, {9545, 9707}, {9780, 37557}, {9827, 11817}, {9833, 34799}, {9871, 9879}, {9911, 20070}, {9912, 20085}, {9917, 20081}, {9918, 20088}, {9921, 12221}, {9922, 12222}, {9924, 40318}, {9968, 41715}, {9971, 19127}, {9972, 41713}, {9979, 42659}, {10046, 14986}, {10095, 13353}, {10110, 13434}, {10282, 34148}, {10329, 20965}, {10330, 12215}, {10355, 13492}, {10528, 26309}, {10529, 26308}, {10539, 11412}, {10541, 10601}, {10564, 15035}, {10641, 11421}, {10642, 11420}, {10721, 12893}, {10722, 39825}, {10723, 39854}, {10733, 13289}, {10984, 15043}, {11064, 15448}, {11078, 41023}, {11092, 41022}, {11130, 14539}, {11131, 14538}, {11134, 36980}, {11137, 36978}, {11141, 11549}, {11142, 11537}, {11381, 11440}, {11402, 11482}, {11416, 44102}, {11438, 15072}, {11441, 17834}, {11442, 31383}, {11449, 13346}, {11451, 43650}, {11456, 37489}, {11459, 15052}, {11464, 13352}, {11472, 41398}, {11550, 23293}, {11557, 40640}, {11574, 41464}, {11643, 34320}, {11671, 15959}, {11793, 43614}, {11809, 26228}, {12111, 26883}, {12160, 14530}, {12254, 12370}, {12307, 31834}, {12319, 32123}, {12359, 16659}, {12364, 12380}, {12384, 14731}, {13175, 20094}, {13203, 32125}, {13222, 20095}, {13336, 15024}, {13339, 13363}, {13349, 41473}, {13350, 41472}, {13366, 21849}, {13367, 13598}, {13391, 22115}, {13445, 15021}, {13451, 15038}, {13470, 43821}, {13474, 15062}, {13754, 14094}, {13851, 15044}, {14173, 37786}, {14174, 25225}, {14179, 37785}, {14180, 25226}, {14262, 38338}, {14611, 25045}, {14669, 38679}, {14671, 15564}, {14703, 34549}, {14704, 15753}, {14705, 15754}, {14805, 34513}, {14906, 30541}, {14918, 20625}, {14927, 37643}, {14984, 40114}, {14996, 36740}, {14997, 36741}, {15004, 22234}, {15026, 37471}, {15028, 37515}, {15033, 18475}, {15034, 43574}, {15039, 40111}, {15059, 29323}, {15068, 37494}, {15077, 34438}, {15361, 20126}, {15516, 44107}, {15574, 15589}, {15577, 35260}, {15655, 21448}, {15786, 39371}, {15801, 32379}, {16119, 20084}, {16166, 29011}, {16186, 30510}, {16187, 33879}, {16272, 33925}, {16313, 40896}, {16321, 16990}, {16324, 16989}, {16332, 29831}, {16463, 36211}, {16464, 36210}, {16510, 41720}, {16760, 23217}, {16776, 32154}, {16823, 26261}, {16835, 43689}, {16836, 43584}, {16881, 43845}, {17100, 37762}, {17128, 31078}, {17497, 18617}, {17508, 22112}, {17984, 23962}, {18125, 34437}, {18487, 41358}, {18860, 38704}, {19128, 44084}, {19189, 43768}, {20079, 34207}, {20080, 37491}, {20185, 23096}, {21009, 23406}, {21401, 34424}, {21402, 34425}, {21659, 41482}, {21766, 31884}, {21969, 34986}, {21970, 26869}, {22113, 22657}, {22114, 22656}, {22239, 34168}, {22687, 25233}, {22689, 25234}, {23395, 23862}, {24164, 30117}, {24650, 44123}, {24651, 44124}, {24687, 25344}, {25328, 35218}, {25739, 36253}, {27866, 41671}, {30716, 30737}, {31606, 38337}, {31652, 38862}, {31817, 43609}, {32171, 37495}, {32235, 40291}, {32239, 32257}, {32428, 38552}, {32479, 42008}, {32609, 37496}, {32624, 37761}, {32625, 37763}, {32739, 41323}, {33155, 41230}, {33582, 33861}, {33873, 36213}, {34137, 38356}, {34224, 41587}, {35266, 40112}, {35356, 39099}, {35360, 41204}, {36201, 41603}, {36414, 36417}, {36849, 39644}, {36990, 37638}, {37538, 37685}, {37801, 38971}, {38225, 38611}, {38672, 38678}, {40130, 41413}, {40911, 40917}, {41583, 41721}, {41596, 41732}, {41607, 41742}, {41613, 41744}, {43829, 43838}

X(23) is the {X(22),:ref:X(25) <X(25)>}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(23), click Tables at the top of this page.

X(23) = reflection of X(i) in X(j) for these (i,j): (110,1495), (323,110), (691,187), (858,468)

X(23) = isogonal conjugate of X(67)

X(23) = isotomic conjugate of X(18019)

X(23) = inverse-in-circumcircle of X(2)

X(23) = anticomplement of X(858)

X(23) = anticomplementary conjugate of X(2892)

X(23) = crosspoint of X(111) and X(251)

X(23) = crosssum of X(i) and X(j) for these (i,j): (125,690), (141,524)

X(23) = crossdifference of every pair of points on line X(39)X(647)

X(23) = complement of X(5189)

X(23) = perspector of ABC and reflection of circummedial triangle in the Euler line

X(23) = antigonal image of X(316)

X(23) = trilinear pole of line X(2492)X(6593)

X(23) = reflection of X(858) in the orthic axis

X(23) = reflection of X(110) in the Lemoine axis

X(23) = polar conjugate of isotomic conjugate of X(22151)

X(23) = X(352)-of-circumsymmedial-triangle

X(23) = X(110)-of-1st-anti-Brocard-triangle

X(23) = crosspoint of X(3) and X(2930) wrt both the excentral and tangential triangles

X(23) = inverse-in-circumcircle of X(2)

X(23) = inverse-in-polar-circle of X(427)

X(23) = inverse-in-{circumcircle, nine-point circle}-inverter of X(5)

X(23) = inverse-in-de-Longchamps-circle of X(1370)

X(23) = X(75)-isoconjugate of X(3455)

X(23) = common radical trace of similitude circles of pairs of the Stammler circles

X(23) = one of two harmonic traces of Ehrmann circles; the other is X(6)

X(23) = X(111)-of-anti-McCay-triangle

X(23) = X(691)-of-1st-Parry-triangle

X(23) = X(842)-of-2nd-Parry-triangle

X(23) = X(1296)-of-3rd-Parry-triangle

X(23) = inverse-in-Parry-isodynamic-circle of X(352) (see X(2))

X(23) = X(111)-of-4th-anti-Brocard-triangle

X(23) = similitude center of anti-McCay and 4th anti-Brocard triangles

X(23) = anti-Artzt-to-4th-anti-Brocard similarity image of X(110)

X(23) = intersection of de Longchamps lines of 1st and 2nd Ehrmann circumscribing triangles

X(23) = intersection of orthic axes of antipedal triangles of PU(1)

X(23) = intersection of de Longchamps lines of anticevian triangles of PU(4)

X(23) = circumtangential isogonal conjugate of X(32305)

X(23) = inverse of X(33502) in the Lucas circles radical circle

X(23) = inverse of X(33503) in the Lucas(-1) circles radical circle

X(23) = trilinear pole, wrt 1st Parry triangle, of line X(110)X(1296)

X(23) = X(19)-isoconjugate of X(34897)

X(23) = X(63)-isoconjugate of X(8791)