X(24) = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE¶

Trilinears

\(sec A cos 2A : sec B cos 2B : sec C cos 2C\)

Barycentrics

\(tan A cos 2A : tan B cos 2B : tan C cos 2C\)

\(tan A - sin 2A : tan B - sin 2B : tan C - sin 2C\)

\(a(sec A - 2 cos A) : :\)

\(a^2 (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2)/(a^2 - b^2 - c^2) : :\)

Notes

As a point on the Euler line, X(24) has Shinagawa coefficients (2F, -E - 2F).

Let A’B’C’ be the orthic triangle. Let A″ = inverse-in-circumcircle of A’, and define B’’ and C’’ cyclically. The lines AA″, BB″, CC″ concur in X(24). (Randy Hutson, September 5, 2015)

X(24) = homothetic center of the tangential triangle and the triangle obtained by reflecting X(4) in the sidelines of ABC.

If you have The Geometer’s Sketchpad, you can view X(24). If you have GeoGebra, you can view X(24). X(24) lies on these lines: {1, 1061}, {2, 3}, {6, 54}, {10, 15177}, {11, 9672}, {12, 9659}, {15, 10642}, {16, 10641}, {19, 2337}, {32, 232}, {33, 35}, {34, 36}, {39, 10311}, {40, 9625}, {49, 568}, {51, 578}, {52, 1147}, {53, 8553}, {55, 6197}, {56, 1870}, {58, 3192}, {60, 36742}, {61, 8739}, {62, 8740}, {64, 74}, {66, 34439}, {68, 3580}, {69, 37488}, {70, 34438}, {96, 847}, {98, 1289}, {99, 39828}, {101, 41320}, {102, 41401}, {104, 3435}, {107, 1093}, {108, 915}, {110, 155}, {111, 8428}, {112, 2079}, {113, 12893}, {114, 39825}, {115, 39854}, {125, 12140}, {132, 34217}, {136, 5961}, {137, 23320}, {143, 32171}, {154, 1181}, {156, 6102}, {157, 31381}, {159, 6776}, {161, 6146}, {165, 9591}, {182, 1843}, {183, 1235}, {184, 389}, {185, 1495}, {187, 1968}, {195, 6242}, {197, 11491}, {206, 19161}, {208, 37583}, {220, 26915}, {221, 19368}, {225, 36152}, {230, 27376}, {242, 1602}, {250, 18879}, {254, 393}, {264, 1078}, {275, 19185}, {278, 7742}, {281, 15817}, {317, 7763}, {340, 7796}, {371, 5413}, {372, 5412}, {386, 9570}, {388, 10037}, {394, 11412}, {399, 7722}, {476, 15112}, {477, 22239}, {487, 12972}, {488, 12973}, {497, 10046}, {498, 11392}, {499, 11393}, {511, 1092}, {515, 8185}, {517, 11363}, {539, 41598}, {569, 5422}, {571, 8745}, {572, 44103}, {573, 1474}, {574, 9700}, {575, 8541}, {576, 44102}, {581, 2299}, {584, 44095}, {601, 2212}, {602, 1395}, {648, 6179}, {842, 10423}, {912, 41609}, {917, 26705}, {925, 39437}, {933, 2383}, {935, 14729}, {944, 1610}, {953, 10016}, {958, 9713}, {974, 15647}, {1033, 33630}, {1058, 16541}, {1063, 1775}, {1075, 3186}, {1112, 1511}, {1141, 15959}, {1151, 3092}, {1152, 3093}, {1172, 36744}, {1173, 14528}, {1192, 1498}, {1199, 11402}, {1204, 6000}, {1216, 15066}, {1249, 8573}, {1294, 30249}, {1297, 39417}, {1304, 14264}, {1324, 1603}, {1351, 19118}, {1376, 9712}, {1384, 3172}, {1385, 1829}, {1407, 26914}, {1473, 26877}, {1485, 6530}, {1503, 11457}, {1533, 22962}, {1560, 14655}, {1604, 18283}, {1619, 18913}, {1620, 10606}, {1627, 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{6593, 11477}, {6684, 37557}, {6696, 16621}, {6749, 9606}, {6770, 9916}, {6771, 12142}, {6773, 9915}, {6774, 12141}, {6799, 33643}, {6800, 9730}, {7085, 26878}, {7293, 37534}, {7354, 9658}, {7581, 19006}, {7582, 19005}, {7583, 13884}, {7584, 13937}, {7649, 39225}, {7687, 32607}, {7688, 11471}, {7689, 12162}, {7690, 12298}, {7691, 11444}, {7692, 12299}, {7709, 22655}, {7717, 21151}, {7731, 17847}, {7735, 41361}, {7749, 27371}, {7754, 41676}, {7786, 36794}, {7952, 8069}, {7967, 8192}, {7999, 17811}, {8071, 34231}, {8127, 8131}, {8128, 8132}, {8148, 31948}, {8190, 11843}, {8191, 11844}, {8194, 11846}, {8195, 11847}, {8262, 15069}, {8550, 15582}, {8718, 22948}, {8754, 34218}, {8780, 12164}, {8883, 14518}, {8939, 19424}, {8943, 19425}, {8982, 8996}, {9638, 19354}, {9704, 15087}, {9706, 11422}, {9729, 10984}, {9737, 44099}, {9744, 23208}, {9777, 11426}, {9781, 10982}, {9861, 9862}, {9876, 12243}, {9908, 11411}, {9910, 12246}, {9912, 12247}, {9913, 12248}, {9914, 12250}, {9917, 12251}, {9918, 12252}, {9919, 12244}, {9920, 12254}, {9921, 12256}, {9922, 12257}, {9924, 12283}, {9925, 11271}, {9934, 17854}, {9938, 12278}, {9967, 26206}, {10098, 40119}, {10110, 11424}, {10192, 12233}, {10246, 11396}, {10267, 11383}, {10269, 22479}, {10274, 32352}, {10310, 20872}, {10313, 23115}, {10519, 37485}, {10539, 11441}, {10540, 32139}, {10546, 15056}, {10564, 43898}, {10574, 15053}, {10601, 15024}, {10610, 11576}, {10645, 11475}, {10646, 11476}, {10722, 39841}, {10723, 39812}, {10733, 12302}, {10785, 10829}, {10786, 10830}, {10788, 10790}, {10805, 10834}, {10806, 10835}, {11204, 32062}, {11206, 18909}, {11245, 31804}, {11362, 37546}, {11386, 26316}, {11388, 26341}, {11389, 26348}, {11390, 26492}, {11391, 26487}, {11400, 16203}, {11401, 16202}, {11408, 11486}, {11409, 11485}, {11423, 17809}, {11431, 32621}, {11433, 18925}, {11439, 11454}, {11440, 15305}, {11442, 12134}, {11458, 17813}, {11459, 17814}, {11462, 17819}, {11463, 17820}, {11465, 17825}, {11466, 17826}, {11467, 17827}, {11496, 20988}, {11500, 20989}, {11508, 15500}, {11550, 13419}, {11572, 23325}, {11589, 39268}, {11616, 14273}, {11641, 13200}, {11695, 43650}, {11704, 32345}, {11745, 23292}, {11750, 43817}, {11898, 12325}, {11935, 15110}, {12022, 19467}, {12041, 12133}, {12042, 12131}, {12095, 34756}, {12111, 12163}, {12112, 12315}, {12115, 26309}, {12116, 26308}, {12118, 32048}, {12136, 34862}, {12137, 12619}, {12138, 38602}, {12145, 38624}, {12174, 32063}, {12228, 16222}, {12235, 27365}, {12236, 15317}, {12245, 12410}, {12249, 12411}, {12253, 12413}, {12255, 12414}, {12281, 17835}, {12282, 17836}, {12284, 17838}, {12285, 17839}, {12286, 17842}, {12287, 17840}, {12288, 17843}, {12289, 17845}, {12295, 12901}, {12296, 12984}, {12297, 12985}, {12310, 12383}, {12324, 18931}, {12509, 12978}, {12510, 12979}, {12675, 41611}, {12828, 30714}, {13017, 17841}, {13018, 17844}, {13019, 13061}, {13020, 13062}, {13035, 13055}, {13036, 13056}, {13049, 13051}, {13050, 13052}, {13093, 34469}, {13166, 38608}, {13172, 13175}, {13199, 13222}, {13202, 13293}, {13292, 37644}, {13321, 14627}, {13323, 44092}, {13346, 44079}, {13363, 34513}, {13417, 17701}, {13429, 44193}, {13440, 44192}, {13450, 22261}, {13452, 43719}, {13472, 43908}, {13561, 34514}, {13568, 15448}, {13674, 13680}, {13794, 13800}, {13851, 34786}, {13886, 13889}, {13939, 13943}, {14111, 34835}, {14216, 16659}, {14270, 16230}, {14581, 35007}, {14644, 18394}, {14649, 20410}, {14651, 39832}, {14708, 20773}, {14831, 43844}, {14900, 15562}, {14912, 19459}, {14915, 43604}, {14984, 41616}, {15004, 37505}, {15032, 19347}, {15035, 15472}, {15072, 43601}, {15080, 43584}, {15107, 37483}, {15135, 19362}, {15344, 26706}, {15460, 24650}, {15461, 24651}, {15475, 39606}, {15513, 33842}, {15581, 19596}, {15644, 43652}, {15873, 16657}, {16035, 19173}, {16116, 16119}, {16172, 32132}, {16229, 39537}, {16261, 43613}, {16473, 31760}, {16625, 34986}, {16654, 23328}, {16658, 43607}, {16836, 35268}, {17824, 32339}, {17837, 22535}, {17927, 38903}, {17984, 38907}, {18344, 39227}, {18374, 34117}, {18390, 21659}, {19123, 19132}, {19149, 43617}, {19168, 19180}, {19169, 19192}, {19414, 19430}, {19415, 19431}, {19416, 19454}, {19417, 19455}, {19440, 19446}, {19441, 19447}, {20477, 44131}, {21637, 23042}, {21639, 34788}, {22241, 32830}, {22352, 37515}, {22480, 40108}, {22531, 22656}, {22532, 22657}, {22533, 22658}, {22538, 22978}, {22661, 32123}, {23096, 38545}, {24206, 32348}, {24320, 28731}, {24817, 24822}, {25739, 26917}, {26286, 40985}, {26302, 26381}, {26303, 26405}, {26304, 26439}, {26305, 26440}, {26306, 26441}, {26371, 26398}, {26372, 26422}, {26373, 26498}, {26374, 26507}, {26375, 26516}, {26376, 26521}, {26704, 32706}, {26869, 43808}, {26896, 26909}, {26916, 26953}, {28724, 40801}, {30250, 39439}, {31860, 41448}, {32137, 32210}, {32217, 40929}, {32247, 32262}, {32249, 32276}, {32250, 32305}, {32337, 32357}, {32340, 32401}, {32649, 39265}, {32704, 40101}, {33582, 41425}, {33586, 35602}, {33802, 43976}, {33843, 37512}, {34146, 43896}, {34292, 35719}, {34382, 40318}, {35217, 36990}, {35265, 43605}, {36103, 37817}, {36750, 44097}, {36754, 44105}, {37472, 39522}, {37475, 43597}, {37499, 38852}, {37638, 41171}, {37672, 43572}, {38457, 38906}, {38461, 38900}, {38462, 38901}, {38850, 41503}, {39478, 39534}, {39808, 39820}, {39809, 39831}, {39837, 39849}, {39838, 39860}, {39874, 39879}, {40321, 41400}, {40825, 41363}, {40981, 41371}, {43394, 43823}, {43818, 43829}

X(24) is the {X(3),:ref:X(4) <X(4)>}-harmonic conjugate of X(378). For a list of other harmonic conjugates of X(24), click Tables at the top of this page.

X(24) = reflection of X(4) in X(235)

X(24) = isogonal conjugate of X(68)

X(24) = isotomic conjugate of X(20563)

X(24) = complement of X(37444)

X(24) = complement of complement of X(31304)

X(24) = anticomplement of X(11585)

X(24) = perspector of ABC and reflection of X(4) in orthic triangle

X(24) = {X(3),:ref:X(25) <X(25)>}-harmonic conjugate of X(4)

X(24) = trilinear pole of line X(924)X(6753)

X(24) = circumcircle-inverse of X(403)

X(24) = orthocentroidal circle-inverse of X(1594)

X(24) = orthoptic-circle-of-Steiner-inellipse-inverse of X(37981)