X₂₄

PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE

X₂₄ is the perspector of ABC and orthic-of-orthic triangle.

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

X₂₄ (PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Orthic Triangle △H_aH_bH_c — vertices H_a, H_b, H_c (Pedal triangle of Orthocenter X₄ (altitude feet))

• Derived Triangle: Orthic-of-Orthic Triangle △H'_aH'_bH'_c — vertices H'_a, H'_b, H'_c (Orthic triangle of the orthic triangle △H_a H_b H_c)

• Concurrence Lines: A–H'_a, B–H'_b, C–H'_c concurring at X₂₄

• Central Lines: {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(\sec A \cos 2A : \sec B \cos 2B : \sec C \cos 2C, Tllrilinears \sec A - 2 \cos A : \sec B - 2 \cos B : \sec C - 2 \cos C\)

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

Search Value (6, 9, 13): \(1.591165914392\)

▸ Alternate Coordinates3 forms

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

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

• Barycentrics: \(a^{2} (a^{4} + b^{4} + c^{4} - 2 a^{2} b^{2} - 2 a^{2} c^{2})/(a^{2} - b^{2} - c^{2}) : :\)

▸ Linear Combinations2 combinations

• X₂₄ = \(6\) X₂ + \((J^{2} - 5)\) X₃

• X₂₄ = \((a^{2} - b^{2} - c^{2}) (a^{2} + b^{2} - c^{2}) (a^{2} - b^{2} + c^{2})\) X₃ + \(2 a^{2} b^{2} c^{2}\) X₄

▸ Incident Central Lines627 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, 3162}, {1748, 42700}, {1753, 2077}, {1824, 32613}, {1828, 32612}, {1853, 23294}, {1857, 7040}, {1861, 25440}, {1862, 33814}, {1871, 2355}, {1872, 26285}, {1876, 37582}, {1878, 23961}, {1892, 11374}, {1899, 9833}, {1900, 33862}, {1902, 3579}, {1905, 2646}, {1971, 39643}, {1975, 28706}, {1994, 9545}, {2080, 11380}, {2192, 11461}, {2203, 5752}, {2211, 5017}, {2332, 4262}, {2356, 37576}, {2373, 30251}, {2374, 30247}, {2501, 5926}, {2698, 32542}, {2781, 38851}, {2914, 12316}, {2918, 11387}, {2930, 32234}, {2935, 10721}, {2936, 23235}, {2965, 8746}, {2967, 44162}, {2979, 37486}, {3043, 19504}, {3044, 39810}, {3047, 19456}, {3060, 11449}, {3068, 8276}, {3069, 8277}, {3085, 10831}, {3086, 10832}, {3087, 31400}, {3095, 44089}, {3098, 12294}, {3100, 9645}, {3167, 12160}, {3197, 11460}, {3259, 39479}, {3292, 14531}, {3311, 5411}, {3312, 5410}, {3357, 11381}, {3426, 11270}, {3431, 3527}, {3432, 25044}, {3438, 3442}, {3439, 3443}, {3447, 14385}, {3448, 12412}, {3455, 11623}, {3532, 16835}, {3564, 41584}, {3574, 23358}, {3576, 7713}, {3581, 15068}, {3767, 5523}, {3785, 15574}, {3796, 15045}, {3964, 32001}, {4293, 18954}, {4294, 10833}, {5012, 15043}, {5013, 9609}, {5023, 33885}, {5050, 12167}, {5085, 7716}, {5090, 26446}, {5092, 19124}, {5095, 12584}, {5185, 38599}, {5186, 33813}, {5307, 39578}, {5338, 15931}, {5347, 36745}, {5359, 40938}, {5398, 44113}, {5446, 12038}, {5449, 18474}, {5480, 35228}, {5562, 9306}, {5594, 10784}, {5595, 10783}, {5603, 11365}, {5609, 13148}, {5621, 32274}, {5622, 8549}, {5640, 13434}, {5651, 11793}, {5657, 7718}, {5667, 14673}, {5690, 12135}, {5706, 38879}, {5878, 32111}, {5879, 40082}, {5892, 13336}, {5938, 41377}, {5944, 5946}, {5962, 16391}, {5963, 32762}, {5972, 15473}, {6101, 37494}, {6104, 8737}, {6105, 8738}, {6193, 6515}, {6200, 9683}, {6221, 9694}, {6243, 16266}, {6247, 16655}, {6284, 9673}, {6291, 12974}, {6293, 15139}, {6361, 9911}, {6396, 11474}, {6406, 12975}, {6413, 8954}, {6414, 32589}, {6458, 26886}, {6560, 35776}, {6561, 35777}, {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}

▸ Functional Relations29 relations

• X₂₄ = homothetic center of the tangential triangle and the triangle obtained by reflecting X₄ in the sidelines of ABC.

• X₂₄ = reflection of X₄ in X₂₃₅

• X₂₄ = isogonal conjugate of X₆₈

• X₂₄ = isotomic conjugate of X₂₀₅₆₃

• X₂₄ = complement of X₃₇₄₄₄

• X₂₄ = complement of complement of X₃₁₃₀₄

• X₂₄ = anticomplement of X₁₁₅₈₅

• X₂₄ = perspector of ABC and reflection of X₄ in orthic triangle

• X₂₄ = {X₃,X₂₅}-harmonic conjugate of X₄

• X₂₄ = trilinear pole of line X₉₂₄X₆₇₅₃

• X₂₄ = circumcircle-inverse of X₄₀₃

• X₂₄ = orthocentroidal circle-inverse of X₁₅₉₄

• X₂₄ = orthoptic-circle-of-Steiner-inellipse-inverse of X₃₇₉₈₁

• X₂₄ = orthoptic-circle-of-Steiner-circumellipse-inverse of anticomplement of X₃₇₉₂₉

• X₂₄ = de-Longchamps-circle-inverse of anticomplement of X₃₇₉₅₁

• X₂₄ = X₂₄₉-Ceva conjugate of X₁₁₂

• X₂₄ = X₅₂-cross conjugate of X₄

• X₂₄ = crosspoint of X₁₀₇ and X₂₅₀

• X₂₄ = crosssum of X_i and X_j for these (i,j): (6, 161), (125, 520), (637, 638)

• X₂₄ = X₄-Hirst inverse of X₄₂₁

• X₂₄ = X₄₆-of-orthic-triangle if ABC is acute

• X₂₄ = X₅₆-of-the-tangential triangle if ABC is acute

• X₂₄ = tangential isogonal conjugate of X₁₄₉₈

• X₂₄ = exsimilicenter of circumcircle and tangential circle when ABC is acute (Yuda Chen, November 7, 2021)

• X₂₄ = insimilicenter of circumcircle and tangential circle when ABC is obtuse

• X₂₄ = inverse-in-polar-circle of X₂₀₇₂

• X₂₄ = homothetic center of tangential and circumorthic triangles

• X₂₄ = homothetic center of orthic and Kosnita triangles

• X₂₄ = X_i-isoconjugate of X_j for these (i,j): (75, 2351), (91, 3)

Notes

As a point on the Euler line, X₂₄ has Shinagawa coefficients \((2F, -E - 2F)\).

Let H_aH_bH_c be the orthic triangle. Let A″ = inverse-in-circumcircle of H_a, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₂₄. (Randy Hutson, September 5, 2015)

X₂₄ is the {X₃,X₄}-harmonic conjugate of X₃₇₈.