X₅₄

KOSNITA POINT

Let O be the circumcenter of triangle ABC, and O_A the circumcenter of triangle BOC. Define O_B and O_C cyclically. Then the lines AO_A, BO_B, CO_C concur in X₅₄.

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

X₅₄ (KOSNITA POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Kosnita Triangle △O_AO_BO_C — vertices O_A, O_B, O_C (Circumcenters of △BOC, △COA, △AOB where O = X₃)

• Concurrence Lines: A–O_A, B–O_B, C–O_C concurring at X₅₄

• Related Centers: X₃ (CIRCUMCENTER)

Trilinears: \(\sec(B - C) : \sec(C - A) : \sec(A - B)\)

Barycentrics: \(\sin A \sec(B - C) : \sin B \sec(C - A) : \sin C \sec(A - B)\)

Search Value (6, 9, 13): \(-6.187759052469\)

▸ Alternate Coordinates4 forms

• Trilinears: \(a/(b \cos B + c \cos C) : :\)

• Barycentrics: \(a^{2}/(S^{2} + S_{B} S_{C}) : :\)

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

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

X₅₄ lies on the Jarabek circumhyperbola, the cubics K005, K073, K112, K316, K361, K364, K373, K388, K439, K464, K466, K467, K469, K471, K499, K500, K502, K526, K566, K569, K589, K590, K629, K633, K646, K668, K822, K919, K942, K947, K976, K1107, K1180, the curves Q023, Q029, Q089, Q110, Q141.

▸ Incident Central Lines448 lines

{1, 2599}, {2, 68}, {3, 97}, {4, 184}, {5, 49}, {6, 24}, {10, 9562}, {11, 2477}, {12, 215}, {13, 3206}, {14, 3205}, {15, 10678}, {16, 10677}, {17, 3201}, {18, 3200}, {19, 16031}, {20, 4846}, {22, 36747}, {23, 5446}, {25, 3527}, {26, 3060}, {28, 1243}, {30, 3521}, {32, 9985}, {33, 9638}, {35, 6286}, {36, 73}, {39, 248}, {51, 288}, {52, 1166}, {55, 9666}, {56, 9653}, {59, 5399}, {60, 5396}, {61, 3166}, {62, 3165}, {64, 378}, {65, 1870}, {66, 3541}, {67, 5622}, {69, 95}, {70, 1899}, {71, 572}, {72, 1006}, {74, 185}, {98, 3203}, {99, 39805}, {112, 217}, {113, 3047}, {114, 3044}, {115, 9697}, {118, 3046}, {119, 3045}, {125, 3043}, {136, 6801}, {137, 33545}, {140, 252}, {143, 2070}, {154, 10594}, {155, 7503}, {156, 381}, {186, 389}, {193, 19131}, {206, 14853}, {219, 26915}, {222, 26914}, {251, 37123}, {262, 3202}, {276, 290}, {287, 37125}, {311, 34385}, {323, 1216}, {324, 37127}, {371, 6414}, {372, 6413}, {376, 10984}, {382, 18550}, {394, 7509}, {397, 11134}, {398, 11137}, {399, 33539}, {402, 12797}, {403, 12241}, {411, 34800}, {418, 2055}, {427, 31804}, {436, 8794}, {476, 36159}, {477, 36179}, {493, 12998}, {494, 12999}, {496, 40450}, {511, 1176}, {523, 36161}, {526, 38897}, {542, 18125}, {546, 10540}, {548, 13623}, {549, 34483}, {550, 11803}, {568, 1658}, {575, 895}, {576, 7556}, {577, 26876}, {632, 21357}, {695, 14153}, {826, 879}, {930, 35720}, {953, 36078}, {970, 1798}, {1062, 9637}, {1075, 21449}, {1087, 2595}, {1113, 14374}, {1114, 14375}, {1192, 35472}, {1204, 11270}, {1246, 7554}, {1263, 6343}, {1291, 15907}, {1296, 9226}, {1329, 9702}, {1351, 9715}, {1352, 18124}, {1353, 19129}, {1437, 6905}, {1439, 1443}, {1495, 10110}, {1498, 11455}, {1503, 15321}, {1506, 9696}, {1511, 12006}, {1587, 13440}, {1588, 13429}, {1593, 3426}, {1594, 6145}, {1595, 16659}, {1598, 3531}, {1656, 9703}, {1698, 9586}, {1699, 9587}, {1853, 38433}, {1879, 9378}, {1907, 16658}, {1976, 37334}, {2051, 9563}, {2071, 40647}, {2393, 32367}, {2574, 14709}, {2575, 14710}, {2616, 3737}, {2620, 7136}, {2623, 10097}, {2781, 34437}, {2886, 9701}, {2904, 34438}, {2929, 42016}, {2937, 10263}, {2981, 14818}, {3048, 5512}, {3068, 8995}, {3069, 13986}, {3090, 9306}, {3091, 9544}, {3147, 11433}, {3167, 7395}, {3292, 7550}, {3311, 6416}, {3312, 6415}, {3336, 3468}, {3357, 13452}, {3398, 36214}, {3431, 11438}, {3432, 32409}, {3448, 10116}, {3470, 38933}, {3471, 38935}, {3515, 11432}, {3517, 9777}, {3523, 13336}, {3524, 37515}, {3525, 15605}, {3526, 11935}, {3528, 37480}, {3529, 31371}, {3530, 13339}, {3532, 10605}, {3542, 14457}, {3545, 15749}, {3548, 18911}, {3563, 32692}, {3575, 34397}, {3580, 7542}, {3613, 11816}, {3627, 30531}, {3628, 40111}, {3796, 10323}, {3815, 9603}, {5050, 6391}, {5067, 5651}, {5085, 34817}, {5092, 41435}, {5093, 16195}, {5133, 12134}, {5157, 10519}, {5198, 14530}, {5254, 9604}, {5418, 9676}, {5422, 6642}, {5447, 15246}, {5462, 6153}, {5486, 35486}, {5494, 10693}, {5498, 15061}, {5504, 9730}, {5562, 34986}, {5587, 9622}, {5597, 12480}, {5598, 12481}, {5640, 7506}, {5643, 12099}, {5663, 11559}, {5721, 38850}, {5870, 10262}, {5871, 10261}, {5891, 41597}, {5898, 13363}, {5900, 25563}, {5907, 35500}, {5946, 13368}, {5972, 19481}, {6000, 14865}, {6030, 13391}, {6151, 14819}, {6198, 11429}, {6239, 12231}, {6240, 12233}, {6243, 7502}, {6400, 12232}, {6515, 41594}, {6561, 9677}, {6636, 10625}, {6640, 18952}, {6643, 37645}, {6644, 8907}, {6696, 16623}, {6794, 7765}, {6800, 7387}, {6815, 12318}, {6853, 18123}, {6875, 13323}, {7393, 15066}, {7505, 39571}, {7507, 32402}, {7514, 11444}, {7516, 7998}, {7517, 26881}, {7525, 37484}, {7526, 12111}, {7527, 12162}, {7529, 35264}, {7544, 34116}, {7547, 7699}, {7549, 41608}, {7574, 13470}, {7575, 16881}, {7576, 34782}, {7577, 16000}, {7689, 39242}, {7728, 11805}, {7731, 19362}, {8227, 9621}, {8562, 14380}, {8743, 40823}, {8795, 41204}, {8889, 38442}, {8918, 8930}, {8919, 8929}, {9140, 13561}, {9418, 39283}, {9590, 31760}, {9625, 31757}, {9652, 10895}, {9667, 10896}, {9729, 22962}, {9786, 14528}, {9818, 11441}, {9932, 15045}, {9971, 15582}, {10018, 13567}, {10024, 12370}, {10095, 13621}, {10202, 28787}, {10205, 35729}, {10226, 15055}, {10295, 13568}, {10299, 13347}, {10575, 12086}, {10601, 11465}, {10602, 11458}, {10721, 11744}, {10950, 40437}, {11004, 37478}, {11077, 41335}, {11202, 13472}, {11263, 38535}, {11264, 34826}, {11381, 13596}, {11403, 32063}, {11416, 15074}, {11439, 31861}, {11440, 18570}, {11443, 38263}, {11460, 19350}, {11461, 19354}, {11462, 19355}, {11463, 19356}, {11466, 19363}, {11467, 19364}, {11477, 19127}, {11491, 20986}, {11591, 34864}, {12007, 13622}, {12023, 12024}, {12041, 35498}, {12084, 15072}, {12106, 15019}, {12110, 40643}, {12112, 13474}, {12229, 12509}, {12230, 12510}, {12281, 19457}, {12282, 19458}, {12283, 19459}, {12284, 19456}, {12285, 19461}, {12286, 19462}, {12287, 19463}, {12288, 19464}, {12359, 41730}, {12834, 13365}, {13011, 13035}, {13012, 13036}, {13017, 19465}, {13018, 19466}, {13351, 37813}, {13364, 18369}, {13382, 21663}, {13420, 18368}, {13432, 15720}, {13433, 34565}, {13445, 13491}, {13488, 32111}, {13598, 37925}, {13754, 14118}, {13856, 38618}, {14070, 37493}, {14071, 25150}, {14106, 14111}, {14152, 26897}, {14220, 34210}, {14371, 14379}, {14491, 34417}, {14518, 34756}, {14542, 18533}, {14587, 18114}, {14641, 37944}, {14788, 37649}, {14805, 15091}, {15053, 16867}, {15056, 15068}, {15059, 24572}, {15093, 32448}, {15121, 15124}, {15232, 32381}, {15305, 32139}, {15328, 38936}, {15340, 27371}, {15401, 15537}, {15644, 22352}, {15646, 16665}, {15712, 26861}, {15760, 18433}, {16252, 16657}, {16766, 31674}, {16837, 34449}, {16868, 18390}, {17702, 34007}, {17711, 23329}, {18324, 37490}, {18374, 22336}, {18376, 40276}, {18474, 34799}, {18559, 34785}, {18874, 21308}, {18945, 32393}, {19123, 19125}, {19124, 39874}, {19136, 38005}, {19142, 22829}, {19151, 37473}, {19186, 19408}, {19187, 19409}, {19212, 33971}, {19349, 19368}, {19358, 19414}, {19359, 19415}, {19440, 19502}, {19441, 19503}, {19460, 22535}, {20190, 32599}, {20193, 30551}, {20421, 23040}, {21394, 30504}, {21849, 37939}, {22233, 41448}, {22330, 37953}, {22533, 32375}, {22950, 22972}, {23128, 26216}, {23293, 25738}, {24385, 36837}, {26877, 26889}, {26896, 26898}, {26916, 26920}, {31376, 34837}, {32110, 38448}, {32248, 39562}, {32249, 32251}, {32284, 37784}, {32321, 41715}, {32357, 34207}, {32661, 41334}, {32713, 42873}, {32737, 38394}, {33543, 37483}, {33695, 35909}, {33992, 35728}, {34351, 41596}, {34384, 39287}, {34418, 35467}, {34664, 41615}, {35480, 40242}, {35602, 37514}, {35724, 35885}, {37489, 38444}, {39808, 39810}, {39837, 39839}

▸ Functional Relations45 relations

• X₅₄ = 3 X₂ - 4 X₆₆₈₉, X₃ + 2 X₁₄₉₃, 3 X₃ - X₁₂₃₀₇, 3 X₃ + X₁₂₃₁₆, 2 X₃ + X₁₅₈₀₁, X₄ + 2 X₁₀₆₁₉, X₄ - 4 X₁₂₂₄₂, 3 X₄ - 2 X₃₂₃₄₀, 5 X₅ - 4 X₂₀₅₈₄, X₅ + 4 X₂₀₅₈₅, 3 X₅₁ - 2 X₁₁₈₀₈, 2 X₅₂ + X₁₂₂₂₆, X₆₄ + 3 X₁₇₈₂₄, X₆₄ - 3 X₃₂₃₄₅, X₇₄ + 2 X₂₉₁₄, X₇₄ + 4 X₃₂₂₂₆, X₁₁₀ + 2 X₁₅₀₈₉, 4 X₁₄₀ - X₃₅₁₉, 2 X₁₉₅ + X₇₆₉₁, X₁₉₅ + 2 X₁₀₆₁₀, 3 X₁₉₅ + X₁₂₃₀₇, 3 X₁₉₅ - X₁₂₃₁₆, 3 X₃₈₁ - 2 X₂₂₈₀₄, 4 X₃₈₉ - X₆₂₄₂, 2 X₃₈₉ + X₂₁₆₆₀, X₅₅₀ + 2 X₁₁₈₀₃, 4 X₅₇₅ - X₉₉₇₂, 5 X₆₃₁ + X₁₁₂₇₁, 5 X₆₃₁ - X₁₂₃₂₅, 5 X₆₃₁ - 4 X₃₂₃₄₈, 5 X₆₃₂ - 3 X₂₁₃₅₇, 4 X₉₇₃ - 5 X₃₅₆₇, 4 X₉₇₃ - 3 X₇₇₃₀, 4 X₉₇₃ + X₁₂₂₉₁, 4 X₉₇₃ - X₁₃₄₂₃, X₁₁₄₁ + 2 X₂₇₄₂₃, 3 X₁₁₅₇ - 4 X₁₂₀₆₀, 4 X₁₄₉₃ + X₇₆₉₁, 6 X₁₄₉₃ + X₁₂₃₀₇, 6 X₁₄₉₃ - X₁₂₃₁₆, 4 X₁₄₉₃ - X₁₅₈₀₁, 5 X₁₆₅₆ - 4 X₁₃₅₆₅, X₂₈₈₈ - 4 X₆₆₈₉, 3 X₂₉₁₇ - 5 X₁₇₈₂₁, 3 X₂₉₁₇ - X₁₇₈₄₆, 7 X₃₀₉₀ - 8 X₃₂₃₉₆, 7 X₃₅₂₃ + 2 X₁₃₄₃₁, 11 X₃₅₂₅ - 4 X₁₅₆₀₅, 5 X₃₅₆₇ - 2 X₆₁₅₂, 5 X₃₅₆₇ - 3 X₇₇₃₀, 5 X₃₅₆₇ + 4 X₁₁₅₇₇, 5 X₃₅₆₇ + X₁₂₂₉₁, 5 X₃₅₆₇ - X₁₃₄₂₃, 2 X₃₅₇₄ + X₁₂₂₅₄, 3 X₃₅₇₄ - X₃₂₃₄₀, X₃₆₂₇ - 4 X₃₀₅₃₁, 8 X₅₄₆₂ - 3 X₄₁₇₁₃, X₅₈₈₉ + 2 X₁₂₆₀₆, 3 X₅₈₉₀ - X₃₂₃₃₉, X₅₈₉₈ - 3 X₃₂₆₀₉, 3 X₅₉₄₆ - X₁₃₃₆₈, 3 X₅₉₄₆ + X₁₅₅₃₂, 3 X₆₀₃₀ - 2 X₁₃₅₆₄, 2 X₆₁₀₂ + X₂₂₈₁₅, 2 X₆₁₅₂ - 3 X₇₇₃₀, X₆₁₅₂ + 2 X₁₁₅₇₇, 2 X₆₁₅₂ + X₁₂₂₉₁, 4 X₆₁₅₃ - 3 X₄₁₇₁₃, X₆₂₄₁ + 2 X₁₂₃₀₀, X₆₂₄₁ + 4 X₁₅₇₃₉, X₆₂₄₂ + 2 X₂₁₆₆₀, X₆₂₄₂ + 4 X₄₀₆₃₂, X₆₂₈₈ - 4 X₈₂₅₄, 5 X₆₂₈₈ - 8 X₂₀₅₈₄, X₆₂₈₈ + 8 X₂₀₅₈₅, X₆₂₈₈ + 2 X₃₆₉₆₆, X₆₇₅₉ - 3 X₁₀₂₇₄, 2 X₆₇₅₉ - 3 X₃₂₃₇₉, X₇₆₉₁ - 4 X₁₀₆₁₀, 3 X₇₆₉₁ - 2 X₁₂₃₀₇, 3 X₇₆₉₁ + 2 X₁₂₃₁₆, 3 X₇₇₃₀ + 4 X₁₁₅₇₇, 3 X₇₇₃₀ + X₁₂₂₉₁, 3 X₇₇₃₀ - X₁₃₄₂₃, X₇₉₇₉ + 2 X₉₉₀₅, X₇₉₇₉ - 4 X₁₂₂₆₆, 5 X₈₂₅₄ - 2 X₂₀₅₈₄, X₈₂₅₄ + 2 X₂₀₅₈₅, 2 X₈₂₅₄ + X₃₆₉₆₆, X₈₇₁₈ + 3 X₁₃₄₈₂, X₈₇₁₈ + 4 X₃₇₄₇₂, 3 X₉₇₃₀ - 2 X₁₁₈₀₂, 7 X₉₇₈₁ - 4 X₁₁₅₇₆, 8 X₉₈₂₇ - 11 X₁₅₀₂₄, 4 X₉₈₂₇ - 3 X₄₁₅₇₈, X₉₉₀₅ + 2 X₁₂₂₆₆, X₉₉₃₅ - 4 X₁₀₂₈₂, 4 X₁₀₁₁₅ + X₁₂₂₂₆, 6 X₁₀₆₁₀ - X₁₂₃₀₇, 6 X₁₀₆₁₀ + X₁₂₃₁₆, 4 X₁₀₆₁₀ + X₁₅₈₀₁, X₁₀₆₁₉ + 2 X₁₂₂₄₂, 3 X₁₀₆₁₉ + X₃₂₃₄₀, X₁₁₂₇₁ + 4 X₃₂₃₄₈, 3 X₁₁₄₀₂ + X₃₂₃₃₃, 3 X₁₁₄₀₂ - X₃₂₃₄₁, X₁₁₄₁₂ - 4 X₁₂₃₆₃, 4 X₁₁₅₇₇ - X₁₂₂₉₁, 4 X₁₁₅₇₇ + X₁₃₄₂₃, 4 X₁₂₀₀₇ + X₁₃₆₂₂, 4 X₁₂₂₄₂ + X₁₂₂₅₄, 6 X₁₂₂₄₂ - X₃₂₃₄₀, 3 X₁₂₂₅₄ + 2 X₃₂₃₄₀, X₁₂₂₈₀ - 7 X₁₅₀₄₃, 2 X₁₂₃₀₇ + 3 X₁₅₈₀₁, 2 X₁₂₃₁₆ - 3 X₁₅₈₀₁, X₁₂₃₂₅ - 4 X₃₂₃₄₈, 4 X₁₃₃₆₅ - 5 X₁₅₀₂₆, X₁₃₄₃₂ + 11 X₁₅₇₂₀, 3 X₁₃₄₈₂ - 4 X₃₇₄₇₂, 2 X₁₄₀₄₉ + X₃₃₅₆₅, 4 X₁₄₈₆₅ - X₁₆₈₃₅, 11 X₁₅₀₂₄ - 6 X₄₁₅₇₈, 5 X₁₅₀₃₄ - 2 X₂₅₇₁₄, X₁₅₈₀₀ - 4 X₂₂₀₅₁, 5 X₁₇₈₂₁ - X₁₇₈₄₆, 5 X₁₇₈₂₁ - 6 X₃₂₃₉₁, X₁₇₈₄₆ - 6 X₃₂₃₉₁, X₁₈₃₆₈ + 2 X₃₄₅₆₄, X₂₀₅₈₄ + 5 X₂₀₅₈₅, 4 X₂₀₅₈₄ + 5 X₃₆₉₆₆, 4 X₂₀₅₈₅ - X₃₆₉₆₆, X₃₂₃₅₂ + 2 X₄₀₆₃₂, 2 X₃₂₃₆₉ - 3 X₃₂₃₉₅.

• X₃ = K(X₂₀,X₂) X₄ = K(X(20,X₂₀ X₅ = K(X₄,X₂) X₁₃ = K(X₁₃,X₁) X₁₇ = K(X₁₃,X₃) X₁₈ = K(X₁₄,X₃) X₁₄₀ = K(X₃, X₂) X₂₅₁ = K(X₆, X₆) X₄₈₁ = K(X₁₇₅,X₁) X₄₈₂ = K(X₁₇₆,X₁)

• X₅₄ = midpoint of X_i and X_j for these {i,j}: {1, 9905}, {3, 195}, {4, 12254}, {5, 36966}, {125, 14049}, {389, 40632}, {973, 11577}, {1263, 6343}, {1493, 10610}, {2929, 42016}, {3574, 10619}, {5889, 32338}, {6276, 6277}, {7691, 15801}, {11271, 12325}, {11597, 15089}, {12026, 31675}, {12291, 13423}, {12307, 12316}, {13368, 15532}, {17824, 32345}, {21660, 32352}, {27196, 27423}, {32333, 32341}, {32346, 32354}

• X₅₄ = reflection of X_i in X_j for these {i,j}: {1, 12266}, {3, 10610}, {4, 3574}, {5, 8254}, {52, 10115}, {110, 11597}, {195, 1493}, {265, 11804}, {1141, 27196}, {1209, 6689}, {2888, 1209}, {2914, 32226}, {2917, 32391}, {3519, 21230}, {3574, 12242}, {6145, 32351}, {6152, 973}, {6153, 5462}, {6242, 32352}, {6288, 5}, {7691, 3}, {7728, 11805}, {7979, 1}, {9972, 9977}, {9977, 575}, {11412, 41590}, {12254, 10619}, {12300, 15739}, {12785, 10}, {12797, 402}, {13121, 10066}, {13122, 10082}, {13423, 6152}, {15062, 14130}, {15800, 20424}, {15801, 195}, {20424, 22051}, {21230, 140}, {21660, 40632}, {23061, 15137}, {32196, 143}, {32338, 12606}, {32352, 389}, {32379, 10274}, {33565, 125}, {41590, 12363}

• X₅₄ = isogonal conjugate of X₅

• X₅₄ = isotomic conjugate of X₃₁₁

• X₅₄ = inverse-in-circumcircle of X₁₁₅₇

• X₅₄ = complement of X₂₈₈₈

• X₅₄ = anticomplement of X₁₂₀₉

• X₅₄ = circumcircle-inverse of X₁₁₅₇

• X₅₄ = Brocard-circle-inverse of X₁₈₃₃₅

• X₅₄ = polar conjugate of X₃₂₄

• X₅₄ = antigonal image of X₃₃₅₆₅

• X₅₄ = symgonal image of X₁₁₅₉₇

• X₅₄ = Thomson-isogonal conjugate of X₃₅₈₈₅

• X₅₄ = psi-transform of X₁₄₆₅₆

• X₅₄ = X_i-anticomplementary conjugate of X_j for these (i,j): {1166, 8}, {2216, 2888}

• X₅₄ = X_i-complementary conjugate of X_j for these (i,j): {3432, 10}, {40140, 21231}

• X₅₄ = X_i-Ceva conjugate of X_j for these (i,j): {5, 2120}, {95, 97}, {97, 33629}, {275, 8882}, {288, 6}, {933, 23286}, {1141, 1157}, {1166, 25044}, {8884, 8883}, {14533, 26887}, {14587, 110}, {18315, 2623}, {18831, 15412}, {20574, 1614}, {23286, 19208}, {35196, 2169}, {39287, 95}

• X₅₄ = X_i-cross conjugate of X_j for these (i,j): {3, 96}, {6, 275}, {184, 14533}, {186, 74}, {389, 4}, {523, 110}, {570, 2}, {1199, 1173}, {2594, 1}, {2623, 18315}, {3269, 39181}, {8603, 2981}, {8604, 6151}, {13366, 6}, {13367, 3}, {14533, 97}, {16030, 95}, {16035, 8884}, {19189, 1298}, {21638, 8795}, {21660, 3519}, {23286, 933}, {30451, 4558}, {32352, 6145}, {39199, 109}, {39201, 112}, {39478, 901}, {40632, 13418}, {41218, 654}

• X₅₄ = cevapoint of X_i and X_j for these (i,j): {3, 1147}, {6, 184}, {15, 3200}, {16, 3201}, {32, 3202}, {39, 3203}, {55, 3204}, {58, 9563}, {61, 3205}, {62, 3206}, {215, 2245}, {523, 8901}, {572, 9562}, {654, 41218}, {3270, 9404}

• X₅₄ = crosspoint of X_i and X_j for these (i,j): {1, 3461}, {3, 3463}, {4, 3459}, {5, 2121}, {95, 275}, {3467, 3469}, {3489, 3490}

• X₅₄ = crosssum of X_i and X_j for these (i,j): {1, 3460}, {2, 17035}, {3, 195}, {4, 3462}, {11, 8819}, {51, 216}, {54, 2120}, {61, 8839}, {62, 8837}, {233, 3078}, {288, 38816}, {523, 8902}, {627, 628}, {1953, 7069}, {2600, 41218}, {3336, 3468}, {3470, 38933}, {3471, 38935}, {6368, 35442}, {8918, 8930}, {8919, 8929}, {12077, 41221}, {17434, 41219}, {21011, 21807}

• X₅₄ = trilinear pole of line {50, 647}

• X₅₄ = crossdifference of every pair of points on line {2081, 2600}

• X₅₄ = trilinear pole of line X₅₀X₆₄₇ (the polar of X₃₂₄ wrt polar circle)

• X₅₄ = pole wrt polar circle of trilinear polar of X₃₂₄

• X₅₄ = X₄₈-isoconjugate (polar conjugate) of X₃₂₄

• X₅₄ = X₉₂-isoconjugate of X₂₁₆

• X₅₄ = intersection of tangents to hyperbola {A, B, C,X₄,X₅}} at X₄ and X₃₄₅₉

• X₅₄ = {X₂₅₉₅,X₂₅₉₆}-harmonic conjugate of X₁₀₈₇

• X₅₄ = trilinear product of vertices of circumnormal triangle

• X₅₄ = intersection of tangents at X₃ and X₄ to Neuberg cubic K001

• X₅₄ = exsimilicenter of nine-point circle and sine-triple-angle circle

• X₅₄ = homothetic center of orthocevian triangle of X₃ and circumorthic triangle

• X₅₄ = perspector of ABC and unary cofactor triangle of reflection triangle

• X₅₄ = X₃-of-reflection-triangle-of-X₅

• X₅₄ = perspector of ABC and cross-triangle of ABC and circumorthic triangle

• X₅₄ = perspector of ABC and Hatzipolakis-Moses triangle

• X₅₄ = X₁₉₁-of-orthic-triangle if ABC is acute

• X₅₄ = trilinear product of vertices of X₄-altimedial triangle

• X₅₄ = X_i-isoconjugate of X_j for these (i,j): {1, 5}, {2, 1953}, {6, 14213}, {7, 7069}, {8, 1393}, {10, 18180}, {19, 343}, {25, 18695}, {31, 311}, {37, 17167}, {38, 17500}, {48, 324}, {51, 75}, {52, 91}, {53, 63}, {54, 1087}, {69, 2181}, {76, 2179}, {79, 35194}, {81, 21011}, {86, 21807}, {92, 216}, {94, 2290}, {100, 21102}, {110, 2618}, {143, 2962}, {158, 5562}, {162, 6368}, {163, 18314}, {217, 1969}, {255, 13450}, {293, 39569}, {304, 3199}, {318, 30493}, {326, 14569}, {467, 1820}, {523, 2617}, {561, 40981}, {610, 13157}, {655, 2600}, {656, 35360}, {661, 14570}, {662, 12077}, {811, 15451}, {823, 17434}, {897, 41586}, {920, 8800}, {921, 41587}, {1154, 2166}, {1209, 2216}, {1263, 1749}, {1474, 42698}, {1568, 36119}, {1577, 1625}, {1707, 27364}, {1826, 16697}, {1895, 8798}, {1956, 32428}, {1972, 2313}, {1973, 28706}, {2081, 32680}, {2153, 33529}, {2154, 33530}, {2167, 36412}, {2180, 5392}, {2184, 42459}, {2222, 6369}, {2595, 7135}, {2596, 2603}, {2599, 3615}, {2621, 18114}, {2964, 25043}, {4560, 35307}, {4575, 23290}, {8769, 41588}, {10015, 35321}, {17438, 31610}, {18070, 35319}, {18833, 27374}, {20577, 36148}, {23181, 24006}, {24000, 35442}, {24041, 41221}, {27371, 34055}, {32678, 41078}, {36035, 36831}

• X₅₄ = barycentric product X_i*X_j for these {i,j}: {1, 2167}, {3, 275}, {4, 97}, {6, 95}, {25, 34386}, {32, 34384}, {39, 39287}, {48, 40440}, {63, 2190}, {69, 8882}, {75, 2148}, {83, 16030}, {92, 2169}, {96, 1993}, {99, 2623}, {110, 15412}, {140, 288}, {182, 42300}, {184, 276}, {226, 35196}, {249, 8901}, {252, 1994}, {253, 33629}, {264, 14533}, {287, 19189}, {290, 41270}, {323, 1141}, {338, 14587}, {340, 11077}, {371, 16032}, {372, 16037}, {394, 8884}, {401, 1298}, {520, 16813}, {523, 18315}, {525, 933}, {571, 34385}, {577, 8795}, {578, 37872}, {647, 18831}, {648, 23286}, {662, 2616}, {801, 16035}, {850, 14586}, {1073, 38808}, {1092, 8794}, {1105, 19180}, {1157, 13582}, {1166, 37636}, {1502, 14573}, {1577, 36134}, {1634, 39182}, {2052, 19210}, {2245, 39277}, {2888, 40140}, {2984, 11245}, {3051, 41488}, {3431, 4993}, {3904, 36078}, {4551, 39177}, {6504, 8883}, {6563, 32692}, {7763, 41271}, {11140, 25044}, {13366, 31617}, {14096, 39283}, {14528, 19188}, {14618, 15958}, {15351, 19208}, {15414, 32713}, {19166, 41890}, {19170, 40448}, {19174, 28724}, {20574, 40684}, {22052, 39286}, {35311, 39181}, {37225, 39274}, {39201, 42405}

• X₅₄ = barycentric quotient X_i/X_j for these {i,j}: {1, 14213}, {2, 311}, {3, 343}, {4, 324}, {6, 5}, {15, 33529}, {16, 33530}, {24, 467}, {25, 53}, {31, 1953}, {32, 51}, {41, 7069}, {42, 21011}, {50, 1154}, {51, 36412}, {58, 17167}, {63, 18695}, {64, 13157}, {69, 28706}, {72, 42698}, {95, 76}, {96, 5392}, {97, 69}, {110, 14570}, {112, 35360}, {154, 42459}, {160, 41480}, {163, 2617}, {184, 216}, {186, 14918}, {187, 41586}, {213, 21807}, {232, 39569}, {251, 17500}, {252, 11140}, {275, 264}, {276, 18022}, {288, 40410}, {323, 1273}, {389, 34836}, {393, 13450}, {512, 12077}, {523, 18314}, {526, 41078}, {560, 2179}, {570, 1209}, {571, 52}, {577, 5562}, {604, 1393}, {647, 6368}, {649, 21102}, {654, 6369}, {661, 2618}, {850, 15415}, {933, 648}, {1141, 94}, {1157, 37779}, {1166, 40393}, {1173, 31610}, {1298, 1972}, {1333, 18180}, {1437, 16697}, {1501, 40981}, {1510, 20577}, {1576, 1625}, {1609, 41587}, {1843, 27371}, {1953, 1087}, {1971, 32428}, {1973, 2181}, {1974, 3199}, {1993, 39113}, {2148, 1}, {2167, 75}, {2168, 91}, {2169, 63}, {2174, 35194}, {2190, 92}, {2207, 14569}, {2501, 23290}, {2616, 1577}, {2623, 523}, {2963, 25043}, {2965, 143}, {3049, 15451}, {3053, 41588}, {3124, 41221}, {3202, 40588}, {3269, 35442}, {3284, 1568}, {3518, 14129}, {5063, 5891}, {6748, 14978}, {8648, 2600}, {8739, 6117}, {8740, 6116}, {8770, 27364}, {8795, 18027}, {8882, 4}, {8883, 6515}, {8884, 2052}, {8901, 338}, {9409, 14391}, {10311, 39530}, {10312, 30506}, {11077, 265}, {13338, 13364}, {13342, 27355}, {13366, 233}, {14270, 2081}, {14533, 3}, {14573, 32}, {14575, 217}, {14579, 1263}, {14585, 418}, {14586, 110}, {14587, 249}, {14642, 8798}, {15109, 21230}, {15412, 850}, {15958, 4558}, {16029, 1591}, {16030, 141}, {16032, 34391}, {16034, 1592}, {16035, 13567}, {16037, 34392}, {16813, 6528}, {18315, 99}, {18353, 565}, {18831, 6331}, {19180, 41005}, {19189, 297}, {19208, 39352}, {19210, 394}, {19306, 1749}, {20574, 31626}, {21449, 9291}, {21461, 36300}, {21462, 36301}, {21741, 2599}, {23195, 42445}, {23286, 525}, {25044, 1994}, {26887, 3164}, {32445, 42453}, {32640, 36831}, {32661, 23181}, {32692, 925}, {33629, 20}, {33872, 14845}, {34384, 1502}, {34386, 305}, {34397, 11062}, {34756, 39114}, {35196, 333}, {36078, 655}, {36134, 662}, {37636, 1225}, {38808, 15466}, {39109, 41536}, {39177, 18155}, {39201, 17434}, {39287, 308}, {40440, 1969}, {40633, 13595}, {41213, 41222}, {41270, 511}, {41271, 2165}, {41331, 27374}, {41373, 41481}, {41488, 40016}, {42293, 34983}, {42300, 327}

• X₅₄ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 3460, 2599}, {2, 2888, 1209}, {2, 9545, 1147}, {2, 18912, 26917}, {3, 1493, 15801}, {3, 1993, 11412}, {3, 7592, 5890}, {3, 11402, 7592}, {3, 12161, 5889}, {3, 12316, 12307}, {3, 15087, 6102}, {3, 16266, 2979}, {3, 19210, 97}, {3, 25044, 1157}, {3, 32046, 5012}, {4, 184, 1614}, {4, 275, 4994}, {4, 578, 15033}, {4, 1614, 14157}, {4, 19467, 12289}, {5, 49, 110}, {5, 567, 13434}, {5, 9706, 9705}, {5, 14516, 41171}, {6, 24, 3567}, {6, 2917, 973}, {6, 14533, 8882}, {6, 14585, 10312}, {6, 15073, 8537}, {6, 19189, 9792}, {6, 19357, 24}, {6, 19468, 6152}, {24, 12291, 12380}, {24, 19357, 11464}, {25, 9707, 26882}, {26, 36749, 3060}, {36, 35197, 7356}, {49, 110, 9705}, {49, 567, 5}, {51, 3518, 38848}, {51, 10282, 3518}, {52, 18475, 7488}, {110, 9706, 49}, {110, 13434, 5}, {140, 11245, 26879}, {140, 40631, 252}, {143, 5944, 2070}, {154, 10982, 10594}, {155, 7503, 11459}, {155, 37506, 7503}, {182, 1092, 631}, {184, 275, 26887}, {184, 578, 4}, {184, 3574, 32379}, {184, 11424, 6759}, {184, 15033, 14157}, {185, 3520, 74}, {185, 11430, 3520}, {186, 1199, 389}, {195, 5012, 10203}, {195, 10610, 7691}, {195, 12307, 12316}, {217, 1970, 112}, {275, 8884, 4}, {275, 38808, 8884}, {288, 20574, 1173}, {323, 37126, 1216}, {378, 1181, 6241}, {381, 9704, 156}, {389, 13366, 1199}, {389, 13367, 186}, {389, 21660, 6242}, {394, 7509, 7999}, {394, 37476, 7509}, {427, 31804, 34224}, {569, 1147, 2}, {578, 6759, 11424}, {578, 10274, 3574}, {627, 628, 1273}, {631, 14912, 18916}, {973, 6152, 7730}, {973, 32391, 24}, {1157, 25042, 3}, {1173, 38848, 51}, {1173, 39667, 288}, {1181, 11425, 378}, {1209, 6689, 2}, {1216, 37513, 37126}, {1263, 10285, 31392}, {1493, 12363, 1993}, {1495, 10110, 34484}, {1498, 35502, 11455}, {1511, 36153, 12006}, {1593, 11456, 12290}, {1593, 19347, 11456}, {1594, 6146, 25739}, {1614, 15033, 4}, {1899, 37119, 23294}, {1994, 7488, 52}, {2070, 14627, 143}, {2595, 2596, 1087}, {2917, 13423, 12380}, {2937, 10263, 15107}, {3091, 9544, 10539}, {3518, 37505, 1173}, {3520, 15032, 185}, {3541, 6776, 11457}, {3567, 7730, 973}, {3567, 11464, 24}, {3567, 12291, 6152}, {3567, 13423, 7730}, {3567, 19468, 12380}, {3574, 21659, 32365}, {3796, 37498, 10323}, {5012, 34148, 3}, {5422, 6642, 15024}, {5622, 32245, 32234}, {5889, 11422, 12161}, {5889, 19167, 19194}, {5890, 11423, 7592}, {5946, 15532, 13368}, {6102, 32136, 15087}, {6143, 33565, 14076}, {6146, 23292, 1594}, {6150, 18016, 3}, {6152, 11577, 12291}, {6640, 18952, 26913}, {6644, 36753, 15043}, {6750, 35717, 4}, {6759, 11424, 4}, {7526, 18445, 12111}, {7542, 13292, 3580}, {7547, 18396, 18394}, {7592, 11402, 11423}, {7592, 16030, 19168}, {7592, 32333, 32339}, {7699, 18394, 7547}, {7722, 32607, 74}, {7730, 12291, 13423}, {7730, 13423, 6152}, {8254, 20585, 36966}, {8254, 36966, 6288}, {8882, 14533, 33629}, {9706, 13434, 110}, {9707, 11426, 9781}, {9730, 12038, 22467}, {9781, 26882, 25}, {9818, 11441, 15058}, {9905, 12266, 7979}, {10066, 10082, 1}, {10274, 12254, 1614}, {10282, 37505, 51}, {10605, 35477, 11468}, {10619, 12242, 4}, {10984, 13346, 376}, {11402, 16030, 19170}, {11402, 32333, 32341}, {11425, 17809, 1181}, {11427, 18925, 4}, {11427, 32354, 3574}, {11430, 15032, 74}, {11449, 15043, 6644}, {11464, 13423, 2917}, {12006, 36153, 15037}, {12038, 22467, 15035}, {12161, 12606, 15801}, {12227, 32607, 7722}, {12234, 23358, 32352}, {13121, 13122, 7979}, {13198, 15463, 74}, {13353, 22115, 140}, {13366, 13367, 389}, {13366, 21660, 12234}, {13367, 21638, 19185}, {13367, 32352, 23358}, {13621, 15038, 10095}, {13630, 43394, 3}, {14389, 14516, 5}, {16029, 16034, 6}, {16030, 16035, 3}, {16030, 19170, 19209}, {16031, 16036, 19}, {16032, 16037, 2}, {16035, 19210, 8883}, {17821, 17846, 2917}, {17928, 36752, 15045}, {18388, 21659, 4}, {18570, 34783, 11440}, {18925, 32346, 12254}, {19095, 19096, 6}, {19172, 19180, 19206}, {19459, 39588, 12283}, {21638, 21660, 19207}, {37481, 37814, 15053}

Notes

See John Rigby, “Brief notes on some forgotten geometrical theorems,” Mathematics and Informatics Quarterly 7 (1997) 156-158.

Darij Grinberg, A New Circumcenter Question.

The above construction of X₅₄ generalizes. Suppose that P and Q are points (as functions of a,b,c). Let A' = Q-of-BCP, B' = Q-of-CAP, C' = Q-of-ABP. If the lines AA', BB', CC' concur, the perspector is called the Kosnita(P,Q) point, denoted by K(P,Q). (Randy Hutson, 9/23/2011)

Let N_AN_BN_C be the reflection triangle of X₅. Let O_A be the circumcenter of AN_BN_C, and define O_B and O_C cyclically. Triangle O_AO_BO_C is perspective to ABC at X₅₄, homothetic to the orthic-of-orthocentroidal triangle at X₅₄, and orthologic to the reflection triangle at X₅₄. (Randy Hutson, June 7, 2019)

Let (N_a) be the reflection of the nine-point circle in BC, and define (N_b) and (N_c) cyclically. X₅₄ is the radical center of (N_a), (N_b), (N_c). The tangents at A, B, C to the Napoleon-Feuerbach cubic K005 concur in X₅₄. Let Ref_aRef_bRef_c be the reflection triangle. Let L_a be the trilinear polar of Ref_a, and define L_b and L_c cyclically. Let A″ be L_b∩L_c, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₅₄. (Randy Hutson, July 23, 2015)

let N_a = reflection of X₅ in the line BC, and define N_b and N_c cyclically. The medial triangle of N_aN_bN_c is perspective to ABC, and the perspector is X₅₄. (Dasari Naga Vijay Krishna, June 8, 2021)