X₃₁₄₆

X₂₅₃-CEVA CONJUGATE OF X₂

For P = X₆₉, the parameter k = = -SW S^2 / (SA SB SC), and P' = X₃₁₄₆

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

X₃₁₄₆ (X₂₅₃-CEVA CONJUGATE OF X₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {8, 144}

• Related Centers: X₂₅₃ (-CROSS CONJUGATE OF X₂), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₈ (NAGEL POINT), X₁₄₄ (ANTICOMPLEMENT OF X₇)

Trilinears: \(u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z)\), where \(x : y : z\) = X₂₅₃ and \(u : v : w\) = X₂

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\), where \(f(a,b,c) = u(-u/x + v/y + w/z)\)

Search Value (6, 9, 13): \(-30.594584021172\)

▸ Alternate Coordinates2 forms

• Trilinears: \(4 \cos A - 3 \sin B \sin C : 4 \cos B - 3 \sin C \sin A : 4 \cos C - 3 \sin A \sin B\)

• Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = 5a^{4} - 2a^{2}(b^{2} + c^{2}) - 3(b^{2} - c^{2})^{2} (E\). Danneels)

X₃₁₄₆ lies on the cubics K117, K127, K347, K425, K461, K558, K841.

▸ Incident Central Lines272 lines

{1, 5556}, {2, 3}, {7, 950}, {8, 144}, {10, 9778}, {11, 5265}, {12, 5281}, {13, 5366}, {14, 5365}, {33, 4296}, {34, 3100}, {35, 10590}, {36, 10591}, {40, 3219}, {51, 10574}, {52, 6241}, {55, 5229}, {56, 5225}, {61, 5335}, {62, 5334}, {63, 5175}, {65, 9961}, {69, 11469}, {74, 12295}, {78, 1750}, {84, 3218}, {107, 5896}, {145, 515}, {146, 6193}, {147, 7900}, {148, 2794}, {149, 2829}, {150, 10725}, {151, 10726}, {152, 10727}, {153, 5840}, {154, 5893}, {165, 9780}, {185, 3060}, {193, 1503}, {226, 4313}, {265, 12244}, {279, 4872}, {280, 5081}, {315, 10513}, {316, 3926}, {317, 6527}, {323, 11441}, {329, 3984}, {346, 7270}, {347, 7282}, {355, 4678}, {385, 2996}, {388, 390}, {389, 11002}, {393, 3284}, {485, 6453}, {486, 6454}, {489, 1271}, {490, 1270}, {497, 3304}, {498, 4324}, {499, 4316}, {511, 5921}, {517, 3621}, {519, 9589}, {527, 12625}, {528, 12632}, {542, 8596}, {543, 7758}, {576, 6776}, {578, 11003}, {671, 10991}, {938, 3586}, {942, 11220}, {944, 3623}, {946, 3622}, {971, 3868}, {990, 5262}, {1056, 9655}, {1058, 9668}, {1105, 1629}, {1125, 9779}, {1131, 3068}, {1132, 3069}, {1151, 2671}, {1152, 2672}, {1173, 4846}, {1181, 1994}, {1219, 4514}, {1285, 5305}, {1327, 8960}, {1350, 3620}, {1351, 12174}, {1352, 7929}, {1376, 8165}, {1467, 8544}, {1478, 3746}, {1479, 4293}, {1493, 12254}, {1495, 11449}, {1498, 1993}, {1539, 12121}, {1587, 6419}, {1588, 6420}, {1614, 9545}, {1697, 8545}, {1698, 12512}, {1699, 3616}, {1737, 4333}, {1836, 3486}, {1837, 3474}, {1853, 5894}, {1870, 9538}, {1891, 4329}, {1968, 10313}, {2548, 7756}, {2549, 7747}, {2777, 3448}, {2883, 11206}, {2979, 5907}, {3047, 9934}, {3055, 11742}, {3058, 9657}, {3070, 3592}, {3071, 3594}, {3085, 3585}, {3086, 3583}, {3087, 5158}, {3101, 11471}, {3180, 5869}, {3181, 5868}, {3241, 4301}, {3306, 9841}, {3329, 5395}, {3346, 10152}, {3431, 5944}, {3436, 6253}, {3476, 12701}, {3488, 11036}, {3579, 5818}, {3580, 5925}, {3590, 10147}, {3591, 10148}, {3601, 5226}, {3618, 10541}, {3624, 12571}, {3672, 5716}, {3679, 5493}, {3681, 7957}, {3785, 7802}, {3812, 5918}, {3817, 5550}, {3818, 10519}, {3869, 12688}, {3870, 12651}, {3873, 12680}, {3876, 5927}, {3877, 9856}, {3935, 6769}, {4295, 10572}, {4298, 10580}, {4304, 5703}, {4305, 12047}, {4308, 12053}, {4309, 5270}, {4311, 9614}, {4312, 6738}, {4314, 5290}, {4317, 4857}, {4325, 10072}, {4330, 10056}, {4355, 6744}, {4440, 11851}, {4855, 5748}, {4907, 7273}, {5007, 5286}, {5012, 11424}, {5032, 8550}, {5057, 12679}, {5080, 5537}, {5128, 7285}, {5204, 10589}, {5217, 10588}, {5218, 10895}, {5250, 11372}, {5251, 12511}, {5254, 5304}, {5260, 5584}, {5284, 8273}, {5302, 7964}, {5319, 11648}, {5328, 5438}, {5434, 9670}, {5435, 9581}, {5446, 5890}, {5480, 7864}, {5596, 9968}, {5609, 7728}, {5640, 9729}, {5643, 9815}, {5656, 9716}, {5658, 11015}, {5663, 6243}, {5698, 5794}, {5734, 5882}, {5763, 9963}, {5787, 12246}, {5795, 11530}, {5806, 10167}, {5842, 12667}, {5870, 6280}, {5871, 6279}, {5889, 6000}, {6054, 10992}, {6194, 6248}, {6256, 10528}, {6321, 9862}, {6337, 7773}, {6427, 7581}, {6428, 7582}, {6482, 12818}, {6483, 12819}, {6515, 12324}, {6519, 8981}, {6564, 9540}, {6759, 9544}, {6800, 11425}, {7288, 10896}, {7710, 7783}, {7712, 11430}, {7738, 7745}, {7785, 8721}, {7787, 12203}, {7795, 7842}, {7928, 10516}, {7989, 10164}, {8117, 8122}, {8118, 8121}, {9140, 10990}, {9580, 9785}, {9613, 10624}, {9730, 9781}, {9748, 12110}, {9799, 9965}, {10404, 11038}, {10446, 10454}, {10453, 12545}, {10625, 11459}, {10711, 10993}, {10738, 12248}, {10749, 12253}, {11004, 11456}, {11411, 12293}, {11412, 11455}, {11416, 11470}, {11417, 11473}, {11418, 11474}, {11420, 11475}, {11421, 11476}, {11432, 11820}, {11523, 12536}, {11750, 12897}, {12160, 12315}, {12219, 12292}, {12220, 12294}, {12223, 12298}, {12224, 12299}, {12226, 12300}, {12282, 12290}, {12317, 12902}, {12684, 12690}

▸ Functional Relations22 relations

• X₃₁₄₆ = midpoint of X_i in X_j for these {i,j}: {382, 5073}, {3529, 11541}

• X₃₁₄₆ = reflection of X_i in X_j for these (i,j): (2, 3543), (3, 3627), (4, 382), (8, 5691), (20, 4), (74, 12295), (145, 962), (146, 10721), (147, 10722), (148, 10723), (149, 10724), (150, 10725), (151, 10726), (152, 10727), (153, 10728), (376, 3830), (550, 3853), (944, 12699), (1657, 5), (3448, 10733), (3529, 3), (3869, 12688), (5059, 20), (5189, 10296), (5925, 6247), (5984, 148), (6225, 5895), (6240, 7553), (6241, 52), (6361, 355), (9862, 6321), (9961, 65), (10430, 3586), (10575, 5446), (11001, 381), (11411, 12293), (11412, 12162), (11750, 12897), (12103, 12102), (12111, 11381), (12121, 1539), (12219, 12292), (12220, 12294), (12221, 12296), (12222, 12297), (12223, 12298), (12224, 12299), (12225, 1885), (12226, 12300), (12244, 265), (12246, 5787), (12248, 10738), (12253, 10749), (12279, 185), (12317, 12902), (12383, 7728), (12536, 11523)

• X₃₁₄₆ = isogonal conjugate of X₃₅₃₂

• X₃₁₄₆ = isotomic conjugate of X₃₅₅₁₀

• X₃₁₄₆ = isotomic conjugate of polar conjugate of X₃₃₆₃₀

• X₃₁₄₆ = complement of X₅₀₅₉

• X₃₁₄₆ = anticomplement of X₂₀

• X₃₁₄₆ = X₂₅₃-Ceva conjugate of X₂

• X₃₁₄₆ = crosspoint of X₁₁₃₁ and X₁₁₃₂

• X₃₁₄₆ = crosspoint of PU(i) for these i: 116, 117

• X₃₁₄₆ = crosssum of X₁₁₅₁ and X₁₁₅₂

• X₃₁₄₆ = orthocentroidal-circle-inverse of X₃₈₃₂

• X₃₁₄₆ = intersection of tangents to Kiepert hyperbola at X₁₁₃₁ and X₁₁₃₂

• X₃₁₄₆ = orthocenter of cevian triangle of X₂₅₃

• X₃₁₄₆ = X₁₁₅₃₁-of-orthic-triangle if ABC is acute

• X₃₁₄₆ = polar conjugate of X₃₈₂₅₃

• X₃₁₄₆ = pole wrt polar circle of trilinear polar of X₃₈₂₅₃ (line X₅₂₃X₁₃₄₇₃)

• X₃₁₄₆ = circumcircle-inverse of X₃₇₉₄₁

• X₃₁₄₆ = X₁₉-isoconjugate of X₃₆₆₀₉

• X₃₁₄₆ = exsimilicenter of polar circle and de Longchamps circle

• X₃₁₄₆ = Ehrmann-mid-to-Johnson similarity image of X₃₈₂

• X₃₁₄₆ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (2, 4, 3832), (2, 20, 3522), (2, 3832, 5068), (2, 3854, 5), (2, 5059, 20), (2, 6872, 11106), (3, 4, 3091), (3, 5, 3525), (3, 381, 3628), (3, 382, 3627), (3, 546, 3090), (3, 1657, 12103), (3, 3090, 10303), (3, 3091, 2), (3, 3525, 3523), (3, 3529, 20), (3, 3627, 4), (3, 3628, 631), (3, 3843, 5079), (3, 5072, 632), (3, 5076, 546), (3, 5079, 140), (3, 5198, 1995), (3, 7530, 3518), (3, 12103, 376), (4, 5, 3839), (4, 20, 2), (4, 21, 6870), (4, 24, 6623), (4, 376, 5), (4, 377, 6894), (4, 382, 3543), (4, 411, 6871), (4, 550, 5056), (4, 631, 381), (4, 1593, 7378), (4, 1657, 3523), (4, 3090, 546), (4, 3522, 5068), (4, 3523, 3854), (4, 3524, 3855), (4, 3528, 3545), (4, 3529, 3), (4, 3533, 3858), (4, 3534, 7486), (4, 3545, 3843), (4, 3575, 6995), (4, 3651, 6843), (4, 3855, 3845), (4, 5059, 3522), (4, 6240, 7487), (4, 6622, 10151), (4, 6815, 7394), (4, 6836, 5046), (4, 6850, 6839), (4, 6851, 6840), (4, 6865, 6957), (4, 6868, 6837), (4, 6869, 6838), (4, 6875, 6866), (4, 6876, 6867), (4, 6897, 6849), (4, 6899, 6893), (4, 6903, 6929), (4, 6906, 6844), (4, 6909, 5187), (4, 6916, 6835), (4, 6925, 2475), (4, 6934, 6848), (4, 6938, 6847), (4, 6942, 6968), (4, 6948, 6953), (4, 6988, 7548), (4, 7580, 5177), (4, 7714, 1906), (4, 10299, 3850), (4, 10431, 6895), (4, 10996, 6997), (4, 11001, 631), (4, 11111, 10883), (4, 11541, 3529), (5, 376, 3523), (5, 550, 12100), (5, 1657, 376), (5, 3523, 2), (5, 3627, 12102), (5, 3830, 4), (5, 3839, 3854), (5, 10124, 1656), (5, 12103, 3), (20, 1559, 2060), (20, 3091, 3), (20, 3523, 376), (20, 3543, 4), (20, 3830, 3854), (20, 3839, 3523), (20, 4208, 7411), (20, 5056, 10304), (20, 7396, 11413), (20, 7486, 3528), (20, 10304, 550), (21, 5177, 2), (22, 7378, 2), (23, 12086, 3), (24, 12085, 2071), (25, 7396, 2), (26, 3520, 10298), (55, 5229, 5261), (56, 5225, 5274), (140, 3534, 3528), (140, 3545, 7486), (140, 3843, 3545), (140, 3857, 5079), (140, 3860, 5), (140, 7486, 2), (376, 1657, 20), (376, 3525, 3), (376, 3830, 3839), (376, 3839, 2), (376, 12100, 10304), (376, 12102, 3091), (377, 452, 2), (377, 11114, 452), (378, 7387, 7488), (381, 550, 631), (381, 631, 5056), (381, 3853, 4), (381, 10304, 2), (381, 11001, 10304), (382, 1657, 3830), (382, 11541, 3091), (388, 6284, 390), (404, 6919, 2), (405, 4208, 2), (411, 1012, 4189), (427, 10565, 2), (428, 7386, 7398), (443, 5129, 2), (443, 11113, 5129), (489, 12323, 1271), (490, 12322, 1270), (497, 7354, 3600), (546, 632, 5072), (546, 3090, 3091), (546, 3627, 5076), (546, 5076, 4), (546, 12103, 12108), (546, 12108, 5), (548, 1656, 3524), (548, 3845, 1656), (549, 3851, 5067), (549, 3861, 3851), (550, 631, 10304), (550, 3628, 3), (550, 3853, 381), (550, 11001, 20), (631, 5056, 2), (631, 11001, 550), (631, 12100, 3523), (632, 5072, 3090), (858, 4232, 2), (946, 5731, 3622), (950, 9579, 7), (962, 6223, 5905), (1006, 6993, 2), (1327, 9681, 8960), (1370, 6995, 2), (1479, 10483, 4293), (1532, 6943, 5154), (1597, 11414, 7503), (1656, 3845, 3855), (1657, 3830, 5), (1657, 3854, 3522), (1657, 3860, 3528), (1657, 12102, 3525), (1699, 4297, 3616), (1885, 12173, 4), (1907, 6823, 5133), (2043, 2044, 3524), (2071, 6623, 2), (2475, 6872, 2), (2478, 6904, 2), (2979, 11439, 5907), (3060, 12279, 185), (3070, 6459, 7585), (3071, 6460, 7586), (3090, 10303, 2), (3091, 10303, 3090), (3091, 11541, 5059), (3149, 6909, 4188), (3151, 6994, 2), (3152, 7518, 2), (3515, 10151, 6622), (3518, 7464, 3), (3522, 3832, 2), (3523, 3839, 5), (3523, 10303, 12108), (3524, 3855, 1656), (3525, 3529, 12103), (3525, 12102, 3839), (3525, 12108, 10303), (3526, 3850, 5071), (3526, 8703, 10299), (3528, 3545, 140), (3528, 3843, 7486), (3529, 3627, 3091), (3529, 5076, 10303), (3529, 12102, 3523), (3530, 3858, 5055), (3530, 5055, 3533), (3530, 12101, 3858), (3534, 3830, 3860), (3534, 3843, 140), (3534, 5079, 3), (3543, 3839, 3830), (3543, 5056, 3853), (3543, 5059, 3832), (3543, 10303, 5076), (3545, 3857, 3091), (3583, 4299, 3086), (3585, 4302, 3085), (3586, 4292, 938), (3616, 10248, 1699), (3627, 5073, 11541), (3627, 11541, 20), (3627, 12102, 3830), (3817, 7987, 5550), (3839, 3854, 3832), (3843, 5079, 3857), (3850, 8703, 3526), (3853, 11001, 5056), (3855, 12811, 3091), (3857, 5079, 3545), (4188, 5187, 2), (4189, 6871, 2), (4190, 5046, 2), (4304, 9612, 5703), (4314, 5290, 10578), (5004, 5005, 9909), (5056, 10304, 631), (5071, 10299, 3526), (5189, 7519, 2), (5446, 10575, 5890), (6039, 6040, 10011), (6284, 12943, 388), (6825, 6888, 2), (6826, 6992, 2), (6831, 6932, 5141), (6833, 6960, 2), (6834, 6972, 2), (6837, 6908, 2), (6838, 6847, 2), (6839, 6987, 2), (6848, 6890, 2), (6850, 7491, 6987), (6868, 6923, 6908), (6869, 6938, 20), (6884, 6889, 2), (6885, 6929, 6964), (6891, 6979, 2), (6917, 6930, 6846), (6926, 6953, 2), (6928, 6948, 6926), (6999, 7406, 2), (7000, 7374, 1513), (7354, 12953, 497), (7379, 7390, 2), (7386, 7398, 2), (7391, 7500, 2), (7503, 11414, 6636), (7517, 12084, 186), (7526, 12083, 7512), (7527, 12082, 7492), (7737, 7748, 5286), (7802, 11185, 3785), (9580, 10106, 9785), (11412, 11455, 12162), (12102, 12103, 5), (12102, 12108, 546), (32614, 32615, 110)

Notes

As a point on the Euler line, X₃₁₄₆ has Shinagawa coefficients \((1, -4)\).

The point X₃₁₄₆ exemplifies a theorem based on a discussion in Tran Quang Hung and Peter Moses, Hyacinthos 25668. The theorem (Peter Moses, April 10, 2017) is as follows.

Suppose that P is a point on a circumhyperbola H. Let A'B'C' be the cevian triangle of P and let O' be the circumcircle of A'B'C'. Let A″ be the point other than A' that lies on both O' and line BC, and define B″ and C″ cyclically. The orthocenter of A″B″C″ lies on the isogonal conjugate of H.

Writing P' for the orthocenter of of A″B″C″, examples of the mapping P → P' include the following:

• Jerabek hyperbola → Euler line

• Feuerbach hyperbola → X₁X₃

• Kiepert hyperbola → Brocard axis.

The image of a point P = a^2 / (S^2+k SB SC) : : on the Jerabek hyperbola is the following point on the Euler line:

(S^2 SA SB+S^2 SB^2+S^2 SA SC+4 S^2 SB SC+3 k SA SB^2 SC+S^2 SC^2+3 k SA SB SC^2+2 k SB^2 SC^2) (S^4 SA^3+S^4 SA^2 SB+S^4 SA SB^2-2 k S^2 SA^3 SB^2-k^2 SA^4 SB^3+S^4 SA^2 SC+7 S^4 SA SB SC-2 k S^2 SA^3 SB SC+2 S^4 SB^2 SC+8 k S^2 SA^2 SB^2 SC-3 k^2 SA^4 SB^2 SC+2 k S^2 SA SB^3 SC+k^2 SA^3 SB^3 SC+S^4 SA SC^2-2 k S^2 SA^3 SC^2+2 S^4 SB SC^2+8 k S^2 SA^2 SB SC^2-3 k^2 SA^4 SB SC^2+14 k S^2 SA SB^2 SC^2+3 k^2 SA^3 SB^2 SC^2+2 k S^2 SB^3 SC^2+7 k^2 SA^2 SB^3 SC^2-k^2 SA^4 SC^3+2 k S^2 SA SB SC^3+k^2 SA^3 SB SC^3+2 k S^2 SB^2 SC^3+7 k^2 SA^2 SB^2 SC^3+5 k^2 SA SB^3 SC^3)

For P = X₆₈, see X₁₃₃₂₂.