X₄₀₅

EULER LINE INTERCEPT OF LINE X₁X_6

X₄₀₅ is the {X₂,X₃}-harmonic conjugate of X₄₇₄. For a list of harmonic conjugates of X₄₀₅, click Tables at the top of this page.

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

X₄₀₅ (EULER LINE INTERCEPT OF LINE X₁X_6) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Central Lines: {2, 377}, {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₃₇₇ (EULER LINE INTERCEPT OF LINE X), X₃ (CIRCUMCENTER)

Trilinears: \(bc(a + b + c) + abc \cos A : :\)

Barycentrics: \(b + c + a(1 + \cos A) : :\)

Search Value (6, 9, 13): \(2.922607564013\)

▸ Alternate Coordinates2 forms

• Trilinears: \(bcS + raS_{A} : :\)

• Barycentrics: \(a (a^{3} - a b^{2} - 2 a b c - 2 b^{2} c - a c^{2} - 2 b c^{2}) : :\)

▸ Incident Central Lines359 lines

{1, 6}, {2, 3}, {8, 943}, {10, 55}, {11, 19755}, {12, 10198}, {31, 5711}, {32, 5275}, {34, 1214}, {35, 1376}, {36, 3624}, {38, 19729}, {39, 19758}, {40, 1730}, {46, 3812}, {51, 5752}, {56, 226}, {57, 3916}, {58, 940}, {63, 942}, {65, 1708}, {75, 7283}, {76, 16992}, {78, 3305}, {84, 8726}, {100, 5175}, {104, 5811}, {142, 4292}, {144, 11036}, {145, 6767}, {171, 19730}, {183, 18140}, {191, 5902}, {192, 19851}, {194, 17000}, {198, 5257}, {200, 3697}, {210, 3811}, {212, 2654}, {222, 1935}, {241, 1448}, {274, 1975}, {283, 5398}, {284, 965}, {306, 5814}, {329, 999}, {333, 10449}, {355, 10267}, {386, 4383}, {388, 1617}, {390, 5082}, {480, 6743}, {495, 3436}, {496, 10527}, {497, 19843}, {498, 1329}, {499, 3816}, {511, 19782}, {516, 5584}, {517, 5250}, {519, 3303}, {528, 4309}, {535, 9657}, {551, 3304}, {572, 2360}, {579, 4877}, {580, 2328}, {595, 5710}, {612, 5266}, {614, 19724}, {748, 1193}, {756, 976}, {846, 986}, {908, 11374}, {920, 13750}, {936, 3601}, {938, 5273}, {946, 3428}, {952, 16202}, {962, 5759}, {966, 4254}, {968, 3931}, {970, 5943}, {971, 10884}, {978, 17123}, {988, 5272}, {990, 12689}, {997, 1864}, {1043, 9534}, {1046, 7262}, {1071, 7330}, {1175, 6061}, {1210, 5745}, {1259, 5722}, {1319, 9850}, {1330, 18134}, {1351, 15988}, {1377, 5414}, {1378, 2066}, {1385, 5777}, {1398, 6356}, {1437, 6176}, {1465, 19372}, {1466, 3911}, {1468, 3720}, {1470, 5433}, {1478, 7742}, {1479, 2886}, {1482, 3877}, {1486, 4026}, {1490, 3576}, {1573, 2241}, {1577, 21789}, {1612, 5716}, {1655, 7754}, {1696, 3986}, {1697, 9623}, {1706, 4002}, {1709, 9943}, {1712, 17102}, {1714, 1834}, {1722, 17594}, {1737, 11507}, {1746, 5786}, {1750, 7987}, {1770, 5880}, {1824, 9895}, {1836, 12609}, {1858, 7082}, {1901, 17398}, {1993, 22136}, {2067, 9678}, {2078, 9578}, {2093, 4004}, {2098, 3884}, {2099, 3878}, {2177, 3214}, {2238, 2271}, {2268, 5783}, {2280, 3691}, {2283, 19931}, {2292, 3924}, {2295, 14974}, {2346, 5686}, {2452, 13869}, {2550, 4294}, {2551, 3085}, {2893, 5224}, {2900, 3740}, {2932, 3035}, {3036, 10087}, {3052, 5264}, {3053, 5277}, {3158, 3921}, {3189, 6600}, {3216, 4255}, {3218, 5708}, {3219, 3868}, {3220, 17306}, {3244, 19750}, {3286, 15668}, {3306, 4652}, {3333, 10582}, {3338, 3742}, {3339, 5665}, {3434, 15171}, {3452, 13411}, {3556, 19869}, {3562, 22117}, {3600, 7677}, {3617, 3871}, {3618, 19766}, {3622, 7373}, {3626, 19751}, {3634, 4413}, {3635, 8162}, {3636, 19739}, {3647, 5221}, {3666, 19728}, {3670, 17054}, {3678, 3715}, {3679, 3746}, {3682, 14547}, {3685, 16824}, {3689, 3983}, {3711, 4015}, {3757, 4385}, {3763, 4265}, {3814, 5172}, {3820, 5552}, {3821, 12579}, {3822, 10895}, {3826, 4302}, {3841, 12953}, {3872, 9957}, {3876, 3940}, {3890, 4861}, {3897, 10246}, {3898, 22837}, {3915, 10459}, {3922, 5183}, {3923, 12567}, {3925, 6284}, {3929, 11518}, {3951, 11520}, {3962, 12559}, {4018, 11529}, {4256, 17749}, {4267, 5737}, {4295, 5698}, {4297, 7700}, {4304, 6666}, {4305, 10609}, {4306, 6180}, {4313, 5809}, {4340, 4648}, {4354, 9640}, {4391, 22160}, {4421, 19875}, {4533, 12260}, {4647, 5695}, {4666, 5045}, {4668, 8168}, {4673, 16821}, {4857, 11235}, {5080, 9654}, {5119, 5836}, {5120, 5746}, {5132, 17259}, {5204, 5267}, {5252, 11510}, {5253, 5550}, {5255, 8616}, {5263, 19853}, {5271, 5295}, {5282, 21808}, {5294, 7085}, {5316, 6700}, {5330, 10247}, {5362, 11485}, {5367, 11486}, {5426, 5506}, {5432, 10958}, {5437, 15803}, {5444, 15446}, {5450, 6260}, {5492, 7986}, {5534, 18908}, {5537, 9588}, {5554, 5690}, {5563, 11194}, {5587, 10902}, {5603, 5758}, {5640, 19771}, {5657, 10306}, {5691, 15931}, {5703, 18228}, {5705, 9581}, {5715, 8227}, {5731, 5817}, {5750, 8804}, {5766, 9785}, {5774, 17751}, {5779, 12528}, {5794, 10572}, {5812, 5886}, {5815, 10578}, {5818, 11491}, {5844, 12000}, {5901, 10680}, {5905, 6147}, {5934, 8109}, {5935, 8110}, {5985, 12188}, {6261, 12664}, {6265, 12691}, {6554, 15288}, {6598, 15175}, {6667, 10090}, {6684, 8582}, {6688, 15489}, {6738, 18249}, {6763, 18398}, {6764, 8236}, {6765, 10389}, {6796, 10175}, {7171, 17612}, {7587, 7593}, {7588, 8080}, {7713, 10319}, {7793, 16999}, {8053, 16828}, {8062, 23189}, {8077, 8079}, {8158, 21168}, {8225, 8233}, {8543, 12848}, {8983, 19014}, {9659, 14667}, {9669, 11680}, {9778, 11024}, {9812, 15911}, {9841, 10857}, {9956, 11499}, {10039, 11508}, {10056, 12607}, {10176, 18233}, {10283, 12001}, {10386, 20075}, {10392, 12447}, {10441, 17185}, {10585, 10592}, {10822, 21746}, {10827, 14798}, {10882, 10888}, {10966, 11376}, {11220, 12684}, {11372, 12565}, {11396, 21318}, {11502, 17606}, {12246, 21151}, {12330, 14647}, {12388, 12397}, {12433, 12649}, {12520, 12688}, {12521, 12692}, {12522, 12693}, {12523, 12694}, {12524, 12695}, {12527, 21620}, {12704, 13374}, {12739, 18254}, {13405, 18250}, {13883, 18999}, {13936, 19000}, {13971, 19013}, {15239, 21164}, {15808, 19746}, {16112, 17653}, {16173, 22560}, {16819, 20172}, {16826, 19719}, {16830, 23407}, {16994, 17128}, {16996, 17129}, {17194, 17811}, {17300, 20077}, {17303, 19857}, {17349, 20018}, {17478, 21761}, {17718, 21077}, {18357, 18518}, {18481, 18761}, {19786, 19844}, {19808, 19845}, {19812, 19841}, {19827, 19842}, {19836, 22654}, {19863, 23361}

▸ Functional Relations6 relations

• X₄₀₅ = complement of X₃₇₇

• X₄₀₅ = anticomplement of X₈₇₂₈

• X₄₀₅ = crosssum of X₈₃₈ and X₁₀₁₅

• X₄₀₅ = crossdifference of every pair of points on line X₅₁₃X₆₄₇

• X₄₀₅ = inverse-in-orthocentroidal circle of X₄₄₂

• X₄₀₅ = {X₁,X₉}-harmonic conjugate of X₇₂

Notes

As a point on the Euler line, X₄₀₅ has Shinagawa coefficients \((abc \$a\$ + S^{2}, -S^{2})\).