X₄₀₁

BAILEY POINT

X₄₀₁ = anticomplement of X₂₉₇

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

X₄₀₁ (BAILEY POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {297, 441}, {2, 3} (Euler line)

• Related Centers: X₂₉₇ (-HIRST INVERSE OF X₄), X₄₄₁ (COMPLEMENT OF X₂₉₇), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \([\sin 2B \sin 2C - \sin^{2}(2A)](\csc A) : :\)

Barycentrics: \(\sin 2B \sin 2C - \sin^{2}(2A) : :\)

Search Value (6, 9, 13): \(3.684283855313\)

▸ Alternate Coordinates3 forms

• Barycentrics: \(\tan B + \tan C - \tan A - \cot \omega : :\)

• Barycentrics: \(a^{8} - 2 a^{6} (b^{2} + c^{2}) + a^{4} (b^{4} + b^{2} c^{2} + c^{4}) + b^{2} c^{2} (b^{2} - c^{2})^{2} : :\)

• Barycentrics: \(\tan A - \cot \omega ' : :\), where ω’ is the Brocard angle of the orthic triangle

▸ Linear Combinations1 combination

• X₄₀₁ = \(3 S_{A} S_{B} S_{C} S_{\omega}\) X₂ - \(S^{4}\) X₂₀

▸ Incident Central Lines101 lines

{2, 3}, {6, 3164}, {32, 40814}, {50, 338}, {51, 12110}, {81, 18667}, {94, 32662}, {95, 14767}, {97, 276}, {99, 36212}, {182, 42329}, {184, 11257}, {187, 41254}, {193, 6527}, {194, 1993}, {216, 36794}, {239, 10538}, {248, 290}, {253, 20080}, {264, 577}, {275, 9290}, {287, 511}, {317, 6389}, {323, 525}, {325, 39359}, {340, 15526}, {343, 7750}, {394, 1975}, {524, 39352}, {571, 41760}, {648, 3284}, {925, 2698}, {1073, 38256}, {1297, 6037}, {1503, 14721}, {1632, 2393}, {1899, 36998}, {1916, 15391}, {1968, 2052}, {1994, 7839}, {2782, 40870}, {2896, 34850}, {3087, 40680}, {3098, 42313}, {3100, 4366}, {3260, 4558}, {3289, 39355}, {3292, 23235}, {3329, 22240}, {3580, 14712}, {4296, 6645}, {5422, 7787}, {5480, 36988}, {5641, 35178}, {6515, 20065}, {6709, 36422}, {6748, 17035}, {7738, 11427}, {7925, 35088}, {8591, 40112}, {8859, 23967}, {9308, 15905}, {9512, 23200}, {9863, 11442}, {9873, 11550}, {9983, 18018}, {10329, 23292}, {10341, 10350}, {10420, 43654}, {11064, 15351}, {11078, 41997}, {11092, 41998}, {11596, 21639}, {12117, 13857}, {12215, 25332}, {13366, 32467}, {14570, 22151}, {14919, 16077}, {14920, 39008}, {14961, 41253}, {15066, 34360}, {15262, 41678}, {15595, 29012}, {15988, 18666}, {19571, 36790}, {19585, 32529}, {19924, 41145}, {21243, 32152}, {23582, 23590}, {23583, 37765}, {27377, 41005}, {30227, 37645}, {32428, 39081}, {34396, 39906}, {34545, 41334}, {35278, 42671}, {36748, 43980}, {38382, 39101}, {40897, 41008}, {41194, 41196}, {41195, 41197}

▸ Functional Relations14 relations

• X₄₀₁ = reflection of X₂₉₇ in X₄₄₁

• X₄₀₁ = isogonal conjugate of X₁₉₈₇

• X₄₀₁ = isotomic conjugate of X₁₉₇₂

• X₄₀₁ = anticomplement of X₂₉₇

• X₄₀₁ = circumcircle-inverse of X₃₇₉₁₈

• X₄₀₁ = perspector of conic {A, B, C,X₉₅,X₆₄₈}}

• X₄₀₁ = X₂-Hirst inverse of X₃

• X₄₀₁ = inverse-in-Steiner-circumellipse of X₃

• X₄₀₁ = X_i-Ceva conjugate of X_j for these (i,j): (2, 39081), (287, 2), (511, 385)

• X₄₀₁ = crosspoint of X₂₇₆ and X₂₉₀

• X₄₀₁ = crosssum of X₂₁₇ and X₂₃₇

• X₄₀₁ = crossdifference of every pair of points on line X₅₁X₆₄₇

• X₄₀₁ = {X₂₄₇₉,X₂₄₈₀}-harmonic conjugate of X₃

• X₄₀₁ = crossdifference of PU(157)

Notes

As a point on the Euler line, X₄₀₁ has Shinagawa coefficients \((EF + F^{2} - S^{2}, 2S^{2})\).