X₄₁₂

EULER X₂₉-2nd-SUBSTITUTION POINT

X₄₁₂ = orthocentroida1-circle-inverse of X₅₂₂₄₈

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

X₄₁₂ (EULER X₂₉-2nd-SUBSTITUTION POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {40, 92}

• Related Centers: X₆₅₃ (TRILINEAR POLE OF LINE X₁X₄), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄₀ (BEVAN POINT), X₉₂ (CEVAPOINT OF INCENTER AND CLAWSON POINT)

Trilinears: \((1/v)^{2} + (1/w)^{2} - (1/u)^{2} : (1/w)^{2} + (1/u)^{2} - (1/v)^{2} : (1/u)^{2} + (1/v)^{2} - (1/w)^{2}\), where \(u : v : w\) = X₂₉; e.g., \(u = u(A,B,C) = (\sec A)/(\cos B + \cos C)\)

Barycentrics: \(a[(1/v)^{2} + (1/w)^{2} - (1/u)^{2}] : b[(1/w)^{2} + (1/u)^{2} - (1/v)^{2}] : c[(1/u)^{2} + (1/v)^{2} - (1/w)^{2}]\)

Search Value (6, 9, 13): \(8.765223466970\)

▸ Alternate Coordinates1 form

• Barycentrics: \((a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) \cdot (a^{6} + a^{5} b - 2 a^{4} b^{2} - 2 a^{3} b^{3} + a^{2} b^{4} + a b^{5} + a^{5} c - a^{4} b c - a b^{4} c + b^{5} c - 2 a^{4} c^{2} + 2 a^{2} b^{2} c^{2} - 2 a^{3} c^{3} - 2 b^{3} c^{3} + a^{2} c^{4} - a b c^{4} + a c^{5} + b c^{5}) : :\)

▸ Incident Central Lines57 lines

{1, 7138}, {2, 3}, {7, 7952}, {33, 10884}, {40, 92}, {46, 158}, {57, 1895}, {63, 318}, {64, 5786}, {65, 243}, {84, 7020}, {162, 580}, {165, 39585}, {185, 2659}, {208, 44697}, {225, 775}, {264, 8822}, {273, 7013}, {278, 962}, {515, 5174}, {516, 1838}, {572, 2326}, {579, 8748}, {820, 2655}, {938, 44695}, {946, 17923}, {1071, 1872}, {1118, 3474}, {1148, 36279}, {1155, 1940}, {1158, 1748}, {1445, 1712}, {1698, 39531}, {1715, 1896}, {1754, 8747}, {1784, 3336}, {1785, 3075}, {1788, 1857}, {1859, 9943}, {1861, 34831}, {1887, 10391}, {1897, 3868}, {1902, 7009}, {2322, 46011}, {3188, 36118}, {3362, 23707}, {3579, 39529}, {4313, 34231}, {5307, 11471}, {5435, 40836}, {5706, 41083}, {6198, 18444}, {6684, 39574}, {7282, 18650}, {16318, 45991}, {17606, 42387}, {45141, 45985}

▸ Functional Relations8 relations

• X₄₁₂ = orthocentroida1-circle-inverse of X₅₂₂₄₈

• X₄₁₂ = X₇₀₄₅-Ceva conjugate of X₆₅₃

• X₄₁₂ = X₂₄₀₂₆-Dao conjugate of X₄₄₄₂₆

• X₄₁₂ = cevapoint of X₄ and X₁₇₁₅

• X₄₁₂ = crosssum of X₈₂₂ and X₃₂₇₀

• X₄₁₂ = barycentric product X_i*X_j for these {i,j}: {75, 38860}, {92, 3562}

• X₄₁₂ = barycentric quotient X_i/X_j for these {i,j}: {3562, 63}, {38860, 1}

• X₄₁₂ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 4, 52248}, {3, 4, 29}, {3, 7524, 7531}, {3, 7567, 7572}, {4, 3144, 235}, {4, 4212, 37368}, {4, 4219, 7513}, {4, 5125, 7541}, {4, 5142, 7563}, {4, 7412, 14004}, {4, 7531, 7524}, {4, 7543, 15763}, {4, 7551, 44225}, {4, 7554, 7497}, {4, 37028, 7518}, {4, 37305, 11109}, {4, 37379, 4198}, {4, 37410, 4194}, {4, 37414, 17555}, {4, 37417, 2}, {20, 6838, 37419}, {20, 37437, 48890}, {27, 26003, 379}, {46, 158, 653}, {46, 51282, 158}, {140, 44225, 7551}, {407, 1885, 4}, {1013, 37235, 29}, {1155, 42385, 1940}, {3522, 7518, 37028}, {3559, 5125, 37279}, {6684, 39574, 52412}, {7497, 7554, 31903}, {7524, 7531, 29}, {7567, 30268, 3}, {14018, 37383, 37389}, {42789, 42790, 37115}

Notes

As a point on the Euler line, X₄₁₂ has Shinagawa coefficients \((FS^{2}, -\$bcS_{B}S_{C}\$ - FS^{2})\).