X₄₄₅

EULER X₇₉-10th-SUBSTITUTION POINT

X₄₄₅ = isotomic conjugate of the isogonal conjugate of X₄₄₀₉₅

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

X₄₄₅ (EULER X₇₉-10th-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), {281, 3580}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₂₈₁ (-CROSS CONJUGATE OF X₉), X₃₅₈₀ (INTERSECTION OF LINES X)

Trilinears: \((v + w)\csc 2A : (w + u)\csc 2B : (u + v)\csc 2C\), where \(u : v : w\) = X₇₉; e.g., \(u(a,b,c) = 1/(1 + 2 \cos A)\)

Barycentrics: \((v + w)\sec A : (w + u)\sec B : (u + v)\sec C\)

Search Value (6, 9, 13): \(3.760057866603\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines25 lines

{2, 3}, {81, 18679}, {92, 30690}, {196, 51358}, {264, 18139}, {273, 27186}, {281, 3580}, {317, 5278}, {318, 32858}, {340, 3578}, {648, 42045}, {1844, 31938}, {1990, 37631}, {2322, 2895}, {3219, 7282}, {3936, 31623}, {8747, 26131}, {14165, 40214}, {14581, 50181}, {17907, 19684}, {18026, 40447}, {19742, 27377}, {35201, 40940}, {37635, 41083}, {45038, 48381}

▸ Functional Relations9 relations

• X₄₄₅ = isotomic conjugate of the isogonal conjugate of X₄₄₀₉₅

• X₄₄₅ = polar conjugate of the isogonal conjugate of X₅₀₀

• X₄₄₅ = X_i-isoconjugate of X_j for these (i,j): {1794, 2160}, {2259, 7100}, {2982, 8606}

• X₄₄₅ = X_i-Dao conjugate of X_j for these (i,j): {63, 5249}, {7100, 18591}, {16585, 52381}, {40937, 52388}

• X₄₄₅ = cevapoint of X₅₀₀ and X₄₄₀₉₅

• X₄₄₅ = crosspoint of X₉₂ and X₅₂₄₁₂

• X₄₄₅ = barycentric product X_i*X_j for these {i,j}: {75, 1844}, {76, 44095}, {92, 16585}, {264, 500}, {273, 31938}, {319, 1838}, {340, 45926}, {1841, 33939}, {1859, 52421}, {1865, 34016}, {5249, 52412}, {6734, 7282}

• X₄₄₅ = barycentric quotient X_i/X_j for these {i,j}: {35, 1794}, {442, 52388}, {500, 3}, {942, 7100}, {1838, 79}, {1841, 2160}, {1844, 1}, {1859, 7073}, {1865, 8818}, {5249, 52381}, {6198, 943}, {14547, 8606}, {16585, 63}, {31938, 78}, {44095, 6}, {45926, 265}, {46883, 52375}, {52412, 40435}

• X₄₄₅ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {297, 25986, 2}, {7282, 52412, 3219}

Notes

As a point on the Euler line, X₄₄₅ has Shinagawa coefficients \(((\$aS_{A}\$ + 2abc)F,\$a\$S^{2})\).