X₄₃₁

EULER X₂₈₃-5th-SUBSTITUTION POINT

X₄₃₁ = midpoint of X₄ and X₃₁₃₈₄

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

X₄₃₁ (EULER X₂₈₃-5th-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)

• Related Centers: X₁₈₅₈ (ZOSMA TRANSFORM OF X₂₂₅), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \((v + w)\sec A : (w + u)\sec B : (u + v)\sec C\), where \(u : v : w\) = X₂₈₃; e.g., \(u = u(A,B,C) = (\cos A)/(\cos B + \cos C)\)

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

Search Value (6, 9, 13): \(-10.356504391821\)

▸ Alternate Coordinates1 form

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

▸ Functional Relations6 relations

• X₄₃₁ = midpoint of X₄ and X₃₁₃₈₄

• X₄₃₁ = orthic-isogonal conjugate of X₁₈₅₈

• X₄₃₁ = X_i-Ceva conjugate of X_j for these (i,j): {4, 1858}, {40097, 523}

• X₄₃₁ = barycentric product X_i*X_j for these {i,j}: {19, 18692}, {225, 45206}, {1858, 40149}

• X₄₃₁ = barycentric quotient X_i/X_j for these {i,j}: {1195, 283}, {1858, 1812}, {18692, 304}, {45206, 332}

• X₄₃₁ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {4, 24, 7491}, {4, 403, 6841}, {4, 411, 1885}, {4, 2476, 427}, {4, 3144, 3559}, {4, 3542, 3560}, {4, 6353, 6872}, {4, 6622, 6837}, {4, 6825, 1593}, {4, 6867, 7507}, {4, 6869, 44438}, {4, 7548, 23047}, {235, 37376, 25}, {429, 37982, 442}, {1596, 37406, 4}

Notes

As a point on the Euler line, X₄₃₁ has Shinagawa coefficients \((\$aS_{A}\$F,- \$aS_{A}\$(E - F) - 2abcF). X₄₃₁ lies on these lines: {2, 3}, {11, 40985}, {12, 1824}, {19, 50036}, {119, 135}, {136, 25640}, {225, 7363}, {388, 11401}, {1112, 12826}, {1478, 26377}, {1829, 12047}, {1834, 44113}, {1842, 5521}, {1848, 31936}, {2886, 5130}, {3485, 11396}, {3822, 39579}, {5086, 12135}, {5139, 20621}, {10572, 11363}, {10895, 11391}, {16178, 42422}, {21664, 34338}, {44092, 44545}\)