X₄₂₅

X₄-HIRST INVERSE OF X₂₁

X₄₂₅ is the X₄-hirst inverse of X₂₁.

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

X₄₂₅ (X₄-HIRST INVERSE OF X₂₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {243, 662}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₂₄₃ (-LINE CONJUGATE OF X₇₃), X₆₆₂ (TRILINEAR POLE OF LINE X₁X₂₁)

Trilinears: \((u^{2} - vw)\sec A : (v^{2} - wu)\sec B : (v^{2} - uv)\sec C\), where \(u : v : w\) = X₂₈₃; e.g., \(u = u(A,B,C) = (\cos A)/(\cos B + \cos C)\)

Barycentrics: \((u^{2} - vw)\tan A : (v^{2} - wu)\tan B : (w^{2} - uv)\tan C\)

Search Value (6, 9, 13): \(1.875882662069\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines8 lines

{2, 3}, {107, 2707}, {243, 662}, {653, 2651}, {1098, 1940}, {2501, 23090}, {17923, 25533}, {18020, 23999}

▸ Functional Relations6 relations

• X₄₂₅ = X₄₁₃₄₉-cross conjugate of X₂₃₆₉₅

• X₄₂₅ = X_i-isoconjugate of X_j for these (i,j): {73, 43746}, {656, 2714}

• X₄₂₅ = X₂₇₁₄-Dao conjugate of X₄₀₅₉₆

• X₄₂₅ = barycentric product X_i*X_j for these {i,j}: {92, 23695}, {648, 2798}, {31623, 41349}

• X₄₂₅ = barycentric quotient X_i/X_j for these {i,j}: {112, 2714}, {1172, 43746}, {2798, 525}, {23695, 63}, {41349, 1214}

• X₄₂₅ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {}

Notes

As a point on the Euler line, X₄₂₅ has Shinagawa coefficients \(((E-2F)^{2}FS^{2}+\$abS_{A}S_{B}\$(2E-7F)F +3\$ab\$F^{2}S^{2}, (E-2F)[(E+F)F- S^{2}]S^{2} +3\$abS_{A}S_{B}\$[3(E+F)F-2S^{2}]-\$ab\$(E+F)FS^{2})\).