X₄₂₄

X₄-HIRST INVERSE OF X₄₅₁

X₄₂₄ is the X₄-hirst inverse of X₄₅₁.

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

X₄₂₄ (X₄-HIRST INVERSE OF X₄₅₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {2501, 3700}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₂₅₀₁ (RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT CIRCLE, TAYLOR CIRCLE}), X₃₇₀₀ (com(EXTOUCH TRIANGLE))

Trilinears: \((u^{2} - vw)\sec A : (v^{2} - wu)\sec B : (v^{2} - uv)\sec C\), where \(u : v : w\) = X₁₉₁; e.g., \(u = u(a,b,c) = (b + c - a)(bc + ca + ab) + b^{3} + c^{3} - a^{3}\)

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

Search Value (6, 9, 13): \(1.897303493226\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines2 lines

{2, 3}, {2501, 3700}

▸ Functional Relations8 relations

• X₄₂₄ = polar conjugate of the isotomic conjugate of X₄₄₃₉₆

• X₄₂₄ = polar conjugate of the isogonal conjugate of X₅₁₆₄

• X₄₂₄ = X₅₁₆₄-cross conjugate of X₄₄₃₉₆

• X₄₂₄ = X₇₂₅₄-Dao conjugate of X₄₁₁₇₉

• X₄₂₄ = crossdifference of every pair of points on line {647, 1437}

• X₄₂₄ = barycentric product X_i*X_j for these {i,j}: {4, 44396}, {264, 5164}

• X₄₂₄ = barycentric quotient X_i/X_j for these {i,j}: {5164, 3}, {44396, 69}

• X₄₂₄ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {}

Notes

As a point on the Euler line, X₄₂₄ has Shinagawa coefficients \((4(E+F)^{2}F-3\$abS_{C}\$F+7\$ab\$(E+F)F, -4(E+F)S^{2}+\$abS_{C}^{2}\$-\$abS_{A}S_{B}\$ -\$ab\$[(E+F)^{2}+3S^{2}])\).