X₄₂₁

X₄-HIRST INVERSE OF X₂₄

X₄₂₁ is the X₄-hirst inverse of X₂₄.

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

X₄₂₁ (X₄-HIRST INVERSE OF X₂₄) 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, 2623}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₂₅₀₁ (RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT CIRCLE, TAYLOR CIRCLE}), X₂₆₂₃ (BARYCENTRIC PRODUCT OF PU(73))

Trilinears: \((u^{2} - vw)\sec A : (v^{2} - wu)\sec B : (v^{2} - uv)\sec C\), where \(u : v : w\) = X₄₇; e.g., \(u = u(A,B,C) = \cos 2A\)

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

Search Value (6, 9, 13): \(5.308204605327\)

▸ Alternate Coordinates1 form

• Barycentrics: \((a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) \cdot (a^{8} - 2 a^{6} b^{2} + a^{4} b^{4} - 2 a^{6} c^{2} + a^{4} b^{2} c^{2} - b^{6} c^{2} + a^{4} c^{4} + 2 b^{4} c^{4} - b^{2} c^{6}) : :\)

▸ Incident Central Lines20 lines

{2, 3}, {107, 44084}, {110, 44145}, {136, 14165}, {275, 47328}, {338, 15139}, {847, 52432}, {1632, 47195}, {2052, 44077}, {2501, 2623}, {2970, 34397}, {6531, 14580}, {8794, 41271}, {11547, 14593}, {12133, 48364}, {18020, 44138}, {19128, 46106}, {43976, 44080}, {44089, 47202}, {44375, 51458}

▸ Functional Relations6 relations

• X₄₂₁ = polar-circle-inverse of X₃₆₁₉₀

• X₄₂₁ = polar conjugate of the isotomic conjugate of X₄₄₃₇₅

• X₄₂₁ = crossdifference of every pair of points on line {647, 5562}

• X₄₂₁ = barycentric product X_i*X_j for these {i,j}: {4, 44375}, {2052, 51458}

• X₄₂₁ = barycentric quotient X_i/X_j for these {i,j}: {44375, 69}, {51458, 394}

• X₄₂₁ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {419, 450, 468}, {2970, 34397, 41204}

Notes

As a point on the Euler line, X₄₂₁ has Shinagawa coefficients \(((E + 4F)F,-(E + F)(E + 2F) + 2S^{2})\).