X₄₂₀

X₄-HIRST INVERSE OF X₄₂₇

X₄₂₀ is the X₄-hirst inverse of X₄₂₇.

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

X₄₂₀ (X₄-HIRST INVERSE OF X₄₂₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {232, 3229}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₂₃₂ (-CEVA CONJUGATE OF X₁₃₂), X₃₂₂₉ (ISOGONAL CONJUGATE OF X₃₂₂₅)

Trilinears: \((u^{2} - vw)\sec A : (v^{2} - wu)\sec B : (v^{2} - uv)\sec C\), where \(u : v : w\) = X₃₈; e.g., \(u = u(a,b,c) = b^{2} + c^{2}\)

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

Search Value (6, 9, 13): \(0.834756041461\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines39 lines

{2, 3}, {112, 8623}, {232, 3229}, {240, 17927}, {340, 44102}, {685, 20021}, {1613, 8743}, {1843, 34236}, {1990, 38294}, {2207, 21001}, {2501, 9210}, {2896, 11380}, {3117, 39575}, {3186, 17907}, {3231, 8744}, {3819, 10357}, {3978, 44146}, {5207, 40876}, {6198, 40790}, {6688, 7859}, {7779, 44090}, {9208, 44427}, {9302, 16080}, {9862, 42671}, {12143, 46226}, {14165, 34854}, {16081, 46306}, {17414, 47217}, {18020, 20022}, {18371, 52418}, {19128, 36213}, {19558, 43696}, {26958, 39646}, {28408, 44443}, {32223, 43453}, {33873, 44084}, {39201, 44451}, {39231, 40559}, {40858, 41676}

▸ Functional Relations13 relations

• X₄₂₀ = polar conjugate of X₁₁₆₀₆

• X₄₂₀ = isotomic conjugate of the isogonal conjugate of X₄₄₀₉₀

• X₄₂₀ = polar conjugate of the isotomic conjugate of X₇₇₇₉

• X₄₂₀ = polar conjugate of the isogonal conjugate of X₂₀₇₆

• X₄₂₀ = X₂₀₇₆-cross conjugate of X₇₇₇₉

• X₄₂₀ = X_i-isoconjugate of X_j for these (i,j): {48, 11606}, {63, 46286}, {656, 46970}, {1176, 17957}

• X₄₂₀ = X_i-Dao conjugate of X_j for these (i,j): {69, 39091}, {385, 12215}, {647, 39079}, {1249, 11606}, {3162, 46286}, {17949, 40938}, {40596, 46970}

• X₄₂₀ = cevapoint of X₂₀₇₆ and X₄₄₀₉₀

• X₄₂₀ = trilinear pole of line {9479, 12830}

• X₄₂₀ = crossdifference of every pair of points on line {647, 22352}

• X₄₂₀ = barycentric product X_i*X_j for these {i,j}: {4, 7779}, {76, 44090}, {92, 17799}, {264, 2076}, {427, 40850}, {648, 9479}, {1235, 46228}, {5113, 6331}, {12830, 35142}, {18010, 41676}, {20883, 34054}

• X₄₂₀ = barycentric quotient X_i/X_j for these {i,j}: {4, 11606}, {25, 46286}, {112, 46970}, {427, 17949}, {2076, 3}, {5113, 647}, {7779, 69}, {8290, 12215}, {8864, 8858}, {9479, 525}, {12830, 3564}, {17442, 17957}, {17799, 63}, {18010, 4580}, {34054, 34055}, {40850, 1799}, {44090, 6}, {46228, 1176}

• X₄₂₀ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {25, 5117, 4}, {25, 11331, 5117}, {297, 419, 4}, {297, 468, 419}, {403, 35474, 4}, {3535, 3536, 37187}, {5000, 5001, 40236}, {5112, 44887, 401}, {6353, 52283, 4}, {20854, 21536, 40236}

Notes

As a point on the Euler line, X₄₂₀ has Shinagawa coefficients \((4(E + F)F,-(E + F)^{2} - S^{2})\).