X₄₂₂

X₄-HIRST INVERSE OF X₂₈

X₄₂₂ is the X₄-hirst inverse of X₂₈.

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

X₄₂₂ (X₄-HIRST INVERSE OF X₂₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {162, 242}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁₆₂ (CEVAPOINT OF X₁₀₈ AND X₁₀₉), X₂₄₂ (-LINE CONJUGATE OF X₇₁)

Trilinears: \((u^{2} - vw)\sec A : (v^{2} - wu)\sec B : (v^{2} - uv)\sec C\), where \(u : v : w\) = X₅₈; e.g., \(u = u(a,b,c) = a/(b + c)\)

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

Search Value (6, 9, 13): \(0.909221981878\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines12 lines

{2, 3}, {107, 2699}, {110, 48380}, {162, 242}, {648, 35155}, {685, 16082}, {1632, 8758}, {1824, 46103}, {2203, 31623}, {2501, 4581}, {4601, 5379}, {17790, 17977}

▸ Functional Relations10 relations

• X₄₂₂ = polar conjugate of X₁₁₆₁₁

• X₄₂₂ = polar conjugate of the isotomic conjugate of X₁₉₆₂₃

• X₄₂₂ = polar conjugate of the isogonal conjugate of X₅₀₀₆

• X₄₂₂ = X₅₀₀₆-cross conjugate of X₁₉₆₂₃

• X₄₂₂ = X_i-isoconjugate of X_j for these (i,j): {10, 17971}, {48, 11611}, {71, 17946}, {72, 17954}, {73, 11609}, {306, 17961}, {656, 2703}, {810, 35147}, {1331, 18015}, {3682, 17981}, {4064, 17939}, {4561, 18002}

• X₄₂₂ = X_i-Dao conjugate of X_j for these (i,j): {525, 35079}, {1249, 11611}, {2703, 40596}, {5521, 18015}, {35147, 39062}

• X₄₂₂ = crossdifference of every pair of points on line {647, 22076}

• X₄₂₂ = barycentric product X_i*X_j for these {i,j}: {4, 19623}, {19, 5209}, {27, 17763}, {28, 17790}, {81, 17987}, {264, 5006}, {286, 5291}, {648, 2787}, {5040, 6331}, {5061, 31623}, {6591, 17935}, {17924, 17944}

• X₄₂₂ = barycentric quotient X_i/X_j for these {i,j}: {4, 11611}, {28, 17946}, {112, 2703}, {648, 35147}, {1172, 11609}, {1333, 17971}, {1474, 17954}, {2203, 17961}, {2787, 525}, {5006, 3}, {5040, 647}, {5061, 1214}, {5209, 304}, {5291, 72}, {5317, 17981}, {6591, 18015}, {17763, 306}, {17790, 20336}, {17944, 1332}, {17977, 3998}, {17987, 321}, {19623, 69}

• X₄₂₂ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {415, 423, 468}

Notes

As a point on the Euler line, X₄₂₂ has Shinagawa coefficients \((4(E+F)^{2}FS^{2}-3\$abS_{C}\$F+7\$ab\$(E+F)F, -2(E+F)[(E+F)^{2}-S^{2}]+\$abS_{C}\$(E+F) -\$ab\$[3(E+F)^{2}-2S^{2}])\).