X₄₁₅

X₄-HIRST INVERSE OF X₂₉

X₄₁₅ is the X₄-hirst inverse of X₂₉.

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

X₄₁₅ (X₄-HIRST INVERSE OF X₂₉) 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, 238}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁₆₂ (CEVAPOINT OF X₁₀₈ AND X₁₀₉), X₂₃₈ (-LINE CONJUGATE OF X₃₇)

Trilinears: \((u^{2} - vw)\sec A : (v^{2} - wu)\sec B : (v^{2} - uv)\sec C\), where \(u : v : w\) = X₂₁; e.g., \(u = u(A,B,C) = 1/(\cos B + \cos C)\)

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

Search Value (6, 9, 13): \(1.593028576024\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines16 lines

{1, 2907}, {2, 3}, {107, 2708}, {158, 1247}, {162, 238}, {243, 2652}, {648, 23710}, {774, 26000}, {1098, 1935}, {1861, 36797}, {1937, 2659}, {2501, 17926}, {2651, 17950}, {4620, 18020}, {5993, 14192}, {6626, 30988}

▸ Functional Relations12 relations

• X₄₁₅ = polar conjugate of X₁₁₆₀₈

• X₄₁₅ = polar conjugate of the isotomic conjugate of X₄₀₈₈₂

• X₄₁₅ = polar conjugate of the isogonal conjugate of X₅₀₆₀

• X₄₁₅ = X₂₄₃-Ceva conjugate of X₂₆₅₉

• X₄₁₅ = X_i-cross conjugate of X_j for these (i,j): {1758, 2651}, {5060, 40882}

• X₄₁₅ = X_i-isoconjugate of X_j for these (i,j): {3, 2652}, {48, 11608}, {65, 17973}, {73, 2648}, {656, 2701}, {810, 35154}, {1214, 17963}, {1409, 17947}, {6516, 18000}, {18013, 36059}

• X₄₁₅ = X_i-Dao conjugate of X_j for these (i,j): {525, 35086}, {1214, 39055}, {1249, 11608}, {2652, 36103}, {2701, 40596}, {17973, 40602}, {18013, 20620}, {35154, 39062}

• X₄₁₅ = cevapoint of X₁₇₅₈ and X₁₇₉₈₅

• X₄₁₅ = trilinear pole of line {2785, 41499}

• X₄₁₅ = barycentric product X_i*X_j for these {i,j}: {4, 40882}, {29, 17950}, {92, 2651}, {264, 5060}, {333, 17985}, {648, 2785}, {1758, 31623}, {3064, 17933}, {5075, 6331}, {17942, 46110}, {17966, 44130}, {35145, 41499}

• X₄₁₅ = barycentric quotient X_i/X_j for these {i,j}: {4, 11608}, {19, 2652}, {29, 17947}, {112, 2701}, {284, 17973}, {648, 35154}, {1172, 2648}, {1758, 1214}, {2299, 17963}, {2651, 63}, {2785, 525}, {3064, 18013}, {5060, 3}, {5075, 647}, {17942, 1813}, {17950, 307}, {17966, 73}, {17975, 40152}, {17985, 226}, {40882, 69}, {41499, 8680}, {51643, 51640}

• X₄₁₅ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {4, 1982, 29}, {422, 468, 423}, {851, 52240, 416}, {860, 2074, 447}, {11110, 37383, 29}, {13739, 17555, 29}

Notes

As a point on the Euler line, X₄₁₅ has Shinagawa coefficients 3\(\$bcS_{B}S_{C}\$\)F + (E - 2F)\(FS^{2}\), -(E + F)*\(\$bcS_{B}S_{C}\$\) + (E + F)\(FS^{2}\) - \(S^{4})\).