X₄₁₆

X₃-HIRST INVERSE OF X₂₁

X₄₁₆ is the X₃-hirst inverse of X₂₁.

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

X₄₁₆ (X₃-HIRST INVERSE OF X₂₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {255, 1248}

• Related Centers: X₁₉₃₆ (CROSSDIFFERENCE OF PU(15)), X₂₆₅₁ (CROSSDIFFERENCE OF PU(81)), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₂₅₅ (ISOGONAL CONJUGATE OF X₁₅₈), X₁₂₄₈ (ISOGONAL CONJUGATE OF X₁₀₄₇)

Trilinears: \((u^{2} - vw)\cos A : (v^{2} - wu)\cos B : (v^{2} - uv)\cos C\), where \(u : v : w\) = X₂₉; e.g., \(u = u(A,B,C) = (\sec A)/(\cos B + \cos C)\)

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

Search Value (6, 9, 13): \(2.941460465282\)

▸ Alternate Coordinates1 form

• Barycentrics: \(a (a + b) \cdot (a + c) \cdot (a^{7} b^{2} - a^{6} b^{3} - 2 a^{5} b^{4} + 2 a^{4} b^{5} + a^{3} b^{6} - a^{2} b^{7} + a^{7} b c - a^{6} b^{2} c - a^{5} b^{3} c + 2 a^{4} b^{4} c - a^{3} b^{5} c - a^{2} b^{6} c + a b^{7} c + a^{7} c^{2} - a^{6} b c^{2} + a^{5} b^{2} c^{2} - a^{4} b^{3} c^{2} - a^{3} b^{4} c^{2} + a^{2} b^{5} c^{2} - a b^{6} c^{2} + b^{7} c^{2} - a^{6} c^{3} - a^{5} b c^{3} - a^{4} b^{2} c^{3} + 2 a^{3} b^{3} c^{3} + a^{2} b^{4} c^{3} - a b^{5} c^{3} + b^{6} c^{3} - 2 a^{5} c^{4} + 2 a^{4} b c^{4} - a^{3} b^{2} c^{4} + a^{2} b^{3} c^{4} + 2 a b^{4} c^{4} - 2 b^{5} c^{4} + 2 a^{4} c^{5} - a^{3} b c^{5} + a^{2} b^{2} c^{5} - a b^{3} c^{5} - 2 b^{4} c^{5} + a^{3} c^{6} - a^{2} b c^{6} - a b^{2} c^{6} + b^{3} c^{6} - a^{2} c^{7} + a b c^{7} + b^{2} c^{7}) : :\)

▸ Incident Central Lines7 lines

{2, 3}, {255, 1248}, {296, 2651}, {820, 3562}, {1936, 2660}, {2659, 44354}, {23090, 32320}

▸ Functional Relations7 relations

• X₄₁₆ = X₁₉₃₆-Ceva conjugate of X₂₆₅₁

• X₂₆₅₅-cross conjugate of X₂₆₅₉

• X₄₁₆ = X_i-isoconjugate of X_j for these (i,j): {4, 2660}, {65, 2656}

• X₄₁₆ = X_i-Dao conjugate of X_j for these (i,j): {2656, 40602}, {2660, 36033}

• X₄₁₆ = barycentric product X_i*X_j for these {i,j}: {21, 44354}, {63, 2659}, {333, 2655}

• X₄₁₆ = barycentric quotient X_i/X_j for these {i,j}: {48, 2660}, {284, 2656}, {2655, 226}, {2659, 92}, {44354, 1441}

• X₄₁₆ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {3, 1982, 21}, {851, 52240, 415}

Notes

As a point on the Euler line, X₄₁₆ has Shinagawa coefficients \((abc[(E - 2F)FS^{4} + 2(\$aS_{A}\$)^{2}F]S^{2} + \$aS_{A}\$(\$bcS_{B}S_{C}\$)^{2}, -abc[S^{4} - (E+ F)FS^{2} + 2(\$aS_{A}\$)^{2}F]S^{2} - \$aS_{A}\$[3EFS^{4} + (\$bcS_{B}S_{C}\$)^{2}])\).