X₄₀₉

EULER X₂₁-1st-SUBSTITUTION POINT

X₄₀₉ = isogonal conjugate of X₉₃₉₉

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

X₄₀₉ (EULER X₂₁-1st-SUBSTITUTION POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {65, 1098}

• Related Centers: X₂₆₄₈ (TRILINEAR POLE OF PU(80)), X₂₆₅₁ (CROSSDIFFERENCE OF PU(81)), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₆₅ (ORTHOCENTER OF THE INTOUCH TRIANGLE), X₁₀₉₈ (TRILINEAR SQUARE OF X₂₁)

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

Barycentrics: \(a(u^{2} + vw) : b(v^{2} + wu) : c(w^{2} + uv)\)

Search Value (6, 9, 13): \(2.330340117158\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines32 lines

{1, 1247}, {2, 3}, {10, 37816}, {46, 51290}, {58, 5883}, {65, 1098}, {81, 3924}, {86, 40980}, {100, 21674}, {229, 2607}, {643, 5836}, {662, 2646}, {759, 1125}, {950, 19642}, {993, 47059}, {1054, 34882}, {1104, 2363}, {1178, 1201}, {1958, 10448}, {2648, 9398}, {2886, 52360}, {3304, 36224}, {3754, 5127}, {3897, 37793}, {4653, 27784}, {5260, 27714}, {7354, 52361}, {9275, 30143}, {15792, 35016}, {17104, 30147}, {24619, 26532}, {34195, 37783}

▸ Functional Relations7 relations

• X₄₀₉ = isogonal conjugate of X₉₃₉₉

• X₄₀₉ = X₂₆₄₈-Ceva conjugate of X₂₆₅₁

• X₄₀₉ = X_i-isoconjugate of X_j for these (i,j): {1, 9399}, {65, 9398}

• X₄₀₉ = X_i-Dao conjugate of X_j for these (i,j): {3, 9399}, {9398, 40602}

• X₄₀₉ = barycentric product X₃₃₃*X₂₆₄₇

• X₄₀₉ = barycentric quotient X_i/X_j for these {i,j}: {6, 9399}, {284, 9398}, {2647, 226}

• X₄₀₉ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 2475, 27687}, {2, 11101, 21}, {3, 11116, 35991}, {21, 29, 413}, {21, 13746, 15776}, {21, 17518, 13588}, {21, 17568, 1817}, {21, 35991, 3}, {21, 37294, 17549}, {65, 1098, 2651}, {405, 17512, 21}, {662, 40430, 2646}, {1010, 36011, 21}, {11102, 19259, 21}, {11110, 37227, 21}, {11110, 52244, 7483}, {13725, 14015, 21}, {13739, 37228, 21}, {16418, 37029, 21}, {16865, 37032, 21}

Notes

As a point on the Euler line, X₄₀₉ has Shinagawa coefficients \((\$aS_{A}\$[(\$aS_{A}\$)^{2} - \$bcS_{B}S_{C}\$] + 2abcFS^{2}, -\$aS_{A}\$[(\$aS_{A}\$)^{2} - \$bcS_{B}S_{C}\$] + abc[(\$aS_{A}\$)^{2} + \$bcS_{B}S_{C}\$ - 2FS^{2}])\).