X₃₅₂₄

EULER LINE INTERCEPT OF LINE X₃₅X_1058

X₃₅₂₄ = trisector nearest X₃ of segment X₂X₃

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

X₃₅₂₄ (EULER LINE INTERCEPT OF LINE X₃₅X_1058) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {376, 3545}, {2, 3} (Euler line)

• Related Centers: X₃₇₆ (CENTROID OF THE ANTIPEDAL TRIANGLE OF X), X₃₅₄₅ (EULER LINE INTERCEPT OF LINE X), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(f(A,B,C) = 4 \cos A + \cos B \cos C : :\)

Barycentrics: \(7 a^{4} - 8 a^{2} (b^{2} + c^{2}) + (b^{2} - c^{2})^{2} : :\)

Search Value (6, 9, 13): \(5.398031526960\)

▸ Alternate Coordinates1 form

• Trilinears: \(3 \cos A + \sin B \sin C : :\)

▸ Incident Central Lines17 lines

{2, 3}, {35, 1058}, {36, 1056}, {40, 551}, {69, 3431}, {98, 2482}, {182, 1992}, {187, 1285}, {519, 3158}, {553, 3487}, {597, 1350}, {943, 1466}, {1000, 1319}, {1385, 3241}, {3058, 3086}, {3070, 3316}, {3071, 3317}

▸ Functional Relations8 relations

• X₃₅₂₄ = midpoint of X₃₇₆ in X₃₅₄₅

• X₃₅₂₄ = reflection of X_i in X_j for these (i,j): (4, 3545), (3545, 2)

• X₃₅₂₄ = isogonal conjugate of X₃₅₃₁

• X₃₅₂₄ = crosssum of X₁₁₄₈₅ and X₁₁₄₈₆

• X₃₅₂₄ = circumcircle-inverse of X₃₇₉₄₆

• X₃₅₂₄ = trisector nearest X₃ of segment X₂X₃

• X₃₅₂₄ = X₂-of-Thomson-triangle

• X₃₅₂₄ = {X₃₇₆,X₆₃₁}-harmonic conjugate of X₂

Notes

As a point on the Euler line, X₃₅₂₄ has Shinagawa coefficients \((4, -3)\).