X₃₅₂₉

EULER LINE INTERCEPT OF LINE X₅₁₆X_944

X₃₅₂₉ = anticomplement of X₃₈₂

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

X₃₅₂₉ (EULER LINE INTERCEPT OF LINE X₅₁₆X_944) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Anticomplementary Triangle △K_aK_bK_c — vertices K_a, K_b, K_c (Formed by lines through A, B, C parallel to opposite sides)

• Central Lines: {2, 382}, {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₃₈₂ (REFLECTION OF CIRCUMCENTER IN ORTHOCENTER), X₃ (CIRCUMCENTER)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = 5 \cos A - 3 \sin B \sin C\)

Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)

Search Value (6, 9, 13): \(44.159309809565\)

▸ Alternate Coordinates1 form

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = 5 \cos B \cos C - 2 \sin B \sin C\)

▸ Incident Central Lines6 lines

{2, 3}, {516, 944}, {1056, 3303}, {1058, 3304}, {1249, 3284}, {1770, 3486}

▸ Functional Relations11 relations

• X₃₅₂₉ = reflection of X_i in X_j for these (i,j): (4, 20), (20, 1657), (382, 550), (3146, 3), (3543, 3534)

• X₃₅₂₉ = anticomplement of X₃₈₂

• X₃₅₂₉ = circumcircle-inverse of X₃₇₉₄₈

• X₃₅₂₉ = orthocentroidal-circle-inverse of X₃₈₅₅

• X₃₅₂₉ = Ehrmann-mid-to-Johnson similarity image of X₂₀

• X₃₅₂₉ = homothetic center of cevian triangle of X₃ and mid-triangle of medial and anticomplementary triangles

• X₃₅₂₉ = {X₂,X₄}-harmonic conjugate of X₃₈₅₅

• X₃₅₂₉ = {X₂,X₂₀}-harmonic conjugate of X₅₅₀

• X₃₅₂₉ = {X₃,X₄}-harmonic conjugate of X₃₀₉₀

• X₃₅₂₉ = {X₄,X₃₅₂₈}-harmonic conjugate of X₂

• X₃₅₂₉ = {X₃₈₂,X₅₅₀}-harmonic conjugate of X₂

Notes

As a point on the Euler line, X₃₅₂₉ has Shinagawa coefficients \((2, -5)\).