X₃₅₂₈

EULER LINE INTERCEPT OF LINE X₃₅X_1056

X₃₅₂₈ = pole of van Aubel line wrt conic {{X₁₃,X₁₄,X₁₇,X₁₈,X₂₀}}

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

X₃₅₂₈ (EULER LINE INTERCEPT OF LINE X₃₅X_1056) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {35, 1056}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₃₅ ({X), X₁₀₅₆ (POINT ALCYONE)

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

Search Value (6, 9, 13): \(8.562217509214\)

▸ Alternate Coordinates4 forms

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = \sec A - 4 \sec B \sec C\)

• Trilinears: \(h(A,B,C) : h(B,C,A) : h(C,A,B)\), where \(h(A,B,C) = 5 \cos B \cos C - 4 \sin B \sin C\)

• Trilinears: \(6 \cos A + \cos(B - C) - 3 \sin B \sin C : :\)

• Trilinears: \(\cos A + \cos(B - C) - 5 \cos B \cos C + 2 \sin B \sin C : :\)

▸ Incident Central Lines7 lines

{2, 3}, {35, 1056}, {36, 1058}, {39, 1285}, {40, 3244}, {165, 944}, {519, 3161}

▸ Functional Relations10 relations

• X₃₅₂₈ = reflection of X_i in X_j for these (i,j): (4, 3090), (3090, 3523), (3523, 3)

• X₃₅₂₈ = reflection of X₂ in X₁₅₇₀₀

• X₃₅₂₈ = anticomplement of X₃₈₅₁

• X₃₅₂₈ = orthocentroidal-circle-inverse of X₃₅₄₄

• X₃₅₂₈ = pole of van Aubel line wrt conic {{X₁₃,X₁₄,X₁₇,X₁₈,X₂₀}}

• X₃₅₂₈ = {X₂,X₃}-harmonic conjugate of X₁₀₂₉₉

• X₃₅₂₈ = {X₂,X₄}-harmonic conjugate of X₃₅₄₄

• X₃₅₂₈ = {X₂,X₂₀}-harmonic conjugate of X₃₈₂

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

• X₃₅₂₈ = {X₃₇₆,X₆₃₁}-harmonic conjugate of X₂₀

Notes

As a point on the Euler line, X₃₅₂₈ has Shinagawa coefficients \((4, -5)\).