X₃₅₂₅

EULER LINE INTERCEPT OF LINE X₆₉X_575

X₃₅₂₅ = crosspoint of X₃₃₁₆ and X₃₃₁₇

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

X₃₅₂₅ (EULER LINE INTERCEPT OF LINE X₆₉X_575) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {69, 575}

• Related Centers: X₃₃₁₆ (ISOGONAL CONJUGATE OF X₃₃₁₁), X₃₃₁₇ (ISOGONAL CONJUGATE OF X₃₃₁₂), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₆₉ (SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE), X₅₇₅ (ORTHOGONAL OF X₁₈₇)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = 4 \cos A + 3 \cos B \cos C\)

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

Search Value (6, 9, 13): \(3.384458629163\)

▸ Alternate Coordinates2 forms

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = 3 \sec A + 4 \sec B \sec C\)

• Barycentrics: \(\cot B \cot C - 4 : :\)

▸ Incident Central Lines13 lines

{2, 3}, {69, 575}, {394, 1199}, {498, 1056}, {499, 1058}, {944, 1698}, {1007, 1078}, {1285, 2548}, {1587, 3316}, {1588, 3317}, {3053, 3055}, {3085, 3304}, {3086, 3303}

▸ Functional Relations4 relations

• X₃₅₂₅ = crosspoint of X₃₃₁₆ and X₃₃₁₇

• X₃₅₂₅ = crosssum of X₃₃₁₁ and X₃₃₁₂

• X₃₅₂₅ = intersection of tangents to Kiepert hyperbola at X₃₃₁₆ and X₃₃₁₇

• X₃₅₂₅ = {X₂,X₃}-harmonic conjugate of X₃₀₉₀

Notes

As a point on the Euler line, X₃₅₂₅ has Shinagawa coefficients \((4, -1)\).