X₃₄₁₂

2nd BROCARD-KIEPERT-FERMAT CUSP

X₃₄₁₂ = {X₆,X₃₅₂₆}-harmonic conjugate of X₃₄₁₁

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

X₃₄₁₂ (2nd BROCARD-KIEPERT-FERMAT CUSP) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 3526}

• Related Centers: X₃₄₁₁ (1st BROCARD-KIEPERT-FERMAT CUSP), X₆ (SYMMEDIAN POINT), X₃₅₂₆ (EULER LINE INTERCEPT OF LINE X)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \cos A - 2 \sin B \sin C - 3 \cos(A -\pi /3)\)

Barycentrics: \(h(A,B,C) : h(B,C,A) : h(C,A,B)\), where \(h(A,B,C) = 16S^{2} + 2 \cdot 3^{3/2}a^{2}S + a^{2}S_{A}\)

Search Value (6, 9, 13): \(1.680501398667\)

▸ Alternate Coordinates2 forms

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = \cos A + 4 \sin B \sin C + 3^{3/2}\sin A\)

• Barycentrics: \(h(A,B,C) : h(B,C,A) : h(C,A,B)\), where \(h(A,B,C) = 20S^{2} + 2 \cdot 3^{3/2}S - S_{B}S_{C}\)

X₃₄₁₂ lies on the Brocard-Kiepert quartic, Q073.

▸ Incident Central Lines7 lines

{5, 14}, {6, 3411}, {13, 382}, {15, 20}, {62, 631}, {397, 548}, {629, 3181}

▸ Functional Relations1 relation

• X₃₄₁₂ = {X₆,X₃₅₂₆}-harmonic conjugate of X₃₄₁₁