X₃₁₀₅

OUTER VERTEX OF 2nd BROCARD EQUILATERAL TRIANGLE

There are two 60-rotations of the 1st Brocard point about the 2nd; X₃₁₀₅ is the one farther from X₃.

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

X₃₁₀₅ (OUTER VERTEX OF 2nd BROCARD EQUILATERAL TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {13, 76}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₁₃ (1st ISOGONIC CENTER), X₇₆ (3rd BROCARD POINT)

Trilinears: \(\cos(A + \omega + \pi /6) : \cos(B + \omega + \pi /6) : \cos(C + \omega + \pi /6)\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(-1.426703164281\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\sin(A + \omega - \pi /3) : \sin(B + \omega - \pi /3) : \sin(C + \omega - \pi /3)\)

• Trilinears: \(f(a,b,c)) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = \sqrt{3}(bc \cos^{2} A - a^{2}\cos B \cos C) - ab \sin B - ac \sin C\)

▸ Incident Central Lines5 lines

{3, 6}, {13, 76}, {18, 262}, {194, 617}, {634, 2896}

▸ Functional Relations2 relations

• X₃₁₀₅ = reflection of X_i in X_j for these (i,j): (3104, 39), (3106, 3107)

• X₃₁₀₅ = inverse-in-2nd-Brocard-circle of X₁₅