X₃₁₀₄

INNER VERTEX OF 1st BROCARD EQUILATERAL TRIANGLE

There are two 60-rotations of the 1st Brocard point about the 2nd; X₃₁₀₄ is the one closer to X₃.

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

X₃₁₀₄ (INNER VERTEX OF 1st 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}, {14, 76}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₁₄ (2nd 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): \(4.555662457207\)

▸ 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}, {14, 76}, {17, 262}, {194, 616}, {633, 2896}

▸ Functional Relations2 relations

• X₃₁₀₄ = reflection of X_i in X_j for these (i,j): (3105, 39), (3107, 3106)

• X₃₁₀₄ = inverse-in-2nd-Brocard-circle of X₁₆