X₃₁₀₇

CENTER OF 2nd BROCARD EQUILATERAL TRIANGLE

X₃₁₀₇ is the center of the equilateral triangle introduced at X₃₁₀₅.

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

X₃₁₀₇ (CENTER 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: {3105, 3106}

• Related Centers: X₃₁₀₅ (OUTER VERTEX OF 2nd BROCARD EQUILATERAL TRIANGLE), X₃₁₀₆ (CENTER OF 1st BROCARD EQUILATERAL TRIANGLE)

Trilinears: \(\cos(A + \omega + \pi /3) : \cos(B + \omega + \pi /3) : \cos(C + \omega + \pi /3)\)

Search Value (6, 9, 13): \(0.567418709548\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\sin(A + \omega - \pi /6) : \sin(B + \omega - \pi /6) : \sin(C + \omega - \pi /6)\)

• Trilinears: \(bc \cos^{2} A - a^{2}\cos B \cos C - \sqrt{3}(ab \sin B + ac \sin C)\)

▸ Incident Central Lines5 lines

{3, 6}, {13, 2782}, {14, 262}, {17, 76}, {194, 628}

▸ Functional Relations5 relations

• X₃₁₀₇ = midpoint of X₃₁₀₅ and X₃₁₀₆

• X₃₁₀₇ = reflection of X_i in X_j for these (i,j): (3105, 3106), (3106, 39)

• X₃₁₀₇ = inverse-in-2nd-Brocard-circle of X₆₁

• X₃₁₀₇ = X₁₄-of-X₃PU(1)

• X₃₁₀₇ = X₁₃-of-X₆PU(1)

Notes

The locus of the X₁₃ in a Brocard porism (triangles sharing circumcircle and Brocard inellipse with ABC) is a circle with center X₃₁₀₇. (Randy Hutson, August 29, 2018)