X₃₁₀₆

CENTER OF 1st BROCARD EQUILATERAL TRIANGLE

X₃₁₀₆ is the center of the equilateral triangle introduced at X₃₁₀₄.

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

X₃₁₀₆ (CENTER 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: {3104, 3107}

• Related Centers: X₃₁₀₄ (INNER VERTEX OF 1st BROCARD EQUILATERAL TRIANGLE), X₃₁₀₇ (CENTER OF 2nd BROCARD EQUILATERAL TRIANGLE)

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

Search Value (6, 9, 13): \(2.561540583378\)

▸ 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, 262}, {14, 2782}, {18, 76}, {194, 627}

▸ Functional Relations5 relations

• X₃₁₀₆ = midpoint of X₃₁₀₄ and X₃₁₀₇

• X₃₁₀₆ = reflection of X_i in X_j for these (i,j): (3105, 3107), (3107, 39)

• X₃₁₀₆ = inverse-in-2nd-Brocard-circle of X₆₂

• X₃₁₀₆ = X₁₃-of-X₃PU(1)

• X₃₁₀₆ = X₁₄-of-X₆PU(1)

Notes

The locus of the 2nd Isogonic Center in a Brocard porism (triangles sharing circumcircle and Brocard inellipse with ABC) is a circle with center X₃₁₀₆. (Randy Hutson, August 29, 2018)