X₃₁₀₂

INNER VERTEX OF THE BROCARD SQUARE

The 1st Brocard point (trilinears \(c/b : a/c : b/a)\) and 2nd Brocard point (\(b/c : c/a : a/b)\) are opposing vertices of a square here called the Brocard square. The other two vertices are X₃₁₀₂ and X₃₁₀₃. Both are on the Brocard axis X₃X₆, and X₃₁₀₂ is the closer to X₃.

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

X₃₁₀₂ (INNER VERTEX OF THE BROCARD SQUARE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3103, 39}

• Related Centers: X₃₁₀₃ (OUTER VERTEX OF THE BROCARD SQUARE), X₃₉ (BROCARD MIDPOINT)

Trilinears: \(\cos(A + \omega - \pi /4) : :\)

Barycentrics: \(a^{2} (b^{4} + c^{4} - a^{2} b^{2} - a^{2} c^{2} + 2 S (b^{2} + c^{2})) : :\)

Search Value (6, 9, 13): \(3.291439847441\)

▸ Alternate Coordinates5 forms

• Trilinears: \(\sin(A + \omega + \pi /4) : :\)

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

• Trilinears: \(\sin A + \cos(A + 2\omega ) : :\)

• Trilinears: \(\cos A + \sin(A + 2\omega ) : :\)

• Trilinears: \(\cos(A + \omega ) + \sin(A + \omega ) : :\)

▸ Incident Central Lines6 lines

{3, 6}, {76, 486}, {194, 488}, {262, 485}, {325, 640}, {2782, 3071}

▸ Functional Relations2 relations

• X₃₁₀₂ = reflection of X₃₁₀₃ in X₃₉

• X₃₁₀₂ = inverse-in-2nd-Brocard-circle of X₃₇₂