X₁₀₄₈

POINT ALGIEBA

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

X₁₀₄₈ (POINT ALGIEBA) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 564}, {484, 1046}

• Related Centers: X₁ (INCENTER), X₅₆₄ (INTERSECTION OF LINES X), X₄₈₄ (1st EVANS PERSPECTOR), X₁₀₄₆ (-CEVA CONJUGATE OF X₁)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = -u^{2} + v^{2} + w^{2} + vw + wu + uv\),\(u : v : w\) = X₅

Search Value (6, 9, 13): \(12.576172369677\)

▸ Alternate Coordinates1 form

• Trilinears: \(a^{7} (b^{2} + b c + c^{2}) + a^{6} (b + c) (b^{2} + b c + c^{2}) - a^{5} (2 b^{4} + b^{3} c + b c^{3} + 2 c^{4}) - 2 a^{4} (b + c)^{3} (b^{2} - b c + c^{2}) + a^{3} (b^{6} - b^{5} c - 2 b^{4} c^{2} - b^{3} c^{3} - 2 b^{2} c^{4} - b c^{5} + c^{6}) + a^{2} (b^{7} + 2 b^{6} c - 2 b^{4} c^{3} - 2 b^{3} c^{4} + 2 b c^{6} + c^{7}) + a b c (b^{2} - c^{2})^{2} (b^{2} + c^{2}) - b^{2} c^{2} (b - c)^{2} (b + c)^{3} : :\)

▸ Incident Central Lines2 lines

{1, 564}, {484, 1046}