X₃₄₈₃

INTERSECTION OF LINES X₁X_1157 and X₄X_484

X₃₄₈₃ = isogonal conjugate of X₇₁₆₅

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

X₃₄₈₃ (INTERSECTION OF LINES X₁X_1157 and X₄X_484) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 484}, {1, 1157}

• Related Centers: X₄ (ORTHOCENTER), X₄₈₄ (1st EVANS PERSPECTOR), X₁ (INCENTER), X₁₁₅₇ (INVERSE-IN-CIRCUMCIRCLE OF X₅₄)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a^{3} - (b + c)(b^{2} + c^{2}) + bc(b + c)P/[(b + c)U + aV]\), where \(P = (b + c)[2a^{2} - (b - c)^{2}] - a(b^{2} + c^{2})\), \(U = a^{2} - (b - c)^{2}\), and \(V = a^{2} - b^{2} - c^{2} + bc\) (M. Iliev, 10/27/08)

Barycentrics: \((a (a^{6}+a^{5} b-a^{4} b^{2}-2 a^{3} b^{3}-a^{2} b^{4}+a b^{5}+b^{6}+a^{5} c+a^{4} b c-a b^{4} c-b^{5} c-a^{4} c^{2}+2 a^{2} b^{2} c^{2}-b^{4} c^{2}-2 a^{3} c^{3}+2 b^{3} c^{3}-a^{2} c^{4}-a b c^{4}-b^{2} c^{4}+a c^{5}-b c^{5}+c^{6}))/(-a^{3}-a^{2} b+a b^{2}+b^{3}-a^{2} c-a b c-b^{2} c+a c^{2}-b c^{2}+c^{3}) : :\)

Search Value (6, 9, 13): \(-4.053279537408\)

X₃₄₈₃ lies on the Neuberg cubic.

▸ Incident Central Lines4 lines

{1, 1157}, {4, 484}, {399, 3466}, {3465, 3481}

▸ Functional Relations2 relations

• X₃₄₈₃ = isogonal conjugate of X₇₁₆₅

• X₃₄₈₃ = antigonal conjugate of X₃₄₃₀₀