X₃₀₈₀

DANNEELS POINT OF X₂₅

X₃₀₈₀ = X₂₅-Ceva conjugate of X₁₁₉₆

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

X₃₀₈₀ (DANNEELS POINT OF X₂₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {682, 1196}

• Related Centers: X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₁₁₉₆ (3rd SARAGOSSA POINT OF X₂₅), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₆₈₂ (POINT ARNEB)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a u^{2}(bv + cw)\), where \(u : v : w\) = X₂₅

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(-0.079537446858\)

▸ Incident Central Lines2 lines

{2, 3}, {682, 1196}

▸ Functional Relations2 relations

• X₃₀₈₀ = X₂₅-Ceva conjugate of X₁₁₉₆

• X₃₀₈₀ =X₃₂₆-isoconjugate of X₆₈₃

Notes

As a point on the Euler line, X₃₀₈₀ has Shinagawa coefficients \(((E + F)^{3}F - (E - F)FS^{2}, -(E + F)^{4} + (E + F)(2E - F)S^{2})\).