X₁₃₄₆

INSIMILICENTER(NINE-POINT CIRCLE, ORTHOCENTROIDAL CIRCLE)

X₁₃₄₆ is the insimilicenter(nine-point circle, orthocentroidal circle).

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

X₁₃₄₆ (INSIMILICENTER(NINE-POINT CIRCLE, ORTHOCENTROIDAL CIRCLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {56, 2464}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₅₆ (EXSIMILICENTER), X₂₄₆₄ (EXSIMILICENTER)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = R \cos A + |OH|\cos(B - C) + 4 R \cos B \cos C\)

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

Search Value (6, 9, 13): \(-0.350118036722\)

▸ Linear Combinations1 combination

• X₁₃₄₆ = \(3(2 + |OH|/R)\) X₂ - \((3 + |OH|/R)\) X₃

▸ Incident Central Lines2 lines

{2, 3}, {56, 2464}

▸ Functional Relations1 relation

• X₁₃₄₆ = internal center of similitude of nine-point circle and orthocentroidal circle (Peter J. C. Moses, 4/2003)

Notes

As a point on the Euler line, X₁₃₄₆ has Shinagawa coefficients \((R + |OH|,3R + |OH|)\).