X₁₃₄₇

EXSIMILICENTER(NINE-POINT CIRCLE, ORTHOCENTROIDAL CIRCLE)

X₁₃₄₇ is the exsimilicenter(nine-point circle, orthocentroidal circle).

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

X₁₃₄₇ (EXSIMILICENTER(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)

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

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

Search Value (6, 9, 13): \(7.484237188688\)

▸ Linear Combinations1 combination

• X₁₃₄₇ = \(3(-2 + |OH|/R)\) X₂ + \((3 - |OH|/R)\) X₃

▸ Incident Central Lines1 line

{2, 3}

▸ Functional Relations1 relation

• X₁₃₄₇ = external center of similitude of nine-point circle and orthocentroidal circle (Peter J. C. Moses, 4/2003)

Notes

As a point on the Euler line, X₁₃₄₇ has Shinagawa coefficients \((R - |OH|,3R - |OH|)\).