X₁₅₈₄

POINT CAPELLA II

X₁₅₈₄ = inverse-in-orthocentroidal-circle of X₁₅₉₂

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

X₁₅₈₄ (POINT CAPELLA II) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {6, 494}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₆ (SYMMEDIAN POINT), X₄₉₄ (2nd VAN LAMOEN HOMOTHETIC CENTER)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \csc A - \cos A\)

Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)

Search Value (6, 9, 13): \(1.934025295992\)

▸ Linear Combinations1 combination

• X₁₅₈₄ = \(6R^{2}\) X₂ - \(S\) X₃

▸ Incident Central Lines3 lines

{2, 3}, {6, 494}, {372, 394}

▸ Functional Relations1 relation

• X₁₅₈₄ = inverse-in-orthocentroidal-circle of X₁₅₉₂

Notes

As a point on the Euler line, X₁₅₈₄ has Shinagawa coefficients \((E - S, S)\).