X₂₄₅₅

2nd INTERSECTION(EULER LINE, STEINERINELLIPSE)

Of the two points of intersection, X₂₄₅₅ is the farther from to X₃.

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

X₂₄₅₅ (2nd INTERSECTION(EULER LINE, STEINERINELLIPSE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 2480}, {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₂₄₈₀ (2nd INTERSECTION), X₃ (CIRCUMCENTER)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where f(a,b,c) = bc[\(3a^{2}\)bc cos B cos C - 4(area of ABC)² + 2K] (see X₂₄₅₄)

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

Search Value (6, 9, 13): \(1.582820175162\)

X₂₄₅₅ lies on the Steiner inscribed ellipse.

▸ Incident Central Lines1 line

{2, 3}

▸ Functional Relations4 relations

• X₂₄₅₅ = midpoint of X₂ and X₂₄₈₀

• X₂₄₅₅ = reflection of X₂₄₅₄ in X₂

• X₂₄₅₅ = complement of X₂₄₇₉

• X₂₄₅₅ = projection from Steiner circumellipse to Steiner inellipse of X₂₄₈₀

Notes

As a point on the Euler line, X₂₄₅₅ has Shinagawa coefficients \((S^{2} - 2K, -3S^{2})\).