X₂₄₈₀

2nd INTERSECTION(EULER LINE, STEINERCIRCUMELLIPSE)

Of the two points of intersection, X₂₄₇₉ is the one farther from X₃.

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

X₂₄₈₀ (2nd INTERSECTION(EULER LINE, STEINERCIRCUMELLIPSE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Anticomplementary Triangle △K_aK_bK_c — vertices K_a, K_b, K_c (Formed by lines through A, B, C parallel to opposite sides)

• Central Lines: {2, 2454}, {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₂₄₅₄ (1st INTERSECTION), X₃ (CIRCUMCENTER)

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

Barycentrics: \(K + 3 S_{B} S_{C} - S^{2} : :\), K as at X₂₄₅₄

Search Value (6, 9, 13): \(0.536271557835\)

▸ Incident Central Lines2 lines

{2, 3}, {671, 24007}

▸ Functional Relations5 relations

• X₂₄₈₀ = reflection of X_i in X_j for these (i,j): (2, 2455), (2479, 2)

• X₂₄₈₀ = anticomplement of X₂₄₅₄

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

• X₂₄₈₀ = Steiner-circumellipse-X₃-antipode of X₂₄₇₉

• X₂₄₈₀ = trilinear pole of line X₂X₂₄₀₀₇

Notes

As a point on the Euler line, X₂₄₈₀ has Shinagawa coefficients \((K - S^{2}, 3S^{2})\).