X₂₄₇₉

1st INTERSECTION(EULER LINE, STEINERCIRCUMELLIPSE)

Of the two points of intersection, X₂₄₇₉ is the one closer to X₃.

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

X₂₄₇₉ (1st 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, 2455}, {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₂₄₅₅ (2nd 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): \(4.722466027142\)

▸ Incident Central Lines2 lines

{2, 3}, {671, 24008}

▸ Functional Relations5 relations

• X₂₄₇₉ = anticomplement of X₂₄₅₅

• X₂₄₇₉ = reflection of X_i in X_j for these (i,j): (2, 2454), (2480, 2)

• X₂₄₇₉ = projection from Steiner inellipse to Steiner circumellipse of X₂₄₅₄

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

• X₂₄₇₉ = trilinear pole of line X₂X₂₄₀₀₈

Notes

As a point on the Euler line, X₂₄₇₉ has Shinagawa coefficients \((K + S^{2}, -3S^{2})\).