X2454

1st INTERSECTION(EULER LINE, STEINERINELLIPSE)

Of the two points of intersection, X2454 is the closer to X3.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 2479 }: Euler Line { 2, 3 }: Euler Line A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_2479 = 1st INTERSECTION(EULER LINE, STEINERCIRCUMELLIPSE) (0.3859, -0.1964) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_2454 = 1st INTERSECTION(EULER LINE, STEINERINELLIPSE) (0.4617, 0.0018) A B C X2 X2479 X3 X2454 (a, b, c) = (5, 6, 7) | AB = 1

X2454 (1st INTERSECTION(EULER LINE, STEINERINELLIPSE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc[3a^{2}bc \cos B \cos C - 4(\operatorname{area} of ABC)^{2} - 2K]\), where \(K = (1/2)[a^{8} + b^{8} + c^{8} - S26 + a^{2}b^{2}c^{2}(a^{2} + b^{2} + c^{2})]^{1/2}\), \(S26 = a^{2}(b^{6} + c^{6}) + b^{2}(c^{6} + a^{6}) + c^{2}(a^{6} + b^{6})\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(3.675917409815\)

X2454 lies on the Steiner inscribed ellipse.

▸ Incident Central Lines1 line

{2, 3}

▸ Functional Relations4 relations

Notes

As a point on the Euler line, X2454 has Shinagawa coefficients \((S^{2} + 2K, -3S^{2})\).