X3568

BELTRAMI-EULER POINT

Let P(2) and U(2) be the 1st and 2nd Beltrami points (as indexed at Bicentric Pairs, accessible using the Tables button at the top of ETC), and let P(40) and U(40) be the isogonal conjugates of P(2) and U(2), respectively. Then X3568 is the point of intersection of the lines P(2)P(40) and U(2)U(40). The name “Beltrami-Euler Point” signifies the fact that X3568 lies on the Euler line. See also X2966 and X3569. Contributed by Chris van Tienhoven, June 7, 2009.
< 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, 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_3 = CIRCUMCENTER (0.5000, 0.1021) X_4 = ORTHOCENTER (0.6122, 0.3958) X_3568 = BELTRAMI-EULER POINT (0.5187, 0.1510) A B C X2 X3 X4 X3568 (a, b, c) = (5, 6, 7) | AB = 1

X3568 (BELTRAMI-EULER POINT) 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 \(f(a,b,c) = bc[a^{12} - 2(b^{2} + c^{2})a^{10} + (b^{4} + 4b^{2}c^{2} + c^{4})a^{8} - 2(b^{6} + c^{6})a^{6}\), + (\(3b^{8}\) - \(3b^{6}\)\(c^{2}\) + \(b^{4}\)\(c^{4}\) - \(3b^{2}\)\(c^{6}\) + \(3c^{8})\)\(a^{4}\), - (\(b^{10}\) - \(b^{6}\)\(c^{4}\) - \(b^{4}\)\(c^{6}\) + \(c^{10})\)\(a^{2}\) + \(b^{4}\)\(c^{4}(b^{2} - c^{2})\)2]
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(4.705679250954\)
▸ Incident Central Lines1 line

{2, 3}

Notes

As a point on the Euler line, X3568 has Shinagawa coefficients \(((E + F)^{3}F - (E + F)(E + 10F)S^{2} + 5S^{4}, (E + F)^{4} - 2(E + F)(2E - 7F)S^{2} - 3S^{4})\).