X1337

1st WERNAU POINT

X1337 is the tangential of X15 on the Neuberg cubic.
< 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) { 4, 616 }: Central Line L(4, 616) { 1157, 1338 }: Central Line L(1157, 1338) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_4 = ORTHOCENTER (0.6122, 0.3958) X_616 = ANTICOMPLEMENT OF X(13) (0.4364, 0.0360) X_1157 = INVERSE-IN-CIRCUMCIRCLE OF X(54) (1.5871, 1.5280) X_1338 = 2nd WERNAU POINT (1.0969, 0.1421) X_1337 = 1st WERNAU POINT (2.6910, 4.6498) A B C X4 X616 X1157 X1338 X1337 (a, b, c) = (5, 6, 7) | AB = 1

X1337 (1st WERNAU 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 \(f(a,b,c) = a[x_{1} + 2Dx_{2}/\sqrt{3}]\), where \(D = \operatorname{area}(ABC)\), \(x_{1} = 4a^{2}U^{3} + (a^{2} + V)(a^{2} + W)U^{2} - 7V^{2}W^{2} - 5a^{2}UVW\), \(x_{2} = 3U^{3} + 7a^{2}U^{2} - 6a^{2}VW - 5 UVW\), \(U = (b^{2} + c^{2} - a^{2})/2\), \(V = (c^{2} + a^{2} - b^{2})/2\), \(W = (a^{2} + b^{2} - c^{2})/2\)
Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a^{2}(2DU - 3^{1/2}VW)/(4D + 3^{1/2}a^{2})\), D, U, V, W as above; see Hyacinthos #8874
Search Value (6, 9, 13): \(-13.209317003852\)
▸ Alternate Coordinates2 forms

  • Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a^{2}(2DU - 3^{1/2}VW)/(4D + 3^{1/2}a^{2})\),

  • Barycentrics: a^2*(-2*(-a^2+b^2+c^2)*S+\(\sqrt{3}\)*(a^2-c^2+b^2)*(a^2-b^2+c^2))*(\(\sqrt{3}\)*b^2+2*S)*(\(\sqrt{3}\)*c^2+2*S) : : (César Lozada, December 15, 2019)

X1337 lies on the Neuberg cubic.

▸ Incident Central Lines4 lines

{4, 616}, {15, 2981}, {399, 3441}, {1157, 1338}

▸ Functional Relations4 relations

Notes

Let A'BC be the external equilateral triangle on side BC, and define CB'A and AC'B cyclically. Let (AB'C') be the circle passing through the points A, B', C', and define (BC'A') and (CA'B') cyclically. The three circles concur in X1337. Wernau is a town near Stuttgart, the site of a mathematics olympiad seminar in Spring 2003. (Darij Grinberg; Hyacinthos, April, 2003: #6874, 6881, 6882; coordinates by Jean-Pierre Ehrmann)