X(1337) (1ST WERNAU POINT)

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
Press the keys “+” and “−” to zoom in or zoom out the visualization window and use the arrow keys to translate it.

You can also construct all centers related with this one (as described in ETC) using the “Run Macro Tool”. To do this, click on the icon , select the center name from the list and, then, click on the vertices A, B and C successively.

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This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[x1 + 2Dx2/sqrt(3)], where
D = area(ABC),
x1 = 4a2U3 + (a2 + V)(a2 + W)U2 - 7V2W2 - 5a2UVW,
x2 = 3U3 + 7a2U2 - 6a2VW - 5 UVW,
U = (b2 + c2 - a2)/2, V = (c2 + a2 - b2)/2, W = (a2 + b2 - c2)/2

Barycentrics    g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(2DU - 31/2VW)/(4D + 31/2a2),
D, U, V, W as above; see Hyacinthos #8874

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 X(1337). The Wernau points, X(1337) and X(1338), lie on the Neuberg cubic. 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)

X(1337) lies on these lines: 4,616    1157,1338

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.