X(2671) (1ST GOLDEN ARBELOS 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) = 1/(2 cos A + τ2sin A), where τ = (1 + sqrt(5))/2 = golden ratio
Trilinears           g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec[A - arccot(3 - 51/2)]       (M. Iliev, 5/13/07)

Barycentrics    (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Suppose UV is a segment and 1/2 < s < 1. Let P be the point on UV satisfying |UP|/|VP| = s. Let O be a semicircle having diameter UV. Let H be the semicircle having diameter VP, on the same side of UV as O. Let K be the semicircle having diameter UP, on the same side of UP as O. The three circular arcs form an arbelos, R.

Let L be the line through P perpendicular to BC. The circle tangent to semicircles O and K and line L is the s-Archimedean circle of the arbelos R. Now continuing with the variable s, suppose ABC is an arbitrary triangle. Let A' be the center of the s-Archimedean circle of the outward arbelos on segment BC, and define B' and C' cyclically.

At the Eleventh International Conference on Fibonacci Numbers and Their Applications (TU Braunschweig, Germany, July 2004), Zvonko Cerin established that the lines AA', BB', CC' concur if and only if s = τ. The point of concurrence is X(2671).

X(2671) lies on the Kiepert hyperbola and these lines: 2,2674    6,2672

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