## X(2039) (1ST NINE-POINT-CIRCLE-KIEPERT-ASYMPTOTES INTERSECTION)

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
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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) = e cos A + 2e cos B cos C + cos(A + ω)
Barycentrics    (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Of the points other than X(115) in which the nine-point circle meets the asymptotes of the Kiepert hyperbola, X(2039) is the one nearer to X(3).

The points X(2039) and X(2040) are endpoints of the diameter of the nine-point circle that is parallel to the line X(25)X(394). (M. Iliev, 5/13/07)

X(2039) = midpoint of X(4) and X(1380)
X(2039) = reflection of X(2040) in X(5)
X(2039) = complement of X(1379)

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