## X(389) (CENTER OF THE TAYLOR CIRCLE)

 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           cos A - cos 2A cos(B - C) : cos B - cos 2B cos(C - A) : cos C - cos 2C cos(A - B)
Barycentrics    a[cos A - cos 2A cos(B - C)] : b[cos B - cos 2B cos(C - A)] : c[cos C - cos 2C cos(A - B)]

If ABC is acute then X(389) is the Spieker center of the orthic triangle. Peter Yff reports (Sept. 19, 2001) that since X(389) is on the Brocard axis, there must exist T for which X(389) is sin(A+T) : sin(B+T) : sin(C+T), and that tan(T) = - cot A cot B cot C.

X(389) lies on these lines:
3,6    4,51    24,184    30,143    54,186    115,129    217,232    517,950

X(389) = midpoint of X(I) and X(J) for these (I,J): (3,52), (4,185), (974,1112)
X(389) = reflection of X(1216) in X(140)
X(389) = inverse-in-Brocard-circle of X(578)
X(389) = crosspoint of X(4) and X(54)
X(389) = crosssum of X(I) and X(J) for these (I,J): (3,5), (6,418)

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