## X(165) (CENTROID OF THE EXCENTRAL TRIANGLE)

 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           tan(B/2) + tan(C/2) - tan(A/2) : tan(C/2) + tan(A/2) - tan(B/2) : tan(A/2) + tan(B/2) - tan(C/2)
= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a2 - 2a(b + c) - (b - c)2

Barycentrics    a[tan(B/2) + tan(C/2) - tan(A/2)] : b[tan(C/2) + tan(A/2) - tan(B/2)] : c[tan(A/2) + tan(B/2) - tan(C/2)]

X(165) = centroid of the triangle with vertices X(1), X(8), X(20)
X(165) = centroid of the triangle with vertices X(4), X(20), X(40)

X(165) is the {X(3),X(40)}-harmonic conjugate of X(1). For a list of harmonic conjugates of X(165), click More at the top of this page.

X(165) = isogonal conjugate of X(3062)
X(165) = X(9)-Ceva conjugate of X(1)
X(165) = X(I)-aleph conjugate of X(J) for these (I,J):
(2,169), (9,165), (21,572), (100,101), (188,9), (259,43), (365,978), (366,57), (650,1053)

X(165) = X(I)-beth conjugate of X(J) for these (I,J): (100,165), (643,200)

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