## X(84) (ISOGONAL CONJUGATE OF X(40))

 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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Download all construction files and macros: tc.zip (10.1 Mb).
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/(cos B + cos C - cos A - 1)
Barycentrics    (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let A',B',C' be the excenters. The perpendiculars from B' to AB and from C' to AC meet in a point A". Points B" and C" are determined cyclically. The hexyl triangle, A"B"C", is perspective to ABC, and X(84) is the perspector.

X(84) lies on these lines: 1,221    3,9    4,57    7,946    8,20    21,285    33,603    36,90    46,80    58,990    171,989    256,988    294,580    309,314    581,941    944,1000

X(84) = reflection of X(I) in X(J) for these (I,J): (40,1158), (1490,3)
X(84) = isogonal conjugate of X(40)
X(84) = isotomic conjugate of X(322)
X(84) = X(I)-Ceva conjugate of X(J) for these (I,J): (189,282), (280,1)
X(84) = X(I)-cross conjugate of X(J) for these (I,J): (19,57), (56,1)
X(84) = X(280)-aleph conjugate of X(84)
X(84) = X(I)-beth conjugate of X(J) for these (I,J): (271,3), (280,280), (285,84)

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

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Departamento de Matemática Aplicada -- Instituto de Matemática -- Universidade Federal Fluminense