X(1989) (ISOGONAL CONJUGATE OF X(323))

 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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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           sin2A csc 3A : sin2B csc 3B : sin2C csc 3C
Barycentrics    sin3A csc 3A : sin3B csc 3B : sin3C csc 3C

X(1989) plays a major role in the theory of special isocubics, as presented in Chapter 6 of

Jean-Pierre Ehrmann and Bernard Gibert,, "Special Isocubics in the Triangle Plane," downloadable from

Bernard Gibert, Cubics in the Triangle Plane.

X(1989) is the barycentric product X(13)*X(14) of the Fermat points. The line through X(50) parallel to
the line X(13)X(14) passes through X(1989).

X(1989) lies on these lines:
2,94    6,13    30,50    53,112    67,868    111,230    403,1990    1427,2006

X(1989) = isogonal conjugate of X(323)
X(1989) = complement of X(1272)
X(1989) = X(94)-Ceva conjugate of X(265)
X(1989) = cevapoint of X(I) and X(J) for these (I,J): (53,1990), (115,1637), (395,396)
X(1989) = crosspoint of X(2) and X(1138)
X(1989) = crosssum of X(6) and X(399)
X(1989) = barycentric product of X(13) and X(14)

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

If you have questions or suggestions, please, contact us using the e-mail presented here.

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