X(1689) (INVERSE-IN-BROCARD-CIRCLE OF X(1687))

 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           f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(A + ω) - sin A
= sin(A + ω) - cos A;

Barycentrics    (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1689) is the external center of similitude of the Gallatly circle and the circumcircle. (Peter J. C. Moses, 9/03)

X(1689) lies on these lines: 1,2018    2,2009    3,6    4,2010    55,2007    56,2008    165,2017    958,2013    1344,2015    1345,2016    1376,2014

X(1689) = reflection of X(1690) in X(6)
X(1689) = inverse-in-Brocard-circle of X(1687)
X(1689) = X(262)-Ceva conjugate of X(1690)

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