## X(485) (VECTEN POINT)

 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           sec(A - π/4) : sec(B - π/4) : sec(C - π/4)
=1/(sin A + cos A) : 1/(sin B + cos B) : 1/(sin C + cos C)
Trilinears           sin A + cos(B - C) : sin B + cos(C - A) : sin C + cos(A - B) (Peter J. C. Moses, 8/22/03)

Barycentrics    sin A sec(A - π/4) : sin B sec(B - π/4) : sin C sec(C - π/4)

Erect a square outwardly from each side of triangle ABC. Let A'B'C' be the triangle formed by the respective centers of the squares. The lines AA', BB', CC' concur in X(485). For details, visit Floor van Lamoen's site, Vierkanten in een driehoek: 1. Omgeschreven vierkanten (van Lamoen, 4/26/98) and his article "Friendship Among Triangle Centers," Forum Geometricorum, 1 (2001) 1-6. See also Paul Yiu's papers "Squares Erected on the Sides of a Triangle", and "On the Squares Erected Externally on the Sides of a Triangle".

X(485) lies on these lines: 2,372    3,590    4,371    5,6    69,639    76,491    226,481    489,671

X(485) = reflection of X(488) in X(641)
X(485) = isogonal conjugate of X(371)
X(485) = isotomic conjugate of X(492)
X(485) = complement of X(488)
X(485) = anticomplement of X(641)
X(485) = X(3)-cross conjugate of X(486)
X(485) = internal center of similitude of nine-point circle and 2nd Lemoine circle

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