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Vecten Points


VectenPoint

Consider three squares erected externally on the sides of a triangle DeltaABC. Call the centers of these squares O_A, O_B, and O_C, respectively. Then the lines AO_A, BO_B, and CO_C concur in a point known as the outer vecten point, which is Kimberling center X_(485). It has equivalent triangle center functions

 alpha_(485)=sec(A-1/4pi)
(1)

and

 alpha_(485)=1/(cosA+sinA).
(2)
InnerVectenPoint

Now consider three squares erected internally on the sides of the triangle. Call the centers of these squares I_A, I_B, and I_C, respectively. Then the lines AI_A, BI_B, and CI_C concur in a point known as the inner vecten point, which is Kimberling center X_(486). It has equivalent triangle center functions

 alpha_(486)=sec(A+1/4pi)
(3)

and

 alpha_(486)=1/(cosA-sinA).
(4)

The nine-point center N, outer vecten point X_(485), and inner vector point X_(486) are collinear (J. Montes Valderrama, pers. comm., R. Barroso Campos, Apr. 20, 2004).


See also

Inner Vecten Triangle, Outer Vecten Triangle

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References

Pedret, J. M. "Problema 163." http://www.aloj.us.es/rbarroso/trianguloscabri/sol/sol163ped.htm.van Lamoen, F. "Vierkanten in een driehoek: 1. Omgeschreven vierkanten." http://home.wxs.nl/~lamoen/wiskunde/vierkant.html.van Lamoen, F. "Friendship Among Triangle Centers." Forum Geom. 1, 1-6, 2001.Yiu, P. "Squares Erected on the Sides of a Triangle." http://www.math.fau.edu/yiu/bottema38.pdf.Yiu, P. "On the Squares Erected Externally on the Sides of a Triangle." http://www.math.fau.edu/yiu/square.pdf.

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Vecten Points

Cite this as:

Weisstein, Eric W. "Vecten Points." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VectenPoints.html

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