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Outer Vecten Triangle


TriangleSquareErecting

By analogy with the outer Napoleon triangle, consider the external erection of three squares on the sides of a triangle DeltaABC. These centers form a triangle DeltaO_AO_BO_C that has (exact) trilinear vertex matrix given by

 [-a/2 (a(sinC+cosC))/2 (a(sinB+cosB))/2; (b(sinC+cosC))/2 -b/2 (b(sinA+cosA))/2; (c(sinB+cosB))/2 (c(sinA+cosA))/2 -c/2]
(1)

(E. Weisstein, Apr. 19, 2004). The vertices of this triangle satisfy a number of remarkable properties.

The area of the outer Vecten triangle is

 Delta_O=Delta+1/8(a^2+b^2+c^2)
(2)

(Casey 1888, p. 149, Ex. 12), where Delta is the area of the reference triangle. Its side lengths are

a^'=sqrt((b^2+c^2+Delta)/2)
(3)
b^'=sqrt((a^2+c^2+Delta)/2)
(4)
c^'=sqrt((a^2+b^2+Delta)/2).
(5)

The circumcircle of the outer Vecten circle is the outer Vecten circle.

The following table gives the centers of the outer Vecten triangle in terms of the centers of the reference triangle for Kimberling centers X_n with n<=100.

SquaresonTriangle

Surprisingly, BO_B_|_O_CO_A, and

 BO_B^_=O_CO_A^_
(6)

(Coxeter and Greitzer 1967, pp. 96-97).

VectenPoint

In addition, the lines AO_A, BO_B, and CO_C are concurrent in a point known as the first Vecten point, which is Kimberling center X_(485).


See also

Inner Vecten Triangle, Outer Napoleon Triangle, Outer Vecten Circle, Vecten Points

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References

Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888.Coxeter, H. S. M. and Greitzer, S. L. "Points and Lines Connected with a Triangle." Ch. 1 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 1-26 and 96-97, 1967.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.

Referenced on Wolfram|Alpha

Outer Vecten Triangle

Cite this as:

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

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