Outer Vecten Triangle
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
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(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
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.
Surprisingly, BO_B_|_O_CO_A, and
| BO_B^_=O_CO_A^_ |
(6)
|
(Coxeter and Greitzer 1967, pp. 96-97).
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 PointsExplore with Wolfram|Alpha
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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 TriangleCite this as:
Weisstein, Eric W. "Outer Vecten Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OuterVectenTriangle.html