Inner Vecten Triangle


If the square is instead erected internally, their centers form a triangle DeltaI_AI_BI_C that has (exact) trilinear vertex matrix given by

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

(E. Weisstein, Apr. 25, 2004).

The area of the inner Vecten triangle is


where Delta is the area of the reference triangle. Its side lengths are


The circumcircle of the inner Vecten triangle is the inner Vecten circle.

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


As in the exterior case, the triangles DeltaABC and DeltaI_AI_BI_C are perspective with perspector at the inner Vecten point, which is Kimberling center X_(486).

See also

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

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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." Lamoen, F. "Friendship Among Triangle Centers." Forum Geom. 1, 1-6, 2001.Yiu, P. "Squares Erected on the Sides of a Triangle.", P. "On the Squares Erected Externally on the Sides of a Triangle."

Referenced on Wolfram|Alpha

Inner Vecten Triangle

Cite this as:

Weisstein, Eric W. "Inner Vecten Triangle." From MathWorld--A Wolfram Web Resource.

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