TOPICS

# Inner Vecten Triangle

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

 (1)

(E. Weisstein, Apr. 25, 2004).

The area of the inner Vecten triangle is

 (2)

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

 (3) (4) (5)

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 with .

 center of inner Vecten triangle center of reference triangle triangle centroid triangle centroid circumcenter complement of orthocenter inner Vecten point de Longchamps point anticomplement of

As in the exterior case, the triangles and are perspective with perspector at the inner Vecten point, which is Kimberling center .

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

## Explore with Wolfram|Alpha

More things to try:

## References

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

Inner Vecten Triangle

## Cite this as:

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