By analogy with the outer Napoleon triangle, consider the external erection of three squares on the sides of a triangle 
. These centers form a triangle 
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]](/images/equations/OuterVectenTriangle/NumberedEquation1.svg) |
(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
 |
(2)
|
(Casey 1888, p. 149, Ex. 12), where 
is the area of the reference
triangle. Its side lengths are
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(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 
with 
.
 | center of outer Vecten triangle |  | center of reference triangle |
 | triangle centroid |  | triangle centroid |
 | circumcenter |  | complement of  |
 | orthocenter |  | outer Vecten point |
 | de Longchamps point |  | anticomplement
of  |
Surprisingly, 
,
and
 |
(6)
|
(Coxeter and Greitzer 1967, pp. 96-97).
In addition, the lines 
, 
, and 
are concurrent in a point
known as the first Vecten point, which is Kimberling
center 
.
