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
(1)
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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
(2)
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(Casey 1888, p. 149, Ex. 12), where is the area of the reference triangle. Its side lengths are
(3)
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(4)
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(5)
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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)
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(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 .