Consider three squares erected externally on the sides of a triangle . Call the centers of these squares , ,
and , respectively. Then the lines , , and concur in a point known as the
outer vecten point, which is Kimberling center . It has equivalent triangle
center functions
(1)
and
(2)
Now consider three squares erected internally on the sides of the triangle. Call the centers of these squares , , and , respectively. Then the lines , , and concur in a point known as the
inner vecten point, which is Kimberling center . It has equivalent triangle
center functions
(3)
and
(4)
The nine-point center , outer vecten point , and inner vector point are collinear (J. Montes
Valderrama, pers. comm., R. Barroso Campos, Apr. 20, 2004).